Unit sphere description of liquid-crystal configurations

In any one-dimensional liquid-crystal configuration, the unit director n(z) generates a path on the unit sphere. Paths that satisfy the equilibrium equations are characterized by special properties that are independent of the boundary conditions. This paper explores these properties for nematic liquid crystals, and applies them to delimit the kinds of equilibrium states that are possible and to simplify calculations needed in the design of liquid-crystal displays. The applications include simple approximate formulas for (1) the relation between the director angle at the boundary ϑb and at the midplane ϑm for twisted states, (2) the dependence on elastic constants of the critical value of ϑb below which the 180° twisted state disappears, being replaced by its topologically equivalent planar state [G. Porte and J. P. Jadot, J. Phys. 39, 213 (1978)], and (3) the region of elastic constant space for bistability of topologically equivalent planar and 180° twisted states. The unit sphere is also applied to specify the paths of nonequilibrium states that are useful in stability studies.