Ab initio calculations and thermodynamic modeling for the Fe–Mn–Nb system

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

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Ab initio calculations and thermodynamic modeling for the Fe–Mn–Nb system Shuhong Liu a,b,n, Bengt Hallstedt a, Denis Music a, Yong Du b a b

Materials Chemistry, RWTH Aachen University, Kopernikusstr. 10, D-52074 Aachen, Germany State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, PR China

a r t i c l e i n f o

abstract

Article history: Received 21 November 2011 Received in revised form 7 March 2012 Accepted 7 March 2012 Available online 8 May 2012

The Fe–Nb and Mn–Nb systems have been thermodynamically investigated within the CALPHAD approach by combining available experimental data and the data from our ab initio calculations. Possible nonmagnetic (NM), ferromagnetic (FM) and antiferromagnetic (AFM) ordering for the endmembers of the intermetallic compounds Laves C14 and m were treated during ab initio calculations. It turns out that the local magnetic moment depends on the lattice site. The energetically most stable states of C14 at the stoichiometric compositions NbFe2 and NbMn2 exhibit the FM and NM ordering, respectively. The FM ordering can lower the total energies for most of the end members of m in the Fe–Nb system. The energy of formation for the ‘‘hypothetical’’ end-members of m, due to the sublattice model used for modeling the ternary solubility, was also calculated by ab initio and incorporated into the modeling of the phase. m is predicted to be marginally stable in the Mn–Nb binary system. Compared to the conventional treatment for the end-members of C14, the introduction of physically based parameters from ab initio calculations makes the thermodynamic optimization process simpler, more effective and more reliable in the Mn–Nb binary system. The obtained thermodynamic parameters for Fe–Nb and Mn–Nb systems can describe the reliable experimental data well. The thermodynamic description for the ternary Fe–Mn–Nb system is then extrapolated for the first time from the three binary edges and the data from our ab initio calculations. Several isothermal sections, the liquidus projection and the reaction scheme have been predicted accordingly. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Fe–Mn–Nb phase diagram Thermodynamic modeling Magnetic properties Ab initio calculations CALPHAD

1. Introduction High manganese austenitic steels with addition of the microalloying elements Nb and V have attracted much scientific interest due to their excellent tensile strength–ductility properties and their great potential in applications for structural components in the automotive industry [1–3]. Knowledge of phase diagrams and thermodynamic properties is essential in defining processing conditions for optimal engineering properties. Therefore, a study to provide a precise and reliable thermodynamic description for high Mn steels including the microalloying elements Nb and V is undertaken. Fe–Mn–Nb is one of the basic systems in this work. However, there are no data on phase equilibria or thermodynamic properties reported in literature. In order to construct a self-consistent thermodynamic description for the Fe–Mn–Nb system, first reliable descriptions of the low-order systems are necessary. Among n Corresponding author at: State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, PR China. Tel.: þ86 731 88877300; fax: þ 86 731 88710855. E-mail address: [email protected] (S. Liu).

0364-5916/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.calphad.2012.03.004

the three binary systems, Fe–Nb and Mn–Nb need to be reinvestigated or investigated thermodynamically for two reasons: (i) the available thermodynamic descriptions [4–9] for the Fe–Nb binary system cannot reproduce more recently published experimental data [10–12] reasonably, especially for the homogeneity ranges of the two stable intermetallic compounds, i.e. the C14 Laves phase (e in Fe–Nb and l in Mn–Nb) and the m phase (Fe7Nb6). Recently, there is a renewed interest in Laves phases as they have become candidates for several functional as well as structural applications [13,14], while the m phase is a hard and brittle intermetallic compound so that its presence should be avoided, or at least carefully controlled. Therefore, the stability ranges for these phases should accurately be studied, with the CALPHAD approach which is well adapted to the modeling of multicomponent systems [15]; (ii) there is no thermodynamic description available for the Mn–Nb binary system. Measured thermodynamic properties (such as enthalpy, entropy, heat capacity and activity etc.) for the intermetallic phases are very important in the thermodynamic modeling of the system. To overcome the lack of experimental data, ab initio calculations based on density functional theory (DFT) [16] are used to predict the enthalpy of formation of the intermetallic

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S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

phases. In order to provide ‘‘the best’’ initial values for the endmembers of the intermetallic phases in the thermodynamic modeling, possible magnetic ordering for these phases are studied via ab initio calculations. In general, this paper is aimed to: (i) employ ab initio calculations to calculate the enthalpies of formation for the end-members of the intermetallic phases; (ii) present the thermodynamic modeling of the Fe–Nb and Mn– Nb binary systems using CALPHAD approach based on the results from ab initio calculations and the available experimental data for both the phase diagram and thermodynamic properties; (iii) predict the phase relationships of the Fe–Mn–Nb ternary system by incorporating the thermodynamic description of the constituent binary systems and the results from ab initio calculations.

2. Literature data In this section, all available literature data about the phase equilibria of the binary edges of the Fe–Mn–Nb system are critically evaluated. Table 1 lists the phase diagram and thermodynamic data, which are considered in the present thermodynamic modeling. 2.1. The Fe–Nb binary system This system has been reviewed critically by Paul and Swartzendruber [17] based on the available data up to 1983 [18–26]. The present authors will focus on the phase diagram data after that time and the available data on thermodynamic properties. According to the assessment by Paul and Swartzendruber [17], Table 1 Summary of the experimental phase diagram data and thermodynamic data in the Fe–Nb and Mn–Nb systems. Reference

Remarka

þ  þ þ

liquidus of m Liquidus of (Nb) Fe solid solution in (Nb) Nb solid solution in (dFe) Nb solid solubility in (aFe) (dFe)/(gFe) boundaries (gFe)/(aFe) boundaries (dFe) phase boundary DHmix at 2035 and 1935 K DHmix at 1873 K DHmix at 1960 K DGf for Fe7  yNb2 (y¼ 0, 3) at 1200–1450 K DGf for Fe2Nb (y¼ 0, 3) at 1000–1400 K DHf for NbFe2 and Fe7Nb6 at 298 K Activity of Fe at 1873 K Activity of Nb in (gFe)at 1273 K and 1373 K

[12,19,20,23,27,28] [18] [10–12,28] [12,28] [25] [19,20,23] [22] [12,28] [18] [28] [18] [11,12,27,28] [12] [24] [23,26] [21,26] [20,23] [31] [29] [30] [34] [35] [37] [32] [33]

Mn–Nb Phase diagram with x(Nb) o5 at% Invariant reactions Phase boundary of l DHf for NbMn2 at 298 K

[48] [49] [50] [52]

Type of data Fe–Nb Invariant reactions Homogeneity range of e Homogeneity range of m Liquidus of (dFe) Liquidus of e

there are only two intermetallic compounds, namely the C14 Laves phase e and m (Fe7Nb6). Employing optical metallography (OM), X-ray diffraction (XRD), differential thermal analysis (DTA) and electron microprobe analysis (EPMA), Zelaya Bejarano et al. [27,28] carried out an exhaustive experimental investigation on the phase equilibria of the Fe–Nb system in the composition range from 20 to 100 at% Nb. The results from Zelaya Bejarano et al. [28] confirmed the assessment by Paul and Swartzendruber [17] on the only two stable intermetallic phases in the system. However, they reported a more narrow composition range for the e phase (from 32 to 37 at%) and a peritectic formation of the m phase at 1520 1C, which are in contradiction to the earlier studies [17]. Recently, several investigations [10–12] have been published on the homogeneity ranges of the two intermetallic phases. The homogeneity range of the Laves phase e has been determined ¨ at 1100 1C by Gruner [10] and at 1200 1C by Takeyama et al. [11] using EPMA on quenched samples. Most recently, Voß et al. [12] reinvestigated the phase equilibria of the system using OM, EPMA, and DTA. The temperatures of the invariant reactions and the homogeneity ranges of the two intermetallic phases, e and m, were determined. There are three pieces of work on the enthalpy of mixing of the liquid phase. Using an isoperibol calorimeter developed by themselves, Iguchi et al. [29] measured the enthalpies of mixing of liquid alloys containing 5–25 at% Nb at 1600 1C. Using a calorimeter with an isothermal jacket, Sudavtsova et al. [30] determined the enthalpies of mixing of liquid alloys with the composition less than 20 at% Nb at 1687 1C. Later, Schaefers et al. [31] reported that the enthalpy of mixing of the system is temperature dependent, based on measurements at 1662 1C and 1762 1C with Nb contents up to 51.5 wt.% by levitation alloying calorimetry (LAC). According to the data from literature [29,31], the enthalpy of mixing for the liquid phase becomes more negative with decreasing temperature, as shown in Fig. 1. Since the data from Sudavtsova et al. [30] are conflicting with those from the other two groups [29,31], only the data from literature [29,31] will be used in the present optimization. Ichise and Horikawa [32] have determined the Fe activity in Fe–Nb alloys at 1600 1C by means of Knudsen cell mass

D þ

D þ  þ  þ 

D þ þ þ þ þ    þ

D 

þ  þ þ

a Indicates whether the data are used or not used in the parameter optimization: þ , used; D not used but considered as reliable data for checking the modeling;  , not used.

