Statistical analysis of the disorder of two-dimensional cellular arrays in directional solidification
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I.
INTRODUCTION
I T has long been recognized that the physical properties of a material are strongly influenced by the microstructure which is formed in the solid phase during processing. Consequently, controlled solidification techniques that permit a close control of the microstructure have been extensively developed to produce engineering components. Therefore, it follows that the prediction of the microstructure as a function of the processing conditions is crucial in engineering. Besides, recent theoretical advances have greatly improved the current knowledge of the stability and selection of nonlinear structures in several pattern-forming instabilities observed in different areas of science fj'2m that were previously disconnected. In directional solidification, the morphological instability of the planar solidification front I4~ results in the formation of ceils and dendrites at the solid-liquid interface, which belong to this continuously expanding family. Apart from their outstanding role in practical applications, these patterns have thus received further attention on a more physical standpoint, as archetypes for the study of the nonlinear aspects of morphogenesis. When a binary alloy, whose solute concentration is C~, is solidified in an imposed temperature gradient G, a fiat solid-liquid interface is no longer morphologically stable when the growth velocity V is above a critical value V,. First, the solidification front develops a cellular microstructure. {Sj These cells advance parallel to the thermal gradient and, when a bulk sample is grown, selforganize into an array of columns (Figure l(a)) which, by chemical etching on a transverse section, shows up as a two-dimensional (2-D) array (Figure 1(b)). For usual processing conditions, such arrays are never perfect and always contain a certain amount of topological defects. 161 In comparison with a few structures observed in other pattern-forming instabilities, e.g., Rayleigh-B6nard conB. BILLIA, Senior Scientist and Group Leader, H. J A M G O T C H I A N , Senior Scientist, and H. N G U Y E N THI, Senior Scientist, are with the Laboratoire de Physique Cristalline de I ' U R A CNRS 797. Universit6 d'Aix-Marseille 1II, 13397 Marseille, France. Manuscript submitted December 27, 1990. METALLURGICAL TRANSACTIONS A
vection, 171 Brnard-Marangoni convection, and nematicshear-flow instability, I8J the analysis of the organization of the cellular arrays is just beginning. On lead-30 wt pct thallium alloys grown upward by the Bridgman method, we have recently initiated such an analysis, t61 By using Wigner-Seitz construction, 19j the basic defects (polygons with less, or more, than six sides) have been identified and numbered. The major result was that the percentage of defects is high enough to induce a defectmediated melting of the regular honeycomb, which is the "2-D solid" array predicted by nonlinear theories. Concomitantly, only a short-range order is detected by the correlation functions. The main objective of the present article is the statistical analys
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