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Materials Today: Proceedings xxx (xxxx) xxx
Contents lists available at ScienceDirect
Materials Today: Proceedings
journal homepage: www.elsevier.com/locate/matpr
Calphad-type assessment of the Pd-Yb binary system supported
by first-principles calculations – Part II: Results
Said Kardellass a,b,⇑, Colette Servant c, Isabelle Drouelle c, Fatima Zahra Chrifi-Alaoui a,
Mohamed Idbenali d, Alyen Abahazem f, Aissam Hidoussi e, Amine Bendarma b, Najim Selhaoui a
a
Laboratoire de Thermodynamique et Energétique, LTE, Université Ibn-Zohr, B.P.8106, Agadir, Morocco
Sustainable Innovation and Applied Research Laboratory, Polytechnic School, International University of Agadir, Agadir, Morocco
Laboratoire de Physicochimie de l’Etat Solide, ICMMO, Université Paris-Sud, 91405 Orsay Cedex, France
d
Laboratoire de Mécanique, Procédés de l’Energie et de L’Environnement (LMP2E), ENSA, Agadir, Morocco
e
Laboratoire d’étude Physico-Chimique des Matériaux, Université de Batna 1, Rue Chahid Boukhlouf, 05000 Batna, Algeria
f
Laboratoire Matériaux et Energies Renouvelables (LMER), Faculté des Sciences, Université Ibn Zohr, Agadir, Morocco
b
c
a r t i c l e
i n f o
Article history:
Received 1 June 2019
Received in revised form 11 August 2019
Accepted 20 August 2019
Available online xxxx
Keywords:
Thermodynamic description
Pd-Yb phase diagram
Pd-based alloys
First-principles calculations
Kaptay and Redlich-Kister models
a b s t r a c t
In the present paper, the thermodynamic properties of the Palladium-Ytterbium binary system were optimized using the CALPHAD approach. The liquid phase and the terminal solid solutions: body-centered
cubic for cYb and face-centered cubic for bYb, and (Pd) are described by the substitutional solution model
with the exponential (Kaptay model) and linear (Redlich-Kister polynomials) models for the temperature
dependence of the excess Gibbs energy. The other intermetallic compounds, PdYb3, a-Pd2Yb5, b-Pd2Yb5,
Pd4Yb3, Pd2Yb, a-Pd21Yb10, b-Pd21Yb10, and Pd7Yb are treated as stoichiometric compounds. The thermodynamic properties of the intermetallic compounds at 0 K are predicted using the first-principles
approach. Two sets of self consistent thermodynamic parameters for the Palladium-Ytterbium binary system are obtained. With the present thermodynamic descriptions, the thermochemical properties and the
phase equilibria are well reproduced.
Ó 2019 Elsevier Ltd. All rights reserved.
Selection and peer-review under responsibility of the scientific committee of the International Conference on Plasma and Energy Materials ICPEM2019.
1. Introduction
Widely used in different sectors viz. nuclear energy, metallurgy,
chemical engineering, electronics, computer manufacturing and
numerous medical devices [1,2], REEs fall into two categories,
viz. light rare earths (La to Sm) with varying levels of uses and
demand and heavy rare earths (Eu to Lu, Y) which are less common
and more valuable [3]. The overall chemical and metallurgical
properties of the REEs are due to their outer electrons 5d6s (3d4s
for Sc and 4d5s for Y). For the lanthanides, the normal configuration for the metallic state is 4fn(5d6s)3, that is, trivalent, with only
Eu and Yb having the divalent 4f(n+1)(5d6s)2 configuration. Recovery of REEs is interesting due to its high market prices along with
various industrial applications. Biosorption represents a biotechnological innovation as well as a cost effective excellent tool for
⇑ Corresponding author.
E-mail address: said.kardellass@edu.uiz.ac.ma (S. Kardellass).
the recovery of rare earth metals from aqueous solutions [4]. Alloying with platinum group metals PGMs especially Palladium, which
is used as catalyst in a number of important chemical processes,
they have an significant influence on the structures and properties.
For instance, the palladium-rich RE solid solution alloys are of
interest due to their own physical and metallurgical properties,
and their potential applications as hydrogen diffusion membranes
for purification and isotope separation but also as redox catalysis
and hydrogen absorption (e.g.: hydrogen storage materials in batteries or fuel cells) [5–6]. Moreover, compounds of the PalladiumRE systems were found to exhibit unique optical properties. The
most pronounced optical properties were found just for LuPd3
and LuPd compounds, which are blue (similar to silicon) and
golden, respectively [7].
The purposes of the present work are (1) to evaluate the measured phase diagram data and thermodynamic data available in
the literature (2), to compute the enthalpies of formation (DHf)
for all the intermetallic compounds in this binary system via
https://doi.org/10.1016/j.matpr.2019.08.196
2214-7853/Ó 2019 Elsevier Ltd. All rights reserved.
Selection and peer-review under responsibility of the scientific committee of the International Conference on Plasma and Energy Materials ICPEM2019.
