The Mathematics of Liquid Crystals - Analysis, Computation and Applications

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Liquid crystals (LC) are mesophases or phases of matter with physical properties intermediate between those of conventional solids and conventional liquids. LCs are ubiquitous in modern life and have widespread applications in science and industry e.g. multimedia technology, optical imaging and bio-medicine. The largest LC application area is display technology, with liquid crystal displays (LCDs) occupying almost 90% of the current flat-panel display market. LCDs are preferred whenever compactness, portability and low power consumption are a priority. The performance of a LCD is controlled by an intricate combination of a variety of factors - external influences, optical properties, response to electric and magnetic fields, elastic effects and defects. Of key importance is the underlying microscopic structure that is often poorly understood. In fact, systematic mechanisms for transferring information between microscopic and macroscopic scales is recognized to be a major challenge for modern LC science.The interaction between mathematics and LC science is twofold. On the one hand, mathematics can give fundamental insight into liquid crystal phenomena which, in turn, is crucial for controlling, predicting and even engineering LC properties. On the other hand, the mathematical modelling of LCs and LCDs leads to novel cutting-edge problems in diverse branches of mathematics e.g. theory of partial differential equations, topology, algebraic geometry, multiscale theory and inverse problems. My research programme aims to (a) to address key mathematical questions in the foundational aspects of LC science complimented by novel numerical algorithms, (b) to develop a cross-disciplinary approach to LC science and (c) integrate theory with industrial LC applications. These problems are of fundamental scientific interest and have immediate relevance to a promising class of high-resolution low power consumption displays known as bistable LCDs. Bistable LCDs are distinctive in the sense that they require power only to switch between optically contrasting states but not to support these states individually e.g. Zenithally Bistable Nematic Device and Post Aligned Bistable Nematic Device.There are a hierarchy of mathematical theories for LCs, ranging from the most detailed atomistic theories to the least detailed macroscopic (continuum) theories. Most of the mathematical work in the field has focused on macroscopic theoretical approaches but a number of open questions remain. In my research programme, I will first develop an arsenal of mathematical tools in the macroscopic theoretical framework. The problems of interest include (i) some key questions related to the effect of geometry and material characteristics on bistability and optical properties and (ii) a rigorous mathematical theory for defects in LCs. Defects are regions of local imperfections in a material and liquid crystal samples are typically populated by such defects. Defects play a crucial role in physical phenomena and yet, they are poorly understood. The second step will be to develop new multiscale methodologies that can couple microscopic and macroscopic models together. The proposed multiscale theories will be analytically tractable, computationally efficient and will capture the microscopic origins of macroscopic behaviour. Such methodologies will also have applications to polymer simulations, membrane modelling and modelling of peptides and proteins. These theoretical and numerical tools will constitute a sound theoretical foundation for bistable LCDs. Industrial researchers are interested in understanding the effect of geometry and material properties on (a) the structure and optical properties of physically observable states and (b) the switching characteristics of the bistable devices. These questions will be answered in active collaboration with industry, with a view to optimize modern LCDs and design new devices tailored to specific applications.