The Mathematics of Liquid Crystals - Analysis, Computation and Applications

Lead Research Organisation: University of Bath
Department Name: Mathematical Sciences

Abstract

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.

Planned Impact

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.

Publications

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Canevari G (2019) Minimizers of a Landau-de Gennes energy with a subquadratic elastic energy in Archive for Rational Mechanics and Analysis

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Canevari G (2016) Radial symmetry on three-dimensional shells in the Landau-de Gennes theory in Physica D: Nonlinear Phenomena

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Canevari G (2017) Order Reconstruction for Nematics on Squares and Hexagons: A Landau--de Gennes Study in SIAM Journal on Applied Mathematics

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Chen G (2017) Global existence and regularity of solutions for active liquid crystals in Journal of Differential Equations

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Chen G (2018) Global Weak Solutions for the Compressible Active Liquid Crystal System in SIAM Journal on Mathematical Analysis

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Crespo M (2017) Solution landscapes in nematic microfluidics in Physica D: Nonlinear Phenomena

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D.Henao (2017) Uniaxial versus Biaxial Character of Nematic Equilibria in Three Dimensions. in Calculus of Variations and PDEs

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Kralj S (2014) Order reconstruction patterns in nematic liquid crystal wells in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Related Projects

Project Reference Relationship Related To Start End Award Value
EP/J001686/1 01/10/2011 31/07/2012 £501,887
EP/J001686/2 Transfer EP/J001686/1 01/08/2012 30/09/2016 £437,848
 
Description I have been supported by an EPSRC Career Acceleration Fellowship titled "The Mathematics of Liquid Crystals - Analysis, Computation and Applications" for the period
1st October 2011 - 30th September 2016. Liquid crystals are classical examples of mesogenic phases of matter, somewhere intermediate between classical solids and liquids. Nematic liquid crystals are prototype examples of liquid crystals i.e. they are anisotropic liquids or liquids with a certain degree of orientational ordering and certain preferred directions. The presence of certain distinguished directions leads to unique optical, mechanical and rheological properties and notably, nematics form the backbone of the multi-billion dollar display industry. There is growing interest in the mathematical modelling of nematic liquid crystals and how we can integrate mathematical modelling with other disciplines to make new advances in the field, especially for new materials, nano-science, optical devices, microfluidics, hydrodynamics and even pharmaceutical applications. My research programme has three separate but closely interlinked themes: (i) analysis of different continuum or macroscopic mathematical theories for nematic liquid crystals, (ii) understanding the deep and intricate relations between microscopic and macroscopic models for nematic liquid crystals and (iii) modelling nematic experiments and new nematic-based devices in industry. I have made good progress on all three lines. We have addressed non-trivial mathematical problems within the celebrated Landau-de Gennes theory for nematic liquid crystals, related to the characterization of defects or material singularities. We have derived non-trivial results on the location of defects in three-dimensional geometries with physically relevant boundary conditions in certain asymptotic limits, corresponding to a well-ordered nematic phase for low temperatures; we have mathematically and numerically characterized new previously unreported defect structures in confined systems and have numerically computed phase diagrams that identify parameter regimes where it is energetically preferable to have defects. These theoretical results can greatly assist efficient numerical simulations and guide new experiments. Further, we have undertaken detailed numerical comparisons between microscopic and macroscopic models for nematic liquid crystals which yield interesting quantitative predictions about where the approaches agree and where they can yield different predictions, providing a solid foundation for multiscale hybrid numerical strategies. Such hybrid algorithms hold great promise for soft matter in general. We have successfully applied our theoretical techniques to the numerical modelling of liquid crystal devices, such as the Zenithally Bistable Nematic Device and the Planar Bistable Nematic Device and experiments on nematic liquid crystals confined to two-dimensional geometries or severely confined systems and the modelling has shed new light into the physical problems: the experimentally observed nematic patterns, their stability, the defects and the dynamics of these systems. A theoretical understanding on these lines often paves the way for new experiments and gives experimentalists and engineers greater control on the material and environmental factors to produce desired outcomes. I work with a large network of collaborators in different disciplines around the world and as the academic network grows, our mathematical tools will be applied to diverse physical problems in different fields, opening new doors for interdisciplinary research with mathematics at the forefront and impact within and outside academia.
Exploitation Route My key findings are best summarized in my published papers, which have been submitted separately.
The papers have academic impact and will be used by fellow mathematicians, theoretical physicists and experimentalists in the field of liquid crystals. They will be used to formulate new mathematical problems, to better understand the experimentally observed phenomena, to design new experiments and build new interdisciplinary networks. The numerical methods in some of my co-authored papers can be of use for other modelling problems in materials science. In particular, the mathematics of liquid crystals is of contemporary interest in the Far East with several leading Chinese research groups working on these problems. As such, my academic work also sets the scene for new international networks in mathematics and materials science.
Sectors Manufacturing, including Industrial Biotechology,Other

