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Results are presented from a variety of molecular simulations of phase coexistence using the well established Gay-Berne (GB) liquid crystal model. Firstly, the simulation of bulk phase coexistence using the Gibbs ensemble Monte Carlo technique is presented, both for one and two-component GB systems. The one-component results, using a novel parameterisation of the GB, show a rich phase behaviour, displaying both isotropic and nematic-vapour coexistence, in good comparison with previous studies. A method for arriving at the two-component parameterisation is then discussed, followed by a novel application of the Gibbs ensemble to the isotropic-nematic transition in two-component systems. Results in broad agreement with theoretical predictions, subject to a large finite size effect, are obtained. Secondly, upon the basis of the one-component Gibbs results, results are presented from a series of molecular dynamics simulations of a free standing GB film in equilibrium with its own saturated vapour. The introduction of inhomogeneity is shown to induce a preferred molecular alignment in the nematic film perpendicular to the liquid-vapour interface. At slightly higher temperatures the nematic film is wet by the isotropic phase, displaying an intermediate ordering regime where the formation of short-lived nematic domains within the film is observed. This effect has been analysed using orientational correlation functions, and shown to result from a decoupling of the planar and perpendicular nematic ordering caused by the system inhomogeneity. A system-size analysis of this effect has also been undertaken, showing a definite increase in the range of decay of these orientational correlations with increasing system size.

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The polaron problem is studied, on an infinite lattice, using the continuous-time path-integral quantum Monte Carlo scheme The method is based on the Feynman technique to analytically integrate out the phonon degrees of freedom. The transformed problem is that of a single electron with retarded self-interaction in imaginary time. The Metropolis algorithm is used to sample an ensemble of electron trajectories with twisted (rather than periodic) boundary conditions in imaginary time, which allows dynamic properties of the system to by measured directly. The method is numerically "exact", in the sense that there are no systematic errors due to finite system size, trotter decomposition or finite temperature The implementation of the algorithm in continuous imaginary time dramatically increases computational efficiency compared with the traditional discrete imaginary time algorithms.

Results are presented from a series of simulations undertaken to determine the effect of a novel form of molecular biaxiality upon the phase behaviour of the well established Gay-Berne (GB) liquid crystal model. Firstly, the simulation of a bulk system interacting via the Internally-Rotated Gay-Berne (IRGB) potential, which offers a single-site representation of a molecule rigidly constrained into a zig-zag conformation, is presented. The results of simulations performed for systems of IRGB particles with an aspect ratio of 3:1 confirm that the introduction of biaxiality into the model results in the destabilisation of the orientationally ordered phases. For particles with a sufficiently pronounced zig-zag conformation, this results in the complete destabilisation of the smectic A phase and the smectic B phase being replaced by the tilted smectic J phase. Following these observations, the effect upon the phase behaviour of increasing molecular elongation is also considered, with an increase in the aspect ratio from 3:1 to 4:1 resulting in the nematic and smectic J phases being replaced by smectic A and smectic G phases respectively. Secondly, a version of the IRGB potential modified to include a degree of molecular flexibility is considered. Results obtained from bulk systems interacting via the flexible IRGB for 3:1 and 4:1 molecules show that the introduction of flexibility results in the destabilisation of the smectic A phase and the stabilisation of the nematic and tilted hexatic phases. Finally, the effect upon the phase behaviour of the rigid IRGB model of the inclusion of a longitudinal linear quadrupole is examined. These results show that increasing quadrupole moment results in the destabilisation of the tilted hexatic phase, although the biaxial order parameter is increased with increasing quadrupole moment. There is no clear correlation between quadrupole magnitude and the other observed phase transitions, with the nematic and smectic A phases being variously stabilised and destabilised with increasing quadrupole magnitude. For the 4:1 molecules with large quadrupole moments, buckled smectic layers are observed where some molecules are tilted with respect to a local layer normal. Of all the systems considered here, this buckled structure is the one which most closely resembles the elusive smectic C phase.

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