Experimental studies and thermodynamic assessment of the Ba-Mo-O system by the CALPHAD method

Thermodynamic measurements on BaMoO 4 , BaMoO 3 and BaMo 3 O 10 are reported, that served as input for the development of a thermodynamic model of the Ba-Mo-O system using the CALPHAD methodology. The valence states of molybdenum in BaMoO 4 and BaMoO 3 were confirmed to be VI and IV, respectively, from X-ray Ab- sorption Near Edge Structure Spectroscopy measurements at the Mo K-edge. The heat capacity at low temperatures of these compounds was obtained from thermal-relaxation calorimetry. Phase equilibrium data in the BaMoO 4 -MoO 3 section were also measured, and the transition enthalpy associated with the peritectic decom- position of BaMo 3 O 10 was determined using Differential Scanning Calorimetry. The developed thermodynamic model used the compound energy formalism for intermediate compounds, and an ionic two-sublattice model for the liquid phase. The optimized Gibbs energies were assessed with respect to the known thermodynamic and phase equilibrium data. A good agreement is generally obtained, but a number of ill-defined data were also identified.


Introduction
The chemistry of fission product (FP) elements in irradiated nuclear fuel, and more particularly of volatile and semi-volatile elements, is of paramount importance, as FPs are the main source for the radiological consequences of a severe accident (SA) with release to the environment. Barium and molybdenum are generated with a high fission yield (~11% and ~25%, respectively [1]) during irradiation of the UO 2 or (U,Pu)O 2 ceramic fuel used in Light Water Reactors (LWRs), and represent key elements for the evaluation of the source term. They are classified as semi-volatile fission products, which implies that their release kinetics are dependent on the redox conditions of the surrounding environment, and are determined by the evaporation of the chemical compounds formed in the irradiated fuel [1]. The association of barium with molybdenum in the nuclear fuel in the form of oxide precipitates at grain boundaries was recently confirmed from the FPT2 test of the PHEBUS Fission Product Program [2]. However, the exact chemical form of barium and molybdenum in the fuel is rather complex, and changes with time in relation with the burnup, temperature and oxygen potential conditions. The severe accident at the Fukushima-Daiichi Nuclear Power Station (FDNPS) has initiated a renewed interest in the behaviour of radionuclides with a potential long-term radiological impact such as those of Ba, Sr, U, Pu, and minor actinides. To respond to this necessity, the TCOFF project (Thermodynamic Characterization of Fuel Debris and Fission Products Based on Scenario Analysis of Severe Accident Progression at Fukushima-Daiichi Nuclear Power Station) was launched in 2017 under the auspices of the OECD/NEA [3], and the experimental study of the Ba-Mo-O system as outlined in this work constitutes a contribution to that initiative.
The solubility of barium in the (U,Pu)O 2 fuel matrix is very low due to its large ionic radius. At low oxygen potentials, it is found in the socalled grey phase of general formula (Ba,Sr,Cs)(Zr,U,Mo,RE)O 3 (RE = rare earths) with perovskite structure [4][5][6][7][8], while it is stable in the Ba(Mo,U)O 4 scheelite phase at high oxygen potentials [5,[7][8][9]. For a thorough safety assessment of nuclear fuel behaviour under operation and in accidental conditions, a complete thermodynamic description of the multi-component system Ba-Sr-Cs-Zr-U-Pu-Mo-RE-O is therefore necessary. A number of thermodynamic and thermophysical data on the sufficient information, presumes similar first nearest neighbour coordination spheres in the liquid and bcc phases. The values of those interaction parameters have been derived in order to agree with the maximum solubility in the Ba liquid phase proposed by Massalski [24]. The formalisms used are described in Section 4.

Binary Ba-O system
The description initially present in the TAF-ID database for Ba-O was inherited from the Fuelbase project [23], based on the work by Zimmermann et al. [25]. The current description was built starting from the later assessment by Zhou et al. [26], that allows to better describe the heat capacity data for BaO 2 , following measurements reported subsequently to the work of Zimmermann et al. The description of the condensed phases BaO and BaO 2 by Zhou et al. have been modified to be consistent with the current models in the TAF-ID, however. The O 2− 2 species has not been considered in the model for the liquid phase, but rather has been replaced by a neutral O species. The stability of the liquid phase for compositions richer in oxygen than BaO has not been forced as in the original description, since no experimental information was available on this issue. The solid phases have been considered as stoichiometric. The BaO 2 homogeneity range modelled by Zimmermann et al. and by Zhou et al. is in fact considered too large according to [27] and [28]. This simplification implied a slight modification of the BaO 2 stoichiometric compound description, so as to maintain the agreement with the oxygen pressure data over the BaO-BaO 2 region. The description of the gaseous species in this system, and in particular of the binary species BaO and Ba 2 O, has been extracted from the substance SGTE (Scientific Group Thermodata Europe) database. The thermodynamic function for the gaseous Ba 2

