Stable and metastable phase equilibria in binary Mg-Gd system: A comprehensive understanding aided by CALPHAD modeling

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Stable and metastable phase equilibria in binary Mg-Gd system: A comprehensive understanding aided by CALPHAD modeling Huaijia Si a, Yuxun Jiang a, Ying Tang b,∗, Lijun Zhang a,∗ a State b School

Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China of Materials Science and Engineering, Hebei University of Technology, Tianjin 300130, China

Received 6 March 2019; received in revised form 9 April 2019; accepted 16 April 2019 Available online xxx

Abstract In this paper, a comprehensive understanding of stable and metastable phase equilibria in binary Mg-Gd system was conducted with an aid of the CALculation of PHAse Diagram (CALPHAD) modeling. Firstly, thermodynamic descriptions of all the stable phases in the Mg-Gd system were re-assessed by considering all the experimental data in the literature. The discrepancy between the phase equilibria and thermochemical properties existing in the previous assessments was eliminated, and the better agreement with the experimental data was achieved in the present assessment. Secondly, the Gibbs energies for metastable β”-Mg3 Gd and β’-Mg7 Gd were constructed based on the first-principles and CALPHAD computed results as well as their correlation, and then incorporated into the CALPHAD descriptions. The model-predicted solvuses of (Mg) in equilibrium with the metastable β”-Mg3 Gd and β’-Mg7 Gd compounds showed very good agreement with the limited experimental data. Finally, the presently obtained thermodynamic descriptions of both stable and metastable phases in the binary Mg-Gd system were further validated by realizing the quantitative Scheil-Gulliver solidification simulations of 5 as-cast Mg-Gd alloys, and the successful prediction of the precipitation sequences in Mg-15Gd and Mg-12Gd alloys during the aging process. © 2019 Published by Elsevier B.V. on behalf of Chongqing University. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer review under responsibility of Chongqing University Keywords: Mg-Gd system; Phase equilibria; Metastable phase; CALPHAD; Precipitation sequence.

1. Introduction Magnesium alloys are known for their light weight, specific stiffness and good cast-ability and are thus greatly attractive in the automotive and aerospace industries. However, the current applications of the Mg alloys are still limited due to their relatively low mechanical properties. It was reported [1–3] that the addition of rare earth (RE) elements can effectively improve the creep resistance and mechanical properties of Mg alloys. Among a variety of RE elements, Gd is one of the promising candidates for novel Mg-based alloys with high mechanical properties. For Mg-Gd alloys, the strengthening mechanisms may include both the solid solution strengthening ∗

Corresponding authors. E-mail addresses: [email protected] (Y. Tang), lijun.zhang@ csu.edu.cn (L. Zhang).

[4] and the precipitation hardening [5]. The equilibrium solubility of Gd in (Mg) can reach the maximum value of 23.5 wt.% at 817 K and decreases to 3.82 wt.% at 473 K [6], which provides the advantageous conditions for aging treatment. The precipitation process in Mg-Gd alloys during aging is rather complex because a series of metastable and also stable phases are involved. In order to further improve the mechanical properties of Mg-Gd alloys, the accurate information on the formation of precipitates and their sequence during the aging process is highly required. Therefore, the cognition of stable and metastable phase equilibria over the entire composition and temperature ranges in binary Mg-Gd system is the prerequisite. The CALculation of PHAse Diagram (CALPHAD) is the most powerful approach to obtain the accurate stable and metastable phase equilibria in a complex system by integrating the phase equilibria and thermochemical properties

https://doi.org/10.1016/j.jma.2019.04.006 2213-9567/© 2019 Published by Elsevier B.V. on behalf of Chongqing University. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer review under responsibility of Chongqing University Please cite this article as: H. Si, Y. Jiang and Y. Tang et al., Stable and metastable phase equilibria in binary Mg-Gd system: A comprehensive understanding aided by CALPHAD modeling, Journal of Magnesium and Alloys, https:// doi.org/ 10.1016/ j.jma.2019.04.006

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from the experimental measurements and/or atomistic simulations [7–10]. For the binary Mg-Gd system, its stable phase equilibria have been thermodynamically assessed by several groups [11–15] using the CALPHAD approach. The earlier thermodynamic assessments performed by Cacciamani et al. [11,12] and Guo et al. [13] cannot well reproduce the solubility of Gd in (Mg) as well as the liquidus and eutectic reaction in Mg-rich side. The subsequent CALPHAD assessment is from Hampl et al. [14], who considered their own experimental data. However, according to Hampl et al. [14], there are discrepancies among the invariant reactions, the activities in Mg-Gd alloys and the enthalpies of formation in the literature, and thus employed two sets of thermodynamic parameters to model the Mg-Gd system. Moreover, the stable intermetallic compounds, Mg5 Gd and Mg3 Gd, in the CALPHAD assessments by Refs. [11–14] were simply treated as the stoichiometric ones. Such simple treatment is inconsistent with the very recent experimental data by Das et al. [15], who measured the homogeneity range for the intermetallic compounds, and also performed a new CALPHAD assessment for binary Mg-Gd system. But in the work of Das et al. [15], the discrepancy between the phase equilibria and thermochemical property data in the literature pointed out by Hampl et al. [14] still existed. As for the metastable phase equilibria, no CALPHAD-type thermodynamic descriptions have been reported for the binary Mg-Gd system up to now. Consequently, the main objectives of this work are i) to perform a CALPHAD re-assessment of thermodynamic descriptions for all the stable phases in the binary Mg-Gd system by fully considering all the experimental phase equilibria and thermochemical properties. The major concern is to eliminate the discrepancy between the phase equilibria and thermochemical property data existing in the previous assessments; ii) to evaluate the thermodynamic descriptions for the metastable β’-Mg7 Gd and β”-Mg3 Gd compounds based on the presently established thermodynamic descriptions of stable phases and the existing first-principles calculations. The major concern in this section is to predict the solvuses of (Mg) in equilibrium with metastable β’-Mg7 Gd and β”-Mg3 Gd compounds, and compare with the limited literature data; and iii) to further validate the reliability of the obtained thermodynamic descriptions for both stable and metastable phases by predicting the as-cast microstructure information during solidification and the precipitation sequence during aging in different Mg-Gd alloys.

