Vacancy diffusion coefficient

For an ion to move through the lattice, there must be an empty equivalent vacancy or interstitial site available, and it must possess sufficient energy to overcome the potential barrier between the two sites. Ionic conductivity, or the transport of charge by mobile ions, is a diffusion and activated process. From Fick s Law, J = —D dn/dx), for diffusion of a species in a concentration gradient, the diffusion coefficient D is given by... 

If samples of two metals widr polished faces are placed in contact then it is clear that atomic transport must occur in both directions until finally an alloy can be formed which has a composition showing die relative numbers of gram-atoms in each section. It is vety unlikely that the diffusion coefficients, of A in B and of B in A, will be equal. Therefore there will be formation of an increasingly substantial vacancy concentration in the metal in which diffusion occurs more rapidly. In fact, if chemically inert marker wires were placed at the original interface, they would be found to move progressively in the direction of slowest diffusion widr a parabolic relationship between the displacement distance and time. 

The diffusion coefficient corresponding to the measured values of /ch (D = kn/4nRn, is the reaction diameter, supposed to be equal to 2 A) equals 2.7 x 10 cm s at 4.2K and 1.9K. The self-diffusion in H2 crystals at 11-14 K is thermally activated with = 0.4 kcal/mol [Weinhaus and Meyer 1972]. At T < 11 K self-diffusion in the H2 crystal involves tunneling of a molecule from the lattice node to the vacancy, formation of the latter requiring 0.22 kcal/mol [Silvera 1980], so that the Arrhenius behavior is preserved. Were the mechanism of diffusion of the H atom the same, the diffusion coefficient at 1.9 K would be ten orders smaller than that at 4.2 K, while the measured values coincide. The diffusion coefficient of the D atoms in the D2 crystal is also the same for 1.9 and 4.2 K. It is 4 orders of magnitude smaller (3 x 10 cm /s) than the diffusion coefficient for H in H2 [Lee et al. 1987]. 

We denote by x the distance from the metal surface, and by n x) and rip x) the concentrations of cation vancancies and positive holes in the oxide. Let and Vp be their mobilities, and and Dp their diffusion coefficients. Let F x) be the electrostatic field in the oxide. J, the flux of cation vacancies (number crossing unit area per second), will be expressed by... 

When a crystal is heated, lattice members become more mobile. As a result, there can be removal of vacancies as they become filled by diffusion. Attractions to nearest neighbors are reestablished that result in a slight increase in density and the liberation of energy. There will be a disappearance of dislocated atoms or perhaps a redistribution of dislocations. These events are known to involve several types of mechanisms. However, the diffusion coefficient, D, is expressed as... 

At high temperatures there is experimental evidence that the Arrhenius plot for some metals is curved, indicating an increased rate of diffusion over that obtained by linear extrapolation of the lower temperature data. This effect is interpreted to indicate enhanced diffusion via divacancies, rather than single vacancy-atom exchange. The diffusion coefficient must now be represented by an Arrhenius equation in the form... 

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 both cases the diffusion coefficient of the ions using the vacancy population will be enhanced considerably. 

The result of the oxygen vacancy population is that the diffusion coefficient of oxygen ions is increased by many orders of magnitude compared to a normal oxide. 

An example of a material (Li3Sb) with a very large Wagner factor is shown in Fig. 8.3. The effective chemical diffusion coefficient is compared with the diffusivity as a function of non-stoichiometry. These data were determined by electrochemical techniques (see Section 8.5). An increase of the diffusion coefficient is observed at about the ideal stoichiometry which corresponds to a change in the mechanism from a predominantly vacancy to interstitial mechanism. The Wagner factor W is as large as 70 000 at the ideal stoichiometry. This gives an effective diffusion coefficient which is more typical of liquids than solids. It is a common... 

From the formation reaction of protonic defects in oxides (eq 23), it is evident that protonic defects coexist with oxide ion vacancies, where the ratio of their concentrations is dependent on temperature and water partial pressure. The formation of protonic defects actually requires the uptake of water from the environment and the transport of water within the oxide lattice. Of course, water does not diffuse as such, but rather, as a result of the ambipolar diffusion of protonic defects (OH and oxide ion vacancies (V ). Assuming ideal behavior of the involved defects (an activity coefficient of unity) the chemical (Tick s) diffusion coefficient of water is... 

Figure 16. Self-diffusion coefficient of oxide ion vacancies in different perovskite-type oxides,which equals the chemical water diffusion coefficient in the fully hydrated state (see text). (Figure reproduced with the kind permission from Elsevier.)...

Figure 28. Svensson s macrohomogeneous model for the i— 1/characteristics of a porous mixed-conducting electrode, (a) The reduction mechanism assuming that both surface and bulk diffusion are active and that direct exchange of oxygen vacancies between the mixed conductor and the electrolyte may occur, (b) Tafel plot of the predicted steady-state i— V characteristics as a function of the bulk oxygen vacancy diffusion coefficient. (Reprinted with permission from ref 186. Copyright 1998 Electrochemical Society, Inc.)...

Figure 29. Comparison of the oxygen vacancy diffusion coefficient (Dy) in LSC (x= 0.2) determined from permeation measurements vs that extracted from impedance measurements using the model in Figure 26. Data are from refs 190 and 28. (Adapted with permission from ref 28. Copyright 1998 Elsevier.)...

Let us now consider the crystal MO. If the diffusion takes place by migration of cationic vacancies, the number of atoms that undergo the process depends on the vacancy concentration [Vm] and the thermal state of single atoms M (the jump takes place only whenever atom M in the neighborhood of the vacancy has sufficient energy to perform it). The diffusion coefficient associated with the vacancy migration process is given by... 

The diffusion coefficient for a given ion in a crystal is determined, as we have seen, by the atomistic properties of the ion in the structural sites where the vacancy (or interstitial) participating in the migration process is created (see eq. 4.71). The units of diffusion (and/or self-diffusion ) are usually cm sec . Pick s first law relates the diffusion of a given ion A (Jf) to the concentration gradient along a given direction X ... 

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