Asymptotic and variational methods for the study of defects in liquid crystals.

Abstract

In this thesis, I investigate how the presence of mesoscopic colloidal inclusions distorts the locally-averaged molecular orientation of a thermotropic nematic shell in a state of thermodynamic equilibrium at a constant temperature. In the following analysis, I focus on three problems, utilising methods from different mathematical disciplines.

Firstly, I consider the Robin problem for the Laplace operator in a two-dimensional domain with a small hole of radius ε>0. I approximate the solution to this problem, in the spirit of compound asymptotics, by the sum of the so-called ``interior'' and ``exterior'' functions, which describe the behaviour of the solution in the bulk of the domain and in the neighbourhood of the small hole, respectively. I prove that the approximation converges uniformly to the solution of the Robin problem, and derive the rate of convergence as a function of the radius of the small hole and the Robin parameter.

Secondly, I define a notion of convergence for sequences of integral energy functionals that are such that the pointwise limit of the related stored energy densities is non-integrable. To derive the limit of a given functional sequence, I consider an asymptotic expansion of each functional into the sum of a divergent lower-order term and convergent higher order term, similar to what is known as Γ-asymptotics. Firstly, I derive the subset of functions that minimise the rate at which the lower-order term diverges. As a consequence, I develop a heuristic algorithm predicting the strengths of defects of a thin film of nematic liquid crystal, given realistic boundary data and certain geometric parameters. Secondly, I then seek the Γ-limit of the convergent higher-order term in this subset. The sum of the lower-order term and the corresponding higher-order Γ-limit is referred to as the Γ-asymptotic limit.

Finally, I study the limiting behaviour of the solutions to Poisson's equation subject to inhomogeneous boundary conditions, as the number of non-periodically distributed holes tends to infinity. I generalise existing convergence results for the case of Robin boundary conditions, by deriving the weak limit of the solutions to the inhomogeneous problems and investigate how the boundary data affects the ``strange term'' in the limit description.

Date of Award	2 Nov 2022
Original language	English
Awarding Institution	University of Bath
Supervisor	Kirill Cherednichenko (Supervisor), Jeyabal Sivaloganathan (Supervisor) & Apala Majumdar (Supervisor)