Polymer molecules
Contents
- 1 Common polymers
- 2 Isomerism
- 3 Types of Polymerization
- 4 Polymer dimensionality and structure
- 5 Renormalization and scaling - The random walk polymer
- 6 What is the energy to stretch a polymer from random packing to fully extended?
- 7 How dense is the polymer?
- 8 How does density vary within the polymer chain?
- 9 Self-avoidance - Flory theory
- 10 Universal Ratios
- 11 Estimating the fractal dimension D
- 12 Polymer degradation
Common polymers
Polyethylene – Cheap plastic bags
Polypropylene – Labware, dishwasher safe!
Polystyrene – Plastic cups and Styrofoam
Polyisoprene – Natural rubber
When samples of rubber first arrived in England, it was observed by Joseph Priestley, in 1770, that a piece of the material was extremely good for rubbing out pencil marks on paper, hence the name "rubber".[1]
This can also be made surprisingly well with an organic reaction using the Ziegler-Natta catalyst.
Polybutadiene – Synthetic rubber
Polyethylene oxide
Water soluble, used in paper making process. A commonly used reagent in biological studies is PEG, a low molecular weight polyethylene oxide, used in osmotic pressure experiments and as a means of concentrating virus particles.
Polydimethylsiloxanes
Silicones, PDMS~ used in soft lithography in making microfluidic devices, also part of Silly Putty
Polyesters
Synthetic fibers, toners
Polypeptides
Polypeptides are defined to be chains of amino acids. Proteins are made out of one or more polypeptide chains. In order to form a polypeptide chain, amino acids are connected together using peptide bonds. For example, we can see a tripeptide molecule in the image below.
[Reference: http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/P/Polypeptides.html]
Proteins exhibit four levels of structure. Primary structure is considered to be the chain of amino acids that create a polypeptide chain. Secondary structure is the ordered arrangement of amino acids in localized regions of a polypeptide molecule. Hydrogen bonding plays important role in stabilizing these patterns. Tertiary structure of a polypeptide is a three-dimensional arrangement of the atoms within one single polypeptide chain. Finally, quaternary structure is used to describe proteins made of multiple polypeptides. Hydrophobic interaction is the primary force responsible for stabilizing subunits (polypeptides) in quaternary structure.
Polypeptide have to be terminated in a specific manner. One end has to be amino-terminal (ends in nitrogen group), whereas the other has to be carboxyl-terminal (ends in carbon group). An example is shown below.
[Reference: http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/P/Polypeptides.html]
Isomerism
Isomers are chemical compounds with the same chemical formula, but different structural formulas. Specifically, isomers have the same number of each element, but they are topologically or geometrically distinct.
For a topological example, consider the following three molecules I,II,III:
Each of these are isomers, however we can see that they are topologically distinct, in the sense that neither can be deformed into the other (without breaking bonds). For example, I has an O atom connected to a C atom which is connected to 2 H atoms, II has an O atom connected to a C atom that is connected to 2 other C atoms, and III has an O atom connected to 2 C atoms. Each of these properties is distinct to that isomer.
When two compounds are topologically identical but geometrically distinct, they are called stereoisomers. As an example of what this means, consider the following molecules:
The top and bottom molecules can clearly be deformed into each other by simply rotating the C=C double bond. Thus we would call these stereoisomers, because they are spatially different, even though they have the same connectivity. Because double-bonds can be considered rigid, these are chemically distinct molecules. Stereoisomers can have radically different chemical properties, largely due to the fact that spatial arrangement can change the dipole moment of the molecule significantly. For example, he two stereoisomers of 2-butenedioic acid are so different that they are given different names: maleic acid fumaric acid.
Isomerism is important in polymers. For example, if the monomers in a polymer are not symmetric, then head-to-tail is different than tail to head.
- Atatic polymers
- Syndiotatic polymers
- Isotatic polymers
- Copolymers
- Random
- Diblock and triblock coploymers
- Coplymers of ethylene and propylene are disordered enough to remain liquid at lower temperatures than either homopolymer.
- Diblock and triblock copolymers can (partially) self-associate
Types of Polymerization
(Witten p.45-46)
-Addition polymerization: a catalyst initiates polymerization in a solution of monomers; each chain has a single active end that reacts only with monomers --> molecular uniformity (example: polypeptides and polysaccharides)
-Condensation polymerization: many chains may react with one another (example: polyamide nylon)
-Living polymerization: various chains are free to exchange monomers amongst themselves--> broad distribution of chain lengths (worm-like micelles)
Polymer dimensionality and structure
- Distinctiveness – Length & flexibility?
