UC Berkeley UC Berkeley Electronic Theses and Dissertations

Title Energetics and Defect Interactions of Complex Oxides for Energy Applications

Permalink https://escholarship.org/uc/item/8q8463rn

Author Solomon, Jonathan Michael

Publication Date 2015

Peer reviewed|Thesis/dissertation

eScholarship.org Powered by the California Digital Library University of California Energetics and Defect Interactions of Complex Oxides for Energy Applications by Jonathan Michael Solomon

A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering - Materials Science and Engineering in the Graduate Division of the University of California, Berkeley

Committee in charge:

Professor Mark Asta, Chair Professor Lutgard de Jonghe Profesor Alexandra Navrotsky Professor Jeﬀrey Neaton

Summer 2015 Energetics and Defect Interactions of Complex Oxides for Energy Applications

Abstract Energetics and Defect Interactions of Complex Oxides for Energy Applications by Jonathan Michael Solomon Doctor of Philosophy in Engineering - Materials Science and Engineering University of California, Berkeley Professor Mark Asta, Chair

The goal of this dissertation is to employ computational methods to gain greater insights into the energetics and defect interactions of complex oxides that are relevant for today’s energy challenges. To achieve this goal, the development of novel computational methodologies are required to handle complex systems, including systems containing nearly 650 ions and systems with tens of thousands of possible atomic configurations. The systems that are investigated in this dissertation are aliovalently doped lanthanum orthophosphate (LaPO4) due to its potential application as a proton conducting electrolyte for intermediate- temperature fuel cells, and aliovalently doped uranium dioxide (UO2) due to its importance in nuclear fuel performance and disposal. First we undertake density-functional-theory (DFT) calculations on the relative energetics of pyrophosphate defects and protons in LaPO4, including their binding with divalent dopant cations. In particular, for supercell calculations with 1.85 mol% Sr doping, we investigate the dopant-binding energies for pyrophosphate defects to be 0.37 eV, which is comparable to the value of 0.34 eV calculated for proton-dopant binding energies in the same system. These results establish that dopant-defect interactions further stabilize proton incorporation, with the hydration enthalpies when the dopants are nearest and furthest from the protons and pyrophosphate defects being -1.66 eV and -1.37 eV, respectively. Even though our calculations show that dopant binding enhances the enthalpic favorability of proton incorporation, they also suggest that such binding is likely to substantially lower the kinetic rate of hydrolysis of pyrophosphate defects. We then shift our focus to solid solutions of fluorite-structured UO2 with trivalent rare earth fission product cations (M3+=Y, La) using a combination of ionic pair potential and DFT based methods. Calculated enthalpies of formation with respect to constituent oxides show higher energetic stability for La solid solutions than for Y. Additionally, calculations performed for different atomic configurations show a preference for reduced (increased) oxygen vacancy coordination around La (Y) dopants. The current results are shown to be qualitatively consistent with related calculations and calorimetric measurements of heats of formation in other trivalent doped fluorite oxides, which show a tendency for increasing stability and increasing preference for higher oxygen coordination with increasing size of the trivalent impurity. We expand this investigation by considering a series of trivalent rare earth fission product cations, specifically, Y3+ (1.02 A,˚ Shannon radius with eightfold coordination), Dy3+ (1.03 A),˚ Gd3+ (1.05 A),˚ Eu3+ (1.07 A),˚ Sm3+ (1.08 A),˚ Pm3+ (1.09 2

A),˚ Nd3+ (1.11 A),˚ Pr3+ (1.13 A),˚ Ce3+ (1.14 A)˚ and La3+ (1.16 A).˚ Compounds with ionic radius of the M3+ species smaller or larger than 1.09 A˚ are found to have energetically preferred defect ordering arrangements. Systems with preferred defect ordering arrangements are suggestive of defect clustering in short range ordered solid solutions, which is expected to limit oxygen ion mobility and therefore the rate of oxidation of spent nuclear fuel. 3+ Finally, the energetics of rare earth substituted (M = La, Y, and Nd) UO2 solid solutions are investigated by employing a combination of calorimetric measurements and DFT based computations. The calorimetric studies are performed by Lei Zhang and Pro- fessor Alexandra Navrotsky at the University of Calfornia, Davis, as part of a joint computational/experimental collaborative effort supported through the Materials Science of Actinides Energy Frontier Research Center. Calculated and measured formation enthalpies agree within 10 kJ/mol for stoichiometric oxygen/metal compositions. To better understand the factors governing the stability and defect binding in rare earth substituted urania solid solutions, systematic trends in the energetics are investigated based on the present results and previous computational and experimental thermochemical studies of rare earth substituted fluorite oxides. A consistent trend towards increased energetic stability with larger size mismatch between the smaller host tetravalent cation and the larger rare earth trivalent cation is found for both actinide and non-actinide fluorite oxide systems where aliovalent substitution of M cations is compensated by oxygen vacancies. However, the large exothermic oxidation enthalpy in the UO2 based systems favors compositions with higher oxygen-to-metal ratios where charge compensation occurs through the formation of uranium cations with higher oxidation states. i

To everyone who supported me. ii

Contents

List of Figures v

List of Tables viii

I Introduction and Background 1

1 Introduction 2 1.1 Fuel Cell Technology ...... 2 1.2 Nuclear Technology ...... 3 1.3 Outline ...... 5

2 Theoretical Framework 7 2.1 Atomistic Simulation ...... 7 2.1.1 The Shell Model ...... 8 2.2 First-principles Methods ...... 9 2.2.1 The Schr¨odingerEquation ...... 9 2.2.2 Density Functional Theory ...... 10

3 Addressing Self-Interaction Errors for Strongly Correlated Materials 12 3.1 Hubbard Model ...... 12 3.2 DFT Corrections for Strongly Correlated Materials ...... 12 3.2.1 Hybrid Functionals ...... 13 3.2.2 Hubbard-U Correction (DFT+U) ...... 13

II Results and Discussion 15

4 Energetics and Defect Interactions: Sr Doped LaPO4 16 4.1 Forward ...... 16 4.2 Introduction ...... 17 4.3 Computational Methods ...... 19 4.4 Results ...... 19 4.4.1 Pyrophosphate and Proton Conﬁgurations ...... 19 4.4.2 Superstructures with Defects Charge Compensated by Sr Dopants . 23 iii

4.4.3 Hydration Enthalpy Calculations for Structures with Neighboring and Isolated Dopants/Defects ...... 25 4.5 Summary and Discussion ...... 26 4.6 Acknowledgements ...... 28

5 Energetics and Defect Interactions: Y and La Doped UO2 29 5.1 Forward ...... 29 5.2 Introduction ...... 29 5.3 Computational Methodology ...... 30 5.3.1 Structure Enumeration ...... 31 5.3.2 Classical Pair Potentials ...... 32 5.3.3 First-principles Calculations ...... 32 5.4 Results ...... 34 5.5 Discussion ...... 36 5.6 Conclusions ...... 42 5.7 Acknowledgments ...... 42

6 Energetics and Defect Interactions: Rare-Earth Doped UO2 45 6.1 Forward ...... 45 6.2 Introduction ...... 45 6.3 Computational Methodology ...... 46 6.4 Results and Discussion ...... 48 6.5 Summary ...... 51 6.6 Acknowledgments ...... 52

7 Comparison to Experiment, Intermediate and Stoichiometric Composi- tions: Y, Nd and La Doped UO2 53 7.1 Forward ...... 53 7.2 Introduction ...... 54 7.3 Methods ...... 55 7.3.1 Experimental Procedures ...... 55 7.3.2 Computational Methods ...... 55 7.4 Results ...... 58 7.5 Discussion ...... 61 7.6 Conclusions ...... 67 7.7 Acknowledgments ...... 67

III Concluding Remarks 68

8 Conclusions and Future Work 69 8.0.1 Conclusions ...... 69 8.0.2 Future Work ...... 69

Bibliography 74 iv

A Appendix 87 A.1 Total energy comparison between ionic pair potentials and density functional theory ...... 87 v

List of Figures

1.1 Schematic of a proton conducting fuel cell. Reformatted from Ref. [52]. . . 3 1.2 The unit cell of LaPO4. Lanthanum, oxygen, and phosphorus ions are shown in blue, red, and yellow, respectively...... 4 1.3 The unit cell for UO2. Uranium and oxygen ions are shown in blue and red, respectively...... 5 1.4 Example of a nuclear fuel cycle. Adapted from Ref. [145]...... 5 1.5 The chemical states of ﬁssion products, denoted by the shaded regions. Or- ange, gray, blue, and green shaded regions denote volatile gases, metallic precipitates, solid solution, and oxide precipitates, respectively. Elements with multiple colors have the possibility of exhibiting multiple chemical states, with the top color denoting the most stable state. Reproduced from Ref. [144]...... 6

4.1 The unit cell of LaPO4 is shown on the left. Lanthanum, oxygen, and phosphorus ions are shown in blue, red, and yellow, respectively. The orthophosphate groups in the shaded region are considered in Figs. 4.1A-C. The py- 6− rophosphate is derived from two orthophosphate groups (2PO4 , Fig. 4.1A) by forming an oxygen vacancy (2+ charge, Fig. 4.1B), causing one oxygen ion to be shared and resulting in the overall 4- charge (Fig. 4.1C). The ions labeled with “prime” represent those within the phosphate group that shares its oxygen ion in the pyrophosphate link...... 17 4.2 The partial density of states for undoped LaPO4 systems containing a pyrophosphate (top) and proton (bottom) defect are shown. The charged defects are compensated with background charges. The arrows denote the extra peaks due to the presence of the defects. The valence band is set to 0 eV. . 22 4.3 The 3 x 3 x 3 supercell containing two protons (H1, H2, shown in black) nearest to each of the Sr dopants (Sr1, Sr2, shown in silver) is shown (“both near” structure). The black dotted circle represents the approximate position of the top proton (H2) within the “one far” structure, in which the proton is far from all defects and the bottom proton (H1) is nearest to a dopant. . . 24

5.1 The ideal UO2 cubic ﬂuorite structure with two trivalent dopants on the cation fcc sublattice and one charge compensating oxygen vacancy on an anion simple cubic sublattice is shown...... 31 vi

5.2 Formation enthalpies of U1−xMxO2−0.5x (M=Y,La) structures are shown for the lowest-energy fully relaxed structures of all compositions enumerated in this study. Filled and open symbols correspond to DFT+U and hybrid- functional calculations, respectively. The dotted line connecting the solid points is a guide to the eye...... 37 5.3 The formation enthalpies calculated using DFT+U for all structures computed in this study are shown. Each structure is denoted by the number of trivalent cations in the four tetrahedral nearest neighbor positions surrounding an oxygen vacancy according to the legend in the top left of the ﬁgure. Note that there are two vacancy sites for x=2/3, and the average number of neighbors between the two sites (rounded to the nearest neighbor) is shown. 38 5.4 Calculated formation enthalpies for Y-substituted AO2 ﬂuorite systems are plotted versus the radius of the host A4+ cation radius. Results for zirconia (A=Zr) and thoria (A=Th) systems were taken from Refs. [31] and [8], respectively...... 41 5.5 The formation enthalpies calculated by DFT+U for selected structures containing, one, two, and three trivalent cation nearest neighbors to an oxygen vacancy are shown for Y and La-substituted systems at x=1/2...... 43

6.1 Formation enthalpies of U0.5M0.5O1.75 structures are shown for the lowest- energy fully relaxed structures of all compositions enumerated in this study. Calculations for M=Y and La were taken from Ref. [141] for comparison. . 49 6.2 The formation enthalpies for selected structures containing, one, two, and three trivalent cation nearest neighbors to an oxygen vacancy are shown. Calculations for M=Y and La were taken from Ref. [141] for comparison. . 50 6.3 Structural motifs of low-energy structures containing one (left), two (middle), and three (right) M3+-oxygen vacancy nearest neighbors...... 51

7.1 The substitution of two U4+ by two Ln3+ can be charge compensated by one Ovac, or in more oxidizing environments, charge compensation can occur by oxidation of U4+ to U5+ for every one Ln3+/U4+ substitution, as illustrated in the motifs above. Systems with compositions that are intermediate between fully oxygen vacancy charge-compensated and fully U5+ charge-compensated contain both oxygen vacancies and U5+ ions...... 57 7.2 Formation enthalpies (∆Hf ) of U1−xLnxO2−0.5x+y structures are shown for the calculated lowest-energy fully relaxed structures of all compositions enumerated in this study, in comparison to calorimetric data presented previously[168]. The enthalpies are plotted against oxygen-to-metal ratio (O/M), which is related to x and y according to Eq. 7.3...... 60 7.3 The formation enthalpies calculated for stoichiometric compositions for both calculation (“Comp.”) and experiment (“Exp.”) with respect to Ln dopant cation size are shown...... 61 7.4 The first-principles calculated percent lattice parameter difference relative to fluorite UO2 for the three Ln dopant species considered in this work for stoichiometric compositions with x=1/3 and 1/2 doping levels are shown. . 62 vii

7.5 Formation enthalpies of A1−xLnxO2−0.5x fluorite oxides from LnO1.5 and fluorite AO2 from calorimetric and computational results, as a function of trivalent dopant cation radius, where x = 1/3 in (a) and x = 1/2 in (b). aLee and Navrotsky[91] bSimoncic and Navrotsky[139] cSimoncic and Navrotsky[140] dBogicevic et al.[31] eChen et al.[37] f Navrotsky et al.[108] gBuyukkilic et al.[33] hSolomon et al.[141] iThis work jAizenshtein et al.[7] kAlexandrov et al.[8] lSolomon et al.[142] ...... 63 7.6 Formation enthalpies of A1−xLnxO2−0.5x fluorite oxides (x=1/3, 1/2) from LnO1.5 and fluorite AO2 from calorimetric and computational results, as a function of tetravalent host cation radius, where Ln = Y in (a), Ln =Gd in (b), and Ln = La in (c). Redrawn from data in Fig. 7.5...... 65 7.7 Formation enthalpies of A1−xLnxO2−0.5x fluorite oxides from LnO1.5 and fluorite AO2 from calorimetric and computational results, as a function of size mismatch of trivalent dopant and tetravalent host cations, where x = 1/3 in (a) and x = 1/2 in (b). Redrawn from data in Fig. 7.5. The dashed lines shown in the figure are obtained by fitting only the experimental data. . . . 66

A.1 Total energies of systems with compositions U2Y2O7 (top) and U2La2O7 (bottom)...... 88 A.2 Total energies of systems with compositions U4Y2O11 (top) and U4La2O11 (bottom)...... 89 A.3 Total energies of systems with compositions U4Y2O10 (top) and U2La2O7 (bottom)...... 90 A.4 Total energies of systems with compositions U4Y2O12 (top) and U4La2O12 (bottom)...... 91 A.5 Total energies of systems with compositions U2Y2O8 (top) and U2La2O8 (bottom)...... 92 viii

List of Tables

4.1 Calculated inter-ionic distances (units of A)˚ of the four possible pryophosphate configurations are listed in columns 1-4. Row 1 shows the distance between the two phosphorus ions. Rows 2 and 3 show bond lengths between the phosphorous ions and the shared oxygen ion. The ions labeled with “prime” represent those within the phosphate group that shares its oxygen ion in the pyrophosphate link. Rows 4-9 list the other phosphorus-oxygen ion bond lengths. Column 5 lists inter-ionic distances calculated previously using ionic potential models.a Row 10 shows the relative energies (∆E) in eV of the pyrophosphate configurations ...... 20 4.2 Proton-dopant distances (units of A)˚ for the three configurations of 3 x 3 x 3 supercells (columns 2-4) containing two Sr dopants and two protons (Sr1, Sr2 and H1, H2, respectively) are shown ...... 24

5.1 Formation enthalpies (kJ/mol-cation) of U1−xMxO2−0.5x (M=Y,La) structures are listed for low-energy fully relaxed structures enumerated in this study. The structures listed have formation energies within 3 kJ/mol-cation of the lowest energy structure for each trivalent cation species at each composition considered. The fourth column lists the direction of ordering of oxygen vacancies. The last column lists the number of trivalent cations in the four tetrahedral nearest neighbor positions surrounding an oxygen vacancy. . . . 35

7.1 The compositions considered for the calculations in this work are shown. The ﬁrst through ﬁfth columns represent the formula, number of U4+ ions, number of U5+ ions, number of oxygen vacancies, and number of symmetrically- distinct structures, respectively...... 56 7.2 The cation sizes in eightfold coordination taken from ref. [135]...... 64

8.1 Oxidation enthalpies for actinide (i.e., U, Np, Pu, Am) oxides are shown. U3O8 and UO3 are α and γ phases, respectively. Enthalpies are in units of kJ/mol...... 73 ix

Acknowledgments

First I would like to acknowledge my dissertation committee: Professors Mark Asta, Lut- gard de Jonghe, Alexandra Navrotsky, and Jeﬀrey Neaton.

To my research advisor, Professor Mark Asta, for believing in me. Thank you for your dedication, insight, honesty, and making my PhD experience memorable.

To my research collaborators, Professor Alexandra Navrotsky and Lei Zhang, for mean- ingful discussions and useful feedback.

To the Asta Research Group, for always being gracious and helpful.

To Nicole Adelstein, Vitaly Alexandrov, and Ben Hanken, for your mentorship during the early stages of my graduate career.

To Babak Sadigh, for your patience and dedication during my summer internship at Lawrence Livermore National Laboratory.

To my University of Florida professors, Dr. Scott Perry, and Dr. Gerald Bourne, for helping me make the right decision about graduate school.

Finally, I would like to acknowledge my family for their unwavering support. Alejandra- thank you for taking a leap of faith and joining me on the west coast, and for being there every step of the way.

This work was supported by the Office of Basic Energy Sciences of the U.S. De- partment of Energy as part of the Materials Science of Actinides Energy Frontier Research Center (DE-SC0001089), the U.S. Department of Energy through the Lawrence Livermore National Laboratory (DE-AC52-07NA27344), and the Department of Defense through the National Defense Science & Engineering Graduate Fellowship Program. This work made use of resources of the National Energy Research Scientific Computing Center, supported by the Office of Basic Energy Sciences of the U.S. Dept of Energy (DE-AC02-05CH11231). 1

Part I

Introduction and Background 2

Chapter 1

Introduction

The future global energy landscape mandates that energy sources should reduce greenhouse gas emissions, reduce fossil fuel dependence, and improve energy security. This mandate will likely be fulﬁlled by a portfolio of energy sources, including nuclear and hydrogen (fuel cell). Both nuclear and fuel cell technologies require engineering of complex oxides materials in order to provide optimal properties. The goals and challenges of engineering complex oxide materials for these technologies are discussed subsequently.

1.1 Fuel Cell Technology

A fuel cell converts chemical energy into electrical energy by an electrochemical 1 reaction (i.e., H2 + 2 O2 → H2O). Hydrogen gas (H2) is converted to positive hydrogen ions (protons) at the anode, and the stripped electrons are passed through an external circuit, creating current flow. In the case of proton conducting fuel cells, the protons pass through a membrane (electrolyte) and combine with oxygen at the cathode to form water byproduct (Fig. 1.1). Oxygen ion conducting fuel cells (i.e., solid oxide fuel cells) work under the same principles, except oxygen ions pass through the electrolyte instead of protons. The electrolyte is a key element in the efficiency of a fuel cell. Required properties of solid electrolytes include high ionic conductivity, chemical and mechanical stability, and corrosion resistance at the prescribed operating conditions. Solid oxide fuel cells, which operate most efficiently at 800◦C, suffer from corrosion and mechanical failure around this temperature. It is therefore desirable to develop solid electrolytes that work at lower temperatures (i.e., 300-600◦C). Proton conductors have shown promise as an ideal candidate for this temperature range[111, 164, 173]. Proton conducting electrolytes with the highest conductivities at intermediate temperatures ranges (i.e., 300-600◦C) are perovskite structured barium zirconate and bar- −3 ium cerate (BaZrO3 and BaCeO3, respectively), with conductivities around 10 S/cm at 300◦C[88]. These materials, however, suffer from highly resistive grain boundaries. Phos- phate based materials, such as rare earth orthophosphates (REP), are a promising alterna- tive. Of the different REP compounds, LaPO4 has been particularly well studied for proton conductor applications, due to its stability and proton uptake in humid atmospheres 3

Figure 1.1: Schematic of a proton conducting fuel cell. Reformatted from Ref. [52].

over intermediate temperature ranges[10]. The unit cell for LaPO4 is shown in Fig. 1.2. The protons are bound to an oxygen ion, and proton transport occurs by a hopping mechanism 3− between neighboring PO4 tetrahedra. A discussion of how protons are incorporated into LaPO4 can be found in Chapter 4. A current drawback for REP compounds as a viable candidate for proton conducting electrolyte applications is the relatively low proton conductivity (10−5.2 S/cm at 500◦C for 1 mol % Sr doping[10]). This value is several orders of magnitude lower than what would be considered competitive for current energy production systems[111]. First principles calculations oﬀer the ability to gain insight into the factors that may limit proton conductivity for these systems, including the energetics and kinetics of water uptake/proton incorporation, and the inﬂuence of defect interactions on these properties. This understanding can enable theorists and experimentalists to make better engineering decisions in developing optimal fuel cell technologies.

1.2 Nuclear Technology

Nuclear energy has provided electricity with minimal carbon emissions for sixty years. According to the International Energy Agency, nuclear energy was responsible for 12% of the world’s electricity generation in 2011[4]. The primary material used for nuclear fuel is uranium dioxide (UO2), which releases a large amount of useful heat upon ﬁssion. The unit cell for UO2 is shown in Fig. 1.3. The fabrication, use, recycling, and post-reactor storage of UO2 is referred to as a “fuel cycle.” An example of a fuel cycle is shown in Fig. 4

Figure 1.2: The unit cell of LaPO4. Lanthanum, oxygen, and phosphorus ions are shown in blue, red, and yellow, respectively.

1.4. A significant concern for nuclear technology applications is the generation of fission products, which affects fuel properties and poses a health risk if exposed to the environment. Fission products typically produced in light water reactors are shown in Fig 1.5. As the global demand for energy increases, it is desirable to increase fuel burnup by increasing the lifetime of fuel in the reactor, and thereby increasing the amount of fission products. Predicting fuel behavior under these conditions requires an understanding of the thermochemical properties of UO2 mixed with fission products. Of particular interest are the trivalent rare earth fission products (Y, La, Ce, Pr, Nd, Pm, Sm, Eu) that dissolve as solute cations in the fuel, as they are known to dramatically affect fuel properties (see Ref. [137] for a review). Soluble trivalent rare earth cations are also relevant in the fabrication stage of the fuel cycle, e.g., purposeful addition of burnable “neutron poisons” (e.g., Gd) to control reactivity during the early stages of fuel burnup[122]. Unfortunately, thermochemical studies of trivalent cations in UO2 are sparse, and have mostly been limited to measurements of the oxygen chemical potential (see Ref. [137] for example). The energetics and defect ordering tendencies of these systems remain far less studied compared to that for other fluorite-structured oxides[127, 76, 106, 31, 30, 123, 8, 167, 14, 167]; prior to this work, only one calorimetry study[99] and one ionic pair potential modeling study[104] had been reported. Direct calorimetric measurements have been historically limited by the refractory nature of UO2, and difficulties controlling and measuring the stoichiometry of these trivalent doped systems. Recent efforts of direct calorimetric measurements by Professor Navrotsky 5

Figure 1.3: The unit cell for UO2. Uranium and oxygen ions are shown in blue and red, respectively.