Binary Mo-O system
The Mo-O system is quite complex with five binary oxide compounds reported, i.e. MoO 2 , MoO 3 , Mo 4 O 11 , Mo 8 O 23 , Mo 9 O 26 , also treated as stoichiometric. The optimized parameters for the Mo-O system are reported in the work of Corcoran et al. [29]. The parameters for the liquid phase have been later updated in the PhD thesis of Kauric [30] to prevent the occurrence of a miscibility gap at very high MoO 3 content in ternary systems such as Cs-Mo-O or Na-Mo-O.

Structural data on the ternary Ba-Mo-O phases
The Ba-Mo-O system is rather complex, with a number of ternary phases reported: BaMoO 4 , Ba 2 MoO 5 , Ba 3 8 . The reported crystal structures for each phase are listed in Table 1. The availability of the atomic positions for the crystal structures is also indicated. In the present thermodynamic model, only the hexavalent phases BaMoO 4 , Ba 2 MoO 5 , Ba 3 MoO 6 , BaMo 3 O 10 , BaMo 2 O 7 , and pentavalent phase BaMoO 3 are considered, for which sufficient structural and thermodynamic information are available, and whose existence are most probable. Trivalent BaMo 6 O 10 and the reported mixed valence state solid phases Ba 8 69, respectively), were not included due to the lack of thermodynamic data, and because of the general scarcity of experimental information on the phase diagram equilibria in the regions of composition where such phases should be stable.

Thermodynamic data
Thermodynamic data in the literature are only available to the best of our knowledge on BaMoO 4 , BaMoO 3 , and BaMo 2 O 7 .

BaMoO 4
The enthalpy of formation of BaMoO 4 was determined by various authors using indirect measurement techniques and solution calorimetry. The values reported are summarized in Table 2. Tamman and Westerhold [40] derived the enthalpy of formation from the measurement of the enthalpy of reaction between barium oxide and molybdenum trioxide at T = 560 K:    [44] Direct (solution calorimetry) − (1545.6 ± 1.9) Shukla et al. [45] Direct (solution calorimetry) − (1547.8 ± 3.6) Singh et al. [46] Indirect (EMF measurements) -second law K [41]. However, the review of Cordfunke and Konings [42] (1)). Combining with the enthalpies of formation of BaO and MoO 3 recommended by [42], we obtain an enthalpy of formation or BaMoO 4 of − (1501.4 ± 10.7) kJ ⋅ mol − 1 . O'Hare [44] studied the precipitation of BaMoO 4 (cr) from an ammoniacal solution of BaCl 2 , and derived the enthalpy of formation based on the reaction: Shukla et al. [45] also studied the precipitation of BaMoO 4 (cr), but from an ammoniacal solution of Ba(NO 3 ) 2 , and derived the enthalpy of formation based on the reaction: Singh et al. measured the standard molar Gibbs energy of formation of BaMoO 4 by measuring the emf of the cell {Pt/BaMoO 3 +BaMoO 4 /CSZ/ air (p(O 2 ) = 21.21 kPa, CSZ = 15 mol% CaO stabilized zirconia)} in the temperature range T = (1091-1309) K, and derived the standard enthalpy of formation by second and third law analyses of the data [46]. A more detailed description of the individual measurements can be found in [42], [45] and [46]. The values obtained by direct solution calorimetry measurements are preferred in this work. They are in good agreement and based on two different reaction schemes. Here we recommend the weighted 1 average of the aforementioned two values, i.e. Δ f H o m (BaMoO 4 , cr, The low-temperature heat capacity was measured from 2.02 to 297.19 K using the thermal-relaxation method [49]. The heat capacity and the standard entropy at 298. 15  High temperature enthalpy increment measurements were reported using drop calorimetry in the temperature range T = (986-1732) K by Saha et al. [50], and in the temperature range T = (299-1020.3) K by Singh et al. [51]. From the data of Saha et al. [50], and constraining the fit to C o p,m (BaMoO 4 , cr, 298.15 K) = 122.2 J ⋅ K − 1 ⋅ mol − 1 , Cordfunke and Konings [42] derived the equation: Singh et al. [51] reported the following equation by constraining to the same heat capacity value at 298.15 K, and combining their data with that of Saha et al. [50]: More recently, Saha et al. [52] reported direct measurements of the heat capacity of BaMoO 4 in the temperature range T = (140-870) K using Differential Scanning Calorimetry (DSC) and the step method. In this work, we have refitted the data of [50] and [51] The latter equation is used for the thermodynamic model and compared in Figs. 2a and 7 a to the experimental data, and selected equations by Corfunke and Konings [42] and Singh et al. [51]. BaMoO 4 was reported to melt congruently at T = 1737 K by Dash et al. [53], but the original source is not cited. It was moreover reported to melt at T = 1723 K in the Handbook of Inorganic Compounds [54] based on the review by Knacke et al. [55]. A previous study by Ustinov et al. [56] reported a melting temperature at T = 1273 K, but this is highly   [58], respectively.

