Numerical modeling of multiphase diffusion in the process of homogenizing in binary alloys
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I. INTRODUCTION
THE interdiffusion in which one or more intermediate phases such as an intermetallic compound are formed in the diffusion zone is generally referred to as multiphase diffusion or reaction diffusion. The author has already analyzed multiphase diffusion in a binary system in infinite and semiinfinite media as a two-variable problem and presented the method to numerically obtain the solution.[1,2] Multiphase diffusion proceeding toward uniformity in composition has attracted much interest recently as a method for adjusting the surface composition after the Al impregnation process,[3] a new processing for intermetallic compounds by heating the stacking sequence of elemental thin films,[4,5] and a surface modification by interdiffusion between plating and the substrate.[6,7] On the other hand, a more fundamental understanding of growth behavior of silicide between a silicon wafer and sputtered metallic thin film[8,9] requires knowledge of the preceding multiphase diffusion. The multiphase diffusion toward homogenization is accompanied by complex phenomena such as disappearance of most of the outer phase, a sudden turn from an increase to a decrease in layer thickness for the adjacent phase, and the appearance of a new product phase in the diffusion zone. Such a diffusion phenomenon will not allow the introduction of the Boltzmann variable and can no longer be solved analytically. As for the multiphase binary diffusion that accompanies the disappearance of phases and proceeds toward the homogenization of composition, Heckel and co-workers,[3,10–12] and much more recently Shimozaki et al.,[13,14] have reported numerical analysis methods based on a difference calculus. In the numerical calculation using the finite-difference method to solve such a problem, the concentration-distance curve in each phase must be given as the initial values. Hickl et al. used the analytical solution of the multiphase diffusion in a
S. TSUJI, Professor, is with the Department of Mechanical and Precision System, School of Science and Engineering, Teikyo University, Toyosatodai, Utsunomiya, 320-8551, Japan. Manuscript submitted May 23, 2000.
METALLURGICAL AND MATERIALS TRANSACTIONS A
semi-infinite medium as the initial value. Their method[3,11] is capable of producing an analytical solution for up to three phases, but is not capable of producing a general analytical solution for four or more phases. Shimozaki et al., on the other hand, has solved the preceding problem by obtaining the initial values by the trial and error method.[13] Nevertheless, their method becomes cumbersome in the case where the layer thickness varies drastically with phase. Since grid points along each time row are spaced equally over the entire diffusion zone, there will be a large number of small distance steps that will remarkably extend the calculation time. Moreover, each phase interface is not necessarily situated on a distance column. Engstrom et al.[15] presented a model to treat multiphase diffusion in a multicomponent system. The model has been
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