Two-dimensional finite difference analysis of shape instabilities in plates
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I.
INTRODUCTION
P L A T E - , or near-lamellar, shaped particles are susceptible to shape instabilities as a result of curvature-induced diffusional processes which lead to a diminished interfacial surface area per unit volume. For particles immersed in a vapor, the atomic transport which leads to such shape changes may take place by vapor transport or interfacial (surface) diffusion. If the plate is embedded in a liquid or solid medium, the associated transport occurs by either interfacial diffusion or diffusion through the phase surrounding the plate. Recent work in our laboratory [L2'31has delineated several basic paths by which plates change their shape in response to capillarity. We have called these changes direct cylinderization, boundary splitting, and edge spheroidization; they are illustrated schematically in Figure 1. During direct cylinderization, the plate's transverse crosssectional area evolves into a circle. This happens as a result of mass transfer from the terminations of the plate to its broad faces. The resulting cylinder can then decompose into a row of spheres via a Rayleigh instability process, f4,s'6] Plate splitting may happen when internal boundaries are present within the plate. Boundaries of this kind may be present as a result of recrystallization or recovery of the plate material, or they may form during plate formation, as in directional solidification of lamellar eutectics. Establishment of the equilibrium dihedral angle at the juncture of the plate surface and the internal boundary produces a groove. Mass is transferred from the groove, which has finite curvature, to the nearby flat
surfaces of the plate. The equilibrium dihedral angle is then maintained by recession of the boundary, and the process eventuates in the splitting of the plate. Cylinderization follows plate splitting, if the new aspect ratios of the resulting plates are relatively small. If they are not, a third instability--edge spheroidization (which can also take place in plates not having internal bounda r i e s ) - - c a n intervene. Edge spheroidization is the result of two consecutive processes. During the initial stages of cylinderization, a ridge forms near the plate termination as a result of the short diffusion path from the termination surface to the closely adjacent flat surface. Following ridge formation, the "pseudo" cylinders at the plate edges can decompose into a row of spheres by a process analogous to the Rayleigh instability of a cylinder. In the work referenced, tL2,31experimental observations of these kinds of instabilities were made in two systems: plates of Fe dispersed in Cu and plates of W dispersed in a Ni-W solid solution. In addition, approximate analyses of the respective times for each instability were made in terms of the plate aspect ratio, the equilibrium dihedral angle developed at the juncture between the internal and interphase boundaries, and appropriate thermodynamic and kinetic factors. The dominant instability was assumed to be the one taking the least time to complete. Direc
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