Random walk, Ion

In a time of t s, a random-walking ion covers a mean square distance of or a mean distance of v/cxS. Thus, by choosing the plane Z, to be at a distance 1 from the transit plane, one has ensured that all the ions in the left compartment will cross the transit plane in a time tprovided they are moving in a left-to-right direction. 

The diffusion coefficient D has appeared in both the macroscopic (Section 4.2.2) and the atomistic (Section 4.2.6) views of diffusion. How does the diffusion coefficient depend on the structure of the medium and the interatomic forces that operate To answer this question, one should have a deeper understanding of this coefficient than that provided hy the empirical first law of Tick, in which D appeared simply as the proportionality constant relating the flux / and the concentration gradient dc/dx. Even the random-walk intapretation of the diffusion coefficient as embodied in the Einstein-Smoluchowski equation (4.27) is not fundamental enough because it is based on the mean square distance traversed by the ion after N stqis taken in a time t and does not probe into the laws governing each stq) taken by the random-walking ion. 

Consider one of these random-walking ions. It can be proved (see Appendix 4.1) that the mean square distance traveled by an ion depends on the number N of jumps the ion takes and the mean jump distance I in the following manner (Fig. 4.36) ... 

It has been shown that when random-walking ions are subjected to a directed force F, they acquire a nonrandom, directed componen t of velocity—the drift velocity v. This drift velocity is in the direction of the force F and is proportional to it... 

FIGURE 8.3 Geometric and energetic relationship for electron thermalization by random walk in liquid hexane in the presence of the geminate positive ion. Here = fid). Reproduced from Mozumder and Magee (1967), with the permission of Am. Inst. Phys. . 

Random Walk of an Ion Between Adjacent Active Centres... 

In this relatively simple random walk model an ion (e.g., a cation) can move freely between two adjacent active centres on an electrode (e.g., cathode) with an equal probability A. The centres are separated by L characteristic length units. When the ion arrives at one of the centres, it will react (e.g., undergoes a cathodic reaction) and the random walk is terminated. The centres are, therefore absorbing states. For the sake of illustration, L = 4 is postulated, i.e., Si and s5 are the absorbing states, if 1 and 5 denote the positions of the active centres on the surface, and s2, s3, and s4 are intermediate states, or ion positions, LIA characteristic units apart. The transitional probabilities (n) = Pr[i-, —>, Sj in n steps] must add up to unity, but their individual values can be any number on the [0, 1] domain. 

In the simplest case, the random walk is equi-probable between two adjacent positions, but no jumpover is allowed, i.e., P3i = Pai= =0 and the ion cannot be stagnant, i.e., p22 = Pa = P44 = 0 (however, pu = Pa = 1). Table 1 indicates that the ion eventually reacts at one of the active centres, the probability of reaction at a centre depending on the initial position of the ion. If, e.g., the initial position was, s4, the probability of the ion arriving eventually at the active centre s is %, and at active centre s5, 3/i The final state is reached essentially after about 25 steps (stages). 

Elements of the Transition Probability Matrix of a Unidimensional Unequiprobable Random Walk Model for an Ion Moving on an Electrode Surface 5i... 

This notion of occasional ion hops, apparently at random, forms the basis of random walk theory which is widely used to provide a semi-quantitative analysis or description of ionic conductivity (Goodenough, 1983 see Chapter 3 for a more detailed treatment of conduction). There is very little evidence in most solid electrolytes that the ions are instead able to move around without thermal activation in a true liquid-like motion. Nor is there much evidence of a free-ion state in which a particular ion can be activated to a state in which it is completely free to move, i.e. there appears to be no ionic equivalent of free or nearly free electron motion. 

The prefactor A or At contains many terms, including the number of mobile ions. Of the two equations, Eqn (2.3) is derived from random walk theory and has some theoretical justification Eqn (2.2) is not based on any theory but is simpler to use since data are plotted as log Arrhenius equation are widely used within errors the value of AH that is obtained is approximately the same using either equation in many cases. 

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... 

In the random-walk model, the individual ions are assumed to move independently of one another. However, long-range electrostatic interactions between the mobile ions make such an assumption unrealistic unless n is quite small. Although corrections to account for correlated motions of the mobile ions at higher values of n may be expected to alter only the factor y of the pre-exponential factor Aj., there are at least two situations where correlated ionic motions must be considered explicitly. The first occurs in stoichiometric compounds having an = 1. but a low AH for a cluster rotation the second occurs for the situation illustrated in Fig. 3.6(c). 

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... 

Ramsey theory, 22 201-204 Random-fragmentation model, Szilard-Chalmers reaction and, 1 270 Random-walk process, correlated pair recombination, post-recoil annealing effects and, 1 288-290 Rare-earth carbides, neutron diffraction studies on, 8 234-236 Rare-earth ions energy transfer, 35 383 hydration shell, 34 212-213 Rare gases... 

Fig. 3.2 The first picture at the upper left-hand comer is a 78 K He field ion micrograph of a W (112) surface with a W adatom on it. From top to bottom and then from left to right the rest of the images are 290 K He field ion micrographs of the same surface where diffusion of the adatom can be clearly seen. When the adatom is near the center of the plane, it performs a random walk. However, when it is slightly off the center it is driven toward the plane edge by a field gradient induced driving force, as will be discussed in Chapter 4. These are real time photos each one is separated by about 5 s.

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... 

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