Average Orientation in a Three-Dimensional Layer of Nematic Liquid Crystal

The average orientation of a nematic liquid crystal is calculated from the Ericksen-Leslie theory for a three-dimensional parallelepiped. When strong magnetic, electric or flow fields are present to determine the orientation, existing results show the effect of side walls on orientation is not great. In the absence of these forces, however, the side walls can influence the average orientation in the sample.e.g. when the homeotropic orientation of a side wall causes an orientation perpendicular to the homeotropic orientation of the top and bottom surfaces. Calculations give the average orientation when only surface effects are considered, i.e. in the absence of magnetic, electric and flow fields, for a homeotropic orientation of the liquid crystal on all surfaces. Results are well approximated by where the overbar denotes a transverse average (in the x and y directions), φ is the angle between the vertical z axis and the director. Lz is the thickness of the layer. and Lx is the length of the side of a square layer.