Random Walk Diffusion

One of the simplest models for diffusion is that of the random movement of atoms. The model is generally called a random (or drunkard s) walk.4 A random walk produces a path that is governed completely by random jumps (Fig. S5.3). That is, each individual jump is unrelated to the step before and is governed solely by the probabilities of taking the alternative steps. The application of random walks to diffusion was first made by 

4Random walks are often called Markov random walks. A Markov chain is a sequence of random events described in terms of a probability that the event under scmtiny evolved from a defined predecessor. In effect there is no memory of any preceding step in a Markov chain. Hidden Markov processes involve some short-term memory of preceding steps. 

Einstein to describe Brownian motion.5 The model can be used to derive the diffusion equations and to relate the diffusion coefficient to atomic movements. 

For convenience, only one-dimensional random movement will be considered. In this case, an atom is constrained to jump from one stable site to the next in the x direction, the choice of +x or -x being selected in a random way.6 For example, imagine a diffusion experiment starting with a thin layer of N atoms on the surface of a crystal. 

6The statistics of this process is identical to those pertaining to the tossing of a coin. The mathematics was first worked out with respect to games of chance by de Moivre, in 1733. It is formally described by the binomial distribution. 

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The factor of 2 in Eq. (5.6) arises from the one-dimensional nature of the random walk and, hence, is a result of the geometry of the diffusion process. In the case of random-walk diffusion on a two-dimensional surface ... 

The statistics of the normal distribution can be applied to give more information about random-walk diffusion. The area under the normal distribution curve represents a probability. In the present case, the probability that any particular atom will be found in the region between the starting point of the diffusion and a distance of + J = + v/(2/V) on either side of it is approximately 68% (Fig. 5.6b). The probability that any particular atom has diffused further than this distance is given by the total area under the curve minus the shaded area, which is approximately 32%. The probability that the atoms have diffused further than 2 J, that is, 2V(2Dr t) is equal to about 5%. 

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. 

Figure 6.4 Distribution curve for ionic conductivity random-walk diffusion in the absence of a field is the dashed line. In the presence of the electric field, the distribution splits into two parts, one for cations and one for anions, each shifted with respect to the origin.

The correlation factor,/, is defined by the ratio of the tracer diffusion coefficient to the random-walk diffusion coefficient (Section 5.6) ... 

Surprisingly, in a random-walk diffusion experiment, although the concentration of atoms in the initial layer always remains greater than in the bulk, it does not mean that these atoms have not diffused. They have taken just as many steps as those furthest from the starting layer ]... 

RANDOM WALK DIFFUSION PERSISTENCE TIME TWIDDLING... 

The size of a surface available for field ion microscope study of surface diffusion is very small, usually much less than 100 A in diameter. The random walk diffusion is therefore restricted by the plane boundary. For a general discussion, however, we will start from the unrestricted random walk. First, we must be aware of the difference between the chemical diffusion coefficient and the tracer diffusion coefficient. The chemical diffusion coefficient, or more precisely the diffusion tensor, is defined by a generalized Fick s law as... 

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. 

The effective polarizability of surface atoms can be determined with different methods. In Section 2.2.4(a) a method was described on a measurement of the field evaporation rate as a function of field of kink site atoms and adsorbed atoms. The polarizability is derived from the coefficient of F2 term in the rate vs. field curve. From the rate measurements, polarizabilities of kink site W atoms and W adatoms on the W (110) surface are determined to be 4.6 0.6 and 6.8 1.0 A3, respectively. The dipole moment and polarizability of an adatom can also be measured from a field dependence of random walk diffusion under the influence of a chemical potential gradient, usually referred as a directional walk, produced by the applied electric field gradient, as reported by Tsong et a/.150,198,203 This study is a good example of random walk under the influence of a chemical potential gradient and will therefore be discussed in some detail. 

One can obtain values of both pQ and a from the intercept and the slope of a plot of (2A T/0.867/J7) sinh (/(p) V2(p2 0) against Fc, for brevity referred as a r-plot. All the parameters in the equation can be measured field gradient from desorption voltages of adatoms at different locations on the plane, (p2)0 from a random walk diffusion experiment, and directional walk experiment. For W adatoms on the W (110)... 

How does the ion move on the surface It cannot drift under an electric field because the field at an interface is normal to the electrode surface (Fig. 7.131) and what is under discussion here is motion parallel to the surface plane. The movements are by a random-walk diffusion process in two dimensions, surface diffusion. 

Consider the diffusion of a randomly walking diffusant in the h.c.p. structure, which is composed of close-packed basal planes stacked in the sequence ABABA. The lattice constants are a and c. The probability of a first-nearest-neighbor jump within a basal plane (jump distance = a) is p, and the probability of a jump between basal planes (jump distance = /a2/3 + c2/4) is 1 — p. If axes X and X2 are located in a basal plane, derive the following expressions for the diffusivities Du and D33 ... 

Demonstrate this result for the diffusion of a randomly walking diffusant in an h.c.p. crystal using the information and results in Exercises 8.9 and 8.10. 

Figure 21 Schematic drawing of the random-walk diffusion model of Butler et al. [180]. It is shown that molecular intermediates that do not hit an added row may dissociate at a step, bind to an Ag-atom and insert in an added row.

The simplest model of translational motions in three dimensions is the continuous random walk diffusion (Pick s law) giving a Lorentzian-shaped scattering law,... 

If the Cg is low then a random walk diffusion of independent molecules can be expected, and would be given as... 

Excluded volume and solvent quality. Up till here, the volume taken up by the polymer itself, i.e., n times the volume of a monomer, has been neglected. In other words, such an ideal random chain has no volume, which would imply that two different segments can occupy the same place in the solvent at the same time. This is, of course, physically impossible, which is why the statistics of a real chain are different from those of a random walk (diffusion). Instead of this, a self-avoiding random walk should be considered, and the average conformation then is different, rm being proportional to n to the power 0.6, rather than 0.5. This means that the molecule is more expanded than an ideal chain. 

The mean square diffusion distance x2 for a random-walk diffusion process is x2 = nr2 = 6Dt, where n is the number of jumps and r is the jump distance. Assume that the cascade volume can be modeled by a gas, then n = vt/r where v is the mean speed of the ions in the cascade. Assume that the energy deposited in the cascade is 1 eV atom-1 and that the cascade lifetime is 10-11 s... 

The longitudinal diffusion term (B) This term accounts for the spreading of molecules in both directions from the band center along the length of the column as a result of random-walk diffusion. This occurs primarily in the mobile phase in GC, but significantly in both phases in LC (as the analytes... 

Now a random-walk diffusion starts as indicated by the zig-zag path. Simultaneously, intersystem crossing -> S) transitions may occur, induced by the coupling of the electron spins with nuclear spins (hyperfine coupling). The rate of this intersystem crossing depends on the nuclear spin state. Here it is assumed that it is faster for p nuclei than for a nuclei. 

The first of these classes involves cluster formation by the successive addition of single randomly walking (diffusion) particles onto a seed particle representing a nucleation center at a fixed point (33,35) (diffusion-limited aggregation, OLA) and the resultant structure has D = 1.7 (d = 2) and 2.5 (d = 3). 

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