http://soft-matter.seas.harvard.edu/api.php?action=feedcontributions&user=MichaelPetralia&feedformat=atom Soft-Matter - User contributions [en] 2020-09-26T00:26:51Z User contributions MediaWiki 1.24.2 http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4479 UserMichaelPetralia 2009-01-11T21:31:15Z <p>MichaelPetralia: /* What knobs do we turn to get gels with large swelling ratios and appropriate mechanical properties? */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> The modulus at equilibrium is thus, <br /> <br /> &lt;math&gt; G_{\theta solvent} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\left(\frac{\phi_0^{1/4}}{N^{3/8}}\right)^{1/3} \approx \frac{kT}{N^{9/8}b^3} \phi_0^{3/4}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> The modulus at equilibrium is, <br /> <br /> &lt;math&gt; G_{athermal} \approx \frac{kT}{Nb^3} \phi_0^{0.44} \left( \frac{\phi_0^{1/4}}{N^{0.573}}\right)^{0.56}\approx \frac{kT}{N^{3.32}b^3} \phi_0^{0.58} &lt;/math&gt;<br /> <br /> Note that the larger power for &lt;math&gt;N&lt;/math&gt; will cause the polymer network swollen in an athermal solvent to have a lower modulus at equilibrium (as expected: the more a gel swells, the weaker it becomes.). The smaller power for &lt;math&gt;\phi_0&lt;/math&gt; will have the opposite effect, but it is small compared to the effect of the number of monomers per chain.<br /> <br /> ===What knobs do we turn to get gels with large swelling ratios and appropriate mechanical properties?===<br /> It is clear from the previous section that there are a number of different &quot;knobs&quot; to turn when it comes to swelling ratios and mechanical properties. We use the shear modulus &lt;math&gt;G&lt;/math&gt; as the &quot;strength&quot; of the material (i.e. it's ability to resist deformation). The shear modulus in the swollen state is influenced by the volume fraction during preparation, the number of chains and the size of the monomer. Here we are assuming that we do not have control over &lt;math&gt; kT &lt;/math&gt;. The relative importance of each of these variables will depend on the type of solvent we use. If we want to affect the modulus, but not the equilibrium swelling ratio, it makes sense to consider changing the size of the monomers, &lt;math&gt;b&lt;/math&gt;, which effectively cancels out because it has the same effect on the osmotic pressure as the shear modulus.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4478 UserMichaelPetralia 2009-01-11T21:28:25Z <p>MichaelPetralia: /* What knobs do we turn to get gels with large swelling ratios and appropriate mechanical properties? */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> The modulus at equilibrium is thus, <br /> <br /> &lt;math&gt; G_{\theta solvent} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\left(\frac{\phi_0^{1/4}}{N^{3/8}}\right)^{1/3} \approx \frac{kT}{N^{9/8}b^3} \phi_0^{3/4}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> The modulus at equilibrium is, <br /> <br /> &lt;math&gt; G_{athermal} \approx \frac{kT}{Nb^3} \phi_0^{0.44} \left( \frac{\phi_0^{1/4}}{N^{0.573}}\right)^{0.56}\approx \frac{kT}{N^{3.32}b^3} \phi_0^{0.58} &lt;/math&gt;<br /> <br /> Note that the larger power for &lt;math&gt;N&lt;/math&gt; will cause the polymer network swollen in an athermal solvent to have a lower modulus at equilibrium (as expected: the more a gel swells, the weaker it becomes.). The smaller power for &lt;math&gt;\phi_0&lt;/math&gt; will have the opposite effect, but it is small compared to the effect of the number of monomers per chain.<br /> <br /> ===What knobs do we turn to get gels with large swelling ratios and appropriate mechanical properties?===<br /> It is clear from the previous section that there are a number of different &quot;knobs&quot; to turn when it comes to swelling ratios and mechanical properties. We use the shear modulus $G$ as the &quot;strength&quot; of the material (i.e. it's ability to resist deformation). The shear modulus in the swollen state is influenced by the volume fraction during preparation, the number of chains and the size of the monomer. Here we are assuming that we do not have control over &lt;math&gt; kT &lt;/math&gt;. The relative importance of each of these variables will depend on the type of solvent we use. If we want to affect the modulus, but not the equilibrium swelling ratio, it makes sense to consider changing the size of the monomers, &lt;math&gt;b&lt;/math&gt;, which effectively cancels out because it has the same effect on the osmotic pressure as the shear modulus.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4477 UserMichaelPetralia 2009-01-11T21:24:03Z <p>MichaelPetralia: /* Swelling in athermal solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> The modulus at equilibrium is thus, <br /> <br /> &lt;math&gt; G_{\theta solvent} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\left(\frac{\phi_0^{1/4}}{N^{3/8}}\right)^{1/3} \approx \frac{kT}{N^{9/8}b^3} \phi_0^{3/4}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> The modulus at equilibrium is, <br /> <br /> &lt;math&gt; G_{athermal} \approx \frac{kT}{Nb^3} \phi_0^{0.44} \left( \frac{\phi_0^{1/4}}{N^{0.573}}\right)^{0.56}\approx \frac{kT}{N^{3.32}b^3} \phi_0^{0.58} &lt;/math&gt;<br /> <br /> Note that the larger power for &lt;math&gt;N&lt;/math&gt; will cause the polymer network swollen in an athermal solvent to have a lower modulus at equilibrium (as expected: the more a gel swells, the weaker it becomes.). The smaller power for &lt;math&gt;\phi_0&lt;/math&gt; will have the opposite effect, but it is small compared to the effect of the number of monomers per chain.<br /> <br /> ===What knobs do we turn to get gels with large swelling ratios and appropriate mechanical properties?===<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4476 UserMichaelPetralia 2009-01-11T20:49:43Z <p>MichaelPetralia: /* Swelling in athermal solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> The modulus at equilibrium is thus, <br /> <br /> &lt;math&gt; G_{\theta solvent} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\left(\frac{\phi_0^{1/4}}{N^{3/8}}\right)^{1/3} \approx \frac{kT}{N^{9/8}b^3} \phi_0^{3/4}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> The modulus at equilibrium is, <br /> <br /> &lt;math&gt; G_{athermal} \approx \frac{kT}{Nb^3} \phi_0^{0.44} \left( \frac{\phi_0^{1/4}}{N^{0.573}}\right)^{0.56}\approx \frac{kT}{N^{3.32}b^3} \phi_0^{0.58} &lt;/math&gt;<br /> <br /> Note that both the larger power for &lt;math&gt;N&lt;/math&gt; and the smaller power for &lt;math&gt;\phi_0&lt;/math&gt; will cause the polymer network swollen in an athermal solvent to have a lower modulus at equilibrium (as expected). The more a gel swells, the weaker it becomes.<br /> <br /> ===What knobs do we turn to get gels with large swelling ratios and appropriate mechanical properties?===<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4475 UserMichaelPetralia 2009-01-11T20:47:23Z <p>MichaelPetralia: /* Swelling of polymer gels (adapted from section 7.4 of Rubenstein's Polymer Physics ) */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> The modulus at equilibrium is thus, <br /> <br /> &lt;math&gt; G_{\theta solvent} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\left(\frac{\phi_0^{1/4}}{N^{3/8}}\right)^{1/3} \approx \frac{kT}{N^{9/8}b^3} \phi_0^{3/4}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> The modulus at equilibrium is, <br /> <br /> &lt;math&gt; G_{athermal} \approx \frac{kT}{Nb^3} \phi_0^{0.44} \left( \frac{\phi_0^{1/4}}{N^{0.573}}\right)^{0.56}\approx \frac{kT}{N^{3.32}b^3} \phi_0^{0.58} &lt;/math&gt;<br /> <br /> ===What knobs do we turn to get gels with large swelling ratios and appropriate mechanical properties?