Universal McMillan–Mayer van der Waals Langevin Gel

[Image: see text] We present a theory for a universal gel based on a McMillan–Mayer treatment of a solute–solvent fluid as a generalization of the universal van der Waals equation of state for a pure liquid/vapor system. The elastic resilience of the networked gel is modeled by a universal Langevin function. This combination of van der Waals interactions and nonlinear Langevin elasticity produces an abrupt onset of large-amplitude density fluctuations deep in the interior of the gel at a critical temperature. Then, at a second, lower, critical temperature, the entire swollen gel collapses to a high-density phase. The universal gel has an “upper” critical temperature behavior, meaning that the gel transition to high density occurs on decreasing the temperature. At the cost of loss of universality, the theory is generalized to predict lower critical temperature dependence, whereby an aqueous hydrophobic gel exhibits phase coexistence when the temperature is raised. The theory is consistent with the Gibbs phase rule, suitably generalized to coexisting phases that are not at the same pressure in equilibrium conditions.