Thermodynamic description of the Nd-Fe-B ternary system

Calphad 66 (2019) 101627

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Thermodynamic description of the Nd-Fe-B ternary system T.L. Chen a b

a,b

a,∗

b

b

, J. Wang , C.P. Guo , C.R. Li , Z.M. Du

b,∗∗

a

, G.H. Rao , H.Y. Zhou

T a

School of Materials Science and Engineering & Guangxi Key Laboratory of Information Materials, Guilin University of Electronic Technology, Guilin, 541004, China School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing, 100083, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Permanent magnet Nd-Fe-B ternary system Phase equilibria CALPHAD

The Nd-B binary system was re-assessed thermodynamically based on the available experimental information from literatures using the CALPHAD method. The solution phases including liquid, fcc, bcc and dhcp, were modeled as substitutional solutions and their excess Gibbs energies were expressed by the Redlich-Kister polynomial. Combined with the previous assessments of the Fe-Nd and the Fe-B binary systems, the thermodynamic description of the Nd-Fe-B ternary system was performed on the basis of the reported experimental information. The solubility of third element in the binary intermetallic compounds was not taken into account due to the lack of experimental data. A set of self-consistent thermodynamic parameters formulating the Gibbs energies of various phases in the Nd-Fe-B ternary system was obtained in the present work. The liquidus projection, several isothermal sections and vertical sections of this ternary system were calculated, which are in reasonable agreement with the reported experimental data.

1. Introduction High performance Nd-Fe-B-based permanent magnets have been used widely in many industrial applications such as wind power, electric vehicles, hard disk and nuclear magnetic resonance system [1–3]. To achieve higher coercivity and higher maximum magnetic energy product of permanent magnets, the heavy rare-earth (RE) metals (such as Dy and Tb) were usually added into Nd-Fe-B-based magnets [4]. However, from an economic point of view, the application of cheap and abundant light RE metals (e.g. La, Ce, Y) in Nd-Fe-B-based permanent magnets would be a promising way to reduce their costs [5–8]. To further understand the influence of RE metals on phase formation, phase transition, microstructure evolution and magnetic properties of Nd-Fe-B-based permanent magnets, the phase equilibira and thermodynamic properties of the related RE-Fe-B alloy systems are indispensable. As the sub-binary systems in the RE-Fe-B ternary systems, many Fe-RE binary systems (such as Fe-Pr, Fe-Nd, Fe-Gd, Fe-Sm, Fe-Dy, Fe-Tb and Fe-La, Fe-Ce, Fe-Y) were re-assessed in our previous work [9–11] and the reported literature [12–14], respectively, forming the significant foundation in the development of thermodynamic database of the RE-Fe-B ternary systems. The Nd-Fe-B ternary system as the most important ternary system in Nd-Fe-B-based permanent magnets was assessed by Hallemans et al. [15]. The calculated results agree well with the experimental data reported by Buschow et al. [16,17], Schneider

∗

et al. [18], Landgraf et al. [19], Knoch et al. [20] and Che et al. [21]. Recently, Ende and Jung [22] assessed the Nd-Fe-B ternary system on the basis of the re-optimization of the Fe-Nd, the Nd-B and the Fe-B subbinary systems using the modified quasi-chemical model to describe liquid phase. The problem of the miscibility gap for the Nd-B liquid phase at high temperature in the assessment of Hallemans et al. [15] was resolved. The calculated vertical sections at different composition conditions can reproduce well the phase relations determined experimentally, while the calculated isothermal sections at different temperatures were not shown to compare with the reported experimental results. Subsequently, considering two metastable ternary compounds (Nd2Fe23B3 and Nd2Fe17B), Zhou et al. [23] updated the optimization of the Nd-Fe-B ternary system using the revised thermodynamic parameters of the Nd-B binary system, but did not consider the solid solubility of NdB6. In their optimization, thermodynamic parameters of the Fe-B and the Fe-Nd binary systems were taken directly from the assessed results [15], but not the revised edition [9] with the consideration of the renewed thermochemical properties of liquid phase. It is expected generally that the calculated isothermal sections, vertical sections and liquidus projection are very similar with the calculated results [15], even if the liquid miscibility gap of the Nd-B binary system at high temperature was prevented. As mentioned above, the Fe-Nd binary system were re-assessed by the present authors based on the new experimental results including the

Corresponding author. Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (Z.M. Du).

∗∗

https://doi.org/10.1016/j.calphad.2019.101627 Received 18 March 2019; Received in revised form 4 May 2019; Accepted 20 May 2019 0364-5916/ © 2019 Elsevier Ltd. All rights reserved.

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enthalpy of mixing of liquid phase measured by Ivanov et al. [24] and phase equilibria data determined in our previous work [9]. In addition, in order to obtain the comprehensive thermodynamic description of the Nd-B binary system, the composition range of NdB6 compound and the thermodynamic properties of NdB4 and NdB6 compounds are necessary to be taken into consideration, which were ignored by Hallemans et al. [15] and Zhou et al. [23]. Therefore, in the present work the previous assessments of the Fe-Nd, the Fe-B and the Nd-B sub-binary systems were reviewed first, and then the Nd-B binary system was re-assessed thermodynamically using CALPHAD method. Combined with the reliable assessments of three sub-binary systems, the thermodynamic description of the Nd-Fe-B ternary system was performed finally based on the critical evaluation of the reported phase equilibrium data and thermodynamic properties. 2. Literature information 2.1. The Fe-Nd binary system The Fe-Nd binary was assessed by Hallemans et al. [15], Schneider et al. [25], and Hennmann et al. [26]. Using the modified quasi-chemical model to describe the liquid phase, Ende and Jung [22] optimized the Fe-Nd binary system. The Fe-Nd binary system was determined experimentally and re-assessed in our previous work [9] based on the new experimental data measured by Ivanov et al. [24]. Thermodynamic parameters of the Fe-Nd binary system obtained finally were used directly in this work. The calculated Fe-Nd binary phase diagram is shown in Fig. 1.

