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Comparison of solidification cracking susceptibility between Al-Mg and Al-Cu alloys during welding: A phase-field study
Scripta Materialia 150 (2018) 120–124
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Regular article
Comparison of solidification cracking susceptibility between Al-Mg and Al-Cu alloys during welding: A phase-field study Shaoning Geng a, Ping Jiang a,⁎, Xinyu Shao a, Gaoyang Mi b, Han Wu a, Yuewei Ai a, Chunming Wang b, Chu Han a, Rong Chen a, Wei Liu a a
The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science & Technology, 430074 Wuhan, PR China School of Materials Science and Engineering, Huazhong University of Science & Technology, 430074 Wuhan, PR China
b
a r t i c l e
i n f o
Article history: Received 7 February 2018 Received in revised form 6 March 2018 Accepted 12 March 2018 Available online xxxx Keywords: Phase-filed Solidification cracking Aluminum alloys Welding
a b s t r a c t Based on two-dimensional phase-field simulations, we demonstrated the possible reasons why Al-Mg alloys can have better resistance to solidification cracking than Al-Cu alloys despite their wide freezing temperature range. Using Al-4.0 wt% Cu and Al-4.0 wt% Mg alloy as examples, we found that back-diffusion is negligible due to the relatively high cooling rate, and extensive coalescence occurs at earlier solidification stage in Al-4.0 wt% Mg alloy compared with Al-4.0 wt% Cu alloy. With considering coalescence predicted by phase-field simulations, the calculated solidification cracking index of Al-Mg alloy is reasonably lower than Al-Cu alloy. © 2018 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Aluminum alloys are known to have high solidification cracking susceptibility (SCS) in welding [1–3]. During welding process, a weak semisolid region, called the mushy zone, exists between weld pool and completely solidified weld metal. Solidification cracking (SC) occurs near the end of mushy zone, where the residual liquid due to segregation can act as thin liquid films along grain boundaries, and hence prevent adjacent grains from bonding together firmly to resist cracking under tension. An alloy with a wider freezing range is expected to have a wider weak mushy zone and thus a higher SCS. As we know, Al-Mg alloy has a much wider freezing temperature range than Al-Cu alloy. However, the SCS of Al-Mg alloy is in fact lower than that of AlCu alloy, which is contrary to our expectation and has not been well understood by far [4]. To understand the reason why Al-Mg alloy has lower SCS than Al-Cu alloy, much efforts have been made based on analytical SC model by Kou and co-authors recently [4–6]. They believed that the unusually significant Mg back-diffusion during solidification due to the very high Mg solubility in Al solid can explain the good weldability of Al-Mg alloy. As an analytical model, Kou's model is concise and has provided us much insight into cracking during solidification. Furthermore, Kou proposed to use the maximum |dT/d(fs)1/2| (T is temperature and fs solid fraction) of an alloy as an index for its SCS [6]. The index has well predicted the susceptibility ranking of commercial wrought Al alloys and identified the filler metals for reducing their susceptibility [4–7]. The index, ⁎ Corresponding author. E-mail address: [email protected] (P. Jiang).
https://doi.org/10.1016/j.scriptamat.2018.03.013 1359-6462/© 2018 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
which can include the effect of coalescence and back diffusion, is related to the fs(T) curve which in mean field theory is typically calculated by the Scheil solidification model of no solid-state diffusion or other models with solid-state diffusion. These models can examine the volume fraction but cannot give microstructural details about segregation and grain coalescence, which play significant roles in resisting SC. Therefore, it is essential to consider the realistic microstructural characteristics in order to better understand SC behavior and predict SCS more precisely. The phase-field (PF) model, with the advantage of avoiding interface tracking, is a powerful tool to describe microstructure evolution during solidification [8–12]. Furthermore, with the fast development of computer science, especially parallel computation using graphics processing unit (GPU), quantitative PF model constructed based on the thininterface limit has become a viable tool for simulating large-scale systems which is capable to examine the development of mushy zone in experimentally relevant systems for dendritic growth. Using largescale PF simulations, we can better understand the dendritic coalescence behavior over the whole freezing temperature range in the mushy zone. On the other hand, a more precise fs(T), which is as an input into Kou's model, can be obtained from PF microstructurally complex simulation data according to Wang et al. [11]. In the present work, we aim to demonstrate why Al-Mg alloy has lower SCS than Al-Cu alloy using two-dimensional (2D) PF simulations. Particularly, we choose Al-4.0 wt% Cu and Al-4.0 wt% Mg alloy systems as examples. The paper is organized as following. Firstly, the PF model and its implementation are described briefly. Then the grain coalescence
S. Geng et al. / Scripta Materialia 150 (2018) 120–124