2. Literature review 2.1. Phase equilibria and thermodynamic properties of stable phases The phase equilibria and thermodynamic properties of Mg-Gd binary system were experimentally investigated by Rokhlin and Nikitina [6,14–25], and concisely summarized in Table 1. It should be noted that all the experimental data before the year 1986 [6,16,17,20,23,24] had been critically

reviewed by Nayeb-Hashemi and Clark [26]. Thus, only the experimental phase equilibria data after 1986 are reviewed here. The liquidus temperatures in the Mg-rich side were experimentally measured by several groups [14,18,19] using the differential thermal analysis (DTA) technique. The liquidus data by Hampl et al. [14,18,19] show good agreement with those from the earlier investigations by Manfrinetti and Gschneidner [17], and thus all the data are used in the present assessment. The solid solubility of Gd in (Mg) was investigated by Hampl et al. [14] and Peng et al. [21]. While the temperature for the eutectic reaction (L→(Mg)+Mg5 Gd) was measured by Hampl et al. [14] and Rokhlin and Nikitina [18], and their obtained results are consistent with the previous experimental data [16,17]. The temperature for the peritectic reaction, L+MgGd→Mg2 Gd, was measured to be 1028.2 K by Janssen et al. [19]. Recently, Das et al. [15] and Zheng et al. [22] investigated the homogeneity ranges of Mg5 Gd, Mg3 Gd and Mg2 Gd by means of the diffusion couple method in combination with the electron probe micro analyzer (EPMA) technique. In contrast, the experimental information on thermodynamic properties for binary Mg-Gd system available in the literature was very limited. The enthalpies of formation for stable intermetallic compounds including Mg5 Gd, Mg3 Gd, Mg2 Gd and MgGd were derived from their measured vapor pressure of Mg by Pahlman and Smith [24]. After that, Cacciamani et al. [25] measured the standard enthalpy of formation for MgGd phase at 300 K by using the calorimetric method, which shows good agreement with the derived one by Pahlman and Smith [24]. Thus, the enthalpy of formation reported by Cacciamani et al. [25] was directly used in the present assessment. Ogren et al. [23] and Pahlman and Smith [24] measured the vapor pressure of Mg in a series of the Mg-Gd two-phase alloys from 650 K to 930 K by using the Knudsen effusion technique. In a standard CALPHAD assessment, the vapor pressure cannot be directly used. As proposed by Hampl et al. [14], the activity of Mg relative to its referal l oy/hcp ence state (i.e., hcp phase) (aMg ) in Mg-Gd alloys at a given temperature can be transformed from the vapor pressure data via the following equation,

al l oy/hcp aMg (T )

=

l oy pal Mg (T )

p0Mg (T )

(1)

l oy where pal Mg (T ) is the vapor pressure of Mg in Mg-Gd alloys, and p0Mg (T ) is the vapor pressure of pure Mg. With Eq. (1), the measured vapor pressure of Mg from Ogren et al. [23] and Pahlman and Smith [24] can be transformed to the activity data. Moreover, considering the fact that the samples prepared by Ogren et al. [23] did not reach the equilibrium state as pointed out by Pahlman and Smith [24], only the activities of Mg transferred from the original vapor pressure data from Pahlman and Smith [24] are used in the present assessment, rather than the data from Ogren et al. [23].

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Table 1 Summary of the phase equilibria and thermodynamic properties for stable phases in the binary Mg-Gd system available in the literature [6,13–25]. Type of data

Temperature

Composition

Experimental method

Quoted modea

Ref.

Liquidus, invariants

673–1143 K

0–100 at.% Gd

–

[16]

273–1573 K 273–973 K

0–100 at.% Gd 0–20 at.% Gd 40 at.% Gd 6.38 at.% Gd 85–100 at.% Gd 0–10 at.% Gd 1.57–7.19 at.% Gd 1.69–7.68 at.% Gd 0–100 at.% Gd 0–100 at.% Gd (αGd)+MgGd two–phase region four two-phase regions from Mg5 Gd to (Gd) MgGd

DTA, XRD, Hardness, microhardness DTA, XRD, metallography TA DTA DSC XRD El. Resistivity EDS, XRD SEM, EDX XRD, EPMA OM, SEM, EPMA Knudsen effusion technique Knudsen effusion technique

+ + + + + + – + + – – +

[17] [18] [19] [14] [20] [6] [21] [14] [15] [22] [23] [24]

Calorimetry

+

[25]

Solubility of Mg in (αGd) Solubility of Gd in (Mg)

Homogeneity range of intermetallic phases Vapor pressure of Mg

Standard enthalpy of formation of MgGd

623–973 473–813 573–823 573–823 703–773 773 K 673–913 653–933 300 K

K K K K K K K

TA = Thermal Analysis; OM = Optical Microscopy; DTA = Differential Thermal Analysis; XRD = X-Ray Diffraction; SEM = Scanning Electron Microscope; EDX = Energy Dispersive X-Ray Spectroscopy; DSC = Differential Scanning Calorimeter; EPMA = Electron Probe Micro Analyzer. a indicates whether the data are used in the parameter optimization: +, used; -, not used but employed for comparison.

2.2. Phase equilibria and thermodynamic properties of metastable phases In the literature, there are numerous experimental studies on the precipitation sequence [27–30], crystal structures [31–33], lattice constants [32,33] and chemical compositions [31–33] of different precipitates in binary Mg-Gd system. Rokhlin et al. [27] pointed out that there is no existence of the Guinier Preston (GP) zone along the precipitation sequence of binary Mg-Gd alloys, but only the transformation from the metastable β”-Mg3 Gd and β’-Mg7 Gd phases to the equilibrium β-Mg5 Gd phase. Then, the existence of metastable β’-Mg7 Gd phase was confirmed in the Mg-15 wt.%Gd alloy during the aging process by Liu et al. [34] using the atomic-scale high-angle annular detector darkfield scanning transmission electron microscopy (HAADFSTEM) technique. Zeng [28] studied the precipitation behavior of Mg-12 wt.% Gd alloy during aging, and observed the existence of β”-Mg3 Gd and β’-Mg7 Gd phases at 175 °C, β’-Mg7 Gd phase at 200 °C and β”-Mg3 Gd phase at 225 °C, respectively. Xie et al. [35] employed the HAADF-STEM techniques to study the phase structures of Mg-11.5 wt.% Gd alloy during isothermal aging at 230 °C. It was found that there are two types of precipitates including the long-structure β’ (denoted as β L ’) and short-structure β’ (denoted as β s ’) around the aging peak of the Mg-11.5 wt% Gd alloy. However, β s ’ precipitate distributes in the middle of β L ’ precipitates rather than in (Mg) matrix. Thus, in the present work, only the precipitation of β L ’ (simplified as β’) from (Mg) matrix was considered for simplification. The existence of the metastable β 1 −Mg3 Gd precipitate was observed during the aging process in the Mg-Gd-Zr [36], Mg-Gd-Y [37], MgY-Nd [38–40] and Mg-Gd-Nd [41] ternary alloys. However, the β 1 −Mg3 Gd phase has not been experimentally confirmed during the aging process in Mg-Gd binary alloys. Thus, the

