Thermodynamic Properties of Uranyl-Containing Materials Based on

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Cite This: J. Phys. Chem. C 2018, 122, 5254−5267

Thermodynamic Properties of Uranyl-Containing Materials Based on Density Functional Theory Francisco Colmenero,*,† Ana María Fernández,‡ Joaquín Cobos,‡ and Vicente Timón† †

Instituto de Estructura de la Materia (CSIC), C/Serrano, 113., 28006 − Madrid, Spain Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), Avda/Complutense, 40, 28040 − Madrid, Spain

‡

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S Supporting Information *

ABSTRACT: The thermodynamic properties of uranyl containing materials, including dehydrated schoepite, metastudtite, studtite, soddyite, rutherfordine, and γ-UO3 were studied. These materials are among the most important secondary phases arising from corrosion of spent nuclear fuel under the final geological disposal conditions, and γ-UO3 is the main oxide of hexavalent uranium. The crystal structures of the dehydrated schoepite and metastudtite were determined by means of density functional theory using a new norm-conserving pseudopotential for uranium atom. The resulting structural properties and X-ray powder patterns were found to be in very good agreement with the experimental data. By using the optimized structures of these materials, as well as of those obtained in previous works for studtite and soddyite, the thermodynamic properties of dehydrated schoepite, metastudtite, studtite, and soddyite were determined, including specific heats, entropies, enthalpies and Gibbs free energies. Finally, the computed thermodynamic properties of these materials together with those reported previously for rutherfordine and γ-UO3 were used to determine their enthalpies and free energies of formation and their variation with temperature. The theoretical results for rutherfordine and γ-UO3 are shown to be in excellent agreement with experimental information, even at high temperatures (up to 700 and 900 K, respectively). The results for dehydrated schoepite are in reasonable agreement with the experimental values obtained from the extrapolation of the measured values for UO2·0.77H2O and UO2·0.85H2O to UO2·1H2O, and in good agreement with previous theoretical data. The corresponding temperature dependent functions and the associated reaction constants for metastudtite, studtite, and soddyite, for which there are no experimental data to compare with, were predicted. The overall thermodynamic data obtained in this work by a theoretical approach can be used for the improvement and extension of the nuclear thermodynamic databases for modeling uranium containing systems and its dynamical behavior under different geochemical environments. waste,7 and may be also of importance in analyzing reactions within breeder reactors. The enormous relevance of the thermodynamic information on uranium containing materials in the assessment of the safety of nuclear waste repositories is reflected by the large number of recent experimental works on this topic culminating in large reviews and updates of thermodynamic properties of uranium bearing species8−12 and other systems containing related elements.13 For example, recent studies report the experimental determinations of the thermodynamic properties at the standard state of uranyl peroxide hydrates,14−17 uranyl carbonate minerals,18 uranyl phosphate and orthophosphate minerals19 and uranyl silicates20−25 by means of solubility and calorimetry measurements.

I. INTRODUCTION In nuclear sciences, the fundamental thermodynamic data are the indispensable basis for a dynamic modeling of the chemical behavior of nuclear materials.1−4 At the end of the 80s decade it was noticed that the development of a thermodynamic database devoted to the nuclear field was a necessity, particularly for the applications in safety and severe accident calculations.5,6 Since then, important progress in the quality and reliability of thermodynamic data for complex thermochemical systems has been achieved. This is because the prediction of the behavior of these hazardous materials under diverse environmental conditions is necessary for the long-term safety performance assessment. For example, the knowledge of precise thermodynamic data is needed to evaluate the origin and evolution of uranium ore bodies, in developing programs for the solution mining of uranium deposits or mine dumps, in the study of degradation models of the spent nuclear fuel (SNF) radioactive © 2018 American Chemical Society

Received: December 15, 2017 Revised: February 11, 2018 Published: February 20, 2018 5254

DOI: 10.1021/acs.jpcc.7b12341 J. Phys. Chem. C 2018, 122, 5254−5267

Article

The Journal of Physical Chemistry C

subsection III.2, the thermodynamic properties of the first four materials were determined by using these optimized structures and the previously reported ones for studtite and soddyite.53,54 Finally, in subsection III.3, the computed thermodynamic properties of these materials together with those reported previously39,40 for rutherfordine and γ-UO3 were used in order to determine their enthalpies and free energies of formation as a function of temperature. Section IV contains a detailed discussion of the results given in section III. In the last section V, the main conclusions obtained from this work are presented.

