A van der Waals theory of nematic liquid crystals: a ‘convex peg in a round hole’ potential

The convex peg potential has been incorporated into a van der Waals theory of the (uniaxial) nematic to isotropic liquid crystal transition. In the convex peg model, molecules are envisioned to have a hard (here biaxial) core embedded in a spherically symmetric square well. Anisotropies in the potential are derived from both its repulsive and attractive regions. In accord with generalized van der Waals theories, the repulsive interactions are treated to all orders (albeit approximately using a resummation technique) and the attractive interactions are incorporated to lowest order. Global phase diagrams are determined and these predict the formation of one uniaxial nematic and two isotropic phases (vapour and liquid). From the phase diagram, one observes the coexistence of the nematic-vapour phases, the nematic-isotropic liquid phases, the vapour-liquid phases as well as a nematic-vapour-isotropic liquid triple point. By virtue of the effective anisotropic attractive forces, the first order character of the nematic to isotropic transition is significantly enhanced from that of a hard uniaxial body. However, the influence of biaxiality more than compensates for this enhancement and the resulting isotropic to nematic transition becomes weakly first order. The temperature dependence of the order parameters at fixed density (and also at fixed pressure) compare favourably with the experimental values of PAA.