Physics:Cole–Davidson equation

From HandWiki

The Cole-Davidson equation is a model used to describe dielectric relaxation in glass-forming liquids.[1] The equation for the complex permittivity is

ε^(ω)=ε+Δε(1+iωτ)β,

where ε is the permittivity at the high frequency limit, Δε=εsε where εs is the static, low frequency permittivity, and τ is the characteristic relaxation time of the medium. The exponent β represents the exponent of the decay of the high frequency wing of the imaginary part, ε(ω)ωβ.

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. 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 ε(ω) 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.

The real and imaginary parts are

ε(ω)=ε+Δε(1+(ωτ)2)β/2cos(βarctan(ωτ))

and

ε(ω)=Δε(1+(ωτ)2)β/2sin(βarctan(ωτ))

See also

References

  1. Davidson, D.W.; Cole, R.H. (1950). "Dielectric relaxation in glycerine". Journal of Chemical Physics 18 (10): 1417. doi:10.1063/1.1747496. Bibcode1950JChPh..18.1417D. 
  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. doi:10.1063/1.440530. Bibcode1980JChPh..73.3348L.