Degenerate Diffusion in Complex Amphiphilic Network Structures

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The principal investigator studies complex network structures in amphiphilic mixtures under a framework that includes degenerate diffusion and functionalized Cahn-Hilliard (FCH) energy. Network structures in complex systems have close relations to phase transitions, materials science, and mathematical biology. He develops methods to model and analyze the development of structures in amphiphilic mixtures and their properties. Such systems include biological membranes and polymer materials like polymer electrolyte membranes. Biological membranes play critical roles in biological processes such as protein transportation, drug encapsulation and delivery. Polymer electrolyte membranes are key ingredients of fuel cells that are able to efficiently produce clean energy and are a promising replacement for traditional batteries. Their effectiveness is directly related to their fine structures. In the course of this project, a graduate student and an undergraduate student are trained.The principal investigator concentrates on the influence of degenerate diffusion on the dynamics of complex amphiphilic structures, the variational properties of the functionalized Cahn-Hilliard energy, and their relation to amphiphilic structures. He formulates bilayers, filamentous pores, micelles, and defect structures such as open edges, end caps, and triple junctions, as minimizers of the FCH energy, and relates them to variational properties of the FCH energy under various geometric constraints. By building degenerate diffusion into the FCH equation, he is able to model biologically significant phenomena such as the co-existence of bilayer vesicles of various sizes and shapes, the diffusion of lipid molecules along lipid bilayer membranes, and the impermeability of bilayers. He studies the evolution of the defects and the sensitivity of their energies to parametric variation, and identifies the energetic cost for the defects to form. The tools developed in this project have wide applications in the modeling of other physical and biological systems, and in the study of other problems in applied mathematics related to nonlinear PDEs.