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 2:5 4 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 20 30 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 ; 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 ; 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 t and the interface moves from a position xi to the solute content CL in the liquid at the interface linearly xi þ xi, as shown in Fig. 1, the interfacial temperature increases with time as follows, decreases by Ti due to cooling, and increases by Ti due V G Ci ¼ K t þ C ; ð11Þ to movement of the interface, which are given by L L 0 Ti ¼ ViðtÞ t; ð2Þ where KL is the proportional constant. In this case, the V R i i interface moves with a constant velocity given by TG ¼ G ðtÞ xi; ð3Þ mL where ViðtÞ and GiðtÞ are the cooling velocity and the R ¼ R0 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 TC 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 Ti ¼ m Ci ðtÞ; ð4Þ dCi kG R R C L L S ¼ L 0 : ð Þ 13 dx mL R where mL is the gradient of the liquidus line in the phase diagram, and is taken to be positive. These temperature When the interface moves with a constant velocity, the changes should be balanced to give the following relation, solute content in the initial transient solidification is given by 6) 7) Ti þ Ti ¼ Ti : ð5Þ Tiller et al. and by Smith et al., and it is known that in the V G C early stage of the solidification, the solute content given ) i i i V ðtÞ t þ G ðtÞ xi ¼ mL CLðtÞ: ð6Þ by Tiller et al.6) shows the similar change to the rigorous 7) When the moving velocity of the interface ðdxi=dtÞ is solution given by Smith et al. Therefore, the present represented by RðtÞ, eq. (6) is rewritten as follows, analysis was carried out by using the solution by Tiller 6) i et al., and the solute content in the liquid is given by dxi 1 i dCL RðtÞ¼ ¼ V ðtÞ m : ð7Þ 2 dt GiðtÞ L dt 1 k ðR Þ CL ¼ C0 1 exp k t k DL When the cooling velocity and the temperature gradient are R constant and represented by V0 and GL, respectively, eq. (7) exp ðx R tÞ þ 1 : ð14Þ is rewritten as follows, DL Criterion for Constitutional Supercooling at Solid-Liquid Interface in Initial Transient Solidification 181
From the above equation, the temperature gradient in the 1 liquid ahead of the interface is calculated. Then by comparing γ 1 ,
0 with the real temperature gradient GL, the criterion for the
R 5 stability of the planar interface is obtained as follows, */ 10
R 20 1 k R ðR Þ2 0.5 9 2 α / 10 °C·s/m mLC0 1 exp k t GL; ð19Þ k DL DL where t is the time while the planar interface stably moves, and eq. (19) is changed as follows, 0 0 10 20 1 DL DL k GL 9 2 t ¼ ln 1 : ð20Þ ξ 2 GL/R0, / 10 °C·s/m k ðR Þ mLC0 1 k R
Fig. 2 Change in R =R0 with GL=R0. In eq. (20), the logarithmic term should be positive to give the following restriction. The solute content in the liquid at the interface is obtained by GL mLC0 1 k inserting x ¼ R t in the above equation, and then differ- < ¼ : ð21Þ R D k entiated by time to obtain the changing rate of the solute L content. Finally, by inserting t ¼ 0, the changing rate of the Under this restriction, the distance x (¼ R t ) while the solute content in the liquid at the interface is obtained in a planar interface stably moves is obtained from eq. (20) as very short time after the start of solidification, and is given by follows, i @C C ð1 kÞ 1 DL DL k GL L 0 2 x ¼ : ð Þ ¼ ðR Þ : ð15Þ ln 1 22 @t t¼0 DL k R mLC0 1 k R By substituting eq. (15) into eq. (8), the following quadratic After substitution of R , eq. (22) is changed as follows, equation concerning to R is obtained, R0 m C ð1 kÞ ¼ k x L 0 2 D ðR Þ þ R R0 ¼ 0: ð16Þ L rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi DLGL 1 ¼ 1 þ 1 þ ln 1 1 þ 1 þ ; The positive solution of this equation is given by, 2
2 ð23aÞ R ¼ R0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4mLC0ð1 kÞ R0 where is the normalized distance of x , and 1 þ 1 þ DL GL D k 2k 1 ¼ L ¼ ¼ : ð23bÞ 2 2mLC0 1 k 2 ¼ R0 sffiffiffiffiffiffiffiffiffiffiffiffi; ð17aÞ The normalized distance of the stable growth monotoni- 1 þ 1 þ cally increases with increasing GL=R0 to infinity at =ðk þ 1Þ, as shown in Fig. 3 by using k of 0.14. In the where figure, however, the larger value of the distance is not valid because the model is applicable only for the early stage of the 4mLC0ð1 kÞ GL ¼ ¼ 4k ; ¼ : ð17bÞ solidification. In the present calculation, the changing rate of D R L 0 the solute content was derived at t ¼ 0. Here, the maximum The parameter ð¼ ðmLC0=DLÞð1 kÞ=kÞ is the criterion for distance Xc is defined as the distance at which the changing the constitutional supercooling given by Tiller et al.6) As rate of the solute content decreases by expð "Þ of the initial shown