The Kirkendall Effect

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)]. 

Before we discuss point defect relaxation phenomena which occur during matter transport in inhomogeneous crystals with different sublattices, let us resume the quantitative treatment of diffusional transport in an inhomogeneous single sublattice crystal occupied with components A and B as well as vacancies. 

In a single sublattice crystal (A, B) with a fixed number of lattice sites and a negligible fraction of vacancies, the sum of the fluxes of A and B has to vanish if the number of sites is to be conserved. We just noted that if we formulate the A and B fluxes in the binary system as usual, they will not be equal in opposite directions because of the differing mobilities (bA 4= bB). However, if we have a local production (annihilation) of lattice sites which operates in such a way as to compensate for any differences in the two fluxes by the local lattice shift velocity, vL, we then obtain 

For the external, observer, jA +/B = 0. From this condition and the Gibbs-Duhem relation, the local lattice velocity becomes 

The vacancy flux and the corresponding lattice shift vanish if bA = bB. In agreement with the irreversible thermodynamics of binary systems i.e., if local equilibrium prevails), there is only one single independent kinetic coefficient, D, necessary for a unique description of the chemical interdiffusion process. Information about individual mobilities and diffusivities can be obtained only from additional knowledge about vL, which must include concepts of the crystal lattice and point defects. 

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Evidence concerning the identity of the mobile species can be obtained from observation [406,411—413] of the dispositions of product phases and phase boundaries relative to inert and immobile markers implanted at the plane of original contact between reactant surfaces. Movement of the markers themselves is known as the Kirkendall effect [414], Carter [415] has used pores in the material as markers. Product layer thickness has alternatively been determined by the decrease in intensity of the X-ray fluorescence from a suitable element which occurs in the underlying reactant but not in the intervening product layers [416]. 

An example where one metal melts before the densification process, is the formation of bronze from a 90 10 weight percentage mixture of copper and tin. The tin melts at a temperature of 505 K, and the liquid immediately wets the copper particles, leaving voids in the compact. The tin then diffuses into the copper particles, leaving further voids due to the Kirkendall effect. The compact is therefore seen to swell before the final sintering temperature of 1080 K is reached. After a period of homogenization dictated by the criterion above, the alloy shrinks on cooling to leave a net dilatation on alloy formation of about 1%. 

The Kirkendall effect (8) is time and temperature dependent, and with some metal couples, it takes place even at room temperature. For instance, adhesion of solder to gold is damaged by heating to about 150°C for about 5 minutes, due to the formation of Kirkendall voids. Naturally, the formation of Kirkendall voids is accelerated by increased temperature and dwelling time. 

In physical and chemical metallurgy, the Kirkendall effect, which is closely related to point defect relaxation during interdiffusion, has been studied extensively. It can be quantitatively defined as the internal rate of production or annihilation of vacan-... 

Let us analyze these results one step further and ask about a quantitative measure of the Kirkendall effect. This effect had been detected by placing inert markers in the interdiffusion zone. Thus, the lattice shift was believed to be observable for an external observer. If we assume that Vm does not depend on concentration and local defect equilibrium is established, the lattice site number density remains constant during interdiffusion. Let us designate rv as the production (annihilation) rate of the vacancies. We can derive from cA+cB+cv = l/Vm and jA +/ B +./v = 0 that... 

The Kirkendall effect in metals shows that during interdiffusion, the relaxation time for local defect equilibration is often sufficiently short (compared to the characteristic time of macroscopic component transport) to justify the assumption of local point defect equilibrium. In those cases, the (isothermal, isobaric) transport coefficients (e.g., Dh bj) are functions only of composition. Those quantitative methods introduced in Section 4.3.3 which have been worked out for multicomponent diffusion can then be applied. 

In the Kirkendall effect, the difference in the fluxes of the two substitutional species requires a net flux of vacancies. The net vacancy flux requires continuous net vacancy generation on one side of the markers and vacancy destruction on the other side (mechanisms of vacancy generation are discussed in Section 11.4). Vacancy creation and destruction can occur by means of dislocation climb and is illustrated in Fig. 3.36 for edge dislocations. Vacancy destruction occurs when atoms from the extra planes associated with these dislocations fill the incoming vacancies and the extra planes shrink (i.e., the dislocations climb as on the left side in Fig. 3.36 toward which the marker is moving). Creation occurs by the reverse process, where the extra planes expand as atoms are added to them in order to form vacancies, as on the right side of Fig. 3.36. This contraction and expansion causes a mass flow that is revealed by the motion of embedded inert markers, as indicated in Fig. 3.3 [4]. 

Figure 3.3 Schematic illustration of the Kirkendall effect in a binary crystalline material...

