Vacancy exchange mechanisms

There is a qualitative distinction between these two types of mass transfer. In the case of vapour phase transport, matter is subtracted from the exposed faces of the particles via dre gas phase at a rate determined by the vapour pressure of the solid, and deposited in the necks. In solid state sintering atoms are removed from the surface and the interior of the particles via the various diffusion vacancy-exchange mechanisms, and the centre-to-cenU e distance of two particles undergoing sintering decreases with time. 

The vacancy exchange mechanism is described in Section 8.1.2. 

Considering Eq. 2.21 in a case in which diffusion occurs in a crystal by the vacancy exchange mechanism, there are four components, Cj., c i, c2, and cy. Because the crystal remains fixed during the diffusion, the C-frame is again used for measuring the flux. The system is chemically homogeneous, so... 

Solution. The diffusion of the atoms will be correlated because of the vacancy exchange mechanism and, therefore, using Eq. 7.52,... 

A brief summary of the diffasion formalism of Agren and Andersson and Agren as applied to a disordered substitutional solid solution is useful. Diffasion is assumed to occur by a vacancy exchange mechanism, in which the equilibrium vacancy concentration is maintained. The partial molar volumes of the substitutional species are assumed to be equal. For a given phase, the flux of species i in the z-direction in volume-flxed frame of reference is given by... 

Because of die rigidity and directionality of die covalent bonds die energies of self-diffusion have been found to be higher diaii diose of metals. In die case of silicon, it appears drat a furdier complication is drat an intersti-tialcy mechanism predominates above 1000°C. Below diis teiiiperamre, bodi elements appear to self-diffuse by atom-vacancy exchange as for die metals. 

The problem of taking into consideration the actual vacancy-mediated atomic exchange mechanism (rather than the direct exchange model used in most theoretical treatments) recently received some attention. In particular, possible presence of vacancy segregation at various structural inhomogeneities was discussed. However, the estimates of these effects by various authors disagree notably with each other , and there seems to be no general treatment of this problem available. 

In Fig. 12 we present some results of MFKEbbased simulation of spinodal decomposition with the vacancy-mediated exchange mechanism. We use the same 2D model on a square lattice with the nearest-neighbor interaction and Fp = 0 as in Refs., ... 

Basically, whenever isotopic exchanges occur between different phases (i.e., heterogeneous equilibria), isotopic fractionations are more appropriately described in terms of differential reaction rates. Simple diffusion laws are nevertheless appropriate in discussions of compositional gradients within a single phase— induced, for instance, by vacancy migration mechanisms, such as those treated in section 4.10—or whenever the isotopic exchange process does not affect the extrinsic stability of the phase. 

Figure 4.39 Illustration of (a) vacancy, (b) interstitial, and (c) interchange (exchange) mechanisms in atomic diffusion. From K. M. Ralls, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission John Wiley Sons, Inc.

In many cases, changes in one extensive quantity are coupled to changes in others. This occurs in the important case of substitutional components in a crystal devoid of sources or sinks for atoms, such as dislocations, as explained in Section 11.1. Here the components are constrained to lie on a fixed network of sites (i.e., the crystal structure), where each site is always occupied by one of the components of the system. Whenever one component leaves a site, it must be replaced. This is called a network constraint [1]. For example, in the case of substitutional diffusion by a vacancy-atom exchange mechanism (discussed in Section 8.1.2), the vacancies are one of the components of the system every time a vacancy leaves a site, it is replaced by an atom. As a result of this replacement constraint, the fluxes of components are not independent of one another. 

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... 

Silver ion vacancies move by a simple replacement mechanism in which a lattice silver ion at a nearest neighbor or 110 position (here the braces are used to represent a set of equivalent lattice positions) moves first into an interstitial position and then into the vacancy. Replacement by direct motion along a [110] direction is thought to be a higher energy process [18]. Monovalent impurities like Au+ and Cu+ diffuse like silver ions [18]. Na + and K + diffuse by a vacancy substitution mechanism in which the ion moves into a nearest neighbor vacancy position. Divalent cation impurities diffuse by exchange with an associated vacancy. Trivalent cations require a second vacancy for electrical neutrality. Their diffusion involves the concerted motion of this neutral complex [18]. Finally anions and anion impurities... 

According to the mechanism, the active center is formed by the interaction of aluminum alkyl with an octahedral vacancy around Ti. For or-TiCls catalyst the formation of active center can be represented as shown in Fig. 9.3. To elaborate, the five-coordinated Ti on the surface has a vacant J-orbital, represented by -Q, which facilitates chemisorption of the aluminum alkyl followed by alkylation of the Ti " ion by an exchange mechanism to form the active center TiRCU-Q. The vacant site at the active center can accommodate the incoming monomer unit, which forms a r-complex with the titanium at the vacant inserted into the Ti-alkyl bond. The sequence of steps is shown in Fig. 9.4 using propylene as the monomer. 

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