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)) $$