Analysis of liquid crystal models

Modeling, simulation, analysis, and design of liquid crystal systems and devices raises many fundamental questions of physics and mathematics. The study of appropriate phase transitions, formation and evolution of singularities and defects in nematic liquid crystals leads to challenging mathematical problems. In order to understand these phenomena we rely on analysis of variational problems and nonlinear partial differential equations arising from corresponding physical models.There are several major theories to study nematic liquid crystals: Onsager theory, Maier-Saupe mean field theory, Oseen-Frank director theory, Ericksen theory, and Landau-de Gennes theory. These theories are based on investigation of the free energy of the liquid crystalline system. Minimizers of the free energy correspond to equilibrium states and in order to understand the liquid crystalline system behavior it is enough to study properties of these minimizers.The main goal of this project is to analytically investigate variational problems arising in Onsager-Maier-Saupe and Landau-de Gennes models. We would like to study the qualitative properties of the equilibrium states of liquid crystalline systems by investigating the minimizers of the corresponding energy functionals. In particular we plan to investigate formation and profiles of the singularities in liquid crystalline systems, phase transitions in biaxial nematics, and relation between different liquid crystal theories. We also would like to gain the insight about nonequilibrium phenomena in the liquid crystalline systems by studying the gradient flow (and related) dynamics generated by free energy.

The proposed interdisciplinary research project will establish rigorous links between different liquid crystal models and develop new analytical tools for the mathematical theory of liquid crystals. Deeper understanding of the mathematics of liquid crystals will immensely benefit growing liquid crystalline community in the UK (Bristol, Oxford, Southampton, Strathclyde, etc.) and throughout the world. The methods and results of this project will be applicable in the study of other natural systems like polymers and biophysical systems. The potential beneficiaries of this proposal will also be researchers working in different areas of physics and chemistry of liquid crystals. In order to improve current liquid crystal display technology and to develop new types of liquid crystal displays it is necessary to understand the formation of singularities and phase transitions in liquid crystalline systems. Since one of the main objectives of this research project is to investigate the formation and evolution of singularities in liquid crystals and phase transitions in biaxial nematics, the outcomes of this research will also be beneficial for liquid crystal industry. Similar mathematical problems arise in many areas and the impact of this research proposal will also be felt in the related fields, for instance, micromagnetics and Ginzburg-Landau theory of superconductivity.