Fig. 1. Calculated enthalpies of mixing of liquid Fe–Nb alloys using the present thermodynamic description along with the experimental data [29–31]. The reference state is liquid Fe and Nb.

S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

spectrometry (KCMS). Using an electrochemical cell, Hawkins [33] determined the activity coefficient of Nb along the (gFe) solvus to be 2.1 at both 1000 and 1100 1C. Employing electromotive force (EMF) technique, Drobyshev and Rezukhina [34] and Barbi [35] measured the Gibbs energy of formation of the e phase at the composition of NbFe2 to be 20,453þ4.63T J/mol-atoms in the temperature range from 971 to 1120 1C and  23,681þ 4.561T J/mol-atoms in the temperature range from 727 to 1127 1C, respectively, referring to (gFe) and (Nb). Their results [34,35] are in reasonable agreement with each other. They both used a measuring electrode consisting of (Fe) þ e þNbO, a reference electrode consisting of (Fe) þ‘‘FeO’’ and a solid oxide electrolyte. With this cell, the oxygen chemical potential of the measuring electrode is determined, which can be converted to Nb chemical potential using external data for the Gibbs energies of NbO and ‘‘FeO’’. Considering this and that both (Fe) and e are nonstoichimetric, the error bar of the so determined Gibbs energy of formation of stoichiometric e (NbFe2) is expected to be quite large. If considering the heat capacity 11.37 þ0.023T J/ (mol-atoms K) determined by Rezukhina et al. [36], the same group as the literature [34], the enthalpy of formation for NbFe2 (e phase) at 25 1C can be evaluated to be  16.06 and 19.29 kJ/mol-atoms, respectively. These values have a large error bar and differ significantly from the one recently determined by Meschel and Kleppa [37], who measured the standard enthalpies of formation for NbFe2 (e) and Fe7Nb6 (m) at 25 1C to be 5.3 71.7 and 6.2 71.8 kJ/mol-atoms, respectively, by direct synthesis calorimetry using a high temperature microcalorimeter up to 1200 1C. In recently published work on the Fe–Ta binary system [38], there is a similar situation concerning the enthalpies of formation and thus we accept the data from Meschel and Kleppa [37] in the present work. The data from Hawkins [33] will be considered during the optimization. The activity data from Ichise and Horikawa [32] will not be used during the optimization, but taken as a comparison for the optimization. There are also some investigations on the magnetic properties of m and e. According to the studies by Read et al. [39] based on ¨ magnetic and Mossbauer measurements in the range of 4–400 K, the m phase at the composition of FeNb is antiferromagnetic below 270 K with a large Fe moment (3 mB), while e is weakly ferromagnetic at the composition of NbFe2, with the ferromagnetism becoming stronger on both sides of stoichiometry. In contrast to the report by Read et al. [39], two groups [40,41] pointed out e as paramagnetic at the composition of NbFe2 and four other groups [42–45] described it as a weakly antiferromagnetic phase. In spite of the controversy of magnetic state of e at the composition of NbFe2, the reported ferromagnetic properties for off stoichiometric composition of e are consistent with each other [39,41,46]. Six groups [4–9] have investigated the Fe–Nb system thermodynamically. None of these descriptions can well reproduce the recently published experimental data [10–12]. Based on the review by Paul and Swartzendruber [17], Huang [4] assessed the Fe–Nb binary system first, where e was treated as stoichiometrics and m was given a very narrow homogeneity with congruent melting, which are in disagreement with the later experimental data [12,28]. Subsequently, Coelho et al. [5] proposed an optimization, fitting the experimental data from Zelaya Bejarano et al. [28] with a classical two-sublattice model for e and the model (Nb)6(Fe,Nb)7 for m. Using a three-sublattice model for e, Srikanth and Petric [6] also optimized the system, but with a stoichiometric m not fitting the experimental data. Later, Toffolon and Servant [7] reoptimized the system thermodynamically. They modeled the e phase with two crystal sublattices and the m phase with a three- and a four-sublattice model. In order to make the fcc solution phase (g) become more stable at low temperature so as

45

to improve the agreements for the slopes of solubility products vs. inverse temperature in the Fe–Nb–C–N system, Lee [8] re-optimized the parameters of g and e of the Fe–Nb binary system, based on the work by Huang [4]. Most recently, based on the work by Toffolon and Servant [7], Mathon et al. [9] adjusted the parameters of the m phase with the two-sublattice model (Nb)6(Fe,Nb)7 in order to obtain a thermodynamic description for the Fe–Nb–Ni ternary system. Additionally, they employed ab initio calculations according to DFT [16] to estimate enthalpies of formation for the end members of the e phase under the hypothesis of FM and/or NM effects and those of m phase under the hypothesis of NM, referring to FM (aFe) and NM (Nb). However, none of these data from the ab initio calculations were used in their optimization. 2.2. The Mn–Nb binary system Experimental thermodynamic and phase diagram data of the Mn–Nb system available in the literature are rather sparse. Okamoto [47] has assessed the partial Mn–Nb phase diagram based on the experimental work of [48–51]. It includes 6 phases: (i) there is a congruent melting C14 Laves phase (l) with a fairly wide solid solution range around NbMn2; (ii) liquidus up to about 50 at% Nb and the solubility of Nb in (gMn) and (dMn) have been experimentally observed; (iii) the phase diagram above 50 at% Nb and the solubility of Nb in (bMn) and (aMn) is unknown. The phase equilibria among the (bMn), (gMn), and (dMn) phases are controversial in the literature. Experimental results from Hellawell [48] presented the eutectic formation of (dMn) and l at 1224 1C, eutectoid decomposition of (dMn) to (bMn) and (gMn) at 1120 1C, and peritectic formation of (bMn) from (dMn) and l at 1145 1C. However, the phase diagram from Savitskii and Kopetskii [49] claimed a peritectic formation of (gMn) from Liquid and (dMn) at 1235 1C, eutectic solidification to (gMn) and l at 1220 1C, and peritectic formations of (bMn) and (aMn) at 1135 1C and 800 1C. Fortunately, the subsequent work by Svechnikov and Pet’kov [50] confirmed the phase transformation types given by Hellawell [48]. However, the invariant reaction temperatures from Svechnikov and Pet’kov [50] are lower than those from Hellawell [48], as shown in Table 2. As a result, the Mn-rich end of the phase diagram will reflect the results of Hellawell [48] in the present modeling, consistent with the assessment of Okamoto [47]. The structure of l has been reported by Wallbaum [51] to be hexagonal (MgZn2-type; C14 Laves phase) with lattice parameters a¼0.4865 nm and c¼0.7955 nm. The congruent melting point of l was consistently reported to be 1500 1C [49,50] and the solubility range was determined to be from 30 to 38 at% Nb by Svechnikov and Pet’kov [50]. Besides, a eutectic reaction between l and Nb has been reported to be at about 1400 1C by Svechnikov and Pet’kov [50]. Most recently, Meschel et al. [52] measured the enthalpy of formation for l at the composition of NbMn2 to be 10.472.7 kJ/ mol-atoms at 25 1C by high temperature direct synthesis calorimetry. The value was compared with the predicted values from the Miedema model [53] and with results from ab initio calculation from one of the co-authors. However, there is no detailed information about the ab initio calculation of this phase, such as the magnetic properties of NbMn2 or the total energy of pure Mn. 2.3. The Fe–Mn binary system There are five stable phases in the Fe–Mn system, (aMn), (bMn), g(fcc), d(bcc) and liquid. The system has been assessed by Huang [54,55] twice. The thermodynamic parameters from Huang [55] will be used in this study directly. The calculated phase diagram of Fe–Mn is shown in Fig. 2.