and solute segregation during solidification are investigated, following by comparing the predicted SCS index from PF simulations with that from analytical SC model. Finally conclusions are given. The PF model developed by Ohno and Matsuura [12] is employed in this study. The advantage of this model is that it allows solute diffusion both in solid and liquid phase. The spurious effects at solid–liquid interfaces resulted from the unrealistically thick solid–liquid interface and the unequal diffusivities are eliminated by a phenomenological “antitrapping current”. The side branching is expressed by introducing a fluc! tuating current J [13] of the Gaussian random number with the vari0 ance of 2DlFuq(ϕ)(1 + (1 − k)u)/(ΔtΔxd), where F0u is the constant noise magnitude depending on interface thickness, Δt is the time increment, Δx is the lattice size, and d = 2 is the dimension. Both Al-4.0 wt% Cu and Al-4.0 wt% Mg alloy systems are investigated, and their physical properties are listed in Table 1 [5,11,14]. We used a thermal gradient G = 250 K/mm and an isothermal velocity Vp = 3 mm/s. This leads to a faster cooling rate of 750 K/s compared with Ref. [4], which may appear in arc surfacing on thick plate, high energy beam welding or related hybrid welding [5,15]. A uniform grid spacing Δx = 0.8W0 = 0.1 μm was used for both alloy systems, which satisfies that the interface thickness W0 is one order of magnitude smaller than a characteristic length of the microstructure [16]. The total system size is 96 μm along width direction and 768 μm along growth direction, which can completely cover the whole freezing temperature range of both alloys. The PF equations are discretized by the normal finite-differential method. The time is discretized by the first-order forward difference method. The bottom boundary was initialized as solid phase with ϕ = + 1 and C = kC0. The top boundary kept unchangeable with ϕ = − 1 and C = C0 . The left and right boundary conditions were set to the periodic conditions for all variables. In Total, more than 7,300,000 meshes were used for all calculations. To accelerate the simulations, parallel computations using GPU with the computer unified device architecture (CUDA) programing language were employed. Fig. 1 shows grain and liquid channel morphologies during columnar growth of Al-4.0 wt% Cu and Al-4.0 wt% Mg alloys under the same solidification conditions, with two isotherms at temperatures Ts (solidus) and Te (eutectic) indicated. As for Al-Cu alloy (Fig. 1(a)), the microstructures appear to be cellular. The liquid channels between columnar grains are notably smooth and continuous, extending to temperature below Te. The residual liquids finally form continuous liquid films along grain boundaries at the root of columnar grains. It should be noted that the liquid below Te becomes metastable and will transform to eutectic. By contrast, fine dendritic microstructures with welldeveloped sidebranches are observed for Al-Mg alloy as shown in Fig. 1(b). This is significantly different from Al-Cu alloy, which can be explained by the noise amplification theory proposed by Pieters and Langer [13,17]. The noise amplitude at the dendritic tip for Al-4.0 wt % Mg is easier to be amplified along the side of dendrite compared to that for Al-4.0 wt% Cu, since the capillary length d0,Mg (3.01 × 10−9 m) is smaller than d0,Cu (3.75 × 10−9 m). The liquid channels become significantly undulant due to appearance of sidebranches. Besides, extensive coalescence occurs at temperatures just below Ts and well above Te,
Table 1 Physical properties of Al-4.0 wt% Cu and Al-4.0 wt% Mg alloys used in the PF simulations [5,11,14].
Melting temperature of Al, Tm (K) Liquidus slope, m Partition coefficient, k Liquid diffusion coefficient, Dl (m2/s) Solid diffusion coefficient, Ds (m2/s) Anisotropy of surface energy, ε4 Gibbs-Thompson, Γ (K m)
Al- Cu
Al- Mg
933.47 −2.6 0.14 3.0 × 10−9 1.0 × 10−12 0.01 2.4 × 10−7
933.47 −5.07 0.32 3.0 × 10−9 1.0 × 10−12 0.01 1.3 × 10−7
121
Fig. 1. Simulated grain and liquid channel morphologies of (a) Al-4 wt% Cu and (b) Al-4 wt % Mg under solidification conditions of G = 250 K/mm and Vp = 3 mm/s. Back-diffusion is considered with Ds = 1 × 10−12 m2/s for both Al-Cu and Al-Mg alloy.
forming a dendritic solid skeleton with discontinuous liquid droplet. Also, it can be seen that the amount of residual liquids of Al-Mg alloy is distinctly less than that of Al-Cu alloy, which can be attributed to the larger partition coefficient k and thus lower solute segregation degree. In fact, most of above phenomena have been experimentally observed in 2014 Al (Al-4.4 wt% Cu) and 5086 Al (Al-4.0 wt% Mg) welds using gas tungsten arc welding by Liu et al. [4]. Fig. 2(a) and (b) shows the solute distribution maps during columnar growth of Al-4.0 wt% Cu alloy and Al-4.0 wt% Mg alloy, respectively. For Al-Cu alloy, it is clear that segregation emerges only between the primary dendrites, forming a smooth and continuous liquid channel (Fig. 2(a)). No coalescence occurs due to the strong segregation. However, for Al-Mg alloy, we can see segregation happens not only between primary dendrites but also between secondary dendrites. To better understand segregation behavior, solute distribution along line AA′ in Fig. 2(a) and line BB′ in Fig. 2(b) is displayed in Fig. 2(c). The black
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Fig. 2. Solute distribution within liquid channels in columnar growth of (a) Al-4.0 wt% Cu alloy and (b) Al-4.0 wt% Mg alloy; (c) solute distribution along line AA′ and BB′; Black arrows indicate segregation between primary dendrites, while green arrows indicates segregation between secondary dendrites; (d) Schematic of the segregation and dendrite coalescence for AlCu and Al-Mg alloys.
