Generalised descriptions on orientational order parameters and mean field theory for uniaxial and biaxial nematic liquid crystals

We present a complete set of the orientational and pseudo-vector ( or polar) order parameters and a general potential of mean torque responsible for the formation of a lowest symmetry of triclinic biaxial nematic phases. By adopting the irreducible spherical tensor formalism, twentyfive orientational and nine pseudo-vector ( or polar) order parameters are explicitly expressed in terms of both the Saupe ordering supermatrix and the Wigner rotation matrices. For various biaxial nematic phases and molecules, the corresponding independent order parameters are uniquely classified, including the orthorhombicd (D-2h); monoclinicd (C-2h), and triclinic (C-i) symmetries as well as the uniaxial nematic phase. These works could also describe the order parameters for the smectic C phases and the polar biaxial phases composed of C-1 symmetry.