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Phase field modeling and computation of vesicle growth or shrinkage
Title
Phase field modeling and computation of vesicle growth or shrinkage
Creator
Tang, Xiaoxia
Date
2023
Description
Lipid bilayers are the basic structural component of all biological cell membranes. It is a semipermeable barrier to most solutes, including...
Show moreLipid bilayers are the basic structural component of all biological cell membranes. It is a semipermeable barrier to most solutes, including ions, glucoses, proteins and other molecules. Vesicles formed by a bilayer lipid membrane are often used as a model system for studying fundamental physics underlying complicated biological systems such as cells and microcapsules. Mathematical modeling of membrane deformation has become an important topic in biological and industrial system for a long time. In this thesis, we develop a phase field model for vesicle growth or shrinkage based on osmotic pressure that arises due to a chemical potential gradient. This thesis consists of three main parts.In the first part, we establish a phase field model for vesicle growth or shrinkage without flow. It consists of an Allen-Cahn equation, which describes the evolution of the phase field parameter (the shape of the vesicle), and a Cahn-Hilliard-type equation, which simulates the evolution of the ionic fluid. The model is mass conserved and surface area constrained during the membrane deformation. Conditions for vesicle growth or shrinkage are analyzed via the common tangent construction. We develop the numerical computing in two-dimensional space using a nonlinear multigrid method which is a combination of nonlinear Gauss-Seidel relaxation operator and V-cycles multigrid solver, and perform convergence tests that suggest an $\mathcal{O}(t+h^2)$ accuracy. Numerical results demonstrate the growth and shrinkage effects graphically and numerically, which agree with the conditions analyzed via the common tangent construction.In the second part, we present a model for vesicle growth or shrinkage with flow. The dynamical equations considered are an Allen-Cahn equation, which describes the phase field evolution, a Cahn-Hilliard-type equation, which simulates the fluid concentration, and a Stokes-type equation, which models the flow. The numerical scheme in two-dimensional space includes a nonlinear multigrid method comprised of a standard FAS method for the Allen-Cahn and Cahn-Hilliard part, and the Vanka smoothing strategy for the Stokes part. Convergence tests imply an $\mathcal{O}(t+h^2)$ accuracy. Numerical results are demonstrated under zero velocity boundary condition and with boundary-driven shear flows, respectively.In the last part, we give an unconditionally energy stable and uniquely solvable finite difference scheme for the model established in the first part. The finite difference scheme is based on a convex splitting of the discrete energy and is semi-implicit. One key difficulty associated with the energy stability is due to the fact that some nonlinear energy functional terms in the expansion is neither convex nor concave. To overcome this subtle difficulty, we add auxiliary terms to make the combined term convex, which in turn yields a convex–concave decomposition of the physical energy. As a result, both the unique solvability and energy stability of the proposed numerical scheme are assured. In addition, we show the scheme is stable in the defined discrete norm.
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