CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 74 (2021) 102281

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Calphad
journal homepage: www.elsevier.com/locate/calphad

Thermodynamic modeling of the Te-X (X = Zr, Ce, Eu) systems

Chenchen Dong a, Jiong Wang a, *, Biao Hu b, Longpeng Zhu a, Qing Wu c, Yong Du a
a
Powder Metallurgy Research Institute, Central South University, Changsha, Hunan, 410083, China
b
School of Materials Science and Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001, China
c
Information & Network Center, Central South University, Changsha, Hunan, 410083, China

A R T I C L E I N F O A B S T R A C T

Keywords: Thermodynamic descriptions of the tellurium-zirconium (Te-Zr), tellurium-cerium (Te-Ce) and tellurium-
Te-Zr europium (Te-Eu) systems have been carried out using the CALPHAD (CALculation of PHAse Diagrams)
Te-Ce method based on the experimental phase equilibria data available in the literature and the enthalpies of for­
Te-Eu
mation calculated by first-principles calculations in the present work. The liquid phase was described by the
Phase diagram
CALPHAD
substitutional solution model for the Te-Zr and Te-Eu systems while the associated solution model for the Te-Ce
system with CeTe as associate. The intermetallics Zr3Te, ZrTe, Zr5Te4, ZrTe3, ZrTe5, CeTe, Ce4Te7, CeTe2, Ce2Te5,
CeTe3, Eu4Te7 and Eu3Te7 were treated as stoichiometric compounds while Zr1+xTe2, Ce3-xTe4 and Eu1-xTe were
modeled by the sublattice model Te1/3(Zr, Va)1/3(Zr, Te)1/3, (Ce, Va)3/7Te4/7 and (Eu, Va)0.5Te0.5, respectively,
on the basis of their homogeneity ranges and crystal structures. A set of self-consistent thermodynamic pa­
rameters for the Te-X (X = Zr, Ce, Eu) systems was obtained. Comparisons between the calculated results and
experimental data available in the literature show that most reliable experimental information can be satisfac­
torily accounted for by the present modeling.

1. Introduction highly efficient thermoelectric material systems under high temperature

situations [17] because of their outstanding thermal stability [18]. It has
The chalcogenides attract much attention because of their remark­ been found that Ce3Te4 could be a potential high temperature thermo­
able properties, which enable a wide array of applications including electric material with high efficiency [19,20]. Additionally, the Eu
phase change materials and thermoelectric devices [1–3]. Nonvolatile chalcogenides with NaCl structure are magnetic semiconductors pos­
phase-change random-access memory (PCRAM) is deemed to be a sessing potential applications in the fields of semiconductor [21]. The
leading candidate for next-generation electronic memory hierarchy [4]. knowledge of phases equilibria and thermodynamic properties is vital
PCRAM products currently employed Ge2Sb2Te5 (GST) as the core ma­ for binary and higher-order systems [15]. The design and development
terial for memory programming [5–7]. And many doping elements [6] of chalcogenides materials require accurate information of the phase
have been studied to obtain better properties compared with GST, equilibria and thermodynamic properties of the Te-based systems.
including Sc [5], Zr [8], Cu [9], Sn [10], Ce [11], In Ref. [12], Ag [13] No thermodynamic assessment of the Te-Zr system was reported in
and Ti [14], etc. The design of chalcogenides based novel materials the literature. A thorough assessment of the Te-Zr system is necessary to
needs detailed information about the phase equilibria and thermody­ construct reliable thermodynamic descriptions of the Te-based systems.
namic properties [15]. Yuan et al. [16] have studied the phase diagram Besides, the Te-Ce and Te-Eu systems have been assessed by Philipp et al.
and thermodynamics of As, Si and Co doping providing reliable ther­ [22] and Ghamri et al. [23], respectively. Philipp et al. [22] assessed the
modynamics basis for the design of phase change materials. Zr-doped Te-Ce system by using ChemSage program [24] on the basis of the result
Sb2Te phase change memory application has better thermal properties from Okamoto [25]. However, their models are not consistent with the
(smaller density change and reset current than GST) and refined grain models used in our previous work [16]. Furthermore, they thought it
size in comparison with Sb2Te film [8]. Therefore, the present work was was not possible to adapt the eutectic reactions occurring in the phase
dedicated to study the doping element of Zr after critical literature re­ diagram especially the reaction Liquid → CeTe + Ce3Te4 at 1858 K.
view. Furthermore, rare-earth chalcogenides were viewed as one of Instead, they hypothesized that the Ce3Te4 phase was formed by a

* Corresponding author.
E-mail addresses: wangjionga@csu.edu.cn, wangjionga@gmail.com (J. Wang).

https://doi.org/10.1016/j.calphad.2021.102281
Received 20 January 2021; Received in revised form 4 May 2021; Accepted 9 May 2021
Available online 11 June 2021
0364-5916/© 2021 Elsevier Ltd. All rights reserved.
C. Dong et al. Calphad 74 (2021) 102281

Table 1
Crystal structures of phases in the Te-Zr, Te-Ce and Te-Eu systems.
System Phase Person symbol/Prototype Space group Lattice parameter (Å) Reference

a b c

Te-Zr (αZr) hP2 P63 mmc 3.233 3.233 5.149 [37]

Mg
(βZr) cI2 Im3m 3.59 3.59 3.59 [28]
W
Zr3Te tI* I4 11.3382 11.3382 5.6265 [43]
Ni3P
Zr5Te4 tI18 I4/m 10.763 10.763 3.840 [35]
Ti5Te4
ZrTe hP* P6/mmm 3.763 3.763 3.862 [35]
–
Zr1+xTe2 hP4 P63 /mmc 3.95 3.95 6.63 [40]
NiAs
ZrTe3 mP8 P21 /m 5.89 3.93 10.1 [38]
ZrSe3
ZrTe5 oC24 Cmcm 3.9876 14.502 13.727 [36]
HfTe5
Te-Ce (γCe) cF4 Fm3m 5.1612 5.1612 5.1612 [32]
Cu
(δCe) cI2 Im3m 4.12 4.12 4.12 [29]
W
CeTe cF8 Fm3m 6.36 6.36 6.36 [31]
NaCl
Ce3-xTe4 cI28 I43d 9.57 9.57 9.57 [34]
Th3Te4
Ce4Te7 tP22 P4mbm – – – [30]
–
CeTe2 tP72 P4/nmm 4.653 4.653 9.305 [41]
CeTe2
Ce2Te5 oS28 Cmcm 4.444 44.5 4.444 [34]
Nd2Te5
CeTe3 oS16 Ama2 4.398 4.398 25.99 [34]
CeTe3
(Te) hP3 P31 21 4.44 4.44 5.92 [42]
γSe
Te-Eu (Eu) cI2 Im3m 4.581 4.581 4.581 [33]
W
Eu1-xTe cF8 Fm3m 6.6 6.6 6.6 [44]
NaCl
Eu4Te7 – – – – – [39]
–
Eu3Te7 – – – – – [39]
–

