Phase diagram and orientational order of a system with second and fourth rank interactions

A simple generalized Lebwohl-Lasher model for liquid crystals, where a fourth rank interaction is added to the usual second rank one, is investigated in detail. We have obtained the phase diagram of the system performing extensive Monte Carlo computer simulations for a range of the fourth to the second rank relative strengths and we compare it with the prediction of Mean Field and Two Site Cluster theories. We show that the addition of a non-negligible fourth rank term significantly changes the temperature dependence of the order parameter. Fourth rank contributions larger than 20% worsen the agreement of the model with the typical temperature behavior of the order in nematics.