Exact solution for the extended Debye theory of dielectric relaxation of nematic liquid crystals
Abstract
The exact solution for the transverse (i.e. in the direction perpendicular to the director axis) component α_{⊥}( ω) of a nematic liquid crystal and the corresponding correlation time T_{⊥} is presented for the uniaxial potential of Martin et al. [Symp. Faraday Soc. 5 (1971) 119]. The corresponding longitudinal (i.e. parallel to the director axis) quantities α_{⊥}( ω), T_{⊥} 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 α_{⊥}( ω) is accomplished by expanding the spatial part of the distribution function of permanent dipole moment orientations on the unit sphere in the FokkerPlanck 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_{⊥} which is easily tabulated for all nematic potential barrier heights σ. A simple analytic expression for T_{∥} which consists of the well known MeierSaupe formula [Mol. Cryst. 1 (1966) 515] with a substantial correction term which yields a close approximation to the exact solution for all σ, and the correct asymptotic behaviour, is also given. The effective eigenvalue method is shown to yield a simple formula for T_{⊥} which is valid for all σ. 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_{∥}, T_{⊥}).
 Publication:

Physica A Statistical Mechanics and its Applications
 Pub Date:
 February 1995
 DOI:
 10.1016/03784371(94)00212C
 Bibcode:
 1995PhyA..213..551C