Average effective interdiffusion coefficients and the Matano plane composition
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I.
INTRODUCTION
THE concentration profiles that develop in diffusion couples during isothermal diffusion in binary and multicomponent systems can be represented by error function solutions provided the interdiffusion coefficients are assumed to be independent of composition. In general, the assumption of constant interdiffusion coefficients is limited, and the concentration profiles can reflect appreciable variation in the interdiffusion coefficients with composition in the diffusion zone. In this context, an analysisV.21 was developed for the determination of average effective interdiffusion coefficients as well as integrated interdiffusion coefficients for each component over a given composition range in the diffusion zone of single-phase and multiphase couples. The analysis allows the evaluation of effective penetration depths of the components on either side of the Matano plane from the average effective interdiffusion coefficients determined over the composition ranges on the two sides of the Matano plane. In this article, the composition at the Matano plane is examined in light of the average effective interdiffusion coefficients. Exact relations for the Matano plane composition are derived in terms of these coefficients and characteristic depth parameters for concentration profiles on either side of the Matano plane. The analysis also inter-relates the interdiffusion flux at the Matano plane to the average effective interdiffusion coefficients and the Matano plane composition. An error function model based on the analysis is developed to generate concentration profiles on one side of the Matano plane from a knowledge of concentration profiles on the other side for single-phase diffusion couples. II.
INTEGRATION OF INTERDIFFUSION FLUX
A schematic concentration profile of a component for an isothermal solid-solid diffusion couple is shown in Figure 1. At time t, the interdiffusion flux Ji of component i at
M.A. DAYANANDA, Professor, is with the School of Materials Engineering, Purdue University, West Lafayette, IN 47907-1289. Manuscript submitted August 14, 1995. 2504--VOLUME 27A, SEPTEMBER 1996
section x on the basis of a laboratory-fixed frame is determined directly from the concentration profile by t3,41 Ci(x)
= 2t
(x - Xo) d c ,
[1]
ci
where Xo identifies the Matano plane, C+ and C,r refer to the terminal compositions of the diffusion couple, and corresponds to the concentration of component i at the Matano plane. Since Ci is a function of the Boltzmann parameter A given by (x - Xo)/'~tt,Eq. [1] can be rewritten as Ci(A)
x/7 =
(A - ao) a c ,
[21
ci
The integral on the right-hand side of Eq. [2] is a function of A and has a definite value K(A) for a given A. Hence, a~ = K (A) 2~/7
[3]
at a given concentration level. Hence, an integration of with respect to time t yields t
J ~dt
= K(A) ~/t
o
= 2t " ~
[4]
Equation [4] provides a link between the total interdiffusion flux past a given concentration level in the diffusion zone over time t to the interdiffusion flux at that level at time
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