Coarsening in binary solid-liquid mixtures
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I.
INTRODUCTION
SOLIDIFICATION into an undercooled liquid frequently results in a mixture of solid dendrites in a liquid of nearly equilibrium temperature and composition. Following this initial solidification reaction, the mixture then undergoes an Ostwald ripening or coarsening process, during which the average size/scale of the mixture increases with time via the growth of regions of interface with low interfacial curvature at the expense of regions of interface with high interfacial curvature. As the ripening process is driven simply by the curvature of the solid-liquid interfaces, it occurs in a wide variety of solidification processes and has been studied in great detail.[l,21 The factors controlling the kinetics of the process are reasonably well understood in low volume fraction systems which coarsen via the exchange of one diffusant, typically either heat or mass. For example, the classical Lifshitz-Slyozov (LS) theory describes the coarsening process in a mixture consisting of a nearly zero volume fraction array of spherical solid particles in a liquid under the conditions that coarsening proceeds via the isothermal transport of mass from large to small particles, t31 The LS theory can also be used to describe the coarsening behavior of pure solid-liquid mixtures which ripen via the transport of heat. More recent theories describing coarsening in systems with a finite volume fraction of solid also assume that either heat or mass, but not both, is transported between the coarsening particles and have been recently reviewed. ~4,sl However, many solid-liquid mixtures contain considerable quantities of solute and are not held isothermally. Thus, in principle, both heat and mass transfer must be considered in order to determine the kinetics of the ripening process. Some insight into the effects the coupled heat and mass flow may have on the kinetics of Ostwald ripening is provided by the recent theoretical work on P.W. VOORHEES, Associate Professor, is with the Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208. Manuscript submitted June 5, 1989. METALLURGICAL TRANSACTIONS A
the rate of dendritic growth into an undercooled liquid alloy, t6'7,81These works have shown that the coupling between the flow of heat and mass can have a large effect on the kinetics of growth, with mass transport controlling the growth kinetics of the dendrite in concentrated alloys and heat transport controlling the growth kinetics in alloys with low solute content. However, it is unclear if the kinetics of ripening will vary with the concentration of solute in the same fashion as was observed for dendritic growth in alloys, as a corresponding theory for ripening in solid-liquid systems which specifically accounts for the coupled mass and heat flow which must occur during the coarsening process has not been developed. It is well known that if the ripening process occurs by the isothermal transport of mass, the mean-field concentration of the matrix will approach its equilibrium value as t -1
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