Empty smectic liquid crystals of hard nanorings: Insights from a second-virial theory

Inspired by recent simulations on highly open liquid crystalline structures formed by rigid planar nanorings, we present a simple theoretical framework explaining the prevalence of smectic over nematic ordering in systems of ring-shaped objects. The key part of our study is a calculation of the excluded volume of such nonconvex particles in the limit of vanishing thickness to diameter ratio. Using a simple stability analysis we then show that dilute systems of ring-shaped particles have a strong propensity to order into smectic structures with an unusual antinematic order while solid disks of the same dimensions exhibit nematic order. Since our model rings have zero internal volume, these smectic structures are essentially empty, resembling the strongly porous structures found in simulation. We argue that the antinematic intralamellar order of the rings plays an essential role in stabilizing these smectic structures.