Please cite this article as: S. Kardellass, C. Servant, I. Drouelle et al., Calphad-type assessment of the Pd-Yb binary system supported by first-principles calculations – Part II: Results, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.196
Phases
Thermodynamic model
Liquid
(Pd,Yb)1
2
Thermodynamic parametersb
Redlich-Kister model [17]
bcc_A2
(Pd,Yb)1(Va)*3
fcc_A1
(Pd,Yb)1(Va)*1
PdYb3
(Pd)0.25(Yb)0.75
aPd2Yb5
(Pd)0.285(Yb)0.714
bPd2Yb5
(Pd)0.285(Yb)0.714
aPdYb
(Pd,Yb)0.50(Pd)0.50
Kaptay model [18–20]
0 Liq
LPd:Yb ¼ - 269657:509 þ 7:929 T
1 Liq
LPd:Yb ¼ - 92217:234 þ 16:714 T
2 Liq
LPd:Yb ¼ þ46141:615 - 21:085 T
0 BCC A2
LPd:Yb ¼ þ5010
0 FCC A1
LPd:Yb ¼ - 228337:332 1:315 T
1 FCC A1
LPd:Yb ¼ - 95731:401 þ 8:478 T
A1
A1
0 PdYb3
GPd:Yb 0:250 298:15 Hfcc
0:750 298:15 Hfcc
¼
Pd
Yb
A1
0 fcc A1
þ
0:750
G
¼
52718:199
þ 10:843
þ0:250 0 Gfcc
Pd
Yb
A1
A1
0 aPd2 Yb5
GPd:Yb 0:285 298:15 Hfcc
0:714 298:15 Hfcc
¼
Pd
Yb
A1
0 fcc A1
þ
0:714
G
¼
þ0:285 0 Gfcc
Pd
Yb
0 Liq
LPd:Yb
1 Liq
LPd:Yb
2 Liq
LPd:Yb
¼ - 255693:94 EXPð- 5:628E - 05 TÞ
¼ - 97449:687 EXPð- 4:135E - 04 TÞ
¼ þ 22699:94 EXPð- 5:822E - 04 TÞ
No excess term optimized
T
- 48250:046 þ 0:0501 T
bPdYb
(Pd,Yb)0.50(Pd)0.50
A1
2 Yb5
GbPd
0:285 298:15 Hfcc
0:714 Pd
Pd:Yb
A1
0 fcc A1
þ
0:714
G
¼
þ0:285 0 Gfcc
Pd
Yb
- 56723:003 þ 8:859 T
0
aPdM
GPd:Pd
¼ þ183254:857
0 aPdM
GPd:Yb 0:500 298:15 Hfcc
Pd
A1
þ 0:500 þ0:500 0 Gfcc
Pd
0 aPdM
LPd;Yb:Pd ¼ - 219378:233
(Pd)0.571(Yb)0.428
aPd5Yb3
(Pd,Yb)0.625(Yb)0.375
0
0
fcc
HYb
A1
¼
A1
0:500 298:15 Hfcc
¼
Yb
A1
Gfcc
¼
80832:267
þ 2:56371 T
Yb
GbPdM
¼ 129496:642 - 4:989 T
Pd:Pd
A1
0 bPdM
GPd:Yb 0:500 298:15 Hfcc
0:500 Pd
A1
þ 0:500 þ0:500 0 Gfcc
Pd
0 bPdM
LPd;Yb:Pd ¼ - 157086:906
Pd4Yb3
A1
298:15
Gfcc
Yb
A1
298:15
Hfcc
Yb
A1
¼ - 80831:963 þ 2:56353 T
A1
A1
0 Pd4 Yb3
GPd:Yb 0:571 298:15 Hfcc
0:428 298:15 Hfcc
¼
Pd
Yb
A1
A1
þ 0:428 0 Gfcc
¼ - 77228:898 þ 0:650 T
þ0:571 0 Gfcc
Pd
Yb
Pd5 Yb3
GaYb:Yb
¼ 164644:571 - 2:044 T
fcc A1
A1
Pd5 Yb3
GaPd:Yb
0:625 298:15 HPd
0:375 298:15 Hfcc
¼
Yb
A1
0 fcc A1
þ0:625 0 Gfcc
þ
0:375
G
¼
74419:861
þ
0:650
T
Pd
Yb
0 aPd5 Yb3
LPd;Yb:Yb ¼ - 219975:471
(Pd,Yb)0.625(Yb)0.375
5 Yb3
GbPd
¼ 134825:227 - 3:005 T
Yb:Yb
A1
A1
5 Yb3
GbPd
0:625 298:15 Hfcc
0:375 298:15 Hfcc
¼
Pd
Yb
Pd:Yb
A1
0 fcc A1
þ
0:375
G
¼
72679:697
0:521 T
þ0:625 0 Gfcc
Pd
Yb
0 bPd5 Yb3
LPd;Yb:Yb ¼ - 203743:952
0
Pd2Yb
(Pd)0.667(Yb)0.333
aPd21Yb10
(Pd)0.667(Yb)0.322
bPd21Yb10
(Pd)0.667(Yb)0.322
fcc
0 Pd2 Yb
GPd:Yb 0:667 298:15 HPd
A1
þ
0:333
þ0:667 0 Gfcc
Pd
(Pd)0.750(Pd,Yb)0.250
0
A1
0:333 298:15 Hfcc
¼
Yb
A1
Gfcc
¼
72047:889
þ 0:806 T
Yb
fcc A1
A1
0 aPd21 Yb10
GPd:Yb
0:677 298:15 HPd
0:322 298:15 Hfcc
¼
Yb
A1
0 fcc A1
þ
0:322
G
¼
71444:09
þ
0:895
T
þ0:677 0 Gfcc
Pd
Yb
0
21 Yb10
GbPd
0:677 Pd:Yb
þ0:677 Pd3Yb
A1
0
A1
Gfcc
Pd
298:15
Hfcc
Pd
þ 0:322 0
A1
0:322 A1
Gfcc
Yb
298:15
fcc
HYb
A1
¼
¼ - 71283:910 þ 0:787 T
Pd7Yb
(Pd)0.875(Yb)0.125
A1
0 bPd2 Yb5
GPd:Yb 0:285 298:15 Hfcc
0:714 Pd
A1
0 fcc A1
þ
0:714
GYb
¼
þ0:285 0 Gfcc
Pd
- 45302:168 - 0:179 T
aPdM
GPd:Pd
¼ þ183254:857
0 aPdM
GPd:Yb 0:500 298:15 Hfcc
Pd
A1
þ 0:500 þ0:500 0 Gfcc
Pd
0 aPdM
LPd;Yb:Pd ¼ - 219421:652
A1
þ 0:500 þ0:500 0 Gfcc
Pd
0 bPdM
LPd;Yb:Pd ¼ - 156874:725
Units in J/mol atom and J/mol atom/K. Reference states are fcc_A1for both elements Pd andYbatTa,
a
Since the Neumann–Kopp Rule has been invoked, the formation properties are independent of temperature,
b
Note that the enthalpies and entropies of formation of the intermetallic compounds are listed per mole of atoms, *(Va) for Vacancy.
