![]() (b,c) Normalized concentration ( c / c *) dependence of T 0. ![]() The value of T 0 in each graph is the average of the four samples with different values of p, and the values in parentheses represent the standard deviation. All gray lines that have the same M and c pass through a vanishing temperature T 0 on the T axis, which leads to Eq. ( 7). We obtain each gray line from a least-squares fit of each sample, which is characterized by the three parameters of the precursors: the molar mass M, the concentration c, and the connectivity p. (a) Temperature ( T) dependence of shear modulus G. The existence of a universal function that governs the energy contribution of gel elasticity. Notably, we find that σ E is a significant negative value. The gray solid, blue dashed, and red dashed lines are obtained in the same way as in (a). The gel sample is synthesized by equal-weight mixing of the two kinds of precursors whose molar mass M and concentration c are 20 kg / mol and 60 g / L, respectively. (b) Typical result of the T dependence of the shear stress σ (black symbols) of a polymer gel through rheological measurements under 60% shear strain γ. Similarly, small energy contributions to elasticity are observed in many rubber materials. In the measured temperature range (black symbols), the ratio of σ E to σ is less than 15%. According to Eq. ( 1), we have the entropy contribution σ S (blue dashed line) and the energy contribution σ E (red dashed line), which corresponds to the intercept of the gray solid line. The gray solid line is obtained from a least-squares fit and extrapolated to T = 0 K. (a) Temperature ( T) dependence of the tensile stress σ (black symbols) of vulcanized rubber through stretching measurements under 60% strain. Irrelevant energy elasticity in natural rubber and relevant negative energy elasticity in a rubberlike polymer gel. Additionally, our findings offer a new perspective and stimulate further research on gel elasticity as well as other fields where entropic force plays an important role, such as in entropic gravity theory. Our findings are of great practical importance because gels are used for medical applications at various temperatures. Because the solvent is the critical factor in differentiating gels from rubbers, the negative energy elasticity is considered to be a distinct characteristic of gels. We further argue that the negative energy elasticity is governed by a universal function, and it vanishes when the solvent is removed. We find that a relative temperature change in rigidity is several times greater in gels than that of rubbers because of the negative energy elasticity. We systematically measure how the rigidity of polymer gels with various network structures varies with temperature. Instead, their softness is determined by negative energy elasticity that coexists with entropy elasticity. Here, we experimentally show that this common belief is false for polymer gels. It has been believed for nearly a century that this softness could be explained by entropy elasticity, where an elastic deformation produces a decrease in the specific entropy. Polymer gels and rubbers (similar polymer networks without the solvent) are softer than metals and ceramics by several orders of magnitude. Our study reveals a secret of the softness of polymer gels: networks of cross-linked flexible polymer chains that contain a large amount of solvent. 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. 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. 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., | G E | ≃ 2 G), although the shear modulus of stable materials is generally bound to be positive. 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. 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. ![]() 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.
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