First-Principles Prediction of Ternary Interstitial Hydride Phase Stability

Article pubs.acs.org/jced

First-Principles Prediction of Ternary Interstitial Hydride Phase Stability in the Th−Zr−H System Kelly M. Nicholson and David S. Sholl* School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, 311 Ferst Drive, Atlanta, Georgia 30332-0100, United States S Supporting Information *

ABSTRACT: Computational methods that characterize the thermodynamic properties of metal hydrides that operate at high temperatures, i.e., T > 800 K, are desirable for a variety of applications, including nuclear fuels and energy storage. Ternary hydrides tend to be less thermodynamically stable than the strongest binary hydride that forms from the metals. In this paper we use firstprinciples methods based on density functional theory, phonon calculations, and grand potential minimization to predict the isobaric phase diagram for 0 K ≤ T ≤ 2000 K for the Th−Zr−H element space, which is of interest given that ThZr2Hx ternary hydrides have been reported with an enhanced stability relative to the binary hydrides. We compute free energies including vibrational contributions for Th, Zr, ThH2, Th4H15, ZrH2, ThZr2, ThZr2H6, and ThZr2H7. We develop a cluster analysis method that efficiently computes the configurational entropy for ThZr2H6 interstitial hydride and conclude that the configurational entropy is not a major driver for the enhanced stability of ThZr2H6 relative to the binary hydrides. Density functional theory (DFT) predicted thermodynamic stabilities for the hydride phases are in reasonable agreement with experimental values. ThZr2H6 is stabilized by finite temperature vibrational effects, and ThZr2H7 is not predicted to be stable at any studied temperature or pressure.

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INTRODUCTION Many metals, intermetallics, and alloys react exothermically with hydrogen to form metal hydrides with high hydrogen densities.1−3 Recent research efforts, including computational modeling, have focused on identifying and engineering the properties of metal hydride systems with moderate thermodynamic stability, i.e., those that operate close to ambient temperatures and pressures, for the onboard solid state storage of hydrogen in fuel cell vehicles.1,4−11 Relatively less focus has been placed on the high temperature applications of metal hydrides such as their role as neutron moderators in nuclear reactors12−15 or as thermochemical media in solar energy storage systems.16−19 A convenient measure of the relative thermodynamic stabilities of metal hydrides is the temperature at which a metal hydride is in thermodynamic equilibrium with 1 bar H2 gas, Td. If heated beyond this temperature, the metal hydride will begin to release hydrogen. In solar energy storage services, it is desirable to use metal hydrides with very high stability (Td > 800 °C) so that the systems can operate at high temperature and utilize high efficiency Stirling or Brayton thermodynamic cycles.19,20 High temperature metal hydrides are also being investigated as tritium gettering materials with potential application in the US Department of Energy Next Generation Nuclear Plant project in which candidate materials must have a high affinity for hydrogen at temperatures of T = 1000 K to 1200 K.21,22 © 2014 American Chemical Society

Binary hydrides MHx of transition metals with high thermodynamic stability like thorium, titanium, and zirconium hydrides serve as neutron moderators in hydride nuclear fuels.12 For example, in TRIGA reactors, U−Zr mixed fuel contains uranium particles embedded in a zirconium hydride matrix.23 One motivation for using hydride fuels is that they allow core designs to be made more compact due to their high hydrogen densities relative to that of pressurized water reactor or boiling water reactor coolants.12−15 The thermodynamic stabilities of binary hydrides are well-characterized with both experimental data and theoretical predictions available.2,7,24 Noncomplex ternary hydrides MNyHx, in particular hydrides of transition metal alloys and intermetallics, are typically metastable to mixtures of the binary hydrides and/or base metals and intermetallics, i.e., the hydriding reaction to form the binary hydride from the elements has a more favorable free energy than the reaction to form the ternary hydride.2,3 For ternary hydrides of transition metal solid solutions, it has been empirically observed that the heat of formation is nearly a weighted average of the heats of formation of the binary hydrides.3 As a consequence, noncomplex ternary systems, isolated experimentally, generally form due to kinetic Special Issue: Modeling and Simulation of Real Systems Received: March 15, 2014 Accepted: June 30, 2014 Published: July 10, 2014 3232