in Fig. 2, the moving velocity of the interface R value, where 0 <"<1, and is obtained from eq. (14) as monotonically increases with increasing GL=R0 from zero to follows, rffiffiffiffiffiffiffiffiffiffiffiffi approach the nominal moving velocity R0. D X ¼ L 1 þ 1 þ "; ð24Þ c kR 2.3 Criterion for stable growth of planar interface in 2 0 initial transient solidification or normalized, In this section, the criterion for the stable growth of rffiffiffiffiffiffiffiffiffiffiffiffi kR0 1 the planar interface in the initial transient solidification is ¼ X ¼ 1 þ 1 þ ": ð25Þ c D c 2 discussed following the criterion of the constitutional super- L cooling. The temperature distribution in the liquid ahead of The normalized maximum distance c for " ¼ 1 is included 6) the interface expected from the solute distribution is given by in Fig. 3. In the figure, c is decreased with increasing 1 k ðR Þ2 GL=R0 to converge unity, and the normalized distance of T ¼ T m C 1 exp k t L 0 L 0 k D the stable growth represented by a dashed line exceeds c , L and hence is not valid. R exp ðx R tÞ þ 1 : ð18Þ Additionally, when the interface moves with the nominal D L moving velocity R0, there is a region of the stable growth of 182 H. Kato and Y. Ando
8 Electric furnace α / 109 °Cs/m2 Specimen 6 5 1 2 10 20 Cu cylinder δ * Brass plate 4
Cu plate 2 20 ∆ * (ε = 1) Normalized distance c Blower 1 0 Fig. 4 Setup for unidirectional solidification. 10 20 30 9 2 GL/R0, ξ / 10 °Cs/m
almost all of the liquid droplets disappeared in the solid Fig. 3 Changes in normalized distance and maximum distance c for stable growth with GL=R0. and the interface became planar. At this time, the interface situated at a fixed position with a planar shape, and hereafter referred to as the stationary interface. the planar interface because the pile up of the solute is so (3) After heating for the required time to obtain the small in the beginning of the solidification. In this case, the stationary interface, the electric source was switched distance xnom while the interface stably moves is given by, off to cool the furnace. (4) After cooling for required times (0 45 s), the alumina 1 DL DL k GL x ¼ ln 1 : ð26Þ tube was quenched in water by removing the cupper nom k R m C 1 k R 0 L 0 0 plate. or normalized, Then the alumina tube was sectioned, mounted, polished and etched with a water solution of sodium hydroxide for R0 ¼ k x ¼ log 1 : ð27Þ observation of the microstructure. nom D nom L The temperature measurement was also carried out with alumina tubes with an inner diameter of 1 mm and 2 mm: 3. Experimental Procedure shallow slits are made on the alumina tube at 50 mm, 60 mm, 70 mm, 80 mm from the bottom of the tube to expose a part 3.1 Preparation of specimen of the aluminum alloy, and the K thermocouples of 0.3 mm Slender alumina tubes containing the Al-4 mass% Cu alloy in diameter were fixed with heat-resistant cement. However, were prepared for the solidification experiment. First, the the temperature measurement of the alumina tube with an aluminum alloy was melted in the carbon crucible in the inner diameter of 0.4 mm was not conducted because of the electric furnace heated at 800 C. Then the alumina tube was difficulty of setting the thermocouple. inserted in the molten alloy from the top of the electric furnace, kept for a few minute for preheating, and then the 3.3 Metallography and chemical analysis molten alloy was sucked into the tube by a slight vacuum. The microstructure near the interface was observed by Then the alumina tube containing the molten alloy was using the optical microscope. Then the solute profile in the cooled to room temperature, and was cut into pieces of solid around the interface was analyzed by using the electron 150 200 mm in length for the solidification experiment. probe X-ray micro-analyzer (EPMA) with the acceleration In the present work, alumina tubes of three sizes (inner voltage of 15 kV and the sample current of 0.5 nA. The diameter thickness = 2 mm 1 mm, 1 mm 0.5 mm, analysis was carried out along the longitudinal direction of 0.4 mm 0.3 mm) were used. the alumina tube from the planar interface to a position in the solid 300 mm apart from the interface with an interval of 3.2 Solidification experiment 1 mm, and then the data were averaged at every ten points to The solidification experiment was carried out with a setup reduce scatter of data. schematically shown in Fig. 4, as follows, (1) The alumina tube was inserted in a copper cylinder 4. Results and Discussion placed on a copper plate situated below the electric tubular furnace. 