Fig. 2. The calculated Fe-B binary phase diagram [27].

from the assessed results by Hallemans et al. [27]. Fig. 2 gives the calculated Fe-B binary phase diagram. 2.3. The Nd-B binary system

2.2. The Fe-B binary system

The Nd-B binary system was optimized by Hallemans et al. [15] according to the limited experimental information reported in literature. The calculated results are reasonable except for the ignorance of composition range of NdB6 compound determined experimentally by Storms [30]. Unfortunately, the calculated Nd-B binary phase diagram shows the immiscible gap of liquid phase at high temperature, which was found by Ende and Jung [22]. Using the modified quasi-chemical model to describe liquid phase, the thermodynamic parameters of the Nd-B binary system was assessed by Ende and Jung [22]. Zhou et al. [23] re-optimized the Nd-B binary system using the calculated enthalpies of mixing of Nd-B liquid phase by the Miedema method. The immiscible gap of liquid phase at high temperature was avoided, while the solid solubility of NdB6 compound was not taken into account. The calculated temperatures and compositions of the invariant reactions in the Nd-B binary system [15,22,23] are different from the results reviewed by Liao et al. [31]. Therefore, the Nd-B binary system was reassessed in this work. The phase relations and thermochemical properties of Nd-B alloys in the composition range between NdB4.41 and NdB8.68 were measured by Storms [30]. The composition range of NdB6 compound in the temperature range from 1700 to 2100 K was determined experimentally. The melting temperature of NdB6 compound was measured to be 2883 ± 30 K by Mordovin and Timofeeva [32], which is very different with the experimental value (2813 K) reported by Samsonov and Paderno [33]. Spear and Solovyev [34] measured the temperature of the peritectic reaction related with the formation of NdB66 compound to be 2423 ± 100 K. Blomberg et al. [35] and Salamakha et al. [36] investigated the crystal structures of NdB6 and NdB4 compounds using single-crystal X-ray diffraction (XRD), while Kienlea et al. [37] studied the crystal structures of Nd2B5 and NdB4 compounds using XRD and high resolution transmission electron microscopy examination. The Gibbs energy of formation of NdB4 compound in the temperature range from 996 to 1060 K was determined by Xi et al. [38] using the electromotive force (EMF) method. Muratov et al. [39] measured the heat capacity and the enthalpy of formation of NdB4 in the

The Fe-B binary system was assessed thermodynamically by Hallemans et al. [27] based on the reported experimental results, and then was optimized by Rompaey et al. [28] and Poletti and Battezzati [29]. Ende and Jung [22] re-optimized the Fe-B binary system using the modified quasi-chemical model to describe the liquid phase. The magnetic contributions of Fe2B and FeB compounds were taken into account by Rompaey et al. [28]. Due to the decomposition of Fe2B compound at low temperature in the Fe-B binary phase diagram calculated by Rompaey et al. [28] and Poletti and Battezzati [29], thermodynamic parameters of the Fe-B binary system in the present work were taken

Fig. 1. The calculated Fe-Nd binary phase diagram [9]. 2

3

Nd2Fe14B (τ1)

B5Nd2

B4Nd

B6Nd

B66Nd

Fe17Nd5

Fe17Nd2

FeB

Fe2B

rhombo (β-B)

dhcp (α-Nd)

fcc (γ-Fe)

TCτ1 = β0τ 1 =

1.65T

(

294 2

This work

(continued on next page)

[64,68]

2

[15]

) ⎤⎦ } − 0.00414178T

0.93

) ⎤⎦

294 T

This work

This work

This work

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[9]

[9]

[27]

[27]

[73]

[73]

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[9]

[27]

This work

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[9]

[27]

This work

This work

[9]

[27]

Reference

586

1011.28412 + 9.6821737T RT

(

+ T ln ⎡1 − exp − ⎣

− RT ln ⎡1 + exp − ⎣

{

−5 3 −8 4 ⎧− 61920.055 − 2.8092T − 5.7 × 10 T + 4.64 × 10 T (T < 298.14) 0 ⎨ − 60000 + 11.4375T + 0.714286 0GBrho + 0.285714 G dhcp (T ≥ 298.14) Nd ⎩

−5 3 −8 4 ⎧ −68552.9251 + 3.4776T − 5 × 10 T + 3.8 × 10 T (T < 298.14) ⎪ 2 ⎨ −76000 + 143.0776T − 21.3T ln T − 0.0033T ⎫ (T ≥ 298.14) ⎬ ⎪ +380000T −1 + 9 × 10−8T3 ⎭ ⎩

−5 3 −8 4 ⎧ −49588.0502 − 0.7142T − 5.214 × 10 T + 4.4 × 10 T (T < 298.14) ⎪ 0 0 −3 2 G B6 Nd = 457.1429 + 0.5714T + GBrho LBB:6BNd , Nd = 1585.7143 − 1.5714T ⎨ −54714.2857 + 103.4286T − 16.0714T ln T − 5.07 × 10 T ⎫ (T ≥ 298.14) B : B ⎬ ⎪ +235714.286T −1 + 1.2143 × 10−7T3 ⎭ ⎩

−5 3 −9 4 ⎧− 6365.2463 − 6.4006T − 1.7612 × 10 T − 5.97 × 10 T (T < 298.14) 0 dhcp 0 ⎨ − 7000 + 0.876T + 0.985075 GBrho + 0.014925 GNd (T ≥ 298.14) ⎩