peritectic reaction like other tellurium-rich compounds. Additionally, 2.1. The Te-Zr system
Philipp et al. [22] ignored the homogeneity of Ce3-xTe4 and CeTe2-x for
simplification. Therefore, a more detailed thermodynamic description is The Te-Zr system was firstly reported by Sodeck et al. [38], mainly
needed. As for the Te-Eu system, the result from Ghamri et al. [23] can investigating the temperatures of melting and decomposition of several
smoothly describe the experimental data. In this work, an alternative set tellurium-rich compounds, from which the ZrTe3 phase was considered
of thermodynamic parameters was obtained with the help of to exist under 903 K. However, it was found that ZrTe3 still exists at
first-principles calculations. And Eu1-xTe was modeled by a new sub­ temperatures above 903 K [49,50], which indicates that the reported
lattice model based on the first-principles calculations results. Besides, it decomposition temperatures of the tellurium-rich compounds are
was found that Eu1-xTe has the same crystal structure with βGeTe [26] doubtful. After that, Cordfunke and Konings [51] as well as Chatto­
and HoTe [27], which were successfully described by using vacancy padhyay and Bharadwaj [52] assessed all the available thermochemical
substitution. Thus, Te-Ce and Te-Eu systems are reevaluated in this work data of the Te-Zr system. However, the phase equilibria in the Zr-rich
to keep consistent with the database [16]. The present work is devoted part of the Te-Zr system were lacking. Then de Boer and Cordfunke
to obtain a set of thermodynamic parameters for the Te-X (X = Zr, Ce, [48] reinvestigated this system by X-ray powder diffraction (XRD) and
Eu) systems via a combined approach of CALPHAD and first-principles differential thermal analysis (DTA) measurements. In their work, seven
calculations for enthalpies of formation of intermetallic compounds. compounds, Zr3Te, Zr5Te4, ZrTe, Zr1+xTe2, “O”, ZrTe3 and ZrTe5, were
found and among which only the Zr1+xTe2 has a homogeneity range. It
2. Literature review was worth mentioning that the “O” phase is a compound at 52 at.% Te
and is different from the ZrTe2 phase in crystal structure. In the present
All the information of crystal structure and lattice parameters for work, the “O” phase was excluded because it has only been proposed by
each element and compound in the Te-X (X = Zr, Ce, Eu) systems de Boer and Cordfunke [48] so far, and its crystal structure has not been
[28–44] are listed in Table 1. In order to facilitate reading, the experi­ verified yet. Besides, the melting temperatures of Zr3Te, Zr5Te4, ZrTe3
mental data on the phase diagram [30,38,45–48] of these systems are and ZrTe5 have been determined, and a tentative phase diagram was
briefly summarized in Table 2. Each of the binary systems is critically constructed by de Boer and Cordfunke [48]. And the liquidus between
reviewed as follows. 22 and 65 at.% Te were determined where liquid-liquid phase separa­
tion was proposed based on liquidus temperatures and the fact that

