Geometry of surface mediated interactions

TL;DR

This paper presents a geometric approach to understanding surface-mediated interactions between particles, expressing forces as contour integrals of surface stress, applicable to large deformations and symmetric configurations.

Contribution

It introduces a parameterization-independent method to compute forces between particles on deformable surfaces, valid beyond linear approximations and leveraging surface symmetries.

Findings

Exact force expressions for identical particles on surfaces.

Applicable to large deformations beyond small-gradient regimes.

Force sign often directly inferred from surface geometry.

Abstract

Soft interfaces can mediate interactions between particles bound to them. The force transmitted through the surface geometry on a particle may be expressed as a closed line integral of the surface stress tensor around that particle. This contour may be deformed to exploit the symmetries present; for two identical particles, one obtains an exact expression for the force between them in terms of the local surface geometry of their mid-plane; in the case of a fluid membrane the sign of the interaction is often evident. The approach, by construction, is adapted directly to the surface and is independent of its parameterization. Furthermore, it is applicable for arbitrarily large deformations; in particular, it remains valid beyond the linear small-gradient regime.