Kinetics of heterogeneous nucleation of gas-atomized Sn–5 mass%Pb droplets

Materials Science and Engineering A 473 (2008) 206–212

Kinetics of heterogeneous nucleation of gas-atomized Sn–5 mass%Pb droplets Shu Li a , Ping Wu a , Wei Zhou a , Teiichi Ando b,∗ a

b

Department of Applied Physics, School of Science, Tianjin University, Tianjin 300072, China Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115, USA Received 21 November 2006; received in revised form 15 March 2007; accepted 21 March 2007

Abstract A method for predicting the nucleation kinetics of gas-atomized droplets has been developed by combining models predicting the nucleation temperature of cooling droplets with a model simulating the droplet motion and cooling in gas atomization. Application to a Sn–5 mass%Pb alloy has yielded continuous-cooling transformation (CCT) diagrams for the heterogeneous droplet nucleation in helium gas atomization. Both internal nucleation caused by a catalyst present in the melt and surface nucleation caused by oxidation are considered. Droplets atomized at a high atomizing gas velocity get around surface oxidation and nucleate internally at high supercoolings. Low atomization gas velocities promote oxidation-catalyzed nucleation which leads to lower supercoolings. The developed method enables improved screening of atomized powders for critical applications where stringent control of powder microstructure is required. © 2007 Elsevier B.V. All rights reserved. Keywords: Gas atomization; Droplet nucleation; CCT diagram

1. Introduction Gas atomization provides an essential means for metallic powder production and is the most widely used commercial rapid solidification process (RSP) for constitutionally complex alloys such as tool steels and nickel-based superalloys [1–3] because of its ability to mass-produce high-performance pre-alloyed powders at low cost. Gas atomization, however, inherently produces droplets of non-uniform diameters that must solidify at different cooling rates. Therefore, atomized powders are often sieved to remove powder particles of undesirable mesh sizes in order to guarantee product quality. Such sieving is justified primarily on the basis of empirical relationships between cooling rates and microstructural parameters such as the secondary dendrite arm spacing (SDAS) [4]. This, however, is misleading because it is the prior supercooling, and not necessarily the cooling rate, that determines the extent of rapid solidification during recalescence [5–9]. Droplet supercooling is determined by the kinetics of droplet nucleation that depend upon the complex interplay

∗

Corresponding author. Tel.: +1 617 373 3811; fax: +1 617 373 2921. E-mail address: [email protected] (T. Ando).

0921-5093/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2007.03.109

among the potency and density of nucleation catalysts and droplet cooling conditions [9]. Droplets of the same size may nucleate at different temperatures if produced under different atomizing conditions or from different melt stocks. Therefore, an ability to predict the complex nucleation behavior of atomized droplets is essential in screening atomized powders into products of desired quality. The nucleation of molten alloy droplets has been studied extensively using various techniques including hot-stage droplet dispersion methods [10–12], emulsification methods [13–17], levitation melting methods [18–20] and drop tower methods [21,22]. Most of these studies, however, only address the nucleation of small stationary droplets except for the ones that used drop tower methods. Recently, the nucleation kinetics of traveling Sn–5 mass%Pb droplets have been studied using a controlled capillary jet breakup process [23] that can generate a steam of mono-disperse droplets [24–28]. The determined nucleation kinetics were presented in the form of continuous-cooling transformation (CCT) diagrams specific to the capillary jet breakup process, for both surface and internal heterogeneous nucleation [26,28,29]. The latter models can also predict nucleation temperature for any other droplet solidification process provided that the droplet cooling schedule is known. The present study combines