===<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4474 UserMichaelPetralia 2009-01-11T20:44:54Z <p>MichaelPetralia: /* Swelling in athermal solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> The modulus at equilibrium is thus, <br /> <br /> &lt;math&gt; G_{\theta solvent} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\left(\frac{\phi_0^{1/4}}{N^{3/8}}\right)^{1/3} \approx \frac{kT}{N^{9/8}b^3} \phi_0^{3/4}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> The modulus at equilibrium is, <br /> <br /> &lt;math&gt; G_{athermal} \approx \frac{kT}{Nb^3} \phi_0^{0.44} \left( \frac{\phi_0^{1/4}}{N^{0.573}}\right)^{0.56}\approx \frac{kT}{N^{3.32}b^3} \phi_0^{0.58} &lt;/math&gt;<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4473 UserMichaelPetralia 2009-01-11T20:44:32Z <p>MichaelPetralia: /* Swelling in athermal solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> The modulus at equilibrium is thus, <br /> <br /> &lt;math&gt; G_{\theta solvent} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\left(\frac{\phi_0^{1/4}}{N^{3/8}}\right)^{1/3} \approx \frac{kT}{N^{9/8}b^3} \phi_0^{3/4}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> The modulus at equilibrium is, <br /> <br /> &lt;math&gt; G_{athermal} \approx \frac{kT}{Nb^3} \phi_0^{0.44} \left( \frac{\phi_0^{1/4}}{N^{0.573}}\right)^{0.56}\approx \frac{kT}{N^{3.32}b^3} \phi_0^{0.58}<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4472 UserMichaelPetralia 2009-01-11T20:40:23Z <p>MichaelPetralia: /* Swelling in \theta-solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> The modulus at equilibrium is thus, <br /> <br /> &lt;math&gt; G_{\theta solvent} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\left(\frac{\phi_0^{1/4}}{N^{3/8}}\right)^{1/3} \approx \frac{kT}{N^{9/8}b^3} \phi_0^{3/4}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4471 UserMichaelPetralia 2009-01-11T20:33:37Z <p>MichaelPetralia: /* Swelling in \theta-solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4470 UserMichaelPetralia 2009-01-11T20:31:08Z <p>MichaelPetralia: /* Research questions */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4469 UserMichaelPetralia 2009-01-11T20:27:45Z <p>MichaelPetralia: /* Introduction */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels possess the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating.<br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4468 UserMichaelPetralia 2009-01-11T20:26:14Z <p>MichaelPetralia: /* Swelling in \theta-solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4467 UserMichaelPetralia 2009-01-11T20:25:45Z <p>MichaelPetralia: /* Swelling in athermal solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4466 UserMichaelPetralia 2009-01-11T20:25:32Z <p>MichaelPetralia: /* Swelling in athermal solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta-solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;solvent.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4465 UserMichaelPetralia 2009-01-11T20:25:11Z <p>MichaelPetralia: /* Swelling in athermal solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} \sim \phi_0^{0.44} \phi^{0.56}&lt;/math&gt;<br /> <br /> The osmotic pressure in an athermal solvent is approximated as, <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} &lt;/math&gt; <br /> <br /> So the equilibrium swelling ratio is<br /> <br /> &lt;math&gt; Q_{athermal} \approx \frac{N^{3\left(3\nu-1\right)/4}}{\phi_0^{1/4}} \approx \frac{N^{0.573}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> Comparing &lt;math&gt; Q_{athermal} &lt;/math&gt; to &lt;math&gt; Q_{\theta-solvent} &lt;/math&gt; we find that a network swells more in an athermal solvent than in a &lt;math&gt;\theta&lt;/math&gt;-solvent.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4464 UserMichaelPetralia 2009-01-11T20:15:13Z <p>MichaelPetralia: /* On the fractal nature of polymer conformations (adapted from [5, section 1.4]) */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} &lt;/math&gt;<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4463 UserMichaelPetralia 2009-01-11T20:14:47Z <p>MichaelPetralia: /* A bit on notation */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \cong &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} &lt;/math&gt;<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4462 UserMichaelPetralia 2009-01-11T20:14:14Z <p>MichaelPetralia: /* Swelling in \theta-solvents */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;. <br /> <br /> The osmotic pressure of a semidilute solution in a &lt;math&gt;\theta&lt;/math&gt;-solvent is approximated by <br /> <br /> &lt;math&gt; \Pi \approx \frac{kT}{b^3}\phi^3 &lt;/math&gt;<br /> <br /> So to find the the equilibrium swelling ratio we write, <br /> <br /> &lt;math&gt; \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3} &lt;/math&gt;<br /> <br /> &lt;math&gt; \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}&lt;/math&gt;<br /> <br /> &lt;math&gt; \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}&lt;/math&gt;<br /> <br /> &lt;math&gt; Q \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}&lt;/math&gt;<br /> <br /> ====Swelling in athermal solvents====<br /> For an polymer network in an athermal solvent we use, <br /> <br /> &lt;math&gt; R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> &lt;math&gt; R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} &lt;/math&gt;<br /> <br /> with &lt;math&gt; \nu \cong 0.588 &lt;/math&gt;. This makes the gel modulus, <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{Nb^3} \phi_0^{1/\left[3\left(3\nu-1\right)\right]}\phi^{\left(9\nu-4\right)/\left[3\left(3\nu-1\right)\right]} &lt;/math&gt;<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4461 UserMichaelPetralia 2009-01-11T19:30:14Z <p>MichaelPetralia: /* Swelling of polymer gels (adapted from section 7.4 of Rubenstein's Polymer Physics ) */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_{eq}}{V_{dry}} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch because of the &lt;math&gt; \lambda^2 &lt;/math&gt; dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4460 UserMichaelPetralia 2009-01-11T19:28:49Z <p>MichaelPetralia: /* Swelling of polymer gels (adapted from section 7.4 of Rubenstein's Polymer Physics ) */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edwards tube model to describe the mechanical properties of the network. If we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state, the shear modulus can be written as <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. However, in realty the deformation of entangled networks is more complicated than the Edward's tube model, with the topological confinement significantly diminished upon swelling. <br /> <br /> Instead, let's consider the free energy required to stretch an ideal chain, which is the Flory form of the elastic part of the free energy:<br /> <br /> &lt;math&gt; F\approx kT\frac{R^2}{Nb^2} &lt;/math&gt;<br /> <br /> where &lt;math&gt; k &lt;/math&gt; is the Boltzman constant, &lt;math&gt; R &lt;/math&gt; is the end-to-end distance, &lt;math&gt; N &lt;/math&gt; is the number of monomers, and &lt;math&gt; b &lt;/math&gt; is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance &lt;math&gt; R_0 &lt;/math&gt; due to a stretch &lt;math&gt; \lambda &lt;/math&gt; is &lt;math&gt; R^2=\left(\lambda R_0\right)^2 &lt;/math&gt;. It is also appropriate to define &lt;math&gt; R^2_{ref} &lt;/math&gt; as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form: <br /> <br /> &lt;math&gt; F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } &lt;/math&gt;<br /> <br /> &lt;math&gt; R^2_{ref} &lt;/math&gt; changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus &lt;math&gt; G &lt;/math&gt; of the swollen gel is proportional to the chain number density &lt;math&gt; \nu = \phi / \left(Nb^3\right) &lt;/math&gt; time the elastic free energy per chain, where &lt;math&gt; \phi =V_{dry}/V&lt;/math&gt; is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent). <br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \nu kT \frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} \approx \frac{kT}{b^3} \frac{\phi}{N}\frac{\left(\lambda R_0\right)^2}{ R^2_{ref}} <br /> &lt;/math&gt;<br /> <br /> The modulus can be thought of as the elastic energy per unit volume. This modulus is effectively the modulus we obtain for the affine network model &lt;math&gt; G = \nu kT &lt;/math&gt; where here &lt;math&gt; \nu=n/V &lt;/math&gt; is the number of network chains &lt;math&gt; n &lt;/math&gt; per unit volume with an extra term to account for the effect of swelling on stiffening the network. The &lt;math&gt; \phi &lt;/math&gt; dependent number density of chains in this