Statistical Mechanics of Phase Coarsening
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STATISTICAL MECHANICS OF PHASE COARSENING M.E. GLICKSMAN,* P.W. VOORHEES,** *Department of Materials Engineering, Rensselaer Polytechnic Institute, Troy, New York 12181; **Metallurgy Division, National Bureau of Standards, Washington, D.C. ABSTRACT Phase coarsening, also termed Ostwald ripening, is generally thought to be a slow, diffusion-controlled process which occurs subsequent to phase separation under extremely small under- or over-saturation levels. The theory due to Lifshitz, Slyzov, and Wagner (LSW), which predicts the coarsening kinetics and the particle distribution function is applicable to dilute systems only, in which particle-particle interactions are unimportant. Most practical systems, however, have large enough volume fractions of the dispersed phase to violate the essential assumptions of LSW theory. Recent progress will be described on simulating Ostwald ripening in randomly dispersed, high volume fraction systems. A fast algorithm for solving the multiparticle diffusion problem (MDP) will be described, permitting simulation of coarsening dynamics by cyclic time-stepping and updating the diffusion solution for large random particle arrays. The rate constants, controlling the growth of the average particle, and the particle distribution functions were obtained by numerical simulations up to a volume fraction of 0.55. A new statistical mechanics theory has now been developed which reproduces the MDP simulation data accurately, and finally makes clear how the linear mean-field approximations employed by LSW theory must be modified to describe real systems. The new theory provides a comprehensive approach to understanding microstructural coarsening in two-phase systems. INTRODUCTION Phase coarsening is an irreversible process commencing immediately after the initial precipitation and rapid growth of a particle or second-phase domain. 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 solid-liquid 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 in accord with the Gibbs-Thomson equation, 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. Most often, however, coarsening is observed late in phase transformation processes, when the degree of supersaturation is small relative to that needed for nucleation or net phase growth.
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