Theory of nine elastic constants of biaxial nematics

Abstract

In this paper, a rotational invariant of interaction energy between two biaxial-shaped molecules is assumed and in the mean field approximation, nine elastic constants for simple distortion patterns in biaxial nematics are derived in terms of the thermal average 〈Dmn(l)〉 〈Dm′n′(l′), where D(l)mn is the Wigner rotation matrix. In the lowest order terms, the elastic constants depend on coefficients Γ, Γ′, λ, order parameters Q̄0 = Q0 〈D(2)00 + Q2〈D02(2) + D0-2(2) and Q̄2 = Q0〈D20(2)〉 + Q2 〈D22(2) + D2-2(2)〉. Here Γ and Γ′ depend on the function form of molecular interaction energy vj′j″j(r12) and probability function fk′k″k(r12), where r12 is the distance between two molecules, and λ is proportional to temperature. Q0 and Q2 are parameters related to multiple moments of molecules. Comparing these results with those obtained from Landau-de Gennes theory, we have obtained relationships between coefficients, order parameters used in both theories. In the special case of uniaxial nematics, both results are reduced to a degenerate case where K11 = K33.