dx.doi.org/10.1021/je500250f | J. Chem. Eng. Data 2014, 59, 3232−3241

Journal of Chemical & Engineering Data

Article

are dominated by the hydrogen gas entropy.8 Ong and colleagues have used similar simplified calculations based on grand potential minimization to predict phase diagrams for the Li−Fe−P−O element space as a function of oxygen chemical potential while ignoring the dynamical or vibrational contributions of the solids.30 There are several reasons, however, to include vibrational corrections in phase diagram predictions. Wolverton et al. found that including vibrational corrections reduced the root-mean-square error in calculated vs experimental reaction enthalpies for alkali, alkaline earth, and early transition metal binary hydrides from (19.4 to 14.7) kJ· mol−1 H2 .11 In cases where the agreement weakened, the effect was small. Beyond enthalpic effects, Akbarzadeh et al. also found that several of the preferred reaction pathways in the Li− Mg−N−H phase diagram computed using dynamically corrected free energies had predicted entropy changes in the range of (109 to 137) J·mol−1·K−1 at T = 500 K, deviating significantly from the standard molar entropy of hydrogen, 130.6 J·mol−1·K−1 at T = 300 K.32 These entropic contributions may become more significant for metal hydride systems that operate at high temperature. We previously benchmarked the performance of the DFT projector-augmented wave method with the GGA PW91 exchange-correlation functional for predicting Td for very stable binary hydrides ZrH2, HfH2, TiH2, LiH, and NaH at several levels of theory including anharmonic contributions, quasiharmonic effects that incorporate thermal expansion, simple harmonic calculations that consider vibrational corrections to the free energy at a ground state volume, and constant entropy models.41 We found that the simple harmonic level of theory predicts Td to within 130 K for the alkali hydrides and to within 50 K for TiH2 and ZrH2 transition metal hydrides. Th and Zr have limited miscibility at low temperature but form a metastable ThZr2 bcc solid solution for T > 1150 K.25 Bartscher, Rebizant, and Haschke studied the phase equilibria and thermodynamic properties of the ThZr2−H system experimentally. Although incomplete, the general features of the Th−Zr−H phase diagram are that hydrogen dissolves in the bcc ThZr2 lattice at high temperature up to a composition of ThZr2H0.6 (α-phase), a two phase (α + β) region exists in the range ThZr2H0.6−1.8, and the β-phase cubic Laves ternary hydride forms for ThZr2H1.8 to ThZr2H7−x.12,25 van Houten and Bartram were the first to isolate cubic Laves ThZr2 and hexagonal ThTi2 ternary hydrides in 1971.12 They determined the hydrogen capacity to be ThZr2H7 and, based on interstitial site availability, reasoned that hydrogen occupies 8b and 48f interstitial sites. Bartscher, Rebizant, and Haschke observed cubic Laves hydrides for ThZr2Hx (1.8 ≤ x ≤ 6.6), but only materials with x ≥ 3.9 were stable upon cooling to room temperature at P = 1 bar H2; lower hydrides dissociated to mixtures of other hydride phases.25 Bartscher et al. performed powder neutron diffraction studies at 290 K for ThZr2Dx (x = 3.6, 4.8, 6.0, and 6.3) to locate deuterium within the lattice and found that the deuterium atoms only occupy tetrahedrally coordinated 96g and 32e sites for all compositions, not 8b or 48f sites as proposed by van Houten and Bartram.12,13 The authors note that, based on comparison to their fitted lattice parameters, the compound isolated by van Houten and Bartram corresponds to ThZr2H5.63 not ThZr2H7, which calls into question the hydrogen site occupations reported for that structure. Terrani et al. used XRD and Rietveld refinement to obtain lattice parameters for ThZr2H7−x14 and found good agreement with the lattice