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S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

Table 2 Calculated invariant phase equilibria using the obtained thermodynamic parameters along with literature data. Reaction between u1/u2/u3

Temperature (1C)

Phase composition (at%Nb)

References

u1

u2

u3

1373 7 1 1370 1356 1365 1348 1370 7 1 1372 7 2 1377 1358 1398 1362

8.2 7 0.7 10.6 11.3–11.7 9.8 10.0 11.99 10.6 10.5 10.0 9.67 10.17 10.2

3.2 3.27 7.58 – – 3.27 – 2.6 2.8 3.11 3.21

25.1 32 33.86 – – – – 33.33 33.85 30.52 24.85

Exp. [12] Exp. [28] Exp. [20] Exp. [19] Exp. [18] Exp. [23] Exp. [22] Cal. [4] Cal. [6] Cal.[7] Cal. [this work]

(dFe)2(gFe) þ e

1220 1220 1200 1208 1190 1183 7 15 1190 1187 1178 1199 1179

6.26 – – 1.7 – 1.5 7 0.1 1.56 1.5 1.34 1.62 1.69

1.27 – – 0.99 – 1.1 1 0.95 0.99 1.15 1.1

33.86 – – – – 26.5 32 33.33 33.78 30.81 25.21

Exp. [20] Exp. [21] Exp. [22] Exp. [23] Exp. [26] Exp. [12] Exp. [28] Cal. [4] Cal. [6] Cal. [7] Cal. [this work]

(gFe) þ e2(aFe)

965 989 960 943 7 25 960 949 955 942 944

0.48 – – – 0.43 0.35 0.33 0.26 0.30

34.77 – – 27.6 32 33.33 33.7 31.47 25.96

3.07 – – – 0.73 0.62 0.62 0.47 0.54

Exp. [20] Exp. [21] Exp. [26] Exp. [12] Exp. [28] Cal. [4] Cal. [6] Cal. [7] Cal. [this work]

L2e

1650–1660 1630 1645 7 5 1615 1641 1639 1647

– – 33.33 33.33 33.33 33.33 32.42

– – – 33.33 33.33 33.33 32.42

– – – – – – –

Exp. [20] Exp. [27,28] Exp. [12] Cal. [4] Cal. [6] Cal. [7] Cal. [this work]

L þ e2m

1523 7 2 1520 1523 1520 1524

52 56 54.91 51.45 50.87

37.6 37 36.95 35.03 40.65

46.5 49 47.5 48.33 48.6

Exp. [12] Exp. [28] Cal. [6] Cal. [7] Cal. [this work]

L2m þ(Nb)

1508 7 2 1500 1398 1485 1484 1503

58.6 7 0.5 59 65.6 60.86 59.26 56.19

51 54 46.15 52.5 49.37 50.62

91 93 91.0 93.97 92.29 90.93

Exp. [12] Exp. [28] Cal. [4] Cal. [6] Cal. [7] Cal. [this work]

1224 1200 1223 1223

3.6 – 3.87 3.82

– – 29.52 29.05

Exp. [48] Exp. [50] Cal. [this work] Set 1 Cal. [this work] Set 2

Fe–Nb L2(dFe)þ e

Mn–Nb L2(dMn)þ l

2.4 – 2.16 2.14

L þ (dMn)2(gMn)

1235

–

–

–

Exp. [49]

L2(gMn)þ l

1220

–

–

–

Exp. [49]

(dMn) þ l2(bMn)

1145 1100 1143 1143

– – 1.61 1.51

– – 29.51 29.14

– – 2.33 2.25

Exp. [48] Exp. [50] Cal. [this work] Set 1 Cal. [this work] Set 2

(gMn)þ l2(bMn)

1135

–

–

–

Exp. [49]

(dMn)2(gMn)þ(bMn)

1120 1122 1122

– 0.51 0.51

0.4 0.32 0.32

– 0.79 0.79

Exp. [48] Cal. [this work] Set 1 Cal. [this work] Set 2

S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

47

Table 2 (continued ) Reaction between

Temperature (1C)

Phase composition (at%Nb)

References

u1/u2/u3 u1

u2

u3

(bMn)þ l2(aMn)

800 718 717

– 0.42 0.41

– 31.04 31.54

– 0.71 0.70

Exp. [49] Cal. [this work] Set 1 Cal. [this work] Set 2

L2l

1500 1500 1500 1501

– 33.3 35.13 34.75

– 33.3 35.13 34.75

– – – –

Exp. [49] Exp. [50] Cal. [this work] Set 1 Cal. [this work] Set 2

L2l þ(Nb)

1400 1398 1397

– 57.33 57.26

– 41.22 41.70

– 86.72 86.67

Exp. [50] Cal. [this work] Set 1 Cal. [this work] Set 2

l þ(Nb)2m

987

38.01

95.35

48.01

Cal. [this work] Set 1

3. Thermodynamic modeling 3.1. Unary phases The Gibbs energy function for the pure element i (i ¼Fe, Mn, Nb) in the phase j is expressed by an equation of the form [56] SER ¼ a þ bT þ cT ln T þ dT 2 þ eT 1 þ f T 3 þiT 7 þ jT 9 Gj m ðTÞH i

ð1Þ

HSER i

is the molar enthalpy of the element i at 25 1C and where 1 bar in its standard element reference (SER) state, and T is the absolute temperature in Kelvin. The last two terms in Eq. (1) are used only outside the range of stability [57], the term iT7 is for a liquid below the melting point and jT  9 is for solid phases above the melting point. 3.2. Solution phases The liquid phase has been treated with a substitutional solution model. The Gibbs energy is described by: L

L

L

GLm HSER ¼ xFe 0 G Fe þxMn 0 G Mn þ xNb 0 G Nb þ RTðxFe ln xFe þxMn ln xMn þ xNb ln xNb Þ

Fig. 2. Calculated phase diagram of the Fe–Mn binary system using the thermodynamic parameters from Huang [55].

þxFe xMn LLFe,Mn þxFe xNb LLFe,Nb þxMn xNb LLMn,Nb þxFe xMn xNb LLFe,Mn,Nb

ð2Þ SER SER xFe HSER Fe þ xMn H Mn þ xNb H Nb ,

SER

is an abbreviation for where H xFe, xMn and xNb are the mole fractions of Fe, Mn and Nb, L L L respectively.0 G Fe , 0 G Mn , and 0 G Nb are the Gibbs energies of the pure components Fe, Mn, Nb in the same state, respectively, and R is the gas constant. The binary interaction LLi,j is modeled using Redlich–Kister polynomials [58], L

L

L

L

LLi,j ¼ 0 L i,j þ 1 L i,j ðxi xj Þ þ 2 L i,j ðxi xj Þ2 þ    þ n L i,j ðxi xj Þn

¼ an þbn T þ cn T ln T

ð4Þ

where an, bn and cn are the model parameters to be optimized. The ternary interaction parameter LLFe,Mn,Nb is modeled as: L

L

L

LLFe,Mn,Nb ¼ xFe 0 L Fe þ xMn 0 L Mn þ xNb 0 L Nb 0 L 0 L L Fe , L Mn

ð5Þ 0 L L Nb

i ¼ Fe,Mn,Nb

and may have a temperature where the parameters dependence. The terminal solid solutions j (j ¼fcc_A1 (g), bcc_A2 [(aFe), (dFe), (dMn), (Nb)], cub_A13 (bMn), cbcc_A12 (aMn)), are modeled by a two-sublattice model (Fe,Mn,Nb)1(Va)p. In this model, the first sublattice is for the substitutional species such as Fe, Mn, Nb, and the second sublattice is for interstitial species. The quantity p is the relative number of sites in the interstitial

i ¼ Fe,Mn,Nb

j

j

þxFe xNb LFe,Nb:Va þ xMn xNb LMn,Nb:Va j

j

þxFe xMn xNb LFe,Mn,Nb:Va þ mag G m

ð3Þ

in which n L L i,j

sublattice, which is 3 for bcc_A2, and 1 for fcc_A1, cub_A13 and cbcc_A12. The Gibbs energy for one mole of atoms of j is given by: X X j j xi 0 G i:V a þ RT xi ln xi þ xFe xMn LFe,Mn:V a Gj m¼

ð6Þ

Similar to the case of the liquid phase, the interactions j j Li,j:Va and Li,j,k:V a can also be expanded as Redlich–Kister polynomials [58]. Since Fe and Mn are magnetic, there is a magnetic contribution to the Gibbs energy. This is accounted for by the j j term mag G m in Eq. (6). An expression for mag G m is given by Hillert and Jarl [59] as mag

j

j

G m ¼ RT lnðb þ aÞf ðtÞ

ð7Þ

where b is the average magnetic moment per atom, t ¼ T=T j c , Tj c is the critical temperature for the magnetic transition of the phase j. Both bjand T j c can have a composition dependence. j

3.3. C14 Laves phase (e in Fe–Nb and l in Mn–Nb) The C14 Laves phase (prototype MgZn2, Pearson symbol hP12), with an ideal stoichiometry BA2, contains three sublattices from

