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Report on Panel Discussion I1" Critical Problems in the Mathematics of Ledgewise Growth

C. ATKINSON, M. ENOMOTO, W.W. MULLINS, and R. TRIVEDI Wednesday, October 4, 1989, Indiana Convention Center, Indianapolis, IN.

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 spatially discrete description of atomic transport should be used near an atomic scale sink. Furthermore, on this scale, fluctuations may be important. (3) What boundary condition is appropriate on the riser of a macroledge? If one assumes a constant concentration (e.g., C~qui0, the ledge changes shape as it moves because of the variable gradient along the riser, increasing toward the top comer of the riser. To obtain a ledge of constant shape, corresponding to experimental evidence in some cases, one usually imposes a uniform normal velocity along the riser surface as an ad hoc boundary condition; a close approximation to this condition at low supersaturations is a constant gradient of concentration normal to the riser surface. The same problem arises in the description of the growth of a faceted crystal of constant shape by volume diffusion to the crystal facets. The growth of a facet or macroledge of fixed shape by diffusion can presumably be understood in terms of a diffusion boundary condition for surface elements that is strongly anisotropic or singular at the facet orientation. An explicit development of such a boundary condition in terms of the interface microstructure to replace the ad hoc procedure described above would be desirable. (4) Other factors that have not been adequately taken into account in the diffusion boundary conditions at ledges and kinks are the possible kinetic elevation of concentration at a sufficiently rapidly moving ledge/kink and the effect of interactions (entropic, elastic, etc.) between neighboring ledges/kinks. The interactions affect the diffusion boundary conditions at ledges/kinks, because they generally produce a net force on a ledge in a ledge train of variable spacing; to balance this force, the local equilibrium concentration at the ledge must shift. On a macroscale, the interactions are manifest as a contribution to the orientation dependence of the interfacial free energy. (5) It would be desirable to explore the relation between the detailed theories of growth by ledge motion due to Atkinson et al., based on Green's source function solutions of the diffusion equation and on asymptotic matching, with solutions based on formulations that use a coarse-grained concentration (on a scale of ~ledge spacing) and average boundary conditions related to the interface structure (e.g., Mullins' treatment); in particular, under what conditions can the coarse-grained treatment be derived from the detailed treatments? The coarse-grained theory assumes that the concentration field in the vicinity of the interface sinks (e.g., ledges or kinks) relaxes sufficiently fast so that it is determined by the instantaneous interface configuration and the average

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gradient normal to the interface via Laplace's equation. In the detailed theory, on the other hand, the previous history of ledge cortfigttrations and motions enters as well into the determination of the concentration field at a given instant. The two formulations can be equivalent only when the configurational history of ledges can be ignored. Computer simulation indicates that it is legitimate to ignore this history if the supersaturation is sufficiently low. This is consistent with the fact that Laplacian solutions for the local concentration field (appropriate at low supersaturations) cannot store historical information. (6) The accuracy of computer models (e.g., Enomoto's model) has not been fully evaluated for situations in which different steps move at different speeds and step nucleation and annihilation (or coalescence) occur. It may be necessary to make a detailed comparison of the computer

results with analytic treatments, such as that presented by Atkinson, before the computer simulations can be applied with confidence to these more complex situations. (7) Models to date have used Fick's law for diffusion in the matrix (usually with a constant diffusion coefficient D) and thus have omitted stress-induced diffusion. Furthermore, stress has not been explicitly taken into account in the boundary conditions used at the interface. It would be desirable to incorporate stress effects. (8) The finite difference computer model of ledgewise growth, in its present form, requires computation speeds and memory sizes of the fastest supercomputers in order to process realistic problems in a reasonable time. It would be desirable to develop a more efficient simulation model which could cope with real growth problems on more commonly accessible computers.

1248--VOLUME 22A, JUNE 1991

METALLURGICAL TRANSACTIONS A

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