A thermodynamic description of the Fe–Co–Gd system

L

Journal of Alloys and Compounds 269 (1998) 144–150

A thermodynamic description of the Fe–Co–Gd system Xuping Su, Weijing Zhang*, Zhenmin Du Department of Materials science and Engineering, University of Science and Technology Beijing, Beijing 100083, P.R. China Received 24 November 1997; received in revised form 20 January 1998

Abstract The phase diagram and thermodynamic data of the Fe–Co–Gd system were critically assessed by means of the computer program, using models for the Gibbs energy of individual phases. Previous assessments of the three binary systems were incorporated. The system contains eight different compounds and they are all treated as line compounds. Good agreement is obtained between the calculation and experimental results in the ternary system.  1998 Elsevier Science S.A. THERMO-CALC,

Keywords: Thermodynamic properties; Fe–Co–Gd system; Phase diagram calculations; Gd Compounds

1. Introduction

2. Thermodynamic models

The class of compounds formed between rare-earths and 3d transition metals are of particular interest regarding their outstanding magnetic properties [1], and their reversible absorption of hydrogen gas at room temperature and atmospheric pressure [2]. Gd is an important rare-earth element. Addition of Gd metal will improve the physical and mechanical properties of many alloys, e.g. when a little Gd metal is added to Fe–base alloys, it will enhance the heat resistance and life-span of the alloys. Fe–Co–Gd alloys also have great potential application in the energy and electronic industries. The phase diagrams and selfconsistent thermodynamic descriptions of the alloy system are of great value to alloy design and processing. Phase diagram calculation using thermodynamic models coupled with key experimental measurements is an economic way to obtain the phase diagram. Liu [3] evaluated the thermodynamic properties of the Fe–Co–Gd ternary system from the only section at 1323 K, measured by Atiq et al [4]. Recently, Shen et al. [5] determined five isothermal sections at 1173, 1223, 1273, 1323 and 1473 K, respectively, using the diffusion couple technique. By modeling the Fe–Co–Gd system, a selfconsistent description of the phase relations and thermodynamic data was obtained by means of the CALPHAD technique.

The Gibb energy of individual phases is described by sublattice models [6] relative to the so-called ‘standard element reference’ (SER), i.e. the enthalpies of the pure elements in their defined reference phase at 298.15 K.

2.1. Liquid, fcc, bcc and hcp phases The liquid, fcc, bcc and hcp phases were described with a substitutional regular solution model for which the Gibbs energy expression is G fm 5 x 0Fe G fFe 1 x 0Co G fCo 1 x 0Gd G fGd 1 RT(x Fe ln x Fe 1 x Co ln x Co 1 x Gd ln x Gd ) 1 E G fm 1 mg G fm

(1)

where x i is the mole fraction of element i (i5Fe, Co, Gd) in ternary, and 0 G fi is the molar Gibbs energy in the structure f in non-magnetic state, taken from the work of Dinsdale [7]. E G fm is the excess Gibbs energy, expressed in Redlich–Kister polynomials as: E

OL x O L x O L

G fm 5 x Fe x Gd

i

f Fe,Gd

(x Fe 2 x Gd )i

i

1 x Co

i

f Co,Gd

i

f Fe,Co

Gd

(x Co 2 x Gd )i

i

1 x Fe

Co

(x Fe 2 x Co )i

i

*Corresponding author. 0925-8388 / 98 / $19.00  1998 Elsevier Science S.A. All rights reserved. PII S0925-8388( 98 )00218-7

f 1 x Fe x Co x Gd L Fe,Co,Gd

(2)

X. Su et al. / Journal of Alloys and Compounds 269 (1998) 144 – 150 f where i L fFe,Co , i L fCo,Gd and i L Fe,Gd are the binary interaction parameters taken from previous assessments [3,8], and L fFe,Co,Gd is the ternary interaction parameter set to be zero, as a result of a lack of experimental data. i f L i , j can be temperature dependent and two terms are usually sufficient, i.e. i

L fi, j 5 a 1 bT

(3)

For the fcc, bcc and hcp phases, there is a magnetic transformation. According to Hillert and Jarl [9], the magnetic contribution to the Gibbs energy mg G fm is described by: mg

G fm 5 RT ln( b f 1 1)f(t ), t 5 T /T cf

(4)

f c

where T is the critical (Curie or Neel) temperature and b f the magnetic moment, which is related to the total magnetic entropy by Eq. (5): Smg 5 R ln( b f 1 1)

f f tities, i.e. T cFe,Co,Gd and b Fe,Co,Gd are set to zero, due to lack of experimental data and extremely low solubilities.