new modulus accounts for the weakening of the gel as it swells. These two forces are competing, the effects of which will be shown in the next sections. <br /> <br /> Any gel swells until the modulus and the osmotic pressure &lt;math&gt; \Pi &lt;/math&gt; are balanced. The osmotic pressure is of a semidilute solution of uncrosslinked chains at the same volume fraction as the gel. We define the equilibrium swelling ratio &lt;math&gt; Q &lt;/math&gt; as the ratio of the volume in the fully swollen state to the volume in the dry state when &lt;math&gt; G\approx \Pi &lt;/math&gt;:<br /> <br /> &lt;math&gt; Q \equiv \frac{V_eq}{V_dry} &lt;/math&gt;<br /> <br /> ====Swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents====<br /> In a &lt;math&gt;\theta&lt;/math&gt;-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity: <br /> <br /> &lt;math&gt; R^2_{ref} \approx R^2_0 \approx Nb^2 &lt;/math&gt;<br /> <br /> So the Panyukov form of the modulus reduces to the Flory form for swelling in &lt;math&gt;\theta&lt;/math&gt;-solvents, which we can rewrite in terms of volume fractions &lt;math&gt; \lambda = \left(\phi_0/\phi\right)^{1/3} &lt;/math&gt;:<br /> <br /> &lt;math&gt; G\left(\phi\right) \approx \frac{kT}{b^3}\frac{\phi}{N}\lambda^2 \approx \frac{kT}{b^3}\frac{\phi}{N}\left(\frac{\phi_0}{\phi}\right)^{2/3} \approx \frac{kT}{Nb^3} \phi_0^{2.3}\phi^{1/3}<br /> &lt;/math&gt;<br /> <br /> where &lt;math&gt; \phi_0 &lt;/math&gt; is the volume fraction in the preparation state (when crosslinking was performed). This modulus is weakly dependent on the concentration (volume fraction) of the network. As the concentration is lowered, the number density of strands naturally decreases. At the same time, the chains also stretch, raising the modulus &lt;math&gt; \left(\lambda^2\right) &lt;/math&gt; dependence. The net effect is a weak decrease of the gel modulus upon swelling which goes like &lt;math&gt; \phi^{1/3} &lt;/math&gt;.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4459 UserMichaelPetralia 2009-01-11T17:01:52Z <p>MichaelPetralia: /* Polymer Gels */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> ===Gelation (Notes on Chapter 6 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> ===Swelling of polymer gels (adapted from section 7.4 of Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; )===<br /> Any polymer network will swell (change volume manyfold) in the appropriate solvent. As a first pass, we could use the Edward's tube model if we replace the number of monomers in a chain &lt;math&gt; N &lt;/math&gt; with the number of monomers in an entanglement chain &lt;math&gt; N_e &lt;/math&gt; in the preparation state. <br /> <br /> &lt;math&gt; G_e = \frac{\rho \mathcal{R} T}{N_eM_o} &lt;/math&gt;<br /> <br /> where &lt;math&gt; G_e &lt;/math&gt; is the shear modulus, &lt;math&gt; \mathcal{R} &lt;/math&gt; is the gas constant, &lt;math&gt; T &lt;/math&gt; is the temperature, and &lt;math&gt; M_o &lt;/math&gt; is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [[http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari#PPA180,M1 6]] It represents the effective chain length between crosslinks, where the crosslinks which dominate are now physical entanglements. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> <br />  [http://books.google.com/books?id=Ag8xkqF4XWYC&amp;printsec=frontcover&amp;client=safari Elias, &lt;i&gt; Macromolecules: Volume 3: Physical Structures and Properties &lt;/i&gt; (2008)]<br /> <br /> [#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4207 UserMichaelPetralia 2009-01-08T03:25:38Z <p>MichaelPetralia: /* On the fractal nature of polymer conformations (adapted from [5, section 1.4]) */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{\mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4206 UserMichaelPetralia 2009-01-08T03:24:48Z <p>MichaelPetralia: /* Polymers in general */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> <br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4203 UserMichaelPetralia 2009-01-08T03:06:42Z <p>MichaelPetralia: /* On the fractal nature of polymer conformations (adapted from [5, section 1.4]) */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;/math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4202 UserMichaelPetralia 2009-01-08T03:05:12Z <p>MichaelPetralia: /* Polymers in general */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. The fractal dimension &lt;math&gt; \mathcal{D} &lt;/math&gt; is defined from the relation between the mass and radius of a fractal. <br /> <br /> &lt;math&gt; m \sim r^{mathcal{D}} &lt;/math&gt;<br /> <br /> If the size (radius of a sphere) changes by a factor &lt;math&gt; C_r &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; r_1 = C_r r_2 &lt;/math&gt;<br /> <br /> and the mass inside this sphere changes by a factor &lt;math&gt; C_m &lt;/math&gt;, so we have<br /> <br /> &lt;math&gt; m_1 = C_m m_2 &lt;/math&gt;<br /> <br /> then it is easy to show that the fractal dimension will be <br /> <br /> &lt;math&gt; \mathcal{D} = \frac{\log C_m}{\log C_r} &lt;/math&gt;<br /> <br /> Polymers are random fractals. When we consider their self-similarity, it is not that smaller subsections of the polymer look exactly like the entire polymer chain, but rather only on average (i.e. they have the same statistical properties). The mean square end-to-end distance of an ideal polymer chain is proportional to its degree of polymerization (which will be our proxy for mass), &lt;math&gt;N&lt;math&gt;.<br /> <br /> &lt;math&gt; N \sim \langle R^2 \rangle &lt;/math&gt;<br /> <br /> A similar relation holds for any subsection of the ideal chain containing &lt;math&gt; g &lt;/math&gt; monomers, <br /> <br /> &lt;math&gt; g \sim \langle r^2 \rangle &lt;/math&gt;<br /> <br /> Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is &lt;math&gt; \mathcal{D} =2 &lt;/math&gt;. For non-ideal polymers, the fractal dimension will take on other values, but typically &lt;math&gt; \mathcal{D} &lt; 3 &lt;/math&gt;. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).<br /> <br /> == Polymer Gels==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4176 UserMichaelPetralia 2009-01-08T00:47:21Z <p>MichaelPetralia: /* A bit on notation */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;)<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim \;&lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. <br /> <br /> <br /> == Polymer Gels==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4175 UserMichaelPetralia 2009-01-08T00:46:02Z <p>MichaelPetralia: </p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> ==A bit on notation==<br /> So I read the following convention in Rubenstein's [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc &lt;i&gt; Polymer Physics &lt;/i&gt;]. I'm new to this field, so I didn't know if it was standard. Anyone have any ideas if this is how the following symbols are typically used in the polymer world?<br /> <br /> &lt;math&gt; V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} &lt;/math&gt;<br /> <br /> *&lt;math&gt; \simeq &lt;/math&gt; is used to indicate a numerical approximation (i.e. &lt;math&gt;4\pi/3\simeq 4.2&lt;/math&gt;<br /> *&lt;math&gt; \approx &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. &lt;math&gt;4\pi/3&lt;/math&gt;)<br /> *&lt;math&gt; \sim &lt;/math&gt; is used to indicate that two quantities are proportional to each other up to a dimensional constant (i.e. mass changes like the radius cubed)<br /> <br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> ==Polymers in general==<br /> The mean-square displacement of a random walk from the origin is equal to the mean-square end-to-end vector of a freely jointed chain with the number of monomers &lt;i&gt;N&lt;/i&gt; equal to the number of steps of the walk and the monomer length &lt;i&gt;b&lt;/i&gt; equal to the step size [5, p. 69]:<br /> &lt;math&gt;\langle \overrightarrow{R}^2\rangle =Nb^2&lt;/math&gt;<br /> <br /> ===On the fractal nature of polymer conformations (adapted from [5, section 1.4])===<br /> Certain polymer characteristics are fixed during polymerization. These include the polymer's microstructure (i.e. organization of atoms along the polymer chain), architecture (topology &lt;i&gt;?does someone have a good definition for this?