 
Description British Council Researcher Link Workshop Grant with South Korea
Amount £40,000 (GBP)
Organisation British Council 
Sector Charity/Non Profit
Country United Kingdom
Start 09/2013 
End 03/2014
 
Description Centre de recherches mathematiques research workshop grant
Amount $40,000 (CAD)
Funding ID Research Workshop Grant 
Organisation Center for Mathematical Research 
Sector Academic/University
Country Canada
Start 06/2016 
End 07/2016
 
Description EPSRC-DST scheme for Indo-UK workshops in applied mathematics
Amount £20,000 (GBP)
Funding ID Research Workshop Grant - exact value of grant unknown but enough to provide for 30-40 participants 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Academic/University
Country United Kingdom
Start 10/2015 
End 11/2015
 
Description IPAM Research Workshops. Los Angeles, USA
Amount $50,000 (USD)
Funding ID Research Workshop Grant - exact value of grant unknown but enough to provide for 30-40 participants 
Organisation Institute for Pure and Applied Mathematics 
Sector Academic/University
Country Unknown
Start 01/2016 
End 02/2016
 
Description Royal Society International Exchange Grant
Amount £5,140 (GBP)
Organisation The Royal Society 
Sector Academic/University
Country United Kingdom
Start 02/2013 
End 02/2015
 
Description Chinese Academy of Sciences President's International Fellowship 
Organisation Chinese Academy of Sciences
Department Academy of Mathematics and Systems Science
Country China 
Sector Academic/University 
PI Contribution This is an international fellowship awarded by the Chinese Academy. I plan to visit the Chinese Academy for a month in Beijing to work with Professor Liqun Zhang and Professor Xianmin Xu.
Collaborator Contribution I applied for this fellowship to develop collaborative projects in partnership with leading Chinese researchers.
Impact Expected to lead to peer-reviewed publications and further funding.
Start Year 2015
 
Description Distinguished Grey Fellowship 
Organisation Durham University
Department Department of Mathematical Sciences
Country United Kingdom 
Sector Academic/University 
PI Contribution I have been awarded a Grey Fellowship to collaborate with Dr Patrick Dondl, Professor Paul Sutcliffe and Dr Chakrabarti at Durham nUniversity in March 2016.
Collaborator Contribution This is a visiting fellowship to work on collaborative research projects.
Impact Expected to lead to publications.
Start Year 2015
 
Description European Consortium for Mathematics in Industry Special Interest Group 
Organisation University of Strathclyde
Country United Kingdom 
Sector Academic/University 
PI Contribution I co-lead the Special Interest Group with Professor Nigel Mottram, University of Strathclyde.
Collaborator Contribution Professor Nigel Mottram is a co-leader of the Special Interest Group on Liquid Crystals, Elastomers and Biological Applications SIG.
Impact http://ecmi-softmatter.blogspot.co.uk/p/about.html
Start Year 2012
 
Description OCIAM Visiting Fellowship 
Organisation University of Oxford
Department Mathematical Institute Oxford
Country United Kingdom 
Sector Academic/University 
PI Contribution I co-supervise a graduate student, Alexander Lewis, at the Mathematical Institute, University of Oxford. The team comprises Professor Peter Howell (Mathematics, Oxford), Dr Dirk Aarts (Chemistry, Oxford), myself and Alexander Lewis. We have one joint paper, that was published earlier this year. I collaborate with Professor Radek Erban (Mathematics, Oxford) and Dr Martin Robinson (Mathematics, Oxford) on multiscale modelling for liquid crystals. I collaborate with Dr Ian Griffiths (Mathematics, Oxford) on microfluidic problems for liquid crystals.
Collaborator Contribution Please see above.
Impact Peer-reviewed publications (in publication record). Two further papers are in preparation.
Start Year 2015
 
Description Visiting Professorship 2015-2018 
Organisation Tata Institute of Fundamental Research
Department Centre for Applicable Mathematics
Country India 
Sector Private 
PI Contribution Co-authored the paper Canevari, G., Ramaswamy, M. and Majumdar, A., 2016. Radial symmetry on three-dimensional shells in the Landau-de Gennes theory. Physica D: Nonlinear Phenomena, 314, pp. 18-34.
Collaborator Contribution Canevari, G., Ramaswamy, M. and Majumdar, A., 2016. Radial symmetry on three-dimensional shells in the Landau-de Gennes theory. Physica D: Nonlinear Phenomena, 314, pp. 18-34.
Impact Canevari, G., Ramaswamy, M. and Majumdar, A., 2016. Radial symmetry on three-dimensional shells in the Landau-de Gennes theory. Physica D: Nonlinear Phenomena, 314, pp. 18-34.
Start Year 2014