Negative Energy Elasticity in a Rubberlike 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 originates mostly from entropy change with deformation. Similarly, polymer gels containing a large amount of solvent have also been postulated that the shear modulus $G$, which is a kind of modulus of elasticity, is approximately equivalent to the entropy contribution $G_S$, but this has yet to be verified experimentally. In this study, we measure the temperature dependence of the shear modulus $G$ in a rubber-like (hyperelastic) polymer gel whose polymer volume fraction is at most 0.1. As a result, we find that the energy contribution $G_E=G-G_S$ can be a significant negative value, reaching up to double the shear modulus $G$ (i.e., $\left|G_E\right| \simeq 2G$), although the shear modulus of stable materials is generally bound to be positive. We further argue that the energy contribution $G_E$ 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 highlight the essential difference between rubber elasticity and gel elasticity, which were previously thought to be the same, and push an established field of gel elasticity into a new direction.