Diffusion in crystals

The driving forces necessary to induce macroscopic fluxes were introduced in Chapter 3 and their connection to microscopic random walks and activated processes was discussed in Chapter 7. However, for diffusion to occur, it is necessary that kinetic mechanisms be available to permit atomic transitions between adjacent locations. These mechanisms are material-dependent. In this chapter, diffusion mechanisms in metallic and ionic crystals are addressed. In crystals that are free of line and planar defects, diffusion mechanisms often involve a point defect, which may be charged in the case of ionic crystals and will interact with electric fields. Additional diffusion mechanisms that occur in crystals with dislocations, free surfaces, and grain boundaries are treated in Chapter 9. 

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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). 

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 5.12 Diffusion in crystals of composition MX containing (a) Schottky and (b) Frenkel defects, schematic V, vacancy, i, interstitial, iy, interstitialcy.

Diffusion coefficients in amorphous solids such as oxide glasses and glasslike amorphous metals can be measured using any of the methods applicable to crystals. In this way it is possible to obtain the diffusion coefficients of, say, alkah and alkaline earth metals in silicate glasses or the diffusion of metal impurities in amorphous alloys. Unlike diffusion in crystals, diffusion coefficients in amorphous solids tend to alter over time, due to relaxation of the amorphous state at the temperature of the diffusion experiment. 

Diffusion in crystals means site-exchange of different SE s. If the SE s are chemically different, as in binary or multicomponent inhomogeneous systems, the concentration equalization is called chemical (inter-)diffusion. 

In formulating Eqn. (5.101) and the following flux equations we tacitly assumed that they suffer no restrictions and so lead to the individual chemical diffusion coefficients >(/). If we wish to write equivalent, equations for,/(A) and/(B), and allow that v(A) = = v(B), then according to Eqn. (5.103), /(A) /(B) since Ve(A) = ]Vc(B)j. However, the conservation of lattice sites requires that j/(A) j = /(B), which contradicts the previous statement. We conclude that in addition to the coupling of the individual fluxes, defect fluxes and point defect relaxation must not only also be considered but are the key problems in the context of chemical diffusion. Let us therefore reconsider in more detail the Kirkendall effect which was introduced qualitatively in Section 5.3.1. It was already mentioned that this effect played a prominent role in understanding diffusion in crystals [A. Smigelskas, E. Kirkendall (1947) L.S. Darken (1948)]. 

J.W. Cahn and F.C. Larche. An invariant formulation of multicomponent diffusion in crystals. Scripta MetalL, 17(7) 927-937, 1983. 

A.D. LeClaire and A.B. Lidiard. Correlation effects in diffusion in crystals. Phil. 

Noncrystalline materials exist in many different forms. A huge variety of atomic and molecular structures, ranging from liquids to simple monatomic amorphous structures to network glasses to dense long-chain polymers, are often complex and difficult to describe. Diffusion in such materials occurs by a correspondingly wide variety of mechanisms, and is, in general, considerably more difficult to analyze quantitatively than is diffusion in crystals. 

Equation 10.12 matches diffusivities measured in simple liquids if the characteristic cage diameter, a(Vcnt), is approximately the particle diameter and 7Vcnt is approximated by the particle volume [1]. DL is not thermally activated—it does not exhibit Arrhenius behavior as does, for example, the diffusivity in crystals, because (u) oc T1/2and (fifree) increases approximately linearly with T [4]. Less approximate models for diffusion in liquids have been reviewed by Frohberg [5j. 

The mechanism by which the self-diffusion in the relaxed state occurs is not firmly established at present. However, there are reasons to believe that for certain atoms in glassy systems, self-diffusion occurs by a direct collective mechanism and is not aided by point defects in thermal equilibrium as in the vacancy mechanism for self-diffusion in crystals (Section 8.2.1).2 These reasons include ... 

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