Gmτ1 = −14354.3836 − 2.8534T + 3R

GmB5 Nd2 =

GmB4 Nd =

Nd GBB:6Nd =

GmB66 Nd =

2 0 liq 1 LB, Fe = −133438 + 33.946T LBliq, Fe = 7771 LBliq, Fe = 29739 1 liq 0 liq 2 liq LFe, Nd = −4103.5 + 7.7T LFe , Nd = − 1883.84 + 6.8T LFe, Nd = 153.74 + 1 liq 0 liq 2 LB, Nd = −50000 + 2.0295T LB, Nd = −12000 + 1.4369T LBliq, Nd = −5000 1 0 liq 2 LB, Fe, Nd = −75000 LBliq, Fe, Nd = −15000 LBliq, Fe, Nd = 50000 0 bcc LB, Fe = −47920 + 42.089T 0 bcc LFe, Nd = 72000 0 bcc LB, Nd = −3900 − 6.1186339T 0 bcc LB, Fe, Nd = 90000 + 5T 0 fcc LB, Fe = −62951 + 49.904T 0 dhcp 0 fcc fcc LFe, Nd = 50000 GNd = 20000 + GNd 0 fcc LB, Nd = 75000 0 fcc LB, Fe, Nd = 90000 0 dhcp GNd cited from SGTE database 0 rho GB cited from SGTE database 0 0 bcc GmFe2 B = −26261 + 3.466T + 0.333333 GBbeta + 0.666667 GFe 0 0 bcc GmFeB = −35287 + 5.992T + 0.5 GBbeta + 0.5 GFe 0 dhcp 0 bcc Fe17 Nd2 Gm = −8684.2105 + 4.7127T + 0.894737 GFe + 0.105263 GNd Fe17 Nd2 Fe17 Nd2 TC = 327⋅β0 = 0.0649 0 dhcp 0 bcc GmFe17 Nd5 = −7500 + 3.8131T + 0.772727 GFe + 0.227273 GNd TCFe17 Nd5 = 503 β0Fe17 Nd5 = 0.0558

Liquid

bcc (α-Fe,δ-Fe,β-Nd)

Thermodynamic parameters

Phases

Table 1 Thermodynamic parameters of the Nd-Fe-B ternary system.

T.L. Chen, et al.

Calphad 66 (2019) 101627

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[15]

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[15]

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10−5T3

⎧ −43751.6451 − 13.2692T − 9.03846154 × (T < 298.14) ⎪ +9.06153846 × 10−8T 4 ⎪ τ3 0 rho Gm = −42512.3285 + 7.9552T + 0.461539 GB ⎨ (T ≥ 298.14) ⎪ 0 dhcp 0 bcc ⎪ +0.153846 GFe + 0.384615 GNd ⎩ TCτ3 = 64 β0τ 3 = 0.5

temperature range between 298 and 2300 K. The heat capacity of NdB4 compound in the low temperature range (1.8–300 K) was determined experimentally by Watanuki et al. [40] by PPMS. Bolgar et al. [41] studied the heat capacities of NdB4 and NdB6 compounds from room temperature to their melting temperatures. Using direct synthesis calorimetry, Meschel and Kleppa [42,43] determined the enthalpies of formation of NdB4 and NdB6 compounds, which are slightly different from the experimental data reported by Storms [30]. Using calorimetry, Westrum et al. [44] determined the heat capacity of NdB6 compound in the low temperature range (5–350 K). Using the physical properties measurement system (PPMS), Reiffers et al. [45] determined the heat capacity of NdB6 compound at the low temperature range (2–300 K). Meschel et al. [46] reviewed the experimental enthalpies of formation of NdB4 and Nd2B5 compounds. 2.4. The Nd-Fe-B ternary system The phase equilibira of the Nd-Fe-B ternary system including the liquidus projection, isothermal sections at different temperatures and vertical sections at different composition conditions were investigated extensively. According to the published experimental information, Raghavan and Antia [47] evaluated the Nd-Fe-B ternary system. Subsequently, Raghavan [48] reviewed the isothermal sections, vertical sections and liquidus projection in the Nd-Fe-B ternary system. Based on the calculated results by Ende and Jung [22], Raghavan [49] updated the review of the Nd-Fe-B ternary system and summarized the invariant reactions. As for the stable ternary intermetallic compounds in the Nd-Fe-B ternary system, three intermetallic compounds, Nd2Fe14B (τ1), Nd1.11Fe4B4 (τ2) and Nd5Fe2B6 (τ3), were recommended by Raghavan and Antia [47] and Raghavan [48,49], which were accepted in the present work. Three compounds, Nd2Fe14B, Nd8Fe27B24 and Nd2FeB3, were treated by Che et al. [21] as the stable stoichiometric compounds in the measurement of the isothermal sections. The crystal structures of Nd2Fe14B and Nd8Fe27B24 compounds were determined by Che et al. [21]. Buschow et al. [17] suggested three intermetallic compounds, Nd2Fe14B, NdFe4B4 and Nd2FeB3, as stable intermetallic compounds. Zhang and Luo [50–52] considered Nd2Fe14B, Nd1+xFe4B4 and Nd2FeB3 as three stable intermetallic compounds. Two intermetallic compounds,

Nd5Fe2B6 (τ3)

⎧ −42312.2194 − 4.1712T − 8.7815587 × 10−5T3 (T < 298.14) ⎪ +8.7815587 × 10−8T 4 ⎪ 0 Gmτ2 = ⎨ −43755.3117 + 9.1539T + 0.439078 GBrho (T ≥ 298.14) ⎪ 0 dhcp 0 bcc ⎪ +0.439078 GFe + 0.121844 GNd ⎩ TCτ2 = 10 β0τ 2 = 0.2 Nd1.11Fe4B4 (τ2)

Phases

Table 1 (continued)

Thermodynamic parameters

Fig. 3. The calculated Nd-B binary system with the experimental data [30,32–34] and the calculations [15,22,23].

