Kinks in two-phase lipid bilayer membranes

TL;DR

This paper extends models of two-phase lipid bilayer membranes by incorporating tangent discontinuities and deriving a sharp interface limit, supported by numerical illustrations.

Contribution

It introduces a new energy model that includes bending energy at tangent discontinuities and derives its sharp interface limit for rotationally symmetric membranes.

Findings

The extended model accounts for tangent discontinuities in membranes.

The derived sharp interface limit matches classical models on reasonable membranes.

Numerical examples demonstrate the theoretical findings.

Abstract

Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least $C^1$ across phase boundaries. We consider the spontaneous curvature model for closed rotationally symmetric two-phase membranes without excluding tangent discontinuities at interfaces a priorily. We introduce a family of energies for smooth surfaces and phase fields for the lipid phases and derive a sharp interface limit that coincides with the $\Gamma$-limit on all reasonable membranes and extends the classical model by assigning a bending energy also to tangent discontinuities. The theoretical result is illustrated by numerical examples.