Triezenberg-Zwanzig expression for the surface tension of a liquid drop

TL;DR

This paper derives formulas for the surface tension, Tolman length, and bending rigidities of a liquid drop using the Ornstein-Zernike correlation function, extending the Triezenberg-Zwanzig approach to curved interfaces.

Contribution

It introduces new expressions for curvature-related surface properties of liquid drops based on external field deformation analysis, linking to density functional theory results.

Findings

Derived formulas for Tolman length and rigidity constants.

Showed dependence of rigidity constants on external field form.

Connected new expressions to existing density functional theory results.

Abstract

Formulas, analogous to the Triezenberg-Zwanzig expression for the surface tension of a planar interface, are presented for the Tolman length, the bending rigidity, and the rigidity constant associated with Gaussian curvature. These expressions feature the Ornstein-Zernike direct correlation function and are derived from considering the deformation of a liquid drop in the presence of an external field. This approach is in line with the original analysis by Yvon in 1948. It is shown that our expressions reduce to previous results from density functional theory when a mean-field approximation is made for the direct correlation function. We stress the importance of the form of the external field used and show how the values of the rigidity constants depend on it.