Energetics and Defect Interactions of Complex Oxides for ... · into the energetics and defect...

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


Copyright 2015

by


1

Abstract


by


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 insightsinto the energetics and defect interactions of complex oxides that are relevant for today’senergy challenges. To achieve this goal, the development of novel computational method-ologies are required to handle complex systems, including systems containing nearly 650ions and systems with tens of thousands of possible atomic configurations. The systemsthat 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 importancein nuclear fuel performance and disposal.

First we undertake density-functional-theory (DFT) calculations on the relativeenergetics of pyrophosphate defects and protons in LaPO4, including their binding withdivalent 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, whichis comparable to the value of 0.34 eV calculated for proton-dopant binding energies in thesame system. These results establish that dopant-defect interactions further stabilize protonincorporation, with the hydration enthalpies when the dopants are nearest and furthest fromthe protons and pyrophosphate defects being -1.66 eV and -1.37 eV, respectively. Eventhough our calculations show that dopant binding enhances the enthalpic favorability ofproton incorporation, they also suggest that such binding is likely to substantially lower thekinetic rate of hydrolysis of pyrophosphate defects.

We then shift our focus to solid solutions of fluorite-structured UO2 with trivalentrare earth fission product cations (M3+=Y, La) using a combination of ionic pair potentialand DFT based methods. Calculated enthalpies of formation with respect to constituentoxides show higher energetic stability for La solid solutions than for Y. Additionally, calcula-tions performed for different atomic configurations show a preference for reduced (increased)oxygen vacancy coordination around La (Y) dopants. The current results are shown to bequalitatively consistent with related calculations and calorimetric measurements of heatsof formation in other trivalent doped fluorite oxides, which show a tendency for increasingstability and increasing preference for higher oxygen coordination with increasing size ofthe trivalent impurity. We expand this investigation by considering a series of trivalentrare earth fission product cations, specifically, Y3+ (1.02 Å, Shannon radius with eightfoldcoordination), Dy3+ (1.03 Å), Gd3+ (1.05 Å), Eu3+ (1.07 Å), Sm3+ (1.08 Å), Pm3+ (1.09

2

Å), Nd3+ (1.11 Å), Pr3+ (1.13 Å), Ce3+ (1.14 Å) and La3+ (1.16 Å). Compounds with ionicradius of the M3+ species smaller or larger than 1.09 Å are found to have energetically pre-ferred defect ordering arrangements. Systems with preferred defect ordering arrangementsare suggestive of defect clustering in short range ordered solid solutions, which is expectedto limit oxygen ion mobility and therefore the rate of oxidation of spent nuclear fuel.

Finally, the energetics of rare earth substituted (M3+= La, Y, and Nd) UO2 solidsolutions are investigated by employing a combination of calorimetric measurements andDFT 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 com-putational/experimental collaborative effort supported through the Materials Science ofActinides Energy Frontier Research Center. Calculated and measured formation enthalpiesagree within 10 kJ/mol for stoichiometric oxygen/metal compositions. To better under-stand the factors governing the stability and defect binding in rare earth substituted uraniasolid solutions, systematic trends in the energetics are investigated based on the presentresults and previous computational and experimental thermochemical studies of rare earthsubstituted fluorite oxides. A consistent trend towards increased energetic stability withlarger size mismatch between the smaller host tetravalent cation and the larger rare earthtrivalent cation is found for both actinide and non-actinide fluorite oxide systems wherealiovalent substitution of M cations is compensated by oxygen vacancies. However, thelarge exothermic oxidation enthalpy in the UO2 based systems favors compositions withhigher oxygen-to-metal ratios where charge compensation occurs through the formation ofuranium 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 21.1 Fuel Cell Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Nuclear Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theoretical Framework 72.1 Atomistic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 The Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 First-principles Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 The Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . 10

3 Addressing Self-Interaction Errors for Strongly Correlated Materials 123.1 Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 DFT Corrections for Strongly Correlated Materials . . . . . . . . . . . . . . 12

3.2.1 Hybrid Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Hubbard-U Correction (DFT+U) . . . . . . . . . . . . . . . . . . . . 13

II Results and Discussion 15

4 Energetics and Defect Interactions: Sr Doped LaPO4 164.1 Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.4.1 Pyrophosphate and Proton Configurations . . . . . . . . . . . . . . . 194.4.2 Superstructures with Defects Charge Compensated by Sr Dopants . 23

iii

4.4.3 Hydration Enthalpy Calculations for Structures with Neighboring andIsolated Dopants/Defects . . . . . . . . . . . . . . . . . . . . . . . . 25

4.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Energetics and Defect Interactions: Y and La Doped UO2 295.1 Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 Computational Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3.1 Structure Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . 315.3.2 Classical Pair Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 325.3.3 First-principles Calculations . . . . . . . . . . . . . . . . . . . . . . . 32

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Energetics and Defect Interactions: Rare-Earth Doped UO2 456.1 Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.3 Computational Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 466.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Comparison to Experiment, Intermediate and Stoichiometric Composi-tions: Y, Nd and La Doped UO2 537.1 Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.3.1 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . 557.3.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

III Concluding Remarks 68

8 Conclusions and Future Work 698.0.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.0.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 74

iv

A Appendix 87A.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]. . . 31.2 The unit cell of LaPO4. Lanthanum, oxygen, and phosphorus ions are shown

in blue, red, and yellow, respectively. . . . . . . . . . . . . . . . . . . . . . . 41.3 The unit cell for UO2. Uranium and oxygen ions are shown in blue and red,

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Example of a nuclear fuel cycle. Adapted from Ref. [145]. . . . . . . . . . . 51.5 The chemical states of fission products, denoted by the shaded regions. Or-

ange, gray, blue, and green shaded regions denote volatile gases, metallic pre-cipitates, solid solution, and oxide precipitates, respectively. Elements withmultiple 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 phos-phorus ions are shown in blue, red, and yellow, respectively. The orthophos-phate groups in the shaded region are considered in Figs. 4.1A-C. The py-rophosphate is derived from two orthophosphate groups (2PO4

6−, Fig. 4.1A)by forming an oxygen vacancy (2+ charge, Fig. 4.1B), causing one oxygenion to be shared and resulting in the overall 4- charge (Fig. 4.1C). The ionslabeled with “prime” represent those within the phosphate group that sharesits oxygen ion in the pyrophosphate link. . . . . . . . . . . . . . . . . . . . 17