While the thermodynamic information database of uranium bearing species is very advanced, there are many systems for which the corresponding data are inaccurate due to large experimental uncertainties.2 A large amount of effort has been dedicated to the assignment and correction of these uncertainties by means of the implementation of new statistical methods for hypothesis testing and the improvement of the techniques used for measuring the thermodynamic properties of these systems.13 The need of a comprehensive, internationally recognized, and quality-assured chemical thermodynamic database, that meets the modeling requirements for the safety assessment of radioactive waste disposal systems, prompted to the Radioactive Waste Management Committee (RWMC) of the Organization for Economic Co-operation and Development (OECD) Nuclear Energy Agency (NEA) to launch the Thermochemical Database Project (NEA TDB). The RWMC assigned a high priority to the critical review of relevant chemical thermodynamic data of inorganic species, actinide compounds, and fission products.10,13 Additionally, the range of conditions (i.e., temperature and pressure) for which the thermodynamic properties are known for this kind of materials is quite limited. The temperature dependence of these properties for anhydrous uranium oxides is well-known and, therefore, thermodynamic studies have been carried out for uranium−oxygen and sodium−uranium−oxygen systems.26−36 However, the knowledge of the corresponding information for most of uranyl containing materials, such as the secondary phases arising from alteration of spent nuclear fuel under final geological disposal is extraordinarily poor. While, most of the thermodynamic properties of these phases are known for the standard state (298.15 K and 1 bar), the almost complete absence of temperature-dependent thermodynamic data for these phases precludes the realization of accurate thermodynamic modeling studies for the study and assessment of the spent nuclear fuel geological disposal. These features point out the need of additional methods for determining accurately the temperature dependence of these properties since the current methodology have not been able to provide this type of data. Previous studies7,37−42 suggest that theoretical methods, free of the difficulties associated with the handling of radioactive materials, constitute a good alternative for the determination of the thermodynamic properties of these materials. In this work, the temperature dependence of different thermodynamic properties of uranyl containing materials, including dehydrated schoepite (UO2(OH)2), metastudtite ((UO2) O2·2H2O), studtite ((UO2) O2·4H2O), soddyite ((UO 2 ) 2 (SiO 4 )·2H 2 O), rutherfordine (UO 2 CO 3 ), and gamma uranium trioxide (γ-UO3), has been studied. These materials, containing UO22+, have been recognized to be fundamental components of the paragenetic sequence of secondary phases that arises from the weathering of uraninite ore deposits and corrosion of spent nuclear fuel under the final geological disposal conditions.43−48,14,17 Gamma uranium trioxide, γ-UO3, is the main oxide of hexavalent uranium, which is common throughout the nuclear fuel cycle.49,50,40 The thermodynamic properties of uranium trioxide computed in this paper have been extensively used in subsequent work.51 This paper is organized as follows. In section II, the methods used in this paper are described. Section III contains the main results obtained in this work. The structures of dehydrated schoepite and soddyite were determined by means of density functional theory using plane waves and pseudopotentials52 and the corresponding results are given in subsection III.1. In

II. METHODS II.1. Computational Methods. CASTEP code,55,56 a module of the Materials Studio package,57 was employed to model dehydrated schoepite and metastudtite structures. The generalized gradient approximation (GGA) together with PBE functional58 and Grimme empirical dispersion correction, called the DFT-D2 approach,59 were used. Dispersion corrections were applied in order to improve the description of the hydrogen bond structure within the unit cell of these materials. The pseudopotentials used for all the atoms, except uranium, were standard norm-conserving pseudopotentials60 given in CASTEP code (00PBE-OP type). However, a new normconserving relativistic pseudopotential for uranium atom, generated in a previous work,41,61 was used in these computations. This pseudopotential has been extensively employed in the research of uranyl-containing materials.39−41,53,54,61 Geometry optimization was carried out by using the Broyden−Fletcher−Goldfarb−Shanno optimization scheme52,62 with a convergence threshold on atomic forces of 0.01 eV/Å. The kinetic energy cutoff and k-point mesh63 were selected to ensure good convergence for computed structures and energies. Dehydrated schoepite and metastudtite structures were optimized in calculations with increasing complexity by increasing these parameters. The optimizations performed with a cutoff of 1000 eV and K meshes of 5 × 3 × 5 (18 K points) for dehydrated schoepite and 3 × 3 × 4 (8 K points) for metastudtite gave well converged structures and were used to determine the final results. II.2. Thermodynamic Properties. The phonon spectrum at the different points of Brillouin zone can be determined using Density Functional Perturbation Theory (DFPT) as second order derivatives of the total energy.64−66 The knowledge of the entire phonon spectrum allows the phonon dispersion curves and density of states to be calculated and, from them, the evaluation of several important thermodynamic quantities in the quasi-harmonic approximation, such as free energies, enthalpies, entropies and specific heats.64,67 The thermodynamic properties of dehydrated schoepite and metastudtite were obtained at the optimized geometry using these methods. Similarly, these properties were determined for studtite and soddyite using the optimized geometries determined in previous works.53,54 II.3. Enthalpies and Free-Energies of Formation. The enthalpies and free-energies of formation of these materials at different temperatures were obtained, using our calculated enthalpy and entropy functions, (HT − H298)calc and Scalc T , by means of the following expressions:68 elements

Δf H(T ) = Δf H ° + (HT − H298)calc −

∑

ni(HT − H298)iexp

i

(1) 5255


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The Journal of Physical Chemistry C Table 1. Dehydrated Schoepite and Paulscherrerite (Last Row) Lattice Parameters parameters

a (Å)

b (Å)

c (Å)

α (deg.)

β (deg.)

γ (deg.)

vol. (Å3)

dens. (g/cm3)

this work DFT37 exp.70,71 exp.72 exp.73

4.2309 4.222 4.242 4.2455 4.288

10.3104 10.304 10.302 10.3183 10.270

6.9195 6.938 6.868 6.8648 6.885

90.0 90.0 90.0 90.0 90.0

90.0 90.0 90.0 90.0 90.39

90.0 90.0 90.0 90.0 90.0

301.87 301.83 300.14 300.72 303.20

6.69 6.69 6.73 6.72 6.66

⎧ S calc − Δ f G (T ) = Δ f H (T ) − T ⎨ ⎪ T ⎩ ⎪

elements

∑ i

⎫ ⎬ ni(ST)iexp ⎪ ⎭ ⎪

(2)

In these equations, ΔfH0 is the standard enthalpy of formation of the material (at the temperature of 298.15 K exp and a pressure of 1 bar) and (HT − H298)exp i and (ST)i are the enthalpy and entropy functions of the elements forming part of the material with stoichiometric coefficients ni, respectively. The specific experimental values used for ΔfH0 are indicated in each case. The enthalpy and entropy functions for the elements H, O, C, and Si were taken from JANAF tables,68 and the corresponding functions for U were taken from Barin.69 The equilibrium constants for the formation reactions were determined in terms of the corresponding free energies of formation using the well-known relationship:68 ΔfG(T) = −RT Ln K.

