Kirkendall effect

In the process of vacancy diffusion at a jimction between two metals, one component may diffuse faster than the other. This imbalance of flow can cause the interface to move and 

Because the diffusion rates of two materials may be different, it is necessary to define an effective diffusivity Dba = Dab + (1 — )Dba/ where x is the mole fraction of component B in A, Dab is the diffusion coefficient of A into B, and Dba is the diffusion of B into A. This distinction is only important for nondilute alloy systems. For dilute systems, x is small and Dba — Dba- 

The rate of diffusion is governed by Pick s laws. Pick s first law simply states that the diffusive flux (number of atoms of component B crossing imit area per imit time) is proportional to the concentration gradient Cb of B atoms in the host of A atoms or 

Pick s first law is useful for solving simple problems in which the concentration is known and the instantaneous flux is required, but most problems in diffusion require solving the time-dependent partial differential equation known as Pick s second law (as modified by Darken), i.e.. 

The Dba is included inside the second derivative to take into account a possible spatial variation in the diffusion coefficient. If there is no spatial variation. Pick s second law simplifies to 

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An example where one metal melts before the densihcation 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 dre Kirkendall effect. The compact is therefore seen to swell before the hnal sintering temperature of 1080 K is reached. After a period of homogenization dictated by tire criterion above, the alloy shrinks on cooling to leave a net dilatation on alloy formation of about 1%. 

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]. 

Figure 3.21 Kirkendall effect shown by the movement of inactive markers originally at the interface between two interdiffusing species.

It is particularly helpful that we can take the Cu-Ni system as an example of the use of successive deposition for preparing alloy films where a miscibility gap exists, and one component can diffuse readily, because this alloy system is also historically important in discussing catalysis by metals. The rate of migration of the copper atoms is much higher than that of the nickel atoms (there is a pronounced Kirkendall effect) and, with polycrystalline specimens, surface diffusion of copper over the nickel crystallites requires a lower activation energy than diffusion into the bulk of the crystallites. Hence, the following model was proposed for the location of the phases in Cu-Ni films (S3), prepared by annealing successively deposited layers at 200°C in vacuum, which was consistent with the experimental data on the work function. 

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-... 

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)]. 

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... 

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)]. 

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... 

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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