# Polymer molecules

## Common polymers

• Polyethylene – Cheap plastic bags
• Polypropylene – Labware, dishwasher safe!
• Polystyrene – Plastic cups and Styrofoam
• Polyisoprene – Natural rubber
• Polyethylene oxide – Water soluble
• Polydimethylsiloxanes – Silicones
• Polyesters – Synthetic fibers, toners
• Polypeptides - Proteins

## 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
Witten
• Syndiotatic polymers
Witten
• Isotatic polymers
Witten
• 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

## 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

## How dense is a polymer?

What is the "size" of a polymer? Take any moment of the end-to-end distance:

$\left[ \left\langle r^{p} \right\rangle _{n} \right]^{{1}/{p}\;}=K_{p}n^{{1}/{2}\;}\sim n^{{1}/{2}\;}$

What's the number density?

$\bar{\rho }\sim \frac{n}{\left\langle r \right\rangle _{n}^{3}}\sim \frac{n}{n^{{3}/{2}\;}}\sim n^{-{1}/{2}\;}$

The greater the molecular weight, the more tenuous the polymer.

## How does density vary within the polymer chain?

Probability of each segment at r:

$\left\langle \rho \left( r \right) \right\rangle _{0}=2\int\limits_{0}^{\infty }{i^{-{3}/{2}\;}}f\left( ri^{-{1}/{2}\;} \right)di$

Define a scaled variable: $\tilde{i}=i^{{1}/{2}\;}r^{-1}$

$\left\langle \rho \left( r \right) \right\rangle _{0}=2r^{2}r^{-3}\int\limits_{0}^{\infty }{\tilde{i}^{-{3}/{2}\;}}f\left( \tilde{i}^{-1} \right)d\tilde{i}$

The integral is the local density: $\left\langle \rho \left( r \right) \right\rangle _{0}={\text{constant}}/{r}\;$

The average density is:

$\left\langle M\left( r \right) \right\rangle =\int\limits_{{r}'<r}{\left\langle \rho \left( {{r}'} \right) \right\rangle _{0}}d^{3}{r}'=\int\limits_{{r}'<r}{{r}'^{2}\left\langle \rho \left( {{r}'} \right) \right\rangle _{0}}d{r}'=\left( \text{constant} \right)r^{2}$

Therefore the random walk polymer has a fractal dimension of D = 2.

## Self-avoidance and fractal structure

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: $p\left( n,r \right)=n^{-{3}/{2}\;}f\left( rn^{-{1}/{2}\;} \right)$ with: $p\left( n,r \right)=n^{-\nu d}f\left( rn^{-\nu } \right)$

This preserves the normalization and the end-to-end distance as $n^{-\nu }$

And the previous results are similar: $\left\langle \rho \left( r \right) \right\rangle _{0}=\left( \text{constant} \right)r^{{1}/{\nu -3}\;}$

and $\left\langle M\left( r \right) \right\rangle =\left( \text{constant} \right)r^{{1}/{v}\;}$