Cole–Cole 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

${\displaystyle \varepsilon ^{*}(\omega )=\varepsilon _{\infty }+{\frac {\varepsilon _{s}-\varepsilon _{\infty }}{1+(i\omega \tau )^{1-\alpha }}}}$

where ${\displaystyle \varepsilon ^{*}}$ is the complex dielectric constant, ${\displaystyle \varepsilon _{s}}$ and ${\displaystyle \varepsilon _{\infty }}$ are the "static" and "infinite frequency" dielectric constants, ${\displaystyle \omega }$ is the angular frequency and ${\displaystyle \tau }$ is a time constant.

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

The separation of the complex dielectric constant ${\displaystyle \varepsilon (\omega )}$ was reported in the original paper by Cole and Cole[1] as follows:

${\displaystyle \varepsilon '=\varepsilon _{\infty }+(\varepsilon _{s}-\varepsilon _{\infty }){\frac {1+(\omega \tau )^{1-\alpha }\sin \alpha \pi /2}{1+2(\omega \tau )^{1-\alpha }\sin \alpha \pi /2+(\omega \tau )^{2(1-\alpha )}}}}$

${\displaystyle \varepsilon ''={\frac {(\varepsilon _{s}-\varepsilon _{\infty })(\omega \tau )^{1-\alpha }\cos \alpha \pi /2}{1+2(\omega \tau )^{1-\alpha }\sin \alpha \pi /2+(\omega \tau )^{2(1-\alpha )}}}}$

Upon introduction of hyperbolic functions, the above expressions reduce to:

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

${\displaystyle \varepsilon ''={\frac {1}{2}}(\varepsilon _{0}-\varepsilon _{\infty }){\frac {\cos \alpha \pi /2}{\cosh((1-\alpha )x)+\sin \alpha \pi /2}}}$

Here ${\displaystyle x=\ln(\omega \tau )}$.

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

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