Distribution function Davidson-Cole

In order to obtain better agreement with experimental results, the concept of a distribution of correlation times was introduced in nuclear magnetic relaxation. Different distribution functions, G(i c), such as Gaussian, and functions proposed by Yager, Kirkwood and Fuoss, Cole and Cole, and Davidson and Cole (asymmetric distribution) are introduced into the Eq. (13), giving a general expression for... 

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. 

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) ... 

Fig. 1.9 Relaxation time distribution functions calculated on the basis of Debye law (D), Cole-Cole law (CC), Davidson-Cole law (DC) and...

The expression (1.24) allows obtaining the distribution function of relaxation times for all empirical laws (1.23). In Fig. 1.9, we show the relaxation time distribution functions, obtained in Ref. [31] with the help of Eq. (1.24). The distribution functions have been obtained for the laws of Cole-Cole k = 0.2), Davidson-Cole (P = 0.6) and Havriliak-Nagami at a = 0.42 when it corresponds to KWW law. It is seen that only C-C law leads to symmetric dishibution function while DC and KWW laws correspond to essentially asymmetric one. The physical mechanisms responsible for different forms of distribution functions in the disordered ferroelechics had been considered in Ref. [32]. It has been shown that random electric field in the disordered systems alters the relaxational barriers so that the distribution of the field results in the barriers distribution, which in turn generates the distribution of relaxation times. Nonlinear contributions of random field are responsible for the functions asymmetry, while the linear contribution gives only symmetric C-C function. 

Two other distribution functions are due to Kirkwood and Fuoss [1941] and Davidson and Cole [1951] (see also Davidson [1961]). 

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...

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. 

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. 

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. 

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. 

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. 

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. 

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