Monte Carlo Results for a Discrete-Lattice Model of Nematic Ordering

Abstract

The lattice version of the Maier-Saupe model for the nematic-liquid-crystal transition consists of an array of unit vectors located at the sites of some regular lattice. The energy of interaction of nearby unit vectors is proportional to $P_2 \left({cos\theta }_{\mathrm{ij}}\right)=\frac{1}{2} (3\mathrm{}{{cos}^2\theta }_{\mathrm{ij}}-1)$, where ${\theta }_{\mathrm{ij}}$ is the angle between the vectors at sites $i$ and $j$ which represent the directions of the long axes of the molecules of the liquid. A discrete model which we call the dodecahedral model is defined by restricting the vectors to point in the directions of the faces of a regular dodecahedron. Monte Carlo results are obtained for this model, which is among the models discussed by Potts as interesting generalizations of the Ising model. It is also one of the sequence of models solved in the mean-field approximation and two-cluster approximation by Priest. The Monte Carlo results on a 10×10×10 simple-cubic lattice with periodic boundary conditions determine the transition to be of first order. The order at the transition $\mathrm{M}={\mathrm{P}}_2\left(cos\theta \right)$ is found to be 0.82. Attempts to obtain results on the model with vectors pointing in the six directions of the faces of a cube failed because the 10×10×10 lattice was too small in this case to discriminate between a first- and second-order transition.