Diffusion fields associated with size and shape coarsening of oblate spheroids
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I.
INTRODUCTION
EVER
since Greenwood[1] first considered the mathematical treatment of coarsening kinetics of spherical particles, many investigations have contributed to theoretical and experimental developments of precipitate coarsening in metallic alloys. Precipitate coarsening is one of the most important problems in the thermal stability of precipitatestrengthened alloys serving at elevated ambient temperatures. The systematic coarsening theory developed by Lifshitz and Slyozov[2] and by Wagner[3] (LSW) in 1961 has been the most frequently used approach to analyzing the coarsening kinetics of precipitate particles.[4–17] Although the theory was derived only for spherical particles, it was applied to almost all precipitate shapes, including nonspherical particles such as rods,[5] plates or discs,[6,11–15] and irregular shapes[8] because of the lack of nonspherical coarsening theories. It is very reasonable to argue, however, that the diffusion field or solute concentration distributed around a nonspherical particle would be quite different from that around a spherical one because of the different symmetries. Coarsening kinetics are thus dependent on angular positions for a nonspherical particle and cannot be simply modified on the basis of the LSW theory.[18] In fact, the agreement between the LSW theory and experiments on nonspherical particles is usually not good, and often requires serious modifications, which usually differ from case to case. The formation and lateral movement of growth ledges,[19,20] long recognized as a mechanism of precipitate growth,[21,22] have been suggested to play an important role in coarsening of precipitate particles.[23–31] Aikin and Pli-
YIWEN MOU, formerly with the Department of Materials Science and Engineering, University of Virginia, is with AFE Technologies, Charlottesville, VA 22901. J.M. HOWE, Associate Professor, is with the Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22903. Manuscript submitted May 20, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS A
chta[30] and Aikin et al.[31] modeled hexagonal disc-shaped precipitates as a circular cylinder having growth ledges on the edge or broad faces or on all the surfaces. The growthrate equations were formulated by applying Jones and Trivedi’s results[32,33] for the concentration gradient in front of a single growth ledge. Their analyses predicted much better coarsening behavior for disc-shaped particles than did many modified LSW approaches. However, ledge densities or interledge spacings were used as inputs to their kinetics equations, and these quantities were usually measured from a coarsening kinetics experiment itself. A disc-shaped precipitate can be modeled as an oblate spheroid or flattened sphere, just as a rod or needle-shaped particle can be modeled as a prolate spheroid or elongated sphere.[18] A spheroidal surface is macroscopically smooth everywhere; this feature simplifies the mathematics involved in solving the diffusion-field equation. Microscopically, on
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