Cole-Davidson model

Sturz and DoUe measured the temperature dependent dipolar spin-lattice relaxation rates and cross-correlation rates between the dipolar and the chemical-shift anisotropy relaxation mechanisms for different nuclei in toluene. They found that the reorientation about the axis in the molecular plane is approximately 2 to 3 times slower than the one perpendicular to the C-2 axis. Suchanski et al measured spin-lattice relaxation times Ti and NOE factors of chemically non-equivalent carbons in meta-fluoroanihne. The analysis showed that the correlation function describing molecular dynamics could be well described in terms of an asymmetric distribution of correlation times predicted by the Cole-Davidson model. In a comprehensive simulation study of neat formic acid Minary et al found good agreement with NMR relaxation time experiments in the liquid phase. Iwahashi et al measured self-diffusion coefficients and spin-lattice relaxation times to study the dynamical conformation of n-saturated and unsaturated fatty acids. 

The Cole-Davidson model These kinds of diagram are also symmetrical or non-symmetrical and may be fairly described by an analytical relationship proposed by Davidson and Cole [77] (Eq. 46) ... 

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... 

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. 

When a is dose to unity this again reduces to Debye s model and for a smaller than unity an asymmetric diagram is obtained. The Cole-Cole diagram arise from symmetrical distribution of relaxation times whereas the Cole-Davidson diagram is obtained from a series of relaxation mechanisms of decreasing importance extending to the high-frequency side of the main dispersion. 

In general, plant tissue immittance data are not in accordance with Cole—Cole system models. Nonsymmetrical models of the Davidson Cole and Havriliak—Negami types have... 

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... 

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). 

The memory effect is described by the integral term of Eq. (10-22), which is absent in the standard approach. Equation (10-22) holds for a liquid crystal with a Debye type of relaxation but it can be easily modified or generalized to other relaxation models (such as Cole-Davidson, Havriliak-Negami and other models with different functional forms of a(t — f) [5, 17]). 

These relationships are known as the Debye formulae. The Debye process has a relaxation time distribution, which is symmetrical around /niax= niax/2n and has a full width at half-maximum of 1.14 decades in frequency for the dielectric loss. In most cases, the half width of measured loss peaks is much broader than the predicted by eqn [26] and in addition, their shapes are asymmetric and with a high-frequency tail. This is the non-Debye (or nonideal) relaxation behavior found in many glass formers. In the literature, several empirical model funaions, mostly generalization of the Debye function, have been developed and tested which are able to describe broadened and/or asymmetric loss peaks. Among these empirical model functions, the most important are the Kohlrausch-Williams-Watts (KWW), Cole-Cole (CC), Cole-Davidson (CD), and the Havriliak-Negami (HN) function. The HN function, with two shape parameters, is the most commonly used funaion in the frequency domain. 

The dielectric dispersion for some solvents is poorly modeled by a multiple Debye form. Alternative, e(cu) distributions such as the Davidson-Cole equation or the Cole-Cole equation are often more appropriate. 

Employing the Davidson-Cole model for propylene carbonate and the Cole-Cole model for propionitrile with the appropriate dielectric parameters from the literature, we have predicted C(t) for these polar aprotic solvents according to the dielectric continuum model. The agreement between the predicted and observed C(t) is not nearly as good as the alcohol examples (see below). 

Other models have been proposed which follow the outlines of the equations already discussed. Equations with parameters that vary as a function of temperature, sunlight, and nutrient concentration have been presented by Davidson and Clymer (9) and simulated by Cole (10). A set of equations which model the population of phytoplankton, zooplankton, and a species of fish in a large lake have been presented by Parker (II). The application of the techniques of phytoplankton modeling to the problem of eutrophication in rivers and estuaries has been proposed by Chen (12). The interrelations between the nitrogen cycle and the phytoplankton population in the Potomac Estuary has been investigated using a feed-forward-feed-back model of the dependent variables, which interact linearly following first order kinetics (13). 

Within the complex plane, two circles are obtained. The overlapping of these two circles depends on the vicinity of the relaxation time or relaxation frequency of the two polar groups. This assumption could be applied to more than two polar groups. Are there two isolated Debye s relaxations or a distribution of relaxation times for a single relaxation process If the latter, it is better to use the Cole and Cole or Davidson and Cole models. Results from permittivity measurements are often displayed in this type of diagram. The disadvantage of these methods is that the frequency is not explicitly shown. 

Further Empirical and Semiempirical Models. Although various empirical distributed-element models have already been discussed, particularly in Section 2.2.2.2, the subject is by no means exhausted. Here we briefly mention and discuss various old and new elements which may sometimes be of use in a fitting circuit such as that of Figure 2.2.10. Complex plane plots of IS data by no means always yield perfect or depressed semicircular arcs often the arc is unsymmetric and cannot be well approximated by the ZC. An unsymmetrical impedance plane arc usually exhibits a peak at low frequencies and CPE-like response at sufficiently high frequencies. The reverse behavior is not, however, unknown (Badwal [1984]). An expression originally proposed in the dielectric field by Davidson and Cole [1951] yields ordinary asymmetric behavior. Its h generalization is... 

Hop, C.E., Cole, M.J., Davidson, R.E., et al. (2008) High throughput ADME screening practical considerations, impact on the portfolio and enabler of in silico ADME models. Current Drug Metabolism, 9, 847-853. 

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