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Thermodynamic description of the Er–Fe–Sb system
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
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Thermodynamic description of the Er–Fe–Sb system Wei Wang, Cuiping Guo, Changrong Li, Zhenmin Du ∗ Department of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China
article
abstract
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Article history: Received 1 December 2010 Received in revised form 1 April 2011 Accepted 2 April 2011 Available online 4 May 2011
The Er–Fe, Er–Sb and Er–Fe–Sb systems were optimized by means of the CALPHAD (CALculation of PHAse Diagram) technique. The solution phases, liquid, bcc, fcc, hcp and rhom, were described by the substitutional solution model. The binary compounds, Er2 Fe17 , Er6 Fe23 , ErFe3 , ErFe2 , Er5 Sb3 , α ErSb, β ErSb, ErSb2 and ternary compound Er6 FeSb2 were treated as stoichiometric compounds. The thermodynamic description of the Fe–Sb system was taken from the literature. A self-consistent thermodynamic description of the Er–Fe–Sb system was obtained. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Er–Fe system Er–Sb system Er–Fe–Sb system Phase diagram Thermodynamic properties
1. Introduction The basic family of binary semiconductor compounds forms the skutterudite structure in general formula MX3 , which belongs to the body-centered cubic space group Im3 (where M represents Co, Rh, Ir or other metal atom; X represents P, As, Sb or other pnictide atom). The crystallographic unit cell consists of eight MX3 units, with eight M atoms occupying the c sites and the 24 X atoms situated on the g sites [1,2]. This type of binary skutterudite compound usually has high Seebeck coefficient and electrical conductivity; however, its merit value Z T remains low because of its high thermal conductivity (Z T¯ = σκS T¯ , where S , σ , κ and T are Seebeck coefficient, electrical conductivity, thermal conductivity and average temperature, respectively). An effective method to lower its thermal conductivity is to get filled skutterudite by filling 2a position in the cubic lattice of skutterudite compound with metal elements. Owing to the rattling motion of many filling atoms, which are heavy-ion species especially rare earth elements, with large vibrational amplitudes, the filled skutterudites have much smaller thermal conductivity in comparison with the corresponding binary skutterudite compounds [3–8]. The filled skutterudite compounds RT4 X12 , where R is a rare earth or acticide, T is a transition metal (Fe, Ru, Os) and X is a pnictogen, are potential thermoelectric materials due to their enhanced Seebeck coefficient. The addition of rare earth elements can change the stability and crystal structure of skutterudite compounds. So 2
∗
Corresponding author. Tel.: +86 10 6233 3772; fax: +86 10 6233 3772. E-mail address: [email protected] (Z. Du).
0364-5916/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2011.04.001
the study of the phase diagram and thermodynamic properties of the related systems is very important. In the present work, the Er–Fe, the Er–Sb and the Er–Fe–Sb system were optimized by means of the CALPHAD technique on the basis of the experimental data in the literature. 2. Literature information of binary systems 2.1. Er–Fe system The entire phase diagram of the Er–Fe system was investigated separately by Meyer [9], Buschow and van der Goot [10], and Kolesnikov et al. [11]. Four compounds, ErFe2 , ErFe3 , Er6 Fe23 and Er2 Fe17 , were determined. The phase relations determined by Meyer [9] and Buschow and van der Goot [10] were in good agreement, but the invariant reaction temperatures were different. Kolesnikov et al. [11] reported the existence of the compound ErFe5 , however, Meyer [9] and Buschow and van der Goot [10] did not find it. In this work, the compound ErFe5 has not been taken into account. Meyer [9] investigated the phase diagram of the Er–Fe system by means of differential thermal analysis, microscopic and X-ray. The largest solid solubility of Er in bcc was about 2 at.%, and the temperature of the invariant reaction bcc → liq. + fcc was 1633 K. Buschow and van der Goot [10] investigated the phase relations, crystal structures, magnetic properties and lattice constants of intermetallic compounds in the Er–Fe system by means of X-ray diffraction, metallography and thermal analysis. The solubility of Er and Fe in each other is not significant, and the temperature of the invariant reaction bcc(Fe) → liq. + fcc(Fe)
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
is 1658 K. The investigations performed by Buschow and van der Goot [10] through thermal arrests are in general a slightly higher temperature than the observations performed by Meyer [9]. Buschow and van der Goot [10] carried out the investigations using sintered Al2 O3 crucibles and explained that in the liquid state the erbium contained in the melt may cause to a reduction of small amounts of the crucible materials, and the contamination of the melt causes a lowering of the melting temperatures. Since the samples in the investigations performed by Buschow and van der Goot [10] were about an order of magnitude larger than those used by Meyer [9], the reaction in the crucible material becomes less important. From this point of view, the investigations performed by Buschow and van der Goot [10] were adopted to optimize the Er–Fe system in this work. The structures of the compounds ErFe2 , ErFe3 and Er2 Fe17 were derived from the CaCu5 structure type [12,13], i.e. ErFe2 with MgCu2 -type cubic [14], ErFe3 with PuNi3 -type rhombohedral, and Er2 Fe17 with Ni17 Th2 -type hexagonal. The structure of the compound Er6 Fe23 is Mn23 Th6 -type cubic [10]. All compounds in the Er–Fe system were treated to be stoichiometric compounds in the present work. The enthalpies of formation of intermetallic compounds in the Er–Fe system were investigated by many investigators [15–18]. Norgren et al. [15] determined the enthalpies of formation of ErFe2 and ErFe3 by indirect solution calorimetry in liquid aluminum at 1100 K. Niessen et al. [16] employed the Miedema model to calculate the enthalpies of formation of ErFe5 , ErFe3 , ErFe2 , ErFe, Er2 Fe, Er3 Fe and Er5 Fe. Colinet and Pasturel [17,18] predicted the enthalpy of formation of ErFe2 using a semi-empirical model based on the energy-band theory. The heat capacity of the compound ErFe2 at 10–300 K was determined by Germano et al. [19]. Recently, Zhou et al. [20] performed a thermodynamic assessment of the Er–Fe system on the basis of the phase diagram investigated by Meyer [9], Buschow and van der Goot [10] and Kolesnikov et al. [11], enthalpies of formation of intermetallic compounds at 298 K determined by Norgren et al. [15]. However, the magnetic contribution to the Gibbs energy in the phases bcc(Fe), fcc(Fe), ErFe2 , ErFe3 , Er6 Fe23 and Er2 Fe17 was not considered. The Er–Fe system is re-optimized in this work.
293
re-optimized the Fe–Sb system and the intermetallic compound FeSb was treated by a three-sublattice model (Fe, Va)1/3 (Fe, Va)1/3 (Sb)1/3 . The calculated results using the thermodynamic parameters [32] were in good agreement with the experimental data. The intermetallic compound FeSb has a hexagonal structure of the NiAs prototype [33–35], so it is reasonable to adopt the three-sublattice model. The thermodynamic parameters obtained by Zhang [32] were adopted in the present work. The Gibbs energy per mole of formula unit FeSb is expressed by Zhang [32] as follows: ′ ′′ FeSb ′ ′′ FeSb GFeSb = y′Fe y′′Fe GFeSb m Fe:Fe:Sb + yFe yVa GFe:Va:Sb + yVa yFe GVa:Fe:Sb
′ ′ ′ ′ + y′Va y′′Va GFeSb Va:Va:Sb + 1/3RT (yFe ln yFe + yVa ln yVa ′′ ′′ ′′ ′′ E FeSb + yFe ln yFe + yVa ln yVa ) + Gm
E
(1)
′′ 0 FeSb GFeSb = y′Fe y′Va (y′′Va 0 LFeSb m Fe,Va:Va:Sb + yFe LFe:Va,Fe:Sb )
′ 0 FeSb + y′′Va y′′Fe (y′Fe 0 LFeSb Fe:Va,Fe:Sb + yVa LVa:Va,Fe:Sb )
y′∗
(2)
y′′∗
where and are the site fractions of Fe or Va in the first and the second sublattices. The parameter GFeSb ∗:∗:Sb represents the Gibbs energies of the compound FeSb when the first and the second sublattices are completely occupied by Fe or Va. 0 LFeSb Fe,Va:∗ :Sb and 0 FeSb L∗ :Fe,Va:Sb
are the interaction parameters between Fe and Va in the first and the second sublattices when the third sublattice is occupied by Sb and the other sublattice is occupied by Fe or Va. It 0 FeSb 0 FeSb is reasonable to assume 0 LFeSb Fe,Va:Va:Sb = LFe,Va:Fe:Sb and LFe:Va,Fe:Sb = 0 FeSb LVa:Va,Fe:Sb
in order to decrease the number of thermodynamic parameters and simplify the assessment procedure. 3. Experimental information of the Er–Fe–Sb system The partial isothermal section at 1170 K (Sb ≤ 50 at.%) was determined by Morozkin [36] by means of X-ray powder diffraction, local X-ray spectral analysis, and metallographic analysis. A ternary compound τ -Er6 FeSb2 with Zr6 CoAs2 -type hexagonal was found. Cai et al. [37] used X-ray powder diffraction, optical microscopy and differential thermal analysis to construct the isothermal section at 773 K, in which the ternary compound τ -Er6 FeSb2 was confirmed. On the basis of the experimental data [36,37], Raghavan [14] reviewed the Er–Fe–Sb system. 4. Thermodynamic models
2.2. Er–Sb system 4.1. Unary phases Iandelli [21] determined the compound ErSb with the NaCltype structure; Eatough and Hall [22] found the compound ErSb2 by means of high pressure and high temperature; Borzone and Fornasini [23] determined of the compound Er5 Sb3 with β−Yb5 Sb3 structure; Abdusalyamova [24] reported three compounds Er5 Sb3 , ErSb and ErSb2 . Until 2000, the first complete phase diagram of the Er–Sb system was constructed by Abdusalyamova and Rachmator [25], in which Er5 Sb3 and ErSb2 are formed by the peritectic reactions at 1913 and 923 K respectively, and ErSb melts congruently at 2313 K. However, the phase diagram in the literature [25] is misprinted. In 2002, Abdusalyamova and Rachmator [26] re-investigated the Er–Sb phase diagram. Four intermetallic compounds Er5 Sb3 , α ErSb, β ErSb and ErSb2 and the complete liquidus were determined. The enthalpies of formation of ErSb [27–29], ErSb2 [28,29] and Er5 Sb3 [29] were determined by different authors.
