| Summary: | A liquid crystal (LC) is a substance that exhibits phases intermediate between a crystal and a disordered liquid state. LCs have attracted longstanding research interest because of their potential commercial applications in opto-electronics, pharmaceuticals and surfactants but also because ordered soft matter is prevalent in bio-molecular systems such as DNA and lipid cell membranes. In liquid-crystalline systems, both molecular shape and asymmetric attractive interactions contribute to the formation and ultimate stability of anisotropic phases. The research outlined in this thesis provides a fundamental understanding of these systems by developing theoretical models and undertaking detailed molecular simulation studies. In the first part of this thesis, prototype oblate models for LCs are studied: cut spheres and cylindrical discs. Coupled with a scaled Onsager approach, a general equation of state (EoS) for hard-core discotic LCs is developed that allows for an accurate description of the isotropic and nematic phases of oblate discs by introducing a correction to incorporate the negative contributions from high-order virial coefficients. Combining the above mentioned approach with an extended cell approach, the isotropic-nematic-columnar phase diagram of cut spheres is determined. The accuracy of the EoS is assessed by comparison with the more traditional Parsons-Lee description and existing simulation data. Although the anisotropic athermal hard-body fluid is a reasonable representation of lyotropic or colloidal LCs, for thermotropic LC systems temperature plays a key role. In the second part of this thesis a model of hard-core particles incorporating additional anisotropic attractive interactions is proposed to describe thermotropic LCs. Based on a perturbation theory and the Onsager-Parsons-Lee approach, a van der Waals-type (meanfield level) theory of attractive hard-core particles is formulated in a compact algebraic form. The phase diagrams of model attractive prolate (spherocylinder) and oblate (cylindrical disc) molecules are calculated in order to examine the separate effects of molecular shape and anisotropic attractive interactions. As a practical example, a coarse-grained model comprising an attractive spherocylinder is employed to describe phase behaviour of solutions of the polypeptide poly-(γ-benzyl-L-glutamate) (PBLG) in dimethylformamide (DMF). Quantitative agreement between the results obtained from the EoS and experimental data is obtained. In the final part of the thesis, a detailed Monte Carlo (MC) simulation study of athermal mixtures of hard spherocylinders and hard spheres between two well separated parallel hard walls is performed. A combination of constant volume (canonical ensemble) and constant (normal) pressure (isobaric-isothermal ensemble) simulations are carried out. With these simulations, the bulk phase behaviour as well as surface-induced LC ordering are explored. The phase diagram of binary mixtures of hard spherocylinders and hard spheres is presented and is compared with the predictions of the one-fluid Parsons-Lee and many-fluid theories. Rich phase behaviour is exhibited on the surface of the walls: drying (de-wetting), isotropic wetting, and nematic wetting are all observed. A previously unreported entropy-driven transition from a bulk nematic state to a homeotropic smectic surface ordering (with particles arranged in a perpendicular orientation relative to the surface plane) is seen in for both the pure hard rod system and the mixture of hard rods and hard spheres as the density is increased (high pressure states).
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