Contact lines for fluid surface adhesion

TL;DR

This paper develops a geometric framework to derive boundary conditions for fluid surface adhesion, considering various Hamiltonians with curvature effects, revealing that these conditions are not solely due to local force balance.

Contribution

It introduces a systematic geometric approach to derive adhesion boundary conditions for complex Hamiltonians involving curvature and its gradients, including new results.

Findings

Boundary conditions depend on surface deformation energies.

Higher order derivatives influence boundary behavior beyond force balance.

Framework applies to adhesion between fluids and rigid substrates.

Abstract

When a fluid surface adheres to a substrate, the location of the contact line adjusts in order to minimize the overall energy. This adhesion balance implies boundary conditions which depend on the characteristic surface deformation energies. We develop a general geometrical framework within which these conditions can be systematically derived. We treat both adhesion to a rigid substrate as well as adhesion between two fluid surfaces, and illustrate our general results for several important Hamiltonians involving both curvature and curvature gradients. Some of these have previously been studied using very different techniques, others are to our knowledge new. What becomes clear in our approach is that, except for capillary phenomena, these boundary conditions are not the manifestation of a local force balance, even if the concept of surface stress is properly generalized. Hamiltonians containing higher order surface derivatives are not just sensitive to boundary translations but also notice changes in slope or even curvature. Both the necessity and the functional form of the corresponding additional contributions follow readily from our treatment.