An exact formalism to study the thermodynamic properties of hard-sphere systems under spherical confinement

TL;DR

This paper introduces a new formalism based on a modified grand canonical ensemble to exactly analyze the thermodynamic properties of hard-sphere systems under spherical confinement, including surface effects and curvature dependence.

Contribution

It develops an exact formalism for studying confined hard-sphere systems and derives explicit expressions for surface tension, adsorption, and curvature effects up to first order in density.

Findings

Exact expressions for surface tension and adsorption as functions of density.

Curvature dependence of surface tension and adsorption derived.

Analysis of thermodynamic properties for systems with up to three particles.

Abstract

This paper presents a modified grand canonical ensemble which provides a new simple and efficient scheme to study few-body fluid-like inhomogeneous systems under confinement. The new formalism is implemented to investigate the exact thermodynamic properties of a hard sphere (HS) fluid-like system with up to three particles confined in a spherical cavity. In addition, the partition function of this system was used to analyze the surface thermodynamic properties of the many-HS system and to derive the exact curvature dependence of both the surface tension and adsorption in powers of the density. The expressions for the surface tension and the adsorption were also obtained for the many- HS system outside of a fixed hard spherical object. We used these results to derive the dependence of the fluid-substrate Tolman length up to first order in density.