Cole–Davidson equation

The Cole-Davidson equation is a model used to describe dielectric relaxation in glass-forming liquids. ^[1] The equation for the complex permittivity is

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

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

The Cole–Davidson equation is a generalization of the Debye relaxation keeping the initial increase of the low frequency wing of the imaginary part, ${\displaystyle \varepsilon ''(\omega )\sim \omega }$. 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, ^[2] although nowadays an approximation by the Havriliak-Negami function or exact numerical calculation may be preferred.

Because the slopes of the peak in ${\displaystyle \varepsilon ''(\omega )}$ 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 ${\displaystyle \alpha =1}$.

The real and imaginary parts are

${\displaystyle \varepsilon '(\omega )=\varepsilon _{\infty }+\Delta \varepsilon \left(1+(\omega \tau )^{2}\right)^{-\beta /2}\cos(\beta \arctan(\omega \tau ))}$

and

${\displaystyle \varepsilon ''(\omega )=\Delta \varepsilon \left(1+(\omega \tau )^{2}\right)^{-\beta /2}\sin(\beta \arctan(\omega \tau ))}$

See also

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

References

1. Davidson, D.W.; Cole, R.H. (1950). "Dielectric relaxation in glycerine". Journal of Chemical Physics . 18 (10): 1417. Bibcode:1950JChPh..18.1417D. doi: 10.1063/1.1747496 .
2. Lindsey, C.P.; Patterson, G.D. (1980). "Detailed comparison of the Williams–Watts and Cole–Davidson functions". Journal of Chemical Physics . 73 (7): 3348–3357. Bibcode:1980JChPh..73.3348L. doi:10.1063/1.440530.