A1
¼
A1
0:500 298:15 Hfcc
¼
Yb
A1
Gfcc
¼
75645:441
þ 3:076 T
Yb
Gfcc
Yb
A1
298:15
Hfcc
Yb
A1
¼
¼ - 75602:902 þ 3:051 T
A1
fcc A1
0 aPd5 Yb3
GPd:Yb 0:625 298:15 Hfcc
0:375 298:15 HYb
¼
Pd
A1
0 fcc A1
þ0:625 0 Gfcc
þ
0:375
G
¼
68646:217
þ
0:972 T
Pd
Yb
0 aPd5 Yb3
LPd;Yb:Yb ¼ - 219280:03
5 Yb3
GbPd
¼ þ 138070:331 - 3:005 T
Yb:Yb
fcc A1
A1
0 bPd5 Yb3
GPd:Yb 0:625 298:15 HPd
0:375 298:15 Hfcc
¼
Yb
A1
0 fcc A1
þ
0:375
G
¼
68242:869
þ
0:705 T
þ0:625 0 Gfcc
Pd
Yb
0 bPd5 Yb3
LPd;Yb:Yb ¼ - 196686:219
fcc
0 Pd2 Yb
GPd:Yb 0:667 298:15 HPd
A1
þ
0:333
þ0:667 0 Gfcc
Pd
A1
0
A1
0:333 298:15 Hfcc
¼
Yb
fcc A1
GYb
¼ - 65717 þ 0:679 T
A1
A1
0 aPd21 Yb10
GPd:Yb
0:677 298:15 Hfcc
0:322 298:15 Hfcc
¼
Pd
Yb
A1
0 fcc A1
þ
0:322
G
¼
65094:386
þ
0:738
T
þ0:677 0 Gfcc
Pd
Yb
0
21 Yb10
GbPd
0:677 Pd:Yb
þ0:677 0
A1
Gfcc
Pd
298:15
Hfcc
Pd
þ 0:322 fcc
0 Pd7 Yb
GPd:Yb 0:875 298:15 HPd
A1
þ0:875 0 Gfcc
þ 0:125 Pd
A1
0:125 298:15 Hfcc
¼
Yb
A1
Gfcc
¼ - 40179:490 þ 7:518 T
Yb
Hfcc
Yb
Pd5 Yb3
GaYb:Yb
¼ þ165041:895 - 2:044 T
fcc
0 Pd7 Yb
GPd:Yb 0:875 298:15 HPd
A1
þ0:875 0 Gfcc
þ 0:125 Pd
0
0
298:15
A1
fcc A1
0 Pd4 Yb3
GPd:Yb 0:571 298:15 Hfcc
0:428 298:15 HYb
¼
Pd
A1
fcc A1
þ 0:428 0 GYb
¼ - 71589:817 þ 1:065 T
þ0:571 0 Gfcc
Pd
Pd3 Yb
GPd:Pd
¼ 6013:804
0 Pd3 Yb
GPd:Yb 0:750 298:15 Hfcc
Pd
A1
þ 0:250 þ0:750 0 Gfcc
Pd
0 Pd3 Yb
LPd:Pd;Yb ¼ - 8001:508
A1
A1
0
Pd3 Yb
GPd:Pd
¼ 2601:005
A1
fcc A1
Pd3 Yb
GPd:Yb
0:750 298:15 Hfcc
0:250 298:15 HYb
¼
Pd
A1
0 fcc A1
þ
0:250
G
¼
60717:520
- 2:336 T
þ0:750 0 Gfcc
Pd
Yb
0 Pd3 Yb
LPd:Pd;Yb ¼ - 157086:906
0
T
- 60466:957 þ 15:703 T
GbPdM
¼ þ129496:642 - 4:989 T
Pd:Pd
A1
0 bPdM
GPd:Yb 0:500 298:15 Hfcc
0:500 Pd
¼
0
bPd5Yb3
0 FCC A1
LPd:Yb ¼ - 207603:954E þ 05 EXPð-9:017E - 06 TÞ
1 FCC A1
LPd:Yb ¼ - 63182:40
A1
A1
0 PdYb3
GPd:Yb 0:250 298:15 Hfcc
0:750 298:15 Hfcc
¼
Pd
Yb
A1
0 fcc A1
þ
0:750
G
¼
52992:667
þ 13:275
þ0:250 0 Gfcc
Pd
Yb
A1
A1
0 aPd2 Yb5
GPd:Yb 0:285 298:15 Hfcc
0:714 298:15 Hfcc
¼
Pd
Yb
A1
0 fcc A1
þ
0:714
G
¼
þ0:285 0 Gfcc
Pd
Yb
0
A1
0
A1
0
A1
0:322 fcc A1
GYb
298:15
Hfcc
Yb
A1
¼
¼ - 64013:243 þ 1:46E - 03 T
A1
0:250 298:15 Hfcc
¼
Yb
A1
Gfcc
¼
60144:816
þ 0:574 T
Yb
A1
0:125 298:15 Hfcc
¼
Yb
fcc A1
GYb
¼ - 32239:586 þ 1:315 T
S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
Please cite this article as: S. Kardellass, C. Servant, I. Drouelle et al., Calphad-type assessment of the Pd-Yb binary system supported by first-principles calculations – Part II: Results, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.196
Table 1
Thermodynamic parameters in the Palladium-Ytterbium binary systema.
