What's the difference between elastic energy and entropic elasticity?

Question

I was reading up some articles on elasticity theory to make an essay about elastic energy in rubber bands, but in the first paragraph of this article it is said that rubber bands do not show elastic potential energy, but entropic elasticity. I've never seen this before, and since its fundamentally a thermodynamics thing, it takes the research to another field completely. What is the actual difference between entropic elasticity and ordinary elasticity? Does it matter when studying the potential energy of an elastic body?

Answer 1

We know that all things occur to maximize entropy. (In other words, the Second Law tells us that we more often see outcomes that are more likely to occur—that is, those outcomes with higher entropy).

As outlined here, energy minimization can act as a surrogate for entropy maximization. We can interpret this as favoring both high entropy and strong bonding, as the latter releases energy that provides the entropy benefit of heating the rest of the universe.

This is all encompassed in the Gibbs free energy, which includes both internal energy and entropy terms.

For a standard metal spring, the internal energy term dominates. As typically modeled using a pair potential, the atomic/molecular spacing is shifted slightly from its (minimum) equilibrium position. (This doesn’t affect the entropy much.) The greater energy upon displacement corresponds to a restoring force that imparts springlike behavior, with linear elasticity and a constant stiffness observed for small displacements.

The ideal gas can also resist deformation (specifically, compression), but this material model has no capacity to store energy. To understand the bulk modulus of the ideal gas, we have to return to the Second Law: the drive for entropy maximization produces a restoring force toward the equilibrium configuration of pressure equilibration with the surroundings. In contrast to internal-energy-mediated elasticity, the behavior of an entropic spring is immediately nonlinear (e.g., the isothermal stiffness of the ideal gas is just its pressure), and the stiffness is strongly temperature dependent because the entropy term in the Gibbs free energy has the temperature as its coefficient.

Entropic stiffness can also be important for polymers, whose deformation involves large-scale uncoiling and unkinking, which can appreciably affect the configurational entropy. As described above, two important implications are greater elastic nonlinearity and greater temperature dependence; as discussed here, the temperature dependence of the stiffness is generally opposite between springs based on internal energy minimization (e.g., metal, ceramic, crosslinked polymers) vs. entropy maximization (e.g., the ideal gas, the ideal elastomer).

Please let me know if anything’s unclear in this draft answer.