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A Phase Field Model for Cell Shapes : : Gamma-Convergence and Numerical Simulations


This dissertation studies two phase field energy functionals used in the modeling of vesicle shape: the Shao--Rappel--Levine (SRL) functional and the Seifert functional. In both functionals the vesicle is conceived as phase field or "diffuse interface" surface. The minimizing diffuse interface surfaces of the SRL functional have minimal bending and surface area and satisfy an enclosed volume constraint. On the other hand, minimizing diffuse interface surfaces of the Seifert functional have minimal bending and satisfy both a surface area constraint and an enclosed volume constraint. In the first three chapters, we prove Gamma-convergence of the SRL functional to a sharp interface functional and we prove that minimizers of the phase field functional converge to minimizers of this sharp interface functional. Since the original functional used to find vesicle shape is a sharp interface functional, these analytical results are significant because they establish an equivalence between the phase field and the sharp interface formulations. In the fourth chapter, we run simulations using the Semi-Explicit Fourier Spectral Method to find minimizers of both phase field functionals. We discover and catalog a variety of 2D and 3D minimizing shapes, classify these minimizers with Betti numbers, and make some observations about how the phase field evolves to reach these minimizers. Our numerical work is significant for three reasons: we recover many of the vesicle shapes that have been observed experimentally; we discover several new stable shapes with interesting topology; and our program proves to be a useful tool for finding surfaces with minimal bending energy for given area and volume constraints. In the fifth and final chapter, we discuss future studies

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