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S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

the crystallographic point of view. The accepted practice is to model it using the two-sublattice model (A,B)1(A,B)2, where the boldface B and A mean the normal species (i.e. major species) in the sublattices. In the Fe–Nb and Mn–Nb binary systems, it occurs with an ideal stoichiometry NbMn2 and NbFe2, respectively, so that the appropriate sublattice model for the phase is (Fe,Mn,Nb)1(Fe,Mn,Nb)2. According to the sublattice formalism, the Gibbs energy for one mole of formula units of the C14 phase based on the compound energy formalism [60] is given by C14

C14

C14

1 2 0 1 2 0 1 2 0 GC14 m ¼ yFe yFe G Fe:Fe þ yFe yMn G Fe:Mn þyFe yNb G Fe:Nb

C14

C14

C14

þy1Mn y2Fe 0 G Mn:Fe þ y1Mn y2Mn 0 G Mn:Mn þ y1Mn y2Nb 0 G Mn:Nb C14

C14

C14

þy1Nb y2Fe 0 G Nb:Fe þy1Nb y2Mn 0 G Nb:Mn þ y1Nb y2Nb 0 G Nb:Nb þRT½ðy1Fe lny1Fe þy1Mn lny1Mn þ y1Nb lny1Nb Þ þ 2ðy2Fe lny2Fe C14

þy2Mn lny2Mn þ y2Nb lny2Nb Þ þy1Fe y1Nb y2Fe 0 L Fe,Nb:Fe C14

C14

C14

þy1Fe y1Nb y2Nb 0 G Fe,Nb:Nb 0 L Fe,Nb:Nb þy1Fe y2Fe y2Nb 0 L Fe:Fe,Nb C14

C14

þy1Nb y2Fe y2Nb 0 L Nb:Fe,Nb þ y1Mn y1Nb y2Mn 0 L Mn,Nb:Mn C14

C14

þy1Mn y1Nb y2Nb 0 L Mn,Nb:Nb þ y1Mn y2Mn y2Nb 0 L Mn:Mn,Nb

Fig. 3. Sketch of AFM ordering of Mn and Fe magnetic moments in planes perpendicular to the [0 0 0 1] direction for the end-members of C14. Dashed lines indicate the planes with Nb sites: (a) NbFe2, (b) NbMn2, (c) FeFe2, (d) MnMn2, (e) FeNb2 and (f) MnNb2.

S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

C14

þ y1Nb y2Mn y2Nb 0 L Nb:Mn,Nb 0

C14

G Fe:Mn and

0

3.4. m Phase

ð8Þ

C14

G Mn:Fe in Eq. (8) are assumed to be

The m phase (Fe7Nb6) has been reported [63] to have a rhombohedral structure (Fe7W6 prototype) with a primitive unit cell of 39 atoms, distributed in five crystallographically distinct lattice sites in the ratios 1:2:2:2:6 with 12-,15-,16-,14-,and 12-fold coordination polyhedral. This is a typical Frank–Kasper type phase. In the m phase, the smaller atoms prefer sites with low coordination number whereas the larger atoms prefer sites with high coordination number [64,65]. In the sublattice notation, the ideal atomic distribution can be 15 16 14 12 represented as ðAÞ12 1 : ðBÞ2 : ðBÞ2 : ðBÞ2 : ðAÞ6 (the subscript is the number of sites in that sublattice, and superscript is the coordination number). This represents the ideal stoichiometry of A7B6, corresponding to the composition of 46.2 at% B. According to the literature, m is presented in 11 binary systems, and the homogeneity range may vary from 38.5 to 54 at% B [64]. In order to describe the observed homogeneity ranges of the m phase, it is necessary to allow mixing of atoms in sublattices. This led to the use of a four-sublattice model by Ansara et al. [66], supported by the investigation of Joubert and Dupin [9], where the atom distribution was confirmed to be (A,B)1 (B)4(A,B)2(A,B)6, which we adopt also in this work. In order to predict the extension of the binary phase m into the ternary Fe–Mn–Nb system, we modify this formulation to be (Fe,Mn,Nb)1(Nb)4(Fe,Nb)2 (Fe,Mn,Nb)6, permitting Mn on the first and fourth sublattices, i.e. the sublattices which are mainly occupied by Fe in the Fe–Nb m phase. The Gibbs energy for one mole of m phase is expressed by

1 0 C14 3 G Fe:Fe þ

C14 C14 C14 20 10 20 3 G Mn:Mn and 3 G Mn:Mn þ 3 G Fe:Fe ,

respectively. In the ternary Fe– Mn–Nb system, the phase will cross the system from the Fe–Nb to the Mn–Nb binary edge, and no ternary parameters are needed. The conventional treatment for the Gibbs energy of the ‘‘endmembers’’ is to consider the Wagner–Schottky constraint [61]. Taking l in the Mn–Nb binary system as an example, there are C14 four ‘‘end-members’’. Among them, only 0 G Nb:Mn , corresponding to the stoichiometric NbMn2, can experimentally be determined. Further, the Gibbs energies of the pure elemental constituents are usually given some empirical values in the CALPHAD modelC14 C14 ing. For 0 G Mn:Mn and 0 G Nb:Nb , 1 and 5 kJ/mol-atoms are arbitrarily C14 chosen in the COST507 database [62].0 G Mn:Nb corresponds to the l phase where all sublattices sites are occupied by antistructure atoms and is determined by the Wagner–Schottky constraint [61]: 0

C14

C14

C14

C14

G Mn:Nb ¼ 0 G Mn:Mn þ 0 G Nb:Nb 0 G Nb:Mn

49

ð9Þ

The present work will overcome the arbitrariness in choosing the values of the ‘‘end-members’’ by determining their physically based total energies from ab initio calculations and use them in the modeling of the Gibbs energies of the C14 Laves phase in both the Fe–Nb and Mn–Nb binary systems.

m

m

Gmm ¼ y1Fe y3Fe y4Fe 0 G Fe:Nb:Fe:Fe þ y1Fe y3Fe y4Nb 0 G Fe:Nb:Fe:Nb

Table 3 Magnetic moments per atom at particular sublattices for FM C14 and m phases. The numbers in parentheses give the coordination number of an atom in the corresponding sublattice. Boldface numbers mark the sublattice occupied by Nb. m denotes average magnetic moment per atom. xNb

Mn–Nb

Fe–Nb

m

Magnetic moment (lB/atom)

Magnetic moment (lB/atom)

2a

4f

6h

2a

4f

6h

(12)

(16)

(12)

(12)

(16)

(12)

 0.32 0.65  0.09 0.00

0.56  0.03 3.50 0.00

 0.19 0.36  0.29 0.00

2.04 1.14 0.02 0.00

2.93  0.31 2.89 0.00

2.07 1.41  0.11 0.00

m

(a) C14 0 0.333 0.667 1

End-members

0.04 0.28 1.01 0.00

2.35 0.79 0.91 0.00

m

Magnetic moment (lB/atom) 3a

6c1

6c2

6c3

18h

(12)

(15)

(16)

(14)

(12)

1.72  0.30 1.20  0.23 1.12 0.00 0.02 0.00 0.07  0.09  0.02 0.93 1.05 0.78  0.10 0.08 0.29 0.43

 0.26  0.31  0.08  0.14  0.09 0.00 0.00 0.00  0.03  0.10 0.00 0.09  0.22 0.08  0.03 0.00  0.07 0.03

 0.34  0.14  0.25 0.01  0.20  0.01 0.00 0.00  0.05  0.08 0.00 0.02  0.31 0.03 0.02 0.00 0.16  0.04

2.41 2.41  0.22  0.18 1.06 0.03 0.00 0.00  0.03  0.07 0.00 2.24 1.56 2.31 2.34 0.00  0.21 2.65

1.62 1.57 1.46 1.41  0.05 0.00 0.00 0.00 0.29 0.73 0.00 0.16 1.85 0.23 0.45 0.01 1.40  0.06

(b)

m Fe9Nb4 Fe8Nb5 Fe7Nb6 Fe6Nb7 Fe3Nb10 Fe2Nb11 Fe1Nb12 Nb13 Mn7Nb6 Mn6Nb7 Mn1Nb12 Fe3Mn6Nb4 Fe8Mn1Nb4 Fe2Mn7Nb4 Fe2Mn6Nb5 Fe1Mn6Nb6 Fe6Mn1Nb6 Fe2Mn1Nb10

1.16 1.00 0.68 0.59 0.18 0.00 0.00 0.00 0.12 0.29 0.00 0.51 1.09 0.93 0.55 0.01 0.60 0.41

50

S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

m

m

m

m

þRT½ðy1Fe lny1Fe þ y1Mn lny1Mn þ y1Nb lny1Nb Þ þ 2ðy3Fe lny3Fe þ y3Nb lny3Nb Þ

þ y1Fe y3Nb y4Fe 0 G Fe:Nb:Nb:Fe þy1Fe y3Nb y4Nb 0 G Fe:Nb:Nb:Nb þ y1Nb y3Fe y4Fe 0 G Nb:Nb:Fe:Fe þy1Nb y3Fe y4Nb 0 G Nb:Nb:Fe:Nb m