4.2 The partial density of states for undoped LaPO4 systems containing a py-rophosphate (top) and proton (bottom) defect are shown. The charged de-fects are compensated with background charges. The arrows denote the extrapeaks 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 (“bothnear” structure). The black dotted circle represents the approximate positionof the top proton (H2) within the “one far” structure, in which the proton isfar 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 thecation fcc sublattice and one charge compensating oxygen vacancy on ananion simple cubic sublattice is shown. . . . . . . . . . . . . . . . . . . . . . 31

vi

5.2 Formation enthalpies of U1−xMxO2−0.5x (M=Y,La) structures are shown forthe lowest-energy fully relaxed structures of all compositions enumerated inthis study. Filled and open symbols correspond to DFT+U and hybrid-functional calculations, respectively. The dotted line connecting the solidpoints is a guide to the eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 The formation enthalpies calculated using DFT+U for all structures com-puted in this study are shown. Each structure is denoted by the number oftrivalent cations in the four tetrahedral nearest neighbor positions surround-ing 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 ofneighbors between the two sites (rounded to the nearest neighbor) is shown. 38

5.4 Calculated formation enthalpies for Y-substituted AO2 fluorite systems areplotted versus the radius of the host A4+ cation radius. Results for zirco-nia (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 con-taining, one, two, and three trivalent cation nearest neighbors to an oxygenvacancy 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, andthree 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 oneOvac, or in more oxidizing environments, charge compensation can occur byoxidation of U4+ to U5+ for every one Ln3+/U4+ substitution, as illustrated inthe motifs above. Systems with compositions that are intermediate betweenfully oxygen vacancy charge-compensated and fully U5+ charge-compensatedcontain both oxygen vacancies and U5+ ions. . . . . . . . . . . . . . . . . . 57

7.2 Formation enthalpies (∆Hf ) of U1−xLnxO2−0.5x+y structures are shown forthe calculated lowest-energy fully relaxed structures of all compositions enu-merated in this study, in comparison to calorimetric data presented previously[168].The enthalpies are plotted against oxygen-to-metal ratio (O/M), which is re-lated to x and y according to Eq. 7.3. . . . . . . . . . . . . . . . . . . . . . 60

7.3 The formation enthalpies calculated for stoichiometric compositions for bothcalculation (“Comp.”) and experiment (“Exp.”) with respect to Ln dopantcation size are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.4 The first-principles calculated percent lattice parameter difference relativeto fluorite UO2 for the three Ln dopant species considered in this work forstoichiometric 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 flu-orite AO2 from calorimetric and computational results, as a function of triva-lent dopant cation radius, where x = 1/3 in (a) and x = 1/2 in (b). aLee andNavrotsky[91] bSimoncic and Navrotsky[139] cSimoncic and Navrotsky[140]dBogicevic et al.[31] eChen et al.[37] fNavrotsky et al.[108] gBuyukkilic etal.[33] hSolomon et al.[141] iThis work jAizenshtein et al.[7] kAlexandrov etal.[8] lSolomon et al.[142] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.6 Formation enthalpies of A1−xLnxO2−0.5x fluorite oxides (x=1/3, 1/2) fromLnO1.5 and fluorite AO2 from calorimetric and computational results, as afunction 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 flu-orite AO2 from calorimetric and computational results, as a function of sizemismatch 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 linesshown 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





viii

List of Tables

4.1 Calculated inter-ionic distances (units of Å) of the four possible pryophos-phate configurations are listed in columns 1-4. Row 1 shows the distancebetween the two phosphorus ions. Rows 2 and 3 show bond lengths betweenthe phosphorous ions and the shared oxygen ion. The ions labeled with“prime” represent those within the phosphate group that shares its oxygenion in the pyrophosphate link. Rows 4-9 list the other phosphorus-oxygenion bond lengths. Column 5 lists inter-ionic distances calculated previouslyusing ionic potential models.a Row 10 shows the relative energies (∆E) in eVof the pyrophosphate configurations . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Proton-dopant distances (units of Å) for the three configurations of 3 x 3 x3 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) struc-tures are listed for low-energy fully relaxed structures enumerated in thisstudy. The structures listed have formation energies within 3 kJ/mol-cationof the lowest energy structure for each trivalent cation species at each compo-sition considered. The fourth column lists the direction of ordering of oxygenvacancies. The last column lists the number of trivalent cations in the fourtetrahedral nearest neighbor positions surrounding an oxygen vacancy. . . . 35

7.1 The compositions considered for the calculations in this work are shown. Thefirst through fifth columns represent the formula, number of U4+ ions, numberof 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 ofkJ/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 yourdedication, 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 theearly stages of my graduate career.

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

To my University of Florida professors, Dr. Scott Perry, and Dr. Gerald Bourne, for helpingme 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 thereevery 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 ResearchCenter (DE-SC0001089), the U.S. Department of Energy through the Lawrence LivermoreNational Laboratory (DE-AC52-07NA27344), and the Department of Defense through theNational Defense Science & Engineering Graduate Fellowship Program. This work madeuse of resources of the National Energy Research Scientific Computing Center, supportedby 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 reducegreenhouse gas emissions, reduce fossil fuel dependence, and improve energy security. Thismandate will likely be fulfilled by a portfolio of energy sources, including nuclear and hydro-gen (fuel cell). Both nuclear and fuel cell technologies require engineering of complex oxidesmaterials in order to provide optimal properties. The goals and challenges of engineeringcomplex oxide materials for these technologies are discussed subsequently.

1.1 Fuel Cell Technology

A fuel cell converts chemical energy into electrical energy by an electrochemicalreaction (i.e., H2 +

12O2 → 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 througha 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 sameprinciples, except oxygen ions pass through the electrolyte instead of protons.

The electrolyte is a key element in the efficiency of a fuel cell. Required propertiesof solid electrolytes include high ionic conductivity, chemical and mechanical stability, andcorrosion resistance at the prescribed operating conditions. Solid oxide fuel cells, whichoperate most efficiently at 800◦C, suffer from corrosion and mechanical failure around thistemperature. It is therefore desirable to develop solid electrolytes that work at lower tem-peratures (i.e., 300-600◦C). Proton conductors have shown promise as an ideal candidatefor this temperature range[111, 164, 173].

Proton conducting electrolytes with the highest conductivities at intermediatetemperatures ranges (i.e., 300-600◦C) are perovskite structured barium zirconate and bar-ium cerate (BaZrO3 and BaCeO3, respectively), with conductivities around 10

−3 S/cm at300◦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 forproton 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. Theprotons are bound to an oxygen ion, and proton transport occurs by a hopping mechanismbetween neighboring PO3−4 tetrahedra. A discussion of how protons are incorporated intoLaPO4 can be found in Chapter 4.

A current drawback for REP compounds as a viable candidate for proton con-ducting electrolyte applications is the relatively low proton conductivity (10−5.2 S/cm at500◦C for 1 mol % Sr doping[10]). This value is several orders of magnitude lower thanwhat would be considered competitive for current energy production systems[111].

