Report on panel discussion II: Critical problems in the mathematics of ledgewise growth
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B R I E F presentations were made in the following three areas: setting up models of precipitate growth by the ledge/kink mechanism (R. Trivedi), computer simulation of growth according to the models (M. Enomoto), and analytic treatment of the models (C. Atkinson). During and after the presentations, the following points were made. (1) Many models treat growth by analyzing the motion of some initial distribution of ledges (e.g., a finite group or an infinite train of ledges) on an infinite interface. They, therefore, omit the effect of the production of ledges, either by nucleation or by a source (e.g., an emergent screw dislocation), and also omit the annihilation of ledges (e.g., at precipitate edges). The inclusion of sources dates back to the treatment of crystal growth by the emergent screw dislocation mechanism in the classical work of Burton, Cabrera, and Frank, but the treatment of volume diffusion in the matrix in that work and most subsequent developments of that work is only approximate. It would be desirable to combine a more rigorous treatment of the diffusion-controlled motion of ledges/kinks with a treatment of the production and annihilation of ledges by sources and sinks. Enomoto's computer simulation work on grain boundary allotriomorphs, presented at the conference, is a step in this direction. It takes into account the nucleation of ledges at grain boundaries, as well as ledge coalescence on the precipitate, and thus raises the possibility of connecting grain boundary precipitate morphology with the rate of ledge nucleation and the reaction time. (2) Most mathematical treatments of growth by the ledge mechanism have used continuum diffusion equations and boundary conditions, thereby implicitly assuming that the height of the ledges is large compared to atomic dimensions. In fact, ledges and kinks may range in scale from atomic to multiatomic (macro) dimensions, depending on energetic and/or kinetic factors; not all of these ledges/ kinks are necessarily diffusion sinks. In the vicinity of an atomic scale ledge/kink that does act as a diffusion sink, the concentration (occupation probability) generC. ATKINSON is with the Department of Mathematics, Imperial College of Science and Technology, Queen's Gate, London SW7 2BZ, United Kingdom. M. ENOMOTO is with the National Research Institute for Metals, Tokyo 153, Japan. W.W. MULLINS, Panel Chairman, is with the Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, Pittsburgh, PA 15213. R. TRIVEDI is with Ames Laboratory, Iowa State University, Ames, IA 50011. This paper is based on a presentation made in the symposium "The Role of Ledges in Phase Transformations" presented as part of the 1989 Fall Meeting of TMS-MSD, October I-5, 1989, in Indianapolis, IN, under the auspices of the Phase Transformations Committee of the Materials Science Division, ASM INTERNATIONAL. METALLURGICAL TRANSACTIONS A
ally varies strongly over atomic distances so that the continuum diffusion equation (Fick's law) breaks down. Hence, a sp
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