Thermodynamic modelling of the Er–Pd system

Journal of Alloys and Compounds 299 (2000) 199–204

L

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Thermodynamic modelling of the Er–Pd system Zhenmin Du*, Haifang Yang Department of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China Received 19 October 1999; accepted 11 November 1999

Abstract By means of the CALPHAD technique, the Er–Pd system was critically assessed. The solution phases (liquid, f.c.c. and h.c.p.) are modeled with the Redlich–Kister equation. The intermetallic compounds ErPd 3 and ErPd, which have a homogeneity range, are treated as the phases MPd 3 and ErM, respectively, by a two-sublattice model with Pd in MPd 3 and Er in ErM on one sublattice and M on the other one, where M is used as an abbreviation for a mixture of Er and Pd. The other compounds were treated as stoichiometric. A set of self-consistent thermodynamic parameters of the Er–Pd system was obtained.  2000 Elsevier Science S.A. All rights reserved. Keywords: Er–Pd alloys; Thermodynamic modelling; CALPHAD technique; Thermodynamic properties

1. Introduction

2. Thermodynamic models

The intermetallic compounds formed by rare earth elements (RE) and transition metals are of particular interest regarding their potential usage as high value functional materials, such as permanent magnet and hydrogen storage materials [1,2]. The palladium-rich rare earth solid solution alloys are of very interest because of their potential applications as hydrogen diffusion membranes for purification and isotope separation [3–14]. To understand the physical properties and the technological applications of these compounds, it is necessary to have a better knowledge of the thermodynamic properties of the technically relevant system. This work deals with the assessment of the thermodynamic description of the Er–Pd system, which is a portion of the general programme of the thermodynamic optimizing on the Pd–RE–H(D) systems, by means of the calculation of phase diagram (CALPHAD) technique. In this method, the thermodynamic properties of the alloy systems are studied by using thermodynamic models for the Gibbs energy of individual phases. The thermodynamic parameters involved in the models are optimized from the experimental thermodynamic and phase diagram information.

2.1. Unary phases

*Corresponding author.

The Gibbs energy function 0 G fi (T ) 5 G fi (T )–H SER for i the element i (i5Er, Pd, respectively) in the phase f (f 5liquid, f.c.c. or h.c.p.) is described by an equation of the following form: 0

G fi (T ) 5 a 1 bT 1 cT ln T 1 dT 2 1 eT 3 1 fT 21 1 gT 7 1 hT 29

(1)

SER i

where H is the molar enthalpy of the element i at 298.15 K in its standard element reference (SER) state, h.c.p. for Er and f.c.c. for Pd. The Gibbs energy of the element i, 0 G fi (T ), in its SER state, is denoted by GHSER i , i.e. GHSER Er 5 G h.c.p. (T ) 2 H SER Er Er

(2)

f.c.c. SER GHSER Pd 5 G Pd (T ) 2 H Pd

(3)

In the present work, the Gibbs energy functions are taken from the SGTE compilation by Dinsdale [15].

2.2. Solution phases In the Er–Pd system, there are three solution phases: liquid, f.c.c. and h.c.p. Their Gibbs energies are described by the following expression:

0925-8388 / 00 / $ – see front matter  2000 Elsevier Science S.A. All rights reserved. PII: S0925-8388( 99 )00691-X

Z. Du, H. Yang / Journal of Alloys and Compounds 299 (2000) 199 – 204

200

SER SER f G fm 2 x Er H Er 2 x Pd H Pd 5 x Er 0 G Er (T ) 1 x Pd 0 G fPd (T )

1 RT(x Er ln x Er 1 x Pd ln x Pd ) 1 E G fm 1 mg G mf

(4)

compounds. The Gibbs energy for per mole of formula unit Er m Pd n is expressed as following: 0

in which G fm is the excess Gibbs energy, expressed by the Redlich–Kister polynomial, E

E

O L (x

f m

i

G 5 x Er x Pd

f

Er

2 x Pd )

i

(5)

i

where i L f is the interaction parameter between elements Er and Pd, which is to be evaluated in the present work. Its general form is L f 5 a 1 bT 1 cT ln T 1 dT 2 1 eT 3 1 fT 21

(6)

but there are cases when only the first one or two terms are used. mg G fm is the magnetic contribution to Gibbs energy in solution phases, and described by mg

f m

f

f

G 5 RT ln( b 1 1)f(t )

(7)

where b f 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; t f is defined as T /T fc , and T fc is the critical temperature for magnetic ordering, i.e., the Curie temperature (T c ) for ferromagnetic ´ temperature (T N ) for antiferromagnetordering or the Neel ic ordering. They are described by the following expressions: T cf 5 x Er 0 T fc Er 1 x Pd 0 T fc Pd 1 x Er x Pd L Tfc