- Polymers are ordered and inflexible in 1D
- Polymers are random and flexible in 2 and 3 D
- Polymers are tenuous in 2 and 3 D
- Polymers “self-avoid” in 2 and 3 D
- Some are constrained – DNA and RNA
- Scaling with molecular weight
- Scaling with structure – more difficult
- Scaling of diffusion and flow
Kinks and rings in confined polymer structures
Some polymers naturally tend to form filamentous structures rather than blobs. Many of the biological polymers (or biopolymers) follow that behavior: actin, fibrin, collagen, tubulin, DNA, RNA and others. Carbon nanotubes can also form very strong linear structures, one of their most promising and applicable properties.
Like many filaments, those just mentioned have a finite bending stiffness, which means that they have a characteristic length-scale over which they will appear rigid or straight: the persistence length. Segments shorter than the persistence length will behave like rigid beams, while longer segments will appear more floppy. In the limit of very long segments, the filament can fold on itself and behave more like a blob.
An interesting question arises when these (bio)polymers are forced to grow in a confined space, smaller than their persistence length. This question comes naturally for biological systems like mammalian cells, where the cytoskeleton is comprised of polymeric filaments whose persistence length is on the order of the cell size (e.g. actin) or much larger than the cell (e.g. tubulin). Another example of this is given in an interesting applied math paper by Cohen and Mahadevan (PNAS 2002)[2], where they talk about growing carbon nanotubes confined in air bubbles. A common phenomenon observed with cytoskeletal proteins and nanotubes, is that they tend to form rings or kinks when growing in a confined space; even more interesting is the fact that the angles of the kinks they form typically take discrete values rather than a broad continuous spectrum.
A very simple explanation for a discrete kink-angle lies in the fact that these filamentous structures are composed of multiple thin single chains of monomers and that the interaction between these monomers causes them to want to be spaced out (laterally and longitudinally) at very specific length-scales. The longitudinal spacing is obviously the same or close to the size of the monomer and the lateral spacing between single chains is determined by the monomer size and the strength of the attraction between chains. It is clear that when we introduce bending into these multi-chain filaments, some of the chains will have a greater curvature than others and this will tend to create an offset in the monomer positions and how they fit together. Because there is an energy associated with creating local curvature in a filament, it is better to create one point of high curvature than to create many points of low curvature. This will follow a similar behavior to that described by the Frenkel-Kontorova model, which provides a microscopic explanation for the formation of certain dislocations in solids.
So because these polymers have strong inter-chain interactions and are formed of discrete monomer units, they tend to prefer a rigid rod state locally and will sharply kink to maintain some level of straightness. The paper by Cohen and Mahadevan provides some additional examples of kinks and rings in filamentous structures.
Renormalization and scaling - The random walk polymer
Probability that the end-to-end vector has some, r , for n segments: | <math>p\left( n,r \right)</math> is a probability per volume |
The chain must have some length | <math>\int{p\left( n,r \right)}d^{3}r=1</math> |
If <math>p\left( n,r \right)</math> be known, what be known about <math>p\left( n+1,r \right)</math>?
Assume segment is flexible, but no self-avoidance. Probability depends on magnitude only. | <math>p_{0}\left( r_{1} \right)</math> |
Probability is the product | <math>p\left( n,\vec{r}-\vec{r}_{1} \right)p_{0}\left( \vec{r}_{1} \right)=p\left( n,\vec{r}-\vec{r}_{1} \right)p_{0}\left( r_{1} \right)</math> |
For all possible <math>\vec{r}_{1}</math> that have <math>\vec{r}_{{}}-\vec{r}_{1}</math> is | <math>p\left( n+1,r \right)=\int{p_{0}\left( r_{1} \right)}p\left( n,\vec{r}-\vec{r}_{1} \right)d^{3}r_{1}</math> |
Consider the case of large 'n'.
