Free energy expansions for renormalized systems for colloids

TL;DR

This paper develops free energy bounds for a binary colloid system by integrating out small spheres, leading to a multi-body interaction model for large spheres, and proves cluster expansion convergence under surface-based conditions.

Contribution

It introduces a novel approach to analyze binary colloid systems by focusing on surface interactions and rigorously treating depletion effects.

Findings

Established free energy bounds for colloids with finite size.

Proved convergence of cluster expansion based on surface area.

Provided rigorous treatment of depletion interactions in hard sphere systems.

Abstract

We consider a binary system of small and large spheres of finite size in a continuous medium interacting via a non-negative potential. We work in the canonical ensemble and compute upper and lower bound for the free energy at finite and infinite volume by first integrating over the small spheres and then treating the effective system of the large ones which now interact via a multi-body potential. By exploiting the underlying structure of the original binary system we prove the convergence of the cluster expansion for the latter system and obtain a sufficient condition which involves the surface of the large spheres rather than their volume (as it would have been the case in a direct application of existing methods directly to the binary system). Our result is valid for the particular case of hard spheres (colloids) for which we rigorously treat the depletion interaction.