On the kinetics of diffusion-limited layer growth in solid-solid systems
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I.
INTRODUCTION
T H E formation of a new alloy or intermetallic compound from two precursor solids is a process of central importance in metallurgy. In many cases, an intermediate phase is formed between the two terminal solid solutions. In order to predict the rate at which this layer grows, it is generally assumed that the nucleation step is very rapid, so that the overall rate is limited by interdiffusion of the two (major) chemical components. The conventional approach is to describe the species fluxes in each phase by means of Fick's first law, with a diffusion coefficient that may or may not vary with composition. [~1As will be seen, if the composition units used are mass or mole fractions (or densities), then this approach is valid only if the overall mass or molar density is the same from one phase to the next. In order to avoid this restriction, Fick's law can be written in terms of volume fractions instead; t21 however, this requires that the partial molar volumes of the two components be constant throughout the system. (Otherwise, conservation equations in terms of volume cannot be written.) While this is an excellent approximation in many cases, it tends to fail for systems characterized by bona fide compound formation, due to the existence of strong covalent bonds. The purpose of the present work is to develop an approach that can be used when the conventional methods (involving Fick's law alone) cannot be justified. The analyses that follow will deal with the special case of layer growth between two initially saturated solid solutions, so that diffusion occurs only within the intermediate phase. The net flux of each species is given as the sum of a purely diffusive (Fick's law) term and a contribution from the bulk flow in the intermediate phase. [3[ The only assumption needed (aside from the standard approximation of a constant diffusion coefficient) is that the overall mass or molar density is uniform throughout the newly formed layer. This will be an excellent approximation if this phase has a very restricted composition range, as tends to be true when a stoichio-
RICHARD S. LARSON, Senior Member of Technical Staff, is with Sandia National Laboratories, Livermore, CA 94551. Manuscript submitted October 23, 1989. METALLURGICAL TRANSACTIONS B
metric compound is formed. Since this is precisely the kind of system for which the partial molar volumes are not expected to be constant, the present analysis can be regarded as complementary to previous work, in which the partial molar volumes are assumed constant but the intermediate-phase density may vary. Two separate problems will be addressed in what follows. The first is the dissolution of a sphere of phase a which is imbedded in a continuum of phase/3, the primary quantity of interest being the total dissolution time z. An approximate (pseudo-steady-state) analytical result for ~-was obtained via conventional methods by Judd and Paxton, [4j who were concerned specifically with the formation of austenite from cementite and ferrite. (Actually, a less
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