2.2. Intermetallic compounds There are eight stable intermetallic compounds in the Fe–Co–Gd system. All the intermetallic compounds are treated as line compounds. No ternary compound has been reported. All the intermetallic compounds are modelled by a two-sublattice model with the transition metal elements on one sublattice and the rare-earth element on the other sublattice, i.e. MA GdB , where M is used as an abbreviation for a mixture of Fe and Co. Their Gibbs energies per mole of formula unit MA GdB can be expressed as: 0 MA Gd B A Gd B A Gd B GM 5 y 0Fe G M m Fe:Gd 1 y Co G Co:Gd

1 ART( y Fe ln y Fe 1 y Co ln y Co )

(5)

1 y Fe y Co

f(t ) is the polynomial obtained by Hillert and Jarl [9] based on the magnetic specific heat of iron, and is presented in detailed form in Table 1. T fc and b f are described by:

O x T 1O x x T 5O x b 1O x x b

T fc 5

0 i

f ci

i

bf

i j

f ci, j

f 1 x Fe x Co x Gd T cFe,Co,Gd

(6)

i

f i

f i, j

1 x Fe x Co x Gd b fFe,Co,Gd

(7)

i j

OL i

MA Gd B Fe,Co:Gd

i

( y Fe 2 y Co ) 1

mg

M Gd B

G mA

i

(8) 0

i, j

0 i

145

MA Gd B hbcc G Fe:Gd 5 A0 G Fe 1 B 0 G hhcp Gd 1 a 1 1 b 1 T 1 c 1 T ln T

(9)

i, j

where T fci and 0 b if are unary quantities, and T fci, j and b if, j are the binary quantities. The Fe–Gd and Co–Gd binary f f quantities, i.e. T cFe,Gd and b Fe,Gd , and the ternary quan-

0

MA Gd B hhcp hhcp G Co:Gd 5 A0 G Co 1 B 0 G Gd 1 a 2 1 b 2 T 1 c 2 T ln T

0

(10) MA Gd B Here, 0 G Fe:Gd represents the Gibbs energy of formation of

Table 1 ´ Optimized parameters describing the thermodynamic properties of the Co–Fe system from Fernandez Guillermet [8] The magnetic contribution to Gibbs energy is described by: mg G fm 5 RT ln( b f 1 1)f(t ), t 5 T /T c for t ,1: f(t ) 5 1 2 [79t 21 /(140p) 1 474 / 497(1 /p 2 1)(t 3 / 6 1 t 9 / 135 1 t 15 / 600)] /A for t .1: f(t ) 5 2 (t 25 / 10 1 t 215 / 315 1 t 225 / 1500) /A where A5(518 / 1125)1(11692 / 15975)[1 /p21], p depends on the structure: 0.4 for bcc and 0.28 for the others. 1 Liq Liquid: 298.15 K,T ,6000 K: 0 L Liq Co,Fe 5 29312; L Co,Fe 5 21752 fcc: 298.15 K,T ,6000 K: 0 fcc L Co,Fe 5-8471 1 fcc L Co,Fe 5118121.6544T hcp: 298.15 K,T ,6000 K: 0 hcp L Co,Fe 5 2400 bcc: 298.15,T ,1768 K: SER 23 2 G 0,bcc T 21.7348310 27 T 3 172526.9T 21 1DG mg Co 2H Co 53427.5751132.634T225.0861T ln T22.654738310 Co 1768,T ,600 K: SER 30 29 G 0,bcc Co 2H Co 5 214260.31252.570T240.5T ln T 19.3488310 T where the magnetic contribution is described as indicated above. bcc T bcc 51.35 mB c 51450 K; b The values are given in SI units per mole of formula unit.

X. Su et al. / Journal of Alloys and Compounds 269 (1998) 144 – 150

146

a compound with all sites on the first sublattice filled with Fe and all sites on the second sublattice filled by Gd. 0 MA Gd B hbcc 0 hhcp G Co:Gd has a similar meaning but for Co. 0 G Fe , G Co 0 hhcp and G Gd are the Gibbs energies of the respective pure elements in a hypothetical non-magnetic hcp or bcc structure. y Fe and y Co are the so-called site fractions and represent the mole fractions of Fe and Co in the first sublattice, respectively. The parameters a i , b i and c i were evaluated separately in the Fe–Gd and Co–Gd systems. i MA Gd B L Fe,Co:Gd is the interaction parameter. which can be temperature dependent, and two terms are usually sufficient, i.e. i

MA Gd B L Fe,Co:Gd 5 a3 1 b3T

(11)