2
C. Dong et al. Calphad 74 (2021) 102281

Table 2 by means of X-ray and thermal analysis, Pardo and Flahaut [60] sug­
Summary of the experimental data on phase equilibria in the Te-X (X = Zr, Ce, gested that the Te-rich eutectic occurs instead at 435 ◦ C and 99 at.% Te,
Eu) systems. which is thought to be more accurate. The remaining portion of the
System Experimental technique Temperature/ Quoted Reference diagram is based on the results of Chukalin et al. [30]. Then Okamoto
Composition/ Mode [61] assessed the Te-Ce system based on the work of Moffat [25], whose
Phase range work was in line with Gschneidner et al. [59]. Seven intermediate
studied
phases (CeTe, Ce3Te4, Ce2Te3, Ce4Te7, CeTe2, Ce2Te5 and CeTe3) are
Te-Zr Isopiestic measurement, 712-1970 K + [48] known in the Te-Ce system, in which CeTe and Ce3Te4 form congruently
differential thermal 30-90 at.% Te
[61]. Chukalin et al. [30] reported that solid solutions exist from Ce3Te4
analysis and X-ray Homogeneity
powder diffraction range of the to Ce2Te3 (57.1–60.0 at.% Te) and CeTe1.9 to CeTe2 (66.0 to 66.67 at.%
Zr1+xTe2 Te). The homogeneity of CeTe2 is ignored in the present work
X-ray powder diffraction 718-1921 K + [38]
and differential thermal 10-87 at.% Te
analysis Homogeneity Table 3
range of the Summary of experimental and calculated data used for the enthalpies of the
Zr1+xTe2 formation from first-principles (0 K) and CALPHAD (room temperature).
Te-Ce X-ray powder diffraction, 728-2093 K + [30]
metallographic and 0-100 at.% Te Compound △f H, J/mole- Method Quoted Reference
thermal analysis Homogeneity atom Mode
range of the Ce3-
Zr3Te − 48525 First-principles + This work
xTe4
− 48437 First-principles – [88]
X-ray powder diffraction 708-2093 K + [60]
− 48244 First-principles – [89]
and thermal analysis 65-100 at.% Te
− 48981 CALPHAD This work
Te-Eu Differential thermal 683-1806 K + [46]
Zr5Te4 − 92111 ± 4800 Calorimetry – [43]
analysis and X-ray 44-64 at.% Te
− 82258 First-principles + This work
powder diffraction Homogeneity
− 82304 First-principles – [88]
range of the Eu1-
− 82690 First-principles – [89]
xTe
− 82619 CALPHAD This work
Differential thermal Homogeneity – [47]
ZrTe − 86741 First-principles + This work
analysis, chemical range of the Eu1-
− 86839 First-principles – [88]
analysis and X-ray xTe
− 86646 First-principles – [89]
powder diffraction
− 89000 CALPHAD This work
Quoted Mode, which indicates whether the data are used in the parameter Zr1+xTe2 − 103800 ± 2500 Calorimetry – [43]
optimization: + used; - not used. − 98000 ± 2300 Calorimetry – [57]
− 85010 First-principles + This work
− 84615 First-principles – [88]
liquid-liquid immiscibility region was suggested to showcase in several − 85006 First-principles – [89]
transition-metal tellurium phase systems [48] based on Chattopadhyay − 89589 CALPHAD This work
and Bharadwaj [52]. But it was depicted that there is most probably only ZrTe3 − 69335 First-principles + This work
− 67059 First-principles [88]
one liquid phase existing in the Te-Zr system [52]. Besides, it is found
–
− 69182 First-principles – [89]
that only elements Cr, Mn, Fe, Co, Cu and Ag with Te have been − 70049 CALPHAD This work
determined having an immiscibility region in the liquid [16,52–55], ZrTe5 − 49089 First-principles + This work
most of which belong to the first transitional system. Therefore, both the − 48629 First-principles – [88]
− 48340 First-principles [89]
liquid-liquid phase separation and the “O” phase were excluded in the –
− 49422 CALPHAD This work
present work, which will directly cause higher melting point of Zr5Te4 in CeTe − 168000 ± 4000 Calorimetry – [64]
order to coordinate the liquidus properly. Then, Okamoto [56] assessed − 131786 First-principles + This work
the Te-Zr phase diagram and summarized crystal structure data ac­ − 140486 First-principles – [88]
cording to the work of de Boer and Cordfunke [48]. There are four − 131513 First-principles – [89]
− 131785 CALPHAD This work
peritectic reactions, Liquid + Zr5Te4 → Zr3Te (1627 ± 6 K), Liquid +
Ce3-xTe4 − 163000 ± 4000 Calorimetry – [64]
ZrTe2 → Zr5Te4 (1795 ± 18 K), Liquid + ZrTe2 → ZrTe3 (1231 ± 2 K), − 135096 First-principles + This work
Liquid + ZrTe3 → ZrTe5 (822 K), and one peritectoid reaction Zr5Te4 + − 143574 First-principles – [88]
ZrTe2 → ZrTe (1505 ± 3 K). Besides, there are two eutectic reactions − 134118 First-principles – [89]
Liquid → (Zr) + Zr3Te and Liquid → (Te) + ZrTe5 at 1616 ± 3 K and 718 − 135096 CALPHAD This work
Ce4Te7 − 129519 CALPHAD This work
K [48], respectively. The enthalpy of formation of ZrTe1.843 was CeTe2 − 117326 First-principles + This work
measured by Johnson [57] by means of drop calorimetry and de Boer − 124566 First-principles – [88]
and Cordfunke [43] by solution calorimetry. de Boer and Cordfunke also − 115206 First-principles – [89]
measured the enthalpy of formation of Zr5Te4 phase [43]. The integral − 124980 CALPHAD This work
Ce2Te5 − 104524 First-principles This work
Gibbs energy of formation at four different temperatures (1457, 1472, +
− 108934 First-principles – [88]
1565, and 1644 K) were also estimated from the measured partial Gibbs − 139714 First-principles – [89]
energy over the composition range between ZrTe1.843 and Zr5Te4 [58]. − 106980 CALPHAD This work
CeTe3 − 95706 First-principles + This work
− 97260 First-principles – [88]
2.2. The Te-Ce system − 97163 First-principles – [89]
− 94090 CALPHAD This work
Chukalin et al. [30] studied the Te-Ce system by means of X-ray, Eu1-xTe − 133000 ± Miedema’s – [66]
13000 model
metallographic and thermal analysis using 99.5% pure Ce and 99.99%
− 194974 ± 1674 Calorimetry – [65]
pure Te. However, the reported Te-rich eutectic temperature (455 ◦ C) is − 175504 First-principles + This work
appreciably higher than the accepted melting point of Te (449.6 ◦ C), − 191207 CALPHAD – [23]
indicating that the purity of the Te stock was probably much lower than − 175503 CALPHAD This work
stated [59] which would especially affect the Te-rich portion of the di­ Quoted Mode, which indicates whether the data are used in the parameter
optimization: + used; - not used.
agram. From an examination of this system between 65 and 100 at.% Te

3
C. Dong et al. Calphad 74 (2021) 102281

Table 4
Calculated invariant reactions compared with the literature data in the Te-Zr, Te-Ce and Te-Eu systems.
System Reaction Reaction type T(K) Composition (at.% Te) Method Reference

Te-Zr Liquid → (Zr) + Zr3Te Eutectic 1616 ± 3 21.0 Experiment [48]

Eutectic 1616 20.0 CALPHAD This work
Zr5Te4 + Liquid → Zr3Te Peritectic 1627 ± 6 22.0 Experiment [48]
Peritectic 1621 22.5 CALPHAD This work
“O” + Liquid → Zr5Te4 Peritectic 1795 ± 18 33.0 Experiment [48]
Liquid + Zr1+xTe2 → Zr5Te4 Peritectic 1837 44.4 CALPHAD This work
Zr5Te4 + “O” → ZrTe Peritectoid 1505 ± 3 – Experiment [48]
Zr5Te4 + ZrTe2 → ZrTe Peritectoid 1505 – CALPHAD This work
Liquid + ZrTe2 → ZrTe3 Peritectic 1231 ± 2 92.0 Experiment [48]
Liquid + ZrTe2 → ZrTe3 Peritectic 1231 88.9 CALPHAD This work
Liquid + ZrTe3 → ZrTe5 Peritectic 822 97.0 Experiment [48]
Peritectic 822 97.3 CALPHAD This work
Liquid → (Te) + ZrTe5 Eutectic 718 98.0 Experiment [48]
Eutectic 720 99.2 CALPHAD This work
Te-Ce Liquid → CeTe + (Ce) Eutectic 1033 2.2 Assessed [61]
Eutectic 1033 3.3 Experiment [30]
Eutectic 1035 2.2 CALPHAD This work
Liquid → CeTe Congruent 2093 50.0 Assessed [61]
Congruent 2093 50.0 Experiment [30]
Congruent 2091 50.0 CALPHAD This work
Liquid → CeTe + Ce3Te4 Eutectic 1858 55.0 Assessed [61]
Eutectic 1858 55.0 Experiment [30]
Eutectic 1866 55.0 CALPHAD This work
Liquid → Ce3Te4 Congruent 1918 57.1 Assessed [61]
Congruent 1882 57.1 Experiment [30]
Congruent 1911 57.1 CALPHAD This work
Liquid + Ce3Te4 → Ce4Te7 Peritectic 1618 67.1 Assessed [61]
Peritectic 1613 66.2 Experiment [30]
Peritectic 1616 67.2 CALPHAD This work
Liquid + Ce4Te7 → CeTe2 Peritectic 1523 69.3 Assessed [61]
Peritectic 1523 70.0 Experiment [30]
Peritectic 1524 70.4 CALPHAD This work
Liquid + CeTe2 → Ce2Te5 Peritectic 1143 80.1 Assessed [61]
Peritectic 1146 93.0 Experiment [30]
Peritectic 1142 84.4 CALPHAD This work
Ce2Te5 → CeTe2 + CeTe3 Peritectoid 880 – Assessed [61]
Peritectoid 883.8 – CALPHAD This work
Liquid + Ce2Te5 → CeTe3 Peritectic 1102 83.3 Assessed [61]
Peritectic 1088 94.8 Experiment [30]
Peritectic 1109 85.8 CALPHAD This work
Liquid → CeTe3 + (Te) Eutectic 708 99.0 Assessed [61]
Eutectic 718 100 Experiment [30]
Eutectic 719 98.7 CALPHAD This work
Te-Eu Liquid → (Eu) + EuTe Eutectic 1071 3.98 Experiment [46]
Eutectic 1053 4.98 Experiment [39]
Eutectic 1053.19 2.82 CALPHAD [23]
Eutectic 1053 3.28 CALPHAD This work
Liquid → EuTe Congruent 1806 50.0 Experiment [46]
Congruent 1799 50.0 Experiment [39]
Congruent 1799.09 50.0 CALPHAD [23]
Congruent 1799 50.0 CALPHAD This work
Liquid + EuTe → Eu4Te7 Peritectic 793 91.48 Experiment [46]
Peritectic 793 89.88 Experiment [39]
Peritectic 793.20 92.68 CALPHAD [23]
Peritectic 793 92.35 CALPHAD This work
Liquid + Eu4Te7 → Eu3Te7 Peritectic 703 95.05 Experiment [46]
Peritectic 703 94.01 Experiment [39]
Peritectic 703.09 95.51 CALPHAD [23]
Peritectic 703 95.20 CALPHAD This work
Liquid → (Te) + Eu3Te7 Eutectic 683 97.8 Experiment [46]
Eutectic 683 98.0 Experiment [39]
Eutectic 683.23 96.16 CALPHAD [23]
Eutectic 684 95.87 CALPHAD This work