β 1 −Mg3 Gd phase was not considered either in the present work. The precipitation sequences of Mg-5 wt.% Gd, Mg-10 wt.% Gd and Mg-15 wt.% Gd alloys during the aging process were determined by Vostrý et al. [29] by means of transmission electron microscopy (TEM) and electron diffraction (ED) techniques. The determined precipitation sequence of the Mg-15 wt.% Gd alloy is: S.S.S.S(Mg)→ β”-Mg3 Gd → β’Mg7 Gd → β-Mg5 Gd, while the sequences for both Mg-5 wt.% Gd and Mg-10 wt.% Gd alloys differ from that of Mg15 wt.% Gd alloy with the absence of the β’-Mg7 Gd phase. Later, Zeng [28] also determined the precipitation sequence of Mg-12 wt.% Gd alloy aging at 175, 200 and 225 °C as S.S.S.S(Mg)→ β”-Mg3 Gd → β’-Mg7 Gd → β-Mg5 Gd. Moreover, according to the measurement of the electrical resistivity, the temperature ranges for the existence of the precipitation sequences S.S.S.S(Mg)→ β”-Mg3 Gd→ β’-Mg7 Gd→ β-Mg5 Gd and S.S.S.S(Mg)→ β’-Mg7 Gd→ β-Mg5 Gd were determined to be <519.55 K and 519.55∼618.35 K by Vostrý et al. [29] in Mg-15 wt.% Gd alloy, respectively. The results by Vostrý et al. [29] were further confirmed by Kekule et al. [30] later. The temperatures related to the phase transitions among metastable/stable phases in the Mg-15 wt.% Gd alloy reported in the literature were summarized in Table 2. Based on the selected area electron diffraction (SAED) and three-dimensional atom probe (3DAP) compositional analysis, Honma et al. [31] suggested that the chemical compositions of the metastable β”-Mg3 Gd and β’-Mg7 Gd phases were quite close to Mg3 Gd and Mg5 Gd, respectively. Moreover, the crystal structures of β”-Mg3 Gd and β’-Mg7 Gd phases were constructed to be D019 and base centered orthorhombic (bco) by using the high resolution transmission electron microscopy (HRTEM) technique. By means of the HAADF-STEM techniques, Nishijima et al. [32,33] proposed a new composition of Mg7 Gd for the precipitate β’-Mg7 Gd on the basis of the

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H. Si, Y. Jiang and Y. Tang et al. / Journal of Magnesium and Alloys xxx (xxxx) xxx Table 2 Summary of the temperature ranges for different precipitation sequences in the Mg-15 wt.% Gd alloys during the aging process. Nominal

Actual composition

Temperature range of precipitation sequence (K)

composition

[wt.% (at.%) ]

S.S.S.S(Mg)→β”→β’→β

S.S.S.S(Mg)→β’→β

Mg-15Gd

14.55 (2.57) 15.00 (2.66)

<519.55 <519.65

519.55–618.35 519.65–617.85

same structural model, i.e., bco. The chemical composition of β”-Mg3 Gd was further confirmed to be Mg3 Gd by means of first-principles calculation [42]. Chen et al. [43] computed the formation energies of the Mg15 Gd and the Mg7 Gd by using the first-principles method. It was found that the Mg7 Gd is energetically more stable in comparison with the Mg15 Gd. This result was further confirmed by Gao et al. [44] using the first-principles calculation. Thus, the composition of the β’-Mg7 Gd phase was accepted to be Mg7 Gd in the present work. Furthermore, it is very difficult to obtain the thermodynamic properties of the metastable phase by using the experimental measurement. Under this situation, the first-principles calculations can be applied to compute various thermodynamic properties based on the crystal structural information. The enthalpies of formation for β”-Mg3 Gd and β’-Mg7 Gd phases were calculated by several groups [42–47] using the first-principles approach. However, it is well known that the accuracy of the thermodynamic properties in alloys with RE elements calculated using the first-principles calculations is still unsatisfactory. In 2016, Peng et al. [47] again computed the enthalpies of formation for β”-Mg3 Gd and β’-Mg7 Gd (β L ’) using the first-principles calculations based on the more accurate “f-core potential” for Gd. Thus, the first-principles calculation results by Peng et al. [47] are used in the present work.

Reference

[29] [30]

expressed as v,φ LMg,Gd =Av + Bv · T

(3)

where Av and Bv are the parameters to be optimized based on the reliable experimental phase equilibria and thermochemical data. 3.1.2. Intermetallic compounds There are four stable intermetallic compounds in the MgGd binary system. Based on the experimental solubility ranges of intermetallic compounds reported by Das et al. [15] and Zheng et al. [22]. The compounds, Mg2 Gd, Mg3 Gd and Mg5 Gd, are described using the sub-lattice model in the present work. The intermetallic compound Mg2 Gd is the isotypic one with MgCu2 structure, while the Mg3 Gd is with the BiF3 structure. Mg2 Gd and Mg3 Gd phases were treated as the formula (Mg, Gd)2 (Mg, Gd) and (Mg, Gd)3 (Mg, Gd) (Note: The bold elements are the main elements in that sublattice), respectively. Considering its asymmetric homogeneity range, the Mg5 Gd compound is modeled as Mg5 (Mg, Gd) in the present work. Taking the Mg2 Gd phase for example, its Gibbs energy per mole-atom can be given as 



g2 Gd g2 Gd   0 M g2 Gd GM = yMg · yMg·0 GM m M g:M g + yMg · yGd · GMg:Gd 



g2 Gd   0 M g2 Gd +yGd · yMg·0 GM Gd:Mg + yGd · yGd · GGd:Gd     +2/3 · R · T (yMg · ln yMg + yGd · ln yGd ) 







