Thermodynamic Properties of Uranyl-Containing Materials Based on

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Thermodynamic Properties of Uranyl Containing Materials Based on Density Functional Theory Francisco Colmenero, Ana Maria Fernandez, Joaquín Cobos, and Vicente Timón J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b12341 • Publication Date (Web): 20 Feb 2018 Downloaded from http://pubs.acs.org on February 26, 2018

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The Journal of Physical Chemistry

Thermodynamic Properties of Uranyl Containing Materials Based on Density Functional Theory Francisco Colmeneroa*, Ana María Fernándezb, Joaquín Cobosb and Vicente Timóna

aInstituto

bCentro

de Estructura de la Materia (CSIC). C/ Serrano, 113. 28006 – Madrid, Spain.

de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT). Avda/ Complutense, 40. 28040 – Madrid, Spain.

Orcid Francisco Colmenero: https://orcid.org/0000-0003-3418-0735 Orcid Ana María Fernández: https://orcid.org/0000-0002-8392-0165 Orcid Joaquín Cobos: https://orcid.org/0000-0003-0285-7617 Orcid Vicente Timón: https://orcid.org/0000-0002-7460-7572

*E-mail: [email protected]

1 ACS Paragon Plus Environment

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ABSTRACT

The thermodynamic properties of uranyl containing materials, including dehydrated schoepite, metastudtite, studtite, soddyite, rutherfordine and γ − UO 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 − UO 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 − UO were used to determine their enthalpies and free energies of formation, and their variation with temperature. The theoretical results for rutherfordine and γ − UO are shown to be in excellent agreement with experimental information even at high temperatures (up to 700 K 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 UO · 0.77 H O and UO · 0.85 H O to UO · 1 H O, 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 is 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 modelling uranium containing systems and its dynamical behaviour under different geochemical environments.


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I. INTRODUCTION In nuclear sciences, the fundamental thermodynamic data are the indispensable basis for a dynamic modelling 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 waste,7 and may be also of importance in analyzing reactions within breeder reactors. The enormous relevance of the thermodynamic information of 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. While the thermodynamic information database of uranium bearing species is very advanced, there are many systems for which the corresponding data is inaccurate due to large experimental



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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 modelling 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 systems26-36. 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 modelling 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


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difficulties associated to 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 (UO (OH) ), metastudtite ((UO ) O · 2 H O), studtite ((UO ) O · 4 H O), soddyite ((UO ) (SiO ) · 2H O), rutherfordine (UO CO ) and gamma uranium trioxide (γ − UO ) has been studied. These materials, containing UO , 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 − UO , 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 used extensively 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 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 − UO 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.



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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 norm-conserving relativistic pseudopotential for uranium atom, generated in a previous work,41,61 was used in these computations. This pseudopotential has been employed extensively in the research of uranyl containing materials.3941,53-54,61

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 cut-off 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 cut-off 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.


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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, (𝐻 − 𝐻

and 𝑆

)

, by

means of the expressions:68 ∆ 𝐻(𝑇 ) = ∆ 𝐻 + (𝐻 − 𝐻

)

−

𝑛 (𝐻 − 𝐻

∆ 𝐺 (𝑇) = ∆ 𝐻 (𝑇) − 𝑇 𝑆

−

𝑛 (𝑆 )

)

(1)

(2)

In these equations, ∆ 𝐻 is the standard enthalpy of formation of the material (at the temperature of 298.15 K and a pressure of 1 bar), and (𝐻 − 𝐻

)

and (𝑆 )

are the enthalpy and entropy

functions of the elements forming part of the material with stoichiometric coefficients 𝑛 , respectively. The specific experimental values used for ∆ 𝐻

are indicated in each case. The



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enthalpy and entropy functions for the elements H, O, C and Si were taken form 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 ∆ 𝐺 (𝑇) = − 𝑅 𝑇 𝐿𝑛 𝐾.

III. RESULTS III.1. Crystal structures III.1.a. Dehydrated schoepite, 𝑼𝑶𝟐 (𝑶𝑯)𝟐 The structure of dehydrated schoepite was determined in calculations with increasing complexity (see Section 2.1). 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

Table 1. Dehydrated schoepite and paulscherrerite (last row) lattice parameters. b (Å) c (Å) α (deg.) β (deg.) γ (deg.) Vol. (Å3) Dens. (g/cm3) Parameters a (Å) This work 4.2309 10.3104 6.9195 90.0 90.0 90.0 301.87 6.690 DFT37 4.222 10.304 6.938 90.0 90.0 90.0 301.83 6.69 Exp.70-71 4.242 10.302 6.868 90.0 90.0 90.0 300.14 6.73 72 Exp. 4.2455 10.3183 6.8648 90.0 90.0 90.0 300.72 6.72 Exp.73 4.288 10.270 6.885 90.0 90.39 90.0 303.20 6.66

The computed structure is shown in Figure 1. In Figure 1.A a general view of the unit cell is given. Figure 1.B provides a view of a structural sheet from [010] direction. The hydrogen bond


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structure in dehydrated schoepite is shown in Figure 2. Table S.1 and Table 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.