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the nucleation-temperature computation models [26,28,29] and a model for droplet cooling in gas-atomization [30] to predict the nucleation kinetics of gas-atomized Sn–5 mass%Pb droplets in relation to the droplet diameter, the atomizing gas velocity and the oxygen potential in the atmosphere. The ultimate goal of the study is to provide a theoretical basis for the screening of gas-atomized powders and optimizing process parameters for maximum yields of useful powders. 2. Models The nucleation temperature of an alloy droplet, TN , depends on the droplet size, the cooling condition (schedule) and the density and potency of heterogeneous nucleation catalysts. The density and potency of nucleation catalysts depends on alloy purity and may vary from an alloy melt to another. Oxide formation on the surface of the droplet may also increase the potency for surface nucleation [24,31,32]. In any droplet solidification process, the droplet size and cooling conditions can be varied independently, for example by using different cooling media, such as argon and helium. Even with the same cooling gas, different droplet solidification processes produce different droplet cooling schedules. For example, in centrifugal atomization, droplets, once they are formed, only decelerate while traveling in a stationary gas atmosphere. This causes the convective cooling rate to decrease monotonically with time. In gas atomization, however, the relative velocity between the droplet and the gas (which determines the instantaneous convective cooling rate) has a high initial value but decreases sharply to a minimum as the droplet is accelerated to the gas velocity, and increases again as the gas velocity quickly attenuates while the droplet decelerates more slowly due to inertia effects. This produces complex cooling schedules for droplets in a gas-atomized spray as addressed by Liu et al. in their modeling of the droplet cooling in gas atomization [30]. The nucleation-temperature prediction models presented in Refs. [26,28,29] treat the droplet size, the cooling condition (schedule) and the density and potency of heterogeneous nucleation catalysts as de-coupled parameters, and therefore are applicable to the complex conditions of droplet cooling encountered in gas atomization. These models and Liu et al.’s model for the cooling of gas-atomized droplets [30] are first outlined below and then applied to the nucleation of gas-atomized Sn–5 mass%Pb droplets in the sections that follow. 2.1. Prediction of nucleation temperature 2.1.1. Oxidation-catalyzed surface nucleation The nucleation of a solid in a liquid usually takes place heterogeneously, either within or on the surface of the liquid. Internal nucleation is catalyzed by a heterogeneous nucleant that preexists in the liquid, but surface nucleation may be caused by oxide formation on the surface of the melt [25,31,32]. The earliest model for the prediction of TN for the surface nucleation of a cooling liquid was presented by Dong et al. [26] in which

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continuous-cooling nucleation kinetics are computed from     TN dt axM Q N exp − − dT = 1 2 2 RT dT (T − T ) T (T − T ) TL L L (1) where a is the surface area of the liquid, x the fraction of the surface that is potent for nucleation, Q the activation energy for the diffusion in the liquid, TL the liquidus temperature of the alloy, dt/dT the inverse of the instantaneous cooling rate, and M and N are material-specific constants. The values of M and N are determined for the alloy of interest by substituting experimental TN data and cooling schedules in Eq. (1). Using Eq. (1) for oxidation-catalyzed surface nucleation requires an expression for x which should increase from zero toward unity as oxidation proceeds. Assuming that the oxidation on the liquid surface proceeds by the nucleation and growth of disk-shaped oxide islands, Li et al. obtained [29]:   T   E x = 1 − exp − π PO2 C exp − RTi T0    T    2  dt E dt dTi × G · exp − dT RT dT dTi Ti (2) where T0 is the initial temperature, PO2 the oxygen partial pressure in the gas, E and G are the activation energy and driving force for oxidation, respectively, and C is a constant specific to the oxidation reaction of concern. Substituting Eq. (2) in Eq. (1) yields an equation for oxidation-catalyzed surface nucleation of a droplet that contains four constants, M, N, C and E. The four constants are first determined for the alloy of interest by substituting four sets of experimental TN data obtained for known droplet cooling schedules, i.e., T0 and dt/dT. Eq. (2) then predicts TN for any cooling schedule. 2.1.2. Internal nucleation The nucleation of a solid in a molten droplet may also be catalyzed by an internal nucleant. Wu and Ando re-write Eq. (1) for such internal heterogeneous nucleation as [28]:    TN dt vd M ∗ N∗ Q − dT = 1 (3) exp − 2 2 RT T (TL − T ) dT TL (TL − T ) where vd is the volume of the melt (droplet) and M* and N* are constants whose values are specific to the alloy of interest. Solving Eq. (3) with two sets of experimental nucleation data and cooling schedules yields the values of M* and N* which are then used in Eq. (3) to compute TN for any cooling schedule. 2.2. Droplet flight dynamics and cooling in gas atomization The cooling schedule of the droplets in gas atomization, i.e., T0 and dt/dT in Eqs. (1) and (2), can be calculated with the model presented by Liu et al. [30]. In their model, the gas flow is assumed to be laminar so that the velocity of a droplet traveling

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in the gas is calculated from [33]: Cd πrd2 4 3 dVd 4 πrd = πrd3 (ρd − ρg )g − ρg V |V | 3 dt 3 2 4˜ dV 3 − C A πrd ρp 3 dt

