Mathematics of Liquid Crystal Elastomers

Fully Funded EPSRC PhD Studentship in Applied and Computational Mathematics

Available from October 2022

Liquid crystal elastomers (LCEs) are advanced, stimuli-responsive materials which can convert light or heat into mechanical work, without the need for batteries, electric wires or gears. These extraordinary properties are due to their molecular architecture consisting of cross-linked networks of polymeric chains (rubber) containing liquid crystal mesogens (usually found in liquid-crystal displays). 

Despite their difficult synthetic processes, the intriguing mechanical behaviour of LCEs has been probed extensively in laboratories around the world thanks to recent developments in additive manufacturing (3D printing).

Theoretically, considerable progress is yet to be made in their description by combining rubber elasticity and liquid crystal theories. Even though some success has been achieved in adapting existing hyperelastic models from nonlinear elasticity to match experimental data, predictive models to bridge the gap between laboratory investigation and industrial applications are still to be established.

This PhD project aims to develop a mathematical model that captures the auxeticity observed in a novel nematic LCE where the thickness of a stretched material sample increases at large strain while the material volume remains unchanged. This is in contrast to many other rubber-like materials and nematic elastomers, which, when extended, become thinner in the perpendicular directions. When viewed as a 2D system, this auxetic effect coincides with an apparent sharp rotation of the average mesogenic alignment director. In 3D, the sharp director rotation is accompanied by a gradual decrease and then increase in uniaxial orientational order coupled with the emergence and subsequent loss of biaxial symmetry.

The project involves three main stages: (1) Constructing a model function that captures the mechanical behaviour observed experimentally in auxetic LCE under different deformations; (2) A rigorous mathematical analysis to establish conditions for existence and uniqueness of solution to the corresponding boundary value problems; (3) Computer simulation and numerical analysis (employing molecular dynamics and finite element methods).

Throughout the project, the PhD student will acquire:

·        The ability to identify, investigate and analyse critically relevant information;

·        A deep knowledge in mathematical analysis of partial differential equations;

·        A strong understanding of fundamental nonlinear solid mechanics;

·        Transferable skills in scientific programming in Matlab, Python, etc;

·        Advanced written and spoken communication skills.

Supervisory meetings will be regular, to guarantee that work progresses as planned and the PhD thesis is completed on time, and to monitor and seek to improve the student’s overall academic development and general wellbeing.

The work will further be assessed annually through a progress review conducted by the project supervisors and one other faculty member.

To expand their mathematical knowledge beyond that offered by their specific project, in addition to research work, in the first two years, the student will attend a total of 100 hours of postgraduate lecture courses run by MAGIC (Mathematics Access Grid Instruction and Collaboration), which is jointly organised remotely by 20 UK Universities. Assignments to the MAGIC courses will be assessed by the course lecturers.

For their general intellectual and personal development, the student will also have numerous opportunities to participate in research seminars, workshops, and national or international conferences where they will be encouraged to present their work to various audiences.

At the School of Mathematics, the successful candidate will become member of a vibrant postgraduate research community, which hosts the SIAM-IMA Student Chapter at Cardiff University, formed by the Society for Industrial and Applied Mathematics and supported by the Institute of Mathematics and its Applications. They will be provided with a desk and a computer, and access to state-of-the-art facilities in the School’s purpose-built new home, Abacws, where the School moved in September 2021.

Cardiff School of Mathematics is committed to establishing an intellectual community based on diversity, fairness, and inclusion, and applications from women and other minorities are particularly encouraged.

HOW TO APPLY

Applicants should apply through the Cardiff University online application portal, for a Doctor of Philosophy in Mathematics with an entry point of October 2022.

In the research proposal section of your application, please specify the project title and supervisors of this project. There is no requirement to submit a detailed research proposal.

In the funding section, please select "I will be applying for a scholarship / grant" and specify that you are applying for advertised funding from EPRSC DTP.

ACADEMIC CRITERIA

A 1^st or upper 2^nd class UK Honours degree (or equivalent) or a Master’s degree is required in mathematics or a related subject.

 ENGLISH LANGUAGE REQUIREMENTS

Applicants for whom English is not their first language must demonstrate proficiency by obtaining an IELTS score of at least 6.5 overall, with a minimum of 5.5 in each skills component. Further acceptable qualifications can be found at https://www.cardiff.ac.uk/study/international/english-language-requirements/postgraduate

SUPPORTING DOCUMENTS

·        Curriculum vitae

·        A personal statement

·        Two completed references

·        Degree certificates and transcripts

·        Proof of English language (if applicable)

DEADLINE FOR APPLICATIONS

10 June 2022

ELLIGIBILITY

 EPSRC studentships are available to home and international students. International students will not be charged the fee difference between the UK and international rate. Applicants should satisfy the UKRI eligibility requirements. 

 Cardiff University is committed to supporting and promoting equality and diversity and to creating an inclusive environment for all. We welcome applications from all members of the global community irrespective of age, disability, sex, gender identity, gender reassignment, marital or civil partnership status, pregnancy or maternity, race, religion or belief and sexual orientation.

We welcome applications for both full and part-time study and from candidates with non-traditional academic backgrounds. For further information about modes of study, please contact us.

ASSESSMENT

 Applicants are reminded to submit all relevant documents (transcripts, supporting statement, etc) by the deadline. Incomplete applications will not be considered.

 Short-listed applicants will be invited to interview w/c 20 June 2022.

 As part of the interview process, applicants will be asked to answer a series of panel questions.

Interviews are expected to take place remotely via Zoom on 29 June 2022. Applicants can expect to hear the outcome of their interview within 2 weeks.