Equilibrium thermomechanics of swollen polymer networks.

The equilibrium shear modulus of some swollen polymer networks decreases with increasing temperature, even when the corresponding dry network shows the classical positive temperature coefficient of entropic elasticity. This sign reversal cannot be explained by the conventional Frenkel-Flory-Rehner framework, in which the mixing free energy is assumed independent of deformation. We show that allowing the Flory-Huggins parameter χ to depend on isochoric strain invariants, a thermodynamically general but previously unexplored possibility, resolves this inconsistency. Within continuum thermodynamics, this yields, to leading order under commonly encountered conditions, an affine modulus-temperature relation, Geq = θ(X - Y) +ΘY, which separates the dry-network contribution X from a solvent-mediated contribution Y. The resulting structure defines an experimental strategy: measure dry and swollen networks over temperature and concentration, then test the consistency of Y inferred from slope and intercept, or directly when Θ is known independently. We apply this framework to poly(n-butyl acrylate) swollen in butyl benzoate and poly(ethylene glycol) swollen in dimethylformamide, obtaining Y < 0 and Y > 0, respectively. For the latter system, the fitted Θ agrees with the known crystallization temperature of poly(ethylene glycol) in dimethylformamide. The present work establishes a thermodynamic framework and validation strategy for solvent-deformation coupling in swollen networks, while leaving molecular identification to future study.