considering its narrow and insignificant range. According to Pardo and depicted by Okomoto [61], especially the striking one between the CeTe
Flahaut [60], Ce2Te5 exists only from 880 ± 19–1143 K. Three eutectic and Ce3Te4. Instead, they assumed the Ce3Te4 was formed through a
reactions Liquid → (Ce) + CeTe, Liquid → CeTe + Ce3Te4, and Liquid → peritectic reaction like other tellurium-rich compounds. Additionally,
CeTe3 + (Te) occurring at 1033, 1858 and 708 K, and two congruent Philipp et al. [22] ignored the homogeneity of Ce3-xTe4 and CeTe2 for
reactions Liquid → CeTe and Liquid → Ce3Te4 at 2093 K and 1918 K simplification. The enthalpies of formation of all the phases were
[61], respectively. computed by fixing the value of the CeTe phase as a reference value
Philipp et al. [22] assessed the Te-Ce system by using ChemSage during the modeling in the ChemSage [24].
program [24] on the basis of the result from Okamoto [61]. However, Thermodynamic data of the Te-Ce system are rare due to the high
they thought it was not possible to adjust the eutectic reactions as vapor of Te [62] which is a great obstacle for experimental accuracy and

4
C. Dong et al. Calphad 74 (2021) 102281

feasibility. Thermodynamic data have been experimentally proven for database compiled by Dinsdale [67].
CeTe by means of vapor pressure measurements using direct calorimetry
by Mills et al. [63], who also provides estimated data for Ce2Te3 3.1.2. Solution phases
(Ce3-xTe4). Additionally, Ferro et al. [64] measured the standard All the solution phases, i.e., liquid in the Te-Zr system, (αZr), (βZr),
enthalpy of the CeTe and Ce3-xTe4 phases by direct calorimetry method. (γCe), (δCe), (Eu), and (Te) are described as substitutional solutions. No
experimental data were found in the literature about mutual solubilities
between Zr, Ce, Eu and Te. Thus, the terminal solid solutions were not
2.3. The Te-Eu system taken into account in this work. The Gibbs free energy of the phase can
be expressed as follows:
Okamoto [39] has evaluated this system based on the experimental
(2)
◦ φ ◦ ◦ φ
data from Sadovskaya and Yarembash [46]. Two terminal solution Gm = xA GφA + xTe GφTe + RT[xA ln(xA ) + xTe ln(xTe )] + E Gm
phases, i.e., BCC_A2 (Eu) and HEXAGONAL_A8 (Te), and three inter­
where xA is mole fraction of elements A (A = Zr, Ce, Eu), o GφA and o GφTe
metallic compounds, i.e., Eu1-xTe, Eu4Te7 and Eu3Te7, exist in the Te-Eu
system, in which the Eu1-xTe phase melts congruently and displays a are the molar Gibbs free energy of the pure elements A (A = Zr, Ce, Eu)
solubility range from 50 to 57.1 at.% Te [39,46,47]. There are two and Te, adopting the SER reference state, which is taken from the SGTE
pure element database [67]. R is the gas constant and the term E Gm
φ
eutectic reactions: Liquid → (Eu) + EuTe and Liquid → (Te) + Eu3Te7.
The former was reported to occur at (1053 ± 3) K [39] or 1071 K [46]. presents the excess Gibbs free energy, which is expressed in
Sadovskaya and Yarembash [46] reported the congruent melting tem­ Redlich-Kister polynomial form [68] as:
perature of Eu1-xTe at 1806 K compared to 1799 K assessed by Okamoto [ ]
(3)
E φ Liq Liq n
[39]. In the present work, the latest result assessed by Okamoto [39] was Gm = xA xTe 0 LA,Te + 1 LA,Te (xA − xTe ) + ⋯⋯ + n LLiq
A,Te (xA − xTe )

adopted. The solubility of each element in the Te-Eu terminal solid so­
lutions was not considered because there was no literature reporting the
n Liq
LA,Te = an + bn T (4)
solubility of terminal solid solutions.
There are some reported experimental thermodynamic properties, where n LLiq
A,Te (n = 0, 1, 2 …) is the binary interaction parameter, an and
including the enthalpies of the Eu1-xTe phase from 300 to 1725 K. The bn (n = 0, 1, 2 …) are the coefficients to be evaluated on the basis of
standard entropies, heat capacities at high temperature and Gibbs free available experimental data.
energies of formation were calculated as a function of temperature by
McMasters et al. [65]. Besides, the standard enthalpy of formation of the 3.1.3. Associate solution model
Eu1-xTe phase was also predicted via Miedema’s model by Eichler et al. Solution model was used for the liquid phase to describe the Te-Ce
[66]. Ghamri et al. [23] evaluated the Te-Eu system for the first time phase diagram at first but it was found impossible to reach the
mainly according to the result of Okamoto [39], in which the standard eutectic between the CeTe and Ce3-xTe4 and the asymmetric liquidus.
enthalpy value of Eu1-xTe from McMaster et al. [65] was taken into Then, the associate solution model was adopted with CeTe as the asso­
account and two sublattice model (Eu, Te)0.5Te0.5 was used to describe ciate in the liquid based on the feature that the liquidus is steep and the
the Eu1-xTe phase. stable intermetallic phases own even higher melting temperatures than
Available experimental enthalpies of formation of line compounds the pure elements. The choice of CeTe as the associate can be verified in
are listed in Table 3. Detailed information of all the invariant reactions two aspects. One is a strongly curved liquidus around the CeTe inter­
above can be found in Table 4. mediate phase indicates a high thermodynamic stability around that
composition. The other is the measured enthalpies of mixing for the
3. Methodology liquid phase show an abrupt change around 50 at.% Te. Consequently,
the Gibbs free energies of the binary liquid for 1 mol species are given:
3.1. Thermodynamic models