First principles calculations offer the ability to gain insight into the factors thatmay limit proton conductivity for these systems, including the energetics and kinetics of wa-ter uptake/proton incorporation, and the influence of defect interactions on these properties.This understanding can enable theorists and experimentalists to make better engineeringdecisions in developing optimal fuel cell technologies.

1.2 Nuclear Technology

Nuclear energy has provided electricity with minimal carbon emissions for sixtyyears. According to the International Energy Agency, nuclear energy was responsible for12% of the world’s electricity generation in 2011[4]. The primary material used for nuclearfuel 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-reactorstorage 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 shownin 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 environ-ment. 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 in-creasing the lifetime of fuel in the reactor, and thereby increasing the amount of fissionproducts. Predicting fuel behavior under these conditions requires an understanding of thethermochemical 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 todramatically affect fuel properties (see Ref. [137] for a review). Soluble trivalent rare earthcations are also relevant in the fabrication stage of the fuel cycle, e.g., purposeful additionof burnable “neutron poisons” (e.g., Gd) to control reactivity during the early stages of fuelburnup[122].

Unfortunately, thermochemical studies of trivalent cations in UO2 are sparse, andhave 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 lessstudied 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 potentialmodeling study[104] had been reported.

Direct calorimetric measurements have been historically limited by the refractorynature of UO2, and difficulties controlling and measuring the stoichiometry of these trivalentdoped 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 givenrise to an opportunity to benchmark density functional theory (DFT) calculations. DFTcalculations assist experimental investigations through accurate and controlled studies ofsystematic trends for various trivalent cation species and compositions.

1.3 Outline

The remainder of the paper is organized as follows. In the next chapter, thetheoretical framework of the computational tools used in this work is discussed. Chapter3 addresses the failure of standard DFT for strongly correlated materials (i.e., UO2 in thecurrent work), and discusses methods to circumvent this issue.

Chapter 4 begins the results section, where the energetics and defect interactionsof 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 solu-tion, and oxide precipitates, respectively. Elements with multiple colors have the possibilityof 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 andlargest trivalent rare earth cations, respectively) doped UO2. Chapter 6 extends this workby considering a series of trivalent rare earth cations to explore systematic trends. WhileChapters 5 and 6 focus on oxygen-to-metal ratios in UO2 corresponding to charge com-pensation only by oxygen vacancies, Chapter 7 includes results for compositions that arecharge compensated with U5+, and comparisons to experimental calorimetry measurementsare made.

Chapter 8 offers a summary and conclusions of the current work, and propositionsfor future work.

7

Chapter 2

Theoretical Framework

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

2.1 Atomistic Simulation

For ionic materials classical atomistic simulations typically involve the applica-tion of theories for interatomic interactions based on classical electrostatic models. Thesecalculations generally require less computational effort compared to quantum mechanicalcalculations (discussed in the next section). This means that high-throughput calculationsare possible for systems with relatively large collections of atoms. The accuracy of suchclassical simulations depends on how well the interatomic potential models describe thereal atom interactions.

Properties of ionic solids can often be modeled effectively using interatomic poten-tials that only consider pair interactions (i.e., interactions between two atoms, as opposedto many-body interactions which, in addition to pair interactions, account for simultaneousinteractions 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 summedover all n atoms in the solid to give the electrostatic energy, Ee, expressed as

Ee =1

2

n∑i

n∑j 6=i

qiqjrij

(2.1)

where qi and qj are the chaarges of ions i and j, and rij is the separation between thepositions of ions i and j. The factor of 12 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

Es =1

2

n∑i

n∑j 6=i

(Aexp

[−rijρ

]− C

(rij)6

)(2.2)

where A, ρ, and C are adjustable parameters. The first and second terms represent repul-sion between electron orbitals on the atoms due to Pauli exclusion, and the weak van derWaals attraction between electron orbitals, respectively. The adjustable parameters canbe fitted using quantum mechanical calculations or to physical properties, e.g., the relativepermittivity, elastic constants, lattice parameters, defect energies, etc.

The total energy of the solid is the combination of Coulombic and short-range pairinteractions

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 inEq. 2.3 have been found to disagree with experiment. The disagreement is often attributedto electronic polarizability of the ions. The electronic polarizability was included into ionicpair potential calculations by Dick and Overhauser with the shell model[45]. In this formal-ism, the formal charge of the ion is separated into a pointlike positively charged core and amassless negatively charged shell which surrounds the core. The shell models the electrondensity around the cores, and is displaced with respect to the core upon polarization. Ingeneral the core and shell will be displaced by different amounts due the interactions withthe surrounding ions.

The core and shell are treated as separate charges in Eq. 2.1, and the short-rangepotential Eq. 2.2 now acts between the shells. However, the core and shell are bound bya “spring” which prevents the core and shell from moving very far from one another. Thepotential energy associated with displacement of the shell with respect to the core is givenby the classical equation for an extended spring

Esp =1

2kδr2i (2.4)

where k is a spring constant (which is inversely proportional to the polarizability of theion), and δri is the amount the spring on ion i is stretched, i.e., the distance between thecenters 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 interatomicpotential models was discussed. A more complete understanding of the properties of solids,however, requires a quantum mechanical description of the interactions between nuclei andelectrons (a comparison between atomistic and first principles calculations for systems con-sidered in this dissertation is shown in the Appendix). In many cases, the calculationmethods based on quantum mechanics do not use any fitting parameters from experimentaldata, and are thus known as “first-principles” methods. The only information these methodsrequire is the atomic numbers of the constituent atoms. An introduction to first-principlesmethods and practical application of these methods for solids using density functional the-ory (DFT) is presented in the next sections (note that Hartree atomic units are used in theremaining sections).