(4)

where rd is the droplet radius, ρd the density of the droplet, ρg the density of the gas, g the gravitational acceleration, Cd the ˜ A the added mass effect constant [33], and V drag coefficient, C is the relative velocity defined as the droplet velocity (Vd ) minus the gas velocity (Vg ). With the relative velocity calculated from Eq. (4), the convective heat transfer coefficient, h, is calculated from the Ranz–Marshall equation [34]:   

 kg VDρg 1/2 Cg ηg 1/3 h= (5) 2.0 + 0.6 D ηg Kg where kg , ηg , and Cg are, respectively, the thermal conductivity, viscosity and heat capacity of the gas and D is the diameter of the droplet. The droplet temperature T is then calculated from −vd ρd Cp

dT = a[εδ(T 4 − Tg4 ) + h(T − Tg )] dt

(6)

where ε is the emissivity, δ the Stephan–Boltzmann constant and Tg is the temperature of the gas. 3. Application to gas-atomized Sn–5 mass%Pb droplets 3.1. Droplet motion The above models were combined to predict the nucleation temperatures of helium gas-atomized Sn–5 mass%Pb droplets. The gas velocity, Vg , in gas atomization depends on the atomizing gas pressure and the type of atomizing nozzle used. From pitot-tube pressure measurements and isentropic gas dynamics calculations, Liu et al. [30] were able to estimate Vg for helium gas in ultrasonic gas atomization [35] as a function of the distance from the atomizing nozzle. The results indicated that a supersonic zone existed within about 15 cm of the nozzle where Vg had a high constant value of about 1700 m/s and that beyond this zone Vg rapidly decreased to below 100 m/s. Fig. 1 shows the Vg as a function of the distance from the nozzle reproduced from Liu et al.’s work.

Fig. 1. Calculated velocities of helium gas-atomized Sn–5 mass%Pb droplets of various diameters. The gas velocity is reproduced from Ref. [30] (initial gas velocity = 1700 m/s). Table 1 Thermophysical data required in Eqs. (4)–(6) obtained from Ref. [30] Property

Symbol

Value

Unit

Density of particlea Gas density (helium) Added mass effect coefficient Thermal conductivity (helium) Dynamic viscosity (helium) Heat capacity (helium) Heat capacity of particle Stephan–Boltzmann constant

ρd ρg ˜A C kg ηg Cg Cp δ

7298 0.18 0.5 0.2 2.5 × 10−5 950 176 5.67 × 10−8

kg m−3 kg m−3 – W m−1 K−1 Pa s J m−3 K−1 J m−3 K−1 W m−2 K−4

a

Calculated for Sn–5 mass%Pb by the rule of mixture.

h calculated with Eq. (5) as a function of the distance from the nozzle for Sn–5 mass%Pb droplets of various diameters. Fig. 3 shows the cooling curves of the Sn–5 mass%Pb droplets calculated from Eq. (6) with the values of h and Tg = 298 K. A strong dependence of cooling rate on droplet diameter is apparent.

3.2. Droplet temperature Fig. 1 also shows the velocity of Sn–5 wt.%Pb helium gasatomized droplets, Vd , calculated for various droplet diameters using the profile of Vg in Fig. 1 and the thermophysical data given in Table 1. As expected, Vd increases with decreasing droplet diameter. Also, droplets, regardless of their size, are accelerated initially until their velocity reaches the gas velocity and decelerated as the gas falls behind the droplets. Consequently, the convective heat transfer coefficient, h, has a high initial value but decreases rapidly as the droplet catches up with the gas and sharply increases again as Vd exceeds Vg . Fig. 2 shows the

Fig. 2. Heat transfer coefficient calculated as a function of the distance from atomization nozzle for different Sn–5 mass%Pb droplets (initial gas velocity = 1700 m/s).

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Table 3 Values of material-specific constants in Eqs. (1)–(3) calculated from the data in Table 2 Constant

Value

Unit

M N C E M* N*

2.7268 × 1015 1.8467 × 106 3.0063 × 1011 4.493 × 104 9.618 × 1021 2.82 × 107

K2 m−2 s−1 K3 mol2 m−2 s−3 N−2 PO−1 2 J/mol K2 m−3 s−1 K3

Table 3, are therefore specific to the Sn–5 mass%Pb alloy used in [26,28]. 3.4. Fraction oxidized, x Fig. 3. Calculated cooling curves of the Sn–5 mass%Pb droplets generated by helium gas atomization (initial gas velocity = 1700 m/s).

