Scaled Particle Theory for Hard Sphere Pairs. I. Mathematical Structure

TL;DR

This paper extends scaled particle theory to predict the hard sphere pair correlation function g(r), deriving a nonlinear integral equation that accurately reproduces virial coefficients and offers a systematic density expansion.

Contribution

It introduces a new nonlinear integral equation within scaled particle theory that predicts g(r) and virial coefficients for hard spheres with high accuracy.

Findings

Replicates exact second and third virial coefficients.

Predicted fourth virial coefficient is within 0.6% of the exact value.

Provides a power series solution in density for g(r).

Abstract

We develop an extension of the original Reiss-Frisch-Lebowitz scaled particle theory that can serve as a predictive method for the hard sphere pair correlation function g(r). The reversible cavity creation work is analyzed both for a single spherical cavity of arbitrary size, as well as for a pair of identical such spherical cavities with variable center-to-center separation. These quantities lead directly to prediction of g(r). Smooth connection conditions have been identified between the small-cavity situation where the work can be exactly and completely expressed in terms of g(r), and the large-cavity regime where macroscopic properties become relevant. Closure conditions emerge which produce a nonlinear integral equation that must be satisfied by the pair correlation function. This integral equation has a structure which straightforwardly generates a solution that is a power series in density. The results of this series replicate the exact second and third virial coefficients for the hard sphere system via the contact value of the pair correlation function. The predicted fourth virial coefficient is approximately 0.6 percent lower than the known exact value. Detailed numerical analysis of the nonlinear integral equation has been deferred to the sequel (following paper)