# Analysis of liquid crystal models

Lead Research Organisation: University of Bristol
Department Name: Mathematics

### Abstract

Modeling, simulation, analysis, and design of liquid crystal systems and devices raises many fundamental questions of physics and mathematics. The study of appropriate phase transitions, formation and evolution of singularities and defects in nematic liquid crystals leads to challenging mathematical problems. In order to understand these phenomena we rely on analysis of variational problems and nonlinear partial differential equations arising from corresponding physical models.There are several major theories to study nematic liquid crystals: Onsager theory, Maier-Saupe mean field theory, Oseen-Frank director theory, Ericksen theory, and Landau-de Gennes theory. These theories are based on investigation of the free energy of the liquid crystalline system. Minimizers of the free energy correspond to equilibrium states and in order to understand the liquid crystalline system behavior it is enough to study properties of these minimizers.The main goal of this project is to analytically investigate variational problems arising in Onsager-Maier-Saupe and Landau-de Gennes models. We would like to study the qualitative properties of the equilibrium states of liquid crystalline systems by investigating the minimizers of the corresponding energy functionals. In particular we plan to investigate formation and profiles of the singularities in liquid crystalline systems, phase transitions in biaxial nematics, and relation between different liquid crystal theories. We also would like to gain the insight about nonequilibrium phenomena in the liquid crystalline systems by studying the gradient flow (and related) dynamics generated by free energy.

### Planned Impact

The proposed interdisciplinary research project will establish rigorous links between different liquid crystal models and develop new analytical tools for the mathematical theory of liquid crystals. Deeper understanding of the mathematics of liquid crystals will immensely benefit growing liquid crystalline community in the UK (Bristol, Oxford, Southampton, Strathclyde, etc.) and throughout the world. The methods and results of this project will be applicable in the study of other natural systems like polymers and biophysical systems. The potential beneficiaries of this proposal will also be researchers working in different areas of physics and chemistry of liquid crystals. In order to improve current liquid crystal display technology and to develop new types of liquid crystal displays it is necessary to understand the formation of singularities and phase transitions in liquid crystalline systems. Since one of the main objectives of this research project is to investigate the formation and evolution of singularities in liquid crystals and phase transitions in biaxial nematics, the outcomes of this research will also be beneficial for liquid crystal industry. Similar mathematical problems arise in many areas and the impact of this research proposal will also be felt in the related fields, for instance, micromagnetics and Ginzburg-Landau theory of superconductivity.

### ORCID iD

Valeriy Slastikov (Principal Investigator)

### Publications

10 25 50
Di Fratta G (2015) Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals in Journal of Nonlinear Science

Fatkullin I (2018) Limit Shapes for Gibbs Ensembles of Partitions in Journal of Statistical Physics

Goussev A (2014) Domain wall motion in thin ferromagnetic nanotubes: Analytic results in EPL (Europhysics Letters)

Goussev A (2013) Domain wall motion in magnetic nanowires: an asymptotic approach. in Proceedings. Mathematical, physical, and engineering sciences

Ignat R (2014) Stability of the Melting Hedgehog in the Landau-de Gennes Theory of Nematic Liquid Crystals in Archive for Rational Mechanics and Analysis

Ignat R (2016) Stability of point defects of degree $$\pm \frac{1}{2}$$ ± 1 2 in a two-dimensional nematic liquid crystal model in Calculus of Variations and Partial Differential Equations

Ignat R (2016) Instability of point defects in a two-dimensional nematic liquid crystal model in Annales de l'Institut Henri Poincaré C, Analyse non linéaire

Description New mathematical ideas and analytical tools were developed for a wide class of problems in mathematical liquid crystals. In particular, development of variational and maximum principle methods for problems in Landau-de Gennes theory yielded a string of new interesting results that shed a light on formation and structure of point defects in liquid crystals. Investigation of the generalized Onsager model provided better understanding of the current dynamical models for nematic liquid crystals.
New directions of the research in the area have been identified and new open problems were formulated. In particular, it includes the structure and stability of critical points of Landau-de Gennes energy in two dimensions and modeling of polydisperse nematics using Onsager-type models.
The existing collaborations have been strengthened and new links between myself and various groups working on liquid crystals were established through research visits and research discussions during various seminars, workshops and conferences.
Exploitation Route Development of new mathematical ideas lead to a breakthrough in analytical treatment of liquid crystal models, in particular Landau-de Gennes model. These analytical techniques can be used not only by researchers working in the field of mathematical liquid crystals but also in a wider area of applied analysis, including polymer physics, superconductivity and micromagnetics.
Sectors Education,Electronics,Other

