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Thermodynamic reassessment of the Mo–Hf and Mo–Zr systems supported by first-principles calculations
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 69 (2020) 101766
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Thermodynamic reassessment of the Mo–Hf and Mo–Zr systems supported by first-principles calculations Ruiqi Zhao a, Jiong Wang a, *, Huimin Yuan a, Biao Hu b, Yong Du a, Zhunli Tan c a
Powder Metallurgy Research Institute, Central South University, Changsha, Hunan, 410083, China School of Materials Science and Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001, China c Material Science and Engineering Research Center, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, 100044, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Mo–Hf Mo–Zr Phase diagram First-principles calculations CALPHAD
Based on the experimental phase equilibria and thermodynamic data available in the literature and enthalpies of formation computed from first-principles calculations, the thermodynamic reassessment of the Mo–Hf and Mo–Zr systems was carried out by means of the CALPHAD (CALculation of PHAse Diagram) method. The enthalpies of formation for stable and metastable Laves (C15, C36, C14) phases and enthalpy of mixing for the β(bcc) solid solution phase in dilute solution were predicted via first-principles calculations to supply the necessary ther modynamic data for the modeling in order to obtain the thermodynamic parameters with physical sound meaning. The relative stability of Laves C14, C15 and C36 in the systems was discussed. The solution phases, i.e. liquid, β(bcc) and α(hcp) were described by the substitutional solution model, and all the Laves phases in the systems were described using two sublattice model. A set of self-consistent thermodynamic parameters were obtained for these binary systems, which agrees well with the experimental data in the literature. Based on these results, the trend of the site occupancy fraction of Laves phase changing with temperature was predicted. The isothermal section and the liquidus projection of Mo-Hf-Zr system were also predicted by combing with the Zr–Hf system reported in the literature.
1. Introduction Since the storage density of silicon-based traditional memory is difficult to meet the needs in the era of big data, it is urgent to develop new memory technology. Resistive Random Access Memory (RRAM) features simple structure, good miniaturization, fast operation speed, low power consumption, and is compatible with the complementary metal oxide semiconductor (CMOS) technology [1]. RRAM is regarded as one of the most promising non-volatile memory in the next generation [2]. At present, the widely used RRAM materials are mainly in the sandwich structure based on the Transition Metal (TM)/TMOx/TM, such as Mo/HfOx/Mo [3], Ti/Mo:ZrO/Pt [4] and Cu/HfOx/Pt [5]. Unfortu nately, different combinations of TM and TM oxides may form different stable or metastable second phases in the device during service [6,7], which is limitation for the application. There are laves phases of AB2 which belongs to the class of Frank-Kasper phases in topologically close-packed structures, i.e. cubic C15 (MgCu2, space group Fd-3m, No. 227), hexagonal C14(MgZn2, space group P63/mmc, No. 194) and C36 (MgNi2, space group P63/mmc, No194) phases [8]. Moreover, the C36
phase is an intermediate structure in the arrangement sequence of atoms from C14 to C15 structures [9]. Few authors have investigated the properties of Laves Mo2Hf and Mo2Zr so far [10–13]. The lattice pa rameters of these compounds were measured by Rapp in 1970 [11]. A investigation of the structural, elastic, and lattice dynamical properties for Mo2Zr and Mo2Hf with C14, C15, C36 phases are conducted using density functional total energy calculations by Turkdal et al. [10]. Since the composition, size and distribution of the Laves phases precipitating in alloys have a big influence on the final properties [14], the stability and thermodynamic properties of the systematic Laves phase are of great importance [15]. Nowadays, computational thermodynamics in the framework of CALculation of PHAse Diagram (CALPHAD) approach, which heavily depends on the quality of the thermodynamic descriptions for the target system, has been proved to be an effective way to accelerate the development of new alloys [15]. Therefore, reliable thermodynamic description of TM-based material is essential for the design of the database of RRAM. However, the existing thermodynamic data is not complete, it was found that the study of Pavlů et al. [16] was compatible
* Corresponding author., E-mail addresses: [email protected], [email protected] (J. Wang). https://doi.org/10.1016/j.calphad.2020.101766 Received 17 November 2019; Received in revised form 7 March 2020; Accepted 16 March 2020 Available online 28 March 2020 0364-5916/© 2020 Elsevier Ltd. All rights reserved.
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reaction (L þ βMo2Hf → βHf) rather than the eutectic reaction proposed by Taylor et al. [12], which agreed with experimental results of Brewer and Lamoreaux [22] and Massalski et al. [24]. The data of Rudy [23] at the Hf rich side was adopted. Garg and Ackerman [21] also reported a less detailed phase diagram by gravimetric and pyrometric methods. Considering all the information above, Brewer and Lamoreaux [22] discussed the critical experiments review of this system. Massalski et al. [24] constructed an experimentally assessed phase diagram according to the results mentioned above. For the intermetallic phase, Mo2Hf is a Laves-type phase with a complex sequence of polymorphs [11,20]. A hexagonal βMo2Hf (C36) and cubic αMo2Hf (C15) phase transition were observed by Taylor et al. [12] using X-ray diffraction and metallography and was confirmed by Eremenko et al. [20], who believed that the transition temperature be tween βMo2Hf and αMo2Hf obviously depends on the composition of compounds and a cubic phase will be formed at low temperature. The data of Taylor et al. [12] was adopted. The transition temperature of Mo-rich side was about 2133 K, and that of Hf-rich side was 2053 K. Eremenko et al. [20] also showed that the homogeneity of Mo2Hf ex tends from the stoichiometric composition to the Hf-rich side. In addi tion, Garg and Ackerman [21] measured the solubility of Hf in Mo, which was 28 at% (2453 K). The maximum solid solubility of Mo in (αHf) was reported at 0.8 at% (1503 K) [12]. Taking the thermodynamic data into consideration, no thermody namic data of βMo2Hf was reported. Brewer and Lamoreaux [22] esti mated the enthalpy of formation of αMo2Hf phase using experimental data of Laves phases of similar size ratios and electronic interactions to those of αMo2Hf. Rudy [23] assessed the enthalpy of formation of αMo2Hf phase using DTA. Recently, the enthalpy of formation of αMo2Hf phase was measured by high temperature direct synthesis calorimetry at 1373 � 2 K [25]. Shao [26] evaluated the enthalpies of formation of the end-members for αMo2Hf and βMo2Hf by CALPHAD method, which is incompatible with the parameters of the TM-based database [16–18]. So the first-principles calculations were introduced to calculate enthalpy of formation for all the end-members of Laves phases [27].
Table 1 Calculated and experimental lattice parameters of the SER structures and the relative difference between the calculated and experimental lattice parameters. Phase
Person symbol
Space group
Prototype
Lattice parameter(Å) Expt. Ref.
Calc.
Diff. (%)
(Mo)
cI2
Im-3m
W
3.150
0.095
(Hf)
hP2
P63/mmc
Mg
(Zr)
hP2
P63/mmc
Mg
a 3.147 [49] a 3.23 [49] c 5.12 a 3.238 [49] c 5.189
3.192
1.176
5.050 3.239
1.367 0.031
5.164
0.482
with our systems [17–19] in the process of building database. Consequently, the aims of the present work are (1) to critically evaluate the phase diagram and thermodynamic data in the Mo–Hf and Mo–Zr systems with a desire to develop a set of self-consistent thermo dynamic parameters for thermodynamic extrapolations of related higher order systems, (2) to calculate the total energies for all stable and metastable Laves phases of Mo–Hf and Mo–Zr systems, (3) to predict the site occupancy fraction trend of Laves phase with temperature by means of the CALPHAD approach. 2. Literature review The phase equilibria of the Mo–Hf system have been experimentally investigated by Taylor et al. [12], Eremenko et al. [20], Garg and Ackerman [21], Brewer and Lamoreaux [22] and Rudy [23]. All these works fit the basic features depicted by Taylor et al. [12], except for the detail of the phase equilibrium in Hf-rich side. Thus, this work mainly based on the phase equilibria data of Taylor et al. [12]. DTA (Differential Thermal Analysis) was used by Rudy [23] to investigate the solid-liquid equilibrium in the Hf-rich region and found that alloys were only separated by a very small temperature interval and the peritectic
Table 2 Equilibrium structural parameters of Laves phases in this work. Symbols a and c stand for lattice constants, △ shows the relative difference between the calculated and experimental atomic volume. System
Structure
Mo–Hf
αMo2Hf (C15)
βMo2Hf (C36)
γMo2Hf (C14)
Mo–Zr
αMo2Zr (C15)
βMo2Zr (C36)
γMo2Zr (C14)
k-point mesh
Lattice parameter(Å)
△(%)
a(Å)
c(Å)
c/a
Mo2Hf [50] Mo2Mo Mo2Hf MoHf2 Hf2Hf Mo2Hf [50] Mo2Mo Mo2Hf MoHf2 Hf2Hf Mo2Mo Mo2Hf MoHf2 Hf2Hf
21 23 21 21 21 19 21 19 17 21 23 21 21 21
� 21 � 23 � 21 � 21 � 21 � 19 � 21 � 19 � 17 � 21 � 23 � 21 � 21 � 21
� 21 � 23 � 21 � 21 � 21 �7 �9 �7 �5 �9 � 15 � 13 � 13 � 13
7.557 7.306 7.544 7.824 8.159 5.366 5.153 5.350 5.575 5.703 5.164 5.370 5.561 5.687
– – – – – 17.408 17.016 17.327 17.640 19.084 8.481 8.597 8.771 9.534
1 1 1 1 1 3.24413 3.26237 3.30215 3.23869 3.16413 1.6425 1.6008 1.5773 1.6766
–
Mo2Zr [51] Mo2Mo MoZr2 Mo2Zr Zr2Zr Mo2Mo MoZr2 Mo2Zr Zr2Zr Mo2Mo MoZr2 Mo2Zr Zr2Zr
23 23 21 23 21 21 21 19 21 23 19 21 19
� 23 � 23 � 21 � 23 � 21 � 21 � 21 � 19 � 21 � 23 � 19 � 21 � 19
� 23 � 23 � 21 � 23 � 21 �9 �9 �7 �9 � 15 � 13 � 13 � 13
7.595 7.306 7.868 7.594 8.253 5.153 6.066 5.386 5.792 5.163 5.639 5.413 5.759
– – – – – 17.016 15.227 17.444 19.072 8.481 8.652 8.625 9.606
1 1 1 1 1 3.30215 2.51022 3.23877 3.29282 1.64253 1.53431 1.59339 1.66800
–
2
0.09637 0.00515 0.109783 0.258527 – 0.06096 0.030498 0.210599 0.266827 – – – – 0.10987 0.11176 0.00039 0.283075 – – – – – – – –
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phase. As indicated by Garg and Ackermann [21], it is difficult to accurately determine the solidus-liquidus relationship in refractory Mo-rich alloys. DTA was used by Prima [31] to measure the Mo solidus, while the uncertainty of using DTA to measure the thermal effect of liquid phase was relatively large [21]. Later, Bhatt et al. [30] measured the Mo solidus temperature using the diffusion couple technique and then determined a solidus composition equals to 5 at.% Zr at 2358 K. This agreed very well with the measured Mo solidus data reported by Garg and Ackermann [21]. Thus, the data of Bhatt et al. [30] was adopted. What’s more, the maximum solubility of Zr in (βMo) has been studied by many researches [28,29,32]. It is indicated that the maximum solubility does not exceed 10–12 at% Mo at the peritectic temperature [28,29,32]. Moreover, the solubilities of Zr in (βMo) were determined by Svechnikov and Spektor [29] by X-ray diffraction and metallography, which were in good agreement with the measurements of Zakharov et al. [32]. Thus, the data of Svechnikov and Spektor [29] was adopted. For the intermetallic phase, αMo2Zr is in Cu2Mg structure (C15 Laves phase) [11], The homogeneity range has not yet been accurately determined and different values have been proposed. It was first described by Domagala et al. [28] that the ideal stoichiometric ratio of Mo2Zr was 2:1, while Garg and Ackermann [21] found the composition range to be between 33.3 and 36 at% of Zr for this intermetallic phase. The diffusion study [30] showed that Laves phase is formed between 33.3 and 34.5 at% of Zr at a temperature slightly below the peritectic isotherm. In other literature, it was found that this compound was nearly stoichiometric [11,23,29], Rapp proposed [11] a composition ranging from 33.48 to 33.77 at% of Zr. Thus, the compound was described with a small composition range in this work. For the thermodynamic data, due to the refractory nature of Mo al loys and Zr alloys, no thermodynamic data concerning the liquid and bcc has been found in the literature. So, the estimated enthalpy values (ΔHmix 6000 J/mol of atoms for 0.5 at% Mo) proposed by Boer Mo;Zr ¼ et al. [33] were used in the present work. Brewer and Lamoreaux [34] estimated the thermodynamic properties of αMo2Zr from the measured thermodynamic properties of Cr2Ta and Cr2Nb by Martin et al. [35]. Recently, the enthalpy of formation of αMo2Zr phase was measured by high temperature direct synthesis calorimetry at 1373 � 2 K [25]. The thermodynamic evaluation model of Mo–Zr system has been reported in Zinkevich and Mattern [36], P�erez and Sundman [37]. Because it seems unrealistic to model the liquid and β(bcc) stages with a fairly high excess entropy in the study of Zinkevich and Mattern [36], therefore, part of parameters of P�erez and Sundman [37] were adopted in this work.
