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Two-dimensional modelling and experimental study on microsegregation during solidification of an Al–Cu binary alloy
Acta Materialia 55 (2007) 1523–1532 www.actamat-journals.com
Two-dimensional modelling and experimental study on microsegregation during solidification of an Al–Cu binary alloy Q. Du b
a,*
, D.G. Eskin a, A. Jacot
b,c
, L. Katgerman
d
a Netherlands Institute for Metals Research, Mekelweg 2, 2628CD Delft, The Netherlands Laboratoire de Simulation des Mate´riaux, Ecole Polytechnique Fe´de´rale de Lausanne, CH-1015 Lausanne, Switzerland c Calcom-ESI SA, Parc Scientifique, Ecole Polytechnique Fe´de´rale de Lausanne, CH-1015 Lausanne, Switzerland d Delft University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands
Received 21 April 2006; received in revised form 9 October 2006; accepted 9 October 2006 Available online 20 December 2006
Abstract To explain the experimentally observed relation between the cooling rate and the non-equilibrium eutectic fraction obtained during solidification experiments of Al–Cu binary alloys [Eskin DG, Du Q, Ruvalcaba D, Katgerman L. Mater Sci Eng A 2005;405:1–10], a two-dimensional (2-D) microsegregation model, the pseudo-front tracking (PFT) method [Jacot A, Rappaz M. Acta Mater 2002;50:1909–26, Jacot A, Rappaz M. Acta Mater 2002;50:3971, Du Q, Jacot A. Acta Mater 2005;53:3479–93], is employed. The analysis of simulated solidification kinetics, including the grain morphology evolution, coarsening and back diffusion, reveals the pronounced effect of dendrite coarsening on the final non-equilibrium eutectic fraction. Because the 2-D modelling can describe rigorously how the important microstructure feature – dendrite arm spacing – evolves, its predictions are in good agreement with the trend observed in the experiments, which both the classic Brody–Flemings model and the 1-D PFT model have failed to explain. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Modelling; Microsegregation; Coarsening; Aluminium alloys; Dendritic growth
1. Introduction Microsegregation is the chemical inhomogeneity at the scale of a grain size or a dendrite arm, and is a result of non-equilibrium solidification of alloys. It is a hot research topic and both experimental and modelling approaches are employed to investigate the concentration profile along dendrite arms and the type and fractions of the interdendritic phases, because of their great importance for downstream processing and for mechanical properties [1–11]. It is clear that back diffusion in the solid is important in the formation of microsegregation, and the diffusion in the liquid is insignificant upon normal casting conditions. The other important factor is dendrite coarsening during solidification. Coarsening in Al–Cu alloys is a result of
*
Corresponding author. Tel.: +31 15 2782263. E-mail address: [email protected] (Q. Du).
the flux of Cu from coarse arms towards fine arms coming from the Gibbs–Thomson effect. During coarsening, highly curved solids (fine arms) are dissolved while regions of low or negative curvature (coarse arms) continue to grow. Although the coarsening is expected to reduce microsegregation [5,6], the extent of this effect is unclear (see Ref. [7] for a debate on this topic). It appears that, depending on the cooling rate, the shape of the cooling curve and the phase diagram, the influence of coarsening could be negligible on the solute distribution in the primary phase, but significant on the eutectic fraction formed [7]. Voller and Beckermann proposed an approximate model of microsegregation with coarsening, in which the effect of coarsening on microsegregation measured by the fraction of nonequilibrium eutectic was estimated [8]. They showed that, depending on the Fourier number (defined as a dimensionless solid-state back diffusion parameter calculated by the product of a constant solid diffusion coefficient and the local solidification time divided by the square of the
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.10.035
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characteristic solidification length), the final eutectic fraction could significantly deviate from the one obtained when coarsening was not taken into account. In contrast, Flemings concluded that the effect of coarsening on microsegregation is small, and certainly less than the effect of back-diffusion in the solid [9]. There are some one-dimensional (1-D) models available [10] in which the coarsening process of dendrite arms is described as a function of local solidification time. Yan et al. [11] demonstrated that the proper description of dendrite grain morphology is very important in modelling microsegregation, though, owing to their intrinsic geometrical assumptions, 1-D models cannot do this in a direct way. Certainly 2-D or 3-D direct simulations of complex grain morphology evolution would be more appropriate. Recently, a 2-D pseudo-front tracking (PFT) model has been developed by Jacot, Rappaz and Du [2–4] for normal casting conditions to predict the final concentration profile and the types and fractions of secondary phases formed, in which the evolution of dendrite arms can be explicitly tracked in a physical manner, and ripening as well as coalescence can be described based on their controlling factors, i.e. the combination of curvature effect and diffusion in the liquid, as well as the possible influence of the grain boundary energy [4]. The direct motivation to perform 2-D microsegregation modelling for the solidification of an Al–Cu alloy has been the interpretation of our earlier experimental results showing an unconventional relationship between cooling rate and non-equilibrium eutectic fraction, which initially increases in the range of slow cooling (<1 K/s) and then gradually decreases in the range of 1–20 K/s [1]. Similar experimental results had been reported by other authors, e.g. Novikov and Zolotorevsky [12]. Coarsening, undercooling of the eutectic reaction and efficient back diffusion due to structural refinement at a faster cooling rate were all suggested as potential causes of the observed behaviour. The diffusion in the solid (back diffusion) directly influences the prediction of the final eutectic fraction. The amount of solute transferred by back diffusion is proportional to the local concentration gradient at the periphery of a growing grain, the interfacial area and the solidification time. Obviously, the longer the solidification time, the more solute can diffuse into the solid phase. At the same time, the slower cooling rate accompanied by longer solidification time favors the formation of coarse dendrite arms. Therefore, the back diffusion in the coarse structure will be hindered by the reduced interfacial area and a lower local average concentration gradient. As a result of the opposite effects of the solidification time and the coarse structure, the final eutectic fraction is difficult to predict. Although the 1-D microsegregation models could predict the fraction of non-equilibrium phases, their average nature and the simplified grain morphology assumption make it impossible to describe how the interface dynamics and the spatial distribution of grain morphology affects the non-equilibrium phase formation. Two-dimensional calculations could shed