arrows indicate segregation between primary dendrites, while green arrows indicate segregation between secondary arms. As can be seen, the solute concentration in liquid channels of Al-Cu alloy is significantly higher than that of Al-Mg alloy. On the one hand, this can be attributed to the higher partition coefficient k of Al-Mg alloy. On the other hand, the segregation between secondary arms entraps considerable solute, which reduces the solute amount rejected into the liquid channels between primary dendrites. Fig. 2(d) shows the schematic of segregation and dendrite coalescence of Al-Cu and Al-Mg alloy corresponding to Fig. 2(a) and (b). As for Al-Cu alloy, all solute rejected by solid is accumulated in the smooth liquid channels, which causes strong segregation between primary dendrites and thus resist the coalescence of neighboring dendrites. As for Al-Mg alloy, the solute can be rejected into liquid channels between primary dendrites as well as between secondary dendrites. This leads to weaker segregation between primary dendrites. Meanwhile, the protruding sidebranches is prone to bridge, with rejecting solute to surrounding liquid. This case is corresponding to Zone A in Fig. 2(b). In this way, extensive coalescence can take place in Al-Mg alloy and finally forms a structure of dendritic solid skeleton with discontinuous liquid droplet. As we know, the liquid channel morphologies during terminal solidification have significant effects on the mechanical properties and SC formation in the mushy zone [2]. A solid network with liquid droplets behaves similarly to a coherent solid with slightly reduced strength and hence has a very low SCS. However,
when the residual liquid forms continuous, thin liquid channels, SC is prone to occur. It should be noted that the coalescence of sidebranches in Al-Mg alloy may increase the resistance to liquid feeding and thus may lead to shrinkage voids. However, considering that cracking occurs at the terminal solidification stage near fs = 1, the shrinkage voids have low tendency to result in cracking because a solid network with relatively high strength has formed in Al-Mg alloy (different from the formation of thin liquid films). Therefore, it is not difficult to understand why Al-4.0 wt% Mg alloy has a relatively higher SCS than Al-4.0 wt% Cu alloy from the perspective of liquid channel morphologies under the solidification conditions involved in this study. To quantitatively characterize SCS, the fs(T) curve is required as an input into Kou's model [7]. Kou's model is sensitive to the fs(T) information. As shown in Ref. [5], with the increase of back-diffusion and hence variation in fs(T), the SCS peak shifts significantly. Therefore, obtaining a precise fs(T) is critical. The fs(T) curves can be calculated from either Scheil model or PF simulations. In the present study, we only examine fs(T) at temperatures above Te because any remaining liquid will become metastable and transform eutectic phase when temperature is below Te. Assuming the liquid → eutectic reaction happens quite close to Te and in a short time scale, the eutectic fraction can be estimated by fe = 1 − fs(Te). Fig. 3 compares the fs(T) curves calculated from the Scheil model and from the PF simulations for Al-4.0 wt% Cu (Fig. 3(a)) and Al-4.0 wt% Mg alloys (Fig. 3(b)). Both Scheil model and PF
S. Geng et al. / Scripta Materialia 150 (2018) 120–124
123
Fig. 3. Solid fraction fs vs. temperature T curves calculated from the Scheil model and from the PF simulations for (a) Al-4.0 wt% Cu alloy and (b) Al-4.0 wt% Mg alloy.
simulations predict the similar trend for two alloys. However, PF simulations give lower fs(T) for a given temperature at low fs range but higher fs(T) at high fs range. This indicates the end of primary solidification at higher temperatures (e.g. Al-4.0 wt% Mg), and a lower fraction of eutectic phase (e.g. Al-4.0 wt% Cu). This discrepancy can be attributed to the following two reasons: (1) Scheil predictions were obtained assuming infinitely rapid liquid diffusion, i.e. complete mixing in liquid, while PF simulations used a realistic liquid diffusion coefficient. This leads higher solidification rate and thus higher fs(T) at earlier solidification stage in Scheil model. As solidification proceeds, more solute accumulates in remaining liquid in Scheil model, resulting in higher segregation and lower fs(T). (2) Scheil model assumes planar solid-liquid interface, but PF model takes the realistic interface curvature into consideration. This makes the PF composition at interfaces inside channels deviate from the equilibrium composition and hence influence fs(T) [11]. It is believed that back-diffusion has significant effects on fs(T) curves for Al alloys, especially for Al-Mg alloy [4–6]. To examine the effects of back-diffusion on fs(T), PF simulations with and without considering solid diffusion are performed for two alloys and compared in Fig. 3. It can be seen that back-diffusion has a minimal effect on fs(T) for both alloys. This is inconsistent with previous predictions using analytical Kurtz-Fisher model (back-diffusion is included) by Liu et al. [4,5]. The main reason for this inconsistency can be attributed to the higher cooling rate considered in this study, leading to less time available for back-diffusion. Based on a recent criterion for cracking that considered the solidification shrinkage, strain rate and liquid feeding, Kou proposed a simple index, i.e. the maximum |dT/d(fs)1/2|, for predicting SCS of an alloy [6]. It should be noted that the steepness of the slope of the T − (fs)1/2 curve is |dT/d(fs)1/2|. According to Fig. 3 (fs axis → (fs)1/2 axis), |dT/d (fs)1/2| increases with (fs)1/2 increasing and the maximum value occurs near (fs)1/2 = 1. Here a special point fSB, where extensive coalescence begins to occur, should be defined. At this point fSB, the crack susceptibility of the solidifying alloy is assumed to end because of the extensive coalescence. It is recognized that the fSB