III. RESULTS III.1. Crystal Structures. III.1.a. Dehydrated Schoepite, UO2(OH)2. The structure of dehydrated schoepite was determined by increasing the complexity of the calculations (see section II.1). Table 1 gives the final lattice parameters, volumes, and densities obtained compared with the experimental ones70−72 and those obtained in previous DFT calculations.37 The last row in Table 1 has been included for comparison, and corresponds to the closely related mineral species paulscherrerite,73 which adopts monoclinic symmetry due to the presence of metaschoepite contaminant instead of the orthorhombic symmetry (Cmca space group) exhibited by the synthetic dehydrated schoepite.70−72 The computed structure is shown in Figure 1. In Figure 1A, a general view of the unit cell is given. Figure 1B provides a view of a structural sheet from the [010] direction. The hydrogen bond structure in dehydrated schoepite is shown in Figure 2. Tables S.1 and S.2 of the Supporting Information give a comparison of the more important geometric parameters (bond distances and angles) obtained with the corresponding experimental data. According to the crystal structure of dehydrated schoepite shown in Figure 1, the uranium atom in the structure of this material displays hexagonal bipyramid coordination, UO2(OH)6, the axial positions being occupied by uranyl oxygen atoms and the equatorial ones by hydroxyl ions. These bypiramids share their equatorial vertices with other uranyl polyhedra forming layers that are held together by means of hydrogen bonds. For this reason, the accurate description of van der Waals dispersion interactions is very important in this case, and the introduction of dispersion corrections improved the calculated structure significantly.37 As it can be seen in Figure 2, the uranyl oxygen atoms are hydrogen-bond acceptors with hydroxyl ions from the upper and lower sheets. Three of the oxygen atoms in the equatorial hydroxyl ions are donors of hydrogen bonds directed toward the lower layer and the other

Figure 1. Structure of dehydrated schoepite, UO2(OH)2: (A) General view of the crystal structure; (B) View of a structural layer from [010]. Color code: U, blue; O, red; H, yellow.

Figure 2. Detailed view of the hydrogen bond structure of dehydrated schoepite. Color code: U, blue; O, red; H, yellow.

three are donors of hydrogen bonds directed toward the upper layer. Tables S.1 and S.2 of the Supporting Information give a comparison of the more important geometric parameters (bond distances and angles) obtained with the corresponding experimental data. Dehydrated schoepite contains only one 5256


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Figure 3. X-ray powder pattern of dehydrated schoepite using Cu Kα radiation: Above: X-ray powder pattern computed from calculated geometry; below: X-ray powder pattern computed from experimental geometry.72

Table 2. Metastudtite Lattice Parameters parameters

a (Å)

b (Å)

c (Å)

α (deg.)

β (deg.)

γ (deg.)

vol. (Å3)

dens. (g/cm3)

this work DFT76 exp.75

8.3977 8.45 8.42

8.5688 8.72 8.78

6.8240 6.75 6.51

90.0 90.0 90.0

90.0 90.0 90.0

90.0 90.0 90.0

491.05 497.37 481.3

4.572 4.514 4.67

structurally (symmetrically) identical U atom and two O atoms (the uranyl, Oy, and hydroxyl, O, oxygen atoms). The uranyl ion is linear, with the computed U−Oy distance being 1.79 Å compared with the experimental value of R(U−Oy) = 1.71 Å. The distances from uranium to the equatorial oxygens are R(U−O) = 2.46 Å and R(U−O′) = 2.52 Å, which compare very well with the experimental values of 2.46 and 2.51 Å, respectively. The values obtained by Weck and Kim37 were R(U−Oy) = 1.82 Å, R(U−O) = 2.46 Å, and R(U−O′) = 2.54 Å. The X-ray powder diffractogram of dehydrated schoepite was computed from the experimental72 and computed structures74 using Cu Kα radiation (λ = 1.540598 Å). The most intense lines (I > 10%) are compared in Figure 3. The values of the main reflections are given in Table S.3 of the Supporting Information. III.1.b. Metastudtite, (UO2) O2·2H2O. Table 2 gives the final lattice parameters, volumes, and densities obtained for metastudtite compared with the experimental ones75 and those obtained in previous DFT calculations.76 The computed structure of metastudtite is shown in Figure 4. Views of a 2 × 2 × 1 supercell from [100] and [001] directions are given in Figure 4A and B, respectively. The hydrogen bond structure is shown in Figure 5. Tables S.4 and S.5 of the Supporting Information give a comparison of the more important geometric parameters (bond distances and angles) obtained with those from the previous DFT calculations of Weck et al.76 As in studtite,76 the uranium atom in metastudtite displays hexagonal bipyramid coordination (see Figure 4). The axial oxygen atoms comprise the uranyl UO2+ 2 group with two distinct short axial UO bonds that are about 1.80 and 1.85 Å long and a nearly linear angle (in studtite the UO bond lengths

Figure 4. Structure of metastudtite, (UO2) O2·2H2O: (A) View of a 2 × 2 × 1 supercell from [100]; (B) View of a 2 × 2 × 1 supercell from [001]. Color code: U, Blue; O, Red; H, Yellow.

were equal). The bipyramids are tilted toward each other and linked through peroxo groups forming chains parallel to a axis. The chains are held together by means of a dense network of hydrogen bonds. For this reason, as in dehydrated schoepite, the introduction of dispersion corrections improved the calculated structure significantly.76 The hydrogen bond 5257


