Physics: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
- [math]\displaystyle{ \hat{\varepsilon}(\omega) = \varepsilon_{\infty} + \frac{\Delta\varepsilon}{(1+i\omega\tau)^{\beta}}, }[/math]
where [math]\displaystyle{ \varepsilon_{\infty} }[/math] is the permittivity at the high frequency limit, [math]\displaystyle{ \Delta\varepsilon = \varepsilon_{s}-\varepsilon_{\infty} }[/math] where [math]\displaystyle{ \varepsilon_{s} }[/math] is the static, low frequency permittivity, and [math]\displaystyle{ \tau }[/math] is the characteristic relaxation time of the medium. The exponent [math]\displaystyle{ \beta }[/math] represents the exponent of the decay of the high frequency wing of the imaginary part, [math]\displaystyle{ \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]\displaystyle{ \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,[2] although nowadays an approximation by the Havriliak-Negami function or exact numerical calculation may be preferred.
Because the slopes of the peak in [math]\displaystyle{ \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]\displaystyle{ \alpha=1 }[/math].
The real and imaginary parts are
- [math]\displaystyle{ \varepsilon'(\omega) = \varepsilon_{\infty} + \Delta\varepsilon\left( 1 + (\omega\tau)^{2} \right)^{-\beta/2} \cos (\beta\arctan(\omega\tau)) }[/math]
and
- [math]\displaystyle{ \varepsilon''(\omega) = \Delta\varepsilon\left( 1 + (\omega\tau)^{2} \right)^{-\beta/2} \sin (\beta\arctan(\omega\tau)) }[/math]
See also
References
- ↑ Davidson, D.W.; Cole, R.H. (1950). "Dielectric relaxation in glycerine". Journal of Chemical Physics 18 (10): 1417. doi:10.1063/1.1747496. Bibcode: 1950JChPh..18.1417D.
- ↑ 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. doi:10.1063/1.440530. Bibcode: 1980JChPh..73.3348L.
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