BaMoO 3
The enthalpy of formation of BaMoO 3 was determined using the emf technique [59], oxygen bomb calorimetry [60] and Knudsen mass-loss method [61]. The values reported in the literature are summarized in Table 3. The values obtained by Dash et al. [61] by second and third laws are in good agreement, but about 35 kJ ⋅ mol − 1 lower than in the other studies. Here we prefer to retain the recommendation of Cordfunke and Konings [42] based on the average values of Zharkova et al. [60] and Rezukhina and Levitskii [59]: There are no low-temperature heat capacity data available for this compound in the literature. Cordfunke and Konings [42] have estimated the heat capacity and entropy based on a comparison with BaZrO 3 : Agarwal et al. [63] reported enthalpy increment measurements in the temperature range T = (477.8-1010.8) K using a high temperature Calvet calorimeter. The fitting of this data [63] [65] are compared in Fig. 3a and b. Both equations yield similar values above T = 450 K, but differ largely on the extrapolation down to room temperature.
In this work, we have refitted the data of Agarwal et al. [ The latter equation was implemented in our thermodynamic model. Paschoal et al. [66] reported the decomposition of BaMoO 3 at T = 1653 K into BaMoO 4 (cr), Mo(cr) and BaO(g) according to Differential Thermal Analysis (DTA) measurements under argon and helium atmospheres, i.e. following the equilibrum reaction 3BaMoO 3 (cr) = 2BaMoO 4 (cr) + Mo(cr) + BaO(g).
More recently, Yamanaka et al. [64] measured the melting temperature of BaMoO 3 using a thermal arrest method under a reducing atmosphere and reported T fus (BaMoO 3 ) = 1791 K. No enthalpy of fusion data are available to this date on this compound.

BaMo 2 O 7
Singh et al.  their data, the authors derived the enthalpy of formation at 298.15 K as The heat capacity was not measured to this date.

Phase diagram data in the BaO-MoO 3 pseudo-binary section
There is no source in the literature reporting (to the best of our knowledge) a phase diagram over the complete composition range between BaO and MoO 3 . BaMoO 4 is found on the BaO-MoO 3 pseudobinary section at the composition x(MoO 3 ) = 0.5. Phase equilibrium data in the BaO-BaMoO 4 section were reported by Yanushkevich and Zhukovskii [68], while a sketch of the BaMoO 4 -MoO 3 section was reported by Ustinov et al. [56] and Zhukovskii et al. [69]. However, the latter two sets of data are in rather poor agreement as seen in Fig. 12.
Yanushkevich and Zhukovskii studied the BaO-BaMoO 4 section using X-ray and visual polythermal techniques [68] on BaCO 3 -MoO 3 mixtures after thermal treatment. Two intermediate compounds were identified on the pseudo-binary section, namely Ba 2 MoO 5 and Ba 3 MoO 6 , with transition temperatures T = 1573 K (peritectic) and T~1825 K (congruent melting), respectively. In addition the authors found two eutectic equilibria, i.e. between BaO and Ba 3 MoO 6 around 1808 K, and between Ba 2 MoO 5 and BaMoO 4 around 1493 K.
Ustinov et al. investigated the BaMoO 4 -MoO 3 phase diagram using thermography and X-ray diffraction [56] on (BaO:MoO 3 ) mixtures. The authors reported the existence of an incongruent melting compound of composition BaMo 2 O 7 (with a decomposition temperature around 988 K), and an eutectic equilibrium at x(MoO 3 ) = 0.811 and T = 913 K. The interplanar spacings reported for the BaMo 2 O 7 phase seem to match the BaMo 3 O 10 composition, however, as seen in Fig. 4a, where they are compared to the X-ray diffraction pattern collected in this work for BaMo 3 O 10 . Moreover, the authors also reported the melting temperature of BaMoO 4 to be 1273 K, which is much lower than found in subsequent studies. The liquidus data on the BaMoO 4 rich-side is therefore doubtful based on the latter result.
Zhukovskii et al. used thermal-optical, X-ray diffraction, and thermographic methods on (BaO:MoO 3 ) mixtures for their investigations [69]. The authors also claimed the existence of the BaMo 2 O 7 compound, but with a lower incongruent melting temperature, i.e. (926 ± 3) K. Again, the interplanar spacings listed for BaMo 2 O 7 are compared in Fig. 4b with the X-ray diffraction data collected in this work for BaMo 3 O 10 . Despite a general shift to lower angles, the d spacings seem to match rather well the BaMo 3 O 10 data. An eutectic equilibrium was also reported between BaMo 2 O 7 and MoO 3 at x(MoO 3 ) = 0.8 and T = (897 ± 3) K, in rather good agreement with the data of Ustinov et al. [56]. BaMoO 4 was found stable up to at least 1673 K. The liquidus data found on the BaMoO 4 rich-side were found much higher than in the studies of Ustinov et al., and seem more trustworthy based on the expected melting temperature of the barium molybdate.