&lt;/i&gt;), degree of polymerization (i.e. the number of monomers in the polymer), and chemical composition of heteropolymers. However, the conformation of a polymer (i.e. the spatial structure of a polymer determined by the relative locations of its monomers) is not fixed, and a single flexible macromolecule can adopt many different conformations. We can specify the conformation by a set of &lt;i&gt; n &lt;/i&gt; bond vectors between neighboring backbone atoms. Conformations are dependent on the:<br /> *flexibility of the polymer chain<br /> *interactions between monomers on the chain (can be attractive or repulsive)<br /> *interactions between elements in the chain and the surroundings (either other chains or solvent) <br /> Tuning the balance between these factors can drastically change the polymer's conformation. The difference in linear dimension between polymers with strong attractions between the monomers and long-range electrostatic repulsions is eight orders of magnitude!<br /> <br /> Most polymer conformations are [http://en.wikipedia.org/wiki/Fractal fractal] (i.e. self-similar) over a wide range of length scales. <br /> <br /> <br /> == Polymer Gels==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4174 UserMichaelPetralia 2009-01-07T23:41:06Z <p>MichaelPetralia: /* Notes on Rubenstein's Polymer Physics  */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> <br /> == Polymer Gels ==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4173 UserMichaelPetralia 2009-01-07T23:29:38Z <p>MichaelPetralia: /* Notes on Rubenstein's Polymer Physics  */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> <br /> == Polymer Gels ==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> * When the extent of reaction (&lt;math&gt;p&lt;/math&gt;) reaches approximately twice the gel point (&lt;math&gt;p_c&lt;/math&gt;), nearly all of the monomers are attached to the gel, so there is essentially no sol remaining, and the gel forms a well-developed network.<br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4172 UserMichaelPetralia 2009-01-07T23:26:32Z <p>MichaelPetralia: </p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> <br /> == Polymer Gels ==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;/i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, incipient gel, or in other words, a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> *Gelation is a continuous phase transition, some other examples of which are the continuous change of magnetization at the Curie point, and the continuous change in density at the vapor-liquid transition at the critical temperature. <br /> *The fraction of the material that is part of the gel (&lt;math&gt; P_{gel} &lt;/math&gt;) grows steadily above the gel point (&lt;math&gt; p_c &lt;/math&gt;). &lt;math&gt; p &lt;/math&gt; is the extent of the reaction (fraction of all possible bonds that are formed), and &lt;math&gt; \sigma &lt;/math&gt; and &lt;math&gt; \tau &lt;/math&gt; are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, &lt;math&gt;\tau = 5/2 &lt;/math&gt;, &lt;math&gt; \sigma = 1/2 &lt;/math&gt;, and &lt;math&gt; \beta=1 &lt;/math&gt;. <br /> <br /> &lt;math&gt; P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta &lt;/math&gt; where &lt;math&gt; \beta= \frac{\tau -2}{\sigma} &lt;/math&gt;<br /> <br /> <br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4136 UserMichaelPetralia 2009-01-07T21:37:04Z <p>MichaelPetralia: /* Polymer Gels */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> <br /> == Polymer Gels ==<br /> <br /> ===Notes on Rubenstein's &lt;i&gt; Polymer Physics &lt;/i&gt; ===<br /> *Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains. <br /> *Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.<br /> *We define a &lt;i&gt; sol &lt;/i&gt; as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. <br /> *At some point (as crosslinking continues) a branched polymer will span the entire network. This is the &lt;i&gt; gel point &lt;/i&gt; or &lt;i&gt; sol-gel transition &lt;\i&gt; and it is defined as the first point at which the system contains an &quot;infinite polymer&quot;, or a polymer which spans the entire system. This molecule will not dissolve in the solvent, but may only swell in it. <br /> *Besides physical gelation (which generally involves tangling up the chains in one way or another (lamellar, microcrystals, or intertwined helices)), chemical gelation can be used to create polymer networks. Three types of chemical gelation are:<br /> ** condensation: occurs when some of the starting monomers have functionality 3 or higher. Condensation of such monomers in the melt (no non-reactive diluent present) can be described with the &lt;b&gt; critical percolation &lt;/b&gt; model.<br /> ** vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the &lt;b&gt;mean-field percolation &lt;/b&gt; model. <br /> ** addition polymerization: in this process a free radical transfers from one vinyl monomer to another, leaving behind a rail of chemical bonds. Certain molecules (with two double bonds) can be visited twice by the radical and become the crosslinks. It is described by &lt;b&gt;kinetic gelation&lt;/b&gt; model. <br /> <br /> <br /> <br /> <br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]]. Though, Rubenstein asserts that a thermoreversible gel (like Jello) is a strong physical gel, which is a solid that can only melt and flow when the external conditions change; strong physical gels are analogous to chemical gels.  <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4109 UserMichaelPetralia 2009-01-07T20:02:05Z <p>MichaelPetralia: /* Research questions */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> <br /> == Polymer Gels ==<br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br />  [http://books.google.com/books?id=RHksknEQYsYC&amp;dq=polymer+physics&amp;ei=-QllSc6GApbAM9K_-Cc Rubinstein, &lt;i&gt; Polymer Physics &lt;/i&gt; (2003)]<br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4102 UserMichaelPetralia 2009-01-07T18:12:16Z <p>MichaelPetralia: /* Polymer Gels */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> <br /> == Polymer Gels ==<br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===The polymer gel network===<br /> de Gennes has us think about the gel as a frozen system, very similar to glasses. In such a system we need two types of statistical information [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]]:<br /> *The situation at the moment of preparation (preparative ensemble)<br /> *The situation at the moment of study (final ensemble)<br /> The conditions under which a gel was formed will determine the microscopic structure of the gel. We should exploit this to great materials with the macroscopic properties we desire. As an example, segregation between polymer chains and a poor solvent will not happen quickly enough (because the polymer is already somewhat gelled) to be macroscopic. The phase separation manifests itself in separate microscopic pockets of solvent and polymers. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA136,M1 4]] By controlling the solvent we use, it seems reasonable that we would have some control over the size and frequency of the pockets. Besides the mechanical response, this would help use control the swelling properties of the gel. <br /> <br /> In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, &lt;math&gt; p_c &lt;/math&gt;, at which the polymer network is connected throughout the material. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] We define &lt;math&gt; p &lt;/math&gt; as the fraction of reacted bonds and &lt;math&gt; \Delta p &lt;/math&gt; as some small fraction compared to &lt;math&gt; p_c &lt;/math&gt;. Percolation theory will be used to understand the behavior of gels around the gelation threshold, &lt;math&gt; p_c &lt;/math&gt;. Though the gel fraction (&lt;math&gt; S_\infty &lt;/math&gt;, the fraction of monomers belonging to the infinite polymer network) increases rapidly with &lt;math&gt; \Delta p &lt;/math&gt;, the elastic modulus of the gel (&lt;math&gt; E &lt;/math&gt;) increases much more slowly. <br /> <br /> &lt;math&gt; S_\infty \simeq \Delta p^\beta &lt;/math&gt; where &lt;math&gt; \beta = 0.39 \;&lt;/math&gt; <br /> <br /> &lt;math&gt; E \simeq \Delta p^t &lt;/math&gt; where &lt;math&gt; t \sim 1.7 \;\text{to}\; 1.9 &lt;/math&gt; for three dimensions. <br /> <br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency, but the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems. This system is physically crosslinked, so we must be aware of the limitations. For gels formed with a physical process (i.e. weakly crosslinked), the crosslinks will eventually split under weak stress and the long-time behavior will always be liquid-like. