Modeling of eutectic spacing in binary alloy under high pressure solidification

Journal of Alloys and Compounds 646 (2015) 63e67

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Modeling of eutectic spacing in binary alloy under high pressure solidification W. Jiang a, C.M. Zou b, *, H.W. Wang a, Z.J. Wei a, * a b

School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, China National Key Laboratory of Metal Precision Hot Forming, Harbin Institute of Technology, Harbin 150001, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 May 2015 Received in revised form 9 June 2015 Accepted 11 June 2015 Available online 14 June 2015

A new mathematical model was developed to predict the average eutectic spacing under high pressure solidification. The model can be used in the binary alloy system which contains solid solution and intermetallic compound. Following this model, the change of eutectic spacing under high pressure can be predicted if the volumetric conditions of the phases are known. Application of the model to the binary Mg-20.3wt.%Al solidified under different pressure was conducted. It was found that the results predicted by this model show an excellent agreement with the eutectic spacing of Mg-20.3wt.%Al alloy solidified under high pressure. And during experiment, an ultimate pressure was found, the microstructure tends to grow as divorced eutectic when the solidified pressure is higher than this ultimate pressure, the formula is not applicable. © 2015 Elsevier B.V. All rights reserved.

Keywords: High pressure solidification Eutectic Modeling Eutectic spacing MgeAl alloys

1. Introduction Eutectic alloys have relatively low melting points, excellent fluidity, and good mechanical properties. Up to now, the growth of eutectic especially the irregular eutectic has still attracted extensive attention [1e3]. More and more attention is paid to different aspects of eutectic theses years, such as the dynamic effects [4], nucleation [5], stability of growth [6], simulation [7,8] and mathematical prediction [9]. The eutectic mechanical properties are mainly dependent on the two important parameters: the relative volume fractions of the eutectic phases and the eutectic spacing [10,11]. The distance between two consecutive lamellae of one phase in one eutectic cluster is termed as eutectic spacing. In the irregular eutectic, the spacing between lamellae varies and is nonuniform. Hence, for irregular eutectic, eutectic spacing is often referred as the average spacing. Much attention has been paid to the theoretical model of the eutectic spacing since the most wellknown theory for the eutectic solidification of Jackson and Hunt [12]. Years later, Fisher [13] and Trivedi [14] and Magnin et al. [15,16] modified the JacksoneHunt model. Then Donaghey and Tiller made two simplifications in the JH-model (DT-model) [17]. Trivedi et al. generalized JH-model and developed the theory of

eutectic growth under rapid solidification conditions [18]. A. Ludwig and S. Leibbrandt investigated the relationship between growth velocity, lamellar spacing and interface undercooling based on DT-model [19]. But all of these works took no account of the effects of pressure. Pressure is an essential thermodynamic parameter which can influence the solidification process [20e22]. There is huge difference in microstructure and properties between the metal solidified under high pressure and atmospheric pressure. Similarly, the eutectic spacing can be influenced by the high pressure. But the relationship of eutectic spacing to pressure has not been sufficiently discussed, especially very little comparable progress has been made in mathematical model of the eutectic spacing under high pressure. With the high pressure method is more widely used in the preparation of metallic materials, studying the influence of high pressure on eutectic spacing becomes significant. In the present article, a new mathematical model was developed to predict the average eutectic spacing in the binary alloy system which contains solid solution and intermetallic compound under high pressure. The Mg-20.3wt.%Al solidified under different pressure were investigated to prove the correctness of the model. 2. Mathematical model

* Corresponding authors. E-mail addresses: [email protected] (C.M. Zou), [email protected] (Z.J. Wei). http://dx.doi.org/10.1016/j.jallcom.2015.06.090 0925-8388/© 2015 Elsevier B.V. All rights reserved.

The focus of the present work is to model the change of the average eutectic spacing under high pressure. An average spacing in

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this model is shown in Fig. 1. The average eutectic spacing is defined as: l¼ (l1þl2þþln)/n. The theoretical basis of regular eutectic growth theory has been established by Jackson and Hunt [12]. Since then, several authors [13,14,23] have extended the Jackson and Hunt model to apply in the irregular eutectic growth. The wellknown and quite often quoted in literature is the model developed by Magnin and Kurz in 1987 [12,18,24e27]:

l2 V ¼ f2

Kr Kc



1=2  P þ 2kP P þ 4p2 l V ¼ f Dq 2pð1  kÞ 2

2

(5)