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radiation (λ = 1.540598 Å). The computed pattern is compared in Figure 6 with the experimental pattern of Debets.77,78 The values of the main reflections are given in Table S.6 of the Supporting Information. III.2. Thermodynamic Properties. III.2.a. Dehydrated Schoepite. A phonon calculation was performed at the optimized structure of dehydrated schoepite. From it, the thermodynamic properties were evaluated. Figure 7A−D shows the isobaric heat capacity, entropy, enthalpy, and free energy functions, respectively. Note that all the enthalpy and free energy values have been divided by the temperature to express these properties in the same units as entropy and heat capacity (J·K−1·mol−1). The values of the calculated thermodynamic functions over the temperature range 0−1000 K are given in Tables S.7−S.10 of the Supporting Information. A comparison of the calculated specific heat values at selected temperatures with the empirical values of Hemingway1 and the theoretical values of Weck and Kim37 are given in Table 3. The precise values of the calculated entropy, enthalpy, and free energy functions at selected temperatures are given in Table 4. III.2.b. Metastudtite and Studtite. The isobaric heat capacity, entropy, enthalpy, and free energy functions of metastudtite and studtite were determined from the results of phonon calculations carried out at their respective optimized equilibrium structures. The calculated thermodynamic functions are shown in Figures 8A−D and 9A−D, respectively. The corresponding values of these functions over the temperature range 0−1000 K are given in Tables S.11−S.15 and S.18 of the Supporting Information. A comparison of the specific heat values of metastudtite and studtite at selected temperatures with the computed values of Sassani et al.7 is given in Table 5. Note that similar results to those from Sassani et al.7 for the thermodynamic properties of metastudtite and studtite were recently computed by Weck and Kim.38 The precise values of the computed entropy, enthalpy, and free energy at selected temperatures for these two materials are given in Table 6. III.2.c. Soddyite. The calculated isobaric heat capacity, entropy, enthalpy, and free energy functions of soddyite are shown in Figure 10A−D, respectively. The values of the calculated thermodynamic functions over the temperature

Figure 5. Detailed view of the hydrogen bond structure of metastudtite. Color code: U, Blue; O, Red; H, Yellow.

structure is shown in detail in Figure 5. All water molecules in metastudtite are structural, that is, all molecules form part of uranyl polyhedra. In studtite, one-half of the water molecules are of crystallization.53 Two different parallel chains, having the uranium atoms at the same height, are related by reflection in a xz plane. Water molecules of two different chains are paired and placed one in front of the other. Each water molecule participates in two hydrogen bonds. The first bond links the water molecule with a uranyl oxygen atom and the other with a peroxide oxygen atom. These oxygen atoms belong to upper and lower chains. The two paired water molecules are hydrogen bonded to the same uranyl oxygen atom. However, these two paired water molecules are hydrogen bonded with a different oxygen atom from the same peroxo group. That is, the two paired water molecules of different chains are hydrogen bonded with the same uranyl oxygen atom and with the two oxygen atoms of the same peroxo group. Tables S.4 and S.5 of the Supporting Information give a comparison of the more important geometric parameters (bond distances and angles) obtained. These parameters were compared with the corresponding data from the previous DFT calculations of Weck et al.,76 both results being in good agreement. The X-ray powder diffractogram of metastudtite was computed from the optimized structure74 using Cu Kα

Figure 6. X-ray powder pattern of metastudtite using Cu Kα radiation: Above: X-ray powder pattern computed from calculated geometry; below: Experimental X-ray powder pattern.77,78 5258


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Figure 7. Calculated thermodynamic properties of dehydrated schoepite: (A) Isobaric specific heat; (B) Entropy; (C) Enthalpy; (D) Gibbs free energy. All functions are given as a function of temperature. Note that the experimental data for the specific heat, entropy, and enthalpy functions displayed in these figures are not measured quantities, but extrapolations of the experimental values obtained for UO2·0.77H2O and UO2·0.85H2O to UO2·1H2O (see Hemingway1).

III.3. Enthalpies and Free Energies of Formation. III.3.a. Rutherfordine and Gamma Uranium Trioxide. In order to obtain the enthalpies and free energies of formation of a mineral phase, the standard enthalpy of formation is required (see eq 1). The thermodynamic properties of the uranyl carbonate mineral rutherfordine and gamma uranium trioxide were determined in previous works.39,40 From these results and the experimental values for the standard enthalpies of formation of these materials reported by Hemingway1 and Guillaumont et al.13 (see Table 8), we obtained the enthalpies and free energies of formation and reaction constants as a function of temperature, which are given in Table 9. In this table, the results are compared with the experimental values of the free energies of formation for rutherfordine and γ-UO3 reported by Hemingway1 in the range from 298.15 to 700 K and by Cordfunke and Westrum79 from 298.15 to 900 K, respectively. III.3.b. Dehydrated Schoepite. The thermodynamic properties of dehydrated schoepite were reported in section III.2.a of this work. The enthalpies and free energies of formation and reaction constants of dehydrated schoepite as a function of temperature, presented in Table 10, were obtained from these results and the experimental value for the standard enthalpy of formation reported by Hemingway1 (Table 8). A comparison of the results with the empirical values reported by Hemingway1 from 298.15 to 700 K is shown in Table 10. III.3.c. Metastudtite, Studtite, and Soddyite. From the thermodynamic properties of metastudtite, studtite, and

Table 3. Comparison of Calculated Isobaric Heat Capacity Functions of Dehydrated Schoepitea T (K)

Cpexp

Cpcalc (this work)

Cpcalc (Weck and Kim37)

298.15 400 500 600 700

109.0 126.3 139.3 148.3 153.9

103.85 118.21 127.42 133.65 138.15

107.21 122.31 131.92 137.81 140.75

a The experimental data is from Hemingway.1 All values are given in units of J·K−1·mol−1.

