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Cole–Davidson equation

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The Cole-Davidson equation is a model used to describe dielectric relaxation in glass-forming liquids. The equation for the complex permittivity is

ε ^ ( ω ) = ε ∞ + Δ ε ( 1 + i ω τ ) β , {\displaystyle {\hat {\varepsilon }}(\omega )=\varepsilon _{\infty }+{\frac {\Delta \varepsilon }{(1+i\omega \tau )^{\beta }}},}

where ε ∞ {\displaystyle \varepsilon _{\infty }} is the permittivity at the high frequency limit, Δ ε = ε s − ε ∞ {\displaystyle \Delta \varepsilon =\varepsilon _{s}-\varepsilon _{\infty }} where ε s {\displaystyle \varepsilon _{s}} is the static, low frequency permittivity, and τ {\displaystyle \tau } is the characteristic relaxation time of the medium. The exponent β {\displaystyle \beta } represents the exponent of the decay of the high frequency wing of the imaginary part, ε ″ ( ω ) ∼ ω − β {\displaystyle \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, ε ″ ( ω ) ∼ ω {\displaystyle \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 ε ″ ( ω ) {\displaystyle \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 α = 1 {\displaystyle \alpha =1} .

The real and imaginary parts are

ε ′ ( ω ) = ε ∞ + Δ ε ( 1 + ( ω τ ) 2 ) − β / 2 cos ⁡ ( β arctan ⁡ ( ω τ ) ) {\displaystyle \varepsilon '(\omega )=\varepsilon _{\infty }+\Delta \varepsilon \left(1+(\omega \tau )^{2}\right)^{-\beta /2}\cos(\beta \arctan(\omega \tau ))}

and

ε ″ ( ω ) = Δ ε ( 1 + ( ω τ ) 2 ) − β / 2 sin ⁡ ( β arctan ⁡ ( ω τ ) ) {\displaystyle \varepsilon ''(\omega )=\Delta \varepsilon \left(1+(\omega \tau )^{2}\right)^{-\beta /2}\sin(\beta \arctan(\omega \tau ))}

See also

References

  1. 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.
  2. 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.
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