EXACT SOLUTION FOR THE EXTENDED DEBYE THEORY OF DIELECTRIC-RELAXATION OF NEMATIC LIQUID-CRYSTALS

The exact solution for the transverse (i.e. in the direction perpendicular to the director axis) component alpha(perpendicular to) (omega) of a nematic liquid crystal and the corresponding correlation time T-perpendicular to is presented for the uniaxial potential of Martin et al. [Symp. Faraday Sec. 5 (1971) 119]. The corresponding longitudinal (i.e. parallel to the director axis) quantities alpha(parallel to)(omega),T-parallel to may be determined by simply replacing magnetic quantities by the corresponding electric ones in our previous study of the magnetic relaxation of single domain ferromagnetic particles Coffey et al. [Phys. Rev. E 49 (1994) 1869]. The calculation of alpha(perpendicular to)(omega) is accomplished by expanding the spatial part of the distribution function of permanent dipole moment orientations on the unit sphere in the Fokker-Planck equation in normalised spherical harmonics. This leads to a three term recurrence relation for the Laplace transform of the transverse decay functions. The recurrence relation is solved exactly in terms of continued fractions. The zero frequency limit of the solution yields an analytic formula for the transverse correlation time T-perpendicular to which is easily tabulated for all nematic potential barrier heights sigma. A simple analytic expression for T-parallel to which consists of the well known Meier-Saupe formula [Mol. Cryst. 1 (1966) 515] with a substantial correction term which yields a close approximation to the exact solution for all sigma, and the correct asymptotic behaviour, is also given. The effective eigenvalue method is shown to yield a simple formula for T-perpendicular to which is valid for all sigma. It appears that the low frequency relaxation process for both orientations of the applied field is accurately described in each case by a single Debye type mechanism with corresponding relaxation times (T-parallel to,T-perpendicular to).