Nematic-hexagonal discotic phase transition: Pretransitional enhancement of elastic constants

The nematic-hexagonal discotic phase transition is treated in the framework of the Landau theory of phase transitions. We assume the existence of an intermediate phase, separating the nematic and the hexagonal, characterized by translational invariance and sixfold rotational order around the nematic director. As order parameter for the condensation of the hexagonal phase from this intermediate phase, we use a set of complex parameters describing the onset of a triple mass-density wave in the plane orthogonal to the nematic director. The Landau free energy contains a coupling term between the director and the order parameter, arising from local invariance under simultaneous rotations of the director and of the axis of the liquid tubes. The fluctuations of the order parameter slightly above the weakly first-order phase transition give a critical contribution to the Frank elastic constants. Following the response-function method of Jahnig and Brochard, we find that splay and twist elastic constants behave as ${(T-T^{*})}^{-\frac{1}{2}}$ for $T>\mathrm{}{T}^{*}$ in the mean-field approximation and hydrodynamic regime, while bend is noncritical.