# Cole–Cole equation

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

In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time t is an exponential law exp(-t/τ).

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.

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.

In physics, angular frequencyω is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.

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.

The Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts 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:

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## References

1. Cole, Kenneth S, Robert H (1941). "Dispersion and Absorption in Dielectrics: I - Alternating Current Characteristics". Journal of Chemical Physics. 9 (4): 341–351. Bibcode:1941JChPh...9..341C. doi:10.1063/1.1750906.

Cole, K.S.; Cole, R.H. (1941). "Dispersion and Absorption in Dielectrics - I Alternating Current Characteristics". J. Chem. Phys. 9 (4): 341–352. Bibcode:1941JChPh...9..341C. doi:10.1063/1.1750906.

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Cole, K.S.; Cole, R.H. (1942). "Dispersion and Absorption in Dielectrics - II Direct Current Characteristics". Journal of Chemical Physics. 10 (2): 98–105. Bibcode:1942JChPh..10...98C. doi:10.1063/1.1723677.

Kalmykov, Y.P.; Coffey, W.T.; Crothers, D.S.F.; Titov, S.V. (2004). "Microscopic Models for Dielectric Relaxation in Disordered Systems". Physical Review E. 70 (4): 041103. Bibcode:2004PhRvE..70d1103K. doi:10.1103/PhysRevE.70.041103. PMID   15600393.