Davidson-Cole equation

In the particular application to dielectric relaxation, fit) is the aftereffect function following the removal of a constant field [8]. The solution of Eq. (93) rendered in the frequency domain yields the Cole-Davidson equation [Eq. (10)] [28],... 

Other relationships which have been used to describe dielectric relaxation data include the Cole-Cole and Cole-Davidson equations [29]. These are preferred when a distribution of relaxation times rather than a single relaxation time is more appropriate to describe the data in a given frequency range. Nevertheless, the Debye model in its simple version or multiple relaxation versions works quite well for most of the solvents considered here. 

There were several attempts to generalize the Debye function like the Cole/Cole formula (Cole and Cole 1941) (symmetric broadened relaxation function), the Cole/Davidson equation (Davidson and Cole 1950, 1951), or the Fuoss/Kirkwood model (asymmetric broadened relaxation function) (Fuoss and Kirkwood 1941). The most general formula is the model function of Havriliak and Negami (HN function) (Havriliak and Negami 1966,1967 Havriliak 1997) which reads... 

A number of other related empirical equations have been proposed. The Cole-Davidson equation includes an asymmetric broadening factor (y) ... 

The combination of the Cole-Cole equation (eq. (10.20)) and the Cole—Davidson equation (eq. (10.24)) is, after the inventors, referred to as the Havriliak— Negami equation ... 

Fig. 4.2. The Argand diagrams for the Debye equation (D), the Cole-Cole equation (CC), and the Cole-Davidson equation (CD).

Relation [1] Is the frequency-dependent analogue of a formula proposed by Chasset and Thirion (2, 3) which has since been applied very frequently to relaxation measurements on cured rubbers. The next three equations are Inspired by similar relations In dielectrics (they are not derived from these) Equation [2] by the Cole-Cole and Equation [3] by the Davidson-Cole relation (15, 16). Both are special cases of the most general Equation [4] which contains five parameters (17). 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

The complex permittivity plots in Figure 6 show high frequency (low e ) divergence from the classical semicircular shape. The shapes are Cole-Davidson arcs (31), indicative of two, or more, superimposed simple polarizations (equation (1)). 

D. Fractional Diffusion Equation for the Cole—Davidson and Havriliak-Negami Behavior... 

Another approach to fractionalize the Fokker-Planck equation incorporating Cole-Davidson behavior can now be given by extending a hypothesis of Nigmatullin and Ryabov [28]. They noted that the ordinary first-order differential equation describing an exponential decay... 

For the purpose of using a kinetic equation incorporating the Cole-Davidson mechanism according to the heuristic procedure of Nigmatullin and Ryabov [28], we may replace the partial time derivative in Eq. (95) by a fractional time derivative oDvt. Thus Eq. (95) becomes [cf. Eq. (93)]... 

In like manner, combining the ideas embodied in the fractional diffusion Eq. (90) describing Cole-Cole relaxation and Eq. (96) describing Cole-Davidson relaxation, we may introduce the fractional kinetic equation... 

As q " is strictly independent of the temperature, equation (2) gives in the fast motion (27cfx 1) as well as in the slow motion case (27ifx 1) the refractive index n(T) at the laser wavelength Xq (c.f (9)). In the acoustic relaxation regime D(q ", T) exeeds n(T). In (35) we present different theoretical curves of D(q ", q , T) calculated under the assumption, that the real part of the complex elastic constant c (q, T) can be written in the form c (q, T) = c (T)-Ac/ 1 + 47i (q,T)x (T). For the exponent P<1 this formular describes a Cole davidson function. The relaxation time x was assumed to follow a VFT law. Under these conditions the OADF deviates from n(T) only well above the TGT and... 

The specific case a=l, 1 gives the Debye relaxa tion law, P= 1, a 1 corresponds to the so-called Cole-Cole equation, whereas the case a= 1, P 1 corresponds to the Cole-Davidson formula. Recently, some progress in the understanding of the physical meaning of the empirical parameters (a, P) has been made (7, 8). Using the conception of a self-similar relaxation process it is possible to understand thenature of a nonexponential relaxation of the Cole-Cole, Cole-Davidson, or Havriliak-Negami type. 

Where M2 is the second moment of the NMR lineshape, J the spectral density function, with (Dq the Larmor frequency, and (0i the frequency of the spin-locking field. The spectral density can be written in terms of the molecular correlation time, x, and the overall shape of the Tjp - temperature dispersion and the relatively shallow minima arc due to the correlation time distribution, although the location of the minimum is unaffected by this distribution. We have examined several models for the distribution, all of which give essentially the same results. One of the more simple is the Cole-Davidson function (75), which has also been applied to the analysis of dielectric relaxations. The relevant expression for the spectral density in this case is given by Equation 4. 

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