Table 4. Calculated Entropy, Enthalpy, and Free-Energy Functions of Dehydrated Schoepitea

a

T (K)

Scalc

(HT − H298)calc

(GT − H298)calc

298.15 400 500 600 700

125.18 159.36 188.06 212.97 234.89

0.00 29.74 49.50 64.07 75.24

−125.18 −129.61 −138.56 −148.90 −159.65

All values are given in units of J·K−1·mol−1.

range 0−1000 K are given in Tables S.19−S.22 of the Supporting Information. The precise values of the computed specific heat, entropy, enthalpy and free energy at selected temperatures are given in Table 7. 5259


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Figure 8. Calculated thermodynamic properties of metastudtite as a function of temperature: (A) isobaric specific heat; (B) entropy; (C) enthalpy; (D) Gibbs free energy.

errors in the computed volume and density with respect to experimental data of Deliens and Piret75 are very small (about 2.1%). The agreement in line positions and intensities with the experimental X-ray powder diffractogram of metastudtite (Figure 6) is also very good. IV.2. Thermodynamic Properties. The thermodynamic properties for dehydrated schoepite, metastudtite, studtite, and soddyite were calculated in this work. The thermodynamic properties of rutherfordine and γ-UO3 were determined in previous studies.39,40 IV.2.a. Dehydrated Schoepite. The value obtained for the isobaric specific heat at zero pressure and 298.15 K is Cp = 103.85 J·K−1·mol−1, which may be compared with the experimental value of 109.0 J·K−1·mol−1 from Hemingway1 and the computed value of 107.21 J·K−1·mol−1 from Weck and Kim.37 The agreement is quite good, our value being about 4.7% smaller than the experimental value. It must be noted that the experimental value is not a measured quantity, but an extrapolation of the experimental values obtained for UO2· 0.77H2O and UO2·0.85H2O to UO2·1H2O. The heat capacity function is compared with the empirical function of Hemingway1 and the theoretical function of Weck and Kim37 in Figure 7A in the temperature range of 298−700 K. As it can be appreciated, while the difference with the empirical function increases when the temperature increases, our calculated function is nearly parallel to that of Weck and Kim.37 These results suggest that the empirical procedure used in the

soddyite reported in section III.2, and the experimental values for the standard enthalpies of formation reported by Guo et al.,17 Kubatko et al.,14 and Gorman-Lewis et al.,21 (Table 8), we obtained the enthalpies and free energies of formation and reaction constants of these materials as a function of temperature that are provided in Table 11.

IV. DISCUSSION IV.1. Crystal Structures. In this work, the crystal structures of dehydrated schoepite and metastudtite were optimized. The crystal structures for the other uranyl containing materials studied here were described in previous works.40,53,54,61 These structures were found to be in good agreement with the corresponding experimental structures. As it may be appreciated in Table 1, our results for dehydrated schoepite, obtained by employing pseudopotentials, are very similar to those of the calculations performed by Weck and Kim,37 who used a more complex description of the interaction between valence electrons and ionic cores: the projector augmented wave (PAW) method.80 The theoretical and experimental results are in very good agreement. The calculated volume and density differ from the experimental values by only about 0.6%. The agreement in line positions and intensities with the experimental X-ray powder diffractogram of dehydrated schoepite (Figure 3) is also very good. The theoretical and experimental lattice parameters of metastudtite (Table 2), are in very good agreement. The 5260


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Figure 9. Calculated thermodynamic properties of studtite as a function of temperature: (A) isobaric specific heat; (B) entropy; (C) enthalpy; (D) Gibbs free energy.

contents is not appropriate, because the type of water in dehydrated schoepite is not free and belongs to the structure as hydroxyl ions. At 700 K, the last temperature considered by Hemingway,1 both the calculated and empirical values of Cp, 138.15 and 153.9 J·K−1·mol−1, respectively, are still well below the Dulong-Petit asymptotic value given by Cp = 3nR = 174.6 J· K−1·mol−1. The calculated entropy, enthalpy, and free energy functions are displayed in Figure 7B−D. The calculated functions for the entropy and enthalpy are compared with the empirical functions of Hemingway1 in Figure 7B and C, respectively. As it can be seen, the calculated enthalpy function is in very good agreement with the empirical enthalpy function and the two entropy functions are very closely parallel. IV.2.b. Metastudtite and Studtite. In the case of metastudtite and studtite, as far as we know, there are not experimental values of these thermodynamic properties.

Table 5. Comparison of Calculated Isobaric Heat Capacity Functions of Metastudtite and Studtitea metastudtite

studtite

T (K)

Cpcalc (this work)

Cpcalc (reference)

Cpcalc (this work)

Cpcalc (reference)

298.15 400 500 600 700 800

163.14 185.00 199.18 209.36 217.24 223.69

155.81 178.05 192.60 203.21 211.40 217.99

219.97 255.40 279.99 298.44 313.18 325.51

211.17 246.96 272.41 291.74 307.07 319.74

a

The reference calculated data are from the work of Sassani et al.7 All values are given in units of J·K−1·mol−1.

experimental work to estimate the thermodynamic properties by extrapolating the measurements from different water

Table 6. Calculated Entropy, Enthalpy, and Free Energy Functions of Metastudtite and Studtitea metastudtite T (K) 298.15 400 500 600 700 800 a

