The Mathematics of Liquid Crystals - Analysis, Computation and Applications

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.

The proposed research is distinctive since it addresses questions of fundamental scientific interest, is motivated by industrial applications and will have demonstrable impact on industry. The academic beneficiaries have been described in the previous section. The main non-academic beneficiaries are -
[1] Industrial liquid crystal (LC) research groups: I have a long-standing collaboration with the Displays Media Research Group at Hewlett Packard (HP) Labs, Bristol, UK. This collaborative experience has given me a good understanding of how industrial research groups work and I am well-trained in disseminating my research amongst industrial circles. The research objectives in this proposal have direct relevance to a new class of high-resolution and low power consumption liquid crystal displays (LCDs), known as bistable LCDs. Bistable LCDs are very popular in the modern technological scene and some of my research objectives will be accomplished in active collaborations with HP researchers. These objectives will give theoretical insight into the design of bistable LCDs with optimal properties. In the long run, these insights can contribute to the manufacture of future high performance displays e.g. e-notebooks, organizers etc. The means of engagement will be - collaborative visits, Competitive Award in Science and Engineering (CASE) students, peer-reviewed publications and departmental seminars. I will aim to build new industrial contacts with e.g. LC research groups at Sharpe Labs and ZBD Ltd. These research groups actively work on bistable LCD design and my research programme is of immediate relevance to their scientific themes. I will organize annual LC days in Oxford and invite LC researchers from Sharpe Labs, ZBD Ltd. to these events. I have recently started to work with Dr Dirk Aarts (Department of Chemistry, Oxford) on polymeric LC systems; this project is distinct from this proposal but the proposed methodologies are transferable to problems in polymer science. My collaborator, Dr Dirk Aarts, has close connections with research groups at Unilever that are interested in biological LC applications. I will build connections with the Unilever research group and share my theoretical approaches with them.
Industrial researchers are often not sufficiently aware of the fact that mathematics can not only explain physical phenomena but also identify new physical phenomena. Oxford has many schemes for bridging this gap between
industry and applied mathematics e.g. OCIAM's weekly interdisciplinary workshops, Study Groups with Industry etc. I will exploit these contacts to publicize the importance of my research to a wider audience and identify new application areas for my research methodologies.
[2] Non-academic researchers :Novel multiscale numerical algorithms will be designed and implemented over the course of the fellowship. These algorithms will integrate conventional methods for solving partial differential equations
with Monte Carlo methods, Molecular Dynamics simulations etc. and will have applications to a range of multiscale problems at the physical sciences/engineering interface e.g. reaction diffusion equations in biology, modelling of fractures, cracks and interfaces etc. Examples of non-academic beneficiaries include medical researchers and engineers. I will design a user-friendly project webpage that will signpost the key features of my research agenda and the mutliscale algorithms. These algorithms will be shared with interested non-academic researchers.
Active public engagement is a central part of research impact. I am currently involved in the establishment of an applied mathematics laboratory in the University of Oxford, which will be used for teaching and outreach activities. Further, I will also participate in public science festivals and write review articles on LCs for non-specialist audiences, the overall aim being to raise the general awareness about my research and its practical implications.