The Kirkendall effect alters the structure of the diffusion zone in crystalline materials. In many cases, the small supersaturation of vacancies on the side losing mass by fast diffusion causes the excess vacancies to precipitate out in the form of small voids, and the region becomes porous [11], Also, the plastic flow maintains a constant cross section in the diffusion zone because of compatibility stresses. These stresses induce dislocation multiplication and the formation of cellular dislocation structures in the diffusion zone. Similar dislocation structures are associated with high-temperature plastic deformation in the absence of diffusion [12-14]. 

Consider now the consequences of the pressure difference. If the membrane became free to move, it would move to the left, compressing the left chamber and expanding the right to equilibrate the pressure difference (Fig. 3.6a). However, if the membrane is constrained, the fluid may cavitate in the left chamber to relieve the low pressure, as in Fig. 3.66. This is analogous to the formation of voids in the Kirkendall effect. 

J. Bardeen and C. Herring. Diffusion in alloys and the Kirkendall effect. In J.H. Hol-lomon, editor, Atom Movements, pages 87-111. American Society for Metals, Cleveland, OH, 1951. 

The Kirkendall effect can be studied by embedding an inert marker in the original step-function interface (x = 0) of the diffusion couple illustrated in Fig. 3.4. Show that this marker will move in the F-frame or, equivalently, with respect to the nondiffused ends of the specimen, according to... 

The difference in diffusivities of the components in a growing chemical compound layer is often connected, especially in the literature on physics and metallurgy and especially in relation to intermetallics, with the Kirkendall effect. From historical and scientific viewpoints, in many cases this does not seem to be sufficiently substantiated. 

The Kirkendall effect was described in 1939-1947. Its final formulation was presented in a paper published by E. Kirkendall in collaboration with his student Alice Smigelskas in 1947 (see also Refs 5 and 8). After this,... 

The Kirkendall effect arises from the different values of the self-diffusion coefficients of the components of a substitutional solid solution, determined by Matano s method. Matano s interface is defined by the condition that as much of the diffusing atoms have migrated away from the one side as have entered the other. If DA = DB, its position coincides with the initial interface between phases A and B. If I)A f DB, it displaces into the side of a faster diffusant (see Fig. 1.22c). Note that KirkendalFs discovery only relates to disordered phases. It was indeed a discovery since at that time most reseachers considered the relation l)A = DB to hold for any solid solution of substitutional type. KirkendalFs experiments showed that in fact this is not always the case. 

At the time of E. Kirkendall, his interpretation of the experimental results obtained was severely criticised. Then, as often happens, the situation changed to the contrary. Now, the Kirkendall effect is found even in those cases to which it has no relation. In particular, this is so in the case of formation of chemical compound layers at the interface of initial substances. 

Different diffusional contributions of the components of a chemical compound to the growth process of its layer at the interface between phases A and B should not be regarded as a manifestation or result of the Kirkendall effect since the fact that these contributions are in general different became known far before discovering this effect, the essence of which consists in different diffusivities of the components of a substitutional solid solution. 

The intrinsic diffusion coefficients, Dk and DB, of a binary alloy A-B express the diffusion of the components A and B relative to the lattice planes [7], Therefore, during interdiffusion, a net flux of atoms across any lattice plane is present, where, normally, the diffusion rates of the diffusing particles A and B are different. Subsequently, this interdiffusion process provokes the shift of lattice planes with respect to a fixed axis of the sample, result which is named the Kirkendall effect [9],... 

Now, we describe the Kirkendall effect [9], The flux, as well as the diffusion coefficient, has to be chosen relative to a frame of reference. In Figure 5.4, the laboratory frame of reference, X, which is the observer frame of reference, and the moving frame of reference, x, which moves with the inert markers, are shown. 

A brass (Cu-Zn) bar, wound with molybdenum wire, was plated with copper metal. The specimen was annealed in a series of steps, in which the movements of the molybdenum wires were recorded. The inert markers had moved from the interface towards the brass end of the specimen, which contained the fastest diffuser - zinc. This is now called the Kirkendall effect. A similar marker experiment had actually been performed by Hartley a year earlier while studying the diffusion of acetone in cellulose acetate (Hartley, 1946), but most metallurgists were not familiar with this work (Darken and Gurry, 1953). 

A very recent novel hquid-phase route to hollow nanocrystals of cobalt oxide and cobalt sulfide takes advantage of the Kirkendall effect (Section 6.4.1). Injection of sulfur or oxygen into a colloidal cobalt nanocrystal dispersion created hollow nanocrystals of... 

In order to improve the kinetics of the Li-N-H system, Xie et al. [96] prepared Li2NH hollow nanospheres by plasma metal reaction based on the Kirkendall effect. The special nanostructure showed significantly improved hydrogen storage kinetics compared to that of the Li2NH micrometer particles. The absorption temperature decreased markedly, and the absorption rate was enhanced dramatically because... 

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