2.2.1 The Schrödinger Equation

A quantum mechanical description of the physical properties of solids can be de-scribed in mathematical form by the time-independent Schrödinger equation

ĤΨ(ri, rI) = EΨ(ri, rI) (2.6)

where Ĥ is the Hamiltonian operator, Ψ(ri, rI) is the wave function, and E is the energy ofthe system. The variables ri and rI represent the position coordinates of all the electronsand nuclei in the system, respectively. Eq. 2.6 is simplified by approximating the positionsof the nuclei to be fixed with respect to the electrons, due the much greater mass of thenuclei 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 asystem

Ĥ = EkinI + Ekini + UIi + Uij + UIJ (2.7)

where EkinI and Ekini represent the kinetic energies of the nuclei and electrons, respectively,

and UIi, Uij , UIJ , are the potential energies from nucleus-electron, electron-electron, andnucleus-nucleus interactions, respectively. The Born-Oppenheimer approximation makes Ĥdepend only on energy terms that are a function of the electron positions ri, which simplifiesEq. 2.7 to

Ĥ = Ekini + UIi + Uij (2.8)

The complete expression for Eq. 2.8 is represented as

Ĥ = −12

n∑i

∇2i −N∑I

n∑i

ZI|rIi|

+1

2

n∑i

n∑j 6=i

1

|rij |(2.9)

where n and N are the number of electrons and nuclei in the system, respectively, ZI is thevalence 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 theSchrödinger equation is limited in practice to very simple systems due to the many-bodyquantum effect. The many-body effect arises from the 3n variables that are required tosolve the Schrödinger equation of an n-electron system. A breakthrough in the practicalapplication 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 theground-state energy is a functional of the electron density, and there exists an electrondensity that minimizes the total energy, which is the exact ground state energy. The elec-tron density n(r) is defined as

n(r) =occupied∑

i

|φi(r)|2 (2.10)

Eq. 2.10 says that n(r) is the sum over a set of probability densities of occupied Kohn-Shamorbitals φi(r).

The Kohn-Sham hamiltonian (Eq. 2.11) decomposes the many-body n-electronsystem into an effective one-electron Schrödinger equation, where the interactions with theother electrons is represented by an effective potential Ueff :

ĤKS = Enonkin + Uext + UH + Uxc = −

1

2∇2i + Ueff (2.11)

Enonkin is the non-interacting component of the electron kinetic energy, Uext is the potentialenergy due to the interaction between electrons and nuclei, and UH (known as the Hartreepotential) is the repulsive Coulomb interaction between each electron and the mean field ofall electrons in the system (n(r))

UH =

∫n(r′)

|r − r′|d3r′ (2.12)

and the many-body interactions reside in the exchange-correlation potential Uxc (note thatUH introduces a spurious self interaction, i.e., an electron interacts with n(r), which containsa contribution from the electron itself). If Uxc is known, then the ground state electrondensity (and therefore the ground state energy) of the non-interacting system of n electronsis the same as that for the many-body n-electron system. In practice, however, exactexpressions for Uxc are unknown and approximations are made.

The local density approximation (LDA) approximates the exchange-correlationenergy as an integral over an energy density �homxc [n(r)] that corresponds to a homogeneouselectron gas with the local density n(r). The LDA exchange-correlation energy ELDAxc [n(r)]is defined as

ELDAxc [n(r)] =

∫n(r)�homxc [n(r)]d

3r (2.13)

and the corresponding LDA exchange-correlation potential is

11

ULDAxc [n(r)] =δELDAxcδn(r)

(2.14)

The LDA works fairly well for systems where the electron density varies slowly;however, the generalized gradient approximation (GGA) was developed to produce moreaccurate exchange-correlation functionals. The GGA accounts not only for the local elec-tron density (LDA), but also the gradient at a given point. There are many forms ofthe GGA functional. The most frequently used functional is PBE (Perdew, Burke, andErnzerhof)[117], which is highly accurate and computationally efficient.

12

Chapter 3

Addressing Self-Interaction Errorsfor 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 “electroncorrelations” are present in systems that contain narrow d- and f - orbitals, including rareearth 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 correspondingto localized f states on the uranium sites, and therefore strong electron correlations. Thefollowing sections provide an overview of the theory behind strongly correlated systems andmethods 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 abandgap exists of bands of like character (i.e., 5f -5f bands of uranium ion for UO2). Thisclass of material is not considered in conventional band theory, in which the band gap ina charge transfer insulator exists between bands of different character (e.g. oxygen 2p anduranium 5f bands in UO3).

The existence of Mott insulators was originally proposed by Hubbard in 1963with 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 anelectron already existing at that site. If the overlap integral is sufficiently small (e.g., due tonarrow electron bands), the Coulomb repulsion term dominates and therefore the electronwill 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-correlationfunctional (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 UO2to be metallic, even though it is known to be a Mott insulator. GGA and LDA have alsobeen 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 GGAor 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, thetwo most widely-used methodologies will be discussed, namely, hybrid functionals and theHubbard-U correction.

3.2.1 Hybrid Functionals

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

∆EPBE+αHFXC = αEHFX + (1− α)EPBEX + EPBEC , (3.1)

where αEHFX is the HF exchange, EPBEX and E

PBEC are the PBE exchange and correlation

energies, respectively, and α represents the fraction of HF exchange that replaces the PBEexchange. The value for α used in this work is 0.25, which is representative of the PBE0functional [2]. The hybrid functionals tend to be computationally intensive for large sys-tems or high throughput calculations, however. The “local hybrid functional for correlatedelectrons” (LHFCE) is a less computationally demanding hybrid-functional approach inwhich the exact exchange is applied locally within the atomic spheres only to the correlatedelectrons (i.e., 5f uranium orbitals)[113]. The LHFCE functional is implemented in thecurrent work for UO2 based systems as a comparative approach to the method discussed inthe next section (the results and implementation of the LHFCE functional in the currentwork is discussed in Chapter 5).

3.2.2 Hubbard-U Correction (DFT+U)

The DFT+U approximation adds a correction to standard DFT exchange-correlationfunctionals. In the formalism by Dudarev et al.[50], the energy functional is expressed as

EPBE+U = EPBE +U − J

2

∑σ

(nm,σ − n2m,σ) (3.2)

where U and J are screened Coulomb and exchange parameters, respectively, nm,σ are theelectron occupations for orbital m and spin σ, such that the total number of electrons,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 penaltyfunction forces integer orbital occupations (0 or 1), which is characteristic of insulatingbehavior. 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 ). Thevalue of Ueff which has been most successfully applied to UO2 based systems in terms ofaccurate descriptions of the electronic structure[51, 5], defect formation energies and bulkproperties[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 sufficientaccuracy. However, application of a Hubard-U term can lead to multiple metastable statesfor these systems, corresponding to different orbital occupancies[47]. The issue of metastablestates and the methods we have used to circumvent this problem for the current work isdiscussed 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 itsstability 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 reportresults of density functional theory (DFT) calculations of the relative energetics of py-rophosphate defects and protons in LaPO4, including their binding with divalent dopantcations. Due to the low symmetry of the monazite crystal structure for LaPO4, there existsfour symmetry-distinct pyrophosphate defect configurations; DFT calculations are used toidentify 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 valueof 0.34 eV calculated for proton-dopant binding energies in the same system. These resultsestablish that dopant-defect interactions further stabilize proton incorporation, with thehydration enthalpies when the dopants are nearest and furthest from the protons and py-rophosphate defects being -1.66 eV and -1.37 eV, respectively. Even though our calculationsshow that dopant binding enhances the enthalpic favorability of proton incorporation, theyalso suggest that such binding is likely to substantially lower the kinetic rate of hydrolysisof 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. ©RoyalSociety of Chemistry

17

Figure 4.1: The unit cell of LaPO4 is shown on the left. Lanthanum, oxygen, and phos-phorus ions are shown in blue, red, and yellow, respectively. The orthophosphate groups inthe shaded region are considered in Figs. 4.1A-C. The pyrophosphate is derived from twoorthophosphate groups (2PO4

6−, 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 itsoxygen ion in the pyrophosphate link.