The magnetic contribution to the Gibbs energy is given by Eq. (12): mg

mg

MA Gd B Gm 5 RT ln( b MA Gd B 1 1)f(t ); t 5 T /T cMA Gd B

A Gd B GM m

(12)

f(t ) is the polynomials obtained by Hillert and Jarl [9] based on the magnetic heat capacity of iron and is presented in detailed form in Table 1. In the binary system A Gd B TM and b MA Gd B are taken from the measured critical c temperature and the mean Bohr magnetic moment per mole of formula unit. In the ternary system, they can be represented by: MA Gd B A Gd B A Gd B TM 5 y Fe T M c cFe:Gd 1 y Co T cCo:Gd

1 y Fe y Co

OT i

MA Gd B cFe,Co:Gd

( y Fe 2 y Co )i

(13)

i

MA Gd B MA Gd B b MA Gd B 5 y Fe b Fe:Gd 1 y Co b Co:Gd

1 y Fe y Co

Ob i

MA Gd B Fe,Co:Gd

( y Fe 2 y Co )i

(14)

i

i

MA Gd B Fe,Co:Gd

i

MA Gd B cFe,Co:Gd

b and T experimental data.

and are set to zero, due to lack of

3. Binary systems

3.1. Fe–Co There are four stable phases in the system: liquid, bcc, hcp and fcc. The thermodynamic properties of the Fe–Co ´ system were assessed by Fernandez Guillermet [8]. The model for the solid phases accounts for the magnetic contribution to the Gibbs energy. An expression for the difference in Gibbs energy of the metastable bcc modification of pure cobalt has been presented. The optimized parameters are shown in Table 1. The calculated phase diagram is displayed in Fig. 1.

3.2. Fe–Gd There has been considerable discussion in the literature concerning this system. Liu et al. [3] assessed this system

Fig. 1. The Fe–Co phase diagram calculated from the thermodynamic ´ data by Fernandez Guillermet [8].

recently, based on most recent experimental data. Eight stable phases exist: liquid, bcc, hcp and fcc, and intermetallic compounds Fe 2 Gd, Fe 3 Gd, Fe 23 Gd 6 and Fe 17 Gd 2 . In Table 2, the optimized parameters are given for the equilibrium phase of the Fe–Gd system. The critical magnetic ordering temperatures and magnetic moments from Wallace and Segal [10] are listed in Table 3, which is used in the assessment by Liu et al. [3]. Calculated Fe–Gd phase diagram is shown in Fig. 2.

3.3. Co–Gd In the assessment by Liu et al. [3], 11 phases exist: liquid, bcc, hcp and fcc, and compounds CoGd 3 , Co 3 Gd 4 , Co 2 Gd, Co 3 Gd, Co 7 Gd 2 , Co 5 Gd and Co 17 Gd 2 . Compound Co 5 Gd is unstable at low temperatures. Both Co 5 Gd and Co 17 Gd 2 were reported to have a certain homogeneity region at high temperatures [11,12], but they are treated as stoichiometric compounds, because of limited information. The optimized set of model parameters is listed in Table 4, and the critical magnetic ordering temperatures and magnetic moments from Burzo [13] are listed in Table 5, which is used in the assessment by Liu et al. [3]. The computed Co–Gd phase diagram is shown in Fig. 3.

4. Fe–Co–Gd ternary system

4.1. Assessment procedure Liu evaluated the thermodynamic properties of the Fe– Co–Gd ternary system from the only one section at 1323 K [3], measured by Atiq et al. [4]. Recently, Shen et al. [5] determined five isothermal sections at 1173, 1223, 1273, 1323 and 1473 K, respectively, by the diffusion couple

X. Su et al. / Journal of Alloys and Compounds 269 (1998) 144 – 150

147

Table 2 Optimized parameters describing the thermodynamic properties of the Fe–Gd system from Liu et al. [3] 0

L Liq Fe,Gd 5 227 625117.869T L Liq Fe,Gd 514 59428.894T 0 bcc L Fe,Gd 5 228 758138.096T 0 fcc L Fe,Gd 530 231 0 hcp L Fe,Gd 51 000 000 0 17 Gd 2 G Fe 2 17 0 G hbcc 2 2 0 G hhcp Fe:Gd Fe Gd 5 2218 222198.884T 0 Fe 23 Gd 6 G Fe:Gd 2 23 0 G hbcc 2 6 0 G hhcp Fe Gd 5 2430 0751182.333T 0 Fe 3 Gd G Fe:Gd 2 3 0 G hbcc 2 0 G hhcp Fe Gd 5 261 393126.214T 0 Fe 2 Gd 21 G Fe:Gd 2 2 0 G hbcc 2 0 G hhcp 29 446 3542T 23 Fe Gd 5 246 829163.295T25.917T ln T2353 121T

Liquid

1

bcc fcc hcp Fe 17 Gd 2 Fe 23 Gd 6 Fe 3 Gd Fe 2 Gd

The values are given in SI units per mole of formula unit.

