Diffusion mean jump times

In this expression yn is the proton gyromagnetic moment and the dipolar second moment. This latter quantity is determined by the arrangement of the protons in the crystal structure and can be calculated if structural data are available for a given ternary hydride. It is also possible to derive the dipolar second moment from the NMR line shape. For a random walk diffusion process the mean jump time is related to the diffusion constant D via the expression (Cotts, 1972)... 

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

The macroscopic diffusion coefficient ) is defined in terms of the mean jump distance a and mean time between jumps r as ... 

Additional dividends from NMR will most likely continue to lie in the area of diffusion and kinetics. Newer NMR techniques here are the ultra-slow motion (25) and rotating frame relaxation (26) techniques which allow measurements of very long jump times. Application of these techniques to the exchange region has been reported for water on NaX in this region they offer a means of deducing second moments of the tightly bound species (9, 52). The CIDNP technique should be applicable to the study of radical reactions on surfaces and in zeolites (58). 

Equation 7.35 is a fundamental relationship between the diffusivity and the mean-square displacement of a particle diffusing for a time r. Because diffusion processes in condensed matter are comprised of a sequence of jumps, the mean-square displacement in Eq. 7.31 should be equivalent to Eq. 7.35. This equivalence, as demonstrated below, results in relations between macroscopic and microscopic diffusion parameters. 

The movement of biological stressors have been described as diffusion and/or jump-dispersal processes. Diffusion involves a gradual spread from the site of introduction and is a function primarily of reproductive rates and motility. Jump-dispersal involves erratic spreads over periods of time, usually by means of a vector. The gypsy moth and zebra mussel have spread this way the gypsy moth via egg masses on vehicles and the zebra mussel via boat ballast water. Biological stressors can use both diffusion and jump-dispersal strategies, which makes it difficult to predict dispersal rates. An additional complication is that biological stressors are influenced by their own survival and reproduction. 

The process of molecular diffusion may be viewed conceptionally as a sequence of jumps with statistically varying jump lengths and residence times. Information about the mean jump length /(P and the mean residence time t, which might be of particular interest for a deeper understanding of the elementary steps of catalysis, may be provided by spectroscopic methods, in particular by quasielastic neutron scattering (see next Section) and nuclear magnetic resonance (NMR). 

Combining the PFG self diffusion measurement with a measurement of the correlation time provides a means of determining directly the mean jump distance. 

Fig. 15. Effects of small-amplitude reorientation on 2H NMR stimulated-echo experiments, as calculated by means of RW simulations. The C-2H bonds perform rotational random jumps on the surface of a cone with a full opening angle % = 6°, which are governed by a broad logarithmic Gaussian distribution of correlation times G(lgr) (a = 2.3). (a) Correlation functions m tp — 30 is) for the indicated mean logarithmic time constants lgr 1. The calculated data are damped by an exponential decay, exp[—(tm/rso)] with rSD = 1 s, so as to mimic effects due to spin diffusion. The dotted lines are fits with Fcos(tm tp) = (1—C) expHtm/t/l + Qexp[—Om/rso)]- (b) Amplitude of the decays, 1-C,p, for various t resulting from these fits. The dotted line is the value of the integral in Eq. (12) as a function of rm. (Adapted from Ref. 76).

The only characteristic of the carrier motion embodied in this equation is the mean square displacement per unit time, which gives D with a simple numerical factor dependent on the number of degrees of freedom summarized in x. All other detail of the motion is lost, and solutions of the diffusion equation represent the true evolution of p(x) only for times long compared with the jump time. 

Self-Diffusion Coefficients, Mean Residence Times beween Jumps, and Root Mean Square Jump Lengths for Ethane in NaX"... 

