Static and Dynamic Configurations in Liquid Crystals

This project is focused on mathematical problems arising from the study of liquid crystal materials, especially on understanding static configurations flow patterns. One area of the research is to study the defect and molecule configurations of minimizing solutions of the liquid crystal energies under different (both strong and weak anchoring) boundary conditions and external fields. Such energies include the Oseen-Frank energy for nematic liquid crystals, the Landau-de Gennes energy for smectic ones, etc.. The awardeee will apply the regularity theory for solutions of partial differential equations to these problems. Liquid crystal materials also experience complicated phase transitions under different conditions. A systematic effort will be made to study the stability of the configurations and bifurcation phenomena in different regimes. Finally this research addresses the dynamic properties of different types of liquid crystals. Recent experimental and theoretical results indicate that there is an interesting coupling between the molecule director field and the flow velocity field. Therefore it is important to consider these vector fields simultaneously and study the interaction between them. The proposer will investigate both the theoretical and computational aspects of this coupling problem. The focus is to illustrate the flow effects on the different configurations. The anisotropic characteristics of the material will cause the flow to exhibit non-Newtonian features which in turn corresponds to defects and different molecule patterns. As a special phase of matter, liquid crystals were discovered more than a century ago. Liquid crystal displays (LCD) have been widely used in industries. The fundamental research of such materials is very important. For example, the presence of defects not only affects the manufacturing procedure of the devices, but also provides a new kind of display where the defects contributes to higher accuracy and memory of the material under external fields. This research will focus on mathematical models for liquid crystals, in particular on possible defect configurations and their behavior over time. This study of static and dynamic defect structures will help to improve the modeling and design of such devices. Moreover, an improved understanding of liquid crystals has benefits for broader classes of materials that are important, such as polymeric and biological materials.