Table 3 Measured T c (K) and b ( mB ) values of the compounds in the Fe–Gd system Phases

Tc b

Fe 2 Gd

Fe 3 Gd

Fe 23 Gd 6

Fe 17 Gd 2

782 3.35

728 1.6

468 14.8

472 21.2

The values are given in SI units per mole of formula unit. The magnetic of compound MA GdB contribution to Gibbs energy is described by: mg A Gd B GM 5RT ln( b 11)f(t ), where t 5T /T c ; f(t ) is the polynomial m presented in detailed form in Table 1; p50.28 for the compounds.

Fig. 2. The Fe–Gd phase diagram calculated from the thermodynamic data by Liu et al. [3].

technique, in which (Fe 12x Co x ) 17 Gd 2 , (Fe 12x Co x ) 3 Gd and (Fe 12x Co x ) 2 Gd show complete mutual solid solubility with x ranging from zero to unity; and Fe 23 Gd 6 ,Co 5 Gd and Co 7 Gd 2 have limited, but significant, solid solubilities for cobalt or iron, respectively; the tie-triangles and tie-lines were established. Poldy and Taylor [14] mentioned that the solubility limit of Fe in (Co 12y ,Fe y )Gd 3 is y50.1. They did not identify which other phases appear upon further addition of iron at 700 K. With the structural stability rule [14,15], Liu et al. [3] suggested that the substitution limit of Co by Fe in Co 3 Gd 4 is also around 0.1 at 700 K. No ternary compound has been reported. Most of the experimental data mentioned above are selected in the evaluation of thermodynamic model parameters. Since Fe 5 Gd, Fe 7 Gd 2 , Fe 3 Gd 4 , FeGd 3 and Co 23 Gd 6 are not stable as binary compounds, their magnetic properties were estimated (Table 6), and lattice stabilities were evaluated. The optimization was carried out by using a computer program THERMO-CALC [16]. The phase diagram data were used as input to the program. All the data were first critically reviewed and selected. Each piece of selected information was given a certain weight by personal judgment, and changed by trial and error during the assessment, until most of the selected experimental information is reproduced within the expected uncertainty limits.

Table 4 Optimized parameters describing the thermodynamic properties of the Co–Gd system (from Liu et al. [3]) Liquid

0

Liq L Co,Gd 5 2136 3881324.444T234.273T ln T Liq L Co,Gd 5 29689 0 bcc L Co,Gd 51 000 000 0 fcc L Co,Gd 51 000 000 0 hcp L Co,Gd 51 000 000 0 0 hhcp 0 hhcp 3 G CoGd Co,Gd 2 G Co 2 3 G Gd 5 250 065162.515T26.4326T ln T 0 0 hhcp 0 hhcp 3 Gd 4 G Co 2 3 G 2 4 G Co,Gd Co Gd 5 21264321123.394T211.257TlnT 0 Co 2 Gd 0 hhcp 0 hhcp G Co,Gd 2 2 G Co 2 G Gd 5 265 853118.864T 0 0 hhcp 0 hhcp 3 Gd G Co Co,Gd 2 3 G Co 2 G Gd 5 294 127130.262T 0 Co 7 Gd 2 0 hhcp 0 hhcp G Co,Gd 2 7 G Co 2 2 G Gd 5 2202 973165.526T 0 0 hhcp 0 hhcp 21 5 Gd G Co 2 5 G 2 G Co,Gd Co Gd 5 2108 678137.548T20.74263T ln T2424 406T 0 Co 17 Gd 2 0 hhcp 0 hhcp G Co,Gd 2 17 G Co 2 2 G Gd 5 2282 208188.842T 1

bcc fcc hcp CoGd 3 Co 3 Gd 4 Co 2 Gd Co 3 Gd Co 7 Gd 2 Co 5 Gd Co 17 Gd 2 The values are given in SI units per mole of formula unit.

X. Su et al. / Journal of Alloys and Compounds 269 (1998) 144 – 150

148

Table 5 Measured T c (K) and b ( mB ) values of the compounds in the Co–Gd system Phases CoGd 3 Tc b

a

130 21.5

Co 3 Gd 4

Co 2 Gd

Co 3 Gd

Co 7 Gd 2

Co 5 Gd

Co 17 Gd 2

233 26.0

395 4.95

611 2.3

767 2.6

1020 1.4

1222 14.1

The values are given in SI units per mole of formula unit. a CoGd 3 is antiferromagnetic and its T N and b values need to be multiplied by 23 when put into THERMO-CALC program; p50.28.