S

calc

179.27 230.50 273.41 310.67 343.55 373.00

(HT − H298)

calc

0.00 44.56 74.12 95.86 112.67 126.16

studtite (GT − H298)

calc

−179.27 −185.94 −199.29 −214.80 −230.88 −246.84

S

calc

232.12 302.03 361.81 414.56 461.71 504.36

(HT − H298)calc

(GT − H298)calc

0.00 60.83 102.29 133.52 158.19 178.36

−232.12 −241.20 −259.52 −281.04 −303.52 −326.00

All values are given in units of J·K−1·mol−1. 5261


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Figure 10. Calculated thermodynamic properties of soddyite as a function of temperature: (A) isobaric specific heat; (B) entropy; (C) enthalpy; (D) Gibbs free energy.

specific heat at zero pressure and 298.15 K is Cp = 163.14 J·K−1· mol−1, which may be compared with the computed value of 155.81 J·K−1·mol−1 from Sassani et al.,7 the difference being about 4.7%. For studtite, we obtained Cp = 219.97 J·K−1·mol−1, which is about 4.2% higher than the value from Sassani et al.7 of 211.17 J·K−1·mol−1. As it can be appreciated in Figures 8A and 9A, our calculated functions are nearly parallel to those of Sassani et al.7 For metastudtite, the calculated value of Cp at the temperature of 800 K, 223.69 J·K−1·mol−1, is 18.5% below the Dulong-Petit asymptotic value, Cp = 274.4 J·K−1·mol−1. Similarly, for studtite at 800 K, we obtain Cp = 325.51 J·K−1· mol−1, which is 23.2% below the asymptotic value, Cp = 424.0 J· K−1·mol−1. IV.2.c. Soddyite. For the uranyl silicate mineral soddyite the experimental values of these thermodynamic properties have not been reported. Besides, as far as we know, our theoretical calculations are the first ones carried out for soddyite.54 Nevertheless, we will compare the experimental value of the free energy of formation of soddyite at the standard state with our results in a later section (section IV.3.c). The value obtained for the isobaric specific heat at zero pressure and 298.15 K is Cp = 275.15 J·K−1·mol−1. The calculated value of Cp at the temperature of 1000 K, 378.09 J·K−1·mol−1, is 10.8% below the Dulong-Petit asymptotic value, Cp = 424.0 J·K−1· mol−1. IV.2.d. Recommended Thermodynamic Data. The calculated thermodynamic properties of γ-UO3 were in very close agreement with the experimental data for the full range of temperatures (0−1000 K).40 Therefore, the use of both sets of

Table 7. Calculated Isobaric Heat Capacity, Entropy, Enthalpy and Free-Energy Functions of Soddyitea

a

T (K)

Cpcalc

Scalc

(HT − H298)calc

(GT − H298)calc

298.15 400 500 600 700 800 900 1000

275.15 309.93 331.75 346.60 357.39 365.71 372.45 378.09

315.95 402.07 473.74 535.61 589.90 638.18 681.66 721.20

0.00 74.87 124.15 160.11 187.54 209.30 227.01 241.90

−315.95 −327.20 −349.59 −375.50 −402.35 −428.88 −454.65 −482.83

All values are given in units of J·K−1·mol−1.

Table 8. Experimental Values the Standard Enthalpies of Formation of Rutherfordine, Gamma Uranium Trioxide, Dehydrated Schoepite, Metastudtite, Studtite, and Soddyite material UO2CO3 γ-UO3 UO2(OH)2 (UO2) O2·2H2O (UO2) O2·4H2O (UO2)2 (SiO4)·2H2O

ΔfH0 (kJ·mol−1) −1704.1 −1223.8 −1534.5 −1779.6 −2344.7 −4045.4

± ± ± ± ± ±

2 1.2 4.0 1.9 4.0 4.9

reference Hemingway1 Guillaumont et al.13 Hemingway1 Guo et al.17 Kubatko et al.14 Gorman-Lewis et al.21

However, we can compare our calculated functions with the results from other theoretical calculations performed by Sassani et al.7 For metastudtite, the value obtained for the isobaric 5262


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The Journal of Physical Chemistry C Table 9. Calculated Enthalpies (ΔfH) and Free Energies (ΔfG) of Formation and Reaction Constants (Log K) of Rutherfordine and Gamma Uranium Trioxide as a Function of Temperaturea T (K)

(ΔfH)calc

5 55.2525 105.5051 206.0101 298.15 400 500 600 700

−5912.12 −2055.63 −1860.20 −1744.31 −1704.10 −1684.32 −1675.68 −1672.99 −1674.73

10 50 100 200 298.15 300 400 500 600 700 800 900

−2612.63 −1486.36 −1337.99 −1254.59 −1223.80 −1223.45 −1210.52 −1204.91 −1203.58 −1205.07 −1208.61 −1213.77

(ΔfG)calc

(ΔfG)exp

Rutherfordine −5911.63 −2035.22 −1813.75 −1654.68 −1577.19 −1577.00 −1515.13 −1533.62 −1465.81 −1491.16 −1422.92 −1448.89 −1384.64 −1406.80 γ-UO3 −2611.59 −1474.77 −1309.89 −1199.43 −1144.80 −1145.79 −1143.95 −1145.31 −1105.45 −1119.32 −1074.74 −1093.66 −1048.44 −1068.27 −1024.88 −1043.08 −1003.13 −1018.02 −982.61 −993.03

Table 11. Calculated Enthalpies (ΔfH) and Free Energies (ΔfG) of Formation and Reaction Constants (Log K) of Metastudtite, Studtite, and Soddyite as a Function of Temperature T (K)

error (%)

Log K

−0.01 1.21 1.70 1.79 1.58

61756.82 1924.01 897.95 419.54 276.31 197.85 153.13 123.87 103.32

0.09 0.12 1.24 1.73 1.86 1.74 1.46 1.05

13641.22 1540.64 684.18 313.25 200.56 199.17 144.35 112.27 91.27 76.48 65.50 57.03

5 55.2525 105.5051 206.0101 298.15 360 400 500 600 700 800 5 55.2525 105.5051 206.0101 298.15 360 400 500 600 700 800

a

The calculated results are compared with the corresponding experimental values of Hemingway1 and Cordfunke and Westrum79 (for rutherfordine and uranium trioxide, respectively). The values of ΔfH and ΔfG are in units of kJ·mol−1.

