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Criterion for Constitutional Supercooling at Solid-Liquid Interface in Initial Transient Solidification with Varying Solute Content at Interface

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Criterion for Constitutional Supercooling at Solid-Liquid Interface in Initial Transient Solidification with Varying Solute Content at Interface Materials Transactions, Vol. 52, No. 2 (2011) pp. 179 to 188 #2011 The Japan Institute of Metals Criterion for Constitutional Supercooling at Solid-Liquid Interface in Initial Transient Solidification with Varying Solute Content at Interface Hiroshi Kato and Yukihiko Ando* Division of Mechanical Science and Engineering, Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan A criterion for appearance of the constitutional supercooling at the solid-liquid interface in the initial transient solidification is discussed theoretically and experimentally. First, a relation between the moving velocity of the interface and the solute content was analyzed to derive a moving velocity of the interface under a simple model of the linear change in the solute content at the interface. And, a criterion for appearance of the constitutional supercooling at the planar interface was analyzed to obtain the distance of the stable growth of the interface with the planar shape. Then, the solidification experiment was carried out with the Al-4 mass% Cu alloy: the aluminum alloy was inserted in the alumina tube of 0.4 to 2 mm in inner diameter and heated for 2:54 h under a temperature gradient to obtain the stationary interface, and then the alumina tube was cooled in the furnace for 0 to 45 s. After furnace cooling, the alumina tube was quenched in water to observe the interface. The interface with the planar shape appeared for 2030 s after the start of furnace cooling, and then the columnar structure grew ahead of the interface. Then the solute content in the solid behind the interface was analyzed to show that the solute content in the specimen quenched after furnace cooling was different from that in the specimen quenched without furnace cooling. The experimental results were compared with the theoretical calculations to infer that the interface moved with the planar shape for a short time after the start of furnace cooling, and then the interface became unstable to form the columnar structure. [doi:10.2320/matertrans.M2010253] (Received July 27, 2010; Accepted November 10, 2010; Published December 22, 2010) Keywords: solidification, solid-liquid interface, initial transient, aluminum alloy, morphological stability 1. Introduction i mL dCL R ¼ R0 À ; ð1Þ There are increasing attention to micro-machines and GL dt micro-devices, which accelerated development of production where R0 is the nominal moving velocity of the interface techniques of micro-components, such as lithography, rapid defined by the cooling velocity V0 divided by the temperature prototyping, and so on. In the casting field, also, there have gradient GL, and mL is the gradient of the liquidus line. Huang been many reports1–5) on micro-casting techniques and et al.10) obtained the length of the initial transient region by production of micro-mold, but there are no reports on using the eq. (1). In these reports, it can be said that the dif- formation mechanisms of the solidification structure. In the ferent moving velocity of the interface may largely influence solidification of micro-scale components, the steady state the solute redistribution in the initial transient solidification. solidification is very limited or does not exist, but the initial Also there have been many reports on the morphological and terminal transient solidifications are dominant. There- stability of the solid-liquid interface. On the steady state fore, it is very important to realize the solidification process solidification process, analyses have been reported based on in the transient solidification, especially in the initial transient the constitutional supercooling by Tiller et al.6) and on the solidification, in order to understand the solidification perturbation of the interface by Sekerka et al.,11–13) by structure in the micro-components. Voronkov,14) by Delves.15,16) Also, Nastac8) reported the On the solute redistribution in the initial transient solid- morphological stability of the interface in the initial transient ification, there are many reports, such as the approximate solidification. These analyses have been conducted under analysis by Tiller et al.,6) and the rigorous analyses by constant velocity of the interface, but as mentioned above, Smith7) and by Nastac.8) These analyses have been carried the moving velocity of the interface is different from the out under the constant moving velocity of the solid-liquid nominal velocity in the initial transient solidification, which interface. Kato,9) however, has pointed out that in the initial results in different solute redistributions from that of the