Fluids of hard convex molecules

Abstract

We consider the excess Helmholtz free energy ΔA for a system of hard convex molecules without additional soft interactions. The starting point is the expansion of -ΔA in irreducible graphs of Mayer f bonds. For such a system, differential and integral geometry can be used to obtain a reformulation in which the use of two-body geometry, as implicit in the f function, is replaced by one-body geometry. A graph point with n incident bonds (n-point) requires a set of n-point measures of one-body geometry. An example is an old (1936) result of Santalo and Blaschke, which expresses the second virial coefficient in terms of the 1-point measures volume, surface, and integral mean curvature. We obtain the corresponding result for the next simplest class of graphs, the rings, which contain only 2-points. This results in an enormous reduction in complexity, especially for mixtures. We define the set of 2-point measures required to compute the ring graphs. For graphs which contain n-points with n > 2, the added complexity of multi-point measures countervails the simplification in going from 2-body to 1-body geometry.