some light on the effects of coarsening and structure refinement on the formation of non-equilibrium eutectics. The results of such calculations are presented here below after a brief description of the model and the experimental procedure. 2. Model description The model combines a direct simulation of the primary phase formation based on the PFT method [2,3] and a mixture approach for the description of the formation of secondary phases [4]. In the 1-D PFT model, the solidified microstructure is assumed to consist of plate-like dendrite arms, the spacing of which is determined from the experimental secondary arm spacing; the diffusion in the liquid and solid can be taken into account, but the curvature undercooling is neglected; the calculation domain is half of the experimental measured value. The 2-D PFT model takes into account both diffusion in the liquid and solid phases and curvature undercooling, which together determine the velocity of the solid–liquid interface. Therefore, this model can explicitly track how the grain morphology evolves. The curvature of solid–liquid interface is calculated with the piecewise linear interface calculation technique [13]. As soon as the liquid becomes undercooled for a secondary solid phase, the calculation enters a second stage in which the mixture approach is invoked and aimed at predicting the formation of secondary phases in the remaining interdendritic regions. In this approach, the interdendritic regions are considered as a mixture of liquid and solid phases. The model provides the evolution of the interdendritic mixture as solidification proceeds, taking into account the effects of back diffusion, which continuously modifies the composition and the volume of the interdendritic regions. See Refs. [2–4] for more details about this model. The effect of the eutectic reaction undercooling can only be taken into account by imposing a pre-described undercooling value, since no rigorous description of the eutectic reaction is included in the 2-D PFT model. This choice is dictated by the numerical difficulty to resolve spatially both the grains of the primary dendritic phase, which are on a scale of 100 lm, and the interdendritic eutectic phases which are on a scale of several microns. In this paper, the effect of the eutectic undercooling is neglected. The possible implications of the eutectic reaction undercooling on the eutectic fraction in our experiments are discussed elsewhere [1]. Due to the 2-D nature of the model, some input parameters, such as the surface energy, have to be tuned to reproduce the appropriate average dendrite arm spacing (DAS) observed upon the metallographic examination. When the calculated DAS is close to the measured value, the effect of grain morphology on the eutectic fraction is considered to be properly taken into account. With this idea, simulations of solidification experiments with cooling rates of 0.8 and 16 K/s were performed (for the choice of the
Q. Du et al. / Acta Materialia 55 (2007) 1523–1532
cooling rates, see Section 4). Table 1 lists the calculations parameters. The analytical model proposed in Ref. [8] is employed to compare with the prediction obtained by PFT model. This analytical model is based on the approximation that coarsening can be modeled in a standard microsegregation model as a back-diffusion process characterized by an enhanced diffusion parameter [10] and has the ability to account for both coarsening and back diffusion. For more details about this model please refer to Refs. [8,10]. 3. Experiments A binary Al–2.53% Cu alloy was prepared in an electrical resistance furnace from 99.9% pure aluminium and an Al–48% Cu master alloy. The alloy was grain-refined with a commercial Al–5% Ti–1% B master alloy, the total amount of Ti being 0.03%. No melt treatment was performed. The chemical composition of the alloy was determined by a spark spectrum analysis. The total amount of impurities was less than 0.1%. Samples were melted inside a cylindrical resistance furnace in alumina crucibles of about 4 cm3 volume. Prior to melting, a K-thermocouple was embedded in the solid sample in a specially drilled hole. In order to assure the good contact between the thermocouple and the melt, a steel-sheathed thermocouple assembly with an external diameter of 1 mm was used. Liquid samples were overheated to 710 C and then cooled inside or outside the furnace to the solidus temperature, with still or forced air, oil or water as a cooling medium.
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The samples were quenched in water as soon as they were fully solid. The cooling rate in the solidification range was then calculated from the cooling curve as an average cooling rate between the liquidus and the solidus [1]. The samples were cut in the horizontal plane as close as possible to the position of the thermocouple tip. These samples were then ground, polished and examined in an optical microscope. Samples for calculation of the grain size were additionally oxidized in a 5% water solution of fluoboric acid. The dendrite arm spacing, the volume fraction of the eutectic phase and the grain size were calculated using standard linear intercept method, both manually and using image-analysis software. Statistical analysis of the results was performed. 4. Results and discussion The experimentally determined dependence of the nonequilibrium eutectic fraction on the cooling rate in the tested alloy is shown in Fig. 1. The same tendency that was observed earlier for different non-grain-refined alloys [1] is confirmed now for the grain-refined alloy. The volume fraction of non-equilibrium eutectic increases with the cooling rate in the range of slow cooling, i.e. up to approximately 1 K/s, and then starts to decrease in the range of cooling rates from 1 to 20 K/s. Two points, at 0.8 and 16 K/s, were chosen for computer simulations. Fig. 2 gives microstructures of the alloy cooled at these two cooling rates, within a frame identical to the domain size used in the calculations (see Table 1).
Table 1 Input parameters for the calculations Calculation Alloy System Phase diagram
Slow cooling, 2-D
Slow cooling, 1-D
Fast cooling, 2-D
Fast cooling, 1-D
Nominal composition (atomic fraction)
Al–Cu Linear binary phase diagram. Liquidus slope is 667.45 K. Partition coefficient is 0.14. Eutectic temperature is 821 K. 0.010908
Thermal condition Cooling rate (K/s)
0.8
16
Surface energy Surface tension over entropy change, r0/DSf (K m) Crystallographic parameter e4
6.5 · 107 0.04
5.5 · 107 0.04
Diffusion Solid state Liquid state
DaCu ¼ 0:65 104 e136000=RT DlCu ¼ 3 109
Solid seed Number Size (lm) Position
7 1.2 Random
1 0.5 Left boundary
16 1.2 Random
1 0.11 Left boundary
Space and time Domain size (lm2) Experimental grain size (lm) Number of cells Solidification time (s) Solidus (C) CPU time
524 · 524 198 ± 15 655 · 756 129 547.8 2 months
50 · 1
330 · 330 82.5 ± 4 412 · 476 6.5 547.8 1.2 days
15 · 1
100 · 1 129 547.8 6 min
150 · 1 6.5 547.8 10 min
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Fig. 1. Dependence of the non-equilibrium eutectic fraction on the cooling rate during solidification of a grain-refined Al–2.5% Cu alloy.