This difference should be small: | <math>p\left( n+1,r \right)-p\left( n,r \right)=\int{p_{0}\left( r_{1} \right)}\left[ p\left( n,\vec{r}-\vec{r}_{1} \right)-p\left( n,r \right) \right]d^{3}r_{1}</math> |
A Taylor expansion around <math>\vec{r}_{1}</math> is: | <math>p\left( n,\vec{r}-\vec{r}_{1} \right)=p\left( n,r \right)-\vec{r}_{1}\cdot \nabla _{r}p\left( n,r \right)+\frac{1}{6}r_{1}^{2}\nabla _{r}^{2}p\left( n,r \right)+\cdots </math> |
If <math>\Delta n=1</math> the RHS remains the same but the LHS is: | <math>\underset{n\to \infty }{\mathop{\lim }}\,\frac{p\left( n+\Delta n,r \right)-p\left( n,r \right)}{\Delta n}\to \frac{dp}{dn}</math> |
And we have: | <math>\frac{dp}{dn}=-\nabla _{r}p\left( n,r \right)\cdot \int{\vec{r}_{1}p_{0}\left( r_{1} \right)d^{3}r_{1}}+\frac{1}{6}\nabla _{r}^{2}p\left( n,r \right)\int{r_{1}^{2}p_{0}\left( r_{1} \right)d^{3}r_{1}}+\cdots </math> |
All odd integrals are zero, therefore: | <math>\frac{dp}{dn}\cong \frac{1}{6}\nabla _{r}^{2}p\left( n,r \right)\int{r_{1}^{2}p_{0}\left( r_{1} \right)d^{3}r_{1}}+constant\cdot \nabla _{r}^{4}p+\cdots </math> |
The integral is <math>\left\langle r^{2} \right\rangle _{1}</math>, therefore: | <math>\frac{dp}{dn}\cong \frac{1}{6}\left\langle r^{2} \right\rangle _{1}\nabla _{r}^{2}p\left( n,r \right)+constant\cdot \nabla _{r}^{4}p+\cdots </math> |
Now the first scaling idea is introduced.
The “shape” of the distribution should be independent of n. |
<math>\begin{align} & p\left( n,r \right)=\eta p\left( \lambda n,\mu r \right) \\ & p\left( n,r \right)=\tilde{p}\left( \tilde{n},\tilde{r} \right) \\ \end{align}</math> |
Where: | <math>\tilde{p}=\eta p,\text{ }\tilde{n}=\lambda n\text{ and }\tilde{r}=\mu r</math> |
Hence: | <math>\begin{align}
& \frac{d}{dn}=\lambda \frac{d}{d\tilde{n}} \\ & \nabla _{r}^{2}=\mu ^{2}\nabla _{{\tilde{r}}}^{2}\text{ and }\nabla _{r}^{4}=\mu ^{4}\nabla _{{\tilde{r}}}^{4} \\ \end{align}</math> |
Or: | <math>\frac{d\tilde{p}}{d\tilde{n}}\cong \frac{1}{6}\frac{\mu ^{2}}{\lambda }\left\langle r^{2} \right\rangle _{1}\nabla _{{\tilde{r}}}^{2}\tilde{p}\left( \tilde{n},\tilde{r} \right)+\text{C}\frac{\mu ^{4}}{\lambda }\nabla _{{\tilde{r}}}^{4}\tilde{p}+\cdots </math> |
Now the second scaling idea is introduced.
If n can be arbitrarily large: | <math>\mu =\lambda ^{{1}/{2}\;}</math> |
Consider the normalization: |
<math>\begin{align} & \int{p\left( n,r \right)}d^{3}r=1 \\ & \int{\eta p\left( \lambda n,\mu r \right)}d^{3}r=1 \\ & \frac{\eta }{\mu ^{3}}\int{p\left( \lambda n,\mu r \right)}d^{3}\left( \mu r \right)=1 \\ & \therefore \text{ }\frac{\eta }{\mu ^{3}}=1 \\ \end{align}</math> |
Or: |
<math>\eta =\lambda ^{{3}/{2}\;}</math> |
Since <math>\lambda </math> is arbitrary, it can be set to <math>{1}/{n}\;</math> and: | <math>\tilde{p}=n^{{-3}/{2}\;}p,\text{ }\tilde{n}=1\text{ and }\tilde{r}=n^{{-1}/{2}\;}r</math> |
The distribution | <math>p\left( n,r \right)=n^{-{3}/{2}\;}p\left( \tilde{n},\tilde{r} \right)_{\tilde{n}=1,\tilde{r}=rn^{-{1}/{2}\;}}</math> |
