Exact solutions for liquid crystal configurations and an improved boundary layer model

Abstract

The director configuration in a liquid crystal cell is basic to an understanding of both the optical contrast between states and switching dynamics. This paper treats the voltage dependence of nematic liquid crystal configurations in which the director orientation depends on a single space coordinate z, and there are equal but oppositely tilted boundary conditions on the two cell surfaces at z=±d/2. In the boundary layer model, configurations are approximated by joining together pieces of ‘‘boundary layer solutions,’’ which are solutions of the equations for equilibrium configurations in a half space with the boundary condition θ=constant at only one surface and θ=dθ/dz=0 at z=∞, where θ is measured from the normal to the surface. This paper (1) indicates exact solutions that satisfy the cell boundary conditions, (2) examines their high voltage asymptotes, (3) proves that the boundary layer solutions reproduce the essential features of these asymptotes, and (4) presents a way of fitting boundary layer solutions into a finite cell by joining them where θ=θmin such that θmin is the correct asymptotic function of voltage. Topologically distinct horizontal (H) and vertical (V) states are distinguished by the boundary conditions: if θ=θb at one surface, then at the other surface θ=−θb for V and π−θb for H. The H state can be symmetric (HS) or asymmetric (HA). All of these states are treated, with emphasis on the high voltage asymptotic behavior. Below the HA−HS threshold, HA is nonexistent, and HS is stable; above this threshold, HA is stable and HS is unstable [Julian Cheng, R. N. Thurston, and D. W. Berreman, J. Appl. Phys. 52, 2756 (1981)].