Simulation results presented in Fig. 3 show that the 1D PFT simulations for both slow and fast cooling cases give higher values of the eutectic phase fraction than the experimental measurement. This overestimation could be explained by the plate-like assumption for the grain morphology made in the model. This assumption leads to a constant interfacial area per unit of volume which is certainly less than the real one, owing to the simplified morphology. Although the final microstructure is rather globular (see Fig. 2c), it could originate from a more dendritic morphology and achieves its final form as a result of ripening and coalescence. Thus, the effect of back diffusion, which is proportional to the interfacial area per unit of volume, is underestimated in the 1-D model, resulting in a higher eutectic fraction. Another reason could be the fixed domain size used in the simulation, which is set as half of the secondary arm spacing measured from the final microstructure. This assumption, which neglects coarsening, certainly leads to a smaller Fourier number for the solid-state back diffusion and to the underestimation of the back diffusion since the final DAS reflects the longest characteristic diffusion length. Moreover, the overestimation of the calculated eutectic fraction (as compared with the experimentally determined values) is much larger for the fast cooling case, implying that the neglected factors in the 1-D PFT model have more pronounced effects upon fast cooling. What is even more important is that the 1-D model predicts an increase in the eutectic volume fraction with increasing cooling rate, which contradicts our experimental observations given in Fig. 1. Apparently, a model that describes how DAS evolves during solidification, i.e. takes into account coarsening, is essential for prediction of the final eutectic fraction. Before presenting the 2-D PFT simulation results, let us check the sensitivity of the eutectic volume fraction to coarsening. The model proposed by Voller and Beckermann [8] is employed to predict the final eutectic fractions for our experiment as a function of the coarsening exponent. We used the internet access to the model provided
by Voller [14]. In these calculations, dendrite arm spacing is not constant during solidification, but is a function of solidification time in the following form: k2 ðtÞ ¼
k02
t
tsol
n ð1Þ
in which k2(t) is the DAS during solidification, k02 is the final DAS, t is the running solidification time, tsol is the total local solidification time and n is the coarsening exponent. The copper diffusion coefficient in the solid is set as a constant. It is taken to be equal to the value of the temperature-dependent diffusion coefficient at a temperature of 580 C (Table 1). The results are shown in Fig. 4. It is interesting to note that when coarsening is absent, i.e. n = 0, the calculated eutectic fraction is almost twice as large as the one calculated with n = 0.33, which is regarded as a typical coarsening exponent for aluminium alloys. According to this model, coarsening makes a significant difference for the final eutectic fraction formed. For the same coarsening exponent, the eutectic fraction predicted for a faster cooling is always higher than the one obtained for a slower cooling. This result seemingly contradicts our experimental observations. But should the coarsening exponent be held as the same value for fast and cooling solidification conditions? In other words, does the coarsening exponent depend on cooling rate? Diepers et al. [15] showed that the change of coarsening conditions from diffusion- to convection-controlled increased the coarsening exponent from 0.33 to 0.5. Given the complex circumstances in which coarsening may occur during solidification and different coarsening mechanisms (local melting, ripening, coalescence and fragmentation), it seems that there is no reason to restrict the coarsening exponent to a constant value. If, in the fast cooling case, the coarsening exponent is slightly higher than the one at the slower cooling rate, then the experimentally observed tendency could be explained, as illustrated in Fig. 4. But this explanation is certainly not complete without a physically based reason to adjust the coarsening exponent.
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Fig. 2. Microstructures of a grain-refined Al–2.5% Cu alloy obtained after solidification at 0.8 K/s (a) and 16 K/s (b, c). The image size is 524 · 524 lm for (a) and 330 · 330 lm for (b) and (c), and corresponds to the domain size in the 2-D PFT calculations (see Table 1). Samples in (a) and (b) are etched slightly to reveal interdendritic eutectics, while the sample in (c) is oxidized and viewed under polarized light to reveal grain structure.
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Since the 2-D PFT model can release the grain morphology restriction made in the 1-D PFT model and track the grain morphology evolution and dendrite arm coarsening, 2-D PFT simulations were performed for our solidification experiments. The results on the final eutectic fraction are given in Fig. 3. The calculated DAS was 118 and 35 lm for the slow and fast cooling cases, respectively. The simulated DAS is slightly larger than those experimentally determined for the two tested cases, at 99.5 lm at slow cooling and 29.7 lm for fast cooling. However, the relative deviation of the calculated DAS from the experimental one is approximately the same for the two cases (about 15% higher in the simulation than in the micrographs). Therefore, one can consider that the 2-D PFT simulations correctly reproduce the observed microstructure and can therefore be used to investigate further the influence of the cooling rate upon the amount of interdendritic eutectics. The 2-D PFT model indeed predicts the decrease in the eutectic fraction with increasing cooling rate that is observed in the experiments but could not be predicted in the 1-D calculations. The computational results also allow us to follow the evolution of the structure, which can be instrumental in understanding the observed phenomena. Fast cooling leads to more dendritic grain morphology compared with slow cooling, as shown experimentally in Fig. 2a and b, and by computer simulations in Figs. 5 and 6. The direct consequence of such a structure refinement in the fast cooling case is a larger area of solid/liquid interface per unit volume, Sv. Fig. 7 shows that in the fast cooling condition Sv is about three times larger than for the slow cooling case during the whole process of solidification. A direct consequence of this could be the reduced extent of microsegregation in the faster cooled structure by means of back diffusion. The comparison of Fourier numbers calculated by setting the characteristic solidification length to the DAS obtained from the completely solidified microstructure for both cases (0.009 and 0.0165 for the fast and slow cooling, respectively) tells us, however, that, in terms of reducing the extent of microsegregation, the structure refinement in the fast cooling case is not as effective as the long solidification time in the slow cooling case. Therefore, the structure refinement due to the increased cooling rate cannot be the main contribution to the observed trend in the eutectic fraction variation. Nonetheless, the predicted evolution of the interfacial concentration is much more realistic in 2-D simulations as compared with the constant value given by 1-D simulations. The declining stage in the evolution of the interfacial area, Sv, with respect to the solid fraction reflects the coarsening and coalescence of dendrite arms. This process is more pronounced for the finer structure, i.e. at a higher cooling rate. The other important parameter that can be extracted from the simulation results and could affect the final eutectic fraction is the evolution of DAS, i.e. coarsening. A rough estimation of DAS can be made through its relation
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Fig. 3. Experimental (E) and simulated (1-D and 2-D) eutectic fractions for the slow and fast cooling rates. The absolute errors in the experimental measurements are 0.16% and 0.06% for the slow and fast cooling rates, respectively.
Fig. 4. The mass eutectic fraction in an Al–2.5% Cu alloy as a function of the coarsening exponent, as predicted by the approximate microsegregation model [4,13] for two given cooling rates.
with the Sv, which is the output parameter of calculations. Both parameters are related to the cubic root of the local solidification time during continuous cooling [16]. Therefore, the mathematical expression for the conversion of Sv into DAS is DAS ¼
C Sv
ð2Þ
where C is a fitting parameter. Please note that the evolution of DAS during solidification cannot be measured experimentally within the range of techniques used in this work. Fig. 7 shows that at the beginning of the solidification, when the solid fraction is quite low, DAS is large, reflecting the fact that side branching has not yet started. The decrease of DAS during solidification is an indication that dendrite branch multiplication is dominant over ripening. With further increases in the solid fraction, dendrites begin to impinge upon each other, and ripening and coalescence takes over the side branching. As a result, DAS starts to increase.