3.3. Material-specific constants The calculated cooling curves (schedules) in Fig. 3 give the values of dt/dT as a function of T and thus permit calculating the nucleation temperature for gas-atomized Sn–5 mass%Pb droplets from Eqs. (1) and (2) for oxidation-catalyzed surface nucleation, or from Eq. (3) for internal nucleation. The values of the constants, M, N, C and E, or M* and N*, however, need to be determined first. This can be achieved by substituting known values of TN and dt/dT in Eqs. (1) and (3) or Eq. (3) and solving the equations for the constants. Such nucleation data were obtained in previous studies with a high-purity Sn–5 mass%Pb alloy containing less than 0.25 mass% impurities [24–27] in which the nucleation kinetics of mono-size droplets generated by controlled capillary jet breakup [23] were determined by calorimetric measurements [24] and droplet in-flight cooling simulation [36]. The reported nucleation temperatures, compiled in Table 2, indicate that the nucleation of Sn–5 mass%Pb droplets is catalyzed by surface oxidation in the presence of oxygen in the atmosphere (Experiments 1–4) but is triggered by an internal catalyst when a reducing atmosphere was used (Experiments 5 and 6). Therefore, the surface oxide gives rise to a higher nucleation potency than the internal catalyst in this alloy. Li et al. [29] and Wu and Ando [28] used the nucleation data and cooling schedule simulations to calculate the constants M, N, C and E for surface nucleation and M* and N* for internal nucleation. The calculated values of the constants, listed in

With the value of C in Table 3, the fraction of the melt surface that is oxidized (and has hence become potent to cause surface nucleation) can be calculated using Eq. (2). Fig. 4 shows the variations of fraction oxidized, x, during cooling calculated for a 500 ␮m Sn–5 mass%Pb droplet at various oxygen concentrations. The dependence of x on the oxygen concentration is apparent. It should be noted that Eq. (2) does implicitly involve the droplet diameter since the calculation of dt/dT from Eqs. (4)–(6) requires to specify the droplet diameter. Since the cooling rate decreases with increasing droplet diameter, larger droplets will have larger values of x than smaller droplets when compared at the same oxygen concentration. 3.5. Prediction of nucleation kinetics With the above scheme, nucleation temperatures were computed for Sn–5 mass%Pb droplets atomized at initial gas velocities of 1700, 1000, 500 and 200 m/s. These initial velocities were chosen to cover more ordinary atomizing conditions as well as the extreme ones that may be encountered in highpressure gas atomization. Since measured values of Vg are

Table 2 Nucleation temperatures and supercoolings of Sn–5 mass%Pb droplets produced by capillary jet breakup in different gas atmospheres Exp. no.

Droplet dia. (␮m)

Atmosphere

TN (K)

T (K)

Reference

1 2 3 4 5 6

185 185 155 155 185 155

N2 –166 ppm O2 N2 –35 ppm O2 N2 –166 ppm O2 N2 –35 ppm O2 N2 –2% H2 N2 –2% H2

472 425 421 393 353 334

27 74 78 106 146 165

[26] [24] [26] [24] [26] [27]

Fig. 4. Fraction oxidized x as a function of temperature at different oxygen concentrations calculated for 500 ␮m Sn–5 mass%Pb droplets helium gas-atomized at an initial gas velocity of 1700 m/s.

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Fig. 5. Gas velocity attenuation curves for initial velocities of 200, 500, 1000 and 1700 m/s used in the computation of nucleation kinetics in the present study. The curves for 200, 500 and 1000 m/s were generated with exponential functions that mimic the curve for 1700 m/s determined by Liu et al. [30].

lacking, the attenuations of Vg from the initial values of 1000, 500 and 200 m/s were mathematically generated with exponential decaying functions mimicking the one for the initial value of 1700 m/s determined by Liu et al. Fig. 5 shows the corresponding attenuations of Vg used in the computation of nucleation temperatures. The artificial curves for the initial gas velocities 1000, 500 and 200 m/s are given to exhibit no initial constant zone. For more precise computations, actual attenuation curves need to be determined for the specific atomizing technology and conditions of concern as done by Liu et al. for their ultrasonic gas atomization experiment [30]. Figs. 6–9 show continuous-cooling transformation (CCT) diagrams for the nucleation of helium gas-atomized Sn–5 mass%Pb droplets calculated for the initial gas velocities of 1700, 1000, 500 and 200 m/s, respectively. The CCT curves are obtained as the trajectories of the points on droplet cooling

Fig. 6. Calculated CCT curves for the surface and internal nucleation of Sn–5 mass%Pb droplets helium gas-atomized at an initial gas velocity of 1700 m/s. The droplet cooling curves were calculated with Liu et al.’s model [30].

