Morphological Thermodynamics of Fluids: Shape Dependence of Free Energies

TL;DR

This paper demonstrates that the thermodynamic potential of a fluid in a container depends on only four shape measures if certain invariances hold, simplifying the understanding of shape effects on fluid free energies.

Contribution

It introduces a morphometric approach showing that thermodynamic quantities depend linearly on mean and Gaussian curvatures under specific constraints, simplifying previous assumptions.

Findings

Numerical verification with hard spheres confirms the morphometric dependence.

Only four shape measures are needed to describe the influence of container shape.

The approach refines the understanding of surface tension and curvature effects.

Abstract

We examine the dependence of a thermodynamic potential of a fluid on the geometry of its container. If motion invariance, continuity, and additivity of the potential are fulfilled, only four morphometric measures are needed to describe fully the influence of an arbitrarily shaped container on the fluid. These three constraints can be understood as a more precise definition for the conventional term "extensive" and have as a consequence that the surface tension and other thermodynamic quantities contain, beside a constant term, only contributions linear in the mean and Gaussian curvature of the container and not an infinite number of curvatures as generally assumed before. We verify this numerically in the entropic system of hard spheres bounded by a curved wall.