value is not universal but is likely to depend on the alloy involved. The selection of fSB value is critical and can significantly affect the SCS prediction. In analytical SC models, however, the fSB value is empirically assumed because the Scheil model cannot predict the coalescence behavior. In the model of Clyne and Davies [18] the crack susceptibility was assumed to end at fs = 0.99, that is fSB = 0.99. In RDG model [19] and Kou's model [6] it is assumed that fSB = 0.98. Since extensive coalescence occurs at fSB to end the crack susceptibility, beyond fSB the steepness |dT/d(fs)1/2| is no longer relevant. Therefore, if the steepness is still increasing beyond fSB, the steepness at fSB is taken as the maximum steepness. If the maximum steepness occurs at Te before fSB, the steepness at fS(Te) is taken as the maximum steepness. Based on above criterion, Kou predicted the SCS of Al-Cu alloy well, and predicted SCS of Al-Mg alloy well with considering Mg
back-diffusion [6]. They believed that Mg back-diffusion plays a significant role on SCS [4–6]. However, from our PF simulations, it is found that the Mg back-diffusion is negligible under the solidification conditions involved in this study. Instead, we observe that extensive coalescence occurs at lower fs due to the lower segregation and bridging of sidebranches in Al-4.0 wt% Mg alloy. Particularly, we estimate fSB = 0.93 instead of fSB = 0.98 for Al-4.0 wt% Mg alloy from our PF simulations. As for Al-4.0 wt% Cu alloy, extensive coalescence does not occur above Te and thus the steepness at fS(Te) is taken as the maximum steepness, which is consistent with fSB = 0.98. Fig. 4 compares the crack susceptibility index calculated from different cases. As for Al-4.0 wt% Cu, the predicted index from PF simulation is similar to that from Scheil model with fSB = 0.98 and its value is approximately 2500 K. However, for Al-4.0 wt% Mg alloy, it can be seen that the predicted index from PF simulations is notably lower than that from Scheil model, which can be explained by the discrepancy in fs(T) curves as shown in Fig. 3. Moreover, the predicted index at fSB = 0.93, which is determined by our PF simulations, is distinctly lower than that at fSB = 0.98, which is assumed empirically, for Al-4.0 wt% Mg, indicating the significant influence of the selection of fSB value. Particularly, the predicted index (about 550 K) from PF simulation at fSB = 0.93 for Al-4.0 wt% Mg is significantly lower than that for Al-4.0 wt% Cu, showing the better SC resistance of Al-4.0 wt% Mg alloy. This agrees well with above analysis of liquid channel morphologies and the fact that Al-Mg alloy has better weldability than Al-Cu alloy. It should be noted that the coalescence behavior may vary with change in alloy composition and solidification conditions, and the fSB value employed to evaluate SCS should be selected
Fig. 4. Comparison of the crack susceptibility index calculated from Scheil model and PF simulations with different fSB for Al-4.0 wt% Cu and Al-4.0 wt% Mg alloys.
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accordingly. Therefore, understanding how alloy composition and solidification conditions affect the coalescence behavior and thus fSB value is of crucial importance for SCS prediction. This will be discussed in our future work. In summary, we demonstrated the possible reasons why Al-Mg alloys can have lower SCS than Al-Cu alloys at relatively high cooling rate (750 K/s) using 2D PF simulations. We found that the backdiffusion is negligible due to the relatively high cooling rate, and coalescence is easier to occur in Al-4.0 wt% Mg alloy than in Al-4.0 wt% Cu alloy due to the lower solute segregation between primary dendrites and the easier bridging of protruding secondary dendrites. This extensive coalescence facilitates forming a firm structure of dendritic solid skeleton with discontinuous liquid droplet. Furthermore, we compared the SC index calculated from PF simulations with that calculated from analytical model. With considering coalescence predicted by PF simulations, the calculated SC index of Al-4.0 wt% Mg alloy is reasonably lower than Al-Cu alloy. Acknowledgements This research has been supported by the National Basic Research Program (973 Program) of China under grant No. 2014CB046703, the National Natural Science Foundation of China (NSFC) under Grant No. 51721092 and the opening project of State Key Laboratory of Digital Manufacturing Equipment and Technology (Huazhong University of
Science & Technology) under grant No. DMETKF2018001. The calculation of this work was performed on TianHe-2 and thanks for the support of National Supercomputer Center in Guangzhou (NSCC-GZ). The authors also would like to thank the anonymous referees for their valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
M.C. Flemings, Solidification Processing, Wiley, 1974 Online Library. S. Kou, Welding Metallurgy, 2nd edition John Wiley and Sons, Hoboken, NJ, 2003. J. Campbell, Castings, Butterworth-Heinemann, 2003. J. Liu, H.P. Duarte, S. Kou, Acta Mater. 122 (2017) 47–59. J. Liu, S. Kou, Acta Mater. 100 (2015) 359–368. S. Kou, Weld. J. 94 (2015) 374–388. S. Kou, Acta Mater. 88 (2015) 366–374. D. Tourret, A. Karma, Acta Mater. 82 (2015) 64–83. B. Echebarria, R. Folch, A. Karma, M. Plapp, Phys. Rev. E 70 (2004), 61604. A. Karma, Phys. Rev. Lett. 87 (2001), 115701. L. Wang, N. Wang, N. Provatas, Acta Mater. 126 (2017) 302–312. M. Ohno, K. Matsuura, Phys. Rev. E 79 (2009), 31603. B. Echebarria, A. Karma, S. Gurevich, Phys. Rev. E 81 (2010), 21608. M. Gündüz, J.D. Hunt, Acta Metall. 37 (1989) 1839–1845. A. Farzadi, M. Doquang, S. Serajzadeh, A.H. Kokabi, G. Amberg, Model. Simul. Mater. Sci. 16 (2008) 65005. H. Xing, X. Dong, H. Wu, G. Hao, J. Wang, C. Chen, K. Jin, Sci. Rep. 6 (2016), 26625. R. Pieters, J.S. Langer, Phys. Rev. Lett. 56 (1986) 1948–1951. T.W. Clyne, G.J. Davies, British Foundryman 74 (1981) 65–73. M. Rappaz, J.M. Drezet, M. Gremaud, Metall. Mater. Trans. A 30 (1999) 449–455.
