Degree: Doctor of Philosophy, Mathematics
Hometown: Danang, Vietnam
Research Overview: My research with my advisor concentrates on nonlinear partial differential equations and applied analysis, with applications in physical, biological, and materials science. The goal is the modeling and analysis of complex amphiphilic structures and the functionality of nanoparticles/proteins and other bio-inspired materials. Our mathematical models are based on the framework of the Cahn-Hilliard and the Functionalized Cahn-Hilliard equations. The Cahn-Hilliard equation arises as a phenomenological model for isothermal phase separation in a binary alloy. Our work shows a bifurcation phenomenon of the Cahn-Hilliard equation determined by the boundary value and a parameter that describes the thickness of a transition layer separating two phases of an underlying system of binary mixtures. It has applications in materials science as it describes important qualitative features of two-phase systems related with phase separation processes. It also has physical applications in many areas of scientific fields such as diblock copolymer, image inpainting, multiphase fluid flows, microstructures with elastic inhomogeneity, tumor growth simulation, and topology optimization. The Functionalized Cahn-Hilliard energy is a phenomenological model describing the free energy of amphiphilic mixtures that supports codimension one bilayer interfaces separating two identical phases by a thin region of another phase. Our work shows that the Functionalized Cahn-Hilliard equation has a nonnegative weak solution, with a specific form of the degenerate mobility. Results for the Functionalized Cahn-Hilliard free energy include the existence of bilayer structures and some analysis of their bifurcation structure, for instance, the pearling bifurcation that can be observed in amphiphilic polymer blends. The analysis from our research will provide new ideas for the structures of amphiphilic mixtures, hence will help advance the understanding of the functionality of nanoparticles, bio-inspired materials, and biological membranes. It helps us understand essential processes like protein transportation, drug encapsulation, and drug delivery. It will also provide new methods and techniques for the modeling and analysis of properties of other complex structures in polymeric materials. With the collaboration with my advisor’s collaborators in biochemistry, the research outcomes will have influence not only the applied mathematics community, but also the materials science and mathematical biology community.