Analysis of Precipitation Kinetics on the Basis of Particle-Size Distributions
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THE kinetics of heterogeneous precipitation reactions in dilute systems is frequently described in terms of nucleation and growth of solute-rich precipitates, however, often departing from different and thus inconsistent thermodynamic bases for nucleation and growth.[1] Most of these models originate from the Kampmann–Wagner numerical (KWN) approach[2]: In this approach, the evolution of the particle-size distribution (PSD) is modeled for discrete time steps and discrete particle-size classes adopting the classical theory of nucleation[3,4] and long-range solute diffusion controlled growth of (spherical) particles.[5,6] The changes in particle-number densities and particle sizes during one time step are calculated by numerical integration of the rate of nucleation and the rates of growth of the individual size classes which are functions of the (time-dependent) mean matrix composition. Such models can thus be classified as mean-field models for precipitation kinetics, constituted by two types of coupled kinetic rate equations, a nucleation rate and a growth rate, which are numerically solved on a discrete time–particle-size grid. The inconsistent nature of the thermodynamic treatments employed for nucleation and growth originates from the thermodynamic stability consideration for the particle–matrix system being treated separately and differently for nucleation and for growth: For nucleation, the nucleation barrier for formation of a stable
BASTIAN RHEINGANS, Research Scientist, is with the Institute for Materials Science, University of Stuttgart, Stuttgart, Germany. Contact e-mail: [email protected] ERIC JAN MITTEMEIJER, Professor, is with the Institute for Materials Science, University of Stuttgart, and is also Director at the Max Planck Institute for Intelligent Systems, Heisenbergstrasse 3, 70569 Stuttgart, Germany. Manuscript submitted January 14, 2015. METALLURGICAL AND MATERIALS TRANSACTIONS A
second-phase particle is evaluated from the release in chemical energy upon formation of the second-phase particle (the chemical driving force for nucleation) and the increase of energy due to formation of an interface between matrix and precipitate particle, and other energy contributions such as an elastic energy contribution in case of a volume misfit between precipitate and matrix. For growth, in order to describe particle coarsening in later stages of the precipitation, the Gibbs–Thomson effect is taken into account, i.e., the compositions of particle and matrix at the particle– matrix interface are taken as functions of the size of the particle, which leads to the dissolution of smaller particles once the value of the mean matrix composition falls below the size-dependent matrix composition at the particle–matrix interface. Inconsistency of the thermodynamic descriptions arises if the nucleation barrier and the Gibbs–Thomson effect are evaluated on the basis of different thermodynamic models, e.g., involving differing descriptions for the chemical Gibbs energies of the matrix phase and the precipitate phase, or
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