Polydisperse systems: Statistical thermodynamics, with applications to several models including hard and permeable spheres

The statistical thermodynamics of polydisperse systems of particles is investigated. A Gibbs–Duhem relation is obtained and the equilibrium conditions relevant to a two-phase system are derived. Systems of hard spheres, and hard spheres with Kac tails, are treated as illustrative examples with analytic results given in the context of scaled-particle (Percus–Yevick) theory as well as the polydisperse generalization of the thermodynamic approximation of Mansoori et al. An exact treatment of the analogous one-dimensional systems is also given. Quantitative results using a Schultz distribution of diameters are presented. A model of interpenetrable particles introduced previously by one of us—the permeable-sphere model—is also considered. Its thermodynamics and pair distribution functions are shown to be exactly obtainable in the context of the Percus–Yevick approximation. For this model, polydispersivity in both particle size and particle impenetrability is considered analytically. The pair potential for this model is discontinuous at the interparticle diameter; generalization of the model for which the pair potential is continuous is also introduced as a model of an effective polymer–polymer potential.