# UserMichaelPetralia

## Contents

- 1 About me
- 2 A bit on notation
- 3 Fun facts on soft matter
- 4 Polymers in general
- 5 Polymer Gels
- 5.1 Introduction
- 5.2 Gelation (Notes on Chapter 6 of Rubenstein's
*Polymer Physics*[5]) - 5.3 The polymer gel network
- 5.4 Swelling of polymer gels (adapted from section 7.4 of Rubenstein's
*Polymer Physics*[5]) - 5.5 What knobs do we turn to get gels with large swelling ratios and appropriate mechanical properties?
- 5.6 Making interesting gel structures
- 5.7 Research questions

## About me

I am working on soft mechanical systems at the 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 *artificial muscles*, 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.

## A bit on notation

So I read the following convention in Rubenstein's * Polymer Physics *. 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?

<math> V = 4\pi/3R^3\simeq 4.2R^3\approx R^3 \sim \text{mass} </math>

- <math> \cong </math> is used to indicate a numerical approximation (i.e. <math>4\pi/3\simeq 4.2</math>)
- <math> \approx </math> is used to indicate that two quantities are proportional to each other up to a dimensionless prefactor of order unity (i.e. <math>4\pi/3</math>)
- <math> \sim \;</math> 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)

## Fun facts on soft matter

Typically, we think of changing the degree/type of cross-linking in order to change the mechanical properties of a polymer. Lengthening of a polymer is accomplished by aligning the polymer chains. It seems that there are polymers where the monomer-monomer interactions are not covalently bonded and thus the lengths of the polymer chains themselves can change.

“. . . in other long-chain objects the subunits are joined not by covalent bonds, but by physical ones. Examples of this are the giant worm-like micelles formed in some amphiphile solutions, and the long chains of compact protein molecules which constitute, for example, actin ﬁlaments. Such objects are sometimes called ‘living polymers’; their characteristic is that they can change their length in response to changes in the environment. This contrasts with the more usual covalently linked polymers, in which the length of the molecules, or the distribution of lengths, is ﬁxed during the polymerization process.” page 73, JonesSoft Condensed Matter

Additionally, we do not need to be so rigid in our thoughts on cross-linking.

“Linear polymers may be connected by physical, rather than chemical, bonds, giving a thermoreversible gel such as a gelatin. ” page 95, JonesSoft Condensed Matter

"...the crosslinks need not be produced by chemical reaction. Any physical process that favors association between certain (but not all) points on different chains may also lead to gels." page 133, de GennesScaling Concepts in Polymer Physics

## Polymers in general

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 *N* equal to the number of steps of the walk and the monomer length *b* equal to the step size [5, p. 69]:

<math>\langle \overrightarrow{R}^2\rangle =Nb^2</math>

### On the fractal nature of polymer conformations (adapted from [5, section 1.4])

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 * n * bond vectors between neighboring backbone atoms. Conformations are dependent on the:

- flexibility of the polymer chain
- interactions between monomers on the chain (can be attractive or repulsive)
- interactions between elements in the chain and the surroundings (either other chains or solvent)

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!

Most polymer conformations are fractal (i.e. self-similar) over a wide range of length scales. The fractal dimension <math> \mathcal{D} </math> is defined from the relation between the mass and radius of a fractal.

<math> m \sim r^{\mathcal{D}} </math>

If the size (radius of a sphere) changes by a factor <math> C_r </math>, so we have

<math> r_1 = C_r r_2 </math>

and the mass inside this sphere changes by a factor <math> C_m </math>, so we have

<math> m_1 = C_m m_2 </math>

then it is easy to show that the fractal dimension will be

<math> \mathcal{D} = \frac{\log C_m}{\log C_r} </math>

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), <math>N</math>.

<math> N \sim \langle R^2 \rangle </math>

A similar relation holds for any subsection of the ideal chain containing <math> g </math> monomers,

<math> g \sim \langle r^2 \rangle </math>

Directly from these relations, we can see that ideal polymer chain is self-similar and its fractal dimension is <math> \mathcal{D} =2 </math>. For non-ideal polymers, the fractal dimension will take on other values, but typically <math> \mathcal{D} < 3 </math>. The fractal dimension is useful, because it tells us how the length scales with the degree of polymerization (mass).

## Polymer Gels

### Introduction

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.

Jones tells us

“A gel is a material composed of subunits that are able to bond with each other in such a way that one obtains a network of macroscopic dimensions, in which all the subunits are connected by bonds. If one starts out with isolated subunits and successively adds bonds, one goes from a liquid (a sol) to a material with a non-zero shear modulus (a gel). A gel has the mechanical properties characteristic of a solid, even though it is structurally disordered and indeed may contain a high volume fraction of liquid solvent.” page 95, JonesSoft Condensed Matter[3]

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 [1]. This phase change is a result of a shift in which forces dominate (entropic, attractive, repulsive).