In the PFT simulation, the kinetics of coarsening is rigorously considered since solid–liquid interface curvature and the diffusion in the liquid are modeled, which are the two governing factors for coarsening. Coarsening is apparently more severe upon faster cooling. This can be readily observed in the simulated microstructures shown in Figs. 5 and 6. In the fast cooling case, tertiary arms are induced and they experience quick and severe re-melting and resolidification (ripening). The surviving tertiary arms finally coalesce. For example, in the microstructure bounded by a small rectangle in Fig. 5, at the beginning there are six tertiary arms. As solidification proceeds, some of these continue to grow while others stop. Finally, only three of them survive. I contrast, in the slow cooling case, only secondary arms are induced and all of them survive. At the late stage of the solidification, coalescence occurs and leads to a reduction in the Sv and an increase in DAS (Fig. 7). The final volume fraction of eutectic is, of course, a measure of the amount of liquid that is retained, due to the non-equilibrium regime of solidification, at the temperature of the eutectic reaction. It is worth noting that coarsening starts to affect the volume fraction of liquid at rather high temperatures. Fig. 8 demonstrates that the volume fractions of liquid in the fast and slow cooling cases cross over at about 646 C, or at a liquid fraction of 0.46. The severe coarsening in the fast cooling case implies that fast cooling must be accompanied by a higher coarsening exponent, as defined in Eq. (1), as compared with the slow cooling rate. Although it is tempting to use Eq. (1), which relates the final DAS to the solidification time at a given coarsening exponent, to recalculate the coarsening exponent with the data obtained from the 2-D calculations about the evolution of DAS (Fig. 7), this cannot be done easily since Eq. (1) only gives a description of ripening, while, as shown in Fig. 7, dendrite multiplication during the first stage of solidification is present in both of these two calculations. Then the recalculation of DAS has to be related to a value at which ripening begins to dominate, which is a rather artificial choice. However, in the range
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Fig. 5. Simulated microstructure evolution during fast cooling (16 K/s). Note the coarsening process in the frame superimposed on the pictures.
where ripening is dominant and Eq. (1) can be applied (solid fraction larger than 0.9), Fig. 7 shows that the difference between DAS in the two cases decreases progressively, which means that in the fast cooling case dendrite branches coarsen faster than in the slow cooling case, implying a larger coarsening exponent in the fast cooling case. Therefore, the difference in coarsening kinetics at different cooling rates can produce the results that are observed experimentally, as suggested by Voller’s model. Further supporting evidence to this homogenization caused by coarsening is the comparison between the concentration profiles along the secondary and tertiary dendrite arms. Due to the active diffusion in the tertiary arms, the solute concentration in the core of such an arm should be higher than that in the core of a secondary arm. Fig. 9a shows the Cu concentration contour at a solidification time of 1.2 s. The dendrite envelope corresponds approximately to the contour line of 7 at.% Cu. The secondary arm, which goes from bottom left to right top of the picture, produces some tertiary arms. Even at this very early stage of solidification the concentration at the core of the tertiary arms is higher than in the secondary arm (2.97 vs. 1.62 at.%) because the former are induced later, and therefore formed at a lower temperature. In Fig. 9b, the solidification is fully completed. As solidification proceeds, the core concentration increases from 2.97
to 3.42 at.% in the tertiary arms while it remains almost unchanged in the secondary arm due to a longer diffusion length. This enrichment in the core of tertiary arms results in a smaller non-equilibrium eutectic fraction. One of the commercial solidification processes in which the range of cooling rate is similar to that discussed in this paper is direct-chill (DC) casting. In a commercial extrusion billet of 200 mm in diameter the cooling rates in the centre and at the periphery of the billet are approximately 3–5 and 15–20 K/s, respectively [17]. The volume fraction of eutectics, however, shows a minimum at the centre of the billet [18], where the cooling rate is the lowest, which contradicts the results and discussion presented in the current paper. We can suggest the following line of logic for the case of DC casting. First, it is well known that the velocity of the solidification front (which is the basic parameter in structure formation rather than the cooling rate, which is a technological parameter) is distributed unevenly across the billet diameter, being the function of the slope of the solidification front with respect to the billet vertical axis. As a result, the maximum velocity of the solidification front occurs in the centre of the billet [17]. Therefore, the solidification time can be shorter in the central part of the billet, where the minimum of the eutectic occurs. This is in accordance with our current results. Secondly, the solidification during DC casting occurs under
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Fig. 6. Simulated microstructure evolution during slow cooling (0.8 K/s).
Fig. 7. Sv and DAS computed by 2-D PFT model for two cooling cases.
conditions where convection is rather active in the transition region of the billet. According to Diepers et al. [15], convection can significantly increase the coarsening exponent, affecting the coarsening kinetics, as shown in Fig. 4. The width of the transition region is maximal at the centre of the billet. As a result, the coarsening may occur there much more efficiently, and could effectively decrease the amount of eutectics. Finally, the negative macrosegregation in the centre of the billet can further contribute to
Fig. 8. Evolution of the liquid fraction during solidification for two cooling cases. The arrow shows the point where two curves intersect.
the decreased amount of eutectics, though it alone cannot explain the variation in the eutectic volume fraction across the billet diameter. The correct trend in the dependence of the eutectic fraction in the range of moderate cooling rates predicted by 2-D PFT calculations is indicative of the important role played by the structure morphology.
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Fig. 9. (a) Cu concentration contour at solidification time = 1.2 s and (b) Cu concentration contour after fully solidification (solidification time = 6.54 s).
In situ observations of the structure evolution during solidification could provide a valuable insight into coarsening kinetics. Recently, such experimental results have begun to emerge thanks to the new technique of X-ray microscopy. Mathiesen and Arnberg reported results on structure evolution in Al–Cu alloys during columnar growth [19]. Fig. 4 in their paper [19] indeed shows significant coarsening of dendrite arms accompanied by dissolution and fragmentation during primary solidification. It will be quite exciting to employ this technique further in the study of coarsening kinetics, since the conventional solidify-and-quench technique can only produce the final microstructure, and cannot show the solidification process.
5. Conclusion By performing 2-D microsegregation modelling, dendrite coarsening during primary solidification has been identified as the main reason for the lower final eutectic fraction formed under faster cooling, as observed in our experiments. Because the 2-D model can describe rigorously how the important microstructure feature – dendrite arm spacing – evolves, its predictions are in good agreement with the trend observed in the experiments, which both the classic Brody–Flemings model and the 1-D PFT model have failed to explain [1]. The coarsening exponent is expected to depend on the cooling rate, but the exact dependence has not been clearly established. It is worth
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investigating how DAS evolves by in situ X-ray microscopy to give some clues about the dependence of the coarsening exponent on cooling rate. Acknowledgements The authors thank Dr. Rene Kieft and Erik-Paul van Klaveren from Corus in helping to run some of the calculations. Special thanks go to Demian G. RuvalcabaJimenez for the experimental data. This work was performed within the framework of the research program of the Netherlands Institute for Metals Research (www. nimr.nl), Project MC4.02134. References [1] Eskin DG, Du Q, Ruvalcaba D, Katgerman L. Mater Sci Eng A 2005;405:1–10. [2] Jacot A, Rappaz M. Acta Mater 2002;50:1909–26. [3] Jacot A, Rappaz M. Acta Mater 2002;50:3971.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
Du Q, Jacot A. Acta Mater 2005;53:3479–93. Mortensen A. Metall Trans A 1989;20A:247–53. Voller VR. J. Cryst. Growth 1999;197:333–40. Kraft T, Chang YA. Metall Mater Trans A 1998;28A:2447–9. Voller VR, Beckermann C. Metall Mater Trans A 1999;30:3016–9. Flemings MC. Mater Trans 2005;46:895–900. Voller VR, Beckermann C. Metall Mater Trans A 1999;30A:2183–9. Yan XY, Xie FY, Chu M, Chang YA. Mater Sci Eng A 2001;302:268–74. Novikov II, Zolotorevsky VS. Dendritnaya likvatsiya v splavakh [Dendritic segregation in alloys]. Moscow: Nauka; 1966. Kothe DB, Rider WJ, Mosso SJ, Brock JS. Internal report, Los Alamos Laboratories; 1996. Available at:. Diepers HJ, Beckermann C, Steinbach I. Acta Mater 1999;47:3663–78. Marsh SP, Glicksman ME. Metall Mater Trans A 1996;27A:557–67. Eskin DG, Zuidema K, Savran VI, Katgerman L. Mater Sci Eng A 2004;384:232–44. Suyitno, Eskin DG, Savran VI, Katgerman L. Metall Mater Trans A 2004;35A:3551–61. Mathiesen RM, Arnberg L. Acta Mater 2005;53:947–56.
