A geometrically-based mean-field theory of polydisperse hard-sphere mixtures

We present a mean-field description of a system of polydisperse hard-spheres. The theory is based on the postulate that the excess statistical properties of a general polydisperse mixture are a function of the number, the mean diameter, surface area, and volume of the constituent particles. Within this model a corresponding states relationship holds between a general polydisperse system and a suitably chosen two-component mixture. This equivalence is used to derive approximate expressions for the free energy and pressure of polydisperse crystal and fluid phases. Quantitative results are presented for the case of a Schultz distribution of diameters. These free energies are used to calculate the solid–fluid phase diagram as a function of diameter polydispersity. We find a terminal polydispersity of 8.3% above which the polydisperse fluid remains stable at all densities. In contrast with recent simulations we find no evidence for a substantial fractionation in diameters between the coexisting fluid and solid phases.