Solution Landscapes and Inverse Problems in the Landau-de Gennes theory for Liquid Crystals

About the Project

Liquid crystals (LCs) are beautiful smart materials that combine fluidity and softness with the structural order of solids. At a basic level, LCs are anisotropic or directional soft materials with distinguished material directions, referred to as "directors". There are a multitude of LC phases: nematic LCs that are directionally ordered complex fluids; cholesteric LCs which are twisted or helical NLCs and smectic LCs that are layered LCs, along with more ordered and exotic LC phases such as columnar, twist-bend, splay-bend phases. LCs have direction-dependent responses to external stimuli such as external fields, mechanical stress, incident light etc. and this intrinsic directionality makes LCs the working material of choice for a variety of electro-optic devices, notably the multi-billion dollar liquid crystal display industry.

The mathematics of LCs is very rich and cuts across analysis, topology, mechanics, partial differential equations and scientific computing, to name a few. Modern LC applications rely heavily on accurate and efficient mathematical modelling of confined LC systems. Typical questions are - can we theoretically predict physically observable LC configurations for a given physical system; can we design a system to stabilise LC configurations with desired properties and can we reconstruct material properties from experimental data on LC systems?

In this project, we will partially address these questions and study prototype LC systems e.g., LCs inside shells, cylinders, cuboids etc. within the celebrated Landau-de Gennes theory for LCs. The Landau-de Gennes theory is a variational theory and the physically admissible configurations are modelled in terms of solutions of appropriately defined boundary-value problems for systems of nonlinear partial differential equations. We will study the qualitative properties of the solution landscapes of these systems of partial differential equations, including both the stable and unstable solutions. In particular, we will use topology and shape optimisation methods to compute the optimal domain shapes that can stabilise solutions with desired/prescribed properties. We will use methods from inverse problems to reconstruct material properties from given solutions in the Landau-de Gennes framework or experimental data, with an overarching view to understand and quantify relationships between LC material properties, geometry, topology and solution landscapes. These relationships ultimately hold pivotal clues to designer futuristic LC technologies.

Supervisors: Professor Apala Majumdar (apala.majumdar@manchester.ac.uk) and Dr Joel Daou (joel.daou@manchester.ac.uk)

Eligibility

Applicants should have, or expect to achieve, at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant mathematical sciences or engineering related discipline. Background knowledge in continuum mechanics, theory of partial differential equations, calculus of variations and numerical methods for differential equations is desirable.

Funding

The successful candidate will join the PhD programme of the Department of Mathematics at the University of Manchester. This 42-month PhD position (funded by the University of Manchester) is a full scholarship which will cover tuition fees for UK-based students and provide an annual stipend in line with EPSRC recommended levels (£20,780 for 2025/26), with an expected start date of 1st January 2026. This scholarship is available only to UK students eligible for home fee status.

Before you apply

We strongly recommend that you contact the supervisors for this project before you apply. Please include details of your current level of study, academic background and any relevant experience and include a paragraph about your motivation to study this PhD project.

How to apply

Apply online through our website: https://uom.link/pgr-apply-2425

When applying, you’ll need to specify the full name of this project, the name of your supervisor, if you already having funding or if you wish to be considered for available funding through the university, details of your previous study, and names and contact details of two referees.

Your application will not be processed without all of the required documents submitted at the time of application, and we cannot accept responsibility for late or missed deadlines. Incomplete applications will not be considered.

After you have applied you will be asked to upload the following supporting documents:

• Final Transcript and certificates of all awarded university level qualifications
• Interim Transcript of any university level qualifications in progress
• CV
• Supporting statement: A one or two page statement outlining your motivation to pursue postgraduate research and why you want to undertake postgraduate research at Manchester, any relevant research or work experience, the key findings of your previous research experience, and techniques and skills you’ve developed. (This is mandatory for all applicants and the application will be put on hold without it).
• Contact details for two referees (please make sure that the contact email you provide is an official university/work email address as we may need to verify the reference)
• English Language certificate (if applicable)

If you have any questions about making an application, please contact our admissions team by emailing FSE.doctoralacademy.admissions@manchester.ac.uk.