4

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Table 2 The invariant reactions in the Nd-B binary system. Invariant reactions

Type

Temperature (K)

Composition ( xBL )

Reference

L ↔ NdB6

congruent melting

L+NdB6 ↔ NdB4

peritectic

L+NdB6 ↔ NdB66

peritectic

L ↔ NdB66+β-B

eutectic

L+NdB4 ↔ Nd2B5

peritectic

L ↔ Nd2B5+β-Nd

eutectic

β-Nd ↔ Nd2B5+α-Nd

peritectic

2882.1 2879 2891.8 2883 2883 2623.1 2634 2533.0 ∼2673 2623 2422.6 2421 2423.6 2423 2348.0 2348 2348.0 2323 2273.0 2386 2273.7 ∼1623 1262.8 1281 1263.2 1283 1127.6 1128 1127.9

0.862 0.857 0.857 0.857 – 0.587 0.62 – 0.73 – 0.999 0.998 – 0.98 0.9999 – – 0.9999 0.335 0.39 – 0.65 0.013 0.004 – ∼0.008 – – –

This work [15] [22] [31] [74] This work [19] [22] [31] [74] This work [15] [22] [31] This work [15] [22] [31] This work [15] [22] [31] This work [15] [22] [31] This work [15] [22]

The liquidus projection of the Nd-Fe-B ternary system in the Fe-Ndrich part (< 50 at. % B) was studied experimentally by Matsuura et al. [56]. Considering the low purity of the starting materials in their experiments, the experimental results [56] were given relatively low weight during the present optimization, while the determined types of invariant reactions were considered. Schneider et al. [18] investigated the liquidus projection with high Fe content (> 50 at.% Fe) and the reaction scheme in the Nd-Fe-B ternary system using metallography. Henig et al. [57] studied the Fe-rich liquidus projection and determined the invariant reactions related to fcc-Fe, Fe2B, Nd2Fe14B and NdFe4B4 phases. The liquidus projection of the Nd-Fe-B ternary system determined by Matsuura et al. [56] was revised by Tsai et al. [58] based on the metallographic results. Knoch et al. [20] determined the liquidus projection of the Nd-Fe-B ternary system in the Fe-rich and Nd-rich corners. The isothermal section in the Fe-rich part (> 30 at.% Fe) at 1173 K was investigated by Buschow et al. [16] by means of metallography and XRD. Che et al. [21] studied systematically the Nd-Fe-B ternary system with less than 50 at.% B content and determined the isothermal section at 298 K. Using DTA, SEM and metallography, Zhang and Luo [50–52] investigated the isothermal section at 298 K, which is in good agreement with the experimental results [21]. Two isothermal sections at 973 and 1173 K were measured by Buschow et al. [17] using XRD, DTA and magnetic measurements. The phase relations at 973 and 1173 K are similar to the experimental results determined by Che et al. [21]. Schneider et al. [18] determined the isothermal section at 1273 K at the Fe-rich side. Using diffusion couple method, Hao and Xu [59] determined the isothermal section at 1273 K. The phase relations including fcc-Fe, Fe2B, Nd2Fe14B and Nd1+xFe4B4 (labeled as Nd2Fe7B6 in Ref. [59]) are contrary to the experimental results [18]. Landgraf et al. [19] studied the isothermal section at 1273 K, which agrees with the experimental results [18]. Recently, Fu et al. [60] determined experimentally the isothermal sections at 873 and 1073 K using XRD and SEM/EDS. The vertical sections at 6 at.% B was determined experimentally by Che et al. [21]. With differential thermal analysis (DTA) measurements,

Fig. 4. The calculated enthalpy of formation of intermetallic compounds at 298 K in the Nd-B binary system with the experimental data [30,39,43,46].

Nd2Fe14B and Nd1+xFe4B4, were formed by the peritectic reactions. Using neutron powder diffraction, Herbst et al. [53] determined firstly the crystal structure of Nd2Fe14B. Givord et al. [54] determined the accurate crystal structure information of Nd1.11Fe4B4 (marked as Nd8Fe27B24 by Che et al. [21]) in the experimental study of the crystal structure of Fe4RE1+xB4 compounds (RE=Nd, Gd) using XRD. Zhao et al. [55] studied the structure orientation of Nd1+xFe4B4 compounds using electron diffraction and XRD. 5

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Fig. 6. The calculated enthalpies of NdB4 referred to its enthalpy at 298 K (H 0 (T ) − H 0 (298) ) with the experimental data [39].

(τ1) compound around the Curie temperature range using the differential scanning calorimetry (DSC). The Curie temperature obtained by Luis et al. [64] is in good agreement with the experimental data determined by Sinnema et al. [65] and Algarabel et al. [66]. The Gibbs energy of formation of Nd2Fe14B (τ1) and Nd1.11Fe4B4 (τ2) compounds in the temperature range from 1000 to 1100 K was determined by Ji and Xi [67] using the electromotive force (EMF) method. Herbst and Hector [68] derived the enthalpy of formation of Nd2Fe14B compound to be -101 kJ/mol from the calculated total electronic energy by means of the density functional theory. 3. Thermodynamic models 3.1. Solution phases Based on the random mixing of the constituent atoms, the solution phase ϕ (ϕ =liquid, bcc-Fe, fcc-Fe, dhcp-Nd, bcc-Nd and rhombo-B) in the Nd-Fe-B ternary system is described using the substitutional solution model [69]. The molar Gibbs energy of the solution phase ϕ can be expressed as follows:

Gmϕ =

∑

x i0 Giϕ + RT

i = B, Fe, Nd ex

Fig. 5. The calculated heat capacity of (a) NdB4 and (b) NdB6 with the experimental data [39–41,44,45].

