Cole–Davidson equation explained
The Cole-Davidson equation is a model used to describe dielectric relaxation in glass-forming liquids.[1] The equation for the complex permittivity is
\hat{\varepsilon}(\omega)=\varepsiloninfty+
| \Delta\varepsilon |
| (1+i\omega\tau)\beta |
,
where
is the
permittivity at the high frequency limit,
\Delta\varepsilon=\varepsilons-\varepsiloninfty
where
is the static, low frequency permittivity, and
is the characteristic
relaxation time of the medium. The exponent
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,
[2] although nowadays an approximation by the
Havriliak-Negami function or exact numerical calculation may be preferred.
Because the slopes of the peak in
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
.
The real and imaginary parts are
\varepsilon'(\omega)=\varepsiloninfty+\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
Notes and References
- Davidson . D.W. . Cole . R.H. . 1950 . Dielectric relaxation in glycerine . . 18 . 10 . 1417 . 10.1063/1.1747496. 1950JChPh..18.1417D . free .
- Lindsey . C.P. . Patterson . G.D. . Detailed comparison of the Williams–Watts and Cole–Davidson functions . 1980 . . 73 . 7 . 3348–3357 . 10.1063/1.440530. 1980JChPh..73.3348L .