0
First-principles calculated the total energy differences, Δ EL SER , between the Laves phases and the SER states calculated in this work, and also compared with the calculated [53,54,58,59] and enthalpies of formation of Mo2Hf and Mo2Zr compounds [21,25,34] and assessed [26,37,58,59] values available in literature. The values marked by an asterisk (*) were obtained with the help of the energy difference EHCP - EFCC from Wang et al. [53], as the value of energy of formation of Laves phase obtained in Sluiter [54] was related to the FCC structure (BCC structure is employed as the SER structure for Mo in Sluiter [54]). Ab initio re sults published in Ref. [53,54] were calculated using the generalized gradient approximation. Composition
Mo–Hf
αMo2Hf
This work Refs.
Mo2Mo 35.44 35.20 [54] 5 [26]
βMo2Hf (C36)
This work Refs.
35.67 5 [26]
9.6 [26]
γMo2Hf (C14)
This work Refs.
36.77
14.58
Mo–Zr
(C15)
Δ0 EL
αMo2Zr
This work Refs.
βMo2Zr (C36) γMo2Zr (C14)
This work This work Refs.
(C15)
SER
(kJ mol
1
atom 1) MoHf2 67.36 30 [26]
Hf2Hf 37.89 35.82* [53,54] 5 [26]
63.21
34.09
40.02 [26] 60.54
5 [26]
–
– MoZr2 67.91 7.24 [37]
35.67
Mo2Zr 12.99 13.31 [58] 12.35 [59] 9.30 [34] 7.4 � 4.2 [25] 7.24 [37] 12.14
30.92* [53,54] Zr2Zr 27.36 26.49* [53,54] 5 [37]
58.25
22.80
36.77
10.64
58.99
20.48
–
19.69* [53,54]
36.50 [54] Mo2Mo 35.44 35.2 [54] 5 [37]
36.5 [54]
Mo2Hf 16.69 16.4 [58] 15.53 [59] 11 � 1.4 [21] 6.4 � 4.3 [25] 10.6 [26] 16.06
–
32.03
3. Methodology
Mo–Zr system has been reviewed for many times in the literature [11,21,23,28–32]. Domagala et al. [28], who have conducted the first study of the Mo–Zr phase equilibria (thermal analysis, metallographic observation) and presented a rough phase diagram containing peritectic reaction (Liquid þ βMo → αMo2Zr), eutectic reaction (Liquid → βZr þ αMo2Zr) and a eutectoid decomposition (βZr → αZr þ αMo2Zr). Later, more phase diagram information was determined by Svechnikov and Spektor [29], Rudy [23], Prima [31], Garg and Ackerman [21]. All these works were mainly consistent with the phase boundary given by Domagala et al. [28] except the eutectic reaction was replaced by a peritectic reaction (Liquid þ αMo2Zr→ βZr) at the Zr-rich side. While it was refuted by Domagala et al. [28] and Garg and Ackerman [21] who found typical eutectic patterns in the microstructures of solidified alloys. The phase boundary data of Domagala et al. [28] were adopted. The temperature-composition data of liquidus in the Mo–Zr system from 30 to 90 at% Zr has been accurately determined by Garg and Ackermann [21] by combining the gravimetric and pyrometric methods with elec tron probe microanalysis (EPMA), which agreed well with the data of Svechnikov and Spektor [29] and Domagala et al. [28]. Therefore, the data of Garg and Ackermann [21] was adopted for Zr-rich side in liquid
3.1. First-principles calculations First-principles calculations on the basis of the Density Functional Theory (DFT) [38,39] are performed using the Projector Augmented Wave (PAW) [40,41] pseudo-potential as implemented in VASP [42,43], with the generalized gradient approximation (GGA) refined by Perdew, Burke and Ernzerhof (PBE) method [44] to calculate the total energies differences of all three Laves phase structures (C14, C15 and C36) and the SER (Standard Element Reference) structures. To maintain consis tency, the cutoff energy of 450 eV was selected for all elements and compounds after convergence tests. All structures are completely relaxed in terms of shape, volume and atomic coordinates. The electron self-consistent energy convergence criterion is 10 6 eV/atom. According to the Monkhorst-Pack scheme [45], the sampling amount of each reciprocal atom of k-point exceeds 10000. The spin polarization is not included in the calculations. Based on the total energy obtained from the beginning, the enthalpy of formation of the terminal compound was calculated as follows: ΔHði:jÞ ¼ Etotði:jÞ 3
1 3E
=
System
totðiÞ
2 3E
=
Table 3
totðjÞ
(1)
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Fig. 1. Calculated Mo–Hf phase diagram Compared to the experimental data (a), a magnify detail of Laves phases (b), and a magnify detail about the phase equilibrium between the βMo2Hf and αMo2Hf phases (c). Full lines represents the optimization in this work, the dashed lines were redrawn from Shao [26] and the experimental points were taken from Refs. [12,21,23,24].
phases multiplied by their molar fractions:
where i and j represents the components of Mo, Hf, Zr. Etotði:jÞ and EtotðiÞ is the total energies for the end-members, and pure elements in SER structure respectively. In order to get more accurate thermodynamic descriptions for bcc phase in the Mo–Hf system, the mixing enthalpy of Hf in Mo-rich side was calculated by dilute solution model. In the calculations, the 54-atom supercell is employed to study bcc phases with compositions of x ¼ 1/ 54; the 16-atom supercell for bcc phases with compositions of x ¼ 1/16. In the present work, supercells for bcc was generated with respect to the standard crystallographic unit cell, and the 2 � 2 � 2 and 3 � 3 � 3 supercell were generated with respect to a two-atom cubic cell for bcc lattice.
(2)
E Gα ¼ Gref þ ΔGid mix þ G
In the Mo–Hf and Mo–Zr system, the Gibbs energy for the solution phase (Liquid, β, α) was described by sub-regular solution with the Redlich-Kister polynomial for the excess Gibbs energy [46]. Gα ¼
X
xi Goi þ RT
i¼A;B
n X
X xi Lnxi þ xA xB i¼A;B
j LA;B ðxA
xB Þj
(3)
j¼0
in which Goi means the contribution of the mechanical mixing of pure elements to the total Gibbs energy, xi is the mole fraction of component, R is the gas constant, T is the absolute temperature. L represents the interaction between Mo and X (Zr or Hf) atoms in each sublattice, temperature dependence of L parameters is given by:
3.2. Thermodynamic models 3.2.1. Solution phases As mentioned above, the CALPHAD method requires modeling of Gibbs energy in each phase as a function of temperature and pressure, and then all other thermodynamic properties can be obtained by ther modynamic correlation. The total molar Gibbs energy of the whole system can be defined as the sum of the Molar Gibbs energies of all
L j ¼ Aj þ Bj T
(4)
where the interaction coefficients Aj and Bj are constants, which can be evaluated in the optimization process.
4
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� � Hfy8α ; Moy8Moα ; Zry8Zrα Hf
1=3
� � Hfy16d ; Moy16d ; Zry16d Mo Zr Hf
2=3
(9)
where ynm indicates the site occupancy fraction of the components m (Hf, Mo, Zr) in the sublattice n (8a or 16d), which is the unknown model parameter. For a given alloy HfxHf MoxMo ZrxZr , where xi ði ¼ Hf; Mo; ZrÞ represents the mole fraction of element in the Laves phase alloy. According to composition normalization and mass balance, Eqs. (10)–(15) can be obtained: xHf þ xMo þ xZr ¼ 1
(10)
8a 8a y8a Hf þ yMo þ yZr ¼ 1
(11)
16d 16d y16d Hf þ yMo þ yZr ¼ 1
(12)
1 3y8a
þ 2 3y16d Hf ¼ xHf
(13)
1 3y8a
þ 2 3y16d Mo ¼ xMo
(14)
= =
0
0
0
0
0
0
0
α α þ y’A y’B y’’B 0 LαA;B:B þ y’A y’’A y’’B LαA;A:B þ y’B y’’A y’’B LB;A:B GE ¼ y’A y’B y’’A LA;B:A
=
(6)
0
=
=
the ideal mixing Gibbs energy id Gα is equal to: � 0 0 0 0 � 00 00 00 0 0 �� ΔGid mix ¼ RT yA ln yA þ yB ln yB þ yA ln yA þ yB ln yB
0
=
(5)
0
Zr
Zr
¼ xZr
(15)
The Gibbs energy of the Cu2Mg-type intermetallic 32Hf-64Mo–4Zr (in at. %) was established as a mathematical model from stable pure elements. According to the characteristics of thermodynamic function, the quantity is determined by the initial state and the final state, and is independent of the intermediate process. We adopted another method to calculate the Gibbs energy of the above-mentioned information, as shown in equation (16). The formation of the alloy can be described with the hypothetical end-members and the alloys are formed from the endmembers, i.e., (1) ¼ (2) þ (3). The Gibbs energy of the phase can be calculated from the pure elements that are stable at room temperature.