þ6ðy4Fe lny4Fe þ y4Mn lny4Mn þ y4Nb lny4Nb Þ

m

þ y1Nb y3Nb y4Fe 0 G Nb:Nb:Nb:Fe þ y1Nb y3Nb y4Nb 0 G Nb:Nb:Nb:Nb m

According to Eq. (10), there are 8 Fe–Nb, 3 Mn–Nb and 7 ternary end-members for m. The total energy for all these 18 endmembers will be calculated by ab initio calculations in order to provide reasonable start values for the thermodynamic modeling. Because parameters for the end-members are sufficient to describe the phase, no interaction parameters will be used during the optimization.

m

þ y1Mn y3Nb y4Mn 0 G Mn:Nb:Nb:Mn þy1Mn y3Nb y4Nb 0 G Mn:Nb:Nb:Nb m þ y1 y3 y4 0 G Nb Nb Mn

Nb:Nb:Nb:Mn

m þ y1 y3 y4 0 G Fe Fe Mn

Fe:Nb:Fe:Mn

m m þ y1Fe y3Nb y4Mn 0 G Fe:Nb:Nb:Mn þ y1Mn y3Fe y4Fe 0 G Mn:Nb:Fe:Fe m

m

þ y1Mn y3Fe y4Mn 0 G Mn:Nb:Fe:Mn þ y1Mn y3Fe y4Nb 0 G Mn:Nb:Fe:Nb m

m

þ y1Mn y3Nb y4Fe 0 G Mn:Nb:Nb:Fe þ y1Nb y3Fe y4Mn 0 G Nb:Nb:Fe:Mn Table 4 Calculated total energy and structural parameters for the end-members of C14 and m with NM, FM and AFM magnetic ordering based on DFT [16]. The values in boldface show the most stable magnetic configurations. Total energies are in ˚ equilibrium volume V0 per kJ/mol-atoms, lattice parameters a and c are in A, formula unit is in A˚ 3. Structure

Mag. order

Total energy

a

C14

NM FM AFM NM FM AFM NM FM AFM NM NM FM AFM NM FM AFM NM FM AFM

 913.923  913.901  913.907  854.018  881.577  873.673  855.725  855.779  855.808  953.191  831.655  850.677  849.822  864.646  864.814  864.894  757.194  777.902  772.012

4.802 4.802 4.806 5.438 5.471 5.432 4.547 4.547 4.547 5.467 4.766 4.822 4.797 5.385 5.423 5.341 4.531 4.677 4.560

7.930 7.942 7.937 8.428 8.524 9.135 7.358 7.358 7.357 8.869 7.811 7.839 7.891 8.258 8.587 8.482 7.288 7.615 7.471

158.4 158.6 158.7 215.8 221.0 233.45 131.7 131.7 131.7 229.6 153.6 157.9 157.3 207.4 218.7 209.6 129.6 144.3 134.5

NM FM NM FM NM FM NM FM NM FM NM FM NM FM NM NM FM NM FM NM FM NM FM NM FM NM FM NM FM NM FM NM FM NM FM

 883.407  886.911  857.875  862.847  846.786  854.819  895.586  898.090  894.162  896.049  904.040  904.083  942.395  942.567  952.231  922.885  923.104  925.325  925.547  947.811  947.797  917.391  917.413  882.774  885.582  890.167  892.903  853.531  859.403  904.895  908.957  889.653  892.031  888.363  591.435

4.883 4.855 4.879 4.853 4.710 4.844 5.015 4.970 5.437 5.418 5.450 5.452 5.483 5.424 5.456 4.826 4.854 4.971 4.974 5.456 5.465 4.836 4.856 4.743 4.726 4.914 4.875 4.767 4.839 5.367 5.458 4.837 4.845 4.756 4.762

26.247 27.000 25.628 26.657 25.794 25.432 26.103 27.027 27.368 27.534 28.522 28.488 27.437 28.139 28.825 27.349 27.056 27.167 27.169 27.855 27.784 27.199 26.964 25.968 26.100 25.934 26.757 25.341 25.537 27.057 26.357 26.493 26.671 25.876 26.228

542.0 551.2 528.3 543.7 495.6 516.9 568.5 578.3 700.3 700.8 733.6 733.4 714.6 717.0 743.0 551.7 552.2 581.4 582.1 718.1 718.7 550.8 550.7 505.9 504.9 542.4 550.7 498.8 517.9 675.0 686.47 536.9 542.3 507.0 515.0

NbMn2

MnNb2

MnMn2

NbNb2 NbFe2

FeNb2

FeFe2

m

Fe7Nb6 Fe8Nb5 Fe9Nb4 Fe6Nb7 Fe3Nb10 Fe2Nb11 Fe1Nb12 Nb13 Mn7Nb6 Mn6Nb7 Mn1Nb12 Fe1Mn6Nb6 Fe3Mn6Nb4 Fe6Mn1Nb6 Fe8Mn1Nb4 Fe2Mn1Nb10 Fe2Mn6Nb5 Fe2Mn7Nb4

c

ð10Þ

V0

4. Ab initio calculations The calculations were done in the framework of DFT [16] by means of the Vienna ab initio simulation package (VASP) [67,68] with the projector augmented wave potential construction [69,70]. An energy cutoff of 400 eV was chosen for all calculations. For the exchange correlation functional, the generalized gradient approximation (GGA) of Perdew and Wang [71] was applied. According to Monkhorst and Pack [72], the Brillouin-zone integration was performed on suitable k-point grids (from 9  9  5 to 11  11  5). Optimization of geometrical parameters (atomic positions, lattice parameters and c/a) was achieved by minimizing forces and total energies. According to the thermodynamic modeling in section 3, there are 7 and 18 end-members for C14 Laves and m phases in the Fe– Mn–Nb system, respectively. As C14(e) is reported with controversial magnetic properties [36–39] in the Fe–Nb binary system and Mn-based Laves-phases XMn2 (X being an early transition metal atom) are of interest for their (known or predicted) magnetic properties [73], it is necessary to study the possible magnetic properties for these phases via ab initio calculations based on DFT. Therefore, in the present work, possible magnetic ordering was studied by performing calculations for NM (nonmagnetic), FM (ferromagnetic) and selected AFM (antiferromagnetic) spin arrangements in suitable supercells in order to provide initial values for the end-members of C14 and m in the thermodynamic modeling. For NbFe2 and NbMn2 of C14 structure, two different crystallographic Fe (or Mn) sites are present: (6h) sites with 6-fold local coordination in planes with a triangular arrangement of Fe (or Mn) atoms and (2a) sites with 2-fold coordination. For each lattice site, spin up and spin down ordering is possible resulting in a different magnetic configuration. We have chosen four layerwise AFM initial arrangements: AFM1 (up/down and down/up arrangement at 2a/6h layers); AFM2 (up/up and down/ down arrangement at 2a/6h layers, see Fig. 3a); AFM3 (up/down at 2a layers); AFM4 (up/down at 6h layers, see Fig. 3b). For FeFe2 and MnMn2, three different Fe (or Mn) sites are crystallographically distinguished: the (2a), (4f) and (6h) sites with 2-, 4- and 6-fold coordination. During the calculation, five layerwise AFM initial arrangements have been probed: AFM10 (alternating up/down in each 2a, 4f, and 6h layers, see Fig. 3c); AFM20 (alternating up/down in 2a layers); AFM30 (alternating up/down in 4f layers); AFM40 (alternating up/up and down/down in 4f layers, see Fig. 3d); AF50 (alternating up/down in 6h layers). For FeNb2 and MnNb2 of C14 structure, there is only one site, namely 4f with 4-fold coordination so that three AFM layerwise arrangements have been treated: AFM100 (alternating up/down and down/up, see Figs. 3e and f); AFM200 (alternating up/up and down/down, see Fig. 3d); AFM300 (alternating up/down). For the end-members of the m phase, only NM and FM calculations have been performed due to the enormous CPU requirements for AFM calculations, which is not feasible with our resources.

