Non-uniform liquid--crystalline phases of parallel hard rod-shaped particles: From ellipsoids to cylinders

In this article we consider systems of parallel hard {\it superellipsoids}, which can be viewed as a possible interpolation between ellipsoids of revolution and cylinders. Superellipsoids are characterized by an aspect ratio and an exponent $\alpha$ (shape parameter) which takes care of the geometry, with $\alpha=1$ corresponding to ellipsoids of revolution, while $\alpha=\infty$ is the limit of cylinders. It is well known that, while hard parallel cylinders exhibit nematic, smectic, and solid phases, hard parallel ellipsoids do not stabilize the smectic phase, the nematic phase transforming directly into a solid as density is increased. We use computer simulation to find evidence that for $\alpha\ge\alpha_c$, where $\alpha_c$ is a critical value which the simulations estimate to be in the interval 1.2--1.3, the smectic phase is stabilized. This is surprisingly close to the ellipsoidal case. In addition, we use a density-functional approach, based on the Parsons--Lee approximation, to describe smectic and columnar ordering. In combination with a free--volume theory for the crystalline phase, a theoretical phase diagram is predicted. While some qualitative features, such as the enhancement of smectic stability for increasing $\alpha$, and the probable absence of a stable columnar phase, are correct, the precise location of coexistence densities are quantitatively incorrect.