Solution of the mixed Dirichlet–Neumann problem for molecular orientation in liquid crystals

The orientation of the nematic director field under the action of an external time‐dependent field is theoretically investigated as a mixed Dirichlet–Neumann boundary‐value problem. This mathematical problem represents the situation in which a nematic liquid crystal sample is limited by two inhomogeneous flat surfaces, separated by a distance d, on which the anchoring is weak. By considering the one‐constant approximation and a parabolic approximation for the surface energy, the initial conditions and boundary‐value problem for the profile of the tilt angle can be analytically solved even in the case in which the surfaces are not identical, which represents the more general situation. The results are valid for small deviations from the homeotropic orientation and for θ?Θ?1, where θ is the actual tilt angle and Θ characterizes the easy direction imposed by the surface, and can be relevant to investigation of the molecular orientation in a nematic cell submitted to a small external voltage.