Diffusion-controlled precipitate growth in ternary systems I
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P R E C I P I T A T I O N phenomena are of considerable p r a c tical significance to the metallurgist. Although the m a jority of c o m m e r c i a l alloys contain g r e a t e r than two elements, most theoretical analyses of precipitate growth relate only to binary systems. This is a r e a s onable development since one must first establish an understanding of binary phenomena before attempting the generalization to more complex multicomponent phenomena. A few w o r k e r s ~-1~ have considered various aspects of the problem of diffusion controlled ferrite and c a r bide precipitation in the system F e - C - X where X is a substitutional alloying element. These studies attempted to gain an insight into the hardenability problem through an understanding of the role of alloying elements in the austenite-decomposition reactions. The present paper is a generalized treatment of diffusion-controlled precipitation in isothermal t e r n a r y systems of a r b i t r a r y constitution. It is of particular interest to establish if and how a system can simultaneously accommodate the interface m a s s conservation conditions with a local equilibrium condition at the interface. Not surprisingly this accommodation is most intricate (and, as will be demonstrated, impossible in many cases) when one independent component diffuses much m o r e rapidly than the other. Therefore, especially careful consideration is given to this extreme. ASSUMPTIONS AND MATHEMATICAL FORMULATION Consider a t e r n a r y system 1-2-3 and a r b i t r a r i l y select 1 and 2 as the independent components (i.e., consider solutes 1 and 2 in solvent 3). At temperature T~ a specimen of uniform composition (CIB, C2B) is in a single-phase /3 field. This specimen undergoes an instantaneous quench to a t e m p e r a t u r e To for which the D. E. COATESis Assistant Professor, Department of Metallurgy, University of British Columbia, Vancouver8, B. C., Canada. Manuscript submitted September 16, 1971. METALLURGICALTRANSACTIONS
composition (C1B, C2B) is in a two-phase ~ +/3 field. The problem is to investigate the diffusional growth of c~ precipitate particles in the/3 matrix. The following assumptions a r e made: The To isotherm of the equilibrium phase diagram for the t e r n a r y system 1-2-3 is known. The diffusion fields around the various precipitate particles do not impinge. This, of course, is equivalent to considering the growth of a single particle in an infinite medium of initial composition (C,B, C2B). T e r n a r y diffusional interaction can be ignored. That is to say, in the flux equations, Ji = -DilVC1 - DiaVC2 (i = 1, 2), the t e r m s DI2VC2 and D2tVC, a r e assumed to be negligible with respect to DuVC1 and D22VC~ r e s p e c tively. Clearly this assumption implies that the diffusion field of component 1 does not interact with that of component 2. Accordingly these fields a r e solutions of the binary equation
a Ci = DiV2Ci at
(i = 1, 2)
[1]
where it is assumed the diffusion coefficients D~ and /92 a r e not functions of concentration. A
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