The Gibbs energy function for the element i (i = Er, Fe, Sb) in the phase φ (φ = liquid, bcc (α Fe or δ Fe), fcc (γ Fe), hcp(Er) or rhombohedral (Sb)) is described as follows, φ
φ
Gi (T ) = 0 Gi (T ) − HiSER (298.15 K)
= a + bT + cT ln T + dT 2 + eT 3 + fT −1 + gT 7 + hT −9 (3) where HiSER (298.15 K) is the molar enthalpy of the element i at
298.15 K in its standard element reference (SER) state, i.e. hcp for Er, bcc for Fe, and rhombohedral for Sb. The Gibbs energy of the φ element i, Gi (T ), in its SER state is denoted by GHSERi , φ
GHSERi = 0 Gi (T ) − HiSER (298.15 K).
(4)
In the present work, the Gibbs energy functions are taken from the SGTE (Scientific Group Thermodata Europe) pure elements database compiled by Dinsdale [38] and listed in Table 1.
2.3. Fe–Sb system
4.2. Solution phases
The Fe–Sb system was optimized by Pei et al. [30] and Boa et al. [31], in which the intermetallic compound FeSb was described by a two-sublattice model Fe(Fe, Sb). Recently, Zhang [32]
In the Er–Fe–Sb system, there are five solution phases, liquid, bcc, fcc, hcp and rhom, which were modeled by the substitutional solution model. Their Gibbs energies are given by:
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W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
Table 1 Thermodynamic parameters of the Er–Fe–Sb systema . Phase
Thermodynamic parameters
Reference
Er
GHSEREr = −8489.1 + 116.6989T − 28.3847T × ln(T ) + 9.9579 × 10−4 T 2 − 9.5256 × 10−7 T 3 + 9581T −1 (298 K < T < 1802 K); −445688.2 + 2233.1021T − 298.1351T × ln(T ) + 0.06595T 2 − 3.041405 × 10−6 T 3 + 1.2397 × 108 T −1 (1802 K < T < 3200 K)
[38]
G(liquid, Er) = 10893.0 + 106.4571T − 28.3847T × ln(T ) + 9.9579 × 10−4 T 2 − 9.5256 × 10−7 T 3 + 9581T −1 (298 K < T < 500 K); 17912.7 + 0.3556T − 12.0762T × ln(T ) − 0.01441T 2 + 1.3165 × 10−6 T 3 − 528122T −1 ; (500 K < T < 1802 K); +747.1 + 187.6230T − 38.7020T × ln(T ) (1802 K < T < 3200 K)
[38]
G(bcc, Er) = −3889.1 + 114.2046T − 28.3847T × ln(T ) + 9.9579 × 10−4 T 2 − 9.5256 × 10−7 T 3 + 9581T −1 (298 K < T < 1802 K); −441088.2 + 2230.6078T − 298.1351T × ln(T ) + 0.06595T 2 − 3.0414 × 10−6 T 3 + 1.2397 × 108 T −1 (1802 K < T < 3200 K)
[38]
Fe
Sb
Liquid
G(hcp, Er) = GHSEREr (298 K < T < 3200 K)
[38]
GHSERFe = 1225.7 + 124.1340T − 23.5143T × ln(T ) − 0.004398T 2 − 5.8927 × 10−8 T 3 + 77359T −1 (298 K < T < 1811 K); −25383.6 + 299.3126T − 46T × ln(T ) + 2.2960 × 1031 T −9 (1811 K < T < 6000 K)
[38]
G(liquid, Fe) = 13265.9 + 117.5756T − 23.5143T × ln(T ) − 0.004398T 2 − 5.8927 × 10−8 T 3 + 77359T −1 − 3.6752 × 10−21 T 7 (298 K < T < 1811 K); −10838.8 + 291.3020T − 46T × ln(T ) (1811 K < T < 6000 K)
[38]
G(bcc, Fe) = GHSERFe (298 K < T < 6000 K)
[38]
Tcbcc = 1043 β bcc (µB ) = 2.22 G(hcp, Fe) = −2480.08 + 136.725T − 24.6643T × ln(T ) − 0.00375752T 2 − 5.8927 × 10−8 T 3 + 77359T −1 (298 K < T < 1811 K); −29340.776 + 304.561559T − 46T × ln(T ) + 2.7885 × 1031 T −9 (1811 K < T < 6000 K)
[38]
G(fcc, Fe) = −236.7 + 132.4160T − 24.6643T × ln(T ) − 3.7575 × 10−3 T 2 − 5.8927 × 10−8 T 3 + 77359.0T −1 (298 K < T < 1811 K); −27097.3963 + 300.2526T − 46.0T × ln(T ) + 2.7885 × 1031 T −9 (1811 K < T < 6000 K) TCfcc = −201.0 β fcc (µB ) = −2.1
[38]
GHSERSb = −9242.9 + 156.1547T − 30.5131T × ln(T ) + 7.7488 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1 (298 K < T < 904 K); −11738.8 + 169.4859T − 31.3800T × ln(T ) + 1.6168 × 1027 T −9 (904 K < T < 2000 K)
[38]
G(liquid, Sb) = 10579.5 + 134.2315T − 30.5131T × ln(T ) + 7.7487 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625T −1 − 1.7485 × 10−20 T 7 (298 K < T < 904 K); 8175.4 + 147.4560T − 31.38T × ln(T ) (904 K < T < 2000 K)
[38]
G(bcc, Sb) = 10631.1 + 141.0547T − 30.5131T × ln(T ) + 7.7487 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1 (298 K < T < 904 K); 8135.2 + 154.3859T − 31.3800T ln(T ) + 1.6168 × 1027 T −9 (904 K < T < 2000 K)
[38]
G(fcc, Sb) = 10631.1 + 142.4547T − 30.5131T × ln(T ) + 7.7487 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1 (298 K < T < 904 K); 8135.2 + 155.7859T − 31.3800T ln(T ) + 1.6168 × 1027 T −9 (904 K < T < 2000 K) G(hcp, Sb) = 10631.1 + 143.1547T − 30.5131T × ln(T ) + 7.7487 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1 (298 K < T < 904 K); 8135.2 + 156.4859T − 31.3800T × ln(T ) + 1.6168 × 1027 T −9 (904 K < T < 2000 K)
[38]
GHSERSb_low = −9242.9 + 156.1547T − 30.5131T × ln(T ) + 7.7488 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1
[32]
G(rhomb, Sb) = GHSERSb
[38]
Model (Er, Fe, Sb)1 0 liq. LEr,Fe = −30954.4 + 6.1787T
This work
1 liq. LEr,Fe
= −19702.4 + 6.0693T
This work
0 liq. LEr,Sb
= −255338.9 + 27.3140T
This work
1 liq. LEr,Sb
= −9620.1
This work
2 liq. LEr,Sb
= 33504.6
This work
0 liq. LFe,Sb
= −16500 + 9.8000T
[32]
1 liq. LFe,Sb
= −9000.0 + 5.2000T
[32]
2 liq. LFe,Sb
= +5500.0
1 liq. LEr,Fe,Sb
hcp
fcc
bcc
[38]
= −130000.0
Model (Er, Fe, Sb)1 0 hcp LEr,Fe = 326856.9
[32] This work This work
0 hcp LEr,Sb
= −198550.7 + 59.0647T
This work
1 hcp LEr,Sb
= −51302.7
This work
Model (Er, Fe, Sb)1 Gfcc Er = GHSEREr + 5000.0
This work
0 fcc LEr,Fe
= 269923.1
This work
0 fcc LFe,Sb
= 3950.0 + 14.0000T
This work
1 fcc LFe,Sb
= −15000.0
This work
Model (Er, Fe, Sb)1 0 bcc LEr,Fe = 64508.4
This work
0 bcc LFe,Sb
= 3512.0 + 21.0000T
[32]
1 bcc LFe,Sb
= −29000.0
[32] (continued on next page)
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
295
Table 1 (continued) Phase
Thermodynamic parameters
Reference
rhom
Grhom = GHSERFe + 5000.0 Fe
This work
Grhom = GHSEREr + 5000.0 Er
This work
= +100000.0
This work
0 rhom LFe,Sb
Er2 Fe17
Er Fe GEr2:Fe 17
= 2GHSEREr +17GHSERFe −256293.9 + 106.9154T
This work
Er6 Fe23
GEr6:Fe 23 = 6GHSEREr +23GHSERFe −522957.4 + 187.3962T
This work
ErFe3
GEr:Fe3 = GHSEREr +3GHSERFe −75097.8 + 24.3123T
ErFe2
GEr:Fe2 = −47682.8 + 466.0492T − 88.2387T ln(T ) − 2.8553 × 10−3 T 2 − 2640 ln(T )
This work
Er5 Sb3
Er Sb GEr5:Sb 3
This work
α ErSb
GαErErSb :Sb = GHSEREr + GHSERSb − 229057.1 + 31.9665T
β ErSb
GEr:Sb = GHSEREr + GHSERSb − 193820.8 + 15.0496T
ErSb2
GEr:Sb2 = GHSEREr + 2GHSERSb − 248823.7 + 52.5233T
FeSb
Model (Fe, Va)1/3 (Fe, Va)1/3 (Sb)1/3 GFeSb Fe:Fe:Sb = 2/3GHSERFe + 1/3GHSERSb − 6950.0 + 5.3000T
[32]
2 −5 3 GFeSb T + 15000.0T −1 Fe:Va:Sb = 1/3GHSERFe + 1/3GHSERSb_Low − 7100.0 + 2.8000T + 0.3373T ln T − 0.01014T + 2.3027 × 10
[32]
GFeSb Va:Fe:Sb = 1/3GHSERFe + 1/3GHSERSb − 1590.0 − 2.1000T
[32]
GFeSb Va:Va:Sb = 1/3GHSERSb + 5522.6 + 1.3000T
[32]
Er Sb ErFe
This work
ErFe
= 5GHSEREr + 3GHSERSb − 709849.6 + 83.7812T
This work
β ErSb
This work
ErSb
This work
0 FeSb LSb:Fe,Va:Va
= 0 LFeSb Sb:Fe,Va:Fe = −6200.0 − 4.0000T
[32]
0 FeSb LSb:Fe:Va,Fe
= 0 LFeSb Sb:Va:Va,Fe = −3196.2 − 1.5573T
[32]
FeSb2
0
FeSb GFe:Sb2
Er6 FeSb2
0
GEr6:Fe:Sb2 = 6GHSEREr + GHSERFe + 2GHSERSb − 423873.8 − 37.3728T
a
= 1/3GHSERFe + 2/3GHSERSb − 12195.0 + 4.6000T
This work
In J/mol of the formula unit.