Figure 1. Structure of dehydrated schoepite, UO (OH) : 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, ORed, H-Yellow.



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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, UO (OH) , 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 which 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 towards the lower layer and the other three are donors of hydrogen bonds directed towards the upper layer. Table S.1 and Table 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 structurally (symmetrically) identical U atom and two O atoms (the uranyl, Oy, and hydroxyl, O, oxygen atoms). The uranyl ion is linear, 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 CuKα radiation (λ=1.540598 Å). The most intense lines (I > 10%)


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are compared in Figure 3. The values of the main reflections are given in Table S.3 of the

[111]

Supporting Information.

80

Calculated Experimental

[020]

100

[222]

[151]

[133]

[151]

100

20 20

25

30

35

40

[220][113]

[200] [042]

40

[040] [131]

[002]

[022]

60

0 15

[133] [222]

[113]

[111]

0

[220]

[200] [042]

[040]

20

80

[131]

[022]

[002]

40

[020]

-1

60

Intensity / counts sec

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


45

50

55

2

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

III.1.b. Metastudtite, (𝑼𝑶𝟐 ) 𝑶𝟐 · 𝟐 𝑯𝟐 𝑶 Table 2 gives the final lattice parameters, volumes and densities obtained for metastudtite compared with the experimental ones,75 and those obtained in previous DFT calculations.76 Table 2. Metastudtite lattice parameters. b (Å) c (Å) α (deg.) β (deg.) γ (deg.) Vol. (Å3) Dens. (g/cm3) Parameters a (Å) This work 8.3977 8.5688 6.8240 90.0 90.0 90.0 491.05 4.572 DFT76 8.45 8.72 6.75 90.0 90.0 90.0 497.37 4.514 Exp.75 8.42 8.78 6.51 90.0 90.0 90.0 481.3 4.67

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 Figures 4.A and 4.B, respectively. The hydrogen bond structure is shown in Figure 5. Table S.4 and Table S.5 of the Supporting Information give a



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comparison of the more important geometric parameters (bond distances and angles) obtained with those from the previous DFT calculations of Weck et al.76.

Figure 4. Structure of metastudtite, (UO ) O · 2 H O: 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.

Figure 5. Detailed view of the hydrogen bond structure of metastudtite. Color code: U-Blue, O-Red, HYellow.

As in studtite,76 the uranium atom in metastudtite displays hexagonal bipyramid coordination (see Figure 4). The axial oxygen atoms comprise the uranyl UO

group with two distinct short


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axial UO bonds which are about 1.80 Å and 1.85 Å long and a nearly linear angle (in studtite the UO bond lengths were equal). The bipyramids are tilted towards 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 improves the calculated structure significantly.76 The hydrogen bond 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. The hydrogen bonds with the peroxide oxygen atom of two paired water molecules are with a different oxygen atom of 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. Table S.4 and Table 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 CuKα 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

[011]

the Supporting Information.

Calculated Experimental

100

[232]

[203]

[130] [040]

[230]

[212]

20 0

[221] [022] [031]

[002]

[210]

[020]

40

[011]

15

20

25

30

35

40

45

[241]

[232] [203]

[040]

[130]

20 0

[221] [031] [022]

[002]

40

[210]

[020]

60

[212] [230]

80 [201]

-1

60

[201]

80

[241]

100

Intensity / counts sec

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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50

55

2

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

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. Figures 7.A, 7.B, 7.C and 7.D show 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 Table S.7 to Table 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


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Kim37 is given in Table 3. The precise values of the calculated entropy, enthalpy and free energy functions at selected temperatures are given in Table 4.

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 UO · 0.77 H O and UO · 0.85 H O to UO · 1 H O (see Hemingway1).



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Table 3. Comparison of calculated isobaric heat capacity functions of dehydrated schoepite. The experimental data is from Hemingway.1 All values are given in units of J·K-1·mol-1. T(K) 298.15 400 500 600 700

Cpexp Cpcalc (this work) Cpcalc (Weck and Kim37) 109.0 103.85 107.21 126.3 118.21 122.31 139.3 127.42 131.92 148.3 133.65 137.81 153.9 138.15 140.75

Table 4. Calculated entropy, enthalpy and free-energy functions of dehydrated schoepite. All values are given in units of J·K-1·mol-1.

T(K) 298.15 400 500 600 700

Scalc (HT-H298)calc (GT-H298)calc 125.18 0.00 -125.18 159.36 29.74 -129.61 188.06 49.50 -138.56 212.97 64.07 -148.90 234.89 75.24 -159.65

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 8.A to 8.D and Figures 9.A to 9.D, respectively. The corresponding values of these functions over the temperature range 0-1000 K are given in Table S.11 to Table S.14 and Table S.15 to Table 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.


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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.