3. Thermodynamic models

+1/3R · T · (yMg · ln yMg + yGd · ln yGd )

3.1. Stable phases

M g2 Gd   0 M g2 Gd +yMg · yGd · (yMg·0 LM g,Gd:M g + yGd · LMg,Gd:Gd )





3.1.1. Solution phases The Gibbs energy of the solution phase φ (αGd, βGd, (Mg) or liquid) is described by the substitutional solution model, and its molar Gibbs energy can be given as





M g2 Gd   0 M g2 Gd +yMg · yGd · (yMg ·0 L M g:M g,Gd + yGd · LGd:Mg,Gd ) (4) 

where yi and yi denote the site fractions of element g2 Gd 0 M g2 Gd i in the first and second sublattices.0 GM GMg:Gd , M g:M g ,

(2)

g2 Gd 0 M g2 Gd 0 Mg2 Gd GM GGd:Gd GGd:Gd represent the Gibbs enerGd:Mg and gies of four end-members, which are relative to the enthalpies of pure Mg and Gd in their standard element reference (SER) states. The reciprocal relation g2 Gd 0 M g2 Gd 0 M g2 Gd 0 M g2 Gd (0 GM M g:M g + GGd:Gd = GGd:Mg + GMg:Gd ) is applied to evalu-

in which and are the Gibbs energies of pure Mg and Gd in the phase state of φ, respectively. xMg and xGd represent the mole fractions of elements Mg and Gd, respectively, and R (8.314 J mol−1 K−1 ) is the gas constant. The last term in Eq. (2) denotes for the excess Gibbs energy, which can be v,φ described using the Redlich-Kister polynomial [48]. LMg,Gd is the vth order interaction parameter, which can be typically

g2 Gd 0 M g2 Gd ate the metastable end-member 0 GM Gd:Mg , LM g,Gd:M g is the interaction parameter between the elements Mg and Gd on the first sublattice when Mg occupies the second sublattice completely, and needs to be optimized based on the corresponding experimental data. Due to the limited experimental information with the compound Mg2 Gd, the following M g2 Gd 0 M g2 Gd 0 M g2 Gd 0 M g2 Gd relations (0 LM g,Gd:M g= LMg,Gd:Gd and LM g:M g,Gd = LGd:Mg,Gd ) are applied in the present assessment for simplification. The

0,φ Gφ (T ) = xMg · G0,φ Mg + xGd · GGd

+R · T · (xMg · ln xMg + xGd · ln xGd )  v,φ v +xMg · xGd · v LMg,Gd · (xMg − xGd ) G0,φ Mg

G0,φ Gd

0

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analogous Gibbs energies as Eq. (4) can be also written for Mg3 Gd and Mg5 Gd. The intermetallic compound MgGd is with the B2 structure, but the homogeneity range of MgGd compound is negligible according to the experimental data by Das et al. [15]. Thus, the MgGd phase is treated as one stoichiometric compound. Its molar Gibbs energy can be given as 0,hcp GMgGd = 0.5 · G0,hcp +A+B·T Mg + 0.5 · GGd

(5)

Here, the coefficients A and B need to be optimized according to the corresponding experimental data. 3.2. Metastable compounds The experimental phase equilibria and thermodynamic data for the metastable compounds, β”-Mg3 Gd and β’-Mg7 Gd, are very limited in the literature. Thus, the compounds β”-Mg3 Gd and β’-Mg7 Gd are just treated as the stoichiometric phases for simplification. The molar Gibbs energy of metastable phase θ (β”-Mg3 Gd or β’-Mg7 Gd) is described as Gθ (T ) = H θ (T ) − T · S θ (T ) = A + B · T + C · T · ln T + D · T 2 + F · T −1

(6)

The coefficients A, B, C, D and F in Eq. (6) need to be determined to obtain the complete thermodynamic description for metastable compound θ . 4. Results and discussion 4.1. Thermodynamic descriptions of stable phases The optimization of the thermodynamic parameters for stable phases in the Mg-Gd binary system was carried out by the PARROT module incorporated in Thermo-calc software package [49] following the assessment strategy in the Co-Si system by Zhang et al. [50]. The optimization process worked by minimizing the square sum of the differences between the measured and calculated phase equilibria/thermochemical properties. In the present assessment procedure, each piece of the experimental information was given a certain weight. The weights were changed systematically during the assessment until most of the selected experimental information can be reproduced within the expected uncertainty limits. The finally obtained thermodynamic parameters for all the stable phases in binary Mg-Gd system are listed in Table 3. The calculated phase diagram of the binary Mg-Gd system according to the presently obtained thermodynamic parameters is shown in Fig. 1. Its comparison with all the experimental phase equilibria [6,14,15,17–22] and the enlarged parts in Mg-rich region are presented in Fig. 2(a) and (b). As can be seen in Fig. 2, the presently calculated phase equilibria are in good agreement with all the experimental data [6,14,15,17–21]. The calculated homogenization ranges of the intermetallic compounds, Mg5 Gd, Mg3 Gd and Mg2 Gd well reproduce the experimental data measured by Das et al. [15]. Besides, the calculated phase equilibria due to the thermodynamic parameters by Hampl et al. [14] are also superimposed

Fig. 1. (color on web) Calculated stable phase diagram of Mg-Gd binary system according to the presently obtained thermodynamic parameters.

