On the thermodynamic theory of curvature-dependent surface tension

TL;DR

This paper derives an exact equation for the Tolman length as a function of radius, introduces a new curvature-dependent surface tension equation, and provides a computational method to analyze surface tension effects in droplets and bubbles.

Contribution

It presents a novel exact equation for the Tolman length, a new curvature-dependent surface tension equation, and a computational procedure for analyzing surface tension in metastable systems.

Findings

Dependence of Tolman length and surface tension on radius for droplets and bubbles.

A new curvature-dependent surface tension equation related to the spinodal point.

Exact Kelvin-type equation valid over the entire metastability region.

Abstract

An exact equation for determining the Tolman length (TL) as a function of radius is obtained and a computational procedure for solving it is proposed. As a result of implementing this procedure, the dependences of the TL and surface tension on radius are obtained for the drop and bubble cases and various equations of state. As one of the results of the thermodynamic study, a new equation for the dependence of surface tension on radius (curvature effect), alternative to the corresponding Tolman equation and associated with the spinodal point, is obtained. The Kelvin type equation for the equimolecular radius is shown to be exact over the entire metastability region and serves as the basis for the TL equation. The expansions of surface tension near the spinodal and binodal points show that the correction to Rusanov s linear asymptotics in the first case is a series in cubes of the radius, whereas a series in curvature holds in the second case. As a result of the analysis of these expansions, the fundamental impossibility to determine the curvature effect analytically from the binodal point is established; the computational procedure determines it from the spinodal point. It is shown that just the characteristics of the system on the spinodal, mainly the TL value at zero radius, determine the curvature effect. In general, the theory reveals a close connection between surface and bulk properties.