Structural properties of additive binary hard-sphere mixtures. III. Direct correlation functions

TL;DR

This paper investigates the direct correlation functions of binary additive hard-sphere mixtures, revealing their monotonic or non-monotonic behavior and discontinuities, using rational-function approximation and WM schemes, advancing understanding of their structural properties.

Contribution

It provides a detailed analysis of the direct correlation functions in binary hard-sphere mixtures, highlighting their monotonicity, polynomial representation, and discontinuities, which were not fully characterized before.

Findings

$c_{ss}(r<\sigma_s)$ and $c_{bb}(r<\sigma_b)$ are monotonic and polynomially representable.

$c_{sb}(r<rac{1}{2}(\sigma_b+\sigma_s))$ is non-monotonic with a well-defined minimum.

Second derivative $c_{sb}''(r)$ has a jump discontinuity at $r=rac{1}{2}(\sigma_b-\sigma_s)$, related to contact values.

Abstract

An analysis of the direct correlation functions $c_{ij} (r)$ of binary additive hard-sphere mixtures of diameters $\sigma_s$ and $\sigma_b$ (where the subscripts $s$ and $b$ refer to the "small" and "big" spheres, respectively), as obtained with the rational-function approximation method and the WM scheme introduced in previous work [S.\ Pieprzyk \emph{et al.}, Phys.\ Rev.\ E {\bf 101}, 012117 (2020)], is performed. The results indicate that the functions $c_{ss}(r<\sigma_s)$ and $c_{bb}(r<\sigma_b)$ in both approaches are monotonic and can be well represented by a low-order polynomial, while the function $c_{sb}(r<\frac{1}{2}(\sigma_b+\sigma_s))$ is not monotonic and exhibits a well defined minimum near $r=\frac{1}{2}(\sigma_b-\sigma_s)$, whose properties are studied in detail. Additionally, we show that the second derivative $c_{sb}''(r)$ presents a jump discontinuity at $r=\frac{1}{2}(\sigma_b-\sigma_s)$ whose magnitude satisfies the same relationship with the contact values of the radial distribution function as in the Percus-Yevick theory.