3.2.2. Intermetallic compounds To simplify consistency with multi-component databases, illustrate the range of uniformity, and maintain the consistency with the model reported in the corresponding literature [26,37], two sublattice model (A, B)2 (A, B)1 were adopted for the Laves structure. And the Gibbs E energy can also be formulated by Eq. (2), but Gref , ΔGid mix and G are expressed differently compared with Eq. (3): 0
Mo
1 3y8a þ 2 3y16d
Fig. 2. Calculated enthalpies of formation of αMo2Hf compound at 0 K, compared with the experimental data [21,25,58,59].
Gref ¼ yA yA0 o GαA:A þ yB yB0 o GαB:B þ yA yB0 o GαA:B þ yB0 yA o GαB:A
Hf
(7)
where yA and yA0 refers to the site fractions of component in the first and 0
0
the second sublattice, respectively, 0 L represents the interaction parameter between the species. In general, the fewer the orders of pa rameters are, the better the optimization is. The four end-members o Gαi:j α
(16) The Gibbs energy of formation from the room temperature stable pure elements can be calculated by
(i ¼ A, B; j ¼ A, B) represents the Gibbs energies of the end elements A2A, B2B, A2B, AB2. These end-members represent all the possibilities that each sublattice is occupied by the single element. The total Gibbs energy of the Laves phase is a function of the temperature and composition of the two sublattices, describing the energy advantage of the element in the second sublattice or the first sublattice. One of them is described as below, for example: o
Gαi:j ¼ 2o Gαi þ o Gαj þ A þ B � T
ΔG ¼ ΔH
TΔS
(17)
where ΔS is the corresponding entropy, which can be obtained during the optimization procedure. ΔH can be calculated by the equation: X X 16d E ΔH ¼ y8a (18) k yl ΔHðk:lÞ þ Δ H
(8)
k¼Hf ;Mo;Zrl¼Hf ;Mo;Zr
Gαi and Gαj are the Gibbs energies of i and j, respectively, adopting the
where ΔHðk:lÞ represents the enthalpy of formation of the end-member compound from the pure elements Hf, Mo, Zr (0 K) in Laves phase, while the enthalpy difference of formation between 0 K and a defined temperature calorific content is neglected in current work. The excess enthalpy of formation ΔE H is obtained during the optimization. When in equilibrium state, ΔG reaches a minimum value, so there will be following constraint equations:
SER reference state. In this paper, these end-members are determined by first-principles calculations, overcoming the arbitrariness of selecting the Gibbs energy values of these end-members. 3.3. Site fraction of laves phase The prototype of the C15 Laves phase of αMo2Hf and αMo2Zr is Cu2Mg structure [11]. Using the general two-sublattice model, it can be described in the form as [47].
∂ΔG ∂ΔG ∂ΔG ∂ΔG ∂ΔG ∂ΔG ¼ 8a ¼ 8a ¼ 16d ¼ 16d ¼ 16d ¼ 0 ∂y8a ∂yMo ∂yZr ∂yHf ∂yMo ∂yZr Hf
5
(19)
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Calphad 69 (2020) 101766
Fig. 3. Calculated Mo–Zr phase diagram Compared to the experimental data (a), a magnify detail about the phase equilibrium between the αMo2Zr and other phases (b), and a magnify detail of αMo2Zr (c). Full lines represents the optimization in this work, the dashed lines were taken from P� erez and Sundman [37] and the experiment points were taken from Refs. [11,21,23,28–30,32].
structure of Cr2Zr [52] structure was relaxed by the similarity of the electronic structure of Mo2Hf, the most stable structure was obtained after relaxation [16]. The metastable phase data of C36 phase of Mo2Zr was predicted by the same method. The comparison between the calculated lattice parameters of SER structures and experimental values are listed in Table 1, those for the C14, C15 and C36 structures are listed in Table 2, the calculated results are in good agreement with the experimental data [50,51]. The total energy difference Δ0 EL SER between the Laves phase and the SER state are then given in Table 3, all stable and metastable phases of the Laves phase of the Mo–Hf and Mo–Zr systems were calculated, which are compared to the results of other authors. It is found that the calculated energies of the Mo2Mo, Hf2Hf, and Zr2Zr are in good agree ment with the calculated value by Wang et al. [53] and Sluiter [54], and are far more positive than those reported by Shao [26] and P�erez and Sundman [37]. The stability of the Mo2X (X ¼ Hf, Zr) Laves phase configuration is sequentially increased in the sequences of C14, C36 and C15. It can be seen from the calculation that the C15 phase should be the most stable in the Laves phases of Mo–Zr and Mo–Hf systems (Table 3). And the stability of C14 and C36 structures at higher temperature is facilitated by the effect of entropy. Based on the reported experimental phase diagrams and
Once the value of ΔHðk:lÞ is determined, the relationship among site oc cupancy fractions and composition and temperature can be obtained: i yW Ei ¼ f ðxi ; TÞ
(20)
In order to handle the complex system of partial differential equations, the software package Thermo-Calc [48] was used to calculate the site occupancy fractions. Once the field occupancy at different temperatures is calculated, the order-to-disorder transition can be studied systematically. 4. Results and discussion In Mo-X (X ¼ Hf, Zr) systems, the structure parameters for SER states of pure Zr, Hf and Mo were taken from Pearson’s Handbook [49]. The experiment structure parameters of A2A, B2B A2B, B2A (A ¼ Mo, B ¼ Hf, Zr) configurations of C15 structures were obtained from the experiments on Mo2Hf [50] and Mo2Zr [51], and all structural parameters of C36 phases were obtained from the experimental data of Mo2Hf [50]. As there are no C14 lattice parameters, in order to complete the informa tion of the Laves phase, the first-principles method was used to calculate the C14 Laves metastable phase data for future reference. The C14 6
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Calphad 69 (2020) 101766
Table 4 Optimized thermodynamic parameters describing the Laves phases in Mo–Hf system. The calculated values shown in boldface were taken from Table 3. Phase
Sublattice model
Evaluated parameters
Liquid
(Mo, Hf)1
0 Liquid LMo;Hf
β(bcc)
α(hcp)
(Mo, Hf)1(Va)
(Mo, Hf)1(Va)
¼ 46500
2 Liquid LMo;Hf
¼
1 βðbccÞ LMo;Zr:Va
¼ 65515
2 βðbccÞ LMo;Zr:Va
¼ 2159
0 αðhcpÞ LMo;Hf:Va
¼
(Mo, Hf)1/3(Mo, Hf)2/3
2 αðhcpÞ LMo;Hf:Va o αMo Hf
GHf:Hf 2
o o
2:270T
[26]
24872 þ 8:905T 29:900T 5:847T
500
[26]
6500
o
¼ 37899 þ
Ghcp Hf
o
αMo Hf
2 GMo:Mo ¼ 35441 þ Gbcc Mo
Mo2 Hf GαHf:Mo ¼ o
16688 þ 1:068T þ o
bcc 1=3 Ghcp Hf þ 2=3 GMo
Fig. 4. Calculated enthalpies of formation of αMo2Zr compound at 0 K, compared with the experimental data [25,34,58,59].
o
αMo2 Hf
GMo:Hf
2=3
thermodynamic data, the thermodynamic parameters were evaluated using the computer optimization program PARROT module of the Thermo-Calc software package [55]. This study used a stepwise opti mization process described by Du et al. [56]. Experimental phase dia gram data and thermodynamic properties were used as input to the program. According to the uncertainty of the experimental data, a certain weight was set for each data selected. During the evaluation process, these uncertainties can be changed by trial and error until the selected experimental information is reproduced within the expected uncertainty range. The optimization of this study was carried out in three steps. In the first step, the Gibbs free energy of the Laves phase end member and the Gibbs free energy from the SGTE [57] pure element were calculated by first-principles. In the second step, the parameters of the β(bcc) phase (Mo–Hf system only) were optimized using the calcu lated dilute solution model data. Finally, the phase diagram data, ther modynamic data and first-principles calculations data were used to evaluate the Laves phase parameters. When optimizing the Mo–Hf and Mo–Zr systems, the Gibbs energy of Laves phase was modeled at the beginning using the data of the total energy difference in Table 3. Combined with the experimental data, the parameter L of non-ideal mixing excess Gibbs energy is obtained. After optimizing the Laves phase, as the peritectic point on the Mo-rich side (Liquid þ βMo → βMo2Hf) was not in good agreement in Mo–Hf system, the β(bcc) phase of the Mo–Hf system was optimized. Considering that there is no experimental and computational data of the β(bcc) phase in the literature, the dilute solution model was used to calculate Hf-rich data of β(bcc) phase so as to optimize. It can be seen from Fig. 1 that β(bcc) phase is in good agreement with experiment data. What’s more, the phase boundary of β(Mo) and liquid phase in the Zr-rich side were not satisfied in Mo–Zr system, thus the parameters of β(bcc) and liquid were reoptimized. As can be seen from Fig. 3, the parameters of β(bcc) and liquid phases agree well with the experimental data after optimi zation. Figs. 2 and 4 show the calculated enthalpy of formation of αMo2Hf and αMo2Zr at 0 K in the present work compared with both firstprinciples databases [58,59] and experiment data [25]. The experi mental data was measured at 1373 K while the calculated data was at 0 K. The results of the present calculations fit well with the databases. All values [34] calculated and assessed by first-principles are lower than the experimental data. The first-principles value is selected in order to guarantee the energetic consistency [16]. The thermodynamic param eters of other phases such as Liquid, α(hcp) in Mo–Hf system and α(hcp)
(βMo2Hf)
(Mo, Hf)1/3(Mo, Hf)2/3
o
Ghcp Hf
2 ¼ LMo:Hf;Mo ¼
0 αMo Hf 2 LHf;Mo:Hf
2 ¼ LHf;Mo:Mo ¼
0 αMo Hf 0 αMo Hf
o
βMo Hf GHf:Hf2
¼ 34086 þ
o
βMo2 Hf GMo:Mo
¼ 35667 þ Gbcc Mo
o
2 Hf GβMo Hf:Mo ¼
3
o
Ghcp Hf
o
6352 2930
Ghcp Hf
o
16000 þ 0:742T þ 1= o
þ 2=3 Gbcc Mo
βMo2 Hf GMo:Hf o 1=3 Gbcc Mo
(Mo, Hf)1/3(Mo, Hf)2/3
þ 1=3
Gbcc Mo
0 αMo Hf 2 LHf:Hf;Mo
o
(γMo2Hf)
2:030T þ
¼ 67360 o
o
¼ 63205 þ 2=3 Ghcp Hf þ
0 βMo Hf 2 LHf:Hf;Mo
2 ¼ LMo:Hf;Mo ¼
0 βMo Hf 2 LHf;Mo:Hf
2 ¼ LHf;Mo:Mo ¼ 97712
o o o
γMo Hf
GHf:Hf2
0 βMo Hf
o
¼ 32030 þ Ghcp Hf o
2 GMo:Mo ¼ 36771 þ Gbcc Mo 2 Hf GγMo Hf:Mo ¼
o
14581 þ 0:172T þ o
bcc 1=3 Ghcp Hf þ 2=3 GMo
o
3129
0 βMo Hf
γMo Hf
γMo Hf
2 GMo:Hf
o
1=3 Gbcc Mo
This work This work This work [26] [26]
¼ 27000 ¼
[26] [26]
28:000T
0:500T
¼
0 βðbccÞ LMo;Hf:Va
1 αðhcpÞ LMo;Hf:Va
(αMo2Hf)
23414
¼
1 Liquid LMo;Hf
Ref.
