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I.

INTRODUCTION

A first-order phase transformation ultimately results in a mixture of two phases. In such a system, a large amount of interracial area is usually present as a result of the polydisperse nature of the mixture. Thus, the total energy of the system can be lowered by decreasing the amount of interfacial area. Reduction of the interfacial area results in an increase in the size scale of the second-phase precipitates. The transformation process that produces these morphological changes is known as Ostwald ripeningU.21 or precipitate coarsening. Due to the small energies associated with the precipitate-matrix interfaces as compared with the energies associated with precipitate growth or other transformation processes, Ostwald ripening typically occurs near the end of a first-order transformation process. The size scale of the system increases by the dissolution of small precipitates and the growth of large ones via a diffusive mass flow from shrinking to growing precipitates. The first comprehensive theory of Ogtwald ripening was given by Lifshitz and Slyozov[3J (LS) and in a related work by Wagnert41 (LSW). Here, the precipitates are assumed to be spherical and fixed in space. The analysis is limited to dilute binary alloys in which the coarsening proceeds by the transport of a single independent chemical component. Additionally, the mean-field description used to derive the expression for the growth rate of a precipitate in the LSW theory requires the system to have a vanishingly small volume fraction of precipitate. Following the work of LSW, there were many attempts to verify the predictions made by the LSW theory. There is experimental evidence to support the time dependence of the average precipitate size predicted by LSW, but there is little correlation between the predicted precipitate distributions and the experimental measurements. In order to exC.J. KUEHMANN, R & D Scientist, is with the BIRL Industrial Research Laboratory, Evanston, IL 60201. P.W. VOORHEES, Professor, is with the Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208. Manuscript submitted March 13, 1995. METALLURGICAL AND MATERIALS TRANSACTIONS A

plain these discrepancies, many researchers have attempted to remove the mean-field restrictions required by the LSW theory (as in the recent review by VoorheestS]). One of the reasons why the LSW theory can only describe coarsening in binary alloys is due to the form of the Gibbs-Thompson equation used in the theory. The GibbsThompson equation is central to any theory of coarsening, as it relates the composition at the precipitate-matrix interface to the interfacial curvature. For a binary alloy, the composition of the precipitate-matrix interface follows directly from the local equilibrium conditions, as given by the equality of the chemical potentials at the interface. In contrast, in higher order alloys, local equilibrium is no longer sufficient to determine these compositions. In a ternary system, four compositions are needed to describe th