Lipid membranes with an edge

TL;DR

This paper derives boundary conditions for lipid membranes with free edges using variational principles, revealing new conditions beyond symmetric cases and discussing effects of Gaussian rigidity.

Contribution

It provides a general derivation of boundary conditions for lipid membranes with edges without assuming symmetry, including new conditions not previously identified.

Findings

Derived boundary conditions for membranes with edges.

Identified an additional boundary condition beyond symmetric cases.

Discussed the impact of Gaussian rigidity on membrane behavior.

Abstract

Consider a lipid membrane with a free exposed edge. The energy describing this membrane is quadratic in the extrinsic curvature of its geometry; that describing the edge is proportional to its length. In this note we determine the boundary conditions satisfied by the equilibria of the membrane on this edge, exploiting variational principles. The derivation is free of any assumptions on the symmetry of the membrane geometry. With respect to earlier work for axially symmetric configurations, we discover the existence of an additional boundary condition which is identically satisfied in that limit. By considering the balance of the forces operating at the edge, we provide a physical interpretation for the boundary conditions. We end with a discussion of the effect of the addition of a Gaussian rigidity term for the membrane.