S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

inactive, which is different from the proposal by Chen et al. [74] for ZrMn2. The energetically most stable AFM ordering for FeFe2 is AFM10 , as shown in Fig. 3c. However, Mn atoms for MnMn2 prefer the ordering of AFM40 , as shown in Fig. 3d. For both FeNb2 and MnNb2, the AFM ordering with the lowest total energy is AFM100 , as shown in Figs. 3e and f. It is also worth noting that the local magnetic moments of Mn may also exceed that of Fe (see Figs. 3e and f), which is a known phenomenon [75]. The calculated total energies and structural parameters for C14 and m crystal structure with their NM, FM, and selected AFM phases are listed in Table 4. To be able to compare our results with the experimental data, the energies of formation must be determined with respect to the reference state of pure elements. At 0 K, because there is no entropy contribution, the energy of formation is derived in the standard way by the energy difference:

5. Result and discussion 5.1. Ab initio calculations The local and total magnetic moments for the Fe and Mn sites of the FM C14 and m are compiled in Table 3. According to our calculation, C14 and m in the Fe–Mn–Nb system should be spinpolarized in most cases at low temperature. Stabilization effects due to magnetic ordering were not found in the following endmembers: Nb3, Nb13, Mn1Nb12, Fe2Nb11, Fe1Nb12, and Fe1Mn6Nb6. Fig. 3 shows the energetically most stable AFM ordering for the C14 structure. From the results shown in Table 3 and Fig. 3, we can conclude that the distribution of the local magnetic moments is dependent on the lattice site. For the selected four AFM arrangements for NbFe2 and NbMn2, the energetically most stable ones turned out to be AFM2 and AFM4, respectively, as shown in Figs. 3a and b. The Mn atoms in the 2a layers are magnetically

m Bn EAf m Bn ¼ EAtotal mEAtotal nEBtotal

Mag. order

Etotal

a

c

V0

References

(aFe)

FM

 792.614  798.413

2.832 2.822

– –

22.7 22.5

This work [78]

(gMn)

NM

 857.308  857.655  859.626

3.496 3.502 3.550

– – –

42.7 43.0 44.7

This work [78] This work

AFM bctMn

AFM

 860.422

3.610

3.433

44.7

This work

(Nb)

NM

 969.727  969.385

3.320 3.322

– –

36.6 36.7

This work [78]

ð11Þ

m Bn where EAtotal , EAtotal and EBtotal are the total energy of the compound AmBn and constituents A and B at the low-temperature ground state, respectively. The calculations for body-centered cubic (bcc) Fe and Nb are straightforward. However, the ground state of Mn is difficult to treat due to the complicated magnetic ordering of (aMn). One can, however, make use of the existence of the high temperature (gMn) phase, which can be quenched down to room temperature. The g phase consists of AFM layers in the [0 0 1] direction of a body-centered tetragonal (bct) structure, and can be easily calculated by a standard DFT approach. In our present calculation, the energy difference between AFM (gMn) to g bct Mn is 1.013 kJ/mol-atoms, which is quite close to the predicted value (about 1.254 kJ/mol-atoms) by the LMTO_ASA method [76]. If considering the work of Hafner and Hobbs [77], who reported the energy difference between AFM (gMn) and (aMn) to be

Table 5 Calculated total energy, structural parameters of Fe, Mn, and Nb, comparing with the literature data. Total energies Etotal are in kJ/mol-atoms, lattice parameters a ˚ equilibrium volume V0 per formula unit is in A˚ 3. and c are in A, Phase

51

Table 6 Enthalpies of formation (kJ/mol-atoms) for the end members of C14 and m under NM, FM, and AFM ordering with respect to FM (aFe), AFM (aMn) and NM (Nb) from ab initio calculations, along with the data from literature. The values in boldface show the most stable magnetic ordering. Compounds

Enthalpy of formation(kJ/mol-atoms) This work (DFT)

C14

NbMn2 MnNb2 MnMn2 NbNb2 FeFe2 NbFe2 FeNb2

m

Fe7Nb6 Fe8Nb5 Fe9Nb4 Fe6Nb7 Fe3Nb10 Fe2Nb11 Fe1Nb12 Nb13 Mn7Nb6 Mn6Nb7 Mn1Nb12 Fe1Mn6Nb6 Fe3Mn6Nb4 Fe6Mn1Nb6 Fe8Mn1Nb4 Fe2Mn1Nb10 Fe2Mn6Nb5 Fe2Mn7Nb4

Literature

NM

FM

AFM

Exp.

DFT

Miedema

 13.29 81.16 10.37 16.54

 13.27 53.60 10.31 –

 13.27 61.50 7.28 –

 10.4 7 2.7 [52] – – –

 5 [53] – – –

14.71

20.60

–

 12.99 79.03

 13.16 60.01

 13.24 60.87

 5.3 7 1.7 [37] –

 15.2 [52] – 9.0 [79] 15.4 [79] 15.6 [9] 31.1 [79] 38.8 [9]  14.5 [9] 2.0 [9]

 9.05 2.86 0.32  7.60 34.69 38.44 13.71 17.50

 12.55  2.11  7.71  10.11 32.81 38.40 13.54 –

– – – – – – – –

 6.2 7 1.8 [37] – – – – – – –

 8.96  3.43 13.94  9.12  1.75  10.16  0.77 29.61 4.99  1.69

 9.18  3.65 13.96  9.14  4.56  12.89  6.64 25.55 2.62  4.76

– – – – – – – – – –

– – – – – – – – – –

35.42

 13.5 [9] – – – – – – 18.0 [79] 16.1 [9] – – – – – – – – – –

– –  21 [53]  24 [53] – – – – – – – – – – – – – – – – –

52

S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

Table 7 Optimized thermodynamic parameters for the binary and ternary systems. Phase Fe–Nb Liquid (L)

bcc_A2 (dFe, aFe,(Nb)) fcc_A1(gFe) C14(e)

m

Mn–Nb Liquid (L)

bcc_A2(dMn, (Nb)) fcc_A1(gMn) cub_A13(bMn)

cbcc_A12(aMn)

C14(l)

m

Fe–Mn–Nb C14

Thermodynamic parameters (J/mol)

Remarka

0 L L Fe,Nb ¼ 73,554:99 þ 104:28T9:984T ln T 1 L L Fe,Nb ¼ 20,336213:525T 0 Bcc_A2 L Fe,Nb ¼ 17,116:9922:054T 1 Bcc_A2 L Fe,Nb ¼ 19,211:28215:937T 0 Fcc_A1 L Fe,Nb ¼ 39,429:64227:084T Bcc_A2 0 C14 G Fe:Fe ¼ 44,130 þ 30 G Fe 0 C14 0 Bcc_A2 G Nb:Nb ¼ 49,620 þ 3 G Nb Bcc_A2 Bcc_A2 0 C14 G Nb:Fe ¼ 32,757:785 þ 20 G Fe þ 0 G Nb 0 C14 0 Bcc_A2 0 Bcc_A2 G Fe:Nb ¼ 180,030 þ G Fe þ 2 G Nb C14 0 C14 L Fe,Nb:Fe ¼ 0 L Fe,Nb:Nb ¼ 49,713:116 Bcc_A2 Bcc_A2 0 m G Fe:Nb:Fe:Fe ¼ 96,407:54 þ 90 G Fe þ 40 G Nb m Bcc_A2 Bcc_A2 0 G Fe:Nb:Fe:Nb ¼ 426,530 þ 30 G Fe þ 100 G Nb Bcc_A2 Bcc_A2 0 m G Fe:Nb:Nb:Fe ¼ 168,016:8 þ 21:029T þ 70 G Fe þ 60 G Nb m Bcc_A2 Bcc_A2 0 G Fe:Nb:Nb:Nb ¼ 176,020 þ 0 G Fe þ 120 G Nb Bcc_A2 0 m 0 Bcc_A2 G Nb:Nb:Fe:Fe ¼ 27,430 þ 8 G Fe þ 50 G Nb m Bcc_A2 Bcc_A2 0 G Nb:Nb:Fe:Nb ¼ 499,200 þ 20 G Fe þ 110 G Nb m Bcc_A2 Bcc_A2 0 G Nb:Nb:Nb:Fe ¼ 85,181:18 þ 60 G Fe þ 70 G Nb m Bcc_A2 0 G Nb:Nb:Nb:Nb ¼ 227,500 þ 130 G Nb

Opt.