(8)

f f b f 5 x Er 0 b Er 1 x Pd 0 b Pd 1 x Er x Pd L fb

(9)

where L fT c and L bf are the magnetic interaction parameters between elements Er and Pd, here set to be zero due to lack of experimental data in the Er–Pd system. The f(t ) represents the polynomials obtained by Hillert and Jarl [16] based on the magnetic specific heat of iron, i.e., for t ,1:

SER SER m Pd n G Er 5 G mEr m Pd n 2 mH Er 2 nH Pd m

5 mGHSER Er 1 nGHSER Pd 1 DG fEr m Pd n m Pd n 1 mg G Er m

(12)

in which DG fEr m Pd n is the Gibbs energy of formation for per mole of formula unit Er m Pd n . Owing to lack of experimental measurements, it is assumed that the Neumann–Kopp rule applies to the heat capacity, i.e., m Pd n DCp50. Thus, DG Er can be given by the following f expression: m Pd n DG Er 5 a 1 bT f

(13)

where the parameters a and b were evaluated in the present work. The magnetic contribution to the Gibbs energy mg Er m Pd n Gm is included in the Gibbs energy 0 G mEr m Pd n , because the Curie temperature of all compounds in the Er–Pd system is much below 298.15 K.

2.4. ErPd3 and ErPd compounds The intermetallic compounds ErPd 3 and ErPd, which have some homogeneity range, are treated as the phases MPd 3 and ErM, respectively, by a two-sublattice model [17,18] with Pd in MPd 3 and Er in ErM on one sublattice and M on the other one, where M is used as an abbreviation for a mixture of Er and Pd. The Gibbs energy for per mole of formula unit MPd 3 and ErM are given by following expressions, respectively: 0

3 3 G MPd 5 G MPd 2 H SER m m MPd 3

MPd 3 MPd 3 5 y Er 0 G Er:Pd 1 y Pd 0 G Pd:Pd 1 RT( y Er ln y Er

1 y Pd ln y Pd ) 1 y Er y Pd

OL i

MPd 3 Er,Pd:Pd

( y Er 2 y Pd )i

i

f(t ) 5 1

(14)

2 [79t 21 /(140p) 1 474 / 497(1 /p 2 1)(t 3 / 6 9

15

1 t / 135 1 t / 600)] /A

(10)

G ErM 5 G mErM 2 H SER m ErM 0 ErM 5 y Er 0 G ErM Er:Er 1 y Pd G Er:Pd 1 RT( y Er ln y Er

for t .1: f(t ) 5 2 (t 25 / 10 1 t 215 / 315 1 t 225 / 1500) /A

0

(11)

1 y Pd ln y Pd ) 1 y Er y Pd

OL i

ErM Er:Er,Pd

( y Er 2 y Pd )i

i

where A5519 / 1125111692 / 15975(1 /p21) and p depends on the structure, 0.4 for b.c.c. structure and 0.28 for the others.

2.3. Stoichiometric intermetallic compounds The intermetallic compounds ErPd 7 , Er 10 Pd 21 , Er 2 Pd 3 , Er 3 Pd 4 , Er 3 Pd 2 , and Er 5 Pd 2 are treated as stoichiometric

(15) SER SER where H SER MPd 3 and H ErM are the abbreviation of y Er H Er 1 SER SER SER (3 1 y Pd )H Pd and (1 1 y Er )H Er 1 y Pd H Pd , and y Er and y Pd are the site fraction of Er and Pd on the M sublattice, respectively. The four parameters denoted 0 G **:* (also called compound energies) are expressed relative to the Gibbs energies of pure h.c.p. Er and f.c.c. Pd at the same

Z. Du, H. Yang / Journal of Alloys and Compounds 299 (2000) 199 – 204

temperature. The i L represents the interaction parameter between elements Er and Pd on the M sublattice.