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA135,M1 4]] <br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> ===Research questions=== <br /> * We can control the pore size inside of gels, which allows us to decrease the swelling times dramatically, but it seems this would always be at the expense of mechanical strength. Can we decrease swelling time while maintaining mechanical integrity?<br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4100 UserMichaelPetralia 2009-01-07T16:36:22Z <p>MichaelPetralia: </p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> &quot;...the crosslinks need not be produced by chemical reaction. Any physical process that favors <br /> association between certain (but not all) points on different chains may also lead to gels.&quot;<br /> [http://books.google.com/books?id=ApzfJ2LYwGUC&amp;pg=PA131&amp;dq=crosslinking+in+special+matrix&amp;ei=M85kSfDGJ4nIMvXHtIkL#PPA133,M1page page 133, de Gennes &lt;i&gt; Scaling Concepts in Polymer Physics&lt;/i&gt;]<br /> <br /> <br /> == Polymer Gels ==<br /> ===Introduction===<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;] <br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden intentionally omitted their consideration. [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ===Making interesting gel structures===<br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. Such macroscopic solutions will work, but efficiency or functionality will suffer. In systems where part of the volume expansion is constrained, the gel will have to exert energy to bend the substrate. In exploiting different diffusive time scales, we lose a level of control and we limit the length of time we can use the device. <br /> <br /> Microscopic solutions may be more appropriate. By controlling the conformation of the polymer chains while they are cross-linking, we should be able to create gels with the mechanical properties we desire. de Gennes suggests that we incorporate polymer chains in various liquid crystalline systems, using them as a mold of sorts (e.g. incorporating hydrophillic polymers in the water layer of a lamellar phase of a lipid/water solution, cross-linking the chains, and washing out the lipid). [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA131,M1 4]] <br /> <br /> de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA133,M1 4]]: At high temperatures the solution is a liquid (simple solution of chains, &lt;i&gt;sol&lt;/i&gt;). Upon cooling, the solution gels. Raising the temperature again, the sol is recovered, but temperature at which this happens is higher than the temperature necessary for gelation. Hysteresis robs us of energy and thus decreases efficiency. Though, the ability to create a gel which is stable over a large temperature range and can reversibly transition between liquid and gel, has potential in soft robotic systems.<br /> <br /> It is possible to create block copolymers, let's say BAB, and place them in a solve good for A but not for B. Then the B portions will tend to coalesce, the nodules that form either being a solid state (liquid crystalline) or a fluid state (micelles) depending on the temperature. [[http://books.google.com/books?id=ApzfJ2LYwGUC&amp;printsec=frontcover#PPA134,M1 4]] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel. <br /> <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br />  [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 de Gennes, &lt;i&gt;Scaling Concepts in Polymer Physics &lt;/i&gt; (1979)]<br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4060 UserMichaelPetralia 2009-01-06T23:54:28Z <p>MichaelPetralia: /* Final Project */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material <br /> with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;, 3]<br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] intentionally omitted their consideration. Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a second and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things.<br /> <br /> ==Ideas without a home==<br /> <br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4059 UserMichaelPetralia 2009-01-06T23:52:06Z <p>MichaelPetralia: /* Final Project */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. Gels are materials that ﬁt somewhere between a solid and a liquid, consisting of a polymer network swollen with an interstitial ﬂuid. The properties of the gel are deﬁned by the polymer network, the interstitial ﬂuid, and the interaction between them. <br /> <br /> Jones tells us<br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid—a sol —to a material <br /> with a non-zero shear modulus—a gel. A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;, 3]<br /> <br /> All gels process the unique ability to undergo abrupt changes in volume, often as a result of small changes in external conditions such as temperature, pH, electric ﬁelds, and solvent and ionic composition [[http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf 1]]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).<br /> <br /> There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 2]] intentionally omitted their consideration. Madden claimed that the response time is typically slow (anywhere from seconds to minutes--I've seen it as short as fraction of a seconds and as long as weeks) and they are relatively weak (~100 kPa--I'm assuming he means this is the tensile stress). Despite these short comings, I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. What I lack is a good understanding of the physics and chemistry at work in polymer gels necessary to judge whether this technology is worth investigating. <br /> <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things. <br /> <br /> <br /> ==Ideas without a home==<br /> <br /> The change in physical shape of polymer gels is dominated by diffusion, and over long time scales will be isotropic. In order to create useful motions, it is likely that the gels will be placed in systems which constrain part of their volume expansion, or geometries which will create anisotropic swelling over short time scales. <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/1/16/MIT_A.I._Memo_1994_Brock.pdf Brock, MIT A.I. Memo No. 1330 (1994)]<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=File:MIT_A.I._Memo_1994_Brock.pdf&diff=4058 File:MIT A.I. Memo 1994 Brock.pdf 2009-01-06T23:32:08Z <p>MichaelPetralia: Review of artificial muscles based on contractile polymers</p> <hr /> <div>Review of artificial muscles based on contractile polymers</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4057 UserMichaelPetralia 2009-01-06T23:23:25Z <p>MichaelPetralia: /* Fun facts on soft matter */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 1]] intentionally omitted their consideration. The response time is typically slow (anywhere from seconds to weeks) and they are relatively weak (QUANTIFY). I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. The change in physical shape of polymer gels <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things. I decided that for my final project, I'd do a critical reading of <br /> <br /> Jones tells us<br /> <br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid—a sol —to a material <br /> with a non-zero shear modulus—a gel. A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4056 UserMichaelPetralia 2009-01-06T23:22:57Z <p>MichaelPetralia: /* Fun facts on soft matter */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> Additionally, we do not need to be so rigid in our thoughts on cross-linking.