4.1 Microstructure at interface (2) The furnace was heated up to and kept at 800 C for Solid-liquid interfaces in the alloy specimen quenched 2:5 4 h, during which the copper plate was cooled by after heating for 2:5 4 h are shown in Fig. 5, in which the air with a blower. interface with a planar shape was observed. In the present By the preliminary experiment, it was found that the work, the solid-liquid interfaces under different situations are amount of the liquid droplet was decreased in the solid discussed. In the preceding section, the interface situated at behind the interface with increasing heating time, and a fixed position under the steady temperature profile was after heating of alumina tubes with a inner diameter of defined as ‘‘the stationary interface’’. And, in Fig. 5, the 0.4 mm and 2 mm for 2.50 h and 4 h, respectively, interface was observed in the specimen after quenching, but Criterion for Constitutional Supercooling at Solid-Liquid Interface in Initial Transient Solidification 183
Liquid
Solid
0.5 mm Fig. 5 Stationary interface in specimens. (a) d : 2 mm (b) d : 1 mm (c) d : 0.4 mm
in the length of the columnar structure with the elapsed time 800 Liquidus temp. after the start of furnace cooling. In the figure, the columnar
/ °C structure shows a finite length in a short time after the start of T 700 Position, h (mm) furnace cooling. This is because the length of fine structures F.T. 600 appeared on the interface was included in the length of the 80 70 columnar structure. The fine structure was observed on the 500 60 interface in all specimens quenched with/without the furnace
Temperature, Temperature, cooling, and hence it was thought to be formed during 50 400 quenching. Therefore, it was thought that the columnar 0 100 200 300 400 structure appeared when the coarser structure than the fine Elapsed time, t / s structure was observed on the interface, namely 20 30 s after the start of furnace cooling, and then grew with a consid- Fig. 6 Change in temperatures at different positions in specimen of 1 mm erable rate. In the figure, dashed lines are the estimated diameter during furnace cooling. F.T. means a temperature of atmosphere lengths of the structure being assumed to develop with the in the furnace. nominal growth velocity R0, and gradients of the estimated length show a good agreement with those of the experimental might be different from one before quenching. Therefore, this results except the initial stage. interface is called as ‘‘the observed interface’’. Finally, the To verify the relation between the delay time before interface existed or moving in the specimen before quenching appearance of the columnar structure and the specimen is called as ‘‘the interface’’. diameter, the temperature change was precisely examined In Fig. 6, temperature changes during the furnace cooling just after the start of furnace cooling. Figure 11 shows the of the specimen with a diameter of 1 mm are shown. temperature change after the start of furnace cooling at a Assuming that the initial interfacial temperature was coin- position where the stationary interface might situated (here- cident with the liquidus temperature (647 C) of the Al- after referred to as the initial interface position). Although 4 mass% Cu alloy,18) the position of the stationary interface the temperature was not measured at the initial interface was estimated to be 60 mm 70 mm from the specimen position, the temperature at each position decreased with the bottom. The specimen of 2 mm in diameter also showed the same rate as shown in Fig. 6, and hence the temperature same temperature change, but took slightly lower temper- change at the initial interface position was estimated by atures than those in the specimen of 1 mm in diameter. interpolating those obtained by thermocouples situated just Figures 7–9 show microstructures in the specimen below and above the initial interface position. Although the quenched after furnace cooling of different times. In a short delay time before the appearance of the columnar structure time after the start of furnace cooling, the interface of the was different for two specimens, the temperature at which planar shape was observed. After 20 30 s, the cellular or the columnar structure appeared was almost the same, and it columnar structure appeared on the observed interface. With was thought that the difference in the delay time was mainly the passage of the time, the columnar structure developed and due to difference in the cooling velocity. changed into the dendrite with side arms. In addition, in Next discussion is on the considerably long delay time Fig. 7(b), the specimen was broken just above the observed before appearance of the columnar structure. As shown in interface, and in the other specimens, cracks appeared above Fig. 11, the temperature was almost constant for a while and the observed interface, which are caused by sudden shrinkage then decreased. Therefore it might be thought that the of the specimen during quenching. Figure 10 shows changes columnar structure appeared when the temperature drop 184 H. Kato and Y. Ando