Where q is constant, which is a collection of all fixed parameters. According to the definition of Peclet coefficient:

(1) P ¼ lV=2D

l is the average eutectic spacing, V is the growth rate, f is a constant for a given system, Kr and Kc can be evaluated from phase diagram and thermodynamic data Where [28]. jma jjmB j 1  k $ $ Kc ¼ jma j þ jmB j 2pD h

According to the Kurz's work [28], P is separated from the function, the Eqn. (4) is obtained as：

2p=P i1=2  1 þ 2k 1 þ ð2p=PÞ2

(2)

(6)

By inserting Eqn. (6) into (5), the Eqn. (5) is obtained as:

 l2 ¼ f2 q

l2 4

2

(3)

Where ma and mB are liquid slopes, k is the distribution coefficient, G is Gibbs-Thomson coefficient, Q is wetting angle, f is a phase volume fraction, P is Peclet coefficient, D is diffusion coefficient in liquid. By inserting Eqns. (2) and (3) into Eqn. (1), the is Eqn. (1) is obtained as:

h i1=2 1 þ ð2p=PÞ2  1 þ 2k 2 2 4pD $ $ l V ¼f f ð1  f Þ 2p=P f jma jGB sin QB þ ð1  f ÞjmB jGa sin Qa jma jjmB jð1  kÞ

(4)

1=2

 2l þ kl

(7)

2pð1  kÞ

There are considerable difference in the order of magnitude between D~1010 m2/s [28], V~10-2 m/s [28] and l~10-6 m/s [28]. 2

2ð1  f ÞjmB jGa sin Qa þ 2f jma jGB sin QB Kr ¼ f ð1  f ÞðjmB j þ jma jÞ

2

þ 4pV 2D

Because l4 >>4pV 2D , so this term 4pV 2D can be omitted. The Eqn. (7) is obtained after simplifying the equation:

l ¼ f2 q

2

2

2

2

k 2pð1  kÞ

(8)

According to the definition of distribution coefficient, k is written as:

k¼

Cs Cl

(9)

Because a phase is solid solution, Cas is solid solubility. According to the contribution given by L. Kaufman and Y. Minamino [29], who set up a mathematical model for solid solubility under high pressure, obtaining:

X a ðPÞ ¼ X a ð0Þ þ

vX a vP

Xa ¼



V B V a X B X a

PdX a dP a

 vV vX a

(10)



RT

(11)

Where superscripts a and B indicate the solid solution and second phase. The term X(P) is the mole fraction of solid solubility at pressure P, Xa is equal to Cs; V is the molar volume; R is the gas constant; and T is the temperature. The Cs can be defined as a function of pressure

Cs ¼ f ðpÞ

(12)

The distribution coefficient can be expressed as

k¼

f ðpÞ Cl

(13)

As the eutectic phase grows, redundant solute ejected from the primary phase is used to generate the secondary phase. The increase of Cs of a phase only slightly affects the composition of the liquid. Therefore, the composition of liquid can be considered as constant. By inserting Eqn. (13) into (8), the Eqn. (8) is obtained as

l ¼ f2 q

Fig. 1. The schematic illustration of irregular eutectic growth and eutectic spacing l.

f ðPÞ 2pðCl  f ðPÞÞ

(14)

Take a derivative of Eqn. (14) with respect to pressure, the result is

W. Jiang et al. / Journal of Alloys and Compounds 646 (2015) 63e67

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predicted if the volumetric conditions of the phases in binary alloy are known. 0

dl f ðPÞCl ¼ dp ðCl  f ðPÞÞ2

(15)

Where f’(P) is derivative of f(P) with respect to pressure. This equation reveals the relationship between the eutectic spacing and pressure in an irregular eutectic. Following this equation, the change of eutectic spacing under high pressure can be

Fig. 2. XRD patterns of Mg-20.3 wt. % Al alloy solidified under different pressures.

