Contact values of the radial distribution functions of additive hard-sphere mixtures in d dimensions: A new proposal

TL;DR

This paper introduces a universal function approach to predict contact values of radial distribution functions in additive hard-sphere mixtures across various dimensions, improving accuracy and consistency with known results.

Contribution

It proposes a new universal function model for contact values, valid across dimensions and consistent with known theories and exact results, enhancing understanding of hard-sphere mixtures.

Findings

The model agrees well with numerical data for dimensions 2 to 5.

It reduces to exact results in one-dimensional systems.

The approach unifies different theoretical approximations under a common framework.

Abstract

The contact values $g_{ij}(\sigma_{ij})$ of the radial distribution functions of a $d$-dimensional mixture of (additive) hard spheres are considered. A `universality' assumption is put forward, according to which $g_{ij}(\sigma_{ij})=G(\eta, z_{ij})$, where $G$ is a common function for all the mixtures of the same dimensionality, regardless of the number of components, $\eta$ is the packing fraction of the mixture, and $z_{ij}$ is a dimensionless parameter that depends on the size distribution and the diameters of spheres $i$ and $j$. For $d=3$, this universality assumption holds for the contact values of the Percus--Yevick approximation, the Scaled Particle Theory, and, consequently, the Boublik--Grundke--Henderson--Lee--Levesque approximation. Known exact consistency conditions are used to express $G(\eta,0)$, $G(\eta,1)$, and $G(\eta,2)$ in terms of the radial distribution at contact of the one-component system. Two specific proposals consistent with the above conditions (a quadratic form and a rational form) are made for the $z$-dependence of $G(\eta,z)$. For one-dimensional systems, the proposals for the contact values reduce to the exact result. Good agreement between the predictions of the proposals and available numerical results is found for $d=2$, 3, 4, and 5.