(a) 15 s (b) 25 s (c) 30 s
(d) 32.5 s (e) 35 s (f) 45 s
Fig. 7 Change in solid-liquid interface in specimens of 2 mm in diameter with elapsed time. 0.5 mm
(a) 15 s (b) 20 s (c) 22.5 s
(d) 25 s (e) 27.5 s (f) 30 s
Fig. 8 Change in solid-liquid interface in specimens of 1 mm in diameter with elapsed time. 0.5 mm
started. However, the delay time was not coincident with the Mizukami et al.19–21) studied the initial solidification stationary time while the temperature was almost constant, process of 18 mass% Cr-8 mass% Ni stainless steel quenched but the columnar structure appeared several seconds after the on a copper substrate, and reported that a large supercooling start of temperature drop. As the reason of this discrepancy, occurred on the specimen surface before the start of solid- following phenomena may be pointed out: ification, and then the cellular structure was formed on the (1) A supercooling occurred following the rapid cooling substrate. In the present work, however, the solidification to give the driving force for forming the columnar started from the stationary interface with a relatively slow structure. rate, and no need of a large supercooling for the start of (2) The interface stably moved with the planar shape, and solidification. Also, even if the supercooling occurred, there then became unstable to form the columnar structure. was only a small difference in the specimens quenched Criterion for Constitutional Supercooling at Solid-Liquid Interface in Initial Transient Solidification 185
10 s 15 s 20 s
25 s 30 s 45 s
Fig. 9 Change in solid-liquid interface in specimens of 0.4 mm in diameter with elapsed time. 0.1mm
2.5 650 V : 0.38 °C/s Exp R0 Diameter 0 1.5 2 0.4 mm / °C 1 mm T 640 / mm 1.5 2 mm V0: 0.53 °C/s 1 (mm) colum colum L 1 2 mm dia. 0.5 L 630 1 mm dia. Temperature, Temperature, L. columnar struct., 0.5 0 0 10 20 30 40 50 0 10 20 30 40 50 Elapsed time, t / s Elapsed time, t / s Fig. 11 Change in temperature at position of solid-liquid interface with elapsed time. Fig. 10 Change in length of columnar structures Lcolum with elapsed time of furnace cooling.
micrograph, the right side of the figure was coincident with with/without the furnace cooling. Therefore, in the case (1), the fine structure appeared on the observed interface, and the the furnace cooling did not considerably affect the solute steep increment of the solute content a few mm before the redistribution in the solid behind the interface. To the right side of the profile is due to the second phase Al2Cu in contrary, when the interface stably moved with the planar the fine structure. shape before the columnar structure appeared as the case (2), In the present work, the solidification started from the the solute concentration gradually increased from the initial stationary interface, and as schematically shown in Fig. 13, content before quenching in water. In this case, existence/ there was a solute distribution following the solidus line in absence of the furnace cooling before quenching might affect the solid behind the stationary interface,18) and with the the solute profile in the solid behind the interface. movement of the interface, the solute content increased in To specify the reason of the delayed appearance of the the liquid at the interface. Following this situation, the columnar structure, the solute profile in the solid behind the position of the stationary interface was fixed as follows. observed interface, namely in the lower region below the First, the solidus line was plotted to fit the solute profile in observed interface, was analyzed. Figure 12 compares the the solid apart from the observed interface as shown by solute distributions behind the observed interface in the dashed-dotted lines in Fig. 12. Then the stationary interface specimen quenched without the furnace cooling (hereafter was fixed at a position where the line crosses the solute referred to as the specimen-NF) and one quenched after the profile. Following this procedure, the stationary interface furnace cooling (the specimen-F). The optical micrograph at was assumed to be at a position 100 150 mm behind the the interface was also included in the figure. As shown in the observed interface. 186 H. Kato and Y. Ando
Observed planar interface C
200 µm Moving of interface
i C L
C0
CS
Solid Liquid kC0 1 Distance 0.4 mm in mold diameter xA Stationary interface 0.9 (a) no furnce cooling Fig. 13 Change in solute content with moving of interface from stationary CS 0.8 interface.