<br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 1]] intentionally omitted their consideration. The response time is typically slow (anywhere from seconds to weeks) and they are relatively weak (QUANTIFY). I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. The change in physical shape of polymer gels <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things. I decided that for my final project, I'd do a critical reading of <br /> <br /> Jones tells us<br /> <br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid—a sol —to a material <br /> with a non-zero shear modulus—a gel. A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4055 UserMichaelPetralia 2009-01-06T23:22:03Z <p>MichaelPetralia: /* Final Project */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 1]] intentionally omitted their consideration. The response time is typically slow (anywhere from seconds to weeks) and they are relatively weak (QUANTIFY). I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. The change in physical shape of polymer gels <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things. I decided that for my final project, I'd do a critical reading of <br /> <br /> Jones tells us<br /> <br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid—a sol —to a material <br /> with a non-zero shear modulus—a gel. A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4054 UserMichaelPetralia 2009-01-06T23:20:38Z <p>MichaelPetralia: </p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;printsec=frontcover#PPA73,M1 page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 1]] intentionally omitted their consideration. The response time is typically slow (anywhere from seconds to weeks) and they are relatively weak (QUANTIFY). I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. The change in physical shape of polymer gels <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things. I decided that for my final project, I'd do a critical reading of <br /> <br /> Jones [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 2] tells us<br /> <br /> “A gel is a material composed of subunits that are able to bond with each other in such a way that one<br /> obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one <br /> starts out with isolated subunits and successively adds bonds, one goes from a liquid—a sol —to a material <br /> with a non-zero shear modulus—a gel. A gel has the mechanical properties characteristic of a solid, even<br /> though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” <br /> <br /> “Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible <br /> gel such as a gelatin. ” <br /> [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA95&amp;dq=connected+by+physical,+rather+than+chemical&amp;ei=_OZjSeiFK4qEywSGtPXRDQ page 95, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;]<br /> <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4052 UserMichaelPetralia 2009-01-06T22:43:10Z <p>MichaelPetralia: /* Final Project */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;<br /> <br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 1]] intentionally omitted their consideration. The response time is typically slow (anywhere from seconds to weeks) and they are relatively weak (QUANTIFY). I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. The change in physical shape of polymer gels <br /> So I picked up de Gennes classic &lt;i&gt; [hhttp://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things. I decided that for my final project, I'd do a critical reading of <br /> <br /> <br /> Jones tells us <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br />  [http://books.google.com/books?id=Hl_HBPUvoNsC&amp;pg=PA2&amp;dq=jones+soft+matter&amp;lr=&amp;ei=ft1jSaHtNoagyASm7qjJCw#PPP1,M1 Jones, &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4051 UserMichaelPetralia 2009-01-06T22:41:28Z <p>MichaelPetralia: /* Final Project */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;<br /> <br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden [[http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf 1]] intentionally omitted their consideration. The response time is typically slow (anywhere from seconds to weeks) and they are relatively weak (QUANTIFY). I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. The change in physical shape of polymer gels <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things. I decided that for my final project, I'd do a critical reading of <br /> <br /> <br /> Jones tells us <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br /> <br />  [http://soft-matter.seas.harvard.edu/images/5/5c/IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf Madden, J.D.W. et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)]<br /> <br /> [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE#PPP1,M1 Jones, R.A.L., &lt;i&gt;Soft Condensed Matter&lt;/i&gt; (2002)]<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=File:IEEE_Journal_of_Oceanic_Engineering_2004_Madden.pdf&diff=4049 File:IEEE Journal of Oceanic Engineering 2004 Madden.pdf 2009-01-06T22:37:29Z <p>MichaelPetralia: Review of artificial muscle technology</p> <hr /> <div>Review of artificial muscle technology</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4048 UserMichaelPetralia 2009-01-06T22:33:32Z <p>MichaelPetralia: </p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> Typically, we think of changing the degree/type of cross-linking in order <br /> to change the mechanical properties of a polymer. Lengthening of a polymer is <br /> accomplished by aligning the polymer chains. It seems that there are polymers <br /> where the monomer-monomer interactions are not covalently bonded and thus <br /> the lengths of the polymer chains themselves can change. <br /> <br /> “. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. <br /> Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long <br /> chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are <br /> sometimes called ‘living polymers’; their characteristic is that they can change their length in response <br /> to changes in the environment. This contrasts with the more usual covalently linked polymers, in which <br /> the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.”<br /> page 73, Jones &lt;i&gt;Soft Condensed Matter&lt;/i&gt;<br /> <br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden  intentionally omitted their consideration. The response time is typically slow (anywhere from seconds to weeks) and they are relatively weak (QUANTIFY). I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. The change in physical shape of polymer gels <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things. I decided that for my final project, I'd do a critical reading of <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br />  Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4046 UserMichaelPetralia 2009-01-06T22:23:34Z <p>MichaelPetralia: /* Final Project */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> <br /> == Final Project ==<br /> One 'soft' actuation technology I'm looking into are polymer gels. There is some criticism in the soft robotics community about the usefulness of polymer gels for artificial muscle type technology. A recent review article by Madden  intentionally omitted their consideration. The response time is typically slow (anywhere from seconds to weeks) and they are relatively weak (QUANTIFY). I believe they still have merit because of the breadth of stimuli that can be used to activate them (light, heat, pH, electric and magnetic fields, ionic strength) and the control we have over their swelling properties. The change in physical shape of polymer gels <br /> So I picked up de Gennes classic &lt;i&gt; [http://books.google.com/books?hl=en&amp;lr=&amp;id=ApzfJ2LYwGUC&amp;oi=fnd&amp;pg=PA13&amp;dq=%22de+Gennes%22+%22Scaling+Concepts+in+Polymer+Physics%22+&amp;ots=JaX57jIV_4&amp;sig=ExNF9hdD1sr9PapK2Te8Bs6HrXE Scaling Concepts in Polymer Physics] &lt;/i&gt; to see if I could learn a few things. I decided that for my final project, I'd do a critical reading of <br /> <br /> ----<br /> &lt;b&gt;References:&lt;/b&gt;<br />  Madden et al., IEEE Journal of Oceanic Engineering &lt;b&gt;29&lt;/b&gt;, 706 (2004)<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4031 UserMichaelPetralia 2009-01-06T22:01:16Z <p>MichaelPetralia: /* Fun facts on soft matter */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> <br /> == Final Project ==<br /> <br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4030 UserMichaelPetralia 2009-01-06T22:00:57Z <p>MichaelPetralia: /* About me */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> <br /> <br /> [[#top | Top of Page]]<br /> == Final Project ==<br /> <br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=UserMichaelPetralia&diff=4029 UserMichaelPetralia 2009-01-06T22:00:43Z <p>MichaelPetralia: /* About me */</p> <hr /> <div>[[Main Page|Home]]<br /> == About me ==<br /> I am working on soft mechanical systems at the [[http://micro.seas.harvard.edu Harvard Microrobotics Laboratory]]. My research focuses on creating changes in stiffness and damping properties, macroscopic motion, and force generation using 'soft' materials. The buzz word is &lt;i&gt;artificial muscles&lt;/i&gt;, but the scope is much broader. I'm thinking about how we can create compliant devices that out perform the typical 'hard' systems engineers usually design.