Stability of Columnar Phase and Smectic A-Crystalline Phase Transition in the System of Parallel Identical Hard Rods

Abstract

It is shown in the 3-rd virial approximation that the smectic A-crystalline phase transition occurs in perfectly aligned identical rigid cylinders (or discs) with diameter D and length L , by using the method of symmetry breaking potential. The smectic A phase and a hexagonal crystalline phase are shown to co-exist for 0.39<η<0.53, where η denotes the packing fraction. In the co-existence region, the crystalline lattice constants \(a \doteqdot 1.35D\) in the hexagonal plane and \(c \doteqdot 1.26L\) along the normal to the plane and the smectic layer spacing \(l \doteqdot 1.35L\) are obtained. The hexagonal columnar phase is shown to be more stable than the smectic A phase at higher density than \(\eta _{\rm SmA-Col}\doteqdot 0.78\) but always less stable than the crystalline phase. These results are in dependent of the ratio D / L due to the assumption of perfectly parallel alignment of molecules.