0.7 / mass%
C moved with the planar shape with gradually increasing the 0.6 solute content, and then during quenching, the interface moved additionally at some distance to follow the increase (b) furnace cooling xB 0.9 for 15 s in the solute content. Therefore, the distance xB from the Estimated with observed interface to the stationary interface in the specimen- Cu content, 0.8 eq. (12) F was larger than the distance xA in the specimen-NF. Dashed lines in the figures are the solute content in the 0.7 solid ahead of the stationary interface estimated with eq. (13) and will be explained precisely in the next section. 0.6 -100 0 100 Difference in the solute distributions behind the observed Distance from stationary int., Dst / µm interface supports the case (2), and it was thought that after the start of furnace cooling, the interface stably moved with 1 the planar shape for 20 30 s, and then the morphological 2 mm in mold diameter change of the interface occurred into the cellular or columnar xA 0.9 (c) no furnce cooling structure. In the next section, the experimental results were compared with the theoretical calculations. 0.8 4.2 Comparison between experimental results and the- 0.7
/ mass% oretical calculations C 0.6 In this section, the solidification parameters, such as the moving velocity of the interface and so on, are calculated (d) furnace cooling x 0.9 for 30 s B with the thermal conditions. In the present experiment, as shown in Fig. 11, just after the start of furnace cooling, the Cu content, 0.8 temperature at the interface changed very slowly, and then decreased with a constant cooling velocity. Therefore, the 0.7 temperature change was approximated by dashed-dotted lines in Fig. 11; the temperature was constant for the 0.6 stationary time of 13 s and 11.5 s, and then cooled with a -100 0 100 constant velocity V ¼ 0:38 C/s and 0.53 C/s, for speci- Distance from stationary int., D / µm 0 st mens of 2 mm and 1 mm in diameter, respectively. Fig. 12 Comparison of copper content profiles in specimens of 0.4 mm in Then, by using values of material properties and thermal mold diameter (a) and (b), and 2 mm in mold diameter (c) and (d) with/ conditions tabulated in Table 1, the solidification parameters, without furnace cooling. such as the moving velocity R , the moving distance x and the time t of the stable movement of the interface, and so on The solute distribution from the stationary interface to the were calculated as shown in Table 2. observed interface was slightly different for both cases. Even In Table 2, for the specimen of 2 mm in diameter, the in the specimen-NF, the solute concentration largely de- moving velocity R of 9.64 mm/s is far lower than the creased from the observed interface to the minimum solute nominal growth rate R0 ¼ 60:8 mm/s, and the moving content. This was thought that during cooling in the distance x of 22.4 mm is comparable with the difference in quenching, the interface moved keeping the planar shape at the distance (xB xA) of about 60 mm and far longer than the some distance to follow the increase of the solute content nominal distance xnom ¼ 0:56 mm. Moreover, the calculated from the minimum solute content of kC0. On the contrary, delay time tdelay before appearance of the columnar structure in the specimen-F, during the furnace cooling, the interface (¼ tst þ t ) was 15.3 s for the specimen of 2 mm in diameter Criterion for Constitutional Supercooling at Solid-Liquid Interface in Initial Transient Solidification 187
Table 1 Material properties and thermal conditions in experiment.
Material properties Values (Ref. No.) Partition coefficient, k 0.14 18) 2 1 9 Diffusion constant in liquid, DL/m s 2:6 10 10) 1 Gradient of liquidus line, mL/ C (mass%) 2.6 18) Constant, / C s m 2 13:5 109 Constant, /m2 ( C) 1 s 1 0:203 10 9 Constant, / C s m 2 24:1 109 Diameter of specimen, d/mm Thermal conditions 21 1 3 3 Temperature gradient around 647 C, GL/ C m 6:25 10 7:63 10 1 Cooling velocity around 647 C, V0/ C s 0.38 0.43 2 9 9 GL=R0, / C s m 0:103 10 0:135 10 1 Nominal velocity of interface, R0/mm s 60.8 56.4
Distance of stable movement with velocity R0, xnom/mm 0.56 0.79
Stationary time, tst/s 13 11.5
Table 2 Comparison of calculated and measured solidification parameters.