Figure 1.4: Example of a nuclear fuel cycle. Adapted from Ref. [145]. and her team at the Peter A. Rock Thermochemistry Laboratory at UC Davis have given rise to an opportunity to benchmark density functional theory (DFT) calculations. DFT calculations assist experimental investigations through accurate and controlled studies of systematic trends for various trivalent cation species and compositions.

1.3 Outline

The remainder of the paper is organized as follows. In the next chapter, the theoretical framework of the computational tools used in this work is discussed. Chapter 3 addresses the failure of standard DFT for strongly correlated materials (i.e., UO2 in the current work), and discusses methods to circumvent this issue. Chapter 4 begins the results section, where the energetics and defect interactions of Sr doped LaPO4 and its impact on proton conductivity are discussed. Chapter 5 dis- 6

Figure 1.5: The chemical states of ﬁssion products, denoted by the shaded regions. Orange, gray, blue, and green shaded regions denote volatile gases, metallic precipitates, solid solution, and oxide precipitates, respectively. Elements with multiple colors have the possibility of exhibiting multiple chemical states, with the top color denoting the most stable state. Reproduced from Ref. [144]. cusses the energetics and defect interactions for yttrium and lanthanum (the smallest and largest trivalent rare earth cations, respectively) doped UO2. Chapter 6 extends this work by considering a series of trivalent rare earth cations to explore systematic trends. While Chapters 5 and 6 focus on oxygen-to-metal ratios in UO2 corresponding to charge compensation only by oxygen vacancies, Chapter 7 includes results for compositions that are charge compensated with U5+, and comparisons to experimental calorimetry measurements are made. Chapter 8 oﬀers a summary and conclusions of the current work, and propositions for future work. 7

Chapter 2

Theoretical Framework

Computational calculations enable us to explore systematic energetic trends for systems with controlled compositions and systems where experimental investigation is dif- ﬁcult. In some cases we can employ simple classical interatomic potential models to screen through thousands of structures, or to perform ﬁnite-temperature molecular dynamics or Monte-Carlo atomistic simulations. For accurate descriptions of the structure and energetics of the systems considered in this work, however, quantum-mechanical theory in the form of density functional theory is often required. In the following sections, the theoretical framework underlying the computational methods used in this work are presented.

2.1 Atomistic Simulation

For ionic materials classical atomistic simulations typically involve the application of theories for interatomic interactions based on classical electrostatic models. These calculations generally require less computational eﬀort compared to quantum mechanical calculations (discussed in the next section). This means that high-throughput calculations are possible for systems with relatively large collections of atoms. The accuracy of such classical simulations depends on how well the interatomic potential models describe the real atom interactions. Properties of ionic solids can often be modeled eﬀectively using interatomic potentials that only consider pair interactions (i.e., interactions between two atoms, as opposed to many-body interactions which, in addition to pair interactions, account for simultaneous interactions between N atoms where N > 2). For ionic materials, ions interact via Coulom- bic forces (i.e., long-range electrostatic interactions). The pair interactions are summed over all n atoms in the solid to give the electrostatic energy, Ee, expressed as

1 n n q q E = X X i j (2.1) e 2 r i j6=i ij where qi and qj are the chaarges of ions i and j, and rij is the separation between the 1 positions of ions i and j. The factor of 2 removes double-counting from the summation. Another term that is added to the ionic pair potential accounts for short-range 8 interactions. One commonly used expression to account for these interactions is the Buck- ingham potential, expressed as ! 1 n n −r C E = X X Aexp ij − (2.2) s 2 ρ (r )6 i j6=i ij where A, ρ, and C are adjustable parameters. The ﬁrst and second terms represent repulsion between electron orbitals on the atoms due to Pauli exclusion, and the weak van der Waals attraction between electron orbitals, respectively. The adjustable parameters can be ﬁtted using quantum mechanical calculations or to physical properties, e.g., the relative permittivity, elastic constants, lattice parameters, defect energies, etc. The total energy of the solid is the combination of Coulombic and short-range pair interactions Etotal = Ee + Es (2.3) The equilibrium total energy corresponds to the positions of the atoms that minimize Eq. 2.3.

2.1.1 The Shell Model Calculations of dielectric and elastic properties of a ionic crystal using the terms in Eq. 2.3 have been found to disagree with experiment. The disagreement is often attributed to electronic polarizability of the ions. The electronic polarizability was included into ionic pair potential calculations by Dick and Overhauser with the shell model[45]. In this formalism, the formal charge of the ion is separated into a pointlike positively charged core and a massless negatively charged shell which surrounds the core. The shell models the electron density around the cores, and is displaced with respect to the core upon polarization. In general the core and shell will be displaced by diﬀerent amounts due the interactions with the surrounding ions. The core and shell are treated as separate charges in Eq. 2.1, and the short-range potential Eq. 2.2 now acts between the shells. However, the core and shell are bound by a “spring” which prevents the core and shell from moving very far from one another. The potential energy associated with displacement of the shell with respect to the core is given by the classical equation for an extended spring 1 E = kδr2 (2.4) sp 2 i where k is a spring constant (which is inversely proportional to the polarizability of the ion), and δri is the amount the spring on ion i is stretched, i.e., the distance between the centers of mass of the core and shell of ion i. The total energy of the solid now becomes

Etotal = Ee + Es + Esp (2.5) 9

2.2 First-principles Methods

In the previous section, a method for describing solids using classical interatomic potential models was discussed. A more complete understanding of the properties of solids, however, requires a quantum mechanical description of the interactions between nuclei and electrons (a comparison between atomistic and first principles calculations for systems considered in this dissertation is shown in the Appendix). In many cases, the calculation methods based on quantum mechanics do not use any fitting parameters from experimental data, and are thus known as “first-principles” methods. The only information these methods require is the atomic numbers of the constituent atoms. An introduction to first-principles methods and practical application of these methods for solids using density functional theory (DFT) is presented in the next sections (note that Hartree atomic units are used in the remaining sections).

2.2.1 The Schr¨odingerEquation A quantum mechanical description of the physical properties of solids can be described in mathematical form by the time-independent Schr¨odingerequation

Hˆ Ψ(ri, rI ) = EΨ(ri, rI ) (2.6) where Hˆ is the Hamiltonian operator, Ψ(ri, rI ) is the wave function, and E is the energy of the system. The variables ri and rI represent the position coordinates of all the electrons and nuclei in the system, respectively. Eq. 2.6 is simpliﬁed by approximating the positions of the nuclei to be ﬁxed with respect to the electrons, due the much greater mass of the nuclei compared to the electrons. This is known as the Born-Oppenheimer approximation. This approximation reduces the variables in Eq. 2.6 to the position of the electrons ri. The Hamiltonian operator is the sum of the kinetic and potential energies of a system

ˆ kin kin H = EI + Ei + UIi + Uij + UIJ (2.7) kin kin where EI and Ei represent the kinetic energies of the nuclei and electrons, respectively, and UIi, Uij, UIJ , are the potential energies from nucleus-electron, electron-electron, and nucleus-nucleus interactions, respectively. The Born-Oppenheimer approximation makes Hˆ depend only on energy terms that are a function of the electron positions ri, which simpliﬁes Eq. 2.7 to

ˆ kin H = Ei + UIi + Uij (2.8) The complete expression for Eq. 2.8 is represented as

1 n N n Z 1 n n 1 Hˆ = − X ∇2 − X X I + X X (2.9) 2 i |r | 2 |r | i I i Ii i j6=i ij where n and N are the number of electrons and nuclei in the system, respectively, ZI is the valence of each nucleus, and rIi is the distance between a nucleus and an electron. 10

2.2.2 Density Functional Theory Even with the approximation discussed in the previous section, the solution to the Schrödingerequation is limited in practice to very simple systems due to the many-body quantum effect. The many-body effect arises from the 3n variables that are required to solve the Schrödingerequation of an n-electron system. A breakthrough in the practical application of quantum mechanical theory to solids came from Hohenberg and Kohn[71] and Kohn and Sham[82] with a scheme called density functional theory (DFT). The Hohenberg-Kohn theorems, which form the basis of DFT, state that the ground-state energy is a functional of the electron density, and there exists an electron density that minimizes the total energy, which is the exact ground state energy. The electron density n(r) is defined as

occupied X 2 n(r) = |φi(r)| (2.10) i Eq. 2.10 says that n(r) is the sum over a set of probability densities of occupied Kohn-Sham orbitals φi(r). The Kohn-Sham hamiltonian (Eq. 2.11) decomposes the many-body n-electron system into an effective one-electron Schrödingerequation, where the interactions with the other electrons is represented by an effective potential Ueff : 1 Hˆ = Enon + U + U + U = − ∇2 + U (2.11) KS kin ext H xc 2 i eff non Ekin is the non-interacting component of the electron kinetic energy, Uext is the potential energy due to the interaction between electrons and nuclei, and UH (known as the Hartree potential) is the repulsive Coulomb interaction between each electron and the mean field of all electrons in the system (n(r))

Z n(r0) U = d3r0 (2.12) H |r − r0|

and the many-body interactions reside in the exchange-correlation potential Uxc (note that UH introduces a spurious self interaction, i.e., an electron interacts with n(r), which contains a contribution from the electron itself). If Uxc is known, then the ground state electron density (and therefore the ground state energy) of the non-interacting system of n electrons is the same as that for the many-body n-electron system. In practice, however, exact expressions for Uxc are unknown and approximations are made. The local density approximation (LDA) approximates the exchange-correlation hom energy as an integral over an energy density xc [n(r)] that corresponds to a homogeneous LDA electron gas with the local density n(r). The LDA exchange-correlation energy Exc [n(r)] is deﬁned as Z LDA hom 3 Exc [n(r)] = n(r)xc [n(r)]d r (2.13) and the corresponding LDA exchange-correlation potential is 11

δELDA U LDA[n(r)] = xc (2.14) xc δn(r) The LDA works fairly well for systems where the electron density varies slowly; however, the generalized gradient approximation (GGA) was developed to produce more accurate exchange-correlation functionals. The GGA accounts not only for the local electron density (LDA), but also the gradient at a given point. There are many forms of the GGA functional. The most frequently used functional is PBE (Perdew, Burke, and Ernzerhof)[117], which is highly accurate and computationally eﬃcient. 12

Chapter 3

Addressing Self-Interaction Errors for Strongly Correlated Materials

Although the GGA functional has been successfully applied to many systems, a major limitation occurs for systems with strongly localized orbitals. These “electron correlations” are present in systems that contain narrow d- and f- orbitals, including rare earth elements and compounds as well as transition metal oxides and actinide oxides[15, 47]. A primary focus of this dissertation concerns UO2, which has narrow 5f bands corresponding to localized f states on the uranium sites, and therefore strong electron correlations. The following sections provide an overview of the theory behind strongly correlated systems and methods within DFT that are used to improve their description.

3.1 Hubbard Model

UO2 is an example of a Mott insulator. Mott insulators are materials in which a bandgap exists of bands of like character (i.e., 5f-5f bands of uranium ion for UO2). This class of material is not considered in conventional band theory, in which the band gap in a charge transfer insulator exists between bands of diﬀerent character (e.g. oxygen 2p and uranium 5f bands in UO3). The existence of Mott insulators was originally proposed by Hubbard in 1963 with the Hubbard model[72]. The model represents the competition between an electron’s “overlap integral,” which represents the ability to hop from one atomic site to another (i.e., conduction) and an electron’s “onsite repulsion” due to the Coulombic interaction with an electron already existing at that site. If the overlap integral is suﬃciently small (e.g., due to narrow electron bands), the Coulomb repulsion term dominates and therefore the electron will not hop to another site, making the solid an insulator.

3.2 DFT Corrections for Strongly Correlated Materials

As mentioned previously, standard approximations for the exchange-correlation functional (i.e., LDA, GGA) are insuﬃcient to accurately describe the nature of strongly 13

correlated systems. For example, GGA and LDA predict the electronic structure of UO2 to be metallic, even though it is known to be a Mott insulator. GGA and LDA have also been shown to incorrectly reproduce defect formation energies and reaction energies[47]. This discrepancy is due to large self-interaction errors that are not canceled with GGA or LDA functionals. Several methods have been developed to correct for these discrep- ancies in computed and experimental properties, including hybrid functionals[2, 69, 126], self-interaction correction (SIC)[119], addition of a Hubbard term (DFT+U)[16, 94, 50], and dynamical mean ﬁeld theory (DFT+DMFT)[57, 84]. In the subsequent sections, the two most widely-used methodologies will be discussed, namely, hybrid functionals and the Hubbard-U correction.

3.2.1 Hybrid Functionals Hybrid functionals are a modiﬁcation of the standard DFT exchange-correlation functional (e.g., PBE) obtained by replacing a fraction of exchange from the standard DFT functional with exchange from Hartree-Fock (HF) theory. The hybrid exchange-correlation energy functional has the form

PBE+αHF HF PBE PBE ∆EXC = αEX + (1 − α)EX + EC , (3.1) HF PBE PBE where αEX is the HF exchange, EX and EC are the PBE exchange and correlation energies, respectively, and α represents the fraction of HF exchange that replaces the PBE exchange. The value for α used in this work is 0.25, which is representative of the PBE0 functional [2]. The hybrid functionals tend to be computationally intensive for large systems or high throughput calculations, however. The “local hybrid functional for correlated electrons” (LHFCE) is a less computationally demanding hybrid-functional approach in which the exact exchange is applied locally within the atomic spheres only to the correlated electrons (i.e., 5f uranium orbitals)[113]. The LHFCE functional is implemented in the current work for UO2 based systems as a comparative approach to the method discussed in the next section (the results and implementation of the LHFCE functional in the current work is discussed in Chapter 5).

3.2.2 Hubbard-U Correction (DFT+U) The DFT+U approximation adds a correction to standard DFT exchange-correlation functionals. In the formalism by Dudarev et al.[50], the energy functional is expressed as

U − J X 2 EPBE+U = EPBE + (nm,σ − nm,σ) (3.2) 2 σ

where U and J are screened Coulomb and exchange parameters, respectively, nm,σ are the electron occupations for orbital m and spin σ, such that the total number of electrons, P Nσ = σ nm,σ. Dudarev’s simpliﬁed expression for the GGA+U energy functional can be described as a penalty function for fractional orbital occupations (i.e., between 0 or 1). This penalty function forces integer orbital occupations (0 or 1), which is characteristic of insulating behavior. The magnitude of the penalty for fractional orbital occupancy depends on the 14

value of U − J, which is commonly referred to as an effective Hubbard-U (Ueff ). The value of Ueff which has been most successfully applied to UO2 based systems in terms of accurate descriptions of the electronic structure[51, 5], defect formation energies and bulk properties[47], and oxidation enthalpies[64] is approximately 4.0 eV. The DFT+U approximation is the most widely used method for addressing self- interaction errors in UO2 based systems due to its computational efficiency and sufficient accuracy. However, application of a Hubard-U term can lead to multiple metastable states for these systems, corresponding to different orbital occupancies[47]. The issue of metastable states and the methods we have used to circumvent this problem for the current work is discussed in Chapter 5. 15

Part II

Results and Discussion 16

Chapter 4

Energetics and Defect Interactions: Sr Doped LaPO4

LaPO4 has been actively studied for proton conductor applications, due to its stability and proton uptake in humid atmospheres over intermediate temperature ranges. An important process underlying the application of this and related materials for proton- conductor applications is the hydrolysis of pyrophosphate defects. In this chapter we report results of density functional theory (DFT) calculations of the relative energetics of pyrophosphate defects and protons in LaPO4, including their binding with divalent dopant cations. Due to the low symmetry of the monazite crystal structure for LaPO4, there exists four symmetry-distinct pyrophosphate defect conﬁgurations; DFT calculations are used to identify the most stable conﬁguration, which is 0.24 eV lower in energy than all others. Further, from supercell calculations with 1.85 mol% Sr doping, we investigate the dopant- binding energies for pyrophosphate defects to be 0.37 eV, which is comparable to the value of 0.34 eV calculated for proton-dopant binding energies in the same system. These results establish that dopant-defect interactions further stabilize proton incorporation, with the hydration enthalpies when the dopants are nearest and furthest from the protons and pyrophosphate defects being -1.66 eV and -1.37 eV, respectively. Even though our calculations show that dopant binding enhances the enthalpic favorability of proton incorporation, they also suggest that such binding is likely to substantially lower the kinetic rate of hydrolysis of pyrophosphate defects.

4.1 Forward

The work presented in this chapter was published by J. M. Solomon, N. Adelstein, L. C. De Jonghe, and M. Asta, in J. Mater. Chem. A, vol. 2, issue 4, pages 1047-1053 (2014), and is reproduced here with permission of the co-authors and publishers. ©Royal Society of Chemistry 17

Figure 4.1: The unit cell of LaPO4 is shown on the left. Lanthanum, oxygen, and phosphorus ions are shown in blue, red, and yellow, respectively. The orthophosphate groups in the shaded region are considered in Figs. 4.1A-C. The pyrophosphate is derived from two 6− orthophosphate groups (2PO4 , Fig. 4.1A) by forming an oxygen vacancy (2+ charge, Fig. 4.1B), causing one oxygen ion to be shared and resulting in the overall 4- charge (Fig. 4.1C). The ions labeled with “prime” represent those within the phosphate group that shares its oxygen ion in the pyrophosphate link.

4.2 Introduction

Rare-earth phosphates (REP) are known proton conductors that are of interest for a variety of applications including intermediate-temperature fuel cell electrolytes, as well as hydrogen separation membranes and sensors[111, 10, 83, 79, 112, 9, 55, 155]. Of the diﬀerent REP compounds, LaPO4 has been particularly well studied for proton conductor applications, due to its stability and proton uptake in humid atmospheres over intermediate temperature ranges (300-800◦C). Commercial viability of this and related REP systems is limited by low proton conductivity, however. For example, the highest reported proton con- −5.2 −3.5 ◦ ductivities for LaPO4 are 10 -10 S/cm at 500-925 C, for 1 mol % Sr doping[10]. These values are several orders of magnitude lower than what would be considered competitive for current energy production systems[111]. Amezawa and co-workers[83] have established a defect chemistry model for pro- 2+ ton incorporation in Sr-doped LaPO4. In this model, divalent Sr dopant cations (Sr ) 2+ are introduced to LaPO4 in the form of strontium pyrophosphate (Sr2P2O7), where Sr 3+ 4− is substituted for La , resulting in a charge-balancing pyrophosphate defect (P2O7 ), according to the following defect reaction:

1 0 1 Sr P O → Sr + (P O )•• (4.1) 2 2 2 7 La 2 2 7 2PO4

In the LaPO4 compound, the pyrophosphate defect (Fig. 4.1C) is derived from two or- 6− thophosphate anions (2PO4 , Fig. 4.1A) by forming an oxygen vacancy (2+ charge, Fig. 4.1B), causing one oxygen ion to be shared and resulting in the overall 4- charge. In the model of Amezawa et. al[83], protons are incorporated into Sr-doped LaPO4 in the form of hydrogen phosphate groups through the following defect reaction: 18

1 1 (P O )•• + H O(g) )* (HPO )• (4.2) 2 2 7 2PO4 2 2 4 PO4 Equation 4.2 is widely accepted as the primary pathway for proton incorporation for LaPO4 doped with divalent metals in humid environments with ambient atmosphere[10, 83, 79, 9]. Equation 4.2 implies that proton incorporation proceeds through the hydrolysis of positively-charged oxygen vacancies present as pyrophosphate defects, to a degree controlled by the partial pressure of water and the enthalpy of this hydration reaction (∆Hhyd). Based on the framework provided by the model of Amezawa et. al, recent studies have begun to provide insights into the microscopic factors limiting proton conductivity in LaPO4. Considering first the thermodynamic factors underlying proton uptake (and therefore carrier density), the hydration enthalpy of the reaction given by Eq. 4.2 has been reported previously from experimental measurements, with values ranging between -0.86 eV and -2.07 eV[55, 112]. The same quantity was also considered in the computational work by Bjorheim et al. who employed density-functional theory (DFT) to calculate a value of -1.34 eV[26]. Although the values for the hydration enthalpy reported in these studies vary by several tenths of an eV, they are in agreement in establishing that proton incorporation is highly exothermic, such that this process is driven by energetics at low temperatures and moderate water partial pressures. We note that the previous computational study of hydration enthalpies[26] did not examine the effects associated with dopant-defect binding, which has been shown in recent work to be on the order of 0.3 eV for the case of protons in LaPO4 calculations[26, 153]. Whether such proton-dopant binding effects are important for hydration thermodynamics depends on the relative magnitudes of proton-dopant and of pyrophosphate-dopant binding energies; the latter is currently unknown for LaPO4, to the best of our knowledge. In addition to its impact on the thermodynamics of proton incorporation, dopant- defect interactions may also limit the mobility of protons and pyrophosphate defects. DFT calculations of the activation energies for proton migration in LaPO4 yield values on the order of 0.7-0.8 eV[153, 166]. Proton-dopant binding energies from recent calculations are of comparable magnitude to the activation energies, namely, 0.2-0.7 eV[3, 26, 153, 121]. The results thus suggest that proton-dopant binding may contribute significantly to the measured values of the activation energies for proton conduction in doped LaPO4. To date, the effect of dopant-defect binding has not been investigated for pyrophosphate defects. The incorporation of protons by the mechanism corresponding to Eq. 4.2 requires diffusion of pyrophosphate defects to the sample surface, which enables interactions with water and proton incorporation. Strong binding between the aliovalent dopant and pyrophosphate may impede this process. While such binding effects have not been quantified for rare-earth phosphate compounds, a previous related computational study[97] for perovskite-structured SrCeO3 reported calculated binding energies between oxygen vacancies and aliovalent dopants of 0.75 to 1.25 eV. Another study[68] reported a calculated binding energy of 0.18 eV for oxygen vacancies in Gd-doped BaCeO3. The oxygen vacancy for these structures is analogous to the pyrophosphate for the LaPO4 system under consideration in this study, and it is thus of interest to understand if dopant-pyrophosphate binding energies are of comparable magnitude, and thus of similar importance in limiting pyrophosphate mobilities. 19

In the current work we employ DFT calculations to study the atomic and electronic structure of proton and pyrophosphate defects in LaPO4, the relative magnitude of the binding energies between these defects and Sr dopants, and the overall eﬀect of dopant- defect binding on hydration enthalpies. The remainder of the paper is organized as follows. In the next section the details of the calculations are described. The calculated results are presented in Section 4.4 and compared to previous related computational studies. Finally, the results are discussed in the context of the thermodynamic and kinetic factors governing proton uptake and conductivities in Section 4.5.

