Davidson-Cole function

The dipolar component arises from diffusion of bound charge or molecular dipole moments. The frequency dependence of the polar component may be represented by the Cole-Davidson function ... 

Lindsey, C. P., Patterson, G. D. Detailed comparison of the Williams-Watts and Cole-Davidson functions, J. Chem. Phys. 73, 3348 (1980)... 

Figure 4.3 Frequency-dependence of the imaginary (loss) part of the dielectric relaxation function for PDE at different temperatures. The lines are fits by the Cole-Davidson function, Eq. (4-2), with cu = 2nf and temperature-dependent exponent given in Fig. 4-4. (Reprinted from Physica, A201 318, Stickel et al. (1993), with kind permission from Elsevier Science - NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

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... 

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]. 

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. 

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 ... 

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. 

As mentioned above, the frequency dependence of the complex dielectric permittivity (e ) of the main relaxation process of glycerol [17,186] can be described by the Cole-Davidson (CD) empirical function [see (21) with a = 1, 0 < Pcd < 1], Now Tcd is the relaxation time which has non-Arrhenius type temperature dependence for glycerol (see Fig. 23). Another well-known possibility is to fit the BDS spectra of glycerol in time domain using the KWW relaxation function (23) < )(t) (see Fig. 24) ... 

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.

Figure 4.7 Cole-Davidson exponent as a function of temperature for glycerol (circles), propylene glycol (squares), and propylene carbonate (triangles). is the temperature at which the crossover from Arrhenius to VFTH behavior is observed, while is the best fit to the critical temperature of the mode-coupling theory (see Section 4.6). (From Schonhals et al. 1993, reprinted with permission from the American Physical Society.)...

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],... 

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. 

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. 

Many materials display non-Debye dielectric behavior by a broader asymmetric loss peak. This non-Debye a.c response can be described by a combination of Cole-Cole [23] and Davidson-Cole [24] functions, and an empirical expression proposed by Havrilink-Negami [25]. The natural gum Arabica is found to obey a non-Debye type of response [25,26] and may be described [27] by the Havrilink-Negami function. 

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... 

HBA. The T p values display minima, coincident with the temperatures of the y and p relaxations previously observed by dielectric and dynamic mechanical analysis. The Tjp - temperature data have been fitted to the Cole-Davidson distribution function, which indicates a broad distribution of correlation times. The activation energies obtained from the fitting are higher than from dielectric data, but, in a qualitative sense support the contention that the y relaxation is associated with the motion of HBA units, and the p relaxation with the motion of HNA units. 

Figure 4. Proton Tjp versus reciprocal temperature for 73R (O), 73M (A) and 30R (0). The lines through the data are fits to the Cole-Davidson distribution function.

Table I. Width Parameters, 8, Activation Energies, E and Pre-Exponential Factors, Xq Obtained From the Cole-Davidson Distribution Function...

Parametric fitting of the data to the sum of a Debye relaxation for the solvent, and to a Cole-Davidson distribution (LiC104, NaClO. in THF) and to a Debye function (other systems) for the solute, is reported. The solute relaxation is interpreted as due to rotational relaxation of the dipolar species present in solution. Structural information frm IR-Raman spectra and electrical conductance data from the literature are used to substantiate the above interpretation. 

Assumption b) is known to be a good approximation for small molecules. However studies of polymers and glass forming materials by dielectric and mechanical loss methods have frequently been interpreted by assuming that molecular motions are best described by a distribution of correlation times. This has resulted in the formulation of a number of well-known distribution functions such as the Cole-Cole (symmetric) and Cole-Davidson (asymmetric) functions, which have been used to fit dielectric data. It is reasonable to suppose that magnetic relaxation times are also subject to the possible presence of distributions, and a number of modifications of Eq, (4) have been made [16 —i 9] on this basis. 

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

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