Vacancy mechanism

We do not discuss all these mechanisms here. However, the most important mechanisms, the vacancy mechanism and the interstitial mechanism, are described. 

As is well known, in thermal equilibrium, every crystal, at a temperature above absolute zero encloses a certain number of vacant lattice sites, where the probability, P, of finding a vacancy in a solid at equilibrium at an absolute temperature, T, is given by 

Kv = e R is a pre-exponential factor, in which, SV represents the entropy of formation of a vacancy 

For a vacancy mechanism, in a self-diffusion process, the jump frequency, T, of an atom to a given adjacent site is given by [4] 

If there are z adjacent sites, then the total jump frequency is given by 

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1 Vacancies will always exist in equilibrium in a crystal because their enthalpy of formation can always be compensated by a configurational entropy increase at finite temperatures (see the derivation of Eq. 3.65). Therefore, vacancies function as a component that occupies substitutional sites. 

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In pure and stoichiometric compounds, intrinsic defects are formed for energetic reasons. Intrinsic ionic conduction, or creation of thermal vacancies by Frenkel, ie, vacancy plus interstitial lattice defects, or by Schottky, cation and anion vacancies, mechanisms can be expressed in terms of an equilibrium constant and, therefore, as a free energy for the formation of defects, If the ion is to jump into a normally occupied lattice site, a term for... 

It should be realized that unlike the study of equilibrium thermodynamics for which a model is often mapped onto Ising system, elementary mechanism of atomic motion plays a deterministic role in the kinetic study. In an actual alloy system, diffusion of an atomic species is mainly driven by vacancy mechanism. The incorporation of the vacancy mechanism into PPM formalism, however, is not readily achieved, since the abundant freedom of microscopic path of atomic movement demands intractable number of variational parameters. The present study is, therefore, limited to a simple spin kinetics, known as Glauber dynamics [14] for which flipping events at fixed lattice points drive the phase transition. Hence, the present study for a spin system is regarded as a precursor to an alloy kinetics. The limitation of the model is critically examined and pointed out in the subsequent sections. 

There are a number of differences between interstitial and substitutional solid solutions, one of the most important of which is the mechanism by which diffusion occurs. In substitutional solid solutions diffusion occurs by the vacancy mechanism already discussed. Since the vacancy concentration and the frequency of vacancy jumps are very low at ambient temperatures, diffusion in substitutional solid solutions is usually negligible at room temperature and only becomes appreciable at temperatures above about 0.5T where is the melting point of the solvent metal (K). In interstitial solid solutions, however, diffusion of the solute atoms occurs by jumps between adjacent interstitial positions. This is a much lower energy process which does not involve vacancies and it therefore occurs at much lower temperatures. Thus hydrogen is mobile in steel at room temperature, while carbon diffuses quite rapidly in steel at temperatures above about 370 K. 

FIGURE 25.3 Schematic representation of ionic motion by (a) a vacancy mechanism and (b) an interstitial mechanism. (From Smart and Moore, 1996, Fig. 5.4, with permission from Routledge/Taylor Francis Group.)... 

Point defects in solids make it possible for ions to move through the structure. Ionic conductivity represents ion transport under the influence of an external electric field. The movement of ions through a lattice can be explained by two possible mechanisms. Figure 25.3 shows their schematic representation. The first, called the vacancy mechanism, represents an ion that hops or jumps from its normal position on the lattice to a neighboring equivalent but vacant site or the movement of a vacancy in the opposite direction. The second one is an interstitial mechanism where an interstitial ion jumps or hops to an adjacent equivalent site. These simple pictures of movement in an ionic lattice, known as the hopping model, ignore more complicated cooperative motions. 

When this system was studied over time, it was found that the marker wires move toward each other. This shows that the most extensive diffusion is zinc from the brass (an alloy of zinc and copper) outward into the copper. If the mechanism of diffusion involved an interchange of copper and zinc, the wires would not move. The diffusion in this case takes place by the vacancy mechanism described later, as zinc moves from the brass into the surrounding copper. As the zinc moves outward, vacancies are produced in the... 

