Orientation dependence of primary dendrite spacing
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I.
INTRODUCTION
OBSERVATIONS of growing solid-liquid interface in transparent analogues have led to a better understanding of planar interface breakdown and microstructure formation during solidification. In particular, it is possible to measure steady-state dendrite tip radius and tip temperature, primary dendrite spacing, and sidebranch spacing. Such measurements are complementary to those carried out with metallic alloys,v-8] Growth models have also been developed to predict the features of dendritic microstructures which are of primary importance for the control of the mechanical properties of cast alloys. The existing models of cellular and primary dendrite arm spacings, A~, relate dendrite tip radius and the length of the mushy zone t9-12] or analyze the stability of an array of dendrites,v3a4] The results of the analytical modelst*-~2] follow the relationship Aicx ATo 9 v -b 9 G -c
[la]
where AT0 is the freezing range of the alloy, v the velocity of the dendrite tip, and G the thermal gradient. The theoretical predictions of Hunt[g] and Kurz and Fished ~~ give a = b = 0.25 and c = 0.5. Experimental values of the positive coefficients b and c are found to vary from 0.19 to 0.75 and 0.3 to 0.56, respectively. Few results for systems in which convection effects are minimizedt121 show the coefficient a to be close to 0.25.E2.41 Most of the observations and theories dealing with constrained growth apply to the case of dendrites aligned with the heat flow direction. The only study concerning the effect of off-axis heat flow on the primary dendrite spacing was carried out by Grugel and ZhoutSl in the succinonitrile-water system. They noticed an increase of A~ with an increase in the orientation angle, 0, between the primary dendrite trunk and the thermal gradient directions. This was also noticed by Gandin and RappazVS] when studying the selection of grains operating in a directionally solidified succinonitrile-l.3 wt pct acetone alloy, t4,51It has
CH.-A. GANDIN, Postdoctoral Fellow, is with the Laboratoire de M~tallurgie Physique, Ecole Polytechnique F~drrale de Lausanne, CH1015 Lausanne, Switzerland. M. ESHELMAN, Postdoctoral Fellow, and R. TRIVEDI, Professor, are with the Ames Laboratory, United States Department of Energy, Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011. Manuscript submitted May 2, 1995. METALLURGICAL AND MATERIALSTRANSACTIONS A
been pointed out that at a "divergent" grain boundary, the secondary and tertiary branching mechanism is responsible for the creation of new primary dendrite spacings. t5,71 This mechanism will be discussed in more detail subsequently. If vL is the external velocity, or the velocity of the liquidus isotherm, the velocity along the direction of the dendrite axis is given by (vL/cos 0). Thus, from Eq. [ 1a], the primary spacing will be altered by (cos 0)b or (cos 0)0.25so that one would expect primary spacing to decrease as the angle 0 is increased. Since the primary spacing increases with the orientation,[8.~51 the orientation
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