# Equilibrium Liquid Crystal Configurations: Energetics, Singularities and Applications

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

The research concerns the mathematical description of liquid crystals, an intermediate state of matter between solids and liquids. Liquid crystals are of technological importance because they are the working material of many electronic displays. A uniaxial nematic liquid crystal consists of rod-like molecules which at each point tend to align in a particular direction, like matches in a matchbox. This direction can be influenced by electric fields. The direction is determined by solving the governing partial differential equations that describe the mechanics of the liquid crystal. These equations depend in particular on the form that is assumed for the energy density of the liquid crystal. The proposal has various aims. The first is to examine when an arrow can be consistently assigned to the preferred direction. Such an arrow is not physical, since the molecules can be inverted end-to-end without changing their mechanical properties. However the existence of such an arrow is assumed in some widely-used theories, and we will study how the predictions of such theories compare to other theories in which no such arrow is assigned. The second aim is to investigate whether forms of the energy density different from those usually used could help to describe the behaviour of liquid crystals at defects, that is at points where the alignment direction is not uniquely defined. Related to this is the question of how the energy density can be determined experimentally. The third aim is to study when the equations have equilibrium solutions, in the context of theories which do not assign an unphysical arrow to the preferred direction, and whether these solutions have defects. Finally we will apply our work to the study of prototype new display devices, such as the Post Aligned Bistable Nematic (PABN) Device being developed at Hewlett-Packard laboratories.

### Publications

Ball J
(2017)

*Mathematics and liquid crystals*in Molecular Crystals and Liquid Crystals
BALL J
(2012)

*ON UNIQUENESS FOR TIME HARMONIC ANISOTROPIC MAXWELL'S EQUATIONS WITH PIECEWISE REGULAR COEFFICIENTS*in Mathematical Models and Methods in Applied Sciences
Ball J
(2008)

*Orientable and Non-Orientable Line Field Models for Uniaxial Nematic Liquid Crystals*in Molecular Crystals and Liquid Crystals
Ball J
(2017)

*Mathematical Thermodynamics of Complex Fluids*
Ball J
(2010)

*Nematic Liquid Crystals: From Maier-Saupe to a Continuum Theory*in Molecular Crystals and Liquid Crystals
Ball J
(2017)

*Partial regularity and smooth topology-preserving approximations of rough domains*in Calculus of Variations and Partial Differential Equations
Ball J
(2011)

*Orientability and Energy Minimization in Liquid Crystal Models*in Archive for Rational Mechanics and Analysis
Capdeboscq Y
(2013)

*On one-dimensional inverse problems arising from polarimetric measurements of nematic liquid crystals*in Inverse Problems
Majumdar A
(2009)

*Landau-De Gennes Theory of Nematic Liquid Crystals: the Oseen-Frank Limit and Beyond*in Archive for Rational Mechanics and AnalysisDescription | The grant concerned the Landau - de Gennes theory of nematic liquid crystals. There were several significant advances concerning how this theory corresponds to the more widely used Oseen-Frank theory, and a new bulk energy function for the Landau - de Gennes theory was proposed. |

Exploitation Route | There is the possibility that the research could lead to new effective computational tools for predicting liquid crystal behaviour in displays. |

Sectors | Digital/Communication/Information Technologies (including Software),Electronics |

Description | The findings have been taken up by various researchers in the theory of liquid crystals. In particular the PI has a large ERC grant in which liquid crystals is one of the main topics, with various students and postdocs working in the area. However so far there have been no known impacts outside academia. |

Description | Advanced Investigator Grant |

Amount | € 2,006,998 (EUR) |

Funding ID | 291053 |

Organisation | European Research Council (ERC) |

Sector | Public |

Country | Belgium |

Start | 03/2012 |

End | 03/2017 |