Density Functional Theory of a Curved Liquid-Vapour Interface: Evaluation of the rigidity constants

TL;DR

This paper develops density functional theory formulas for the bending and Gaussian rigidity constants of curved liquid-vapor interfaces, analyzing their dependence on the dividing surface and evaluating their values for specific molecular interactions.

Contribution

It introduces new formulas for rigidity constants within density functional theory and examines their sensitivity to the choice of dividing surface, especially the equimolar surface.

Findings

Rigidity constants are sensitive to the dividing surface choice.

For short-range interactions, k is negative (~-0.5 to -1.0 kT) and k_bar is positive.

For dispersion forces, a log(R)/R^2 term replaces rigidity constants.

Abstract

It is argued that to arrive at a quantitative description of the surface tension of a liquid drop as a function of its inverse radius, it is necessary to include the bending rigidity k and Gaussian rigidity k_bar in its description. New formulas for k and k_bar in the context of density functional theory with a non-local, integral expression for the interaction between molecules are presented. These expressions are used to investigate the influence of the choice of Gibbs dividing surface and it is shown that for a one-component system, the equimolar surface has a special status in the sense that both k and k_bar are then the least sensitive to a change in the location of the dividing surface. Furthermore, the equimolar value for k corresponds to its maximum value and the equimolar value for k_bar corresponds to its minimum value. An explicit evaluation using a short-ranged interaction potential between molecules, shows that k is negative with a value around minus 0.5-1.0 kT and that k_bar is positive with a value which is a bit more than half the magnitude of k. Finally, for dispersion forces between molecules, we show that a term proportional to log(R)/R^2 replaces the rigidity constants and we determine the (universal) proportionality constants.