A Possible Field-Theoretical Model for the Nematic-Isotropic Phase Transition in Liquid Crystals
We intend to study some aspects of the well-known nematic-isotropic phase transition in liquid crystals, by exploiting the nonlinear features of the system. Treating the molecules as hard rods and regarding the fact that the heads and the tails of the molecules cannot be distinguished in the nematic phase and using the translational invariance along the z-axis ( which is parallel to the director $\hat n$ in the nematic phase), we arrive at a model that displays the essential features of the system. If we assume that the potential is periodic and write it as, for example, -αcosβϕ, the system undergoes a phase transition like that we found in the sine-Gordon model at β2=8π1. Thus, the critical temperature (NI) is given by T =2πLKθ/ KB where L is the length of the system in the z-direction, Kθ is the elastic torsion constant and KB is the Boltzmann constant.
It is necessary to ellaborate this general conjecture by continuing the development of the Statistical Mechanics of the system. But this is a hard problem ( the model is in [2+1] dimensions). Despite of this, some interesting questions may arise. We mention, for example, the possibility of using the soliton solutions of the model and to investigate their role in the transition. On the other hand we may ask if it is possible to relate this picture of a liquid crystal to some manifestations of the Coulomb gas. The problem is how to do it if we attain a positive answer.