Two-dimensional modelling and experimental study on microsegregation during solidification of an Al–Cu binary alloy Q. Du b
a,*
, D.G. Eskin a, A. Jacot
b,c
, L. Katgerman
d
a Netherlands Institute for Metals Research, Mekelweg 2, 2628CD Delft, The Netherlands Laboratoire de Simulation des Mate´riaux, Ecole Polytechnique Fe´de´rale de Lausanne, CH-1015 Lausanne, Switzerland c Calcom-ESI SA, Parc Scientifique, Ecole Polytechnique Fe´de´rale de Lausanne, CH-1015 Lausanne, Switzerland d Delft University of Technology, Mekelweg 2, 2628CD Delft, The Netherlands
Received 21 April 2006; received in revised form 9 October 2006; accepted 9 October 2006 Available online 20 December 2006
Abstract To explain the experimentally observed relation between the cooling rate and the non-equilibrium eutectic fraction obtained during solidification experiments of Al–Cu binary alloys [Eskin DG, Du Q, Ruvalcaba D, Katgerman L. Mater Sci Eng A 2005;405:1–10], a two-dimensional (2-D) microsegregation model, the pseudo-front tracking (PFT) method [Jacot A, Rappaz M. Acta Mater 2002;50:1909–26, Jacot A, Rappaz M. Acta Mater 2002;50:3971, Du Q, Jacot A. Acta Mater 2005;53:3479–93], is employed. The analysis of simulated solidification kinetics, including the grain morphology evolution, coarsening and back diffusion, reveals the pronounced effect of dendrite coarsening on the final non-equilibrium eutectic fraction. Because the 2-D modelling can describe rigorously how the important microstructure feature – dendrite arm spacing – evolves, its predictions are in good agreement with the trend observed in the experiments, which both the classic Brody–Flemings model and the 1-D PFT model have failed to explain. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Modelling; Microsegregation; Coarsening; Aluminium alloys; Dendritic growth
1. Introduction Microsegregation is the chemical inhomogeneity at the scale of a grain size or a dendrite arm, and is a result of non-equilibrium solidification of alloys. It is a hot research topic and both experimental and modelling approaches are employed to investigate the concentration profile along dendrite arms and the type and fractions of the interdendritic phases, because of their great importance for downstream processing and for mechanical properties [1–11]. It is clear that back diffusion in the solid is important in the formation of microsegregation, and the diffusion in the liquid is insignificant upon normal casting conditions. The other important factor is dendrite coarsening during solidification. Coarsening in Al–Cu alloys is a result of
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the flux of Cu from coarse arms towards fine arms coming from the Gibbs–Thomson effect. During coarsening, highly curved solids (fine arms) are dissolved while regions of low or negative curvature (coarse arms) continue to grow. Although the coarsening is expected to reduce microsegregation [5,6], the extent of this effect is unclear (see Ref. [7] for a debate on this topic). It appears that, depending on the cooling rate, the shape of the cooling curve and the phase diagram, the influence of coarsening could be negligible on the solute distribution in the primary phase, but significant on the eutectic fraction formed [7]. Voller and Beckermann proposed an approximate model of microsegregation with coarsening, in which the effect of coarsening on microsegregation measured by the fraction of nonequilibrium eutectic was estimated [8]. They showed that, depending on the Fourier number (defined as a dimensionless solid-state back diffusion parameter calculated by the product of a constant solid diffusion coefficient and the local solidification time divided by the square of the
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.10.035
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characteristic solidification length), the final eutectic fraction could significantly deviate from the one obtained when coarsening was not taken into account. In contrast, Flemings concluded that the effect of coarsening on microsegregation is small, and certainly less than the effect of back-diffusion in the solid [9]. There are some one-dimensional (1-D) models available [10] in which the coarsening process of dendrite arms is described as a function of local solidification time. Yan et al. [11] demonstrated that the proper description of dendrite grain morphology is very important in modelling microsegregation, though, owing to their intrinsic geometrical assumptions, 1-D models cannot do this in a direct way. Certainly 2-D or 3-D direct simulations of complex grain morphology evolution would be more appropriate. Recently, a 2-D pseudo-front tracking (PFT) model has been developed by Jacot, Rappaz and Du [2–4] for normal casting conditions to predict the final concentration profile and the types and fractions of secondary phases formed, in which the evolution of dendrite arms can be explicitly tracked in a physical manner, and ripening as well as coalescence can be described based on their controlling factors, i.e. the combination of curvature effect and diffusion in the liquid, as well as the possible influence of the grain boundary energy [4]. The direct motivation to perform 2-D microsegregation modelling for the solidification of an Al–Cu alloy has been the interpretation of our earlier experimental results showing an unconventional relationship between cooling rate and non-equilibrium eutectic fraction, which initially increases in the range of slow cooling (<1 K/s) and then gradually decreases in the range of 1–20 K/s [1]. Similar experimental results had been reported by other authors, e.g. Novikov and Zolotorevsky [12]. Coarsening, undercooling of the eutectic reaction and efficient back diffusion due to structural refinement at a faster cooling rate were all suggested as potential causes of the observed behaviour. The diffusion in the solid (back diffusion) directly influences the prediction of the final eutectic fraction. The amount of solute transferred by back diffusion is proportional to the local concentration gradient at the periphery of a growing grain, the interfacial area and the solidification time. Obviously, the longer the solidification time, the more solute can diffuse into the solid phase. At the same time, the slower cooling rate accompanied by longer solidification time favors the formation of coarse dendrite arms. Therefore, the back diffusion in the coarse structure will be hindered by the reduced interfacial area and a lower local average concentration gradient. As a result of the opposite effects of the solidification time and the coarse structure, the final eutectic fraction is difficult to predict. Although the 1-D microsegregation models could predict the fraction of non-equilibrium phases, their average nature and the simplified grain morphology assumption make it impossible to describe how the interface dynamics and the spatial distribution of grain morphology affects the non-equilibrium phase formation. Two-dimensional calculations could shed
