Mean-field phase diagram of a coupledXY-Ising model for discotic liquid crystals

A simple model Hamiltonian is derived to describe the hexagonal disordered Dhd and hexagonal ordered Dₕₒ columnar phases of discotic liquid crystals with a restricted set of parameters. Characterizing a rigid helical column by an XY-like angular variable and by an Ising-like helicity variable reduces the three-dimensional problem to a coupled XY-Ising Hamiltonian on a triangular lattice. A Landau-type mean-field free energy expansion for two kinds of order parameter is described. The period-3 phase diagram exhibits many types of phases including those with total Ising (helicity) disorder with XY order. This feature is proposed as the structure of the hexa(hexylthio)triphenylene (HHTT) compounds in the Dhd phase. The phase diagram also shows phases with two columns of opposite helicity to the third one and phases where 1/3 of the columns have helicity disorder. These helicity states are relevant to the Dₕₒ phase of (HHTT) compounds.