Experimental and numerical study of pattern formation in faceted cellular array growth
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INTRODUCTION
D E P E N D I N G on the entropy of fusion of the material and the crystallographic orientation of the interface, a growing crystal can exhibit either a faceted or a nonfaceted solid/liquid interface. [q It has been recognized that, generally, most metals and organic compounds are nonfaceting materials, while most semiconductors, semimetals, and some organic compounds are faceting crystals, although there exists a transition from faceting to norffaceting as the interface undercooling, or the growth rate, is increased. Nonfaceting crystals have been studied for quite some time, whereas faceting crystals have just recently begun to receive much attention, due to the technological importance of semiconductors. [2-81 When a crystal, either faceting or nonfaceting, is grown from its melt unidirectionally in a temperature field, initially a planar solid/liquid interface will be formed and will be maintained, if the interface is stable, against natural perturbations. However, if the interface is unstable, due to, for example, an increase in the growth rate, V, or the melt solute concentration, Co, or a decrease in the temperature gradient, G, in the melt ahead of the interface, instabilities will develop on the interface, starting from the intersections of the interface with grain boundaries. These instabilities will eventually evolve into an array of cells, faceted or nonfaceted. Further increases in the instability of the interface will lead to the formation of a dendritic array, which again can be faceted or nonfaceted. The dimension of these cells or dendrites, i.e., the cellular/dendritic spacing, A, determines to a great extent the physical, chemical, and mechanical properties of the grown crystal.
DONGKAI SHANGGUAN, formerly Assistant Research Engineer, The University of Alabama, is now Manufacturing Engineer, Electronics Division, Ford Motor Company, Dearborn, MI 48121-6010. JOHN D. HUNT, Reader in Physical Metallurgy, is with the Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom. This paper is based on a presentation made in the symposium "Growth and Configurations of Faceted Solid-Liquid Interfaces" presented during the TMS annual meeting, New Orleans, Louisiana, February 17-21, 1991, under the auspices of the TMS Solidification Committee. METALLURGICAL TRANSACTIONS A
It has long been a subject of interest to predict the nonfaceted cell/dendrite spacing as a function of the growth conditions (V, G, and Co) and material parameters. Due to difficulties in handling a time-dependent problem, most theoretical treatments, t9-~51 analytical or numerical, have assumed steady-state growth. Almost all of the theoretical models, regardless of the variety of assumptions made and solution techniques involved, have come to the same conclusion that there is a very wide range of steady-state solutions for a given growth condition for a given alloy. That is, there seems to be no unique correspondence between the steady-state cell/ dendrite spacing and the external growth condition. How
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