Closure-based perturbative density-functional theory of hard-sphere freezing: properties of the bridge functional

The study of freezing using perturbative classical density-functional theory is revisited, using a bridge functional approach to resum all terms beyond second order in the free energy expansion. More precisely, the first-order direct correlation function of the solid phase is written as a functional expansion about the homogeneous liquid phase, and the sum of all higher-order terms is represented as a functional of the second-order term. Information about the shape and uniqueness of this bridge functional for the case of hard spheres is obtained via an inversion procedure that employs Monte Carlo fluid-solid coexistence data from the literature. The parametric plots obtained from the inversion procedure show very little scatter in certain regions, suggesting a unique functional dependence, but large scatter in other regions. The scatter is related to the anisotropy of the solid lattice at the particle scale. Interestingly, the thermodynamic properties of the phase transition are quite insensitive to the regions where the scatter is large, and several simple closures (i.e., analytical forms of the bridge function) reproduce exactly the liquid-solid coexistence densities and Lindemann parameter from simulation. The form of these closures is significantly different from the usual closures employed in liquid-state integral equation theory.