The Method of Lines for Solving the Poisson Boundary Problem for Smectic SmC* an an External Electric Field

Abstract
The theoretical model of the unwound smectic liquid crystal (LC) chiral thin slab’s behaviour in the «bookshelf» geometry under the influence of the various tilted electric fields to the smectic layers in the one-constant approximation K_i = K, (i = 1,2,3) when all Frank elastic constants are equal to each other was proposed. For the solution of the director n distribution problem in the smectic LC specimen, which is described by the two-dimensional nonlinear Poisson equation of the elliptic kind with periodic boundary conditions, the numeric semianalytic method of lines was applied. This approach uses the finite difference method for all variables in the problem under consideration except one permitting the analytical integration of it. The method of linesbased on numeric calculus and used for parabolic kind of the partial differential equations (PDE) with the use of computer integration systems was adopted for this purpose. Our equation belongs to the elliptic kind of the PDE and approach mentioned above requires some development. The false transient method of lines for elliptic equations gives one possibility to avoid this restriction by introducing the fictive time variable. Then the time additional derivative transfers the PDE into the parabolic kind. The director field relaxation process towards its stable state is illustrated on the basis of such approach. The commensurate macrostructures appearance in the bulk of the ferroelectric LC possessing the spontaneous polarization was demonstrated.

Keywords: ferroelectric liquid crystal, «bookshelf» geometry, commensurate structures, periodic structures, method of lines, semianalytic method