Radial distribution function for hard spheres from scaled particle theory, and an improved equation of state

Abstract

Previous applications of scaled particle theory have been limited to the calculation of thermodynamic properties of fluids rather than structure. In the present paper, the theory is expanded so that it is capable of yielding the radial distribution function. The method is first illustrated by applying it to one- dimensional fluids of hard rods where, as in other theories, the radial distribution function is obtained exactly. It is then applied to a fluid of hard spheres where a closure condition is necessary. This condition is supported by recent work in scaled particle theory dealing with the thermodynamics of boundary layers. It is used to calculate the radial distribution function around a λ-cule of varying size, including one of the size of a typical hard sphere solvent molecule. The results are very good, yielding (at the highest density at which numerical solutions have been possible, ρ=0.372ρ0, where ρ0 is the density of close packing) results for the contact radial distribution function and the equation of state which were somewhat better than the original scaled particle and Percus-Yevick theories. These theories yield results which are very close to the predictions of Monte Carlo calculations. As in the original scaled particle theory directed at the contact distribution function alone, exact results (exact closure) are possible for λ-cules of radii corresponding to a point or smaller. Our numerical method becomes unstable for densities higher than ρ=0.372ρ0, and we are attempting to improve the procedure.