One type of diffusion mechanism is known as the interstitial mechanism because it involves movement of a lattice member from one interstitial position to another. When diffusion involves the motion of a particle from a regular lattice site into a vacancy, the vacancy then is located where the site was vacated by the moving species. Therefore, the vacancy moves in the opposite direction to that of the moving lattice member. This type of diffusion is referred to as the vacancy mechanism. In some instances, it is possible for a lattice member to vacate a lattice site and for that site to be filled simultaneously by another unit. In effect, there is a "rotation" of two lattice members, so this mechanism is referred to as the rotation mechanism of 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... 

Diffusion in the extrinsic region can readily be modified by doping, although knowledge of the mechanism by which the diffusion takes place is important if this is to be immediately successful. For example, sodium chloride structure materials that conduct by a vacancy mechanism can have the cation conductivity enhanced by doping with divalent cations, as these generate compensating cation vacancies. The inclusion of cadmium chloride into sodium chloride can be written ... 

The question raised by Anderson (1970,1971) and Anderson et al (1973) as to whether anion point defects are eliminated completely by the creation of extended CS plane defects, is a very important one. This is because anion point defects can be hardly eliminated totally because apart from statistical thermodynamics considerations they must be involved in diffusion process. Oxygen isotope exchange experiments indeed suggest that oxygen diffuses readily by vacancy mechanism. In many oxides it is difficult to compare small anion deficiency with the extent of extended defects and in doped complex oxides there is a very real discrepancy between the area of CS plane present which defines the number of oxygen sites eliminated and the oxygen deficit in the sample (Anderson 1970, Anderson et al 1973). We attempt to address these issues and elucidate the role of anion point defects in oxides in oxidation catalysis (chapter 3). 

Atoms in the free surface of solids (with no neighbors) have a higher free energy than those in the interior and surface energy can be estimated from the number of surface bonds (Cottrell 1971). We have discussed non-stoichiometric ceramic oxides like titania, FeO and UO2 earlier where matter is transported by the vacancy mechanism. Segregation of impurities at surfaces or interfaces is also important, with equilibrium and non-equilibrium conditions deciding the type of defect complexes that can occur. Simple oxides like MgO can have simple anion or cation vacancies when surface and Mg + are removed from the surface,... 

FIGURE 5.4 Schematic representation of ionic motion by (a) a vacancy mechanism and (b) an interstitial mechanism. 

In a molecular dynamic simulation147 of bulk atomic diffusion by a vacancy mechanism, two atoms may occasionally jump together as a pair. The temperature of the simulation is close to the melting point of the crystal. In FTM studies of single atom and atomic cluster diffusion, the temperature is only about one tenth the melting point of the substrate. All cluster diffusion, except that in the (1 x 1) to (1 x 2) surface reconstruction of Pt and Ir (110) surfaces already discussed in Section 4.1.2(b), are consistent with mechanisms based on jumps of individual atoms.148,149 In fact, jumps of individual atoms in the coupled motion of adatoms in the adjacent channel of the W (112) surface can be directly seen in the FTM if the temperature of the tip is raised to near 270 K.150... 

Vacancy mechanism. If there is a vacancy in a lattice, il may be possible for an adjacent ion of the type that is missing, normally a cation, to migrate into it, the difficulty of migration being rdated to the sizes of the migrating ion and the ions that surround it and tend to impede il. 

Vacancy mechanism, 266 Valence bond (VB) theory, 139-153. 391-394.474 Valence shell electron pair repulsion (VSEPR) model. 203-206. 217-218... 

As an illustration, consider the isothermal, isobaric diffusional mixing of two elemental crystals, A and B, by a vacancy mechanism. Initially, A and B possess different vacancy concentrations Cy(A) and Cy(B). During interdiffusion, these concentrations have to change locally towards the new equilibrium values Cy(A,B), which depend on the local (A, B) composition. Vacancy relaxation will be slow if the external surfaces of the crystal, which act as the only sinks and sources, are far away. This is true for large samples. Although linear transport theory may apply for all structure elements, the (local) vacancy equilibrium is not fully established during the interdiffusion process. Consequently, the (local) transport coefficients (DA,DB), which are proportional to the vacancy concentration, are no longer functions of state (Le., dependent on composition only) but explicitly dependent on the diffusion time and the space coordinate. Non-linear transport equations are the result. 

