The Growth of a Single Cell/Dendrite in a Directional Solidification Process
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DUE to the pressing need to optimize properties of a material, microstructure control through solidification processing has become increasingly important; therefore, research focus has been on modeling the formation and lengthscale selection of solidification microstructures. Depending on solidification conditions (growth velocity (V), temperature gradient (G), and alloy composition (Co)), the solid/liquid interface can assume a planar or nonplanar (cellular or dendritic) pattern. For the cellular/ dendritic morphology, the shape of a growing tip and the spacing selection have been studied both theoretically and experimentally.[1–8] For the evolution of a dendrite tip of a pure substance, Ivantsov[1] first solved the equations governing the growth from its undercooled melt and found that the tip should be a paraboloid of revolution and the tip radius should be controlled by the undercooling of the melt bath. The smooth region of a dendrite tip in both undercooled growth and directional growth of alloys is also found to be a paraboloid of revolution.[2,6] However, the shape of a smooth cell in the directional growth of alloys is much more complicated:[5–11] the tip region may be fit into a Saffman–Taylor finger and the tail part usually follows the Scheil equation. A criterion to match these two regions remains unclear and some mathematical treatments have been proposed, but all lack the clear physics.[7,12] On the other hand, the complex cell shape can be studied through numerical calculations[4,5,13] and phasefield simulations,[14–16] and the calculated shape can be S. LU, Professor, is with the School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China. S. LIU, Associate Scientist, is with the Materials and Engineering Physics Program, Ames Laboratory-USDOE, Ames, IA 50011, USA. Contact e-mail: [email protected] This article is based on a presentation made in the symposium entitled ‘‘Solidification Modeling and Microstructure Formation: In Honor of Prof. John Hunt ’’, which occurred March 13–15, 2006, in the TMS Spring meeting in San Antonio, TX, under the auspices of the TMS Materials Processing and Manufacturing Division, Solidification Committee. Article published online March 31, 2007. 1378—VOLUME 38A, JULY 2007
directly compared with experiments. Hunt et al.[4,5] have pioneered the computation of cell shapes in a directional solidification process and significant progress has been made ever since the first finite-difference work by McCartney and Hunt.[4] Hunt et al. have not only studied the shape evolution of a single cell but the competitive growth of an array of cells as well.[13,17] The work on the array stability has been well appreciated and verified through precisely controlled experiments;[18–21] however, for the simpler case where the growth of a single cell has been simulated,[5,13] critical experimental results that may verify the calculation are still lacking. This is because growth of a single cell/dendrite requires the use of a capillary tube as
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