limitations in forming the most stable mixture at the low temperatures of interest for hydrogen storage applications, and ternary hydrides dissociate at lower Td than the strongest binary hydride that forms from the base M and N metals. LaNi5Hx, for example, forms upon the reaction of hydrogen with LaNi5 due to the kinetic limitations associated with separating the metals into the more favorable LaH2 and Ni phases at ambient conditions.2 One expects that diffusion barriers to form the most thermodynamically stable state are reduced at high temperatures. An apparent exception to this metastability rule is the ThZr2−H system, in which stable ternary interstitial metal hydrides ThZr2Hx have been isolated and characterized experimentally.12,13,25 ThZr2H6 dissociates at Td ≈ 1250 K15,25 compared with Td ≈ 1150 K (637 K) for ZrH226,27 and ThH2 (Th4H15).28 This system is of interest because the U−Th−Zr hydride fuel absorbs more hydrogen at high temperature than U−Zr hydride fuels.23 It has also been studied as a proxy for hydride systems of actinide materials such as the cubic Laves Pa3Hx (4 ≤ x ≤ 5)25 and 237Np, 241Am, and 243 Am hydride targets used in transmutation applications.29 If computational tools can be used to accurately predict the thermodynamic properties of ternary hydrides that operate at high temperature, it may be possible to identify other systems with this enhanced thermodynamic stability relative to the binary hydrides or to predict operating parameters for these systems of interest without expensive experimental studies. In this paper, we use first-principles density functional theory (DFT) and grand potential minimization methods to predict the relative thermodynamic stabilities of Zr and Th binary hydrides, ThZr2H6, and ThZr2H7, as a function of hydrogen pressure and temperature to serve as a proof of principle calculation to judge whether or not the ternary hydride is a thermodynamically preferred phase. To our knowledge, no other first-principles study of the Th−Zr−H sytem has of yet been performed. A challenge in hydrogen storage applications is to predict the temperature, hydrogen pressure, and composition-dependent stable phases and equilibrium reaction pathways for multicomponent systems. Phase diagrams detailing thermodynamic equilibria of components can provide valuable information regarding relevant reactions and processing conditions.30 While equilibrium hydrogen release pathways can be determined manually by enumerating all possible reactions, this can be cumbersome and prone to error for multicomponent systems.31 Akbarzadeh, Ozolins, and Wolverton developed a grand canonical linear programming (GCLP) method to systematically and automatically compute all thermodynamically favored reactions in a multicomponent system open to a hydrogen atmosphere based on a library of compound compositions and total energies determined through first-principles and dynamical calculations.32 The method identified all experimentally observed reaction pathways in addition to previously unobserved high temperature decomposition pathways in the Li−Mg−N−H system. It has since been applied to other bulk hydrides8,11,31,33−38 and extended to incorporate nanoclusters39 with general good agreement between theory and experiment in terms of computed thermodynamic properties. A similar method that accounts for multiple gaseous species has recently been applied to predict phase diagrams in the Li−Ca−B−N−H system.40 Alapati et al. performed a large-scale screening analysis of destabilized metal hydride systems using simplified calculations that assume changes in the free energy at finite temperatures 3233



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parameters obtained by Bartscher et al.13 Further analysis via SEM indicated that the ternary hydride was the dominant phase in the system, strengthening the argument for enhanced stability relative to the binary hydrides.14 Binary hydrides of Zr and Th include face-centered tetragonal (fct) ε-ZrH2 and ThH2. A face-centered cubic (fcc) phase also exists in the hydrogen deficient δ-ZrHx (x = 1.5 to 1.8), while ThH2−x remains fct. Additionally, there is a higher hydride, Th4H15, that crystallizes with body-centered cubic (bcc) symmetry.12 Konashi et al. calculated the enthalpy of formation for ThZr2Hx (x = 4 and 6) using the semiempirical Miedema model based on geometric considerations and the GriessenDriessen model based on electronic band structure.15 They found that the Griessen model predicts van’t Hoff plots in good agreement with the experimental result, predicting a Td of 1210 K (1380 K) for ThZr2H6 (ThZr2H4). Pudjanto et al. then extended the Miedema and Griessen models to predict hydrogen partial pressures for theoretical actinide Np−Zr−H, Am−Zr−H, and Pu−Zr−H systems for A5B, A2B, AB, AB2, and AB5 intermetallics.42 In this study, we use more rigorous DFTbased methods to predict the isobaric phase diagram of the Th−Zr−H system with methods that could be extended to identify other hydrides with enhanced stability or to inform the operational design of such very stable hydride systems.

