Cellular and dendritic growth: Part II. Theory
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I.
INTRODUCTION
THE importance of directional solidification studies has been well recognized since the early systematic scientific investigations carried out by Chalmers and co-workers ~ to understand the solidification characteristics of alloys. A majority of alloys grow under conditions which give rise to dendritic interfaces. For this reason, considerable theoretical and experimental attempts have been made to understand the characteristics of dendritic growth carried out under controlled solidification conditions. Unfortunately, no satisfactory treatment is available as yet which can quantitatively explain all the available experimental data and which correctly incorporates all the physics of the problem. It is the purpose of this paper to present a detailed theoretical model of cellular and dendritic growth under controlled solidification conditions. An approximate model of dendritic interface growth under controlled solidification conditions was given by Bower et al. 2 They predicted that the undercooling, T, at the dendritic tip is given by the expression AT -
GLD R
[1]
where GL is the temperature gradient in liquid, D the solute diffusion coefficient in liquid, and R the growth rate. Such a formulation is valid only under large temperature gradient and low velocity values. Their treatment does not consider the variation of dendrite tip radius or sidewise diffusion near the dendrite tip region. The three-dimensional solute diffusion problem was treated by Burden and Hunt, 3 and their results provided an excellent qualitative understanding of the variation of tip temperature as a function of GL and R. They assumed that, for a given temperature gradient, tip undercooling passes through a minimum as velocity is increased. A formulation was developed by Trivedi4 under the condition which ensures that the dendrite tip is stable so that it can grow in steady-state fashion. A constitutional supercooling parameter is introduced, assuming that the tendency for perturbation to grow is directly proportional to the parameter if there is any perturbation in the dendrite shape. YASUNORI MIYATA and TOSHIO SUZUKI are Associate Professors with the Department of Mechanical Engineering, Faculty of Engineering, The Technological University of Nagaoka, 949-54, Nagaoka, Japan. Manuscript submitted June 25, 1984. METALLURGICALTRANSACTIONS A
Another approach was proposed by Kurz and Fisher, s where the radius of curvature of a dendrite tip was assumed to be proportional to the wave-length of perturbation in a planar interface. Their prediction provides a close quantitative understanding of the radius of curvature of the dendrite tip as a function of GL and R. The approaches given by Trivedi and by Kurz and Fisher suggest an essential importance of the condition that the dendrite tip is stable in shape. The purpose of this paper is to develop a consistent theory of dendritic growth which takes into account the heat flow and the solute one around the dendritic front, and which correctly predicts the limiting behavior of a plan
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