Ostwald Ripening in Ternary Alloys C.J. KUEHMANN and P.W. VOORHEES A theory of coarsening in an isothermal, ternary alloy is developed in an effort to understand the effects of a third chemical component on the ripening behavior of a two-phase system. The analysis is valid for a general, nonideal, nondilute solution, but is limited to extremely small volume fractio- of the coarsening phase and neglects off-diagonal terms in the diffusion matrix. The Gibbs-Thomp- son equation in a ternary system undergoing coarsening reveals that the concentrations at the parti- clelmatrix interface are dependent on the far-field supersaturations as well as on the particle radius. In addition, the capillary length depends on the diffusivities of the two components. An asymptotic analysis shows that the exponents of the temporal power laws for the average particle radius, number of particles per unit volume, and the matrix supersaturations are the same as that found in the binary limit; however, the amplitudes of the power laws are modified. We find that the trajectory of the matrix supersaturation must lie along a tie-line, but the trajectory of the particle composition does not. An expression for the effect of dilute ternary additions to the coarsening rate of a binary alloy is also given. I. INTRODUCTION plain these discrepancies, many researchers have attempted A first-order phase transformation ultimately results in a mixture of two phases. In such a system, a large amount of interfacial area is usually present as a result of the po- lydisperse 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 precipi- tates. The transformation process that produces these mor- phological changes is known as Ostwald ripening['"] 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 trans- formation processes, Ostwald ripening typically occurs near the end of a fn-st-order transformation process. The size scale of the system increases by the dissolution of small precipitates and the growth of large ones via a difisive mass flow from shrinking to growing precipitates. The first comprehensive theory of Ostwald ripening was given by Lifshitz and Slyozovr31(LS) and in a related work by Wagnefi41(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 vol- ume 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 distri- butions and the experimental measurements. In order to ex- ;o remove the mean-field restrictions required by the LSW theory (as in the recent review by Voorhee~[~I). One of the reasons why the LSW theory can only de- scribe coarsening in binary alloys is due to the form of the Gibbs-Thompson equation used in the theory. The Gibbs- Thompson equation is central to any theory of coarsening, as it relates the composition at the precipitate-matrix inter- face to the interfacial curvature. For ; binary alloy, the composition of the precipitate-matrix interface follows di- rectly 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 ter- nary system, four compositions are needed to describe the composition of the ma&x and precipitate at the precipitate- matrix interface; however, only three equilibrium condi- tions are available, one for each chemical component. Thus, in a ternary alloy, the fourth condition is furnished by either a global mass conservation condition or one of the equa- tions used to describe the growth kinetics of the precipitate. In the case of Ostwald ripening, the latter condition is em- ployed, and thus, the coarsening behavior in higher order alloys may be significantly different fiom that in binaries. There is a body of work on diffusive transformations in higher order allois which provides a sound background for this study. Coate~,[~,~] in his two-part work, developed the kinetic equations for precipitate growth in a ternary system, a necessary first step in developing a theory of coarsening. However, he did not consider the effects of curvature of the precipitate interface on the interfacial compositions. Bjorklund et al.r81 and Slyozov and Sagalo~ich[~J~.~~1 con- sidered the problem of coarsening in multicomponent sys- tems. but restricted themselves to dilute solutions and did not allow the precipitate composition to vary with the pre- - - cipitate radius. Recent work by Umantsev and 01son[lfi on multicomponent coarsening removed the limitations of di- C.J. KIEHMANN, R & D Scientist, is with the BIRL Industrial lute solution thermodynamics, but did not consider the ef- Research Laboratory, Evanston, IL 60201. P.W. VOORHEES, Professor, fects ofinterfacial curvature on the precipitate composition^ is with the Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208. Morral and Purdy developed a general theory for coarsen- Manuscript submitted March 13, 1995. ing in n-component all0ys.['~~'~1 However, they only pre- METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 27A, APRIL 1996-437