Relaxation frequencies

Turning from chemical exchange to nuclear relaxation time measurements, the field of NMR offers many good examples of chemical information from T, measurements. Recall from Fig. 4-7 that Ti is reciprocally related to Tc, the correlation time, for high-frequency relaxation modes. For small- to medium-size molecules in the liquid phase, T, lies to the left side of the minimum in Fig. 4-7. A larger value of T, is, therefore, associated with a smaller Tc, hence, with a more rapid rate of molecular motion. It is possible to measure Ti for individual carbon atoms in a molecule, and such results provide detailed information on the local motion of atoms or groups of atoms. Levy and Nelson " have reviewed these observations. A few examples are shown here. T, values (in seconds) are noted for individual carbon atoms. 

The high frequency relaxation is attributed in part to the modulation of intermolecular dipolar interactions by the translational diffusion. The cutoff frequency (60 MHz at 55°C) corresponds to the local diffusive jump frequency that is estimated from measurements of the diffusion coefficient (D 10"6 cm2/sec at 55°) (19, 21). This cutoff frequency also varies in temperature with the same activation energy (Eact 0.25 eV) as the diffusion frequency. 

Soil-Water Mixtures Figure 2 presents the dielectric spectra of clay-water mixtures (kaolinite and montmorillonite) after the influence of dc conductivity is removed. In addition to the orientational polarization of bulk water at 20GHz and adsorbed water at 10MHz, a low frequency polarization is observed at kHz to MHz frequencies. It should be noted that this low-frequency relaxation process cannot be generalized for all mineral-water... 

With the large number of motional theories being touted the need for multi-frequency relaxation studies becomes critical. At one frequency most theories can satisfactorily predict the behavior because of the many adjustable parameters. By initiating multifield and multitemperature and NOEF studies, more subtle features of molecular motion can be probed. Although the motional model used by us is adequate, it may not be the best model. Indeed, Howarth has had better results with our preliminary data using internal librational motion. This enforces the need for measuring as many relaxation parameters as possible, under as many different conditions as possible. 

A dipole moment of type (c) in a flexible side chain, e.g. as in poly(methyl methacrylate), imposes a dipole moment which can be resolved into a rigid, perpendicular component and a rotatable component. One might have supposed that two separate relaxation processes might then have occurred, but in practice only one, high-frequency, relaxation is observed and one concludes that in solution the side-group rotation and segmental motion in the main chain are combined in a single, fast process (North, 1972). 

The Ta high-frequency relaxation domain the transition region between the rubbery and glassy regions... 

Other mathematical forms may be used to describe the high frequency relaxation. These various equations, either phenomenological or based on diiiusion defect models lead to a characteristic relaxation time xhp of the glass transition (Ta) domain of the same order of magnitude. 

Figure 3.89. The two relaxations for a colloidal particle. The low-frequency relaxation is determined by diffusion in the far field, that at high frequency by Maxwell conduction in the double layer. The imaginary part "(fl)) has peaks at the two relaxations. Molecular relaxations, as considered in figs. 1.4,7 and 1.4.8 come on top of these and are observable at higher frequencies.

Relaxation in Insulating Crystals. The low-frequency relaxation of ionic materials consists typically of a set of simple decays arising from the motion of defects. - Some of these have freedom of extensive motion and contribute to the conduction current others are constrained to a few ndghbouring sites. Except at high temperature, the defects are remnants of the previous thermal and mechanical history of the sample. Their movement is a thermally activated process and relaxation times normally vary with temperature as exp AjkT). The e qierimental picture may be much complicated by the motion of electrons loosely bound to crystal defects. 

The Argand diagram of the 2.2 M solution of Et NCl shows three relaxation processes typical for aqueous electrolyte solutions (1) ion-pair relaxation (r l = tip), (2) low frequency relaxation (rj, as 8 ps) of water, (3) high frequency relaxation a 1 ps) of water, in contrast to that of the 2 M solution of Bu NBr where the relaxation process (2) splits up into two processes. Figure 7 shows the concentration dependence of the... 

Figure 7 Frequency dependence of the low frequency relaxation time r l of water in EttNCl solutions (curve 3) and splitting of in ButNBr solutions into (curve 1) and (curve 2), c. f. fig. 6.

Figure 4.6 l/logio(/oo//p) versus temperature for propylene carbonate. Here logjQ foo = InjToo -13.11, where too (in sec) is the high-frequency relaxation time in Eq. (4-3). Ta is the crossover temperature between Arrhenius and VFTH behavior. The prediction of the mode-coupling theory MCT (see Section 4.6) is also shown, where 7). is the critical temperature. (From Schbnhals et al. 1993, with permission.)... 

Caution should be used when interpreting the electrolyte-resistance-corrected Bode plots. As seen in equation (16.20), nonzero values for Rg — Rg t) can give the appearance of an additional high-frequency relaxation process. When possible, an eissessment of Rg i should be made independently of the regression. 

Figure 12. High frequency relaxation of oxidized anionic p ystyrene (1 hr at 254 nm in 400 torr 0 j. Effect of temperature.

Finally, it is apparent that between the low-frequency and very high-frequency bands, at some values of model parameters, a third band exists in the dielectric loss spectra (see, e.g., Fig. 30). This band is due to the high-frequency relaxation modes of the dipoles in the potential wells (without crossing the potential barrier) which will always exist in the spectra even in the noninertial limit (see Section III.B). Such relaxation modes are generally termed the intrawell modes. The characteristic frequency of this band depends on the barrier height v and the anomalous exponent a. 

The molecular weight dependence enters Equation 29 through tiq For this reason It Is convenient to define the reduced frequency relaxation spectrum as ... 

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