Self-consistent forms of the chemical rate theory of Ostwald ripening

The chemical rate theory of Ostwald ripening introduced by A. D. Brailsford and P. Wynblatt (Act. Metall.27 (1979) 498) determines the mean growth rate of particles of a particular size class by solving the diffusion equations for a representative particle (radius r) surrounded by a shell of matrix (the averaging sphere, radius rA) outside which there is a homogeneous effective medium averaging the emission and absorption of solute atoms by the remainder of the particles. Brailsford and Wynblatt set r = rA, in effect removing the matrix shell. It is argued herein that the feature of the theory so omitted is a very important one and we therefore use it to develop and extend the theory to make it self-consistent in the sense that the mean ratio of the particle and averaging sphere volumes is equal to the volume fraction of particles. Three self-consistent versions are developed, two of which have rA relatively constant for small particles and slowly increasing for particles greater than approximately average size. These were motivated by the observation from numerical simulations that small particles are little influenced by their neighbours whereas larger particles are much more strongly affected by the environment. Analytical expressions in terms of experimentally observable variables are given for the probability distributions for particle sizes, and tables of the parameters required to evaluate the distribution functions as a function of volume fraction are provided. It is concluded that the properties of the Brailsford and Wynblatt effective medium are closely reproduced by the alternative analytical theories, but that the idea of a matrix shell round the representative particle is unique to the chemical rate theory. It is argued that this feature makes the theory flexible and adaptable. This adaptability could be used to reproduce the results of sophisticated numerical simulations in a form which would be computationally efficient to include in wider simulations involving, say, the effect of particle growth on long term mechanical properties.