Negative energy elasticity in a rubber-like gel

Rubber elasticity is the archetype of the entropic force emerging from the second law of thermodynamics; numerous experimental and theoretical studies on natural and synthetic rubbers have shown that the elasticity $G$ is approximately equivalent to the entropy contribution $G_S$. Similarly, polymer gels containing a large amount of solvent have also been assumed to exhibit $G\simeq G_S$, but this has yet to be verified experimentally. Here, we measure the temperature dependence of the shear modulus in a hyperelastic polymer gel and find that the energy contribution $G_E=G-G_S$ can be a negative value, reaching up to double the $G$ (i.e., $\left|G_E\right| \simeq 2G$), although the elasticity of stable materials is generally bound to be positive. We further argue that $G_E$ is governed by a vanishing temperature that is a universal function of the normalized polymer concentration, and $G_E$ vanishes when the solvent is removed. Our findings would stimulate a re-examination of a vast amount of research on gel elasticity.