4.2 Introduction

Rare-earth phosphates (REP) are known proton conductors that are of interest fora variety of applications including intermediate-temperature fuel cell electrolytes, as wellas hydrogen separation membranes and sensors[111, 10, 83, 79, 112, 9, 55, 155]. Of thedifferent REP compounds, LaPO4 has been particularly well studied for proton conductorapplications, due to its stability and proton uptake in humid atmospheres over intermediatetemperature ranges (300-800◦C). Commercial viability of this and related REP systems islimited by low proton conductivity, however. For example, the highest reported proton con-ductivities for LaPO4 are 10

−5.2-10−3.5 S/cm at 500-925◦C, for 1 mol % Sr doping[10]. Thesevalues are several orders of magnitude lower than what would be considered competitivefor current energy production systems[111].

Amezawa and co-workers[83] have established a defect chemistry model for pro-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

2+

is substituted for La3+, resulting in a charge-balancing pyrophosphate defect (P2O74−),

according to the following defect reaction:

1

2Sr2P2O7 → Sr

′La +

1

2(P2O7)

••2PO4 (4.1)

In the LaPO4 compound, the pyrophosphate defect (Fig. 4.1C) is derived from two or-thophosphate anions (2PO4

6−, 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 themodel of Amezawa et. al [83], protons are incorporated into Sr-doped LaPO4 in the form ofhydrogen phosphate groups through the following defect reaction:

18

1

2(P2O7)

••2PO4 +

1

2H2O(g) ⇀↽ (HPO4)

•PO4 (4.2)

Equation 4.2 is widely accepted as the primary pathway for proton incorporationfor 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 ofpositively-charged oxygen vacancies present as pyrophosphate defects, to a degree controlledby 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 studieshave begun to provide insights into the microscopic factors limiting proton conductivityin LaPO4. Considering first the thermodynamic factors underlying proton uptake (andtherefore carrier density), the hydration enthalpy of the reaction given by Eq. 4.2 has beenreported previously from experimental measurements, with values ranging between -0.86 eVand -2.07 eV[55, 112]. The same quantity was also considered in the computational workby 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 varyby several tenths of an eV, they are in agreement in establishing that proton incorporationis highly exothermic, such that this process is driven by energetics at low temperaturesand moderate water partial pressures. We note that the previous computational study ofhydration 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 protonsin LaPO4 calculations[26, 153]. Whether such proton-dopant binding effects are importantfor hydration thermodynamics depends on the relative magnitudes of proton-dopant and ofpyrophosphate-dopant binding energies; the latter is currently unknown for LaPO4, to thebest 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. DFTcalculations of the activation energies for proton migration in LaPO4 yield values on theorder of 0.7-0.8 eV[153, 166]. Proton-dopant binding energies from recent calculations areof 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 themeasured values of the activation energies for proton conduction in doped LaPO4.

To date, the effect of dopant-defect binding has not been investigated for pyrophos-phate defects. The incorporation of protons by the mechanism corresponding to Eq. 4.2requires diffusion of pyrophosphate defects to the sample surface, which enables interac-tions with water and proton incorporation. Strong binding between the aliovalent dopantand pyrophosphate may impede this process. While such binding effects have not beenquantified for rare-earth phosphate compounds, a previous related computational study[97]for perovskite-structured SrCeO3 reported calculated binding energies between oxygen va-cancies and aliovalent dopants of 0.75 to 1.25 eV. Another study[68] reported a calculatedbinding energy of 0.18 eV for oxygen vacancies in Gd-doped BaCeO3. The oxygen vacancyfor these structures is analogous to the pyrophosphate for the LaPO4 system under con-sideration in this study, and it is thus of interest to understand if dopant-pyrophosphatebinding energies are of comparable magnitude, and thus of similar importance in limitingpyrophosphate mobilities.

19

In the current work we employ DFT calculations to study the atomic and electronicstructure of proton and pyrophosphate defects in LaPO4, the relative magnitude of thebinding 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 arepresented 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 governingproton uptake and conductivities in Section 4.5.

4.3 Computational Methods

All calculations have been performed using DFT within the generalized gradientapproximation (GGA) due to Perdew, Burke, and Ernzerhof (PBE)[117]. We have employedthe 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 calculationsare those specified as La, P, Sr, H and O in the VASP PBE library. The PAW potentialsuse 11 valence electrons for La (5s25p66s25d1), 5 for P (3s23p3), 10 for Sr (4s24p65s2), 1for H (1s1), and 6 for O (2s22p4). A plane-wave cutoff energy of 500 eV is employed, andreciprocal 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 areconverged to within 10 meV/Å. From convergence checks with respect to plane-wave cutoffand k-point sampling we estimate the binding energies and bond lengths reported below tobe converged at the level of approximately a few meV and 0.01 Å, respectively. The cal-culation 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 con-taining four formula units, and features monoclinic symmetry (P21/n space group)[110].The current calculated values for the lattice parameters of 6.93 Å, 7.15 Å, 6.54 Å, and103.72o for a, b, c, and β, respectively, agree to within 1.5% of experimentally measuredroom-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 symmet-rically 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 wereinvestigated by removing each of the four oxygen ions from a specific phosphate group in a 2x 2 x 2 supercell and allowing the cell to relax. Dopants were not considered for these initialcalculations, and a doubly negative background charge was implemented to compensate thedoubly 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 Å) of the four possible pryophosphateconfigurations are listed in columns 1-4. Row 1 shows the distance between the two phos-phorus ions. Rows 2 and 3 show bond lengths between the phosphorous ions and the sharedoxygen ion. The ions labeled with “prime” represent those within the phosphate group thatshares its oxygen ion in the pyrophosphate link. Rows 4-9 list the other phosphorus-oxygenion bond lengths. Column 5 lists inter-ionic distances calculated previously using ionicpotential models.a Row 10 shows the relative energies (∆E) in eV of the pyrophosphateconfigurations