o
¼ 60542 þ 2=3 Ghcp Hf þ
This work This work This work This work This work This work This work This work This work This work This work This work This work This work This work This work
in Mo–Zr system are respectively taken from Shao [26] and P�erez and Sundman [37], because no new information is included on these phases. The thermodynamic parameters describing the Mo–Hf and Mo–Zr sys tems are listed in Tables 4 and 5. The model described in Section 3.2 provides a good description of the systems. The calculated phase dia grams of Mo–Hf and Mo–Zr systems are shown in Figs. 1 and 3. It can be clearly seen that the first-principles calculations of end-member of Laves phases well describe the phase diagram of Mo–Hf and Mo–Zr systems (experimental point), and the Laves phase boundary is especially in good agreement with the experimental data. The reasonable physical signif icance of the parameters of the Gibbs energy of the Laves phase makes the optimization process simpler, more effective and more reliable and these parameters can be used in the extrapolation of higher-order systems. The present model and method described in section 3.3 can predict the relationship between the site occupancy fraction and composition at 7
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Calphad 69 (2020) 101766
Table 5 Optimized thermodynamic parameters describing the Laves phases in Mo–Zr system. The calculated values shown in boldface were taken from Table 3. Phase
Sublattice model
Evaluated parameters
Ref.
Liquid
(Mo, Zr)1
0 Liquid LMo;Zr
¼
1 Liquid LMo;Zr
¼ 17990
2 Liquid LMo;Zr
¼
This work This work This work This work This work [37]
β(bcc)
(Mo, Zr)1(Va)
α(hcp)
(Mo, Zr)1(Va)
(αMo2Zr)
(Mo, Zr)1/3(Mo, Zr)2/3
27196 þ 8:823T 6:812T
5416 þ 9:460
0 βðbccÞ LMo;Zr:Va
¼
2768 þ 13:411T
0 βðbccÞ LMo;Zr:Va
¼
9152 þ 6:856T
0 αðhcpÞ LMo;Zr:Va ¼ 26754 þ 4:560T o o αMo2 Zr GMo:Mo ¼ 35441 þ Gbcc Mo o
o
Mo2 Zr GαZr:Zr ¼ 27356 þ Ghcp Zr
o
Mo2 Zr GαZr:Mo ¼ o
12994 þ 2:670T þ o
hcp 2=3 Gbcc Mo þ 1=3 GZr o
αMo2 Zr
GMo:Zr o
1=3
(βMo2Zr)
(Mo, Zr)1/3(Mo, Zr)2/3
¼ 67914
Gbcc Mo
þ 2=3
o
0
Ghcp Zr
2:130T þ
0 αMo2 Zr LMo:Mo;Zr
Mo2 Zr ¼ LαZr:Mo;Zr ¼ 14616
0 αMo2 Zr LMo;Zr:Mo
Mo2 Zr ¼ LαMo;Zr:Zr ¼
0
2 Zr GβMo ¼ 22800 þ Ghcp Zr:Zr Zr
o
2 Zr bcc GβMo Mo:Mo ¼ 35667 þ GMo
o
2 Zr GβMo Zr:Mo ¼
o
o
12142 þ 2:550T þ o
bcc 1=3 Ghcp Zr þ 2=3 GMo
o
(γMo2Zr)
(Mo, Zr)1/3(Mo, Zr)2/3
2 Zr GβMo Mo:Zr o bcc 1=3 GMo o γMo2 Zr GZr:Zr
This work
¼ 20484 þ Ghcp Zr
o
This work This work This work
o
γMo2 Zr GZr:Mo ¼
o
10643 þ 1:850T þ o
bcc 1=3 Ghcp Zr þ 2=3 GMo
o
γMo2 Zr GMo:Zr ¼ 58991 þ 2=3 Ghcp Zr þ o
1=3
Gbcc Mo
Fig. 5. Site fraction of elements Zr in sublattice in different components as the temperature changes (the solid line represents the interaction parameter L and b value for the end-members are included, and the dashed line represents no interaction parameter L and b value are included).
This work This work This work This work This work
o
γMo2 Zr GMo:Mo ¼ 36771 þ Gbcc Mo
o
This work
¼ 58246 þ 2=3 Ghcp Zr þ
o
o
3702
o
o
This work This work This work
treatment temperature. At a temperature of higher than 1100 K, Zr atoms completely occupies the 8a sublattice regardless of the xMo/xHf. This is of great significance for the optimization of ternary systems. If Zr atom completely occupies the 8a sublattices, it means that the endmember parameters of the 16d sublattices can be ignored, the opti mized parameters can be reduced. The present model and method can reliably predict the thermody namic data and phase diagram data of higher-order systems. No exper imental phase equilibrium data and thermodynamic data of Mo-Hf-Zr have been reported in the literature, and only Bla�zlna et al. [60] measured the unit cell parameters, cell volumes and hardness of αMo2HfxZr1-x (x ¼ 0–0.75) alloy at 1873 K by X-ray and metallography. The experimental results show that bcc solid solutions (βMo, βHf, βZr) and MgCu2 type (αMo2HfxZr1-x) phase exist at 1873 K, and the hardness of Mo2Hf decreases gradually with the increase of Hf content, among which the highest solubility of Hf in Mo2Zr is 33.3%. Thus, the ther modynamic parameters are extrapolated from this work and Zr–Hf system from Hu et al. [61]. The calculated isothermal sections of the Mo-Hf-Zr system were shown in Fig. 6. Comparisons between the calculated phase diagrams and the experiments show that the thermo dynamic description in this paper can reproduce the experimental data well. The liquidus projection of the Mo-Hf-Zr system with isotherm is calculated using the present thermodynamic parameters and is shown in Fig. 7. It can be seen from Fig. 7 that four primary solidification fields (βMo, βHf, βZr), βMo2Hf, αMo2HfxZr1-x, (βMo) and two invariant reac tion L þ βMo2Hf þ (βMo) → αMo2HfxZr1-x at 2404 K and L þ βMo2Hf → αMo2HfxZr1-x þ (βMo,βHf, βZr) at 2148 K are calculated. The corre sponding reaction scheme of the liquidus projection is given in Fig. 8.
This work
any temperature. For stoichiometric Mo2Hf (Mo2Zr), Hf (Zr) atom completely occupies sublattice 8a, while Mo atom completely occupies sublattice 16d, the site occupancy behavior is independent of tempera ture. But when Zr atom is added into C15-type Mo2Hf-based Mo-Hf-Zr alloy, the site occupancy fraction changes, as shown in Fig. 5. When xMo/xHf keeps 2:1, Zr occupies 8a sublattice and 16d sublattice partially below 1500 K. From 500 to 1500 K, Zr occupies the 8a sublattice rapidly and Zr completely occupies 8a sublattice up to 1500 K. When xMo/xHf is less than 2:1, Zr atoms prefer to occupy 16d sublattice at temperature below 400 K while they also tend to occupy 8a sublattice with the temperature increasing. And Zr atoms fully occupy 8a sublattice when the temperature is higher than 1000 K. When xMo/xHf is over 2:1, Zr atoms mainly occupy 8a sublattice at temperature below 1000 K, only a small portion occupies the 16d sublattice. When the temperature is higher than 1000 K, the Zr atoms completely occupy the 8a sublattice. The dashed line in Fig. 5 also shows the site occupancy without inter action parameters. It can be seen that the trend between the two is roughly the same, with minor differences. The one containing interac tion parameter L and entropy value b are more rigorous in thermody namic properties. In this paper, the solid line containing interaction parameter L and entropy value b are preferred. From what have been discussed above, the results show that the positional preference of alloying element Zr varies with the change of xMo/xHf and heat
5. Conclusion In this study, the stable and metastable Laves phase of Mo–Hf and Mo–Zr systems were studied. The relative stability of the Laves phase in these systems is revealed by comparing the energy calculated by the first-principles of these structures with the total energy of the ideal element mixture. On the other hand, the formation of enthalpy and mixed enthalpy of the solid phase Mo2Hf Laves phase in the dilute so lution was predicted by the first-principles calculations method. The unavailable thermochemical information in this experiment provides meaningful values for the parameterization of the Gibbs energy func tion. The obtained descriptions of Mo–Zr and Mo–Hf were then 8
R. Zhao et al.
Calphad 69 (2020) 101766
Fig. 6. Calculated isothermal section of the Mo-Hf-Zr system: (a) 1173 K, (b) 1573 K, (c) 1873K with experimental data from Bla�zlna et al. [60], (d) 2273 K.
9
R. Zhao et al.
Calphad 69 (2020) 101766
Fig. 7. Calculated liquidus projection for the whole Mo-Hf-Zr system. The full lines represent the monovariant lines, and the dashed lines are isotherms.
Fig. 8. The reaction scheme for the Mo-Hf-Zr system according to the present calculation with temperature in K.
combined with that of Zr–Hf system to form the basis of Mo-Hf-Zr ternary system. The isothermal section and the liquidus projection of Mo-Hf-Zr system were also predicted by combing with the Zr–Hf system reported in the literature. Comparisons between the calculated and measured phase diagrams show that most of experimental data are satisfactorily reproduced by the present thermodynamic descriptions. Based on these results, by using the method of two-sublattice model and first-principles calculations, it is found that the occupation of ternary Mo-Hf-Zr alloy in C15 phase Zr atom tends to occupy the 8a sublattice with the increase of temperature.
work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgements The financial supports from the National Key Research and Devel opment Program of China (Materials Genome Initiative: 2017YFB0701700) and the National Natural Science Foundation for Youth of China (Grant Nos. 51601228, 51429101 and 51602351) are greatly acknowledged.
