On the Variation of the K13Nematic Surface-Like Volume Energy and the Nematic Surface Energy of Mada

Abstract

Two possible ways of the one-dimensional variation of the K₁₃ nematic surface-like volume energy are discussed. The first way of variation when the deformation angle θ and its derivative θ’ are considered as independent functions at the boundary shows that the K₁₃ problem has no solution. The second way of variation when θ and θ’ are considered as dependent functions at the boundary shows that the problem can be solved by introducing of the inverse function and variation of a functional with a movable boundary. The three-dimensional solution for a variational problem including a nematic energy density containing both first and second spatial gradients has been found on the basis of Ericksen & Toupin variational arguments. The surface and body forces as well as the generalized surface and body forces were obtained. The surface molecular field including the K₁₃ term was obtained in explicit form. This field for the one-dimensional case unambiguously confirms the validity of our boundary conditions previously obtained (J. Physique 38, 1013, 1977). On the basis of our unpublished theoretical calculations and experimental results obtained by other researchers for some electrooptical effects, we have estimated that the value of K₁₃ for the nematic MBBA at room temperature is slightly smaller than half of the splay elastic coefficient K'₁₁. In analogy with the solution of the K₁₃ elastic problem and using the variational arguments of Ericksen & Toupin we have obtained the surface and body forces as well as the generalized surface and body forces for the case of Mada elastic energy.