in Fig. 2 for a direct comparison. It should be noted that the blue dashed lines are calculated due to Model I (denoted as I) from Hampl et al. [14], while the green dashed line are due to Model II (denoted as II). As shown in Fig. 2(a), the presently calculated liquidus in the Mg-rich side and solubility range of (αGd) show better agreement with the experimental data [14,15,17–20] than the previous assessments by Hampl et al. [14] (both Model I and Model II). Moreover, the calculated eutectic reaction, L→(Mg)+Mg5 Gd, the solid solubility of Gd in (Mg) and liquidus temperatures in the Mg-rich side also agree very well with the experimental data [6,14,15,17,18,21,22], as shown in Fig. 2(b). It is well known that the accurate solubility information in the Mg-rich side plays an important role in the design of the novel Mg alloys. Fig. 3 displays the presently calculated logarithm of the solubility of Gd in (Mg) vs. reciprocal absolute temperature, compared with the experimental data [6,14,15,21,22]. It can be seen in the figure that the present calculations are in good agreement with most of the experimental data [6,14,15,21,22] except for one experimental point (at 473 K) from Rokhlin and Nikitina [6]. The maximum solubility of Gd in (Mg) obtained in this work is 4.5 at.%, which agrees well with the experimental ones (i.e., 4 ± 0.3 at.% Gd [17] and 4.53 at.% Gd [6]). What’s more, at high temperatures, both the calculated results in this work and those from Hampl et al. [14] can reproduce the experimental data very well. While at low temperatures, the predicted solubility of Gd in (Mg) by Hampl et al. [14] with Model I decreases rapidly, and shows an obvious deviation from the experimental data around 560 K [6,14,21]. Again, it also indicates in the figure that the present result is as good as that from Model II by Hampl et al. [14]. Table 4 summarizes the calculated temperatures and compositions for all the invariant reactions in the binary Mg-Gd system, compared with the experimental data [14,16–19] together with those calculations from the previous assessments [13–15]. It shows that the presently calculated

Please cite this article as: H. Si, Y. Jiang and Y. Tang et al., Stable and metastable phase equilibria in binary Mg-Gd system: A comprehensive understanding aided by CALPHAD modeling, Journal of Magnesium and Alloys, https:// doi.org/ 10.1016/ j.jma.2019.04.006

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H. Si, Y. Jiang and Y. Tang et al. / Journal of Magnesium and Alloys xxx (xxxx) xxx Table 3 Summary of the finally obtained thermodynamic parameters of stable and metastable phases in the Mg-Gd system.a . Phases

Thermodynamic parameters

Liquid: (Mg, Gd)

LMg,Gd = −12381 − 33.0 · T

0,L iquid 1,L iquid

LMg,Gd = 67231 − 22.9 · T 2,L iquid

LMg,Gd = −29659 + 17.1 · T (αGd)/(Mg)-hcp: (Mg, Gd)

0,hcp

LMg,Gd = 3827 − 29.7 · T 1,hcp

LMg,Gd = −11674 + 26.9 · T 2,hcp

LMg,Gd = −15122 − 18.9 · T 3,hcp

LMg,Gd = 27388 (βGd)-bcc: (Mg, Gd)

0,bcc LMg,Gd = −28500 − 15.4 · T 1,bcc LMg,Gd = 27590 − 6.2 · T

MgGd: (Mg)1/2 (Gd)1/2

0,hcp GMgGd = 1/2 · G0,hcp Mg (T ) + 1/2 · GGd (T ) − 15100 − 2.8 · T

Mg2 Gd: (Mg,Gd)2/3 (Mg, Gd)2/3

g2 Gd 0,hcp 0,hcp GM Mg:Gd = 2/3 · GMg (T ) + 1/3 · GGd (T ) − 19831 + 1.8 · T g2 Gd 0,hcp GM M g:M g = GMg (T ) + 13499 g2 Gd 0,hcp GM Gd:Gd = GGd (T ) + 3100 g2 Gd 0,hcp 0,hcp GM Gd:Mg = 1/3 · GMg (T ) + 2/3 · GGd (T ) + 36430 − 1.8 · T M g2 Gd LMg,Gd: ∗ = −2968 g2 Gd L∗M:Mg,Gd = −25307

Mg3 Gd: (Mg,Gd)3/4 (Mg, Gd)1/4

g3 Gd 0,hcp 0,hcp GM Mg:Gd = 3/4 · GMg (T ) + 1/4 · GGd (T ) − 20130 + 3.1 · T M g Gd

GM g:3M g = G0,hcp Mg (T ) + 5533 g3 Gd 0,hcp GM Gd:Gd = GGd (T ) + 9985 g3 Gd 0,hcp 0,hcp GM Gd:Mg = 1/4 · GMg (T ) + 3/4 · GGd (T ) + 35648 − 3.1 · T M g3 Gd LMg,Gd: ∗ = −17109 M g Gd

3 L∗ :Mg,Gd = −8220

Mg5 Gd: (Mg)5/6 (Mg, Gd)1/6

M g Gd

0,hcp GMg:5Gd = 5/6 · G0,hcp Mg (T ) + 1/6 · GGd (T ) − 15832 + 2.2 · T M g Gd

GM g:5M g = G0,hcp Mg (T ) + 3187 

β”-Mg3 Gd: (Mg)3/4 (Gd)1/4

Gβ −M g3 Gd = −27134.97 + 123.42 · T − 23.6648 · T · ln (T ) −0.004829 · T 2 + 92379.96 · T −1

β’-Mg7 Gd: (Mg)7/8 (Gd)1/8

Gβ −M g7 Gd = −18021.40 + 129.08 − 24.2894 · T · ln (T ) −0.003774 · T 2 + 77416.97 · T −1



a

Gibbs energy in J/mol-atoms; Temperature in Kelvin.

results agree well with the experimental data, and also the calculated results of the invariant reactions temperature and the phase compositions from the previous assessments [13–15]. Fig. 4 shows the presently calculated enthalpies of formation in binary Mg-Gd system at 298 K over the entire composition range, compared with the experimental data [24,25] as well as the theoretical predictions [45,51]. Moreover, the calculated results due to the previous assessments [13–15] are also superimposed in Fig. 4 for a direct comparison. As shown in Fig. 4, the presently calculated enthalpies of formation show very good agreement with the only experimental data by Cacciamani et al. [25], the experimental ones evaluated from the data of vapor pressure [24] and the ones predicted by using the modified analytical embedded atom method (EAM) [51], but show significant deviations from the first-principles

computed data by Tao et al. [45]. Comparing with the previous assessment results [13–15] in Fig. 4, the result in this work can reproduce the experimental data [24,25] better, indicating that the presently obtained thermodynamic descriptions are more reliable than those from the previous assessments [13–15]. The presently calculated logarithm values of activities of Mg in different two-phase regions (i.e., (αGd)+MgGd, MgGd+Mg2 Gd, Mg2 Gd+Mg3 Gd and Mg3 Gd+Mg5 Gd) of the binary Mg-Gd system along with the reciprocal of the temperature are shown in Fig. 5. The activity data transformed from the experimental vapor pressure of Mg [23,24] via Eq. (1) as well as the calculated results due to the previous assessment [14] are appended in Fig. 5. As can be seen in the figure, the present calculation results agree well with all the activity data [23,24], and show better agreement than the

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Fig. 3. (color on web) Calculated solid solubility of Gd in (Mg) along with the experimental data [6,14,15,21,22] and the calculated results due to the previous assessment [14]. I and II represent the calculated results duo to Model I and Model II in Ref. [14], respectively.