Contents lists available at ScienceDirect
Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat
Regular article
Comparison of solidification cracking susceptibility between Al-Mg and Al-Cu alloys during welding: A phase-field study Shaoning Geng a, Ping Jiang a,⁎, Xinyu Shao a, Gaoyang Mi b, Han Wu a, Yuewei Ai a, Chunming Wang b, Chu Han a, Rong Chen a, Wei Liu a a
The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science & Technology, 430074 Wuhan, PR China School of Materials Science and Engineering, Huazhong University of Science & Technology, 430074 Wuhan, PR China
b
a r t i c l e
i n f o
Article history: Received 7 February 2018 Received in revised form 6 March 2018 Accepted 12 March 2018 Available online xxxx Keywords: Phase-filed Solidification cracking Aluminum alloys Welding
a b s t r a c t Based on two-dimensional phase-field simulations, we demonstrated the possible reasons why Al-Mg alloys can have better resistance to solidification cracking than Al-Cu alloys despite their wide freezing temperature range. Using Al-4.0 wt% Cu and Al-4.0 wt% Mg alloy as examples, we found that back-diffusion is negligible due to the relatively high cooling rate, and extensive coalescence occurs at earlier solidification stage in Al-4.0 wt% Mg alloy compared with Al-4.0 wt% Cu alloy. With considering coalescence predicted by phase-field simulations, the calculated solidification cracking index of Al-Mg alloy is reasonably lower than Al-Cu alloy. © 2018 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Aluminum alloys are known to have high solidification cracking susceptibility (SCS) in welding [1–3]. During welding process, a weak semisolid region, called the mushy zone, exists between weld pool and completely solidified weld metal. Solidification cracking (SC) occurs near the end of mushy zone, where the residual liquid due to segregation can act as thin liquid films along grain boundaries, and hence prevent adjacent grains from bonding together firmly to resist cracking under tension. An alloy with a wider freezing range is expected to have a wider weak mushy zone and thus a higher SCS. As we know, Al-Mg alloy has a much wider freezing temperature range than Al-Cu alloy. However, the SCS of Al-Mg alloy is in fact lower than that of AlCu alloy, which is contrary to our expectation and has not been well understood by far [4]. To understand the reason why Al-Mg alloy has lower SCS than Al-Cu alloy, much efforts have been made based on analytical SC model by Kou and co-authors recently [4–6]. They believed that the unusually significant Mg back-diffusion during solidification due to the very high Mg solubility in Al solid can explain the good weldability of Al-Mg alloy. As an analytical model, Kou's model is concise and has provided us much insight into cracking during solidification. Furthermore, Kou proposed to use the maximum |dT/d(fs)1/2| (T is temperature and fs solid fraction) of an alloy as an index for its SCS [6]. The index has well predicted the susceptibility ranking of commercial wrought Al alloys and identified the filler metals for reducing their susceptibility [4–7]. The index, ⁎ Corresponding author. E-mail address: [email protected] (P. Jiang).
