Fast and slow motions

The rate at which the temperature excess d changes on the new timescale is given by dd/dT = g(cc, d)/c. Because e has been defined as a small parameter, this will be a large quantity unless the function g(oc, d) is close to zero. Thus we can expect that the temperature excess will try to adjust very quickly (i.e. while a changes very little) until. 

This is the fast motion of the system. We can draw this nullcline , so called because one of the rates of change is zero along this curve, in the a-d plane in Fig. 5.7. It has a maximum and a minimum whose coordinates are relatively easily located as a function of y, as we will see below (they also correspond to the maximum and minimum in the stationary-state locus). 

As a varies according to its own rate eqn (5.61), the slow motion of the system, we can expect the trajectory in the phase plane to try to stay as close as possible to the g(a, d) = 0 nullcline. We can also draw the /(a, d) = 0 nullcline on to the phase plane. This is a curve on which da/d Tis zero, so the concentration of A passes through a maximum or a minimum in time whenever a trajectory crosses it. There is one point where the two nullclines intersect. Here both time derivatives vanish simultaneously this is the sta- 

The continuous motion around the circuit ABCD which thus ensues gives the relaxation oscillation shown schematically in Fig. 5.8(d). 

This qualitative description only holds strictly in the limit e - 0. Given this qualification, however, we can use the condition for a unique stationary-state 

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At temperatures around 50-60°C the three-site jump model is not a good approximation to the multi-site jump model, because the motion is not sufficiently rapid to be in the fast motion limit. However, the calculated spectra are fairly fitted with the observed ones. This is because the calculated spectrum is a superposition of constituent spectra whose rates are spread over several orders, so that the resultant spectrum is governed by the constituent spectra in the fast and slow motion limits having greater intensity than that in the intermediate exchange regime. 

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

Fig. 12.16 Full and partial synchronization of the fast and slow motions between two interacting nephrons (/ = 13.5 s, a = 30.0 and y = 0.06). Full synchronization is realized when both the fast nj and slow ns rotation numbers equal to 1. To the right in the figure there is an interval where only the slow modes are synchronized. The delay / in the loop of Henle for the second nephron is used as a parameter.

In Table 1, S2 is the general order parameter describing bond motions at the ps-ns time scale, te is the characteristic correlation time for these motions and Rex is the conformational exchange term representing motions on the ps-ms time scale. Internal motions on the ps-ns time scale are considered as the superposition of fast and slow motions characterized by S2 and if as well as Ss2 and ts. Here, S2 = Ss2Sf2 and t, [Pg.52]

Based on random fields, relaxation theory R2 decreases as molecular tumbling gets faster and more effectively averages the residual dipolar broadening. Both in the fast and slow motion limits, R2 is proportional to tc. ... 

The adiabatic approximation, which is to be applied to bound-state dynamics, involves the separation of fast and slow motions. However, unlike the original application of the Bom-Oppenheimer approximation,77 in which fast electronic motion is separated from the slower nuclear one, there is no clear separation of time scales for nuclear motions. Nevertheless, there is now ample evidence to show that there are always states, to be termed adiabatic, which are accurately described by an adiabatic approximation.66,69-71 78... 

Here J/(coo) and /,(wo) represent the reduced spectral density functions describing the fast and slow motions, respectively. The parameter S is often referred to as an order parameter and is given by the average ... 

The nested TCP expressions can be readily applied to the separation of fast and slow motions in calculations of collisional energy transfer. For example, in a case involving rotational and vibrational motions, the slow variables are the Euler angles Tand the fast variables are the displacements u, one would first calculate the inner TCP for vibrational motions while the orientations of the molecules are kept fixed. The result would then be weighted by a time-dependent function of the orientation angles describing the effect of rotations on the vibrational TCP for each initial vibrational state. 