φ
φ
φ
Gφm = xEr GEr (T ) + xFe GFe (T ) + xSb GSb (T )
A Bn
where 1Gf m
+ RT (xEr ln xEr + xFe ln xFe + xSb ln xSb ) + E Gφm (5) where R is the ideal gas constant; xEr , xFe and xSb are the mole φ
fractions of the pure elements Er, Fe and Sb, respectively; E Gm is the excess Gibbs energy expressed by the Redlich–Kister–Muggianu formalism [39,40], E
[32]
Er FeSb
φ
Gm = xEr xFe
−
j φ LEr,Fe
(xEr − xFe )
j
−
j φ LEr,Sb
(xEr − xSb )j
+ xFe xSb
j φ LFe,Sb
φ
(xFe − xSb )j + xEr xFe xSb LEr,Fe,Sb
(6)
j
φ
φ
φ
where j LEr,Fe , j LEr,Sb and j LFe,Sb are the binary interaction parameters between elements Er and Fe, Er and Sb, and Fe and Sb, respectively, and their general form is, Lφ = aφ + bφ T + c φ T ln T + dφ T 2 + eφ T 3 + f φ T −1 .
(7)
In most cases the first one or two terms are used according to φ the temperature dependence of the experimental data. LEr,Fe,Sb is the ternary interaction parameter and is expressed as: φ
φ
φ
φ
LEr,Fe,Sb = xEr 0 LEr,Fe,Sb + xFe 1 LEr,Fe,Sb + xSb 2 LEr,Fe,Sb j φ LEr,Fe,Sb
φ
Cp = a + bT + cT −1 .
2 GErFe = m
j
−
unit Am Bn ; GAmm Bn is the magnetic contribution to the Gibbs energy, which will be discussed in Section 4.4. According to the experimental values, the heat capacity of ErFe2 [19], could be fit by the following formula, (10)
The Gibbs energy per mole of formula unit ErFe2 is expressed as following:
j
+ xEr xSb
is the Gibbs energy of formation per mole of formula
mg
φ
φ
(8) φ
where = aj + bj T (j = 0, 1, 2). aj and bj are the parameters to be optimized in this work.
non-mg
non-mg
2 2 GErFe + mg GErFe m m
2 GErFe = p + qT − aT ln(T ) − m
(11) 1 2
bT 2 + c ln(T )
(12)
where the parameters a, b and c in Eq. (12) are obtained from Eq. (10); the parameters p and q should be evaluated in the present ErFe work; mg Gm 2 is the magnetic contribution to the Gibbs energy, which will be discussed in Section 4.4. All the binary compounds in the ternary Er–Fe–Sb system have no significant solubility of the third component at 773 [37] and 1070 K [36], so the thermodynamic models of the binary compounds in the ternary Er–Fe–Sb system are consistent with the models in the binary Er–Fe, Er–Sb and Fe–Sb systems. The Gibbs energy of the ternary compound Er6 FeSb2 per mole of formula unit is expressed: Er6 FeSb2
Er6 FeSb2 Gm = 6GHSEREr + GHSERFe + 2GHSERSb + 1Gf
(13)
Er
where 1Gf 6 FeSb2 is the Gibbs energy of formation per mole of formula unit Er6 FeSb2 .
4.3. Stoichiometric intermetallic compounds The compounds Er2 Fe17 , Er6 Fe23 , ErFe3 and ErFe2 in the Er–Fe system and the compounds Er5 Sb3 , α ErSb, β ErSb and ErSb2 in the Er–Sb system are treated as stoichiometric compound Am Bn in the present work. The Gibbs energy per mole of formula unit Am Bn except ErFe2 is expressed as follows: A Bn
GAmm Bn = mGHSERA + nGHSERB + 1Gf m
+ mg GAmm Bn
(9)
4.4. The magnetic contribution to the Gibbs energy The magnetic contribution to the Gibbs energy scribed by mg
Gm = RT ln(β + 1)f (τ )
mg
Gm is de(14)
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W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
Fig. 1a. Calculated Er–Fe phase diagram with the experimental data [9–11].
Fig. 2. Calculated enthalpy of formation of the compounds at 298 K in the Er–Fe system and comparison with the experimental data [15], the values predicted by the Miedema model [16] and calculated by Colinet and Pasturel [17,18].
Fig. 1b. Enlarged section of Fig. 1a.
where β is a quantity related to the total magnetic entropy, which in most cases is set equal to the Bohr magnetic moment per mole of atoms; τ is defined as T /Tc , and Tc is the critical temperature for magnetic ordering, i.e. the Curie temperature (Tc ) for ferromagnetic ordering and the Néel temperature (TN ) for antiferromagnetic ordering; f (τ ) represents the polynomials obtained by Hillert and Jarl [41] based on the magnetic specific heat of iron, i.e. for τ < 1: f (τ ) = 1 −
79τ −1
A
×
1
τ3 6
140p
+
τ9 135
+
+
474 497
τ 15
1 p
φ
φ
(15)
τ −25 + (16) A 10 315 1500 518 11692 1 where A = 1125 + 15975 − 1 and p depends on the structure, p 1
τ5
+
τ −15
0.4 for bcc structure and 0.28 for the others.
φ
φ
φ
φ
β = xEr βEr + xFe βFe +
for τ > 1: f (τ ) = −
For the solid solution phase φ (φ = bcc, fcc) in the Er–Fe sysφ tem, TC and β φ are described as: φ
TC = xEr 0 TC Er + xFe 0 TC Fe + xEr xFe LTC
−1
600
Fig. 3. Calculated heat capacity of ErFe2 in the Er–Fe system and comparison with the experimental data determined by Germano et al. [19].
0
φ
0
φ
φ xEr xFe Lβ
(17) (18)
where LTC and Lβ are the magnetic interaction parameters between Er and Fe. Due to a lack of magnetic data and the very small solubility of Er in the solid solution phases [10], both the magφ φ netic interaction parameters LTC and Lβ are set to zero in present work. For the intermetallic compounds Erm Fen (i.e. Er2 Fe17 , Er6 Fe23 , Er Fe ErFe3 and ErFe2 ), TC m n and β Erm Fen are taken from Buschow and van der Goot [10] as listed in Table 2.
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
297
Fig. 4. Calculated (HT0 − H00 ) of ErFe2 in the Er–Fe system and comparison with the experimental data determined by Germano et al. [19].
Fig. 6. Calculated standard enthalpy of formation of the compounds at 298 K in the Er–Sb system and comparison with the experimental data [27–29]. The reference states are hcp for Er and rhombohedral for Sb.
Fig. 5. Calculated Er–Sb phase diagram with the experimental data [26].
Fig. 7. Phase diagram of the Fe–Sb system calculated by Zhang [32].
Table 2 Measured TC and β (µB ) values in the Er–Fe system [10]a .
on the basis of the experimental information available in the literature [14,36,37].
Phase
Er2 Fe17
Er6 Fe23
ErFe3
ErFe2
TC (K)
310 16.2
495 7.2
555 3.2
590 4.8
β (µB ) a
In SI units per mole of formula unit.
5. Assessment procedure A general rule for selection of the adjustable parameters is that only those coefficients determined by the experimental values should be adjusted [42]. The assessment is carried out by means of the optimization module PARROT of the thermodynamic software Thermo-Calc [43], which can deals with various kinds of experimental information. The thermodynamic optimization of the Er–Fe, the Er–Sb and the Er–Fe–Sb systems were carefully performed in this work. The thermodynamic parameters of the Er–Fe–Sb system are optimized
Er Fe liq. hcp 0 liq. 0 bcc LEr,Fe GEr2:Fe 17 LEr,Fe 1 LEr,Fe 0 LEr,Fe 0 Lfcc Er,Fe ErFe GEr:Fe2 in the Er–Fe–Sb system were from the
,
The parameters Er Sb GEr6:Fe 23
ErFe GEr:Fe3
,
,
,
,
,
, and thermodynamic description of the Er–Fe system optimized by this Er Sb liq. liq. liq. hcp hcp work. The parameters 0 LEr,Sb , 1 LEr,Sb , 2 LEr,Sb , 0 LEr,Sb , 1 LEr,Sb , GEr:5Sb 3 , β ErSb
2 GαErErSb :Sb , GEr:Sb and GEr:Sb in the Er–Fe–Sb system were also from the Er–Sb system optimized by this work. The parameliq. liq. liq. 1 fcc 0 bcc 1 bcc 0 rhom ters 0 LFe,Sb , 1 LFe,Sb , 2 LFe,Sb , 0 Lfcc Fe,Sb , LFe,Sb , LFe,Sb , LFe,Sb , LFe,Sb and
0
ErSb
FeSb
FeSb FeSb FeSb 0 FeSb 0 FeSb GFe:Sb2 , GFeSb Fe:Fe:Sb , GFe:Va:Sb , GVa:Fe:Sb , GVa:Va:Sb , LSb:Fe,Va:Va , LSb:Fe,Va:Fe ,
0 FeSb LSb:Fe:Va,Fe
and 0 LFeSb Sb:Va:Va,Fe in the Er–Fe–Sb system were from the Fe–Sb system optimized by Zhang [32]. For the ternary interaction liq. parameters of the liquid phase, only 1 LEr,Fe,Sb is optimized in order to avoid the miscibility gap of liquid at high temperature, and liq. 0 liq. LEr,Fe,Sb and 2 LEr,Fe,Sb are set to be 0. The Gibbs energy of the ternary Er FeSb2
compound Er6 FeSb2 Gm 6
is optimized in the present work.