4.3 Computational Methods

All calculations have been performed using DFT within the generalized gradient approximation (GGA) due to Perdew, Burke, and Ernzerhof (PBE)[117]. We have employed the projector-augmented-wave method (PAW) as implemented in the Vienna Ab-initio Sim- ulation Package (VASP) code[85, 85]. The PAW potentials[27] used in these calculations are those specified as La, P, Sr, H and O in the VASP PBE library. The PAW potentials use 11 valence electrons for La (5s25p66s25d1), 5 for P (3s23p3), 10 for Sr (4s24p65s2), 1 for H (1s1), and 6 for O (2s22p4). A plane-wave cutoff energy of 500 eV is employed, and reciprocal space is sampled with a single k-point (Γ) due to the large size of the supercells (at least 32 formula units per cell). Atomic positions are optimized until the forces are converged to within 10 meV/A.˚ From convergence checks with respect to plane-wave cutoff and k-point sampling we estimate the binding energies and bond lengths reported below to be converged at the level of approximately a few meV and 0.01 A,˚ respectively. The calculation of formation energies of individual point defects was not considered in this work, since the primary focus was on quantification of dopant-defect binding energies.

4.4 Results

4.4.1 Pyrophosphate and Proton Conﬁgurations

LaPO4 forms in the monazite crystal structure, which possesses a unit cell containing four formula units, and features monoclinic symmetry (P21/n space group)[110]. The current calculated values for the lattice parameters of 6.93 A,˚ 7.15 A,˚ 6.54 A,˚ and 103.72o for a, b, c, and β, respectively, agree to within 1.5% of experimentally measured room-temperature values[110] and to within 0.2% of previous GGA calculations[3, 166]. In order to calculate a defect structure of LaPO4 containing pyrophosphate defects, one must first determine the most stable defect configuration. LaPO4 has four symmetrically distinct oxygen ions labeled O1, O2, O3, and O4 (as represented in Fig. 4.1A), consistent with the notation used by Ni et al.[110]. All possible pyrophosphate defects were investigated by removing each of the four oxygen ions from a specific phosphate group in a 2 x 2 x 2 supercell and allowing the cell to relax. Dopants were not considered for these initial calculations, and a doubly negative background charge was implemented to compensate the doubly positive charge of the oxygen vacancy. In Fig. 4.1, ions labeled with “prime” (e.g., O3’) belong to the phosphate group that shares its oxygen ion in the pyrophosphate link 20

Table 4.1: Calculated inter-ionic distances (units of A)˚ of the four possible pryophosphate configurations are listed in columns 1-4. Row 1 shows the distance between the two phosphorus ions. Rows 2 and 3 show bond lengths between the phosphorous ions and the shared oxygen ion. The ions labeled with “prime” represent those within the phosphate group that shares its oxygen ion in the pyrophosphate link. Rows 4-9 list the other phosphorus-oxygen ion bond lengths. Column 5 lists inter-ionic distances calculated previously using ionic potential models.a Row 10 shows the relative energies (∆E) in eV of the pyrophosphate configurations DFT Calculations From This Work O1 Removed O2 Removed O3 Removed O4 Removed Ionic Potentialsa P-P’ (A)˚ 3.04 P-P’ 3.07 P-P’ 3.19 P-P’ 3.05 P1-P2 3.18 P-O3’ 1.70 P-O4’ 1.71 P-O3’ 1.71 P-O1’ 1.73 P1-Ob 1.72 P’-O3’ 1.73 P’-O4’ 1.68 P’-O3’ 1.69 P’-O1’ 1.70 P2-Ob 1.72 P-O2 1.53 P-O1 1.52 P-O1 1.52 P-O1 1.52 P1-O2 1.55 P-O3 1.53 P-O3 1.52 P-O2 1.53 P-O2 1.52 P1-O3 1.56 P-O4 1.51 P-O4 1.51 P-O4 1.51 P-O3 1.51 P1-O4 1.55 P’-O1’ 1.52 P’-O1’ 1.52 P’-O1’ 1.52 P-O2’ 1.53 P2-O2 1.55 P’-O2’ 1.52 P’-O2’ 1.54 P’-O2’ 1.53 P-O3’ 1.53 P2-O3 1.55 P’-O4’ 1.51 P’-O3’ 1.53 P’-O4’ 1.52 P-O4’ 1.51 P2-O4 1.57 ∆E (eV) 0.24 0.41 0.00 0.24 aPhadke et al., J. Mater. Chem., 2012, 22, 25388 21 with the neighboring phosphate anion. The sequential frames in Figs. 4.1A-C illustrate the process by which the pyrophosphate link is formed in the calculations. Specifically, in Fig. 4.1B, the oxygen ion is removed; in Fig. 4.1C, the oxygen-deficient phosphate group moves towards the neighboring phosphate group which rotates to form a bridging oxygen bond. It is interesting to note that the pyrophosphate group forms spontaneously in the relaxation of a geometry initialized with an oxygen vacancy; this spontaneous formation of pyrophosphate defects was also observed in previous first-principles calculations of LaPO4, LaAsO4, and LaVO4 systems initialized with an oxygen vacancy[26]. Analysis of bond lengths and inter-ionic distances is performed to check the consis- tency of the calculated results for the pyrophosphate defect, based on the GGA functional, with known structural information for pyrophosphate anions in related crystal structures. Columns 1-4 of Table 4.1 list inter-ionic distances of the four possible pryophosphate configurations. Row 1 lists the distance between the two phosphorus ions. Rows 2 and 3 list bond lengths between the phosphorous ions and the shared oxygen ion. Note the shared oxygen ion is dependent on which ion is originally removed, with O3’, O4’, O3’, and O1’ shared for O1, O2, O3, O4, removal, respectively. Rows 4-9 list the other phosphorus-oxygen ion bond lengths, which are similar for all four configurations. Column 5 shows inter-ionic distances calculated previously using classical ionic- potential models[121]. It is unclear which of the four different pyrophosphate configurations was considered in those calculations, and the ion labeling is not necessarily consistent with our work . Nevertheless, the inter-ionic distances obtained from the ionic-potential models are in reasonable agreement with the current first principles calculations. Our calculated results show that the P-O bonds that form the pyrophosphate link (i.e., bridging bonds) are relatively long, while the non-bridging P-O bonds are relatively short. This trend is consistent with experimental bond lengths reported for pyrophosphate compounds such as Mg2P2O7[35], Cu2P2O7[131], Ca2P2O7[36], and Zn2P2O7[34]. The shorter bond length for non-bridging P-O bonds is attributed to delocalization of π-bonding which gives double-bond character, whereas the bridging P-O bonds have mainly single-bond character[43]. Row 10 in Table 4.1 shows the relative energies (∆E) of the pyrophosphate configurations. The configuration resulting from O3 removal gives the lowest energy compared to O1, O2, and O4 removal by 0.24 eV, 0.41 eV, and 0.24 eV, respectively. The atomic coordinates of the lowest-energy defect configuration are used as initial positions for the subsequent 3 x 3 x 3 supercell defect structure calculations including dopant cation defects explicitly, as described in the next sub-section. We consider next results for proton defects. The position of the lowest energy configuration for the proton in undoped LaPO4 was computed previously[166]. Specifically, the proton forms a hydroxyl bond with O4 and, upon optimization with DFT, forms a hydrogen bond with O1 on an adjacent orthophosphate group. Adelstein et al.[3] confirmed this to also be the lowest energy position when the proton is positioned near a dopant for Ba-doped LaPO4. The OH···O bond lengths for the protonic defect obtained in the current calculations for undoped LaPO4 agree well (to within 0.01 A)˚ with previously reported DFT calculations[166, 3]. 22

Figure 4.2: The partial density of states for undoped LaPO4 systems containing a pyrophosphate (top) and proton (bottom) defect are shown. The charged defects are compensated with background charges. The arrows denote the extra peaks due to the presence of the defects. The valence band is set to 0 eV. 23

The electronic structure for LaPO4 systems containing a pyrophosphate and a proton defect are shown as partial density of states at the top and bottom of Fig. 4.2, respectively. Formation of the pyrophosphate defect leads to split-off O 2s and 2p bands (Fig. 4.2 focuses on the O 2p bands, and the split-off peak is denoted by the arrow). As the hydration reaction proceeds, the pyrophosphate ion is replaced with protons, with an electronic structure corresponding to the bottom of Fig. 4.2. The presence of proton defects again leads to a split-off peak, that is, in this case, the result of hybridization between the proton and oxygen ion that form the hydroxyl bond. The effect of the electronic structure due to dopant-defect interactions is discussed below.

4.4.2 Superstructures with Defects Charge Compensated by Sr Dopants To investigate the interaction between pyrophosphates, protons and dopants, we employ 3 x 3 x 3 supercells (648 ions) with defect concentrations consistent with overall charge neutrality. We substitute two La ions with Sr, giving a dopant concentration of 1.85% (2/108), sufficiently dilute to approximate the reported 1.9% experimental solubility limit[10]. The large supercell considered in this work provides an opportunity to study relatively long-ranged dopant-defect interactions. Two geometries were calculated to determine the Sr-pyrophosphate binding energy; namely, structures in which one of the dopants is nearest and furthest from the pyrophosphate defect, such that the nearest dopant-defect distances for the two cases are 2.81 A˚ and 12.60 A,˚ respectively. The binding energy computed from the difference in energy for these two configurations is 0.37 eV. In order to directly compare the relative energetics pertaining to the pyrophosphate and protonic defects, a 3 x 3 x 3 superstructure with two dopants and two protons was considered. Three configurations were calculated for this defect structure, namely, each proton nearest to a dopant (“both near”), one proton nearest to a dopant and the other proton far from all defects (“one far”), and both protons far from all defects (“both far”). A representation of the 3 x 3 x 3 “both near” structure with two Sr dopants and two protons is shown in Fig. 3. The black dotted circle represents the approximate position of the top proton of the “one far” structure in which it is far from all other defects. Table 4.2 shows the proton-dopant distances for each configuration and their energies relative to the lowest-energy configuration (∆E). The ∆E values of the “one far” and “both far” structures relative to the “both near” structure (i.e., binding energies) are 0.34 eV and 0.65 eV (0.325 eV/proton), respectively. The energy to separate one proton from a dopant (i.e., 0.34 eV) is slightly larger than the value of 0.31 eV for the binding energy obtained by Toyoura et al.[153] This small discrepancy is likely due to the longer range proton-Sr interaction that is explored by using a larger superstructure in the present work. The values of ∆E for the “one far” and “both far” structures suggests that proton- dopant binding effects for this system are approximately pairwise additive, i.e., the ∆E value for the “both far” configuration is roughly twice as large in magnitude as that for the “one far” configuration. 24

Table 4.2: Proton-dopant distances (units of A)˚ for the three conﬁgurations of 3 x 3 x 3 supercells (columns 2-4) containing two Sr dopants and two protons (Sr1, Sr2 and H1, H2, respectively) are shown Both Near One Far Both Far H1-Sr1 (A)˚ 2.62 2.62 10.07 H2-Sr2 2.62 10.98 10.29 ∆E (eV) 0.00 0.34 0.65

Figure 4.3: The 3 x 3 x 3 supercell containing two protons (H1, H2, shown in black) nearest to each of the Sr dopants (Sr1, Sr2, shown in silver) is shown (“both near” structure). The black dotted circle represents the approximate position of the top proton (H2) within the “one far” structure, in which the proton is far from all defects and the bottom proton (H1) is nearest to a dopant. 25

4.4.3 Hydration Enthalpy Calculations for Structures with Neighboring and Isolated Dopants/Defects In previous computational work the energetic preference for proton incorporation has been quantiﬁed by the value of the hydration enthalpy (∆Hhyd)[26, 68]. Following Hermet et al.[68], we will deﬁne this quantity for systems with dopants as:

∆Hhyd = E(LaP O4 + 2Sr + 2H) − E(LaP O4 + 2Sr + 2VO) − E(H2O) (4.3)

In our calculation, E(LaPO4 + 2Sr + 2H) and E(LaPO4 + 2Sr + 2VO) are the total DFT energies of 3 x 3 x 3 LaPO4 supercells with two lanthanum-substituted strontium dopants, containing two protons and one pyrophosphate defect, respectively. E(H2O) is the DFT energy of an isolated water molecule. ∆Hhyd is defined as the hydration enthalpy at zero temperature and pressure (ignoring zero-point energy corrections). The use of the standard GGA functionals is known to strongly overestimate the energy of the O2 molecule in oxidation enthalpy calculations[159]. This raises a potentially important issue concerning the possible errors in the calculated hydration enthalpies that could be introduced through the use of the GGA. In the current work, the primary error associated with potential overbinding for the water molecule would arise in our reported values for the hydration enthalpies, which involves the difference in energy between H2O in the gas versus hydroxide bonds in the crystal. It is thus interesting to note that in a previously published study by Baranek et al., the hydration enthalpy for MO + H2O (g) = M(OH)2 (M=Mg, Ca) was considered using Hartree-Fock, LDA, GGA, and hybrid (B3LYP) functionals[19]. The results show that the GGA hydration enthalpies agree with hybrid calculations to within 28 meV for both M=Mg and Ca, suggesting a reasonable level of accuracy of the GGA relative to higher-order theoretical calculations for hydration enthalpies. The hydration enthalpies when the dopants are nearest and furthest from the other defects are all exothermic, -1.66 eV and -1.37 eV, representing the most stable and least stable cases, respectively. It is worth noting the latter enthalpy agrees to within three percent with the value obtained in a previous DFT study[26] that used a charged-defect supercell methodology to calculate the hydration enthalpy for the limiting case corresponding to non-interacting defects (-1.34 eV). The negative sign of the hydration enthalpies is consistent with the observations of Amezawa et. al which led to the conclusion that the water uptake reaction is exothermic[10, 83]. Their experimental results demonstrated proton uptake at relatively low temperatures and low water vapor partial pressures, which suggests that the uptake is enthalpically driven. Furthermore, two independent experimental studies have reported quantitative estimates for the hydration enthalpies of -0.86 eV[55] and -2.07 eV[112]. To compare our calculated results to the experimental estimates, we must account for the temperature dependence of the enthalpies of reactants and products in Eq. 4.2. These corrections include two main contributions (neglecting the difference in heat capacities between LaPO4 with pyrophosphate versus proton defects): (1) The zero point energies for the protons in a water molecule versus hydroxyl bonds in LaPO4. An analysis of the relative magnitudes of the zero-point-energy corrections for water versus hydroxide bonds in two different hydroxide compounds was presented by 26

Ozolins et al. in Ref. [115]. The authors showed that the correction to the hydration enthalpy was on the order of 0.02 eV per hydroxide bond, which is roughly two orders smaller in magnitude than the hydration enthalpies considered in this work. As a consequence we will neglect this contribution in what follows. (2) The temperature-dependence of the enthalpy of water vapor. The enthalpy of water vapor (H=E+PV) can be approximated as H = E0 + 4 RT, where E0 is the zero- temperature energy, and R is the ideal gas constant. The 4RT contribution comes from equipartition of the energy of a water molecule (ignoring vibrational states), and assumes ideal gas behavior. This term is consistent with the experimental heat capacity (Cp) of water vapor, which is near 4R over the relevant temperature range for which the hydration reaction is favored (300-600oC). The finite-temperature corrections to the enthalpy of water range between approximately +0.2 eV and +0.3 eV in this temperature range. Accounting for the second correction, the finite-temperature corrected calculated hydration enthalpy ranges between -1.57 eV to -1.67 eV between 300oC and 600oC for hydration involving protons that are not bound to dopants in LaPO4. When binding to divalent Sr dopant cations is accounted for, these values shift to -1.86 eV to -1.96 eV over the same temperature range. The calculated values that account for dopant-defect binding are closest to the experimentally-derived estimate of -2.07 eV obtained in Ref. [112], while calculated results with and without dopant-defect binding are considerably more negative than the value of -0.86 eV reported in Ref. [55]. Further insight into the nature of dopant-defect interactions and their effect on the hydrolysis reaction in Eq. 4.2 can be obtained by analyzing changes to the electronic structure as the defects are moved towards the dopant cations. First considering the proton defect, the distance between the split-off defect state highlighted in Fig. 4.2 and the oxygen 2p band is calculated to increase by 0.27 eV when protons are close to dopants; this downward shift in the defect state is consistent with the relatively large proton-dopant binding energy and can be attributed largely to an electrostatic effect. We can look into the relative energies of forming an oxygen vacancy (i.e., pyrophosphate) by comparing the electronic structure of an oxygen ion near and far from the Sr dopant. The band energies for the oxygen ion near to the dopant are higher relative to those far from the dopant, which can be understood by electrostatic repulsive interactions between the oxygen ion and the effectively negatively charged dopant. Thus it is more favorable to form a vacancy (in other words, less likely to fill a vacancy) near to a dopant, which is again consistent with the relatively large binding energy of the pyrophosphate defect.

4.5 Summary and Discussion

In summary, DFT-GGA calculations have been applied in a computational investigation of the relative energetics of protons and pyrophosphate defects and their interactions with dopant cations in Sr-doped LaPO4. The most stable pyrophosphate configuration for LaPO4 was calculated to be 0.24 eV lower than the next stable configuration. The bond lengths for all four pyrophosphate configurations are in reasonable agreement with previous calculations based on classical ionic-potential models[121]. The pyrophosphate- and proton- dopant binding energies were calculated to be 0.37 eV and 0.34 eV, respectively. Proton 27 incorporation is favored (exothermic) when the dopants are both nearest and furthest from the other defects, with calculated zero-temperature values of the hydration enthalpies being -1.66 eV, and -1.37 eV, respectively. It is interesting to compare the current results with previous calculations that have explored dopant-proton binding energies[3]. Specifically, the Sr-proton binding energy calculated here is 0.14 eV larger in magnitude than the previously-calculated Ba-proton binding energy of 0.2 eV. It is interesting to note that this trend is consistent with experimentally measured activation energies for proton conduction, which are 0.19 eV larger in Sr than Ba-doped LaPO4[10]. The larger binding energy for Sr versus Ba dopants is likely to be a result of reduced displacement of the neighboring oxygen ions that surround the Sr dopant. Specifically, there is less size mismatch between the Sr dopant and smaller La host (+8%) compared to the Ba dopant (+21%). Reduced ionic size mismatch correlates in the calculations with less distortion in the oxygen ion bond lengths around the dopant, which corresponds to higher binding energies presumably because of the strain in OH···O bond lengths arising from the dopant-induced displacements in the neighboring oxygen-ion positions. We consider next the calculated results for the effect of dopant-defect binding on the hydration enthalpy. The reason the hydration enthalpy for the case in which dopants are nearest to the other defects (i.e., protons and pyrophosphate) is approximately 0.3 eV lower compared to the case where the dopants are furthest from the other defects can be understood by considering the relative magnitudes of the pyrophosphate and proton binding energies with the dopants. Referring to Eq. 4.3, one defect structure has two protons, and the other has one pyrophosphate. The pyrophosphate binding energy is only slightly larger than the proton binding energy, which effectively cancels the energetic contribution arising from binding of the dopant for one proton and one pyrophosphate. Therefore, the only significant contribution to the hydration enthalpy for the case of nearest-neighbor dopant- defect interactions (with respect to the case of far-removed dopant-defect pairs) derives from the other proton which has a binding energy on the order of 0.3 eV. Overall, dopant-defect neighboring interactions lead to a change in the hydration enthalpy that further favors proton incorporation. While dopant-defect binding energies are found to favor proton incorporation thermodynamically, strong binding between the pryophosphate defect and the Sr dopants could give rise to kinetic limitations associated with the rate of proton uptake. Specifically, pyrophosphate-Sr binding is expected to hinder the mobility of the pyrophospate defects from the bulk to the surface, which limits proton incorporation at the surface and lowers the rate at which proton concentrations in the bulk of the material can equilibrate. Overall, the present calculated results suggest competing thermodynamic and kinetic contributions of dopant-defect binding energies underlying the process of hydrolysis of LaPO4, which may be present in other rare earth orthophosphate compounds as well. Since the calculated effects are relatively large on the scale of hydration enthalpies and migration energies calculated in previous studies[3, 26, 153, 121, 166], dopant-defect interactions should be considered in further detail in future computational studies aimed at identifying strategies for optimizing the performance of REP compounds for proton conductor applications. 28

4.6 Acknowledgements

This work was supported by the Office of Basic Energy Sciences, Materials Sciences and Engineering Division, of the U.S. Department of Energy under Contract No. DE-AC02- 05CH11231, and by the Department of Defense through the National Defense Science and Engineering Graduate Fellowship Program. This work made use of resources of the National Energy Research Scientific Computing Center, supported by the Office of Basic Energy Sciences of the U.S. Dept of Energy (DE-AC02-05CH11231). The authors would like to thank Hannah L. Ray for useful discussions. 29

Chapter 5

Energetics and Defect Interactions: Y and La Doped UO2

The energetics and defect ordering tendencies in solid solutions of fluorite-structured 3+ UO2 with trivalent rare earth cations (M =Y, La) are investigated computationally using a combination of ionic pair potential and density functional theory (DFT) based methods. Calculated enthalpies of formation with respect to constituent oxides show higher energetic stability for La solid solutions than for Y. Additionally, calculations performed for different atomic configurations show a preference for reduced (increased) oxygen vacancy coordination around La (Y) dopants. The current results are shown to be qualitatively consistent with related calculations and calorimetric measurements of heats of formation in other trivalent doped fluorite oxides, which show a tendency for increasing stability and increasing preference for higher oxygen coordination with increasing size of the trivalent impurity. The implications of these results are discussed in the context of the effect of trivalent impurities on oxygen ion mobilities in UO2, which are relevant to the understanding of experimental observations concerning the effect of trivalent fission products on oxidative corrosion rates of spent nuclear fuel.