5 55.2525 105.5051 206.0101 298.15 360 400 500 600 700 800 900

Table 10. Calculated Enthalpies (ΔfH) and Free Energies (ΔfG) of Formation and Associated Reaction Constants (Log K) of Dehydrated Schoepite as a Function of Temperaturea T (K)

(ΔfH)calc

(ΔfG)calc

5 55.2525 105.5051 206.0101 298.15 360 400 500 600 700 800

−5318.41 −1847.94 −1673.87 −1570.85 −1534.50 −1522.12 −1516.68 −1509.05 −1507.07 −1508.98 −1513.68

−5317.92 −1826.85 −1624.51 −1473.99 −1395.56 −1354.15 −1330.16 −1276.73 −1229.48 −1186.36 −1146.07

(ΔfG)exp

−1404.90 −1360.60 −1317.28 −1274.25 −1231.56

Log K 55554.52 1727.02 804.26 402.02 244.49 196.48 173.70 133.38 107.03 88.52 74.83

ΔfH (kJ·mol−1)

ΔfG (kJ·mol−1)

Metastudtite −7332.09 −7331.46 −2237.47 −2204.89 −1983.38 −1906.70 −1832.83 −1678.59 −1779.60 −1556.66 −1761.10 −1491.58 −1752.95 −1453.66 −1741.50 −1368.64 −1738.43 −1292.94 −1740.89 −1223.39 −1747.30 −1158.17 Studtite −9636.46 −9635.63 −2939.58 −2891.23 −2607.07 −2491.61 −2412.18 −2175.54 −2344.70 −1998.43 −2321.09 −1900.87 −2310.72 −1843.12 −2296.28 −1711.54 −2292.73 −1592.42 −2296.47 −1481.81 −2305.45 −1377.39 Soddyite −13631.08 −13630.00 −4840.12 −4790.81 −4398.86 −4284.45 −4137.73 −3910.95 −4045.40 −3720.31 −4013.52 −3621.19 −3999.57 −3564.31 −3979.87 −3438.65 −3974.61 −3328.70 −3979.05 −3229.12 −3990.36 −3136.61 −4006.97 −3049.27

Log K 76589.26 2084.41 943.97 425.60 272.71 216.42 189.82 142.98 112.56 91.29 75.62 100660.19 2733.24 1233.54 551.60 350.11 275.80 240.68 178.80 138.63 110.57 89.93 142388.07 4529.02 2121.14 991.61 651.77 525.41 465.44 359.22 289.78 240.95 204.79 176.97

procedure from different water contents1 and, since the use of this procedure appears to provide inaccurate results, we recommend the theoretical results as the best thermodynamic data available for this material. For the remaining uranyl containing materials, metastudtite, studtite and soddyite, there are only experimental data available for the enthalpies of formation at the standard state. Therefore, the theoretical results are the unique temperature dependent thermodynamic data available. Although the accuracy of the thermodynamic data seems to be good enough, the results may be improved by using more complete theoretical treatments, such as advanced density functionals with more demanding calculation parameters. However, this would require much larger computational resources. IV.3. Enthalpies and Free Energies of Formation. IV.3.a. Rutherfordine and Gamma Uranium Trioxide. The results obtained for the enthalpies and free-energies of formation of rutherfordine at different temperatures are

The experimental values are from Hemingway.1 The values of ΔfH and ΔfG are in units of kJ·mol−1. a

data, experimental and theoretical, can be recommended because they have the same level of accuracy. In the case of rutherfordine,39 there were experimental data only from 300 to 700 K, and our theoretical results are in very good agreement with them. However, we recommend the use of the theoretical results because they extend the temperature range in which these properties are known from 0 to 700 K. For dehydrated schoepite, the available experimental data were empirically determined by using an extrapolation 5263


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Figure 11. Calculated free energies of formation and reaction constants of rutherfordine, γ-UO3, dehydrated schoepite, metastudtite, studtite, and soddyite as a function of temperature.

compared with the experimental values of Hemingway1 in Table 9. The results are also displayed in Figure 11A where the results are shown to be in excellent agreement with the experimental data. For comparison, the experimental value of the standard free-energy of formation1 is ΔfG0 = −1577.0 ± 4.2 kJ·mol−1, which is essentially the same as the calculated value, and the errors remain very small up to 700 K. The error is about 1.6% at 700 K. Furthermore, as in our previous work,39 the theoretical results extend the range of temperature in which these properties are known to the full range of thermodynamic stability of rutherfordine, from 0 to 700 K. For uranium trioxide, the results obtained for the enthalpies and free energies of formation at different temperatures are compared with the experimental results of Cordfunke and Westrum79 in Table 9 and Figure 11B. The results are again in excellent agreement with the experimental data. For comparison, the experimental value of the standard free energy of formation12 is ΔfG0 = −1145.739 ± 1.207 kJ·mol−1, which differs from the calculated value by 0.1%. The errors remain very small up to 900 K and, at this temperature, the error is about 1.0%.

