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Computational Study of Surface Characteristics and Thermal Conductivity in Oxide Materials

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Title:
Computational Study of Surface Characteristics and Thermal Conductivity in Oxide Materials
Creator:
Lee, Chan-Woo
Place of Publication:
[Gainesville, Fla.]
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University of Florida
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english
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1 online resource (147 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Materials Science and Engineering
Committee Chair:
Sinnott, Susan B.
Committee Members:
Wachsman, Eric D.
Nino, Juan C.
Phillpot, Simon R.
Asthagiri, Aravind
Graduation Date:
8/7/2010

Subjects

Subjects / Keywords:
Atoms ( jstor )
Cathodes ( jstor )
Conceptual lattices ( jstor )
Electronics ( jstor )
Electrons ( jstor )
Energy ( jstor )
Ions ( jstor )
Oxides ( jstor )
Oxygen ( jstor )
Thermal conductivity ( jstor )
Materials Science and Engineering -- Dissertations, Academic -- UF
abo3, cathode, computationalmaterials, densityfunctionaltheory, feo2, fluorite, fuelcell, lafeo3, lao, lscf, lsf, moleculardynamics, nuclearfuel, orr, oxides, oxygenreductionreaction, perovskite, sofc, surface, surfacephasediagram, surfacestoichiometry, thermalconductivity, uo2
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Electronic Thesis or Dissertation
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
Materials Science and Engineering thesis, Ph.D.

Notes

Abstract:
Solid oxide fuel cells (SOFCs) with their high efficiency of energy conversion, low emissions, and ability to use commercial fuels have the potential to provide a new energy landscape. Although SOFCs do not require noble metal catalysts, it has the disadvantage of requiring an operating temperature above 800 C, due to the slow oxygen reduction reaction (ORR) on its cathode surface. Nuclear fission has been focused as one of possible technologies that provide more efficient energy system. UO2 is the fuel utilized in most of nuclear reactors. Despite its advantages as nuclear fuel (e.g. high melting temperature and chemical and structural stability), low thermal conductivity is one of the disadvantages of UO2 as the fuel. This dissertation is focused on understanding the effect of surface structures of LaFeO3 and lattice voids of UO2 on the properties for their energy-related applications which will enable improve the performance of the devices. The approach used here involves electronic structure calculations, molecular dynamics, and microscopic thermodynamics. For lanthanum ferrite (LaFeO3, LFO) which is the base material of cathodes for SOFCs, the phase diagram of the LFO (010) surface was developed to predict its stable surface stoichiometry at different operating conditions. The surface stoichiometries of LaO- and FeO2- type terminations of LFO (010) surface was understood from electronic and structural perspectives. Based on the developed surface phase diagram, the ORR was investigated. Oxidized and stoichiometric FeO2-type surfaces are chosen to represent the stable terminations below and above 800 K. The behavior of surface oxygen vacancy was found to be crucial factor in limiting ORR. When the concentration of oxygen vacancy is high, the rate-limiting step was the elementary step with the highest activation energy. This rate-limiting step was different between the stoichiometric and oxidized surfaces. By contrast, in low concentration of oxygen vacancy, the ORR was limited by site availability for the incorporation reaction of oxygen species (so called entropy barrier). For fluorite-structured UO2, the contribution of bubbles and lattice voids on thermal conductivity of UO2 was investigated. Both bubbles and pores decreased the thermal conductivity of UO2. In particular, the dependence of thermal conductivity on the size of the void was quantified. Additionally, using He atoms in the void as a model bubble, it was validated that the penetration of He atoms reduces the thermal conductivity of UO2. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2010.
Local:
Adviser: Sinnott, Susan B.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31
Statement of Responsibility:
by Chan-Woo Lee.

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Applicable rights reserved.
Embargo Date:
8/31/2011
Resource Identifier:
004979818 ( ALEPH )
706497202 ( OCLC )
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1 COMPUTATIONAL STUDY OF SURFACE CHARACTERISTIC S AND THERMAL CONDUCTIVITY IN OXIDE MATERIALS By CHAN WOO LEE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMEN TS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010

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2 2010 Chan Woo Lee

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3 TO MY FAMILY WITH LOVE

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4 ACKNOWLEDGMENTS Looking back on my Ph.D. venture, I want to take the time to pay gratitude to the numerous peop le who helped in numerous ways. First and foremost, I would like to thank my advisor Dr Susan Sinnott for her undivided instruction and auspices throughout this endeavor Her support, guidance, patience, and trust were established from the moment I firs t spoke with her on the phone for an interview and were unwavering ever since. She taught me valuable lessons from think ing critically as a scientist to work ing as a professional. Most importantly, she assisted me in overcoming any adversities that I encou ntered during my Ph.D study. Furthermore, I would like to thank my co advisor Dr. Simon Phillpot who furthered my growth as a scientist by allowing me to think logically and decide independently with great patience. His encouragement always came from hi s heart and always motivated me. Thank you, Dr. Phillpot. I will miss your big smile, immense sense of humor, and that blackboard in your office always filled with various figures and equations My gratitude also goes out to my previous adviso rs, Dr. Chan ghee Lee, Dr. Kyung Jong Lee, and Dr. Yong Chae Chung, whose help and guidance has been greatly appreciated. Additionally, I thank my committee member s, Dr. Eric Wachsman, Dr. Juan Nino, and Dr. Aravind Asthagiri Dr. Wachsman and his group (especially, Dr Dongjo Oh and Dr. Heesung Yoon ) kept my perspective on experimental data unbiased Dr. Nino provided critical and helpful comments on my dissertation while Dr. Asthagiri offered helpful advice on my research. Although we did work together, he never hesit ated to give advice on my projects, which I deeply appreciate.

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5 Moreover, I would like to mention my dear friends in the group, especially Rakesh Behera who has been my mentor since my second day in Gainesville. I enjoyed the numerous discussions with him as well as the lessons learned from those discussions. With his help and advice, I prevailed though many obstacles that I experienced during my Ph.D. period. Besides all the members and alumni of the Sinpot group, I want to acknowledge Dilpuneet Aidhy (DP) Peter Barry, Patrick Chiu, Minki Hong, Inkook Jang, Donghyun Kim, Donghwa Lee, Tao Liang, Priyank Shukla, and Taku Watanabe Furthermore, I would like to thank my friends and colleagues outside of the group: Hosang Ahn, Jinsoo Ahn, Sanghyun Eom, Junghun Jang, Dowon Jung, Jaeyoung Jung, Seyeon Jung, Doyoung Kim, Jinwook Kim, Seonhoo Kim, Byungwook Lee, Jaewon Lee, Sungwook Mhin, Seungyong Son, Junhan Yuh and all of the other Korean students and alumni of the department. My Ph.D. research was facilitated b y multitudes of collaborations and such collaborations w ere a great privilege to be a part of. To work and discuss with Drs. In Tae Bae, Ram Devanathan, Jung Bae Kim, Yoon Suk Kim, Hendrik Monkhorst Satoshi Okamoto, German Samolyuk, Roger Stoller, Sak ong Sung, and Doris Vogtenhuber I will be truly thankful. I now want to acknowledge my family I truly believe that I am a lucky individual to have such great parents. Although I chose a divergent and difficult life path from most people, they always told me You can do it. Without their continuous support and encouragement, my Ph.D. period would have been undoubtedly tougher. Never forgetting to send me weekly e mails throughout my entire Ph.D. period, I always felt their love and support. I lov e you and thank you, Mom and Dad. I also truly

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6 appreciate the love and constant support from my mother in law she always cheered me up with her kindness. I want to thank her and I am proud that she can officially call me Dr. Lee (Yi Baksa in Korean ) from n ow on. Additionally, I am extremely glad that I can show my diploma to my grandfather. Even though I cannot guarantee that I can be as rich as he wants me to be, I can promise him that I will be prosperous in all aspects of my life In addition to the p arents, I would like to share this achievement with the rest of my family, who provided me additional love and support: my two younger sisters, three sisters in law, four brothers in law, one nephew and one niece (for now) Last but most certainly not lea st, my wife, Ha Jung deserves special recognition for having been with me throughout the years, always supporting me and loving me unconditionally. Without the consistent encouragement, cooperation, and patience of my wife, I am certain that I could never have completed my Ph.D. program. Ha Jung, I am truly a lucky man to have married such a beautiful, humorous, and generous woman like you. Thank you and I love you (more than the universe).

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7 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 9 LIST OF FIGURES ................................ ................................ ................................ ........ 10 L IST OF ABBREVIATIONS ................................ ................................ ........................... 14 ABSTRACT ................................ ................................ ................................ ................... 15 CHAPTER 1 BACKGROUND ................................ ................................ ................................ ...... 17 1.1 Motivation ................................ ................................ ................................ ......... 17 1. 2 Solid Oxide Fuel Cell s (SOFC s ) ................................ ................................ ....... 17 1.2.1 Operation and Components of SOFCs ................................ .................... 17 1.2.2 Intermediate Temperature (IT) SOFCs ................................ .................... 19 1. 2 3 Oxygen Reduction Reaction (ORR) ................................ ......................... 22 1. 3 Perovsk ite Oxides ................................ ................................ ............................. 23 1. 3 .1 Structure of Perovskite Oxides ................................ ................................ 24 1. 3 .2 Properties of Perovskite Oxides ................................ .............................. 25 1. 3 .3 Lanthanum Ferrite ( LaFeO 3 ) ................................ ................................ ... 26 1.4 Uranium Dioxide (UO 2 ) ................................ ................................ ..................... 26 1.4.1 UO 2 as Nuclear Fuel ................................ ................................ ................ 27 1.4.2 Lattice Void and He Bubble in UO 2 ................................ .......................... 28 1. 5 Objectives ................................ ................................ ................................ ......... 29 2 COMP UTATIONAL DETAILS ................................ ................................ ................. 39 2.1 Density Functional Theory (DFT) ................................ ................................ ...... 39 2.1.1 Exchange and C correlation ................................ ................................ ..... 39 2. 1.2 Details of DFT ................................ ................................ ......................... 40 2 1.3 Local Density Approximation (LDA) and Generalized Gradient Approximation (GGA) for Exchange and Correlation ................................ .... 42 2.2 Ab Initio Atomistic Thermodynamics ................................ ................................ 43 2.2.1 Surface Gibbs Free Energy of LFO Surface ................................ ............ 44 2 2 .2 Chemical Potential of Oxygen as a Function of p(O 2 ) and T ................... 47 3 STABILIZATION MECHANISMS OF the LANTHANUM FERRITE (010) SURFACE ................................ ................................ ................................ ............... 50 3.1 Introduction ................................ ................................ ................................ ....... 50 3 .2 Bulk L aFeO 3 ................................ ................................ ................................ ...... 50

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8 3.3 LaFeO 3 (010) Surfaces with a Variation of Surface Oxygen Stoichiometry ...... 51 3.4 Charge Compensation of LaFeO 3 (010) Surface ................................ .............. 53 3.5 Stabilization Mechanisms of the L a F e O 3 (010) Surface ................................ .... 56 3.5.1 Contribution of Atomic Species to Charge Compensation ....................... 56 3.5.2 Contributions of Sublayers to Charge Compensation .............................. 62 3.5.3 Contribution of Atomic Relaxation to Charge Compensation ................... 64 4 SURFACE PHASE DIAGRAM OF LANTHANUM FERRITE (010) SURFACE ....... 79 4.1 Introduction ................................ ................................ ................................ ....... 79 4.2 Secondary Phases of LFO ................................ ................................ ................ 80 4.3 Surface Phase Diagram of LFO (010) Surfaces ................................ ............... 83 4.4 Contributions of E lectronic R edistribution and C hemical P otential in S urface S tability ................................ ................................ ................................ ................ 87 4.5 Discussion ................................ ................................ ................................ ........ 89 4.5.1 Relation to Experimental Results ................................ ............................ 89 4.5.2 Comparison with LaMnO 3 S urface ................................ .......................... 92 5 OXYGEN REDUCTION REACTIONS ON LANTANUM FERRITE (010) SURFACES ................................ ................................ ................................ .......... 106 5.1 Introduction ................................ ................................ ................................ ..... 106 5.2 Computational Details ................................ ................................ ..................... 106 5.3 Asymmetric Surface Slabs ................................ ................................ .............. 107 5.4 Elementary Steps for ORRs ................................ ................................ ............ 108 5.4.1 Elementary Reaction Steps on FeO 2 +0.5O low Surface Termination ...... 109 5.4.2 Elementary Reaction Steps on Stoichiometric FeO 2 Surface Termination ................................ ................................ ................................ 110 5.5 Discussion ................................ ................................ ................................ ...... 111 5.5.1 ORR on the FeO 2 +0.5O low Surface ................................ ....................... 111 5.5.2 ORR on the Stoichiometr ic FeO 2 Surface ................................ ............. 112 5.5.3 Relation to Experimental Results ................................ .......................... 113 6 THE EFFECTS OF LATTICE VOIDS AND HELIUM GAS BUBBLES ON THE THER MAL CONDUCTIVITY OF UO 2 ................................ ................................ ... 121 6 .1 Introduction ................................ ................................ ................................ ..... 121 6.2 Potential Models ................................ ................................ ............................. 121 6.3 Thermal Expansion ................................ ................................ ......................... 122 6.4 Thermal Conductivity of Single Crystal UO 2 ................................ ................... 122 6.4.1 UO 2 with Lattice Voids ................................ ................................ ........... 124 6.4.2 UO 2 with Lattice Voids and He Bubbles ................................ ................ 125 7 CONCLUSIONS ................................ ................................ ................................ ... 135 LIST O F REFERENCES ................................ ................................ ............................. 140 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 147

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9 LIST OF TABLES Table page 1 1 Comparison of v alence stability and O 2 reaction among La 1 x Sr x MnO 3 La 1 x Sr x CoO 3 and La 1 x Sr x FeO 3 ................................ ................................ ............... 31 1 2 Various properties of perovskite oxides ................................ .............................. 31 3 1 Structural information of bulk L a F e O 3 ................................ ................................ 66 3 2 Possible surface terminations of LaFeO 3 (010) plane ................................ ........ 66 3 3 Comparison bet ween the sum of surface layer charges ( ) and half of bulk layer charge ( ).. ................................ ................................ .................... 67 4 1 Calculated formation enthalpy of considered secondary p hases of bulk LaFeO 3 ................................ ................................ ................................ ............... 94 5 1 Adsorption enegy of O and O 2 on Fe sites of symmetric slab with 15 atomic layers and asymmetric slabs with 11 and 9 atomic layers.. .............................. 115 6 1 Parameters of interatomic potentials ................................ ................................ 128 6 2 Lattice constants of UO 2 unitcell ................................ ................................ ....... 128 6 3 T hermal conductivities of UO 2 ................................ ................................ .......... 128 7 1 Summary of stabilization mechanisms of LaFeO 3 (010) surface ...................... 139

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10 LIST OF FIGURES Figure page 1 1 World Energy Consumption by (a) region, and (b) type of energy resource. Adapted from reference 1 ................................ ................................ ................... 3 2 1 2 Energy conversion steps of (a) typic al power plant and (b) fuel cell. ................. 32 1 3 Schematic description of operation principle of solid oxide fuel cell ................... 32 1 4 The condu ctivities of electrolytes of fuel cells: phosphoric acid fuel cell (PAFC), polymer electrolyte fuel cell (PEFC), molten carbonate fuel cell (MCFC), and solid oxi de fuel cells (SOFC) ................................ ......................... 33 1 5 Sep arated area specific resistance (ASR) in a conventional SOFC with La 1 x Sr x MnO 3 cathode, Y 2 O 3 stabilized ZrO 2 (YSZ) electrolyte, and Ni YSZ anode 33 1 6 Schematic description of possible elemen tary reaction steps of oxygen reduction reaction of two different classes of cathode materials: (a) electronic conductor and (b) Mixed ionic electronic conducting (MIEC) cathodes. ............. 34 1 7 Chemi cal elements that can occupy A and B sites in perovskites. Adapted from reference 8. ................................ ................................ ................................ 35 1 8 Space group maps of (a) A 2+ B 4+ O 3 and (b) A 3+ B 3+ O 3 Adapted from reference 8. ................................ ................................ ................................ ........................ 35 1 9 (a) Schematic of LaFeO 3 unit cell. (b) Relative direction of orthorhombic unit cell (solid line) of LaFeO 3 to pseudo cubic cell (dotted line).. ............................. 36 1 10 Coordination number (CN) of (a) La, (b) Fe, and (c) O(1), (d) O(2) in bulk LaFeO3. Here, O(1) and O(2) have 8d and 4c as multiplicity and Wyckoff letter, respectively. ................................ ................................ .............................. 36 1 11 Schemat ic description of available spin combinations for antiferromagnetic (AFM) state within ideal perovskite unitcell. Each corner is occupied by B site cations. ................................ ................................ ................................ ............... 37 1 12 Schematic of the fluorite s tructure UO 2 Blue and red spheres depict uranium and oxygen, respectively. ................................ ................................ ................... 37 1 13 Phase diagram of U O system ................................ ................................ ............ 38 1 1 4 (a) SEM image of a (111) surface of UO 2 crystal (b) Oblique angle scanning electron microscopy image of cross section through one of bright symmetrical features in (a) shows that they are microscopic voids ................... 38

