Cole-Davidson plot

Figure 2 For packed samples of crushed ice particles, (a) Temperature dependence of the dielectric relaxation time rand (h) Cole-Cole plots of the sample (after 400 h of annealing at -1 °C, 2-4 mm particle size, 536 kg/m packed density) at -10 °C. The dielectric dispersion is of the Davidson-Cole type (a=0.97, f=0.85).

It was also found that the variations of l/Ti required a Davidson-Cole distribution function for best fit. In the mixed alkali glasses the MT plots at high temperatures become symmetrical and broader. The activation energies determined from NMR, Enmr, and the E from conductivity measurements have also been compared. Since Li NMR senses only the lithium ion and not the other alkali, the Enmr in mixed alkali regions Eire observed to be lower than Ea. 

The well-known Cole-Cole plot (0single line in this a, jS plane see Figure 29 (on page 264). The Kirkwood-Fuoss as well as the Gaussian functions are essentially coincident with the Cole-Cole function. The Cole-Davidson function Os]8 l.O is also a single line in this plane and forms one of the bounds that is perpendicular to the Cole-Cole line. The stretched exponential is represented by a hyperbolic-like line in the a, )3 plane and takes the form ... 

Comparison between Cole-Cole plots for the Debye, Cole-Cole and Davidson-Cole equations is made in Fig. 4.2. The arc corresponding to the Cole-Cole equation is symmetrical and forms a portion of a circle, the centre of which is below the 6 -axis. The corresponding distribution function of relaxation times is symmetric, although there is no closed expression for F t) which would give the Cole-Cole equation. The Davidson-Cole arc is a skewed one, and reflects strongly asymmetric distribution of relaxation times. The distribution is peaked at the critical relaxation time Tq, with a decaying tail of shorter relaxation times. There is an exact expression for the autocorrelation function leading to the Davidson-Cole equation. 

Figure 21. Light scattering (LS) spectra of 2-picoline for different temperatures T > Tg (indicated as numbers) plotted as a function of rescaled frequency vxa solid line interpolation by a Cole-Davidson (CD) susceptibility, dotted line power-law indicating locus of the minimum (T > Tg), cf. Eq. 35 (adapted from [183]) below, say 175 K, the evolution of the spectra changes, i.e. the a-scaling fails.

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)). 

Examples of the complex plane plots obtained for fractal electrodes are presented in Fig. 33. With a decrease in parameter ([), the semicircles become deformed (skewed). The complex plane impedance plots obtained from Eq. (183) are formally similar to those found by Davidson and Cole " in their dielectric studies. Kinetic analysis of the hydrogen evolution reaction on surfaces displaying fractal ac impedance behavior was... 

Like the Cole-Davidson function, the Williams-Watts approach gives asymmetric plots in the complex e plane. A detailed comparison of these two forms has been made by Lindsey and Patterson [1980]. 

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... 

Such slow motions have a different mechanism from that operating in the normal low viscosity liquid, leading to a greatly increased value for Qapp (" 200 kJ mol ) and a broadened loss curve. They also studied mixtures such as 43.5 mol % chlorobenzene in pyridine, 42.5 mol % bromobenzene in pyridine, 36.8 mol % 1-chloronaphthalene in pyridine, 37.5 mol % 1-chloronaphthalene in tetrahydrofuran, 43.4 mol % bromobenzene in tetrahydrofuran and 43.8 mol % toluene in pyridine. They observed a and 6 processes as before. There was no evidence of structure in the a peaks, indicating that both dipolar components in the mixture move cooperatively in one general a process - a result previously obtained for mixtures of alkyl halides (54). Also the a process was asymmetric in the Cole-Davidson sense, as had been found for the alkyl halidees C52, 54-57). As one example, figure 2 shows the plot of... 

Figure 5 shows the vs - log f plot for phthalic anhydride in o-terphenyl 66), just above the Tg of the solution. This process has a large apparent activation energy (Qapp 270 kJ mol"l) and the loss curve is asymmetric in the Cole-Davidson sense. Williams and coworkers (66-68) observed a similar well-defined a-relaxation for the following solutes phthalic anhydride, enthrone, nitrobenzene, 1-chloronaphthalene, benzophenone, cyclohexanone, fenchone, camphor and fluorenone. It was shown that the a-process did not relax all of Since Sq and could... 

The results in Figure 28 can be mapped into contour maps of constant /, or c as functions of a and /3 see Figures 30 and 31 (on pages 265 and 266). These plots are important because they permit us to pick of the values of c and / for experimental values of a and )3. In Figures 30 and 31 we have plotted the values of / and c for the Cole-Cole and Cole-Davidson functions, respectively. In addition there are a large number of theoretical correlation functions that form lines in this plane. 

Figure 30. Plot of / as a function of or or that meet the Cole-Cole or the Cole-Davidson condition.

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