On the sharp interface limit of a phase field model for near-spherical two phase biomembranes

TL;DR

This paper analyzes the sharp interface limit of a phase field model for near-spherical biomembranes, deriving the limiting energy and equations governing membrane composition and shape, relevant to lipid raft formation.

Contribution

It introduces a reduced diffuse interface energy, derives its Gamma-limit, and connects phase field dynamics to sharp interface evolution for biomembranes.

Findings

Gamma-limit of the diffuse interface energy matches the Euler-Lagrange equations.

Derived a coupled system of gradient flow equations for membrane shape and composition.

Connected phase field dynamics to geodesic curvature flow and a fourth-order free boundary PDE.

Abstract

We consider sharp interface asymptotics for a phase field model of two phase near spherical biomembranes involving a coupling between the local mean curvature and the local composition proposed by the first and second authors. The model is motivated by lipid raft formation. We introduce a reduced diffuse interface energy depending only on the membrane composition and derive the $\Gamma-$limit. We demonstrate that the Euler-Lagrange equations for the limiting functional and the sharp interface energy coincide. Finally, we consider a system of gradient flow equations with conserved Allen-Cahn dynamics for the phase field model. Performing a formal asymptotic analysis we obtain a system of gradient flow equations for the sharp interface energy coupling geodesic curvature flow for the phase interface to a fourth order PDE free boundary problem for the surface deformation.