4.2. Results The thermodynamic description of the Fe–Co–Gd system obtained in the present work is listed in Table 7. In Figs. 4–9, the calculated isothermal sections at 1173, 1223, 1273, 1323, 1473 and 700 K are shown in comparison with experimental data [4,5], respectively. The calculated solubility limits of M 17 Gd 2 , M 7 Gd 2 and M 23 Gd 6 are in good agreement with the experiments. The calculated tie-lines

Fig. 3. The Co–Gd phase diagram calculated from the thermodynamic data by Liu et al. [3].

Table 6 Estimated T c (K) and b ( mB ) values of the compounds in the Fe–Gd and Co–Gd system [3] Phases

Tc b

Fe 5 Gd

Fe 7 Gd 2

Fe 3 Gd 4

FeGd 3

Co 23 Gd 6

470 4.86

560 4.23

681 10.57

476 9.16

836 7.97

The values are given in SI units per mole of formula unit.

The optimization was carried out by steps. All the parameters were finally evaluated together to give the best description of the system.

Fig. 4. The isothermal section of the Fe–Co–Gd system at 1173 K with experimental data [5].

Table 7 Optimized parameters describing the thermodynamic properties of the Fe–Co–Gd system M 17 Gd 2 M 5 Gd M 23 Gd 6 M 7 Gd 2 M 3 Gd M 2 Gd M 3 Gd 4 MGd 3 The values are given in SI units per mole of formula unit.

0

17 Gd 2 LM Fe,Co:Gd 5 290 822287.431T Fe 5 Gd 17 Gd 2 23 Gd 6 G Fe:Gd 5 0.125 0 G Co 1 0.125 0 G Fe 1 53 944 Fe:Gd Fe:Gd 0 M 5 Gd 2 L Fe,Co:Gd 5 2128 31824.262T 0 Co 7 Gd 2 Co 5 Gd 23 Gd 6 G Co 5 2.3333 0 G Co:Gd 1 1.3333 0 G Co:Gd 1 4382 Co:Gd 0 M 23 Gd 6 L Fe,Co:Gd 5 2187 801 0 0 Fe 23 Gd 6 Fe 3 Gd 7 Gd 2 G Fe 1 0.8 0 G Fe:Gd 1 79 111 Fe:Gd 5 0.2 G Fe:Gd 0 M 7 Gd 2 L Fe,Co:Gd 5 2177 786233.873T 0 M 3 Gd L Fe,Co:Gd 5 254 506 0 M 2 Gd L Fe,Co:Gd 5 229 771 0 Fe 3 Gd 4 0 hhcp 2 Gd G Fe:Gd 5 1.5 0 G Fe Fe:Gd 1 2.5 G Gd 1 43 553 0 M 3 Gd 4 L Fe,Co:Gd 5 270 871 0 0 Fe 2 Gd 0 hhcp 3 G FeGd Fe:Gd 5 0.5 G Fe:Gd 1 2.5 G Gd 1 7421 0

X. Su et al. / Journal of Alloys and Compounds 269 (1998) 144 – 150

Fig. 5. The isothermal section of the Fe–Co–Gd system at 1223 K with experimental data [5].

149

Fig. 8. The isothermal section of the Fe–Co–Gd system at 1473 K with experimental data [5].

Fig. 6. The isothermal section of the Fe–Co–Gd system at 1273 K with experimental data [5]. Fig. 9. The isothermal section of the Fe–Co–Gd system at 700 K with experimental data [5].

agree generally well with the measured tie-lines [5]. Fig. 10 is a vertical section of the M 17 Gd 2 compound.

5. Conclusions

Fig. 7. The isothermal section of the Fe–Co–Gd system at 1323 K with experimental data [4,5].

The thermodynamic properties of the Fe–Co–Gd system were evaluated from the experimental information available in the literature. A consistent set of thermodynamic parameters was derived. The calculated phase equilibria agree well with the literature data. With the thermodynamic description available, one can now make various calculations of practical interest. However, this description should be viewed as a preliminary one. More experimental work on this system may be necessary to improve the description.

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150

References

Fig. 10. Vertical section of M 17 Gd 2 .

Acknowledgements This work was supported by the National Natural Science Foundation of China. The authors would like to express their gratitude to Royal Institute of Technology, Sweden, for supplying the Thermo-Calc software.

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