Free energy power expansion for orientationally ordered phases: energy and entropy

We propose a new approach for description of orientational phase transitions that utilizes the following specific features of the orientational energy $E$ and entropy $S$: (a) $S$ possesses an additional symmetry in comparison with $E$, being invariant under rotation of the molecular frame; and (b) $E$ contributes only to the second order terms because the pair molecular interaction is dominant. The approach is based on minimization of the scaled orientational free energy $\bar{F}=F/T=E/T-S$ instead of $F$ because $\bar{F}$ obeys the standard assumption of the Landau theory that only the second order terms are temperature dependent. We apply the approach to build a model for nematic phases in materials with non-polar parallelepiped-type molecules with symmetry$\ D_{2h}$. The presented model introduces complex OPs, generalizes the Landau-de Gennes (LdeG) theory and predicts the existence of a biaxial nematic phase for the forth order expansion of $\bar{F}$.