298
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
Fig. 8. Calculated isothermal section of the Er–Fe–Sb system at 773 K by the present thermodynamic description with the experimental data [37]. The dotted lines are the tie-line.
Fig. 10a. Calculated projection of the liquidus surfaces in the Er–Fe–Sb system using the present thermodynamic description.
Fig. 10b. Enlarged section of Fig. 10a. Fig. 9. Calculated isothermal section of the Er–Fe–Sb system at 1170 K by the present thermodynamic description with the experimental data [36]. The dotted lines are the tie-line.
6. Results and discussions The thermodynamic description of the Er–Fe–Sb system obtained in the present work is listed in Table 1. Fig. 1a presents the calculated phase diagram of the Er–Fe system in this work. Fig. 1b shows the enlarged section of Fig. 1a. The invariant reactions of the Er–Fe system are listed in Table 3. The temperature discrepancy between calculations and experimental data for the invariant reactions liq. + fcc(Fe) → Er2 Fe17 , liq. + ErFe2 → ErFe3 , liq. + ErFe3 → Er6 Fe23 and liq. → Er6 Fe23 + Er2 Fe17 is 9 K, 7 K, 3 K, 12 K, respectively. The calculated phase equilibrium results are similar to that of Zhou et al. [20], while the magnetic contributions are considered in the present work. Fig. 2 is the calculated enthalpy of formation of the compounds in the Er–Fe system at 298 K and the comparison with the experimental data [15], the values predicted using the Miedema model [16] and the calculated results using tight banding model by Colinet and Pasturel [17,18].
Fig. 10c. Enlarged section of Fig. 10a.
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
299
Fig. 11. Predicted reaction scheme of the Er–Fe–Sb system.
The calculated heat capacity and (HT0 − H00 ) of the compound ErFe2 in comparison with the experimental data in the temperature range 10–300 K determined by Germano et al. [19] are illustrated in Figs. 3 and 4, respectively. The Cp expression as the polynomial a + bT + cT −1 (not usually T −2 ) is used to decease the discrepancy between calculations and experiments. Due to the limitation of the polynomial expression, as shown in the Figs. 3 and
4, the calculated values in very low temperature (<30 K) were not fit very well with the experiments. Fig. 5 is the calculated Er–Sb phase diagram with the experimental data. Fig. 6 illustrates the calculated standard enthalpy of formation of the compounds in the Er–Sb system at 298 K with the experimental data [27–29], and the reference states are hcp for Er and rhombohedral for Sb. The invariant reactions of the Er–Sb
300
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
Table 3 Invariant reactions of the Er–Fe system. Reaction
Present work
Experimental data [10]
T (K) bcc → liq. + fcc liq. + fcc → Er2 Fe17 liq. + ErFe2 → ErFe3 liq. + ErFe3 → Er6 Fe23 liq. → Er6 Fe23 + Er2 Fe17 liq. → hcp + ErFe2 Er2 Fe17 + fcc → bcc
1658 1619 1611 1600 1600 1189 1186
x (Fe) 0.9997 0.8947 0.7647 0.8177 0.8186 0.2806 —
0.9179 1.0000 — — — 0.0001 1.0000
T (K) 1.0000 — — — — — 1.0000
1658 1628 1618 1603 1588 1188 1185
x (Fe) 0.989 0.880 0.755 0.800 0.835 0.300 —
0.900 1.000 — — — — 1.000
0.995 — — — — — 1.000
Table 4 Invariant reactions of the Er–Sb system. Reaction
Present work
Experimental data [26]
T (K) liq. → hcp + Er5 Sb3 liq. + α ErSb → Er5 Sb3 liq. → β ErSb liq. + α ErSb → ErSb2 liq. → ErSb2 + rhom α ErSb → β ErSb
1443 1913 2313 923 893 2083
x (Sb) 0.1396 0.3005 — 0.9627 0.9728 —
0.0242 — — — — —
Table 5 Invariant reactions in the Er–Fe–Sb system. Reaction liq. + α ErSb + τ bcc + liq. + α ErSb liq. + ErFe2 + τ liq. + hcp + τ liq. + α ErSb + FeSb liq. + α ErSb → τ + Er5 Sb3 liq. + bcc → α ErSb + fcc liq. + fcc → α ErSb + Er2 Fe17 liq. + ErFe2 → ErFe3 + τ liq. + τ → α ErSb + ErFe3 liq. + ErFe3 → α ErSb + Er6 Fe23 liq. → α ErSb + Er2 Fe17 + Er6 Fe23 liq. → τ + hcp + Er5 Sb3 liq. → α ErSb + FeSb + bcc liq. + τ → ErFe2 + hcp liq. + FeSb → α ErSb + FeSb2 liq. + α ErSb → FeSb2 + ErSb2 liq. → FeSb2 + ErSb2 + rhombohedral
T (K) — — — —
1443 1913 2313 923 893 2083
x (Sb) 0.155 — — 0.976 0.994 —
> 0.01 — — — — —
— — — —
have been performed. The present optimized thermodynamic parameters can be used in various thermodynamic calculations of practical interest or in the assessment of high-order systems.
Present work Type
T (K)
C1 C2 C3 C4 C5 U1 U2 U3 U4 U5 U6 E1 E2 E3 U7 U8 U9 E4
1958 1783 1626 1533 1296 1910 1658 1618 1608 1602 1598 1598 1442 1279 1189 1013 921 890
system are listed in Table 4. Satisfactory agreement is obtained between the calculations and experiments. Fig. 7 is the calculated phase diagram of the Fe–Sb system using the thermodynamic parameters optimized by Zhang [32]. Figs. 8 and 9 show the calculated isothermal sections of the Er–Fe–Sb system at 773 K and 1170 K in comparison with the experimental data [37,36], respectively. These two calculated isothermal sections are in good agreement with the investigations [36,37]. Figs. 10a–10c are the predicted projection of the liquidus surfaces of the Er–Fe–Sb system according to the present thermodynamic description. Fig. 11 shows the predicted reaction scheme in the Er–Fe–Sb system based on the calculated results in the present work. Table 5 shows the invariant reactions in the Er–Fe–Sb system. 7. Conclusion A thermodynamic description of the Er–Fe–Sb system was evaluated on the basis of the experimental information. A set of self-consistent thermodynamic parameters describing the Gibbs energy of each individual phase as a function of composition and temperature was derived. The predicted projection of the liquidus surfaces and the reaction scheme of the Er–Fe–Sb system
Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 50934011 and 50971027) and the National Doctorate Fund of the State Education Ministry of China (No. 20090006120029). References [1] T. Liu, X. Tang, W. Xie, Y. Yan, Q. Zhang, J. Rare Earths 25 (2007) 739. [2] G.S. Nolas, D.T. Morelli, T.M. Tritt, Annu. Rev. Mater. Sci. 29 (1999) 89. [3] E. Alleno, D. Bérardan, C. Godart, M. Puyet, B. Lenoir, R. Lackner, E. Bauer, L. Girard, D. Ravot, Physica B 383 (2006) 103. [4] K. Yang, H. Cheng, H.H. Hng, J. Ma, J.L. Mi, X.B. Zhao, T.J. Zhu, Y.B. Zhang, J. Alloys Compd. 467 (2009) 528. [5] Y.Z. Pei, S.Q. Bai, X.Y. Zhao, W. Zhang, L.D. Chen, Solid State Sci. 10 (2008) 1422. [6] G. Rogl, A. Grytsiv, E. Bauer, P. Rogl, M. Zehetbauer, Intermetallics 18 (2010) 57. [7] K. Tanaka, D. Kikuchi, Y. Kawahito, M. Ueda, H. Aoki, K. Kuwahara, H. Sugawara, Y. Aoki, H. Sato, Physica B 403 (2008) 866. [8] K. Nouneh, A.H. Reshak, S. Auluck, I.V. Kityk, R. Viennois, S. Benet, S. Charar, J. Alloys Compd. 437 (2007) 39. [9] A. Meyer, J. Less-Common Met. 18 (1969) 41. [10] K.H.J. Buschow, A.S. van der Goot, Phys. Status Solidi 35 (1969) 515. [11] V.E. Kolesnikov, V.F. Trekhova, E.M. Savitskii, Izv. Akad. Nauk SSSR Neorg. Mater. 7 (1971) 433. [12] D.T. Cromer, A. Larson, Acta Crystallogr. 12 (1959) 855. [13] J.V. Florio, N.C. Baenzinger, R.E. Rundle, Acta Crystallogr. 9 (1956) 367. [14] V. Raghavan, J. Phase Equilib. Diff. 29 (2008) 445. [15] S. Norgren, F. Hodaj, P. Azay, C. Colinet, Metall. Mater. Trans. A 29A (1998) 1367. [16] A.K. Niessen, F.R. de Boer, R. Boom, P.F. de Châtel, W.C.M. Mattens, A.R. Miedema, CALPHAD 7 (1983) 51. [17] C. Colinet, A. Pasturel, CALPHAD 11 (1987) 335. [18] C. Colinet, A. Pasturel, CALPHAD 11 (1987) 323. [19] D.J. Germano, R.A. Butera, K.A. Gschneidner, J. Solid State Chem. 37 (1981) 383. [20] G.J. Zhou, Z.W. Liu, D.C. Zeng, Z.P. Jin, Physica B 405 (2010) 3590. [21] A. Iandelli, Accad. Naz. Lincei Estratto Rend. Cl. Sci. Fis., Mat. 37 (1964) 160. [22] L.N. Eatough, H.T. Hall, Inorg. Chem. 8 (1969) 1439. [23] G. Borzone, M.L. Fornasini, Acta Crystallogr. 46 (1990) 2456. [24] M.N. Abdusalyamova, J. Alloys Compd. 202 (1993) L15. [25] M.N. Abdusalyamova, O.I. Rachmatov, J. Alloys Compd. 299 (2000) L1. [26] M.N. Abdusalyamova, O.I. Rachmatov, Z. Naturforsch. A 57 (2002) 98. [27] J.N. Pratt, K.S. Chua, Final Report, 2027/048/RL, Department of Physical Metallurgy and Science of Materials, University of Birminghan, UK, 1970. [28] V.I. Goryacheva, Ya.I. Gerasimov, V.P. Vasil’ev, Zh. Fiz. Khim. 55 (1981) 1080. [29] G. Cacciamani, G. Borzone, N. Parodi, R. Ferro, Z. Met.kd. 87 (1996) 562. [30] B. Pei, B. Björkman, B. Sundman, B. Janson, CALPHAD 19 (1995) 1. [31] D. Boa, S. Hassam, G. Kra, K.P. Kotchi, J. Rogez, CALPHAD 32 (2008) 227.