Table 5. Comparison of calculated isobaric heat capacity functions of metastudtite and studtite. 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. T(K) 298.15 400 500 600 700 800

Cpcalc

Metastudtite Studtite calc calc (this work) Cp (reference) Cp (this work) Cpcalc (reference) 163.14 155.81 219.97 211.17 185.00 178.05 255.40 246.96 199.18 192.60 279.99 272.41 209.36 203.21 298.44 291.74 217.24 211.40 313.18 307.07 223.69 217.99 325.51 319.74



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Table 6. Calculated entropy, enthalpy and free energy functions of metastudtite and studtite. All values are given in units of J·K-1·mol-1. T(K) 298.15 400 500 600 700 800

Metastudtite Scalc (HT-H298)exp (GT-H298)exp 179.27 0.00 -179.27 230.50 44.56 -185.94 273.41 74.12 -199.29 310.67 95.86 -214.80 343.55 112.67 -230.88 373.00 126.16 -246.84

Studtite Scalc (HT-H298)calc (GT-H298)exp 232.12 0.00 -232.12 302.03 60.83 -241.20 361.81 102.29 -259.52 414.56 133.52 -281.04 461.71 158.19 -303.52 504.36 178.36 -326.00

Figure 9. Calculated thermodynamic properties of studtite as a function of temperature.: A) Isobaric specific heat; B) Entropy; C) Enthalpy; D) Gibbs free energy.


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III.2.c. Soddyite The calculated isobaric heat capacity, entropy, enthalpy and free energy functions of soddyite are shown in Figures 10.A, 10.B, 10.C and 10.D, respectively. The values of the calculated thermodynamic functions over the temperature range 0-1000 K are given in Table S.19 to Table 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.

Figure 10. Calculated thermodynamic properties of soddyite as a function of temperature: A) Isobaric specific heat; B) Entropy; C) Enthalpy; D) Gibbs free energy.



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Table 7. Calculated isobaric heat capacity, entropy, enthalpy and free energy functions of soddyite. All values are given in units of J·K-1·mol-1. T(K) 298.15 400 500 600 700 800 900 1000

Cpcalc 275.15 309.93 331.75 346.60 357.39 365.71 372.45 378.09

Scalc (HT-H298)calc (GT-H298)calc 315.95 0.00 -315.95 402.07 74.87 -327.20 473.74 124.15 -349.59 535.61 160.11 -375.50 589.90 187.54 -402.35 638.18 209.30 -428.88 681.66 227.01 -454.65 721.20 241.90 -482.83

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 Equation (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 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 − UO reported by Hemingway1 in the range 298.15 to 700 K and by Cordfunke and Westrum79 from 298.15 to 900 K, respectively. Table 8. Experimental values the standard enthalpies of formation of rutherfordine, gamma uranium trioxide, dehydrated schoepite, metastudtite, studtite and soddyite. Material UO CO γ − UO UO (OH) (UO ) O · 2 H O (UO ) O · 4 H O (UO ) (SiO ) · 2H O

∆𝒇 𝑯𝟎 (𝐤𝐉 · 𝐦𝐨𝐥 𝟏 ) Reference −1704.1 ± 2 Hemingway1 −1223.8 ± 1.2 Guillaumont et al.13 −1534.5 ± 4.0 Hemingway1 −1779.6 ± 1.9 Guo et al.17 −2344.7 ± 4.0 Kubatko et al.14 −4045.4 ± 4.9 Gorman-Lewis et al.21


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Table 9. Calculated enthalpies (∆ 𝐻) and free-energies (∆ 𝐺) of formation and reaction constants (Log K) of rutherfordine and gamma uranium trioxide as a function of temperature. 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 ∆ 𝐻 and ∆ 𝐺 are in units of kJ·mol-1. T(K)

(∆𝒇 𝑯)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

(∆𝒇 𝑮)𝒄𝒂𝒍𝒄 (∆𝒇 𝑮)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

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

III.3.b. Dehydrated schoepite The thermodynamic properties of dehydrated schoepite were reported in Section 3.2.1 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).



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A comparison of the results with the empirical values reported by Hemingway1 from 298.15 to 700 K is shown in Table 10.

Table 10. Calculated enthalpies (∆ 𝐻) and free-energies (∆ 𝐺) of formation and associated reaction constants (Log K) of dehydrated schoepite as a function of temperature. The values of ∆ 𝐻 and ∆ 𝐺 are in units of kJ·mol-1. (∆𝒇 𝑯)𝒄𝒂𝒍𝒄 T(K) 5 -5318.41 55.2525 -1847.94 105.5051 -1673.87 206.0101 -1570.85 298.15 -1534.50 360 -1522.12 400 -1516.68 500 -1509.05 600 -1507.07 700 -1508.98 800 -1513.68

(∆𝒇 𝑮)𝒄𝒂𝒍𝒄 -5317.92 -1826.85 -1624.51 -1473.99 -1395.56 -1354.15 -1330.16 -1276.73 -1229.48 -1186.36 -1146.07

(∆𝒇 𝑮)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

III.3.c. Metastudtite, studtite and soddyite From the thermodynamic properties of metastudtite, studtite and soddyite reported in Section 3.2 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 which are provided in Table 11.


