Acta mater. 49 (2001) 3821–3828 www.elsevier.com/locate/actamat A RANDOM WALK APPROACH TO OSTWALD RIPENING M. SCHWIND† and J. A ˚ GREN Department of Materials Science and and Engineering KTH, SE-10044 Stockholm, Sweden ( Received 9 January 2001; received in revised form 9 May 2001; accepted 24 June 2001 ) Abstract—We investigate a numerical model of Ostwald ripening in which matter is discretised. The model is based on transport by random walk and certain rules at the phase interfaces which mimic the continuous solution of the diffusion and moving boundary problem. In the limit of low volume fractions the results from our simulations agree quantitatively with the results predicted by the Lifshitz–Slyosov–Wagner theory. In contrast to the LSW model, the volume fraction of particles enters our model as part of the diffusion problem and it may therefore be confidently used to investigate what happens when the volume fraction of particles increases.  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Properties and phenomena; Theory & modeling; Kinetics; Transport; Diffusion 1. BACKGROUND A phase transformation is usually divided into three different stages: nucleation, growth and coarsening or Ostwald ripening. Ostwald ripening is the process by which a system of particles embedded in a matrix phase increases its average radius; large particles tend to grow and small ones tend to shrink and finally dis- solve. In contrast to the growth stage the volume frac- tion of the particles remains almost constant during Ostwald ripening since the system is close to equilib- rium. The driving force for the transformation is pro- vided by internal interfaces in the material; a coarse structure has lower specific interfacial area than a fine one and is therefore favoured. In most models of Ostwald ripening, local equilib- rium conditions are imposed at the phase interfaces between the particles and the matrix. The interfacial energy is included in the thermodynamics of the two phases and gives rise to a concentration difference between the matrix outside small and large particles. This concentration difference drives the diffusion through the matrix, and the growth or shrinkage of the particles is governed by a flux balance condition at the phase interface. Intuitively one would like to use a thermodynamic description of the phases under consideration, include the interfacial energy and finally solve the diffusion problem by for instance finite element methods. Unfortunately, the strategy illustrated in Fig. 1 is very demanding from a computational point of view and † To whom all correspondence should be addressed. E-mail address: martin@met.kth.se (M. Schwind) 1359-6454/01/$20.00  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII:S1359-6454(01)00273-7 is not feasible with today’s computers. Some simpli- fications are therefore necessary. By far the most cited model for Ostwald ripening was elaborated some forty years ago by Lifshitz and Slyosov [1] and Wagner [2]. The model applies to a binary system A–B in which the matrix phase M is mainly A and the particle phase P is mainly B. They stated that each particle is spherical and surrounded by a concentration field in the matrix‡ x B M given by the solution of the Laplace equation in spherical coor- dinates x M B (r) = C 1 r + C 2 (1) where the constants C 1 and C 2 are given by the com- positions at the phase interface and the average com- position of the system far away from the particle. They also assume that the matrix phase is an ideal, dilute solution, and the composition x M/P B (R) of the matrix phase outside a particle of radius R is therefore given by the Gibbs–Thomson equation x M/P B (R)-x M/P B,eq = a R (2) where x M/P B,eq is the composition at equilibrium when ‡ Throughout this work the variable x denotes mole frac- tion (i.e. x A = N A /N i ) and c is used for concentration expressed in the unit number/volume.