Interface mediated interactions between particles -- a geometrical approach

TL;DR

This paper presents a geometrical framework to exactly compute forces between particles on deformable interfaces, incorporating additional surface degrees of freedom and providing explicit force-distance relations.

Contribution

It introduces a divergence-free surface stress tensor formalism that accurately captures interface-mediated interactions without smallness assumptions.

Findings

Exact force expressions derived for various surface Hamiltonians.

Force-distance relations obtained in the linear regime.

Sign of forces clarified in specific symmetric configurations.

Abstract

Particles bound to an interface interact because they deform its shape. The stresses that result are fully encoded in the geometry and described by a divergence-free surface stress tensor. This stress tensor can be used to express the force on a particle as a line integral along any conveniently chosen closed contour that surrounds the particle. The resulting expression is exact (i.e., free of any "smallness" assumptions) and independent of the chosen surface parametrization. Additional surface degrees of freedom, such as vector fields describing lipid tilt, are readily included in this formalism. As an illustration, we derive the exact force for several important surface Hamiltonians in various symmetric two-particle configurations in terms of the midplane geometry; its sign is evident in certain interesting limits. Specializing to the linear regime, where the shape can be analytically determined, these general expressions yield force-distance relations, several of which have originally been derived by using an energy based approach.