[ Liq ] E Liq
o
GLiq = yLiq o Liq Liq o Liq Liq o Liq Liq Liq
Ce GCe (T) + yTe GTe (T) + yCeTe GCeTe (T) + RT yCe ln(yCe ) + yCeTe ln(yCeTe ) + yTe ln(yTe ) + Gm (5)

Gibbs energy of all the phases existing in the Te-X (X = Zr, Ce, Eu)
systems will be introduced in this section. o
GLiq o Liq o Liq
CeTe (T) = GCe + GTe (T) + a + bT (6)
3.1.1. Pure elements Liq Liq Liq
For pure components, its Gibbs energy is just related to temperature
E
Gm = 0 LCe, ​ CeTe yLiq Liq 0 Liq Liq
Ce yCeTe + LCeTe,Te yCeTe yTe

and pressure which features the lattice stability. The relationship be­ Liq
+ 1 LCe,CeTe yLiq Liq
(7)
Ce yCeTe ( yCe − yCeTe )
tween Gibbs free energy and temperature can be expressed as follows:
◦ φ
GA (T) = GφA (T) − HiSER = a + bT + cTlnT + dT 2 + eT 3 + fT − 1 + gT 7 where o GLiq o Liq o Liq
Ce , GTe and GCeTe are the molar Gibbs free energies of pure

+ hT − 9
(1) liquid Ce, Te, and CeTe, respectively; yLiq
Ce , yTe and yCeTe are the site
Liq Liq

Liq Liq Liq

fractions of species in the liquid; 0 LCe, ​ CeTe , 0 LCeTe, ​ Te and 1 LCe, ​ CeTe are
where T is the absolute temperature, and a, b, c, d, e, f, g and h are co­ the binary interaction parameters to be evaluated.
efficients. Generally, the enthalpy of stable component is assumed to
0 under 298.15 K and 101325 Pa. The standard element reference (SER) 3.1.4. Sublattice model
state for Zr, Ce, Eu and Te are HCP_A3, FCC_A1, BCC_A2 and HEX­ The phase Zr1+xTe2 with NiAs-type structure [40] exhibits a
AGONAL_A8, respectively. In this work, the values of these coefficients considerable solubility range, it was described using a three sublattice
for each element A (A = Zr, Ce, Eu and Te) in the φ phase were taken model, Te1/3(Zr, Te)1/3(Zr, Va)1/3, out of the consideration of its crystal
from the SGTE (Scientific Group Thermodata Europe) pure element structure and aimed homogeneity range. The defects in the Zr atom

5
 C. Dong et al. Calphad 74 (2021) 102281

Table 5
Thermodynamic parameters for the Te-Zr, Te-Ce and Te-Eu systems.
System Phase Model Parameter

Te-Zr Liquid (Zr, Te) 0 Liq

LZr,Te = − 161604 − 39.994T
1 Liq
LZr,Te = − 57002 − 21.796T
2 Liq
LZr,Te = − 23537 + 10T
Zr3Te (Zr)0.75(Te)0.25 o
GZr3 Te o Hex A8
+ 0.75o GHcp_A3
Zr:Te = − 48525 + 2.072T + 0.25 GTe Zr
Zr5Te4 (Zr)5/9(Te)4/9 o
GZr5 Te4
= − 82619 + 2.383T + 4/9 o Hex A8
GTe + 5/9o GHcp_A3
Zr:Te Zr
ZrTe (Zr)0.5(Te)0.5 o
GZrTe = − 88239 + 2.547T + 2/3 o
GHex A8
+ 1/3o GHcp_A3
Zr:Te Te Zr
Zr1+xTe2 (Te)1/3(Zr, Te)1/3(Zr, Va)1/3 o
GZr1+x Te2
= − 41358 + 3.400T + 1/3 o Hex A8
GTe + 2/3o GHcp_A3
Te:Zr:Zr Zr
o
GZr1+x Te2
Te:Te:Zr = − 89589 + 1.950T + 2/3 o Hex A8
GTe + 1/3o GHcp_A3
Zr
o
GZr1+x Te2
Te:Te:Va = + 42985 + 2/3 o Hex A8
GTe
0 Liq
LTe:Zr,Te:Va = − 58261 + 4.761T
0 Liq
LTe:Zr,Te:Zr = − 79898 + 4.441T
1 Liq
LTe:Zr,Te:Zr = − 26322 + 4.075T
ZrTe3 (Zr)0.25(Te)0.75 o
GZrTe o Hex A8
+ 0.25o GHcp_A3
Zr:Te = − 70660 + 0.210T + 0.75 GTe
3
Zr
ZrTe5 (Zr)1/6(Te)5/6 o
GZrTe 5
= − 49422 + 1.737T + 5/6 o
GHex A8
+ 1/6o GHcp_A3
Zr:Te Te Zr
Te-Ce Liquid (Ce, CeTe, Te) 0 Liq
LCe,CeTe = + 3004 + 1.996T
1 Liq
LCe,CeTe = + 5979 − 1.122T
0 Liq
LCeTe,Te = − 126985 − 17.604T
1 Liq
LCeTe,Te = − 7879 − 9.917T
2 Liq
LCeTe,Te = − 8000 + 15T
CeTe (Ce)0.5(Te)0.5 o
GCeTe o Hex A8
+ 0.5o GFcc_A1
Ce:Te = − 131785 + 3.612T + 0.5 GTe Ce
Ce3-xTe4 (Ce, Va)3/7(Te)4/7 o
GCe 3− x Te4
= − 135096 + 2.175T + 4/7 o Hex A8
GTe + 3/7o GFcc_A1
Ce:Te Ce
o
GCe 3− x Te4
Va:Te = + 27369 − 10.230 + 3/7 o Hex A8
GTe
0 Ce3− x Te4
LCe,Va:Te = − 42710 + 9.895T
1 Ce3− x Te4
LCe,Va:Te = − 52854 + 1.996T
Ce4Te7 (Ce)4/11(Te)7/11 o
GCe 4 Te7
= − 129519 + 3.120T + 7/11o GHex A8
+ 4/11o GFcc_A1
Ce:Te Te Ce
CeTe2 (Ce)1/3(Te)2/3 o
GCeTe 2
= − 124980 + 4.300T + 2/3 o
GHex A8
+ 1/3o GFcc_A1
Ce:Te Te Ce
Ce2Te5 (Ce)2/7(Te)5/7 o
GCe 2 Te5
= − 106980 + 1.620T + 5/7 o Hex A8
GTe + 2/7o GFcc_A1
Ce:Te Ce
CeTe3 (Ce)0.25(Te)0.75 o
GCeTe 3
= − 94090 + 0.3T + 0.25 o
GHex A8
+ 0.75o GFcc_A1
Ce:Te Te Ce
Te-Eu Liquid (Eu, Te) 0 Liq
LEu,Te = − 280003 − 71.500T
1 Liq
LEu,Te = + 20587 + 50.066T
2 Liq
LEu,Te = + 19852
Eu1-xTe (Eu, Va)0.5Te0.5 o
GEu 1− x Te
= − 176515 + 26.480T + 0.5o GBcc_A2 + 0.5o GHex_A8
Eu:Te Eu Te
o
GEu 1− x Te
Va:Te = + 39618 + 0.5o GHex_A8
Te
0 Eu1− x Te
LEu,Va:Te = − 81502 + 14.536T
1 Eu1− x Te
LEu,Va:Te = − 45000
Eu4Te7 Eu4/11Te7/11 o
GEu 4 Te7 o Bcc_A2
+ 7/11o GHex_A8
Eu:Te = − 144024 + 26.716T + 4/11 GEu Te
Eu3Te7 Eu0.3Te0.7 o
GEu 4 Te7 o Bcc_A2
+ 7/11o GHex_A8
Eu:Te = − 144024 + 26.716T + 4/11 GEu Te

