Density functional theory of statistical mechanics: Equilibrium vacancy concentration and freezing in polydisperse systems

Two extensions of the density functional theory of freezing are presented: the prediction of the concentration of vacancies in the coexisting crystal at phase coexistence and the prediction of the freezing characteristics of a polydisperse system of hard spheres. The vacancy density calculated for a Lennard-Jones system (freezing into a face-centered cubic crystal) from this density functional theory is quite comparable with the vacancy density measured in medium-mass rare gas crystals. The method is applied to the hard sphere system using the Percus-Yevick approximation, and is solved analytically.; The density functional theory is also applied to polydisperse hard spheres freezing to both face-centered cubic and hexagonally close-packed crystals. Polydisperse hard sphere systems with both a gamma distribution of diameters and a Gaussian distribution of diameters are studied. Two approximate approaches are stated and discussed. It is determined that when the standard deviation of the distribution of sizes (for either distribution type) exceeds about 5%, the system no longer freezes into a crystalline array.