Thermodynamic modelling assessment
Dash et al. [53] reported a thermodynamic model for the Ba-Mo-O system using the SOLGASMIX-PV program [70]. The authors considered 9 compounds in this system: BaMoO 3 24 . The Gibbs energy functions for the ternary phases were estimated using the additive oxide method for the standard entropy and enthalpy of formation, and using the rule of Neumann-Kopp for the heat capacity, when no experimental data were available in the literature. Only the computed isothermal section of the Ba-Mo-O phase diagram at T = 700 K is shown in their work. The authors reported O 2 (g), BaO(g), BaMoO 4 (g) and BaMoO 3 (g) to be the predominant vapour species in all ternary phase fields, and over pure compounds. The authors also observed that the calculated oxygen potentials over the ternary phase fields and pure compounds were not affected much by the change of the Gibbs energy functions of the ternary phases. However, this had a more pronounced effect on the partial pressures of Ba(g), BaO(g), BaMoO 4 (g) and BaMoO 3 (g) (one to two orders of magnitude). Experimental investigations of the stable ternary phase fields are needed to confirm the predictions of the computed phase diagram by [53], which are unfortunately not available to this date. The purity of the synthesized materials was checked using X-ray diffraction. No secondary phases were detected, and the samples' purities is expected to be better than 99% 4 .

Powder X-ray diffraction (XRD)
X-ray diffraction patterns were collected at room temperature using a PANalytical X'Pert PRO X-ray diffractometer mounted in the Bragg-Brentano configuration with a Cu anode (0.4 mm × 12 mm line focus, 45 kV, 40 mA), and the X-ray scattered intensities were measured with a real time multi strip (RTMS) detector (X'Celerator). The data were collected by step scanning in the angle range 10 • ≤2θ ≤ 120 • with a step size of 0.008 • (2θ); total measuring time was about 8 h. Structural analysis was performed by the Rietveld method with the FullProf suite [71].

X-ray Absorption Near Edge Structure Spectroscopy (XANES)
XANES measurements were performed at the BM26A-DUBBLE BeamLine (Dutch-Belgian Beamline) of the European Synchrotron Radiation Facility (ESRF, Grenoble, France). Small amounts (10-20 mg) of powdered sample were mixed with boron nitride (BN), and pressed into pellets for the measurements. The storage ring operating conditions were 6.0 GeV and 170-200 mA. A double crystal monochromater mounted with a Si(111) crystal coupled to collimating and focusing Pt coated mirrors was used.
XANES spectra were collected at room temperature in transmission mode at the Mo K-edge. A step size of 1 eV was used in the edge region.
The energy E 0 of the edge absorption threshold position was taken at the inflection point of the spectrum by using the first node of the second derivative. The position of the pre-peak was selected from the first node of the first derivative. Several acquisitions were performed on the same sample and summed up to improve the signal-to-noise ratio. Before averaging the scans, each spectrum was aligned using the XANES spectrum of a metallic molybdenum reference foil measured before and after the series of samples under investigation (i.e. MoO 2 , MoO 3 , BaMoO 3 , and BaMoO 4 ). The ATHENA software [72] was used to normalize the spectra.

Low-temperature heat capacity
Low-temperature heat capacity measurements were performed on BaMoO 4 (m = 12.25 (5) 3 , respectively), so as to improve the heat transfer with the sample platform of these oxide materials. The heat capacity contribution of the Stycast was subtracted from the recorded data. This technique is based on a relaxation method, which was critically assessed by Lashley et al. [73]. The contributions of the sample platform, wires, and grease were deduced by a separate measurement of an addenda curve. Based on the experience acquired on this instrument with standard materials and other compounds, and the error associated with the encapsulation procedure in Stycast [74], the uncertainty was estimated at about 1-2% in the middle range of acquisition (from 10 to 70 K), and reaching about 3% at the lowest temperatures and near room temperature [73,74].

