Dissolution of particles in binary alloys: part I. computer simulations
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I.
INTRODUCTION
H O M O G E N I Z A T I O N and solution heat treatments are very important heat-treatment operations carried out during metal processing. Homogenization is the commonly used term for the treatment resulting in a decrease of the microsegregation formed during solidification, while solution heat treatment refers to the treatment where second-phase particles become unstable and dissolve into the matrix phase. Very often these two processes take place simultaneously, and this is probably the reason for a somewhat inconsistent phrasing in the literature. It is also quite common to use the term homogenization as a general term for high-temperature annealing of castings. This is also inadequate, because a heterogenization instead of a homogenization occurs quite often during high-temperature annealing. The term soaking is an appropriate term for such high-temperature annealing. In spite of their obvious industrial importance, the homogenization and solution heat-treatment reactions have received rather limited scientific attention. On the other hand, the reverse reaction, namely the nucleation and growth of second-phase particles in a homogeneous supersaturated solution, has been under extensive investigation for a long time. The reason for this difference in scientific interest probably reflects the difference in the central problems in the two cases. For the latter reaction, the central problem is related to the physics of the reactions (nucleation and growth), while for the former reaction, the central problem is related to the complexity of the mathematical description.
ULF H. TUNDAL, Research Scientist, is with the Division of Metallurgy, SINTEF, N-7034 Trondheim, Norway. NILS RYUM, Professor, is with the Department of Metallurgy, The Norwegian Institute of Technology, N-7034 Trondheim, Norway. Manuscript submitted April 10, 1991. METALLURGICAL TRANSACTIONS A
The first attempt to develop mathematical models for diffusion-controlled dissolution of second-phase particles in binary alloys was made by Thomas and Whelan, m Aaron, tzl and Whelan. c31 These early models were limited to the dissolution of one single particle in an infinite matrix. Thomas and Whelan approximated the dissolution process to the reverse growth process, while Aaron's model was a one-dimensional model and thus cannot describe the dissolution of small particles. Whelan derived the correct differential equation for the dissolution of a spherical particle in an infinite matrix and also gave the general solution to this equation. The dissolution process was found to be much more complicated to describe mathematically than the growth process (or the reversed growth process). Whelan also discussed some limiting cases where the solution could be given in closed forms. In all of these early attempts to describe the dissolution process mathematically, several assumptions were made. (1) The rate of dissolution is limited by long-range diffusion; hence, equilibrium is always established at the interface between the particle and the matri
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