Description The immediate impact is in the research area of mathematical liquid crystals and it has been described in the Key Findings section. Apart from this, the results of the research project have been used in education by improving the presentation of the mathematical theory of variational problems to both undergraduate and graduate students.
First Year Of Impact 2012
Sector Education,Other

Description Liquid Crystals Defects in Landau-de Gennes Theory
Amount £155,927 (GBP)
Funding ID RPG-2014-226
Organisation The Leverhulme Trust
Sector Charity/Non Profit
Country United Kingdom
Start 01/2015
End 01/2018

Description Mathematical Analysis of Domain Wall Motion in Nanowires
Amount £303,615 (GBP)
Funding ID EP/K02390X/1
Organisation Engineering and Physical Sciences Research Council (EPSRC)
Sector Public
Country United Kingdom
Start 09/2013
End 09/2016

Description Research in Pairs
Amount € 1,000 (EUR)
Organisation Mathematical Research Institute of Oberwolfach
Country Germany
Start 04/2014
End 05/2014

Description Research in Pairs
Amount € 1,000 (EUR)
Organisation International Centre for Mathematical Meetings (CIRM)
Country France
Start 06/2014
End 07/2014

Description Research in Paris
Amount € 1,500 (EUR)
Organisation Henri Poincaré University
Country France
Start 08/2014
End 09/2014

Description Mathematical Analysis of Landau-de Gennes Model
Organisation Princeton University
Country United States
PI Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Collaborator Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Impact The outcomes of this collaboration include several scientific papers published in top mathematical journals (10.1137/130948598, 10.1007/s00205-014-0791-4, 10.1016/j.crma.2013.07.012) and further funding obtained, including Research in Pairs grants from Oberwolfach and CIRM, Research in Paris grant from IHP and Leverhulme Trust research grant.
Start Year 2010

Description Mathematical Analysis of Landau-de Gennes Model
Organisation University Paris Sud
Country France
PI Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Collaborator Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Impact The outcomes of this collaboration include several scientific papers published in top mathematical journals (10.1137/130948598, 10.1007/s00205-014-0791-4, 10.1016/j.crma.2013.07.012) and further funding obtained, including Research in Pairs grants from Oberwolfach and CIRM, Research in Paris grant from IHP and Leverhulme Trust research grant.
Start Year 2010

Description Mathematical Analysis of Landau-de Gennes Model
Organisation University of Sussex
Country United Kingdom
PI Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Collaborator Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Impact The outcomes of this collaboration include several scientific papers published in top mathematical journals (10.1137/130948598, 10.1007/s00205-014-0791-4, 10.1016/j.crma.2013.07.012) and further funding obtained, including Research in Pairs grants from Oberwolfach and CIRM, Research in Paris grant from IHP and Leverhulme Trust research grant.
Start Year 2010

Description Mathematical Analysis of Onsager Model
Organisation University of Arizona
Country United States
PI Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Collaborator Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Impact One published research paper (10.3934/dcdss.2015.8.323) and development of a numerical code.
Start Year 2006

Description Singularities and phase transitions: statics and dynamics
Organisation Northumbria University
Country United Kingdom
PI Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Collaborator Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Impact Several papers published in top interdisciplinary journals (10034752, 10083932, 84881123523). Further funding obtained from EPSRC (EP/K02390X/1).
Start Year 2009

Description Variational models in micromagnetics and liquid crystals
Organisation International School for Advanced Studies
Country Italy
PI Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Collaborator Contribution This is a research collaboration and the main contribution is an expertise and a joint research.
Impact Submitted research paper (arXiv:1311.4435)
Start Year 2006

Description Applied Mathematics seminar, University of Strathclyde
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Seminar talk.

Research discussions.
Year(s) Of Engagement Activity 2012

Description Colloquium talk, Worcester Polytechnic Institute
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Colloquium talk.

Research discussions.
Year(s) Of Engagement Activity 2013