Davidson-Cole distribution

Cole-Davidson distribution parameter j3, and generalized order parameter S. ... 

Table4.5-1 Reorientational correlation times rat 357 K and fit parameters activation energy Ea, Cole-Davidson distribution parameter p, and generalized order parameter S. ...

Usually, spin-lattice relaxation experiments are performed at one or a few magnetic fields. The spectral density can thus be determined at only a few Larmor frequencies, so that a detailed analysis of its temperature variation is not possible. Here, an analysis of the spin-spin relaxation times, T, can provide further information about the spectral density, since T2 1 S2(to = 0). Often, the Cole-Davidson distribution Gco(lnT2) [34] is chosen to interpolate the relaxation around the maximum. However, one has to keep in mind that the spectral density close to Tg contains additional contributions from secondary relaxations, such as the excess wing and/or the (i-process discussed in the following sections. In Section IV.C we give an example of a quantitative description of 7) (T) at T > 7 obtained by approximating the spectral density S2(co) using dielectric data. 

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. 

For NaClOi 0.05 M and O.IM in THF (13) also a Cole-Davidson distribution with parameters 6=0.9 and 6 = 0.8 respectively describes the solute relaxation. 

The Cole-Davidson distribution with the parameter 7 0.4—0.7 giving the dependence T(7 /ti) wdiich is the closest to the linear dependence over a wide range of T/n (at r/rj > (1—10) X 10 (K cP), agrees reasonably well with experimental data on PL The box distribution (in the log t scale) of Frohlich s type where Tmta distributions ranging from t=0 to t but r idly decreasing at... 

The largest correlation times, and thus the slowest reorientational motion, were shown by the three C- Fl vectors of the aromatic ring, with values of between approximately 60 and 70 ps at 357 K, values expected for viscous liquids like ionic liquids. The activation energies are also in the typical range for viscous liquids. As can be seen from Table 4.5-1, the best fit was obtained for a combination of the Cole-Davidson with the Lipari-Szabo spectral density, with a distribution parame-... 

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... 

The inverse (logarithmic scale) distribution of the Cole-Davidson type yields... 

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. 

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. 

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. 

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. 

As one can see in Figure 14 the relaxation spectra of the isotropic phase of all substances studied have maxima of losses above 100 MHz, so our HP setup can cover only a low-frequency part of the absorption bands. Therefore, to obtain the relaxation times vs. T or p we had to extrapolate the measured spectra to higher frequencies in order to find the critical frequency / = 1/(27tt ). According to Parneix et al. the Cole-Davidson skewed arcs should be used for that purpose. However, the recent measurements carried out by Gestblom and co-workers with the use of the TDS method have shown that the spectra of the isotropic phase of 5CB and 5PCH could be well described by the Cole-Cole equation with a symmetric distribution of the relaxation times. 

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. 

Cole and Davidson s continuous distribution of correlation times [9] has found broad application in the interpretation of relaxation data of viscous liquids and glassy solids. The corresponding spectral density is ... 

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. 

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. 

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