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11 3 1 Electronic density of states of bulk LaFeO 3 Electrons in Fe atom have high spin configuration in which electrons are aligned in the upward direction (majority). E F is set to zero (vertical dotted line). ................................ ................ 68 3 2 Total energy comparison of bulk L aFe O 3 (LFO) with different space groups relative to space group Pnma and magnetic ground state [G type antiferromagnetic state, G AF ] which is chosen to be the zero. ......................... 68 3 3 Top view of (a) FeO 2 and (b) LaO surfaces. (c) and (d) show top view and side view of FeO 2 0.5 and FeO 2 +0.5O low surface terminations, respectively. ..... 69 3 4 (a) Line diagram for charges ( ) in the surface layers; (b) Line diagram of of top layers. ................................ .................. 70 3 5 Spatial variations of the elec tric field, E and of the electrostatic potential, V in a macroscopic surface models cut along a polar direction. ............................ 71 3 6 Averaged relative Bader charge ( ) of surface atoms for relaxed LaO+ x O/O low surface terminations. ................................ ........................ 72 3 7 Averaged relative Bader charge ( ) of surface atoms for relaxed FeO 2 + x O/O low surface terminations. ................................ ....................... 72 3 8 Schematics of charge compensation mechanisms of LaO type surface terminations [(a) and (b)] and FeO 2 type sur face terminations [(c) and (d)] ........ 73 3 9 Electronic density of states for LaO+ x O low terminated surfaces ( x = 1.0, 0.5, 0.0, 0.5, 1.0, 1.5, and 2.0). The Fermi energy is set to zero (vertical dotted line). ................................ ................................ ................................ .................... 74 3 10 Electronic density of states for FeO 2 + x O low terminated surfaces ( x = 2.0, 1.5, 1.0, 0.5, 0.0, 0.5, 1.0, 1.5, and 2.0). The Fermi energy ( E F ) is set to zero (vertical dotted line). ................................ ................................ ................... 75 3 11 Schematic descriptions of (a) LaO+1.5O low (b) LaO+2.0O low (c) FeO 2 +1.5O low and (d) FeO 2 +2.0O low surface terminations ................................ 76 3 12 Percentage of change in charges in the first layer before and a fter electronic redistribution and layer charge based on bulk LaFeO 3 ................................ ....... 76 3 13 Relative interlayer spacing of LaO type and FeO 2 type surface terminations with respect to bulk interlayer spacing on (010) direction. ................................ .. 77 3 14 Relative interlayer spacing of LaO type and FeO 2 type surface terminations with respect to bulk interlayer spacing on (010) direction. ................................ .. 78

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12 4 1 Structural and magnetic ground states of secondary phases of bulk LaFeO 3 (T<1800 K ). ................................ ................................ ................................ ........ 95 4 2 Calculated super cells of (a) La, (b) La 2 O 3 (c) Fe, (d) F eO, (e) Fe 2 O 3 and (f) Fe 3 O 4 ................................ ................................ ................................ ................. 96 4 3 Side view of surface phase diagram of LFO (010) surfaces. White area describes positive surface Gibbs free energy, ................................ ............... 97 4 4 (a) Surface phase diagram, and (b) relative chemical potential of oxygen ( O ) as a function of T ( K ) and p(O 2 ) ( atm ) ................................ ....................... 98 4 5 Line diagrams that show the most stable surface oxygen stoichiometry of (a) FeO 2 and (b) LaO type surface terminatio ns as a function of temperature ....... 99 4 6 Surface phase diagram at (a) 773 K (b) 1073 K and (c) 1223 K with p (O 2 ) 0.21 atm The gray highlighted regions in (a), (b) and (c) represents stability region of bulk L a F e O 3 .. ................................ ................................ ..................... 100 4 7 Sketch of surface stabilities of (a) FeO 2 type and (b) LaO type surfaces at 77 3 K 1073 K and 1223 K Each surface symbol does not describe actual configurations of oxygen species on the surface. ................................ ............. 101 4 8 Relative electronic contribution terms of LaO and FeO 2 type surf aces as a function of surface oxygen stoichiometry. The electronic contribution terms of stoichiometric LaO and FeO 2 surfaces are set to zero. ................................ .... 102 4 9 Surface Gibbs free energies described by electronic and environmental contributions at 773 K and 1223 K with p (O 2 ) 0.21 atm .. ................................ 103 4 10 (a) Surface phase diagram, and (b) relative chemical potential of oxygen ( O ) as a function of T ( K ) and p(O 2 ) ( atm ) The labels on the lines in (b) represent p(O 2 ) of high vacuum condition (10 6 10 12 atm ). ............................ 104 4 11 Temperature programmed desorption of O 2 for LaFeO 3 cal cined at (a) 973 K and (b) 1123 K ................................ ................................ ................................ .. 105 5 1 Schematic description of nudged elastic band ( NEB ) method. ......................... 115 5 2 Atomic structure o f asymmetric 9 layer slab with FeO 2 termination. Only 5 layers above bottom 4 layers are fully relaxed. ................................ ................ 116 5 3 Atomic displacements in z direction of symmetric slab nineteen atomic layers an d asymmetric slab with thirteen atomic layers. ................................ ... 116 5 4 Schematic description of considered reactions for each elementary step of FeO2+0.5O low surface termination and their relative energy prof iles. ............... 117

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13 5 5 Schematic description of considered reactions for each elementary step of stoichiometric FeO 2 surface termination and their relative energy profiles.. ..... 118 5 6 Predicted oxygen reduction reaction path of FeO 2 +0.5O low surface. ................ 119 5 7 Predicted oxygen reduction reaction paths of FeO 2 surface.. ........................... 120 6 1 Normalized lattice parameter as a function of temperature from the Grimes potential and experiment. ................................ ................................ ................. 129 6 2 Schematic of 10x10x40 sim ulation cell of UO 2 with lattice void which contains He atoms ................................ ................................ ................................ .......... 129 6 3 Simulation setup of the non equilibrium molecular dynamics for calculating thermal conductivity. The heat source and heat sink are located at a quarter of the cell length away from the cent er of the system (gray regions) ................ 130 6 4 Calculated thermal conductivity as a function of a radius of spherical void. .. 130 6 5 The same thermal conductivity data as in Figure 6 2 in relative thermal conductivity ( ) and effective cross sectional area ( ) perpendicu lar to heat flux direction. ................................ ................................ ........................ 131 6 6 Calculated thermal conductivities of sphere voids with different radii as a function of He bubble density. Numbers on each data point are the number of He atoms corresponding to each bubble density. ................................ ............. 132 6 7 Change in thermal conductivity [ ] as a function of bubble density. 0 is thermal conductivity at each void without He bubble. .................. 132 6 8 Cross x10 28 atoms/m 3 ). Number of He atoms in the bubble, n He is 31. He atoms are distributed near the interface between the bubble and UO 2 lattice. ................................ .................. 133 6 9 Radial distributions of U, O, and He atoms from the center of the void within UO 2 simulation cell (r=12.0 ). ................................ ................................ ......... 133 6 10 No. of He atoms penetrated into UO 2 lattice, and the ratio of He atoms penetrated into UO 2 lattice over total No. o f He atoms at UO 2 with voids of r=12.0 ................................ ................................ ................................ ........... 134

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14 LIST OF ABBREVIATION S AFM Antiferromagnetic ASR Area specific resistance DFT Density functional theory DOS Density of states GGA Generalized gradient approximation IT SOFC Intermediate temperature Solid oxide fuel cell MCFC Molten carbonate fuel cell MIEC Mixed ionic and electronic conductor NEB Nudged elastic band method LDA Local density approximation LSC La 1 x Sr x Co O 3 LSF La 1 x Sr x FeO 3 LSGM La 1 x Sr x Ga 1 y Mg y O 3 LSM La 1 x Sr x MnO 3 LFO LaFeO 3 MD Molecular dynamics ORR Oxygen reduction reaction PAFC Phosphoric acid fuel cell PEFC Polymer electrolyte fuel cell SOFC Solid oxide fuel cell TPD Temperature programmed desorption TPR Temperature programmed reduction YSZ Y 2 O 3 stabilized ZrO 2

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15 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMPUTATIONAL STUDY OF SURFACE CHARACTERISTIC AND THERMAL CONDUCTIVITY IN OXIDE MATERIALS By C han W oo L ee August 2010 Chair: Susan B. Sinnott Major: Materials Science and Engineering S olid oxide fuel cell s (SOFC s ) with their high efficiency of energy conversion, low emissions, and ability to use com mercial fuels ha ve the potential to provide a new energy landscape. A lthough SOFC s do not require noble metal catalysts it has the disadvantage of requiring an operating temperature above 800 C due to the slow oxygen reduction reaction (ORR) on its cath ode surface. N uclear fission has been focused as one of possible technologies that provide more efficient energy system UO 2 is the fuel utilized in most of nuclear reactors. Despite its advantages as n uclear fuel (e.g. high melting temperature and chemic al and structural stability), low thermal conductivity is one of the disadvantages of UO 2 as the fuel. This dissertation is focused on understanding the effect of surface structures of LaFeO 3 and lattice voids of UO 2 on the properties for their energy rela ted applications which will enable improve the performance of the devices The approach used here involves e lectronic structure calculations, molecular dynamics, and microscopic thermodynamics.

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16 For l anthanum ferrite (LaFeO 3 LFO ) which is the base materia l of cathodes for SOFC s the phase diagram of the LFO (010) surface was developed to predict its stable surface stoichiometry at different operating conditions The surface stoichiometries of LaO and FeO 2 type terminations of LFO (010) surface was unders tood from electronic and structural perspectives Based on the developed surface phase diagram, the ORR was investigated. O xidized and stoichiometric FeO 2 type surfaces are chosen to represent the stable termination s below and above 800 K T he behavior of surface oxygen vacancy was found to be crucial factor in limit ing ORR. When the concentration of oxygen vacancy is high, the rate limiting step was the elementary step with the highest activation energy T his rate limiting step was different between the st oichiometric and oxidized surfaces. By contrast, in low concentration of oxygen vacancy the ORR was limited by site availability for the incorporation reaction of oxygen species (so called entropy barrier). For f luorite structured UO 2 the contribution of bubbles and lattice voids on thermal conductivity of UO 2 was investigated. B oth bubbles and pores decrease d the thermal conductivity of UO 2 In particular, the dependence of thermal conductivity on the size of the void was quantified Additionally, using He atoms in the void as a model bubble, it was validated that the penetration of He atoms reduce s the thermal conductivity of UO 2

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17 CHAPTER 1 BACKGROUND 1 .1 Motivation There are different resources for producing energy, e.g., liquids, coal, natural gas, s olar, and hydroelectric As shown in Figure 1 1 it is predicted that world energy consu mption will grow by over 50% within the next 25 years 1 In particular contribution of countries outside of Organization for Economic Cooperation and Development (OECD) to the growth is more significant than that of OECD countries. Also, the portion of fossil fuels (liquids, coal, and natural gas) for the energ y consumption is much higher than other resources (over 80%). While fossil fuels are so indispensable, they are limited 2 Also, the emission of CO 2 gas from fossil fuels is the main s ource of greenhouse gases which cause global warming 3, 4 Consequently, it is important to develop new energy resources to reduce the contribution of fossil fuels to global energy consumption. 1 2 Solid Oxide Fuel Cell s (SOFC s ) 1 2 .1 Operation and Components of SOFC s In a conventional power plant, chemical energy from the fuel is converted into electrical energy by the processes of heat generation, conversion of thermal energy into mechanical energy, and conversion of mechanical energy in to electrical energy ( Fig ure 1 1 ( a)). Thus, the efficiency is limited by the Carnot efficiency at the thermal step and there are additional losses via radiation of heat and friction. A SOFC on the other hand, can directly convert c hemical energy in fuels into electrical energy (see Figure 1 2 (b) ) T he power generating efficiency of the fuel cell is not limited by the Carnot efficiency, therefore overall efficiency is much high er (<9 0%

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18 with c ombined heat and power generation ) than ty pical power plant (~30% for gasoline engine) 5 Also as there are no moving parts in the device the fuel cell can provide quieter and clean er power generation source As fuels for SOFCs, in addit ion to hydrogen, hydrocarbon s such as methane (CH 4 ) and propane (C 3 H 8 ) can be used after fuel reforming which can produce s hydrogen from the these common fuels. The overall reaction for the SOFC is spontaneous (Figure 1 3 ) Once air and fuel gas are deliv ered to the anode and cathode respectively, net chemical driving force for oxygen transport is generated by an oxygen partial pressure ( p(O 2 ) ) gradient across the electrolyte. Using Krger Vink notation 6 r eactions at the cathode and anode can be written as follows: (Cathode) ( 1 1 ) (Anode) ( 1 2 ) where and describe oxygen vacanc y and oxygen anion in the lattice respectively Combining Equations ( 1 1) a nd ( 1 2 ), the overall reaction for the system is ( 1 3 ) The conduction of oxygen ions from the cathode to the anode through the electrolyte must b e balanced by the electronic charge flow through an external circuit thus providing electricity. As with any electrochemical device, SOFC s consist of three essential components a cathode, an electrolyte and an anode as indicated in Figure 1 3 ; and each single cell is connected by an interconnect to form the SOFC stack Both anode and cathode, (the electrodes ) are typically porous in order to increase reaction areas and improve

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19 mass transport of reactants They sandwich a dense, gas tight electrolyte, wh ich is an electronic insulator that only allows the conduction of oxide ions. A c onventional SOFC consists of yttria (Y 2 O 3 ) stabilized zirconia (ZrO 2 ) (YSZ) electrolyte, La 1 x Sr x MnO 3 (LSM) cathode, and Ni YSZ cermet anode. These high temperature SOFCs ope rate in the temperature range of 800 1100 C and this temperature limitation comes from ionic conductivity of YSZ which become s an ionic conductor above a temperature of about 800 C (Figure 1 4 ). 1 2 .2 Intermediate T emperature (IT) SOFCs T his high operati ng temperature of SOFCs provides advantages ; for instance, the heat required for reforming process can be supplied from the exhaust heat of the SOFCs. For similar reason s the SOFCs can be developed as hybrid systems with gas turbines. Importantly precio us metal e lectrocatalysts like platinum are not required for operating However, due to th e cost associated with material requirements at high operating temperatures bringing to market these technologically proven devices have been slow 7 Also, as trend s shift from large to small SOFCs by technological requests (e.g. auxiliary power units for automotive applications ) 8, 9 the combined cycle applications are no longer required Furthermore the development of these smaller intermediate temperature SOFC s (IT SOFC s ) is being stimulated by a liberalization of the established energy ma rkets in developed countries 7 Consequently, there are incentives to lowering the operating temperature as far as possibl e without compromising the performance of the device. By decreasing the operating temperature range to 500 800 C for IT SOFCs the following technological benefits can be achieved :

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20 M etal interconnects which show severe corrosio n at the high temperature r ange can replace LaCrO 3 based oxide interconnects. Radiation heat loss becomes less significant as the temperature decreases. Therefore, management of heat becomes easier at lower temperatures 1 0 S tart up tim e for the device can be shortened However, there are also technological challenges for IT SOFCs: As seen from Figure 1 4 ionic conductivity of YSZ, electrolyte for the high temperature SOFCs decreases rapidly as temperature is lowered. So, it is essential to discover new electrolyte materials havin g fast oxide ion conductivities One of the notable candidates for electrolyte materials of IT SOFCs is doped lanthanum gallate (La 1 x Sr x Ga 1 y Mg y O 3 LSGM) 11, 12 which shows oxide ion conductivity at 800 C comparable to YSZ at 1000 C (Figure 1 4 ). For the LSM cathode for the conventional SOFCs, Cr poisoning is severe. The Cr species can reduce catalytic activity of LSM by blocking its electrochemically active sites 13 and this poisoning becomes signifi cant as operating temperature decrease s Activit y of electrodes also decreases significantly as temperature is decreased. In particular the reactivity of oxygen molecule s on LSM cathode surface is reduced significantly by temperature decease For anode materials, Ni YSZ is still the best choice for IT SOFCs. However, as the LSGM has reaction with NiO 14 Ni based anodes are limited in their manufacture in combination with LSGM One possible solution is the development of perovskite based anodes (e.g. Sr 2 MgMoO 6 and La 0.75 Sr 0.25 Cr 0.5 Mn 0 .5 O 3 ) 15, 16

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21 Even though lowering the temperature of the conventional SOFC s reduces the ionic conductivity of YSZ, the most significant drawback from decreasing the operating t emperature is the reduced catalytic activity of the LSM cathode surface. From Figure 1 5 the main barri er for acceptable performance of the high temperature SOFCs is the voltage loss from the LSM cathode [ increase in area specific resistance (ASR)] Speci fically, the plot of ASR as a function of temperature at Figure 1 5 shows that ASR of the LSM cathode is incre asing exponentially as temperature decreases The resistance increase from the LSM is about 50% of the total ASR increase of the SOFCs at 800 C. F or these reasons a high performance cathode is required for IT SOFCs 1 2 2 1 La 1 x Sr x Fe 1 y O 3 (LSF) as c athode for IT SOFCs LSF and La 1 x Sr x CoO 3 (LSC) categorized as mixed ionic and electronic conductor s ( MIEC ) are potential materials for a cathode of IT SOFCs. Table 1 1 summarizes the valence stabilities and corresponding O 2 reactions of LSM, LSC, and LSF. For the cathode materials with the formula of ABO 3 oxygen vacancies are formed by the reduction of B site cations on the substitution of Sr 2+ to the La 3+ on A site ( ). In spite of that, the oxygen vacancy formation is competing with the oxidation of the transition metal ions in B sites, and this oxidation depends on the valence stability of t he B site cations. As shown i n T able 1 1, Mn is stable as Mn 4+ in the LSM. So, Sr substitution gives rise to the oxidation of Mn (Mn 3+ Mn 4+ ) and the formation of oxygen vacancy is not energetically feasible This valence stability also influences the reaction of oxygen species on th e LSM surface; that is the reason why only triple phase boundary are catalytically active sites for the LSM cathode. Similar explanations can be applied to the MIEC properties of LSC and LSF.