<br /> <br /> == Fun facts on soft matter ==<br /> <br /> <br /> [[#top | Top of Page]]<br /> == Final Project ==<br /> <br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> [[Main Page|Home]]</div> MichaelPetralia http://soft-matter.seas.harvard.edu/index.php?title=Emulsions_and_foams&diff=3892 Emulsions and foams 2009-01-05T23:58:21Z <p>MichaelPetralia: /* Emulsions */</p> <hr /> <div>[[Main_Page | Home]]<br /> == Topics ==<br /> <br /> {| cellspacing = &quot;0&quot; border = &quot;1&quot; style=&quot;margin: 1em auto 1em auto&quot;<br /> |- valign = &quot;center&quot; align = &quot;center&quot;<br /> ! width=50% | [[ Emulsions ]]<br /> ! width=50% | [[Image:Clayton_Fig_10.jpg |thumb| 200px | center | W. Clayton, ''The theory of emulsions and emulsification''; J.&amp;A. Churchill:London, '''1923''', p. 89.]]<br /> |- valign = &quot;center&quot; align = &quot;center&quot;<br /> ! width=50% | [[ Foams ]]<br /> ! width=50% | [[Image:JeanSimeonChardin.png |thumb| 200px | center | Jean Simeon Chardin 1699 - 1779]]<br /> |}<br /> <br /> {| cellspacing = &quot;0&quot; border = &quot;1&quot; style=&quot;margin: 1em auto 1em auto&quot;<br /> |- valign = &quot;center&quot; align = &quot;left&quot;<br /> | width=50% | [[Image:IceCream.jpg |thumb| 300px | center | ]]<br /> | width=50% | Ice cream is a:<br /> <br /> Foam of air bubbles,<br /> <br /> Stabilized with small (yellow) oil drops,<br /> <br /> In a matrix that is,<br /> <br /> An emulsion of more oil drops<br /> <br /> And a suspension of (blue) ice crystals,<br /> <br /> In a continuous (grey) phase of surfactants, micelles, and solutes<br /> <br /> in water<br /> <br /> In a sugar cone.<br /> <br /> |}<br /> <br /> {| cellspacing = &quot;0&quot; border = &quot;1&quot; style=&quot;margin: 1em auto 1em auto&quot;<br /> |- valign = &quot;center&quot; align = &quot;center&quot;<br /> ! width=50% | Phase 1<br /> ! width=50% | Phase 2<br /> |- valign = &quot;center&quot; align = &quot;center&quot;<br /> | Droplet<br /> | Serum<br /> |- valign = &quot;center&quot; align = &quot;center&quot;<br /> | Dispersed<br /> | Medium<br /> |- valign = &quot;center&quot; align = &quot;center&quot;<br /> | Discontinuous<br /> | Continuous<br /> |- valign = &quot;center&quot; align = &quot;center&quot;<br /> | Internal<br /> | External<br /> |}<br /> <br /> === Cake batter is an emulsion and foam===<br /> <br /> A cake mixture is a complex, multi-component system, being simultaneously a foam, an emulsion and a complex colloidal dispersion. Of the main components of a cake batter - egg, flour, fats and sugar - only the sugar is non-colloidal. The process of transformation of these ingredients into cake (i.e. solid foam) is not completely understood, although it is known that for the success of the recipe it is vital to retain the air bubbles within the cooked batter.<br /> <br /> [[Image:batter.jpg]]<br /> [[Image:cake.jpg]]<br /> <br /> To better understand the 'foamy' nature of cakes, let us break up the cake batter into its most essential components: <br /> <br /> 1) '''Flour''', which is mainly composed of gluten and starch. ''Gluten'' was named by the Chinese 'the muscle of flour' due to its elastic nature. Gluten is composed of long protein molecules ( gliadins and glutenins) which are responsible for the elastic behavior of dough. Upon kneading and stretching of dough, gluten proteins unfold and align. The coiled, spring-like structure of proteins can unfold and store some of the energy of stretching, but when the stretching is stopped, the molecules spring back to their compact coiled form. This is macroscopically manifested when stretched dough creeps up to its original shape. Although bread preparation benefits from a strong, elastic gluten, excessive amounts of the protein are not desirable in puffy pastries and raised cakes. Ways to limit gluten presence in batter are the use of low-protein flours as well as adding water in the dough, which dilutes the gluten proteins and limits their bonds.<br /> <br /> ''Starch'' makes up 70% of the flour weight. Therefore it is a structural component of doughs, especially the low-gluten cake batters. Along with water, starch interpenetrates the gluten network and break it up tenderizing the dough. During the baking of bread and cake, starch granules absorb water and set to form the rigid bulk of the walls that surround the bubbles of carbon dioxide. At the same time, their swollen rigidity stops the expansion of bubbles and forces the water vapor inside to pop the bubbles and escape, turning the foam of separate bubbles into a continuous network of connected holes. If this didn't happen, then at the end of baking the cooling water vapor would contract and cause the cake to collapse.<br /> <br /> 2)''' Eggs''', which contain proteins, fats and emulsifiers. The proteins in eggs coagulate during cooking and supplement the gluten structure. The fats and emulsifiers in eggs work like starch, weakening the gluten network and stabilizing the bubbles in the dough.<br /> <br /> 3) '''Fats''', contained in oils and shortening. Fats are an important component in pie crusts and puff pastries, where layers of solid fats separate thin layers of dough from each other so that they cook into separate layers of pastry. In cakes, fat and oil molecules bond to parts of the gluten protein coils and prevent the proteins from forming a strong gluten. This is the reason why in making bread, which requires a strong gluten, flour and water are mixed alone.<br /> <br /> 4) '''Gas bubble'''s make up as much as 80% of a cake's volume. These weaken the gluten and starch network and divide it into millions of delicate sheets that form the bubble walls. Baking yeasts and chemical leavenings are routinely used to fill baked goods with gas bubbles. A common misconception is that these products create new bubbles: in fact, the carbon dioxide in yeast is released into the water phase of the batter, diffuses to the pre-existing air bubbles and enlarges them. This is why the initial aeration of dough and batter through kneading, strongly influences the final texture of baked goods.<br /> <br /> Information adapted from:<br /> <br /> * Ian W. Hamley, 'Introduction to soft matter', John Wiley &amp; Sons (2007)<br /> <br /> *Harrold McGee, 'On food and cooking: the science and lore of the kitchen', Scribner NY (2004)<br /> <br /> <br /> <br /> [[#top | Top of Page]]<br /> ----<br /> == Bibliography ==<br /> N.H Hyam, &quot;Quantitative Evaluation of Factors Affecting the Sensitivity of Penetrant Systems&quot;. Materials Evaluation, pp. 31-38, February 1972.<br /> <br /> A. V. Rode, &quot;Electronic and magnetic properties of carbon nanofoam produced by high-repetition-rate laser ablation&quot;. Applied Surface Science 197–198: 644–649. 2002.