Gj(T ) ≈ Fj(T ) = E0, j + Fjvib(T ) − TS jconf

where the free energy is the sum of the ground state electronic energy E0, the vibrational contribution to the Helmholtz free energy Fvib including zero point and finite temperature effects, and configurational entropy corrections Sconf, if any. We determine E0 using DFT geometry relaxations and Fvib using calculations of the material’s vibrational density of states. We refer to these below as phonon calculations for succinctness. Sconf is defined using the Boltzmann definition as S conf = kB ln(Nc)

ϕ ̅ (T , P , x Th , x Zr , μH ) 2

vib

≈

THEORETICAL METHODS Hydrogen Grand Potential. Details of the GCLP method for predicting the equilibrium mixture of metals, metal hydrides, and alloys from a library of compositions and energies at a given hydrogen chemical potential are described in ref 31. A key constraint is that we cannot make predictions about the stability of compounds or phases that are not included in the library. Since we have used pymatgen43 software to perform the grand potential minimization and phase diagram predictions, we build upon the nomenclature of Ong, Ceder, and colleagues.30,44 The difference in our calculation and theirs is that we incorporate vibrational and configurational entropy corrections to the free energies of the condensed phases. Phase equilibria for systems dominated by condensed phases and open to a hydrogen atmosphere are governed by minimization of the overall system free energy, specifically the minimization of the hydrogen grand potential. In an isothermal and isobaric Th−Zr−H system, this hydrogen grand potential ϕ can be written as

(5)

(6)

where P0 is a reference pressure defined as 1 bar and R is the universal gas constant. The corresponding van’t Hoff plots provide a useful means for comparing theoretical and experimental metal hydride stabilities. ThZr2H6 Configurational Entropy. ThZr2Hx crystallizes in the cubic Laves structure with compositions that range up to an experimentally determined x = 7.12−14,25 Each thorium atom is surrounded by 4 thorium and 12 zirconium atoms.13 The metal lattice forms three types of tetrahedrally coordinated interstitial sites: 32e (coordinated to one Th and three Zr atoms), 96g (coordinated to two Th and two Zr atoms), and 8b (coordinated to four Zr atoms). Additionally, there are 48f triangularly coordinated sites. As previously mentioned, Bartscher et al. determined that hydrogen atoms occupy only 32e and 96g interstitial sites.13 In the case of ThZr2D6, the hydrogen occupancy among the 32e and 96g sites was 0.598 ± 0.035 and 0.301 ± 0.012, respectively. In a unit cell with composition Th8Zr16H48, the hydrogen loading is then approximately 19 and 29 H atoms to the 32e and 96g sites, respectively.

2

2

(1)

where T, P, N, and μH2 are temperature, pressure, number of atoms, and the chemical potential given by the total energy of a hydrogen molecule, respectively. The Gibbs free energy is defined as G(T , P) = U + PV − TS = F + PV