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CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
Thermodynamic description of the Er–Fe–Sb system Wei Wang, Cuiping Guo, Changrong Li, Zhenmin Du ∗ Department of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China
article
abstract
info
Article history: Received 1 December 2010 Received in revised form 1 April 2011 Accepted 2 April 2011 Available online 4 May 2011
The Er–Fe, Er–Sb and Er–Fe–Sb systems were optimized by means of the CALPHAD (CALculation of PHAse Diagram) technique. The solution phases, liquid, bcc, fcc, hcp and rhom, were described by the substitutional solution model. The binary compounds, Er2 Fe17 , Er6 Fe23 , ErFe3 , ErFe2 , Er5 Sb3 , α ErSb, β ErSb, ErSb2 and ternary compound Er6 FeSb2 were treated as stoichiometric compounds. The thermodynamic description of the Fe–Sb system was taken from the literature. A self-consistent thermodynamic description of the Er–Fe–Sb system was obtained. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Er–Fe system Er–Sb system Er–Fe–Sb system Phase diagram Thermodynamic properties
1. Introduction The basic family of binary semiconductor compounds forms the skutterudite structure in general formula MX3 , which belongs to the body-centered cubic space group Im3 (where M represents Co, Rh, Ir or other metal atom; X represents P, As, Sb or other pnictide atom). The crystallographic unit cell consists of eight MX3 units, with eight M atoms occupying the c sites and the 24 X atoms situated on the g sites [1,2]. This type of binary skutterudite compound usually has high Seebeck coefficient and electrical conductivity; however, its merit value Z T remains low because of its high thermal conductivity (Z T¯ = σκS T¯ , where S , σ , κ and T are Seebeck coefficient, electrical conductivity, thermal conductivity and average temperature, respectively). An effective method to lower its thermal conductivity is to get filled skutterudite by filling 2a position in the cubic lattice of skutterudite compound with metal elements. Owing to the rattling motion of many filling atoms, which are heavy-ion species especially rare earth elements, with large vibrational amplitudes, the filled skutterudites have much smaller thermal conductivity in comparison with the corresponding binary skutterudite compounds [3–8]. The filled skutterudite compounds RT4 X12 , where R is a rare earth or acticide, T is a transition metal (Fe, Ru, Os) and X is a pnictogen, are potential thermoelectric materials due to their enhanced Seebeck coefficient. The addition of rare earth elements can change the stability and crystal structure of skutterudite compounds. So 2
∗
Corresponding author. Tel.: +86 10 6233 3772; fax: +86 10 6233 3772. E-mail address: [email protected] (Z. Du).
0364-5916/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2011.04.001
the study of the phase diagram and thermodynamic properties of the related systems is very important. In the present work, the Er–Fe, the Er–Sb and the Er–Fe–Sb system were optimized by means of the CALPHAD technique on the basis of the experimental data in the literature. 2. Literature information of binary systems 2.1. Er–Fe system The entire phase diagram of the Er–Fe system was investigated separately by Meyer [9], Buschow and van der Goot [10], and Kolesnikov et al. [11]. Four compounds, ErFe2 , ErFe3 , Er6 Fe23 and Er2 Fe17 , were determined. The phase relations determined by Meyer [9] and Buschow and van der Goot [10] were in good agreement, but the invariant reaction temperatures were different. Kolesnikov et al. [11] reported the existence of the compound ErFe5 , however, Meyer [9] and Buschow and van der Goot [10] did not find it. In this work, the compound ErFe5 has not been taken into account. Meyer [9] investigated the phase diagram of the Er–Fe system by means of differential thermal analysis, microscopic and X-ray. The largest solid solubility of Er in bcc was about 2 at.%, and the temperature of the invariant reaction bcc → liq. + fcc was 1633 K. Buschow and van der Goot [10] investigated the phase relations, crystal structures, magnetic properties and lattice constants of intermetallic compounds in the Er–Fe system by means of X-ray diffraction, metallography and thermal analysis. The solubility of Er and Fe in each other is not significant, and the temperature of the invariant reaction bcc(Fe) → liq. + fcc(Fe)
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
is 1658 K. The investigations performed by Buschow and van der Goot [10] through thermal arrests are in general a slightly higher temperature than the observations performed by Meyer [9]. Buschow and van der Goot [10] carried out the investigations using sintered Al2 O3 crucibles and explained that in the liquid state the erbium contained in the melt may cause to a reduction of small amounts of the crucible materials, and the contamination of the melt causes a lowering of the melting temperatures. Since the samples in the investigations performed by Buschow and van der Goot [10] were about an order of magnitude larger than those used by Meyer [9], the reaction in the crucible material becomes less important. From this point of view, the investigations performed by Buschow and van der Goot [10] were adopted to optimize the Er–Fe system in this work. The structures of the compounds ErFe2 , ErFe3 and Er2 Fe17 were derived from the CaCu5 structure type [12,13], i.e. ErFe2 with MgCu2 -type cubic [14], ErFe3 with PuNi3 -type rhombohedral, and Er2 Fe17 with Ni17 Th2 -type hexagonal. The structure of the compound Er6 Fe23 is Mn23 Th6 -type cubic [10]. All compounds in the Er–Fe system were treated to be stoichiometric compounds in the present work. The enthalpies of formation of intermetallic compounds in the Er–Fe system were investigated by many investigators [15–18]. Norgren et al. [15] determined the enthalpies of formation of ErFe2 and ErFe3 by indirect solution calorimetry in liquid aluminum at 1100 K. Niessen et al. [16] employed the Miedema model to calculate the enthalpies of formation of ErFe5 , ErFe3 , ErFe2 , ErFe, Er2 Fe, Er3 Fe and Er5 Fe. Colinet and Pasturel [17,18] predicted the enthalpy of formation of ErFe2 using a semi-empirical model based on the energy-band theory. The heat capacity of the compound ErFe2 at 10–300 K was determined by Germano et al. [19]. Recently, Zhou et al. [20] performed a thermodynamic assessment of the Er–Fe system on the basis of the phase diagram investigated by Meyer [9], Buschow and van der Goot [10] and Kolesnikov et al. [11], enthalpies of formation of intermetallic compounds at 298 K determined by Norgren et al. [15]. However, the magnetic contribution to the Gibbs energy in the phases bcc(Fe), fcc(Fe), ErFe2 , ErFe3 , Er6 Fe23 and Er2 Fe17 was not considered. The Er–Fe system is re-optimized in this work.