5.1 Forward

The work presented in this chapter was published by J. M. Solomon, V. Alexan- drov, B. Sadigh, A. Navrotsky, and M. Asta, in Acta Materialia, vol. 2, issue 4, pages 1047-1053 (2014), and is reproduced here with permission of the co-authors and publishers. ©Elsevier

5.2 Introduction

Fluorite-structured oxide compounds can often be doped with high concentrations of trivalent cations. Under reducing conditions this doping leads to the formation of charge compensating oxygen vacancy defects. Recent calorimetry studies on trivalent-cation doped ZrO2[92], HfO2[91], CeO2[38] and ThO2[7] provide evidence for strong defect association 30 in such systems. The experimental findings have been shown to be consistent with recent computational work on some of these same systems, which demonstrate energetic binding/clustering tendencies between cations and oxygen vacancies, to a degree that has been found to correlate strongly with ionic radii[127, 76, 106, 31, 30, 123, 8, 167]. Strong oxygen vacancy-cation association reduces the mobility of oxygen ions and therefore the ionic conductivity in aliovalently-doped fluorite oxides, which can be detrimental for their use in fuel cell electrolyte and oxygen sensor applications[73]. Similar effects have also been discussed in the context of the electrochemical behavior of spent nuclear fuel composed of fluorite-structured UO2 with dissolved trivalent fission products. Specifically, recent studies have attributed rare earth (i.e., trivalent) fission product dopant-oxygen vacancy clusters to the decreased rate of oxidative dissolution with increasing burnup in spent fuel[67]. The defect clusters are believed to reduce the concentration of mobile oxygen vacancies which facilitate oxidation and eventual dissolution of the fuel, in a manner similar to that by which rare earth doping impedes air oxidation of the cubic fluorite structure to orthorhombic U3O8[150, 39, 40, 101]. Oxidation and dissolution of nuclear fuel in aqueous environments is of particular concern at the back end of the fuel cycle, i.e., waste disposal[151, 138]. Because of the significant changes in oxidation and corrosion behavior of nuclear fuel associated with rare earth fission products, understanding the thermochemical properties of these defects in UO2 is of particular interest. The effect of rare earth fission product doping on the oxygen chemical potential in UO2 has been studied extensively[137]. How- ever, the energetics and defect ordering tendencies of these systems remain far less studied compared to other fluorite-structured oxides[127, 76, 106, 31, 30, 123, 8, 167]. Specifically, only one calorimetry study[99] and one ionic pair potential modeling investigation[104] have been reported to date. The aim of the current work is to provide further insight into the thermochemical behavior of UO2 doped with trivalent cations through the application of density functional theory (DFT) based calculations in which we explore stability and defect ordering/clustering trends with respect to relative cation sizes. It is expected that UO2 substituted with trivalent cations will contain oxygen vacancies or localized electron holes (i.e., U5+) as charge compensating defects. The current work focuses only on systems with oxygen vacancies (oxygen deficiency rather than oxygen excess). We consider the smallest 3+ and largest trivalent rare earth-fission product cations that are soluble in UO2, namely, Y (1.02 A˚ ionic radius in eightfold coordination) and La3+ (1.16 A)[135].˚ The paper is organized as follows. In the next section the details of the calculations are described. This is followed by a presentation of the results for formation energetics and defect interactions in Section 5.4. In Section 5.5 the results are discussed in the context of the previous models proposed for explaining the effect of cation size on the stability and defect ordering behavior in aliovalently-doped fluorite structures.

5.3 Computational Methodology

The computational approach employed in this work is similar to that applied in studies of doped ZrO2 and ThO2 by Bogicevic et al.[31, 30] and Alexandrov et al.[8], respectively. The approach involves calculations of a relatively large number of ordered 31 defect-dopant conﬁgurations as a basis for exploring trends in energetic stability and defect ordering. The following subsections provide detailed descriptions of the structural enumeration techniques (Sec. 5.3.1), the screening of the structures using ionic pair potential models (Sec. 5.3.2), and the DFT based calculations (Sec. 5.3.3) implemented in this work.

5.3.1 Structure Enumeration

In order to determine the most stable defect structures for ﬂuorite-structured UO2 doped with trivalent cation (M3+) oxides, we investigate many hypothetical cation-vacancy arrangements. The cation sublattice contains host and dopant species (i.e., U4+ and M3+, respectively), and the anion sublattice contains host and vacancy species (i.e., O2− and Ovac); the substitution of two U4+ by two M3+ can be charge-compensated by one Ovac, as illustrated in Fig. 5.1. We employ a structure enumeration technique developed for cluster expansion based studies of alloy thermodynamics, considering supercells consisting of up to six formula units, employing an algorithm by Hart and Forcade[65] that is implemented in the alloy theoretic automatic toolkit (ATAT)[157, 156]. The compositions considered are U4M2O11 (x=1/3, six formula units: 4 U ions, 2 M ions, 11 O ions, and 1 Ovac), U2M2O7 (x=1/2, four formula units: 2 U ions, 2 M ions, 7 O ions, and 1 Ovac), and U2M4O10 (x=2/3, six formula units: 2 U ions, 4 M ions, 10 O ions, and 2 Ovac). The structure enumeration yielded 117, 27, and 710 symmetry-distinct structures for x=1/3, 1/2, and 2/3, respectively.

Figure 5.1: The ideal UO2 cubic ﬂuorite structure with two trivalent dopants on the cation fcc sublattice and one charge compensating oxygen vacancy on an anion simple cubic sublattice is shown. 32

5.3.2 Classical Pair Potentials The energies of all enumerated structures were calculated using ionic pair-potential models as a ﬁrst “screening” step, with the intent of eliminating from consideration by more computationally demanding DFT based calculations the electrostatically unfavorable structures for a given composition and M3+ species. In these calculations the potential energy between ions i and j is modeled through pair potentials of the following form:

qiqj −rij C E(rij) = + A exp ( ) − 6 (5.1) rij ρ rij

The ﬁrst term represents the coulombic interaction where qi is the respective ionic charge and rij is the distance between ions i and j. The latter two repulsive and attractive terms represent the short range Buckingham potential, where A, ρ, and C are system-dependent parameters. In addition, polarizabilities are introduced for individual ionic species according to the shell model as formulated by Dick and Overhauser[45]. The Buckingham and shell model parameters used in this work are given in Refs. [25, 32, 60]. Geometry relaxations and energy minimizations were performed using the General Utility Lattice Program[54].

5.3.3 First-principles Calculations All structures for a given composition and M3+ species were ranked according to their total energy based on the ionic pair potential calculations described above. Initially DFT calculations were performed for the resulting six lowest energy structures for a given composition. If any of the five higher energy structures were found to be lower than the ground state predicted by the pair potential models, additional structures were considered in an effort to ensure that the lowest energy configuration was identified by the DFT based methods. Most of the results reported below were obtained using DFT within the formalism of the projector augmented-wave (PAW) method[27, 87] and the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)[117, 118] as implemented in the Vienna ab initio simulation package (VASP)[85, 86]. In these calculations, a Hubbard-U correction was implemented within the formalism of Dudarev et al.[50] to account for the self- interaction error for localized 5f electrons in uranium. The value of the parameter Ueff = U-J in the DFT+U formalism was chosen to be 4.00 eV for this study; this value is similar to that found to give optimal results in comparisons to spectroscopic data for UO2 by Du- darev et al. (3.99 eV)[50], and was found to provide good agreement between calculated and measured oxidation enthalpies of UO2[64]. The PAW potentials used in this work are U, O, Y sv, and La in the VASP PBE library. The PAW potentials use 14 valence electrons for U (6s26p65f36d17s2), 6 for O (2s22p4), 11 for Y (4s24p65s24d1), and 11 for La (5s25p66s25d1). We employ a plane-wave cutoff energy of 500 eV, and the Brillouin zone is sampled using the Monkhorst-Pack scheme using a 4 x 4 x 4 k-point mesh for all structures. Atomic positions were relaxed with no symmetry constraints until residual forces were below approximately 20 meV/A.˚ From convergence checks with respect to plane-wave cutoff and k-point sampling we estimate the formation enthalpies to be converged at the level of a few tenths of a kJ/mol-cation. The results of the first-principles calculations presented below made use of 33 a scalar-relativistic approximation for valence electrons. We also performed calculations of the lowest energy Y and La-substituted structures for x=1/2 including spin-orbit coupling, which was found to change the formation enthalpies by less than 1 kJ/mol-cation. The formation enthalpy for a solid solution with respect to its constituent oxides is defined as

∆Hf = E[U1−xMxO2−0.5x] − (1 − x)E[UO2] − xE[MO1.5], (5.2) where E[U1−xMxO2−0.5x] denotes the energy per cation of a solid solution with cation dopant concentration x, E[UO2] denotes the energy per cation of fluorite-structured UO2 (calculated here using a 12-atom unit cell with 1k antiferromagnetic order), and E[MO1.5] denotes the energy per cation of Y2O3 or La2O3 in the experimentally observed (i.e., room temperature) cubic bixbyite (C-type sesquioxide) or hexagonal (A-type) structure, respectively. Both ferromagnetic and antiferromagnetic arrangements for the uranium ion magnetic moments were considered within the solid solution defect structures, and the latter were found to be lower in energy. Because there are four uranium atoms in the structures associated with x=1/3, there are three unique antiferromagnetic spin arrangements (i.e., up-up-down- down, up-down-up-down, up-down-down-up). We found little difference in the calculated formation enthalpies with respect to each of these magnetic configurations (i.e., within 0.2 kJ/mol-cation). In calculations that make use of the DFT+U methodology, it is well documented in the literature that one can converge to multiple metastable states corresponding to different orbital occupancies for the localized f electrons[47, 170, 171]. Thus, a practical issue in such calculations is to ensure that the computed energy corresponds to the true electronic ground state (or close to it)[136, 90]. Several different methods have been proposed to address this issue for UO2[56, 171, 46, 49, 102]. In this work we employ the approach by Meredig et al.[102], involving a gradual “ramping” up of the Hubbard-U parameter. In the current implementation, Ueff starts at zero and is increased stepwise by 0.1 eV in order to localize the U 5f electrons. This methodology offers sufficient computational efficiency for treating many compounds with relatively large unit cells. In order to be confident of the accuracy of the “U-ramping” method, we compared results obtained by this approach with those derived by the occupation matrix control (OMC) method, which has been applied to plutonium oxides (PuO2 and Pu2O3)[75] and subsequently UO2[46, 49]. The OMC approach involves initializing calculations with many distinct imposed values of the diagonal and off-diagonal occupation matrices for the U 5f electrons, and taking the lowest of the resulting calculated energies as the ground-state. For comparison of the “U-ramping” and OMC methods we performed static DFT+U calculations of a simple defect-fluorite structure, namely, ULa2O5, which contains a single oxygen vacancy. We used the OMC approach discussed in Ref. [46] in which a total of 61 occupation matrices were sampled (21 diagonal, 40 off-diagonal). The ramping method gave the lowest total energy by approximately 2.4 kJ/mol-cation compared the lowest total energy determined using OMC. A more exhaustive sampling of matrices in the OMC approach could potentially result in energies that are closer to (or perhaps even lower than) that resulting from the “U-ramping” approach; however, it is encouraging that none of the energies derived from the OMC method were significantly lower than the value obtained from the ramping approach. 34

Separate DFT calculations using a hybrid functional[23] were implemented to compare results to formation enthalpies calculated using the DFT+U method. The hybrid- functional methodology has been shown to accurately describe lattice constants and electronic structure of UO2[89, 125, 132, 160, 161]. The “local hybrid functional for correlated electrons” (LHFCE), which is also referred to as “exact exchange for correlated electrons” (EECE), is a hybrid-functional approach in which the exact exchange is applied locally within the atomic spheres only to the correlated electrons (i.e., 5f uranium orbitals in the current work). The LHFCE functional is less computationally demanding compared to the full hybrid functional, and has been shown to accurately describe lattice constants, electronic structure, and defect formation energies of UO2[74, 42]. The LHFCE calculations were performed using the all-electron full- potential lin- earized augmented plane wave plus local orbitals basis (FP-L/APW+lo) method as implemented in the WIEN2k code. Details of the LHFCE method implementation in WIEN2k are discussed in Ref. [113]. The hybrid exchange-correlation energy functional used in this work has the form

PBE+αHF HF PBE PBE ∆EXC = αEX + (1 − α)EX + EC , (5.3) HF PBE PBE where αEX is the HF exchange, EX and EC are the PBE exchange and correlation energies, respectively, and α represents the fraction of HF exchange that replaces the PBE exchange. The value for α used in this work is 0.25, which is representative of the PBE0 functional [2]. In the FP-L/APW+lo method the convergence of the basis set is controlled by a cutoff parameter RrmtKmax, where Rrmt is the smallest atomic sphere radius in the cell and Kmax is the magnitude of the largest wavenumber of the plane wave expansion. Additional local orbitals were added for U 6s, 6p, 4d, 5f and for O 2s states. The Rrmt values were set to 2.3 Bohr, 1.6 Bohr, 2.26 Bohr, and 2.3 Bohr for U, O, Y, and La, respectively, and RrmtKmax = 8. The Brillouin zone was sampled with 32 k-points and the integrations were conducted by the tetrahedron method of Blöchl, Jepsen, and Andersen[27]. The calculations were performed spin polarized with antiferromagnetic configurations for the structures containing uranium atoms. The energy, charge, and force convergence criteria used for these computations are 1 meV, 0.0001 and 26 meV/A.˚

5.4 Results

The results for all calculations presented below were computed using the GGA+U method, unless explicitly stated otherwise. We ﬁrst investigate the energetic stability of U1−xMxO2−0.5x (M=Y, La) through consideration of the formation enthalpy ∆Hf with respect to constituent oxides, as deﬁned in Eq. 5.2. Results for the lowest energy structures of Y and La-substituted UO2 are shown in Fig. 5.2 and Table 5.1. The positive values of ∆Hf for Y-containing solid solutions indicate that phase separation is favored at low temperatures, whereas the negative values of ∆Hf for La-containing solid solutions indicate a compound forming tendency. The formation enthalpy values for calculations with the hybrid functional are consistent with the DFT+U calculations for both Y and La-substituted systems, with respect 35

Table 5.1: Formation enthalpies (kJ/mol-cation) of U1−xMxO2−0.5x (M=Y,La) structures are listed for low-energy fully relaxed structures enumerated in this study. The structures listed have formation energies within 3 kJ/mol-cation of the lowest energy structure for each trivalent cation species at each composition considered. The fourth column lists the direction of ordering of oxygen vacancies. The last column lists the number of trivalent cations in the four tetrahedral nearest neighbor positions surrounding an oxygen vacancy. # M-Vac

x M ∆Hf Direction Neighbors 2/3 La -1.6 h110i 2 Y 10.8 h110i 3 11.3 h211i 3 12.3 h211i 3 12.6 h211i 3 12.6 h110i 3 1/2 La -1.3 h110i 1 1.4 h110i 1 2.1 h110i 1 Y 11.5 h110i 3 1/3 La -5.2 h211i 0 Y 7.5 h211i 2 8.3 h211i 2 8.9 h211i 2 9.2 h110i 3 36

to the order of magnitude and sign of ∆Hf , and the trend with substitution level, as shown in Fig. 5.2. A comparison of the filled and open symbols, corresponding respectively to GGA+U and hybrid-functional results, show agreement at the level of a few kJ/mol-cation. This good level of agreement between calculated results obtained using different codes and functionals is encouraging. We consider next the relative ordering of the cation-vacancy configurations giving the lowest formation enthalpies. Table 5.1 lists, for the most stable configurations, the formation enthalpy and structural features including the direction of oxygen vacancy ordering and the number of trivalent cations (per vacancy site) that are nearest neighbors to an oxygen vacancy (for structures with x=2/3 that contain two oxygen vacancies, the reported number of nearest-neighbor cations is an average over both vacancy sites). In the fluorite lattice, each oxygen vacancy has four cation nearest neighbors, so that the maximum number of M3+ nearest neighbors to the vacancy site is four. We observe that the oxygen vacancies prefer to align as either second or sixth nearest neighbors along the h110i and h112i directions, respectively. In the lowest-energy structures for Y-substituted systems, a higher number of M3+ neighbors around the vacancies are preferred compared to the La-substituted systems for all compositions. The formation enthalpies for all structures computed by the GGA+U method in this study are shown in Fig. 5.3. The wide range of enthalpies for each composition illustrates the magnitude of the dependence of the energy on the defect configuration. Y- substituted systems for x=1/3 and 2/3 show several nearly degenerate low energy structures, whereas the lowest-energy configuration for La-substituted systems shows a larger gap between the lowest and next-lowest energy structures for all three compositions. The results in Fig. 5.3 again highlight the stronger energetic preference for M3+ binding with oxygen vacancies in the Y- versus La-substituted systems.

5.5 Discussion

To gain additional insights into the factors governing the stability of trivalent impurities in UO2, we compare below the main results of the present work with those derived in previous computational and experimental studies of the energetics of aliovalently doped fluorite-structured oxide solid solutions[8, 31, 38, 7]. Additionally, the nature and energetics of cation-vacancy ordering derived from the present results are analyzed, and the consequences of these findings for oxygen ion mobilities are discussed. The formation enthalpies presented above for UO2-based solid solutions are lower than those computed in a previous DFT study of ThO2-based systems[8]. Considering that the ionic radius of U4+ (1.00 A)˚ is smaller than that of Th4+ (1.05 A),˚ these comparisons suggest a trend towards higher energetic stability (lower formation enthalpies) with decreasing size of the 4+ host cation, for a given M3+ impurity. This trend is also observed if the present calculated values of ∆Hf for Y-doped UO2 are compared with those derived in previous DFT calculations for Y-doped ZrO2[31]; the latter are found to be significantly lower in energy, consistent with the smaller ionic radius of Zr4+ (0.84 A).˚ The effect of the 4+ cation size is summarized for Y-doped solid solutions in Fig. 5.4 (similar behavior is also found for La-doping, although fewer results are available in the 37

Figure 5.2: Formation enthalpies of U1−xMxO2−0.5x (M=Y,La) structures are shown for the lowest-energy fully relaxed structures of all compositions enumerated in this study. Filled and open symbols correspond to DFT+U and hybrid-functional calculations, respectively. The dotted line connecting the solid points is a guide to the eye. 38

Figure 5.3: The formation enthalpies calculated using DFT+U for all structures computed in this study are shown. Each structure is denoted by the number of trivalent cations in the four tetrahedral nearest neighbor positions surrounding an oxygen vacancy according to the legend in the top left of the figure. Note that there are two vacancy sites for x=2/3, and the average number of neighbors between the two sites (rounded to the nearest neighbor) is shown. 39 literature). This trend is consistent with observations derived from calorimetric experiments by Chen and Navrotsky[38] and Aizenshtein, Shvareva and Navrotsky[7]. In an analysis of the measured formation enthalpies across trivalent-doped ThO2, CeO2, ZrO2 and HfO2, these authors have argued that the enhanced stability of systems with smaller 4+ cations is associated with a preference for reduced oxygen ion coordination. Specifically, the formation of a fluorite-structured solid solution from AO2 and M2O3 constituents necessarily leads to a decrease in the average oxygen coordination surrounding the A4+ cation, relative to the eightfold coordination characteristic of the pure AO2 constituent. For small cations such as Zr4+ and Hf4+, which are known to prefer sevenfold coordination, this effect is expected to be stabilizing. We consider next the trends in ∆Hf with respect to the size of the trivalent cation, for a fixed 4+ cation species. Specifically, the trend towards more positive formation energies for the smaller Y versus the larger La trivalent cations is consistent with both calorimetric and computational results in CeO2 and ThO2 systems[38, 7, 8]. It has been suggested[38, 7] that this trend can be attributed to the smaller lattice distortions that have been observed to occur when larger trivalent cations are substituted into oxygen-vacancy charge-compensated fluorite-structured oxide compounds. Specifically, pair potential calculations of La- and Y- substituted CeO2 show much smaller changes in the nearest neighbor cation-cation nearest neighbor distances for La- relative to Y-substituted structures[66]. We find similar trends in the present work: La-U and U-U distances in U0.5La0.5O1.75 structures are computed to be at most 0.08 A˚ different from the nearest neighbor spacing on the cation sublattice of pure UO2 (the differences in bond distances are due to both volume change and local bond distortions), while for U0.5Y0.5O1.75 structures these bond-length distortions (i.e., Y-U, U- U) are as large as 0.20 A.˚ The DFT calculations are performed at 0 K, whereas the calorimetric measurements, for which comparisons in systematic trends are made, correspond to room temperature. At finite temperature, additional contributions to the formation enthalpy arise (e.g., due to vibrational excitations) which have not been considered in the present work. To quantify the relative importance of these contributions, we consider a previous study[22] on Y-doped UO2. In this study it was reported that the heat capacities for the solid-solution samples showed approximately 30% deviation from Kopp’s law, i.e., from a concentration- weighted linear combination of the heat capacities of the constituent oxide components. We fit the heat capacity versus temperature data reported in [22] from 0-298 K to a third order polynomial, and integrated over this temperature range to compute a conservative estimate of the thermal contributions to the enthalpy at room temperature. From this value we subtracted the contribution to the enthalpies of the constituent compounds, given by inte- grating the Kopp’s law heat capacity. The resulting estimate of the thermal contribution to the formation enthalpy at 298 K is approximately 0.3 kJ/mol-cation. This contribution is roughly an order of magnitude smaller than the formation enthalpies reported in this work. Therefore, we conclude that thermal effects lead to minimal contributions to the measured formation enthalpies at room temperature, and that their neglect in the DFT calculations is warranted for the purposes of the present study. An additional consideration that should be addressed in the comparison of DFT calculations to measured formation enthalpies relates to the nature of the configurational 40 order in experiments versus calculations. For the systems considered in this study, the experimental samples are known to be solid solutions, and lack long-range order characteristic of the compounds considered in the DFT calculations. Comparisons between such DFT calculations and experiments can be justified if the latter display appreciable short- range order. While substantial ordering effects have been reported in experimental studies on similar fluorite-structured solid solutions[7, 92, 91, 38], it is difficult to quantify whether the effects are sufficiently pronounced for direct comparison with DFT calculations on fully ordered compounds. Nevertheless, it is assumed that the trends with respect to trivalent cation species should be comparable regardless of the extent of order/disorder, as was found to be the case in previous studies for La and Y substituted ThO2. Specifically, it was found that the energetic trends for ThO2 with respect to dopant size are consistent whether based on DFT results for ordered systems[8], experimental measurements for samples with some degree of configurational short-range order, or estimates of the energetics of fully disor- dered solid solutions derived from experimental data[7]. Since such trends are the focus of the current work, we believe comparisons between experimental measurements and DFT calculations are justified. We consider finally an analysis of the nature of the energetically favorable cation- vacancy ordering tendencies for the UO2-M2O3 structures, considering first the lowest- energy structure of composition U2La2O7 (x= 1/2), which is computed to have a negative formation enthalpy in the present study, suggesting a tendency towards stable compound formation. We focus on this particular stoichiometry, as it is characteristic of the pyrochlore structure (A2B2O7) which is found to be a thermodynamically stable compound for many fluorite-based systems[147]. The pyrochlore structure is a derivative of fluorite where the A and B cations are ordered along the h110i direction, and the tetrahedrally-coordinated oxygen vacancies are surrounded by adjacent A4+ cations. The configuration of the pyrochlore structure requires eight formula units (4 U ions, 4 La ions, 14 O ions, and 2 Ovac), and was therefore not considered in our structure enumerations, which considered compounds with four formula units for x=1/2. However, we performed an additional calculation of the formation enthalpy for pyrochlore-structured U2La2O7, obtaining a value of ∆Hf =+5.8 kJ/mol-cation, which is roughly 7 kJ/mol-cation higher in energy than the lowest-energy configuration for ∆Hf identified in this work. The relatively high energy of the pyrochlore structure for this system is consistent with arguments suggested previously based on the B3+/A4+ cation size ratios required for pyrochlore stability[147]. To gain insight into the nature of the preferred interactions between the trivalent dopants and charge-compensating oxygen vacancies in UO2-M2O3 solid solutions, we plot in Fig. 5.5 formation enthalpies for three structures featuring different number of trivalent cation/oxygen vacancy nearest neighbors, for both M=La and Y. The three structures considered have the same vacancy ordering pattern (i.e., aligned along h110i) and one, two or three M3+-vacancy neighbors. The configurations chosen are 1) the lowest energy Y-substituted structure (three M3+-Ovac neighbors), 2) the lowest energy La substituted structure (one M3+-Ovac neighbor), and 3) a structure with two M3+-vacancy neighbors that is within 4 kJ/mol-cation of the lowest energy structure for both La- and Y-substituted systems. The results in Fig. 5.5 demonstrate that Y and La dopants display qualitatively 41

Figure 5.4: Calculated formation enthalpies for Y-substituted AO2 fluorite systems are plotted versus the radius of the host A4+ cation radius. Results for zirconia (A=Zr) and thoria (A=Th) systems were taken from Refs. [31] and [8], respectively. 42 different preferences; the formation enthalpies for Y- and La- substituted structures decrease (more exothermic) and increase (more endothermic), respectively, with respect to the number of M3+-vacancy neighbors. These results suggest a significant energetic preference for M3+-oxygen vacancy and U4+-oxygen vacancy clustering in Y and La containing systems, respectively. In both cases, these clustering tendencies would be expected to lower oxygen-ion mobility, as discussed previously in the context of ionic conductivity in fluorite- structured solid electrolytes[73]. The current results thus support the suggestion from Ref. [67] that the effect of increased burnup (leading to higher concentrations of soluble trivalent fission products) on oxidative corrosion rates in spent nuclear fuel can be attributed to lower oxygen mobility due to defect association.