Diam. specimen, d/mm Solidification parameters 21 Moving velocity of interface defined by eq. (13), R /mm s 1 9.64 10.12 Distance of stable movement with velocity R , x /mm 22.4 24.8 Distance between planar interface and stationary interface for 91 96 specimen quenched without furnace cooling, xA/mm Distance between planar interface and stationary interface for 154 132 specimen quenched after furnace cooling, xB/mm Elapsed time of stable movement with velocity R , t /s 2.3 2.4 Calculated delay time before appearance of columnar structure, 15.3 13.9 tdelay (Calculated)/s Experimental delay time before appearance of columnar structure, 30 23 tdelay (Experiment)/s Gradient of solute content in solid ahead of stationary interface, i 1 0.0018 0.0019 dCS=dx/mass% mm
and was longer than the delay time tdelay of 13.9 s for the specimen of 1 mm in diameter, and this magnitude relation 5. Conclusion was consistent with that of the experimental delay time. i Finally, the changing rate (dCs=dx) of the solute content In the initial transient solidification, due to the change in in the solid at the interface following the movement of the solute content at the interface, the moving velocity of the the interface was estimated by using eq. (13) to be 0.0018 interface was less than the nominal velocity R0. Assuming mass%/mm and 0.0019 mass%/mm for the specimens of that the temperature gradient GL and the cooling velocity V0 2 mm and 1 mm in diameter, respectively. As shown in are constant and also that the solute content linearly increases Fig. 12, the estimated solute content ahead of the stationary with time, the moving velocity of the interface R is obtained interface (dashed lines in the figures) showed a rather good as follows, agreement with the measured solute profile, but was slightly 2 4mLC0ð1 kÞ higher than the measured one. R ¼ R0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ : 1 þ 1 þ ðR =G Þ D From these discussions, the experimental results were 0 L L relatively in good agreement with the theoretical calculations And, the distance x while the planar interface stably moves carried out with a simple model, and it was inferred that the is obtained as follows, interface stably moved with the planar shape for a short time rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi DL in the furnace cooling, and then became unstable to result in x ¼ 1 þ 1 þ ln 1 1 þ 1 þ ; 2kR the morphological change into the cellular or columnar 0 structure. where ¼ 2k= and ¼ GL=R0. 188 H. Kato and Y. Ando
Then the solidification experiment was carried out: the 72–78. (in Japanese) Al-4 mass% Cu alloy inserted in the alumina cylinder with 21) H. Mizukami, T. Suzuki and T. Umeda: Tetsu-to Hagane 78 (1992) 95–102. (in Japanese) different inner diameters was heated for 2:50 4 h under a temperature gradient to obtain the stationary solid-liquid interface, and then cooled in the furnace. After required Appendix: Nomenclature times, the alumina tube was quenched in water for observa- tion of the interface. The interface with the planar shape was CL: Solute content in the liquid i observed for 20 30 s after the start of furnace cooling, and CL: Solute content in the liquid at the solid-liquid interface i then the columnar structure appeared and grew ahead of the CS: Solute content in the solid at the interface interface. In the solid behind the observed interface, the C0: Average solute content in the alloy solute content gradually decreased with the distance apart DL: Diffusion constant of the solute in the liquid i from the interface, took a local minimum and then gradually TC: Temperature change at the interface following the increased, and the stationary interface was thought to be change in the solute content, appearing in eq. (4) i at the position of the minimum solute content 100 150 mm TG: Temperature change at the interface following the behind the interface. The distance between the observed change in the interface position, appearing in eq. (3) i interface and the stationary interface was larger for the TV: Temperature change at the interface following specimen quenched after furnace cooling than that quenched cooling at the interface, appearing