∑

x i ln x i +

ex

Gmϕ +

mag

Gmϕ

i = B, Fe, Nd j

(1)

j

Gmϕ = xB xFe ∑j = 0 LBϕ, Fe (xB − xFe ) j + xB xNd ∑j = 0 LBϕ, Nd (xB − xNd ) j j

ϕ j + xFe xNd ∑j = 0 LFe , Nd (xFe − xNd ) 0

Schneider et al. [18] investigated three vertical sections at 4 at.% B, 80 at.% Fe, 73.3 at.% Fe as well as the Fe-Nd2B vertical section. Zhang and Luo [50–52] determined two vertical sections at 5.88 at.% B and 11.76 at.% Nd. Using DTA, XRD and metallography, Tsai et al. [58] measured the Nd-Fe14B vertical section. Landgraf et al. [19] studied the vertical section at 60 at.% Nd by means of DTA. Knoch et al. [20] determined two vertical sections at 30 at.% Nd and 4 at.% B as well as the Nd73Fe27-Fe77B23 vertical section. The heat capacity of Nd2Fe14B (τ1) compound at low temperature range was determined experimentally by Fujii et al. [61] and Pique et al. [62,63]. Luis et al. [64] measured the heat capacity of Nd2Fe14B

1

2

+ xB xFe xNd (xB LBϕ, Fe, Nd + xFe LBϕ, Fe, Nd + xNd LBϕ, Fe, Nd ) mag

Gmϕ = RT ln(β0 + 1) g (τ )

(2) (3)

where xi means the mole fraction of the element i, T stands for the 0 absolute temperature in Kelvin, and R is the gas constant. Giϕ is the molar Gibbs energy of element i in structure of ϕ referred to the enthalpy of its stable state at 298.15 K and 1 bar. In this work, the molar 0 0 ϕ 0 ϕ Gibbs energies of the elements, GBϕ , GFe and GNd , are taken from the Scientific Group Thermodata Europe (SGTE) compiled by Dinsdale j ex [70]. Gmϕ is the excess molar Gibbs energy of ϕ . In Equ.(2), LBϕ, Fe , 6

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Fig. 7. The calculated liquidus projection of the Nd-Fe-B ternary system with the calculations [15,22]. j

0

j

1

2

ϕ ϕ ϕ ϕ LBϕ, Nd , LFe , Nd and LB, Fe, Nd , LB, Fe, Nd , LB, Fe, Nd , are the binary and

ternary interaction parameters, respectively.

j

LBϕ, Fe

and

taken directly from the assessed results [9,27], while 0

1 2 LBϕ, Fe, Nd , LBϕ, Fe, Nd , LBϕ, Fe, Nd ,

j

ϕ LFe , Nd j ϕ LB, Nd

GmNd5 B2 =

were and

(10)

are the parameters to be optimized in this

work, which are given as:

GmNdB66 =

j

LBϕ, Nd = aj + bj T

j

LBϕ, Fe, Nd

C2 + D2 T + E2 T 3 + F2 T 4 (T < 298.14) ⎧ 0 dhcp 0 rho ⎨ H2 + I2 T + 0.714286 GB + 0.285714 GNd (T ≥ 298.14) ⎩

(4)

C3 + D3 T + E3 T 3 + F3 T 4 (T < 298.14) ⎧ 0 dhcp ⎨ H3 + I3 T + 0.985075 0GBrho + 0.014925 GNd ( T ≥ 298.14) ⎩ (11)

= cj + dj T

(5)

Considering the composition range of B in NdB6 compound determined by Storms [30], the sublattice model (B)6(B, Nd)1 was used in this work and thus its molar Gibbs energy is given as:

mag

In Equ.(3), Gmϕ is the magnetic contribution to the molar Gibbs energy of the magnetic phase ϕ and modified by Hillert and Jarl [71]. τ is the normalized temperature, which can be expressed as: τ = T TCϕ , and

II II NdB6 II II II II 6 GmNdB6 = YNd GBNdB : Nd + YB GB : B + 0.142857RT (YB ln YB + YNd ln YNd )

TCϕ is the Curie temperature of the phase ϕ . β0 represents the Bohr magnetons. Based on the equation suggested by Hillert and Jarl [71], g (τ) is expressed as: g (τ ) = 1 −

0

II 6 + YBII YNd LBNdB : B, Nd

(12)

1 ⎡ 79τ −1 474 ⎛ 1 τ3 τ9 τ 15 ⎞ ⎤ + ⎜ − 1⎞⎟ ⎛ + + (τ ≤ 1) M⎢ 140 p 497 p 6 135 600 ⎠⎥ ⎝ ⎠⎝ ⎣ ⎦ ⎜

C4 + D4 T + E4 T 3 + F4 T 4 (T < 298.14) ⎧ 6 GBNdB : Nd = −1 2 3 ⎨ ⎩ H4 + I4 T + J4 T ln T + K 4 T + L4 T + M4 T (T ≥ 298.14) (13)

⎟

(6)

g (τ ) = −

1 ⎛ τ −5 τ −15 τ −25 ⎞ + + (τ ≤ 1) M ⎝ 10 315 1500 ⎠ ⎜

0

rho 2 6 GBNdB : B = C5 + D5 T + E5 T ln T + F5 T + GB

⎟

(7)

0

M=

518 11692 ⎛ 1 + ⎜ − 1⎞⎟ 1125 15975 ⎝ p ⎠

(8)

(14)

6 LBNdB : B, Nd = C6 + D6 T

(15)

II where YBII and YNd denote the site fractions of B and Nd in the second sublattice, respectively. On the other hand, the ternary intermetallic compounds in the NdFe-B ternary system including Nd2Fe14B (τ1), Nd1.11Fe4B4 (τ2) and Nd5Fe2B6 (τ3) are treated as stoichiometric compounds. The heat capacity of Nd2Fe14B was determined by Fujii et al. [61], Pique et al. [62,63] and Luis et al. [64], and according to Chen and Sundman [72], the molar Gibbs energy of Nd2Fe14B is expressed as:

where the parameter p is determined by the crystal structure of the phase ϕ (p=0.4 for bcc and 0.28 for other structures). 3.2. Intermetallic compounds In the Nd-B binary system, the intermetallic compounds, Nd2B5, NdB4 and NdB66, are treated as stoichiometric compounds in this work. The heat capacities of NdB4 and NdB6 were determined experimentally by Muratov et al. [39], Watanuki et al. [40], Bolgar et al. [41], Westrum et al. [44] and Reiffers et al. [45]. The molar Gibbs energies of NdB4, Nd2B5 and NdB66 are described as:

