The effects of elastic stress on the kinetics of ostwald ripening: The two-particle problem
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I.
INTRODUCTION
T H E kinetics of Ostwald ripening processes often are described by relationships between an average length scale of the mixture and a temporal power law with a positive exponent. These scaling laws can be derived from an assumption of self-similarity of the microstructure with time l~l or from a kinetic equation that describes the growth rate of a second-phase particle with respect to another, t21 For example, Lifshitz and Slyozov (LS) 12'31use a kinetic equation appropriate for an infinitely dilute array of spherical particles in a stress-free matrix to predict that the average particle radius should increase as t ~/3 where t is time. The LS theory assumes that the mechanism responsible for the transformation process is the diffusion of mass from regions of high interfacial curvature to regions of low interfacial curvature. Such a morphological evolution process is consistent with a dimunition of the total interfacial area (and, hence, total interfacial energy with time) and is called Ostwald ripening. In many systems, however, the ripening process may not be driven entirely by the interfacial curvature as in the LS case; and, as a consequence, both the temporal evolution of the system and the magnitude of the exponent of the classical power law, which describes the time dependence of the average particle radius, can be different. An example is provided by the hydrodynamic motion accompanying spinodal decomposition in fluids. In this case, during the late stages of spinodal decomposition, the curvature of the interfaces between the two phases induces fluid flow through the dependence of the pressure at the interface on the curvature of the interface. WILLIAM C. JOHNSON, Associate Professor, is with the Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, Pittsburgh, PA 15213-3890. P.W. VOORHEES, Associate Professor, is with the Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208. D.E. ZUPON, Metallurgist, is with the Boeing Military Aircraft Company, Wichita, KS 67277-7730. Manuscript submitted July 26, 1988. METALLURGICAL TRANSACTIONS A
This flow results in the size scale of the mixture increasing as t instead of the usual t 1/3 law. 14,51 As the mechanism responsible for this flow is generic to the nature of a fluid, one would not expect to observe the classical temporal power laws for late-stage processes in these twophase fluids. In this work, we examine the influence of another phenomenon which is generic to a large class of twophase systems, namely, elastic stress in two-phase solids, on particle growth rates during ripening. As did LS, we assume that the dynamical nature of the system is governed by isothermal mass flow through the matrix phase. However, unlike LS, we relax the assumption of a vanishingly small volume fraction of the second phase and allow for the presence of stress generated by misfit strains. To identify the potential magnitude of the elastic effects, we consider the problem of determining t
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