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22 Even though both LSF and LSC have similar ionic conductivit y L SF is chemically stable in the presence of an YSZ electrolyte whereas LSC is unstable with the YSZ and can form SrZrO 3 at its interface with the electrolyte Moreover, LSC shows large thermal expansion mismatch with YSZ ; these factors make it an unsuitable material for use in SOFC cathode 17 1 2 3 Oxygen Reduction Reaction (ORR ) The overall ORR on the SOFC cathode can be written as: ( 1 4) where is an oxygen vacancy, is an electron, and represents an oxide ion in the cathode and/or electrolyte lattices. Practically, the overall ORR is composed of numerous elementary reaction steps such as adsorption, dissociation, electron transfer, surface diffusion, and inco rporation ( Figure 1 6 ) Depending on the type of SOFC cathode (electronic conductor or MIEC), the ORR is achieved via one or more pathways to reach the electrolyte. Figure 1 6 sketches possible scenarios of ORR for (a) an electronic conducting cathode that takes only a surface pathway and (b) a MIEC cathode having both surface and bulk pathways. Each elementary step has its own activation energy barrier ( E a ) and the step with the highest barrier becomes the rate determining step. As this rate determining s tep acts as a bottle neck for the entire SOFC performance the reduction of the energy barrier for rate determining step will lead to the accelerated development of high performance IT SOFCs. Even though a number of papers have been devoted to the mecha nistic study of ORR 18 23 24 there still remains uncertainty and disagreement abou t the rate limiting

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23 step In particular it is reported that the rate limiting step can be changed depending on operating temperatures and p(O 2 ) For LSM, et al. examined La 0.8 Sr 0.2 MnO 3 microelectrodes at p(O 2 ) mixed bulk and triple phase boundary charge transfer is the rate limiting step below 700C with E a = 1.18 0.08 2.46 0.07 eV They also determined that surface chemical reactions, including the dissociation and adsorption o f O 2 molecules are the rate limiting step above 700 C with E a of 1.71 1.88 0.02 eV 22 In addition, Kan et al. used isotopic tracers to determine that incorporation and dissociative adsorption are the dominant rate limiting step on (La 0.8 Sr 0.2 ) 0.98 MnO 3 pow ders at 600C and 800C, respectively, with E a = 0.41 eV between 600 to 650C and E a = 0.90 eV between 700 to 800C 21 Similarly, Fieig et al. examined the rate limiting step dependence on p(O 2 ) of (La 0.8 Sr 0.2 ) 0.92 MnO 3 thin films and found that surface related re actions and ionic transport are the rate limiting step at low p(O 2 ) ( m atm range) and high p(O 2 ) respectively 20 For La 1 x Sr x Fe 1 y Co y O3 (LSCF) surfaces, on La 0.6 Sr 0.4 Co 0.6 Fe 0.4 O 3 surfaces with temperatures of 650 to 982C, ten Elshof et al suggested that ORRs are controlled by surface exchange including the adsorption of O 2 molecules in p(O 2 ) 0.0 1 0.03 atm and by diffusion in p(O 2 ) 0. 1 1.0 atm respectively 23 Consequently, to push the envelope of SOFC development progress in more fully understanding the underlying mechanisms of ORR is needed 1 3 Perovskite Oxides P erovs kites have been essential materials for SOFCs as they can be ideally used as all components of SOFCs : LSM and LSF for ca thode, LSGM for electrol yte, Sr 2 MgMoO 6 and La 0.75 Sr 0.25 Cr 0.5 Mn 0.5 O 3 for anode and LaCrO 3 based interconnect

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24 Therefore, it is meaningful to review fundamental information of perovskite materials including their structures and properties. 1 3 .1 Structure of Perovskite Oxides Per ovskite is categorized as a mixed oxide consisting of two or more different cations with different oxidation states, ionic radii, and coordination numbers. This diversity results in various different properties in comparison with those of simple oxides. The chemical formula of ideal perovskite is ABO 3 A and B denote two different cations, where A site cations are typically larger than the B site cations and have similar size to the anions. Almost all elements excluding inert gases can occupy either A or B sites (Figure 1 7 ). The stability and space group are mainly determined by the ratio of the ionic radii of the A and B site cations as well as their electronic structures. I deal perovskite has a cubic structure ( #221). SrTiO 3 is commonly regarded as typical cubic pero vskite as its structure approaches closely that of the i deal perovskite. The c oordination numbers of A, B, and O in ideal perovskite are 12, 6, and 6 (A site cations:4 and B site cati ons:2 ): as B O bonding is highly covalent, its coordination number is lower than 6. Although a few compounds have the ideal structures the majority of perovskites have distorted variants with lower symmetry To understand the d erivations from the ideal cubic structure, firstly, the perovskites are regarded as purely ionic systems. For the ideal perovskite, the radii of the A, B, and O ions have the following relationship: ( 1 5)

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25 Therefor e, the deviation from the ideal structure in perovskites can be described by so called tolerance factor, t 25 which is observed by Goldschmidt : ( 1 6) In perovskite oxides, t lies between approximately 0.80 and 1.10. For t > 1 .0 the A O distance is large and the B atoms are too small for the oxygen octahedral. I n the case of t < 1 .0 the A site cation is small in comparison to the space available within the oxygen octahedral, thereby resulting in a lower coordination number for the A site cation Spe cifically, i f t is slightly less tha n ideal value then rotations and tilting of the oxygen octahedral is typically observed. If t is much lower than one the structure will be distorted to have six fold coordination for the A site cation F igure 1 8 shows the space groups of perovskites with A 2+ B 4+ O 3 and A 3+ B 3+ O 3 combinations as functions of ionic radii of A and B site cations In the F igure 1 8 as t decreases, the space group of perovskite is shifted from cubic to triclinic. However it must be emphasiz ed that even the though tolerance factor t might be rough guide as to whether a given set of ions is stable in a perovskite structure or not the value of t is not a definite indicator of the space group of given perovskite This is because factors other t han ionic radii, e.g. covalency, Jahn Teller distortion and lone pair effects, play a role in determining the space group 26 1 3 .2 Properties of Perovskite Oxides Perovskite oxides possess various properties due to their varied structures and chemical compositions. As well as f erroe lectricity and superconductivity, electrical conductivity, ionic conductivity, mixed ionic and electronic conductivity, catalytic activity, ferromagnetis m and multiferroicity are possible (Table 1 2)

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2 6 1 3 .3 Lanthanum Ferrite ( LaFeO 3 ) L a F e O 3 (LFO) is categorized as a A 3+ B 3+ O 3 type perovskite oxides. LFO has the Pnma structure ( P 2 1 /n 2 1 /m 2 1 /a space group #62) within an orthorhombic unit cell (Figure 1 9 (a )) At approximately 1250 K LFO experiences an orthorhombic to rhombohedral phase transition and has R3c structure (space group #161) 27 The Pnma structure is derived from the ideal cubic perovskite with structure (#221). In the Pnma structure, the oxygen octahedra are tilted with respect to their neighbors in a re peating pattern. 26 This octahedral tilting both breaks the symmetry of the <001> directions and increase s the size of the unit cell from one formula unit (5 atoms) to four formula units (20 atoms). The [100] and [001] directions of the Pnma unit cell are aligned with the and [101] directions in the cubic perovskite structure. The [010] directions in the tw o structures are the same as illustrated in Figure 1 9 (b) the coordination numbers of La, Fe, and O are 8, 6, and 6 respectively (Figure 1 10 ) Coordination number of A site cation for Pnma can be 12, 10, 9 and 8 depending on the magnitude of tilt 26 The m agnetic ground state of LFO is G type antiferromagnetic in which there is anti ferro type arrangement of spins in all three directions with N el temperature of 740 K 28 F igure 1 1 1 shows available spin combinations of antiferromagnetic state within ideal cubic perovskite. 1 4 Uranium Dioxide (UO 2 ) Uranium Dioxide (UO 2 ) has fluorite structure with space group of (F4/m 2/m) with lattice parameter of 5.4698 at 300 K (see Figure 1 1 2 ). U atoms occupy the face centered cubic sites (Wycoff position 4 a ) and O atoms sit at the tetrahedral

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27 sites (Wycoff position 8 c ). The coordination numbers of U an d O atoms are 8 and 4, respectively; the ionic radii of U 4+ and O 2 are r(U 4+ )=1.00 and r(O 2 )=1.38 29 The electronegativities of U and O a re 1.38 and 3.44 respectively, thus the difference is 2.06. This implies that the U O bonds in UO 2 have approximately 60% ionic character. UO 2 ha s almost a single phase up to its melting point, 2865 C with broad range of s toichiometry as seen in Figure 1 1 3 The fluorite structure is stable at O/U <2.25, with U 4 O 9 oxide at O/U>2.25. Other oxides including U 3 O 7 U 3 O 8 and UO 3 exist at low temperature (<400 C ). 1 4 .1 UO 2 as Nuclear Fuel Uranium (U) has been a technologically important element for more tha n half century 30 especially as a nuclear fuel in fission reactors for generating electrical power. Perhaps the most important motivation that U is used for the reactors is the natural occurrence of 235 U. Even though the portion of the 235 U in natural U is 0.7% which is very small with respect to predominant isotope 238 U (99.3%), it is the only fi ssile isotope which in found in significant quantity in nature. U is generally found as triuranium octooxide (U 3 O 8 ) ore. It is existed as olive green to black solid without odor. However, for the reactor fuel, U is used in the form of UO 2 rather than the U 3 O 8 by following reasons: High melting temperature 2875 C Chemical stability with cladding materials and coolant in the reactor Fluorite structure which can relatively easily accommodate fission products 30 However, UO 2 also has disadvantages as the reactor fuel: Low thermal conductivity (~5 W/mK at ~530 C )

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28 Lower density of fissile elements than its metallic form, which can decrease thermal con ductivity 31 Relat ive difficulty of processing in comparison with its metallic form 30 UO 2 normally exists under steep thermal gradient (10 4 C /cm ) 32 Low thermal conductivity of UO 2 makes issues caused by the therm al gradient worse. 1 4 .2 Lattice V oid and He B ubble in UO 2 Fission products can be defined as the atomic species generated by fission reactions. An archetypal fission reaction is as follows: ( 1 7) Here, a heavy atom (in this case, U) is hit by a neutron, n; the heavy atom becomes unstable after absorbing the neutron and splits into fast moving lighter elements and extra neutrons, which can induce additional fission reactions. The fission reactions generate various fis sion products as well as lattice defects. The fission products can be defined as atomic species generated after a fission reaction, and include rare gases, metallic and oxide inclusions, and volatiles 33 As well as the fission products, UO 2 also contains lattice voids. These voids are also known as cavities, pores, and bubbles 34 Figure 1 1 4 shows typical scanning electron microscopy (SEM) image of lattice voids in UO 2 There are two types of voids in UO 2 nuclear fuels. The voids generated by precipitation of fission gases are generally called bubbles and usua lly small; for example, a recent experimental study on UO 2 single crystals reported that the radius of He and Xe bubbles are ~250 and 30 50 respectively. These bubbles contain a fission gas at high pressure (e.g. ~300 atm at r=100 ) 32 The other type of void which is often called pore comes from incomplete

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29 densification of the nuclear fuel during it s manufacture. These fabrication pores are generally large (r> 1 m) and contain gas at low pressure (~3 atm at 10 m). Experimentally, the % volume of void of UO 2 varies from about 5 10% 35 Bubbles are filled with fission products. However, pores are primarily filled with He with small amounts of CO, CO 2 and fission gases. Both bubbles and pores decrease the thermal conductivity of t he UO 2 and thereby lead to fuel temperatures higher than dense fuel. The density of the fission gas in the bubble (No. of atoms/volume of voids) depends on the size of the bubbles. In particular Xe bubble has densit ies of ~ 1.0x10 28 atoms/m 3 and ~ 1.0x10 27 atoms/m 3 at r=10 and 10 3 respectively 32 Helium (He) is one of fission gases existed in the UO 2 fi llet which can be produced by decay of actinide elements (e.g. ) particle consists of and As He has relatively high thermal conductivity than other gases (~0.3 W/mK at 800 K ) He is important in the perspective of thermal conductivity of nuclear fuel. He is used to fill the space between the UO 2 fuel rods and the cladding materials to increase thermal conductivity of nuclear fuel. Also, both bubbles and pores contain He atoms. Because of its low solubility in UO 2 He atoms ha ve tendenc y to precipitate, form ing bubbles, or to be released from the UO 2 However, release of He from the fuel is generally not significant due to its low mobility 36, 37 Instead, He atoms tend to be trapped in lattice voids 38 1. 5 Objectives The aim of this work is to answer the open questions in energy related applications of oxide materials LFO and UO 2 In engineering LFO based cathode

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30 materials of IT SOFCs for optimal performance at lower operating tempera tures, following questions must be answered: 1) What is the stable surface stoichiometry of LFO at SOFC operating conditions? 2) What is the origin of the stoichiometric change ? 3) What is the rate limiting step of ORR in LFO surface ? 4) Does the rate limi ting step change as a function of surface stoichiometry of LFO (in other words, SOFC operating condition)? In UO2 nuclear fuel, to investigate the contribution of bubbles and lattice voids which can decrease the thermal conductivity of the UO 2 following questions must be addressed: 5 ) W hat is relationship between lattice void size in UO 2 and its thermal conductivity ? 6 ) He atom itself has high thermal conductivity. So, does He bubble in UO 2 also increase its thermal conductivity?

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31 Table 1 1 Comparison o f valence stability and O 2 reaction among La 1 x Sr x MnO 3 La 1 x Sr x Co O 3 and La 1 x Sr x Fe O 3 39 La 1 x Sr x MnO 3 La 1 x Sr x Fe O 3 La 1 x Sr x Co O 3 Valence stability Mn 4+ Stable Fe 4+ Unstable Co 4+/3+ Unstable Surface O 2 reaction Slow Fast Fast Reaction sites Triple phase boundary Surface Surface Table 1 2 Various properties of perovskite oxides Propert y Compounds Ferroelectricity BaTiO 3 PbTiO 3 BiFeO 3 Piezoelectricity Pb(Zr,Ti)O 3 (Bi,Na)TiO 3 Electrical conductivity SrFeO 3 LaCoO 3 LaNiO 3 LaCrO 3 Superconductivity (La Sr)CuO 3 YBa 2 Cu 3 O 7 Ionic conductivity CaTiO 3 (La,Sr)(Ga,Mg)O 3 BaZrO 3 SrZrO 3 BaCeO 3 Magnetism LaMnO 3 LaFeO 3 BaCuO 3 Catalytic property LaCoO 3 LaMnO 3 LaFeO 3 BaCuO 3 Multiferroicity BiFeO 3 BiMnO 3

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32 Figure 1 1. World Energy Consumption by (a) region, and (b) type of energy resource. Adapted from reference 1 Figure 1 2 En ergy conversion steps of (a) typical power plant and (b) fuel cell. Figure 1 3 Schematic description of operation principle of solid oxide fuel cell

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33 Figure 1 4 The conductivit ies of electrolytes of fuel cells: phosphoric acid fuel cell (PAFC), polymer electrolyte fuel cell (PEFC), molten carbonate fuel cell (MCFC), and solid oxide fuel cells (SOFC). Adapted from reference 8 Figure 1 5 Separated a rea specific resista nce (ASR) in a conventional SOFC with L a 1 x S r x M n O 3 cathode, Y 2 O 3 stabilized ZrO 2 ( YSZ ) electrolyte, and Ni YSZ anode Adapted from p.281 of reference 40

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34 Figure 1 6 Schematic description of possible elementary reaction steps of oxygen reduction reaction and possible pathways of two different classes of cathode materials: (a) electronic conductor and (b) Mixed ionic electronic conducting ( MIEC ) cathodes.