( )} − RT ln ⎡1 + exp ( − ) ⎤⎦ ⎣

Gmτ1 = C7 + D7 T + 3R

{

E7 2

E

+ T ln ⎡1 − exp − T7 ⎤ + F7 T 2 + ⎣ ⎦

mag

Gmτ1

H7 + I7 T RT

(16)

GmNdB4

C1 + D1 T + E1 T 3 + F1 T 4 (T < 298.14) =⎧ −1 2 3 ⎨ ⎩ H1 + I1 T + J1 T ln T + K1 T + L1 T + M1 T (T ≥ 298.14)

Due to the lack of heat capacity data, using Neumann-Kopp rule, the molar Gibbs energies of Nd1.11Fe4B4 (τ2) and Nd5Fe2B6 (τ3) are expressed as:

(9) 7

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Fig. 8. Calculated reaction scheme in the Nd-Fe-B ternary system. (Unit: K). mag τ2 3 4 Gm (T < 298.14) ⎧ C8 + D8 T + E8 T + F8 T + ⎪ 0 bcc + 0.121844 (T ≥ 298.14) Gmτ2 = H8 + I8 T + 0.439078 GFe ⎨ 0 0 dhcp ⎪ GNd + 0.439078 GBrho + magGmτ2 ⎩

(17)

mag τ3 3 4 Gm (T < 298.14) ⎧ C9 + D9 T + E9 T + F9 T + ⎪ 0 bcc = H9 + I9 T + 0.153846 GFe + 0.384615 (T ≥ 298.14) ⎨ 0 0 dhcp ⎪ GNd + 0.461539 GBrho + magGmτ3 ⎩

(18)

Gmτ3

4. Results and discussion Based on the experimental information in the reported literature, the thermodynamic parameters of various phases in the Nd-B binary system and the Nd-Fe-B ternary system were optimized using the PARROT module in the Thermo-calc® software package developed by Sundman et al. [73]. The PARROT module works by minimizing the square sum of the differences between the experimentally determined data and the calculated values. During the optimizing procedure, each set of the experimental data is given a certain weight according to the reliability and compatibility of the experimental data. Finally, thermodynamic parameters of all phases in the Nd-Fe-B ternary system were obtained and were summarized in Table 1. The calculated results in the following sections are compared with the experimental results.

The parameters Ci, Di, Ei, Fi, Hi, Ii, Ji, Ki, Li and Mi (i=1,2,3,4,5,6,7,8,9) are to be optimized in this work. The magnetic contributions of three ternary compounds, magGmτ1 , magGmτ2 and magGmτ3 , are expressed by Equ.(3). Parameter p for all the three phases is 0.28. According to the reported results [15,64,68], the Curie temperatures and Bohr magnetrons of Nd2Fe14B (τ1), Nd1.11Fe4B4 (τ2) and Nd5Fe2B6 (τ3) are 586 K and 0.93 μB, 10 K and 0.2 μB, 64 K and 0.5 μB, respectively.

4.1. The Nd-B binary system Fig. 3 is the calculated Nd-B binary phase diagram with the experimental data [30,32–34] and the calculations [15,22,23]. As can be 8

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Table 3 The invariant reactions in the Nd-Fe-B ternary system. Invariant reactions

Type

L+NdB6 ↔ FeB+NdB4

U1

L+NdB6 ↔ FeB+NdB66 L ↔ FeB+β-B+NdB66 L+FeB ↔ NdB4+Fe2B

U2 E1 U3

L+NdB4 ↔ τ2 L+NdB4 ↔ Fe2B+τ2

m1 U4

L+NdB4 ↔ τ2+Nd2B5 L+γ-Fe ↔ τ1 L+γ-Fe ↔ τ1+Fe17Nd2

U5 m2 U6

L+τ2+Nd2B5 ↔ τ3 L ↔ τ1+τ2

P1 m3

L+τ2 ↔ τ1+ Fe2B

U7

L ↔ τ1+γ-Fe+Fe2B

E2

L+Nd2B5 ↔ τ3+β-Nd

U8

β-Nd ↔ L+α-Nd+τ3 L+Fe17Nd2 ↔ τ1+Fe17Nd5

ME U9

L+τ3↔τ2+α-Nd

U10

L+τ2↔τ1+α-Nd

U11

L ↔ τ1+α-Nd+Fe17Nd5

E3

Temperature (K)

1900.7 1653.8 1806.7 1782.1 1672.2 1661.2 1556 1645.9 1625.8 1648.2 1550.5 1603.2 1454.1 1451.7 1448.6 1443.2 1453 1403 1450.8 1435 1439.3 1394.2 ∼1368 1383.0 1382 ∼1368 1368 1381.5 1381 ∼1378 1389 1373.5 ∼1403 1148.7 ∼1393 1127.4 1070.0 1071.1 978.2 ∼1018 975 1015.3 966.1 983 958 ∼983 982.7 ∼938 958.0 955.4

Composition (at.%)

Reference

xBL

L xNd

47.52 42.75 61.87 63.31 32.53 32 38.87 27.00 28.47 31 38.49 – 11.12 8.72 6.79 6 – – 5.72 – 9.97 15.48 ∼17 16.04 16.3 ∼20 – 15.76 16 ∼17 – 19.35 ∼19 0.65 ∼3 0.52 0.13 0.05 0.14 – 1 0.58 0.12 – 1 ∼2 0.41 7 0.05 0.02

0.09 5.00 0.00005 0 1.02 2 5.01 8.37 3.59 3 5.25 – 51.57 16.40 20.57 23 – – 24.7 – 54.41 12.04 ∼12 6.74 7.4 ∼8 – 5.83 6 ∼7 – 5.18 ∼7 92.57 ∼94 91.43 70.88 70.54 79.74 – 77 82.88 78.80 – 76 ∼75 79.91 67 78.23 78.28