S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
Fig. 1a. Calculated Palladium-Ytterbium phase diagram (continuous line) using the
thermodynamic parameters optimized in the present work used Redlich-Kister
model [17] along with the experimental data from literature, [11–13] and [14].
Fig. 1b. Calculated Palladium-Ytterbium phase diagram (continuous line) using the
thermodynamic parameters optimized in the present work used Kaptay model [18–
20] along with the experimental data from literature, [11–13] and [14].
first-principles calculations to supply the experimental information, and (3) to obtain a set of self-consistent thermodynamic
parameters for the Pd-Yb binary system by means of the CALPHAD
approach.
2. Assessment procedure
Although there are a large number of thermodynamic assessments published, there is a rather small number of publications
discussing the process of assessment. Recently, in the hope to help
researchers in Calphad modelling field, Tang [8] published very
important work in which he discussed the assessment procedure
3
Fig. 2a. Enlarged part of the calculated Pd-Yb phase diagram using the thermodynamic parameters optimized with Redlich-Kister model showing the congruent
melting of the b-Pd5Yb3, Pd4Yb3 and b-PdYb phases.
Fig. 2b. Enlarged part of the calculated Pd-Yb phase diagram using the thermodynamic parameters optimized with Kaptay model showing the congruent melting
of the b-Pd5Yb3, Pd4Yb3 and b-PdYb phases.
and described how to use the PARROT module in Thermo-Calc
[9,10] to optimize thermodynamic parameters. Most of the above
experimental information was selected for the evaluation of the
thermodynamic model parameters. In the present work, the phase
relation and transformation temperatures are based on recent version of phase diagram of Pd-Yb system determined by [11–14]. An
assessed value of the parameters in the Gibbs free energy expressions was carried out by means of the computer optimization program: the PARROT module of the Thermo-Calc software. The
stability constraint was enforced by requiring that the Gibbs
energy of the liquid phase had positive curvature,
2
2
i.e.d GLiq =dxi 0, at all compositions and temperatures up to
Please cite this article as: S. Kardellass, C. Servant, I. Drouelle et al., Calphad-type assessment of the Pd-Yb binary system supported by first-principles calculations – Part II: Results, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.196
4
S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
10,000 K. The program allows simultaneous treatment of all the
various types of experimental data. Each piece of information is
given a certain weight, chosen by personal judgment, which is a
measure of its probable accuracy. This was done in order to reproduce the phase diagram correctly. The detailed description of the
assessment procedure can refer the book written by Lukas et al.
[15]. The optimization strategy is to start with the liquid phase
when we have enough data to fix its variables. Part of other phases
will depend on the liquid. Subsequently, intermetallic compounds
and the other solution phases were introduced into the modeling
one by one. Parameters ai and bi mentioned in
i a
LPd;Yb ¼ ai þ bi T (i = 0, 1, 2) were adjusted to describe the
enthalpies of formation, liquidus and invariant reactions. Afterwards, parameters for the solution phases were introduced to
describe the solubility ranges according to the measured terminal
solubilities and the invariant reactions in which they are involved.
When the calculated invariant temperatures were adjusted within
the claimed error ranges, sublattice models were introduced to
describe the composition ranges of three phases exist in the PdYb system. In the light of the discussion envisaged in stoichiometric compounds with homogeneity range part, we note here that the
parameters obtained from the first treatment were used as the
starting values for the second treatment. An optimization with
all variables selected may lead to (i) errors, (ii) local minima, (iii)
inconsistencies. To have a self-consistent set, the last optimizevariable in PARROT is done over all variables (generally without
those related to Cp). Fortunately, in the present assessment all
parameters for different phases were optimized simultaneously
allowing to achieve a globally self-consistent thermodynamic
description.
3. Results and discussion
In the Table 1, the thermodynamic parameters optimized in the
present work using two formalisms are listed. The calculated phase
diagram using the presently obtained parameters with RedlichKister and Kaptay formalisms are depicted in Figs. 1a and b, respectively, along with reported experimental data published by
[11–14]. Enlarged parts at high temperature are shown in Figs.
2a and b to bring to light the congruent meltings. As shown in
the figures, a satisfactory agreement between the calculated phase
diagrams and the experimental data has been achieved except for
the PdYb HT $ Liq and Pd2 Yb $ Pd21 Yb10 LT þ Pd5 Yb3 HT reactions which are calculated at temperature too low, 1445 °C (with
Redlich-Kister formalism) and 1438 °C (with Kaptay formalism)
instead of 1460 °C [11,12], and 1150 °C (with Redlich-Kister formalism) and 1141 °C (with Kaptay formalism) instead of 1195 °C
[11,12], respectively. In fact, these discrepancies between the
experiment and calculation are due to the different weights given
to the experimental data during the optimization. In both cases,
small visible deviation from experimental data can be detected
on the range of liquidus extends from 0.30 to 0.60 at.Yb. Our result
makes the liquidus around the most stable compound more symmetric. Furthermore, the Kaptay model gives a better account of
the eutectic reaction Pd5 Yb3 HT þ Pd4 Yb3 $ Liq than the RedlichKister model. For both diagrams, the catatectic reaction
bYb $ Liq þ cYb is well reproduced. However, on the Yb rich side,
the bYb domain calculated is too extended while [11,12] reported
no extension. We have tried to eliminate this extension by adding
fcc A1
. However, we have not succeeded in
a positive term to 0 GYb:VA
reproducing the eutectic bYb þ PdYb3 $ Liqin a satisfying way.
