Curvature dependence of the electrolytic liquid-liquid interfacial tension

TL;DR

This paper investigates how the curvature of liquid droplets affects the interfacial tension in electrolyte solutions, deriving analytical expressions and identifying different regimes of electrostatic influence.

Contribution

It provides a new analytical framework for understanding curvature effects on liquid-liquid interfacial tension in electrolytes, including symmetry relations and scaling regimes.

Findings

Electrostatic contribution to interfacial tension depends on droplet radius.

Symmetry relations hold in the low-curvature limit, affecting the Tolman length.

A low-curvature expansion is valid when droplet size exceeds the Debye length.

Abstract

The interfacial tension of a liquid droplet surrounded by another liquid in the presence of microscopic ions is studied as a function of the droplet radius. An analytical expression for the interfacial tension is obtained within a linear Poisson-Boltzmann theory and compared with numerical results from non-linear Poisson-Boltzmann theory. The excess liquid-liquid interfacial tension with respect to the pure, salt-free liquid-liquid interfacial tension is found to decompose into a curvature-independent part due to short-ranged interfacial effects and a curvature-dependent electrostatic contribution. Several curvature-dependent regimes of different scaling of the electrostatic excess interfacial tension are identified. Symmetry relations of the interfacial tension upon swapping droplet and bulk liquid are found to hold in the low-curvature limit, which, e.g., lead to a sign change of the excess Tolman length. For some systems a low-curvature expansion up to second order turns out to be applicable if and only if the droplet size exceeds the Debye screening length in the droplet, independent of the Debye length in the bulk.