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

The exact solution for the transverse (i.e. in the direction perpendic ular to the director axis) component alpha(perpendicular to) (omega) o f a nematic liquid crystal and the corresponding correlation time T-pe rpendicular 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 quantit ies by the corresponding electric ones in our previous study of the ma gnetic relaxation of single domain ferromagnetic particles Coffey et a l. [Phys. Rev. E 49 (1994) 1869]. The calculation of alpha(perpendicul ar to)(omega) is accomplished by expanding the spatial part of the dis tribution function of permanent dipole moment orientations on the unit sphere in the Fokker-Planck equation in normalised spherical harmonic s. This leads to a three term recurrence relation for the Laplace tran sform of the transverse decay functions. The recurrence relation is so lved exactly in terms of continued fractions. The zero frequency limit of the solution yields an analytic formula for the transverse correla tion time T-perpendicular to which is easily tabulated for all nematic potential barrier heights sigma. A simple analytic expression for T-p arallel 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 corr ect asymptotic behaviour, is also given. The effective eigenvalue meth od is shown to yield a simple formula for T-perpendicular to which is valid for all sigma. It appears that the low frequency relaxation proc ess for both orientations of the applied field is accurately described in each case by a single Debye type mechanism with corresponding rela xation times (T-parallel to,T-perpendicular to).