Cole–Davidson equation

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

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

where $\varepsilon _{\infty }$ is the permittivity at the high frequency limit, $\Delta \varepsilon =\varepsilon _{s}-\varepsilon _{\infty }$ where $\varepsilon _{s}$ 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, $\varepsilon ''(\omega )\sim \omega ^{-\beta }$.

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

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

The real and imaginary parts are

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

and

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

See also

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