3. Results and discussion 3.1. Application to a binary MgeAl alloy An MgeAl alloy with 20.3wt.%Al was prepared by conventional casting from 99.99 wt.% pure Mg and Al. The samples for highpressure solidification were cut into cylinders 20 mm in diameter and 18 mm in length. The experiments were carried out using a sixanvil apparatus with pressures of 1, 2 and 3 GPa. The phases were characterized by X-ray diffraction (XRD) using a Panalytical Empyrean X-ray diffractometer with monochromatic Cu Ka radiation. Morphology was examined by scanning electron microscopy (SEM) using Quanta 200FEG. Fig. 2 presents the XRD patterns of Mg-20.3wt.%Al alloy solidified under different pressures. The Mg-20.3wt.%Al alloy solidified under normal pressure contains a-Mg and g-Mg17Al12. The reflections of a-Mg become stronger and the reflections of the gphase decrease with increasing pressure. Meanwhile the peaks of a-Mg phase shift to higher angle, indicating that the lattice parameters decrease with pressure. A quantitative analysis of the diffraction patterns by the Rietveld method shows that the weight percent values of g-Mg17Al12 phase is 33.7% under normal pressure, and when the pressure increase from 1 GPa, 2 GPae3 GPa, the calculated weight percent values of g-Mg17Al12 are 15.1%, 9.2% and 3.1% respectively. Fig. 3 shows the change of morphology of (aþg) eutectic after high pressure solidification. It can be seen that the microstructure of Mg-20.3wt.%Al alloy solidified under normal pressure consists of primary a-Mg and reticulated eutectic (gþa) phase. The dark contrast part of eutectic is a-Mg phase. Around the perimeter of the

Fig. 3. Morphology of eutectic solidified under different pressures: (a) normal pressure, (b) 1 GPa, (c) 2 GPa (d) 3 GPa.

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W. Jiang et al. / Journal of Alloys and Compounds 646 (2015) 63e67

Table 1 The average eutectic spacing (AES) under different pressure. Pressure

Normal pressure

1 GPa

2 GPa

3 GPa

AES(mm)

0.5866

1.7598

2.6816

e

eutectic organization there is a circle of g phase halo, which completely isolates the primary a-Mg from the a-Mg of eutectic. g phase connected to the halo. This shows that g phase is the primary phase of MgeMg17Al12 eutectic. Increase the solidification pressure to 1 and 2 GPa, leading to a different morphology of the eutectic. High pressure increases the eutectic mesh diameter and the thickness of both two phases, decreases the mesh density and the continuous degree of network eutectic, the g halo disappear, g phase and a phase alternately contact with primary a phase. When the pressure is 3 GPa, the reticulated eutectics disappear, g phase is irregularly distributed at the grain boundary of primary a-Mg phase, and this growth pattern is like divorced eutectic. The average eutectic spacing (AES) under different pressure is shown in Table 1. In Fig. 4, lattice parameter data for solution phase are shown according to known reports [30e32] and our own work. With the pressure increase, the solid solubility of Al increase. And with the

solid solubility of Al increase, lattice parameter of a-Mg decrease. According to J. L. Murray's data about MgeAl alloy system [33], we can know Va/Xa<0 and Vb>Vg>Va，it means VB > Va, XB > Xa, so the data in Fig. 4 is in good agreement with the Eqn. (11). Pressure can increase the solid solubility of Al in a-Mg. So Cs increases with the increasing of pressure. For MgeAl system, f(P) is an increasing function 0

dl f ðPÞCl ¼ >0 dp ðCl  f ðPÞÞ2

(16)

It is shown that the eutectic spacing is increasing with increase of solidification pressure for MgeAl system. But the formula is applicable only to certain conditions. As is shown in Fig. 5, when the pressure increases from atmosphere to 2 GPa, the eutectic spacing increases obviously. But when the pressure increases to 3 GPa, the eutectic morphology tends to grow as divorced eutectic, the formula is not applicable. This is presumably due to the existence of an ultimate pressure Px that when the solidification pressure is higher than Px, the amount of a phase will be greater than B phase. This condition is similar to what is shown in Fig. 1, where the peaks of g phase cannot be visible when the pressure reaches 3 GPa, and the generation of divorced eutectic is preferred. This condition also applies to other alloys whose microstructural feature will drastically change under high pressure, such as AleNieY alloy [34] whose rod-eutectic feature will transform to netted-eutectic under high pressure. In such case, this formula is no longer applicable for calculating eutectic spacing variations.

3.2. Discussion of different alloy systems As indicated by equations, the change in the eutectic spacing caused by high pressure is closely related to the solid solubility, and the solid solubility is mainly influenced by the volumetric

Fig. 4. The relationship of Lattice Parameter and solid solution of Al in a-Mg.

Fig. 5. The relationship between eutectic spacing and pressure.

