We present a study of the theory and simulation of Liquid Crystals. A general introduction\ud to the field is given, then the essential features of the Monte Carlo (MC)\ud sampling algorithm are described and explained, along with some of the practical\ud considerations in the implementation of MC. Several quantitative measures used to\ud describe liquid crystalline systems are outlined, including the second rank order tensor,\ud in addition to some of those from elastic theory and density functional theory.\ud Monte Carlo Simulations were performed in bulk geometry in the canonical ensemble\ud in order to calculate the Frank elastic constants of hard spherocylinders, hard\ud platelets and hard cut-spheres at three thicknesses. Onsager’s density functional\ud theory was also performed to yield the elastic constants for hard platelets, and this\ud amounts to using a virial expansion in the free energy, truncated at second order.\ud Our collaborators for O’Brien et al.  provided results for the elastic constants\ud from a calculation of the higher order virial coefficients. All of the results from theory\ud are compared to simulation, with some experimental determinations available.\ud All three elastic constants compared well with the high-order virial theory, there\ud is quantitative agreement with the experimental values, and the effect of increasing\ud thickness of discs was found to improve the agreement of the ratio of K1/K3.\ud Aspects of translationally ordered phases are studied in the context of constrained\ud non-equilibrium systems. Monte Carlo was also performed for platelets confined in\ud wedge geometry, with several choices for the types of wall. A local approximation is\ud utilised to yield the depletion force and potential as a function of the wall separation, as well as the adsorption between the walls. The adsorption for large separations\ud exhibited general qualitative agreement with theory and Gibbs Ensemble simulations.\ud The two different wall boundary conditions produced different orientational\ud structural features, with repulsive and attractive depletion potentials measured, and\ud a planar surface phase that does not appear in the bulk is categorised
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