 Cole-Cole Equation
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Cole%E2%80%93Cole Equation

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

It is given by the equation

$\varepsilon ^{*}(\omega )=\varepsilon _{\infty }+{\frac {\varepsilon _{s}-\varepsilon _{\infty }}{1+(i\omega \tau )^{1-\alpha }}}$ where $\varepsilon ^{*}$ is the complex dielectric constant, $\varepsilon _{s}$ and $\varepsilon _{\infty }$ are the "static" and "infinite frequency" dielectric constants, $\omega$ is the angular frequency and $\tau$ is a time constant.

The exponent parameter $\alpha$ , which takes a value between 0 and 1, allows to describe different spectral shapes. When $\alpha =0$ , the Cole-Cole model reduces to the Debye model. When $\alpha >0$ , the relaxation is stretched, i.e. it extends over a wider range on a logarithmic $\omega$ scale than Debye relaxation.

The separation of the complex dielectric constant $\varepsilon (\omega )$ was reported in the original paper by Cole and Cole as follows:

$\varepsilon '=\varepsilon _{\infty }+(\varepsilon _{s}-\varepsilon _{\infty }){\frac {1+(\omega \tau )^{1-\alpha }\sin \alpha \pi /2}{1+2(\omega \tau )^{1-\alpha }\sin \alpha \pi /2+(\omega \tau )^{2(1-\alpha )}}}$ $\varepsilon ''={\frac {(\varepsilon _{s}-\varepsilon _{\infty })(\omega \tau )^{1-\alpha }\cos \alpha \pi /2}{1+2(\omega \tau )^{1-\alpha }\sin \alpha \pi /2+(\omega \tau )^{2(1-\alpha )}}}$ Upon introduction of hyperbolic functions, the above expressions reduce to:

$\varepsilon '=\varepsilon _{\infty }+{\frac {1}{2}}(\varepsilon _{0}-\varepsilon _{\infty })\left[1-{\frac {\sinh((1-\alpha )x)}{\cosh((1-\alpha )x)+\cos \alpha \pi /2}}\right]$ $\varepsilon ''={\frac {1}{2}}(\varepsilon _{0}-\varepsilon _{\infty }){\frac {\cos \alpha \pi /2}{\cosh((1-\alpha )x)+\sin \alpha \pi /2}}$ Here $x=\ln(\omega \tau )$ .

These equations reduce to the Debye expression when $\alpha =0$ .

Cole-Cole relaxation constitutes a special case of Havriliak-Negami relaxation when the symmetry parameter (?) is equal to 1, that is, when the relaxation peaks are symmetric. Another special case of Havriliak-Negami relaxation (?<1, ?=1) is known as Cole-Davidson relaxation. For an abridged and updated review of anomalous dielectric relaxation in disordered systems, see Kalmykov.