Ostwald ripening of solid-liquid Pb-Sn dispersions

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I.

INTRODUCTION

DURING the last decade, numerous theoretical approaches have been attempted to improve the classical theory of self-similar coarsening or Ostwald ripening of dispersions developed by Lifshitz and Slyozovtl] and Wagnert21 in 1961 referred to here as LSW theory. In their analysis of the coarsening problem, LSW give an explicit relation for the time dependence of the mean particle radius in a dispersion: -

-

R3 - R3 -

8

Do'l~c~

9 kBT (c~ -

t = KLswt c~)

[1]

with D the diffusion coefficient of solute in the matrix, ~r the interfacial tension, ~ the atomic volume, and c= and c~ the equilibrium concentrations of solute in the matrix and the particle at a planar interface, respectively. Lifshitz and Slyozov and Wagner also showed that in the limit of long annealing times (called asymptotic regime), the size distribution separates into a product of a time-dependent function and a time-independent one if the particle sizes are scaled properly. The scaling length was shown to be the mean particle radius R such that scaled lengths are p = R/-R. It is predicted that an asymptotic state is reached which is independent of the initial particle size distribution. This classical theory has, however, the drawback that it is only valid for a vanishingly small volume fraction qb of dispersed phase. In experimental situations, however, this assumption is very often violated. In order to remove the zero volume fraction assumption, it is essential to take into account interactions between the diffusion fields around the particles. There have been several approaches to solving this problem Asimov,[31 Heckel,[ 41 Ardell,] 51 and Davies e t al.[61), but only recently have realistic models of the coarsening process at finite volume fractions been proposed (Brailsford and Wynblatty] Voorhees and Glicksman,Is,9,j~ Marquese and Ross,V~,~2] Enomoto and c o - w o r k e r s , ]13-~6]

1. SEYHAN, Postdoctoral Student, and L. RATKE, Senior Scientist, are with the Institute for Space Simulation, 51140 Cologne, Germany. W. BENDER, Postdoctoral Student, and P.W. VOORHEES, Professor, are with the Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208. Manuscript submitted June 16, 1995. 2470--VOLUME 27A, SEPTEMBER [996

Marsh, u7) Marder, t~s] Yao e t al., u9] and Akaiwa and Voorheest2~ In all of these treatments, the particles are assumed to be spherical and fixed in space. Diffusion of solute within the matrix determines the growth and shrinkage of the particles or dispersoids. Many of these theories solve the multiparticle diffusion equation (i.e., Laplace's equation) for solute by placing point sources or sinks of solute at the center of each particle. The strength of the point sources is determined by the boundary conditions at the interface (the Gibbs-Thomson effect) through a solution to a set of linear equations. Each of the new theoretical developments employs a different procedure to determine the statistically averaged growth rate d R / d t or the averaged source/sink strength of

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