2

NTh + NZr

⎛ −ΔG(T ) ⎞ P = exp⎜ ⎟ ⎝ RT ⎠ P0

= G(T , P , NTh , NZr , μH ) − μH NH2 2

E0 + F (T ) − TS conf − μH (T , P)NH2

Specifying the chemical potential of H2 for a T and P of interest as well as the relevant energies for a library of compounds that can form from a combination of Th, Zr, and H, one can determine the equilibrium mixture of compounds that minimizes the grand potential as a function of composition. Using the methodology of Ozolins and colleagues,32 shifts in the mixture of stable phases for a given P can be tracked as a function of T to identify relevant reactions and to construct a phase diagram. Finally, the hydrogen overpressure for a metal− hydrogen system at a given temperature is computed from the metal hydride decomposition reaction free energy ΔG using the van’t Hoff relation9,45

ϕ(T , P , NTh , NZr , μH ) 2

(4)

where kB is the Boltzmann constant, and Nc is the number of unique arrangements of the system in question. Sconf is evaluated for ThZr2H6 in which Nc represents the number of ways to arrange hydrogen among the interstitial sites to meet the partial occupancy requirements of the ternary hydride. Because Nc is nontrivial to compute in this system, we discuss a number of approximations for Nc in the text below. If the fraction of component j is written xj = Nj(NTh + NZr)−1, then the normalized grand potential becomes

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(T , P , NTh , NZr , μH )

(3)

(2)

where U, S, and F are the internal energy, entropy, and Helmholtz free energies, respectively. If we assume that Δ(PV)solids ≈ 0 for reactions that primarily involve condensed phases, we can write the free energy for a given solid j within the harmonic approximation as 3234



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Random assignment of H atoms to 32e and 96g sites to fulfill this partial occupancy requirement for ThZr2H6 gives

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COMPUTATIONAL METHODS

Plane-wave DFT calculations were carried out via the Vienna ab initio simulation package (VASP)47−51 using the projector augmented wave method with the GGA PW91 exchangecorrelation functional.52−54 Experimental crystal structures of relevant compounds in the Th−Zr−H phase space were taken from the Inorganic Crystal Structure Database (ICSD).55,56 These include the low temperature hcp and high temperature bcc forms of Zr, fcc Th, cubic Th4H15, bct ThH2, and bct εZrH2. The cubic Laves ThZr2H7 structure from van Houten and Bartram was studied with completely filled 48f and 8b sites.12 The cubic Laves ThZr2H6 structure was taken from Bartscher et al., which includes site occupancy factors of approximately 0.598 (0.301) for the 32e (96g) sites or, in the unit cell, approximately 19 (29) H atoms in 32e (96g) sites.13 Additionally, a hypothetical ordered cubic ThZr2 phase was studied by removing H atoms from the ThZr2H6 structure. We make no attempt to account for unsaturated hydride phases or lower ThZr2Hx hydrides in this work. However, it has been observed that enthalpies of formation of ThZr2Hx tend to increase with decreasing hydrogen to metal ratios indicating that lower hydrides will be more thermodynamically stable than ThZr2H6.57 This observation is in agreement with the experimental van’t Hoff plots for ThZr2Hx for 1.5 ≤ x ≤ 6.0.25 The crystallographic unit cell for each compound was relaxed using the conjugate gradient method until forces on each atom were less than 10−4 eV·Å−1 and electronic steps were converged to within 10−7 eV. For ThZr2H6, we randomly assigned H atoms to 96g sites and 32e sites until 15 unique configurations were found that simultaneously met the partial occupancy and Westlake minimum separation requirements. We then fully relaxed each structure. The average predicted lattice parameter for the distorted cubic structures was a = 9.124 ± 0.005 Å with α, β, and γ = 89.9 ± 0.3°. E0 was largely independent of the arrangement of H atoms across the 15 configurations with the largest difference in ground state energy of 0.043 eV·f.u−1 (formula unit) or 0 is also shown in units of kJ·mol−1 H 2.