https://doi.org/10.1016/j.scriptamat.2018.03.013 1359-6462/© 2018 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
which can include the effect of coalescence and back diffusion, is related to the fs(T) curve which in mean field theory is typically calculated by the Scheil solidification model of no solid-state diffusion or other models with solid-state diffusion. These models can examine the volume fraction but cannot give microstructural details about segregation and grain coalescence, which play significant roles in resisting SC. Therefore, it is essential to consider the realistic microstructural characteristics in order to better understand SC behavior and predict SCS more precisely. The phase-field (PF) model, with the advantage of avoiding interface tracking, is a powerful tool to describe microstructure evolution during solidification [8–12]. Furthermore, with the fast development of computer science, especially parallel computation using graphics processing unit (GPU), quantitative PF model constructed based on the thininterface limit has become a viable tool for simulating large-scale systems which is capable to examine the development of mushy zone in experimentally relevant systems for dendritic growth. Using largescale PF simulations, we can better understand the dendritic coalescence behavior over the whole freezing temperature range in the mushy zone. On the other hand, a more precise fs(T), which is as an input into Kou's model, can be obtained from PF microstructurally complex simulation data according to Wang et al. [11]. In the present work, we aim to demonstrate why Al-Mg alloy has lower SCS than Al-Cu alloy using two-dimensional (2D) PF simulations. Particularly, we choose Al-4.0 wt% Cu and Al-4.0 wt% Mg alloy systems as examples. The paper is organized as following. Firstly, the PF model and its implementation are described briefly. Then the grain coalescence
S. Geng et al. / Scripta Materialia 150 (2018) 120–124
and solute segregation during solidification are investigated, following by comparing the predicted SCS index from PF simulations with that from analytical SC model. Finally conclusions are given. The PF model developed by Ohno and Matsuura [12] is employed in this study. The advantage of this model is that it allows solute diffusion both in solid and liquid phase. The spurious effects at solid–liquid interfaces resulted from the unrealistically thick solid–liquid interface and the unequal diffusivities are eliminated by a phenomenological “antitrapping current”. The side branching is expressed by introducing a fluc! tuating current J [13] of the Gaussian random number with the vari0 ance of 2DlFuq(ϕ)(1 + (1 − k)u)/(ΔtΔxd), where F0u is the constant noise magnitude depending on interface thickness, Δt is the time increment, Δx is the lattice size, and d = 2 is the dimension. Both Al-4.0 wt% Cu and Al-4.0 wt% Mg alloy systems are investigated, and their physical properties are listed in Table 1 [5,11,14]. We used a thermal gradient G = 250 K/mm and an isothermal velocity Vp = 3 mm/s. This leads to a faster cooling rate of 750 K/s compared with Ref. [4], which may appear in arc surfacing on thick plate, high energy beam welding or related hybrid welding [5,15]. A uniform grid spacing Δx = 0.8W0 = 0.1 μm was used for both alloy systems, which satisfies that the interface thickness W0 is one order of magnitude smaller than a characteristic length of the microstructure [16]. The total system size is 96 μm along width direction and 768 μm along growth direction, which can completely cover the whole freezing temperature range of both alloys. The PF equations are discretized by the normal finite-differential method. The time is discretized by the first-order forward difference method. The bottom boundary was initialized as solid phase with ϕ = + 1 and C = kC0. The top boundary kept unchangeable with ϕ = − 1 and C = C0 . The left and right boundary conditions were set to the periodic conditions for all variables. In Total, more than 7,300,000 meshes were used for all calculations. To accelerate the simulations, parallel computations using GPU with the computer unified device architecture (CUDA) programing language were employed. Fig. 1 shows grain and liquid channel morphologies during columnar growth of Al-4.0 wt% Cu and Al-4.0 wt% Mg alloys under the same solidification conditions, with two isotherms at temperatures Ts (solidus) and Te (eutectic) indicated. As for Al-Cu alloy (Fig. 1(a)), the microstructures appear to be cellular. The liquid channels between columnar grains are notably smooth and continuous, extending to temperature below Te. The residual liquids finally form continuous liquid films along grain boundaries at the root of columnar grains. It should be noted that the liquid below Te becomes metastable and will transform to eutectic. By contrast, fine dendritic microstructures with welldeveloped sidebranches are observed for Al-Mg alloy as shown in Fig. 1(b). This is significantly different from Al-Cu alloy, which can be explained by the noise amplification theory proposed by Pieters and Langer [13,17]. The noise amplitude at the dendritic tip for Al-4.0 wt % Mg is easier to be amplified along the side of dendrite compared to that for Al-4.0 wt% Cu, since the capillary length d0,Mg (3.01 × 10−9 m) is smaller than d0,Cu (3.75 × 10−9 m). The liquid channels become significantly undulant due to appearance of sidebranches. Besides, extensive coalescence occurs at temperatures just below Ts and well above Te,
Table 1 Physical properties of Al-4.0 wt% Cu and Al-4.0 wt% Mg alloys used in the PF simulations [5,11,14].
Melting temperature of Al, Tm (K) Liquidus slope, m Partition coefficient, k Liquid diffusion coefficient, Dl (m2/s) Solid diffusion coefficient, Ds (m2/s) Anisotropy of surface energy, ε4 Gibbs-Thompson, Γ (K m)
Al- Cu
Al- Mg
933.47 −2.6 0.14 3.0 × 10−9 1.0 × 10−12 0.01 2.4 × 10−7
933.47 −5.07 0.32 3.0 × 10−9 1.0 × 10−12 0.01 1.3 × 10−7
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Fig. 1. Simulated grain and liquid channel morphologies of (a) Al-4 wt% Cu and (b) Al-4 wt % Mg under solidification conditions of G = 250 K/mm and Vp = 3 mm/s. Back-diffusion is considered with Ds = 1 × 10−12 m2/s for both Al-Cu and Al-Mg alloy.
forming a dendritic solid skeleton with discontinuous liquid droplet. Also, it can be seen that the amount of residual liquids of Al-Mg alloy is distinctly less than that of Al-Cu alloy, which can be attributed to the larger partition coefficient k and thus lower solute segregation degree. In fact, most of above phenomena have been experimentally observed in 2014 Al (Al-4.4 wt% Cu) and 5086 Al (Al-4.0 wt% Mg) welds using gas tungsten arc welding by Liu et al. [4]. Fig. 2(a) and (b) shows the solute distribution maps during columnar growth of Al-4.0 wt% Cu alloy and Al-4.0 wt% Mg alloy, respectively. For Al-Cu alloy, it is clear that segregation emerges only between the primary dendrites, forming a smooth and continuous liquid channel (Fig. 2(a)). No coalescence occurs due to the strong segregation. However, for Al-Mg alloy, we can see segregation happens not only between primary dendrites but also between secondary dendrites. To better understand segregation behavior, solute distribution along line AA′ in Fig. 2(a) and line BB′ in Fig. 2(b) is displayed in Fig. 2(c). The black
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Fig. 2. Solute distribution within liquid channels in columnar growth of (a) Al-4.0 wt% Cu alloy and (b) Al-4.0 wt% Mg alloy; (c) solute distribution along line AA′ and BB′; Black arrows indicate segregation between primary dendrites, while green arrows indicates segregation between secondary dendrites; (d) Schematic of the segregation and dendrite coalescence for AlCu and Al-Mg alloys.