Simulations of C NMR lineshapes have shown that experimental spectra that appear to result from a superposition of two different lines (cf. Fig. 15) can be explained by the above-mentioned molecular jump model. Analogous conclusions were drawn from macroscopic sorption kinetic data (82). From the experimental C NMR lineshapes, a mean residence time tj of 20 and 150 p-s for a concentration of six molecules per u.c. at 250 and 200 K, respectively, was derived. Provided that these jumps detected in C NMR spectroscopy are accompanied by a translational motion of the molecules, it is possible to derive self-diffusivities D from the mean residence times. Assuming the diffusion path of a migrating molecule as a sum of individual activated jumps, for isotropic systems the relation (P) = 6Dtj is valid, where (P) denotes the mean square jump length. Following experimental and theoretical studies on the preferential sorption sites of benzene molecules in the MFI framework (83-90), in our estimate the mean distance between adjacent sorption sites is assumed to be 1 nm. 

This elastic scattering term is known as the elastic incoherent structure factor. It decreases from unity at (2 = 0 to 0 at large Q. As the area of S" (Q, (o) in the 00 direction is unity, there is an additional quasi-elastic component that increases from 0 at (2 = 0 to unity at large Q. The form of the quasi-elastic component depends on the nature of the localised diffusion. In the simplest case, where the jumping is between two trapping sites, the quasi-elastic term is a Lorentzian with a (2-independent width which is just 1/t where x is the mean residence time on either site. Two specific models will be noted here (a) random jumping round a ring of sites, the Barnes model [38] and (b)... 

The diffusivity of DNA in this matrix, measured by both the jump-time distribution and the growth of the mean-squared displacement, indicates an exponential decay with respect to molecular weight, although the amount of data is very limited. If these data are fit by a power law, one obtains D -i i o.7 Although this is very close to inverse dependence on molecular weight predicted by the Rouse model, the dynamics are very different. It is clear from these experiments that direct visualization is necessary to determine the true mode of migration [34]. 

If a dielectric material is suddenly placed in an electric field, the permanent molecular dipoles in the dielectricum will attempt to orientate. The orientation occurs by a random process, that is, via diffusion or jumps. The applied electrical field, of course, influences the mean orientation more than the reorientation of the individual molecular dipoles. Since molecular dipolar reorientation is coupled with reorientation of molecules or molecular groups, the time required for macroscopic reorientation corresponds to that for the reorientation of the molecules or groups. 

Anomalous diffusion is often caused by memory effects and Levy-type statistics [185, 53], Specifically, superdiffusion is observed for random walks with heavytailed jump length distributions and subdiffusion for heavy-tailed waiting time distributions, see Sect. 3.4. The latter type of distribution can be caused by traps that have an infinite mean waiting time [185]. For reviews of anomalous diffusion see, e.g., [298,299, 229]. 

So far we have discussed random walks with a finite mean waiting time and a finite variance of the jump length. These models lead to the classical parabolic scaling X x/e, t t/e. The governing macroscopic equation for the density p becomes the standard diffusion equation. Let us now consider two cases for which the scaling is anomalous and the mean-field equations for p are fractional diffusion equations. 

We conclude that as long as the mean waiting time and the variance of the jumps are finite, parabolic scaling leads to the Brownian motion in the limit e 0. The macroscopic equation for the density of particles is a scale-invariant diffusion equation. Infinite variance of jumps in the domain of attraction of a stable law leads to Ldvy processes, Levy flights. In the limit e 0, the particle position X (t) becomes self-similar with exponent 1/a. Recall that the random process X(t) is self-similar, if there exists a scaling exponent H such that X t) and e X(t/e) have the same distributions for any scaling parameter e. In this case we write... 

In summary, the critical reaction rate r depends on the dispersal kernel and on the mean waiting time only, i.e., r does not depend on the shape of the waiting time PDF. If the reaction rate is larger than the jump frequency, the invasion front never fails, independent of the shape of the waiting time PDF, even if the dispersal kernel is maximally biased. Finally, the diffusion approximation always underestimates r. ... 

Figure 8 shows that sorption and NMR diffiision coefficients decrease widi increasing sorbate concentration. Assuming diffusion proceeds via activated jumps there are two mechanisms to explain this rinding (i) a reduction of the mean molecular jump distance and/or (ii) an increase in the mean residence time between two succeeding molecular jumps as the concentration increases. 

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