Note: Gibbs energy in J/mol-atom, temperature (T) in Kelvin and pressure (P) in Pa. The Gibbs energies for the pure elements are from the SGTE compilation.

layers cause a reduction in overall Te-Zr bonding interactions lead to of formation is not consistent at 66.66 at.% Te. There is a difference lies
NiAs-type according to the approach of Hückel calculations [69], which in the homogeneity range among the systems. In the Te-Zr system the
means that the Zr layers are occupied by vacancies. Besides, after range is 54-66.66 at.% Te while in the Te-Co and Co-Sb systems is
reviewing all the NiAs-type compounds in literature [70], only the 55.4-64.2 at.% Te. Therefore, the model of Te1/3(Zr, Va)1/3(Va, Zr)1/3
βCoTe and CoSb in the Te-Co and Co-Sb systems have been assessed seems impossible to reproduce such wider range. Therefore, in the
thermodynamically with a three sublattice model Te1/3(Co, Va)1/3(Va, present work we employed the model of Te1/3(Te, Zr)1/3(Va, Zr)1/3 to
Co)1/3 [16] and Sb1/3(Co, Va)1/3(Va, Co)1/3 [71]. Thus, a three sub­ modify the homogeneity range of Zr1+xTe2. According to the general
lattice model Te1/3(Zr, Va)1/3(Va, Zr)1/3 was used in this work. How­ sublattice model [72,73], the Gibbs free energy of the phase is
ever, the result cannot reproduce the homogeneity well and the enthalpy expressed:

6
C. Dong et al. Calphad 74 (2021) 102281

′′′ ′′′ ′′′ ′′′

o
GZr1+x Te2 = y′′Zr yVa o GZr 1+x Te2 ′′ o Zr1+x Te2 ′′ o Zr1+x Te2 ′′ o Zr1+x Te2
Te:Zr:Va + yZr yZr GTe:Zr:Zr + yTe yZr GTe:Te:Zr + yTe yVa GTe:Te:Va
/ ( ′′ ′′′ ′′′ ′′′ ′′′ ) Zr Te (8)
+1 3RT yZr lny′′Zr + y′′Te lny′′Te + yZr lnyZr + yVa lnyVa + E Gm 1+x 2

( ′′′ ) ( )
E Zr Te2 Zr1+x Te2 ′′′ Zr1+x Te2 ′′′ ′′′ Zr1+x Te2 Zr1+x Te2
Gm 1+x = y′′Zr y′′Te yZr 0 LTe:Zr,Te:Zr + yVa 0 LTe:Zr,Te:Va + yVa yZr y′′Zr 0 LTe:Zr:Zr,Va + y′′Te 0 LTe:Te:Zr,Va
( ′′′ )( )
(9)
Zr1+x Te2 ′′′ Zr1+x Te2
+y′′Zr y′′Te yZr 1 LTe:Zr,Te:Zr + yVa 1 LTe:Zr,Te:Va y′′Zr − y′′Te
( ) ( ′′′ ′′′ )
′′′ ′′′ Zr1+x Te2 Zr1+x Te2
+yVa yZr y′′Zr 1 LTe:Zr:Zr,Va + y′′Te 1 LTe:Te:Zr,Va yZr − yVa ​

tively, in the first sublattice; v LCe 3− x Te4

Ce,Va:Te (v = 1, 2, …) is the binary inter­

action parameter, which can be written as v LCe 3− x Te4

Ce,Va:Te = av + bv T, av and
bv (v = 1, 2, …) are the parameters to be evaluated on the basis of
available experimental data. Since the experimental data involving Ce3-
xTe4 are limited, it would be difficult to assess the quantities in Eq. (10)
directly. In the present work, Ce3-xTe4 was modeled as stoichiometric
compound at first, then treated by a two-sublattice model (Ce, Va)3/
7(Te)4/7.

