Vacancy diffusivity

Kitamura N, Lagaiiy M G and Webb M B 1993 Reai-time observations of vacancy diffusion on Si(100)-(2 1) by scanning tunneiing microscopy Phys. Rev. Lett. 71 2082... 

STM has not as yet proved to be easily applicable to the area of ultrafast surface phenomena. Nevertheless, some success has been achieved in the direct observation of dynamic processes with a larger timescale. Kitamura et al [23], using a high-temperature STM to scan single lines repeatedly and to display the results as a time-ver.sn.s-position pseudoimage, were able to follow the difflision of atomic-scale vacancies on a heated Si(OOl) surface in real time. They were able to show that vacancy diffusion proceeds exclusively in one dimension, along the dimer row. 

An account of the mechanism for creep in solids placed under a compressive hydrostatic suess which involves atom-vacancy diffusion only is considered in Nabano and Hemirg s (1950) volume diffusion model. The counter-movement of atoms and vacancies tends to relieve the effects of applied pressure, causing extension normal to the applied sU ess, and sluinkage in the direction of the applied sU ess, as might be anticipated from Le Chatelier s principle. The opposite movement occurs in the case of a tensile sU ess. The analysis yields the relationship... 

The second mechanism is that of vacancy diffusion. When zinc diffuses in brass, for example, the zinc atom (comparable in size to the copper atom) cannot fit into the interstices - the zinc atom has to wait until a vacancy, or missing atom, appears next to it before it can move. This is the mechanism by which most diffusion in crystals takes place (Figs. 18.7 and 10.4). 

Several authors " have suggested that in some systems voids, far from acting as diffusion barriers, may actually assist transport by permitting a dissociation-recombination mechanism. The presence of elements which could give rise to carrier molecules, e.g. carbon or hydrogen , and thus to the behaviour illustrated in Fig. 1.87, would particularly favour this mechanism. The oxidant side of the pore functions as a sink for vacancies diffusing from the oxide/gas interface by a reaction which yields gas of sufficiently high chemical potential to oxidise the metal side of the pore. The vacancies created by this reaction then travel to the metal/oxide interface where they are accommodated by plastic flow, or they may form additional voids by the mechanisms already discussed. The reaction sequence at the various interfaces (Fig. 1.87b) for the oxidation of iron (prior to the formation of Fe Oj) would be... 

Note that the cheurged vacancy diffuses as one of the reacting species to form the defect compound. This situation is quite common in the solid state chemistry of compounds conteiining multivalent cations. The trivalent Ni3+ also gives rise to a new compound, N1A104. Yet the same... 

Figure 8.7 Frames (23 by 35 A) of an STM movie taken at 65 K at close to a complete monolayer of hydrogen adatoms at Pd(l 11) showing vacancy diffusion. The images (b) and (c) show the aggregation of two nearest neighbour vacancies, which has the appearance of a three lobed object due to the rapid diffusion of one H atom next to the vacancy dimer. (Reproduced from Ref. 24).

When the random-walk model is expanded to take into account the real structures of solids, it becomes apparent that diffusion in crystals is dependent upon point defect populations. To give a simple example, imagine a crystal such as that of a metal in which all of the atom sites are occupied. Inherently, diffusion from one normally occupied site to another would be impossible in such a crystal and a random walk cannot occur at all. However, diffusion can occur if a population of defects such as vacancies exists. In this case, atoms can jump from a normal site into a neighboring vacancy and so gradually move through the crystal. Movement of a diffusing atom into a vacant site corresponds to movement of the vacancy in the other direction (Fig. 5.7). In practice, it is often very convenient, in problems where vacancy diffusion occurs, to ignore atom movement and to focus attention upon the diffusion of the vacancies as if they were real particles. This process is therefore frequently referred to as vacancy diffusion... 

Figure 5.7 Vacancy diffusion in the (001) plane of a cubic crystal. The unit cell is shaded. An atom can jump to a neighboring vacancy but not to an already occupied position. Diffusion parallel to one of the axial directions will be by a zigzag route.

The path that the diffusing atom takes will depend upon the structure of the crystal. For example, the 100 planes of the face-centered cubic structure of elements such as copper are identical to that drawn in Figure 5.7. Direct diffusion of a tracer atom along the cubic axes by vacancy diffusion will require that the moving atom must squeeze between two other atoms. It is more likely that the actual path will be a dog-leg, in <110> directions, shown as a dashed line on Figure 5.7. 

In the case of interstitials—self-interstitials, impurities, or dopants—two diffusion mechanisms can be envisaged. In the simplest case, an interstitial can jump to a neighboring interstitial position (Fig. 5.8a). This is called interstitial diffusion and is sometimes referred to as direct diffusion to distinguish it from vacancy diffusion (indirect diffusion). 

Substitutional impurities can move by way of a number of mechanisms. The most usual is the vacancy mechanism described above. Diffusion studies on semiconductors have suggested that a number of additional mechanisms might hold. As well as vacancy diffusion, an impurity can swap places with a neighboring normal atom, exchange diffusion, while in ring diffusion cooperation between several atoms is... 

When Schottky defects are present in a crystal, vacancies occur on both the cation and anion sublattices, allowing both cation and anion vacancy diffusion to occur (Fig. 5.12a). In the case of Frenkel defects interstitial, interstitialcy, and vacancy diffusion can take place in the same crystal with respect to the atoms forming the Frenkel defect population (Fig. 5.12b). 

Figure 5.13 Diffusion of a cation-anion divacancy within the (100) plane of a sodium chloride structure crystal (a-c) shows diffusion by way of individual cation and anion vacancy diffusion.

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... 

Figure 5.17 Correlated motion during vacancy diffusion (a) vacancy can jump to any surrounding position and its motion follows a random walk (b, c) the motion of a tracer atom is correlated, as a jump into a vacancy (b) is most likely to be followed by a jump back again (c).

However, the diffusion of a tracer atom by the mechanism of vacancy diffusion is different. A tracer can only move if it is next to a vacancy, and in this case, the tracer can only jump to the vacancy (Fig. 5.17b). The possibility of any other jump is excluded. Similarly, when the tracer has made the jump, then it is equally clear that the most likely jump for the tracer is back to the vacancy (Fig. 5.17c). The tracer can only jump to a new position after the vacancy has diffused to an alternative neighboring position. 

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. 

Suppose that vacancy diffusion is the principal mechanism involved in atom transport. An expression for the fraction of vacancies in a pure crystal is [Eq. (2.6)]... 

The same analysis can be applied to more complex situations. Suppose that cation vacancy diffusion is the predominant migration mechanism, in a sodium chloride structure crystal, of formula MX, which contains Schottky defects as the major type of intrinsic defects. The relevant defect concentration [ii] is [Eq. (2.11)]... 

Derive the equation for interstitial diffusion in a pure material equivalent to that for vacancy diffusion ... 

An example of a layer structure mixed conductor is provided by the cathode material L CoC used in lithium batteries. In this solid the ionic conductivity component is due to the migration of Li+ ions between sheets of electronically conducting C0O2. The production of a successful mixed conductor by doping can be illustrated by the oxide Cei-jPxx02- Reduction of this solid produces oxygen vacancies and Pr3+ ions. The electronic conductivity mechanism in these oxides is believed to be by way of electron hopping between Pr4+ and Pr3+, and the ionic conductivity is essentially vacancy diffusion of O2- ions. 

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.)...

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