Fig. 7. Calculated CCT curves for the surface and internal nucleation of Sn–5 mass%Pb droplets helium gas-atomized at an initial gas velocity of 1000 m/s. The droplet cooling curves were calculated with Liu et al.’s model [30].

Fig. 8. Calculated CCT curves for the surface and internal nucleation of Sn–5 mass%Pb droplets helium gas-atomized at an initial gas velocity of 500 m/s. The droplet cooling curves were calculated with Liu et al.’s model [30].

Fig. 9. Calculated CCT curves for the surface and internal nucleation of Sn–5 mass%Pb droplets helium gas-atomized at an initial gas velocity of 200 m/s. The droplet cooling curves were calculated with Liu et al.’s model [30].

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curves calculated with Eqs. (4)–(6) at which nucleation is predicted to takes place. Many cooling curves were calculated for different droplet diameters to produce smooth CCT curves but only five of such cooling curves, representing 150, 250, 350, 500 and 750 ␮m droplets, are shown in the figures for a clear view of the computation results. The CCT curves were computed for both oxidation-catalyzed surface nucleation and internal nucleation using the values of the constants shown in Table 3 in Eqs. (1) and (2) or Eq. (3) and therefore are specific to the Sn–5 mass%Pb alloy used in the previous studies [24–27]. An observation common to all of the four gas velocity conditions is that the oxidation-catalyzed surface nucleation occurs at much higher temperatures than the internal nucleation, indicating a higher nucleation potency of the surface oxide. Therefore, large supercoolings of gas-atomized Sn–5 mass%Pb droplets would be obtained if surface oxidation were delayed or suppressed. The CCT diagram calculated for 1700 m/s, Fig. 6, shows that surface oxidation would not catalyze nucleation in Sn–5 mass%Pb droplets smaller than about 650 ␮m even at a very high oxygen concentration of 20,000 ppm (2%). This suggests that oxidation-catalyzed surface nucleation of gas-atomized Sn–Pb droplets may be circumvented almost regardless of the oxygen concentration in the gas if the gas velocity is sufficiently high. This is understood from Eq. (2) where the factor dt/dT, i.e., the inverse of the cooling rate, decreases with increasing gas velocity, thereby decreasing the extent of oxidation, x. Low initial gas velocities, however, increase the chance of oxidation-catalyzed surface nucleation as seen in Figs. 7–9 where the CCT curves calculated at 500, and 200 ppm O2 intersect with the cooling curve of 750 ␮m droplets. The critical droplet diameter above which nucleation is catalyzed by surface oxidation progressively decreases with decreasing initial gas velocity. Eq. (2) also suggests that the chance of oxidation-catalyzed surface nucleation decreases with decreasing oxygen concentration in the gas. In fact, large Sn–5 mass%Pb droplets, even if they were atomized at a low initial gas velocity of 200 m/s, would still be free of surface nucleation if the oxygen concentration were controlled below 200 ppm, Fig. 9. Therefore, it would relatively be easy to suppress oxidation-catalyzed surface nucleation in gas-atomized droplets of this particular Sn–5 mass%Pb alloy. However, alloys that are more susceptible to oxidation would require a higher gas velocity and/or a lower oxygen concentration, or even a reducing atmosphere, to prevent premature nucleation due to oxidation. Fig. 10 shows the oxidation-limited supercooling attainable in helium gas atomization with this Sn–5 mass%Pb alloy at initial gas velocities of 1700, 1000, 500 and 200 m/s and oxygen contents of 200, 500 and 20,000 ppm O2 for the diameter range where surface nucleation may result. Strong effects of gas velocity and oxygen content on the oxidation-limited supercooling are predicted, particularly toward the lower end of the diameter range shown. An important implication, which also applies to more readily oxidizing alloys, is that particles sieved from an atomized powder may not be the same microstructurally as those of comparable mesh sizes from another powder atom-