293
re-optimized the Fe–Sb system and the intermetallic compound FeSb was treated by a three-sublattice model (Fe, Va)1/3 (Fe, Va)1/3 (Sb)1/3 . The calculated results using the thermodynamic parameters [32] were in good agreement with the experimental data. The intermetallic compound FeSb has a hexagonal structure of the NiAs prototype [33–35], so it is reasonable to adopt the three-sublattice model. The thermodynamic parameters obtained by Zhang [32] were adopted in the present work. The Gibbs energy per mole of formula unit FeSb is expressed by Zhang [32] as follows: ′ ′′ FeSb ′ ′′ FeSb GFeSb = y′Fe y′′Fe GFeSb m Fe:Fe:Sb + yFe yVa GFe:Va:Sb + yVa yFe GVa:Fe:Sb
′ ′ ′ ′ + y′Va y′′Va GFeSb Va:Va:Sb + 1/3RT (yFe ln yFe + yVa ln yVa ′′ ′′ ′′ ′′ E FeSb + yFe ln yFe + yVa ln yVa ) + Gm
E
(1)
′′ 0 FeSb GFeSb = y′Fe y′Va (y′′Va 0 LFeSb m Fe,Va:Va:Sb + yFe LFe:Va,Fe:Sb )
′ 0 FeSb + y′′Va y′′Fe (y′Fe 0 LFeSb Fe:Va,Fe:Sb + yVa LVa:Va,Fe:Sb )
y′∗
(2)
y′′∗
where and are the site fractions of Fe or Va in the first and the second sublattices. The parameter GFeSb ∗:∗:Sb represents the Gibbs energies of the compound FeSb when the first and the second sublattices are completely occupied by Fe or Va. 0 LFeSb Fe,Va:∗ :Sb and 0 FeSb L∗ :Fe,Va:Sb
are the interaction parameters between Fe and Va in the first and the second sublattices when the third sublattice is occupied by Sb and the other sublattice is occupied by Fe or Va. It 0 FeSb 0 FeSb is reasonable to assume 0 LFeSb Fe,Va:Va:Sb = LFe,Va:Fe:Sb and LFe:Va,Fe:Sb = 0 FeSb LVa:Va,Fe:Sb
in order to decrease the number of thermodynamic parameters and simplify the assessment procedure. 3. Experimental information of the Er–Fe–Sb system The partial isothermal section at 1170 K (Sb ≤ 50 at.%) was determined by Morozkin [36] by means of X-ray powder diffraction, local X-ray spectral analysis, and metallographic analysis. A ternary compound τ -Er6 FeSb2 with Zr6 CoAs2 -type hexagonal was found. Cai et al. [37] used X-ray powder diffraction, optical microscopy and differential thermal analysis to construct the isothermal section at 773 K, in which the ternary compound τ -Er6 FeSb2 was confirmed. On the basis of the experimental data [36,37], Raghavan [14] reviewed the Er–Fe–Sb system. 4. Thermodynamic models
2.2. Er–Sb system 4.1. Unary phases Iandelli [21] determined the compound ErSb with the NaCltype structure; Eatough and Hall [22] found the compound ErSb2 by means of high pressure and high temperature; Borzone and Fornasini [23] determined of the compound Er5 Sb3 with β−Yb5 Sb3 structure; Abdusalyamova [24] reported three compounds Er5 Sb3 , ErSb and ErSb2 . Until 2000, the first complete phase diagram of the Er–Sb system was constructed by Abdusalyamova and Rachmator [25], in which Er5 Sb3 and ErSb2 are formed by the peritectic reactions at 1913 and 923 K respectively, and ErSb melts congruently at 2313 K. However, the phase diagram in the literature [25] is misprinted. In 2002, Abdusalyamova and Rachmator [26] re-investigated the Er–Sb phase diagram. Four intermetallic compounds Er5 Sb3 , α ErSb, β ErSb and ErSb2 and the complete liquidus were determined. The enthalpies of formation of ErSb [27–29], ErSb2 [28,29] and Er5 Sb3 [29] were determined by different authors.
The Gibbs energy function for the element i (i = Er, Fe, Sb) in the phase φ (φ = liquid, bcc (α Fe or δ Fe), fcc (γ Fe), hcp(Er) or rhombohedral (Sb)) is described as follows, φ
φ
Gi (T ) = 0 Gi (T ) − HiSER (298.15 K)
= a + bT + cT ln T + dT 2 + eT 3 + fT −1 + gT 7 + hT −9 (3) where HiSER (298.15 K) is the molar enthalpy of the element i at
298.15 K in its standard element reference (SER) state, i.e. hcp for Er, bcc for Fe, and rhombohedral for Sb. The Gibbs energy of the φ element i, Gi (T ), in its SER state is denoted by GHSERi , φ
GHSERi = 0 Gi (T ) − HiSER (298.15 K).
(4)
In the present work, the Gibbs energy functions are taken from the SGTE (Scientific Group Thermodata Europe) pure elements database compiled by Dinsdale [38] and listed in Table 1.
2.3. Fe–Sb system
4.2. Solution phases
The Fe–Sb system was optimized by Pei et al. [30] and Boa et al. [31], in which the intermetallic compound FeSb was described by a two-sublattice model Fe(Fe, Sb). Recently, Zhang [32]
In the Er–Fe–Sb system, there are five solution phases, liquid, bcc, fcc, hcp and rhom, which were modeled by the substitutional solution model. Their Gibbs energies are given by:
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W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
Table 1 Thermodynamic parameters of the Er–Fe–Sb systema . Phase
Thermodynamic parameters
Reference
Er
GHSEREr = −8489.1 + 116.6989T − 28.3847T × ln(T ) + 9.9579 × 10−4 T 2 − 9.5256 × 10−7 T 3 + 9581T −1 (298 K < T < 1802 K); −445688.2 + 2233.1021T − 298.1351T × ln(T ) + 0.06595T 2 − 3.041405 × 10−6 T 3 + 1.2397 × 108 T −1 (1802 K < T < 3200 K)
[38]
G(liquid, Er) = 10893.0 + 106.4571T − 28.3847T × ln(T ) + 9.9579 × 10−4 T 2 − 9.5256 × 10−7 T 3 + 9581T −1 (298 K < T < 500 K); 17912.7 + 0.3556T − 12.0762T × ln(T ) − 0.01441T 2 + 1.3165 × 10−6 T 3 − 528122T −1 ; (500 K < T < 1802 K); +747.1 + 187.6230T − 38.7020T × ln(T ) (1802 K < T < 3200 K)
[38]
G(bcc, Er) = −3889.1 + 114.2046T − 28.3847T × ln(T ) + 9.9579 × 10−4 T 2 − 9.5256 × 10−7 T 3 + 9581T −1 (298 K < T < 1802 K); −441088.2 + 2230.6078T − 298.1351T × ln(T ) + 0.06595T 2 − 3.0414 × 10−6 T 3 + 1.2397 × 108 T −1 (1802 K < T < 3200 K)
[38]
Fe
Sb
Liquid
G(hcp, Er) = GHSEREr (298 K < T < 3200 K)
[38]
GHSERFe = 1225.7 + 124.1340T − 23.5143T × ln(T ) − 0.004398T 2 − 5.8927 × 10−8 T 3 + 77359T −1 (298 K < T < 1811 K); −25383.6 + 299.3126T − 46T × ln(T ) + 2.2960 × 1031 T −9 (1811 K < T < 6000 K)
[38]
G(liquid, Fe) = 13265.9 + 117.5756T − 23.5143T × ln(T ) − 0.004398T 2 − 5.8927 × 10−8 T 3 + 77359T −1 − 3.6752 × 10−21 T 7 (298 K < T < 1811 K); −10838.8 + 291.3020T − 46T × ln(T ) (1811 K < T < 6000 K)
[38]
G(bcc, Fe) = GHSERFe (298 K < T < 6000 K)
[38]
Tcbcc = 1043 β bcc (µB ) = 2.22 G(hcp, Fe) = −2480.08 + 136.725T − 24.6643T × ln(T ) − 0.00375752T 2 − 5.8927 × 10−8 T 3 + 77359T −1 (298 K < T < 1811 K); −29340.776 + 304.561559T − 46T × ln(T ) + 2.7885 × 1031 T −9 (1811 K < T < 6000 K)
[38]
G(fcc, Fe) = −236.7 + 132.4160T − 24.6643T × ln(T ) − 3.7575 × 10−3 T 2 − 5.8927 × 10−8 T 3 + 77359.0T −1 (298 K < T < 1811 K); −27097.3963 + 300.2526T − 46.0T × ln(T ) + 2.7885 × 1031 T −9 (1811 K < T < 6000 K) TCfcc = −201.0 β fcc (µB ) = −2.1
[38]
GHSERSb = −9242.9 + 156.1547T − 30.5131T × ln(T ) + 7.7488 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1 (298 K < T < 904 K); −11738.8 + 169.4859T − 31.3800T × ln(T ) + 1.6168 × 1027 T −9 (904 K < T < 2000 K)
[38]
G(liquid, Sb) = 10579.5 + 134.2315T − 30.5131T × ln(T ) + 7.7487 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625T −1 − 1.7485 × 10−20 T 7 (298 K < T < 904 K); 8175.4 + 147.4560T − 31.38T × ln(T ) (904 K < T < 2000 K)
[38]
G(bcc, Sb) = 10631.1 + 141.0547T − 30.5131T × ln(T ) + 7.7487 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1 (298 K < T < 904 K); 8135.2 + 154.3859T − 31.3800T ln(T ) + 1.6168 × 1027 T −9 (904 K < T < 2000 K)
[38]
G(fcc, Sb) = 10631.1 + 142.4547T − 30.5131T × ln(T ) + 7.7487 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1 (298 K < T < 904 K); 8135.2 + 155.7859T − 31.3800T ln(T ) + 1.6168 × 1027 T −9 (904 K < T < 2000 K) G(hcp, Sb) = 10631.1 + 143.1547T − 30.5131T × ln(T ) + 7.7487 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1 (298 K < T < 904 K); 8135.2 + 156.4859T − 31.3800T × ln(T ) + 1.6168 × 1027 T −9 (904 K < T < 2000 K)