respectively. However, the competing reaction ThZr2H7 → ThZr2H6 + 0.5H2 has ΔE0 = −67.3 kJ·mol−1 H2 and remains exothermic for all T, indicating that it is more favorable to decompose the ordered higher hydride into a mixture of H2 gas and the lower hydride ThZr2H6 with partial occupancy of interstitial sites. The hexagonal form of Zr is predicted to be the most stable form of the pure metal at all T. This disagrees with experimental evidence that cubic Zr becomes stable for T > 1135 K at low pressures.59 The difference in the overall free energies for these two phases is small, within 10 kJ·mol−1 for all T, and, therefore, using the free energy of hcp Zr rather than bcc Zr for reactions at high temperature will not have a large effect on the results, in particular relative rankings of thermodynamic stability. Phase Diagram. Figure 3a−c presents three phase diagrams predicted using the GCLP method at P = 1 bar H2 with values for the solid phase Helmholtz free energies that increase in computational complexity. Figure 3a displays the relative stabilities of the studied phases as a function of T and metal species composition based only on ground state DFT energies and the chemical potential of H2, i.e., all phonon contributions are ignored. This simplified calculation correctly reproduces the basic decomposition pathways of the binary hydrides with the higher thorium hydride decomposing to the dihydride at moderate temperature and the thorium and zirconium dihydrides having comparable thermodynamic stabilities. However, at this level of theory, ternary hydrides are not predicted to form. Rather, the reaction ThZr2H6 + 7/ 8H2 → 1/4Th4H15 + 2ZrH2 proceeds exothermically with ΔE0 = −121.1 kJ·mol−1 H2 at low T, and a similar reaction can be written at T > 887 K for decomposition to a mixture of the binary dihydrides. Figure 3b includes vibrational effects computed using phonon calculations for the condensed phases but no configurational entropy correction for ThZr2H6. The ternary

Figure 1. Helmholtz vibrational free energy contributions for solid phases in Th−Zr−H element space computed within the harmonic approximation.

the Th−Zr−H element space. As expected, vibrational effects are more significant for the hydride phases due to the high frequencies associated with modes involving hydrogen. Zero point energies are 45.2 (36.1) kJ·mol−1 H2 for ZrH2 (ThH2) compared with just 2.1 (1.3) kJ·mol−1 for hexagonal Zr (Th) phases. Other calculated zero point energies are (33.9, 40.2, and 38.1) kJ·mol −1 H 2 for Th 4 H 15 ThZr2 H 6 , and ThZr2 H 7 compared with (1.5 and 1.5) kJ·mol−1·atom−1 for Zr(bcc) and ThZr2, respectively. Of note, the vibrational contribution to ThZr2H6 is intermediate to the binary dihydrides. Figure 2 shows the overall contributions of Fvib(T) of the solid phases to the reaction free energy relative to the T > 0 H2 contribution for key reactions listed in Table 1. For hydride dissociation reactions 1 to 5 and 8, the net vibrational contribution of the solids to the reaction free energy is nearly constant for T < 800 K. Over the entire temperature range, the net vibrational effect for all metal hydride dissociation reactions is less than 50 kJ·mol−1 H2. At low temperatures, the net vibrational contribution tends to destabilize the metal hydride, and at temperatures greater than this, gaseous H2 thermodynamics dominate. ThZr2H7 decomposition reactions 1 and 2 are both highly endothermic with ΔE0 = 152.7 and ΔE0 = 161.3 kJ·mol−1 H2, 3237



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Figure 4. Computed van’t Hoff plots for stable metal hydrides in the Th−Zr−H space.

