Structure of two-dimensional rods confined by a line boundary

In two dimensions, a nematic liquid crystal displays a two-fold rotational symmetry about the normal to the embedding surface. A confining line boundary that restricts the system within an area can geometrically frustrate the system, creating a defective structure in orientation. On the basis of the generalized Onsager model, we present the free energy as a function of the particle density distribution for systems under a square or circular confinement. The numerical solution of the model explains the coupled orientational and directional order observed experimentally.