Fig. 2. (color on web) Calculated stable phase diagram of Mg-Gd binary system, compared with the experimental data [6,14,15,17–22] and the calculated results due to the previous assessment [14]: (a) over entire composition domain, and (b) in Mg-rich side. I and II represent the calculated results due to Model I and Model II in Ref. [14], respectively.

previous one [14], which further indicates the reliability of the presently obtained thermodynamic parameters. 4.2. Thermodynamic descriptions of metastable β”-Mg3 Gd and β’-Mg7 Gd phases The calculated enthalpies of formation of the stable Mg5 Gd and Mg3 Gd phases based on the presently obtained thermodynamic descriptions are shown as squares in Fig. 6. The first-principle computed enthalpies of formation for the stable Mg3 Gd phase due to Tao et al. [45] and metastable β”Mg3 Gd phase due to Peng et al. [47] are presented as red and blue circles in Fig. 6, respectively. The difference between the presently calculated enthalpy of formation of the stable Mg3 Gd phase (in red square) and the first-principles

computed value (in red circle) is marked as H1 , which can be considered as the systematic deviation between the CALPHAD and first-principles calculated enthalpies of formation for the metastable β”-Mg3 Gd phase. Then, based on the first-principles calculated enthalpy of formation of the metastable β”-Mg3 Gd phase (denoted in blue circle) from Peng et al. [47] and H1 , the “true” enthalpy of formation for the metastable β”-Mg3 Gd phase can be obtained, i.e., HβFP − H1 , which is denoted as the inverted triangle. For the Mg-Gd alloys with x(Gd)0.5, one can make a further assumption that the enthalpy of formation in binary Mg-Gd system is related to the Mg-Gd bond energy, and linearly proportional to the fraction of Mg-Gd bonds, which are thus linearly correlated with the composition of Gd. Thus, the systematic deviation (H2 ) can be used as the difference between the first-principles computed enthalpy of formation of the metastable β’-Mg7 Gd phase and the “true” enthalpy of formation, as marked in Fig. 6. Based on the first-principle computed enthalpy of formation of the metastable β’-Mg7 Gd phase (denoted as green star) by Peng et al. [47] and the deviation H2 , one can easily estimate the “true” enthalpy of formation of the metastable β’-Mg7 Gd phase. Besides the above-estimated “true” enthalpies of formation of the metastable β’-Mg7 Gd and β”-Mg3 Gd phases at 298 K, one also needs the temperature-dependent entropy and/or heat capacity data, which are missing in the literature, in order to construct the Gibbs energies of the metastable β’-Mg7 Gd and β”-Mg3 Gd phases over the wide temperature range. In this case, one can further assume that S(β”-Mg3 Gd) = S(Mg3 Gd) and S(β’-Mg7 Gd) = S(Mg5 Gd) for simplification. Then, the Gibbs energy expressions for the metastable β”-Mg3 Gd and β’-Mg7 Gd phases can be then obtained, and also listed in

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H. Si, Y. Jiang and Y. Tang et al. / Journal of Magnesium and Alloys xxx (xxxx) xxx Table 4 Summary of the calculated invariant reactions in comparison with the experimental data [14,16–19] and the calculated results due to the previous assessments [13–15]. Reaction

T (K)

Phase compositions (at.% Gd)a

Reference∗

L→(Mg)+Mg5 Gd

820.2 ± 1 818.2 ± 7 821.2 ± 2 815.2 ± 2 827.2 820.2 821.2 822.3 820.6 913.2 ± 7 931.2 ± 2 915.2 ± 2 929.2 930.2 822.2 911.4 931.6 1013.2 ± 7 993.2 ± 2 993.2 1017.2c 928.2 996.9 994.2 1083±7 1029±2 1028.2 1029.2 1045.2 1033.2 1034.8 1027.8 1141.2 ± 2 1141.2 1142.2 1142.2 1142.9d 1140.5 973.2 ± 2 973.2 973.2 957.2 956.3 974.0

– 5.7 8.8 9.5 7.6 7.5 8.3 7.4 7.8 ∼8

Exp. [14] Exp. [16]b Exp. [17] Exp. [18] Cal. [13] Cal. [14] I Cal. [14] II Cal. [15] Cal. -this work Exp. [16] Exp. [17] Exp. [18] Cal. [13] Cal. [14] I Cal. [14] II Cal. [15] Cal. -this work Exp. [16] Exp. [17] Cal. [13] Cal. [14] I Cal. [14] II Cal. [15] Cal. -this work Exp. [16] Exp. [17] Exp. [19] Cal. [13] Cal. [14] I Cal. [14] II Cal. [15] Cal. -this work Exp. [17] Cal. [13] Cal. [14] I Cal. [14] II Cal. [15] Cal. -this work Exp. [17] Cal. [13] Cal. [14] I Cal. [14] II Cal. [15] Cal. -this work

L+Mg3 Gd→Mg5 Gd

L+Mg2 Gd→Mg3 Gd

L+MgGd→Mg2 Gd

L+(βGd)→MgGd

(βGd)→MgGd+(αGd)

a b c d ∗

4 ± 0.3

16.0 15.8 14.5 14.1 16.2 ∼20 23.9 26.2 18.9 22.2 25.0

29.2 32.3 31.3 27.6 33.6 ∼49 46.3 49.1 47.9 51.7 49.6 74 76.5 73.9 72.2 79.2 74.6

3.8 4.9 4.6 4.6 4.5

16.6 16.6 16.2 16.3

25 25 21.9 23.1

16.6 16.6 16.6 16.7

33.3 33.3 30.3 30.0

25 25 27.0 26.4

50 50 48.7 50 63.4 63.5 63.2 61.3 66.6 63.4

33.3 33.3 34.5 35.4

49.9 50 50 50.0 50

86.3 88.1 87.5 86.3 86.6

49.0 50 50 49.5 50

Phase compositions are shown in the sequence of the reactions. Mg5 Gd was reported as Mg9 Gd in Ref. [16]. The invariant reaction was calculated as a eutectic one L→Mg2 Gd+Mg3 Gd inModel I by Ref. [14]. The invariant reaction was calculated as a eutectic one L→(βGd)+MgGd by Ref. [15]. I and II represent Model I and Model II in Ref. [14], respectively.