[38]
GHSERSb_low = −9242.9 + 156.1547T − 30.5131T × ln(T ) + 7.7488 × 10−3 T 2 − 3.0034 × 10−6 T 3 + 100625.0T −1
[32]
G(rhomb, Sb) = GHSERSb
[38]
Model (Er, Fe, Sb)1 0 liq. LEr,Fe = −30954.4 + 6.1787T
This work
1 liq. LEr,Fe
= −19702.4 + 6.0693T
This work
0 liq. LEr,Sb
= −255338.9 + 27.3140T
This work
1 liq. LEr,Sb
= −9620.1
This work
2 liq. LEr,Sb
= 33504.6
This work
0 liq. LFe,Sb
= −16500 + 9.8000T
[32]
1 liq. LFe,Sb
= −9000.0 + 5.2000T
[32]
2 liq. LFe,Sb
= +5500.0
1 liq. LEr,Fe,Sb
hcp
fcc
bcc
[38]
= −130000.0
Model (Er, Fe, Sb)1 0 hcp LEr,Fe = 326856.9
[32] This work This work
0 hcp LEr,Sb
= −198550.7 + 59.0647T
This work
1 hcp LEr,Sb
= −51302.7
This work
Model (Er, Fe, Sb)1 Gfcc Er = GHSEREr + 5000.0
This work
0 fcc LEr,Fe
= 269923.1
This work
0 fcc LFe,Sb
= 3950.0 + 14.0000T
This work
1 fcc LFe,Sb
= −15000.0
This work
Model (Er, Fe, Sb)1 0 bcc LEr,Fe = 64508.4
This work
0 bcc LFe,Sb
= 3512.0 + 21.0000T
[32]
1 bcc LFe,Sb
= −29000.0
[32] (continued on next page)
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
295
Table 1 (continued) Phase
Thermodynamic parameters
Reference
rhom
Grhom = GHSERFe + 5000.0 Fe
This work
Grhom = GHSEREr + 5000.0 Er
This work
= +100000.0
This work
0 rhom LFe,Sb
Er2 Fe17
Er Fe GEr2:Fe 17
= 2GHSEREr +17GHSERFe −256293.9 + 106.9154T
This work
Er6 Fe23
GEr6:Fe 23 = 6GHSEREr +23GHSERFe −522957.4 + 187.3962T
This work
ErFe3
GEr:Fe3 = GHSEREr +3GHSERFe −75097.8 + 24.3123T
ErFe2
GEr:Fe2 = −47682.8 + 466.0492T − 88.2387T ln(T ) − 2.8553 × 10−3 T 2 − 2640 ln(T )
This work
Er5 Sb3
Er Sb GEr5:Sb 3
This work
α ErSb
GαErErSb :Sb = GHSEREr + GHSERSb − 229057.1 + 31.9665T
β ErSb
GEr:Sb = GHSEREr + GHSERSb − 193820.8 + 15.0496T
ErSb2
GEr:Sb2 = GHSEREr + 2GHSERSb − 248823.7 + 52.5233T
FeSb
Model (Fe, Va)1/3 (Fe, Va)1/3 (Sb)1/3 GFeSb Fe:Fe:Sb = 2/3GHSERFe + 1/3GHSERSb − 6950.0 + 5.3000T
[32]
2 −5 3 GFeSb T + 15000.0T −1 Fe:Va:Sb = 1/3GHSERFe + 1/3GHSERSb_Low − 7100.0 + 2.8000T + 0.3373T ln T − 0.01014T + 2.3027 × 10
[32]
GFeSb Va:Fe:Sb = 1/3GHSERFe + 1/3GHSERSb − 1590.0 − 2.1000T
[32]
GFeSb Va:Va:Sb = 1/3GHSERSb + 5522.6 + 1.3000T
[32]
Er Sb ErFe
This work
ErFe
= 5GHSEREr + 3GHSERSb − 709849.6 + 83.7812T
This work
β ErSb
This work
ErSb
This work
0 FeSb LSb:Fe,Va:Va
= 0 LFeSb Sb:Fe,Va:Fe = −6200.0 − 4.0000T
[32]
0 FeSb LSb:Fe:Va,Fe
= 0 LFeSb Sb:Va:Va,Fe = −3196.2 − 1.5573T
[32]
FeSb2
0
FeSb GFe:Sb2
Er6 FeSb2
0
GEr6:Fe:Sb2 = 6GHSEREr + GHSERFe + 2GHSERSb − 423873.8 − 37.3728T
a
= 1/3GHSERFe + 2/3GHSERSb − 12195.0 + 4.6000T
This work
In J/mol of the formula unit.
φ
φ
φ
Gφm = xEr GEr (T ) + xFe GFe (T ) + xSb GSb (T )
A Bn
where 1Gf m
+ RT (xEr ln xEr + xFe ln xFe + xSb ln xSb ) + E Gφm (5) where R is the ideal gas constant; xEr , xFe and xSb are the mole φ
fractions of the pure elements Er, Fe and Sb, respectively; E Gm is the excess Gibbs energy expressed by the Redlich–Kister–Muggianu formalism [39,40], E
[32]
Er FeSb
φ
Gm = xEr xFe
−
j φ LEr,Fe
(xEr − xFe )
j
−
j φ LEr,Sb
(xEr − xSb )j
+ xFe xSb
j φ LFe,Sb
φ
(xFe − xSb )j + xEr xFe xSb LEr,Fe,Sb
(6)
j
φ
φ
φ
where j LEr,Fe , j LEr,Sb and j LFe,Sb are the binary interaction parameters between elements Er and Fe, Er and Sb, and Fe and Sb, respectively, and their general form is, Lφ = aφ + bφ T + c φ T ln T + dφ T 2 + eφ T 3 + f φ T −1 .
(7)
In most cases the first one or two terms are used according to φ the temperature dependence of the experimental data. LEr,Fe,Sb is the ternary interaction parameter and is expressed as: φ
φ
φ
φ
LEr,Fe,Sb = xEr 0 LEr,Fe,Sb + xFe 1 LEr,Fe,Sb + xSb 2 LEr,Fe,Sb j φ LEr,Fe,Sb
φ
Cp = a + bT + cT −1 .
2 GErFe = m
j
−
unit Am Bn ; GAmm Bn is the magnetic contribution to the Gibbs energy, which will be discussed in Section 4.4. According to the experimental values, the heat capacity of ErFe2 [19], could be fit by the following formula, (10)
The Gibbs energy per mole of formula unit ErFe2 is expressed as following:
j
+ xEr xSb
is the Gibbs energy of formation per mole of formula
mg
φ
φ
(8) φ
where = aj + bj T (j = 0, 1, 2). aj and bj are the parameters to be optimized in this work.
non-mg
non-mg
2 2 GErFe + mg GErFe m m
2 GErFe = p + qT − aT ln(T ) − m
(11) 1 2
bT 2 + c ln(T )
(12)
where the parameters a, b and c in Eq. (12) are obtained from Eq. (10); the parameters p and q should be evaluated in the present ErFe work; mg Gm 2 is the magnetic contribution to the Gibbs energy, which will be discussed in Section 4.4. All the binary compounds in the ternary Er–Fe–Sb system have no significant solubility of the third component at 773 [37] and 1070 K [36], so the thermodynamic models of the binary compounds in the ternary Er–Fe–Sb system are consistent with the models in the binary Er–Fe, Er–Sb and Fe–Sb systems. The Gibbs energy of the ternary compound Er6 FeSb2 per mole of formula unit is expressed: Er6 FeSb2
Er6 FeSb2 Gm = 6GHSEREr + GHSERFe + 2GHSERSb + 1Gf
(13)
Er
where 1Gf 6 FeSb2 is the Gibbs energy of formation per mole of formula unit Er6 FeSb2 .
4.3. Stoichiometric intermetallic compounds The compounds Er2 Fe17 , Er6 Fe23 , ErFe3 and ErFe2 in the Er–Fe system and the compounds Er5 Sb3 , α ErSb, β ErSb and ErSb2 in the Er–Sb system are treated as stoichiometric compound Am Bn in the present work. The Gibbs energy per mole of formula unit Am Bn except ErFe2 is expressed as follows: A Bn
GAmm Bn = mGHSERA + nGHSERB + 1Gf m
+ mg GAmm Bn
(9)
4.4. The magnetic contribution to the Gibbs energy The magnetic contribution to the Gibbs energy scribed by mg
Gm = RT ln(β + 1)f (τ )
mg
Gm is de(14)
296
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
Fig. 1a. Calculated Er–Fe phase diagram with the experimental data [9–11].
Fig. 2. Calculated enthalpy of formation of the compounds at 298 K in the Er–Fe system and comparison with the experimental data [15], the values predicted by the Miedema model [16] and calculated by Colinet and Pasturel [17,18].
Fig. 1b. Enlarged section of Fig. 1a.
where β is a quantity related to the total magnetic entropy, which in most cases is set equal to the Bohr magnetic moment per mole of atoms; τ is defined as T /Tc , and Tc is the critical temperature for magnetic ordering, i.e. the Curie temperature (Tc ) for ferromagnetic ordering and the Néel temperature (TN ) for antiferromagnetic ordering; f (τ ) represents the polynomials obtained by Hillert and Jarl [41] based on the magnetic specific heat of iron, i.e. for τ < 1: f (τ ) = 1 −
79τ −1
A
×
1
τ3 6
140p
+
τ9 135
+
+
474 497
τ 15
1 p
φ
φ
(15)
τ −25 + (16) A 10 315 1500 518 11692 1 where A = 1125 + 15975 − 1 and p depends on the structure, p 1
τ5
+
τ −15
0.4 for bcc structure and 0.28 for the others.
φ
φ
φ
φ
β = xEr βEr + xFe βFe +
for τ > 1: f (τ ) = −
For the solid solution phase φ (φ = bcc, fcc) in the Er–Fe sysφ tem, TC and β φ are described as: φ
TC = xEr 0 TC Er + xFe 0 TC Fe + xEr xFe LTC
−1
600
Fig. 3. Calculated heat capacity of ErFe2 in the Er–Fe system and comparison with the experimental data determined by Germano et al. [19].
0
φ
0
φ
φ xEr xFe Lβ
(17) (18)
where LTC and Lβ are the magnetic interaction parameters between Er and Fe. Due to a lack of magnetic data and the very small solubility of Er in the solid solution phases [10], both the magφ φ netic interaction parameters LTC and Lβ are set to zero in present work. For the intermetallic compounds Erm Fen (i.e. Er2 Fe17 , Er6 Fe23 , Er Fe ErFe3 and ErFe2 ), TC m n and β Erm Fen are taken from Buschow and van der Goot [10] as listed in Table 2.