The Yb-rich eutectic was usually difficult to reproduce. We think
it may be caused by probable contamination with other lanthanides, or inaccuracies in SGTE description of Yb. We usually
neglected these discrepancies. The calculated phase boundaries
by using both descriptions of liquid phase and solid solutions are
in good agreement with all measured data. Especially, no liquid
miscibility gap was observed within the defined temperature range
of the SGTE database (PURE5) (298.15–104 K) [16]. The invariant
equilibria in the Pd-Yb binary system are displayed in Table 2. As
shown in this table, satisfactory agreement is obtained between
the calculations by using two models and experiments, where
the largest uncertainty in the temperature of all invariant reactions
is about 15 K. In view of the estimated experimental errors (about
1–2 at.% Pd), 20 of 22 experimental invariant reaction compositions in the Palladium-Ytterbium binary system are well reproduced. The present optimization result is the compromise
Table 2
Invariant reactions of the Palladium-Ytterbium binary system.Comparison between the experimental data by [11–14] and the calculated values after the current optimization
with.
Reactions
Reaction Type
Literature [11–14]
T/°C
bYb $ aYb þ Liq
bYb þ PdYb3 $ Liq
PdYb3 $ Pd2 Yb5 LT þ Liq
Pd2 Yb5 HT $ PdYb LT þ Liq
Pd2 Yb5 LT $ Pd2 Yb5 HT
PdYb HT $ Liq
PdYb LT $ PdYb HT þ Liq
PdYb HT $ PdYb LT þ Liq
PdYb LT þ Pd4 Yb3 $ Liq
Pd4 Yb3 $ Liq
Pd5 Yb3 HT þ Pd4 Yb3 $ Liq
Pd5 Yb3 HT $ Liq
Pd5 Yb3 LT $ Pd4 Yb3 þ Pd5 Yb3 LT
Pd5 Yb3 HT $ Pd2 Yb þ Pd5 Yb3 LT
Pd5 Yb3 HT þ Pd21 Yb10 HT $ Liq
Pd2 Yb $ Pd21 Yb10 LT þ Pd5 Yb3 HT
Pd21 Yb10 LT $ Pd21 Yb10 HT
Pd21 Yb10 HT $ Pd3 Yb þ Liq
Pd3 Yb $ Liq
Pd3 Yb þ fcc A1 $ Liq
Pd7 Yb $ Pd3 Yb þ fcc A1
Liq $ ðPdÞ
Peritectoid
Eutectic
Peritectic
Peritectic
Transition
Congruent
Transition
Transition
Eutectic
Congruent
Eutectic
Congruent
Transition
Transition
Eutectic
Peritectoid
Transition
Peritectic
Congruent
Eutectic
Peritectoid
Melting
765
615
670
695
685
1460
1435
1390
1390
1415
1345
1360
1155
1185
1340
1195
1195
1380
1700
1350
449
1555
Composition at.%Yb
95.00
85.10
77.80
73.60
71.40
50.00
49.20
45.20
45.40
42.80
39.30
37.50
37.20
37.50
36.90
33.30
32.20
35.20
25.00
13.40
12.50
0 to 12.2
Current work
Redlich-Kister Formalism [17]
Kaptay Formalism [18–20]
T/°C
Composition, at.%Yb
T/°C
Composition, at.%Yb
775
633
656
699
689
1445
1437
1394
1394
1401
1360
1361
1161
1185
1348
1150
1196
1365
1697
1346
451
1555
94.72
83.33
79.24
72.00
71.40
50.00
51.85
44.83
44.84
42.80
38.87
37.50
37.20
37.50
35.70
33.30
32.20
34.29
25.0
14.68
11.48
0
760
621
667
693
682
1438
1432
1392
1392
1402
1348
1343
1154
1185
1348
1141
1195
1376
1702
1352
433
1555
95.08
85.11
77.66
72.21
71.40
50.00
51.20
44
44.74
42.80
38.35
37.50
37.20
37.50
36.70
33.30
32.20
35.32
25.0
14.05
21.04
0
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5
S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
Table 3
Comparison between the enthalpies, entropies, and the Gibbs energies of formation of the intermetallic compounds in the Palladium-REE binary systems derived from the present
optimization and taken from literature.
Intermetallic
compounds
Enthalpies of formation, J.
