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
:$\varepsilon^*(\omega) = \varepsilon_\infty + \frac$
where $\varepsilon^*$ is the complex dielectric constant, $\varepsilon_s$ and $\varepsilon_\infty$ are the "static" and "infinite frequency" dielectric constants, $\omega$ is the angular frequency and $\tau$ is a time constant.

The exponent parameter $\alpha$, which takes a value between 0 and 1, allows the description of different spectral shapes. When $\alpha=0$, the Cole-Cole model reduces to the Debye model. When $\alpha>0$, the relaxation is ''stretched''. That is, it extends over a wider range on a logarithmic $\omega$ scale than Debye relaxation.

The separation of the complex dielectric constant $\varepsilon(\omega)$ was reported in the original paper by Kenneth Stewart Cole and Robert Hugh Cole as follows:

$\varepsilon' = \varepsilon_\infty+ (\varepsilon_s-\varepsilon_\infty)\frac$

$\varepsilon''=\frac$

Upon introduction of hyperbolic functions, the above expressions reduce to:

\varepsilon'=\varepsilon_\infty+\frac(\varepsilon_0-\varepsilon_\infty)\left -\frac\right/math>

$\varepsilon''=\frac(\varepsilon_0-\varepsilon_\infty)\frac$

Here $x = \ln(\omega\tau)$.

These equations reduce to the Debye expression when $\alpha =0$.

Cole–Cole relaxation constitutes a special case of Havriliak–Negami relaxation when the symmetry parameter $\beta=1$, that is, when the relaxation peaks are symmetric. Another special case of Havriliak–Negami relaxation where $\beta<1$ and $\alpha=1$ is known as Cole–Davidson relaxation. For an abridged and updated review of anomalous dielectric relaxation in disordered systems, see Kalmykov.

References

Further reading

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{{DEFAULTSORT:Cole-Cole equation
Electric and magnetic fields in matter