Freedericksz thresholds in nematics as eigenvalues

Abstract

ABSTRACT In this paper there is demonstrated that the threshold external field magnitudes for Freedericksz transition in nematicliquid crystals can be considered in the form of the boundary eigenvalue problem for differential operator d2/dx2 withdifferent boundary conditions. Three cases have been investigated due to three boundary conditions at the planar nematicslayer surfaces: strong anchoring of molecules parallel to the surface, week anchoring of molecules at the surface and stronganchoring ofmolecules tilted to the surface.Keywords: nematic liquid crystals, Freedericksz transition, mathematical modelling, boundary eigenvalue problem. 1. INTRODUCTION The static deformations in nematics can be described by the three-dimensional unit vector field n called director,characterising the average local orientation of molecules. In the frame of linear theory of elasticity (i.e. in the frame of smallstrains) they are completely characterised by the director and the gradient of it (i.e. the derivatives of the director componentswith respect to the co-ordinates2 ). Every deformation can be represented as the superposition of the three elementarydeformations: splay, bend and twist; the elastic potential2 F (i.e. the free energy) has the densityf:(1)