Ternary Diffusion. Solutions with Diffusion Coefficients Linearly Dependent on Concentrations
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I.
INTRODUCTION
IN a previous paper, ~ one of us showed that in an interdiffusing ternary ideal solid solution, the measurable partial diffusion coefficients (/~3's) are approximately linearly related to the concentrations of the components for suitably restricted values of the tracer diffusion coefficients. The restrictions are as follows: (1) the tracer diffusion coefficients must be constant; (2) the ratios of the latter must not be very much different from unity. The purpose of this paper is to calculate concentration-penetration curves and diffusion paths for typical diffusion couples in this system. A solution by Stokes 2 is available for a binary diffusing system in which the diffusion coefficient is a linear function of the concentration. It is possible to generalize his solution to the case of ternary diffusion, and we have in fact done so. 3 However, since the linear relationship between an individual diffusion coefficient and a concentration is simpler in our case than that adopted by Stokes, we have been able to use a simpler and more straightforward approach. The latter is outlined below.
In the equations above, cl and c2 have been chosen as the independent concentrations, and the/53's, ( i , j = 1,2), are the corresponding measurable partial diffusion coefficients. The variable A is the Boltzmann transformation variable, such that
where x is the position coordinate and t is the time. The applicable boundary conditions are as follows: cl = Cll,
C2 = C21,
A = --o~
14]
C 1 ~-- C12,
C2 ~-" C22,
A = q'-~
[5]
The special case of interest is one in which the /)3's depend linearly on the concentrations, as follows: ~ b~, = D* - (O* - D*)c,
[6]
D~2 = - ( D *
- D*)c,
[71
~3
- D*)c2
[ 81
(O* - D*)cz
[91
= _(D*
D32 = D* -
II.
DIFFERENTIAL EQUATIONS FOR TERNARY DIFFUSION
We wish to obtain solutions for a special case of the following set of the generalized Fick's differential equations: - ( A / 2 ) ( d c , / d A ) = ( d / d A ) {[/)~,(cl, c2)] ( d c , / d A ) } + ( d / d A ) {[/~2(c~, c2)] ( d c J d A ) }
In the above equations, Di* is the tracer diffusion coefficient of component i. Substituting Eqs. [6] through [91 into Eqs. [l] and [2], we get - ( A / 2 ) (dc,/dA) = ( d / d a ) {[D~* - (V* - D * ) c , ] ( d c , / d A ) } -
( d / d a ) [(D* - V*)c~(dc2/dA)]
[101
[1] - ( A / 2 ) (dc2/dA) = - ( d / d A ) [(D* - D*)c2 ( d c , / d A ) ]
- ( A / 2 ) (dc2/dA) = ( d / d A ) {[/5~,(c,, c2)] ( d c , / d A ) } + ( d / d A ) {[/532(c,, c2)] (dc2/dA)}
[31
A = xt -I/2
[2]
+ ( d / d A ) {[D* - (D* - D*)c2] (dc2/dA)}
Ill] LOUIS S. CASTLEMAN is Professor of Physical Metallurgy, Department of Physical and Engineering Metallurgy, Polytechnic Institute of New York, 333 Jay Street, Brooklyn, NY 11201. GUOJING WO, Visiting Scholar at Polytechnic Institute of New York, is with Baiyin Mining and Metallurgical Research Institute, Lanzhou, Gansu Province, People's Republic of China. Manuscript submitted December 9, 1983. METALLURGICAL TRANSACTIONS A
It is convenient to define a new variable, A', and
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