Ostwald ripening in a system with a high volume fraction of coarsening phase
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I.
INTRODUCTION
OSTWALD ripening, or coarsening, is a common phenomenon occurring during liquid phase sintering. Coarsening usually involves the flow of solute atoms from the small particles to the large particles, although more generally the flow may be comprised of a flux of enthalpy (pure solidliquid systems), solvent atoms (pure liquid-vapor systems), or vacancies (void coarsening). The fundamental driving force for the interparticle diffusion flow is well understood to be the chemical potential differences established among the particle or domain surfaces, which requires that the chemical potential of a curved interface differs from that of a flat interface. The process of interparticle diffusion causes the smaller particles to become still smaller and larger ones to grow, with the overall process resulting in an increase in the average particle radius. In principle, phase coarsening can occur any time when interfaces with disparate curvatures are in close enough proximity to allow the chemical potential gradient to drive a significant flow. An especially interesting aspect of coarsening is that after a sufficiently long time the distribution of particle sizes enters an asymptotic state, wherein the form of the distribution function normed to the (moving) average particle radius, Ray, becomes independent of time. The asymptotic state is also characterized by an average radius that increases as the cube root of time. It is indeed remarkable P.W. VOORHEES is with the Metal Science and Standards Division, National Bureau of Standards, Washington, DC 20234. M.E. GLICKSMAN is Chairman, Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12181. This paper is based on a presentation delivered at the symposium "Activated and Liquid Phase Sintering of Refractory Metals and Their Compounds" held at the annum meeting of the AIME in Atlanta, Georgia on March 9, 1983, under the sponsorship of the TMS Refractory Metals Committee of AIME. METALLURGICALTRANSACTIONS A
that a polydispere system of particles randomly distributed throughout a matrix will spontaneously coarsen such that the system collectively tends toward a fixed size distribution (relative to the average) and will display an average which grows as the cube root of time. The first major advance in the theoretical description of the coarsening behavior of systems with a dispersed second phase was made by Lifshitz and Slyozov 1and Wagner.2 This theory, often referred to as the LSW theory, treats the coarsening particles in the limit of infinite separation, so the volume fraction of coarsening phase, fv, tends to zero. Two important predictions of the LSW theory are: (1) the existence and specific shape of an asymptotic particle size distribution and (2) that R3AVis proportional to time with the constant of proportionality, commonly called the rate constant, dependent only on material parameters. Not surprising, virtually all experimental observations in coarsening systems indicate distribution functions that are broader than that predict
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