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35 Figur e 1 7 Chemical elements that can occupy A and B sites in perovskite s. Adapted from reference 8 Figure 1 8 Space group map s of (a) A 2+ B 4+ O 3 and (b) A 3+ B 3+ O 3 Adapted from reference 8

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36 Fig ure 1 9 (a) Schematic of LaFeO 3 unit cell. (b) Relative di rection of orthorhombic unit cell (solid line) of LaFeO 3 to pseudo cubic cell (dot ted line). The oxygen octahedra are shaded light blue Figure 1 10 Coordination number (CN) of (a) La, (b) Fe, and (c) O(1), (d) O(2) in bulk LaFeO3 Here, O(1) and O(2) have 8d and 4c as multiplicity and Wyckoff letter, respectively.

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37 Figure 1 1 1 Schematic description of available spin combinations for antiferromagnetic (AFM) state within ideal perovskite unitcell. Each corner is occupied by B site cations. Figur e 1 1 2 Schematic of the fluorite structure UO 2 Blue and red spheres depict uranium and oxygen, respectively.

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38 Figure 1 1 3 Phase diagram of U O system. The S and L in subscripts are used to indicate the solid and liquid phase. Adapted from reference 41 The number in each phase is related t o the equation for p(O 2 ) in Table 2 of the reference, which is used to calculate thermodynamic properties of the phase. Figure 1 1 4 (a) SEM image of a (111) surface of UO 2 crystal (b) Oblique angle scanning electron microscopy image of cross section t hrough one of bright symmetrical features in (a) shows that they are microscopic voids ( Reproduced from Reference 34 ).

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39 CHAPTER 2 COMPUTATIONAL DETAIL S 2.1 Density Functional Theory (DFT) D ensity functional theory (D FT ) was developed by Hohenberg, Kohn and Sham 42, 43 The main objective of DFT is to replace complicated many body electronic wave function with the electronic density Whereas the many body wave function is dependent on 3 N variables which are three spatial variables for each of the N electrons, the density is only a function of three variables. The density is a simpler quantity to deal with both conceptually and practically. function, which means that the energy has a functional depe ndence on single electron 2.1.1 Exchange and C orrelation To move forward to DFT, understanding of exchange and correlation is essential. Electrons repel each other due to th e Coulomb interaction between their charges. The Coulomb energy of a system of electrons can be reduced by keeping the electrons spatially separated However, this has to be balanced against the kinetic energy cost of deforming the electronic wave functi ons in order to separate the electrons. The effects of electron electron interaction are briefly described below. Exchange effect: t he w ave function of a many electron system produces a spatial separation between electrons that have the same spin and thus reduces the Coulomb energy of the electronic system. The reduction in the energy of the electronic system due to the spatial separation is called exchange energy. It is straightforward to include exchange effect in total energy calculations, and this is generally referred to as the Hartree Fock approximation ( E HF ).

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40 Correlation effect: i f electrons of opposite spins are spatially separated, the Coulomb energy of the electronic system is reduced at the cost of increasing the kinetic energy of the electrons The correlation energy ( E corr ) is described as (2 1) Here, E total is many electron total energy. E corr of many electron system is extremely difficult to be calculated as wave function of the system depends on the coordinates of all the n electrons. Undoubtedly, from the view of computational efficiency, the computational cost of calculating E corr grows exponentially with the number of electrons. 2 1.2 Details of DFT DFT provide s new efficient method of describing the effect of exchange and correlation DFT focus on the single electron density i.e. a quantity related to wave function, ( 2 2 ) The Kohn Sham total energy functional can be written ( 2 3 ) The first term on the right hand side is the energy for interaction of the electrons with the external pote ntial V The second term is the Coulomb energy of a density distributio n The quantity is a universal functional of the which means that it is uniquely specified by the density of the interacting electrons and does not depend on the particular V acting on the electrons. contains the many electron kinetic energy of the electrons and the exchange and correlation energy functional ( ). Definite

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41 expression for is needed to perform practical total energy calculations and this is only possible if an approxi mation is made Kohn and Sham introduced the concept of imaginary non interacting electrons having the same density of the true interacting electrons. Non interacting electrons are described by single particle wave functions an d their density is given by: ( 2 4 ) Here the factor 2 accounts for spin degeneracy. In terms of the total energy functional can be written as: ( 2 5 ) The third term on the right hand side of this expression is the single particle kinetic energy of the non interacting electrons. The functional is called the exchange correlation energy functional. Here, rather than directly minimizing with respect to we have to minimize it with respect to the The condition for minimum total energy functional corresponds to the following Euler Lagrange equation: ( 2 6 ) And Eq uation 2 5 becomes: ( 2 7 )

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42 Here, is local potential called the exchange a nd correlation potential. Eq uation 2 7 is called the Kohn Sham equation. It defines an exact mapping b etween a system of interacting electrons and an imaginary system of non interacting electrons having the same density of the interacting electrons. After solving Eq uation 2 7 the electr on density is obtained from Eq uation 2 4 and the corresponding ground state energy (of the interacting electrons) is calculated from Eq uation 2 5 2 1.3 Local D ensity A pproximation (LDA) and G eneralized G radient A pproximation (GGA) for E xchange and C orrelation Though detail calculation procedure of DFT has been explained specif ication of and is not provided yet To specify and approximations must be used as we do not know exact functional form of exchange and c orrelation terms. The basic approximation is the LDA 43 We can define the exchange and correlation energy per electron as: ( 2 8 ) The exact is a non local functional of the density, i.e. at point r depends on the density at all other points in space. In the LDA we assume instead that depends only on the density at point r itself. In other words, becomes local functional of the density: ( 2 9 ) This seems a reasonable approximation for systems where the density varies slowly in space. However, LDA tends to overestimate, often substantially (even by 20

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43 percent or more) the cohesive energies of solids and the atomization energies of molecules. To overcome the inaccuracy, generalized gradient approximat ion (GGA) 44 was deve loped. In this approach, Eq uation 2 8 is written as: (2 10 ) This semi local approximation for depends not only on the local density as in the LDA, but also on the local density gradient In general, t he GGA improves substantially over the LDA, particularly for cohesive properties a nd atomization energies. GGA bond lengths and is on average slightly better than the already good LDA results. Usually, LDA underestimates bond lengths by 1 2 percent, overestimate them. Overall the predictive power of GGA is rath er good. 2.2 Ab Initio Atomistic Thermodynamics Energies calculated from DFT do not include temperature and pressure effects. In other words, all physical quantities from DFT are only valid at T = 0 K and p = 0 atm In studying a (T, p) ensemble, the key q uantity is the Gibbs free energy G (2 11) The first term is total energy which is calculated from DFT. The second term represents vibrational contribution which incl udes vibrational energy and vibrational entropy term Here, contains zero point energy. The third term contains configurational entropy The final term is pV term.

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44 The general concept of the atomistic thermodynamics will be exemplified by the calculation of surface Gibbs free energy of LFO surface. 2.2.1 Surface Gibbs F ree E nergy of LFO S urface The surface Gibbs free ene rgy for surface termination i i is: ( 2 12 ) where is the Gibbs free energy of surface slab i is number of each type j atom within the slab and is the chemical potential of each atomic component j in bulk LFO, for j = La, Fe, and O. The chemical potentials term is the Gibbs free energy of the materials in the slab if they were reversibly inserted into a materials reserv oir. 45 The factor of comes from the use of a symmetric surface slab that contains two identical surface terminations. From Eq uation 2 12 i is defined as the excess Gibbs free energy due to the presence of the surface. The term, which is the number of off stoi chiometric atoms of component n with respect to component m in each surface of slab i is: (2 13) where the values of and are 1, 1, and 3 resp ectively. Taking Fe as m, Eq uation 2 12 can be combined with Eq uation 2 13 to form: ( 2 14 )

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45 The i can be expressed by defining T p ( O 2 ), and the chemical potentials of either of the cations. Here we choose La as the reference; choosing Fe instead does not change the result s In order to evaluate i Fe is determined as: ( 2 15 ) This states that the chemical potential of lanthanum ferrite is equal to the sum of the chemical potentials of each atomic type in LFO crystal. As the surface of each slab must be in equilibrium with bulk LFO, is identical to the Gibbs free energy of bulk LFO: ( 2 16 ) By combining with Eq uations 2 15 and 2 16 Eq uation 2 14 can be simplified to: ( 2 17 ) There are boun dary conditions which restrict the physical range of and First, in order to prevent the spontaneous degradation of LFO surfaces, i must be positive. Second, since the system does not precipita te into metals and oxides (for this case, La, Fe, La 2 O 3 FeO, Fe 2 O 3 and Fe 3 O 4 ) the following conditions must be satisfied: ( 2 18 ) ( 2 19 ) ( 2 20 ) ( 2 21 ) ( 2 22 )

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46 ( 2 23 ) The total energ ies of the metals and oxides can be calculated using their respective atomic structures and magnetic ground states at 0 K The oxygen rich condition is defined as the point beyond which gaseous O 2 starts to condense on the surface. However, condensed O 2 does not exist in the SOFC processing conditions (gaseous O 2 condenses to liquid at 90.2 K ). T his condition provides an upper limit of : ( 2 24 ) Rather than using chemical potentials themselves as variables we can use deviations of chemical potential s from their reference state s as variables 46, 47 In particular, instead of and we use and as variables with and as reference states for La and O, respectively. In addition, we approximate the Gibbs free energies with total energies from DFT calculations as follows: ( 2 25 ) Consequently, Eq uations ( 2 16 ) ( 2 24 ) can be rew ritten as follows: ( 2 26 ) ( 2 27 ) ( 2 28 ) ( 2 29 )

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47 ( 2 30 ) ( 2 31 ) ( 2 32 ) ( 2 33 ) ( 2 34 ) where f is formation energy of the corresponding phases which are calculated from DFT. In this formalism, the vibrational contribution to the surface Gibbs free energy (the difference in vibrational modes and absolute vibrational contribution which comes from off stoichiometric atoms) is neglected. Typically, this vibrational contribution is found to be 10meV/ 2 which corresponds to 0.3 eV per surface unit cell (31.46 2 ) 48 ; this value In needed, accurate vibrational contribution to surface G ibbs free energies could be determined from the phonon density of states. 2 2 .2 Chemical P otential of O xygen as a F unction of p(O 2 ) and T Under equilibrium condition s between bulk and gas phases the chemical potential of oxygen in the LFO bulk is identi cal to the chemical potential of oxygen in the gas phase: (2 35 ) In addition, under the ideal gas approximation, O is directly related to p(O 2 ) and T by

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48 (2 36 ) where k is the Boltzman n constant and p 0 is the reference pressure, 0.21 atm The can be written as ( 2 37 ) We determ ine the in Eq uation (2 37 ) using thermodynamic data from the NIST JANAF thermochemical tables 49 However, from these tables i s different from its counterpart in Eq uation (2 37 ) because the reference T and p(O 2 ) of 298.15 K and 0.98 atm respectively, are not identical to those of from the formalism which is calculated at 0 K W e therefore define (2 38 ) to quantify the difference between these two definitions. The Gibbs free energy of the oxygen i s expressed in terms of the enthalpy and entropy of O 2 : (2 39 ) In this case, is directly extracted from the NIST JANAF tables ( meV /O 2 ), while is determined by the enthalpies of the

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49 M x O y oxides and M metals (in this case, M =La and Fe), and heats of formation for the oxides. In particular, the enthalpy of an M x O y oxide can be written as ( 2 40 ) Here, enthalpies of the oxide and of metal can be approximated by the total energies for these materials from DFT (Eq uation 2 25 ), while the heats of formation of the oxide, come from the NIST JANAF tables 49 We can use averaged value s of from all the considered oxides. For specifically, value averaged from secondary phases of LFO is 0.11 eV with standard deviation of 0.28 eV This deviation comes, at least in large part, from the omission of the vibrational energy contributions of the oxides in our calculations (Eq uation 2 25 ). In particular, as the pV term can be negligible 48 we assume that the major contributions to the enthalpy are the total energy and vibrati onal energy including zero point energy. As the vibrational energies of the materials are not identical, the deviation in is inevitable. This thermodynamic formalism for surface Gibbs free energy will be used in developing phase diagram of LFO (010) surface in Chapter 4.

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50 CHAPTER 3 STABILIZATION MECHAN ISMS OF THE LANTHANUM FERRITE (010) SURFACE 3 .1 Introduction Because of t he effect of surface composition on oxidation reduction reactions, 50 it is necessary to characterize the stability of various cathode surfaces including non ide al off stoichiometric surface terminations that may be stabilized by the operating conditions of the SOFCs. In particular, it is possible that surface terminations with low coordinated surface oxygen atoms ( O low ) may be preferentially stabilized at interm ediate oper ating temperatures. Indeed, it has been reported for several metal oxides that ideal stoichiometric surface terminations are favorable only at low and/or high temperatures. 48, 51 52 In this chapter, t he structure of the L a F e O 3 (LFO) (010) (1x1) surfaces with various terminations are determined, with particular attention paid to the role of electronic charge transfer and surface atomic relaxation on surface stability. D ifference s in the compensation mechanisms of surface terminations with and without O low are also analyzed 3 .2 Bulk L aFeO 3 Table 3 1 provides structural information for bulk LFO as calculated with DFT The calculated lattice constants agree well with experimental values and a previous DFT study 53, 54 In comparison with experimental lattice constants, the largest deviation is an overestimate by 1.725% for the lattice parameter in the a direction Such overestimates are typic al for calculations based on the GGA functional. 55 Fig ure 3 1 il lustrates the electronic density of states ( DOS ) of bulk LFO. The r adii for Wigner Seitz cells for all DOS results are determined based on Shannon ionic radii :

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51 r La =1.16, r Fe =0.65 (high spin configurati on) and r O =1.4 ; 29 from Fig ure 3 1 it can be seen that the Fe ions have electrons in high spin configurations. The calculated Kohn Sham band gap is 1.0 eV which is less than the experimental band gap of 2.1 eV 56 Nonetheless, the DFT calculations correctly predict LFO to be insulating, which is consistent with the findings of previous DFT studies. 57, 58 To establish that DFT can correctly determine the stru ctural and magnetic ground states of the system, I compare the total energies of bulk LFO with the and Pnma space groups, with various magnetic states including ferromagnet ic, A type antiferromagnetic ( A AF ), C type antiferromagnetic ( C AF ), and G type antiferrom agnetic ( G AF ) (see F ig ure 3 2 ). For P6 3 cm, A AF G AF and ferromagnetic states are considered based on the magnetic orderings of YMnO 3 which has same space group. 59 DFT predicts the structural and magnetic ground state of bulk LFO to be Pnma with G AF which is consistent with reported experimental results. 58, 60, 61 3 .3 LaFeO 3 (010) Surfaces with a Variation of Surface Oxygen Stoichiometry LFO (010) is good model plane to investigate surface stability. This is because stoichiometric terminations of the (010) plane LaO and FeO 2 have w eak surface polarity in comparison with those of other low Miller index planes (e.g. (110)) In particular, b ased on ionic charges (La: +3 e Fe: +3 e and O: 2 e ) the LaO and FeO 2 terminations of the LFO (010) plane have +1 e and 1 e as their layer charge s respectively. However, layer charges for stoichiometric terminations of the (110 ) plane (LaO 3 and Fe) are = 3 e for LaO 3 and =3 e for Fe, respectively The surface polarity is proportional to its layer charge, and large surface polarity is the source of surface instability.

