Theory of nine elastic constants of biaxial nematics

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 \langle D_mn^(l) \rangle \langle D_m'n'^(l') \rangle , where D_mn^(l) is the Wigner rotation matrix. In the lowest order terms, the elastic constants depend on coefficients \varGamma, \Gamma', \lambda , order parameters \bar Q_0 = Q_0 \langle D_00^(2) \rangle + Q_2 \langle D_02^(2) + D_0 - 2^(2) \rangle and \bar Q_2 = Q_0 \langle D_20^(2) \rangle + Q_2 \langle D_22^(2) + D_2 - 2^(2) \rangle . Here \varGamma and \varGamma' depend on the function form of molecular interaction energy v_j'j''j ( r_12 ) and probability function f_k'k''k ( r_12 ), where r_12 is the distance between two molecules, and \lambda is proportional to temperature. Q_0 and Q_2 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 K_11 = K_33.