Integrating factor

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, by adding two exponential parameters to the Debye equation:

${\hat {\varepsilon }}(\omega )=\varepsilon _{\infty }+{\frac {\Delta \varepsilon }{(1+(i\omega \tau )^{\alpha })^{\beta }}},$

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

The storage part $\varepsilon '$ and the loss part $\varepsilon ''$ of the permittivity (here: ${\hat {\varepsilon }}(\omega )=\varepsilon '(\omega )-i\varepsilon ''(\omega )$) can be calculated as

$\varepsilon '(\omega )=\left(1+2(\omega \tau )^{\alpha }\cos(\pi \alpha /2)+(\omega \tau )^{2\alpha }\right)^{-\beta /2}\cos(\beta \phi )$

and

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

with

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

The maximum of the loss part lies at

$\omega _{\rm {max}}=\left({\sin \left({\pi \alpha \over 2(\beta +1)}\right) \over \sin \left({\pi \alpha \beta \over 2(\beta +1)}\right)}\right)^{1/\alpha }\tau ^{-1}$

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

${{\hat {\varepsilon }}(\omega )-\epsilon _{\infty } \over \Delta \varepsilon }=\int _{\tau _{D}=0}^{\infty }{1 \over 1+i\omega \tau _{D}}g(\ln \tau _{D})d\ln \tau _{D}$

with the distribution function

$g(\ln \tau _{D})={1 \over \pi }{(\tau _{D}/\tau )^{\alpha \beta }\sin(\beta \theta ) \over ((\tau _{D}/\tau )^{2\alpha }+2(\tau _{D}/\tau )^{\alpha }\cos(\pi \alpha )+1)^{\beta /2}}$

where

$\theta =\arctan \left({\sin(\pi \alpha ) \over (\tau _{D}/\tau )^{\alpha }+\cos(\pi \alpha )}\right)$

if the argument of the arctangent is positive, else

$\theta =\arctan \left({\sin(\pi \alpha ) \over (\tau _{D}/\tau )^{\alpha }+\cos(\pi \alpha )}\right)+\pi$

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

$\langle \ln \tau _{D}\rangle =\ln \tau +{\Psi (\beta )+{\rm {Eu}} \over \alpha }$

where $\Psi$ is the digamma function and ${\rm {Eu}}$ the Euler constant.

The inverse Fourier transform of the Havriliak-Negami function (the corresponding time-domain relaxation function) can be numerically calculated. It can be shown that the series expansions involved are special cases of the Fox-Wright function. Unfortunately, there are no numerical algorithms available for the computation of such functions with the required parameters.

References

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See also

• Cole–Cole equation
• Dielectric spectroscopy
• Dipole