Wetting and Minimal Surfaces

TL;DR

This paper investigates minimal surfaces in wetting phenomena, deriving boundary equations and diagrammatic rules to compute non-linear corrections to the energy, and explores the quasi-local nature of perturbations and singularities in transformations.

Contribution

It introduces a new diagrammatic approach and simple algorithm for calculating minimal surfaces and energies in wetting, highlighting the quasi-locality of perturbations and singularities in transformations.

Findings

Derived coupled boundary equations for contact lines.

Developed a diagrammatic method for non-linear energy corrections.

Demonstrated the quasi-locality of geometric length scales.

Abstract

We study minimal surfaces which arise in wetting and capillarity phenomena. Using conformal coordinates, we reduce the problem to a set of coupled boundary equations for the contact line of the fluid surface, and then derive simple diagrammatic rules to calculate the non-linear corrections to the Joanny-de Gennes energy. We argue that perturbation theory is quasi-local, i.e. that all geometric length scales of the fluid container decouple from the short-wavelength deformations of the contact line. This is illustrated by a calculation of the linearized interaction between contact lines on two opposite parallel walls. We present a simple algorithm to compute the minimal surface and its energy based on these ideas. We also point out the intriguing singularities that arise in the Legendre transformation from the pure Dirichlet to the mixed Dirichlet-Neumann problem.