is now: | <math>p\left( n,r \right)=n^{-{3}/{2}\;}f\left( rn^{-{1}/{2}\;} \right)</math> |
The moments of the distribution are: | <math>\begin{align}
& \left\langle r^{p} \right\rangle _{n}=\int{r^{p}\cdot n^{-{3}/{2}\;}}f\left( rn^{-{1}/{2}\;} \right)d^{3}r \\ & \text{ }=n^{{p}/{2}\;}n^{-{3}/{2}\;}n^{{3}/{2}\;}\int{\left( rn^{-{1}/{2}\;} \right)^{p}\cdot }f\left( rn^{-{1}/{2}\;} \right)d^{3}\left( rn^{-{1}/{2}\;} \right) \\ & \text{ }=n^{{p}/{2}\;}\int{\left( {\hat{r}} \right)^{p}\cdot }f\left( {\hat{r}} \right)d^{3}\hat{r} \\ \end{align}</math> |
Since the integral is a constant: |
<math>\left[ \left\langle r^{p} \right\rangle _{n} \right]^{{1}/{p}\;}=K_{p}n^{{1}/{2}\;}</math> |
Now the solution to | <math>\frac{dp}{dn}=\frac{1}{6}\left\langle r^{2} \right\rangle _{1}\nabla _{{}}^{2}p\left( n,r \right)</math> |
is: | <math>p\left( n,r \right)=\left[ {2\pi n\left\langle r^{2} \right\rangle _{1}}/{3}\; \right]^{-{3}/{2}\;}\exp \left[ -\frac{3r^{2}}{2n\left\langle r^{2} \right\rangle _{1}} \right]</math> |
What is the energy to stretch a polymer from random packing to fully extended?
The only component is entropy, hence | _{initial}^{final}=\frac{3}{2}kT\frac{r^{2}}{\left\langle r^{2} \right\rangle }</math> |
How dense is the polymer?
Remember - No liquid has been mentioned!!
What is the "size" of a polymer?
Take any moment of the end-to-end distance: <math>\left[ \left\langle r^{p} \right\rangle _{n} \right]^{{1}/{p}\;}=K_{p}n^{{1}/{2}\;}\sim n^{{1}/{2}\;}</math>
What's the number density? <math>\bar{\rho }\sim \frac{n}{\left\langle r \right\rangle _{n}^{3}}\sim \frac{n}{n^{{3}/{2}\;}}\sim n^{-{1}/{2}\;}</math>
The greater the molecular weight, the more tenuous the polymer.
How does density vary within the polymer chain?
Probability of each segment at r: | <math>\left\langle \rho \left( r \right) \right\rangle _{0}=2\int\limits_{0}^{\infty }{i^{-{3}/{2}\;}}f\left( ri^{-{1}/{2}\;} \right)di</math> |
Define a scaled variable: <math>\tilde{i}=i^{{1}/{2}\;}r^{-1}</math> | <math>\left\langle \rho \left( r \right) \right\rangle _{0}=2r^{2}r^{-3}\int\limits_{0}^{\infty }{\tilde{i}^{-{3}/{2}\;}}f\left( \tilde{i}^{-1} \right)d\tilde{i}</math> |
The integral is the local density: | <math>\left\langle \rho \left( r \right) \right\rangle _{0}={\text{constant}}/{r}\;</math> |
The average density is: | <math>\left\langle M\left( r \right) \right\rangle =\int\limits_{{r}'<r}{\left\langle \rho \left( {{r}'} \right) \right\rangle _{0}}d^{3}{r}'=\int\limits_{{r}'<r}{{r}'^{2}\left\langle \rho \left( {{r}'} \right) \right\rangle _{0}}d{r}'=\left( \text{constant} \right)r^{2}</math> |
Therefore the random walk polymer has a fractal dimension of D = 2.
Self-avoidance - Flory theory
Purely locally or global models do not change the scaling of density.
Assume that self-avoidance occurs on all length scales.
On each and every length scale the polymer is expanded by a factor b.
