Relaxation-dispersion curves

Several methods have been proposed to improve the measurement of the relaxation rate in the rotating frame Rip that is used to obtain the exchange contribution through the registration of the relaxation dispersion curves (dependence on the amplitude and frequency of the applied rf field). 

Typical spin-lattice relaxation dispersion curves... 

Figures 28, 29, 30, 31, 32, 33, and 34 show a series of typical spin-lattice relaxation dispersion curves. The technique has been applied to melts, solutions, and networks of numerous polymer species. As experimental parameters, the temperature, the molecular weight, the concentration, and the cross-link density were varied. For control and comparison, the studies are partly supplemented by rotating-frame spin-lattice relaxation data and, of course, by high-field data of the ordinary spin-lattice relaxation time. Furthermore, the deuteron spin-lattice relaxation was employed for identifying the role different spin interactions are playing for relaxation dispersion.

In the plot shown in Fig. 31e, spin-lattice relaxation dispersion curves of permanently cross-linked PDMS are shown. With decreasing mesh length, the chain modes appear to be shifted to lower frequencies. This is indicated by lower values of the relaxation times while the dispersion slopes of regions I and II are retained. On the other hand, the effect on the crossover frequency is minor. 

For spin-f nuclei, dipolar interactions may be modulated by intramolecular (DF, reorientation etc.) and/or intermolecular (TD) processes. In general, the intra- and inter-molecular processes can produce quite different Tj frequency dispersion curves. In practice, NMR field cycling experiments are often needed to extend the frequency domain from those employed in conventional spectrometers to a lower frequency range (i.e., the kHz regime) for unambiguous separation (and identification) of different relaxation mechanisms. The proton spin relaxation by anisotropic TD in various mesophases has been considered by Zumer and Vilfan.131 133,159 In the nematic phase, Zumer and Vilfan found the following expression for T ... 

This mechanism is identical to the one arising from the contact interaction between an unpaired electron and a nuclear spin (41). In that case, the hyperfine coupling (generally denoted by Asc or A and exists only if the electron density is non-zero at the considered nucleus, hence the terminology of contact ) replaces the J coupling and the earlier statement (i) may be untrue because it so happens that T becomes very short. In that case, dispersion curves provide some information about electronic relaxation. These points are discussed in detail in Section II.B of Chapter 2 and I.A.l of Chapter 3. 

At ambient temperature, H, and 0 relaxation is in the extreme narrow range and dispersion curves are perfectly flat (see Fig. 9 bottom) precluding any correlation time determination. Furthermore as inter- and intramolecular contributions to proton relaxation cannot be easily separated and as the deuterium and 0 quadrupole coupling constants are not known with sufficient accuracy, there is a real problem for determining a meaningful correlation time. This problem was solved only in the early 1980s by resorting to the cross-relaxation rate which is purely intramolecular... 

The dependence of T (B) on the field B was soon nicknamed as the Ti dispersion curve or, more recently, as the Nuclear Magnetic Relaxation Dispersion (NMRD) profile. The first experimental curve of this type (Pig. 1) was published in 1950 by Ramsey and Pound (15,16). 

Since this is essentially an engineering chapter, we shall dwell only on the last point. Prom the BPP formula, it was already qualitatively clear that, in order to become efficient and useful tools, the dispersion curves must extend over a wide interval of relaxation field values (preferably several orders of magnitude). 

FIGURE 4.9 Relaxation titration curves (left) and NMRD dispersion curves (right) of ligands CPw3 ( ) and CPwl7 ( ) in anhydrous acetonitrile. 

Relaxation dispersion measurements are applied to systems that are undergoing an exchange process on the microsecond to millisecond timescale. Measurement of the f 2 relaxation rate in a series of experiments where the pulsing frequency is varied results in additional intensity for resonances of nuclei involved in the exchange process [18,19]. The resulting dispersion curve, showing as a function of 1/tqp can be fitted to functions such as [19,20]... 

Fig. 1.6. 15 N i 2 relaxation dispersion profile for Argl24 of pKID recorded at 800 (filled circles) and 500MHz (open circles). Dispersion curves for ImM [15N]-pKID in the presence of 0.95, 1.00, 1.05, and l.lOmM KIX are shown...

Relaxations observed in polymers show broader dispersion curves and lower loss maxima than those predicted by the Debye model, and the (s" s ) curve falls inside the semicircle. This led Cole and Cole (1941) to suggest the following semi-empirical equation for dielectric relaxations in polymers ... 

In most other cases the dispersion curves are flattened in comparison with pure Debye functions as presented in (10). This indicates the existence of more than one relaxation time. A rather obvious explanation is based on an ellipsoidal shape of the protein molecule with components of the permanent dipole along each of the three principal axes. In general, this should result in three corresponding relaxation times. Applying this concept to practical examples, it turns out that the assumption of an ellipsoid of revolution with only two relaxation times is quite sufficient for a quantitative fit of the experimental curves. In such a way, the ratio of the axes can be determined. It must be emphasized, however, that the data really do not permit an unambiguous discrimination between two or more individual relaxation times. 

More complicated dielectric properties may be described by superimposing such terms with different relaxation times and the corresponding values of Aco- This always results in a flattening of the dispersion curves (i.e. e and e plotted vs. log to) in comparison with the basic Debye functions of (10). 

Relaxation times r obtained from experimental dispersion curves in Figs. 59-61 by using Eq. (81) can be compared with molecular weights M of the same polymer samples. The dependence of r on M for poIy(butyl isocyanate) is shown in Fig. 62. The experimental points fit a curve the slope of which decreases with increasing M from 2.7 to 1.5. Taking into account Eq. (82) this can be expressed by... 

As mentioned in seetion 1, the MD approach can be utilized to obtain parameters sueh as frequency-dependent relaxation times and phonon dispersion relations (the dependence of the frequency w on the wave vector k). From the phonon dispersion curves, we can compute parameters such as the phonon group veloeity v , density of states D(w), and specific heat C . The phonon group velocities and relaxation times ean then be input into the BTE (Eq. 2.1). 

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