Several points are to be noted. Firstly, pores and changes of sample dimension have been observed at and near interdiffusion zones [R. Busch, V. Ruth (1991)]. Pore formation is witness to a certain point defect supersaturation and indicates that sinks and sources for point defects are not sufficiently effective to maintain local defect equilibrium. Secondly, it is not necessary to assume a vacancy mechanism for atomic motion in order to invoke a Kirkendall effect. Finally, external observers would still see a marker movement (markers connected by lattice planes) in spite of bA = bB (no Kirkendall effect) if Vm depends on composition. The consequences of a variable molar volume for the determination of diffusion coefficients in binary systems have been thoroughly discussed (F. Sauer, V. Freise (1962) C. Wagner (1969) H. Schmalzried (1981)]. 

Considering their possible applications in fuel cells, hydrogen sensors, electro-chromic displays, and other industrial devices, there has been an intensive search for proton conducting crystals. In principle, this type of conduction may be achieved in two ways a) by substituting protons for other positively charged mobile structure elements of a particular crystal and b) by growing crystals which contain a sufficient amount of protons as regular structure elements. Diffusional motion (e.g., by a vacancy mechanism) then leads to proton conduction. Both sorts of proton conductors are known [P. Colomban (1992)]. 

Figure 8.2 Vacancy mechanism for diffusion of substitutional atoms.

Migration of atoms that occupy substitutional sites may occur through a range of mechanisms involving either vacancy- or interstitial-type defects. In f.c.c., b.c.c., and hexagonal close-packed (h.c.p.) metals, self-diffusion occurs predominantly by the vacancy mechanism [4, 5]. However, in some cases self-diffusion by the... 

Self-Diffusion by the Vacancy Mechanism in the F.C.C. Structure. Each site on an f.c.c. lattice has 12 nearest-neighbors, and if vacancies occupy sites randomly and have a jump frequency IV,... 

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... 

An accurate determination of f can be obtained by considering all contributing vacancy trajectories to determine (cos ) by use of Eq. 8.29 [13]. For f.c.c., the accurate value of f is found to be 0.78 thus, correlations affect the diffusivity value by about 22% in Eq. 7.52.7 Correlations can have a considerably larger effect on the diffusivity for substitutional solute atoms by the vacancy mechanism. 

Diffusion of Solute Atoms by Vacancy Mechanism in Close-Packed Structure. Diffusion of substitutional solutes in dilute solution by the vacancy mechanism is more complex than self-diffusion because the vacancies may interact with the solute atoms and no longer be randomly distributed. If the vacancies are attracted to the solute atoms, any resulting association will strongly affect the solute-atom diffusivity. 

Self-Diffusion by the Interstitialcy Mechanism. If their formation energy is not too large, the equilibrium population of self-interstitials may be large enough to contribute to the self-diffusivity. In this case, the self-diffusivity is similar to that for self-diffusion via the vacancy mechanism (Eq. 8.19) with the vacancy formation and migration energies replaced by corresponding self-interstitial quantities. The... 

Intrinsic Crystal Self-Diffusion. A simple example of intrinsic self-diffusion in an ionic material is pure stoichiometric KC1, illustrated in Fig. 8.11a. As in many alkali halides, the predominant point defects are cation and anion vacancy complexes (Schottky defects), and therefore self-diffusion takes place by a vacancy mechanism. For stoichiometric KC1, the anion and cation vacancies are created in equal numbers because of the electroneutrality condition. These vacancies can be created... 

The cation self-diffusivity due to the vacancy mechanism varies as the one-sixth power of the oxygen pressure at constant temperature and the activation energy is... 

It has sometimes been claimed that the observation of a Kirkendall effect implies that the diffusion occurred by a vacancy mechanism. However, a Kirkendall effect can be produced just as well by the interstitialcy mechanism. Explain why this is so. 

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