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52 T he sixteen different (010) surface terminations are constructed ( summarized in Table 3 2 ) : seven s urface terminations based on the stoichiometric LaO surface termination and nine based on the stoichiometric FeO 2 surface termination T he nomenclature used to describe the various surface s is b ased on the number of O atoms that are added to or removed from the topmost LaO and FeO 2 layers In particular, terminations are denoted LaO+ x O and FeO 2 + x O where x describes the number of added or removed O atoms at the surface per one LaO/FeO 2 molecular unit. As there are two FeO 2 or LaO units in each layer of the (1x1) surface unit cell if there is one O low on the FeO 2 layer, for example, the corresponding surface termination is designated as FeO 2 +0.5O low The initial positions for added and removed O atoms are determined from the Wyckoff positions of the corresponding O atoms in bulk LFO Fi g ures 3 3 ( a ) 3 3 ( b ) and 3 3 ( c ) illustrate the LaO, LaO+0.5O low and LaO+1.5O low surface terminations Fi g ures 3 3 ( d ) 3 3 ( e ) and 3 3 ( f ) show the FeO 2 FeO 2 +0.5O low and FeO 2 +1.5O low surface terminations Hereafter, surfaces with LaO x O/O low and FeO 2 x O/O low surface terminations will be simply designated as LaO type and FeO 2 type surface terminations, res pectively. In all cases the topmost FeO 2 or LaO layer in contact with the vacuum region is designated as the first layer. For surface terminations with an overlayer of O low atoms, the O low atoms are designated as being in layer zero. The surface slab model s are built with a (1x1) surface unit cell within the plane of the surface. To determine the energy dependence on surface unit cell size, I compare the total energies of the LaO surface termination with (1x1) and (2x2) surface unit cells, where each slab h as a thickness of 11 atomic layers. The resulting surface energy difference between the two systems is 0 1 meV/ 2 Considering that surface energy is

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53 on the order of an 1 .0x10 3 m eV/ 2 and the difference s in the surface energies of the structures consider ed is larger than 0.2 m eV/ 2 within same LaO or FeO 2 type surface terminations this difference is sufficiently small to justify the use of the (1x1) surface unit cell in the work shown here. The s urface energy is d efined in the usual manner as : ( 3 1) where G slab is the Gibbs free energy of the surface slab, which is equivalent to the total energy from the DFT calculations in these zero Kelvin and zero pressure calculations in the absence o f v ibrational contributions. In Eq uation ( 3 1 ) A is the surface area; the index i runs over all atomic species in the system (in this case La, Fe, and O), i is the chemical potential of the i th component in the bulk unit from which surface is constructed, and N i is the number of each atomic species. The factor of arises because the symmetric system slab contains two equivalent surfaces. The chemical pote ntial of oxygen is determined based on section 2.2.2. Each surface termination considered is built as a symmetric surface slab with a thickness of more than nine atomic layers The positions of all of the atoms in the system are fully relaxed with the exc eption of the atoms in the center layer of e ach slab, which are held fixed. 3 .4 Charge Compensation of LaFeO 3 (010) S urface It is instructive to construct line diagrams in which the top layers of stoichiometrically different surface terminations are align ed by their layer charges as is done in Fi g ure 3 4 First, I consider a simple line diagram Fig ure 3 4 ( a ) for the top

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54 layer charges ( ) based on formal ionic charges (La: +3 e Fe: +3 e and O: 2 e ). Surface terminations are cate gorized as terminations with positive or negative All reduced surface terminations have positive or zero and, all oxidized surface terminations have negative or zero Stoichio metric LaO and FeO 2 surface terminations have of +1.0 e and 1.0 e respectively The LFO (010) surface is categorized as a type III polar surface 62 because LaO and FeO 2 layers have non zero formal ionic charges ( 0) and each repeat unit ( LaO FeO 2 bilayer s) bear s a non zero dipole moment ( 0) ; t ype I surfaces have zero layer charge ( =0) and zero dipole momen t ( = 0 ) and type II surfaces have non zero layer charge ( 0) and zero dipole moment ( = 0 ). Type III polar surfaces with ideal layer charge s are unstable because the electrostatic potential V between the repeat unit monotonically increases with surface th ickness due to the non zero electrostatic field per repeat unit The c orresponding electrostatic energy is significant and is the source of surface instability The contribution of the electrostatic potential to surface instability can be understood by con sidering a macroscopic surface model which is illustrated in F ig ure 3 5 63, 64 In particular, I consider polar surface s consisting of repeat units of bilayers with + and as the ir layer charge per unit area, respectively as is done in Fig ure 3 2 a. The assumptions for the macroscopic surface model are that t he discrete atomic structure inside each layer paral lel to the surface can be neglected and that t he electron density in th e layers is localized with no charge overlap between the layers.

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55 Using this model, the a veraged electrostatic field per repeat unit is and is non zero. As a result V increase s by = per repeat unit of thickness R and increases monotonic ally with increasing surface thickness which leads to system instability Consequently, must be modified as zero to suppress the monotonic increase of V This is ava ilable by surface charge compensation Here, it is assume d that only top layer experience charge compensation. This is reasonable as top layers show the most significant electronic redistribution. C harge compensation of the first layer charge causes to be equal to which goes to zero when 1st = ( 3 2 ) where 1st and B are first layer charge after charge compensation and layer charge of bulk system. Under these condition s, which are illustrated in Fig ure 3 5 ( b ) there i s no potential across the slab. The electronic redistribution in the top surface layer is largest but charge compensation in lower layers can still occur, leading Equation ( 3 2) to the more general form 63, 64 : ( 3 3 ) where i is the layer charge associated with the i th layer, and m is the number of outer layers for which the layer charges are modified 63, 64 This analysis makes clear that is more useful than for understanding surfa ce charge compensation as F ig ure 4 3 ( b ) also

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56 indicates Here, is the layer charge from bulk LFO using formal ionic charges I f is positive then additional electrons with magnitude of are needed to achieve charge compensation. On the other hand, if is negative additional holes that are equivalent in charge to are needed. 3 .5 Stabilization M echanisms of the L a F e O 3 (010) S urface Three factors will be examined to investigate how the LFO (010) surface is stabilized : (i) the c ontribution of each ato mic species to charge compensation; (ii) the contribution of sub layers ; and (iii) the contribution of atomic relaxation. 3 .5.1 Contribution of Atomic Species to Charge Compensa t ion Bader topological analysis : Bader topological analysis 65 is performed on the calculated charge densities of bulk LFO and all surface terminations. The calculated Bader charges of La, Fe, and O atom s in bulk LFO are found to be = 2.09 e = 1.66 e and = 1.25 e respectively. The l ayer charge s are calculated as the sum of Bader charges of atoms in the layer For example, of first layer of LaO+0.5O low surface terminat ion is As seen from Table 3 3 all surface terminations sa tis fy Eq uation ( 3 3 ) with a deviation of about 5% from one half of the Bader charge value of a bulk layer. To investigate the contribution of each atomic species (La, Fe and O) to charge compensation, Bader charges of each atomic species are analyzed in terms of where q and q bulk LFO are Bader atomic charge at the surface and the corresponding Bader atomic charge from bulk LFO ; < > denotes the average over the

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57 surface layer. Atoms with negative have additional electrons relative to their bulk counterparts, whi le atoms with positive have fewer electrons. F ig ures 3 6 and 3 7 illustrate for the cations in the first and second layers, and for the anions in the zeroth and first layers of the relaxed LaO type and FeO 2 type surface terminations respectively For reduced surface terminations and stoichiometric LaO surface terminations where electrons are needed for charge compensation (see Fig ure 3 4 ( b ) ), the contribution of La and Fe cations is dominant. As indicated in Fig ure 3 6 the La and Fe cations in the relaxed stoichiometric LaO terminated surface are slightly nega tive relative to cations in the bulk. This is because there are fewer oxygen anions to accept electrons. In particular, the coordination number of the La in the first layer of the stoichiometric surface is 5, while the coordination of bulk La is 8; the cor responding surface and bulk values for Fe are both 6 When the surface is reduced, there are even fewer oxygen anions available, and so the charge on the surface cations becomes even more negative relative to the bulk, i.e., the surface La and Fe ions are less ionized than in the bulk. This is illustrated by a reduction in coordination to 3 for La and 5 for Fe for the most reduced surface. The surface La and Fe ions achieve a minimum value of about 0.6 e In other words, the reduced and stoichiometric LaO t ype surfaces achieve neutrality by countering positive with a redistribution of charge such that the surface is electron rich. For relaxed oxidized LaO surface terminations, there is an excess of oxygen ions at the surface and t heir relative Bader charges are more positive than those of oxygen anions in the bulk, as indicated in Fig ure 3 6 This is because the oxygen ions are

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58 undercoordinated to cations and thus have accepted fewer electrons than their bulk counterparts. In the c ase of the zeroth layer O low ions, they are slightly more undercoordinated than the first layer oxygen ions ( their coordination numbers are 2 and 3, respectively). Therefore their relative Bader charges are the same as, or more positive than, the oxygen i ons in the first layer. Stated another way, the oxidized LaO type surfaces achieve neutrality by countering negative with a redistribution of charge such that the surface is hole rich. These effects are similar to the proposed c harge compensation mechanism for the surface reconstruction (RT5) of the LaAlO 3 (001) su rface, which contains La vacanc i es in the La 5 O 5 (V La La 4 O 5 ) surface unit cell. 66 The charge distribution at the V La La 4 O 5 surface is found to be dominated by oxygen ions that are less negatively charged than bulk oxygen ions T he atomic ratio of the La and O atoms in the top layer, V La La 4 O 5 is analogous to the LaO+0.25O low surface te rmination under consideration here. In the case of the fully stoichiometric FeO 2 surface termination, the relative charges on the Fe and La ions are approximately zero relative to the bulk, as indicated in Fig ure 3 7 ; the surface as a whole is slightly po sitive relative to the bulk which can be e xplained by line diagram at Figure 3 4 ( b ) When the surface is reduced, however, there is a substantial difference in the responses of the Fe and La surface cations. The La cations are slightly negative as they are unable to donate all of their electrons to oxygen ions; they achieve a minimum relative charge of about 0. 2 e However the relative charges on the surface Fe cations decrease precipitously to achieve an average value of about 1.6 e when the surface is fu lly reduced. This difference is due to the multivalent

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59 nature of the Fe, which exists as Fe +3 in LFO but can convert to Fe +2 under these types of reducing conditions. As a result, the Fe cations at the reduced surfaces donate even fewer electrons to oxygen ions than bulk Fe. In other words, the reduced surfaces again achieve neutrality by countering positive with a redistribution of charge that leads to an electron rich surface. In the case of the oxidized FeO 2 type surface termi nations the charges on the first layer oxygen ions are only slightly positive relative to bulk oxygen ions because, even under fully oxidized conditions, they are sufficiently coordinated by cations (their coordination number is 4) The O low ions, however are more undercoordinated than in the case of the LaO type surface terminations ( their coordination number is 1 ), and thus have relative Bader charges that are high relative to bulk oxygen ions. Stated differently, charge compensation to counter negative is achieved that results in a hole rich surface. The O low relative charges for relaxed, oxidized FeO 2 type surface terminations are so much higher than in the case of their LaO type surface termination counterparts because the s urface region at LaO surface termination is more electron rich than that at FeO 2 surface termination Figure 3 4 ( b ) illustrated that the FeO 2 surface termination has fewer electrons than the LaO surface termination Therefore, when additional undercoordina ted O low ions are added to the FeO 2 terminated surface during oxidation, there are fewer electrons available to them from the cations. This discrepancy between surface electronic redistribution s in oxidized LaO and FeO 2 type surface terminations are sch ematically sketched in Figure 3 8.

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60 DOS of LaFeO 3 (010) s urfaces : In order to interpret the charge compensation of LaO type and FeO 2 type surface terminations in terms of their electronic structures, the DOS for these terminations are calculated and analyze d in F igures 3 9 and 3 10 respectively. Surface electronic redistribution of surface terminations with x 0.0 mainly comes from Fe atoms in surface region. In other words, the electronic redistribution of Fe atoms at surface region is crucial for charge compensation at surface terminations with x 0.0 In detail, LaO type surface terminations with x 0.0 exper ience metallization mainly by electronic redistribution of second layer Fe atoms Similarly, FeO 2 type surface terminations with x 0.0 also show metallization by top layer Fe atoms As x increase s the DOS of the La, Fe, and O atoms at surface region beco me similar to those in bulk LFO and the amount of hole s associated with the surface O atoms increase s However, O low atoms have a DOS that differs from that of the O atoms at surface region Also, as surface is oxidized, O low atoms change their forms from charged O atoms to charged O 2 molecule. In detail, f or 0.0< x 1.0, new O 2p states appear near the Fermi level ( E F ) More 2p states are occupied in the LaO type surface terminations relative to the FeO 2 type surface terminations. As surfaces are oxidized more ( x 1.5 ), i n both LaO and FeO 2 type surface terminations the O low atoms have electronic structure s close to charge O 2 molecules. For LaO+1.5O low and LaO+2.0O low surface terminations, the and states are filled from 6.0 eV below E F The FeO 2 +1.5O low and FeO 2 +2.0O low surface terminations have similar orb ital states with a band shift but show relatively small state filling of O low atoms. This indicates that O low

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61 atoms in LaO type surface terminations have more electrons than O low atoms in FeO 2 type surface terminations. In addition, when O low atoms are present, holes on surface O atoms are observed near the E F for both LaO type and FeO 2 type surface terminations, although they are most significant for LaO type surface terminations From the calculated DOS, it is observed that some O low atoms have intera ct ion with each other as surfaces are oxidized To investigate atomic structures of the O low atoms, oxidized surface structure are analyzed (see F ig ure 3 1 1 ). For both LaO type and FeO 2 type surface terminations, as the number of O low atoms is increased (o xidized) the O low atoms start interacting with each other. In particular, in the case of LaO+1.5O low surface termination, two of three O low atoms are interacting; their interatomic separation i s 2.12 The O low atoms in LaO+2.0 O low surface termination all interact with each other, as = 1.41 In case of the FeO 2 +1.5O low surface termination, two of three O low atoms interact with each other. However, the for inter acting O low atoms is 1.25 and is smaller than the LaO+1.5O low surface termination by 0.87 Similar to the LaO+2.0O low surface termination, all O low atoms in FeO 2 +2.0O low surface termination interact with each other and = 1.26 Considering that calculated bond length of O 2 superoxide ( ), and peroxide ( ) are 1.23 1.36 and 1.42 respectively (experimental bond length is 1.21, 1.33, and 1.49 for O 2 and ), it is expected that LaO type surface terminations should have interacting O low atoms which are similar to peroxide only at x =2.0: O low atoms for surface terminations with 0.0< x 1.5 have weak int eraction with each other and exist as

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62 ions. However, FeO 2 type surface terminations have interacting O low atoms from x =1.5 which are similar to neutral O 2 molecule form with weak charge transfer. These trends of charge transfer at the surface region reflec t the insulating character of LFO. Transition metal oxides can be categorized as either charge transfer (CT) or Mott Hubbard insulators depending on how their band gaps are defined. Although there is no strict distinction between the two types of insulator s, LFO is supposed to be close to the CT insulator The band gap of CT insulator is defined as an energy difference between the top of the filled p bands of ligand anions (in this case, O 2p ) and the bottom of unfilled upper Hubbard band ( in this case, Fe 3d ) ( F igure 3 1 ). On the contrary, the band gap of Mott Hubbard insulator is determined by splitting of occupied lower Hubbard b and and unoccupied upper Hubbard band 56 Consequently, holes are predominantly generated from the O 2 p states, and electrons prefer to populate Fe 3d states. 3 .5.2 Contributions of Subl ayers to Charge Compensation To investigate the contributions of sub surface layers (sublayers) to charge redistribution and compensation after surface relaxation the Bader charges of all of the surface terminations are analyzed in terms of and where and are the charges of the first layer without and with the electronic changes relative to the bulk respectively and i s the layer charge based on bulk LFO ( see Fi g ure 3 1 2 ). In particular, is the charge required to achieve charge compensation and is the change of the charge within the first

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63 layer after electronic rela xation. (%) can be defined as a measure of the contribution of the first layer to the electronic relaxation and redistribution relative to the bulk Figure 3 1 2 indicates that reduced and stoichiometric LaO type surface terminati ons have that is less than 40% while the reduced FeO 2 type surface terminations have that range from 50% to 70% over the same range of values This i ndicates that the contribu tion of sublayers to the optimized charges at the surface is more important for LaO type than FeO 2 type surface terminations. For oxidized surfaces the of LaO type surface terminations is consistently less than or equal to those of FeO 2 type surface terminations with the same Th is contribution of electrons from the sublayers to the LaO terminated surfaces lessens the extent to which the relative Bader charges of the O low ions are positive relative to bulk oxygen ions. However, based on F igure 3 4 ( b ) the contribution of the sublayers to charge compensation of the FeO 2 surface termination with O low atoms is expected to be larger in magnitude than that of the LaO surface termination with O low atoms. The r eason is because the surface region of the stoichiometric FeO 2 surface termination is electron poor relative to the bulk region after charge redistribution In summary the contribution of sublayers to charge compensation depends on surface stoichiometr y The c ontribution of sublayers to charge compensation is larger in reduced LaO type surface terminations than in reduced FeO 2 type surface terminations (with positive ) For oxidized surface s ( with negative ) the

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64 contribution of sublayers to charge compensation in LaO type surface terminations is greater than or equal to that in FeO 2 type surface terminations. 3 .5.3 Contribution of Atomic Relaxation to Charge Compensation Atomic relaxation is described b y the relative change in the interlayer spacing, Here, is the distance between the i th and j th layers along [010] direction ; = 1.972 is the interlayer spacing along the [01 0] in bulk LFO As illustrated i n F ig ure 3 1 3 the of LaO type surface terminations are larger than those of the FeO 2 type surface terminations. Considering that the contribution of the sublayers to the charge compensation of red uced LaO type surface terminations is larger than that of reduced FeO 2 type surface terminations it can be deduced that the contributions of the sublayers to charge compensation is facilitated by atomic relaxation. For oxidized surface terminations, the i nterpretation of the contribution of atomic relaxation to surface charge compensation is not straightforward ( see F i g ure 3 1 4 ). For surface terminations with = 0.5 e and 2.5 e the of LaO type surface t erminations are larger than those of FeO 2 type surface terminations with the same For surface terminations with = 1.5 e and 3.5 e of LaO type surface terminations are simila r to those of FeO 2 type surface terminations. Consequently, atomic relaxation is generally larger for LaO type surface terminations than for FeO 2 type surface terminations. The s trong atomic relaxation of LaO surface termination can be understood by a con sideration of atomic packing and bond breaking at the surface Based on Shannon radii 29 the LaO layer h as a low er packing ratio ( 0.66) than t he FeO 2 layer ( 0.84). Also,

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65 t he stoichiometric LaO terminated surface loses three bonds relative to the bulk when the surface is created (coordination number of La is reduced from 8 to 5) compared to the FeO 2 terminated surface, which loses one bond relat ive to the bulk (coordination number of Fe is reduced from 6 to 5) Therefore, I would expect the LaO terminated surface to relax more which is what the calculations yield. Nevertheless the contribution of atomic relaxation is not always large for the La O type surface terminations For instance, surfaces with = 1.5 e and 3.5 e ( x =1.0 and 2.0), the of LaO type and FeO 2 type surface terminations are similar. Figures 3 1 4 b and 3 1 4 d indicate that some s urfaces with a higher number of O low ions have less surface relaxation, which suggests that the O low configurations may be influencing the relaxation process. However, additional study is required to fully determine this relationship.

