View source for Havriliak–Negami relaxation

{{Short description|Model in electromagnetism}}
{{Use American English|date = February 2019}}
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 [[polymer]]s,<ref>
{{cite journal
 |last1=Havriliak |first1=S.
 |last2=Negami |first2=S.
 |year=1967
 |title=A complex plane representation of dielectric and mechanical relaxation processes in some polymers
 |journal=[[Polymer (journal)|Polymer]]
 |volume=8 |pages=161–210
 |doi=10.1016/0032-3861(67)90021-3
}}</ref> by adding two [[Exponential function|exponential]] parameters to the Debye equation:

:<math>
\hat{\varepsilon}(\omega) = \varepsilon_{\infty} + \frac{\Delta\varepsilon}{(1+(i\omega\tau)^{\alpha})^{\beta}},
</math>

where <math>\varepsilon_{\infty}</math> is the [[permittivity]] at the high frequency limit, <math>\Delta\varepsilon = \varepsilon_{s}-\varepsilon_{\infty}</math> where <math>\varepsilon_{s}</math> is the static, low frequency permittivity, and <math>\tau</math> is the characteristic [[relaxation time]] of the medium. The exponents <math>\alpha</math> and <math>\beta</math> 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 <math>\beta = 1</math> the Havriliak–Negami equation reduces to the [[Cole–Cole equation]], for <math>\alpha=1</math> to the [[Cole–Davidson equation]].

== Mathematical properties ==

=== Real and imaginary parts ===

The storage part <math>\varepsilon'</math> and the loss part <math>\varepsilon''</math> of the permittivity (here: <math> \hat{\varepsilon}(\omega) = \varepsilon'(\omega) - i \varepsilon''(\omega) </math> with <math> (\pm i)^2=-1 </math>) can be calculated as

:<math>
\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)
</math>

and

:<math>
\varepsilon''(\omega) = \Delta\varepsilon\left( 1 + 2 (\omega\tau)^\alpha \cos (\pi\alpha/2) + (\omega\tau)^{2\alpha} \right)^{-\beta/2} \sin (\beta\phi)
</math>

with

:<math>
\phi = \arctan \left( { (\omega\tau)^\alpha \sin(\pi\alpha/2) \over
1 + (\omega\tau)^\alpha \cos(\pi\alpha/2) } \right)
</math>

=== Loss peak ===

The maximum of the loss part lies at

:<math>
\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}
</math>

=== Superposition of Lorentzians ===

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

:<math>
{ \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
</math>

with the real valued distribution function

:<math>
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} }
</math>

where

:<math>
\theta = \arctan \left( { \sin (\pi\alpha) \over ( \tau_D / \tau )^{\alpha} + \cos (\pi\alpha) } \right)
</math>

if the argument of the arctangent is positive, else<ref>
{{cite journal
 | last = Zorn |first= R.
 | year = 1999
 | title = Applicability of Distribution Functions for the Havriliak–Negami Spectral Function
 | journal = [[Journal of Polymer Science Part B]]
 | volume = 37 |issue=10 | pages = 1043–1044
 | bibcode =1999JPoSB..37.1043Z
 | doi = 10.1002/(SICI)1099-0488(19990515)37:10<1043::AID-POLB9>3.3.CO;2-8
}}</ref>

:<math>
\theta = \arctan \left( { \sin (\pi\alpha) \over ( \tau_D / \tau )^{\alpha} + \cos (\pi\alpha) } \right) + \pi
</math>

{{off topic|date=January 2022}}
Noteworthy, <math> g ( \ln \tau ) </math> becomes imaginary valued for

:<math>
{ \hat{\varepsilon}(\omega) - \epsilon_\infty \over \Delta\varepsilon } = { (i \omega \tau)^{\alpha \beta} \over (1 + (i \omega \tau)^{\alpha})^{\beta} }
</math>

and complex valued for

:<math>
{ \hat{\varepsilon}(\omega) - \epsilon_\infty \over \Delta\varepsilon } = { 1 \over (1 - (\omega \tau)^{2\alpha})^{\beta} }
</math>

=== Logarithmic moments ===

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

:<math>
\langle \ln\tau_D \rangle = \ln\tau + { \Psi(\beta) + {\rm Eu} \over \alpha }
</math>

where <math>\Psi</math> is the [[digamma function]] and <math>{\rm Eu}</math> the [[Euler constant]].<ref>
{{cite journal
 | last = Zorn |first= R.
 | year = 2002
 | title = Logarithmic moments of relaxation time distributions
 | journal = [[Journal of Chemical Physics]]
 | volume = 116 |issue= 8| pages = 3204–3209
 | bibcode = 2002JChPh.116.3204Z
 | doi=10.1063/1.1446035
| url= http://juser.fz-juelich.de/record/1954/files/10418.pdf}}</ref>

=== Inverse Fourier transform ===

The inverse Fourier transform of the Havriliak-Negami function (the corresponding time-domain relaxation function) can be numerically calculated.<ref>{{cite journal
 | author = Schönhals, A.
 | year = 1991
 | title = Fast calculation of the time dependent dielectric permittivity for the Havriliak-Negami function
 | journal = [[Acta Polymerica]]
 | volume = 42
 | pages = 149–151
  }}</ref> It can be shown that the series expansions involved are special cases of the [[Fox–Wright function]].<ref>{{cite journal
 | author = Hilfer, J.
 | year = 2002
 | title = ''H''-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems
 | journal = [[Physical Review]] E
 | volume = 65
 | pages = 061510
 | doi=10.1103/physreve.65.061510
  |bibcode = 2002PhRvE..65f1510H }}</ref> In particular, in the time-domain the corresponding of <math>\hat{\varepsilon}(\omega)</math> can be represented as

:<math>
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}) ,
</math>

where <math>\delta(t)</math> is the Dirac delta function and

:<math>
E_{\alpha,\beta}^{\gamma}(z) = \frac{1}{\Gamma(\gamma)} \sum_{k=0}^{\infty} \frac{\Gamma(\gamma+k) z^{k}}{k! \Gamma(\alpha k + \beta)}
</math>
is a special instance of the [[Fox–Wright function]] and, precisely, it is the three parameters [[Mittag-Leffler function]]<ref>{{cite book
 | first = Rudolf | last = Gorenflo
 | first2 = Anatoly A. | last2 =  Kilbas
 | first3 = Francesco | last3 = Mainardi 
 | first4 = Sergei V. | last4 = Rogosin 
 | year = 2014
 | title = Mittag-Leffler Functions, Related Topics and Applications
 | editor = Springer
 | ISBN=978-3-662-43929-6
  }}</ref> also known as the Prabhakar function. The function <math>E_{\alpha,\beta}^{\gamma}(z) </math> can be numerically evaluated, for instance, by means of a Matlab code
.<ref>{{cite web
 | last = Garrappa
 | first = Roberto
 | title = The Mittag-Leffler function
 | url = http://www.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function
 | access-date = 3 November 2014 
}}</ref>

==See also==
* [[Dielectric#Debye relaxation|Debye relaxation]]
* [[Cole–Cole equation]]
* [[Cole–Davidson equation]]
* [[Curie–von Schweidler law]]
* [[Dielectric spectroscopy]]
* [[Dipole]]

==References==
{{reflist}}

{{DEFAULTSORT:Havriliak-Negami relaxation}}
[[Category:Electric and magnetic fields in matter]]