Diffusion with Correlated Jumps

The calculated root-mean-square displacement for a general sequence of jumps has two terms in Eq. 7.31. The first term, NT(r2), corresponds to an ideal random walk (see Eq. 7.47) and the second term arises from possible correlation effects when successive jumps do not occur completely at random. 

For walks with correlations,5 a correlation factor, f, can be defined 

For a random walk, f = 1 because the double sum in Eq. 7.49 is zero and Eq. 7.50 reduces to the form of Eq. 7.47. In principle, f can have a wide range of values corresponding to physical processes relating to specific diffusion mechanisms. This is readily apparent in extreme cases of perfectly correlated one-dimensional diffusion on a lattice via nearest-neighbor jumps. When each jump is identical to its predecessor, Eq. 7.49 shows that the correlation factor f equals NT.6 Another extreme is the case of f = 0, which occurs if each individual jump is exactly opposite the previous jump. However, there are many real diffusion processes that are nearly ideal random walks and have values of f 1, which are described in more detail in Chapter 8. 

The relationship between the macroscopic isotropic diffusivity, D, and microscopic jump processes can be evaluated in three dimensions. The equivalence of Eqs. 7.31 and 7.35 means that 

Substitution of Eq. 7.49 into Eq. 7.51 yields the relation between the macroscopic isotropic diffusivity and microscopic parameters 

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Eq. 9 leads to spin diffusion with the flip-flop rate limiting the observed T. Below 10 s, the relaxation for the A spin is limited by the relaxation sink, which corresponds to the bonded methylene carbon. The effect of the siqall deviation In the coefficients from 1.0 and 0.0 near t s 10 s is non-exponential recovery, with the negative coefficient leading to a curve which is concave downward. For the M spin, the decay Jumps from theX curve for correlation times <10 " s to the Xg curve for correlation times >10 s. 

The intensity auto-correlation method has long been in use at high temperature to study dynamics in solutions [88]. Intensity fluctuations can arise from spectral jumps or changes, from rotational or translational diffusion with respect to the exciting beam, and from any process which can modulate the emitted intensity. Since correlation of single molecule fluorescence works at liquid helium temperature as well as at room temperature (see Section 2.1), it probably can cover the whole intermediate... 

Many NMR dynamics studies attempt to correlate the observed data with models of molecular motion. Some of the more successful are the Hall-Helfand [189,190], the Dejean-Laupretre-Monnerie (DEM) [189,191, 192], and the Williams-Watts [189,194,195] models. They predict the relaxation times and NOEs expected from specific types of motion such as unrestricted diffusion or discrete jumps. Very often, distinct parts of the molecule are be best modeled by different functions. For a poly(isobutyl methacrylate) solu-... 

As will be described below, diffusion is often measured by using tracer atoms and one then obtains values of the diffusion coefficient of the tracer atoms. Depending on the diffusion mechanism the tracer diffusion is in most cases not completely random, but is to some extent correlated with previous jumps. This will be further discussed later on. 

Specific models for internal motions can be used to interpret heteronuclear relaxation, such as restricted diffusion and site-jump models. However, model-free formal methods are preferable, at least for the initial analysis, since available experimental data generally are insufficient to completely characterize complex internal motions or to uniquely determine a specific motional model. The model-free approach of Lipari and Szabo for the analysis of relaxation data has been used for proteins and even for peptides. It attempts to reproduce relaxation rates by a weighted product of spectral density functions with different correlation times The weighting factors are identified as order parameters for the molecular rotational correlation time and optional further local correlation times r. The term (1-S ) would then be proportional to the amplitude of the corresponding internal motion. However, the Lipari-Szabo approach is based on the assumption that molecular and local correlation times are not coupled, i.e. they should be distinct enough (e.g. differing by at least a factor of 10 in time) to allow for this separation. However, in small molecules the rates of these different processes are of the same order of magnitude, and the requirements of the Lipari-Szabo approach may not be fulfilled. Molecular dynamics simulation provide a complementary approach for the interpretation of relaxation measurements. 