5.6 Conclusions

In this study DFT based computational methods were employed to study formation energetics and defect ordering tendencies in UO2 compounds substituted with Y and La cations, which are common soluble fission products in nuclear fuel. We consider substitutional configurations that are charge-compensated with oxygen vacancies, finding that phase separation is energetically favored for all compositions considered for Y-substituted UO2, whereas compound formation is favored for La-substituted UO2. The calculations are thus characterized by a trend towards increasing energetic stability of solid solutions with the larger La impurity relative to the smaller Y impurity. This trend is consistent with previous computational and calorimetric studies[38, 7, 8] for fluorite-structured solid solutions based on CeO2 and ThO2. We observe in general that oxygen vacancies prefer to align as either second or sixth nearest neighbors on the oxygen sublattice along the h110i and h112i directions, respectively, suggesting that the energetically preferred defect ordering configurations are characterized by a repulsive interaction between nearest neighbor oxygen vacancies. In the Y-substituted systems, structures with an enhanced number of vacancies around the Y3+ ions are found to be favored energetically, while the opposite is found for La-substituted systems. The pronounced energetic preferences for dopant cation-vacancy or host cation- vacancy clustering are expected to have an important effect on oxygen ion mobilities and therefore the oxidative corrosion rates in spent nuclear fuel.

5.7 Acknowledgments

J.M. Solomon was supported by the Oﬃce of Basic Energy Sciences of the U.S. De- partment of Energy as part of the Materials Science of Actinides Energy Frontier Research Center (DE-SC0001089) for initial DFT+U calculations, the U.S. Department of Energy through the Lawrence Livermore National Laboratory (DE-AC52-07NA27344) for the hybrid calculations, and the Department of Defense through the National Defense Science & Engineering Graduate Fellowship Program for the remainder of the work. V. Alexandrov, A. Navrotsky, and M. Asta were supported by the U.S. Department of Energy as part of the Materials Science of Actinides Energy Frontier Research Center (DE-SC0001089). 43

Figure 5.5: The formation enthalpies calculated by DFT+U for selected structures containing, one, two, and three trivalent cation nearest neighbors to an oxygen vacancy are shown for Y and La-substituted systems at x=1/2. 44

B. Sadigh was supported by the U.S. Department of Energy through the Lawrence Liver- more National Laboratory (DE-AC52-07NA27344). This work made use of resources of the National Energy Research Scientiﬁc Computing Center, supported by the Oﬃce of Basic Energy Sciences of the U.S. Dept of Energy (DE-AC02-05CH11231). The authors would like to thank B. E. Hanken and L. Zhang for useful discussions. 45

Chapter 6

Energetics and Defect Interactions: Rare-Earth Doped UO2

Trends in the energetics and defect clustering tendencies for UO2 compounds substituted with trivalent rare earth cations (M3+) are investigated computationally using methods based on density functional theory. Higher energetic stability of U1−xMxO2−0.5x solid solutions relative to constituent oxides and increased preference for higher oxygen coordination around the trivalent cation are found with increasing size of the M3+ species. The implications of the computational results for the eﬀect of trivalent ﬁssion products on oxygen ion mobility in spent fuel are discussed.

6.1 Forward

The work presented in this chapter was published by J. M. Solomon, A. Navrot- sky, and M. Asta, in Journal of Nuclear Materials, vol. 457, pages 252-255 (2015), and is reproduced here with permission of the co-authors and publishers. ©Elsevier

6.2 Introduction

A key concern of the nuclear materials community is the assurance of safe long- term storage of spent nuclear fuel. In this context, an issue that has received considerable attention is the effect that the substitution of U4+ cations by soluble trivalent rare earth fission product cations (M3+) has on the rate of oxidation and oxidative corrosion in uranium oxide (UO2)[129, 128, 101, 78, 150, 100, 6]. In particular, recent studies[67, 129] attributed the trend towards decreased oxidative dissolution rates with increasing burnup, to the interactions between cations and charge-compensating defects (i.e., oxygen vacancies) in the formation of stable defect clusters. The authors in Ref. [129] found evidence of such defect clusters with increasing trivalent rare earth doping level using Raman spectroscopic techniques. These clusters may form by the thermodynamically-favorable association of trivalent substitutional atoms (M3+) with charge-compensating oxygen vacancies. The formation of such stable clusters has been hypothesized to reduce the concentration of mo- 46 bile (unbound) oxygen vacancies, and thus is expected to lead to a lowering of oxygen ion diffusivity[129, 128, 67]. Cluster formation may thus be expected to reduce the rate of oxidation and eventual dissolution of the fuel in repository waste storage containers for both thermodynamic (by stabilizing the solid solution) and kinetic (by reducing oxygen mobility) reasons. While the effect of rare earth substitution on oxygen chemical potentials, lattice constants and thermal conductivity in UO2 has been well studied (e.g., [137, 53] and references therein), and initial experimental and computational studies of formation energetics in urania-rare earth solid solutions have been recently reported[99, 141], explicit investigations of defect-clustering behavior has been addressed in relatively few studies[141, 116, 104]. Computational modeling based on classical ionic pair potentials[104] and defect chemistry models[116] conclude that defect clustering is energetically favorable (as it must be in order for clustering to occur, because clustering decreases the entropy); however, a detailed study of systematic trends in energetics and clustering tendencies with varying concentrations and types of rare earth trivalent cations (M3+) has not been reported to the best of our knowledge. As a consequence, understanding the effects of defect clustering on solid-solution stability and oxygen vacancy mobilities, which is key in understanding fuel oxidation/corrosion behavior, remains incomplete. Previous studies[106, 31, 30] on trivalent-doped fluorite systems concluded that oxygen vacancy mobility is highest when there is little energetic preference for a particular defect ordering arrangement. In a previous computational study by the authors[141] employing density-functional-theory-based methods, different energetic properties and defect clustering tendencies were obtained for oxygen-vacancy-charge-compensated UO2 systems (U1−xMxO2−0.5x) containing the smallest and largest rare earth trivalent cations (M), Y and La, respectively. That work suggested that the magnitude and nature of vacancy interactions with trivalent cations are influenced by the ionic radius of the M3+ species. However, the results were based on calculations for only two rare earth cations, and further work is warranted to better establish and understand such trends. In the present work, the energetics are further investigated by considering a series of trivalent rare earth cations substituted in UO2 solid solutions with charge compensating oxygen vacancies. Specifically, 3+ we consider systems of composition U0.5M0.5O1.75, with M being Y (1.02 A,˚ Shannon radius with eightfold coordination[135]), Dy3+ (1.03 A),˚ Gd3+ (1.05 A),˚ Eu3+ (1.07 A),˚ Sm3+ (1.08 A),˚ Pm3+ (1.09 A),˚ Nd3+ (1.11 A),˚ Pr3+ (1.13 A),˚ Ce3+ (1.14 A)˚ and La3+ (1.16 A).˚ The results reveal clear trends that are discussed in the context of models that have been proposed for other trivalent-doped fluorite-structured oxide systems. We note that in the current calculations we do not consider uranium in oxidation states higher than +4, so the total vacancy concentration is fixed by the amount of rare earth substitution, with no oxygen interstitials present.

6.3 Computational Methodology

The computational approach employed in this work has been described previously[141], and is summarized briefly in this section. The lowest-energy structure for a given concentration of a particular M3+ species (and corresponding number of oxygen vacancies) 47 is determined by calculation of many hypothetical structures with different configurations of U4+ and M3+ on the cation sublattice, and oxygen ions and vacancies on the anion sublattice of the fluorite structure. This involves the use of ionic pair potential models[59, 105, 106, 32, 146] to screen out configurations with large energies, and the use of DFT-based methods to determine energies of the lowest-energy structures. The DFT energies were obtained within the formalism of the projector augmented-wave (PAW) method[27, 87] and the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)[117, 118] as implemented in the Vienna ab initio simulation package (VASP)[85, 86]. A Hubbard-U correction in the DFT+U formalism of Dudarev et al.[50] with Ueff = U-J = 4.00 eV was employed for the treatment of the uranium 5f orbitals in this study. We employ the approach by Meredig et al. [102] involving a gradual ramping up of the Hubbard-U parameter. In our previous work[141] we compared this ramping method to the Occupa- tion Matrix Control (OMC) method by Dorado et al.[46], in which we sampled 61 different occupation matrices for a trivalent doped fluorite UO2 system charged-compensated with oxygen vacancies, and found that the ramping method gave rise to a configuration with a lower total energy than that derived using OMC. The PAW potentials used in this work are U, O, Dy 3, Gd 3, Eu 3, Sm 3, Pm 3, Nd 3, Pr 3, and Ce 3 in the VASP PBE library, where the “ 3” denotes that the occupied 4f orbitals in the trivalent oxidation state of the rare-earth ions are treated as core electrons. We employ a plane-wave cutoff energy of 500 eV, and the Brillouin zone is sampled using the Monkhorst-Pack scheme with 4 x 4 x 4 k-point meshes. Atomic positions were relaxed with no symmetry constraints until residual forces were below approximately 20 meV/A.˚ From convergence checks with respect to plane-wave cutoff and k-point sampling we estimate the formation enthalpies are converged at the level of a few tenths of a kJ/mol-cation. The formation enthalpy for the U0.5M0.5O1.75 compounds considered here, with respect to its constituent oxides, is defined as

∆Hf = E(U0.5M0.5O1.75) − [E(UO2) − E(MO1.5)]/2, (6.1) where the energies (E) are expressed per cation. In Eq. 6.1, E(UO2) is for fluorite- structured UO2 (calculated here using a 12-atom unit cell with 1k antiferromagnetic order), and E(MO1.5) is for M2O3 in the experimentally observed (i.e., room temperature) cubic bixbyite (C-type sesquioxide) or hexagonal (A-type) structure. The experimentally observed C-type sesquioxides are Y2O3, Dy2O3, Gd2O3, Eu2O3, Sm2O3, and Pm2O3, and the A-type sesquioxides are Nd2O3, Pr2O3, Ce2O3, and La2O3[172]. In this study we investigated M3+ substitution levels that are high relative to typical fission product yields. This was done for several reasons. It allows for the practical application of high throughput calculations for systems with various dopant species. It also allows for direct comparison with previous results that considered M3+= Y3+ and La3+, which were compared to other fluorite oxide systems where high M3+ substitution levels are common[141]. 48

6.4 Results and Discussion

We begin by analyzing the calculated formation enthalpies for the lowest-energy 3+ structures of U0.5M0.5O1.75 plotted as a function of the Shannon ionic radius of the M cation for eightfold coordination (Fig. 6.1). Positive values of ∆Hf indicate that phase separation into UO2 and M2O3 is favored (endothermic) at low temperatures, whereas negative values of ∆Hf indicate a compound forming tendency (exothermic). Figure 6.1 illustrates that compound formation is preferred (weakly) only for the largest M3+ cation sizes, while the formation enthalpy is increasingly positive as the M3+ ionic radius decreases. The magnitudes of the formation enthalpies are relatively small in comparison to the magnitude of the configurational entropy contribution to the Gibbs free energy at temperatures relevant to the processing and service of nuclear fuels. For example, for a U1−xMxO2−0.5x system with x = 0.5 the ideal configurational entropy leads to a contribution of -12 kJ/mol-cation to the Gibbs free energy of formation at T=1000 K, which is larger than the magnitude of ∆Hf for all the trivalent species plotted in Fig. 6.1. Conse- quently the formation free energy is expected to be negative at such temperatures for all of the systems considered in this work. This result is consistent with the fact that rare-earth fission products typically show solubility in UO2 over wide composition and temperature ranges[80]. We consider now the defect-clustering tendencies in UO2-MO1.5 systems by exam- ining the nature of preferred interactions between M3+ and oxygen vacancies. The formation enthalpies of three low-energy structures with one, two, and three M3+-oxygen vacancy nearest neighbors averaged over all M3+ ions in the system (two total) are compared for each M3+ species in Fig. 6.2, denoted by the blue, green, and red circles, respectively. The structural motifs of the three structures are represented in Fig. 6.3. The results in Fig. 6.2 demonstrate that the small and large M cations display qualitatively different preferences; the formation enthalpies of systems containing smaller and larger M cations decrease (more exothermic) and increase (more endothermic), respectively, with increasing number of M3+-vacancy neighbors. For an M cation size of 1.09 A˚ (M=Pm), the formation enthalpies for the three configurations are nearly degenerate, i.e. there is little energetic preference for M3+-oxygen vacancy and U4+-oxygen vacancy nearest neighbors. Similar observations of a trivalent cation size for which there is an apparent lack of preferred clustering tendencies has been found in previous calculations for CeO2[106, 14] and ZrO2[31, 167] fluorite structures. The trends in Figs. 6.1 and 6.2 are consistent with models presented in the literature to explain similar results in other fluorite-structured oxide systems[38, 7]. We first consider the trend of lower formation enthalpy (more exothermic) with increasing size of the M3+ rare earth ion in Fig. 6.1. U4+ prefers an eightfold oxygen ion coordination environment. While of similar size to U4+, the rare-earth M3+ cations considered in this work generally prefer a lower coordination environment due to their lower valence. Upon 4+ formation of the defected fluorite compound U1−xMxO2−0.5x,U in general must lower its coordination environment relative to its coordination in pure UO2 due to the formation of charge compensating oxygen vacancies; concomitantly, M3+ must increase its coordination environment relative to pure M2O3. Increasing energetic stability of the U1−xMxO2−0.5x compounds with increasing size of the M3+ cation can then be understood as resulting 49

Figure 6.1: Formation enthalpies of U0.5M0.5O1.75 structures are shown for the lowest-energy fully relaxed structures of all compositions enumerated in this study. Calculations for M=Y and La were taken from Ref. [141] for comparison. 50

Figure 6.2: The formation enthalpies for selected structures containing, one, two, and three trivalent cation nearest neighbors to an oxygen vacancy are shown. Calculations for M=Y and La were taken from Ref. [141] for comparison. 51

Figure 6.3: Structural motifs of low-energy structures containing one (left), two (middle), and three (right) M3+-oxygen vacancy nearest neighbors. from the lower energetic penalty associated with increasing the oxygen coordination around larger rare earth ions. The trend in Fig. 6.2 shows that the preference for oxygen vacancy nearest neighbors decreases with increasing size of the M3+ cations, which is due to larger ions preferring higher oxygen ion coordination. The preference for a particular defect ordering arrangement, i.e. for rare earth cations with size smaller or larger than the radius at the crossover point, is suggestive of defect clustering in short range ordered solid solutions. This clustering effect is expected to lower oxygen ion mobility, as discussed previously in the context of ionic conductivity in fluorite-structured solid electrolytes[73], by limiting the number of oxygen vacancies available for oxygen to diffuse into the solid. This kinetic limitation of oxygen incorporation due to clustering is believed to reduce oxidation and subsequent dissolution of spent fuel[67, 129]. The work in Ref. [129] which found evidence of stable defect clusters and reduced fuel dissolution with increasing M=Gd and Dy dopants is consistent with our calculations which show strong preference for ordering/clustering for these systems (Fig. 6.2), supporting the hypothesis that clustering is present and that it kinetically limits fuel corrosion.

6.5 Summary

A systematic DFT study of UO2 compounds substituted with trivalent rare earth 3+ cations (M ) found higher energetic stability of U1−xMxO2−0.5x solid solutions relative to constituent oxides with increasing size of the M3+ species. Furthermore, an investigation of the defect ordering tendencies shows an increasing preference for higher oxygen coordination around the rare earth ions with increasing size of these M3+ species. Compounds with ionic radius of the M3+ species smaller or larger than 1.09 A˚ are found to have energetically preferred defect ordering arrangements. Systems with preferred defect ordering arrangements are suggestive of defect clustering in short range ordered solid solutions, which is expected to limit oxygen ion mobility and therefore the rate of oxidation of spent nuclear fuel. 52

6.6 Acknowledgments

J.M.S. was supported by the Office of Basic Energy Sciences of the U.S. Depart- ment of Energy as part of the Materials Science of Actinides Energy Frontier Research Center (DE-SC0001089) for initial DFT+U calculations, and the Department of Defense through the National Defense Science & Engineering Graduate Fellowship Program for the remainder of the work. A.N. and M.A. were supported by the U.S. Department of Energy as part of the Materials Science of Actinides Energy Frontier Research Center (DE- SC0001089). This work made use of resources of the National Energy Research Scientific Computing Center, supported by the Office of Basic Energy Sciences of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. 53

Chapter 7

Comparison to Experiment, Intermediate and Stoichiometric Compositions: Y, Nd and La Doped UO2

The energetics of rare earth doped UO2 solid solutions (U1−xLnxO2−0.5x+y, where Ln = La, Y, and Nd) are investigated employing a combination of calorimetric measurements and density functional theory based computations. Calculated and measured formation enthalpies are found to agree within 10 kJ/mol for stoichiometric oxygen/metal compositions. To better understand the factors governing the stability and defect binding in rare earth doped urania solid solutions, systematic trends in the energetics are investigated based on the present results and previous computational and experimental thermochemical studies of rare earth doped fluorite oxides (A1−xLnxO2−0.5x, where A = Hf, Zr, Ce, and Th). A trend towards increased energetic stability with larger size mismatch, between the smaller host tetravalent fluorite cation and the larger rare earth dopant trivalent cation, was found to be consistent for both actinide and non-actinide fluorite oxide systems where aliovalent substitution of Ln cations is compensated by oxygen vacancies. Compared to the other fluorite systems, however, the large exothermic oxidation enthalpy in the UO2 based systems favors oxygen rich compositions where charge compensation occurs through the formation of uranium cations with higher oxidation states.

7.1 Forward

The work presented in this chapter was authored by L. Zhang, J. M. Solomon, M. Asta, and A. Navrotsky, and has been submitted to Acta Materialia and is reproduced here with permission of the co-authors. L. Zhang performed the calorimetry measurements reported in this chapter under the guidance of A. Navrotsky. All of the computational results were led by J. M. Solomon under the supervision of M. Asta. Both L. Zhang and J. M. Solomon led the analysis of literature data, with input from A. Navrotsky and M. Asta. 54

7.2 Introduction

The structural and thermochemical properties of rare earth doped UO2 solid solutions have been studied extensively due to their relevance in optimizing nuclear fuel performance[162]. Trivalent rare earth cations are common fission products that are soluble with UO2 over wide compositions ranges[61]. Both experimental[63, 143, 95] and thermodynamic modeling studies[61, 109, 137] have been applied to better understand and predict thermochemical and thermophysical properties of these systems. In particular, oxygen partial molar free energies[63], phase relations[143, 95, 20, 21, 134], composition dependent lattice parameters[70, 114, 152], and electronic properties[163, 17, 107] have been previously reported for a variety of different rare-earth substitutions. Despite extensive previous work in rare earth urania solid solutions, direct experimental measurements of the energetics of these materials have become available only relatively recently[99, 168]. In a recent paper[168] Zhang and Navrotsky reported the formation enthalpies of La-, Y-, and Nd-doped uranium oxides over a wide range of dopant concentrations and oxygen stoichiometries (mostly hyperstoichiometric) studied by high temperature oxide melt solution calorimetry. The oxidation enthalpies of LnxU1−xO2−0.5x+y were found to be similar to that of pure UO2. The strongly exothermic formation enthalpies of these oxygen-rich rare earth doped uranium oxides explains the difficulties in preparing samples in the oxygen deficient region. The enthalpies of formation from constituent oxides become increasingly exothermic with increasing doping level. The enthalpies also show small variations with oxygen content in the oxygen excess (hyperstoichiometric) region. Computational investigations of the energetics of rare earth substituted urania solid solutions have also been reported only relatively recently[141, 142]. In contrast to the experimental investigations, which focused on stoichiometric and hyperstoichiometric oxygen to metal ratios, the computational work published to date investigated hyperstoichiometric compositions where charge compensation occurs through oxygen vacancies. The focus on such compositions in the computational work was motivated by the desire to understand trends in the binding of rare-earth cations and vacancies, which has been described as a central issue in understanding the influence of burnup on the rate of oxidative corrosion in spent nuclear fuel[129, 67]. In the present work the computational studies are extended to consider compositions with oxygen stoichiometries varying from the limit of oxygen vacancy compensation up to stoichiometric oxygen/metal ratios. We note that due to the importance of oxygen clustering at hyperstoichiometric compositions[98, 13, 12, 11] the phase space underlying the energetics of rare earth substituted urania solid solutions in this composition range is not readily modeled by first-principles calculations. Never- theless, the extension of the computational studies to stoichiometric oxygen/metal ratios enables direct comparisons between calculated and measured data near these compositions. By combining the calculated results with the experimental data, which are limited to near stoichiometric to hyperstoichiometric compositions, a more complete picture of the effect of oxygen composition on the energetic stability of rare earth urania solid solutions can be obtained. The goal of this paper is, for the first time, to combine computational and experimental methods to study the energetics of rare-earth substituted urania solid solutions, using theory for compositions where experiment is difficult, and vice versa, and making 55 comparisons for compositions where theory and experiment overlap by extending calculations to more oxygen-rich compositions. We also investigate systematic trends based on previous computational[141, 142, 31, 8] and experimental[91, 37, 108, 33, 7, 140, 139] energetic studies of rare earth doped fluorite oxides, with results of the current work, in order to improve the overall understanding of the factors governing the stability of these systems. The key findings are as follows: (1) Very favorable agreement between computation and experiment is found at compositions where direct comparisons can be made. (2) Trends for compositions that are fully oxygen vacancy compensated are consistent with other fluorite systems and show a clear dependence on size mismatch between host and rare earth cation size. (3) UO2 is distinguished from other fluorite systems by the very large and exothermic oxidation enthalpy due to the energetic stability of the higher oxidation states of uranium.

