Model in electromagnetism
The Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts 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:
![{\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon _{\infty }+{\frac {\Delta \varepsilon }{(1+(i\omega \tau )^{\alpha })^{\beta }}},}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
Real and imaginary parts
The storage part
and the loss part
of the permittivity (here:
with
) can be calculated as
![{\displaystyle \varepsilon '(\omega )=\varepsilon _{\infty }+\Delta \varepsilon \left(1+2(\omega \tau )^{\alpha }\cos(\pi \alpha /2)+(\omega \tau )^{2\alpha }\right)^{-\beta /2}\cos(\beta \phi )}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
![{\displaystyle \varepsilon ''(\omega )=\Delta \varepsilon \left(1+2(\omega \tau )^{\alpha }\cos(\pi \alpha /2)+(\omega \tau )^{2\alpha }\right)^{-\beta /2}\sin(\beta \phi )}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with
![{\displaystyle \phi =\arctan \left({(\omega \tau )^{\alpha }\sin(\pi \alpha /2) \over 1+(\omega \tau )^{\alpha }\cos(\pi \alpha /2)}\right)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Loss peak
The maximum of the loss part lies at
![{\displaystyle \omega _{\rm {max}}=\left({\sin \left({\pi \alpha \over 2(\beta +1)}\right) \over \sin \left({\pi \alpha \beta \over 2(\beta +1)}\right)}\right)^{1/\alpha }\tau ^{-1}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Superposition of Lorentzians
The Havriliak–Negami relaxation can be expressed as a superposition of individual Debye relaxations
![{\displaystyle {{\hat {\varepsilon }}(\omega )-\epsilon _{\infty } \over \Delta \varepsilon }=\int _{-\infty }^{\infty }{1 \over 1+i\omega \tau _{D}}g(\ln \tau _{D})d\ln \tau _{D}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with the real valued distribution function
![{\displaystyle g(\ln \tau _{D})={1 \over \pi }{(\tau _{D}/\tau )^{\alpha \beta }\sin(\beta \theta ) \over ((\tau _{D}/\tau )^{2\alpha }+2(\tau _{D}/\tau )^{\alpha }\cos(\pi \alpha )+1)^{\beta /2}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
![{\displaystyle \theta =\arctan \left({\sin(\pi \alpha ) \over (\tau _{D}/\tau )^{\alpha }+\cos(\pi \alpha )}\right)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
if the argument of the arctangent is positive, else[2]
![{\displaystyle \theta =\arctan \left({\sin(\pi \alpha ) \over (\tau _{D}/\tau )^{\alpha }+\cos(\pi \alpha )}\right)+\pi }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Noteworthy,
becomes imaginary valued for
![{\displaystyle {{\hat {\varepsilon }}(\omega )-\epsilon _{\infty } \over \Delta \varepsilon }={(i\omega \tau )^{\alpha \beta } \over (1+(i\omega \tau )^{\alpha })^{\beta }}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and complex valued for
![{\displaystyle {{\hat {\varepsilon }}(\omega )-\epsilon _{\infty } \over \Delta \varepsilon }={1 \over (1-(\omega \tau )^{2\alpha })^{\beta }}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Logarithmic moments
The first logarithmic moment of this distribution, the average logarithmic relaxation time is
![{\displaystyle \langle \ln \tau _{D}\rangle =\ln \tau +{\Psi (\beta )+{\rm {Eu}} \over \alpha }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is the digamma function and
the Euler constant.[3]
Inverse Fourier transform
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
![{\displaystyle X(t)=\varepsilon _{\infty }\delta (t)+{\frac {\Delta \varepsilon }{\tau }}\left({\frac {t}{\tau }}\right)^{\alpha \beta -1}E_{\alpha ,\alpha \beta }^{\beta }(-(t/\tau )^{\alpha }),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is the Dirac delta function and
![{\displaystyle E_{\alpha ,\beta }^{\gamma }(z)={\frac {1}{\Gamma (\gamma )}}\sum _{k=0}^{\infty }{\frac {\Gamma (\gamma +k)z^{k}}{k!\Gamma (\alpha k+\beta )}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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]
See also
References
- ^ 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.
- ^ 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.
- ^ Zorn, R. (2002). "Logarithmic moments of relaxation time distributions" (PDF). Journal of Chemical Physics. 116 (8): 3204–3209. Bibcode:2002JChPh.116.3204Z. doi:10.1063/1.1446035.
- ^ Schönhals, A. (1991). "Fast calculation of the time dependent dielectric permittivity for the Havriliak-Negami function". Acta Polymerica. 42: 149–151.
- ^ 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.
- ^ 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.
- ^ Garrappa, Roberto. "The Mittag-Leffler function". Retrieved 3 November 2014.