Distribution of correlation times

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 ... 

Pulsed deuteron NMR is described, which has recently been developed to become a powerftd tool for studying molectdar order and dynamics in solid polymers. In drawn fibres the complete orientational distribution function for the polymer chains can be determined from the analysis of deuteron NMR line shapes. By analyzing the line shapes of 2H absorption spectra and spectra obtained via solid echo and spin alignment, respectively, both type and timescale of rotational motions can be determined over an extraordinary wide range of characteristic frequencies, approximately 10 MHz to 1 Hz. In addition, motional heterogeneities can be detected and the resulting distribution of correlation times can directly be determined. 

Fig. 13. Calculated 2H solid echo spectra for log-Gaussian distributions of correlation times of different widths. Note the differences of the line shapes for fully relaxed and partially relaxed spectra. The centre of the distribution of correlation times is given as a normalized exchange rate a0 = 1/3tc. For deuterons in aliphatic C—H bonds the conversion factor is approximately 4.10s sec-1...

Polycarbonate (PC) serves as a convenient example for both, the direct determination of the distribution of correlation times and the close connection of localized motions and mechanical properties. This material shows a pronounced P-relaxation in the glassy state, but the nature of the corresponding motional mechanism was not clear 76 80> before the advent of advanced NMR techniques. Meanwhile it has been shown both from 2H NMR 17) and later from 13C NMRSI) that only the phenyl groups exhibit major mobility, consisting in 180° flips augmented by substantial small angle fluctuations about the same axis, reaching an rms amplitude of 35° at 380 K, for details see Ref. 17). 

Fig. 23. Experimental and calculated methyl-deuteron spectra of polycarbonate for different temperatures and different evolution times Tr For the definition of cf. Fig. 13. The width of the distribution of correlation times is 2.7 decades...

Fig. 29. Observed and calculated 2H NMR spectra for the mesogenic groups of a) the nematic (m = 2), b) the smectic (m = 6) liquid crystalline polymer in the glassy state, showing the line shape changes due to the freezing of the jump motion of the labelled phenyl ring. The exchange frequency corresponds to the centre of the distribution of correlation times. Note that the order parameters are different, S = 0.65 in the frozen nematic, and S = 0.85 in the frozen smectic system, respectively...

H NMR 38,39,42, 50-55 Hole capacity constant 205 Homogeneous distribution of correlation times 37 Humic acids 17 Humidity of plastics 119 Humins 17 Hydrogen bond 200 Hydrophilic 191, 194, 206... 

The relaxation data for the anomeric protons of the polysaccharides (see Table II) lack utility, inasmuch as the / ,(ns) values are identical within experimental error. Obviously, the distribution of correlation times associated with backbone and side-chain motions, complex patterns of intramolecular interaction, and significant cross-relaxation and cross-correlation effects dramatically lessen the diagnostic potential of these relaxation rates. 

For folded proteins, relaxation data are commonly interpreted within the framework of the model-free formalism, in which the dynamics are described by an overall rotational correlation time rm, an internal correlation time xe, and an order parameter. S 2 describing the amplitude of the internal motions (Lipari and Szabo, 1982a,b). Model-free analysis is popular because it describes molecular motions in terms of a set of intuitive physical parameters. However, the underlying assumptions of model-free analysis—that the molecule tumbles with a single isotropic correlation time and that internal motions are very much faster than overall tumbling—are of questionable validity for unfolded or partly folded proteins. Nevertheless, qualitative insights into the dynamics of unfolded states can be obtained by model-free analysis (Alexandrescu and Shortle, 1994 Buck etal., 1996 Farrow etal., 1995a). An extension of the model-free analysis to incorporate a spectral density function that assumes a distribution of correlation times on the nanosecond time scale has recently been reported (Buevich et al., 2001 Buevich and Baum, 1999) and better fits the experimental 15N relaxation data for an unfolded protein than does the conventional model-free approach. 

It was soon realized that a distribution of exponential correlation times is required to characterize backbone motion for a successful Interpretation of both carbon-13 Ti and NOE values in many polymers (, lO). A correlation function corresponding to a distribution of exponential correlation times can be generated in two ways. First, a convenient mathematical form can serve as the basis for generating and adjusting a distribution of correlation times. Functions used earlier for the analysis of dielectric relaxation such as the Cole-Cole (U.) and Fuoss-Kirkwood (l2) descriptions can be applied to the interpretation of carbon-13 relaxation. Probably the most proficient of the mathematical form models is the log-X distribution introduced by Schaefer (lO). These models are able to account for carbon-13 Ti and NOE data although some authors have questioned the physical insight provided by the fitting parameters (], 13) ... 

A. Jones, Clark University, Massachusetts When the methyl group is attached to the backbone or even to a side chain it has a whole distribution of correlation times associated either with the backbone or the side chain in addition to its methyl group rotation. Our assumption in these models is that they are independent so that the methyl group Ti would be calculated on the... 

Relaxors where there is no macroscopic symmetry breaking and where, in view of site and charge disorder, there is an extremely broad distribution of correlation times. The longest correlation time diverges at the freezing transition whereas other correlation times are still finite [e.g., Pb (Mgi/3Nb2/3) O3]. 

Relaxation times for water filling the pores of an NaX specimen have been fitted to a model with the following assumptions (a) coupling, as above, of molecular diffusion and rotation (b) the median jump time r is governed by a free volume law (allows the curvature in the plots of jump rate, (3r) x vs. 10S/T in Figure 5), and (c) a broad distribution of correlation times (allows a better fit to the data, accounts for an apparent two-phase behavior in T2 (31, 39), and is reasonable in terms of the previous discussion of Pi(f) and r). 

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... 

Spin-lattice relaxation times of carbon-13 in different polypropylene stereosequences differ slightly while nuclear Overhauser enhancements are almost identical (1.8-2.0) [533] isotactic sequences display larger Tx values than the syndiotactic stereoisomers. Other vinyl polymers behave correspondingly [534]. Carbon-13 spin-lattice relaxation times further indicate that dynamic properties in solution depend on configurational sequences longer than pentads. The ratio 7J(CH) 7J(CH2) varies between 1.6 to 1.9 thus, relaxation can be influenced by anisotropic motions of chain segments or by unusual distributions of correlation times [181],... 

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