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Table 11. Calculated enthalpies (∆ 𝐻) and free-energies (∆ 𝐺) of formation and reaction constants (Log K) of metastudtite, studtite and soddyite as a function of temperature. T(K) 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 5 55.2525 105.5051 206.0101 298.15 360 400 500 600 700 800 900

Log K ∆𝒇 𝑯 (kJ·mol-1) ∆𝒇 𝑮 (kJ·mol-1) Metastudtite -7332.09 -7331.46 76589.26 -2237.47 -2204.89 2084.41 -1983.38 -1906.70 943.97 -1832.83 -1678.59 425.60 -1779.60 -1556.66 272.71 -1761.10 -1491.58 216.42 -1752.95 -1453.66 189.82 -1741.50 -1368.64 142.98 -1738.43 -1292.94 112.56 -1740.89 -1223.39 91.29 -1747.30 -1158.17 75.62 Studtite -9636.46 -9635.63 100660.19 -2939.58 -2891.23 2733.24 -2607.07 -2491.61 1233.54 -2412.18 -2175.54 551.60 -2344.70 -1998.43 350.11 -2321.09 -1900.87 275.80 -2310.72 -1843.12 240.68 -2296.28 -1711.54 178.80 -2292.73 -1592.42 138.63 -2296.47 -1481.81 110.57 -2305.45 -1377.39 89.93 Soddyite -13631.08 -13630.00 142388.07 -4840.12 -4790.81 4529.02 -4398.86 -4284.45 2121.14 -4137.73 -3910.95 991.61 -4045.40 -3720.31 651.77 -4013.52 -3621.19 525.41 -3999.57 -3564.31 465.44 -3979.87 -3438.65 359.22 -3974.61 -3328.70 289.78 -3979.05 -3229.12 240.95 -3990.36 -3136.61 204.79 -4006.97 -3049.27 176.97



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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 can be found 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 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 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 -UO3 were determined in previous studies.39,40


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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 UO · 0.77 H O and UO · 0.85 H O to UO · 1 H O. The heat capacity function is compared with the empirical function of Hemingway1 and the theoretical function of Weck and Kim37 in Fig. 7.A 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 experimental work to estimate the thermodynamic properties by extrapolating the measurements from different water 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 7.B to Figure 7.D. The calculated functions for the entropy and enthalpy are compared with the empirical functions of Hemingway1 in Figure 7.B and Figure 7.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.



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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. 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 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 obtain Cp=219.97 J·K-1·mol-1, which is about 4.2% higher than the value from Sassani al.7 of 211.17 J·K-1·mol-1. As it can be appreciated in Figures 8.A and 9.A, 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·K1·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 4.3.3). 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.


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IV.2.d. Recommended thermodynamic data The calculated thermodynamic properties of γ − UO 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 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 were 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 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 requires much larger computational resources.



Page 28 of 44

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 compared with the experimental values of Hemingway1 in Table 9. The results are also displayed in Figure 11.A where it is shown that the results are in excellent agreement with the experimental data. For comparison, the experimental value of the standard free-energy of formation1 is ∆ 𝐺 = −1577.0 ± 4.2 kJ · mol , 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 11.B. The results are again in excellent agreement with the experimental data. For comparison, the experimental value of the standard free-energy of formation12 is ∆ 𝐺 = −1145.739 ± 1.207 kJ · mol , 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 UO (OH) at different temperatures are presented in Table 10 and shown graphically in Figure 11.C. For comparison, the experimental value of the standard free-energy of formation1 is ∆ 𝐺 = −1404.9 ± 4.0 kJ ·


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mol , 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 UO · 0.77 H O and UO · 0.85 H O to UO · 1 H O.

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



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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 11.D, 11.E and 11.F. For comparison, the experimental value of the standard free-energy of formation of soddyite21 is ∆ 𝐺 = −3652.2 ± 4.2 kJ · mol , 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 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 modelling uranium containing systems and its dynamical behavior under different geochemical environments.

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 modelling 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


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schoepite, metastudtite, studtite, soddyite, rutherfordine and γ − UO 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 − UO , are among the main secondary phases arising from corrosion of spent nuclear fuel under final geological disposal conditions.43-48,14,17 γ − UO is the main oxide of hexavalent uranium which is common through 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 soddyite53-54. 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 − UO

36-37

were used to determine their enthalpies and free

energies of formation as a function of temperature. The theoretical results obtained for rutherfordine and γ − UO 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 UO · 0.77 H O and UO · 0.85 H O to UO · 1 H O, 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 is 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



Page 32 of 44

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.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Phone number: +34 915616800 Ext. 941033 ORCID. Francisco Colmenero: 0000-0003-3418-0735

ACKNOWLEDGEMENT 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 CETA-CIEMAT, 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 to Dr. Rafael Escribano for reading the document and many helpful comments.


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SUPPORTING INFORMATION The Supplementary Information associated with this article contains the following data: 

Table S.1 and Table 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.



Table S.4 and Table 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.

REFERENCES (1) Hemingway, B. S. Thermodynamic Properties of Selected Uranium Compounds and Aqueous Species at 298.15 K and 1 Bar and at Higher Temperatures - Preliminary Models for the Origin of Coffinite Deposits; USGS Open-File Report 82–619: 1982. (2) Langmuir, D. Aqueous Environmental Geochemistry; Prentice-Hall: New Jersey, 1997; pp 486–557. (3) Casas, I.; Bruno, J.; Cera, E.; Finch, R. J.; Ewing, R. C. Kinetic and Thermodynamic Studies of Uranium Minerals Assessment of the Long-Term Evolution of Spent Nuclear Fuel; SKB Technical Report 94-16; Swedish Nuclear Fuel and Waste Management Co.: Stockholm, Sweden, 1994.



