Fréedericksz transitions in nematic liquid crystals as boundary eigenvalue problems

Abstract

Fréedericksz transitions between stationary planar deformation states, induced by cover surface interactions and by static external electric or magnetic field in a flat-parallel nematic liquid crystal cell, are studied theoretically in one-dimensional approximation. There is demonstrated that for initial stages of transition, when the deviation of director field from a boundary-induced state is small, the system of Euler-Lagrange equations for free-energy functional can be linearised and the threshold external field magnitudes can be deduced from eigenvalues of differential operator of second derivative for boundary eigenvalue problems with different boundary conditions. Different combinations of electric or magnetic field, parallel or perpendicular to the nematics layer boundaries, and different boundary conditions (including both strong and weak anchoring of nematics molecules) are analysed by using the same formalism. The considerations are based on Frank theory of static deformations of nematic liquid crystals. The formulae for the threshold fields, as depending on nematics and nematics-substrate coupling parameters, are derived. There are stated possible applications of these formulae to determining material parameters by exploiting threshold field magnitudes experimentally found.