mol1. at1
Pd3RE
Sc
Y
Eu
Gd
Dy
Er
Yb
Pd2RE5
Y
Eu
Dy
Yb
PdRE
Sc
Y
Eu
Gd
Dy
Er
Yb
Pd4RE3
Y
Gd
Dy
Er
Yb
Pd21RE10
Er
Yb
Pd5RE3
Pd2RE
Yb
Sc
Y
Eu
Gd
Dy
Yb
82174
96080
95334
78078
79821
94070
60717.52
60144.82
64755.16
68000 ± 2000
21084
68548
44766
63190
48250.046
56723
60466.96
45302.17
56818.93
60350
89250
90583
78545
85265
80946
90231
80832.26
80831.96
75645.44
75602.902
54000
90954
66000 ± 3000
83450
92300
82523
81371
92573
77228.89
71589.82
92562.04
73000 ± 3000
81181
94070
71283.91
71444.09
64013.24
65094.38
80776.67
75000 ± 4000
75630
72679.697
74419.861
68242.869
68646.217
88681.57
75000 ± 3000
81703
84788
94330
104630
80498
81376
72047.88
65717
80776.67
T(K)
Entropies of formation,
J. mol1. at1. K1
Gibbs free Energies,
J. mol1. at1
Technique Used
References
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
819–
1240
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
819–
1240
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
819–
1240
298.15
298.15
298.15
298.15
298.15
298.15
298.15
819–
1240
298.15
298.15
298.15
298.15
298.15
298.15
819–
1240
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
18.13
12.51
10.83
7.63
7.11
5.723
2.336
0.574
–
–
87579.45
99809.85
98562.96
80352.88
81940.84
95776.31
60021.04
60315.96
–
–
Optimization
Optimization
Optimization
Optimization
Optimization
Optimization
OptimizationR-K
OptimizationK
Prediction
KEML/KEMS*
[29]
[28]
[24]
[25]
[26]
[27]
Current work
Current work
[22]
[12]
–
13.686
13.95
11.72
0.0501
8.859
15.703
0.179
–
–
13.61
4.145
8.461
8.684
3.500
0.862
2.5637
2.5635
3.076
3.051
–
–
–
–
72628.48
40606.80
66684.31
48264.98
59364.31
65148.81
45248.80
–
–
93307.82
91818.83
81067.64
87854.13
81989.52
89973.99
81596.627
81596.267
76562.549
76512.557
–
–
–
Ab-initio
Optimization
Optimization
Optimization
OptimizationR-K
Current work
[28]
[24]
[26]
Current work
OptimizationK
Current work
Prediction
Ab-initio
Optimization
Optimization
Optimization
Optimization
Optimization
Optimization
OptimizationR-K
[22]
Current work
[29]
[28]
[24]
[25]
[26]
[27]
Current work
OptimizationK
Current work
Prediction
Prediction
KEML/KEMS*
[21]
[22]
[12]
–
4.396
5.790
2.966
0.549
0.650
1.065
–
–
–
93610.66
84249.28
82255.31
92409.31
77422.68
71907.35
–
–
Ab-initio
Optimization
Optimization
Optimization
Optimization
OptimizationR-K
OptimizationK
Prediction
KEML/KEMS*
Current work
[28]
[25]
[26]
[27]
Current work
Current work
[22]
[12]
–
5.723
0.787
0.896
0.00146
0.738
–
–
–
95776.31
71518.55
71711.23
64013.675
65314.415
–
–
Ab-initio
Optimization
OptimizationR-K
Current work
[27]
Current work
OptimizationK
Current work
Prediction
KEML/KEMS*
[22]
[12]
–
0.522
0.650
0.705
0.972
–
–
–
72835.33
74613.65
68453.06
68936.018
–
–
Ab-initio
OptimizationR-K
Current work
Current work
OptimizationK
Current work
Prediction
KEML/KEMS*
[22]
[12]
–
16.175
8.189
9.236
6.446
5.200
0.806
0.679
–
–
89610.576
96771.550
107383.713
82419.875
82926.38
72288.189
65919.443
–
Ab-initio
Optimization
Optimization
Optimization
Optimization
Optimization
OptimizationR-K
OptimizationK
Prediction
Current work
[29]
[28]
[24]
[25]
[26]
Current work
Current work
[22]
(continued on next page)
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S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
Table 3 (continued)
Intermetallic
compounds
Enthalpies of formation, J.
mol1. at1
PdRE3
Y
Yb
Pd7RE
Y
Eu
Gd
Dy
Er
Yb
62254
52718.199
52992.667
49609.80
52818
58188
51176
48335
48326
55755
40179.49
32239.586
32071.91
T(K)
Entropies of formation,
J. mol1. at1. K1
Gibbs free Energies,
J. mol1. at1
Technique Used
References
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
298.15
13.89
10.843
13.275
–
–
12.943
6.200
9.402
8.268
7.360
7.518
1.315
–
66395.30
55951.03
56950.608
–
–
62046.95
53024.53
51138.20
50791.10
57949.38
42420.981
32631.653
–
Optimization
OptimizationR-K
OptimizationK
Prediction
Ab-initio
Optimization
Optimization
Optimization
Optimization
Optimization
OptimizationR-K
OptimizationK
Prediction
[28]
Current work
Current work
[22]
Current work
[28]
[24]
[25]
[26]
[27]
Current work
Current work
[22]
*KEML: Knudsen Effusion Mass Loss, KEMS: Knudsen Effusion Mass Spectrometry.
between phase diagram and thermodynamic data. The calculated
standard enthalpies, entropies and Gibbs free energies of formation
of the intermetallic compounds according to the present description along with reported measurements (at various T), ab-initio,
and theoretically predicted values for comparison are summarized
in Table 3 and presented in Fig. 3a. Error bars are plotted according
to the reported experimental uncertainties for each piece of experimental data. This figure shows that the computed values are in the
range of experimental data. The enthalpy variation vs. composition
is correctly reproduced. The present optimization of the thermodynamic parameters by using the Kaptay and Redlich-Kister models
was based on the assumption that the enthalpy and entropy of formation (the reference state are solid: fcc_A1 for Pd and fcc_A1 for
Yb) are independent of the temperature and, hence, the parameters
obtained from fits to high-temperature data are used for calculating standard enthalpies at 298.15 K. Compared with experimental
values measured by Ciccioli et al. [12] and our obtained ab-initio
data, overall agreement is achieved for Pd3Yb, a- and b-Pd21Yb10,
Fig 3a. Comparison of the calculated enthalpies J. mol1. at1 (HMR) of the
intermetallic compounds in the Pd-Yb binary system with the experimental data in
J. mol1. at1. HMR calculated with the present database: using Redlich-Kister
model [17] ( ,
), using Kaptay model ( ,
) [18–20], theoretically
predicted [22] (
), experimental measured by [12] ( with
is Error bars), and
ab-initio data (This work).
