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 （Dmn^（l）） （Dm＇n＇^（l＇））, where Dmn^（l） is the Wigner rotation matrix. In the lowest order terms, the elastic constants depend on coefficients Γ,Γ＇, λ, order parameters Q0 = Q0（D00^（2）） ＋Q2（D02^（2）＋D0-2^（2）） and Q2 = 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 （τ12） and probability function fk′k″k （τ12）, 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.

Theory of nine elastic constants of biaxial nematics

In this paper, a rotational invariant of interaction energy betweentwo biaxial-shaped molecules is assumed and in the mean fieldapproximation, nine elastic constants for simple distortion patternsin 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. Inthe lowest order terms, the elastic constants depend on coefficients$Ga$, ${Ga'}$, $lambda $, order parameters $bar {Q}_0 = Q_0langle {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 $Ga $ and ${Ga'}$ depend on the function form of molecularinteraction energy $v_{j'j'j} ( {r_{12} } )$ and probabilityfunction $f_{k'k'k} ( {r_{12} } )$, where $r_{12} $ is the distancebetween two molecules, and $lambda $ is proportional to temperature.$Q_0 $ and $Q_2 $ are parameters related to multiple moments ofmolecules. Comparing these results with those obtained fromLandau--de Gennes theory, we have obtained relationships betweencoefficients, order parameters used in both theories. In the specialcase of uniaxial nematics, both results are reduced to a degeneratecase where $K_{11} = K_{33}$.