This work [22] This work This work This work [15] [22] This work This work [15] [22] [52] This work This work This work [15] [18] [20] [22] [52] This work This work [55] This work [15] [18] [20] This work [15] [18] [20] [22] [56] This work [56] This work This work [22] This work [20] [15] [22] This work [20] [15] [19] [22] [56] This work [22]

eutectoid reaction, β-Nd (bcc-Nd) ↔ Nd2B5+α-Nd, agrees well with the calculated values [15,22]. Compared with the calculations [15,22,23], the calculated liquidus are acceptable and the solid solubility of B in NdB6 compound was well reproduced in this work. It should be pointed out that the melting temperatures of intermetallic compounds and the temperatures of some invariant reactions in the Nd-B binary system are very high (> 2000 K) and thus are very difficult to be determined experimentally. Further experiments are still needed to measure the Nd-B binary system. Fig. 4 shows the calculated enthalpies of formation of intermetallic compounds in the Nd-B binary system at 298 K with the experimental data [30,39,43,46]. The calculated enthalpy of formation of NdB6 agrees well the experimental data [30], while the calculated enthalpy of formation of NdB4 is much more negative than the experimental data [30,39,43,46]. On the other hand, Fig. 5 presents the calculated heat capacities of NdB4 and NdB6 compounds with the experimental data measured by Muratov et al. [39], Watanuki et al. [40], Bolgar et al.

seen, the calculated Nd-B binary phase diagram agrees well with the reported experimental phase diagram data. Especially, the calculated solid solution boundary of NdB6 reproduces well the experimental results [30]. The calculated temperatures and compositions of all invariant reactions in the Nd-B binary system with the calculated values [15,22,31,74] were shown in Table 2. The calculated melting temperature (2882.1 K) of NdB6 is in good agreement with the calculated values [15,22,31] and the data recommended in Ref. [74]. The calculated temperatures of three peritectic reactions, L+NdB6 ↔ NdB4, L +NdB6 ↔ NdB66 and L+NdB4 ↔ Nd2B5, are 2623.1, 2422.6 and 2273.0 K, respectively, which are generally consistent with the data recommended in Ref. [74] and the calculated values [15] and are slightly deviation from the calculated values [22,31]. The calculated temperatures (2348.0 and 1262.8 K) of two eutectic reactions (L ↔ NdB66+β-B and L ↔ Nd2B5+dhcp-Nd (α-Nd)) are in good accordance with the calculated values [15,22] but different with the reported data [31]. In addition, the calculated temperature (1127.6 K) of the 9

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Fig. 9. The calculated isothermal sections of the Nd-Fe-B ternary system at different temperatures with the experimental data [16–18,21,38,50,51,60]. (a) 298 K, (b) 873 K, (c) 973 K, (d) 1073 K, (e) 1173 K and (f) 1273 K.

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Fig. 11. The calculated vertical sections of (a) Fe-Nd2B and (b) Nd2Fe14B (τ1)Nd1.11Fe4B4 (τ2) with the experimental data [18,52] and the calculations [15,22,23].

4.2. The Nd-Fe-B ternary system Fig. 7 is the calculated liquidus projection of the Nd-Fe-B ternary system with the calculations [15,22]. The calculated reaction scheme related to the liquid phase in the Nd-Fe-B ternary system is shown in Fig. 8. The comparison of the invariant reactions with corresponding experimental results is given in Table 3. As can be seen, there are three saddle points, m1 (8.37 at.% Nd, 27.00 at.% B), m2 (16.40 at.% Nd, 8.72 at.% B) and m3 (12.04 at.% Nd, 15.48 at.% B), respectively, which are successively related to three reactions, L+NdB4 ↔ τ2 at 1645.9 K, L +γ-Fe ↔ τ1 at 1451.7 K and L ↔ τ1+τ2 at 1394.2 K. Two eutectic reactions, E2 (5.83 at.% Nd, 15.76 at.% B), L ↔ τ1+γ-Fe+Fe2B at 1381.5 K and E3 (78.23 at.% Nd, 0.05 at.% B), L ↔ τ1+α-Nd+Fe17Nd5 at 958.0 K, are located at the Fe-Nd side, while the eutectic reaction, E1

Fig. 10. The calculated vertical sections of (a) 4 at.% B, (b) 5.88 at.% B and (c) 6 at.% B with the experimental data [18,20,21,52] and the calculations [15,22].

[41], Westrum et al. [44] and Reiffers et al. [45]. Fig. 6 is the calculated enthalpy (H 0 (T ) − H 0 (298) ) of NdB4 compound with the experimental data [39]. As can be seen, the calculated heat capacities and enthalpies are in good agreement with the experimental data [39–41,44,45].

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Fig. 13. The calculated vertical section of (a) 30 at.% Nd and (b) 60 at.% Nd with the experimental data [19,20] and the calculations [15,22].

Fig. 12. The calculated vertical sections of (a) 73.3 at.% Fe and (b) 80 at.% Fe with the experimental data [18] and the calculations [15,22].

compound, τ2+bcc-Fe and τ1+τ2, in Fig. 9 (a) are slightly different from the experimental results [21]. It should be mentioned that the three-phase region of τ1+Fe17Nd2+Fe17Nd5 was originally determined to be that of τ1+Fe17Nd2+α-Nd because Fe17Nd5 was not found by Zhang and Luo [50,51] and was confirmed later by Landgraf et al. [19]. In addition, the composition of τ2 compound as Nd8Fe27B24 reported by Che et al. [21] and Zhang and Luo [50,51] is slightly different with the recommended composition of τ2 compound as Nd1.11Fe4B4 [47] used in this work. The calculated isothermal sections at 873 and 1073 K are compared with the experimental results determined by Fu et al. [60] as given in Fig. 9 (b) and 9(d). The only difference between the calculated and determined isothermal section at 873 K is the phase boundary of NdB6. Nonetheless, the calculated results can reproduce well the phase

