Interdiffusion in Intermetallics
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nt paper,[1] Tiwari and Mehrotra (TM) make the assertion that a number of well-known diffusion mechanisms [the six-jump cycle (6JC), the triple defect (TD), and the antistructural bridge (ASB) mechanisms] that have been postulated for many years to operate in binary intermetallics play no role in the process of Kirkendall shift, i.e., interdiffusion, and that only vacancies can provide for such a process. Their paper has a number of serious misconceptions, errors of fact, and apparent ignorance of the pertinent literature and classical diffusion theory that need to be corrected. This is the purpose of the present paper. Tiwari and Mehrotra introduce the Einstein–Smoluchowski Equation for the diffusion coefficient D (their Eq. [1]). For convenience, we reproduce it here: D ¼ hX2 i=2t
½1
2
where hX i is the mean square displacement of the diffusing particles in the x-direction and t is the diffusion time. TM make the quite extraordinary comment that Eq. [1] implies no correlation between successive jumps of the diffusing particles. In other words, it is correct only for a pure (uncorrelated) random walk. This comment is incorrect. Equation [1] is perfectly general for chemically homogeneous systems and does indeed cover correlated random walks. This is shown in its standard expansion for the hopping model in most texts on solid state diffusion, see, for example, References 2 to 4. In their discussion of the Kirkendall shift occurring in interdiffusion in a binary diffusion couple, TM make the
IRINA V. BELOVA and GRAEME E. MURCH, Professors, are with the Centre for Mass and Thermal Transport in Engineering Materials, University of Newcastle, Callaghan, NSW 2308, Australia. Contact e-mail: [email protected] Manuscript submitted May 6, 2013. Article published online July 31, 2013 METALLURGICAL AND MATERIALS TRANSACTIONS A
further extraordinary comment that the ‘‘nonconservative flux of vacancies implies the presence of vacancies over and above the equilibrium concentration, which is the necessary condition for the creation of a vacancy flux.’’ This statement is also incorrect and appears to result in the later fallacious arguments of the paper. It also suggests a misunderstanding of how the vacancy flux in interdiffusion comes about. This subject has been dealt with at length in numerous texts, see, for example, References 2 and 3, and it would be pedantic to give it here. But, we can provide a qualitative description of how the vacancy flux arises in interdiffusion. In Figure 1(a), we show a vacancy with an A atom on the left and a B atom on the right. Let us assume that the A atom–vacancy exchange rate wA is greater than the B atom exchange rate wB. The vacancy will clearly preferentially exchange with the A atom. As a result, when averaged over many such situations, in the AB diffusion couple, there will be a net flux of vacancies which must inevitably be from right to left (Figure 1(b)). Simply put, this is the origin of the vacancy flux in interdiffusion. For a net flux of vacancies to be maintained, the vacancies must b
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