Cole-Davidson equation dielectric relaxation

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. 

Although these equations give very close agreement with experiment over a wide frequency range, they do not provide for a finite value of steady-state compliance y at low frequencies or long times as observed for example in the data of Fig. 13-3. The most precise data show small deviations from the BEL equations and are better fitted by an empirical equation which is reminiscent of the equation used by Davidson and Cole to represent dielectric relaxation ... 

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... 

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. 

A common approach to model the dielectric response, typically used for impedance spectroscopy, is based on equivalent circuits consisting of a number of resistors, capacitors, constant phase elements, and others. Alternatively, the dielectric response can be modeled by a set of model relaxation functions like the Debye function or more generalized (semiempirical) Cole-Cole, Cole-Davidson, or Dissado-Hill equation (Kremer and Schonhals 2002). 

Equations (1.23a), (1.23b) and (1.23c) are, respectively, Cole-Cole (C-C) (0Davidson-Cole (D-C) (0Havriliak-Negami (0empirical laws. The calculations of permittivity on the base of Eq. (1.22) with relaxation function corresponding to KWW law (see Eq. 1.20) yield Eq. (1.23c) with y8 = a - [30]. Expression (1.23c) delivers pretty good description of experimental data obtained by dielectric spectroscopy, radiospectroscopy and quasielastic neutron scattering. It can be shown, that the physical mechanism, underlying the expressions (1.23) is the distribution of relaxation times in a system. Namely, Equation (1.23) can be derived by the averaging of simple Debye response (1.21) with properly tailored distribution function of relaxation times F(x) ... 

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