  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, in polymer gels containing a
large amount of solvent, it has also been postulated that the shear modulus
(the modulus of rigidity) $G$, 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 rubberlike (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 the established field
of gel elasticity into a new direction.
