### Thermodynamic theory of polymer elasticity

**Rubber elasticity**, a well-known example of hyperelasticity, describes the mechanical behavior of many polymers, especially those with Cross-link.

## Contents

## Thermodynamics

Temperature affects the elasticity of elastomers in an unusual way. Heating causes them to contract, and cooling causes expansion.^{[1]}
This can be observed with an ordinary rubber band. Stretching a rubber band will cause it to release heat (press it against your lips), while releasing it after it has been stretched will lead it to absorb heat, causing its surroundings to become cooler. This phenomenon can be explained with Gibb's Free Energy. Rearranging Δ*G*=Δ*H*−*T*Δ*S*, where *G* is the free energy, *H* is the enthalpy, and *S* is the entropy, we get *T*Δ*S*=Δ*H*−Δ*G*. Since stretching is nonspontaneous, as it requires external work, *T*Δ*S* must be negative. Since *T* is always positive (it can never reach absolute zero), the Δ*S* must be negative, implying that the rubber in its natural state is more entangled (with more microstates) than when it is under tension. Thus, when the tension is removed, the reaction is spontaneous, leading Δ*G* to be negative. Consequently, the cooling effect must result in a positive Δ*G*, so Δ*S* will be positive there.^{[2]}^{[3]}

The result is that an elastomer behaves somewhat like an ideal monatomic gas, inasmuch as (to good approximation) elastic polymers do **not** store any potential energy in stretched chemical bonds or elastic work done in stretching molecules, when work is done upon them. Instead, all work done on the rubber is "released" (not stored) and appears immediately in the polymer as thermal energy. In the same way, all work that the elastic does on the surroundings results in the disappearance of thermal energy in order to do the work (the elastic band grows cooler, like an expanding gas). This last phenomenon is the critical clue that the ability of an elastomer to do work depends (as with an ideal gas) only on entropy-change considerations, and not on any stored (i.e., potential) energy within the polymer bonds. Instead, the energy to do work comes entirely from thermal energy, and (as in the case of an expanding ideal gas) only the positive entropy change of the polymer allows its internal thermal energy to be converted efficiently (100% in theory) into work.

## Models

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.^{[4]} 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 $b\; \backslash ,$ is the length of a rigid segment, $n\; \backslash ,$ is the number of segments of length $b\; \backslash ,$, $r\; \backslash ,$ is the distance between the fixed and free ends, and $L\_c\; \backslash ,$ is the "contour length" or $nb\; \backslash ,$. Above the glass transition temperature, the polymer chain oscillates and $r\; \backslash ,$ changes over time. The probability of finding the chain ends a distance $r\; \backslash ,$ apart is given by the following Gaussian distribution:

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

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

- $\backslash begin\{align\}$

\langle r\rangle &= 0 \\ \langle r^2\rangle &= nb^2 \\ \langle r^2\rangle^\frac{1}{2} &= \sqrt{n} b

\end{align}

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.

- $\backslash begin\{align\}$

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

\end{align}

Note that the elastic coefficient $\backslash frac\{3\; k\_B\; T\}\{L\_c\; b\}$ is temperature dependent. If we increase the rubber temperature, the elastic coefficient also rises. This is the reason why rubber under constant strain shrinks when its temperature increases.

## 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 $L\_p\; \backslash ,$, the persistence length, replaces $b\; \backslash ,$. Then, the force follows this equation:

- $F\; \backslash approx\; \backslash frac\{k\_B\; T\}\{L\_p\}\; \backslash left\; (\; \backslash frac\{1\}\{4\; \backslash left(\; 1-\; \backslash frac\{r\}\{L\_c\}\; \backslash right\; )^2\}\; -\; \backslash frac\{1\}\{4\}\; +\; \backslash frac\{r\}\{L\_c\}\; \backslash right\; )\; \backslash ,$

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 ($r=L\_c\; \backslash ,$), an infinite force is required, which is intuitive. Graphically, the force begins at the origin and initially increases linearly with $r\; \backslash ,$. The force then plateaus but eventually increases again and approaches infinity as the chain length approaches $L\_c\; \backslash ,$