transient solidification, the solute content at the interface nominal moving velocity. Recently Yao et al.17) discussed changes with the movement of the interface resulting in the the nucleation in the initial transient solidification in change in the interfacial temperature, which means even consideration of the change in the solute content by using though the cooling velocity and the temperature gradient are the eq. (1), but they did not discuss the criterion for the constant, the moving velocity of the interface changes in the morphological stability in the initial transient solidification. initial transient solidification. Then he derived a relation In the present work, a moving velocity of the interface was between the moving velocity of the interface R and the solute derived in the initial transient solidification, and then the i content in the liquid at the interface CL, by equating the criterion for appearance of the constitutional supercooling at temperature distributions in the liquid ahead of the interface the interface was discussed to obtain a distance of the stable described by the fixed coordinate system and the moving growth of the planar interface. Then, the solidification coordinate system with the interface as follows, experiment was carried out by using the Al-4 mass% Cu alloy to observe the solidification structure, and the solute *Graduate Student, Saitama University. Present address: Calsonic Kansei content profile was analyzed in the solid around the interface. Corporation, Saitama 331-0823, Japan Finally, the experimental results were compared with the 180 H. Kato and Y. Ando T C t i mL dCL RðtÞ¼R0 À ; ð8Þ t + ∆t GL dt Phase Diagram where R is the nominal moving velocity of the interface C 0 L m Temperature Change - given by V0=GL. Equation (8) is the same as eq. (8)–(10) in 0 T Ti the ref. 9). Since the moving velocity RðtÞ takes a positive = ∆T i i C L ∆T T V ∆ i C i value, the right side of eq. (8) should be positive to give the T G L following relation, ∆ i T C Change in Solute Content i ∆C i dC R0GL L L : ð9Þ dt mL T S C0 X Temp ∆ By integrating eq. (9) in consideration of the condition: at T0 xi i t ¼ 0, CL ¼ C0, Fig. 1 Change in solute content and temperature change in initial transient R G solidification. i 0 L CL t þ C0: ð10Þ mL theoretical calculation to discuss the stable movement of the This equation gives the upper limit of the solute content in interface in the initial transient solidification. the liquid at the interface at time t in the initial transient solidification. 2. Mathematical Analysis In the present paper, the discussion is limited in the initial transient solidification, but eqs. (7) and (8) are also appli- 2.1 Relationship between moving velocity of interface cable to the unsteady solidification, such as the terminal and solute content in initial transient solidification transient solidification. In this section, the movement of the planar interface is analyzed in the initial transient solidification of a binary alloy 2.2 Moving velocity of interface in initial transient with a solute content of C0 and a partition coefficient of k.In solidification the present discussion, the partition coefficient k is assumed In this section, a very short time after the start of to be constant and less than unity. Symbols used in the solidification is considered, and it is assumed that the cooling present paper are summarized in Appendix. velocity and the temperature gradient are constant to be V0 At the beginning of the solidification, the solid with the and GL, respectively. Also, in the preceding section, it was solute content of kC0 appears neighboring the liquid of Co to shown that there is the upper limit in the changing rate of the form the solid-liquid interface. When the time passes from t solute content at the interface, and hence it is assumed that i to t þ Át and the interface moves from a position xi to the solute content CL in the liquid at the interface linearly xi þ Áxi, as shown in Fig. 1, the interfacial temperature increases with time as follows, decreases by ÁTi due to cooling, and increases by ÁTi due V G Ci ¼ K t þ C ; ð11Þ to movement of the interface, which are given by L L 0 ÀÁTi ¼ViðtÞÁt; ð2Þ where KL is the proportional constant. In this case, the V RÃ i i interface moves with a constant velocity given by ÁTG ¼ G ðtÞÁxi; ð3Þ Ã mL where ViðtÞ and GiðtÞ are the cooling velocity and the R ¼ R0 À KL: ð12Þ GL temperature gradient in the liquid ahead of the interface at time t, respectively. Also, the interfacial temperature de- And, when the planar interface stably moves, the solute i i creases by ÁTC with increasing solute content in the liquid at content CS in the solid at the interface changes with the the interface Ci as follows, following rate, L ÀÁTi ¼m ÁCi ðtÞ; ð4Þ dCi kG R À RÃ C L L S ¼ L 0 : ð Þ Ã 13 dx mL R where mL is the gradient of the liquidus line in the phase diagram, and is taken to be positive.
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