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.

It is given by the equation

where is the complex dielectric constant, and are the "static" and "infinite frequency" dielectric constants, is the angular frequency and is a time constant.

The exponent parameter , which takes a value between 0 and 1, allows to describe different spectral shapes. When , the Cole-Cole model reduces to the Debye model. When , the relaxation is stretched, i.e. it extends over a wider range on a logarithmic 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, for an abridged and updated review of anomalous dielectric relaxation in dissored systems see Kalmykov.


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. 

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: 041103. Bibcode:2004PhRvE..70d1103K. doi:10.1103/PhysRevE.70.041103.