0 L L Mn,Nb ¼ 15,322:786 1 L L Mn,Nb ¼ 12,376:04 0 Bcc_A2 L Mn,Nb ¼ 16,895:03 0 Fcc_A1 L Mn,Nb ¼ 6305:5 Bcc_A2 0 cub_A13 G Nb:Va ¼ 22,000 þ 0 G Nb cub_A13 0 L Mn,Nb:Va ¼ 10,485:26 Bcc_A2 0 cbcc_A12 G Nb:Va ¼ 17,600 þ 0 G Nb 0 cbcc_A12 L Mn,Nb:Va ¼ 10,485:26 Cbcc_A12 0 C14 G Mn:Mn ¼ 21,840 þ 30 G Mn Cbcc_A12 Bcc_A2 0 C14 G Nb:Mn ¼ 32,983:815 þ 20 G Mn þ 0 G Nb 0 C14 0 Cbcc_A12 0 Bcc_A2 G Mn:Nb ¼ 160,800þ G Mn þ 2 G Nb C14 0 C14 L Mn:Mn,Nb ¼ 0 L Nb:Mn,Nb ¼ 5249:5 Cbcc_A12 0 C14 G Mn:Mn ¼ 3000 þ 30 G Mn C14 Bcc_A2 0 G Nb:Nb ¼ 15,000 þ 30 G Nb C14 Cbcc_A12 Bcc_A20 0 G Nb:Mn ¼ 33,354:052 þ 20 G Mn þ 0 G Nb C14 C14 C14 0 0 0 GC14 Mn:Nb ¼ G Mn:Mn þ G Nb:Nb  G Nb:Mn 0 C14 0 C14 L Mn:Mn,Nb ¼ L Nb:Mn,Nb ¼ 48,832:70 C14 0 C14 L Mn,Nb:Mn ¼ 0 L Mn,Nb:Nb ¼ 23,322:74 Cbcc_A12 Bcc_A2 0 m G Mn:Nb:Nb:Mn ¼ 119,340 þ 70 G Mn þ 60 G Nb 0 m 0 Cbcc_A12 0 Bcc_A2 G Nb:Nb:Nb:Mn ¼ 47,450 þ 6 G Mn þ 7 G Nb Cbcc_A12 Bcc_A2 0 m G Mn:Nb:Nb:Nb ¼ 181,220 þ 0 G Mn þ 120 G Nb

Opt.

0

C14

C14

m

C14 C14 ¼ 13 0 G Mn:Mn þ 230 G Fe:Fe

Bcc_A2 Cbcc_A12 G Fe:Nb:Nb:Mn ¼ 118,820 þ G Fe þ 60 G Mn þ6 m Bcc_A2 Cbcc_A12 Bcc_A2 0 G Fe:Nb:Fe:Mn ¼ 59,280 þ 30 G Fe þ 60 G Mn þ 40 G Nb Bcc_A2 Cbcc_A12 Bcc_A2 0 m G Mn:Nb:Nb:Fe ¼ 167,570 þ 60 G Fe þ 0 G Mn þ 60 G Nb 0 m 0 Bcc_A2 0 Cbcc_A12 0 Bcc_A2 G Mn:Nb:Fe:Fe ¼ 86,320 þ 8 G Fe þ G Mn þ 4 G Nb Bcc_A2 Cbcc_A12 Bcc_A2 0 m G Mn:Nb:Fe:Nb ¼ 332,150 þ 20 G Fe þ 0 G Mn þ 100 G Nb 0 m 0 Bcc_A2 0 Cbcc_A12 0 Bcc_A2 G Nb:Nb:Fe:Mn ¼ 34,060 þ 2 G Fe þ 6 G Mn þ 5 G Nb Bcc_A2 Cbcc_A12 Bcc_A2 0 m G Mn:Nb:Fe:Mn ¼ 61,880 þ 20 G Fe þ 70 G Mn þ 40 G Nb 0

Opt. Opt. Emp. Emp. Opt. Emp. Opt. Opt. Emp. Opt. Emp. Emp. Emp. Opt. Emp.

Opt. Opt. Opt. Emp. Opt. Emp. Emp. Emp. (Set 1) Opt. (Set 1) Emp. (Set 1) Opt. (Set 1) Emp. (Set 2) Emp. (Set 2) Opt. (Set 2)

Opt. (Set 2) Opt. (Set 2) Emp. (Set 1) Emp. (Set 1) Emp. (Set 1)

Emp.

C14

G Fe:Mn ¼ 0 G Fe:Mn þ 23 0 G Mn

0 C14 G Mn:Fe

Opt. Opt.

0

Emp. Emp. Emp. Emp. Emp. Emp. Emp. Emp.

a ‘‘Emp.’’ means the values for the parameters are from ab initio calculations or empirical data. ‘‘Opt.’’ stands for values that have been optimized.

S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

6.464 kJ/mol-atoms without relaxing the structure to bct, the energy difference between g bct Mn and (aMn) in the present calculations would be 5.45 kJ/mol-atoms. In order to further verify our calculations, the calculated total energies for NM (gMn) and (Nb) as well as FM (aFe) are compared with the data from Wang et al. [78], as shown in Table 5. Table 6 presents the enthalpies of formation for the endmembers of C14 and m structures with respect to FM (aFe), AFM (aMn) and NM (Nb). The values indicated in bold are the most stable one for the end-members. According to our present calculations, the NM ground state is the lowest energy state for NbMn2. Comparing with the calculations by Meschel et al. [52] and the Miedema model [53] for NbMn2, our results is closer to the measured one [52]. The energetically most stable state for NbFe2 is found to be AFM. The energy difference between FM and AFM is less than 0.1 kJ/mol-atoms, consisting with the reported ‘‘weak AFM’’ by [42–45]. This is maybe the reason for that several groups found it to be FM [39,46]. The magnetic properties of C14 are strongly dependent on the composition in both the Mn–Nb and Fe–Nb binary systems. The ferromagnetism becomes stronger on both sides of NbFe2 in the Fe–Nb system, which is confirmed in literature [39,41,46]. Under the present calculations, the total energy of FM configuration is more negative than NM for Fe7Nb6. Compared to the measured enthalpy of formation by high temperature direct synthesis calorimetry from Meschel and Kleppa [37], the predicted values for both NbFe2 and Fe7Nb6 in the present work are too negative. However, our calculations for NbFe2, Fe7Nb6, FeFe2 and NbNb2 are in good agreement with other ab initio calculations [9,79]. The reason for the large difference between the calorimetric data and the data from ab initio calculations could be that the enthalpy of formation is generally temperature-dependent. 5.2. Thermodynamic optimization Evaluation of the model parameters is attained by recurrent runs of the PARROT module of the Thermo-Calc program [80], which works by minimizing the square sum of the differences between measured and calculated values. The step-by-step optimization procedure described by Du et al. [81] was utilized in the present assessment.

53

Fe6Nb7and Fe9Nb4 were optimized to fit the experimental data [12,28,37]. Subsequently, fcc_A1 was involved in the optimization by considering the experimental phase diagram data on the fcc_A1 solvus boundary [21,23,26]. The activity data from Hawkins [33] have not been considered during the optimization because the phase diagram in the fcc_A1 region will become worse if we try to fit these activity data. Finally, the thermodynamic parameters for all phases were optimized simultaneously by taking into account all of the selected phase diagram and thermodynamic data. These optimized parameters are listed in Table 7. In Fig. 1 the calculated enthalpies of mixing for liquid using the present thermodynamic description are compared to the experimental data [29–31]. The present parameters describe the data from literature [29,31] well. The calculated enthalpy of mixing for the liquid alloys at 1960 K is also present in the figure, confirming the large error of the experimental data from Sudavtsova et al. [30], which were not used during the optimization. Fig. 4 shows the calculated enthalpies of formation with the presently obtained parameters along with the experimentally measured values [34,35,37] as well as the data from ab initio calculations from both this work and the literature [9]. The proposed description reproduces the data from Meschel and Kleppa [37] reasonably. According to the present calculations, the lowest enthalpy of formation for the solid compounds lies at the composition of Fe7Nb6 instead of NbFe2 with the highest congruent melting temperature. The case is similar to that in the Fe–Ta binary system [38]. Fig. 5 shows the comparison between experimentally measured [32] and calculated activities of iron in Fe–Nb alloys at 1600 1C. It should be mentioned that these data have not been used in our optimization but the comparison shows the present description can predict the experimental data [32] well. Finally, Fig. 6 shows the calculated phase diagram of the Fe–Nb binary system along with the experimental data [10–12,18–28]. Fig. 6a is the calculated complete phase diagram in comparison with experimental data. The data from Goldschmidt [18] are too low in temperature so that they were not used in the optimization, as shown in Table 1. However, they are included in the figure as a comparison because they are the only data available for the

5.2.1. The Fe–Nb binary system The optimization was started with the liquid phase. For liquid, at least a0 and b0 in Eq. (4) can be adjusted because there are enthalpies of mixing with xNb o0.4 and liquidus measured accurately nearly over the whole composition range. Since the experimental enthalpies of mixing are dependent of temperature as shown in Fig. 1, c0 is optimized accordingly. As for the entropy, it is impossible to tell in advance from the experimental phase diagram data whether more than one coefficient is necessary or not. Consequently, the optimization was started with one coefficient b0. In the present case, it was found that a1 and b1 should also be introduced in order to describe the properties of the liquid phase satisfactorily. Secondly, bcc_A2 was involved in the optimization with the experimental phase diagram data from [11,12,19,20,23,26–28]. A two-term Redlich–Kister equation [58] was adopted for the bcc_A2 phase. Thirdly, the calculated enthalpies of formation for the FeFe2, NbNb2 and FeNb2 of e (C14 Laves) from the ab initio calculations were incorporated into the sublattice model for the e phase, while that for NbFe2 was used as start value for the optimization to fit the experimental phase diagram and thermodynamic data [10–12,28,37]. The interaction parameters 0 C14 L Fe,Nb:Fe

C14

¼ 0 L Fe,Nb:Nb were used to obtain a better reproduction of the experimental data. Then the m phase was optimized in a similar way as e and the parameters of the end-members Fe7Nb6,

Fig. 4. Calculated enthalpies of formation for the solid Fe–Nb alloys at 25 1C using the present thermodynamic description, comparing with the experimental data [34,35,37] and the data from ab initio calculations from the present work and literature [9].