Pd2Yb, a- and b-Pd5Yb3, Pd4Yb3, a- and b-Pd2Yb5, PdYb3, and
Pd7Yb phases. The agreement is less satisfactory for a- and
b-PdYb equiatomic compound. For that phase, we pointed out that
the enthalpy of formation given by [21] is less negative than optimized in the present study. The calculated values of the enthalpies
of formation by using Kaptay model are more negative than the
ones calculated by using Redlich-Kister model (see Table 3). In
addition, in both cases, these calculated values are less negative
than the predicted ones by [22]. This later, generally predict more
exothermic values of DHf than are indicated by experiment and
their values differ more from experimental results than the other
predictions. Our calculated formation enthalpies from CALPHAD
approach are in good agreement with our DFT GGA-PBE calculations, especially for Kaptay model, only a minor difference (several
kJ) which can be attributed to the fact that CALPHAD enthalpies are
given at 298.15 K, and DFT calculations are done at 0 K, in addition
to the convergence accuracy in our DFT calculations (±1 kJ/mol.
at.). We mention that GGA-PBE doesn’t take into account the correlation of the 4f states of Yb element, which can be enhanced
using Hubbard-DFT methods (GGA + U). Unfortunately, Pd-Yb
compounds are conductors without a band gap, and it is very difficult to keep a physical meaning by fitting the appropriate value
for the Hubbard parameter U to other properties rather than the
band gap. Although these discrepancies, DFT (GGA-PBE) calculations without employing the Hubbard parameter U still provide
an accurate results with the experimental methods, and more
accurate than the semi-empirical values for similar systems. The
Redlich-Kister model’s minimum formation enthalpy is at a,bPdYb, while the experiment’s minimum [12] is at Pd5Yb3Pd21Yb10 (xPd = 0.625–0.677). Our results for this phase are in the
range of the values reported in [23] for the equimolar phases PdRE,
from 76 to 95 kJ/mol at., much more exothermic than one estimated by [12]. As it is rather difficult to assess the actual accuracy
of such values, the agreement is considered satisfactory. The calculated Gibbs’ energy change during the intermetallic formation from
the solid compounds at 298 K (dotted line with + for Kaptay model
and r for Redlich-Kister model) and at 1000 K (dashed line with
+ for Kaptay model and r for Redlich-Kister model) are also shown
in Fig. 3b. Moreover, the enthalpies of formation and the Gibbs free
energies of the PdpREq compounds extracted from the literature
[12,24–29] next to the experimental data, the values obtained from
thermodynamic optimization, and theoretical models are compared in Table 3. We note that the variation with uncertainty of
those enthalpies increase with the atomic number. It will be apparent from these figures that while there is reasonable agreement in
the order of magnitude between experimental and predicted values, there are also some significant differences. Our optimized values agree well with the noted variation of the (Sc, Y, La-Lu) with Pd
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S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
7
Fig. 3b. Calculated Gibbs free energies of the intermetallic compounds in the Pd-Yb binary system at T = 298 K, using Redlich-Kister model (
), using Kaptay model
(
) and at T = 1000 K, using Redlich-Kister model (
), using Kaptay model (
). The reference states for the elements are fcc_A1 for Pd and fcc_A1 for Yb.
Fig. 4a. Calculated integral (HMR) and partial (HPd and HYb) enthalpies of mixing of liquid phase at temperatures of 1000 K (continous and dashed lines), 2000 K (dash dotted
line), and 3000 K (dotted line) using two models Redlich-Kister (R-K) and Kaptay (K) along with experimental data of Ivanov et al. [21] in 0.63 < xYb < 1 range at T = 1300 K.
enthalpy of formation. As mentioned above, no experimental thermodynamic property of liquid has been reported in the literature.
In order to provide a reasonable prediction of the properties of liquid, we have plotted on the liquidus of the REEs–Pd binary systems. Moreover, the calculated enthalpies and entropies of
mixing of liquid phase are plotted, respectively. The calculated
integral and partial enthalpies of mixing of the Pd-Yb binary
system at T = 1300 K, compared with those found by Ivanov et al.
[30] using calorimetry for Pd-Eu melts are shown in Fig. 4a. We
noticed a good agreement in the Eu rich corner for both calculated
properties by applying two models with those measured in [30].
The value calculated by Entall Miedema Calculator developed by
De˛bski et al. [22] is in good agreement with our calculated values
in both cases. We have plotted the excess Gibbs energies of mixing
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S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
of the liquid phase at xYb = 0.1, 0.2, 0.3, and 0.4 as function of temperature from 0 to 4.104 K for 0L by applying the two models as
follows:
Redlich Kister model : DGExYb
¼ xYb ð1 xYb Þ ð269657:509 þ 7:929 TÞ
Kaptay model : DGExYb
¼ xYb ð1 xYb Þ ð255693:94 EXPð- 5:628E - 05 TÞÞ
As can be seen in the Fig. 4b, at high temperatures interval:
For Kaptay model: in accordance with the rule of Lupis and
Elliott proposed for the first time in 1966 [31] and recenly reformulated by Kaptay [98] as,‘‘Real solid, liquid and gaseous solutions (and pure gases) gradually approach the state of an ideal
solution (perfect gas) as temperature increases at any fixed
pressure and composition.’’, the excess Gibbs energy function
tends towards zero.
However, for Redlich-Kister model: this function changes its
sign from negative to positive and goes to more and more positive values which is probably unphysical [32].