7.3 Methods

7.3.1 Experimental Procedures

The rare earth doped uranium oxide solutions (LnxU1−xO2−0.5x+y, where Ln = La, Y, and Nd) were prepared by coprecipitation method followed by sintering at 1100 1450 C under reducing atmosphere. The sintered materials were characterized carefully to obtain a single-phase homogeneous fluorite oxide. Their phases were examined by XRD to confirm only a single fluorite phase, chemical compositions determined by electron micro- probe analysis to confirm homogeneous phases, and uranium oxidation states measured by Ce(IV)-Fe(II) back-titration after dissolution of the samples in warm acid mixtures added with excess Ce(IV) solutions. The drop solution enthalpies were measured by dropping samples into a molten oxide solvent (3NaO·4MoO3) in a high temperature calorimeter at 700 C and the enthalpies of formation at room temperature from oxides were calculated. A detailed description of the preparation, characterization, and calorimetry of these materials can be found in ref. [168]. The calorimetric study showed that the formation enthalpies of the rare earth doped uranium oxides from rare earth oxide and a mixture of UO2 and UO3 that gives the observed oxygen content are independent of the oxygen content in the oxygen excess region. This allows simple and accurate estimation of the enthalpy of formation of a sample with any given rare earth content and oxidation state in the oxygen excess region.

7.3.2 Computational Methods The computational approach has been described previously in work focused on the study of vacancy-compensated urania rare earth oxide solid solutions[141]. This approach is summarized brieﬂy in this section, with an emphasis on describing the extensions of the methods to consider higher oxygen concentrations where charge compensation involves the formation of higher oxidation states of uranium, taken as U5+. The lowest energy structure for a given concentration of a particular Ln3+ species (and corresponding number of oxygen vacancies) is determined by calculation of many hypothetical arrangements of the uranium and rare earth species on the cation sublattice and oxygen and vacancies on the anion sublattice. The cation sublattice contains host and rare earth species U4+,U5+ and Ln3+, and the anion sublattice contains host and vacancy 56

2− 4+ species, O and Ovac. At hypostoichiometric compositions the substitution of two U 3+ by two Ln can be charge compensated by one Ovac. For a stoichiometric oxygen/metal ratio equal to 2, charge compensation can occur by oxidation of U4+ to U5+ for every one Ln3+/U4+ substitution, as illustrated in Fig. 7.1. Systems with compositions that are intermediate between fully oxygen vacancy charge - compensated and fully U5+ charge - compensated contain both oxygen vacancies and U5+ ions. We employ a structure enumeration technique developed for cluster expansion based studies of alloy thermodynamics, considering supercells consisting of up to eight formula units, employing an algorithm by Hart and Forcade[65] that is implemented in the alloy theoretic automatic toolkit (ATAT)[157, 156]. We consider three different oxygen to metal ratios, namely reduced (fully charge-compensated with oxygen vacancies), stoichiometric (fully charge-compensated with U5+ ions), and intermediate (charge-compensated by 5+ both oxygen vacancies and U ions). Specifically, the reduced regime considers U4Ln2O11 4+ (six formula units: 4 U ions, 2 Ln ions, 11 O ions, and 1 Ovac) and U2Ln2O7 (four formula 4+ units: 2 U ions, 2 Ln ions, 7 O ions, and 1 Ovac), yielding 117 and 27 symmetry - distinct structures, respectively. The stoichiometric regime considers U4Ln2O12 (six formula units: 4+ 5+ 2 U ions, 2 U ions, 2 Ln ions, and 12 O ions) and U2Ln2O8 (four formula units: 2 U5+ ions, 2 Ln ions, and 8 O ions), yielding 100 and 5 symmetry-distinct structures, re- 4+ spectively. The intermediate regime considers U4Ln4O15 (eight formula units: 2 U ions, 5+ 2 U ions 4 Ln ions, 15 O ions, and 1 Ovac), yielding 7296 symmetry-distinct structures. The compositions considered for the calculations in this work are summarized in Table 7.1. Table 7.1: The compositions considered for the calculations in this work are shown. The first through fifth columns represent the formula, number of U4+ ions, number of U5+ ions, number of oxygen vacancies, and number of symmetrically-distinct structures, respectively. 4+ 5+ Formula U U Ovac Structures Reduced U4Ln2O11 4 0 1 117 U2Ln2O7 2 0 1 27 Stoichiometric U4Ln2O12 2 2 0 100 U2Ln2O8 0 2 0 5 Intermediate U4Ln4O15 2 2 1 7296

After enumerating the different possible configurations at the different oxygen concentrations we use ionic pair potential models[59, 105, 106, 32, 146, 64] to screen out configurations with large energies, and employ density functional theory (DFT) based methods to determine energies of the lowest-energy structures. The DFT energies were obtained within the formalism of the projector augmented wave (PAW) method[27, 87] and the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)[117, 118] as implemented in the Vienna ab initio simulation package (VASP)[85, 86]. A Hubbard-U correction in the DFT+U formalism of Dudarev et al.[50] with Ueff = U-J = 4.00 eV was employed for the treatment of the uranium 5f orbitals in this study. We employ the approach 57

Figure 7.1: The substitution of two U4+ by two Ln3+ can be charge compensated by one Ovac, or in more oxidizing environments, charge compensation can occur by oxidation of U4+ to U5+ for every one Ln3+/U4+ substitution, as illustrated in the motifs above. Systems with compositions that are intermediate between fully oxygen vacancy charge-compensated and fully U5+ charge-compensated contain both oxygen vacancies and U5+ ions. 58 by Meredig et al.[102] involving a gradual ramping up of the Hubbard-U parameter. In our previous work[141] we compared this ramping method to the Occupation Matrix Control (OMC) method by Dorado et al.[46] and found that the ramping method gave rise to a configuration with a lower total energy than that derived using OMC for the 61 matrices that were sampled. The PAW potentials used in this work are U, O, Y sv, La and Nd 3 in the VASP PBE library, where the “ 3” denotes that the occupied 4f orbitals in the trivalent oxidation state of the rare-earth ions are treated as core electrons. We employ a plane-wave cutoff energy of 500 eV, and the Brillouin zone is sampled using the Monkhorst-Pack scheme with 4 x 4 x 4 k-point meshes. Atomic positions were relaxed with no symmetry constraints until residual forces were below approximately 20 meV/A.˚ From convergence checks with respect to plane-wave cutoff and k-point sampling we estimate the formation enthalpies are converged at the level of a few tenths of a kJ/mol-cation.

7.4 Results

For rare earth doped uranium oxide solid solutions formed from constituent oxides, the formation reaction can be written as x Ln O + (1 − x − y)UO + yUO → U Ln O (7.1) 2 2 3 2 3 1−x x 2−0.5x+y The corresponding formation enthalpy to be calculated is thus deﬁned as

∆Hf = H[U1−xLnxO2−0.5x+y] − (1 − x − y)H[UO2] − xH[LnO1.5] − yH[UO3] (7.2) where x is the Ln3+ substitution fraction, H[A] is the enthalpy per formula unit of compound A, and y is the oxygen content, which is related to the oxygen to metal ratio (O/M) by

y = O/M + 0.5x − 2 (7.3) such that for a given Ln substitution level x, y is directly proportional to the oxygen to metal ratio. For “reduced” compositions which are fully charge compensated by oxygen vacancies (i.e. y = 0), Eq. 7.2 becomes

∆Hf = H[U1−xLnxO2−0.5x] − (1 − x)H[UO2] − xH[LnO1.5] (7.4) and for stoichiometric compositions which are fully charge compensated by U5+ compensation (i.e. y = 0.5x), Eq. 7.2 becomes 3x x ∆H = H[U Ln O ] − (1 − )H[UO ] − xH[LnO ] − H[UO ] (7.5) f 1−x x 2 2 2 1.5 2 3

In Eq. 7.2, H[UO2] is for ﬂuorite-structured UO2, and H[LnO1.5] is for Ln2O3 in the experimentally observed (i.e., room temperature) cubic bixbyite (C-type sesquioxide) or hexagonal (A-type) structure. The experimentally observed C-type sesquioxides are Y2O3 59

and Nd2O3, and the A-type sesquioxide is La2O3[172]. In the calculations, all enthalpies are approximated by zero-temperature total energies. The calculated and experimental formation enthalpies for 1/2 and 1/3 Ln substitution levels where Ln=Y, Nd, and La are plotted against oxygen- to-metal ratio (O/M) in Fig. 7.2. We find that for the experimental data (denoted by filled circles), compositions that are near-stoichiometric (O/M = 2) or hyper-stoichiometric (O/M > 2) are highly exothermic and tend to show very little dependence on oxygen content in this regime. In the hypo-stoichiometric (O/M < 2) regime, the calculations (denoted by open circles) show the formation enthalpy is most endothermic for fully oxygen vacancy compensated systems, and becomes more exothermic with increasing oxygen content. The large energetic change with oxygen content in this regime compared to the minimal change in formation enthalpy for O/M > 2 can be attributed to different defect mechanisms in the two regimes. For O/M < 2, as the oxygen content increases, oxygen vacancies are filled and U4+ is oxidized to U5+ and/or U6+; for O/M > 2, oxygen vacancies are filled and no longer contribute to the energetics, and increasing oxygen content is accommodated through the formation of oxygen interstitials and related defect clusters. The computational results for U1−xLnxO2−0.5x materials suggest their lack of stability (slightly positive ∆Hf ), contrast- ing to the much stabilized stoichiometric U1−xLnxO2 materials, which explains why it has been extremely difficult to prepare samples in the vacancy regime even with substantial rare earth doping as seen in ref. [168]. Next we make direct comparisons with calculation and experiment at similar oxygen compositions. Specifically, we compare the formation enthalpies with respect to Ln dopant cation size in Fig. 7.3. The compositions considered for the calculations represented in Fig. 7.3 are fully stoichiometric (O/M = 2) with exactly one half or one third Ln substitution (Ln/(Ln+U) 0.5 or 0.33). The experimental Ln substitution levels correspond exactly or are very close to the substitution levels used in the calculations, for each dopant species and substitution level were obtained by averaging the experimental data points in Fig. 7.2, as we have shown that they are consistent over the hyperstoichiometric regime[168]. The calculations and experiments show agreement for all compositions within 10 kJ/mol. This level of agreement is viewed to be highly reasonable considering the influence on the energetics of different approximations used in the first principles calculations. For example, in our previous work comparisons between formation energies computed with DFT+U and hybrid methods were reported[141], with the latter leading to results that were more exothermic by approximately 5 - 6 kJ/mol. The present comparisons between experiment and theory may suggest that the hybrid methods would lead to improved agreement with measurements, although further work would be needed at the stoichiometric oxygen concentrations to draw definitive conclusions. The calculations allow us to look at systematic trends in the lattice parameter change with respect to fluorite UO2 for the various Ln dopant species and doping levels, as represented in Fig. 7.4 for stoichiometric oxygen to metal compositions. Fig. 7.4 shows that for all three Ln dopant species, increasing doping level gives a greater lattice parameter mismatch, although this increase is the least for Ln = Nd. Furthermore, for Ln = Y, the lattice mismatch is negative and becomes increasingly negative with doping level, which indicates a lattice contraction upon doping, whereas for the larger Nd and La cations, lattice 60

Figure 7.2: Formation enthalpies (∆Hf ) of U1−xLnxO2−0.5x+y structures are shown for the calculated lowest-energy fully relaxed structures of all compositions enumerated in this study, in comparison to calorimetric data presented previously[168]. The enthalpies are plotted against oxygen-to-metal ratio (O/M), which is related to x and y according to Eq. 7.3. 61

Figure 7.3: The formation enthalpies calculated for stoichiometric compositions for both calculation (“Comp.”) and experiment (“Exp.”) with respect to Ln dopant cation size are shown. expansion is expected. The largest lattice parameter mismatch occurs for Ln = Y, which is a possible explanation for why it is the least stable solid solution energetically. All these ﬁndings are qualitatively consistent with experimental observations[168].

7.5 Discussion

In ﬂuorite structured AO2 compounds, with A=Hf, Zr, Ce, and Th, substitution of the host (A) cation with trivalent rare earths is observed to lead to the formation of charge-compensating oxygen vacancies[91, 37, 108, 33, 7]. In the limiting case where the oxygen/metal ratio corresponds to complete charge compensation by oxygen vacancies the host (A) remains primarily tetravalent and the defect chemistry can be described as follows:

0 3 1 xLnO AO→2 xLn + xOX + xV •• (7.6) 1.5 A 2 O 2 O Since experimentally it has proven diﬃcult to prepare rare earth urania solid solution samples with the reduced U1−xLnxO2−0.5x oxygen stoichiometries, ﬁrst principles calculations will be used as a basis for comparing results for urania solid solutions with those in the re- 62

Figure 7.4: The first-principles calculated percent lattice parameter difference relative to fluorite UO2 for the three Ln dopant species considered in this work for stoichiometric compositions with x=1/3 and 1/2 doping levels are shown. lated fluorite compounds based on tetravalent Zr, Hf, Ce and Th in this hypostoichiometric regime. Specifically, we make direct comparison of the formation enthalpies (see Eq. 7.7) of A1−xLnxO2−0.5x fluorite oxides from constituent oxides at the similar dopant levels with calorimetric and computational results in Fig. 7.5.

∆Hf = H[A1−xLnxO2−0.5x] − (1 − x)H[AO2] − xH[LnO1.5] (7.7) As shown in Figs. 7.5a and 7.5b, the dopant concentrations selected for comparison are 1/3 and 1/2, which are the exact x value for calculations, and the same or very similar values for experiments. The radii of the tetravalent host cations (A = Hf, Zr, Ce, U, Th) and trivalent dopant cations (Ln = Yb, Y, Dy, Gd, Sm, Nd, La) in eightfold coordination are listed in Table 7.2. It should be noted here we are comparing the formation enthalpies of these doped fluorite oxides from fluorite AO2 and LnO1.5 in the cubic bixbyite (C-type sesquioxide) or hexagonal (A-type) structure, where YbO1.5, YO1.5, DyO1.5, GdO1.5, and SmO1.5 are C-type sesquioxides, NdO1.5 is dimorphic and LaO1.5 is A-type[172]. Since neither ZrO2 nor HfO2 has the fluorite structure at room temperature, and the reported calorimetric formation enthalpies of doped zirconia and doped hafnia are from LnO1.5 and monoclinic AO2[91, 140, 139], the energetics of their transformation to fluorite structure needs to be accounted for. After such correction we obtain the formation enthalpies of A1−xLnxO2−0.5x (A=Hf, Zr) that are shown in Fig. 7.5 by

∆Hf = ∆Hf,ox[A1−xLnxO2−0.5x] − (1 − x)∆Htrans[AO2] (7.8)

The transformation enthalpies of HfO2 and ZrO2 from the monoclinic to cubic ﬂuorite structure used here (8.8 ± 3.4 kJ/mol and 26 ± 1.0 kJ/mol respectively) are taken from Simoncic and Navrotsky[140]. 63

Figure 7.5: Formation enthalpies of A1−xLnxO2−0.5x ﬂuorite oxides from LnO1.5 and ﬂuorite AO2 from calorimetric and computational results, as a function of trivalent dopant cation radius, where x = 1/3 in (a) and x = 1/2 in (b). aLee and Navrotsky[91] bSimoncic and Navrotsky[139] cSimoncic and Navrotsky[140] dBogicevic et al.[31] eChen et al.[37] f Navrotsky et al.[108] gBuyukkilic et al.[33] hSolomon et al.[141] iThis work jAizenshtein et al.[7] kAlexandrov et al.[8] lSolomon et al.[142] 64

Table 7.2: The cation sizes in eightfold coordination taken from ref. [135]. 4+ 3+ A rA4+ (A)˚ Ln rLn3+ (A)˚ Hf 0.83 Yb 0.985 Zr 0.84 Y 1.019 Ce 0.97 Dy 1.027 U 1.0 Gd 1.053 - - Nd 1.109 - - La 1.16

As displayed in Fig. 7.5, when trivalent dopant cation size increases, the formation enthalpies of different doped AO2 fluorite oxides monotonically decrease, for a given fixed cation substitution level. Both the experimental and computational results display this stabilizing effect when adding larger trivalent dopants. Specifically, the computational data for doped urania and thoria show the decreasing formation enthalpies with increasing dopant radius, and a similar trend is shown in the experimental data for doped hafnia, zirconia, and ceria. Concomitantly, as the tetravalent host cation radius decreases (Fig. 7.6), the doped fluorite oxide becomes more energetically favorable. When direct comparisons can be made between calculations and experiments in Figs. 7.5 and 7.6, the enthalpies agree within 10 kJ/mol. Next we combine what we learned from Figs. 7.5 and 7.6 to explore how size differences between the host and dopant cations affect the energetics. Instead of using the cation radii as the variables as shown in Figs. 7.5 and 7.6, here we introduce the relative size mismatch between the dopant (Ln3+) and host (A4+) cations,

rA4+ − rLn3+ ∆r = (7.9) rA4+

We redraw Fig. 7.5 by converting the x-axis of rLn3+ to ∆r, and thus obtain Fig. 7.7. Thereby we can compare all the doped fluorite oxide systems directly. The dashed lines shown in Fig. 7.7 are obtained by fitting the experimental calorimetric formation enthalpies. As, ∆r increases, ∆Hf decreases, leading to enhanced thermodynamic stability. As the size mismatch reaches 10 to 15 %, the formation enthalpy changes from slightly endothermic to increasingly exothermic. The computational results on doped ZrO2, UO2, and ThO2 are consistent with the general trends seen in the experimental data. This change in energetics with size mismatch can be attributed to defect clustering tendencies for each doped fluorite oxide system. The enhanced stability of systems with larger size mismatch (i.e., smaller host A4+ cation, larger Ln3+ cation) is interpreted to be associated with a preference for reduced (increased) oxygen ion coordination of the host (dopant) cation[141, 142]. Specifically, the formation of a fluorite-structured solid solution from AO2 and Ln2O3 constituents necessarily leads to a decrease (increase) in the average oxygen coordination surrounding the A4+ (Ln3+) cation, relative to the 8-fold (6-fold/7-fold) coordination characteristic of the pure AO2 (Ln2O3). Therefore, systems with small host cations which prefer lower coordination (e.g., Zr4+, Hf4+) and large dopant cations which 65

Figure 7.6: Formation enthalpies of A1−xLnxO2−0.5x ﬂuorite oxides (x=1/3, 1/2) from LnO1.5 and ﬂuorite AO2 from calorimetric and computational results, as a function of tetravalent host cation radius, where Ln = Y in (a), Ln =Gd in (b), and Ln = La in (c). Redrawn from data in Fig. 7.5. 66

Figure 7.7: Formation enthalpies of A1−xLnxO2−0.5x fluorite oxides from LnO1.5 and fluorite AO2 from calorimetric and computational results, as a function of size mismatch of trivalent dopant and tetravalent host cations, where x = 1/3 in (a) and x = 1/2 in (b). Redrawn from data in Fig. 7.5. The dashed lines shown in the figure are obtained by fitting only the experimental data. 67 prefer higher coordination (e.g., Sm3+, Nd3+, La3+) are expected to be more energetically favorable.

7.6 Conclusions

A combined experimental and computational investigation of the energetic properties of rare earth doped UO2 systems is presented, based on calorimetric measurements and DFT-based calculations. For compositions where direct comparisons between theory and experiment are possible, agreement within 10 kJ/mol is obtained. Systematic trends based on previous experimental and computational thermochemical studies of rare earth doped fluorite oxides with oxygen vacancy compensation, and results of the current work are presented. Consistent trends are found for all systems considered (including actinide and non-actinide oxide systems), demonstrating that larger size mismatch between the smaller host fluorite cation (A4+) and the larger rare earth dopant cation (Ln3+) gives rise to more energetically stable solid solutions. Doped UO2 is unique compared to the other doped fluorite oxides considered, however, due to a strong energetic preference for higher oxygen compositions, regardless of the rare earth dopant species and doping level.

7.7 Acknowledgments

This work was supported as part of the Materials Science of Actinides, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award No. DE-SC0001089. While L.Z., A.N, and M.A. were supported by that project, J.M.S. also received a Department of Defense National Defense Science Engineering Graduate Fellowship. This work made use of resources of the National Energy Research Scientific Computing Center, supported by the Office of Basic Energy Sciences of the U.S. Department of Energy under Contract No. DE-AC0205CH11231. 68

Part III

Concluding Remarks 69

Chapter 8

Conclusions and Future Work

8.0.1 Conclusions In this dissertation, we have employed computational methods to improve our understanding of the energetics and defect behavior of complex ionic compounds for energy applications. The main conclusions are summarized below. In Chapter 4, Sr-doped LaPO4, a candidate for proton conducting fuel cell electrolytes, was found to have a strong energetic preference for proton incorporation; however, proton uptake was found to be limited kinetically due to strong binding between the Sr dopant and the pyrophosphate defect. In Chapters 5 and 6, solid solutions of UO2 (the most widely used nuclear fuel) with soluble trivalent rare earth fission products, were found to be more stabilizing with increasing size of the rare earth cation. Furthermore, an investigation of the defect ordering tendencies shows an increasing preference for higher oxygen coordination around the rare earth ions with increasing size of the rare earth cation. This preference for a particular oxygen coordination environment is suggestive of defect clustering in short range ordered solid solutions, and is expected to limit oxygen ion mobility and therefore the rate of oxidation and eventual corrosion of spent nuclear fuel. In Chapter 7, calculated formation enthalpies for stoichiometric solid solutions of Y, Nd, and La doped UO2 were found to agree with experiment to within 10 kJ/mol. The calculations and experiments suggest that these rare earth doped systems favor oxygen rich compositions where charge compensation occurs through the formation of uranium cations with higher oxidation states. It was also found that for vacancy compensated compositions the UO2 systems follow generally the same trend as other fluorites of increasing stability with increasing size difference between host and trivalent cation.

8.0.2 Future Work This dissertation presents some of the first investigations of fundamental energetic properties for actinide oxide solid solutions with aliovalent cation substitutions. To derive a more complete understanding of the thermochemistry of nuclear fuels, many ar- eas remain to be studied. Our primary focus was on rare earth fission product cations in UO2, and we developed simple coordination arguments based on ionic models to explain 70 trends in the energetics and defect ordering tendencies. An interesting question to ask is 3+ whether these trends hold up for UO2 substituted with trivalent actinides, e.g. Am and 3+ 3+ 3+ Np . The energetics of solid solutions of Am and Np in UO2 are relevant due to their promise as blanket fuels in fast neutron reactors, in order to transmute the highly radiotoxic Am3+/Np3+ while providing power to the grid[124]. Furthermore, it is of interest to un- 3+ derstand the energetic stability of Am in UO2 to gain insight on thermodynamic driving forces for fuel separations, which is critical for optimizing the fuel recycling processes. A recent first principles study found unexpected covalency in some actinide oxides, including americium oxide[126]. This covalent behavior may manifest different energetic and defect 3+ ordering behavior for Am in UO2 compared to the rare earth elements of similar size. The Department of Energy’s Materials Science of Actinides Energy Frontier Re- search Center, which has funded a large portion of this work, has begun to investigate the energetic properties of plutonium oxide systems using experimental calorimetric studies. This provides a collaborative opportunity to combine computational and experimental methods in the study of the thermochemistry of these materials, in a manner similar to that presented above for UO2. The remainder of this section provides a literature review of recent work that sets the stage for further investigations.