some light on the effects of coarsening and structure refinement on the formation of non-equilibrium eutectics. The results of such calculations are presented here below after a brief description of the model and the experimental procedure. 2. Model description The model combines a direct simulation of the primary phase formation based on the PFT method [2,3] and a mixture approach for the description of the formation of secondary phases [4]. In the 1-D PFT model, the solidified microstructure is assumed to consist of plate-like dendrite arms, the spacing of which is determined from the experimental secondary arm spacing; the diffusion in the liquid and solid can be taken into account, but the curvature undercooling is neglected; the calculation domain is half of the experimental measured value. The 2-D PFT model takes into account both diffusion in the liquid and solid phases and curvature undercooling, which together determine the velocity of the solid–liquid interface. Therefore, this model can explicitly track how the grain morphology evolves. The curvature of solid–liquid interface is calculated with the piecewise linear interface calculation technique [13]. As soon as the liquid becomes undercooled for a secondary solid phase, the calculation enters a second stage in which the mixture approach is invoked and aimed at predicting the formation of secondary phases in the remaining interdendritic regions. In this approach, the interdendritic regions are considered as a mixture of liquid and solid phases. The model provides the evolution of the interdendritic mixture as solidification proceeds, taking into account the effects of back diffusion, which continuously modifies the composition and the volume of the interdendritic regions. See Refs. [2–4] for more details about this model. The effect of the eutectic reaction undercooling can only be taken into account by imposing a pre-described undercooling value, since no rigorous description of the eutectic reaction is included in the 2-D PFT model. This choice is dictated by the numerical difficulty to resolve spatially both the grains of the primary dendritic phase, which are on a scale of 100 lm, and the interdendritic eutectic phases which are on a scale of several microns. In this paper, the effect of the eutectic undercooling is neglected. The possible implications of the eutectic reaction undercooling on the eutectic fraction in our experiments are discussed elsewhere [1]. Due to the 2-D nature of the model, some input parameters, such as the surface energy, have to be tuned to reproduce the appropriate average dendrite arm spacing (DAS) observed upon the metallographic examination. When the calculated DAS is close to the measured value, the effect of grain morphology on the eutectic fraction is considered to be properly taken into account. With this idea, simulations of solidification experiments with cooling rates of 0.8 and 16 K/s were performed (for the choice of the
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cooling rates, see Section 4). Table 1 lists the calculations parameters. The analytical model proposed in Ref. [8] is employed to compare with the prediction obtained by PFT model. This analytical model is based on the approximation that coarsening can be modeled in a standard microsegregation model as a back-diffusion process characterized by an enhanced diffusion parameter [10] and has the ability to account for both coarsening and back diffusion. For more details about this model please refer to Refs. [8,10]. 3. Experiments A binary Al–2.53% Cu alloy was prepared in an electrical resistance furnace from 99.9% pure aluminium and an Al–48% Cu master alloy. The alloy was grain-refined with a commercial Al–5% Ti–1% B master alloy, the total amount of Ti being 0.03%. No melt treatment was performed. The chemical composition of the alloy was determined by a spark spectrum analysis. The total amount of impurities was less than 0.1%. Samples were melted inside a cylindrical resistance furnace in alumina crucibles of about 4 cm3 volume. Prior to melting, a K-thermocouple was embedded in the solid sample in a specially drilled hole. In order to assure the good contact between the thermocouple and the melt, a steel-sheathed thermocouple assembly with an external diameter of 1 mm was used. Liquid samples were overheated to 710 C and then cooled inside or outside the furnace to the solidus temperature, with still or forced air, oil or water as a cooling medium.
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The samples were quenched in water as soon as they were fully solid. The cooling rate in the solidification range was then calculated from the cooling curve as an average cooling rate between the liquidus and the solidus [1]. The samples were cut in the horizontal plane as close as possible to the position of the thermocouple tip. These samples were then ground, polished and examined in an optical microscope. Samples for calculation of the grain size were additionally oxidized in a 5% water solution of fluoboric acid. The dendrite arm spacing, the volume fraction of the eutectic phase and the grain size were calculated using standard linear intercept method, both manually and using image-analysis software. Statistical analysis of the results was performed. 4. Results and discussion The experimentally determined dependence of the nonequilibrium eutectic fraction on the cooling rate in the tested alloy is shown in Fig. 1. The same tendency that was observed earlier for different non-grain-refined alloys [1] is confirmed now for the grain-refined alloy. The volume fraction of non-equilibrium eutectic increases with the cooling rate in the range of slow cooling, i.e. up to approximately 1 K/s, and then starts to decrease in the range of cooling rates from 1 to 20 K/s. Two points, at 0.8 and 16 K/s, were chosen for computer simulations. Fig. 2 gives microstructures of the alloy cooled at these two cooling rates, within a frame identical to the domain size used in the calculations (see Table 1).
Table 1 Input parameters for the calculations Calculation Alloy System Phase diagram
Slow cooling, 2-D
Slow cooling, 1-D
Fast cooling, 2-D
Fast cooling, 1-D
Nominal composition (atomic fraction)
Al–Cu Linear binary phase diagram. Liquidus slope is 667.45 K. Partition coefficient is 0.14. Eutectic temperature is 821 K. 0.010908
Thermal condition Cooling rate (K/s)
0.8
16
Surface energy Surface tension over entropy change, r0/DSf (K m) Crystallographic parameter e4
6.5 · 107 0.04
5.5 · 107 0.04
Diffusion Solid state Liquid state
DaCu ¼ 0:65 104 e136000=RT DlCu ¼ 3 109
Solid seed Number Size (lm) Position
7 1.2 Random
1 0.5 Left boundary
16 1.2 Random
1 0.11 Left boundary
Space and time Domain size (lm2) Experimental grain size (lm) Number of cells Solidification time (s) Solidus (C) CPU time
524 · 524 198 ± 15 655 · 756 129 547.8 2 months
50 · 1
330 · 330 82.5 ± 4 412 · 476 6.5 547.8 1.2 days
15 · 1
100 · 1 129 547.8 6 min
150 · 1 6.5 547.8 10 min
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Fig. 1. Dependence of the non-equilibrium eutectic fraction on the cooling rate during solidification of a grain-refined Al–2.5% Cu alloy.
