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

:$\hat(\omega) = \varepsilon_ + \frac,$

where $\varepsilon_$ is the permittivity at the high frequency limit, $\Delta\varepsilon = \varepsilon_-\varepsilon_$ where $\varepsilon_$ 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, $\varepsilon''(\omega) \sim \omega^$.

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

:$\varepsilon'(\omega) = \varepsilon_ + \Delta\varepsilon\left( 1 + (\omega\tau)^ \right)^ \cos (\beta\arctan(\omega\tau))$

and

:$\varepsilon''(\omega) = \Delta\varepsilon\left( 1 + (\omega\tau)^ \right)^ \sin (\beta\arctan(\omega\tau))$

References

{{DEFAULTSORT:Cole-Davidson equation
Equations
Glass
Liquids
Electric and magnetic fields in matter