Computation of the nonhomogeneous equilibrium states of a rigid-rod solution

The nonhomogeneous equilibrium phase behavior of a solution of rigid rods is analyzed for a periodic one-dimensional system. Stable and unstable equilibrium solutions for the distribution function are computed as extrema of the free energy of the system expressed by the nonhomogeneous generalization of Onsager's [Ann. N.Y. Acad. Sci. 51, 627 (1949)] theory, which models interaction between rods on the scale of a single rod length. Biaxial equilibrium solutions are computed in a periodic system by discretizing the Euler-Lagrange nonlinear integral equation by the finite-element method and using Newton's method to solve the resulting set of nonlinear equations. Stable states for isotropic-nematic coexistence are computed in a periodic system rather than the semi-infinite system used in previous calculations. The density and order parameter profiles evolve monotically from the isotropic phase to the nematic phase. Unstable, nonhomogeneous, equilibrium states are also computed for concentrations of rods that exceed the value for spinodal decomposition. These nonhomogeneous states are characterized by combinations of bend, twist, and splay distortions in physical space and correspond to unstable attractors in the dynamic process of isotropic-nematic spinodal decomposition. For large systems, the nonhomogeneous states develop wide, bulklike nematic regions separated by thin regions with sharp gradients in orientation. The free energy formulation was also used to compute the accurate neutral stability curve; this curve shows the limits of applicability of the low-wave-number approximations frequently used in the study of spinodal decomposition.