<br /> ----<br /> == Ice Cream Floats ==<br /> <br /> One interesting point building on the look into ice cream is the role it plays when you make an ice cream float. If you've ever made one, you'll remember that when the ice cream is added to the soda that it begins to rapidly foam up. You can see this in the image below:<br /> <br /> [[Image:Float.jpg|center]]<br /> <br /> Looking at this, it begs the question of why the ice cream causes an increase in foam in the soda. The reason is simple. As mentioned above, ice cream is an emulsion of various things like fat and sugar. As it melts, these tiny particles are released into the soda. As you know, soft drinks are liquids that have been supersaturated with carbon dioxide. Soda fizz comes from the nucleation of the carbon dioxide. The particles from the ice cream alter the solubility of the carbon dioxide and enable the creation of foam nucleated on the small particles. The amount of foam created is directly proportional to how much CO2 is diluted in the liquid. Furthermore its the fat serving as the instigators of nucleation that further the stability of the foam produced.<br /> <br /> References:<br /> http://www.newton.dep.anl.gov/askasci/chem00/chem00313.htm<br /> <br /> <br /> ----<br /> == The Mentos-Coke experiment ==<br /> <br /> We have all surely heard, by now, of the famous Mentos in Diet Coke experiment (http://en.wikipedia.org/wiki/Diet_Coke_and_Mentos). I thought it worthy to mention here, because it seems to fit in the category of soft matter in numerous ways:<br /> <br /> 1) First of all, and specifically related to the topic of emulsions and foams, when the Mentos are dropped into the soda bottle, CO2 bubbles rapidly nucleate and grow on the rough surface of the candy, thus causing rapid expansion of an emulsion (CO2 gas in liquid soda) and allowing for very interesting science experiments, such as soda fountains, soda jets or soda rockets!<br /> <br /> 2) Interesting to note that mint Mentos work better than fruit Mentos, which have a soft surface coating that slows down nucleation of CO2 bubbles.<br /> <br /> 3) Interesting as well to note that Diet Coke is used preferably to Coke. The soda, in ths science experiment only provides highly dissolved carbon dioxide at a first glance. But from what has been reported in the TV series &quot;Mythbusters&quot; is that the replacement sugar used in Diet Coke, aspartame, is also responsible for the nice jetting regime that we see in this fun experiment. So although I haven't tried it myself, it would seem that regular Coke would not do so well.<br /> <br /> 4) One website (http://www.newscientist.com/article/dn14114-science-of-mentosdiet-coke-explosions-explained.html) claims that surface tension of soda with the sugar aspartame is lower than that of soda with regular sugar. And because lower surface tension would help bubbles grow faster, Diet Coke works better.<br /> <br /> 5) Moreover, the presence of other surfactants in Mentos may dissolve with the rest of the candy and help reduce the surface tension of the Diet Coke even further.<br /> <br /> <br /> So we've got here a combination of surface geometry, surface tension, rapid foam formation, phase transition (from dissolved CO2 to foam), surface coating and others that really fit into the themes of soft condensed matter! That is perhaps why one of Harvard's former soft condensed matter post-docs Dan Blair, now assistant professor at Georgetown, was asked to comment on this phenomenon on NECN a while back (http://www.seas.harvard.edu/newsandevents/newsarchives/index.html?d=2006.06).<br /> <br /> <br /> ==Aluminum Foam==<br /> Aluminum foam has applications to a variety of industries, including automotive, aviation, railway and engine building industries. It is an extremely stiff material but also low weight. It also has high energy absorption. Aluminum foam has specifically be used in the following common applications: fluid flow control devices, energy/impact absorption, compact heat exchangers, lightweight composite panels, and trabecular bone simulation (aluminum foam actually micmics bone tissue!) Some specific physical characteristics of aluminum foam are, &lt;br&gt;<br /> &lt;center&gt;Compression Stretching 2.53 MPa&lt;br&gt;<br /> Elasticity Modulus (of compression) 93.08 MPa&lt;br&gt;<br /> Specific Heat .895 J/gC&lt;br&gt;<br /> Melting Point 660 C (quite high)&lt;/center&gt;&lt;br&gt;<br /> To produce aluminum foam, powdered aluminum is mixed with a product that releases gas at a high temperature. It is then compacted and put into a mold form. It is heated until the aluminum starts to foam. Then it is taken out of the heated environment so it stays in its desired shape. &lt;br&gt;&lt;br&gt;<br /> Aluminum foam is preferably used over other polymeric foams (such as honeycomb and polymer foams), because it is recyclable, it has material stability over time, environmental and temperature conditions. It is also very cheap to make as compared to other commonly used foams. Aluminum foam also do not give off noxious fumes when they decompose, and they are stronger than other polymer foams.&lt;br&gt;&lt;br&gt;<br /> Aluminum foam will be used more and more in the future!&lt;br&gt;<br /> &lt;center&gt;[[Image:Aluminium-foam.jpg]]&lt;/center&gt;<br /> <br /> ==Extended Reading==<br /> ===Foams===<br /> *Exerowa <br /> **Chapter 1. Formation and structure of foams; <br /> ***It is not possible to obtain stable (long-living) foams from pure liquid. Stable polyhedral foams are only formed in the presence of an appropriate surfactant (or surfactant mixture).&quot; p. 1<br /> ***Foams can be formed by condensation (decreasing pressure or increasing temperature, or using a chemical reaction) or dispersion (injecting bubbles into a liquid by various methods). p. 4<br /> **Chapter 3. Physical chemistry of foam films; <br /> **Chapter 11. Black foam films: Application in medicine. <br /> *Mysels<br /> **Chapters 1 – 7<br /> *Prud’homme <br /> **Chapter 1. Thin liquid film physics<br /> **Chapter 2. Structure, drainage, and coalescence of foams and concentrated emulsions<br /> **Chapter 3. Foam rheology<br /> ***&quot;Liquid foams may be viewed as structured fluids possessing a complex rheological behavior strongly dependent upon local structure and physiochemical constitution.&quot; p. 189 <br /> **Chapter 4. Experimental results on foam rheology <br /> *Norde <br /> **Chapter 18. Emulsions and foams<br /> ***Emulsions and foams are thermodynamically unstable dispersions. p. 366<br /> ***There dispersed phase is liquid, which means that their particles are deformable, the interface between the particles and the continuous phase is deformable, and the particles may coalesce. p. 366 <br /> *Weaire<br /> **Chapter 1. Introduction<br /> **Chapter 2. Local equilibrium rules<br /> **Chapter 3. Quantitative description of foam structures<br /> **Chapter 8. Rheology<br /> **Chapter 12 Foam collapse<br /> <br /> ===Emulsions===<br /> *Becher<br /> **Chapter 1. Introduction and definitions<br /> **Chapter 3. Physical properties of emulsions<br /> ***&quot;In all probability, in the majority of cases emulsion droplets less than about 0.2 microns are not common; the largest droplets in an emulsion may well be of the order of a hundred times this value.&quot; p. 65 <br /> **Chapter 4 - 6. Emulsion stability<br /> **Chapter 7 – 8. Emulsion instability<br /> *McClements<br /> **Chapter 6. Emulsion formation <br /> **Chapter 7. Emulsion stability <br /> **Chapter 8. Emulsion rheology <br /> *Sjöblom <br /> **Chapter 1. An experimental and theoretical approach to the dynamic behavior of emulsions; <br /> ***&quot;Practically all colloidal interactions are of a short range, almost never extending over distances greater than the size of the particle. Hence they have little influence over the transport of the particles, although they are crucial in determining the collision efficiency.&quot; p. 6<br /> ***&quot;The universal attractive forces between atoms and molecules, known as van der Waals forces, also operate between macroscopic objects and play a very important role in the interaction of colloidal particles. Indeed, without these forces, aggregations of particles would usually be prevented by the hydrodynamic interaction.&quot; p. 8 <br /> **Chapter 2. Spontaneous emulsification <br /> **Chapter 3 Symmetric thin liquid films with fluid interfaces <br /> ----<br /> [[Main_Page | Home]]</div> MichaelPetralia