A new model for diffusion-controlled precipitation reactions using the extended volume concept

Abstract

In this work a new model for diffusion-controlled precipitation reactions is derived, IntroductionDiffusion-controlled precipitation reactions are important in a wide variety of commercially important materials. It is important to have available an accurate model for the kinetics of the reaction as heat treatment can determine a range of properties.Preferably such a kinetic model should be accurate, transparent, avoid computationally expensive implementations, and lead to analysis methods that are widely applicable.The objective of this work is to derive and test a new model for diffusion-controlled precipitation reactions that meets these criteria.Diffusion-controlled precipitation reactions can be thought of as being the combination of 4 overlapping processes: nucleation, growth, soft impingement and coarsening.Several attempts at providing a computationally friendly suitable framework for predicting the progress of diffusion-controlled reactions incorporating a treatment of impingement have been published (see e.g. [1,2,3]), and some less computationally friendly attempts at models have been published more recently [4]. The present paper focuses primarily on the treatment of soft impingement.One group of existing modelling approaches is based on direct consideration of the diffusion flux at the interface. For instance, the numerical method formulated by Kampmann and Wagner [5], as applied by several authors (e.g. [6,7,8,9]), treats the growth of individual spherical particles following the equation:where R is the radius of a growing particle, D is the diffusion constant, c p is the solute concentration in the precipitate, c m is the solute concentration at the precipitate/matrix interface that is evaluated by the Gibbs-Thompson relationship [10,11], ) (t c is the mean concentration of the matrix. A characteristic of this approach is that the interaction of the diffusional growth of the various particles is drastically simplified: a spherical geometry of diffusion field is assumed and interaction is described through a single parameter, the mean concentration ) (t c . We can term this a mean field approach.Published as: Starink, M.J. (2014) Thermochimica Acta, 596, pp. 109-119 3 Some of the users of KW model have described this treatment of impingement as 'a simple response equation ' [12], i.e. it is an approximation for which the accuracy is as yet unproven and it has been suggested that the mean field approach underestimates impingement because it ignores a geometrical component to the impingement process [13]. The contribution of a geometrical component has been investigated in [13,14].Benefits of the KW approach are that basic capillarity effects can be included and that, in principle and at some computational costs, the numerical scheme can be implemented to model different groups of precipitates and (in principle 1 ) incorporate coarsening [7,8]. The models described by Svoboda, Fisher and co-workers [15,16] applying the thermodynamic extremal principle (first described by Onsager [17]) empl...

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