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

$\varepsilon^*(\omega) - \varepsilon_\infty = \frac{\varepsilon_s - \varepsilon_\infty}{1+(i\omega\tau)^{1 - \alpha}}$

where ε ^* is the complex dielectric constant, ε_s and $\varepsilon_\infty$ are the "static" and "infinite frequency" dielectric constants, ω is the angular frequency and τ is a time constant.

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

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=0) is known as Cole-Davidson relaxation.

References

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

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