DFT Calculations From This Work

O1 Removed O2 Removed O3 Removed O4 Removed Ionic Potentialsa

P-P’ (Å) 3.04 P-P’ 3.07 P-P’ 3.19 P-P’ 3.05 P1-P2 3.18P-O3’ 1.70 P-O4’ 1.71 P-O3’ 1.71 P-O1’ 1.73 P1-Ob 1.72P’-O3’ 1.73 P’-O4’ 1.68 P’-O3’ 1.69 P’-O1’ 1.70 P2-Ob 1.72P-O2 1.53 P-O1 1.52 P-O1 1.52 P-O1 1.52 P1-O2 1.55P-O3 1.53 P-O3 1.52 P-O2 1.53 P-O2 1.52 P1-O3 1.56P-O4 1.51 P-O4 1.51 P-O4 1.51 P-O3 1.51 P1-O4 1.55P’-O1’ 1.52 P’-O1’ 1.52 P’-O1’ 1.52 P-O2’ 1.53 P2-O2 1.55P’-O2’ 1.52 P’-O2’ 1.54 P’-O2’ 1.53 P-O3’ 1.53 P2-O3 1.55P’-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 pyrophos-

phate 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 phos-phate group which rotates to form a bridging oxygen bond. It is interesting to note that thepyrophosphate group forms spontaneously in the relaxation of a geometry initialized withan oxygen vacancy; this spontaneous formation of pyrophosphate defects was also observedin previous first-principles calculations of LaPO4, LaAsO4, and LaVO4 systems initializedwith 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 config-urations. Row 1 lists the distance between the two phosphorus ions. Rows 2 and 3 list bondlengths between the phosphorous ions and the shared oxygen ion. Note the shared oxygenion is dependent on which ion is originally removed, with O3’, O4’, O3’, and O1’ sharedfor O1, O2, O3, O4, removal, respectively. Rows 4-9 list the other phosphorus-oxygen ionbond 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 configurationswas considered in those calculations, and the ion labeling is not necessarily consistent withour work . Nevertheless, the inter-ionic distances obtained from the ionic-potential modelsare 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 relativelyshort. This trend is consistent with experimental bond lengths reported for pyrophosphatecompounds such as Mg2P2O7[35], Cu2P2O7[131], Ca2P2O7[36], and Zn2P2O7[34]. Theshorter bond length for non-bridging P-O bonds is attributed to delocalization of π-bondingwhich gives double-bond character, whereas the bridging P-O bonds have mainly single-bondcharacter[43].

Row 10 in Table 4.1 shows the relative energies (∆E) of the pyrophosphate con-figurations. The configuration resulting from O3 removal gives the lowest energy comparedto O1, O2, and O4 removal by 0.24 eV, 0.41 eV, and 0.24 eV, respectively. The atomiccoordinates of the lowest-energy defect configuration are used as initial positions for thesubsequent 3 x 3 x 3 supercell defect structure calculations including dopant cation defectsexplicitly, as described in the next sub-section.

We consider next results for proton defects. The position of the lowest energyconfiguration 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 ahydrogen bond with O1 on an adjacent orthophosphate group. Adelstein et al.[3] confirmedthis to also be the lowest energy position when the proton is positioned near a dopant forBa-doped LaPO4. The OH· · ·O bond lengths for the protonic defect obtained in the currentcalculations for undoped LaPO4 agree well (to within 0.01 Å) with previously reported DFTcalculations[166, 3].

22

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

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The electronic structure for LaPO4 systems containing a pyrophosphate and aproton 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). Asthe hydration reaction proceeds, the pyrophosphate ion is replaced with protons, with anelectronic structure corresponding to the bottom of Fig. 4.2. The presence of proton defectsagain leads to a split-off peak, that is, in this case, the result of hybridization between theproton and oxygen ion that form the hydroxyl bond. The effect of the electronic structuredue 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, weemploy 3 x 3 x 3 supercells (648 ions) with defect concentrations consistent with overallcharge neutrality. We substitute two La ions with Sr, giving a dopant concentration of1.85% (2/108), sufficiently dilute to approximate the reported 1.9% experimental solubilitylimit[10]. The large supercell considered in this work provides an opportunity to studyrelatively long-ranged dopant-defect interactions.

Two geometries were calculated to determine the Sr-pyrophosphate binding en-ergy; namely, structures in which one of the dopants is nearest and furthest from thepyrophosphate defect, such that the nearest dopant-defect distances for the two cases are2.81 Å and 12.60 Å, respectively. The binding energy computed from the difference inenergy for these two configurations is 0.37 eV.

In order to directly compare the relative energetics pertaining to the pyrophosphateand protonic defects, a 3 x 3 x 3 superstructure with two dopants and two protons wasconsidered. Three configurations were calculated for this defect structure, namely, eachproton nearest to a dopant (“both near”), one proton nearest to a dopant and the otherproton far from all defects (“one far”), and both protons far from all defects (“both far”). Arepresentation of the 3 x 3 x 3 “both near” structure with two Sr dopants and two protonsis shown in Fig. 3. The black dotted circle represents the approximate position of the topproton 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 en-ergies 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 0.34eV and 0.65 eV (0.325 eV/proton), respectively. The energy to separate one proton froma dopant (i.e., 0.34 eV) is slightly larger than the value of 0.31 eV for the binding energyobtained by Toyoura et al.[153] This small discrepancy is likely due to the longer rangeproton-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 valuefor the “both far” configuration is roughly twice as large in magnitude as that for the “onefar” configuration.

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Table 4.2: Proton-dopant distances (units of Å) for the three configurations of 3 x 3 x 3supercells (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 (Å) 2.62 2.62 10.07H2-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) nearestto each of the Sr dopants (Sr1, Sr2, shown in silver) is shown (“both near” structure). Theblack 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.

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4.4.3 Hydration Enthalpy Calculations for Structures with Neighboringand Isolated Dopants/Defects

In previous computational work the energetic preference for proton incorporationhas been quantified by the value of the hydration enthalpy (∆Hhyd)[26, 68]. FollowingHermet et al.[68], we will define this quantity for systems with dopants as:

∆Hhyd = E(LaPO4 + 2Sr + 2H)− E(LaPO4 + 2Sr + 2VO)− E(H2O) (4.3)

In our calculation, E(LaPO4 + 2Sr + 2H) and E(LaPO4 + 2Sr + 2VO) are the total DFTenergies 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 DFTenergy of an isolated water molecule. ∆Hhyd is defined as the hydration enthalpy at zerotemperature and pressure (ignoring zero-point energy corrections).