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66 Table 3 1 Structur al information of bulk L a F e O 3 Lattice constants() This study (GGA) GGA 54 Experimental 53 a 5.661 5.599 5.565 b 7.888 7.888 7.862 c 5.556 5.535 5.556 Table 3 2 Possible surface terminations o f L a F e O 3 (010) plane Type of surface terminations LaO type FeO 2 type FeO 2 2.0O FeO 2 1.5O LaO 1.0O FeO 2 1.0O LaO 0.5O FeO 2 0.5O LaO FeO 2 LaO+0.5O low FeO 2 +0.5O low LaO+1.0O low FeO 2 +1.0O low LaO+1.5O low FeO 2 +1.5O low LaO+2.0O low FeO 2 +2.0O low

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67 Table 3 3 Comparison between the sum of surface layer charges ( ) and half of bulk layer charge ( ). x indicates the stoi chiometry of the surface, and m 1 is the number of layers over whi ch the sum is carried out. All charges are determined by Bader analysis. For x>0.0, the layer number is counted from zero as layers consist ing of O low atoms are designated as zeroth layers. For x > 0.0, m =a means there are a+1 surface layers before center bulk layer. For x 0.0, m=a means there are a number of surface layers. LaO type surface terminations x m Difference (%) 1.0 6 0.413 0.413 < 0.1 0.5 6 0.420 0.418 0.425 0.0 5 0.439 0.439 0.034 0 .5 6 0.422 0.423 0.243 1.0 4 0.428 0.428 < 0.1 1.5 6 0.407 0.429 5.215 2.0 5 0.413 0.413 < 0.1 FeO 2 type surface terminations x m Difference (%) 2.0 6 0.450 0.449 < 0.1 1.5 6 0.443 0.448 1.166 1.0 5 0.411 0.412 < 0.1 0.5 6 0.431 0.426 1.227 0.0 5 0.421 0.422 0.207 0.5 6 0.418 0.417 0.210 1.0 5 0.429 0.428 0.123 1.5 6 0.420 0.420 < 0.1 2.0 6 0.420 0.420 0.024

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68 Fig u re 3 1 Electronic density of states of bulk L a F e O 3 Electrons in Fe atom have high spin configuration in which electrons are aligned in the up ward direction (majority) E F is set to zero (vertical dotted line). Fig ure 3 2 Total energy comparison of bulk L aFe O 3 (LFO) with different space groups relative to space group Pnma and magnetic ground state [G type antiferromagnetic state, G AF ] which is chosen to be the zero.

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69 Fig ure 3 3 Top view of (a) FeO 2 and (b) LaO surfaces. (c) and (d) show top vie w and side view of FeO 2 0.5 and FeO 2 +0.5O low surface terminations, respectively.

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70 Fig ure 3 4 (a) Line diagram for charges ( ) in the surface layers ; (b) Line diagram of of top layers F or (b) need electrons (holes means additional electrons (holes) are required to compensate top layer charge to =0.

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71 Fig ure 3 5 Spatial variations of the electric field E and of the electrostatic potential V in a macroscopic su rface models cut along a polar direction. (a) When all layers bear without charge compensation the electrostatic potential increases monotonically through the sample. (b) The layer charge of top layers is modified by : the electrostatic potential no longer show s monotonic increase through the surface model 63, 64

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72 Fig ure 3 6 Averaged relative Bader charge ( ) of surface atoms for relaxed LaO+ x O/O low surface terminations. Fig ure 3 7 Averaged relative Bader charge ( ) of surface atoms for relaxed FeO 2 + x O/O low surface terminations.

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73 Fig ure 3 8 Schematics of charge comp ensation mechanisms of LaO type surface terminations [(a) and (b)] and FeO 2 type surface terminations [(c) and (d)]. (b) and (d) include the low coordinated O low atoms ( x >0.0). It is assumed that surface electronic structures are mainly redistributed at fi rst and second layers. Band shift is somewhat exaggerated.

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74 Fig ure 3 9 Electronic density of states for LaO+ x O low terminated surfaces ( x = 1.0, 0.5, 0.0, 0.5, 1.0, 1.5, and 2.0). The Fermi energy is set to zero (vertical dotte d line)

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75 Fig ure 3 10 Electronic density of states for FeO 2 + x O low terminated surfaces ( x = 2.0, 1.5, 1.0, 0.5, 0.0, 0.5, 1.0, 1.5, and 2.0). The Fermi energy ( E F ) is set to zero (vertical dotte d line)

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76 Fig ure 3 1 1 Schematic descriptions of (a) LaO+1.5O low (b) LaO+2.0O low (c) FeO 2 +1.5O low and (d) FeO 2 +2.0O low surface terminations Fig ure 3 1 2 Percentage of change in charges in the first layer before and after electronic redistribution and layer charge based on bulk LaFeO 3 All charges are determined by Bade r analysis. If is 100%, charge compensation is achieved only by the first layer.

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77 Fig ure 3 1 3 Relative interlayer spacing of LaO type and FeO 2 type surface terminations with respect to bulk interlayer spacing on (010) directi on ( ) where is interlayer spacing between i th and j th layers and is interlayer spacing of bulk L a F e O 3 1.972 ). Positive (negative) relaxation means expansion (contraction) of interlayer spacing. (a) Surface terminations with =2.5 e (b) Surface terminations with =1.5 e (c) Surface terminations with =0.5 e X axis shows layer numbers used for calculatin g For example, y value for x=12 is for first and second layers.

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78 Fig ure 3 1 4 Relative interlayer spacing of LaO type and FeO 2 type surface terminations with respect to bulk interlayer spacing on (0 10) direction Same setting from Fig ure 2 13 is applied. (a) Surface terminations with = 0.5 e (b) Surface terminations with = 1.5 e (c) Surface terminations with = 2.5 e (d) Surfa ce terminations with = 3.5 e

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79 CHAPTER 4 SURFACE PHASE DIAGRA M OF LANTHANUM FERRI TE (010) SURFACE 4 .1 Introduction In the previous chapter stabilization mechanisms of LFO (010) surface as a function of surface oxygen stoichi ometr y were proposed. However, these stabilization mechanisms are solely related to surfac e electronic redistribution and the contribution of environmental param eters, p(O 2 ) and temperature( T ) to surface stability must be considered to determine the stabl e surface structure of LFO (010) surface as a function of the environmental parameters. As SOFCs operates at oxygen partial pressures p(O 2 ) atm and T varying from 500C to 1000C, it is probable that the surface structure of the SOFC cathode is diff erent from its stoichiometric one. In particular due to the ambient condition s in SOFCs change in surface oxygen stoichiometry (or surface oxygen coverage) at the cathode is expected. Therefore investigating the contribution of environmental parameters t o the surface structure of SOFC cathode materials is imperative to understand stable surface structure s that are directly related to oxygen reduction reactions. 50 Despite the merit of electronic structure calculations to directly compare material stabilities, 67, 68 re latively few studies on SOFC cathode materials ha ve been devoted to surface stabili ty as a function of oxygen stoichiometry For instance, Mastrikov et al. showed that the MnO 2 termination with adsorbed oxygen is the most stable form of (010) surface for c ubic LaMnO 3 at p(O 2 ) atm and T= 1200 K but o nly MnO 2 and LaO terminated surfaces with and without adsorbed oxygen atoms we re considered 69 In addition, the stabilization mechanisms due to electron redistribution within LFO (010)

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80 surface s with different surface oxygen stoichiometr ies we re explored in Chapter 3 In particular, it wa s predicted that oxygen reduction should occur more favorably on the stoichiometric FeO 2 surface than on its stoichiometric counterpart with LaO type terminations due to the multivalent character of the Fe ion. It wa s also predicted that the oxidation of the LaO surface should b e energetically more favorable than oxidation of the FeO 2 surface; however, no attempt was made to correlate these stabilization mechanisms to the environmental conditions. The present chapter thus attempts to extensively investigate sta ble surface oxyge n stoichiometr ies of LFO at SOFCs operating conditions using a combin ation of DFT and thermodynamics. There are two broad aims for this chapter The first is to provide a surface phase diagram of the LFO (010) surface I choose LFO (010) surface as our mod el surface because th is crystallographic plane ha s been examined experimentally and theoretically for various other perovskite systems including BaTiO 3 SrTiO 3 and LaAlO 3 and BaZrO 3 46, 47, 66, 70 73 The second aim is to gain insight into the variation in the surface stability of the LFO (010) surface as a function of environmental parameters. The surface stability is described with respect to two contributions: surface electronic redistribution and the chemical pote ntials of off stoichiometric surface atoms. By comparing the two terms, I will show the significance of the chemical potential contribution in determining the stability of various surface termination s I dentical surface slab models and corresponding nomenc lature as in Chapter 3 are used. 4 .2 Secondary P hases of LFO There are seven different secondary phases of LFO : La, La 2 O 3 Fe, FeO, Fe 2 O 3 and Fe 3 O 4 They have different structural and magnetic ground states as a function of T

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81 i) La: La has three different structural ground states at atmospheric pressure: 74 double HCP (P6 3 /mm c) below 609 K ; FCC ( ) from 609 K to 1138 K ; BCC ( ) from 1138 K to melting temperature, 1191 K ii) La 2 O 3 : Even though La exhibits two oxidation states, + 2 and + 3 the latter being much more stable. So, La 2 O 3 is the only available oxide derived from La and O. It has double HCP structure ( P6 3 /mmc ). iii) Fe: Fe has two different phases at atmospheric pressure: Fe (BCC, ) below 1173 K Fe (FCC, ) from 1173 K to 1665 K and Fe (BCC) from 1665 K to melting temperature. For magnetic ground states, Fe is FM below T C 1043 K and becomes PM above the temperatu re. iv) FeO : FeO (Wustite) has melting temperature of 1650 K B e low T N 470 K Wustite has AFM magnetic state with Rhombohedral structure ( ). After AFM to PM transition at the N el temperature, Wustite has cubic v) Fe 2 O 3 : There are four different fo r m s of Fe 2 O 3 Fe 2 O 3 (Hematite) Fe 2 O 3, Fe 2 O 3 (Maghemite) and Fe 2 O 3 However, only Fe 2 O 3 is stable and others are metastable. Melting temperature of Fe 2 O 3 is 1838 K. I t has Rhombohedral structure ( ). M agnetic states of the Fe 2 O 3 are somewhat complicated: AFM below Morin transition temperature ~260 K and weak FM or canted (non collinear) AFM between the Morin temperature and T N 950 K

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82 vi) Fe 3 O 4 : Below so called Verwey tra nsition temperature 120 K Fe 3 O 4 (Magnetite) is orthorhombic ( Pmca ). Above the Verwey transition temperature, Magnetite has cubic structure ( ). It s T C for transition from FIM to PM is 858 K The structural and magnetic informati on of the secondary phases of bulk LFO are summarized at Figure 4 1 The total energies of the phases provide boundary conditions in which the range of and are restricted : they are calculated using th eir respective structural and magnetic ground states at 0 K Their atomic structures optimized by DFT calculations are depicted in Figure 4 2. Also, Table 4 1 shows calculated formation enthalpies of bulk LFO and its secondary phases. Additionally, as con densed O 2 does not exist under SOFC processing conditions (gaseous O 2 condenses to liquid at 90.2 K ), the oxygen rich condition is defined as the point beyond which gaseous O 2 starts to exist as condensed O 2 on the surface ; this provides an upper limit of : ( 4 1) Due to the periodic boundary conditions that are applied in our DFT calculations, 75 the distance between an O 2 molecule and its periodic image is 20 which corresponds to an O 2 density that is higher than that of liquid oxygen. Consequently, in Equation ( 4 1) c orresponds to the total energy of condensed O 2 molecules at 0 K without the vibrational contribution (zero point energy). Even though the calculated binding energy of O 2 ( 5.66 eV ) is consistent with the results of other DFT calculations with the GGA PBE fu nctional 76 it is higher than the experimental binding energy of 5.23 eV 77 This overestimation in the binding energy is a

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83 well known weakness of DFT. However, this intrinsic error does not affect our results as is well above the range in which I am interested. N onetheless calculating accurate total energ ies of O 2 can be crucial for applications at high p(O 2 ) and/or low T 4 .3 Surface Phase Diagram of LFO (010) Surfaces The s urface phase diagram of the LFO (010) surfaces as a function of La O and surface Gib bs free energy ( ) is shown at Figure 4 3. Even though surface phase diagram is widely used term to describe a diagram showing relative stability of various surface terminations 78 82 this does not imply that surfac e in the diagram is treated as independent thermodynamic phase Therefore, the surface phase diagram does not follow Gibbs phase rule. However, due to its complexity, the surface phase diagram is generally displayed as a top view of the diagram 46, 83 Figure 4 4 ( a ) which is the to p view of Fig ure 4 3 indicates the most stable stru cture of the LFO (010) surface as a function of La and O Figure 4 4 ( b ) illustrates O as a function of p(O 2 ) and T based on the ideal gas approximation. The stability region of bulk LFO which is indicated by the hashed region in Figure 4 4 ( a ) La O for surf a ce calculations. Outside of th is range, spontaneous precipitation of secondary phases including La, Fe, and their oxides is thermodynamically favorable. Hence, only the surface phase diagram within the stability region is physically meaningful here. In thi s work, p(O 2 ) is fixed as 0.21 atm (red solid line in Fig ure 4 4 ( b ) ), and three different temperatures [ 773 K C ), 1073 K C ) and 1223 K C )] are chosen to represent different aspects of SOFC operating conditions. Specifically, the temperatures 773 K and 1073 K represent the minimum and maximum operating

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84 temperatures of intermediate temperature S OFCs, respectively, while 1223 K is typical of high temperature SOFCs. It should be noted that even though I focus here on SOFC operating conditions, this surface phase diagram can be used to predict the most stable surface at any p(O 2 ) and T Figure 4 4 ( a ) indicates that FeO 2 type surfaces are most prevalent as the stability region of bulk LFO significantly overlaps with th ese termination s However, LaO type surfaces are also present within the stability region. Thus, the stabilities of both FeO 2 and LaO type surfaces are considered One of the important results from the diagram is the temperature associated with several stoichiometric transitions The transition temperatures of surface oxygen stoichiometry for LaO and FeO 2 type surfaces are summarized in the form of a line diagram in Figure 4 5 N o reduced LaO type surfaces are predicted to have preferred stability within the surface phase diagram. Thus, for both FeO 2 and LaO type surfaces, reduced terminations are not the most stable under SOFC operat ing condition s, which is not surprising given the relatively high oxygen partial pressures and temperatures. It should be pointed out that even though reduced FeO 2 surfaces are predicted to be the most stable above 2370 K these transitions occur out side t he stability region of bulk LFO ( the hashed region in Figure 4 4 ( a ) ) and so are not predicted to be relevant To summarize, under SOFC operating conditions the most stable surface structure changes from FeO 2 +1.5O low (at 773 K ) to stoichiometric FeO 2 (at 1073 K and 1223 K ). For LaO type surfaces, the oxidized termination is always the most stable [ LaO+1.5O low at 773 K to LaO+0.5O low at 1073 K and 1223 K ]

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85 Figure 4 6 illustrates the surface Gibbs free energies of FeO 2 and LaO type surfaces at fixed O with La and the surface Gibbs free energy, as variables. Each surface phase diagram can be considered to be a cross section along each parallel dotted line in Figure 4 4 Figure 4 6 indicates that at 773 K the stoichiometric FeO 2 and oxidized FeO 2 ty pe surfaces have similar stabilities and congregate in a 0.1 eV range in the region of lower surface Gibbs free energies As this range is smaller than the vibrational contribution both stoichiometric and oxidized FeO 2 type terminations can co exist. The energy difference between the most stable FeO 2 +1.5O low and reduced FeO 2 type surfaces is large (>2.48 eV ), which indicates that reduced FeO 2 type surfaces will be unfavorable at 773 K For LaO type surfaces, oxidized terminations have similar stabilities: the width of their surface Gibbs free energies is 0.24 eV The energy difference between the most stable LaO+1.5O low and stoichiometric LaO is 1.73 eV ; the difference between the LaO+1.5O low and reduced LaO is more than 6.34 eV Consequently, based on the energy range accessible by the vibrational contribution, only oxidized LaO type surfaces are predicted to be present at 773 K A t 1073 K the stoichiometric FeO 2 surface becomes the most stable. In addition the slightly oxidized surface (FeO 2 +0.5O low ) ma y be present, as it is only 0.3 eV higher in energy, which is on the order of the uncertainty from the neglect of vibrational contribution s In contrast, other FeO 2 type surfaces including highly oxidized and reduced versions are predicted to be absent Among the LaO type surfaces, slightly oxidized LaO+0.5O low is the most stable at this temperature As the energy difference between the most stable LaO+0.5O low and LaO+1.0O low is similar to the uncertainty

