Ternary diffusion: The ( 2 ~D) matrix of an ideal solid solution
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I.
INTRODUCTION
E X P E R I M E N T A L I S T S who study ternary diffusion usually set themselves the goal of providing a phenomenological description of one-dimensional, isothermal, isobaric diffusion in such systems. They do so by evaluating the "measurable" partial interdiffusion coefficients. (In this paper, we adopt the terminology of Manning ~for the various diffusion coefficients.) These coefficients comprise the elements of the (2/)) matrix of the diffusing system and are obtained from the set of generalized Fick's equations below: jl = --/)~l(OCl/Ox)
-- /)~2(0C2/Ox)
[la]
J2 = - / ) 3 1 ( 0 c l / 0 x )
- /)32(0c2/0x)
[lb]
where jl andj2 are the fluxes in the x-direction and c~ and c2 are the concentrations of the first and second components, respectively. In a previous paper2 which we shall refer to as I, we described the standard procedure whereby Eqs. [1] are derived from the set of three generalized Fick's equations, whose coefficients are the partial interdiffusion coefficients /)ik(i, k = 1,2, 3). These equations are as follows: 3
ji = - ~,/)ik(OCk/OX)
(i = 1,2, 3)
[2]
k=l
The measurable partial interdiffusion coefficients /)3 (i, k = 1,2) are related to the partial interdiffusion coefficients/)~g(i, k = l, 2, 3) by the following equations: /)3 = /),k -- /)/3
(i, k = 1,2)
[3]
The determination of the (2/)) matrix at only one composition point on the Gibbs triangle involves a laborious procedure in which two diffusion couples must be selected whose diffusion paths intersect at the composition point of interest.3 Thus, it is no easy matter to characterize diffusion throughout an entire ternary diffusing system. Once the/)~k's are obtained for a diffusing system, how does one go about interpreting the data? By its very nature, a phenomenological theory ignores the details of atom migration, and so one is restricted to look for relationships among the /)~k'S and the composition variables. Nevertheless, there is a natural tendency among investigators to attempt to gain insight into the atomistic processes from the 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. Manuscript submitted June 2, 1982.
METALLURGICAL TRANSACTIONS A
perceived peculiarities in the phenomenological data interrelationships. As an example, suppose that at a given composition in a ternary system one finds that b~l is negative.4 Does this result stem from the thermodynamics of the ternary solid solution, is it a consequence of a large difference in the jump frequencies of the different atom species, or is it due to a specific combination of thermodynamic and kinetic factors? It is obvious that without a detailed a p r i o r i understanding of the atomistic mechanisms operating in a given system, speculations of the kind described above are fruitless. A more logical way to proceed is to derive the values for the/)~k'S from atomistic parameters by using a kinetic theory whose assumptions do not
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