Studies of Liquid–Solid Transitions Using a Thermodynamic Perturbation Method with Modified Weighted Density Approximation

Abstract

Despite intensive investigations based on statistical mechanics, liquid–solid transition is far from complete understanding. The reason for this difficulty lies in the fact that any conventional theory including the density functional theory cannot handle the attractive part that exists in any interaction potential. Thus far, a density functional theory incorporated with the modified weighted density approximation (MWDA) has been developed to predict the liquid–solid transition of a hard-sphere system. In this approach, the free energy and direct correlation function (DCF) in a liquid state were calculated using the analytical solution of the Percus–Yevick (PY) equation. The coexisting density of solid and liquid states given by this approach has agreed with those obtained by computer simulations within 0.7%. However, this method cannot be applied to realistic systems where constituents interact mutually through potential energy with an attractive part. This is because there are no analytical expressions available for the free energy and DCF. In this short note, we apply an alternative approach to analyzing liquid–solid transitions for atoms whose interaction potential has an attractive part. We employ a thermodynamic perturbation method developed by Rascon et al. The method has been applied successfully to the analysis of systems with nonmonotonic potential energy such as the Lennard-Jones, square well, and Dzugutov potential. The required information in this approach is only the potential function and the structure of a solid. Exploiting the theory developed by Weeks, Chandler, and Andersen (WCA), we first expand the free energy of the liquid and solid states. Namely, we split the potential energy into a hard-sphere reference potential and a perturbation potential. Thus, the free energy can be written as F1⁄2 ðrÞ 1⁄4 FHS1⁄2 ðrÞ þ 2 N 2 V Z 1