AmO2 and AmO2-UO2

Suzuki et al[149] performed electronic structure calculations on AmO2, Am2O3, UO2, and Am0.5U0.5O2, using the all-electron Wien2K code. Hubbard-U parameters of 4.5 eV and 4.8 eV were chosen for the U f and Am f orbitals, respectively. Spin orbit interactions were also considered. They found that the antiferromagnetic arrangement was most stable for both AmO2 and A-type Am2O3. It should be noted that C-type Am2O3, the stable phase as room temperature, was not considered in this study. The stable valence states for the mixed oxide Am0.5U0.5O2 were found to be 3+ for Am and 5+ for U. Lu et al.[96] performed electronic structure calculations on ﬂuorite AmO2 using VASP, using a 12 atom unit cell with collinear antiferromagnetic order along [001]. A Hubbard-U parameter of 4.0 eV (Ueff with the Dudarev formalism) was chosen, and a quasi- annealing procedure[56] was used to avoid metastable stables, ﬁnding a 1.35 eV lowering in energy when this method was used. The GGA+U calculations reasonably reproduced the experimental lattice constants, band gap, and magnetic moments. Spin orbit coupling was used for all calculations. The local spin moment on the Am cation was found to be 5.15µB, which is similar to results obtained from HSE calculations. Calculations for atomic charges showed that 2.26 electrons transfer from Am to O atoms, compared to 2.56 for UO2, 2.48 for NpO2, and 2.40 for PuO2[169, 158]. The authors concluded that increasing the atomic number of the actinide elements decreases the ionicity of the bond between the actinide and oxygen atoms. Accompanying lower ionicity, the bond lengths between the two species of atoms decrease with increasing actinide atomic number. 71

PuO2 Recent works using first principles calculations have accurately determined the electronic structure of plutonium oxide systems[119, 169, 48, 148, 75, 29], which provides a good starting point for the energetic studies. For example, Zhang et al.[169] used the GGA+U approach with the Dudarev formalism, and found an effective Hubbard-U of around 4 eV to accurately reproduce the experimental lattice constants and electronic structure. The antiferromagnetic arrangement was found to be most stable, with a magnetic moment of 4.17µB. Spin orbit coupling was not used for this study. Bo et al.[29] calculated the oxidation energy from Pu2O3 to PuO2 with respect to the effective Hubbard U, although comparison to experimental oxidation enthalpies was not provided.

NpO2 Wang et al.[158] found that the GGA+U scheme accurately reproduces the experimental lattice parameter and electronic structure of NpO2. An eﬀective Hubbard U of 4.0 eV was chosen for the Np f atoms, and spin orbit coupling was not considered. The ferromagnetic arrangement is most stable, although the energy diﬀerence with respect to the antiferromagnetic state was found to be negligible, and the magnetic moment was found to be 3.09µB. Unlike AmO2 and PuO2, where there is covalency due to strong hybridization between f states with oxygen p states, the relative covalency of NpO2 is very weak, with an electronic structure more similar to that of UO2[126].

Suggestions for Future Work

In light of the strong covalency of PuO2 relative to UO2[126], computing the formation enthalpies of PuO2 with trivalent rare earth substitutions would be of interest. 5+ PuO2 does not in general form hyperstoichiometric compositions (i.e., Pu is not stable in the oxide), as shown in the Pu-O phase diagram[62]. However, oxygen vacancy-compensated compositions could be considered and compared directly to our UO2 studies. Specifically, one could test whether the same trend for size difference between Pu4+ and Ln3+ versus formation enthalpies would hold. If these systems do not follow the same trend, it could imply that the increased covalency in the bonding is playing an important role in governing the thermodynamic stability of the solid solutions. UO2-PuO2 solids solutions have been widely studied and are relevant in the context of mixed-oxide fuels that have been used in commercial reactors since the 1980s[1]. Hanken et al.[64] performed DFT-based calculations the energetics of on UO2-CeO2 solid solutions, where CeO2 is used both computationally and experimentally as a surrogate for PuO2. The study found that charge transfer process of U4+ and Ce4+ to U5+ and Ce3+ was found to be weakly disfavorable energetically, concluding that entropic contributions to the free energy would cause charge transfer to be favorable at typical in-pile reactor temperatures. A natural next step would be to conduct a similar investigation for UO2-PuO2 to determine whether CeO2 is an acceptable surrogate for thermochemical studies. NpO2, which contains a smaller cation size than in UO2, provides an interesting case study. Specifically, NpO2 is relatively ionic, and has more similar bonding character to ThO2 and UO2, and less similar to PuO2 and AmO2, which have more covalent 72 character[126]. One could therefore surmise that the energetic trends with trivalent rare earth substitutions found in NpO2 would be similar to UO2 and ThO2 due to similar bonding character. If the trends are found to be different from ThO2 and UO2, then it suggests that degree of covalency is not the only governing factor in the energetic behavior of these systems. Investigation of the Np-O phase diagram[130] shows stability of Np2O5 and therefore stability of Np5+ in the oxide. This enables calculations of the energetics of rare earth 5+ substituted NpO2 solid solutions containing Np for comparative study to our UO2 calculations. Specifically, comparing the stabilization effect due to increased oxygen content for these defected NpO2 and UO2 solid solutions would provide a better understanding of the relative oxidation behavior of these systems. Americium in oxide form can be stable in both 3+ and 4+ oxidation states according to the Am-O phase diagram[58]. For AmO2, which was found to have strong covalent character[126], it would be of interest to investigate if formation enthalpies for solid solutions with trivalent rare earths follow the same trend with size mismatch as the other fluorite oxides studied in this dissertation, including actinide oxides UO2 and ThO2; once again one would be designing the calculations to look at effects of covalency on energetic stability of solid solutions. The second investigation would be to look at americium’s behavior as a trivalent impurity in UO2, to determine whether the formation enthalpies for UO2-AmO1.5 follow the same trends as the rare earths.

Foreseeable Challenges All of the aforementioned proposed works bring about new foreseeable challenges. One such challenge is choosing an appropriate Hubbard U parameter for the f electron- containing actinides, namely, Am, Np, and Pu. Past experience suggests that fitting the U parameter to the oxidation enthalpies of the actinide oxides gives DFT+U calculated formation enthalpies of the rare earth substituted solid solutions that agree well with experiment[64, 168]. The room temperature formation enthalpies of AmO2 and Am2O3 1 are reported in ref. [58]; therefore, the oxidation enthalpy for Am2O3 and 2 O2 forming 2AmO2 can be readily determined. Room temperature formation enthalpies for PuO2 and Pu2O3 are found in ref. [62]. The room temperature formation enthalpies for NpO2 and 1 Np2O5 are available in ref. [93], such that the oxidation enthalpy of 2NpO2 and 2 O2 forming Np2O5 can be determined. Experimental oxidation enthalpies for UO2, NpO2, PuO2, and AmO2 are shown in Table 8.1. The methods described in this dissertation involve initial screening of many hypothetical structures using classical interatomic potential models. Interatomic Buckingham potentials (i.e., pair interactions with O2− and the respective cation) are available in consistent sets for (a) U4+, Pu4+ (b) U4+, Np4+, Pu4+, Am4+, and Gd3+, and (c) U4+, Am4+, and Am3+. Sets (b) and (c) include Morse parameters to account for covalency effects. All potential parameters are reviewed in ref. [44]. The most recent consistent potentials for actinide oxides (i.e., ThO2, UO2, NpO2, PuO2, AmO2) which includes many-body effects has been developed by Robin Grimes and co-workers and can be found with publication references at http://abulafia.mt.ic.ac.uk/potentials/actinides/. Potential parameters for Np5+ are presently unavailable, however, they can be fit 73

Table 8.1: Oxidation enthalpies for actinide (i.e., U, Np, Pu, Am) oxides are shown. U3O8 and UO3 are α and γ phases, respectively. Enthalpies are in units of kJ/mol. Reaction ∆Ho(298 K) Ref.

3UO2 + O2 → U3O8 -319.8 [41] 1 UO2 + 2 O2 → UO3 -138.8 [41] 1 2NpO2 + 2 O2 → Np2O5 -14.7 [93] 1 P u2O3 + 2 O2 → 2P uO2 -464.4 [62] 1 Am2O3 + 2 O2 → 2AmO2 -170.7 [58] to experimental lattice and elastic constants, dielectric constants, or DFT+U calculated formation enthalpies (e.g., if we consider the composition Np2M2O8, where both Np atoms are 5+, there are only 5 hypothetical structures and therefore require minimal computational time for ﬁrst-principles calculations). 74

Bibliography

[1] http://www.world-nuclear.org/info/Nuclear-Fuel-Cycle/Fuel-Recycling/Mixed- Oxide-Fuel-MOX/.

[2] C. Adamo and V. Barone. Toward reliable density functional methods without adjustable parameters: The pbe0 model. The Journal of Chemical Physics, 110(13):6158–6170, 1999.

[3] N. Adelstein, J.B. Neaton, M. Asta, and L.C. De Jonghe. First-principles studies of proton-ba interactions in doped lapo4. J. Mater. Chem., 22:3758–3763, 2012.

[4] International Energy Agency. Key world energy statistics. 2013.

[5] J.A. Aguiar, Q.M. Ramasse, M. Asta, and N. D. Browning. Investigating the electronic structure of ﬂuorite-structured oxide compounds: comparison of experimental eels with ﬁrst principles calculations. Journal of Physics: Condensed Matter, 24(29):295503, 2012.

[6] E.A. Aitken. Oxidation kinetics of solid solutions of urania-yttria. Journal of Nuclear Materials, 19(3):248 – 266, 1966.

[7] M. Aizenshtein, T.Y. Shvareva, and A. Navrotsky. Thermochemistry of lanthana- and yttria-doped thoria. Journal of the American Ceramic Society, 93(12):4142–4147, 2010.

[8] V. Alexandrov, N. Grønbech-Jensen, A. Navrotsky, and M. Asta. First-principles computational study of defect clustering in solid solutions of tho2 with trivalent oxides. Phys. Rev. B, 82:174115, 2010.

[9] K. Amezawa, S. Kjelstrup, T. Norby, and Y. Ito. Protonic and native conduction in Sr-substituted LaPO4 studied by thermoelectric power measurements. J. Electrochem. Soc., 145(10):3313–3319, 1998.

[10] K. Amezawa, Y. Tomii, and N. Yamamoto. High temperature protonic conduction in lapo4 doped with alkaline earth metals. Solid State Ionics, 176(12):135 – 141, 2005.

[11] D.A. Andersson, G. Baldinozzi, L. Desgranges, D.R. Conradson, and S.D. Conradson. Density functional theory calculations of uo2 oxidation: Evolution of uo2+x, u4o9y, u3o7, and u3o8. Inorganic Chemistry, 52(5):2769–2778, 2013. 75

[12] D.A. Andersson, F.J. Espinosa-Faller, B.P. Uberuaga, and S.D. Conradson. Stability and migration of large oxygen clusters in uo2+x: Density functional theory calculations. The Journal of Chemical Physics, 136(23), 2012.

[13] D.A. Andersson, J. Lezama, B.P. Uberuaga, C. Deo, and S.D. Conradson. Cooper- ativity among defect sites in ao2+x and A4o9 (a = U, Np, P u): Density functional calculations. Phys. Rev. B, 79:024110, 2009.

[14] D.A. Andersson, S.I. Simak, N.V. Skorodumova, I.A. Abrikosov, and B. Johans- son. Optimization of ionic conductivity in doped ceria. Proceedings of the National Academy of Sciences of the United States of America, 103(10):3518–3521, 2006.

[15] V.I. Anisimov, F. Aryasetiawan, and A.I. Lichtenstein. First-principles calculations of the electronic structure and spectra of strongly correlated systems: the lda + u method. Journal of Physics: Condensed Matter, 9(4):767, 1997.

[16] V.I. Anisimov, J. Zaanen, and O.K. Andersen. Band theory and mott insulators: Hubbard u instead of stoner i. Phys. Rev. B, 44:943–954, 1991.

[17] Y. Arita, S. Hamada, and T. Matsui. High-temperature heat capacities and electrical conductivities of uo2 doped with simulated ﬁssion products for 210 atThermochimica Acta, 247(2):225 – 236, 1994.

[18] S.P.S. Badwal and D.J.M. Bevan. Urania-yttria solid solution electrodes for high- temperature electrochemical applications. Journal of Materials Science, 14(10):2353– 2365, 1979.

[19] P. Baranek, A. Lichanot, R. Orlando, and R. Dovesi. Structural and vibrational properties of solid mg(oh)2 and ca(oh)2 performances of various hamiltonians. Chemical Physics Letters, 340(34):362 – 369, 2001.

[20] S.F. Bartram, E.F. Juenke, and E.A. Aitken. Phase relations in the system uo2-uo3 y2o3. Journal of the American Ceramic Society, 47(4):171–175, 1964.

[21] R.J. Beals, J.H. Handwerk, and B.J. Wrona. Behavior of uraniarare-earth oxides at high temperatures. Journal of the American Ceramic Society, 52(11):578–581, 1969.

[22] M. Beauvy, T. Duverneix, C. Berlanga, R. Mazoyer, and C. Duriez. Actinide trans- mutation: new investigation on some actinide compounds. Journal of Alloys and Compounds, 271273(0):557 – 562, 1998.

[23] A. D. Becke. Density-functional thermochemistry. III. The role of exact exchange. , 98:5648–5652, 1993.

[24] R.K. Behera, C.S. Deo, and H. Xu. Eﬀect of the substitution of f-electron elements on the structure and elastic properties of uo2. Journal of Nuclear Materials, 433(13):504 – 513, 2013. 76

[25] D.J. Binks and R.W. Grimes. Incorporation of monovalent ions in zno and their influence on varistor degradation. Journal of the American Ceramic Society, 76(9):2370– 2372, 1993. [26] T.S. Bjorheim, T. Norby, and R. Haugsrud. Hydration and proton conductivity in laaso4. J. Mater. Chem., 22:1652–1661, 2012. [27] P.E. Blöchl. Projector augmented-wave method. Phys. Rev. B, 50:17953–17979, 1994. [28] P.E. Blöchl, O. Jepsen, and O.K. Andersen. Improved tetrahedron method for brillouin-zone integrations. Phys. Rev. B, 49:16223–16233, 1994. [29] S. Bo and Z. Ping. First-principles local density approximation (lda) + u and generalized gradient approximation (gga) + u studies of plutonium oxides. Chinese Physics B, 17(4):1364, 2008. [30] A. Bogicevic and C. Wolverton. Nature and strength of defect interactions in cubic stabilized zirconia. Phys. Rev. B, 67:024106, 2003. [31] A. Bogicevic, C. Wolverton, G.M. Crosbie, and E.B. Stechel. Defect ordering in aliovalently doped cubic zirconia from first principles. Phys. Rev. B, 64:014106, 2001. [32] G. Busker, A. Chroneos, R.W. Grimes, and I. Chen. Solution mechanisms for dopant oxides in yttria. Journal of the American Ceramic Society, 82(6):1553–1559, 1999. [33] S. Buyukkilic, T. Shvareva, and A. Navrotsky. Enthalpies of formation and insights into defect association in ceria singly and doubly doped with neodymia and samaria. Solid State Ionics, 227(0):17 – 22, 2012. [34] C. Calvo. The crystal structure and phase transitions of -zn2p2o7. Canadian Journal of Chemistry, 43(5):1147–1153, 1965. [35] C. Calvo. The crystal structure of -mg2p2o7. Acta Crystallographica, 23(2):289–295, 1967. [36] C. Calvo. Crystal structure of .alpha.-calcium pyrophosphate. Inorganic Chemistry, 7(7):1345–1351, 1968. [37] W. Chen, T.A. Lee, and A. Navrotsky. Enthalpy of formation of yttria-doped ceria. Journal of Materials Research, 20:144–150, 2005. [38] W. Chen and A. Navrotsky. Thermochemical study of trivalent-doped ceria systems: Ceo2mo1.5 (m = la, gd, and y). Journal of Materials Research, 21:3242–3251, 2006. [39] J. Choi, R.J. McEachern, P. Taylor, and D.D. Wood. The effect of fission products on the rate of u3o8 formation in simfuel oxidized in air at 250c. Journal of Nuclear Materials, 230(3):250 – 258, 1996. [40] J. Cobos, D. Papaioannou, J. Spino, and M. Coquerelle. Phase characterisation of simulated high burn-up uo2 fuel. Journal of Alloys and Compounds, 271273(0):610 – 615, 1998. 77

[41] J.D. Cox, D.D. Wagman, and V.A. Medvedev. CODATA Key Values for Thermody- namics. Hemisphere Publishing Corp., New York, 1989.

[42] J. Crocombette, D. Torumba, and A. Chartier. Charge states of point defects in uranium oxide calculated with a local hybrid functional for correlated electrons. Phys. Rev. B, 83:184107, 2011.

[43] D.W.J. Cruickshank. The role of 3d-orbitals in small pi-bonds between (a) silicon, phosphorus, sulphur, or chlorine and (b) oxygen or nitrogen. J. Chem. Soc., pages 5486–5504, 1961.

[44] C. Deo, D. Andersson, C. Battaile, and B. Uberuaga. Atomistic Calculations of the Eﬀect of Minor Actinides on Thermodynamic and Kinetic Properties of UO 2+-x. Nuclear Energy University Programs, 2012.

[45] B.G. Dick and A.W. Overhauser. Theory of the dielectric constants of alkali halide crystals. Phys. Rev., 112:90–103, 1958.

[46] B. Dorado, B. Amadon, M. Freyss, and M. Bertolus. DFT + U calculations of the ground state and metastable states of uranium dioxide. Phys. Rev. B, 79:235125, 2009.

[47] B. Dorado, M. Freyss, B. Amadon, M. Bertolus, G. Jomard, and P. Garcia. Advances in ﬁrst-principles modelling of point defects in uo2 : f electron correlations and the issue of local energyminima. Journal of Physics: Condensed Matter, 25(33):333201, 2013. And references therein.

[48] B. Dorado and P. Garcia. First-principles dft + u modeling of actinide-based alloys: Application to paramagnetic phases of uo2 and (u,pu) mixed oxides. Phys. Rev. B, 87:195139, 2013.

[49] B. Dorado, G. Jomard, M. Freyss, and M. Bertolus. Stability of oxygen point defects in uo2 by ﬁrst-principles DFT + U calculations: Occupation matrix control and jahn- teller distortion. Phys. Rev. B, 82:035114, 2010.

[50] S.L. Dudarev, G.A. Botton, S.Y. Savrasov, C.J. Humphreys, and A.P. Sutton. Electron-energy-loss spectra and the structural stability of nickel oxide: an lsda+u study. Phys. Rev. B, 57:1505–1509, 1998.

[51] S.L. Dudarev, M.R. Castell, G.A. Botton, S.Y. Savrasov, C. Muggelberg, G.A.D. Briggs, A.P. Sutton, and D.T. Goddard. Understanding stm images and eels spectra of oxides with strongly correlated electrons: a comparison of nickel and uranium oxides. Micron, 31(4):363 – 372, 2000.

[52] J. Dura. http://www.nist.gov/mml/msed/functionalpolymer/fuelcell.cfm.

[53] S. Fukushima, T. Ohmichi, A. Maeda, and H. Watanabe. The eﬀect of yttrium content on the thermal conductivity of near-stoichiometric (u,y)o2 solid solutions. Journal of Nuclear Materials, 102(12):30 – 39, 1981. 78

[54] J.D. Gale and A.L. Rohl. The general utility lattice program (gulp). Molecular Simulation, 29(5):291–341, 2003.

[55] S. Gallini, M. Hansel, T. Norby, M.T. Colomer, and J.R. Jurado. Impedance spectroscopy and proton transport number measurements on Sr-substituted LaPO4 prepared by combustion synthesis. Solid State Ion., 162(SI):167–173, 2003. 11th Inter- national Conference on Solid State Protonic Conductors.

[56] H.Y. Geng, Y. Chen, Y. Kaneta, M. Kinoshita, and Q. Wu. Interplay of defect cluster and the stability of xenon in uranium dioxide from density functional calculations. Phys. Rev. B, 82:094106, 2010.

[57] A. Georges, G. Kotliar, W. Krauth, and M.J. Rozenberg. Dynamical mean-ﬁeld theory of strongly correlated fermion systems and the limit of inﬁnite dimensions. Rev. Mod. Phys., 68:13–125, 1996.

[58] P. Gotcu-Freis, J.-Y. Colle, C. Guneau, N. Dupin, B. Sundman, and R.J.M. Kon- ings. A thermodynamic study of the puamo system. Journal of Nuclear Materials, 414(3):408 – 421, 2011.

[59] R.W. Grimes, D.. Binks, and A.B. Lidiard. The extent of zinc oxide solution in zinc chromate spinel. Philosophical Magazine A, 72(3):651–668, 1995.

[60] R.W. Grimes, G. Busker, M.A. McCoy, A. Chroneos, J.A. Kilner, and S.P. Chen. An x-ray diﬀraction study on the kinetics of the cation redistribution in the spinel nial2o4. Berichte der Bunsengesellschaft fur physikalische Chemie, 101(9):1204–1210, 1997.

[61] R.W. Grimes and C.R.A. Catlow. The stability of ﬁssion products in uranium dioxide. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 335(1639):609–634, 1991.

[62] C. Guneau, C. Chatillon, and B. Sundman. Thermodynamic modelling of the pluto- niumoxygen system. Journal of Nuclear Materials, 378(3):257 – 272, 2008.

[63] K. Hagemark and M. Broli. Equilibrium oxygen pressures over solid solutions of urania-yttria and urania-lanthana at 1100 to 14ooc. Journal of the American Ceramic Society, 50(11):563–567, 1967.

[64] B.E. Hanken, C.R. Stanek, N. Grønbech-Jensen, and M. Asta. Computational study of the energetics of charge and cation mixing in u1-xcexo2. Phys. Rev. B, 84:085131, 2011.

[65] G.L.W. Hart and R.W. Forcade. Algorithm for generating derivative structures. Phys. Rev. B, 77:224115, 2008.