W. Jiang et al. / Journal of Alloys and Compounds 646 (2015) 63e67

conditions, so the change in the eutectic spacing of different alloy systems depends on the following cases. 1) When VB > Va, XB > Xa and vVa/vXa<0, pressure increases the solid solubility noticeably. The value of f(P) in Eq. (12) is positive and is expected to be larger. So the value of Eq. (15) is positive too, the eutectic spacing will be lager with the pressure increasing. This condition means that the molar volume of the B phase is larger than that of the a, the molar volume of a phase decreases with the increase of solid solubility and the mole fraction of solute in phase B is more than it in phase a. When VB < Va, XB < Xa and vVa/vXa<0, the value of f(P) in Eq. (12) is expected to be larger too, so pressure can also largen the eutectic spacing. 2) When VB < Va, XB > Xa and vVa/vXa>0, there should be a large reduction of solid solubility caused by pressure. These conditions are the reverse of the first two cases. The value of f(P) in Eq. (12) is decreasing and is expected to be smaller. So the value of Eq. (15) is decreasing too, the eutectic spacing will be smaller with the pressure increasing. When VB > Va, XB < Xa and vVa/ vXa>0, pressure also decreases the solid solubility. The value of f(P) in Eq. (12) is expected to be smaller too, so pressure can also reduce the eutectic spacing. 3) When the VB > Va, XB > Xa and vVa/vXa>0 or VB < Va, XB < Xa and vVa/vXa>0 or VB > Va, XB < Xa and vVa/vXa<0 or VB < Va, XB > Xa and vVa/vXa<0, directions of the changes in the value of f(P) in V B V a  vV a , vX a X B X a V V and vV a . So vX a X B X a

Eq. (12) caused by pressure depend on the value of B

a

because there is an offset between the values of the change in eutectic spacing under high pressure is harder to predict. 4. Conclusion

Based on the JH-model modified by Magnin and Kurz and considering the effect of pressure, a new mathematical model (Eqn. (15)) describing the eutectic growth was proposed in this article. The model can be used in the binary alloy system which contains solid solution and intermetallic compound. This equation reveals the relationship between the eutectic spacing and pressure in an irregular eutectic. Following this equation, the change of eutectic spacing under high pressure can be predicted if the volumetric conditions of the phases in binary alloy are known. The correctness of this equation is verified by applying this equation to the MgeAl alloy, where the predicted result derived from this equation fits very well with the experimental observation. And an ultimate pressure for MgeAl alloy was found, the microstructure tends to grow as divorced eutectic when the solidified pressure is higher than this ultimate pressure, the formula is not applicable. Acknowledgments The authors gratefully acknowledge the support of the Natural Science Foundation of China (No. 51171054, No.51001041) References
lu, G. Faivre, A theory of thin [1] S. Akamatsu, S. Bottin-Rousseau, M. S¸erefog lamellar eutectic growth with anisotropic interphase boundaries, Acta Mater. 60 (2012) 3199e3205.

