This thesis concerns two distinct problems from the mathematical analysis of complex materials. One of the central themes is variational principles, which are widely used to understand physical phenomena, and one of the central aims of this type of work is to provide rigorous mathematical support to known results from physical literature. The first problem studied originates from micromagnetic theory (continuum theory for magnetism) and is that of domain wall motion in ferromagnetic nanowires. The magnetization dynamics is governed by a partial differential equation known as the Landau-Lifshitz-Gilbert equation. In this thesis, we present a new asymptotic method for understanding the dynamics of a domain wall in a thin (effectively one-dimensional) magnetic nanowire under the influence of an applied magnetic field or electronic spin current. We describe two flavours of solution: travelling waves, and oscillating solutions, as well as the transitions between them, by the use of perturbation expansions. We also provide a rigorous proof of the existence of travelling-wave solutions to this problem by the application of the implicit function theorem. We then present and analyze a heuristically derived model for the motion of vortex domain walls containing a micromagnetic singularity in thicker wires where the one-dimensional approximation fails. The second problem studied is that of phase transitions in biaxial nematic liquid crystals. Here, we study a variational theory based on the Onsager functional: a free energy functional derived from statistical mechanics of a liquid composed of anisotropic particles. We characterize the phase diagram of a biaxial liquid by considering three separate regimes: the weak-biaxial interaction regime (where the potential interaction between neighbouring molecules is close to uniaxial), the low-temperature (strong interaction) regime, and the near-isotropic regime. The global phase diagram is then inferred from these three analyses.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:658635 |
| Date | January 2014 |
| Creators | Lund, Ross G. |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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