Table 3. Fig. 7 presents the calculated Gibbs energies of the metastable β”-Mg3 Gd and β’-Mg7 Gd phases as well as those of the stable Mg5 Gd and Mg3 Gd phases according to the presently obtained thermodynamic descriptions for both stable and metastable phases. As can be seen, the Gibbs energy of the stable Mg3 Gd phase is slightly lower than that of the metastable β”-Mg3 Gd phase, which ensures the stability of the Mg3 Gd phase over the entire temperature range.

Fig. 8 shows the calculated Mg-Gd binary phase diagram containing metastable phases in Mg-rich region according to the presently obtained thermodynamic parameters listed in Table 3. While Fig. 9 presents the logarithmic values of solvuses of (Mg) in equilibrium with the stable Mg3 Gd, the metastable β’-Mg7 Gd and the metastable β”-Mg3 Gd phases, along with the corresponding experimental data [6,14,15,21,22,29,30]. As can be seen in Figs. 8 and 9, the presently calculated solvuses with the metastable β”-Mg3 Gd

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Fig. 4. (color on web) Calculated enthalpies of formation in the binary Mg-Gd system at 298 K in comparison with the experimental data [24,25], first-principle calculations [45], EAM calculations [51] and those due to the previous CALPHAD assessments [13–15]. I and II represent the calculated results duo to Model I and Model II in Ref. [14], respectively.

Fig. 5. (color on web) Calculated activities of Mg in different two-phase MgGd alloys, compared with the transformed data from the experimental vapor pressure of Mg [23,24] and the results calculated from the previous assessment [14].  1 ,  2 ,  3 ,  4 denote the two-phase regions of αGd+MgGd, MgGd+Mg2 Gd, Mg2 Gd+Mg3 Gd and Mg3 Gd+Mg5 Gd, respectively. I and II represent the calculated results duo to Model I and Model II in Ref. [14], respectively.

and β’-Mg7 Gd phases reproduce the experimental data from Vostrý et al. [29] and Kekule et al. [30] very well. Moreover, the measured solubility of Gd in (Mg) at 473 K by Rokhlin et al. [6] shows an obvious deviation from the presently calculated stable solvus of (Mg) (solid line), but agrees with the presently calculated solvuses of (Mg) in equilibrium with the metastable β’-Mg7 Gd phase. The possible reason may be that the precipitate phase is not the stable Mg5 Gd after aging at 473 K for 200 h but the metastable β’-Mg7 Gd precipitate.

Fig. 6. (color on web) Schematic diagram for derivation procedure of enthalpies of formation for metastable β”-Mg3 Gd and β’-Mg7 Gd phases at 298 K (blue triangles) in the present work. The enthalpies of formation for stable Mg5 Gd and Mg3 Gd due to the present CALPHAD descriptions (squares), as well as that for stable Mg3 Gd and those for metastable β”-Mg3 Gd and β’-Mg7 Gd phases (circles and pentagon) due to the firstprinciples calculations [45,47] are also superimposed.

4.3. Validation of the thermodynamic descriptions of stable and metastable phases In order to verify the reliability of thermodynamic descriptions of the stable phases in Mg-Gd binary system, the Scheil solidification simulations were performed in a series of MgGd alloys. The microstructures of 5 as-cast Mg-Gd alloys

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Fig. 7. (color on web) Temperature-dependent Gibbs energies for β-Mg5 Gd, Mg3 Gd, β”-Mg3 Gd and β’-Mg7 Gd phases at the stoichiometric compositions obtained in the present work.

Fig. 9. (color on web) Calculated solid solubilities of Gd in (Mg) in equilibrium with stable Mg5 Gd, metastable β”-Mg3 Gd and β’-Mg7 Gd phases, compared with the experimental data [6,14,15,21,22,29,30].

were investigated by Zhong et al. [52], Kubásek and Vojtˇech [53] and Vlˇcek et al. [54], respectively. The compositions of these 5 alloys are summarized in Table 5. Moreover, the area fractions of phases were determined by image processing and data analysis software. The solidification sequences of the 5 as-cast samples were then simulated by using the Scheil solidification model based on the presently obtained thermodynamic descriptions, and were thus compared with the experimental results [52–54]. The Scheil solidification model, which is based on the assumption of fast diffusion in the liquid phase but no diffusion in the solid, is much closer to the actual casting conditions, compared with the

equilibrium solidification [55]. According to the present simulation results, the solidification sequences of the Mg6 wt.%Gd, Mg-8 wt.%Gd, Mg-10 wt.%Gd and Mg-22 wt.%Gd alloys are as follows: liquid → liquid+(Mg) → (Mg)+Mg5 Gd, while the solidification sequence for Mg-1 wt.%Gd is as follow: liquid → liquid+(Mg) → (Mg), which are consistent with the experimental observations [52–54]. Fig. 10 shows the model-predicted and measured fractions of the Mg5 Gd and (Mg) phases. In Fig. 10, the model-predicted volume fraction is equal to the experimental area fraction for each phase along the diagonal line. As can be seen from Fig. 10(a) and

Fig. 8. (color on web) Calculated Mg-Gd binary phase diagram in Mg-rich region containing metastable β”-Mg3 Gd and β’-Mg7 Gd phases, compared with the experimental data [6,14,15,21,22,29,30]. Please cite this article as: H. Si, Y. Jiang and Y. Tang et al., Stable and metastable phase equilibria in binary Mg-Gd system: A comprehensive understanding aided by CALPHAD modeling, Journal of Magnesium and Alloys, https:// doi.org/ 10.1016/ j.jma.2019.04.006