The use of the standard GGA functionals is known to strongly overestimate theenergy of the O2 molecule in oxidation enthalpy calculations[159]. This raises a potentiallyimportant issue concerning the possible errors in the calculated hydration enthalpies thatcould be introduced through the use of the GGA. In the current work, the primary errorassociated with potential overbinding for the water molecule would arise in our reportedvalues for the hydration enthalpies, which involves the difference in energy between H2Oin the gas versus hydroxide bonds in the crystal. It is thus interesting to note that ina 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 withhybrid calculations to within 28 meV for both M=Mg and Ca, suggesting a reasonablelevel of accuracy of the GGA relative to higher-order theoretical calculations for hydrationenthalpies.

The hydration enthalpies when the dopants are nearest and furthest from the otherdefects are all exothermic, -1.66 eV and -1.37 eV, representing the most stable and leaststable cases, respectively. It is worth noting the latter enthalpy agrees to within threepercent with the value obtained in a previous DFT study[26] that used a charged-defect su-percell methodology to calculate the hydration enthalpy for the limiting case correspondingto non-interacting defects (-1.34 eV).

The negative sign of the hydration enthalpies is consistent with the observations ofAmezawa 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 temperaturesand low water vapor partial pressures, which suggests that the uptake is enthalpically driven.Furthermore, two independent experimental studies have reported quantitative estimatesfor the hydration enthalpies of -0.86 eV[55] and -2.07 eV[112].

To compare our calculated results to the experimental estimates, we must accountfor 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 capacitiesbetween LaPO4 with pyrophosphate versus proton defects):

(1) The zero point energies for the protons in a water molecule versus hydroxylbonds in LaPO4. An analysis of the relative magnitudes of the zero-point-energy correctionsfor water versus hydroxide bonds in two different hydroxide compounds was presented by

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Ozolins et al. in Ref. [115]. The authors showed that the correction to the hydrationenthalpy was on the order of 0.02 eV per hydroxide bond, which is roughly two orders smallerin magnitude than the hydration enthalpies considered in this work. As a consequence wewill neglect this contribution in what follows.

(2) The temperature-dependence of the enthalpy of water vapor. The enthalpy ofwater 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 fromequipartition of the energy of a water molecule (ignoring vibrational states), and assumesideal gas behavior. This term is consistent with the experimental heat capacity (Cp) ofwater vapor, which is near 4R over the relevant temperature range for which the hydrationreaction is favored (300-600oC). The finite-temperature corrections to the enthalpy of waterrange between approximately +0.2 eV and +0.3 eV in this temperature range.

Accounting for the second correction, the finite-temperature corrected calculatedhydration enthalpy ranges between -1.57 eV to -1.67 eV between 300oC and 600oC forhydration involving protons that are not bound to dopants in LaPO4. When binding todivalent Sr dopant cations is accounted for, these values shift to -1.86 eV to -1.96 eV overthe same temperature range. The calculated values that account for dopant-defect bindingare closest to the experimentally-derived estimate of -2.07 eV obtained in Ref. [112], whilecalculated results with and without dopant-defect binding are considerably more negativethan the value of -0.86 eV reported in Ref. [55].

Further insight into the nature of dopant-defect interactions and their effect onthe hydrolysis reaction in Eq. 4.2 can be obtained by analyzing changes to the electronicstructure as the defects are moved towards the dopant cations. First considering the pro-ton defect, the distance between the split-off defect state highlighted in Fig. 4.2 and theoxygen 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-dopantbinding energy and can be attributed largely to an electrostatic effect. We can look intothe relative energies of forming an oxygen vacancy (i.e., pyrophosphate) by comparing theelectronic structure of an oxygen ion near and far from the Sr dopant. The band energiesfor 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 andthe effectively negatively charged dopant. Thus it is more favorable to form a vacancy (inother words, less likely to fill a vacancy) near to a dopant, which is again consistent withthe relatively large binding energy of the pyrophosphate defect.

4.5 Summary and Discussion

In summary, DFT-GGA calculations have been applied in a computational investi-gation of the relative energetics of protons and pyrophosphate defects and their interactionswith dopant cations in Sr-doped LaPO4. The most stable pyrophosphate configuration forLaPO4 was calculated to be 0.24 eV lower than the next stable configuration. The bondlengths for all four pyrophosphate configurations are in reasonable agreement with previouscalculations 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

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incorporation is favored (exothermic) when the dopants are both nearest and furthest fromthe 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 thathave explored dopant-proton binding energies[3]. Specifically, the Sr-proton binding energycalculated here is 0.14 eV larger in magnitude than the previously-calculated Ba-protonbinding energy of 0.2 eV. It is interesting to note that this trend is consistent with exper-imentally measured activation energies for proton conduction, which are 0.19 eV larger inSr than Ba-doped LaPO4[10].

The larger binding energy for Sr versus Ba dopants is likely to be a result of reduceddisplacement of the neighboring oxygen ions that surround the Sr dopant. Specifically, thereis less size mismatch between the Sr dopant and smaller La host (+8%) compared to theBa dopant (+21%). Reduced ionic size mismatch correlates in the calculations with lessdistortion in the oxygen ion bond lengths around the dopant, which corresponds to higherbinding energies presumably because of the strain in OH· · ·O bond lengths arising from thedopant-induced displacements in the neighboring oxygen-ion positions.

We consider next the calculated results for the effect of dopant-defect binding onthe hydration enthalpy. The reason the hydration enthalpy for the case in which dopantsare nearest to the other defects (i.e., protons and pyrophosphate) is approximately 0.3 eVlower compared to the case where the dopants are furthest from the other defects can beunderstood by considering the relative magnitudes of the pyrophosphate and proton bindingenergies with the dopants. Referring to Eq. 4.3, one defect structure has two protons, andthe other has one pyrophosphate. The pyrophosphate binding energy is only slightly largerthan the proton binding energy, which effectively cancels the energetic contribution arisingfrom binding of the dopant for one proton and one pyrophosphate. Therefore, the onlysignificant 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 fromthe other proton which has a binding energy on the order of 0.3 eV. Overall, dopant-defectneighboring interactions lead to a change in the hydration enthalpy that further favorsproton incorporation.

While dopant-defect binding energies are found to favor proton incorporation ther-modynamically, strong binding between the pryophosphate defect and the Sr dopants couldgive rise to kinetic limitations associated with the rate of proton uptake. Specifically,pyrophosphate-Sr binding is expected to hinder the mobility of the pyrophospate defectsfrom the bulk to the surface, which limits proton incorporation at the surface and lowersthe rate at which proton concentrations in the bulk of the material can equilibrate.

Overall, the present calculated results suggest competing thermodynamic and ki-netic contributions of dopant-defect binding energies underlying the process of hydrolysis ofLaPO4, which may be present in other rare earth orthophosphate compounds as well. Sincethe calculated effects are relatively large on the scale of hydration enthalpies and migra-tion energies calculated in previous studies[3, 26, 153, 121, 166], dopant-defect interactionsshould be considered in further detail in future computational studies aimed at identifyingstrategies for optimizing the performance of REP compounds for proton conductor appli-cations.

