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Exact solution for the extended Debye theory of dielectric relaxation of nematic liquid crystals

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  • Coffey, W.T.
  • Crothers, D.S.F.
  • Kalmykov, Yu.P.
  • Waldron, J.T.

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 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⊥ which is easily tabulated for all nematic potential barrier heights σ. A simple analytic expression for T∥ 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 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⊥).

Suggested Citation

  • Coffey, W.T. & Crothers, D.S.F. & Kalmykov, Yu.P. & Waldron, J.T., 1995. "Exact solution for the extended Debye theory of dielectric relaxation of nematic liquid crystals," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 213(4), pages 551-575.
  • Handle: RePEc:eee:phsmap:v:213:y:1995:i:4:p:551-575
    DOI: 10.1016/0378-4371(94)00212-C
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    References listed on IDEAS

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    1. Vollmer, H.D. & Risken, H., 1982. "Eigenvalues and eigenfunctions of the Kramers equation. Application to the Brownian motion of a pendulum," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 110(1), pages 106-127.
    2. Donald R. Kaldor & William E. Saupe, 1966. "Estimates and Projections of an Income-Efficient Commercial-Farm Industry in the North Central States," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 48(3_Part_I), pages 578-596.
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    Cited by:

    1. Sitnitsky, A.E., 2016. "Probability distribution function for reorientations in Maier–Saupe potential," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 452(C), pages 220-228.

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