Slow relaxation mechanisms

Ferry,J.D., Williams,M.L., Stern,D.M. Slow relaxation mechanisms in concentrated polymer solutions. J. Phys. Chem. 58,987-992 (1954). 

Integration over equations 41 and 42 with appropriate limits by equations 19, 23, and 24 of Chapter 3 and addition of vRT/2 in the first two cases provides the viscoelastic functions G(t), G, and G" for the Chompff-Duiser theory. The corresponding curve for the loss compliance J" is included in Fig. 10-7. It extends farther to the low-frequency side than the others, as would be expected from the additional slow relaxation mechanisms. 

Figure 2. The two relaxational mechanisms in hydrogen bonding. F, fast mode S, slow mode B, bending mode TB, thermal bath.

On the other hand, one has to take into account the influence of the surrounding which must induce an irreversible evolution of the H-bond system when its fast mode is excited the fast mode may be directly damped by the medium that is the direct relaxation mechanism. It may be also damped through the slow mode to which it is anharmonically coupled, that is the indirect relaxation mechanism. A schematical illustration of these two damping mechanism is given in Fig. 2. Of course, the role played by damping must be more important for H bonds in condensed phase. 

Region II. If the chemical exchange is slow compared with the relaxation mechanism (which is incorporated in the terms Aa>M or 2m)> that is,... 

It can be seen that, in all cases, relaxation rates are directly proportional to (Aa). Because Aa reflects the anisotropy of the shielding tensor and because the chemical shift originates from the shielding effect, the terminology Chemical Shift Anisotropy is used for denoting this relaxation mechanism. Dispersion may be disconcerting because of the presence of Bq (proportional to cOq) in the numerator of and R2 (Eq. (49)). Imagine that molecular reorientation is sufficiently slow so that coo 1 for all considered values of coo from (49), it can be seen that R is constant whereas R2 increases when Bq increases, a somewhat unusual behavior. 

A variety of relaxation time studies have been performed on toluene. The choice of deuterated toluene avoids certain complicating factors which affect proton NMR studies, such as, dipolar or spin-spin couplings. The dominant relaxation mechanism is quadrupolar and the relaxation times are determined by the reorientation of the C-D bond vector. Relaxation times such as T, are sensitive to the motions of the solvent around the larmor frequency, which is on the order of 14 MHz in this study. T2 measurements may probe slower motions if the molecule undergoes slow and/or anisotropic motion. The relaxation time results presented in Figure 3 are significantly shorter than those found in bulk toluene solutions (18.19). In bulk toluene, the T and T2 values are equal above the melting temperature (1.2.). In this polymer system T2 < T indicative of slow and/or anisotropic reorientation. 

Slow relaxations can be exemplified by the system behaviour corresponding to the adsorption mechanism (8) when the parameters % are close to their bifurcation values. 

In numerical experiments, slow relaxation is distinctly observed if the trajectory approaches the unstable steady state. The system rapidly enters its neighbourhood (after 1 s) and then relatively slowly (during 100 s) moves toward its stable steady state. This phenomenon has been described for the three-step adsorption mechanism. 

Another variant of the explanation is the list of substances and the reaction mechanism is incomplete and so slow relaxations are explained by the slow steps that have not been taken into account. It must be emphasized that slow transition processes can also be caused by slow steps, in those cases in which the steady state rate of a catalytic reaction is high. This can be exemplified by two linear catalytic cycles connected by a slow step... 

Finally, we can suggest a third explanation fast steps can compose a mechanism with slow relaxations. Indeed, nothing suggests that the relaxation time for a set of chemical kinetic equations is directly dependent on the characteristic times of the individual steps. But it cannot be treated as a reason for slow relaxations. It is only a simple indication for the possibility of finding such reasons here. Let us now indicate the reasons according to which fast steps can compose a mechanism with slow relaxations. 

Let us start with some simple considerations and three explicitly integrated examples. We can immediately give two simple mechanisms for the generation of slow relaxations. It is a delay of the motion near the unstable fixed point and (for the systems that are dependent of the parameter) a delay of the motion in the region, where at small variations of the parameter there appears a fixed point. 

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