Declaration of competing interest The authors declared that they have no conflicts of interest to this 10
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Calphad 69 (2020) 101766
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Thermodynamic reassessment of the Mo–Hf and Mo–Zr systems supported by first-principles calculations Ruiqi Zhao a, Jiong Wang a, *, Huimin Yuan a, Biao Hu b, Yong Du a, Zhunli Tan c a
Powder Metallurgy Research Institute, Central South University, Changsha, Hunan, 410083, China School of Materials Science and Engineering, Anhui University of Science and Technology, Huainan, Anhui, 232001, China c Material Science and Engineering Research Center, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, 100044, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Mo–Hf Mo–Zr Phase diagram First-principles calculations CALPHAD
Based on the experimental phase equilibria and thermodynamic data available in the literature and enthalpies of formation computed from first-principles calculations, the thermodynamic reassessment of the Mo–Hf and Mo–Zr systems was carried out by means of the CALPHAD (CALculation of PHAse Diagram) method. The enthalpies of formation for stable and metastable Laves (C15, C36, C14) phases and enthalpy of mixing for the β(bcc) solid solution phase in dilute solution were predicted via first-principles calculations to supply the necessary ther modynamic data for the modeling in order to obtain the thermodynamic parameters with physical sound meaning. The relative stability of Laves C14, C15 and C36 in the systems was discussed. The solution phases, i.e. liquid, β(bcc) and α(hcp) were described by the substitutional solution model, and all the Laves phases in the systems were described using two sublattice model. A set of self-consistent thermodynamic parameters were obtained for these binary systems, which agrees well with the experimental data in the literature. Based on these results, the trend of the site occupancy fraction of Laves phase changing with temperature was predicted. The isothermal section and the liquidus projection of Mo-Hf-Zr system were also predicted by combing with the Zr–Hf system reported in the literature.
1. Introduction Since the storage density of silicon-based traditional memory is difficult to meet the needs in the era of big data, it is urgent to develop new memory technology. Resistive Random Access Memory (RRAM) features simple structure, good miniaturization, fast operation speed, low power consumption, and is compatible with the complementary metal oxide semiconductor (CMOS) technology [1]. RRAM is regarded as one of the most promising non-volatile memory in the next generation [2]. At present, the widely used RRAM materials are mainly in the sandwich structure based on the Transition Metal (TM)/TMOx/TM, such as Mo/HfOx/Mo [3], Ti/Mo:ZrO/Pt [4] and Cu/HfOx/Pt [5]. Unfortu nately, different combinations of TM and TM oxides may form different stable or metastable second phases in the device during service [6,7], which is limitation for the application. There are laves phases of AB2 which belongs to the class of Frank-Kasper phases in topologically close-packed structures, i.e. cubic C15 (MgCu2, space group Fd-3m, No. 227), hexagonal C14(MgZn2, space group P63/mmc, No. 194) and C36 (MgNi2, space group P63/mmc, No194) phases [8]. Moreover, the C36
phase is an intermediate structure in the arrangement sequence of atoms from C14 to C15 structures [9]. Few authors have investigated the properties of Laves Mo2Hf and Mo2Zr so far [10–13]. The lattice pa rameters of these compounds were measured by Rapp in 1970 [11]. A investigation of the structural, elastic, and lattice dynamical properties for Mo2Zr and Mo2Hf with C14, C15, C36 phases are conducted using density functional total energy calculations by Turkdal et al. [10]. Since the composition, size and distribution of the Laves phases precipitating in alloys have a big influence on the final properties [14], the stability and thermodynamic properties of the systematic Laves phase are of great importance [15]. Nowadays, computational thermodynamics in the framework of CALculation of PHAse Diagram (CALPHAD) approach, which heavily depends on the quality of the thermodynamic descriptions for the target system, has been proved to be an effective way to accelerate the development of new alloys [15]. Therefore, reliable thermodynamic description of TM-based material is essential for the design of the database of RRAM. However, the existing thermodynamic data is not complete, it was found that the study of Pavlů et al. [16] was compatible
* Corresponding author., E-mail addresses: [email protected], [email protected] (J. Wang). https://doi.org/10.1016/j.calphad.2020.101766 Received 17 November 2019; Received in revised form 7 March 2020; Accepted 16 March 2020 Available online 28 March 2020 0364-5916/© 2020 Elsevier Ltd. All rights reserved.
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reaction (L þ βMo2Hf → βHf) rather than the eutectic reaction proposed by Taylor et al. [12], which agreed with experimental results of Brewer and Lamoreaux [22] and Massalski et al. [24]. The data of Rudy [23] at the Hf rich side was adopted. Garg and Ackerman [21] also reported a less detailed phase diagram by gravimetric and pyrometric methods. Considering all the information above, Brewer and Lamoreaux [22] discussed the critical experiments review of this system. Massalski et al. [24] constructed an experimentally assessed phase diagram according to the results mentioned above. For the intermetallic phase, Mo2Hf is a Laves-type phase with a complex sequence of polymorphs [11,20]. A hexagonal βMo2Hf (C36) and cubic αMo2Hf (C15) phase transition were observed by Taylor et al. [12] using X-ray diffraction and metallography and was confirmed by Eremenko et al. [20], who believed that the transition temperature be tween βMo2Hf and αMo2Hf obviously depends on the composition of compounds and a cubic phase will be formed at low temperature. The data of Taylor et al. [12] was adopted. The transition temperature of Mo-rich side was about 2133 K, and that of Hf-rich side was 2053 K. Eremenko et al. [20] also showed that the homogeneity of Mo2Hf ex tends from the stoichiometric composition to the Hf-rich side. In addi tion, Garg and Ackerman [21] measured the solubility of Hf in Mo, which was 28 at% (2453 K). The maximum solid solubility of Mo in (αHf) was reported at 0.8 at% (1503 K) [12]. Taking the thermodynamic data into consideration, no thermody namic data of βMo2Hf was reported. Brewer and Lamoreaux [22] esti mated the enthalpy of formation of αMo2Hf phase using experimental data of Laves phases of similar size ratios and electronic interactions to those of αMo2Hf. Rudy [23] assessed the enthalpy of formation of αMo2Hf phase using DTA. Recently, the enthalpy of formation of αMo2Hf phase was measured by high temperature direct synthesis calorimetry at 1373 � 2 K [25]. Shao [26] evaluated the enthalpies of formation of the end-members for αMo2Hf and βMo2Hf by CALPHAD method, which is incompatible with the parameters of the TM-based database [16–18]. So the first-principles calculations were introduced to calculate enthalpy of formation for all the end-members of Laves phases [27].
Table 1 Calculated and experimental lattice parameters of the SER structures and the relative difference between the calculated and experimental lattice parameters. Phase
Person symbol
Space group
Prototype
Lattice parameter(Å) Expt. Ref.
Calc.
Diff. (%)
(Mo)
cI2
Im-3m
W
3.150
0.095
(Hf)
hP2
P63/mmc
Mg
(Zr)
hP2
P63/mmc
Mg
a 3.147 [49] a 3.23 [49] c 5.12 a 3.238 [49] c 5.189
3.192
1.176
5.050 3.239
1.367 0.031
5.164
0.482
with our systems [17–19] in the process of building database. Consequently, the aims of the present work are (1) to critically evaluate the phase diagram and thermodynamic data in the Mo–Hf and Mo–Zr systems with a desire to develop a set of self-consistent thermo dynamic parameters for thermodynamic extrapolations of related higher order systems, (2) to calculate the total energies for all stable and metastable Laves phases of Mo–Hf and Mo–Zr systems, (3) to predict the site occupancy fraction trend of Laves phase with temperature by means of the CALPHAD approach. 2. Literature review The phase equilibria of the Mo–Hf system have been experimentally investigated by Taylor et al. [12], Eremenko et al. [20], Garg and Ackerman [21], Brewer and Lamoreaux [22] and Rudy [23]. All these works fit the basic features depicted by Taylor et al. [12], except for the detail of the phase equilibrium in Hf-rich side. Thus, this work mainly based on the phase equilibria data of Taylor et al. [12]. DTA (Differential Thermal Analysis) was used by Rudy [23] to investigate the solid-liquid equilibrium in the Hf-rich region and found that alloys were only separated by a very small temperature interval and the peritectic
Table 2 Equilibrium structural parameters of Laves phases in this work. Symbols a and c stand for lattice constants, △ shows the relative difference between the calculated and experimental atomic volume. System
Structure
Mo–Hf
αMo2Hf (C15)
βMo2Hf (C36)
γMo2Hf (C14)
Mo–Zr
αMo2Zr (C15)
βMo2Zr (C36)
γMo2Zr (C14)
k-point mesh
Lattice parameter(Å)
△(%)
a(Å)
c(Å)
c/a
Mo2Hf [50] Mo2Mo Mo2Hf MoHf2 Hf2Hf Mo2Hf [50] Mo2Mo Mo2Hf MoHf2 Hf2Hf Mo2Mo Mo2Hf MoHf2 Hf2Hf
21 23 21 21 21 19 21 19 17 21 23 21 21 21
� 21 � 23 � 21 � 21 � 21 � 19 � 21 � 19 � 17 � 21 � 23 � 21 � 21 � 21
� 21 � 23 � 21 � 21 � 21 �7 �9 �7 �5 �9 � 15 � 13 � 13 � 13
7.557 7.306 7.544 7.824 8.159 5.366 5.153 5.350 5.575 5.703 5.164 5.370 5.561 5.687
– – – – – 17.408 17.016 17.327 17.640 19.084 8.481 8.597 8.771 9.534
1 1 1 1 1 3.24413 3.26237 3.30215 3.23869 3.16413 1.6425 1.6008 1.5773 1.6766
–
Mo2Zr [51] Mo2Mo MoZr2 Mo2Zr Zr2Zr Mo2Mo MoZr2 Mo2Zr Zr2Zr Mo2Mo MoZr2 Mo2Zr Zr2Zr
23 23 21 23 21 21 21 19 21 23 19 21 19
� 23 � 23 � 21 � 23 � 21 � 21 � 21 � 19 � 21 � 23 � 19 � 21 � 19
� 23 � 23 � 21 � 23 � 21 �9 �9 �7 �9 � 15 � 13 � 13 � 13
7.595 7.306 7.868 7.594 8.253 5.153 6.066 5.386 5.792 5.163 5.639 5.413 5.759
– – – – – 17.016 15.227 17.444 19.072 8.481 8.652 8.625 9.606
1 1 1 1 1 3.30215 2.51022 3.23877 3.29282 1.64253 1.53431 1.59339 1.66800
–
2
0.09637 0.00515 0.109783 0.258527 – 0.06096 0.030498 0.210599 0.266827 – – – – 0.10987 0.11176 0.00039 0.283075 – – – – – – – –
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phase. As indicated by Garg and Ackermann [21], it is difficult to accurately determine the solidus-liquidus relationship in refractory Mo-rich alloys. DTA was used by Prima [31] to measure the Mo solidus, while the uncertainty of using DTA to measure the thermal effect of liquid phase was relatively large [21]. Later, Bhatt et al. [30] measured the Mo solidus temperature using the diffusion couple technique and then determined a solidus composition equals to 5 at.% Zr at 2358 K. This agreed very well with the measured Mo solidus data reported by Garg and Ackermann [21]. Thus, the data of Bhatt et al. [30] was adopted. What’s more, the maximum solubility of Zr in (βMo) has been studied by many researches [28,29,32]. It is indicated that the maximum solubility does not exceed 10–12 at% Mo at the peritectic temperature [28,29,32]. Moreover, the solubilities of Zr in (βMo) were determined by Svechnikov and Spektor [29] by X-ray diffraction and metallography, which were in good agreement with the measurements of Zakharov et al. [32]. Thus, the data of Svechnikov and Spektor [29] was adopted. For the intermetallic phase, αMo2Zr is in Cu2Mg structure (C15 Laves phase) [11], The homogeneity range has not yet been accurately determined and different values have been proposed. It was first described by Domagala et al. [28] that the ideal stoichiometric ratio of Mo2Zr was 2:1, while Garg and Ackermann [21] found the composition range to be between 33.3 and 36 at% of Zr for this intermetallic phase. The diffusion study [30] showed that Laves phase is formed between 33.3 and 34.5 at% of Zr at a temperature slightly below the peritectic isotherm. In other literature, it was found that this compound was nearly stoichiometric [11,23,29], Rapp proposed [11] a composition ranging from 33.48 to 33.77 at% of Zr. Thus, the compound was described with a small composition range in this work. For the thermodynamic data, due to the refractory nature of Mo al loys and Zr alloys, no thermodynamic data concerning the liquid and bcc has been found in the literature. So, the estimated enthalpy values (ΔHmix 6000 J/mol of atoms for 0.5 at% Mo) proposed by Boer Mo;Zr ¼ et al. [33] were used in the present work. Brewer and Lamoreaux [34] estimated the thermodynamic properties of αMo2Zr from the measured thermodynamic properties of Cr2Ta and Cr2Nb by Martin et al. [35]. Recently, the enthalpy of formation of αMo2Zr phase was measured by high temperature direct synthesis calorimetry at 1373 � 2 K [25]. The thermodynamic evaluation model of Mo–Zr system has been reported in Zinkevich and Mattern [36], P�erez and Sundman [37]. Because it seems unrealistic to model the liquid and β(bcc) stages with a fairly high excess entropy in the study of Zinkevich and Mattern [36], therefore, part of parameters of P�erez and Sundman [37] were adopted in this work.