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86 range, the latter is also predicted to be present Other LaO type surfaces will not be present as their energ ies are much larger than th ose of the most favorable surface s At 1223 K because there is a significant energy difference between the most stable FeO 2 and other less stable FeO 2 type surfaces (> 0.5 7 eV ), only the FeO 2 surfa ce should be present. Even at th is highest temperature, the reduction of the stoichiometric FeO 2 is not predicted to occur, as the energy difference between the stoichiometric and reduced surfaces is more than 1.89 eV Among th e LaO type surfaces, the LaO+0.5O low surface should be observed because it is significantly more stable than other LaO type terminations. For both LaO and FeO 2 type terminations the energies of the reduced surfaces approach those of the oxidized surface s as the temper ature is increased. However, because of the relatively high oxygen partial pressures, th is stabilization is not sufficient to make the reduced surfaces the most stable. Thus surface oxygen vacanc ies are predicted to not form on either LaO or FeO 2 type surfaces within th is temperature range at this oxygen partial pressure The trends from Fig ures 4 4 and 4 6 are sch ematically summarized in Fig ure 4 7. Within the stability region of bulk LFO (grey regions in Figure 4 6 ), the surface Gibbs fr ee energy difference between the most stable LaO and FeO 2 type surfaces varies by La For example, at 773 K the energy difference between the most stable LaO type (LaO+1.5O low ) and the most stable FeO 2 type (FeO 2 +1.5O low ) surfaces is 0.17 eV at La = 8.40 eV which is a maximum. The difference increases to 1.25 eV at a minimum La of 8.94 eV This means that the increases in La decrease the energy difference between the most stable LaO and FeO 2 type terminations. At 773 K it is probable that both LaO+1.5O low and FeO 2 +1.5O low can co exist since the vibrational

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87 contribution ( 0 .3 eV ) is larger than the energy difference of 0.17 eV However, as the temperature increases (Figures 4 6 (b) and (c)), the energy difference between LaO and FeO 2 type surfaces also increases. Specifically, the difference increase s from 0.17 eV at 773 K to 0.28 eV at 1073 K and to 0.32 eV at 1223 K This implies that FeO 2 type terminations become more stable than the LaO type terminations as the temperature increases. 4 .4 Contributions of E lectronic R edistribution and C hemical P otential in S urface S tabil ity The s urface Gibbs free energy is determined by both electronic and environmental contributions. For example, let us assume that there is one surface slab in which the numbers of La, Fe, and O are N La N Fe and N O respectively. Their chemical potential s are and with their counterparts in bulk LFO being Using Eq uations 2 15 and 2 16 and in Eq uation 2 12 can be re written as follows: ( 4 3 ) ( 4 4 ) Here, is the number of stoichiometric LFO units in th e slab. By these expressions Eq uation ( 2 7 ) can be written as: ( 4 5 )

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88 Consequently, the surface Gibbs free energy of slab i is described by a) the difference between the chemical potentials of the slab and those of the bul k, and b) the number of off stoichiometric atoms. In a), electronic redistribution in the surface make the chemical potentials of the slab different from the bulk, and t he chemical potentials of the bulk (specifically, that of oxygen) is controlled by envi ronmental parameters of gas phase which is in equilibrium with the bulk. These are defined as the electronic contribution and the environmental contribution respectively for the rest of th is discussion. To quantify these two contributions, the total ener gies of bulk La ( E La ), bulk Fe ( E Fe ), and the O 2 molecule ( ) at 0 K without vibrational contributions are used as the reference energies of La, Fe, and O respectively. Even though their chemical potentials in the bulk LFO are mor e suitable, they cannot be calculated within the DFT scheme. Electronic contribution: Since describes the Gibbs free energy of off stoichiometric atoms in slab i in Eq uation ( 2 7 ) quantif ies the energ y change of off stoichiometric atoms with reference to total energies of corresponding atomic component. Figure 4 8 illustrates the electronic contribution for all the LaO and FeO 2 type surfaces. As the electronic contribution terms of stoichiometric s u rfaces are set to zero in the F igure 4 8 the value s along the y axis represent the electronic contribution due to the change in stoichiometry. The LaO and FeO 2 type surfaces are plotted by their oxygen stoichiometries x to compare their preferences in o xidation and reduction. In the

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89 case of reduced surfaces ( x <0.0), the stoichiometric FeO 2 ha s high preference for oxygen reduction with respect to the LaO However, i n the case of oxidized surfaces ( x >0.0), the LaO is preferred over the FeO 2 These results directly verify the stabilization mechanisms of LFO (010) surfaces proposed in previous chapter. This will be discussed in detail at Conclusion chapter, Chapter7 84 Environmental contribution: As and are th e chemical potentials of La and O in bulk LFO which are in equilibrium with gas phase (see Eq uation 2 5 ), the last term in Eq uation 2 6 describes the change in chemical potentials of off stoichiometric atoms as a function of envi ronmental parameters with respect to their reference energies. A combination of the e lectronic and environmental contributions at 773 K and 1223 K are compared i n Fig ure 4 9 At 773 K for LaO type terminations (Figure 4 9 ( a ) ), the surface Gibbs free ener gies of the oxidized surfaces are almost identical to each other as the change in the electronic contribution is similar to that of the environmental contribution. This is the same trend observed for FeO 2 type surfaces (Figure 4 9 ( b ) ). As the temperature increase s to 1223 K the environmental contribution becomes larger than that at 773 K Therefore, the overall stability of certain surface is mostly guided by the environmental contribution at high temperatures. 4 .5 Discussion 4 .5.1 Relation to E xperiment a l Results It is interesting to see to what extent the surface phase diagrams can be validated (Figures 4 4 to 4 7 ) against results from temperature programmed desorption (TPD) and temperature programmed reduction (TPR) experiments. 85 90

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90 As TPD and TPR results represent polycrystalline LFO su rface which consists of various planes, it is possible that there is a discrepancy between the predictions from the surface phase diagrams and the experiments. However, as the LF O (010) plane has relatively high stability than other planes due to its weak surface polarity (see Section 3 3), it is probable that the significant portion of polycrystalline LFO surface has (010) plane Consequently, qualitative comparison between the s urface phase diagrams and the TPD and TPR results is available. In addition, the kinetic factors in the TPD and TPR results were not considered during the development of our surface phase diagrams However, it is reasonable to expect that the desorption p eak s of so called and oxygen atoms which are ascribed to adsorbed (O low ), and lattice oxygen atoms respectively will correlate with the change in surface oxygen stoichiometry in the diagrams 39, 88, 91 Here, lattice oxygen implies oxygen atom s in the first or subsurface layers. Therefore, i compare the temperatures where the surface oxygen stoichiometry is predicted to change with oxygen desorption trend s reported from experiment Even though most experiment s are performed under high vacuu m condition s (10 6 to 10 12 atm ), the p(O 2 ) in each experiment is slightly different. Because the T corresponding to a specific O varies with p(O 2 ) predicted transition temperatures for LaO+1.5O low LaO+0.5O low FeO 2 +2.0O low FeO 2 +1.5O low and FeO 2 +1.5O low FeO 2 which are available stoichiometric transition s within the stability of bulk LFO have a range of 377.5 601.5 K under high vacuum condition s ( see Figure 4 10 ).

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91 Considering lattice oxygen atoms Figure 4 4 predicts that no surface oxygen vacancies should form until 1223 K Also, reduced surfaces can not be the most favorable within the stability of bulk LFO because there is no overlap between the m and the bulk LFO regio n This is consistent with the TPD and TPR result s, most of which show no desorption peak for oxygen up to the highest temperatures probed ( 1273 K ) 85 90, 92 Instead of surface oxygen vacanc ies secondary phases of La 2 O 3 and Fe were observed at 850 1250 K which is also consistent with our predictions. 87, 92 For adsorbed oxygen atoms (O low ) the TPD and TPR experiments on LFO generally show two different trends based on the calcination temperature of the ceramic. In the case of LFO calcined below 923 K desorption f or oxygen was clearly observed. In particular Tascn et al. observed oxygen desorption at 600 850 K at a calcination temperature of 923 K 87 Similarly, Kim et al detected oxygen desorption from 573 K ( there was a significant peak around 973 K ) after calcination at 973 K 85 Finally, Wachowski et al. reported two desorption steps at 520 580 K and 720 870 K respectively when LFO was calcined at 773 K 92 In contrast, a t a high calcination temperature of 1123 K most of the experiments reported no desorption peak of oxygen 88 90 even though weak peak s below 773 K were observed by Nitadori et al. 86 For convenience, in the following discussion temperatures of 973 K or less will be defined as low calcination temperature and those at 1123 K or higher will be defined as high calcination te mperature The p redicted transition temperature range of 377 601 K from the surface phase diagram illustrated in Figure 4 10 agrees well with T P D results for samples processed at low calcination temperatures 85, 87 92 The effect of calcination temperature on the

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92 desorption trend of adsorbed oxygen O low can be interpreted as the relative stability of LaO to FeO 2 type surfaces as a function of temperature. From Figure 4 6 it is predicted that FeO 2 type surface s become more stable than LaO type termination s as the temperature increases. Considering that the degree of oxidation is weak er at FeO 2 type terminations than at LaO type terminations ( as indicated in Figure 4 7 ), a t high calcination temperature s LFO surfac e s that contain substantial portion s of FeO 2 type terminations will experience little desorption of O low However, f o r low calcination temperature s the fraction of LaO type surfaces will increase and a relatively strong desorption peak will be detected. T his can also explain the experimental result that the oxidative non stoichiometry of LFO decreases with increas ing calcination temperature. 93 4 .5.2 Comparison with LaMnO 3 S urface Recently, Mastrikov et al. 69 calculated surface pha se diagram s of orthorhombic and cubic LaMnO 3 (010) using combined approach of DFT and thermodynamics with a similar approach to tha t described in Chapter 3 They developed surface phase diagram with six different (010) and (110) surfaces [ MnO 2 MnO 2 +0.125O low LaO, an d LaO +0.125O low surfaces for (010); and O and O 2 terminations for (110)]. The surface Gibbs free energies are compared at 1200 K with p(O 2 ) = 0.21 atm which is similar to the conditions analyzed in Figure 4 6 Even though stoichiometric and sl ightly oxidized (010) surfaces were only considered in their work it is probable that consideration of highly oxidized surface terminations cannot modify the surface phase diagram of LaMnO 3 at high SOFC operating temperatures ( 1073 K an d 1223 K in this st udy). This is because i t is experimentally reported by TPD and TPR measurements that desorption of oxygen on LaMnO 3 surfaces takes place at a low temperature ( 500 K ), which is similar to the

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93 LFO 88, 90 This implies that highly oxidized LaMnO 3 (010) surfaces may be less favorable than slightly oxidized and stoic hiometric terminations at high SOFC operating temperatures. However, it must be pointed out that highly oxidized surfaces in LaMnO 3 can be stable at low temperature ranges (for this case, 773 K ), which is also the case for the LFO (see Figure 4 5 ). For ort horhombic and cubic LaMnO 3 (010), t he difference s in surface Gibbs free energies between the most stable MnO 2 type surface s and LaO type counterpart are about 0. 75 eV and 1. 25 eV respectively at 1200 K at maximum La : this values are varied La By considering the magnitude of the vibrational contribution (~0.3 eV ), the MnO 2 type surfaces can be preferred at this temperature. However, t he difference is much smaller in LFO (010) 0.28 eV, which is similar to the v ibrational contribution to the energy Therefore, at 1200 K both LaO and FeO 2 type terminations can be possible for LFO Previous studies on LaMnO 3 and LFO (010) surfaces mainly focus ed on MnO 2 and FeO 2 type surfaces with little consideration on LaO type terminations. It is probable that th e energy difference between the most stable MnO 2 type and LaO type surfaces decreases as temperature is lowered. However, the effect of temperature on the energy difference is not expected to be significant. This is because temperature only affects the O and there are only small number of off stoichiometric oxygen atoms in the LaMnO 3 (010) surfaces.

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94 Table 4 1. C alculated formation enthalpy of considered secondary phases of bulk LaFeO 3 Formation enthalpy ( eV ) [De viation from experime n tal data (%) ] LaFeO 3 13.91 [0.52] La 2 O 3 18.95 [1.90] FeO 2.37 [ 15.86] Fe 2 O 3 7.61 [ 11.11] Fe 3 O 4 11.33 [ 2.46] T=298.15 K

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95 Figure 4 1. Structural and magnetic ground states of secondary phases of bulk L a F e O 3 (T<1800 K ).

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96 Figure 4 2. Calculated super cells of (a) La, (b) La 2 O 3 (c) Fe, (d) FeO, (e) Fe 2 O 3 and (f) Fe 3 O 4

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97 Figure 4 3. Side view of surface phase diagram of LFO (010) surfaces. White area describes positive surface Gibbs free energy,

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98 Fig ure 4 4. (a) Surface phase diagram, and (b) relative chemical potential of oxygen ( O ) as a function of T ( K ) and p(O 2 ) ( atm ). The labels on the lines in (b) represent p(O 2 ) in atm unit. The numbers in circles indicate solid lines whe re metals or their oxides start to precipitate: (1) La, (2) La 2 O 3 (3) Fe, (4) FeO, (5) Fe 2 O 3 and (6) Fe 3 O 4

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99 Fig ure 4 5. Line diagrams that show the most stable surface oxygen stoichiometry of (a) FeO 2 and (b) LaO type surface terminations as a function of temperature. Solid parallel lines indicate the most stable oxygen stoichiometry and dotted vertical lines indicate transition temperatures.

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100 Fig ure 4 6. Surface phase diagram at (a) 773 K (b) 1073 K and (c) 1223 K with p(O 2 ) atm The gray highlighted regions in (a), (b) and (c) represents stability region of bulk L a F e O 3 The right hand side figures for (a), (b) and (c) illustrate the magnified view of the rectangular sections shown in the main figures.

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101 Fi gure 4 7. S ketch of surface stabilities of (a) FeO 2 type and (b) LaO type surface s at 773 K 1073 K and 1223 K Each surface symbol does not describe actual configurations of oxygen species on the surface.

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102 Fig ure 4 8. Relative electronic contribution terms of LaO and FeO 2 type surface s as a function of surface oxygen stoichiometry. The electronic contribution terms of stoichiometric LaO and FeO 2 surfaces are set to zero.

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103 Fi gure 4 9. Surface Gibbs free energies described by electronic and environmental contributions at 773 K and 1223 K with p (O 2 ) atm Red and blue arrows indicate most stable surface oxygen stoichiometry at 773 K and 1223 K respectively.

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104 Fig ure 4 10. (a) Surface phase diagram, and (b) r elative chemical potential of oxygen ( O ) as a function of T ( K ) and p(O 2 ) ( atm ) The labels on the lines in (b) represent p(O 2 ) of high vacuum condition (10 6 10 12 atm ) Dot ted lines describe the range of transition temperatures for surface oxygen sto ichiometries within the stability region of bulk L a F e O 3 (hashed region).