[66] H. Hayashi, R. Sagawa, H. Inaba, and K. Kawamura. Molecular dynamics calculations on ceria-based solid electrolytes with diﬀerent radius dopants. Solid State Ionics, 131(34):281 – 290, 2000. 79

[67] H. He, P.G. Keech, M.E. Broczkowski, J.J. Nol, and D.W. Shoesmith. Characteri- zation of the influence of fission product doping on the anodic reactivity of uranium dioxide. Canadian Journal of Chemistry, 85(10):702–713, 2007. [68] J. Hermet, F. Bottin, G. Dezanneau, and G. Geneste. Thermodynamics of hydration and oxidation in the proton conductor gd-doped barium cerate from density functional theory calculations. Phys. Rev. B, 85:205137, 2012. [69] J. Heyd, G.E. Scuseria, and M. Ernzerhof. Hybrid functionals based on a screened coulomb potential. The Journal of Chemical Physics, 118(18):8207–8215, 2003. [70] D.C. Hill. Phase relations and crystal chemistry in the system uranium oxidelan- thanum oxide. Journal of the American Ceramic Society, 45(6):258–263, 1962. [71] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev., 136:B864–B871, 1964. [72] J. Hubbard. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 276(1365):238–257, 1963. [73] H. Inaba and H. Tagawa. Ceria-based solid electrolytes. Solid State Ionics, 83(12):1 – 16, 1996. [74] F. Jollet, G. Jomard, B. Amadon, J.P. Crocombette, and D. Torumba. Hybrid functional for correlated electrons in the projector augmented-wave formalism: Study of multiple minima for actinide oxides. Phys. Rev. B, 80:235109, 2009. [75] G. Jomard, B. Amadon, F. Bottin, and M. Torrent. Structural, thermodynamic, and electronic properties of plutonium oxides from first principles. Phys. Rev. B, 78:075125, 2008. [76] M.S. Khan, M.S. Islam, and D.R. Bates. Cation doping and oxygen diffusion in zirconia: a combined atomistic simulation and molecular dynamics study. J. Mater. Chem., 8:2299–2307, 1998. [77] J.C. Killeen. The effect of additives on the irradiation behaviour of uo2. Journal of Nuclear Materials, 58(1):39 – 46, 1975. [78] J. Kim, Y. Ha, S. Park, K. Jee, and W. Kim. Effect of a trivalent dopant, gd3+, on the oxidation of uranium dioxide. Journal of Nuclear Materials, 297(3):327 – 331, 2001. [79] N. Kitamura, K. Amezawa, Y. Tomii, T. Hanada, N. Yamamoto, T. Omata, and S. Otsuka-Yao-Matsuo. Electrical conduction properties of sr-doped lapo4 and cepo4 under oxidizing and reducing conditions. J. Electrochem. Soc.,, 152(4):A658–A663, 2005. [80] H. Kleykamp. The chemical state of the fission products in oxide fuels. Journal of Nuclear Materials, 131(23):221 – 246, 1985. 80

[81] H. Kleykamp. The solubility of selected ﬁssion products in uo2 and (u, pu)o2. Journal of Nuclear Materials, 206(1):82 – 86, 1993.

[82] W. Kohn and L.J. Sham. Self-consistent equations including exchange and correlation eﬀects. Phys. Rev., 140:A1133–A1138, 1965.

[83] A. Koji, M. Hideki, T. Yoichi, and Y. Naoichi. Protonic conduction and defect structures in sr-doped lapo4. Solid State Ion., 145(14):233 – 240, 2001.

[84] G. Kotliar, S.Y. Savrasov, K. Haule, V.S. Oudovenko, O. Parcollet, and C.A. Mari- anetti. Electronic structure calculations with dynamical mean-ﬁeld theory. Rev. Mod. Phys., 78:865–951, 2006.

[85] G. Kresse and J. Furthmuller. Eﬃcient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B, 54(16):11169–11186, 1996.

[86] G. Kresse and J. Furthmller. Eﬃciency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Computational Materials Science, 6(1):15 – 50, 1996.

[87] G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B, 59:1758–1775, 1999.

[88] K.D. Kreuer. Aspects of the formation and mobility of protonic charge carriers and the stability of perovskite-type oxides. Solid State Ionics, 125(14):285 – 302, 1999.

[89] K.N. Kudin, G.E. Scuseria, and R.L. Martin. Hybrid density-functional theory and the insulating gap of uo2. Phys. Rev. Lett., 89:266402, 2002. [90] P. Larson, W.R.L. Lambrecht, A. Chantis, and M. van Schilfgaarde. Electronic structure of rare-earth nitrides using the LSDA + u approach: Importance of allowing 4f orbitals to break the cubic crystal symmetry. Phys. Rev. B, 75:045114, 2007.

[91] T.A. Lee and A. Navrotsky. Enthalpy of formation of cubic yttria-stabilized hafnia. Journal of Materials Research, 19:1855–1861, 2004.

[92] T.A. Lee, A. Navrotsky, and I. Molodetsky. Enthalpy of formation of cubic yttria- stabilized zirconia. Journal of Materials Research, 18:908–918, 2003.

[93] R.J. Lemire. Chemical Thermodynamics of Neptunium and Plutonium. Elsevier, Oxford, 2001.

[94] A.I. Liechtenstein, V.I. Anisimov, and J. Zaanen. Density-functional theory and strong interactions: Orbital ordering in mott-hubbard insulators. Phys. Rev. B, 52:R5467– R5470, 1995.

[95] T.B. Lindeme and J.O. Brynestad. Review and chemical thermodynamic representation of u1zcezo2x and u1zlnzo2x); ln = y, la; nd, gd. Journal of the American Ceramic Society, 69(12):867–876, 1986. 81

[96] Y. Lu, Y. Yang, F. Zheng, B. Wang, and P. Zhang. Electronic, mechanical, and thermodynamic properties of americium dioxide. Journal of Nuclear Materials, 441(13):411 – 420, 2013.

[97] G.C. Mather and M.S. Islam. Defect and dopant properties of the srceo3-based proton conductor. Chem. Mater., 17(7):1736–1744, 2005.

[98] T. Matsui and K. Naito. Electrical conductivity measurement and thermogravimetric study of lanthanum-doped uranium dioxide. Journal of Nuclear Materials, 138(1):19 – 26, 1986.

[99] L. Mazeina, A. Navrotsky, and M. Greenblatt. Calorimetric determination of energetics of solid solutions of uo2+x with cao and y2o3. Journal of Nuclear Materials, 373(13):39 – 43, 2008.

[100] R.J. McEachern, D.C. Doern, and D.D. Wood. The eﬀect of rare-earth ﬁssion products on the rate of u3o8 formation on uo2. Journal of Nuclear Materials, 252(12):145 – 149, 1998.

[101] R.J. McEachern and P. Taylor. A review of the oxidation of uranium dioxide at temperatures below 400c. Journal of Nuclear Materials, 254(23):87 – 121, 1998.

[102] B. Meredig, A. Thompson, H.A. Hansen, C. Wolverton, and A. van de Walle. Method for locating low-energy solutions within DFT + u. Phys. Rev. B, 82:195128, 2010.

[103] S.C. Middleburgh, R.W. Grimes, K.H. Desai, P.R. Blair, L. Hallstadius, K. Backman, and P. Van Uﬀelen. Swelling due to ﬁssion products and additives dissolved within the uranium dioxide lattice. Journal of Nuclear Materials, 427(13):359 – 363, 2012.

[104] S.C. Middleburgh, D.C. Parﬁtt, R.W. Grimes, B. Dorado, M. Bertolus, P.R. Blair, L. Hallstadius, and K. Backman. Solution of trivalent cations into uranium dioxide. Journal of Nuclear Materials, 420(13):258 – 261, 2012.

[105] L. Minervini, R.W. Grimes, and K.E. Sickafus. Disorder in pyrochlore oxides. Journal of the American Ceramic Society, 83(8):1873–1878, 2000.

[106] L. Minervini, M.O. Zacate, and R.W. Grimes. Defect cluster formation in m2o3-doped ceo2. Solid State Ionics, 116(34):339 – 349, 1999.

[107] Z.A. Munir. The electrical conductivity of doped and undoped uranium oxide. Inter- national Journal of Thermophysics, 2(2):177–186, 1981.

[108] A. Navrotsky, P. Simoncic, H. Yokokawa, W. Chen, and T. Lee. Calorimetric measurements of energetics of defect interactions in ﬂuorite oxides. Faraday Discuss., 134:171–180, 2007.

[109] P.V. Nerikar, X. Liu, B.P. Uberuaga, C.R. Stanek, S.R. Phillpot, and S.B. Sinnott. Thermodynamics of ﬁssion products in uo 2 x. Journal of Physics: Condensed Matter, 21(43):435602, 2009. 82

[110] Y.X. Ni, J.M. Hughes, and A.N. Mariano. Crystak-chemistry of the monazite and xenotime structures. Am. Miner., 80(1-2):21–26, 1995.

[111] T. Norby. The promise of protonics. Nature, 410:877–878, 2001.

[112] T. Norby and N. Christiansen. Proton conduction in ca-substituted and sr-substituted lapo4. Solid State Ion., 77:240–243, 1995. VIIth International Conference on Solid State Protonic Conductors.

[113] P. Novak, J. Kune, L. Chaput, and W.E. Pickett. Exact exchange for correlated electrons. Physica Status Solidi (b), 243(3):563–572, 2006.

[114] T. Ohmichi, S. Fukushima, A. Maeda, and H. Watanabe. On the relation between lattice parameter and o/m ratio for uranium dioxide-trivalent rare earth oxide solid solution. Journal of Nuclear Materials, 102(12):40 – 46, 1981.

[115] V. Ozolins, F. Zhou, and M. Asta. Ruthenia-Based Electrochemical Supercapacitors: Insights from First-Principles Calculations. Acc. Chem. Res., 46(5):1084–1093, 2013.

[116] K. Park and D.R. Olander. Defect models for the oxygen potentials of gadolinium-and europium-doped urania. Journal of Nuclear Materials, 187(1):89 – 96, 1992.

[117] J.P. Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation made simple. Phys. Rev. Lett., 77(18):3865–3868, 1996.

[118] J.P. Perdew, K. Burke, and M. Ernzerhof. Generalized gradient approximation made simple [phys. rev. lett. 77, 3865 (1996)]. Phys. Rev. Lett., 78:1396–1396, 1997.

[119] L. Petit, A. Svane, Z. Szotek, and W.M. Temmerman. First-principles calculations of puo2x. Science, 301(5632):498–501, 2003.

[120] L. Petit, A. Svane, Z. Szotek, W.M. Temmerman, and G.M. Stocks. Electronic structure and ionicity of actinide oxides from ﬁrst principles. Phys. Rev. B, 81:045108, 2010.

[121] S. Phadke, J.C. Nino, and S.M. Islam. Structural and defect properties of the LaPO4 and LaP5O14-based proton conductors. J. Mater. Chem., 22(48):25388–25394, 2012.

[122] S.G. Popov, V.A. Lysenko, and V.N. Proselkov. Thermodynamic simulation of phase equilibria in the uo2-gd2o3 system at high temperatures. High Temperature, 50(2):221–224, 2012.

[123] A. Predith, G. Ceder, C. Wolverton, K. Persson, and T. Mueller. Ab initio prediction of ordered ground-state structures in zro2-y2o3. Phys. Rev. B, 77:144104, 2008.

[124] D. Prieur, P. Martin, F. Lebreton, T. Delahaye, D. Banerjee, A.C. Scheinost, and A. Jankowiak. Accommodation of multivalent cations in ﬂuorite-type solid solutions: Case of am-bearing {UO2}. Journal of Nuclear Materials, 434(13):7 – 16, 2013. Special Section on Spent Nuclear Fuel. 83

[125] I.D. Prodan, G.E. Scuseria, and R.L. Martin. Assessment of metageneralized gradient approximation and screened coulomb hybrid density functionals on bulk actinide oxides. Phys. Rev. B, 73:045104, 2006.

[126] I.D. Prodan, G.E. Scuseria, and R.L. Martin. Covalency in the actinide dioxides: Systematic study of the electronic properties using screened hybrid density functional theory. Phys. Rev. B, 76:033101, 2007.

[127] A.K.A. Pryde, S. Vyas, R.W. Grimes, J.A. Gardner, and R. Wang. Cadmium and indium defects in ceria and their interaction with oxygen vacancies and small polarons. Phys. Rev. B, 52:13214–13222, 1995.

[128] M. Razdan and D.W. Shoesmith. The electrochemical reactivity of 6.0 wtcarbon- ate/bicarbonate solutions. Journal of The Electrochemical Society, 161(4):H225– H234, 2014.

[129] M. Razdan and D.W. Shoesmith. Inﬂuence of trivalent-dopants on the structural and electrochemical properties of uranium dioxide (uo2). Journal of The Electrochemical Society, 161(3):H105–H113, 2014.

[130] K. Richter and C. Sari. Phase relationships in the neptunium-oxygen system. Journal of Nuclear Materials, 148(3):266 – 271, 1987.

[131] B.E. Robertson and C. Calvo. The crystal structure and phase transformation of -cu2p2o7. Acta Crystallographica, 22(5):665–672, 1967.

[132] L.E. Roy, T. Durakiewicz, R.L. Martin, J.E. Peralta, G.E. Scuseria, C.G. Olson, J.J. Joyce, and E. Guziewicz. Dispersion in the mott insulator uo2: A comparison of photoemission spectroscopy and screened hybrid density functional theory. Journal of Computational Chemistry, 29(13):2288–2294, 2008.

[133] H. Runge and M. Greenblatt. Structure and conductivity investigations of alkaline earth substituted uranium oxide, u1xa0;xa0;xmxo2xa0;xa0; (mxa0;=xa0;mg, ca, sr) for solid oxide fuel cell applications. Solid State Ionics, 177(34):269 – 274, 2006.

[134] F. Schleifer, A. Naoumidis, and H. Nickel. Solid solutions and phase equilibria in (u,zr,ln) oxides in the temperature range 12701670 k. Journal of Nuclear Materials, 101(12):150 – 161, 1981.

[135] R.D. Shannon. Revised eﬀective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallographica Section A, 32(5):751– 767, 1976.

[136] A.B. Shick, W.E. Pickett, and A.I. Liechtenstein. Ground and metastable states in -ce from correlated band theory. Journal of Electron Spectroscopy and Related Phenomena, 114116(0):753 – 758, 2001.

[137] D. Shin and T.M. Besmann. Thermodynamic modeling of the (u,la)o2x solid solution phase. Journal of Nuclear Materials, 433(13):227 – 232, 2013. And references therein. 84

[138] D.W. Shoesmith. Fuel corrosion processes under waste disposal conditions. Journal of Nuclear Materials, 282(1):1 – 31, 2000.

[139] P. Simoncic and A. Navrotsky. Energetics of rare-earth-doped hafnia. Journal of Materials Research, 22:876–885, 2007.

[140] P. Simoncic and A. Navrotsky. Systematics of phase transition and mixing energetics in rare earth, yttrium, and scandium stabilized zirconia and hafnia. Journal of the American Ceramic Society, 90(7):2143–2150, 2007.

[141] J.M. Solomon, V. Alexandrov, B. Sadigh, A. Navrotsky, and M. Asta. Computational study of the energetics and defect clustering tendencies for y- and la-doped uo2. Acta Materialia, 78(0):282 – 289, 2014.

[142] J.M. Solomon, A. Navrotsky, and M. Asta. Energetics and defect clustering trends for trivalent rare earth cations substituted in uo2. Journal of Nuclear Materials, 457(0):252 – 255, 2015.

[143] E. Stadlbauer, U. Wichmann, U. Lott, and C. Keller. Thermodynamics and phase relationships of the ternary lanthanum-uranium-oxygen system. Journal of Solid State Chemistry, 10(4):341 – 350, 1974.

[144] C. Stanek. Atomic Scale Disorder in Fluorite and Fluorite Related Oxides. PhD thesis, 2003.

[145] C.R. Stanek, R.W. Grimes, C. Unal, S.A. Maloy, and S.C. Scott. Fundamentals of Materials for Energy and Environmental Sustainability. Cambridge University Press, 2012.

[146] C.R. Stanek, L. Minervini, and R.W. Grimes. Nonstoichiometry in a2b2o7 pyrochlores. Journal of the American Ceramic Society, 85(11):2792–2798, 2002.

[147] M.A. Subramanian, G. Aravamudan, and G.V. Subba Rao. Oxide pyrochlores a review. Progress in Solid State Chemistry, 15(2):55 – 143, 1983.

[148] B. Sun, P. Zhang, and X. Zhao. First-principles local density approximation+u and generalized gradient approximation+u study of plutonium oxides. The Journal of Chemical Physics, 128(8):–, 2008.

[149] C. Suzuki, T. Nishi, M. Nakada, T. Tsuru, M. Akabori, M. Hirata, and Y. Kaji. Dft study on the electronic structure and chemical state of americium in an (am,u) mixed oxide. Journal of Physics and Chemistry of Solids, 74(12):1769 – 1774, 2013.

[150] L.E. Thomas, R.E. Einziger, and H.C. Buchanan. Eﬀect of ﬁssion products on air- oxidation of lwr spent fuel. Journal of Nuclear Materials, 201(0):310 – 319, 1993.

[151] M.E. Torrero, E. Baraj, J. de Pablo, J. Gimnez, and I. Casas. Kinetics of corrosion and dissolution of uranium dioxide as a function of ph. International Journal of Chemical Kinetics, 29(4):261–267, 1997. 85

[152] T. Toshihide, I. Mitsushige, Y. Toshiyuki, and O. Kinji. Effect of cations on lattice constants of (myu1y)o2.00 (m=pu, th, la) at low doped cation concentrations. Journal of Alloys and Compounds, 271273(0):391 – 394, 1998. [153] K. Toyoura, N. Hatada, Y. Nose, I. Tanaka, K. Matsunaga, and T. Uda. Proton- conducting network in lanthanum orthophosphate. J. Phys. Chem. C, 116(36):19117– 19124, 2012. [154] M. Trummer, B. Dahlgren, and M. Jonsson. The effect of y2o3 on the dynamics of oxidative dissolution of uo2. Journal of Nuclear Materials, 407(3):195 – 199, 2010. [155] F. Tyholdt, J.A. Horst, S. Jorgensen, T. Ostvold, and T. Norby. Segregation of Sr in Sr-doped LaPO4 ceramics. Surf. Interface Anal., 30(1):95–97, 2000. [156] A. van de Walle. Multicomponent multisublattice alloys, nonconfigurational entropy and other additions to the alloy theoretic automated toolkit. Calphad, 33(2):266 – 278, 2009. [157] A. van de Walle, M. Asta, and G. Ceder. The alloy theoretic automated toolkit: A user guide. Calphad, 26(4):539 – 553, 2002. [158] B. Wang, H. Shi, W. Li, and P. Zhang. First-principles LDA + u and GGA + u study of neptunium dioxide. Phys. Rev. B, 81:045119, 2010. [159] L. Wang, T. Maxisch, and G. Ceder. Oxidation energies of transition metal oxides within the GGA + U framework. Phys. Rev. B, 73:195107, 2006. [160] X. Wen, R.L. Martin, L.E. Roy, G.E. Scuseria, S.P. Rudin, E.R. Batista, T.M. Mc- Cleskey, B.L. Scott, E. Bauer, J.J. Joyce, and T. Durakiewicz. Effect of spin-orbit coupling on the actinide dioxides AnO(2) (An=Th, Pa, U, Np, Pu, and Am): A screened hybrid density functional study. Journal of Chemical Physics, 137(15):154707, 2012. [161] X. Wen, R.L. Martin, G.E. Scuseria, S.P. Rudin, and E.R. Batista. A screened hybrid dft study of actinide oxides, nitrides, and carbides. The Journal of Physical Chemistry C, 117(25):13122–13128, 2013. [162] W.B. Wilson, C.A. Alexander, and A.F. Gerds. Stabilization of uo2. Journal of Inorganic and Nuclear Chemistry, 20(34):242 – 251, 1961. [163] J.H. Yang, K.S. Kim, K.W. Kang, K.W. Song, and Y.H. Jung. Electrical conductivity and non-stoichiometry in the (u,gd)o2x system. Journal of Nuclear Materials, 340(23):171 – 178, 2005. [164] L. Yang, C. Zuo, S. Wang, Z. Cheng, and M. Liu. A novel composite cathode for low-temperature sofcs based on oxide proton conductors. Advanced Materials, 20(17):3280–3283, 2008. [165] K. Yoshida, T. Arima, Y. Inagaki, K. Idemitsu, M. Osaka, and S. Miwa. Oxygen potential of hypo-stoichiometric la-doped uo2. Journal of Nuclear Materials, 418(13):22 – 26, 2011. 86

[166] R. Yu and L.C. De Jonghe. Proton-transfer mechanism in lapo4. J. Phys. Chem. C, 111(29):11003–11007, 2007.

[167] M.O. Zacate, L. Minervini, D.J. Bradﬁeld, R.W. Grimes, and K.E. Sickafus. Defect cluster formation in m2o3-doped cubic zro2. Solid State Ionics, 128(14):243 – 254, 2000.

[168] L. Zhang and A. Navrotsky. Submitted to Journal of Nuclear Materials.

[169] P. Zhang, B. Wang, and X. Zhao. Ground-state properties and high-pressure behavior of plutonium dioxide: Density functional theory calculations. Phys. Rev. B, 82:144110, 2010.

[170] F. Zhou and V. Ozoli¸nˇs.Obtaining correct orbital ground states in electron systems using a nonspherical self-interaction-corrected method. Phys. Rev. B, 80:125127, 2009.

[171] F. Zhou and V. Ozoli¸nˇs. Crystal ﬁeld and magnetic structure of uo2. Phys. Rev. B, 83:085106, 2011.

[172] M. Zinkevich. Thermodynamics of rare earth sesquioxides. Progress in Materials Science, 52(4):597 – 647, 2007.

[173] C. Zuo, S. Zha, M. Liu, M. Hatano, and M. Uchiyama. Ba(zr0.1ce0.7y0.2)o3 as an electrolyte for low-temperature solid-oxide fuel cells. Advanced Materials, 18(24):3318– 3320, 2006. 87

Appendix A

Appendix

A.1 Total energy comparison between ionic pair potentials and density functional theory

In Figs. A.1 to A.5, we present a comparison between total energies of systems calculated using ionic pair potentials and density functional theory methods (DFT) for a series of compositions with Y and La rare earth substituted cations in UO2, in order to check the reliability of using pair potentials as a high-throughput method for screening out many energetically disfavorable structures (the theory behind the pair potentials and DFT methods are discussed in Chapters 2 and 3, and the details of the methods specific to these calculations can be found in Chapter 5). All structures for a given composition and M3+ species were ranked according to their total energy based on the ionic pair potential calculations. Initially DFT calculations were performed for the resulting six lowest energy structures for a given composition. If any of the five higher energy structures were found to be lower than the ground state predicted by the pair potential models, additional structures were considered in an effort to ensure that the lowest energy configuration was identified by the DFT based methods. In all cases, the ionic pair potentials predict the DFT-calculated lowest energy structure within the first ten structures, and in several cases exactly predicts the lowest energy structure. This analysis supports the methodology of using ionic pair potentials as an screening tool, after which DFT calculations on only a few structures are necessary to accurately determine the total energy of the most stable structure. 88

Figure A.1: Total energies of systems with compositions U2Y2O7 (top) and U2La2O7 (bottom). 89

Figure A.2: Total energies of systems with compositions U4Y2O11 (top) and U4La2O11 (bottom). 90

Figure A.3: Total energies of systems with compositions U4Y2O10 (top) and U2La2O7 (bottom). 91

Figure A.4: Total energies of systems with compositions U4Y2O12 (top) and U4La2O12 (bottom). 92

Figure A.5: Total energies of systems with compositions U2Y2O8 (top) and U2La2O8 (bottom).