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(4) Navrotsky, A.; Shvareva, T. Y.; Guo, X.; Rock, P. A. Thermodynamics of Uranium Minerals and Related Materials; Mineralogical Association of Canada Short Course 43: Winnipeg, MB, May 2013. (5) Chevalier, P.-Y.; Fischer, E.; Cheynet, B. Progress in the Thermodynamic Modelling of the O–U Binary System. J. Nucl. Mat. 2002, 303, 1–28. (6) Konings, R. J. M.; Benes, O.; Kovacs, A.; Manara, D.; Gorokhov, D. S.; Iorish, V. S.; Yungman, V.; Shenyavskaya, E.; Osina, E. The Thermodynamic Properties of the f-Elements and their Compounds. Part 2. The Lanthanide and Actinide Oxides. J. Phys. Chem. Ref. Data 2014, 43, 013101. (7) Sassani, D. C.; Jové-Colón, C. F.; Weck, P. F.; Jerden, J. L.; Frey, K. E.; Cruse, T.; Ebert, W. L.; Buck, E. C.; Wittman, R. S. Used Fuel Degradation: Experimental and Modeling Report, Fuel Cycle Research and Development Report FCRD-UFD-2013-000404; Sandia National Laboratories: Albuquerque, New Mexico, October 17, 2013. (8) Wanner, H.; Forest, I., Eds. Chemical Thermodynamics of Uranium; Elsevier Science Publishers B.V.: Amsterdam, North-Holland, 1992. (9) Murphy, W. M.; Pabalan, R. T. Review of Empirical Thermodynamic Data for Uranyl Silicate Minerals and Experimental Plan; Nuclear Regulatory Commission Contract NRC-02-93005; Center for Nuclear Waste Regulatory Analyses: San Antonio, Texas, June 1995. (10) Grenthe, I; Fuger, J.; Konings, R. J. M.; Lemire, R. J.; Muller, A. B.; Nguyen-Trung, C.; Wanner, H. Chemical Thermodynamics of Uranium; Nuclear Energy Agency Organisation for Economic Co-Operation and Development, OECD: Issy-les-Moulineaux, France, 2004.


Page 35 of 44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(11) Shvareva, T. Y.; Fein, J. B.; Navrotsky, A. Thermodynamic Properties of Uranyl Minerals: Constraints from Calorimetry and Solubility Measurements. Ind. Eng. Chem. Res. 2012, 51, 607– 613. (12) Gorman-Lewis, D.; Burns, P. C.; Fein, J. B. Review of Uranyl Mineral Solubility Measurements. J. Chem. Thermodyn. 2008, 40, 335–352. (13) Guillaumont, N. Y. R.; Fanghänel, T.; Neck, V.; Fuger, J.; Palmer, D. A.; Grenthe, I.; Rand, M. H. Update on the Chemical Thermodynamics of Uranium, Neptunium, Plutonium, Americium, and Technetium; Mompean, F. J., Illemassene, M., Domenech-Orti, C., Ben Said, K., Eds.; OECD Nuclear Energy Agency, Data Bank: Issy-les-Moulineaux, France, 2003. (14) Kubatko, K.-A.; Helean, K. B.; Navrotsky, A.; Burns, P. C. Stability of Peroxide-Containing Uranyl Minerals. Science 2003, 302, 1191–1193. (15) Kubatko, K.-A. Crystallography, Hyerarchy of Crystal Structures, and Chemical Thermodynamics of Selected Uranyl Phases. Ph.D. Thesis, Graduate School of the University of Notre Dame, Illinois, 2005. (16) Armstrong, C. R.; Nyman, M.; Shvareva, T. Y.; Sigmon, G. E.; Burns, P. C.; Navrotsky, A. Uranyl Peroxide Enhanced Nuclear Fuel Corrosion in Seawater. Proc. Natl. Acad. Sci. U.S.A. 2012, 109, 1874–1877. (17) Guo, X.; Ushakova S. V.; Labs, S.; Curtius, H.; Bosbach, D.; Navrotsky, A. Energetics of Metastudtite and Implications for Nuclear Waste Alteration. Proc. Natl. Acad. Sci. U.S.A. 2014 111, 17737–17742.



