A simple model for the stiffness constants of nematic liquid crystals based on distributed harmonic forces between the molecules

Abstract

The stiffness constants of a nematic liquid crystal are calculated for the situation of perfect orientational order with the aid of a model of distributed harmonic forces. Both attractive and repulsive forces are taken as distributed along the molecules, and thus are only important for the parts of two molecules that are in close proximity. For small deviations between the directors of two molecules the potential is assumed to vary quadratically in the deflection. In addition two models for the pair distribution function are considered. In the case of nematic-like pair correlation (molecular centres of neighbours distributed at random around the excluded volume due to the central molecule) K ₃/K ₁ is predicted to vary as L /W ², where L and W are the length and width of the molecule, respectively. This is in agreement with more sophisticated theories and explains the experimental trend observed for molecules without alkyl chains. The case of smectic-like pair correlation (the neighbours have some preference for their centres to be in the same plane) is simulated by taking a gaussian distribution. Increasing the sharpness of this distribution leads to a decrease of K ₃/K ₁, which explains the experimental results for homologous series. In all cases K ₂/K ₁ = 1/3 is predicted, again in agreement with more elaborate theories and in reasonable agreement with the experimental results.