# The Mathematics of Liquid Crystals - Analysis, Computation and Applications

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### 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.

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.

[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.

## People |
## ORCID iD |

Apala Majumdar (Principal Investigator / Fellow) |

### Publications

MAJUMDAR A
(2011)

*The radial-hedgehog solution in Landau-de Gennes' theory for nematic liquid crystals*in European Journal of Applied Mathematics
Majumdar A
(2018)

*Remarks on uniaxial solutions in the Landau-de Gennes theory*in Journal of Mathematical Analysis and Applications
Canevari G
(2016)

*Radial symmetry on three-dimensional shells in the Landau-de Gennes theory*in Physica D: Nonlinear Phenomena
Lewis A
(2017)

*Revisiting the Two-Dimensional Defect-Free Azimuthal Nematic Equilibrium on an Annulus*in SIAM Journal on Applied Mathematics
Chen G
(2018)

*Global Weak Solutions for the Compressible Active Liquid Crystal System*in SIAM Journal on Mathematical Analysis### 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 | The grant EP/J001686/1 comprises roughly the first year of my five-year EPSRC Career Acceleration Fellowship. My fellowship is based around questions in the mathematical theories for nematic liquid crystals, their relationship to other theories in materials science, numerical simulations of prototype liquid crystalline systems and applications of theory to new experiments and liquid crystal devices. The first year of my fellowship set the scene for two key theoretical findings, that have subsequently informed my ongoing research during EP/J001686/2. [1] In joint work with Duvan Henao, I rigorously study the low temperature limit of the Landau-de Gennes theory for nematic liquid crystals. This temperature limit is well-suited for an analytic study of liquid crystal defects, an area which is largely open. We successfully establish a correspondence between vortex structures in the Ginzburg-Landau theory for superconductivity and typical three-dimensional point defects in the Landau-de Gennes theory for nematic liquid crystals, in this temperature limit. Whilst the result is largely theoretical in nature, it is valuable because it beautifully elucidates the overarching mathematical similarities between the Ginzburg-Landau theory and Landau-de Gennes theory along with the underpinning differences, and opens new avenues for consolidating techniques from different mathematical theories in materials science for new mathematical advances. This work was published as Henao, D. and Majumdar, A., 2012. Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystals. SIAM Journal on Mathematical Analysis (SIMA), 44 (5), pp. 3217-3241. [2] In joint work with Peter Howell, Dirk Aarts and Alexander Lewis, we derive explicit estimates for the energies of experimentally observed nematic patterns in prototype two-dimensional geometries. The energy estimates can be related to frequency of observation of states in experiments and can also be used to derive estimates for material-dependent properties e.g. surface extrapolation length. An example of this work can be found in Lewis, A., Garlea, I., Alvarado, J., Dammone, O., Howell, P., Majumdar, A., Mulder, B., Lettinga, M. P., Koenderink, G. and Aarts, D., 2014. Colloidal liquid crystals in rectangular confinement:Theory and experiment. Soft Matter, 39 (10), pp. 7865-7873. |

Exploitation Route | These findings are immediately relevant to applied analysts, modellers, experimentalists and computational researchers in the field. The analytical work on the connections between Ginzburg-Landau theory and Landau-de Gennes theory has already been followed up by graduate students in France, who are building on our results to mathematically characterize admissible defects in liquid crystals. The energy estimates for nematic patterns in prototype geometries can be generalized to more complicated systems, yielding quantitative predictions about the observability and stability of different states that are crucially needed for predictive modelling and effective design and optimization of experimental strategies. |

Sectors | Other |

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 |