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
297
Fig. 4. Calculated (HT0 − H00 ) of ErFe2 in the Er–Fe system and comparison with the experimental data determined by Germano et al. [19].
Fig. 6. Calculated standard enthalpy of formation of the compounds at 298 K in the Er–Sb system and comparison with the experimental data [27–29]. The reference states are hcp for Er and rhombohedral for Sb.
Fig. 5. Calculated Er–Sb phase diagram with the experimental data [26].
Fig. 7. Phase diagram of the Fe–Sb system calculated by Zhang [32].
Table 2 Measured TC and β (µB ) values in the Er–Fe system [10]a .
on the basis of the experimental information available in the literature [14,36,37].
Phase
Er2 Fe17
Er6 Fe23
ErFe3
ErFe2
TC (K)
310 16.2
495 7.2
555 3.2
590 4.8
β (µB ) a
In SI units per mole of formula unit.
5. Assessment procedure A general rule for selection of the adjustable parameters is that only those coefficients determined by the experimental values should be adjusted [42]. The assessment is carried out by means of the optimization module PARROT of the thermodynamic software Thermo-Calc [43], which can deals with various kinds of experimental information. The thermodynamic optimization of the Er–Fe, the Er–Sb and the Er–Fe–Sb systems were carefully performed in this work. The thermodynamic parameters of the Er–Fe–Sb system are optimized
Er Fe liq. hcp 0 liq. 0 bcc LEr,Fe GEr2:Fe 17 LEr,Fe 1 LEr,Fe 0 LEr,Fe 0 Lfcc Er,Fe ErFe GEr:Fe2 in the Er–Fe–Sb system were from the
,
The parameters Er Sb GEr6:Fe 23
ErFe GEr:Fe3
,
,
,
,
,
, and thermodynamic description of the Er–Fe system optimized by this Er Sb liq. liq. liq. hcp hcp work. The parameters 0 LEr,Sb , 1 LEr,Sb , 2 LEr,Sb , 0 LEr,Sb , 1 LEr,Sb , GEr:5Sb 3 , β ErSb
2 GαErErSb :Sb , GEr:Sb and GEr:Sb in the Er–Fe–Sb system were also from the Er–Sb system optimized by this work. The parameliq. liq. liq. 1 fcc 0 bcc 1 bcc 0 rhom ters 0 LFe,Sb , 1 LFe,Sb , 2 LFe,Sb , 0 Lfcc Fe,Sb , LFe,Sb , LFe,Sb , LFe,Sb , LFe,Sb and
0
ErSb
FeSb
FeSb FeSb FeSb 0 FeSb 0 FeSb GFe:Sb2 , GFeSb Fe:Fe:Sb , GFe:Va:Sb , GVa:Fe:Sb , GVa:Va:Sb , LSb:Fe,Va:Va , LSb:Fe,Va:Fe ,
0 FeSb LSb:Fe:Va,Fe
and 0 LFeSb Sb:Va:Va,Fe in the Er–Fe–Sb system were from the Fe–Sb system optimized by Zhang [32]. For the ternary interaction liq. parameters of the liquid phase, only 1 LEr,Fe,Sb is optimized in order to avoid the miscibility gap of liquid at high temperature, and liq. 0 liq. LEr,Fe,Sb and 2 LEr,Fe,Sb are set to be 0. The Gibbs energy of the ternary Er FeSb2
compound Er6 FeSb2 Gm 6
is optimized in the present work.
298
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
Fig. 8. Calculated isothermal section of the Er–Fe–Sb system at 773 K by the present thermodynamic description with the experimental data [37]. The dotted lines are the tie-line.
Fig. 10a. Calculated projection of the liquidus surfaces in the Er–Fe–Sb system using the present thermodynamic description.
Fig. 10b. Enlarged section of Fig. 10a. Fig. 9. Calculated isothermal section of the Er–Fe–Sb system at 1170 K by the present thermodynamic description with the experimental data [36]. The dotted lines are the tie-line.
6. Results and discussions The thermodynamic description of the Er–Fe–Sb system obtained in the present work is listed in Table 1. Fig. 1a presents the calculated phase diagram of the Er–Fe system in this work. Fig. 1b shows the enlarged section of Fig. 1a. The invariant reactions of the Er–Fe system are listed in Table 3. The temperature discrepancy between calculations and experimental data for the invariant reactions liq. + fcc(Fe) → Er2 Fe17 , liq. + ErFe2 → ErFe3 , liq. + ErFe3 → Er6 Fe23 and liq. → Er6 Fe23 + Er2 Fe17 is 9 K, 7 K, 3 K, 12 K, respectively. The calculated phase equilibrium results are similar to that of Zhou et al. [20], while the magnetic contributions are considered in the present work. Fig. 2 is the calculated enthalpy of formation of the compounds in the Er–Fe system at 298 K and the comparison with the experimental data [15], the values predicted using the Miedema model [16] and the calculated results using tight banding model by Colinet and Pasturel [17,18].
Fig. 10c. Enlarged section of Fig. 10a.
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
299
Fig. 11. Predicted reaction scheme of the Er–Fe–Sb system.
The calculated heat capacity and (HT0 − H00 ) of the compound ErFe2 in comparison with the experimental data in the temperature range 10–300 K determined by Germano et al. [19] are illustrated in Figs. 3 and 4, respectively. The Cp expression as the polynomial a + bT + cT −1 (not usually T −2 ) is used to decease the discrepancy between calculations and experiments. Due to the limitation of the polynomial expression, as shown in the Figs. 3 and
4, the calculated values in very low temperature (<30 K) were not fit very well with the experiments. Fig. 5 is the calculated Er–Sb phase diagram with the experimental data. Fig. 6 illustrates the calculated standard enthalpy of formation of the compounds in the Er–Sb system at 298 K with the experimental data [27–29], and the reference states are hcp for Er and rhombohedral for Sb. The invariant reactions of the Er–Sb
300
W. Wang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 292–301
Table 3 Invariant reactions of the Er–Fe system. Reaction
Present work
Experimental data [10]
T (K) bcc → liq. + fcc liq. + fcc → Er2 Fe17 liq. + ErFe2 → ErFe3 liq. + ErFe3 → Er6 Fe23 liq. → Er6 Fe23 + Er2 Fe17 liq. → hcp + ErFe2 Er2 Fe17 + fcc → bcc
1658 1619 1611 1600 1600 1189 1186
x (Fe) 0.9997 0.8947 0.7647 0.8177 0.8186 0.2806 —
0.9179 1.0000 — — — 0.0001 1.0000
T (K) 1.0000 — — — — — 1.0000
1658 1628 1618 1603 1588 1188 1185
x (Fe) 0.989 0.880 0.755 0.800 0.835 0.300 —
0.900 1.000 — — — — 1.000
0.995 — — — — — 1.000
Table 4 Invariant reactions of the Er–Sb system. Reaction
Present work
Experimental data [26]
T (K) liq. → hcp + Er5 Sb3 liq. + α ErSb → Er5 Sb3 liq. → β ErSb liq. + α ErSb → ErSb2 liq. → ErSb2 + rhom α ErSb → β ErSb
1443 1913 2313 923 893 2083
x (Sb) 0.1396 0.3005 — 0.9627 0.9728 —
0.0242 — — — — —
Table 5 Invariant reactions in the Er–Fe–Sb system. Reaction liq. + α ErSb + τ bcc + liq. + α ErSb liq. + ErFe2 + τ liq. + hcp + τ liq. + α ErSb + FeSb liq. + α ErSb → τ + Er5 Sb3 liq. + bcc → α ErSb + fcc liq. + fcc → α ErSb + Er2 Fe17 liq. + ErFe2 → ErFe3 + τ liq. + τ → α ErSb + ErFe3 liq. + ErFe3 → α ErSb + Er6 Fe23 liq. → α ErSb + Er2 Fe17 + Er6 Fe23 liq. → τ + hcp + Er5 Sb3 liq. → α ErSb + FeSb + bcc liq. + τ → ErFe2 + hcp liq. + FeSb → α ErSb + FeSb2 liq. + α ErSb → FeSb2 + ErSb2 liq. → FeSb2 + ErSb2 + rhombohedral
T (K) — — — —
1443 1913 2313 923 893 2083
x (Sb) 0.155 — — 0.976 0.994 —
> 0.01 — — — — —
— — — —
have been performed. The present optimized thermodynamic parameters can be used in various thermodynamic calculations of practical interest or in the assessment of high-order systems.
Present work Type
T (K)
C1 C2 C3 C4 C5 U1 U2 U3 U4 U5 U6 E1 E2 E3 U7 U8 U9 E4
1958 1783 1626 1533 1296 1910 1658 1618 1608 1602 1598 1598 1442 1279 1189 1013 921 890
system are listed in Table 4. Satisfactory agreement is obtained between the calculations and experiments. Fig. 7 is the calculated phase diagram of the Fe–Sb system using the thermodynamic parameters optimized by Zhang [32]. Figs. 8 and 9 show the calculated isothermal sections of the Er–Fe–Sb system at 773 K and 1170 K in comparison with the experimental data [37,36], respectively. These two calculated isothermal sections are in good agreement with the investigations [36,37]. Figs. 10a–10c are the predicted projection of the liquidus surfaces of the Er–Fe–Sb system according to the present thermodynamic description. Fig. 11 shows the predicted reaction scheme in the Er–Fe–Sb system based on the calculated results in the present work. Table 5 shows the invariant reactions in the Er–Fe–Sb system. 7. Conclusion A thermodynamic description of the Er–Fe–Sb system was evaluated on the basis of the experimental information. A set of self-consistent thermodynamic parameters describing the Gibbs energy of each individual phase as a function of composition and temperature was derived. The predicted projection of the liquidus surfaces and the reaction scheme of the Er–Fe–Sb system
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