Phase field modeling and computation of vesicle growth or shrinkage

TL;DR

This paper introduces a phase field model for vesicle growth and shrinkage driven by osmotic pressure, combining Allen-Cahn and Cahn-Hilliard equations, with numerical schemes that accurately capture equilibrium shapes and morphologies.

Contribution

The paper develops a novel phase field model incorporating mass conservation and surface constraints for vesicle dynamics, along with an efficient numerical scheme for equilibrium shape computation.

Findings

Diffuse interface model captures key vesicle growth/shrinkage features.

Large concentration differences lead to circle-like equilibrium shapes.

Shrinking vesicles exhibit diverse finger-like morphologies.

Abstract

We present a phase field model for vesicle growth or shrinkage induced by an osmotic pressure due to a chemical potential gradient. The model consists of an Allen-Cahn equation describing the evolution of phase field and a Cahn-Hilliard equation describing the evolution of concentration field. We establish control conditions for vesicle growth or shrinkage via a common tangent construction. During the membrane deformation, the model ensures total mass conservation and satisfies surface area constraint. We develop a nonlinear numerical scheme, a combination of nonlinear Gauss-Seidel relaxation operator and a V-cycles multigrid solver, for computing equilibrium shapes of a 2D vesicle. Convergence tests confirm an $\mathcal{O}(t+h^2)$ accuracy. Numerical results reveal that the diffuse interface model captures the main feature of dynamics: for a growing vesicle, there exist circle-like equilibrium shapes if the concentration difference across the membrane and the initial osmotic pressure are large enough; while for a shrinking vesicle, there exists a rich collection of finger-like equilibrium morphologies.