Havriliak–Negami relaxation

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Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model, accounting for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers,[1] by adding two exponential parameters to the Debye equation:

where is the permittivity at the high frequency limit, where is the static, low frequency permittivity, and is the characteristic relaxation time of the medium. The exponents and describe the asymmetry and broadness of the corresponding spectra.

Depending on application, the Fourier transform of the stretched exponential function can be a viable alternative that has one parameter less.

For the Havriliak–Negami equation reduces to the Cole–Cole equation, for to the Cole–Davidson equation.

Mathematical properties[edit]

Real and imaginary parts[edit]

The storage part and the loss part of the permittivity (here: ) can be calculated as

and

with

Loss peak[edit]

The maximum of the loss part lies at

Superposition of Lorentzians[edit]

The Havriliak–Negami relaxation can be expressed as a superposition of individual Debye relaxations

with the distribution function

where

if the argument of the arctangent is positive, else[2]

Logarithmic moments[edit]

The first logarithmic moment of this distribution, the average logarithmic relaxation time is

where is the digamma function and the Euler constant.[3]

Inverse Fourier transform[edit]

The inverse Fourier transform of the Havriliak-Negami function (the corresponding time-domain relaxation function) can be numerically calculated.[4] It can be shown that the series expansions involved are special cases of the Fox-Wright function.[5] In particular, in the time-domain the corresponding of can be represented as

where is the Dirac delta function and

is a special instance of the Fox-Wright function and, precisely, it is the three parameters Mittag-Leffler function[6] also known as the Prabhakar function. The function can be numerically evaluated, for instance, by means of a Matlab code .[7]

References[edit]

  1. ^ Havriliak, S.; Negami, S. (1967). "A complex plane representation of dielectric and mechanical relaxation processes in some polymers". Polymer. 8: 161–210. doi:10.1016/0032-3861(67)90021-3. 
  2. ^ Zorn, R. (1999). "Applicability of Distribution Functions for the Havriliak–Negami Spectral Function". Journal of Polymer Science Part B. 37 (10): 1043–1044. Bibcode:1999JPoSB..37.1043Z. doi:10.1002/(SICI)1099-0488(19990515)37:10<1043::AID-POLB9>3.3.CO;2-8. 
  3. ^ Zorn, R. (2002). "Logarithmic moments of relaxation time distributions". Journal of Chemical Physics. 116 (8): 3204–3209. Bibcode:2002JChPh.116.3204Z. doi:10.1063/1.1446035. 
  4. ^ Schönhals, A. (1991). "Fast calculation of the time dependent dielectric permittivity for the Havriliak-Negami function". Acta Polymerica. 42: 149–151. 
  5. ^ Hilfer, J. (2002). "H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems". Physical Review E. 65: 061510. Bibcode:2002PhRvE..65f1510H. doi:10.1103/physreve.65.061510. 
  6. ^ Gorenflo, Rudolf; Kilbas, Anatoly A.; Mainardi, Francesco; Rogosin, Sergei V. (2014). Springer, ed. Mittag-Leffler Functions, Related Topics and Applications. ISBN 978-3-662-43929-6. 
  7. ^ Garrappa, Roberto. "The Mittag-Leffler function". Retrieved 3 November 2014. 

See also[edit]