Phase field modeling and computation of vesicle growth or shrinkage

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.