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Fig. 10. Oxidation-limited droplet supercooling vs. droplet diameter calculated for helium gas-atomized Sn–5 mass%Pb droplets atomized at initial gas velocities of 200, 500, 1000 and 1700 m/s in the presence of 20,000, 500 and 200 ppm O2 .

ized at different gas velocities or oxygen contents, i.e., particle size-based powder screening may not guarantee product quality, particularly for fine RSP powders. Figs. 6–9 also show CCT curves that represent the kinetics of nucleation caused by an internal catalyst. Internal heterogeneous nucleation should depend on the melt volume (i.e., droplet diameter) and the cooling rate as seen in Eq. (3). The computed CCT curves indeed indicate higher nucleation temperatures for larger droplets. However, for a given droplet size, the nucleation time decreases with increasing gas velocity while the nucleation temperature increases only slightly. For example, at an initial gas velocity of 1700 m/s, a 350 ␮m droplet would require only 0.008 s to cool and internally nucleate at 345 K while the same droplet, at an initial gas velocity of 200 m/s, would need 0.0013 s to cool and nucleate at a similar temperature of 355 K. This is because the driving force for nucleation builds up faster in a droplet cooled at a higher rate. Lower nucleation temperatures, and supercoolings well in excess of 150 K, are predicted for smaller droplets. It may even seem that very small droplet supercool with no limitations. The latter, however, would not be seen in reality. Since the CCT curves were computed with the values of M* and N* in Table 3, they refer to a specific catalyst that existed in the Sn–5 mass%Pb alloy used in the previous studies [24–27]. Other catalysts with different potencies may also exist in the alloy. Even if there were no other catalyst, homogeneous nucleation must eventually set in. 4. Summary A method for predicting the nucleation kinetics of gasatomized droplets has been developed by combining models predicting the nucleation temperature of cooling droplets with a model simulating the droplet motion and cooling in gas atomization. Both the internal nucleation caused by a catalyst present in the melt and the surface nucleation caused by oxidation are considered. Application to a 99.75% pure Sn–5 mass%Pb alloy yielded the following knowledge:

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(1) In helium gas atomization with a high gas velocity, Sn–5 mass%Pb droplets cool so fast that surface oxidationcatalyzed nucleation is not an issue unless the oxygen content in the atomizing gas is very high or the droplets are very large. Instead, droplets nucleate internally at high supercoolings. In conventional gas atomization with a low gas velocity, there may be enough time for an oxide to nucleate and grow on the droplet surface causing surface nucleation at low supercoolings if droplets are sufficiently large. (2) Assessment of the supercooling of gas-atomized droplets needs to consider all of the variables that determine the supercooling, namely the droplet size, the cooling condition, and the potency and density of internal and surface catalysts. Gas-atomized powder particles of a given size from different batches may have different microstructures and therefore may not be mixed. The developed model provides an improved method for screening atomized powders for critical applications where powder microstructures need to be stringently controlled. Acknowledgements The authors would like to acknowledge the support of the National Natural Science Foundation of China (50674071), the Tianjin Natural Science Foundation (06YFJZJC01300), the Program for New Century Excellent Talents in University (NCET) and the Platform Project of Tianjin for Innovation in Science and Technology and Environmental Construction (06TXTJJC13900). References [1] I.R. Sare, R.W.K. Honeycombe, Met. Sci. 13 (5) (1979) 269–279. [2] A. F¨olzer, C. Tornberg, Mater. Sci. Forum 426–432 (2003) 4167–4172. [3] J.V. Wood, P.F. Mills, A.R. Waugh, J.V. Bee, J. Mater. Sci. 15 (1980) 2709–2719. [4] R.E. Spear, G.R. Gardner, Trans. AFS 71 (1963) 209. [5] T. Kattamis, M.C. Flemings, Trans. AIME 236 (1966) 1523–1532. [6] C.G. Levi, R. Mehrabian, Metall. Trans. A 13A (1982) 13–23. [7] M.G. Chu, Y. Shiohara, M.C. Flemings, Metall. Trans. A 15A (1984) 1303–1310. [8] W.J. Boettinger, L. Bendersky, J.G. Early, Metall. Trans. A 17A (1986) 781–790.

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