(1.73 × 10−10 at. % Nd, 63.31 at.% B), L ↔ FeB+β-B+NdB66 at 1782.1 K, is very close to the Fe-B side. In addition, there are eleven peritectic reactions from U1 to U11 as shown in Table 3 and Fig. 8. It can be seen that the temperatures and compositions of many invariant reactions in the Nd-Fe-B ternary system are predicated through the calculation and thus are necessary to be confirmed further by the experiments. Fig. 9 displays the comparison of the calculated isothermal sections at different temperatures (298, 873, 973, 1073, 1173 and 1273 K) with the experimental results [16–18,21,38,50,51,60]. As shown in Fig. 9 (a), the calculated isothermal section at 298 K is consistent with the experimental results [21,50,51]. The two-phase regions related to τ2 12

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Fig. 15. The calculated heat capacity of Nd2Fe14B (τ1) with the experimental data [61–64].

results [18,20,21,52] and the calculations [15,22]. The calculated results are in good accordance with the experimental results [18,20,21,52], while the calculated liquidus around τ1 compound is slightly higher than the experimental data [21]. It is noticed that the unexpected phase transformation at around 350 K in the calculation of Ende and Jung [22] can be found in these three vertical sections in Fig. 10. The phase transformation at around 900 K at the Fe-rich corner indicate the calculated phase relations by Ende and Jung [22] can not match the experimental results of isothermal sections at 973, 1073 and 1173 K [16,17,38,60]. The similar situation was also found in the following comparison of the vertical sections. Fig. 11(a) compares the calculated Fe-Nd2B vertical section with the experimental results [18]. It is shown that the temperature of the peritectic reaction, L+γ-Fe ↔ τ1, was calculated to be 1451.7 K, which is in good agreement with the experimental data (1453 K) determined by Schneider et al. [18]. The calculated results in this work are similar with the calculations [15,22]. Fig. 11(b) is the calculated τ1-τ2 vertical section with the experimental results [52]. The calculated liquidus is obviously higher than the experimental data measured by Zhang and Luo [52]. Fig. 12 presents the calculated vertical sections of 73.3 at.% Fe and 80 at.% Fe with the experimental results measured by Schneider et al. [18]. The calculated results are in good agreement with the experimental results [18]. Fig. 13 illustrates the calculated vertical sections at the different Nd compositions (30 and 60 at.% Nd) with the experimental data determined by Knoch et al. [20] and Landgraf et al. [19] and the calculations [15,22]. The calculated vertical section of 30 at.% Nd is in good agreement with the experimental results [20] in Fig. 13(a), while the calculated liquid phase boundary of vertical section of 60 at.% Nd shows slight deviation with the experimental results [19] in Fig. 13(b). This deviation could result from the different thermodynamic parameters of liquid phase in Nd-B binary system used in this work and in Refs. [15,22]. Fig. 14(a) and (b) are the calculated vertical sections of Fe27Nd73-Fe77B23 and Nd-Fe14B with the experimental results [19,20,59], respectively. As can be seen, these two calculated vertical sections agree well with the experimental results measured by Landgraf et al. [19], Knoch et al. [20] and Tsai et al. [58]. Fig. 15 compares the calculated heat capacity of Nd2Fe14B (τ1) with the experimental data [61–64]. The calculated heat capacity is consistent well with the experimental data [61–64], and is better than that

Fig. 14. The calculated vertical sections of (a) Fe27Nd73-Fe77B23 and (b) NdFe14B with the experimental data [19,20,58] and the calculations [15,22].

relation related to NdB6 at 873 and 1073 K, although the phase region with B content lower than 50 at. % at 1073 K were only determined by Fu et al. [60]. Fig. 9 (c) shows the calculated isothermal section at 973 K with the experimental results determined by Buschow et al. [17]. As can be seen, the liquid phase appears in the Nd-rich corner because Fe-Nd alloys would melt above 960 K according to the Fe-Nd binary phase diagram [9]. As shown in Fig. 9 (e), the calculated isothermal section at 1173 K is compared with the experimental results measured by Buschow et al. [16,17]. The calculated results agree well with the experimental results [16,17,38]. Compared with the isothermal section at 1173 K, the phase relation of the isothermal section at 1273 K was changed in Fig. 9 (f) because of the solid reaction (τ1+Fe2B ↔ γ-Fe +τ2) determined by Schneider et al. [18]. Fig. 10 shows the calculated three vertical sections at the different B contents (4 at.% B, 5.88 at.% B and 6 at.% B) with the experimental

13

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reported by Ende and Jung [22].

[20] [21] [22] [23] [24]

5. Conclusions The Nd-B binary system was re-assessed using the CALPHAD method. A set of self-consistent parameters for describing various phases in the Nd-B binary system was obtained, which can reproduce well the reported experimental data including both the phase diagram and the thermodynamic properties. Combined with the previous assessments of the Fe-Nd and the Fe-B binary systems, the thermodynamic description of the Nd-Fe-B ternary system has been developed with the available experimental information on the phase diagram data and thermodynamic data. Using the present thermodynamic parameters obtained finally, the liquidus projection, vertical sections and isothermal sections were calculated, which are in good agreement with the reported experimental data.

[25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

[35]

Acknowledgements

[36]

This work was supported financially by the National Key Research and Development Program of China (2016YFB0700901), the National Basic Foundation of China (2014CB643703), the National Natural Science Foundation of China (51761008, 51461013), the Guangxi Natural Science Foundation (2016GXNSFDA380015, 2014GXNSFBA118235, 2016GXNSFGA380001), the Guangxi Project of Science and Technology (2017AD23031) and Guangxi Key Laboratory of Information Materials, Guilin University of Electronic Technology, China (171005-Z). The author (J. Wang) thanks Dr. Qing Chen, Thermo-Calc Software AB, Stockholm, Sweden, for helpful discussion.

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Appendix A. Supplementary data

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