Diffusion mechanism random walk

The very simple and somewhat ad hoc form of Glarum s assumptions coupled with the one dimensional diffusion model used have led to several extensions of the original treatment these include relaxation by next nearest and other neighboring defects again by diffusion or random walk models and to the three dimensioanl case by such models. These go well beyond the motivation of the original idea to see whether a simple cooperative mechanism could account for marked short time deviations from simple models of rotational relaxation of a single dipole in a mean field approximation. For a review of much of this see B ttcher-Bordevijk (48). Particularly in the three dimensional case however there is an increasing question as to how far one can or should invoke isotropic diffusion processes to relax the component of a molecular electric moment parallel to that at an earlier time. 

There are two other methods in which computers can be used to give information about defects in solids, often setting out from atomistic simulations or quantum mechanical foundations. Statistical methods, which can be applied to the generation of random walks, of relevance to diffusion of defects in solids or over surfaces, are well suited to a small computer. Similarly, the generation of patterns, such as the aggregation of atoms by diffusion, or superlattice arrays of defects, or defects formed by radiation damage, can be depicted visually, which leads to a better understanding of atomic processes. 

The random-walk model of diffusion needs to be modified if it is to accurately represent the mechanism of the diffusion. One important change regards the number of point defects present. It has already been pointed out that vacancy diffusion in, for example, a metal crystal cannot occur without an existing population of vacancies. Because of this the random-walk jump probability must be modified to take vacancy numbers into account. In this case, the probability that a vacancy is available to a diffusing atom can be approximated by the number of vacant sites present in the crystal, d], expressed as a fraction, that is... 

The vacancy will follow a random-walk diffusion route, while the diffusion of the tracer by a vacancy diffusion mechanism will be constrained. When these processes are considered over many jumps, the mean square displacement of the tracer will be less than that of the vacancy, even though both have taken the same number of jumps. Therefore, it is expected that the observed diffusion coefficient of the tracer will be less than that of the vacancy. In these circumstances, the random-walk diffusion equations need to be modified for the tracer. This is done by ascribing a different probability to each of the various jumps that the tracer may make. The result is that the random-walk diffusion expression must be multiplied by a correlation factor, / which takes the diffusion mechanism into account. 

In the case of interstitial diffusion in which we have only a few diffusing interstitial atoms and many available empty interstitial sites, random-walk equations would be accurate, and a correlation factor of 1.0 would be expected. This will be so whether the interstitial is a native atom or a tracer atom. When tracer diffusion by a colinear intersticialcy mechanism is considered, this will not be true and the situation is analogous to that of vacancy diffusion. Consider a tracer atom in an interstitial position (Fig. 5.18a). An initial jump can be in any random direction in the structure. Suppose that the jump shown in Figure 5.18b occurs, leading to the situation in Figure 5.18c. The most likely next jump of the tracer, which must be back to an interstitial site, will be a return jump (Fig. 5.18c/). Once again the diffusion of the interstitial is different from that of a completely random walk, and once again a correlation factor, / is needed to compare the two situations. 

If both ionic conductivity and ionic diffusion occur by the same random-walk mechanism, a relationship between the self-diffusion coefficient, D, and the ionic... 

This equation shows that it is possible to determine the diffusion coefficient from the easier measurement of ionic conductivity. However, Da is derived by assuming that the conductivity mechanism utilizes a random-walk mechanism, which may not true. 

To obtain a more complete description, we need to find an analytic expression for the pre-exponential factor Dq of the diffusion coefficient by considering the microscopic mechanism of diffusion. The most straightforward approach, which neglects correlated motion between the ions, is given by the random-walk theory. In this model, an individual ion of charge q reacts to a uniform electric field along the x-axis supplied, in this case, by reversible nonblocking electrodes such that dCj(x)/dx = 0. Since two... 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /isotropic medium. Hence for an indirect interstitial mechanism, the corresponding mobility is expressed by... 

Since the random-walk approach is successful in molecular diffusion (K5) and Brownian motion studies (C14), it would seem that it might also be useful for the dispersion process. This has been considered by Baron (B2), Ranz (Rl), Reran (B5), Scheidegger (S6), Latinen (L4) and more recently by de Josselin de Jong (D14) and Salfman (SI, S2, S3). The latter two did not strictly use random-walk since a completely random process was not assumed. Methods based on statistical mechanics have been proposed by Evans et al. (E7), Prager (P8), and Scheidegger (S7). 

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. 

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 driving forces necessary to induce macroscopic fluxes were introduced in Chapter 3 and their connection to microscopic random walks and activated processes was discussed in Chapter 7. However, for diffusion to occur, it is necessary that kinetic mechanisms be available to permit atomic transitions between adjacent locations. These mechanisms are material-dependent. In this chapter, diffusion mechanisms in metallic and ionic crystals are addressed. In crystals that are free of line and planar defects, diffusion mechanisms often involve a point defect, which may be charged in the case of ionic crystals and will interact with electric fields. Additional diffusion mechanisms that occur in crystals with dislocations, free surfaces, and grain boundaries are treated in Chapter 9. 

Equation 8.19 contains the correlation factor, f, which in this case is not unity since the self-diffusion of tracer atoms by the vacancy mechanism involves correlation. Correlation is present because the jumping sequence of each tracer atom produced by atom-vacancy exchanges is not a random walk. This may be seen by... 

For simulation, a 3-D random walk algorithm was developed to study diffusion-controlled mixing phenomena [160]. Several assumptions were made, i.e. only EOF carries out fluid transport, only neutral and point-like analytes are present and the transport in each dimension is fully independent. An elastic collision mechanism was applied for molecule-wall collisions. The analyte was introduced as a stream of 200 molecules ms-1. 

The simplest mechanisms leading to the dispersion (spreading) of a zone s molecules can be described by the classical random-walk model [9], as noted in Section 5.3. However this model does not fully account for the complexities of migration. It gives, instead, a simple approximation which inherits the most essential and important properties (foremost of all the randomness) of the real migration process. The random-walk model has been used in a similar first-approximation role in many fields (chemical kinetics, diffusion, polymer chain configuration, etc.) and is thus important in its own right. 

In the frequency domain, Jonscher s power-law wings, when evaluated by ac-conductivity measurements, sometimes reveal a dual transport mechanism with different characteristic times. In particular, they treat anomalous diffusion as a random walk in fractal geometry [31] or as a thermally activated hopping transport mechanism [37]. 

It is possible to obtain expressions for the diffusivity for a particular case, if one knows the microscopic mechanism of diffusion. However, it will be instructive first to derive an expression for D using a probabihstic approach for which no detailed mechanism is assumed - the theory of random walk. A species (atom or ion) starting at its original position, makes n jumps and ends up at a final position that is related to the original position by a vector, designated as / ... 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

As far as the physical mechanism underlying the Cole-Cole equation is concerned, we first remark that Eq. (9) arises from the diffusion limit of a continuous-time random walk (CTRW) [17] (see Section II.A). In this context one should recall that the Einstein theory of the Brownian motion relies on the diffusion limit of a discrete time random walk. Here the random walker makes a jump of a fixed mean-square length in a fixed time, so that the only random... 

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