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The '''Cole–Cole equation''' is a [[relaxation (physics)|relaxation]] model that is often used to describe [[dielectric relaxation]] in [[polymer]]s.
 
It is given by the equation
:<math>
\varepsilon^*(\omega) = \varepsilon_\infty + \frac{\varepsilon_s - \varepsilon_\infty}{1+(i\omega\tau)^{1 - \alpha}}
</math>
where <math>\varepsilon^*</math> is the [[Dielectric constant#Complex permittivity|complex dielectric constant]], <math>\varepsilon_s</math> and <math>\varepsilon_\infty</math> are the "static" and "infinite frequency" dielectric constants, <math>\omega</math> is the [[angular frequency]] and <math>\tau</math> is a [[Relaxation (physics)#Dielectric relaxation time|dielectric relaxation time constant]].

The exponent parameter <math>\alpha</math>, which takes a value between 0 and 1, allows the description of different spectral shapes. When <math>\alpha=0</math>, the Cole-Cole model reduces to the [[Dielectric#Debye relaxation|Debye]] model. When <math>\alpha>0</math>, the relaxation is ''stretched''. That is, it extends over a wider range on a logarithmic <math>\omega</math> scale than Debye relaxation.

The separation of the complex dielectric constant <math>\varepsilon(\omega)</math> was reported in the original paper by [[Kenneth Stewart Cole]] and Robert Hugh Cole<ref name="Cole_1941"/> as follows:

<math>\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)}}</math>

<math>\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)}}</math>

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

<math>\varepsilon'=\varepsilon_\infty+\frac{1}{2}(\varepsilon_0-\varepsilon_\infty)\left[1-\frac{\sinh((1-\alpha)x)}{\cosh((1-\alpha)x)+\sin(\alpha\pi/2)}\right]</math>

<math>\varepsilon''=\frac{1}{2}(\varepsilon_0-\varepsilon_\infty)\frac{\cos(\alpha\pi/2)}{\cosh((1-\alpha)x)+\sin(\alpha\pi/2)}</math>

Here <math>x = \ln(\omega\tau)</math>.

These equations reduce to the Debye expression when <math>\alpha =0</math>.

The Cole-Cole equation's time domain current response corresponds to the [[Curie–von Schweidler law]] and the charge response corresponds to the [[stretched exponential function]] or the Kohlrausch–Williams–Watts (KWW) function, for small time arguments.<ref>{{Cite journal |last=Holm|first=Sverre|title=Time domain characterization of the Cole-Cole dielectric model |url=https://sciendo.com/article/10.2478/joeb-2020-0015 |journal=Journal of Electrical Bioimpedance |year=2020 |volume=11 |issue=1 |pages=101–105|doi=10.2478/joeb-2020-0015|pmid=33584910 |pmc=7851980 }}</ref>

Cole–Cole relaxation constitutes a special case of [[Havriliak–Negami relaxation]] when the symmetry parameter <math>\beta=1</math>, that is, when the relaxation peaks are symmetric. Another special case of Havriliak–Negami relaxation where <math>\beta<1</math> and <math>\alpha=1</math> is known as [[Cole–Davidson_equation|Cole–Davidson relaxation]]. For an abridged and updated review of anomalous dielectric relaxation in disordered systems, see Kalmykov.

==See also==
* [[Dielectric#Debye_relaxation|Debye relaxation]]
* [[Cole–Davidson equation|Cole–Davidson relaxation]]
* [[Havriliak–Negami relaxation]]
* [[Curie–von Schweidler law]]

== References ==
{{Reflist|refs=
<ref name="Cole_1941">{{cite journal |author-last1=Cole |author-first1=Kenneth Stewart |author-last2=Cole |author-link1=Kenneth Stewart Cole |author-first2=Robert Hugh |author-link2=Robert Hugh Cole |date=1941 |title=Dispersion and Absorption in Dielectrics I. Alternating Current Characteristics |journal=Journal of Chemical Physics |volume=9 |issue=4 |pages=341–351 |bibcode=1941JChPh...9..341C |doi=10.1063/1.1750906}}</ref>
}}

== Further reading ==
* {{cite journal |author-last1=Cole |author-first1=Kenneth Stewart |author-link1=Kenneth Stewart Cole |author-last2=Cole |author-first2=Robert Hugh |author-link2=Robert Hugh Cole |title=Dispersion and Absorption in Dielectrics II. Direct Current Characteristics |journal=Journal of Chemical Physics |date=1942 |volume=10 |issue=2 |pages=98–105 |bibcode=1942JChPh..10...98C |doi=10.1063/1.1723677}}
* {{cite journal |author-last1=Kalmykov |author-first1=Y. P. |author-last2=Coffey |author-first2=W. T. |author-last3=Crothers |author-first3=D. S. F. |author-last4=Titov |author-first4=S. V. |title=Microscopic Models for Dielectric Relaxation in Disordered Systems |journal=Physical Review E |date=2004 |volume=70 |issue=4 |pages=041103 |bibcode=2004PhRvE..70d1103K |doi=10.1103/PhysRevE.70.041103 |pmid=15600393 }}

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[[Category:Electric and magnetic fields in matter]]