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4.6 Acknowledgements

This work was supported by the Office of Basic Energy Sciences, Materials Sciencesand 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 andEngineering Graduate Fellowship Program. This work made use of resources of the NationalEnergy Research Scientific Computing Center, supported by the Office of Basic EnergySciences of the U.S. Dept of Energy (DE-AC02-05CH11231). The authors would like tothank Hannah L. Ray for useful discussions.

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Chapter 5

Energetics and Defect Interactions:Y and La Doped UO2

The energetics and defect ordering tendencies in solid solutions of fluorite-structuredUO2 with trivalent rare earth cations (M

3+=Y, La) are investigated computationally usinga combination of ionic pair potential and density functional theory (DFT) based methods.Calculated enthalpies of formation with respect to constituent oxides show higher ener-getic stability for La solid solutions than for Y. Additionally, calculations performed fordifferent atomic configurations show a preference for reduced (increased) oxygen vacancycoordination around La (Y) dopants. The current results are shown to be qualitativelyconsistent with related calculations and calorimetric measurements of heats of formationin other trivalent doped fluorite oxides, which show a tendency for increasing stability andincreasing preference for higher oxygen coordination with increasing size of the trivalentimpurity. The implications of these results are discussed in the context of the effect oftrivalent impurities on oxygen ion mobilities in UO2, which are relevant to the under-standing of experimental observations concerning the effect of trivalent fission products onoxidative 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, pages1047-1053 (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 concentrationsof trivalent cations. Under reducing conditions this doping leads to the formation of chargecompensating oxygen vacancy defects. Recent calorimetry studies on trivalent-cation dopedZrO2[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 recentcomputational work on some of these same systems, which demonstrate energetic bind-ing/clustering tendencies between cations and oxygen vacancies, to a degree that has beenfound to correlate strongly with ionic radii[127, 76, 106, 31, 30, 123, 8, 167]. Strong oxy-gen vacancy-cation association reduces the mobility of oxygen ions and therefore the ionicconductivity in aliovalently-doped fluorite oxides, which can be detrimental for their use infuel cell electrolyte and oxygen sensor applications[73].

Similar effects have also been discussed in the context of the electrochemical be-havior of spent nuclear fuel composed of fluorite-structured UO2 with dissolved trivalentfission products. Specifically, recent studies have attributed rare earth (i.e., trivalent) fissionproduct dopant-oxygen vacancy clusters to the decreased rate of oxidative dissolution withincreasing burnup in spent fuel[67]. The defect clusters are believed to reduce the concen-tration of mobile oxygen vacancies which facilitate oxidation and eventual dissolution of thefuel, in a manner similar to that by which rare earth doping impedes air oxidation of thecubic fluorite structure to orthorhombic U3O8[150, 39, 40, 101]. Oxidation and dissolutionof nuclear fuel in aqueous environments is of particular concern at the back end of the fuelcycle, i.e., waste disposal[151, 138].

Because of the significant changes in oxidation and corrosion behavior of nuclearfuel associated with rare earth fission products, understanding the thermochemical proper-ties of these defects in UO2 is of particular interest. The effect of rare earth fission productdoping 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 studiedcompared 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] havebeen reported to date. The aim of the current work is to provide further insight into thethermochemical behavior of UO2 doped with trivalent cations through the application ofdensity functional theory (DFT) based calculations in which we explore stability and defectordering/clustering trends with respect to relative cation sizes. It is expected that UO2substituted 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 withoxygen vacancies (oxygen deficiency rather than oxygen excess). We consider the smallestand largest trivalent rare earth-fission product cations that are soluble in UO2, namely, Y

3+

(1.02 Å ionic radius in eightfold coordination) and La3+ (1.16 Å)[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 anddefect interactions in Section 5.4. In Section 5.5 the results are discussed in the context ofthe previous models proposed for explaining the effect of cation size on the stability anddefect ordering behavior in aliovalently-doped fluorite structures.

5.3 Computational Methodology

The computational approach employed in this work is similar to that appliedin 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

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defect-dopant configurations as a basis for exploring trends in energetic stability and defectordering. The following subsections provide detailed descriptions of the structural enumera-tion 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 UO2doped with trivalent cation (M3+) oxides, we investigate many hypothetical cation-vacancyarrangements. 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− andOvac); the substitution of two U

4+ by two M3+ can be charge-compensated by one Ovac,as illustrated in Fig. 5.1. We employ a structure enumeration technique developed for clusterexpansion based studies of alloy thermodynamics, considering supercells consisting of up tosix formula units, employing an algorithm by Hart and Forcade[65] that is implemented inthe alloy theoretic automatic toolkit (ATAT)[157, 156]. The compositions considered areU4M2O11 (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 structureenumeration yielded 117, 27, and 710 symmetry-distinct structures for x=1/3, 1/2, and2/3, respectively.

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

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5.3.2 Classical Pair Potentials

The energies of all enumerated structures were calculated using ionic pair-potentialmodels as a first “screening” step, with the intent of eliminating from consideration bymore computationally demanding DFT based calculations the electrostatically unfavorablestructures for a given composition and M3+ species. In these calculations the potentialenergy between ions i and j is modeled through pair potentials of the following form:

E(rij) =qiqjrij

+A exp (−rijρ

)− Cr6ij

(5.1)

The first term represents the coulombic interaction where qi is the respective ionic chargeand rij is the distance between ions i and j. The latter two repulsive and attractive termsrepresent the short range Buckingham potential, where A, ρ, and C are system-dependentparameters. In addition, polarizabilities are introduced for individual ionic species accordingto the shell model as formulated by Dick and Overhauser[45]. The Buckingham and shellmodel parameters used in this work are given in Refs. [25, 32, 60]. Geometry relaxationsand 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 totheir total energy based on the ionic pair potential calculations described above. InitiallyDFT calculations were performed for the resulting six lowest energy structures for a givencomposition. If any of the five higher energy structures were found to be lower than theground state predicted by the pair potential models, additional structures were consideredin an effort to ensure that the lowest energy configuration was identified by the DFT basedmethods.

Most of the results reported below were obtained using DFT within the formalismof the projector augmented-wave (PAW) method[27, 87] and the Perdew-Burke-Ernzerhof(PBE) generalized gradient approximation (GGA)[117, 118] as implemented in the Viennaab initio simulation package (VASP)[85, 86]. In these calculations, a Hubbard-U correc-tion was implemented wit

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