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Nonhomogeneous effects on the phase behavior of rodlike liquid crystals are analyzed through computational simulation on the length scale of a single rod. Rodlike molecules can form a variety of structured phases; the simplest are the isotropic phase (randomly aligned) and the nematic phase (aligned along a common director). A system of rigid rods will go through a spinodal phase transition from isotropic to nematic as the total concentration of rods is increased. Computational methods are developed to use a discretized form of the full nonhomogeneous Onsager intermolecular potential which models interactions on the scale of a single rod length. This potential makes it possible to characterize nonhomogeneous structures and interfaces in terms of the rod length with no adjustable parameters.

Two computational methods are used to study the properties of phase transitions in a one-dimensional system:

(I) A general method for computing equilibrium solutions in a periodic system is developed that discretizes the free-energy expression by the finite element method and uses a parallel Newton's method to solve the resulting dense system of nonlinear equations. Stable states for isotropic-nematic coexistence are computed in a periodic system of finite size. The method is also used to compute the multiple, unstable, nonhomogeneous, equilibrium states in the spinodal regime. These nonhomogeneous equilibrium solutions correspond to unstable attractors in the dynamic process of isotropic-nematic spinodal decomposition. The method is also used to evaluate the impact of hard wall boundaries on the one-dimensional system. The system goes through three phase transition stages as the concentration is increased; the rods at the wall go through a uniaxial-biaxial transition, then a secondary nematic film grows into the interior of the system, and finally the center of the system aligns at high concentrations. The stability limits and the continuity of the final stage are determined by the wall separation.

(II) In addition to the equilibrium calculations, the dynamic process of spinodal decomposition is simulated. The nonhomogeneous Doi equation for the rod distribution function is discretized by the finite element method and integrated forward in time by using a parallel, semi-implicit scheme. The method is applied to isotropic-nematic spinodal decomposition and to the behavior of misaligned nematic grains. The effects of rotational and translational diffusivity ratios are computed, and the mechanisms for alignment and phase separation are analyzed. The initial stages of spinodal decomposition are also simulated in Fourier space in order to study the growth rate of the dominant perturbation. These results mark the first full computation of the evolution of the distribution function for spinodal decomposition in nonhomogeneous rigid-rod systems.

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