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. [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.

### Gelation (Notes on Chapter 6 of Rubenstein's * Polymer Physics * [5])

- Gelation is the transition from liquid to solid by the formation of crosslinks between polymer chains.
- Jello's crosslinks are induced by microcrystallization upon cooling. Epoxy's crosslinks are induced by covalent bonds formed upon mixing.
- We define a
*sol*as the polydisperse mixture of branched polymers obtained as the result of linking linear polymer strands together, since the molecules are still soluble. - At some point (as crosslinking continues) a branched polymer will span the entire network. This is the
*gel point*or*sol-gel transition*and it is defined as the first point at which the system contains an "infinite polymer", 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. - 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:
- 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
**critical percolation**model. - vulcanization: is the process of crosslinking long linear chains that start out strongly overlapping each other. It is well described by the
**mean-field percolation**model. - 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
**kinetic gelation**model.

- 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
- 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.
- The fraction of the material that is part of the gel (<math> P_{gel} </math>) grows steadily above the gel point (<math> p_c </math>). <math> p </math> is the extent of the reaction (fraction of all possible bonds that are formed), and <math> \sigma </math> and <math> \tau </math> are critical exponents determined by the amount of overlap (dimensionality) linking the species being gelled. If there is significant overlap, <math>\tau = 5/2 </math>, <math> \sigma = 1/2 </math>, and <math> \beta=1 </math>.

<math> P_{gel} \sim \left(\frac{p-p_c}{p_c}\right)^\beta </math> where <math> \beta= \frac{\tau -2}{\sigma} </math>

- When the extent of reaction (<math>p</math>) reaches approximately twice the gel point (<math>p_c</math>), nearly all of the monomers are attached to the gel in a single macroscopic network polmyer.

### The polymer gel network

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 [4]:

- The situation at the moment of preparation (preparative ensemble)
- The situation at the moment of study (final ensemble)

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. [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.

In gels which are chemically crosslinked, there is a threshold of the number of crosslinks, <math> p_c </math>, at which the polymer network is connected throughout the material. [4] We define <math> p </math> as the fraction of reacted bonds and <math> \Delta p </math> as some small fraction compared to <math> p_c </math>. Percolation theory will be used to understand the behavior of gels around the gelation threshold, <math> p_c </math>. Though the gel fraction (<math> S_\infty </math>, the fraction of monomers belonging to the infinite polymer network) increases rapidly with <math> \Delta p </math>, the elastic modulus of the gel (<math> E </math>) increases much more slowly.

<math> S_\infty \simeq \Delta p^\beta </math> where <math> \beta = 0.39 \;</math>

<math> E \simeq \Delta p^t </math> where <math> t \sim 1.7 \;\text{to}\; 1.9 </math> for three dimensions.

### Swelling of polymer gels (adapted from section 7.4 of Rubenstein's * Polymer Physics * [5])

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 <math> N </math> with the number of monomers in an entanglement chain <math> N_e </math> in the preparation state, the shear modulus can be written as

<math> G_e = \frac{\rho \mathcal{R} T}{N_eM_o} </math>

where <math> \mathcal{R} </math> is the gas constant, <math> T </math> is the temperature, and <math> M_o </math> is the molar mass of the monomer. An entanglement chain is the section of a polymer chain between two entanglement contacts. [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.

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:

<math> F\approx kT\frac{R^2}{Nb^2} </math>

where <math> k </math> is the Boltzman constant, <math> R </math> is the end-to-end distance, <math> N </math> is the number of monomers, and <math> b </math> is the monomer size. Assuming affine deformations, the mean square end-to-end distance of a polymer chain with original end-to-end distance <math> R_0 </math> due to a stretch <math> \lambda </math> is <math> R^2=\left(\lambda R_0\right)^2 </math>. It is also appropriate to define <math> R^2_{ref} </math> as the mean-square fluctuation of the end-to-end distance of the network strand, and rewrite the free energy in the Panyukov form:

<math> F\approx kT\frac{\left(\lambda R_0\right)^2}{ R^2_{ref} } </math>

<math> R^2_{ref} </math> changes as the network swells (or the quality of the solvent changes), which is reflected in a change in network elasticity. The modulus <math> G </math> of the swollen gel is proportional to the chain number density <math> \nu = \phi / \left(Nb^3\right) </math> time the elastic free energy per chain, where <math> \phi =V_{dry}/V</math> is the volume fraction of a dry gel to total gel volume (i.e. the a measure of the amount of polymer to solvent).

