Multiphase binary diffusion in infinite and semi-infinite media: Part II. On the numerical calculation of the rate const
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I.
INTRODUCTION
g number of experimental data on the formation and growth of intermediate phase layers in binary systems have been reported. The data have shown that the layer thicknesses of the product phases grow with various rates, as one or more of the product phases have much greater rates than those of the other product phases. It also has been reported that only some or none of the possible intermediate phases which are depicted by the equilibrium phase diagram appear in the diffusion zone. ll-41 In discussing the formation of intermediate phases, Seith, tSl Baird, [6l and Nishizawa and Chiba tTj listed such criteria for the layer growth as diffusion coefficient in a phase, formation energy, crystal structure, homogeneity range of an intermediate phase, and so on. The descriptions, however, were not successful in the general validity. Jost [8] derived exact solutions for the layer growth of a single product phase in infinite and semi-infinite media. Gurov et al.[91 developed a method of evaluating the rate constants for layer growth of intermediate phases formed in an infinite medium and obtained successful agreement between the calculated and observed values for the Ag-Zn and Cu-Zn systems. In the model presentedy m the concentration varies linearly with position in each phase layer, including both the terminal phase layers. J~iger and MatauschektJ t] recently found approximate solutions for the layer growth of one and two product phases in a semi-infinite medium on the assumption of the linear concentration in a phase. The solutions may be easily generalized to an n-phase system, and each layer thickness may be calculated from the corresponding simple equation. The extent of error in the results obtained using the methods of Gurov e t a / . [91 and of J~iger and Matauschek, t~u owing to the linear assumption, can be ascertained by comparing them with the corresponding S. TSUJI, Professor, is with the Department of Mechanical and Precision Systems, School of Science and Engineering, Teikyo University, Utsunomiya, Japan 320. Manuscript submitted May 18, 1992. METALLURGICAL AND MATERIALS TRANSACTIONS A
Jost exact solutions, tSl However, no exact solutions have been derived for more than one product phase. The purpose of this article is to present two numerical methods for solving a series of exact relations describing multiphase binary diffusion in infinite and semi-infinite media, respectively, without any limitation on the number of possible phases present in a diffusion couple, t12]
II.
INFINITE MEDIUM
A. Arrangement of Equations We shall consider here that (n - 2) intermediate phases dictated by the equilibrium diagram for a couple in a binary system are formed in the diffusion layer, where n -> 3. Any couple which consists of pure metals, primary solid solutions, intermetallic compounds, or any pair of these is within the scope of this numerical analysis. Under the condition where interdiffusion coefficient in phase j,/9, depends on composition, the average interdiffusion coefficient for phase j, D
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