3.1.5. Stoichiometric compounds

The intermetallic phases Zr3Te, ZrTe, Zr5Te4, ZrTe3, ZrTe5, CeTe,
Ce4Te7, CeTe2, Ce2Te5, CeTe3, Eu4Te7 and Eu3Te7 were treated as stoi­
chiometric compounds with a certain composition of Te. Here, taking
Zr3Te as an example, its Gibbs energy per mole atom can be expressed as
following equation:
o
GZr
m
3 Te SER
− 0.75HZr SER
− 0.25HTe = aZr3 Te + bZr3 Te T + 0.75o GHcp_A3
Zr

+ 0.25o GHex_A8
Te (11)

in which aZr3 Te and bZr3 Te are the parameters to be evaluated during the
optimization process. The two pure components (Zr) and (Te) take the
stable phase o GHcp_A3
Zr and o GHex_A8
Te as the reference states, respectively.
Fig. 1. Calculated Te-Zr phase diagram with the experimental data from de
Boer and Cordfunke [48].

where o GZr1+x Te2

Te:A:B (A = Te, Zr; B = Va, Zr) represents the Gibbs free energy
of the corresponding hypothesis compounds. y′′i and yi are the site
′′′

fractions of component i (i = Zr, Te, Va) on the second and the third
Zr1+x Te2 Zr Te
sublattice, respectively. 0 LTe:A:Va,Zr and 0 LTe:Te,Zr:B
1+x 2
are the interaction
parameters between Zr and Te while the other sublattices are occupied
by Te and B.
Both Ce3-xTe4 and Eu1-xTe show a solubility range, which were
described by the formula (Ce, Va)3/7(Te)4/7 and (Eu, Va)0.5Te0.5.
Because the solution range of this phase extends only on the Te-rich side
of the equiatomic composition and thus defects were considered on the
Ce and Eu sublattices. Besides, Ce3-xTe4 has a structure of Th3P4-type
[34], the same as La3-xTe4, which is modeled as (La, Va)3(Te)4 [74].
Therefore, the model for La3-xTe4 in the La-Te system is used to describe
the Ce3-xTe4 for compatibility. According to the general sublattice model
[72,73], the Gibbs free energy of the phase for 1 mol atom is expressed
as (taking Ce3-xTe4 as an example):
o Ce3− x Te4
/ /
G = yCe o GCe 3− x Te4
Ce:Te + yVa o GCe3− x Te4
Va:Te + 3 7RTyCe lnyCe + 4 7RTyTe lnyTe
∑ Ce Te
+yCe yTe v
LCe,Va:Te
3− x 4
(yCe − yTe )v
v

(10)
Fig. 2. Enthalpies of formation of the Te-Zr system at 298.15 K from the pre­
where yCe , yTe and yVa are the site fractions of Ce, Te and Va, respec­ sent CALPHAD modeling, in comparison with the first-principles calculations
[88,89] and experimental data [43,57].

7
C. Dong et al. Calphad 74 (2021) 102281

Simulation Package (VASP) [79,80] and ALKEMIE [81], with the

generalized gradient approximation (GGA) refined by Perdew, Burke
and Ernzerhof (PBE) [82]. Selected 500 eV as the cutoff energy for all
elements and compounds after convergence tests. All structures are
quite relaxed in the case of shape, volume and atomic coordinates. The
tetrahedron method with Blöchl corrections [83] was employed for ac­
curate total energy and electronic structure in integration of reciprocal
space. The convergence criterion for electrons self-consistent energy and
forces on ions is 10− 5 eV/atom and 0.02 eV/Å, respectively. The sam­
pling amount of each reciprocal atom of k-point exceeds 10000 ac­
cording to the Gamma scheme. The spin polarization is not included in
the calculations. Based on the total energy obtained from the beginning,

Fig. 3. Calculated value of yZr yZr , yTe yZr , yTe yVa and yZr yVa of Zr1+xTe2 phase in
′′′ ′′′ ′′′ ′′′
′′ ′′ ′′ ′′

the entire composition of Te (yM and yN denote the site fraction of M (M = Zr,
′′′
′′

Te) in the second and N (N = Zr, Va) in third sublattice).

Similar expressions like Eq. (11) can be applied to describe other stoi­
chiometric phases in this work. The complete and self-consistent ther­
modynamic parameters of Te-X (X = Zr, Ce, Eu) systems are listed in
Table 5.

3.2. First-principles calculations

First-principles calculations based on the Density Functional Theory

Fig. 5. Mole fraction of species in the liquid as a function of mole fraction of Te
(DFT) [75,76] are performed by using the Projector Augmented Wave
at 2100 K from the present CALPHAD modeling.
(PAW) [77,78] pseudo-potential as implemented in the Vienna Ab initio

Fig. 4. Calculated phase diagram of the Te-Ce system (black solid line) compared with (a) assessed experimental data (maroon open triangle) from Okamoto [61];
(b) result from Philipp et al. [22] (magenta dash line) and assessed result from Okamoto [61] (maroon dotted line).

8
C. Dong et al. Calphad 74 (2021) 102281

experimental data are listed in Table 3.

4. Results and discussion

The thermodynamic parameters of the Te-Zr, Te-Ce and Te-Eu sys­

tems were evaluated based on the available experimental data cited
above and calculated enthalpies of formation in the present work. When
there are no phase transitions for the pure elements and compounds
between 0 K and room temperature, the enthalpy of formation of a stable
or metastable compound, TeaXb, can be approximated via first-principles
calculations [15]. The optimization was performed in the PARROT
module of the Thermo-Calc software [73] which works by reaching the
minimal square sum of the errors. Each piece of experimental informa­
tion was given a certain weight, and the weight was varied systemati­
cally during the optimization process until most of the experimental data
have been accounted for. Finally, a set of optimized model parameters
were obtained, reaching a good agreement with the experimental data
within the expected uncertainty limits. For the Te-Zr system, during the
assessment, the parameters in Eq. (11) for stoichiometric compounds
were optimized by considering the data of enthalpies of formation
calculated by first-principles calculations as well as the phase equilibria
data. Later, the parameters of liquidus of the Te-Zr system were opti­
mized. At last, all the parameters were optimized at the same time based
on the experimental data. The same process was also applied to the
Fig. 6. Enthalpies of formation of the Te-Ce system at 298.15 K in comparison Te-Ce and Te-Eu systems. Finally, a set of self-consistent thermodynamic
with the experimental data [64] and first-principles calculations [88,89]. parameters for the Te-X (X = Zr, Ce, Eu) system were obtained as pre­
sented in Table 5.
the enthalpy of formation of TeaXb (X = Zr, Ce, Eu) was calculated as The step-by-step optimization process, critically described by Du
follows: et al. [86], was used to optimize the Te-Zr system. At first, we regarded
a b the Zr1+xTe2 phase as a line compound with 66.6666 at.% Te so as to
Δf E(Tea Xb ) = E(Tea Xb ) − E(Te) − E(X) (12) acquire a set of reasonable parameters. And then the homogeneity range
a+b a+b
was added to modify and secure the expected range. The computed
where E(Tea Xb ), E(Te) and E(X) were equilibrium energies of the com­ phase diagram of Zr-Te is shown in Fig. 1. The present modeling meets
pound TeaXb and the pure elements of Te and X (X = Zr, Ce, Eu) in their most of the experimental data. Although the liquidus between 22 and 44
standard reference structures. Since the calculations are performed at 0 at.% Te exists a small discrepancy and the temperature of the reaction
K and 0 Pa, the energy of formation is taken to be the enthalpy of for­ ZrTe2 + Liquid → Zr5Te4 deviates the experiment 1795 ± 18 K, it is a
mation. More details please refer to our previous works [84,85]. All the compromise for the exclusion of the liquid-liquid immiscibility. The
calculated enthalpy of formation intermediate phases except for Ce4Te7, calculated enthalpies of formation at 298.15 K is shown in Fig. 2.
Eu4Te7 and Eu3Te7 (due to no crystal structures information) and Clearly, the result of the enthalpy of formation is in excellent consistent
end-members in Te-X (X = Zr, Ce, Eu) systems in comparison with the with the first-principles calculations in this work. Furthermore, we
calculated the effect of different interaction parameters of Zr1+xTe2,