0
First-principles calculated the total energy differences, Δ EL SER , between the Laves phases and the SER states calculated in this work, and also compared with the calculated [53,54,58,59] and enthalpies of formation of Mo2Hf and Mo2Zr compounds [21,25,34] and assessed [26,37,58,59] values available in literature. The values marked by an asterisk (*) were obtained with the help of the energy difference EHCP - EFCC from Wang et al. [53], as the value of energy of formation of Laves phase obtained in Sluiter [54] was related to the FCC structure (BCC structure is employed as the SER structure for Mo in Sluiter [54]). Ab initio re sults published in Ref. [53,54] were calculated using the generalized gradient approximation. Composition
Mo–Hf
αMo2Hf
This work Refs.
Mo2Mo 35.44 35.20 [54] 5 [26]
βMo2Hf (C36)
This work Refs.
35.67 5 [26]
9.6 [26]
γMo2Hf (C14)
This work Refs.
36.77
14.58
Mo–Zr
(C15)
Δ0 EL
αMo2Zr
This work Refs.
βMo2Zr (C36) γMo2Zr (C14)
This work This work Refs.
(C15)
SER
(kJ mol
1
atom 1) MoHf2 67.36 30 [26]
Hf2Hf 37.89 35.82* [53,54] 5 [26]
63.21
34.09
40.02 [26] 60.54
5 [26]
–
– MoZr2 67.91 7.24 [37]
35.67
Mo2Zr 12.99 13.31 [58] 12.35 [59] 9.30 [34] 7.4 � 4.2 [25] 7.24 [37] 12.14
30.92* [53,54] Zr2Zr 27.36 26.49* [53,54] 5 [37]
58.25
22.80
36.77
10.64
58.99
20.48
–
19.69* [53,54]
36.50 [54] Mo2Mo 35.44 35.2 [54] 5 [37]
36.5 [54]
Mo2Hf 16.69 16.4 [58] 15.53 [59] 11 � 1.4 [21] 6.4 � 4.3 [25] 10.6 [26] 16.06
–
32.03
3. Methodology
Mo–Zr system has been reviewed for many times in the literature [11,21,23,28–32]. Domagala et al. [28], who have conducted the first study of the Mo–Zr phase equilibria (thermal analysis, metallographic observation) and presented a rough phase diagram containing peritectic reaction (Liquid þ βMo → αMo2Zr), eutectic reaction (Liquid → βZr þ αMo2Zr) and a eutectoid decomposition (βZr → αZr þ αMo2Zr). Later, more phase diagram information was determined by Svechnikov and Spektor [29], Rudy [23], Prima [31], Garg and Ackerman [21]. All these works were mainly consistent with the phase boundary given by Domagala et al. [28] except the eutectic reaction was replaced by a peritectic reaction (Liquid þ αMo2Zr→ βZr) at the Zr-rich side. While it was refuted by Domagala et al. [28] and Garg and Ackerman [21] who found typical eutectic patterns in the microstructures of solidified alloys. The phase boundary data of Domagala et al. [28] were adopted. The temperature-composition data of liquidus in the Mo–Zr system from 30 to 90 at% Zr has been accurately determined by Garg and Ackermann [21] by combining the gravimetric and pyrometric methods with elec tron probe microanalysis (EPMA), which agreed well with the data of Svechnikov and Spektor [29] and Domagala et al. [28]. Therefore, the data of Garg and Ackermann [21] was adopted for Zr-rich side in liquid
3.1. First-principles calculations First-principles calculations on the basis of the Density Functional Theory (DFT) [38,39] are performed using the Projector Augmented Wave (PAW) [40,41] pseudo-potential as implemented in VASP [42,43], with the generalized gradient approximation (GGA) refined by Perdew, Burke and Ernzerhof (PBE) method [44] to calculate the total energies differences of all three Laves phase structures (C14, C15 and C36) and the SER (Standard Element Reference) structures. To maintain consis tency, the cutoff energy of 450 eV was selected for all elements and compounds after convergence tests. All structures are completely relaxed in terms of shape, volume and atomic coordinates. The electron self-consistent energy convergence criterion is 10 6 eV/atom. According to the Monkhorst-Pack scheme [45], the sampling amount of each reciprocal atom of k-point exceeds 10000. The spin polarization is not included in the calculations. Based on the total energy obtained from the beginning, the enthalpy of formation of the terminal compound was calculated as follows: ΔHði:jÞ ¼ Etotði:jÞ 3
1 3E
=
System
totðiÞ
2 3E
=
Table 3
totðjÞ
(1)
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Fig. 1. Calculated Mo–Hf phase diagram Compared to the experimental data (a), a magnify detail of Laves phases (b), and a magnify detail about the phase equilibrium between the βMo2Hf and αMo2Hf phases (c). Full lines represents the optimization in this work, the dashed lines were redrawn from Shao [26] and the experimental points were taken from Refs. [12,21,23,24].
phases multiplied by their molar fractions:
where i and j represents the components of Mo, Hf, Zr. Etotði:jÞ and EtotðiÞ is the total energies for the end-members, and pure elements in SER structure respectively. In order to get more accurate thermodynamic descriptions for bcc phase in the Mo–Hf system, the mixing enthalpy of Hf in Mo-rich side was calculated by dilute solution model. In the calculations, the 54-atom supercell is employed to study bcc phases with compositions of x ¼ 1/ 54; the 16-atom supercell for bcc phases with compositions of x ¼ 1/16. In the present work, supercells for bcc was generated with respect to the standard crystallographic unit cell, and the 2 � 2 � 2 and 3 � 3 � 3 supercell were generated with respect to a two-atom cubic cell for bcc lattice.
(2)
E Gα ¼ Gref þ ΔGid mix þ G
In the Mo–Hf and Mo–Zr system, the Gibbs energy for the solution phase (Liquid, β, α) was described by sub-regular solution with the Redlich-Kister polynomial for the excess Gibbs energy [46]. Gα ¼
X
xi Goi þ RT
i¼A;B
n X
X xi Lnxi þ xA xB i¼A;B
j LA;B ðxA
xB Þj
(3)
j¼0
in which Goi means the contribution of the mechanical mixing of pure elements to the total Gibbs energy, xi is the mole fraction of component, R is the gas constant, T is the absolute temperature. L represents the interaction between Mo and X (Zr or Hf) atoms in each sublattice, temperature dependence of L parameters is given by:
3.2. Thermodynamic models 3.2.1. Solution phases As mentioned above, the CALPHAD method requires modeling of Gibbs energy in each phase as a function of temperature and pressure, and then all other thermodynamic properties can be obtained by ther modynamic correlation. The total molar Gibbs energy of the whole system can be defined as the sum of the Molar Gibbs energies of all
L j ¼ Aj þ Bj T
(4)
where the interaction coefficients Aj and Bj are constants, which can be evaluated in the optimization process.
4
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� � Hfy8α ; Moy8Moα ; Zry8Zrα Hf
1=3
� � Hfy16d ; Moy16d ; Zry16d Mo Zr Hf
2=3
(9)
where ynm indicates the site occupancy fraction of the components m (Hf, Mo, Zr) in the sublattice n (8a or 16d), which is the unknown model parameter. For a given alloy HfxHf MoxMo ZrxZr , where xi ði ¼ Hf; Mo; ZrÞ represents the mole fraction of element in the Laves phase alloy. According to composition normalization and mass balance, Eqs. (10)–(15) can be obtained: xHf þ xMo þ xZr ¼ 1
(10)
8a 8a y8a Hf þ yMo þ yZr ¼ 1
(11)
16d 16d y16d Hf þ yMo þ yZr ¼ 1
(12)
1 3y8a
þ 2 3y16d Hf ¼ xHf
(13)
1 3y8a
þ 2 3y16d Mo ¼ xMo
(14)
= =
0
0
0
0
0
0
0
α α þ y’A y’B y’’B 0 LαA;B:B þ y’A y’’A y’’B LαA;A:B þ y’B y’’A y’’B LB;A:B GE ¼ y’A y’B y’’A LA;B:A
=
(6)
0
=
=
the ideal mixing Gibbs energy id Gα is equal to: � 0 0 0 0 � 00 00 00 0 0 �� ΔGid mix ¼ RT yA ln yA þ yB ln yB þ yA ln yA þ yB ln yB
0
=
(5)
0
Zr
Zr
¼ xZr
(15)
The Gibbs energy of the Cu2Mg-type intermetallic 32Hf-64Mo–4Zr (in at. %) was established as a mathematical model from stable pure elements. According to the characteristics of thermodynamic function, the quantity is determined by the initial state and the final state, and is independent of the intermediate process. We adopted another method to calculate the Gibbs energy of the above-mentioned information, as shown in equation (16). The formation of the alloy can be described with the hypothetical end-members and the alloys are formed from the endmembers, i.e., (1) ¼ (2) þ (3). The Gibbs energy of the phase can be calculated from the pure elements that are stable at room temperature.