in eq. (2) without the furnace cooling. GiðtÞ: Temperature gradient in the liquid at the interface The experimental results were compared with the theoret- at time t ical calculations, and it was inferred that during the furnace GL: Constant temperature gradient in the liquid at the cooling, the interface stably moved with the planar shape interface from the stationary interface for a short time and then became k: Equilibrium partition coefficient at the interface unstable to change into the columnar structure. KL: Proportional constant for the solute content in the liquid at the interface, appearing in eq. (11) Acknowledgements mL: Gradient of the liquidus line RðtÞ: Moving velocity of the interface at time t, appearing The authors would like to express their thanks to Mr. Y. in eq. (8) Kawada, Saitama University, for his help in preparation of R0: Nominal moving velocity of the interface given by the specimens, and also to Associate Professor K. Kageyama, V0=GL Saitama University, and Dr. T. Okane, National Institute R : Moving velocity of the interface given by R0 of Advaned Industrial Science and Technology (AIST), for ðmL=GLÞKL, appearing in eq. (12) useful discussion with them. t : Time while the planar interface stably moves with the growth rate R , appearing in eq. (20) REFERENCES tst: Time while the temperature was almost constant after start of furnace cooling 1) H. Noguchi, S. Abe and M. Murakawa: J. Japan Soc. Precision Eng. 69 tdelay: Delay time before appearance of columnar structure (2003) 125–129. (in Japanese) after start of furnace cooling 2) S. Abe, H. Nuguchi and M. Murakawa: J. Japan Soc. Precision Eng. 70 (2004) 1407–1411. (in Japanese) T0: Melting temperature of pure metal i 3) J. A. Bardt, G. R. Bourne, T. N. Schmidt, J. C. Ziegert and W. g. T : Temperature at the interface Sawyer: J. Mater. Res. 22 (2007) 339–343. TL: Temperature in the liquid 4) J. H. Park, S. O. Choi, R. Kamath, Y. K. Yoon, M. G. Allen and M. R. ViðtÞ: Cooling velocity at the interface at time t Prausnitz: Biomed. Microdevices 9 (2007) 223–234. V0: Constant cooling velocity at the interface 5) Y. Tang, W. K. Tan, J. Y. H. Fuh, H. Tl. Loh, Y. S. Wong, S. C. H. Thian and L. Lu: J. Mater. Process. Technol. 192–193 (2007) 334–339. x : Distance where the planar interface stably moves with 6) W. A. Tiller, K. A. Jackson, J. W. Rutter and B. Chalmers: Acta Metall. the velocity R , appearing in eq. (22) 1 (1953) 428–437. xnom: Nominal distance where the planar interface stably 7) V. B. Smith, W. A. Tiller and J. W. Rutter: Can. J. Phys. 33 (1955) 723– moves with the nominal velocity R0, appearing in eq. (26) 745. Xc : Maximum distance at which the condition of the early 8) L. Nastac: J. Crystal Growth 193 (1998) 271–284. 9) H. Kato: PhD Thesis (University of Tokyo, 1975) pp. 203–224. stage solidification is satisfied, appearing in eq. (24) ¼ 4mLC0ð1 kÞ ¼ 4k : Constant, appearing in eq. (17b) 10) W. D. Huang, Q. M. Wei and Y. H. Zhou: J. Crystal Growth 100 (1990) DL 26–30. ¼ DL ð k Þ¼2k ¼ 1 : Constant, appearing in eq. (23b) 2mLC0 1 k 2 11) W. W. Mullins and R. F. Sekerka: J. Appl. Phys. 35 (1964) 444–451. : Normalized distance of the stable growth with the 12) R. F. Sekerka: J. Appl. Phys. 36 (1965) 264–268. velocity R , appearing in eq. (23a) 13) R. F. Sekerka: J. Crystal Growth 3 (1968) 71–81. 14) V. V. Voronkov: Soviet Physics—Solid State 6 (1965) 2378–2381. nom: Normalized distance of the stable growth with the 15) R. T. Delves: Physica Status Solidi (b) 16 (1966) 621–632. velocity R0, appearing in eq. (27) m C 16) R. T. Delves: Physica Status Solidi (b) 17 (1966) 119–130. ¼ L 0 1 k: Criterion of the constitutional supercooling DL k 17) X. Yao, A. K. Dahle, C. J. Davidson and D. H. StJohn: J. Mater. Res. 21 given by Tiller et al.6) (2006) 2470–2479. : Parameter given by GL=R0, appearing in eq. (17b) 18) H. Kato and T. Umeda: J. Crystal Growth 38 (1977) 93–102. 19) H. Mizukami, T. Suzuki and T. Umeda: Tetsu-to-Hagane 77 (1991) c : Normalized maximum distance to satisfy the 134–141. (in Japanese) condition of the early stage solidification, appearing in 20) H. Mizukami, T. Suzuki and T. Umeda: Tetsu-to-Hagane 78 (1992) eq. (25)