Mathematical Theories for Liquid Crystals and their Applications

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 comprise asymmetric molecular building blocks: rod-like, disc-like, box-shaped, bent-core molecules etc. These building blocks assemble/self-organise into different LC phases, all of which have special material directions, known as LC directors, but exhibit different levels of positional organisation/order. Nematic LCs (NLCs) are the simplest LCs with nematic directors but no positional order, cholesteric LCs are twisted or helical NLCs, smectic LCs are layered LCs, along with more ordered and exotic LC phases such as columnar, twist-bend, splay-bend phases. LCs have long fascinated scientists with their exceptional physical properties and responsiveness to external stimuli and have diverse applications across the multi-billion dollar display industry, photonics, metamaterials, robotics, healthcare technologies etc.

The mathematics of LCs is very rich and cuts across analysis, topology, mechanics, partial differential equations and scientific computing, to name a few. There are competing LC theories e.g., molecular-level models with molecular-level information, mean-field models that average molecular details by a mean field, and continuum theories that describe the LC phase by a macroscopic order parameter that describes the macroscopic or measurable LC properties. There are several crucial but open questions wherein mathematics can play a key role, e.g., (i) can we mathematically describe the principles of LC phase formation; (ii) can we design LC systems with prescribed properties and (iii) can we use mathematics to engineer the next generation of LC applications?

We will partially address these open questions in this project. We propose to design and analyse new sophisticated multiscale LC models that will combine the accuracy of molecular models, which contain information about molecular shape and interactions, with the efficiency of continuum theories. We will compare the multiscale predictions to conventional continuum predictions to understand the relationships between the different theoretical frameworks. The analysis will be accompanied by detailed numerical computations in the multiscale modelling frameworks, particularly to explore parameter regimes inaccessible to analytic tools. Finally, we plan to work with experimentalists to examine LCs confined in thin-slab geometries, cylinders, shells and various prototype two-dimensional and three-dimensional geometries. Such systems have potential applications to sensors, photonics, metamaterials, and displays.

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 science or engineering related discipline. Background knowledge in continuum mechanics, theory of partial differential equations and calculus of variations 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 October 2025. 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. Please note that when you register your interest, the email will be sent to Prof Apala Majumdar who is based at The University of Strathclyde.

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.