# Polymer forces

## Polymer ordering at surfaces

Israelachvili, Fig. 13.13

PEO has an “inverse” water solubility – it becomes less soluble at higher temperatures.

A general principle:

The less soluble the adsorbed polymer – the less stable the dispersion.

PEO has a nonpolar part but also an oxygen atom that allows for hydrogen bonding. Thus, at lower temperatures, these molecules can interact with water, but as temperature increases, hydrogen bonding weakens, and the nonpolar part of the polymer chain begins to dominate. The molecules become more hydrophobic. And when molecules are hydrophobic and not soluble, they want to form clumps and come closer together. That is why the equilibrium energy decreases with temperature.

## MW and temperature effects

Israelachvili, Fig. 14.5

Polystyrene polymers on mica. Polystyrene is a hydrocarbon chain with phenyl group rings attached to every second carbon, and is therefore hydrophobic.

(a) End-grafted in toluene, (b) Adsorbed from cyclohexane.

Both solubility and bridging effects are possible in (b)

So, when the polymers are put in toulene (a), they are soluble and repel each other more. When they are put in cyclohexane, they are not soluble, and thus minimize energy by clumping together more.

## Polymers at surfaces

Israelacvili, Fig. 14.5
• (a) In solution
• (b) End-grafted
• (d) Adsorbed at low &thetha;
• (e) Adsorbed at high &thetha;
• (f) Bridging

## Polymer Elasticity

Rubber elasticity, also known as hyperelasticity, describes the mechanical behavior of many polymers, especially those with crosslinking. Invoking the theory of rubber elasticity, one considers a polymer chain in a crosslinked network as an entropic spring. When the chain is stretched, the entropy is reduced by a large margin because there are fewer conformations available. Therefore, there is a restoring force, which causes the polymer chain to return to its equilibrium or unstretched state, such as a high entropy random coil configuration, once the external force is removed. This is the reason why rubber bands return to their original state. Two common models for rubber elasticity are the freely-jointed chain model and the worm-like chain model.

Polymers can be modeled as freely jointed chains with one fixed end and one free end (FJC model) where $b \,$ is the length of a rigid segment, $n \,$ is the number of segments of length $b \,$, $r \,$ is the distance between the fixed and free ends, and $L_c \,$ is the "contour length" or $nb \,$. Above the glass transition temperature, the polymer chain oscillates and $r \,$ changes over time. The probability of finding the chain ends a distance $r \,$ apart is given by the following Gaussian distribution:

$P(r,n)dr = 4 \pi r^2\left( \frac{2 n b^2 \pi}{3}\right)^{-3/2} \exp \left( \frac{-3r^2}{2nb^2} \right) dr \,$

Note that the movement could be backwards or forwards, so the net time average $\langle r\rangle$ will be zero. However, one can use the root mean square as a useful measure of that distance.

$\langle r\rangle = 0 \,$
$\langle r^2\rangle = nb^2 \,$
$\langle r^2\rangle^{1/2} = \sqrt{n} b \,$

Ideally, the polymer chain's movement is purely entropic (no enthalpic, or heat-related, forces involved). By using the following basic equations for entropy and Helmholtz free energy, we can model the driving force of entropy "pulling" the polymer into an unstretched conformation. Note that the force equation resembles that of a spring: F=kx.

$S = k_B \ln \Omega \, \approx k_B \ln ( P(r,n) dr ) \,$
$A \approx -TS = -k_B T \frac{3 r^2}{2 L_c b} \,$
$F \approx \frac{-dA}{dr} = \frac{3 k_B T}{L_c b} r \,$

The worm-like chain model(WLC) takes the energy required to bend a molecule into account. The variables are the same except that $L_p \,$, the persistence length, replaces $b \,$. Then, the force follows this equation:

$F \approx \frac{k_B T}{L_p} \left ( \frac{1}{4 \left( 1- \frac{r}{L_c} \right )^2} - \frac{1}{4} + \frac{r}{L_c} \right ) \,$

Therefore, when there is no distance between chain ends (r=0), the force required to do so is zero, and to fully extend the polymer chain ($r=L_c \,$), an infinite force is required, which is intuitive. Graphically, the force begins at the origin and initially increases linearly with $r \,$. The force then plateaus but eventually increases again and approaches infinity as the chain length approaches $L_c \,$.

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