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,  by adding two exponential parameters to the Debye equation: Asymmetry is the absence of, or a violation of, symmetry. Symmetry is an important property of both physical and abstract systems and it may be displayed in precise terms or in more aesthetic terms. The absence of or violation of symmetry that are either expected or desired can have important consequences for a system. A polymer is a large molecule, or macromolecule, composed of many repeated subunits. Due to their broad range of properties, both synthetic and natural polymers play essential and ubiquitous roles in everyday life. Polymers range from familiar synthetic plastics such as polystyrene to natural biopolymers such as DNA and proteins that are fundamental to biological structure and function. Polymers, both natural and synthetic, are created via polymerization of many small molecules, known as monomers. Their consequently large molecular mass relative to small molecule compounds produces unique physical properties, including toughness, viscoelasticity, and a tendency to form glasses and semicrystalline structures rather than crystals. The terms polymer and resin are often synonymous with plastic. In mathematics, an exponential function is a function of the form

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${\hat {\varepsilon }}(\omega )=\varepsilon _{\infty }+{\frac {\Delta \varepsilon }{(1+(i\omega \tau )^{\alpha })^{\beta }}},$ 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. In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium. Accordingly, a charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.

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

For $\beta =1$ the Havriliak–Negami equation reduces to the Cole–Cole equation, for $\alpha =1$ to the Cole–Davidson equation.

The Cole–Cole equation is a relaxation model that is often used to describe dielectric relaxation in polymers.

Mathematical properties

Real and imaginary parts

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

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

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

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

The maximum of the loss part lies at

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

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 

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

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 ${\rm {Eu}}$  In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

Inverse Fourier transform

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.  In particular, in the time-domain the corresponding of ${\hat {\varepsilon }}(\omega )$ can be represented as

$X(t)=\varepsilon _{\infty }\delta (t)+{\frac {\Delta \varepsilon }{\tau }}\left({\frac {t}{\tau }}\right)^{\alpha \beta -1}E_{\alpha ,\alpha \beta }^{\beta }(-(t/\tau )^{\alpha }),$ where $\delta (t)$ is the Dirac delta function and

$E_{\alpha ,\beta }^{\gamma }(z)={\frac {1}{\Gamma (\gamma )}}\sum _{k=0}^{\infty }{\frac {\Gamma (\gamma +k)z^{k}}{k!\Gamma (\alpha k+\beta )}}$ is a special instance of the Fox-Wright function and, precisely, it is the three parameters Mittag-Leffler function  also known as the Prabhakar function. The function $E_{\alpha ,\beta }^{\gamma }(z)$ can be numerically evaluated, for instance, by means of a Matlab code . 

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