<math> 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}} </math>

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 <math> G = \nu kT </math> where here <math> \nu=n/V </math> is the number of network chains <math> n </math> per unit volume with an extra term to account for the effect of swelling on stiffening the network. The <math> \phi </math> 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.

Any gel swells until the modulus and the osmotic pressure <math> \Pi </math> 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 <math> Q </math> as the ratio of the volume in the fully swollen state to the volume in the dry state when <math> G\approx \Pi </math>:

<math> Q \equiv \frac{V_{eq}}{V_{dry}} </math>

#### Swelling in <math>\theta</math>-solvents

In a <math>\theta</math>-solvent, the mean-square end-to-end distance of a free chain is independent of concentration as are the fluctuations that control its elasticity:

<math> R^2_{ref} \approx R^2_0 \approx Nb^2 </math>

So the Panyukov form of the modulus reduces to the Flory form for swelling in <math>\theta</math>-solvents, which we can rewrite in terms of volume fractions <math> \lambda = \left(\phi_0/\phi\right)^{1/3} </math>:

<math> 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} </math>

where <math> \phi_0 </math> 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 <math> \lambda^2 </math> dependence, and the modulus increases. The net effect is a weak decrease of the gel modulus upon swelling which goes like <math> \phi^{1/3} </math>.

The osmotic pressure of a semidilute solution in a <math>\theta</math>-solvent is approximated by

<math> \Pi \approx \frac{kT}{b^3}\phi^3 </math>

So to find the the equilibrium swelling ratio we write,

<math> \frac{kT}{b^3}\phi^3 \approx \frac{kT}{Nb^3} \phi_0^{2/3}\phi^{1/3} </math>

<math> \phi^{8/3} \approx \frac{1}{N} \phi_0^{2/3}</math>

<math> \left(\frac{V_{dry}}{V}\right)^{8/3} \approx \frac{\phi_0^{2/3}}{N}</math>

<math> Q_{\theta solvent} \approx \frac{V}{V_{dry}} \approx \frac{N^{3/8}}{\phi_0^{1/4}}</math>

The modulus at equilibrium is thus,

<math> 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}</math>

#### Swelling in athermal solvents

For an polymer network in an athermal solvent we use,

<math> R_0 \approx bN^{1/2} \phi_0^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} </math>

<math> R \approx bN^{1/2} \phi^{-\left(\nu -1/2\right)/\left(3\nu-1\right)} </math>

with <math> \nu \cong 0.588 </math>. This makes the gel modulus,

<math> 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}</math>

The osmotic pressure in an athermal solvent is approximated as,

<math> \Pi \approx \frac{kT}{b^3}\phi^{3\nu/\left(3\nu-1\right)} </math>

So the equilibrium swelling ratio is

<math> 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}}</math>

Comparing <math> Q_{athermal} </math> to <math> Q_{\theta solvent} </math> we find that a network swells more in an athermal solvent than in a <math>\theta</math>-solvent.

The modulus at equilibrium is,

<math> 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} </math>

Note that the larger power for <math>N</math> 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 <math>\phi_0</math> will have the opposite effect, but it is small compared to the effect of the number of monomers per chain.

### What knobs do we turn to get gels with large swelling ratios and appropriate mechanical properties?

It is clear from the previous section that there are a number of different "knobs" to turn when it comes to swelling ratios and mechanical properties. We use the shear modulus <math>G</math> as the "strength" 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 <math> kT </math>. 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, <math>b</math>, which effectively cancels out because it has the same effect on the osmotic pressure as the shear modulus.

### Making interesting gel structures

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.

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). [4]

de Gennes also mentions the hysteresis associated with the 'melting' and solidification of gelatin (1.5%)/water solutions [4]: At high temperatures the solution is a liquid (simple solution of chains, *sol*). 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. [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. [5]

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. [4] Either way, by changing the solvent, or the temperature, we should be able to drastically affect the mechanical properties of the gel.

### Research questions

- 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?

**References:**

[1] Brock, MIT A.I. Memo No. 1330 (1994)

[2] Madden et al., IEEE Journal of Oceanic Engineering **29**, 706 (2004)

[3] Jones, *Soft Condensed Matter* (2002)

[4] de Gennes, *Scaling Concepts in Polymer Physics * (1979)

[5] Rubinstein, * Polymer Physics * (2003)

[6] Elias, * Macromolecules: Volume 3: Physical Structures and Properties * (2008)