Other classical-path treatments have been formulated for atom-molecule [43,95-101] and molecule-molecule [102-107] collisions however, they were mostly based on internal-state expansions that become computationally impractical as the collision energy increases. Semiclassical calculations have also been implemented by means of time-dependent wavepackets [39], whose propagation can become expensive for motions with widely differing time scales. The following TCF-semiclassical approach encompasses very high densities of internal states as well as fast and slow motions. 

The principle of fast and slow motional averaging (also described in the next chapter) applies similarly to protons and dipolar interactions. IH wide-line spectra are generally without structure, due to the superposition of a large number of dipolar interactions. This makes 2H wide-line investigations far more attractive than those of protons in spite of the lower sensitivity and necessity for isotopic labelling. 

Deuterated spin labels significantly simplify analysis of both fast- and slow-motional spectra The contribution of unresolved hyperfine interaction due to deuterium nuclei is much smaller than that of protons (Fig. 2) due to a smaller nuclear magnetic moment ... 

PEO) with isotactic poly(methyl methacrylate) (PMMA) [25, 32] and labeUed (both random and chain-end) PMMA with poIy(vinylidene fluoride) (PVDF) [33, 34]. These are complex systems because they show partial miscibility and because in each case one of the components is capable of crystallising. Nevertheless, these authors obtained fundamental information on the morphology of the blend and phase separation. In the PEO-PMMA blends, the spectra were of the composite type, and they ascribed the fast- and slow-motion spectra to nitroxides located in PEO-rich regions and PMMA-rich regions respectively. 

Since the CW-ESR spectra provides structural information and dynamics at different time scales, proper account of fast and slow motion of the labeled molecules is required for correct reproduction of the spectra. While the fast motion can be derived from a fast-motional perturbative model, in the slow-motion regime the effects on the spin relaxation processes exerted by the molecular motions requires a more sophisticated theoretical approach. The calculation of rotational diffusion in solution can be tackled by solving the stochastic Liouville equation (SLE) or by longtime-scale molecular dynamics simulations [94—96]. 

In a typical experiment for polymeric systems, FPR spectra are recorded over a wide temperature range and analyzed in terms of the probe dynamics. Quite often (especially for blends), two-component spectra can be observed at certain temperatures that consist of fast- and slow-motional signals (cf. Figure 23,11c), Such a spectrum is indicative of a two-phase morphology and probe partition into different environments. The basic parameters providing information on the examined system are and Ta temperatures which correspond to... 

Here A is an amplitude, x and 1 — x are the relative amounts of two exponential decays, and (t) is a single function that happens to have a rapid decay, an intermediate plateau, and a final long-time decay. WithEq. 11.10 the fast and slow motions are both happening independently at all times. There are thus two modes. In Eq. 11.11 there is a single relaxation of somewhat complex form 4>(t) has fast and slow parts, but the slow part of 4>(r) is actually absent at early times single complex relaxation. 

The Jump model has a fairly simple derivation and illustrates the general distinction between fast and slow motions (relative to the rotational tumbling time). Other models, such as diffusion in an angular cone or in a continuous torsional potential, may be considered for quantitative analysis. One powerful way of representing the general effects of internal motion on NMR relaxation is the model-free approach developed by Lipari and Szabo. Here the potentially complex spectral density is replaced by a two-term expression in which... 

PDS, and 6-PDS using the time-resolved fluorescence depolarization method involving two dissimilar probes, 2,5-dimethyl-l,4-dioxo-3,6-diphenylpyr-rolo[3,4-c]pyrrole (DMDPP) and coumarin 6 (C6). The decay of anisotropy for both probes in all the six micelles has been rationalized on the basis of a two-step model consisting of fast-restricted rotation of the probe and slow lateral diffusion of the probe in the micelle that are coupled to the overall rotation of the micelle. On the basis of the assumption that the fast and slow motions are separable, the experimentally obtained slow and fast reorientation times and Xfajt) are related to the time constants for lateral diffusion (xj, wobbling motion (Xw), and rotation of the micelle as a whole (Xm) by the following relationships ... 

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