Differential Scanning Calorimetry (DSC)
The transition temperatures in the BaMoO 4 -MoO 3 pseudo-binary section of the Ba-Mo-O system were measured using simultaneous Thermogravimetry (TG)-Differential Scanning Calorimetry (DSC) measurements using plate type sensors in the TG-DSC module of the Setaram 96 Line calorimeter.
The temperatures were monitored throughout the experiments by a series of interconnected S-type thermocouples. The temperature on the heating ramp (10 K ⋅ min − 1 ) was calibrated and corrected for the effect of the heating rate by measuring the melting points of standard high purity metals (In, Sn, Pb, Al, Ag, Au) at 2-4-6-8-10-12 K ⋅ min − 1 . The calibration procedure was performed as recommended by Höne et al. [75] and Gatta et al. [76]. The transition temperatures in the BaMoO 4 -MoO 3 phase diagram were derived on the heating ramp as the onset temperature using tangential analysis of the recorded heat flow. The liquidus temperature of mixtures was derived from the peak extremum of the last thermal event. The uncertainty on the measured temperatures is estimated to be ± 5 K for pure compounds and ±10 K for mixtures. BaMoO 4 and MoO 3 were mixed together in the selected stoichometric ratios by grinding at room temperature, and subsequently inserted in the calorimeter for measurement. In some cases, the mixtures were pre-treated under air at 800 K for 20 h before the TG-DSC measurement. The samples were placed in an alumina crucible on top of Table 4 Provenance and purity of the samples investigated in this study. XRD: X-ray diffraction; DSC: Differential Scanning Calorimetry.  Table 5 Refined lattice parameters. X-ray diffraction measurements were performed at room temperature and atmospheric pressure. The derived standard uncertainties are given in parenthesis.  boron nitride powder to avoid chemical interactions with the crucible upon melting (which would lead to the formation of aluminium molybdate). They were measured under oxygen flow to avoid reduction of the MoO 3 to lower valence states molybdenum oxides. It should be noted that with the present measurement configuration, the data could be collected up to a maximum temperature of about 1173 K, due to excessive boron nitride oxidation above the latter temperature, which affected the shape of the heat flow baseline curve. One typical measurement consisted in three heating cycles with 10 K ⋅ min − 1 heating rate. The data collected on the first heating ramp were not considered for the analysis, however. The first heating cycle was used to equilibrate the samples, and the data collected in the subsequent two cycles were used for the analysis. The shape of the heat flow signal was mostly identical on the second and third cycles, indicating that thermodynamic equilibrium conditions were reached. After the TG-DSC measurements, selected samples were analysed using X-ray diffraction to confirm the nature of the observed transitions.
In addition, the enthalpy associated with the peritectic decomposition of BaMo 3 O 10 was determined in this work using the Multi HTC module of the same 96 Line allowing 3D-heat flow measurements. The enthalpy was determined by placing a reference material of well-known transition enthalpy in the reference crucible and measuring both sample and reference materials in the same cycle. This configuration allows to calculate for each individual measurement cycle the detector sensitivity equal to: The detector sensitivity is assumed to remain the same at the temperature of the transition event of the sample, which is a reasonable approximation for two events sufficiently close to each other.

Pure elements
The Gibbs energy functions of the pure elements i at temperature T and in their state φ are given by: where n is an integer (2, 3, -1...). The parameters reported by Dinsdale are used in this work for pure barium, molybdenum, and oxygen [79]. Metallic barium and molybdenum are included in the description of the BCC A2 phase with sublattices (Ba,Mo)(O,Va) 3 (Va being a vacancy). The Gibbs energy functions of the (Ba)(Va) 3 and (Mo)(Va) 3 endmembers are those of Dinsdale [79].

Binary oxides
The binary oxides BaO 2 , MoO 2 , Mo 4 O 11 , Mo 8 O 23 , Mo 9 O 26 , and MoO 3 are described as stoichiometric compounds. The corresponding Gibbs energy functions have the same form as in Eq. (7): where n φ i is the number of atoms of the i th element in the oxide formula. These are taken from the TAF-ID database [10].

Halite phase
The halite BaO is described with a two sublattice model (Ba 2+ ,Va) (O 2− ,Va), where Va are vacancies. The Gibbs energies of the endmembers take the same form as in Eq. (7) and Eq. (8). No interaction parameter was introduced.

Hexavalent ternary molybdates
The hexavalent ternary molybdates (BaMoO 4 , Ba 3  Experimental thermodynamic data are only available for the BaMoO 4 scheelite phase as detailed in Section 2. The Gibbs energy for this phase has been expressed based on the recommended enthalpy of formation, standard entropy and heat capacity, and the enthalpic a and entropic b coefficients were subsequently optimized to fit the reported congruent melting temperature and the rest of the thermodynamic and phase diagram data in the BaO-MoO 3 section. For the other compounds, the Gibbs energies have been expressed as a function of BaMoO 4 and the binary oxides BaO and MoO 3 . The corresponding enthalpies of formation and entropies have been further optimized to fit the available phase diagram data in the BaO-MoO 3 pseudo-binary section.

