The Cole–Cole equation is a relaxation model that is often used to describe dielectric relaxation in polymers.
is the complex dielectric constant,
are the "static" and "infinite frequency" dielectric constants,
is a dielectric relaxation time constant.
, which takes a value between 0 and 1, allows the description of different spectral shapes.
, the Cole-Cole model reduces to the Debye model.
That is, it extends over a wider range on a logarithmic
scale than Debye relaxation.
The separation of the complex dielectric constant
ε ( ω )
was reported in the original paper by Kenneth Stewart Cole and Robert Hugh Cole[1] as follows:
1 + ( ω τ
sin α π
1 + 2 ( ω τ
sin α π
2 + ( ω τ
cos α π
sin α π
Upon introduction of hyperbolic functions, the above expressions reduce to:
cosh ( ( 1 − α ) x ) + sin ( α π
cos ( α π
cosh ( ( 1 − α ) x ) + sin ( α π
x = ln ( ω τ )
These equations reduce to the Debye expression when
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.
[2] Cole–Cole relaxation constitutes a special case of Havriliak–Negami relaxation when the symmetry parameter
, that is, when the relaxation peaks are symmetric.
Another special case of Havriliak–Negami relaxation where
For an abridged and updated review of anomalous dielectric relaxation in disordered systems, see Kalmykov.