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-S_A) lyotropic phase transition at a packing fraction η = η_NA = 0.355 where the smectic layer spacing d is d_c = 1.35L. These values agree with results by computer simulation: η_NA = 0.35 and d_c = 1.27L. Incorporating an attractive potential, we show the system exhibits the second order N-S_A phase transition at a temperature T_NA, and find for T < T_NA the relation (d/d_c - 1)/(1 - T/T_NA) ∼ 10⁻¹ and the layer compression modulus B - 15(kT/v_mol) (1 - T/T_NA), where v_mol is the molecular volume. The stability of the columnar phase due to the hard core repulsion cannot be proved in the second virial approximation.

Key Words:

• statistical theory
• hard rod fluids
• columnar phases
• excluded volume effect
• lyotropic SA phase
• thermotropic SA phase