arrows indicate segregation between primary dendrites, while green arrows indicate segregation between secondary arms. As can be seen, the solute concentration in liquid channels of Al-Cu alloy is significantly higher than that of Al-Mg alloy. On the one hand, this can be attributed to the higher partition coefficient k of Al-Mg alloy. On the other hand, the segregation between secondary arms entraps considerable solute, which reduces the solute amount rejected into the liquid channels between primary dendrites. Fig. 2(d) shows the schematic of segregation and dendrite coalescence of Al-Cu and Al-Mg alloy corresponding to Fig. 2(a) and (b). As for Al-Cu alloy, all solute rejected by solid is accumulated in the smooth liquid channels, which causes strong segregation between primary dendrites and thus resist the coalescence of neighboring dendrites. As for Al-Mg alloy, the solute can be rejected into liquid channels between primary dendrites as well as between secondary dendrites. This leads to weaker segregation between primary dendrites. Meanwhile, the protruding sidebranches is prone to bridge, with rejecting solute to surrounding liquid. This case is corresponding to Zone A in Fig. 2(b). In this way, extensive coalescence can take place in Al-Mg alloy and finally forms a structure of dendritic solid skeleton with discontinuous liquid droplet. As we know, the liquid channel morphologies during terminal solidification have significant effects on the mechanical properties and SC formation in the mushy zone [2]. A solid network with liquid droplets behaves similarly to a coherent solid with slightly reduced strength and hence has a very low SCS. However,
when the residual liquid forms continuous, thin liquid channels, SC is prone to occur. It should be noted that the coalescence of sidebranches in Al-Mg alloy may increase the resistance to liquid feeding and thus may lead to shrinkage voids. However, considering that cracking occurs at the terminal solidification stage near fs = 1, the shrinkage voids have low tendency to result in cracking because a solid network with relatively high strength has formed in Al-Mg alloy (different from the formation of thin liquid films). Therefore, it is not difficult to understand why Al-4.0 wt% Mg alloy has a relatively higher SCS than Al-4.0 wt% Cu alloy from the perspective of liquid channel morphologies under the solidification conditions involved in this study. To quantitatively characterize SCS, the fs(T) curve is required as an input into Kou's model [7]. Kou's model is sensitive to the fs(T) information. As shown in Ref. [5], with the increase of back-diffusion and hence variation in fs(T), the SCS peak shifts significantly. Therefore, obtaining a precise fs(T) is critical. The fs(T) curves can be calculated from either Scheil model or PF simulations. In the present study, we only examine fs(T) at temperatures above Te because any remaining liquid will become metastable and transform eutectic phase when temperature is below Te. Assuming the liquid → eutectic reaction happens quite close to Te and in a short time scale, the eutectic fraction can be estimated by fe = 1 − fs(Te). Fig. 3 compares the fs(T) curves calculated from the Scheil model and from the PF simulations for Al-4.0 wt% Cu (Fig. 3(a)) and Al-4.0 wt% Mg alloys (Fig. 3(b)). Both Scheil model and PF
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Fig. 3. Solid fraction fs vs. temperature T curves calculated from the Scheil model and from the PF simulations for (a) Al-4.0 wt% Cu alloy and (b) Al-4.0 wt% Mg alloy.