Fig. 7. Calculated phase diagram of Te-Eu system with (a) assessed experimental data (magenta open circles) from Okamoto [39]; (b) both the assessed experimental
data (magenta open circles) from Okamoto and calculated result from Ghamri et al. [23] (dotted line).

9
C. Dong et al. Calphad 74 (2021) 102281

the Ce (FCC_A1) and Te (HEAXGONAL_A8) phases, are represented in

Fig. 6 along with the predictions from the first-principles calculations.
The first-principles calculations are taken from OQMD (Open Quantum
Materials Database) [88], Material Project [89] and present work, in
which the result is in excellent agreement with our own calculations but
is not negative enough compared to the result of Ferro et al. [64].
Fig. 7(a) shows the calculated phase diagram of the Te-Eu system
along with the experimental data taken from Ref. [39] and the calcu­
lated result of Ghamri et al. [23], which is almost in accordance with the
assessed experimental data. Fig. 7(b) shows the comparison among the
assessed phase diagram, calculated results from Ghamri et al. [23] and
present modeling results. The eutectic reaction temperature for Liquid
→ (Eu) + EuTe and Liquid → (Te) + Eu3Te7 are 1053 K and 682.1 K,
respectively, while for the remaining two peritectic reactions Liquid +
EuTe → Eu4Te7 and Liquid + Eu4Te7 → Eu3Te7 the temperatures are 793
K and 703 K, respectively. The calculated melting temperature of EuTe is
1799 K. Fig. 8 gives the calculated enthalpies of formation of all the
intermetallic compounds at 298.15 K with the experimental values and
the first-principles calculations in the present work, from which it can
been seen that the result has a better agreement with the first-principles
calculations results but is less positive than the results from Ghamri et al.
[23] and McMaster et al. [65].

5. Conclusion
Fig. 8. Enthalpies of formation of the Te-Eu system at 298.15 K from the
present CALPHAD modeling, in comparison with the first-principles calcula­ All the experimental phase diagram and thermodynamic data
tions in this work, Ghamri et al. [23] and experimental data from Refs. [65,66]. available for the Te-X (X = Zr, Ce, Eu) systems were critically evaluated,
and a set of self-consistent thermodynamic parameters was obtained for
from which we can see the effect of y′′Zr yZr is decreasing while y′′Te yZr each of the Te-X (X = Zr, Ce, Eu) systems using CALPHAD method based
′′′ ′′′

increases, as shown in Fig. 3. And when the composition of Te is more on the critically reviewed literature data and first-principles calculated
than 66.66 at.%, the effect of y′′Zr yVa weakens while y′′Te yVa strengthens. data in the present work. The composition range of phase Zr1+xTe2, Ce3-
′′′ ′′′

The calculated phase diagram using the present parameters in xTe4 and Eu1-xTe have been satisfactorily reproduced via the present
comparison with assessed data from Okamoto [61] are shown in Fig. 4 parameters, and the thermodynamic behaviors of the liquid phase in the
(a) and the compared result of all available data is presented in Fig. 4(b). Te-Ce system are well described by employing the associate solution
As for this system, we tried to realize the phase diagram by using sub­ model. Comprehensive comparisons between the calculated and exper­
lattice model for the liquid phase at first but it was found impossible to imental data indicate that the present thermodynamic descriptions can
reach the eutectic between the CeTe and Ce3-xTe4 and the asymmetric smoothly explain the existing experimental information of the Te-X (X =
liquidus. Then, the phase diagram was reproduced by using associate Zr, Ce, Eu) systems.
model for liquid, a better physical description of the short-range
ordering behavior in the liquid phase than the sub-regular solution Data availability
model [87]. The short-range ordering behavior can be further examined
by plotting the entropy of mixing of the liquid phase. Fig. 5 shows the All the key data to this article can be found in supplementary files.
calculated species fraction in liquid as a function of Te concentration at
2100 K. It is noted that the dominant species in the liquid phase is the Declaration of competing interest
CeTe associate around the mole fraction x(Te) = 0.5, which suggests the
strong ordering interactions between the atoms in the liquid phase The authors declare that they have no known competing financial
around this composition. The calculated invariant temperatures and interests or personal relationships that could have appeared to influence
compositions together with the experimental values are listed in Table 4. the work reported in this paper.
As shown in Table 4, the present calculation accounts for the
invariant temperatures well within experimental errors. The calculated Acknowledgements
melting temperatures of CeTe and Ce3-xTe4 are 2090.9 K and 1911.4 K,
respectively, which are close to the experimental ones (2093 K and 1918 The financial supports from the National Key Research and Devel­
K). In the present work, all the invariant reactions are well reproduced. opment Program of China (Materials Genome Initiative:
Notably, small differences exist in the Te-rich side both the liquidus and 2017YFB0701700), the National Natural Science Foundation of China
the composition of Te of the liquid phase. The obtained liquidus deviates (Grant Nos. 52071002 and 51601228) are greatly acknowledged. First-
the experimental data compared to the result from Okamoto [61] but principles calculations were partly carried out at the High Performance
shows a better result than that of Philipp et al. [22]. Besides, Ce2Te5 Computing of Central South University.
exists between 880 K and 1143 K and this range is calculated to be
883.94-1142.33 K in the present work, which is in good agreement with Appendix A. Supplementary data
the experimental values. As for the homogeneity of the Ce3-xTe4 phase,
the calculated result is 57.1-59.7 at.% Te while the experimental value is Supplementary data to this article can be found online at https://doi.
57.1–60 at.% Te within the accepted deviation. org/10.1016/j.calphad.2021.102281.
The calculated standard enthalpies of formation using the present
thermodynamic modeling parameters of the compounds CeTe, Ce3-xTe4,
Ce4Te7, CeTe2, Ce2Te5 and CeTe3 at 298.15 K, with reference states of

10
C. Dong et al. Calphad 74 (2021) 102281

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