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(18) Kubatko, K.-A.; Helean, K. B.; Navrotsky, A.; Burns, P. C. Thermodynamics of Uranyl Minerals:

Enthalpies

of

Formation

of

Rutherfordine,

UO2CO3,

Andersonite,

Na2CaUO2(CO3)3(H2O)5, and Grimselite, K3NaUO2(CO3)·3H2O. Am. Mineral. 2005, 90, 1284– 1290. (19) Gorman-Lewis, D.; Shvareva, T. Y.; Kubatko, K.-A.; Burns, P. C.; Wellman, D. M.; McNamara, B.; Szymanowsky, J. E. S; Navrotsky, A.; Fein, J. B. Thermodynamic Properties of Autunite, Uranyl Hydrogen Phosphate, and Uranyl Orthophosphate from Solubility and Calorimetric Measurements. Environ. Sci. Technol. 2009, 43, 7416–7422. (20) Nguyen, S. N.; Silva, R. J.; Weed, H. C.; Andrews, L. E. Standard Gibbs Free Energies of Formation at the Temperature 303.15 K of Four Uranyl Silicates: Soddyite, Uranophane, Sodium Boltwoodite, and Sodium Weeksite. J. Chem. Thermodyn. 1992, 24, 259–276. (21) Gorman-Lewis, D.; Mazeina, L.; Fein, J. B.; Szymanowski, J. E. S.; Burns, P. C.; Navrotsky, A. Thermodynamic Properties of Soddyite from Solubility and Calorimetry Measurements. J. Chem. Thermodyn. 2007, 39, 568–575. (22) Prikryl, J. D. Uranophane Dissolution and Growth in CaCl2–SiO2(aq) Test Solutions. Geochim. Cosmochim. Acta 2008, 72, 4508–4520. (23) Shvareva, T. Y.; Mazeina, L.; Gorman-Lewis, D.; Burns, P. C.; Szymanowski, J. E. S.; Fein, J. B.; Navrotsky, A. Thermodynamic Characterization of Boltwoodite and Uranophane: Enthalpy of Formation and Aqueous Solubility. Geochim. Cosmochim. Acta 2011, 75, 5269–5282. (24) Guo, X.; Szenknect, S.; Mesbah, A.; Labs, S.; Clavier, N.; Poinssote; C.; Ushakova, S. V.; Curtius, H.; Bosbach, D.; Ewing, R. C.; Burns, P. C.; Dacheux, N.; Navrotsky, A.


Page 37 of 44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Thermodynamics of Formation of Coffinite, USiO4. Proc. Natl. Acad. Sci. U.S.A. 2014, 112, 6551– 6555. (25) Szenknect, S.; Mesbah, A.; Cordara, T.; Clavier, N.; Brau, H.-P.; Le Goffa, X.; Poinssot, C.; Ewing, R. C.; Dacheux, N. First Experimental Determination of the Solubility Constant of Coffinite. Geochim. Cosmochim. Acta 2016, 81, 36–53. (26) Gueneau, C.; Baichi, M.; Labroche, D.; Chatillon, C.; Sundman, B. Thermodynamic Assessment of the Uranium–Oxygen System. J. Nucl. Mater. 2002, 304, 161–175. (27) Loukusa, H.; Ikonen, T.; Valtavirta, V.; Tulkki, V. Thermochemical Modeling of Nuclear Fuel and the Effects of Oxygen Potential Buffers. J. Nucl. Mater. 2016, 481, 101–110. (28) Piro, M. H. A.; Besmann, T. M.; Simunovic, S.; Lewis, B. J.; Thompson, W. T. Numerical Verification of Equilibrium Thermodynamic Computations in Nuclear Fuel Performance Codes. J. Nucl. Mater. 2011, 414, 399–407. (29) Piro, M. H. A.; Banfield, J.; Clarno, K. T.; Thompson, W. T Coupled Thermochemical, Isotopic Evolution and Heat Transfer Simulations in Highly Irradiated UO2 Nuclear Fuel. J. Nucl. Mater. 2013, 441, 240–251. (30) Piro, M. H. A.; Banfield, J.; Clarno, K. T.; Simunovic, S.; Besmann, T. M. Simulation of Thermochemistry and Isotopic Evolution of Irradiated Nuclear Fuel. Trans. Am. Nucl. Soc. 2013, 108, 367–370. (31) Piro, M. H. A.; Leitch, B. W. Conjugate Heat Transfer Simulations of Advanced Research Reactor Fuel. Nucl. Eng. Des. 2014, 274, 30–43.



Page 38 of 44

(32) Corcoran, E. C.; Lewis, B. J.; Thompson, W. T.; Mouris, J.; He, Z. Controlled Oxidation Experiments of Simulated Irradiated UO2 Fuel in Relation to Thermochemical Modelling. J. Nucl. Mater. 2011, 414, 73–82. (33) Corcoran, E. C.; Kaye, M. H.; Piro, M. H. A. An Overview of Thermochemical Modelling of CANDU Fuel and Applications to the Nuclear Industry. Calphad 2016, 55, 52–62. (34) Besmann, T. M.; Bartel, T. J.; Bertolus, M.; Blanc, V.; Bouineau, V.; Carlot, G.; et al. Stateof-the-Art

Report

on

Multi-Scale

Modelling

of

Nuclear

Fuels.