Simulation results presented in Fig. 3 show that the 1D PFT simulations for both slow and fast cooling cases give higher values of the eutectic phase fraction than the experimental measurement. This overestimation could be explained by the plate-like assumption for the grain morphology made in the model. This assumption leads to a constant interfacial area per unit of volume which is certainly less than the real one, owing to the simplified morphology. Although the final microstructure is rather globular (see Fig. 2c), it could originate from a more dendritic morphology and achieves its final form as a result of ripening and coalescence. Thus, the effect of back diffusion, which is proportional to the interfacial area per unit of volume, is underestimated in the 1-D model, resulting in a higher eutectic fraction. Another reason could be the fixed domain size used in the simulation, which is set as half of the secondary arm spacing measured from the final microstructure. This assumption, which neglects coarsening, certainly leads to a smaller Fourier number for the solid-state back diffusion and to the underestimation of the back diffusion since the final DAS reflects the longest characteristic diffusion length. Moreover, the overestimation of the calculated eutectic fraction (as compared with the experimentally determined values) is much larger for the fast cooling case, implying that the neglected factors in the 1-D PFT model have more pronounced effects upon fast cooling. What is even more important is that the 1-D model predicts an increase in the eutectic volume fraction with increasing cooling rate, which contradicts our experimental observations given in Fig. 1. Apparently, a model that describes how DAS evolves during solidification, i.e. takes into account coarsening, is essential for prediction of the final eutectic fraction. Before presenting the 2-D PFT simulation results, let us check the sensitivity of the eutectic volume fraction to coarsening. The model proposed by Voller and Beckermann [8] is employed to predict the final eutectic fractions for our experiment as a function of the coarsening exponent. We used the internet access to the model provided
by Voller [14]. In these calculations, dendrite arm spacing is not constant during solidification, but is a function of solidification time in the following form: k2 ðtÞ ¼
k02
t
tsol
n ð1Þ
in which k2(t) is the DAS during solidification, k02 is the final DAS, t is the running solidification time, tsol is the total local solidification time and n is the coarsening exponent. The copper diffusion coefficient in the solid is set as a constant. It is taken to be equal to the value of the temperature-dependent diffusion coefficient at a temperature of 580 C (Table 1). The results are shown in Fig. 4. It is interesting to note that when coarsening is absent, i.e. n = 0, the calculated eutectic fraction is almost twice as large as the one calculated with n = 0.33, which is regarded as a typical coarsening exponent for aluminium alloys. According to this model, coarsening makes a significant difference for the final eutectic fraction formed. For the same coarsening exponent, the eutectic fraction predicted for a faster cooling is always higher than the one obtained for a slower cooling. This result seemingly contradicts our experimental observations. But should the coarsening exponent be held as the same value for fast and cooling solidification conditions? In other words, does the coarsening exponent depend on cooling rate? Diepers et al. [15] showed that the change of coarsening conditions from diffusion- to convection-controlled increased the coarsening exponent from 0.33 to 0.5. Given the complex circumstances in which coarsening may occur during solidification and different coarsening mechanisms (local melting, ripening, coalescence and fragmentation), it seems that there is no reason to restrict the coarsening exponent to a constant value. If, in the fast cooling case, the coarsening exponent is slightly higher than the one at the slower cooling rate, then the experimentally observed tendency could be explained, as illustrated in Fig. 4. But this explanation is certainly not complete without a physically based reason to adjust the coarsening exponent.
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Fig. 2. Microstructures of a grain-refined Al–2.5% Cu alloy obtained after solidification at 0.8 K/s (a) and 16 K/s (b, c). The image size is 524 · 524 lm for (a) and 330 · 330 lm for (b) and (c), and corresponds to the domain size in the 2-D PFT calculations (see Table 1). Samples in (a) and (b) are etched slightly to reveal interdendritic eutectics, while the sample in (c) is oxidized and viewed under polarized light to reveal grain structure.
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Since the 2-D PFT model can release the grain morphology restriction made in the 1-D PFT model and track the grain morphology evolution and dendrite arm coarsening, 2-D PFT simulations were performed for our solidification experiments. The results on the final eutectic fraction are given in Fig. 3. The calculated DAS was 118 and 35 lm for the slow and fast cooling cases, respectively. The simulated DAS is slightly larger than those experimentally determined for the two tested cases, at 99.5 lm at slow cooling and 29.7 lm for fast cooling. However, the relative deviation of the calculated DAS from the experimental one is approximately the same for the two cases (about 15% higher in the simulation than in the micrographs). Therefore, one can consider that the 2-D PFT simulations correctly reproduce the observed microstructure and can therefore be used to investigate further the influence of the cooling rate upon the amount of interdendritic eutectics. The 2-D PFT model indeed predicts the decrease in the eutectic fraction with increasing cooling rate that is observed in the experiments but could not be predicted in the 1-D calculations. The computational results also allow us to follow the evolution of the structure, which can be instrumental in understanding the observed phenomena. Fast cooling leads to more dendritic grain morphology compared with slow cooling, as shown experimentally in Fig. 2a and b, and by computer simulations in Figs. 5 and 6. The direct consequence of such a structure refinement in the fast cooling case is a larger area of solid/liquid interface per unit volume, Sv. Fig. 7 shows that in the fast cooling condition Sv is about three times larger than for the slow cooling case during the whole process of solidification. A direct consequence of this could be the reduced extent of microsegregation in the faster cooled structure by means of back diffusion. The comparison of Fourier numbers calculated by setting the characteristic solidification length to the DAS obtained from the completely solidified microstructure for both cases (0.009 and 0.0165 for the fast and slow cooling, respectively) tells us, however, that, in terms of reducing the extent of microsegregation, the structure refinement in the fast cooling case is not as effective as the long solidification time in the slow cooling case. Therefore, the structure refinement due to the increased cooling rate cannot be the main contribution to the observed trend in the eutectic fraction variation. Nonetheless, the predicted evolution of the interfacial concentration is much more realistic in 2-D simulations as compared with the constant value given by 1-D simulations. The declining stage in the evolution of the interfacial area, Sv, with respect to the solid fraction reflects the coarsening and coalescence of dendrite arms. This process is more pronounced for the finer structure, i.e. at a higher cooling rate. The other important parameter that can be extracted from the simulation results and could affect the final eutectic fraction is the evolution of DAS, i.e. coarsening. A rough estimation of DAS can be made through its relation
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Fig. 3. Experimental (E) and simulated (1-D and 2-D) eutectic fractions for the slow and fast cooling rates. The absolute errors in the experimental measurements are 0.16% and 0.06% for the slow and fast cooling rates, respectively.
Fig. 4. The mass eutectic fraction in an Al–2.5% Cu alloy as a function of the coarsening exponent, as predicted by the approximate microsegregation model [4,13] for two given cooling rates.
with the Sv, which is the output parameter of calculations. Both parameters are related to the cubic root of the local solidification time during continuous cooling [16]. Therefore, the mathematical expression for the conversion of Sv into DAS is DAS ¼
C Sv
ð2Þ
where C is a fitting parameter. Please note that the evolution of DAS during solidification cannot be measured experimentally within the range of techniques used in this work. Fig. 7 shows that at the beginning of the solidification, when the solid fraction is quite low, DAS is large, reflecting the fact that side branching has not yet started. The decrease of DAS during solidification is an indication that dendrite branch multiplication is dominant over ripening. With further increases in the solid fraction, dendrites begin to impinge upon each other, and ripening and coalescence takes over the side branching. As a result, DAS starts to increase.
