Mathematical Analysis of Topics from Materials Science

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This project is devoted to the mathematical study of two important classes of contemporary materials that are used in a variety of technologies: liquid crystal materials and block copolymers. While liquid crystal materials are quite common and are used, for example, for liquid crystal displays, the block copolymer materials are less familiar. They possess a remarkable capacity for self-assembly into ordered structures. The goal of this project is to investigate several mathematical models of these materials to better understand their structure, defects, and behavior. The project will provide opportunities for the research training of undergraduate and graduate students. This project will develop analytical and numerical study of some nonlinear partial differential equations (PDE) modeling liquid crystals and block copolymer systems. The principal investigator will address (1) several analytic problems in both static and dynamic configurations in the framework of Q-tensor order parameters, that include defect configurations around a colloid particle, a physical condition and its mathematical role in the hydrodynamic system, loss of initial physicality in all non-co-rotational hydrodynamic system, a gradient flow dynamics generated by a singular potential; (2) the asymptotic behavior and trend to equilibrium of classical solutions of the Doi-Onsager equation that is derived in molecular theory to model rigid rod-like polymer molecules; (3) the global well-posedness and stable, efficient numeric schemes of a gradient flow dynamics modeling the self-assembly of block copolymers during time evolution. The main ingredients and techniques involved in the study include methods from elliptic and parabolic PDE, stability analysis, gradient flow theory in Hilbert and metric spaces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.