Pair correlation function in a fluid with density inhomogeneities: results of the Percus‒Yevick and hypernetted chain approximations for hard spheres near a hard wall

Abstract
Solutions for the pair correlation function and density profile of a system of hard spheres near a hard wall are obtained by using the Percus-Yevick and hypernetted chain approximations, generalized for inhomogeneous fluids. The Percus-Yevick (PY) results are similar in accuracy to those obtained for the bulk fluid. The PY pair correlation function is generally too small near contact but quite good overall. The hypernetted chain (h.n.c.) results are difficult to obtain numerically and are poorer than in the bulk. Often the h.n.c. pair correlations are too small at contact, in contrast to the bulk case where they are too large, although there are configurations where the contact values of the pair correlation function are too large. Nearly always the error in the h.n.c. results is much worse than is the case for the bulk. Both approximations are qualitatively satisfactory in that they predict the correct asymmetries between the values of the pair correlation functions for pairs of hard spheres whose line of centres is parallel or normal to the surface of the wall.