Improving the calculation of interdiffusion coefficients
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I.
INTRODUCTION
THE interdiffusion coefficient is
a quantity that is useful to both researchers and professional engineers. Improvements in the ability to measure diffusion coefficients or modifications in the numerical techniques used to obtain interdiffusion coefficients from raw diffusion data are expected to improve the accuracy of the data in the interdiffusion data base. Such improvements will influence both basic research and practical applications in metallurgy. On the theoretical end, the literature on diffusion currently lists two major theories that relate the interdiffusion coefficient to the system thermodynamics and to the phenomenological coefficients. The older theory by Darken tl] is simpler since it assumes that the off diagonal phenomenological coefficients are zero and that vacancies are in local equilibrium in a diffusion couple. Darken's theory leads to the following expression:
l)=(D*c2+D*ct)
1+
[11
0 In c /
Similarly, the theories of Howard and Lidiard, [21 Manning, [31 and Kirkaldy [4j lead to the following expression: z5 =
+
9
+
(0,n:) 1 +
+
L.B.
[2]
0 In From Eqs. [1] and [2], it is clear that the theories differ only in the additional term that is appended in Eq. [2]. The contribution of this term to the interdiffusion coefficient has been estimated to be of the order of 5 pct for isomorphous binary alloys. TM Since the uncertainty in typical diffusion data can be as high as 20 to 50 pct, it RAKESH R. KAPOOR, Postdoctoral Associate, and THOMAS W. EAGAR, Leaders for Manufacturing Professor of Materials Engineering, are with the Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. Manuscript submitted October 6, 1989. METALLURGICAL TRANSACTIONS A
is not possible to compute the contribution of the second term in Eq. [2] for real systems. Thus, despite the availability of a more sophisticated atomistic theory, one is unable to verify it from diffusivity measurements* due *Typically, Kirkendall velocity measurements are used to discriminate between Eqs. [2] and [1], since one may expect as much as a 28 pct difference in the velocities computed from Darken's theory and those computed from the theory by Manning, Howard and Lidiard, and Kirkaldy.
to the large errors in the diffusivity data. Hence, efforts directed toward improving the overall accuracy of diffusion data are important from a theoretical standpoint. There have been very few changes in the method of obtaining interdiffusion coefficients over the past four to five decades. The Boltzmann-Matano analysis of a diffusion profile, which yields the interdiffusion coefficient as a function of composition, has been extensively used.[4,5] The diffusivity is computed using the following expression:
15(c) = ~
(x(c) - xM) dc
[3]
Thus, the method relies on the slope of the diffusion profile as well as the area between the diffusion profile and the Matano interface. Both of these quantities must be estimated from discrete measurements of the diffusion profile (pairs of C
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