Above room temperature, the mobile 3 d electrons are well described by a random mixture of Fel" and FeB ions with the mobile electrons diffusing from iron to iron, some being thermally excited to FeA ions, but the motional enthalpy on the B sites is AH < kT. As the temperature is lowered through Tc, the Seebeck coefficient shows the influence of a change in mobile-electron spin degeneracy, and at room temperature the Seebeck coefficient is enhanced by correlated multielectron jumps that provide a mobile electron access to all its nearest neighbors. The electron-hopping time xi, = coi = 10" s... 

The motional dynamics of O J adsorbed on Ti supported surfaces has been analyzed over the temperature range 4.2-400 K in a recent paper by Shiotani et al. (66). Of the several types of 02, a species noted as 02 (III), and characterized by gxx = 2.0025, gyy = 2.0092, g12 = 2.0271 at 4.2 K, exhibited highly anisotropic motion. While gxx and gzz varied with increasing temperature and were accompanied by drastic line shape changes, gyy was found to remain constant. This observation indicates that the molecular motion of this 02 can be described by rotation about the y axis perpendicular to the internuclear axis of 02 and perpendicular to the surface with the notation given in Fig. 4. The EPR line shapes were simulated for different possible models and it was found that a weak jump rotational diffusion gave a best fit of the observed spectra below 57.4 K, whereas some of the models could fit the data above this temperature. The rotational correlation time was found to range from 10 5 sec (below 14.5 K) to 10 9 sec (263 K), while the... 

Atomic jumps in random walk diffusion of closely bound atomic clusters on the W (110) surface cannot be seen. A diatomic cluster always lines up in either one of the two (111) surface channel directions. But even in such cases, theoretical models of the atomic jumps can be proposed and can be compared with experimental results. For diffusion of diatomic clusters on the W (110) surface, a two-jump mechanism has been proposed by Bassett151 and by Cowan.152 Experimental studies are reported by Bassett and by Tsong Casanova.153 Bassett measured the probability of cluster orientation changes as a function of the mean square displacement, and compared the data with those derived with a Monte Carlo simulation based on the two-jump mechanism. The two results agree well only for very small displacements. Tsong Casanova, on the other hand, measured two-dimensional displacement distributions. They also introduced a correlation factor for these two atomic jumps, which resulted in an excellent agreement between their experimental and simulated results. We now discuss briefly this latter study. 

There are two possible directions for both the atoms in both the first and second atomic jumps. If the jumping direction is completely random and the two atoms have the same probability of performing a jump, then these atomic jumps are said to be uncorrelated. A correlation factor, /, has been introduced for the two atomic jumps, which is defined as the extra probability that the atom making the first jump will also make the second jump in the forward direction. The rest of the probability, (1 — /), is then shared equally for either of the two atoms jumping in either of the two directions. Two experimental displacement distributions measured at 299 K and 309 K fit best with a Monte Carlo simulation with / = 0.1 and /=0.36, respectively. The correlation factor increases with diffusion temperature as can be expected. It is interesting to note that when/= 1, only a and steps can occur. 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

Relaxation times for water filling the pores of an NaX specimen have been fitted to a model with the following assumptions (a) coupling, as above, of molecular diffusion and rotation (b) the median jump time r is governed by a free volume law (allows the curvature in the plots of jump rate, (3r) x vs. 10S/T in Figure 5), and (c) a broad distribution of correlation times (allows a better fit to the data, accounts for an apparent two-phase behavior in T2 (31, 39), and is reasonable in terms of the previous discussion of Pi(f) and r). 

Solutions for this type of kinetics can only be achieved numerically. Model calculations with constant kinetic parameters have been made [H. Wiedersich, et al. (1979)], however, the modeling of realistic transport (diffusion) coefficients which enter into the flux equations is most difficult since the jump rate vA vB. Also, the individual point defects have limited lifetimes which determine the magnitude of correlation factors (see Section 5.2.2). Explicit modeling for dilute or non-dilute alloys can be found in [A.R. Allnatt, A.B. Lidiard (1993)]. 

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