BaMoO 3 perovskite
The descriptions adopted for the BaUO 3 and BaZrO 3 perovskites in the TAF-ID database is also adopted in this work for BaMoO 3 to enable extrapolations to higher order systems, and the modelling of the multi-

Liquid phase
An ionic two-sublattice model is used to describe the liquid phase [80], with Ba 2+ and Mo 4+ cations on the first sublattice, and MoO 2− 4 , O 2− anions, charged vacancies Va Q− , neutral MoO 3 , and neutral oxygen O on the second sublattice: P and Q are equal to the average charge of the opposite sublattice: where yBa 2+ , yMo 4+ , yMoO 2− 4 , yO 2− , and y Va Q− are the site fractions of barium cations, molybdenum cations, MoO 2− 4 , oxygen anions, and charged vacancies on the second sublattice, respectively. P and Q vary with composition via the site fractions so as to keep the phase electrically neutral.
The Gibbs energy of the liquid phase in this formalism is given by: with A.L. Smith et al.
are the reference terms corresponding to the Gibbs energies of barium molybdate BaMoO 4 (l) (times two), barium oxide BaO(l) (times two), barium metal, Mo 6 O 16 (l), molybdenum oxide MoO 2 (l) (times two), molybdenum metal, MoO 3 (l), and pure oxygen. The Gibbs energy of the liquid phase also contains a configurational entropy term related to mixing of the species on the first and second sublattices. Finally, the excess Gibbs energy interaction parameters terms used in this model are L 0 for the BaO-BaMoO 4 range, and L 0 for the BaMoO 4 -MoO 3 range, respectively.

Gas phase
The gas phase is described by an ideal mixture of (Ba, Ba 2 , BaO, Ba 2 where y i is the fraction of the species i in the gas phase, o G φ i the standard Gibbs energy of the gaseous species i, and P o the standard pressure. The function for BaMoO 4 is taken from the SGTE database [81]. Note that the computed pressures in the ternary system Ba-Mo-O are not commented in this work, and will be the subject of future work in our research group. We refer the reader to the Appendix for the expressions of the gaseous functions of the binary species.

Structural analysis
BaMoO 4 , BaMoO 3 , and BaMo 3 O 10 adopt a tetragonal, cubic, and monoclinic structure, respectively. The scheelite BaMoO 4 crystallizes in space group I4 1 /a. BaMoO 3 shows a perovskite-type structure, in space group Pm3m. BaMo 3 O 10 was reported with the space group P2 1 . The refined cell parameters obtained by the Rietveld method from the XRD data are summarized in Table 5, and the X-ray diffraction patterns are shown in Fig. 5. The refined atomic positions are provided in the Supplementary Information.

Valence state determination by XANES: BaMoO 4 and BaMoO 3
The XANES spectra of BaMoO 4 Table 6. The absorption edge of BaMoO 3 is very close to that of Mo IV O 2 , while that of BaMoO 4 is very well aligned with that of Mo VI O 3 , confirming the tetravalent and hexavalent valence states of molybdenum in those two materials, respectively. A shift of the inflection point to higher energy is observed with increasing valence state, as expected. In addition, the spectrum of BaMoO 4 shows a characteristic pre-edge feature around 20006.1 eV of relatively high intensity. This is due to the presence of short and highly covalent Mo-O bonds in tetrahedral geometry in the latter compound (see Fig. 5a), which enhance 4d-5p mixing through their hybridization with O(2p) [82][83][84]. Similarly, a pre-edge shoulder appears in the spectrum of α-MoO 3 related to the distorted MoO 6 octahedra in the structure (the 1s(Mo) → 4d(Mo) + 2p(O) transition is dipole-forbidden  for a perfectly regular MoO 6 octahedron). A similar pre-peak feature has been reported in the literature for Na 2 MoO 4 , K 2 MoO 4 , and CaMoO 4 [84].