IV.3.b. Dehydrated Schoepite. The results obtained for the enthalpies and free-energies of formation of UO2(OH)2 at different temperatures are presented in Table 10 and shown graphically in Figure 11C. For comparison, the experimental value of the standard free energy of formation1 is ΔfG0 = −1404.9 ± 4.0 kJ·mol−1, which differs from the calculated value by 0.7%. Although the agreement with the temperature dependent free energies of formation given by Hemingway1 is reasonable, it must be remarked that the comparison is not strictly valid since these values are not measured quantities, but extrapolations of the experimental values obtained for UO2· 0.77H2O and UO2·0.85H2O to UO2·1H2O. IV.3.c. Metastudtite, Studtite, and Soddyite. The results obtained for the enthalpies and free energies of formation metastudtite, studtite, and soddyite at different temperatures are reported in Table 11 and displayed in Figures 11D−F. For comparison, the experimental value of the standard free-energy of formation of soddyite21 is ΔfG0 = −3652.2 ± 4.2 kJ·mol−1, which differs from the calculated value by 1.9%. IV.3.d. Applicability of the Theoretical Data in Nuclear Databases. The methods used in a forthcoming paper51 illustrate how the thermodynamic data obtained in this work 5264



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may be used to obtain the Gibbs free energy of a large number of important reactions involving these materials, which are relevant, for example, to evaluate the thermodynamic stability of the secondary phases of spent nuclear fuel under real conditions of a final deep geological disposal. The overall thermodynamic data obtained by the theoretical approach can be used for the improvement and extension of the nuclear thermodynamic databases, which can be employed for modeling uranium containing systems and its dynamical behavior under different geochemical environments.

Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b12341. Tables S.1 and S.2: Calculated bond distances and angles of dehydrated schoepite compared with experimental data. Table S.3: Main reflections in the X-ray powder pattern of dehydrated schoepite. Tables S.4 and S.5: Calculated bond distances and angles of metastudtite compared with the DFT results of Weck et al.76 Table S.6: Main reflections in the X-ray powder pattern of metastudtite. Tables S.7−S.10, S.11−S.14. S.15−S.18, and S.19−S.22: Calculated heat capacity, entropy, enthalpy and free-energy functions of dehydrated schoepite, metastudtite, studtite and soddyite, respectively (PDF).

V. CONCLUSIONS The knowledge of precise thermodynamic data is fundamental for the development of geochemical and fuel degradation models, as well as for the dynamic modeling of the chemical behavior of nuclear materials. In this work, the thermodynamic properties (specific heats, entropies, enthalpies and free energies) of uranyl containing materials, including dehydrated schoepite, metastudtite, studtite, soddyite, rutherfordine, and γUO3 were studied and used to determine their enthalpies and free energies of formation as a function of temperature. The studied uranyl containing materials, except γ-UO3, are among the main secondary phases arising from corrosion of spent nuclear fuel under final geological disposal conditions.43−48,14,17 γ-UO3 is the main oxide of hexavalent uranium, which is common throughout the nuclear fuel cycle.49,50,40 In the first place, the structures of the dehydrated schoepite and metastudtite were obtained by means of density functional theory using a new norm-conserving pseudopotential for uranium atom. The resulting structural properties and X-ray powder patterns were found to be in very good agreement with the experimental data. Then, the thermodynamic properties of the dehydrated schoepite, metastudtite, studtite, and soddyite were determined by using these optimized structures and the previously reported ones for studtite and soddyite.53,54 The thermodynamic properties calculated included specific heats, entropies, enthalpies, and Gibbs free energies. Finally, the computed thermodynamic properties of these materials together with those reported previously for rutherfordine and γ-UO336,37 were used to determine their enthalpies and free energies of formation as a function of temperature. The theoretical results obtained for rutherfordine and γ − UO3 are shown to be in excellent agreement with experimental information. For dehydrated schoepite the results are in reasonable agreement with the values obtained from extrapolation of the measured values for UO2·0.77H2O and UO2·0.85H2O to UO2·1H2O, and in good agreement with previous theoretical results. For metastudtite, studtite and soddyite the corresponding temperature dependent functions and the associated reaction constants, for which there are no experimental data to compare with, were predicted. From these results we conclude that the use of theoretical methods is a safe and accurate method for the determination of the thermodynamic properties of uranyl containing materials and their temperature dependence. These properties are of primary importance for the performance assessment calculations in the context of the radioactive waste disposal. The great usefulness of the thermodynamic properties reported in this paper will be illustrated in a subsequent work,51 in which the enthalpies and free energies of a large series of reactions involving uranyl containing materials were determined and used to analyze the relative thermodynamic stability of these phases under final geological disposal conditions.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +34 915616800, Ext. 941033. ORCID

Francisco Colmenero: 0000-0003-3418-0735 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by ENRESA in the project: No. 079000189 “Aplicación de técnicas de caracterización en el estudio de la estabilidad del combustible nuclear irradiado en condiciones de almacenamiento” (ACESCO) and Project FIS2013-48087-C2-1-P. Supercomputer time by the CETACIEMAT, CTI-CSIC, and CESGA centers are also acknowledged. This work has been carried out in the context of a CSIC−CIEMAT collaboration agreement: “Caracterización experimental y teórica de fases secundarias y óxidos de uranio formados en condiciones de almacenamiento de combustible nuclear”. We also want to thank Dr. Rafael Escribano for reading the document and many helpful comments.

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REFERENCES

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