Search

The Lifshitz-Slezov-Wagner (LSW) theory of Ostwald ripening, limited by bulk diffusion or surface reaction rates, is extended to the case of dislocation or grain-boundary diffusion. The theory predicts a universal particle size distribution, F(R/Re), where RC is the critical radius i.e. the radius of a particle which is instantaneously in equilibrium with the solute concentration in the matrix. Some of the approximations used in the theory are critically examined. The approximation eulR 1 + m/R, used in applying Thomson's equation to the theory, is not always valid and can cause a significant error in the shape of the size distribution function for small values of R. Inhomogeneities in the distribution of interparticle separation can cause different groups of particles in the same solid to ripen independently, thus giving an observed size distribution which does not agree with the simple theory. A comparison of the theory with recent experiments on alkali halides and glass supports the above findings and explains a variety of observations satisfactorily. The observed size distribution of Ag particles in glass is described by two overlapping distributions for the bulk diffusion case. Ag particles in KC1 are described by the narrow distribution obtained for the dislocation diffusion case and the distribution of MnC12.6 NaCl (Suzuki phase) in NaCl by the surface reaction case when allowance is made for the errors due to the approximations discussed above. l. Introduction. The kinetics of precipitation from a supersaturation of solute atoms or defects in a crystal matrix have been extensively discussed by numerous authors e. g. [l-71, and recently applied to the aggregation of F centres into metallic colloids [g]. The precipitation process proceeds until the supersaturation of solute has become small, and most of the excess solute has been incorporated into precipitated particles. The particle size distribution during this process remains fairly narrow. However, this distribution does not persist, because the larger interfacial energy of a large number of small particles drives the system towards a situation where the mean particle size is increased. This process is known as Ostwald ripening, and in principle extends until the particles (*) Permanent address : Defence Research and Development Organization, South Block, New Delhi 11001 1, India. have grown into a single large precipitate or the solute has migrated to the surface of the crystal [9]. The mechanism of Ostwald ripening relies on the transport of solute from small precipitates to larger precipitates, so that the small ones shrink and the large ones grow. These solute currents arise because the solute concentration C, in equilibrium with a particle of radius R is given by Thomson's equation [9]. C , is the solute concentration in equilibrium with a plane surface and a = 2 SZalkT, where S2 is the atomic volume and a the interfacial energy of the particle in the matrix. Since CR is larger for small values of R a concentration gradient will exist so that solute flows from the region near a small particle towards a large particle. During Ostwald ripening a range of particle sizes will be present. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19767103 C7-464 S. C. JAIN AND A. E. HUGHES The theory of Ostwald ripening was formulated by Greenwood [l01 and considerably extended by Lifshitz and Slezov [ll-121 and Wagner [13]. These theories predict that the ripening system evolves into a quasi steady state in which the size distribution F(u) expressed in terms of the reduced variable U = R!R does not change with time and in which the mean radius R grows as t"where n < 1. The exponent n is + when the ripening process is controlled by the rate at which solute evaporates and condenses at the particles (case l), and 4 when the rate-limiting process is bulk diffusion of solute atoms (case 2). Extensive experimental work has been done to test these theories in the case of metal alloys 191, with reasonable success. However, the observed size distributions are usually rather wider than predicted. Very little work has been done on ionic crystals and glasses, although size distributions of colloids and precipitates have been reported [14-181, and some attempt has been made to interpret the ripening results theoretically [19-211. In this paper we report a more extensive analysis of the experimental results, and extend the Lifshitz-SlezovWagner (LSW) theory in two important respects. The first of these is to cover the case where transport of solute between particles takes place by dislocation or grain boundary diffusion (case 3) rather than through the bulk crystal. We also show that it is possible under some conditions for different groups of precipitate particles to ripen almost independently in the same crystal. This leads to an interpretation of features of the observed size distributions which are not explained by the conventional LSW theory. Since the detailed arguments have been presented in another report [22] we shall concentrate only on the main results. 2. Ripening theory and its extension to dislocation and grain boundary diffusion. Suppose that the concentration of solute at a point distant from a precipitate particle is C,, i. e. C, represents the average value of the solute concentration in the bulk of the crystal. The rate of change of the radius of a particle is then given in the steady state approximation (see