Replace: <math>p\left( n,r \right)=n^{-{3}/{2}\;}f\left( rn^{-{1}/{2}\;} \right)</math> with: <math>p\left( n,r \right)=n^{-\nu d}f\left( rn^{-\nu } \right)</math>
This preserves the normalization and the end-to-end distance as <math>n^{-\nu }</math>
And the previous results are similar: <math>\left\langle \rho \left( r \right) \right\rangle _{0}=\left( \text{constant} \right)r^{{1}/{\nu -3}\;}</math>
and <math>\left\langle M\left( r \right) \right\rangle =\left( \text{constant} \right)r^{{1}/{v}\;}</math>
Universal Ratios
In the previous discussion, we have seen that a polymer has a single asymptotic probability distribution function p(n,r) for any random walk. This is true for all polymers, irregardless of the specific details of how the random walk was created. Furthermore, we have explored the fact that all self-repelling polymers exhibit common behavior, independent of the specifics of the repulsion. Since the aforementioned characteristics are seen for any polymer, we call them universal. Still, even in these universal functions, there is still room for choosing an arbitrary scaling coefficients e.g. <math>\mu</math> or <math>\lambda</math>. Therefore, some quantities are not going to be dependent only on p(n,r). But quantity like <math><r^4>/<r^2>^2</math> has both <math><r^4></math> and <math><r^2></math> scaling as <math>\mu^4</math> which cancels out so we can conclude <math><r^4>/<r^2>^2</math> is only dependent on prabability distribution function, and not on any scaling properties. These kinds of ratios independent of scaling are referred to as universal ratios.
[For more information, see Witten p. 72-73]
Estimating the fractal dimension D
Consider the work to expand a polymer from the “ideal” state. |
<math>U\sim kT\frac{r^{2}}{\left\langle r^{2} \right\rangle }\sim kT\frac{r^{2}}{n}</math> |
Think of self-avoiding polymers as ones with slightly repulsive interactions. | |
The number of (non-local) contacts n times the probability that a monomer has a contact: | <math>V\simeq n\left( {\nu \cdot n}/{r^{3}}\; \right)U^{contact}\sim U^{contact}\frac{n^{2}}{r^{3}}</math> |
Equating the energies and re-arranging: |
<math>\begin{align} & kT\frac{r^{2}}{n}\sim U^{contact}\frac{n^{2}}{r^{3}} \\ & n\sim r^{{5}/{3}\;} \\ \end{align}</math> |
Therefore the self-avoiding polymer has a fractal dimension of D = 5/3
What is a fractal dimension, anyway?
This is the wikipedia page for it (since we all know that that is a fantastically reliable resource): Fractal Dimension
For a biological reference, I found this article on the fractal analysis of lysozyme interesting; I had never heard of fractals in bio polymers, so this was a neat one to read through. Here is the reference, since I'm not sure how GNU this is (you can snag it through the Harvard library system, though):
Rigid structure of fractal aggregates of lysozyme G. C. Fadda et al 2000 Europhys. Lett. 52 712-718
Abstract: The aggregation of hen egg-white lysozyme upon salt addition was studied by quasi-elastic light scattering. Our results agree with the fractal structure of the aggregates already reported in the literature. However, we also demonstrate that these aggregates are rigid, since they do not display any fluctuation of internal concentration. Such a rigid internal structure is a key point to reconcile the fractal structure of the aggregates and their colloid-like ordering. Furthermore, this result has to be considered for understanding crystal nucleation.
--BPappas 22:09, 25 October 2008 (UTC)
Polymer degradation
Polymer degradation is a change in the properties—tensile strength, colour, shape, etc.—of a polymer or polymer-based product under the influence of one or more environmental factors, such as heat, light or chemicals. It is often due to the hydrolysis of the bonds connecting the polymer chain, which in turn leads to a decrease in the molecular mass of the polymer. These changes may be undesirable, such as changes during use, or desirable, as in biodegradation or deliberately lowering the molecular mass of a polymer. Such changes occur primarily because of the effect of these factors on the chemical composition of the polymer. Ozone cracking and UV degradation are specific failure modes for certain polymers.
The degradation of polymers to form smaller molecules may proceed by random scission or specific scission. The degradation of polyethylene occurs by random scission—a random breakage of the linkages (bonds) that hold the atoms of the polymer together. When heated above 450°C it degrades to form a mixture of hydrocarbons. Other polymers—like polyalphamethylstyrene—undergo specific chain scission with breakage occurring only at the ends. They literally unzip or depolymerize to become the constituent monomer.
However, the degradation process can be useful from the viewpoints of understanding the structure of a polymer or recycling/reusing the polymer waste to prevent or reduce environmental pollution. Polylactic acid and polyglycolic acid, for example, are two polymers that are useful for their ability to degrade under aqueous conditions. A copolymer of these polymers is used for biomedical applications, such as hydrolysable stitches that degrade over time after they are applied to a wound. These materials can also be used for plastics that will degrade over time after they are used and will therefore not remain as litter.