Nuclear

Science

NEA/NSC/R/(2015)5. Nuclear Energy Agency - Energy Agency Organisation for Economic CoOperation and Development: 2015. (35) Besmann, T. M.; McMurray, J. W.; Simunovic, S. Application of Thermochemical Modeling to Assessment/Evaluation of Nuclear Fuel Behavior. Calphad 2016, 55, 47–51. (36) Smith, A. L.; Gueneau, C.; Fleche, J.-L.; Chatain, S.; Benes, O.; Konings, R. J. M. Thermodynamic Assessment of the Na-O and Na-U-O Systems: Margin to the Safe Operation of SFRs. J. Chem. Thermodyn. 2017, 114, 93–115. (37) Weck, P. F.; Kim, E. Layered Uranium(VI) Hydroxides: Structural and Thermodynamic Properties of Dehydrated Schoepite α-UO2(OH)2. Dalton Trans. 2014, 43, 17191–17199. (38) Weck, P. F.; Kim, E. Uncloaking the Thermodynamics of the Studtite to Metastudtite ShearInduced Transformation. J. Phys. Chem. C 2016, 120, 16553–16560. (39) Colmenero, F.; Bonales, L. J.; Cobos, J., Timón, V. Thermodynamic and Mechanical Properties of Rutherfordine Mineral Based on Density Functional Theory. J. Phys. Chem. C 2017, 121, 5994–6001.


Page 39 of 44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(40) Colmenero, F.; Bonales, L. J.; Cobos, J.; Timón, V. Density Functional Theory Study of the Thermodynamic and Raman Vibrational Properties of γ − UO Polymorph. J. Phys. Chem. C 2017, 121, 14507−14516. (41) Colmenero, F. Characterization of Secondary Phases of Spent Nuclear Fuel under Final Geological Disposal Conditions. Ph. D. Thesis, Universidad Autónoma de Madrid, Madrid, 2017, 443 pp. (42) Beridze, G.; Kowalski, P. M. Benchmarking the DFT+U Method for Thermochemical Calculations of Uranium Molecular Compounds and Solids. J. Phys. Chem. A 2014, 118, 11797– 11810. (43) Frondel, C. Mineral Composition of Gummite. Am. Mineral. 1956, 41, 539–568 (44) Frondel, C. Systematic Mineralogy of Uranium and Thorium. U.S. Geol. Surv. Bull. 1958, 1064. (45) Finch, R. J.; Ewing, R. C. The Corrosion of Uraninite under Oxidizing Conditions. J. Nucl. Mater. 1992, 190, 133–156. (46) Pearcy, E. C.; Prikryl, J. D.; Murphy, W. M.; Leslie, B. W. Alteration of Uraninite from the Nopal I Deposit, Peña Blanca District, Chihuahua, Mexico, Compared to Degradation of Spent Nuclear Fuel in the Proposed US High-Level Nuclear Waste Repository at Yucca Mountain, Nevada. Appl. Geochem. 1994, 9, 713–732. (47) Wronkiewicz, D. J.; Bates, J. K.; Gerding, T. J.; Veleckis, E.; Tani, B. S. Uranium Release and Secondary Phase Formation During Unsaturated Testing of UO2 at 90°C. J. Nucl. Mater. 1992, 190, 107–127.



Page 40 of 44

(48) Wronkiewicz, D. J.; Bates, J. K.; Wolf, S. F.; Buck, E. C. Ten-Year Results from Unsaturated Drip Tests with UO2 at 90°C: Implications for the Corrosion of Spent Nuclear Fuel. J. Nucl. Mater. 1996, 238, 78–95. (49) Benedict, M.; Pigford, T. H.; Levi, H. W. Nuclear Chemical Engineering; McGraw-Hill: New York, 1981. (50)

Schulz,

W.

W.

Uranium

Processing;

in

Encyclopedia

Britannica,

https://

global.britannica.com/technology/uranium-processing (accessed November 30, 2017). (51) Colmenero, F.; Fernández, A. M.; Cobos, J.; Timón, V. Temperature Dependent Free Energies of Reaction of Uranyl Containing Materials Based on Density Functional Theory. J. Phys. Chem. C, submitted. (52) Payne, M. C.; Teter, M. P.; Ailan, D. C.; Arias, A.; Joannopoulos, J. D. Iterative Minimization Techniques for Ab Initio Total-Energy Calculations: Molecular Dynamics and Conjugate Gradients. Rev. Mod. Phys. 1992, 64, 1045–1097. (53) Colmenero, F.; Bonales, L. J.; Cobos, J.; Timón, V. Study of the Thermal Stability of Studtite by In Situ Raman Spectroscopy and DFT Calculations. Spectrochim. Acta. A 2017, 174, 245–253. (54) Colmenero, F.; Bonales, L. J.; Cobos, J.; Timón, V. Structural, Mechanical and Vibrational Study of Uranyl Silicate Mineral Soddyite by DFT Calculations. J. Solid. State Chem. 2017, 253, 249−257. (55) Clark, S. J.; Segall, M. D.; Pickard, C. J.; Hasnip P. J.; Probert, M. I. J.; Refson, K.; Payne, M. C. First Principles Methods Using CASTEP. Z. Kristallogr. 2005, 220, 567–570.


