The Molecular Potential of Biaxial Molecules

The Maier-Saupe theory assmnes high symmetry for molecules forming liquid crystals. In reahty, this is usually not the case and the theory has been extended [3.18] to lath-like molecules. The order parameter tensor S is given by Eq. (3.8) for a biaxial molecule in a uniaxial phase. In the principal axis x y z) system of 5, only two order parameters, Szz and D = Sxx — Syy, are needed, which are related to the Wigner matrices according to Eq. (2.43)  

The potential of mean torque V(cos 0) = — l Pl) Fl(cos 6) must now be altered, since for biaxial molecules their orientation is specified by the 

40) is equivalent to that for cylindrically symmetric molecules, and the second term results from the molecular biaxiality D 0). 2,0 and 2,2 can be written in terms of order parameters (c q q) and (do,2cos20) and expansion coefficients U2mm U202 — U220 for pure systems) which are similar to U2 in Eq. (3.29), 

The self-consistency equations for the determination of the temperature dependences of order parameters are given by [3.18] 

the self-consistency equations ensure that the Helmholtz free energy 

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