[School of Natural Sciences PhD Scholarships] Solution Landscapes and Inverse Problems in the Landau-de Gennes theory for Liquid Crystals

About the project

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. This project is expected to start in September 2027. Before you apply: To apply, please contact the supervisors: Professor Apala Majumdar ( apala.majumdar@manchester.ac.uk ) and Dr Joel Daou ( joel.daou@manchester.ac.uk ). 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: To be considered for this project you must complete a formal application through our online application portal. If you already have an applicant account this link will directly open an application for PhD School of Natural Sciences Scholarships . If you don’t already have an applicant account, please follow the instructions here . When applying, please specify the full title and supervisor/s of the project, details of your previous study, and names and contact details of two referees. You must also upload a Supporting Statement describing your motivation to apply to the project, your CV and transcripts of awarded and in-progress university qualifications . Please note late or incomplete applications will not be considered. Equality, diversity and inclusion are fundamental to the success of The University of Manchester and central to all our activities. A diverse research community strengthens creativity, productivity and quality, while increasing the societal and economic impact of our work. We welcome applicants from all career paths, backgrounds and sections of the community, regardless of age, disability, ethnicity, gender, gender expression, sexual orientation or transgender status. We welcome applications from candidates returning to study after a career break or experience in other roles. Flexible study arrangements may be available, including part-time study at 50%, 60% or 80%, subject to the requirements of the project and funder. Eligibility : The standard academic entry requirement for this PhD is an upper second-class (2:1) honours degree (or international equivalent) in a relevant mathematical sciences or engineering related discipline OR any upper-second class (2:1) honours degree and a Master’s degree at merit (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. This project will remain open until filled. If your application is submitted by 1st November 2026, you can expect a decision by 18th December 2026. If your application is submitted by 15th January 2027, you can expect a decision by 30th March 2027. Self or externally funded students can also be considered for this project. FSESoNS

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