Statistical Theory of Smectic A and Columnar Phases in Hard Rod Fluids

Abstract The free energy of smectic A and columnar phases in parallel hard cylinders with length L and width D are calculated as functions of the order parameters. In the third vial approximation we find the second order nematic to smectic A (N-SA) lyotropic phase transition at a packing fraction η = ηNA = 0.355 where the smectic layer spacing d is dc = 1.35L. These values agree with results by computer simulation: ηNA = 0.35 and dc = 1.27L. Incorporating an attractive potential, we show the system exhibits the second order N-SA phase transition at a temperature TNA, and find for T < TNA the relation (d/dc - 1)/(1 - T/TNA) ∼ 10−1 and the layer compression modulus B - 15(kT/vmol) (1 - T/TNA), where vmol is the molecular volume. The stability of the columnar phase due to the hard core repulsion cannot be proved in the second virial approximation.