The effect of convection on disorder in primary cellular and dendritic arrays
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I. INTRODUCTION
ONE of the important microstructural length scales of directionally solidified alloys is the primary spacing, or the periodicity, of the cellular and dendritic structures. This periodicity often controls the microsegregation profiles and governs the formation of a second phase in the intercellular or interdendritic regions and, consequently, influences the properties of the material. Several theoretical and experimental studies have, thus, been carried out to relate primary spacing to the growth rate, temperature gradient, and composition.[1–16] Although the average spacing is most often reported in the literature, experimental studies show that a significantly broad spectrum of primary spacings is present.[6] This disorder in spacings can occur due to two factors: a weak spacing-selection criterion, which causes a range of spacings to be present, and the effect of convection in large or bulk samples. Fundamental studies on the distribution of primary spacings under diffusive-growth conditions have been carried out theoretically as well as experimentally. Warren and Langer[2] have developed a diffusive-boundary-layer model which predicted that the primary dendrite spacing would not have a unique value under given growth conditions. Rather, a band of primary spacings would be stable. A similar conclusion was reached by Hunt and Lu[4] through a detailed numerical solution of the primary-spacing problem. They concluded that the maximum spacing would be less than a factor of 2 from the minimum spacing; otherwise, a new cell or dendrite R. TRIVEDI, Professor, is with the Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011. Contact e-mail: [email protected] P. MAZUMDER, formerly Graduate Student, Department of Mechanical Engineering, Iowa State University, is Research Scientist with Corning Inc., Corning, NY 14831. S.N. TEWARI, Professor, is with the Chemical Engineering Department, Cleveland State University, Cleveland, OH 44115. Manuscript submitted June 1, 1999. METALLURGICAL AND MATERIALS TRANSACTIONS A
with a minimum spacing would be created. Thus, the spacing distribution under given growth conditions should, at most, be within a factor of 2 if diffusive growth conditions are present. Detailed experimental studies, carried out in thin samples of a transparent material in which the primary spacing is controlled by the diffusion process, have confirmed the presence of a finite but small range of spacings under given growth conditions.[6] Through studies of the dynamics of spacing adjustments, it was shown that cells can translate laterally to minimize the disorder, and a new cell or dendrite was created when the spacing exceeded about 1.5 times the minimum stable spacing.[6] In contrast to the previous results, experimental studies in bulk metallic and organic systems have clearly shown the presence of a wide range of spacings or a highly disordered spatial arrangement of cells and dendrites. Convection effects have been assumed to be responsible for this large d
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