Cole–Davidson equation

Template:Technical The Cole-Davidson equation is a model used to describe dielectric relaxation in glass-forming liquids. The equation for the complex permittivity is

$\hat{\epsilon }(\omega )={\epsilon }_{\mathrm{\infty }}+\frac{\mathrm{\Delta }\epsilon }{(1+i\omega \tau {)}^{\beta }},$

where ${\epsilon }_{\mathrm{\infty }}$ is the permittivity at the high frequency limit, $\mathrm{\Delta }\epsilon ={\epsilon }_s-{\epsilon }_{\mathrm{\infty }}$ where ${\epsilon }_s$ is the static, low frequency permittivity, and $\tau$ is the characteristic relaxation time of the medium. The exponent $\beta$ represents the exponent of the decay of the high frequency wing of the imaginary part, ${\epsilon }^{″}(\omega )\sim {\omega }^{-\beta }$.

The Cole–Davidson equation is a generalization of the Debye relaxation keeping the initial increase of the low frequency wing of the imaginary part, ${\epsilon }^{″}(\omega )\sim \omega$. Because this is also a characteristic feature of the Fourier transform of the stretched exponential function it has been considered as an approximation of the latter, although nowadays an approximation by the Havriliak-Negami function or exact numerical calculation may be preferred.

Because the slopes of the peak in ${\epsilon }^{″}(\omega )$ in double-logarithmic representation are different it is considered an asymmetric generalization in contrast to the Cole-Cole equation.

The Cole–Davidson equation is the special case of the Havriliak-Negami relaxation with $\alpha =1$.

The real and imaginary parts are

${\epsilon }^{\prime }(\omega )={\epsilon }_{\mathrm{\infty }}+\mathrm{\Delta }\epsilon {(1+(\omega \tau {)}^2)}^{-\beta /2}\cos (\beta \arctan (\omega \tau ))$

and

${\epsilon }^{″}(\omega )=\mathrm{\Delta }\epsilon {(1+(\omega \tau {)}^2)}^{-\beta /2}\sin (\beta \arctan (\omega \tau ))$

See also

• Debye relaxation
• Cole-Cole relaxation
• Havriliak–Negami relaxation
• Curie–von Schweidler law