3. Experimental information The complete phase diagram of the Er–Pd system was first measured by Loebich and Raub [19]. They found seven intermetallic compounds in the Er–Pd system, and called them ErPd 3 , ErPd 2 , Er 2 Pd 3 , Er 4 Pd 5 , ErPd, Er 3 Pd 2 and Er 5 Pd 2 , in which ErPd 3 , Er 4 Pd 5 , ErPd and Er 5 Pd 2 melted congruently, others formed by peritectic reactions. Later Er 4 Pd 5 was modified to Er 3 Pd 4 based on crystal structure data reported by Palenzona and Iandelli [20]. And ErPd 2 was changed to Er 10 Pd 21 assuming similarity to the Sm–Pd system [21], in which the subsequent single-crystal structure investigated by Fornasini et al. [22] clarified that low-symmetry SmPd 2 [19] is actually Sm 10 Pd 21 . The compound ErPd 7 was found to exist by Sakamoto et al. [23]. The homogeneity range of the intermetallic compounds ErPd 3 , ErPd and the solubility of Er in f.c.c.(Pd) were measured by Loebich and Raub [19]. They also indicated the presence of polymorphic transformation of the intermetallic compounds ErPd and Er 4 Pd 5 which was identified to be Er 3 Pd 4 by Palenzona and Iandelli [20]. Palenzona and co-workers [24,25] measured the enthalpies of formation of Er 2 Pd 5 and ErPd by differential calorimetric method, meanwhile the enthalpy of formation of Er 3 Pd 2 was predicted [25]. Ramaprabhu and Weiss [26] reported the enthalpies of formation of Er 3 Pd 2 and Er 3 Pd 4 . Recently Guo and Kleppa [27] have determined the enthalpies of formation of Er 3 Pd 4 , ErPd and ErPd 3 by direct synthesis calorimetry. In addition, the enthalpies of formation of the intermetallic compounds in the Er–Pd system were calculated by Kubaschewski [28] depending on the effective coordination number of each metal during alloying and the heat of sublimation of each pure metal. Ramaprabhu and Weiss [26] and Guo and Kleppa [27] predicated the enthalpies of formation of compounds in the Er–Pd system using the Miedema model [29–31], respectively.

4. Assessment procedure The optimization was carried out by means of the Thermo-Calc software [32], which can handle various kinds of experimental data. The program works by minimizing an error sum where each of the selected data values is given a certain weight. The weight is chosen by personal judgement and changed by trial and error during the work until most of the selected experimental information is reproduced within the expected uncertainty limits. All of the phase diagram and crystal structure experimental information determined by Loebich and Raub

201

[19], Palenzona and Iandelli [20], Fornasini et al. [22] and Sakamoto et al. [23] has been selected for the evaluation of the thermodynamic model parameters. It exists great diversity in the enthalpies of formation of intermetallic compounds determined or calculated by several investigators [24–31]. The experimental data measured by Guo and Kleppa [27] are given a larger weight in the present work. The optimization was carried out by two steps. In the first treatment, ErPd 3 and ErPd are assumed to be stoichiometric compounds; in the second treatment, they are treated as MPd 3 and ErM by a two-sublattice model, which is described in Section 2.4. The parameters obtained from the first treatment were used as starting values for the second treatment. In the f.c.c.(Pd), the temperature dependence is well established. The coefficients a 0 and b 0 can be reliably obtained from the experimental data. No solubility is assumed in the h.c.p. solution, which is realized by assigning a large positive interaction parameter, i.e., 0

5 21 L h.c.p. Er,Pd 5 10 J mol

(16)

For liquid, at least a 0 and b 0 in Eq. (4) can be adjusted because the liquidus was measured accurately over the whole composition range. In the present work, it was found that a 1 and b 1 should also be introduced in order to describe the properties of the liquid satisfactorily. 0 ErM 3 The quantities 0 G MPd Pd:Pd in Eq. (14) and G Er:Er in Eq. (15) describe a metastable form of pure Pd with ErPd 3 structure and pure Er with ErPd structure, respectively. a ErM The former is taken from Du et al. [33], and 0 G Er:Er is optimized to be a positive a 0 above the GHSER Er . But for 0 b ErM G Er:Er , the parameters a 0 and b 0 are to be optimized because the polymorphic transformation of the intermetallic compounds ErPd is determined well [19].

5. Results and discussions The thermodynamic description of the Er–Pd system obtained in the present work is listed in Table 1. The Er–Pd phase diagram calculated by means of the thermodynamic parameters is presented in Fig. 1 with experimental data [19,23]. Fig. 2 is the enlarged section of Fig. 1. The calculations agree well with the experimental results. The invariant equilibria in the Er–Pd system are listed in Table 2. As shown in the table, very good agreement is obtained between the calculations and experiments, in which the largest uncertainty is about 28C. In view of the estimated experimental errors (about 1–2 at.%), 21 of the 22 experimental invariant reaction compositions in the Er–Pd system are well reproduced. Fig. 3 shows the calculated enthalpies of formation in the Er–Pd system with the experimental data [24–27] and

