Downloaded By: [Trinity College] At: 15:41 12 March 2007 LIQUID CRYSTALS, 1993, VOL. zyxwv 14, No. 4, 1227-1236 zyxw On the calculation of the dielectric relaxation times of a nematic liquid crystal from the non-inertial Langevin equation by W. T. COFFEY* School of Engineering, Trinity College, Dublin 2, Ireland and YU. P. KALMYKOV Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Sq 1, Fryazino, Moscow Region, 141120 Russia The theory of dielectric relaxation of uniaxial nematic liquid crystals is developed without recourse to the Fokker-Planck equation by direct averaging of the non-inertial Langevin equation for the rotational brownian motion of the linear molecule in a mean-field nematic potential. The non-inertial equation is regarded as a non-linear Stratonovich stochastic differential equation. The molecular equations for the average values of the dipole moment components so obtained involve both the nematic field and a suddenly applied weak DC measuring field. The equations are linearized in the DC field so that the AC response may be found by linear response theory. The Laplace transform of the equations is closed by a procedure which corresponds exactly to the effective eigenvalue method. It allows one to obtain formulae valid for all barrier heights for the longitudinal zyxw z ,, and transverse zy zI relaxation times for an arbitrary uniaxial nematic potential in terms of the order parameter. The complex susceptibility induced by a weak AC field applied parallel and perpendicular to the axis of symmetry is also calculated. 1. Introduction The theory of dielectric relaxation of nematic liquid crystals due to Martin zy et al. [ 11 proceeds from the Fokker-Planck equation without explicit reference to the underly- ing Langevin equation. Their aim is to extend the Debye theory of dielectric relaxation of assemblies of non-interacting polar molecules subjected to a weak alternating (AC) field to include the effects of a strong intermolecular potential giving rise to the nematic state. The AC response is usually obtained indirectly from linear response theory [2] by considering the response to a small DC step field. The essence of the diffusion equation method [3] is to write down the particular form of the Fokker-Planck equation known as the Smoluchowski equation, for the transition probability of orientations of dipoles in configuration space. This is solved [ 11by the method of separation of the variables. The separation procedure gives rise to an equation of Sturm-Liouville type [3] in the space variable which is related to Legendre’s equation. The reciprocal of the lowest eigenvalue of the Sturm-Liouville equation yields the longest relaxation time of the probability density of orientations. Furthermore on expanding the dipole moment as a series of eigenfunctions of the Sturm-Liouville equation and averaging over the distribution function the orient- ational polarization may be expressed [3] as an infinite set of discrete Debye type relaxation mechanisms with relaxation times and amplitudes determined by the eigenvalues of the Sturm-Liouville equation. Approximate analytic solutions for the * Author for correspondence. 0267-8292/93 $1000 zyxwvu 0 1993 Taylor & Francis Ltd.