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Table 5 List of compositions of 5 as-cast Mg-Gd alloys, the volume fractions of phases predicted using Scheil solidification simulation, compared with the experimental data [52–54]. Nominal composition Mg-6Gd Mg-8Gd Mg-10Gd Mg-1Gd Mg-22Gd a

Actual Composition wt.% Gd (at.% Gd) 5.36(0.87) 8.14(1.35) 9.20(1.54) 0.570(0.0885) 22.9 ± 0.3(4.389)

Experimental area fraction (%)

Calculated volume fraction (%)

Mg5 Gd

(Mg)

Mg5 Gd

(Mg)

99.81 99.77 99.59 100.00 93.26

1.3 × 10−5

∼100.00 99.90408 99.8224 100.00 93.77

0.19 0.23 0.41 0.00 6.74

0.09592 0.1776 0.00 6.23

Ref.

[52] [52] [52] [53] [54]

a

Mg5 Gd was reported as a composition of Mg46 Gd9 in Ref. [54].

Fig. 11. (color on web) Calculated molar Gibbs energies of (Mg), Mg5 Gd, Mg3 Gd, β”-Mg3 Gd and β’-Mg7 Gd phases at 473 K and their equilibrium compositions according to the presently obtained thermodynamic descriptions.

Fig. 10. (color on web) Model-predicted volume fractions for (a) Mg5 Gd, and (b) (Mg) in the 5 as-cast Mg-Gd alloys by using the Scheil solidification simulation based on the presently obtained thermodynamic descriptions, compared with the experimental data [52–54]. Along the diagonal line, the model-predicted volume fraction is equal to the experimental area fraction.

(b), the predicted values of Mg5 Gd and (Mg) correspond well with the experimental data [52–54]. It further indicates the thermodynamic descriptions for the stable phases in Mg-Gd binary system obtained in present work are reliable. Fig. 11 shows the Gibbs energies of the (Mg), Mg5 Gd, Mg3 Gd, β”-Mg3 Gd and β’-Mg7 Gd phases varying with composition at 473 K according to the presently obtained thermodynamic descriptions for stable and metastable phases. It should be noted here that because the metastable β”Mg3 Gd and β’-Mg7 Gd were treated as the stoichiometric compounds, their isothermal Gibbs energies are denoted as points. As shown in the magnification diagram in Fig. 11, (Mg) reaches equilibrium with the stable βMg5 Gd, metastable β”-Mg3 Gd and β’-Mg7 Gd phases at 0.114 at.%Gd (0.74 wt.%Gd), 0.736 at.%Gd (4.56 wt.%Gd), and 1.423 at.%Gd (8.50 wt.% Gd), respectively. The precipitation sequence of Mg-15 wt.%Gd alloy at 473 K was

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determined experimentally by Vostrý et al. [29] and Kekule et al. [30] as: S.S.S.S(Mg)→ β”-Mg3 Gd → β’-Mg7 Gd → βMg5 Gd. Moreover, the same precipitation sequence was determined experimentally by Zeng [28] for the Mg-12 wt.%Gd alloy at 473 K. The compositions of both Mg-15 wt.%Gd and Mg-12 wt.%Gd alloys are larger than 1.423 at.%Gd (8.50 wt.%Gd) as shown as the black dotted line in Fig. 11. According to Fig. 11, β”-Mg3 Gd, β’-Mg7 Gd and β-Mg5 Gd should appear in sequence during the aging process of Mg15 wt.%Gd and Mg-12 wt.%Gd at 473 K, and the precipitation sequence can be predicated as: S.S.S.S(Mg) → β”Mg3 Gd → β’-Mg7 Gd → β-Mg5 Gd, which is in agreement with the reported ones [28–30]. This fact also further verifies the reliability of the presently obtained thermodynamic descriptions of stable and metastable phases in binary Mg-Gd system. 5. Conclusion • A CALPHAD thermodynamic re-assessment of all the stable phases in the binary Mg-Gd system was performed by fully considering all the experimental phase equilibria and thermochemical properties. The discrepancy between the phase equilibria and thermochemical property data existing in the previous assessments was successfully eliminated. Moreover, a comprehensive comparison between the calculated phase equilibria/thermochemical properties and the experimental data indicated that the present assessment can give the better agreement with most of the experimental data than the previous assessments. • Based on the CALPHAD results of stable phases and firstprinciples results of both stable/metastable phases as well as the simplified correlation between the CALPHAD and first-principles results, the Gibbs energies for metastable β”-Mg3 Gd and β’-Mg7 Gd phases were constructed, and incorporated into the CALPHAD thermodynamic descriptions, from which the model-predicted solvuses of (Mg) in equilibrium with the metastable β’-Mg7 Gd and β”-Mg3 Gd compounds agree well with the limited experimental data in the literature. • Furthermore, the Scheil solidification simulations of 5 ascast Mg-Gd alloys were conducted to further verify the reliability of the established thermodynamic descriptions for the stable phases in the Mg-Gd binary system. While the stability of different precipitates of Mg-15 wt.% Gd and Mg-12 wt.% Gd alloys at 473 K were analyzed, and their precipitation sequence was predicted to be the same as the experimental observations, indicating that the presently obtained thermodynamic descriptions of both stable and metastable phases are reliable, and can be directly applied in the further design of novel Mg-Gd based alloys. Acknowledgements The financial support from the National Key Research and Development Program of China (Grant no. 2016YFB0301101), the Hunan Provincial Science and

Technology Program of China (Grant no. 2017RS3002)Huxiang Youth Talent Plan, the Youth Talent Project of Innovation-driven Plan at Central South University (Grant no. 2019XZ027) and the Hebei Provincial Science and Technology Program of China (Grant no. BJ2018026)-Outstanding Young Talents Plan is acknowledged. Ying Tang acknowledges the financial support from the Yuanguang fellowship released by Hebei University of Technology.

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Please cite this article as: H. Si, Y. Jiang and Y. Tang et al., Stable and metastable phase equilibria in binary Mg-Gd system: A comprehensive understanding aided by CALPHAD modeling, Journal of Magnesium and Alloys, https:// doi.org/ 10.1016/ j.jma.2019.04.006