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 $\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 $K_{11} = K_{33}$.
Received: 03 July 2006
Revised: 09 August 2007
Accepted manuscript online:
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