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Title: Mathematical models of liquid crystals and related materials
Author: Taylor, Jamie M.
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2016
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In the broadest sense, the focus of this thesis concerns the role of entropy in macroscopic systems related to liquid crystals in equilibrium. We be- gin by defining a broad generalisation of the Ball-Majumdar potential, applicable to a wide variety of order parameter models relevant to liquid crystals. First we provide a geometric characterisation of the set of ad- missible order parameters for general systems. We demonstrate the key properties of convexity, regularity, and blow up at the boundary of the singular potential. This is then followed by a discussion of applications to Onsager type models and enforcing physicality constraints in Landau expansion-type models. Next we consider the recently proposed Onsager-type model of Zheng, Taylor and Palffy-Muhoray. Unlike Onsager's free energy density, the non-trivial solutions to the new model all have non-trivial support, implying two-sided variations cannot find solutions. In order to overcome this issue, we apply similar heuristics to the previous chapter to split the minimisation into two steps, a strictly convex minimisation problem with linear constraints and a finite dimensional, C1 minimisation problem, the latter of which provides an Euler-Lagrange-type equation satisfied by all Lp -local minimisers. We also perform a more direct analysis on the model to obtain the qualitative features of its phase behaviour. Finally, we consider an extension of the Kuhn-Grün model of polymer elasticity, adapted for use in the modelling of main-chain nematic elas- tomers. This is a particular case of the generalised singular potential, where the anisotropy of the polymer network is included as a further parameter in the energy. This shows the expected energetic coupling between the anisotropy of the system and the extension of a single chain. Furthermore, the polymer chain is finitely extensible, with the energy blowing up as the chain reaches the taut-chain limit, and an implicit force/extension relationship is provided. We conclude with a discussion of network theo-ries for describing macroscopic deformations, with particular focus on the rigidity properties that arise from the growth rate of the chain energy.
Supervisor: Ball, John Sponsor: European Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available