Translational ordering in systems of rods and discs: The effect of discretizing the orientational and/or the translational degrees of freedom

Off-lattice, freely rotating hard rods are a reasonable representation of certain lyotropic liquid crystalline systems. The translational ordering of these rods is compared with that of rods which can point along only one of the three coordinate axes and that of rods on a lattice, using bifurcation analysis of a free energy expression truncated at the second virial coefficient level. Whereas off-lattice freely rotating hard rods form a smectic phase, long rods on a lattice form a columnar phase. Even for off-lattice rods, restricting their orientations changes their behaviour qualitatively, at least if the rods are very long. Disc-like molecules are shown to be affected similarly by the imposition of a lattice. In general, it is not possible to study translationally ordered, liquid-crystalline phases of off-lattice systems by examining their lattice analogues.