Page 41 of 44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(56) Milman, V.; Refson, K.; Clark, S. J.; Pickard, C. J.; Yates, J. R.; Gao, S. P.; Hasnip, P. J.; Probert, M. I. J.; Perlov, A.; Segall, M. D. Electron and Vibrational Spectroscopies Using DFT, Plane Waves and Pseudopotentials: CASTEP Implementation. J. Mol. Struct. Theochem 2010, 954, 22–35. (57)

MaterialsStudio,

http://accelrys.com/products/collaborativescience/biovia-materials-

studio/, accessed November 30, 2017. (58) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. (59) Grimme, S. Semiempirical GGA-type Density Functional Constructed with a Long-Range Dispersion Correction. J. Comput. Chem. 2006, 27, 1787–1799. (60) Troullier, N.; Martins, J. L. Efficient Pseudopotentials for Plane-Wave Calculations. Phys. Rev. B 1991, 43, 1993–2006. (61) Bonales, L. J.; Colmenero, F.; Cobos, J.; Timón, V. Spectroscopic Raman Characterization of Rutherfordine: a Combined DFT and Experimental Study. Phys. Chem. Chem. Phys. 2016, 18, 16575–16584. (62) Pfrommer, B. G.; Cote, M.; Louie, S. G.; Cohen, M. L. Relaxation of Crystals with the Quasi-Newton Method. J. Comput. Phys. 1997, 131, 233–240. (63) Monkhorst, H. J.; Pack, J. D. Special Points for Brillonin-zone Integration. Phys. Rev. B 1976, 13, 5188–5192. (64) Baroni, S.; de Gironcoli, S.; Dal Corso, A. Phonons and Related Crystal Properties from Density-Functional Perturbation Theory. Rev. Mod. Phys. 2001, 73, 515–562.



Page 42 of 44

(65) Gonze, X.; Lee, C. Dynamical Matrices, Born Effective Charges, Dielectric Permittivity Tensors, and Interatomic Force Constants from Density-Functional Perturbation Theory. Phys. Rev. B 1997, 55, 10355–10368. (66) Refson, K.; Tulip, P. R.; Clark, S. J. Variational Density-Functional Perturbation Pheory for Dielectrics and Lattice Dynamics. Phys. Rev. B 2006, 73, 155114. (67) Lee, C.; Gonze, X. Ab Initio Calculation of the Thermodynamic Properties and Atomic Temperature Factors of SiO2 α-Quartz and Stishovite. Phys. Rev. B 1995, 51, 8610–8613. (68) Chase, M. W.; Davies, C. A.; Downey, J. R.; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. JANAF Thermochemical Tables. Third Edition. J. Phys. Chem. Ref. Data 1985, 14, Suppl. 1, 1–1856. (69) Barin, I. Thermochemical Data of Pure Substances; Third edition; VCH: Weinheim, 1995. (70) Taylor, J. C. The Structure of the α Form of Uranyl Hydroxide. Acta Crystallogr. B 1971, 27, 1088–1091. (71) Taylor, J. C.; Hurst, H. J. The Hydrogen-Atom Locations in the α and β Forms of Uranyl Hydroxide. Acta Crystallogr. B 1971, 27, 2018–2022. (72) Taylor, J. C.; Kelly, J. W.; Downer, B. A Study of the β→α Phase Transition in UO (OH) by Dilatometric, Microcalorimetric and X-ray Diffraction Techniques. J. Solid State Chem. 1972, 5, 291–299. (73) Brugger, J.; Meisser, N.; Etschmann, B.; Ansermet, S.; Pring, A. Paulscherrerite from the Number 2 Workings, Mt. Painter Inlier, Northern Flinders Ranges, South Australia: “Dehydrated schoepite” Is a Mineral After All. Am. Mineral. 2011, 96, 229–240.


Page 43 of 44 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(74) Downs, R. T.; Bartelmehs, K. L.; Gibbs, G. V.; Boisen, M. B. Interactive Software for Calculating and Displaying X-Ray or Neutron Powder Diffractometer Patterns of Crystalline Materials. Am. Mineral. 1993, 78, 1104–1107. (75) Deliens, M.; Piret, P. Metastudtite, UO · 4 H O, a New Mineral from Shinkolobwe, Shaba, Zaire. Am. Mineral. 1963, 68, 456–458. (76) Weck, P. F.; Kim, E.; Jové-Colón, C. F.; Sassani, D. C. Structures of Uranyl Peroxide Hydrates: A First-Principles Study of Studtite and Metastudtite. Dalton Trans. 2012, 41, 9748– 9752. (77) Debets, P. C. X-ray Diffraction Data on Hydrated Uranium Peroxide. J. Inorg. Nucl. Chem. 1963, 25, 727–730. (78) Sattonnay, G.; Ardois, C.; Corbel, C.; Lucchini, J. F.; Barthe, M.-F.; Garrido, F.; Gosset, D. Alpha-Radiolysis Effects on UO Alteration in Water. J. Nucl. Mater. 2001, 288, 11–19. (79) Cordfunke, E. H. P.; Westrum, E. F. The Thermodynamic Properties of β-UO3 and γ-UO3. Thermochim. Acta 1988, 124, 285–296. (80) Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953–17979; Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 1999, 59, 1758–1775.



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TABLE OF CONTENTS GRAPHIC


Thermodynamic Properties of Uranyl-Containing Materials Based on

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