Moreover, at low-T interval, the Redlich-Kister polynomial provides more negative excess Gibbs energy compared to the Kaptay
formalism. This behavior changes at middle-T interval, the Kaptay
formalism provides more negative excess Gibbs energy compared
to the Redlich-Kister polynomial. With our optimization, in both
cases, the evolutions of Gibbs free energy of mixing for the liquid
phase as a function of temperature T, show that when T is increasing up to 104 K, the Gibbs free energy for the liquid phase increases
in absolute value. The single liquid phase is stable at very high
temperatures (if gas phase is suspended). As we can see, no spinodal points were detected until a temperature of 104 K, and no formation of an inverted miscibility gap was detected during the
optimization in the region of validity of parameters. Theoretically,
as explained by [19,33], the appearance of the high-T artefact
Fig. 5a. Calculated heat contents (green continous line) for PdYb along with the
experimental values (
) [34] and least-squares fitted to the experimental data
(
) [34].(For interpretation of the references to colour in this figure legend, the
reader is referred to the web version of this article.)
34
Fig. 5b. Calculated heat capacity in J.(mol K)–1 (continous line) of the PdYb
intermetallic compounds along with least-squares fitted to the experimental data
(
) [34]. Calculated heat capacity in J.(mol K) –1(dashed line) of the Pd (
) and
Yb (
), respectively.
U
U
( b0 T >> aU
0 ) if the linear model is applied is a0 < 0 and
U
b0 > 2 R
Fig. 4b. Calculated excess Gibbs energies of the liquid phase in different temperature (0 – 40,000 K) and concentrations (xYb = 0.1, 0.2, 0.3, and 0.4) for Pd-Yb system
using Redlich-Kister (continous line) [17] and Kaptay formalisms (dashed line)
[18–20].
(sU
0 < 16:629).
In
the
case
of
our
sys-
U
temaU
0 ¼ - 269657:509 J/mol.at. and b0 ¼ - 7:929 J/mol.at.K (the
reader is referred to the Table 1), thus no miscibility gaps arises
using the linear model.
Heat contents growth of the intermediate valence compound
YbPd as a function of T are displayed in Fig. 5a. The calculated heat
contents increased continuously together with the experimentally
determined data and least squared fitted ones by [34] in the range
400 to 1200 K.
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S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
9
Fig. 6a. Calculated partial pressure lnpYb using two Redlich-Kister (dotted line) and Kaptay (dashed dotted line) formalisms along with experimental ones determined by
Ciccioli et al. [12] (continous line).
Fig. 6b. Calculated chemical activities of the Palladium and Ytterbium at different temperature T = 1000, 1500, 2000 and 3000 K by using Kaptay model [18–20].
A satisfactory agreement is noted. The calculated temperature
dependencies of specific heat for PdYb are shown in Fig. 5b
together with least squared fitted values [34]. As we can see the
transition temperature calculated in this study was 553 K. Furthermore, our calculated heat capacities are approximately higher and
they are slowly increasing when the amount of Yb increases in the
PdYb compound. A discrepancy is not higher. Ciccioli et al. [12]
described Yb gas properties by a simple formula log10pYb (bar) = 5
– 7340 / T (K). In Fig. 6a the lnpYb are plotted versus the tempera-
tures. The experimental values are also reported. In the temperature range and within the error margins, the variation of the
partial pressures be approximated by a linear function of the temperatures. A satisfactory agreement was achieved for exponential
model. The calculated evolution of the activities of Yb and Pd with
Kaptay formalism at different temperatures T = 1000, 1500, 2000,
and 3000 K are shown in Fig. 6b. The activities of palladium in
the ytterbium alloys are found to have a negative deviation from
Raoultian behavior over most of the composition range. Same
Please cite this article as: S. Kardellass, C. Servant, I. Drouelle et al., Calphad-type assessment of the Pd-Yb binary system supported by first-principles calculations – Part II: Results, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.196
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S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
Fig. 7a. Gibbs energies of the liquid phase and solid phases in the Pd-Yb binary system at T = 298.15 K using Redlich-Kister formalism.
Fig. 7b. Gibbs energies of the liquid phase and solid phases in the Pd-Yb binary system at T = 298.15 K using Kaptay formalism.
behevior was observed for the activities of ytterbium in palladium
alloys. Although the exponential model is proven to be an efficient
tool to avoid high-T artifacts (artificial inverted miscibility gap),
Schmid-Fetzer [33] claimed that it can lead to a low-T artefact,
i.e. to the artificial low-T re-stabilization of the liquid solution.
The results are shown in Figs. 7a, and b, respectively. As we
can see the exponential and linear models for the solid phases
provides much more negative Gibbs energy values compared to
the Gibbs energies of the liquid solution. Hence we can conclude
that our optimisation does not lead to any low-T artefact. The
Gibbs energy values of the liquid solution and solid phases are
close in palladium rich corner in the case of exponential model.
4. Summary
The phase relations and thermodynamic description of the
Palladium-Ytterbium binary system were critically evaluated from
the experimental information available in the literature for the first
time. A good agreement is obtained between experiments and
Please cite this article as: S. Kardellass, C. Servant, I. Drouelle et al., Calphad-type assessment of the Pd-Yb binary system supported by first-principles calculations – Part II: Results, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.196
S. Kardellass et al. / Materials Today: Proceedings xxx (xxxx) xxx
calculations. It is suggested that new and more reliable thermodynamic measurements are necessary and would help to refine the
present thermodynamic description.
Acknowledgements
The author of this work wants to thank M. A. Shevchenko from
Frantsevich Institute for Problems of Materials Science, Kiev,
Ukraine, actually Ph.D in the University of Queensland, Pyrometallurgy Research Centre, Brisbane, Australia for all help and advices.
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Please cite this article as: S. Kardellass, C. Servant, I. Drouelle et al., Calphad-type assessment of the Pd-Yb binary system supported by first-principles calculations – Part II: Results, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2019.08.196
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