Molecular Description of Nematic Liquid Crystals

Liquid crystals are considered as gases of long barlike molecules. Variables describing the position and orientation of individual molecules are introduced. These are used to arrive at a microscopic definition of the order in liquid crystals in terms of a symmetric traceless tensor. The Frank and Leslie-Ericksen directors are shown to be equivalent to the unit eigenvector associated with the largest eigenvalue of this symmetric traceless tensor. Attention is then restricted to nematics. It is shown that the Frank free energy gives a valid description of equilibrium properties of nematics to order $\frac{{\chi }^a{H}^2}{\rho \kappa T{m}^{-1\mathrm{}}}$, where ${\chi }^a$ is the anisotropic magnetic susceptibility, $H$ the external magnetic field, $\rho$ the mass density, $T$ the temperature, $k$ the Boltzmann constant, and $m$ the molecular mass. Microscopic justification is given for the linearized Leslie-Ericksen hydrodynamic theory for nematics. A microscopic response-function description of nematics is then presented, and contact is made between the general frequency- and wave-number-dependent response functions and those calculated from the phenomenological hydrodynamic theory. Applications to light scattering and nuclear magnetic resonance are considered. $\frac{1}{T_1}$ calculated here differs from that calculated from the Pincus theory by a factor of $S^2$, where $S$ is the parameter measuring the degree of nematic order.