A molecular theory for the freezing of hard spheres

Abstract

Using statisical mechanics and a series of well-defined approximations we present a calculation of the equilibrium liquid to solid transition for hard spheres. No computer simulation results are used. The transition is located from structural information about the liquid using a first-principles order parameter theory of freezing. The order parameters are the coefficients of a Fourier expansion of the spatially varying single-particle density ρ(r) in terms of the reciprocal lattice vectors of the solid. The thermodynamic and structural properties are calculated from a perturbation expansion in ρ(r). The effects of second order terms in this expansion and intermediate wavelength properties are investigated for the first time. The theory predicts that the equilibrium freezing transition occurs from a liquid with density 0.976 to an fcc solid of density 1.035 (in units of σ3, where σ is the hard sphere diameter), in good agreement with Monte Carlo simulations which find empirically that the liquid at a density 0.939–0.948 has the same pressure and free energy as the solid at a density 1.036–1.045.