Z. Du, H. Yang / Journal of Alloys and Compounds 299 (2000) 199 – 204

202

Table 1 Thermodynamic parameters of the Er–Pd system a Liquid

0

L liq. 5 2 368991 1 24.014 T L liq. 5 1 144011 2 10.082 T 0 G f.c.c. 2 GHSER Er 5 1 344620 Er 0 f.c.c. L 5 2 859094 1 37.316 T 0 h.c.p. L 5 100000 5 Pd 2 DG Er 5 2 438832 1 29.007 T f 3 Pd 2 DG Er 5 2 432499 1 37.993 T f 0 ErM G aEr:Er 2 2GHSER Er 5 1 256845 0 ErM G aEr:Pd 2 GHSER Er 2 GHSER Pd 5 2 183153 1 1.551 T 0 a ErM L Er:Er,Pd 5 2 306264 0 b ErM G Er:Er 2 2GHSER Er 5 1 304466 1 24.901 T 0 b ErM G Er:Pd 2 GHSER Er 2 GHSER Pd 5 2 180463 2 1.725 T 0 b ErM L Er:Er,Pd 5 2 369289 DG fEr 3 Pd 4 5 2 648017 2 3.848 T DG af Er 2 Pd 3 5 2 466614 1 3.240 T 2 Pd 3 DG bEr 5 2 458747 2 2.288 T f 10 Pd 21 DG Er 5 2 2929281 1 106.496 T f 0 MPd 3 G Er:Pd 2 GHSER Er 2 3GHSER Pd 5 2 376278 1 22.892 T 0 3 G MPd Pd:Pd 2 4GHSER Pd 5 1 10492 0 MPd 3 L Er,Pd:Pd 5 2 52533 DG fErPd 7 5 2 446044 1 58.926 T 1

f.c.c. h.c.p. Er 5 Pd 2 Er 3 Pd 2 aErM

bErM

Er 3 Pd 4 a Er 2 Pd 3 b Er 2 Pd 3 Er 10 Pd 21 MPd 3

ErPd 7 a

In J /(mol of formula units).

Fig. 2. Enlarged section of Fig. 1.

predicted values from Miedema model [30,31], respectively. Because it has much different in the enthalpies of formation of intermetallic compounds reported by several investigators [24–31], the experimental data measured by Guo and Kleppa [27] are given a larger weight in the present work. Satisfactory agreement is shown between the calculations and experiments.

6. Conclusions The phase relations and thermodynamic description in the Er–Pd system were evaluated from the experimental information available in the literature. A consistent set of

Fig. 1. The Er–Pd phase diagram calculated by present thermodynamic description with the experimental data [19,23].

Z. Du, H. Yang / Journal of Alloys and Compounds 299 (2000) 199 – 204

203

Table 2 Invariant reactions in the Er–Pd system Reaction

Liq.→h.c.p.(Er)1Er 5 Pd 2 Liq.→Er 5 Pd 2 Liq.→Er 5 Pd 2 1Er 3 Pd 2 Er 3 Pd 2 1bErPd→aErPd Liq.1bErPd→Er 3 Pd 2 Liq.→bErPd Liq.→bErPd1Er 3 Pd 4 a bErPd→aErPd1Er 3 Pd 4 Liq.→Er 3 Pd 4 Liq.1Er 3 Pd 4 →bEr 2 Pd 3 Liq.→bEr 2 Pd 3 1Er 10 Pd 21 b bEr 2 Pd 3 →aEr 2 Pd 3 Liq.1ErPd 3 →Er 10 Pd 21 Liq.→ErPd 3 Liq.→ErPd 3 1f.c.c.(Pd) f.c.c.(Pd)1ErPd 3 →ErPd 7 c

Present work

Loebich and Raub [19]

T (K)

x(Pd)

1190 1212 1196 838 1265 1813 1703 821 1722 1596 1566 1423 1608 1983 1553 723

0.2417 — 0.3198 — 0.3640 0.4994 0.5507 0.5000 — 0.6215 0.6318 — 0.6458 0.7501 0.8563 0.8913

0.000 — — 0.4965 0.4774 — 0.5000 0.5000— — — — 0.7500 — 0.7882 0.7815

— — 0.4942 — — — — — — 0.8815 —

T (K)

x(Pd)

1188 1213 1196 838 1264 1813 1703 821 1723 1596 1568 1423 1608 1983 1553 723

0.265 — 0.315 — 0.350 0.500 0.535 0.500 — 0.626 0.635 — 0.659 0.750 0.860 0.900

0.000 — — 0.491 0.491 — 0.500 0.500 — — — — 0.750 — 0.795 0.777

— — 0.491 — — — — — — 0.870 —

a

Er 4 Pd 5 is modified to Er 3 Pd 4 by Palenzona and Iandelli [20]. ErPd 2 is changed to Er 10 Pd 21 by Borzone et al. [21]. c The experimental data are taken from Sakamoto et al. [23]. b

thermodynamic parameters was derived. With the thermodynamic description available, one can now make various calculations of practical interest.

Acknowledgements The authors would like to express their gratitude to Thermo-Calc AB for supplying the Thermo-Calc software. This work was supported by National Natural Science Foundation of China (NSFC) (Grant No. 59871008).

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Fig. 3. Calculated enthalpy of formation in the Er–Pd system with the experiments [24–27] and predicted values [26,27] using the Miedema model [30] and the Miedema model revised by Niessen et al. [31].

[15] [16] [17]

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