Evaluation of the methods for calculating the concentration-dependent diffusivity in binary systems
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DUCTION
LINEAR solutions to Fick’s second law require constant diffusivity, D. However, in reality, the diffusion coefficient may vary due to a range of parameters such as temperature, pressure, composition, and crystal orientation. The variation of diffusivity with concentration becomes particularly important when the concentration difference across a diffusion couple is large. In order to address the problem of a concentration-dependent diffusivity, Boltzmann[1] developed the theory and Matano,[2] the experimental procedure, now known collectively as the Boltzmann–Matano (B–M) method. From their analysis, one does not obtain a diffusion solution for concentration as a function of time. Instead, one calculates the concentration-dependent diffusivity, D(C ), at any specified time. The result, D(C ), is given by the integrodifferential expression
1 2Z e xdC C8
1 dx D(C ) 5 2 2t dC
[1]
C8 C2
The distances, x, are measured from the so-called Matano interface (x 5 0). The Matano interface defines a plane in the diffusion zone about which there is a balance of mass. In other words, the amount of material lost by diffusion on one side of the Matano interface is equal to the amount of material gained by diffusion on the other side of the Matano interface for the component in consideration. Graphically, this can be represented by the solid line in the inset in Figure 1 that makes the two hatched areas equal (i.e., A1 5 A2). Mathematically, this condition can be written as
SRIDHAR K. KAILASAM, Graduate Student, JEFFREY C. LACOMBE, Postdoctoral Research Associate, and MARTIN E. GLICKSMAN, Professor, are with the Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590. Manuscript submitted December 4, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
C8
C1
C2
C8
e xdC 5 e xdC
[2]
where C2 and C+ represent the left- and the right-hand-side concentrations of a component at the ends of the diffusion couple. Once this plane is located, it is designated as x 5 0, and all distances on the penetration profiles are recalculated with respect to the Matano interface. It has been pointed out that the procedure of finding the location of the Matano interface can be tedious and inaccurate. Inasmuch as distances are measured with respect to the Matano interface, errors in its determination result in errors in all subsequent calculations. The other drawback of this method is that large errors arise in the calculation of diffusivity near the ends of the penetration curves. Toward the ends of the penetration curve, the integral in Eq. [1] becomes very small. Also, the concentration gradient, dC/ dx, vanishes, and its inverse, dx/dC, becomes unbounded. Because the numerical evaluation of the integral and the inverse of the slope toward the ends of the penetration profile are difficult to perform accurately, large uncontrolled errors are introduced in the determination of the diffusivity. Another commonly used assumption is that the Matano interface, determined for any one component in the
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