Onsager's model of isotropic-nematic phase transition and its extensions

We study Onsager’s model of isotropic–nematic phase transition with orientation parameter on a circle and sphere. We show the axial symmetry and derive explicit formulae for all critical points. Using the information about critical points we investigate a theory of orientational order in nematic liquid crystals which interpolates between several distinct approaches based on the director field (Oseen and Frank), order parameter tensor (Landau and de Gennes), and orientation probability density function (Onsager). As in density-functional theories, the free energy is a functional of spatially-dependent orientation distribution, however, the spatial variation effects are taken into account via phenomenological elastic terms rather than by means of a direct pair-correlation function. As a particular example we consider a simplified model with orientation parameter on a circle and illustrate its relation to complex Ginzburg-Landau theory.