simulations predict the similar trend for two alloys. However, PF simulations give lower fs(T) for a given temperature at low fs range but higher fs(T) at high fs range. This indicates the end of primary solidification at higher temperatures (e.g. Al-4.0 wt% Mg), and a lower fraction of eutectic phase (e.g. Al-4.0 wt% Cu). This discrepancy can be attributed to the following two reasons: (1) Scheil predictions were obtained assuming infinitely rapid liquid diffusion, i.e. complete mixing in liquid, while PF simulations used a realistic liquid diffusion coefficient. This leads higher solidification rate and thus higher fs(T) at earlier solidification stage in Scheil model. As solidification proceeds, more solute accumulates in remaining liquid in Scheil model, resulting in higher segregation and lower fs(T). (2) Scheil model assumes planar solid-liquid interface, but PF model takes the realistic interface curvature into consideration. This makes the PF composition at interfaces inside channels deviate from the equilibrium composition and hence influence fs(T) [11]. It is believed that back-diffusion has significant effects on fs(T) curves for Al alloys, especially for Al-Mg alloy [4–6]. To examine the effects of back-diffusion on fs(T), PF simulations with and without considering solid diffusion are performed for two alloys and compared in Fig. 3. It can be seen that back-diffusion has a minimal effect on fs(T) for both alloys. This is inconsistent with previous predictions using analytical Kurtz-Fisher model (back-diffusion is included) by Liu et al. [4,5]. The main reason for this inconsistency can be attributed to the higher cooling rate considered in this study, leading to less time available for back-diffusion. Based on a recent criterion for cracking that considered the solidification shrinkage, strain rate and liquid feeding, Kou proposed a simple index, i.e. the maximum |dT/d(fs)1/2|, for predicting SCS of an alloy [6]. It should be noted that the steepness of the slope of the T − (fs)1/2 curve is |dT/d(fs)1/2|. According to Fig. 3 (fs axis → (fs)1/2 axis), |dT/d (fs)1/2| increases with (fs)1/2 increasing and the maximum value occurs near (fs)1/2 = 1. Here a special point fSB, where extensive coalescence begins to occur, should be defined. At this point fSB, the crack susceptibility of the solidifying alloy is assumed to end because of the extensive coalescence. It is recognized that the fSB value is not universal but is likely to depend on the alloy involved. The selection of fSB value is critical and can significantly affect the SCS prediction. In analytical SC models, however, the fSB value is empirically assumed because the Scheil model cannot predict the coalescence behavior. In the model of Clyne and Davies [18] the crack susceptibility was assumed to end at fs = 0.99, that is fSB = 0.99. In RDG model [19] and Kou's model [6] it is assumed that fSB = 0.98. Since extensive coalescence occurs at fSB to end the crack susceptibility, beyond fSB the steepness |dT/d(fs)1/2| is no longer relevant. Therefore, if the steepness is still increasing beyond fSB, the steepness at fSB is taken as the maximum steepness. If the maximum steepness occurs at Te before fSB, the steepness at fS(Te) is taken as the maximum steepness. Based on above criterion, Kou predicted the SCS of Al-Cu alloy well, and predicted SCS of Al-Mg alloy well with considering Mg
back-diffusion [6]. They believed that Mg back-diffusion plays a significant role on SCS [4–6]. However, from our PF simulations, it is found that the Mg back-diffusion is negligible under the solidification conditions involved in this study. Instead, we observe that extensive coalescence occurs at lower fs due to the lower segregation and bridging of sidebranches in Al-4.0 wt% Mg alloy. Particularly, we estimate fSB = 0.93 instead of fSB = 0.98 for Al-4.0 wt% Mg alloy from our PF simulations. As for Al-4.0 wt% Cu alloy, extensive coalescence does not occur above Te and thus the steepness at fS(Te) is taken as the maximum steepness, which is consistent with fSB = 0.98. Fig. 4 compares the crack susceptibility index calculated from different cases. As for Al-4.0 wt% Cu, the predicted index from PF simulation is similar to that from Scheil model with fSB = 0.98 and its value is approximately 2500 K. However, for Al-4.0 wt% Mg alloy, it can be seen that the predicted index from PF simulations is notably lower than that from Scheil model, which can be explained by the discrepancy in fs(T) curves as shown in Fig. 3. Moreover, the predicted index at fSB = 0.93, which is determined by our PF simulations, is distinctly lower than that at fSB = 0.98, which is assumed empirically, for Al-4.0 wt% Mg, indicating the significant influence of the selection of fSB value. Particularly, the predicted index (about 550 K) from PF simulation at fSB = 0.93 for Al-4.0 wt% Mg is significantly lower than that for Al-4.0 wt% Cu, showing the better SC resistance of Al-4.0 wt% Mg alloy. This agrees well with above analysis of liquid channel morphologies and the fact that Al-Mg alloy has better weldability than Al-Cu alloy. It should be noted that the coalescence behavior may vary with change in alloy composition and solidification conditions, and the fSB value employed to evaluate SCS should be selected
Fig. 4. Comparison of the crack susceptibility index calculated from Scheil model and PF simulations with different fSB for Al-4.0 wt% Cu and Al-4.0 wt% Mg alloys.
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accordingly. Therefore, understanding how alloy composition and solidification conditions affect the coalescence behavior and thus fSB value is of crucial importance for SCS prediction. This will be discussed in our future work. In summary, we demonstrated the possible reasons why Al-Mg alloys can have lower SCS than Al-Cu alloys at relatively high cooling rate (750 K/s) using 2D PF simulations. We found that the backdiffusion is negligible due to the relatively high cooling rate, and coalescence is easier to occur in Al-4.0 wt% Mg alloy than in Al-4.0 wt% Cu alloy due to the lower solute segregation between primary dendrites and the easier bridging of protruding secondary dendrites. This extensive coalescence facilitates forming a firm structure of dendritic solid skeleton with discontinuous liquid droplet. Furthermore, we compared the SC index calculated from PF simulations with that calculated from analytical model. With considering coalescence predicted by PF simulations, the calculated SC index of Al-4.0 wt% Mg alloy is reasonably lower than Al-Cu alloy. Acknowledgements This research has been supported by the National Basic Research Program (973 Program) of China under grant No. 2014CB046703, the National Natural Science Foundation of China (NSFC) under Grant No. 51721092 and the opening project of State Key Laboratory of Digital Manufacturing Equipment and Technology (Huazhong University of
Science & Technology) under grant No. DMETKF2018001. The calculation of this work was performed on TianHe-2 and thanks for the support of National Supercomputer Center in Guangzhou (NSCC-GZ). The authors also would like to thank the anonymous referees for their valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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