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
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.
The separation of the complex dielectric constant was reported in the original paper by Cole and Cole as follows:
Upon introduction of hyperbolic functions, the above expressions reduce to:
Here.
These equations reduce to the Debye expression when.
Cole–Cole relaxation constitutes a special case of Havriliak–Negami relaxation when the symmetry parameter is equal to 1, that is, when the relaxation peaks are symmetric. Another special case of Havriliak–Negami relaxation is known as Cole–Davidson relaxation. For an abridged and updated review of anomalous dielectric relaxation in disordered systems, see Kalmykov.