only a slight rise in the predicted Td when accounting for vibrational effects. This phase diagram indicates that ThZr2H6 decomposes to a mixture of hydrogen and the pure metals at high temperature and that a high temperature bcc solid solution forms for T ≥ 1588 K. This is our best estimate of the thermodynamic stability of ThZr2H6 relative to a simplified set of products that can form in the Th−Zr−H element space. van’t Hoff Plots. Figure 4 shows the computed van’t Hoff plots for the metal hydride phases that are predicted to form. Fitting this data to lines, we obtained the enthalpies and entropies of formation for the three stable high temperature metal hydrides shown in Table 3. Experimentally derived thermodynamic data are included for comparison. ThZr2H6 and ThH2 have nearly identical entropies of formation at ΔS° = −133 J·K−1·mol−1 H2, which is very close to the ideal entropy of H2 gas, 130.6 J·mol−1·K−1 at T = 300 K.32 The entropy of formation for Zr indicates a stronger influence of the vibrational free energies of the solid materials. This is confirmed in Figure 2 reactions 1, 4, and 5. ΔFvib solids for the ZrH2 decomposition reaction has a larger magnitude over the entire T range than either the ThH2 or ThZr2H6 decomposition reactions. Bartscher et al. integrated P−C isotherms for 1183 K ≤ T ≤ 1321 K to obtain the ΔH° and ΔS° for ThZr2H6 from ThZr2.25 Since continuous isotherms across the composition range were not available, they estimated equilibrium pressures via extrapolation for ThZr2Hx (x > 3.9) and T > 1183 K. Our predicted enthalpies and entropies overestimate the enthalpy and entropy of formation for ThZr2H6 compared with these experimental extrapolated values. The net effect is that ThZr2H6 decomposes to the elements and hydrogen gas at Td = 1168 K compared with the experimental value of Td ≈ 1236 ± 150 K based on the equilibrium relation.25 Thus, our calculations predict the overall thermodynamic stability of the ternary hydride with reasonable accuracy. For P > 150 bar H2, the GCLP method predicts ThZr2H6 decomposes via the reaction ThZr2H6 → ThZr2 + 3H2 rather than ThZr2H6 → Th + 2Zr + 3H2. For P < 0.01 bar H2, ThZr2H6 is predicted to dissociate via ThZr2H6 → Th + 2ZrH2 + H2. Due to the subtle differences in stabilities of these materials, the calculated decomposition reaction pathways are dependent on the hydrogen overpressure, imposed via the hydrogen chemical potential. More detailed methods are needed to differentiate among these reaction paths as a

Figure 3. Predicted phase diagrams for Th−Zr−H element space for P = 1 bar H2. (a) Fsolids = E0, (b) Fsolids = E0 + Fvib(T), and (c)Fsolids = E0 conf + Fvib(T) − TSThZr . Experimental values for metal hydride 2H6 decomposition temperatures, Td, are 1250 K (ThZr2H6),15 1154 K (ZrH2),26,27 1154 K (ThH2),71 and 636 K (Th4H15)71

hydride ThZr2H6 is predicted to be thermodynamically stable for 609 K ≤ T ≤ 1122 K, and ThZr2H7 does not form at any T. At 609 K, Th4H15 and ZrH15 release hydrogen and form ThZr2H6. This is consistent with the general experimental behavior in which a mixture of Th4H15 and ZrH2 was isolated at T < 1180 K with the ternary hydride forming at high T.25 The ternary hydride and binary hydrides decompose to the elements at high T with the ternary hydride nominally more thermodynamically stable than ZrH2. However, the difference in their predicted Td’s is only about 10 K or ΔΔH ≈ 1.3 kJ· mol−1 H2 assuming a constant ΔS = 0.130 J·mol−1·K−1, given that at thermodynamic equilibrium ΔG = 0 = ΔH − TΔS. This value is on the order of the error expected from numerical convergence issues such as k-point sampling, finite cutoff energy, or q-point sampling. Therefore, a conservative statement is, rather, that the ternary and binary hydrides are of comparable thermodynamic stability. Additionally, at this level of theory, the theoretical binary alloy ThZr2 is predicted to form from a mixture of the elements for T ≥ 1122 K. Again, this is generally in agreement with the experimental observation that a bcc solid solution of ThZr2 is stable for T ≥ 1150 K. In this paper, we consider only an ordered ThZr2 solid. Methods that further account for the thermodynamic mixing effects of the bcc solid solution of ThZr2 may refine the details of the high temperature products.60 In Figure 3c we include our best estimate for configurational entropy effects due to partial hydrogen occupancy of the interstitial sites for the ThZr2H6 ternary hydride. Here, TSconf is 5.4 kJ·mol−1 H2 at 1000 K. The effect of the configurational entropy on the stability of the ternary hydride is minor with 3238