3.2.2. Intermetallic compounds To simplify consistency with multi-component databases, illustrate the range of uniformity, and maintain the consistency with the model reported in the corresponding literature [26,37], two sublattice model (A, B)2 (A, B)1 were adopted for the Laves structure. And the Gibbs E energy can also be formulated by Eq. (2), but Gref , ΔGid mix and G are expressed differently compared with Eq. (3): 0
Mo
1 3y8a þ 2 3y16d
Fig. 2. Calculated enthalpies of formation of αMo2Hf compound at 0 K, compared with the experimental data [21,25,58,59].
Gref ¼ yA yA0 o GαA:A þ yB yB0 o GαB:B þ yA yB0 o GαA:B þ yB0 yA o GαB:A
Hf
(7)
where yA and yA0 refers to the site fractions of component in the first and 0
0
the second sublattice, respectively, 0 L represents the interaction parameter between the species. In general, the fewer the orders of pa rameters are, the better the optimization is. The four end-members o Gαi:j α
(16) The Gibbs energy of formation from the room temperature stable pure elements can be calculated by
(i ¼ A, B; j ¼ A, B) represents the Gibbs energies of the end elements A2A, B2B, A2B, AB2. These end-members represent all the possibilities that each sublattice is occupied by the single element. The total Gibbs energy of the Laves phase is a function of the temperature and composition of the two sublattices, describing the energy advantage of the element in the second sublattice or the first sublattice. One of them is described as below, for example: o
Gαi:j ¼ 2o Gαi þ o Gαj þ A þ B � T
ΔG ¼ ΔH
TΔS
(17)
where ΔS is the corresponding entropy, which can be obtained during the optimization procedure. ΔH can be calculated by the equation: X X 16d E ΔH ¼ y8a (18) k yl ΔHðk:lÞ þ Δ H
(8)
k¼Hf ;Mo;Zrl¼Hf ;Mo;Zr
Gαi and Gαj are the Gibbs energies of i and j, respectively, adopting the
where ΔHðk:lÞ represents the enthalpy of formation of the end-member compound from the pure elements Hf, Mo, Zr (0 K) in Laves phase, while the enthalpy difference of formation between 0 K and a defined temperature calorific content is neglected in current work. The excess enthalpy of formation ΔE H is obtained during the optimization. When in equilibrium state, ΔG reaches a minimum value, so there will be following constraint equations:
SER reference state. In this paper, these end-members are determined by first-principles calculations, overcoming the arbitrariness of selecting the Gibbs energy values of these end-members. 3.3. Site fraction of laves phase The prototype of the C15 Laves phase of αMo2Hf and αMo2Zr is Cu2Mg structure [11]. Using the general two-sublattice model, it can be described in the form as [47].
∂ΔG ∂ΔG ∂ΔG ∂ΔG ∂ΔG ∂ΔG ¼ 8a ¼ 8a ¼ 16d ¼ 16d ¼ 16d ¼ 0 ∂y8a ∂yMo ∂yZr ∂yHf ∂yMo ∂yZr Hf
5
(19)
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Fig. 3. Calculated Mo–Zr phase diagram Compared to the experimental data (a), a magnify detail about the phase equilibrium between the αMo2Zr and other phases (b), and a magnify detail of αMo2Zr (c). Full lines represents the optimization in this work, the dashed lines were taken from P� erez and Sundman [37] and the experiment points were taken from Refs. [11,21,23,28–30,32].
structure of Cr2Zr [52] structure was relaxed by the similarity of the electronic structure of Mo2Hf, the most stable structure was obtained after relaxation [16]. The metastable phase data of C36 phase of Mo2Zr was predicted by the same method. The comparison between the calculated lattice parameters of SER structures and experimental values are listed in Table 1, those for the C14, C15 and C36 structures are listed in Table 2, the calculated results are in good agreement with the experimental data [50,51]. The total energy difference Δ0 EL SER between the Laves phase and the SER state are then given in Table 3, all stable and metastable phases of the Laves phase of the Mo–Hf and Mo–Zr systems were calculated, which are compared to the results of other authors. It is found that the calculated energies of the Mo2Mo, Hf2Hf, and Zr2Zr are in good agree ment with the calculated value by Wang et al. [53] and Sluiter [54], and are far more positive than those reported by Shao [26] and P�erez and Sundman [37]. The stability of the Mo2X (X ¼ Hf, Zr) Laves phase configuration is sequentially increased in the sequences of C14, C36 and C15. It can be seen from the calculation that the C15 phase should be the most stable in the Laves phases of Mo–Zr and Mo–Hf systems (Table 3). And the stability of C14 and C36 structures at higher temperature is facilitated by the effect of entropy. Based on the reported experimental phase diagrams and
Once the value of ΔHðk:lÞ is determined, the relationship among site oc cupancy fractions and composition and temperature can be obtained: i yW Ei ¼ f ðxi ; TÞ
(20)
In order to handle the complex system of partial differential equations, the software package Thermo-Calc [48] was used to calculate the site occupancy fractions. Once the field occupancy at different temperatures is calculated, the order-to-disorder transition can be studied systematically. 4. Results and discussion In Mo-X (X ¼ Hf, Zr) systems, the structure parameters for SER states of pure Zr, Hf and Mo were taken from Pearson’s Handbook [49]. The experiment structure parameters of A2A, B2B A2B, B2A (A ¼ Mo, B ¼ Hf, Zr) configurations of C15 structures were obtained from the experiments on Mo2Hf [50] and Mo2Zr [51], and all structural parameters of C36 phases were obtained from the experimental data of Mo2Hf [50]. As there are no C14 lattice parameters, in order to complete the informa tion of the Laves phase, the first-principles method was used to calculate the C14 Laves metastable phase data for future reference. The C14 6
R. Zhao et al.
Calphad 69 (2020) 101766
Table 4 Optimized thermodynamic parameters describing the Laves phases in Mo–Hf system. The calculated values shown in boldface were taken from Table 3. Phase
Sublattice model
Evaluated parameters
Liquid
(Mo, Hf)1
0 Liquid LMo;Hf
β(bcc)
α(hcp)
(Mo, Hf)1(Va)
(Mo, Hf)1(Va)
¼ 46500
2 Liquid LMo;Hf
¼
1 βðbccÞ LMo;Zr:Va
¼ 65515
2 βðbccÞ LMo;Zr:Va
¼ 2159
0 αðhcpÞ LMo;Hf:Va
¼
(Mo, Hf)1/3(Mo, Hf)2/3
2 αðhcpÞ LMo;Hf:Va o αMo Hf
GHf:Hf 2
o o
2:270T
[26]
24872 þ 8:905T 29:900T 5:847T
500
[26]
6500
o
¼ 37899 þ
Ghcp Hf
o
αMo Hf
2 GMo:Mo ¼ 35441 þ Gbcc Mo
Mo2 Hf GαHf:Mo ¼ o
16688 þ 1:068T þ o
bcc 1=3 Ghcp Hf þ 2=3 GMo
Fig. 4. Calculated enthalpies of formation of αMo2Zr compound at 0 K, compared with the experimental data [25,34,58,59].
o
αMo2 Hf
GMo:Hf
2=3
thermodynamic data, the thermodynamic parameters were evaluated using the computer optimization program PARROT module of the Thermo-Calc software package [55]. This study used a stepwise opti mization process described by Du et al. [56]. Experimental phase dia gram data and thermodynamic properties were used as input to the program. According to the uncertainty of the experimental data, a certain weight was set for each data selected. During the evaluation process, these uncertainties can be changed by trial and error until the selected experimental information is reproduced within the expected uncertainty range. The optimization of this study was carried out in three steps. In the first step, the Gibbs free energy of the Laves phase end member and the Gibbs free energy from the SGTE [57] pure element were calculated by first-principles. In the second step, the parameters of the β(bcc) phase (Mo–Hf system only) were optimized using the calcu lated dilute solution model data. Finally, the phase diagram data, ther modynamic data and first-principles calculations data were used to evaluate the Laves phase parameters. When optimizing the Mo–Hf and Mo–Zr systems, the Gibbs energy of Laves phase was modeled at the beginning using the data of the total energy difference in Table 3. Combined with the experimental data, the parameter L of non-ideal mixing excess Gibbs energy is obtained. After optimizing the Laves phase, as the peritectic point on the Mo-rich side (Liquid þ βMo → βMo2Hf) was not in good agreement in Mo–Hf system, the β(bcc) phase of the Mo–Hf system was optimized. Considering that there is no experimental and computational data of the β(bcc) phase in the literature, the dilute solution model was used to calculate Hf-rich data of β(bcc) phase so as to optimize. It can be seen from Fig. 1 that β(bcc) phase is in good agreement with experiment data. What’s more, the phase boundary of β(Mo) and liquid phase in the Zr-rich side were not satisfied in Mo–Zr system, thus the parameters of β(bcc) and liquid were reoptimized. As can be seen from Fig. 3, the parameters of β(bcc) and liquid phases agree well with the experimental data after optimi zation. Figs. 2 and 4 show the calculated enthalpy of formation of αMo2Hf and αMo2Zr at 0 K in the present work compared with both firstprinciples databases [58,59] and experiment data [25]. The experi mental data was measured at 1373 K while the calculated data was at 0 K. The results of the present calculations fit well with the databases. All values [34] calculated and assessed by first-principles are lower than the experimental data. The first-principles value is selected in order to guarantee the energetic consistency [16]. The thermodynamic param eters of other phases such as Liquid, α(hcp) in Mo–Hf system and α(hcp)
(βMo2Hf)
(Mo, Hf)1/3(Mo, Hf)2/3
o
Ghcp Hf
2 ¼ LMo:Hf;Mo ¼
0 αMo Hf 2 LHf;Mo:Hf
2 ¼ LHf;Mo:Mo ¼
0 αMo Hf 0 αMo Hf
o
βMo Hf GHf:Hf2
¼ 34086 þ
o
βMo2 Hf GMo:Mo
¼ 35667 þ Gbcc Mo
o
2 Hf GβMo Hf:Mo ¼
3
o
Ghcp Hf
o
6352 2930
Ghcp Hf
o
16000 þ 0:742T þ 1= o
þ 2=3 Gbcc Mo
βMo2 Hf GMo:Hf o 1=3 Gbcc Mo
(Mo, Hf)1/3(Mo, Hf)2/3
þ 1=3
Gbcc Mo
0 αMo Hf 2 LHf:Hf;Mo
o
(γMo2Hf)
2:030T þ
¼ 67360 o
o
¼ 63205 þ 2=3 Ghcp Hf þ
0 βMo Hf 2 LHf:Hf;Mo
2 ¼ LMo:Hf;Mo ¼
0 βMo Hf 2 LHf;Mo:Hf
2 ¼ LHf;Mo:Mo ¼ 97712
o o o
γMo Hf
GHf:Hf2
0 βMo Hf
o
¼ 32030 þ Ghcp Hf o
2 GMo:Mo ¼ 36771 þ Gbcc Mo 2 Hf GγMo Hf:Mo ¼
o
14581 þ 0:172T þ o
bcc 1=3 Ghcp Hf þ 2=3 GMo
o
3129
0 βMo Hf
γMo Hf
γMo Hf
2 GMo:Hf
o
1=3 Gbcc Mo
This work This work This work [26] [26]
¼ 27000 ¼
[26] [26]
28:000T
0:500T
¼
0 βðbccÞ LMo;Hf:Va
1 αðhcpÞ LMo;Hf:Va
(αMo2Hf)
23414
¼
1 Liquid LMo;Hf
Ref.
