Primary spacing in directional solidification
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I.
INTRODUCTION
THE control of the morphology and characteristic microstructural spacings by directional solidification of alloys is a very powerful tool in solidification processing. Theoretical models and extensive experimental studies have now well established the morphology transitions from planarto-cellular-to-dendritic microstructure as a function of growth velocity.[1–4] However, the corresponding variation of the primary spacing, a very important microstructural feature in directional solidification, can not yet be satisfactorily described. In current analytical models, the dendritic envelope is normally imagined to be represented by a smooth interface with elliptical[4] or parabolic shape.[5,6] The curvature radius at the tip of the ellipsoid or paraboloid is assumed to be equal to the dendrite tip radius R. Therefore, the primary spacing l can be determined by geometrical consideration between R and l. As the solidification velocity V is increased from Vc, the critical velocity for planar front growth, the spacing l is generally predicted to increase first, then go through a maximum and finally decrease. Measurements[7–10] have shown a more complex l behavior. After breakdown of the planar front, the primary spacing first delivers a minimum during the cellular growth, then increases to a maximum, corresponding to the cellulardendritic transition, and finally decreases again within the purely dendritic growth regime. This l behavior cannot be reasonably described by the current analytical models. The experimental results also show that there is a range of velocities over which both dendritic and cellular structures can be present.[9] Recently, a numerical model has been developed by Lu et al. to describe the array growth of cells and dendrites.[11,12] The model was established for solving the solute transport problem in the liquid using a time-dependent finite difference method. Solution to the diffusion problem and the prediction of the self-consistent shape were made using a fully implicit control volume method. The model predicts a range of spacings for cells and a separate range DEXIN MA, Research Associate, is with ACCESS e.V., D-52056 Aachen, Germany. PETER R. SAHM, University Professor and Director, is with Giesserei-Institut RWTH Aachen, D-52056 Aachen, Germany. Manuscript submitted December 12, 1995. METALLURGICAL AND MATERIALS TRANSACTIONS A
for dendrites. By suitable nondimensionalization, Hunt and Lu tried to give the analytic expressions to fit the numerical results.[13] These analytic expressions allow experimental results to be easily compared with theory and allow the sensitivity of the spacing to the values of the physical constants to be gaged. This model was originally set up to describe cells but was extended when it appeared to be making relevant predictions for dendrites. So the treatment is limited because dendrite arms are not molded. Here, we suggest a new analytical model to give a simple description of the primary spacing. The model is based on the analyses of a typical dendrite
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