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# Rubber 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.

### Freely-Jointed 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:

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

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

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

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.

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

### Worm-Like Chain Model

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:

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

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 $L_c ,$