o
¼ 60542 þ 2=3 Ghcp Hf þ
This work This work This work This work This work This work This work This work This work This work This work This work This work This work This work This work
in Mo–Zr system are respectively taken from Shao [26] and P�erez and Sundman [37], because no new information is included on these phases. The thermodynamic parameters describing the Mo–Hf and Mo–Zr sys tems are listed in Tables 4 and 5. The model described in Section 3.2 provides a good description of the systems. The calculated phase dia grams of Mo–Hf and Mo–Zr systems are shown in Figs. 1 and 3. It can be clearly seen that the first-principles calculations of end-member of Laves phases well describe the phase diagram of Mo–Hf and Mo–Zr systems (experimental point), and the Laves phase boundary is especially in good agreement with the experimental data. The reasonable physical signif icance of the parameters of the Gibbs energy of the Laves phase makes the optimization process simpler, more effective and more reliable and these parameters can be used in the extrapolation of higher-order systems. The present model and method described in section 3.3 can predict the relationship between the site occupancy fraction and composition at 7
R. Zhao et al.
Calphad 69 (2020) 101766
Table 5 Optimized thermodynamic parameters describing the Laves phases in Mo–Zr system. The calculated values shown in boldface were taken from Table 3. Phase
Sublattice model
Evaluated parameters
Ref.
Liquid
(Mo, Zr)1
0 Liquid LMo;Zr
¼
1 Liquid LMo;Zr
¼ 17990
2 Liquid LMo;Zr
¼
This work This work This work This work This work [37]
β(bcc)
(Mo, Zr)1(Va)
α(hcp)
(Mo, Zr)1(Va)
(αMo2Zr)
(Mo, Zr)1/3(Mo, Zr)2/3
27196 þ 8:823T 6:812T
5416 þ 9:460
0 βðbccÞ LMo;Zr:Va
¼
2768 þ 13:411T
0 βðbccÞ LMo;Zr:Va
¼
9152 þ 6:856T
0 αðhcpÞ LMo;Zr:Va ¼ 26754 þ 4:560T o o αMo2 Zr GMo:Mo ¼ 35441 þ Gbcc Mo o
o
Mo2 Zr GαZr:Zr ¼ 27356 þ Ghcp Zr
o
Mo2 Zr GαZr:Mo ¼ o
12994 þ 2:670T þ o
hcp 2=3 Gbcc Mo þ 1=3 GZr o
αMo2 Zr
GMo:Zr o
1=3
(βMo2Zr)
(Mo, Zr)1/3(Mo, Zr)2/3
¼ 67914
Gbcc Mo
þ 2=3
o
0
Ghcp Zr
2:130T þ
0 αMo2 Zr LMo:Mo;Zr
Mo2 Zr ¼ LαZr:Mo;Zr ¼ 14616
0 αMo2 Zr LMo;Zr:Mo
Mo2 Zr ¼ LαMo;Zr:Zr ¼
0
2 Zr GβMo ¼ 22800 þ Ghcp Zr:Zr Zr
o
2 Zr bcc GβMo Mo:Mo ¼ 35667 þ GMo
o
2 Zr GβMo Zr:Mo ¼
o
o
12142 þ 2:550T þ o
bcc 1=3 Ghcp Zr þ 2=3 GMo
o
(γMo2Zr)
(Mo, Zr)1/3(Mo, Zr)2/3
2 Zr GβMo Mo:Zr o bcc 1=3 GMo o γMo2 Zr GZr:Zr
This work
¼ 20484 þ Ghcp Zr
o
This work This work This work
o
γMo2 Zr GZr:Mo ¼
o
10643 þ 1:850T þ o
bcc 1=3 Ghcp Zr þ 2=3 GMo
o
γMo2 Zr GMo:Zr ¼ 58991 þ 2=3 Ghcp Zr þ o
1=3
Gbcc Mo
Fig. 5. Site fraction of elements Zr in sublattice in different components as the temperature changes (the solid line represents the interaction parameter L and b value for the end-members are included, and the dashed line represents no interaction parameter L and b value are included).
This work This work This work This work This work
o
γMo2 Zr GMo:Mo ¼ 36771 þ Gbcc Mo
o
This work
¼ 58246 þ 2=3 Ghcp Zr þ
o
o
3702
o
o
This work This work This work
treatment temperature. At a temperature of higher than 1100 K, Zr atoms completely occupies the 8a sublattice regardless of the xMo/xHf. This is of great significance for the optimization of ternary systems. If Zr atom completely occupies the 8a sublattices, it means that the endmember parameters of the 16d sublattices can be ignored, the opti mized parameters can be reduced. The present model and method can reliably predict the thermody namic data and phase diagram data of higher-order systems. No exper imental phase equilibrium data and thermodynamic data of Mo-Hf-Zr have been reported in the literature, and only Bla�zlna et al. [60] measured the unit cell parameters, cell volumes and hardness of αMo2HfxZr1-x (x ¼ 0–0.75) alloy at 1873 K by X-ray and metallography. The experimental results show that bcc solid solutions (βMo, βHf, βZr) and MgCu2 type (αMo2HfxZr1-x) phase exist at 1873 K, and the hardness of Mo2Hf decreases gradually with the increase of Hf content, among which the highest solubility of Hf in Mo2Zr is 33.3%. Thus, the ther modynamic parameters are extrapolated from this work and Zr–Hf system from Hu et al. [61]. The calculated isothermal sections of the Mo-Hf-Zr system were shown in Fig. 6. Comparisons between the calculated phase diagrams and the experiments show that the thermo dynamic description in this paper can reproduce the experimental data well. The liquidus projection of the Mo-Hf-Zr system with isotherm is calculated using the present thermodynamic parameters and is shown in Fig. 7. It can be seen from Fig. 7 that four primary solidification fields (βMo, βHf, βZr), βMo2Hf, αMo2HfxZr1-x, (βMo) and two invariant reac tion L þ βMo2Hf þ (βMo) → αMo2HfxZr1-x at 2404 K and L þ βMo2Hf → αMo2HfxZr1-x þ (βMo,βHf, βZr) at 2148 K are calculated. The corre sponding reaction scheme of the liquidus projection is given in Fig. 8.
This work
any temperature. For stoichiometric Mo2Hf (Mo2Zr), Hf (Zr) atom completely occupies sublattice 8a, while Mo atom completely occupies sublattice 16d, the site occupancy behavior is independent of tempera ture. But when Zr atom is added into C15-type Mo2Hf-based Mo-Hf-Zr alloy, the site occupancy fraction changes, as shown in Fig. 5. When xMo/xHf keeps 2:1, Zr occupies 8a sublattice and 16d sublattice partially below 1500 K. From 500 to 1500 K, Zr occupies the 8a sublattice rapidly and Zr completely occupies 8a sublattice up to 1500 K. When xMo/xHf is less than 2:1, Zr atoms prefer to occupy 16d sublattice at temperature below 400 K while they also tend to occupy 8a sublattice with the temperature increasing. And Zr atoms fully occupy 8a sublattice when the temperature is higher than 1000 K. When xMo/xHf is over 2:1, Zr atoms mainly occupy 8a sublattice at temperature below 1000 K, only a small portion occupies the 16d sublattice. When the temperature is higher than 1000 K, the Zr atoms completely occupy the 8a sublattice. The dashed line in Fig. 5 also shows the site occupancy without inter action parameters. It can be seen that the trend between the two is roughly the same, with minor differences. The one containing interac tion parameter L and entropy value b are more rigorous in thermody namic properties. In this paper, the solid line containing interaction parameter L and entropy value b are preferred. From what have been discussed above, the results show that the positional preference of alloying element Zr varies with the change of xMo/xHf and heat
5. Conclusion In this study, the stable and metastable Laves phase of Mo–Hf and Mo–Zr systems were studied. The relative stability of the Laves phase in these systems is revealed by comparing the energy calculated by the first-principles of these structures with the total energy of the ideal element mixture. On the other hand, the formation of enthalpy and mixed enthalpy of the solid phase Mo2Hf Laves phase in the dilute so lution was predicted by the first-principles calculations method. The unavailable thermochemical information in this experiment provides meaningful values for the parameterization of the Gibbs energy func tion. The obtained descriptions of Mo–Zr and Mo–Hf were then 8
R. Zhao et al.
Calphad 69 (2020) 101766
Fig. 6. Calculated isothermal section of the Mo-Hf-Zr system: (a) 1173 K, (b) 1573 K, (c) 1873K with experimental data from Bla�zlna et al. [60], (d) 2273 K.
9
R. Zhao et al.
Calphad 69 (2020) 101766
Fig. 7. Calculated liquidus projection for the whole Mo-Hf-Zr system. The full lines represent the monovariant lines, and the dashed lines are isotherms.
Fig. 8. The reaction scheme for the Mo-Hf-Zr system according to the present calculation with temperature in K.
combined with that of Zr–Hf system to form the basis of Mo-Hf-Zr ternary system. The isothermal section and the liquidus projection of Mo-Hf-Zr system were also predicted by combing with the Zr–Hf system reported in the literature. Comparisons between the calculated and measured phase diagrams show that most of experimental data are satisfactorily reproduced by the present thermodynamic descriptions. Based on these results, by using the method of two-sublattice model and first-principles calculations, it is found that the occupation of ternary Mo-Hf-Zr alloy in C15 phase Zr atom tends to occupy the 8a sublattice with the increase of temperature.
work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgements The financial supports from the National Key Research and Devel opment Program of China (Materials Genome Initiative: 2017YFB0701700) and the National Natural Science Foundation for Youth of China (Grant Nos. 51601228, 51429101 and 51602351) are greatly acknowledged.
Declaration of competing interest The authors declared that they have no conflicts of interest to this 10
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Calphad 69 (2020) 101766
References
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