Cole–Davidson equation
 | This article may be too technical for most readers to understand. (February 2022) |
The Cole-Davidson equation is a model used to describe dielectric relaxation in glass-forming liquids.<ref> 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.</ref> The equation for the complex permittivity is
- <math>
\hat{\varepsilon}(\omega) = \varepsilon_{\infty} + \frac{\Delta\varepsilon}{(1+i\omega\tau)^{\beta}}, </math>
where <math>\varepsilon_{\infty}</math> is the permittivity at the high frequency limit, <math>\Delta\varepsilon = \varepsilon_{s}-\varepsilon_{\infty}</math> where <math>\varepsilon_{s}</math> is the static, low frequency permittivity, and <math>\tau</math> is the characteristic relaxation time of the medium. The exponent <math>\beta</math> represents the exponent of the decay of the high frequency wing of the imaginary part, <math>\varepsilon(\omega) \sim \omega^{-\beta}</math>.
The Cole–Davidson equation is a generalization of the Debye relaxation keeping the initial increase of the low frequency wing of the imaginary part, <math>\varepsilon(\omega) \sim \omega</math>. 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,<ref> 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.</ref> although nowadays an approximation by the Havriliak-Negami function or exact numerical calculation may be preferred.
Because the slopes of the peak in <math>\varepsilon(\omega)</math> 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 <math>\alpha=1</math>.
The real and imaginary parts are
- <math>
\varepsilon'(\omega) = \varepsilon_{\infty} + \Delta\varepsilon\left( 1 + (\omega\tau)^{2} \right)^{-\beta/2} \cos (\beta\arctan(\omega\tau)) </math>
and
- <math>
\varepsilon(\omega) = \Delta\varepsilon\left( 1 + (\omega\tau)^{2} \right)^{-\beta/2} \sin (\beta\arctan(\omega\tau)) </math>