Standard entropy determination: BaMoO 4 and BaMoO 3
The low-temperature heat capacity data of BaMoO 4 and BaMoO 3 measured in the absence of magnetic field are shown in Figs. 2b and 8  In the low-temperature region (below T = 15.7 K for BaMoO 4 and T = 26.6 K for BaMoO 3 in this case), the phonon contribution can be modelled using an harmonic-lattice model [85], as given by Eq. (14), where the number of required terms augments with the high temperature limit of the fit: where n = 3, 5, 7, 9...
The corresponding coefficients for BaMoO 4 and BaMoO 3 are listed in Table 7.
The electronic contribution of the conduction electrons at the Fermi surface are expressed with a linear term γT [86]. The electronic specific heat of BaMoO 4 is zero, as could be expected for such an insulating material. An electronic contribution of 7.71 mJ ⋅ mol − 1 ⋅ K − 2 is found for BaMoO 3 . This is in good accordance with the study of Hayashi and Aoki [87], who reported metallic conductivity in the range (2.5 to 300) K based on resistivity measurements.
In the high-temperature region, the lattice contribution is modelled using a combination of Debye and Einstein functions [88], as expressed in Eq. (15). This method was used in the literature for several classes of inorganic materials: iron phosphates [89][90][91], zirconolite [92], calcium titanate [93], dicesium molybdate [94], double molybdates [95,96], alkali uranate and neptunate [97,98]. Two Einstein functions were used in combination with a Debye function. The fitted parameters are listed in Table 7. The sum (n D + n E1 + n E2 ) is 5.97 and 5.08 for BaMoO 4 and BaMoO 3 , respectively. The same procedure applied to the 22.85 mg pellet of BaMoO 4 gave a sum (n D + n E1 + n E2 ) equal to 5.35, hence an underestimation compared to the 6 atoms expected from the formula unit. The measurement of very insulating materials as is the case for BaMoO 4 is not always straightforward using the thermal-relaxation technique. The encapsulation procedure in Stycast has proved 2.15708 ⋅ 10 − 12 θ E2 /K 721.21 n D + n E1 + n E2 /mol 5.08 A.L. Smith et al. effective to improve the thermal coupling with the sample platform. Nevertheless, a too large weight in case of an insulator can lead to strong thermal inertia and long relaxation times, which reduces the accuracy of the results above ~250 K. Because the fitting results obtained on the 12.25 mg pellet were found closer to the expected 6 atoms per formula unit than the results obtained on the 22.85 mg pellet, the former data were considered to be more reliable and were selected in this work. As a consequence, the retained heat capacity and entropy data for the thermodynamic assessment are higher than reported in the work of Morishita et al. [49]. It should be pointed out that the optimized standard entropy for BaMoO 4 in the thermodynamic model (see Section 6) is in very good agreement with the data selected herein. In fact, optimization of the thermodynamic model while constraining the heat capacity and standard entropy to values closer to that of Morishita et al. was also tested, but did not give as satisfactory results.
where D(θ D ), E(θ E1 ) and E(θ E2 ) are the Debye and Einstein functions, respectively, as written in equations (16) and (17). θ D , θ E1 , and θ E2 are the characteristic Debye and Einstein temperatures. n D , n E1 , andn E2 are adjustable parameters, whose sum (n D + n E1 + n E2 ) should be approximately equal to the number of atoms in the formula unit (i.e., 6 and 5 in this case).
where R is the universal gas constant.

Phase diagram measurements in the BaMoO 4 -MoO 3 pseudo-binary section
The transition temperatures in the BaMoO 4 -MoO 3 pseudo-binary section measured in this work by TG-DSC and DSC are listed in Table 8, and shown in Figs. 12 and 13 . The corresponding thermograms are shown in Figs. A.1 and A.2 in the Appendix. No noticeable weight loss was observed from the thermogravimetry results, thus the initial Table 8 Equilibrium data collected by TG-DSC and DSC at pressure (0.10 ± 0.01) MPa on the heating ramp using a 10 K ⋅ min − 1 heating rate. Mixtures that were pre-treated under air at 800 K for 20 h before the TG-DSC measurements are indicated with an *. The corresponding compositions in the BaO-MoO 3 pseudo-binary phase diagram is given by x(MoO 3 ). ** Indicates a very minor impurity contamination. I indicates an unidentified impurity.  targeted composition should not have shifted during the measurement due to vaporization processes. Taken into account the uncertainty associated with the temperature calibration procedure, the final uncertainty on the measured temperatures is estimated to be ± 10 K.

Transition enthalpy determination: BaMo 3 O 10
The decomposition temperature and enthalpy associated with the peritectic decomposition of BaMo 3 O 10 was determined using Differential Scanning Calorimetry. The average transition temperature over four heating cycles yielded (918 ± 5) K. In addition, the enthalpy of transition was estimated by measuring BaMo 3 O 10 together with Cs 2 TeO 4