67

[2] J.H. Li, X.D. Wang, T.H. Ludwig, Y. Tsunekawa, L. Arnberg, J.Z. Jiang, P. Schumacher, Modification of eutectic Si in Al-Si alloys with Eu addition, Acta Mater. 84 (2015) 153e163. [3] J.J. Xu, Y.Q. Chen, Steady spatially-periodic eutectic growth with the effect of triple point in directional solidification, Acta Mater. 80 (2014) 220e238. [4] S. Liu, J. Lee, R. Trivedi, Dynamic effects in the lamellar-rod eutectic transition, Acta Mater. 59 (2011) 3102e3115. [5] S.D. McDonald, K. Nogita, A.K. Dahle, Eutectic nucleation in Al-Si alloys, Acta Mater. 52 (2004) 4273e4280. [6] A. Parisi, M. Plapp, Stability of lamellar eutectic growth, Acta Mater. 56 (2008) 1348e1357. [7] M. Perrut, A. Parisi, S. Akamatsu, S. Bottin-Rousseau, G. Faivre, M. Plapp, Role of transverse temperature gradients in the generation of lamellar eutectic solidification patterns, Acta Mater. 58 (2010) 1761e1769. [8] M. Zhu, L. Zhang, H. Zhao, D.M. Stefanescu, Modeling of microstructural evolution during divorced eutectic solidification of spheroidal graphite irons, Acta Mater. 84 (2015) 413e425. [9] W. Zhang, L. Xiao, A Numerical model of lamellar spacing of Al-Si Eutectic, Acta Metall. Sin. 14 (2009) 291e295. € [10] M. Gündüz, H. Kaya, E. Çadırlı, A. Ozmen, Interflake spacings and undercoolings in Al-Si irregular eutectic alloy, Mater. Sci. Eng. A 369 (2004) 215e229. [11] A. Karma, M. Plapp, New insights into the morphological stability of eutectic and peritectic coupled growth, JOM 56 (2004) 28e32.  (Ed.), [12] K.A. Jackson, J.D. Hunt, Lamellar and rod eutectic growth, in: P. Pelce Dynamics of Curved Fronts, Academic Press, San Diego, 1988. [13] D.J. Fisher, W. Kurz, A theory of branching limited growth of irregular eutectics, Acta Metall. 28 (1980) 777e794. [14] R. Trivedi, J.T. Mason, J.D. Verhoeven, W. Kurz, Eutectic spacing selection in lead-based alloy systems, Metall. Trans. A 22 (1991) 2523e2533. [15] P. Magnin, J.T. Mason, R. Trivedi, Growth of irregular eutectics and the Al-Si system, Acta Metall. 39 (1991) 469e480. [16] P. Magnin, R. Trivedi, Eutectic growth: A modification of the Jackson and Hunt theory, Acta Metall. 39 (1991) 453e467. [17] L.F. Donaghey, W.A. Tiller, On the diffusion of solute during the eutectoid and eutectic transformations, Mater. Sci. Eng. 3 (1968) 231e239. [18] R. Trivedi, P. Magnin, W. Kurz, Theory of eutectic growth under rapid solidification conditions, Acta Metall. 35 (1987) 971e980. [19] A. Ludwig, S. Leibbrandt, Generalised ‘Jackson-Hunt’ model for eutectic solidification at low and large Peclet numbers and any binary eutectic phase diagram, Mater. Sci. Eng. A 375e377 (2004) 540e546. [20] H. Fu, Q. Peng, J. Guo, B. Liu, W. Wu, High-pressure synthesis of a nanoscale YB12 strengthening precipitate in Mg-Y alloys, Scr. Mater. 76 (2014) 33e36. [21] B. Han, S. Wu, Microstructural evolution of high manganese steel solidified under superhigh pressure, Mater. Lett. 70 (2012) 7e10. [22] P. Ma, C.M. Zou, H.W. Wang, S. Scudino, K.K. Song, M. Samadi Khoshkhoo, Z.J. Wei, U. Kühn, J. Eckert, Structure of GP zones in Al-Si matrix composites solidified under high pressure, Mater. Lett. 109 (2013) 1e4. [23] T. Sato, Y. Sayama, Completely and partially co-operative growth of eutectics, J. Cryst. Growth 22 (1974) 259e271. [24] P. Magnin, W. Kurz, An analytical model of irregular eutectic growth and its application to Fe-C, Acta Metall. 35 (1987) 1119e1128. [25] R.W. Series, J.D. Hunt, K.A. Jackson, The use of an electric analogue to solve the lamellar eutectic diffusion problem, J. Cryst. Growth 40 (1977) 221e233. [26] J.N. Clark, J.T. Edwards, R. Elliott, Eutectic solidification, Metall. Mater. Trans. A 6 (1975) 232e233. [27] J.N. Clark, R. Elliott, Interlamellar spacing measurements in the Sn-Pb and AlCuAl2 Eutectic Systems, Metall. Trans. A 7 (1976) 1197e1202. [28] W. Kurz, D.J. Fisher, Fundamentals of Solidification, fourth ed., Trans Tech Publications Ltd, Trans Tech House, Aedermannsdorf, 1986. [29] Y. Minamino, T. Yamane, H. Araki, N. Takeuchi, Y.S. Kang, Y. Miyamoto, T. Okamoto, Solid solubilities of manganese and titanium in aluminum at 0.1 MPa and 2.1 GPa, Metall. Mater. Trans. A 22 (1991) 783e786. [30] K. Detert, L. Thomas, Investigation of precipitation in aluminium magnesium alloys by transmission electron microscopy, Z. Met. 55 (1964) 83e87. [31] G.V. Raynor, The lattice spacings of the primary solid solutions in magnesium of the metals of Group IIIB and of tin and lead, Proc. R. Soc. Lond. Ser. A 180 (1942) 107e121. [32] R.S. Busk, Lattice parameters of magnesium alloys, Trans. AIME 188 (1950) 1460e1464. [33] J.L. Murray, The Al-Mg (Aluminum-Magnesium) system, Bull. Alloy Phase Diagrams 3 (1982) 60e74. [34] R. Xu, The effect of high pressure on solidification microstructure of Al-Ni-Y alloy, Mater. Lett. 59 (2005) 2818e2820.