54

S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

liquidus on the Nb-rich side. Figs. 6b and c show the calculated Fe-rich corner of the phase diagram and the g loop including available experimental data from literature. From these figures one may conclude that the present thermodynamic description can well reproduce the experimental data with respect to the composition of phases and the temperature of invariant reactions. Table 2 gives an overview of the invariant equilibria in the Fe–Nb binary system including the calculated temperatures and compositions. Available experimental data are included for comparison.

Fig. 5. Calculated activities of Fe and Nb in Fe–Nb alloys at 1600 1C using the present thermodynamic description, along with the experimental data from Ichise and Horikawa [32].

5.2.2. The Mn–Nb binary system To perform an optimization, one needs both the phase equilibrium data and thermodynamic data since the thermodynamic properties of the system cannot be unique defined by phase equilibrium alone. Therefore the optimization for the Mn–Nb system was started from the l phase (C14 Laves) based on the limited experimental data [48,50,52] and the data from the present ab initio calculations. Then the liquid, bcc_A2, fcc_A1, cub_A13 and cbcc_A12 phases were included in the optimization. Only temperature independent parameters were used owing to the limited amount of experimental data. The lattice stability of Nb in the structure of cub_A13 and cbcc_A12 is not known. It is impossible to derive them from the not so well established phase equilibrium information in the Mn-rich side. However, Sluiter [79] has calculated the structural enthalpy of Nb in bcc_A2, cub_A13 and cbcc_A12 relative to fcc_A1 using ab initio

Fig. 6. Calculated phase diagram of the Fe–Nb system using the present thermodynamic description, along with the experimental data [10–12,18–28]. (a) The whole phase diagram; (b) partial one in the temperature range from 900 to 1600 1C in the Fe-rich side; (c) interrupted g loop in the Fe–Nb phase diagram.

S. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 38 (2012) 43–58

calculations. Considering the data from Sluiter [79], two values, 0

cub_A13

bcc_A2

0 G Nb

cbcc_A12

bcc_A2

and 0 G Nb 0 G Nb ¼ 22,000 J=mol-atoms, were fixed in the present work. Then the G Nb

¼ 17,600 J=mol-atoms cub_A13

interaction parameter of 0 L Mn,Nb:Va was determined by considering the two invariant reactions including (bMn). Since the experimental data from Savitskii and Kopetskii [49] are regarded as less reliable, there are no experimental data available for (aMn). 0 cbcc_A12 L Mn,Nb:Va

cub_A13

¼ 0 L Mn,Nb:Va will be tentatively chosen in this work.

Two sets of parameters were obtained for the l phase in the present work. The first treatment (Set 1) was to fix the parameters of

C14 0 C14 G Mn:Mn ,0 G Nb:Nb

and

0 C14 G Mn:Nb

in Eq. (8) with the data from the C14

present ab initio calculations. For the parameter of 0 G Nb:Mn , the

55

date from the present ab initio calculations was used as a start value and optimized to obtain the final value according to the available experimental phase diagram and thermodynamic data [48,50,52]. As mentioned in section 4, Mn has been considered in the sublattice model of m in order to check the extension of the phase into the ternary Fe–Mn–Nb system. Due to the lack of experimental data on the compounds in the Mn–Nb binary system, the values from the ab initio calculations were used m

m

m

directly for 0 G Mn:Nb:Nb:Mn ,0 G Mn:Nb:Nb:Nb and 0 G Nb:Nb:Nb:Mn . The second treatment (Set 2) used the conventional Wagner–Schottky C14

constraint for the l phase. Only the parameters for 0 G Nb:Mn were optimized according to the available experimental data C14

C14

[48,50,52]. The parameters of 0 G Mn:Mn and 0 G Nb:Nb in Eq. (8) were fixed with the empirical values 3000 J/mol-atoms and 15,000 J/ mol-atoms from the COST507 database [62], respectively, and 0

C14

G Mn:Nb was deduced by considering the Wagner–Schottky constraint [61]. In Set 2, parameters for fcc_A1, bcc_A2, cub_A13, and cbcc_A12 from Set 1 were fixed, no more adjustment has been performed in order to compare the two treatments on the l phase directly. The obtained two sets of parameters are listed in Table 7. Using the parameters, the phase diagram of the Mn–Nb system was calculated and compared with the experimental data. As shown in Fig. 7, both sets of parameters can reproduce phase diagram satisfactorily. However, the number of parameters in Set 1 is smaller than in Set 2, as indicated in Table 7. Introducing physically based parameters is advantageous as it makes the optimization process simpler, more effective and more reliable. According to the present calculation, m is marginally stable in the Mn–Nb binary system. But it is ambiguous since no experimental data is available with the temperature range lower than 1000 1C in the Nb rich side and there is a deviation for the date obtained by ab initio calculations from the experimental date [52] for NbMn2, as shown in Fig. 8. Further experimental investigation of the possible existence of m in the Mn–Nb system is necessary.

Fig. 7. Calculated phase diagram of the Mn–Nb system using the present thermodynamic description, along with the experimental data [48–50]. (a) The whole phase diagram; (b) partial one in the temperature range from 1060 to 1260 1C in the Mn-rich side. The temperatures are shown in the figure are from the calculation based on the parameters of Set 1. Note that m in (a) is not included in Set 2.

Fig. 8. Calculated enthalpies of formation for the solid Mn–Nb alloys at 25 1C using the present thermodynamic description, comparing with the experimental data [52] and the data from ab initio calculations from the present work and literature [52].

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Fig. 9. Calculated isothermal sections at 600, 800, 1000, 1300 1C of the Fe–Mn–Nb system: (a) 600 1C, (b) 800 1C, (c) 1000 1C and (d) 1300 1C.

the experimental data, the results from the present ab initio calculations for the ternary end-members of m were integrated in the sublattice model directly. Using these parameters, several isothermal sections for the ternary Fe–Mn–Nb system were predicted, as shown in Fig. 9. From the calculations we can see the homogeneity ranges of C14 and m will become wider with increasing temperature. Figs. 10 and 11 are the proposed liquidus projection and the reaction scheme of the ternary system. The calculated invariant reactions are also presented in Table 2.

6. Conclusions

Fig. 10. Calculated liquidus projection of the Fe–Mn–Nb system.

5.2.3. Fe–Mn–Nb ternary system Based on the presently obtained parameters for the Fe–Nb (Set 1) and Mn–Nb systems as well as the parameters for the Fe–Mn system from Huang [55], the thermodynamic description for the Fe–Mn–Nb ternary system was extrapolated. Due to the lack of

Possible NM, FM and AFM ordering for the end-members of the intermetallic compounds C14 Laves and m in the Fe–Mn–Nb system have been studied by ab initio calculations. It turns out that the local magnetic moment depends on lattice site. By incorporating ab initio calculations for the total energy into the CALPHAD approach, the Fe–Nb and Mn–Nb systems have thermodynamically been investigated based on the available experimental data. m is predicted to be marginally stable in the Mn–Nb binary system. Compared to the conventional treatment for the end-members of C14 in the Mn–Nb binary system, the introduction of physically based total energies by ab initio calculations makes the optimization process simpler, more effective and more reliable. The obtained thermodynamic parameters for the Fe–Nb and Mn–Nb systems can describe the reliable experimental data

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57

Fig. 11. Proposed reaction scheme of the Fe–Mn–Nb system.

well. Thermodynamic description for the Fe–Mn–Nb system is then extrapolated from the three binary systems and the data from the present ab initio calculations. Several isothermal sections and the liquidus projection as well as the reaction scheme are predicted.

Acknowledgment The financial support from Research Fund for Coal and Steel of the European Union (Grant agreement no. RFSR-CT-2010-00018) is gratefully acknowledged. Yong Du would like to thank the financial support from the Creative Research Group of the National Natural Science Foundation of China (Grant no. 51021063).

Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.calphad.2012.03.004.

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