Table 10
Summary of the thermodynamic data for pure elements and oxides selected in the present work. SER refers to the phase of the element stable at 298.15 K. The optimized coefficients are marked in bold.
This work  +4.32346667 ⋅ 10 − 7 T 3 + 1172955T − 1 , while the uncertainty was calculated using √ as recommended in [48] considering that the individual measurements are independent source of data since fresh materials and crucibles were used each time.     reference material, whose enthalpy of transition was determined in another work in our research group as Δ tr H o m (Cs 2 TeO 4 , cr, T tr ) = (2.67 ± 0.14) kJ ⋅ mol − 1 [99]. Cs 2 TeO 4 was chosen as it displays a transition temperature close to the peritectic transition of BaMo 3 O 10 , but without overlapping. Moreover, both reference and sample being oxide materials, they are expected to show similar thermal properties (heat capacity and thermal conductivity). The curve of the recorded heat flow versus temperature for the latter measurement is shown in Fig. 9.
The first single peak corresponds to the polymorphic transition of Cs 2 TeO 4 (from an orthorhombic α phase to an hexagonal β phase [99]) and the second peak to the peritectic decomposition of BaMo 3 O 10 . Note that the opposite directions for both (endothermic) events is due to the different positioning in the reference and sample crucibles, respectively.  Two measurements were performed on fresh material and averaged, yielding an enthalpy of transition equal to Δ tr H o m (BaMo 3 O 10 , cr, T tr ) = (82.8 ± 20.0) 6 kJ ⋅ mol − 1 . The results of the individual cycles are listed in Table 9. The reported uncertainties on the individual measurements are standard uncertainties. They include the uncertainty on the transition enthalpy of the reference material (Cs 2 TeO 4 ) combined with the uncertainty associated with the choice of the baseline for the peak integration (linear, spline, or tangential sigmoid).

Optimized parameters
A summary of the optimized parameters for the Ba-Mo-O system is given in Table 10.

Thermodynamic data
The standard enthalpies of formation and standard entropies at 298.15 K optimized in this work are listed in Table 11. The values for BaMoO 4 are in good agreement with the selected values from the literature review (see Section 2) considering the reported uncertainties. The optimized enthalpy of formation and standard entropy of BaMoO 3 are higher than recommended from the literature review, however. Such high values appeared necessary to match the decomposition temperature of this compound (calculated at T = 1660 K and reported at T = 1653 K by Paschoal et al. [66]). Since neither the enthalpy of formation nor the decomposition temperature for this compound are known with certitude, complementary measurements would be very valuable for the improvement of the thermodynamic model.
The calculated enthalpy increments and heat capacities of the same compounds are shown in Figs. 10a, 10 b, 11 a and 11 b. The agreement with the experimental data is very good. Note that the reported heat capacity equations are only valid above 298.15 K, and match the imposed constraints on C o p,m (298.15 K), as selected from the literature review.

Phase diagram data
The calculated pseudo-binary section BaO-MoO 3 is shown in Fig. 12and compared with the available phase diagram data.
The agreement with the data of Yanushkevich et al. [68] in the BaO-BaMoO 4 section, and the data measured in the BaMoO 4 -MoO 3 section in this work and by Zhukovskii et al. [69], is generally good. The calculated invariant equilibria are listed in Table 12. Ba 3 MoO 6 and BaMoO 4 melt congruently at T = 1833 K and T = 1734 K, respectively. Ba 2 MoO 5 undergoes a peritectic decomposition at T = 1572 K, and BaMo 3 O 10 a peritectic decomposition at T = 917 K. BaMo 2 O 7 is suggested to undergo a peritectoid decomposition at T = 892 K. The crystal structure of this compound remains unknown, however, and the post-XRD characterizations did not seem to indicate the presence of such a phase, by contrast with the measured calorimetric heat flow signals. Its existence thus still needs confirmation. The liquidus line between BaO and BaMoO 4 follows rather well the data of Yanushkevich et al. [68]. The situation on the BaMoO 4 -MoO 3 section is more complex. The liquidus line is in good agreement with the results of Zhukovskii et al. [69] up to x(MoO 3 )~0.8. In the MoO 3 -rich region of the phase diagram, the liquidus line falls in between our data and that of Zhukovskii et al. [69] and Ustinov et al. [56]. Complementary measurements in the BaMoO 4 -MoO 3 section, and more specifically of the liquidus line would be very valuable to ascertain the phase equilibria in this region and solve the discrepancy between the three sets of data.
Finally, isothermal sections calculated at T = 300, 700, 1200, and 1700 K are shown in Figs. 14a-d, respectively. The calculated equilibrium ternary phase fields should be confirmed experimentally to ascertain the predictions of our model. The isothermal section computed at T = 700 K is in rather good agreeement with that of Dash et al. [53], although the comparison is not straightforward. Dash

Conclusions
The Ba-Mo-O system is rather complex, with a number of ternary barium molybdate phases reported in the literature. The thermodynamic modelling assessment for this system presented in this work includes BaMoO 4 , Ba 3  known and uncertain, with quite large discrepancies between past literature studies and this work. The vapour pressures in the ternary phase fields and above pure compounds were not discussed, and will be the subject of future studies.