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Table 3. Comparison of Predicted and Experimental Enthalpies and Entropies of Formation for High-Temperature Metal Hydridesa DFT prediction

a

experimental reference

ΔH°

ΔS°

ΔH°

ΔS°

metal hydride

kJ·mol−1 H2

J·K−1·mol−1 H2

kJ·mol−1 H2

J·K−1·mol−1 H2

ThZr2H6 ZrH2

−156.1 −161.7

−133.3 −145.1

−111.3 ± 6.7 −134.4 ± 0.2

ThH2

−145.4

−133.3

−137.7 ± 8.3b −162.8 ± 1.3c −169.5c −146 ± 8d

−129 ± 7

DFT values derived from eq 6. bAt T = 1250 K.25 cAt T = 298.15 K qtd. in Zuzek, Abriata, and San-Martin.72 dAt T = 1000 K.28

function of pressure.61 Further improvements to computed phase boundaries could be made with a more rigorous treatment of the high temperature ThZr2 bcc solid solution free energy. While an unambiguous ranking of the relative thermodynamic stabilities of the ThZr2H6, ZrH2, and ThH2 is not reached at this level of theory, Figure 4 clearly shows that the ternary hydride is thermodynamically viable at high temperature and across a wide range of pressures.

experiments are inconvenient or component materials are scarce.

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ASSOCIATED CONTENT

S Supporting Information *

Details regarding convergence of DFT and phonon calculations in addition to a description of the cluster analysis algorithm used to compute Sconf for ThZr2H6. This material is available free of charge via the Internet at http://pubs.acs.org.

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CONCLUSION First-principles methods capable of predicting the thermodynamic properties of metal hydrides that operate at high temperature will be useful in several applications such as informing the optimal operational conditions for hydride nuclear fuels. In this paper, we consider the Th−Zr−H element space because it contains an experimentally observed ternary interstitial metal hydride with lower H2 overpressures than that of the competing binary hydrides, an anomaly for metallic hydrides. We predict the phase diagram for a mixture of Th and Zr open to a hydrogen atmosphere using the GCLP method and DFT to compute free energies of relevant compounds, including saturated ThZr2H7 and ThZr2H6 with partial H occupancies. We developed a cluster analysis method that efficiently samples the ThZr2H6 32e−96g interstitial site configuration space and computes the number of arrangements that simultaneously meet the hydrogen partial occupancy requirement as well as the Westlake minimum H−H separation distance. We drastically reduced the overestimate of the configurational entropy based on complete H disorder for ThZr2H6 from 79 to 16 J·mol−1·K−1 and conclude that the configurational entropy is not a major driver for the enhanced stability of ThZr2H6 relative to the binary hydrides. The ThZr2H7 ternary hydride of van Houten and Bartram12 does not form for any temperature or pressure, and the relaxed lattice constants deviate significantly from the experimental structure. To our knowledge, ThZr2H7 has yet to be reproduced experimentally by another group and further work should be completed to verify the saturation hydrogen loading for the ThZr2Hx metal hydride. We computed van’t Hoff plots for the stable binary and ternary hydrides and estimated enthalpies and entropies of formation. Our predicted Td of 1168 K for ThZr2H6 is in reasonable agreement with the experimental extrapolated Td ≈ 1236 ± 150 K. The thermodynamic stabilities of the very stable metal hydrides, particularly ThZr2H6 and ZrH2, are nearly degenerate at this level of theory. Nevertheless, this study shows that first-principles methods can successfully reproduce general phase stability behavior of a simple ternary metal− hydrogen system that operates at high temperature. These methods may be useful in metal hydride applications where

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Funding

This research was performed using funding received from the DOE Office of Nuclear Energy’s Nuclear Energy University Programs. Notes

The authors declare no competing financial interest. K.M.N. e-mail: [email protected].

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REFERENCES

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