In the PFT simulation, the kinetics of coarsening is rigorously considered since solid–liquid interface curvature and the diffusion in the liquid are modeled, which are the two governing factors for coarsening. Coarsening is apparently more severe upon faster cooling. This can be readily observed in the simulated microstructures shown in Figs. 5 and 6. In the fast cooling case, tertiary arms are induced and they experience quick and severe re-melting and resolidification (ripening). The surviving tertiary arms finally coalesce. For example, in the microstructure bounded by a small rectangle in Fig. 5, at the beginning there are six tertiary arms. As solidification proceeds, some of these continue to grow while others stop. Finally, only three of them survive. I contrast, in the slow cooling case, only secondary arms are induced and all of them survive. At the late stage of the solidification, coalescence occurs and leads to a reduction in the Sv and an increase in DAS (Fig. 7). The final volume fraction of eutectic is, of course, a measure of the amount of liquid that is retained, due to the non-equilibrium regime of solidification, at the temperature of the eutectic reaction. It is worth noting that coarsening starts to affect the volume fraction of liquid at rather high temperatures. Fig. 8 demonstrates that the volume fractions of liquid in the fast and slow cooling cases cross over at about 646 C, or at a liquid fraction of 0.46. The severe coarsening in the fast cooling case implies that fast cooling must be accompanied by a higher coarsening exponent, as defined in Eq. (1), as compared with the slow cooling rate. Although it is tempting to use Eq. (1), which relates the final DAS to the solidification time at a given coarsening exponent, to recalculate the coarsening exponent with the data obtained from the 2-D calculations about the evolution of DAS (Fig. 7), this cannot be done easily since Eq. (1) only gives a description of ripening, while, as shown in Fig. 7, dendrite multiplication during the first stage of solidification is present in both of these two calculations. Then the recalculation of DAS has to be related to a value at which ripening begins to dominate, which is a rather artificial choice. However, in the range
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Fig. 5. Simulated microstructure evolution during fast cooling (16 K/s). Note the coarsening process in the frame superimposed on the pictures.
where ripening is dominant and Eq. (1) can be applied (solid fraction larger than 0.9), Fig. 7 shows that the difference between DAS in the two cases decreases progressively, which means that in the fast cooling case dendrite branches coarsen faster than in the slow cooling case, implying a larger coarsening exponent in the fast cooling case. Therefore, the difference in coarsening kinetics at different cooling rates can produce the results that are observed experimentally, as suggested by Voller’s model. Further supporting evidence to this homogenization caused by coarsening is the comparison between the concentration profiles along the secondary and tertiary dendrite arms. Due to the active diffusion in the tertiary arms, the solute concentration in the core of such an arm should be higher than that in the core of a secondary arm. Fig. 9a shows the Cu concentration contour at a solidification time of 1.2 s. The dendrite envelope corresponds approximately to the contour line of 7 at.% Cu. The secondary arm, which goes from bottom left to right top of the picture, produces some tertiary arms. Even at this very early stage of solidification the concentration at the core of the tertiary arms is higher than in the secondary arm (2.97 vs. 1.62 at.%) because the former are induced later, and therefore formed at a lower temperature. In Fig. 9b, the solidification is fully completed. As solidification proceeds, the core concentration increases from 2.97
to 3.42 at.% in the tertiary arms while it remains almost unchanged in the secondary arm due to a longer diffusion length. This enrichment in the core of tertiary arms results in a smaller non-equilibrium eutectic fraction. One of the commercial solidification processes in which the range of cooling rate is similar to that discussed in this paper is direct-chill (DC) casting. In a commercial extrusion billet of 200 mm in diameter the cooling rates in the centre and at the periphery of the billet are approximately 3–5 and 15–20 K/s, respectively [17]. The volume fraction of eutectics, however, shows a minimum at the centre of the billet [18], where the cooling rate is the lowest, which contradicts the results and discussion presented in the current paper. We can suggest the following line of logic for the case of DC casting. First, it is well known that the velocity of the solidification front (which is the basic parameter in structure formation rather than the cooling rate, which is a technological parameter) is distributed unevenly across the billet diameter, being the function of the slope of the solidification front with respect to the billet vertical axis. As a result, the maximum velocity of the solidification front occurs in the centre of the billet [17]. Therefore, the solidification time can be shorter in the central part of the billet, where the minimum of the eutectic occurs. This is in accordance with our current results. Secondly, the solidification during DC casting occurs under
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Fig. 6. Simulated microstructure evolution during slow cooling (0.8 K/s).
Fig. 7. Sv and DAS computed by 2-D PFT model for two cooling cases.
conditions where convection is rather active in the transition region of the billet. According to Diepers et al. [15], convection can significantly increase the coarsening exponent, affecting the coarsening kinetics, as shown in Fig. 4. The width of the transition region is maximal at the centre of the billet. As a result, the coarsening may occur there much more efficiently, and could effectively decrease the amount of eutectics. Finally, the negative macrosegregation in the centre of the billet can further contribute to
Fig. 8. Evolution of the liquid fraction during solidification for two cooling cases. The arrow shows the point where two curves intersect.
the decreased amount of eutectics, though it alone cannot explain the variation in the eutectic volume fraction across the billet diameter. The correct trend in the dependence of the eutectic fraction in the range of moderate cooling rates predicted by 2-D PFT calculations is indicative of the important role played by the structure morphology.
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Fig. 9. (a) Cu concentration contour at solidification time = 1.2 s and (b) Cu concentration contour after fully solidification (solidification time = 6.54 s).
In situ observations of the structure evolution during solidification could provide a valuable insight into coarsening kinetics. Recently, such experimental results have begun to emerge thanks to the new technique of X-ray microscopy. Mathiesen and Arnberg reported results on structure evolution in Al–Cu alloys during columnar growth [19]. Fig. 4 in their paper [19] indeed shows significant coarsening of dendrite arms accompanied by dissolution and fragmentation during primary solidification. It will be quite exciting to employ this technique further in the study of coarsening kinetics, since the conventional solidify-and-quench technique can only produce the final microstructure, and cannot show the solidification process.
5. Conclusion By performing 2-D microsegregation modelling, dendrite coarsening during primary solidification has been identified as the main reason for the lower final eutectic fraction formed under faster cooling, as observed in our experiments. Because the 2-D model can describe rigorously how the important microstructure feature – dendrite arm spacing – evolves, its predictions are in good agreement with the trend observed in the experiments, which both the classic Brody–Flemings model and the 1-D PFT model have failed to explain [1]. The coarsening exponent is expected to depend on the cooling rate, but the exact dependence has not been clearly established. It is worth
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investigating how DAS evolves by in situ X-ray microscopy to give some clues about the dependence of the coarsening exponent on cooling rate. Acknowledgements The authors thank Dr. Rene Kieft and Erik-Paul van Klaveren from Corus in helping to run some of the calculations. Special thanks go to Demian G. RuvalcabaJimenez for the experimental data. This work was performed within the framework of the research program of the Netherlands Institute for Metals Research (www. nimr.nl), Project MC4.02134. References [1] Eskin DG, Du Q, Ruvalcaba D, Katgerman L. Mater Sci Eng A 2005;405:1–10. [2] Jacot A, Rappaz M. Acta Mater 2002;50:1909–26. [3] Jacot A, Rappaz M. Acta Mater 2002;50:3971.
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