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105 Figure 4 11. Temperature programmed desorption of O 2 for L a F e O 3 calcined at (a) 973 K and (b) 1123 K (reproduced from the references 85, 9 0 )

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106 CHAPTER 5 OXYGE N REDUCTION REACTIONS ON LANTANUM FERRITE (010) SURFACES 5.1 Introduction This chapter aims to gain insight into ORR on the LFO (010) surface and to determine the associated rate limiting step s T here have been many DFT studies o f thes e reactions o n LaMnO 3 based materials 69, 83, 94 97 but they have mainly focus ed on ideal stoichiometric surfaces. In this work, the objective is to elucidate the contribution of surface oxygen stoichiometry to ORR by comparing the details of these reactions o n stoichiometrically different LFO (010) surfaces. Additionally, the concept of an entropy barrier is employed to offer a new interpretation of rate determining step of these reactions on the LFO surface 19 In particular, based on the developed surface phase diagram shown in Chapter 5, the ORR of FeO 2 and FeO 2 +0.5O low surface terminations are examined : at fixed p(O 2 ) 0.21 atm the stoichiometric FeO 2 surface is stable above 800 K, while the FeO 2 +0.5O low surface represents oxidized surfaces which are stable below 800 K ( see Figure 4 5). Considering that the operating temperature of IT SOFCs is about T =773 1073 K the F eO 2 +0.5O low and FeO 2 represent stable surfaces at the minimum and maximum of the associated temperature range fo r these devices 5.2 Computational D etails The DFT calculations utilized in this work uses identical setting s as those discussed in Chapters 4 and 5 The exception is that here a 1 x 1 x1 k point Monkhorst Pack mesh ( point) is used for the integrations over the Bri llouin zone of the LFO surface. The ionic relaxation is performed until the Hellmann Feynman forc e on each atom is less than 0.0 3 eV /

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107 The a ctivation energy barriers ( E a ) for reactions are calculated using the nudged elastic band (NEB) method 98 99 100 This is a method for finding saddle point and minimum energy path between reactant and product (both need to be determined in advance) (Figure 5 1). In particular after linearly interpolating a set of images between initial and final states ( as a guess at the minimum energy path) each image is optimized to find the lowest energy possible w hile maintaining equal spacing to neighboring images: this constraint is applied by adding spring forces along the band b etween the images. Specifically, e ach image corresponds to a specific geometry of the atoms on their way from the initial to the final state This image can be understood as a snapshot along the reaction path Thus, o nce the ene rg y of this string of images has been minimized, the true minimum energy path is calculate d. 5.3 Asymmetric S urface S labs Unlike previous chapters (Chapters 4 and 5), asymmetric surface slabs with nine atomic layers with a ( 2 x 2 ) surface unit cell are employed. The b ottom four layers are fixed to mimic bulk LFO and five atomic layers above the bulk region are fully relaxed (Figure 5 2). To verify the validity of these asymmetric slab models atomic displacements in the direction normal to the surfaces of asymmetric slabs ( containing 11 or 9 atomic layers) are compared with th ose of symmetric slab s of 15 atomic layers (Figure 5 3) The symmetric slab contain ed seven active layers on either side of a fixed center layer, while the asymmetric 11 layer slab h as seven active layers and four f ixed layers ; thus, both slabs have a comparable number of active layers Here, all tested slabs have FeO 2 terminations. The m aximum deviation between atomic displacements of the symmetric and asymmetric slabs is predicted to 0.07 and it is clear that the asymmetric slab with

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108 9 atomic layers describes the surface atomic structure accurately in comparison with its symmetric counterpart. Additionally, the adsorption energies of O and O 2 on the symmetric and asymmetric slabs are compared in Table 5 1. The se energies on the asymmetric 9 layer slab are higher by 0.05 eV and 0.1 eV respectively in comparison with the values on the symmetric 15 layer slab Considering that the average error associated with the use of the GGA functional is o n the order of 0.3 eV / molecule for chemical reaction s 101 it is deemed that the asymmetric slab with 9 atomic layers adequately describe s the reaction of oxygen species on these surfaces. 5.4 Elementary S teps for ORRs The pathways leading from gaseous oxygen molecule s to o xygen ions in the bulk region of the LFO system can consist of several reaction steps that can be combined in different combinations to describe various ORRs 69 Therefore it is possible to focus o n and compare a few elementary reaction steps to predict the energetically favorable reaction paths and corresponding rate determining steps of the FeO 2 and FeO 2 +0.5O low surfaces. These elementary steps can be categorized as follows: (a) Adsorption: adsorption of O 2 (b) Dissociation: dissociation of the adsorbed O 2 (c) Surface diffusion: Diffusion of the dissoc iated O ion or oxygen vacancy with in the first atomic layer (d) Incorporation: Incorporation of the O ion into oxygen vacancy in first layer (e) Bulk diffusion: Transp ort of O ion in first layer to oxygen vacancy in second layer. Figures 5 4 and 5 5 show the considered reaction steps and corresponding energy profiles for the FeO 2 +0.5O low and FeO 2 surfaces, respectively.

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109 5.4.1 Elementary R eaction S teps on FeO 2 +0.5O low S urface T ermination (a) Adsorption: In the case of the FeO 2 +0.5O low surface (Figure 5 4), adsorption of an O 2 molecule is tested in both end on (oblique to the surface) and side on (parallel to the surface) configurations. The adsorption energy ( E ad ) of O 2 is calculated using : (6 1) T he E ad of end on O 2 is predicted to be 1.49 eV which is lower than that of side on O 2 by 0.33 eV As t his is within the uncertainty limits of DFT calculations it can be expected that the sta bility of the end on O 2 is comparable to that of the side on O 2 (b) Dissociation: The dissociated O ion can be moved to i) a Fe site that is covered by an other O low ion or ii) to clean a Fe site. The latter has E a = 1.10 eV which is lower than the former by 1.94 eV This implies that adsorbed O 2 prefers to dissociate on a clean Fe site. (c) Surface diffusion: To incorporate into surface oxygen vacancy sites, the dissociated O ion must have contact with a surface oxygen vacancy This can be made either by i) hopping of the O ion or ii) transport of surface oxygen vacancy The E a of the second option is 0.45 eV which is significantly lower than that of the first by 2.29 eV Consequently, it is predicted that surface diffusion depends predominantly on the tra nsport of surface oxygen vacancy Similar DFT result on LaMnO 3 surface has been reported by other s 83 (d) Incorporation: Pairing of O species with oxygen vacancy is followed by incorporation of O species into a n oxygen vacancy site. The E a of incorporating O is lower than the incorporation of O 2 by 0.19 eV

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110 (e) Bulk diffusion: The i ncorporated O ion must diffuse into the bulk LFO region to have ionic conduction. The E a of bulk diffusion is 0.43 eV and energy level of the product is hi gher than that of reactant. This implies that oxygen vacancy is more stable in the first atomic layer than in the second atomic layer. Here, the topmost FeO 2 layer in contact with the vacuum region is designated as the first layer. 5.4.2 Elementary R eacti on S teps on S toichiometric FeO 2 S urface T ermination (a) Adsorption: Similar to FeO 2 +0.5O low surface, a dsorbed O 2 on the FeO 2 surface is stable with end on configuration with E a = 1.48 eV ( see Figure 5 5) (b) Dissociation: Even though O 2 is stable in its molecular form, the dissociation of O 2 has no activation energy Interestingly, the O 2 is stabilized by the dissociatio n: there is no stabilization by dissociation at FeO 2 +0.5O low surface. (c) Surface diffusion: The h oping of an O ion from one Fe site to anothe r has E a =1.62 eV. This barrier is higher than that for surface diffusion of oxygen vacancy in first layer by 0.81 eV As this difference in E a is smaller than that on the FeO 2 +0.5O low surface ( E a =2.29 eV ), the contribution of O ion in surface diffus ion is predicted to be substantially high er on the FeO 2 surface than on the FeO 2 +0.5O low surface. Additionally, even though surface diffusion of oxygen vacancy is more energetically favorable than diffusion of surface O ion on both the FeO 2 and FeO 2 +0.5O lo w surfaces, E a for surface diffusion of V O is highest in the case of the FeO 2 surface (d) Incorporation : T he incorporation of both O and O 2 into oxygen vacancy has no barriers. In particular, in the case of the incorporation of an O ion, no local minima ha ve been identified by our DFT calculations. This may imply that the hopping of oxygen

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111 vacancy adjacent to surface O ion is followed by spontaneous incorporation of the O ion to the oxygen vacancy (e) Bulk diffusion: The E a of bulk diffusion is 0.31 eV such that the energy level of the product is equal to that of the reactant within the uncertainty levels of the calculations This implies that oxygen vacancy may be equally stable at the first and second layers. 5.5 Discussion 5.5.1 ORR o n the FeO 2 +0.5O low S urface As dissociation and surface diffusion of O ion are energetically unfavorable on the FeO 2 +0.5O low surface, the hopping of surface oxygen vacancy makes the incorporation of O species possible. In addition th is surface oxygen vacancy Is also pred icted to promote the dissociation of O 2 (Figure 5 6) To determine the rate determining step of the ORR the concentration of oxygen vacancy as well as elementary steps with the highest E a must be considered 19 In particular, even though surface oxygen vacancy allows incorporation of O, the frequency of the event is determined by the number of oxygen vacancy on the surface. Despite the fact that the surface phase diagrams shown in Chapter 5 predict that [V O ] is expected to be very low in oxidized FeO 2 type surfaces, it is possible that [V O ] is increased by hetero cations including Sr and Co (see Section 1.2.1 of Chapter 1 for more detail explanation) Consequently, both high and low oxygen vacancy concentration c ases will be discussed here High oxygen vacancy concentration : With enough oxygen vacancy on surface, O 2 molecules do not need to wait to be paired with oxygen vacancy So, the ORRs will be mainly determined by reaction steps with the highest E a Consequ ently, both surface

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112 diffusion of oxygen vacancy and bulk diffusion will be rate determining step s on the FeO 2 +0.5O low surface. Low oxygen vacancy concentration : In this case, the entire reaction process is limited by site availability for incorporation re action (i.e. how often surface O can encounter oxygen vacancy ). Specifically, this entropy barrier will be a bottleneck to ORR that accompanies a highly improbable statistical configuration 19 As this barrier is not related to a Boltzmann activation factor, ORR in the low oxygen vacancy concentration is no longer limited by the E a of surface diffusion of oxygen vacancy and/or bulk diffusion. 5 .5.2 ORR on the S toichiometric FeO 2 S urface The dissociation of O 2 i s more favorable on the FeO 2 surface than on the FeO 2 +0.5O low surface. The difference between them can be interpreted by electron transfer from the surface region to the adsorbed oxygen With Bader topological analysis 65, 102 the c harges of the adsorbed O 2 (O_1: O which has direct bonding with Fe site and O_2: O which has bonding with only O_1) are = 0.13 e and = 0.06 e on the FeO 2 +0.5O low surface Almost no electrons are transferr ed to the O 2 Even though additional electrons are transferred to O_2 after dissociation, the electron transfer is insignificant ( = = 0.12 e ). In the case of the stoichiometric FeO 2 surface, the adsorbed O 2 has = 0.54 e and = 0.16 e However, after dissociation of the O 2 and becomes 0.52e and = 0.53 e This is due to hole rich characteristic of FeO 2 type surfaces which is more significant at oxidized FeO2 type surfaces (see Chapter 3).

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113 Considering ORR (Figure 5 7) the d issociation of O 2 is followed by surface diffusion of oxygen vacancy which enables incorporation of O species In addition, r eactio n without dissociation is also possible The dissociation does not affect the reaction on this surface as its E a is zero and incorporations of O and O 2 have no barriers. Unlike FeO 2 +0.5O low only surface diffusion of oxygen vacancy with its high E a will im pact the reactions. High oxygen vacancy concentration : In this case, the O will not wait to be paired with oxygen vacancy Consequently, the rate determining step of the ORR will be the reaction step with the highest E a which is surface diffusion of oxygen vacancy Low oxygen vacancy concentration : At low oxygen vacancy concentration similar to FeO 2 +0.5O low case, site availability for incorporation reaction will limit ORR of the FeO 2 surface. In summary the rate determining step of the ORR depend s greatl y on oxygen vacancy concentration S urface diffusion and bulk diffusion can be rate determining step s at high oxygen vacancy concentration In contrast, at low oxygen vacancy concentration the rate limiting step is mainly related to site availability for incorporation Thus, u nder SO FC operating conditions where oxygen vacancy concentration of LFO (010) surface is low ( see Figures 4 4 and 4 5), the ORR at both FeO 2 and FeO 2 +0.5O low surfaces will be dominated by this entropy barrier that involves an statist ical configuration of oxygen vacancy which is unfavorable for incorporation. 5.5.3 Relation to Experimental Results It is meaningful to see to what extent our predictions can be validated against experimental findings This is somewhat complicated by the fact that e xperimental studies o f ORRs have mostly focused on LSF and LSCF surfaces 19, 21, 23, 24, 103, 104

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114 However, it is possible to discuss similarities and differences in the results. In particular, ten Elshof et al. 19, 23 proposed that the rate determining step of LSCF at T 923 1285 K involved molecular O 2 and surface oxygen vacancy This result is consistent with our predictions on both FeO 2 and FeO 2 +0.5O low terminations with high oxygen vacancy concentration in which the rate limiting step contains surface diffusion of oxy gen vacancy Interestingly, their E a =0.62 0.72 eV for ORRs is located between calculated E a in FeO 2 +0.5O low and FeO 2 surfaces with high oxygen vacancy concentration Similarly, Jiang reported that both surface process (e.g. surface diffusion) and bulk diffusion play important roles in determining ORRs on LSCF 24 These results are also consistent with our predictions at FeO 2 +0.5O low with high oxygen vacancy concentration However, there are also discrepancy between calculations and experimental counterparts 21, 103, 104 For instance, Kan et al. reported that dissociative adsorption is rate determining step of LSCF 21 and Liu et al. suggested that rate limiting step is the first electron transfer step(in this case, adsorption of O 2 ) 104 To take a closer look at the reactions of O species on the LFO (010) surface LaO type surfaces as wel l as additional surface stoichiometry (e.g. fully oxidized surface) must be considered.

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115 Table 5 1. Adsorption enegy of O and O 2 on Fe sites of symmetric slab with 15 atomic layers and asymmetric slabs with 11 and 9 atomic layers. Numbers in braket s are difference in adsorption energy with respect to the symmetric slab. Adsorption energy E ad (eV) Symm etric 15 layers Asymm etric 1 1 layers Asymm etric 9 layers O 6.53 6.51 ( 0.02) 6.4 8 ( 0.05) O 2 11.6 7 11.6 2 ( 0.05) 11.56 ( 0.10) Figure 5 1 Schematic description of nudged elastic band ( NEB ) method.

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116 Figure 5 2. Atomic structure of asymmetric 9 layer slab with FeO 2 termination. Only 5 layers above bottom 4 layers are fully relaxed. Figure 5 3 Atomic displacements in z direction of symmetric slab nineteen atomic layers and asymmetric slab with thirteen atomic layers.

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117 Figure 5 4. Schematic description of considered reactions for each elementary step of FeO2+0.5O low surface termination and their relative energy pro files. Star and smiling face represent initial and final positions of oxygen species, respectively. Blue dotted lines indicate reaction paths of the oxygen species.

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118 Figure 5 5. Schematic description of considered reactions for each elementary step of s toichiometric FeO 2 surface termination and their relative energy profiles. Star and smiling face represent initial and final positions of oxygen species, respectively. Blue dotted lines indicate reaction paths of the oxygen species.

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119 Figure 5 6. Predic ted oxygen reduction reaction path of FeO 2 +0.5O low surface. Activation energy barrier ( E a ) of surface diffusion is the highest and is almost identical to that of bulk diffusion.

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120 Figure 5 7. Predicted oxygen reduction reaction paths of FeO 2 surface. I) contains dissociation step [(b) i] but II) does not. In both oxygen reduction reaction paths, activation energy barrier ( E a ) of surface diffusion is the highest ( E a = 0.81 eV ).

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121 CHAPTER 6 THE EFFECTS OF LATTICE VOIDS AND HELIUM GAS BUBBLES ON THE THERMAL CO NDUCTIVITY OF UO 2 6 .1 Introduction The thermal conductivity of nuclear fuel materials is one of their most essential performance metrics. Considering one of the disadvantages of UO 2 as a nuclear fuel in fission reactors is its low thermal conductivity, inv estigating the effect of lattice voids and fission products on the thermal conductivity of UO 2 is important. As fission gas can influence thermal conductivity of nuclear fuel 32, 105 and they are produced in abundance (e.g. ~14% of fission products at UO 2 in pressurized water reactor) 33 it will be critical to characterize the effects of fission gas on the thermal conduction in UO 2 In t his study, helium is chosen as a representative of fission gas 6 .2 Potential Models The interactions between the U, O, and He atoms are described by long range electrostatic component and short range interactions which is mostly repulsive in nature The electrostatic interactions are calculated by the direct summation method to avoid the computational expense associated with the Ewald method. In this work, the short range interaction model by Grimes et al. 106 is used to describe interatomic behavior. The Grimes potential is a mix of Buckingham type potential (Equation 6 1) and Lennard Jones type potential (Equation 6 2) : (6 1) (6 2)

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122 The Buckingham potential describes O O, U O, U U, and He He interactions, whi le the Lennard Jones potential describes He O and He U interactions (Table 6 1). Calculated lattice constant of the UO 2 is compared with other theoretical and experimental values in Table 6 2. The Grimes potential gives good values for the lattice paramete r. 6 .3 Thermal Expansion The thermal expansion of UO 2 is calculated using Andersen s constant pressure scheme. The supercell consists of 6x6x6 unit cells. The calculations are run for 7.5 ps with time step of 0.25 fs The temperature of the system is incr eased from 0 K to 1000 K with intervals of 100 K The lattice constant at each temperature is measured by averaging the volume of the system from the last 5.5 ps of the calculation. Figure 6 1 shows the normalized lattice parameter as a function of tempera ture. The Grimes potential gives systematically lower thermal expansion at all temperatures than experimental counterparts. In particular, the thermal expansion coefficient from the MD calculations is =6.2x10 6 K 1 compared to the experimental coefficient, =11.8 x10 6 K 1 6 .4 Thermal C onductivity of S ingle C rystal UO 2 The effects of lattice voids and helium bubbles on the thermal conductivity of single crystal UO 2 are calculated by non equilibrium M D simulation. This method is also called as direct method 107 The simulation cell used in this method is a square cylinder which is long in the z direction, and relatively narrow in the x and y directions (Figure 6 2). Before the actual thermal transport calculation, the cell is equilibrated at the target temperature using a constant t emperature and constant pressure simulation algorithm

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123 (NPT) for thermal and strain equilibration I determined that 12.5 ps is sufficient to equilibrate the system. After the equilibration two regions in the cell are designated as the heat source and heat sink (Figure 6 3) The cell dimensions are fixed with respect to the thermal expansion at the target temperature, and the heat source and heat sink are turned on to provide a steady state thermal flow. In particular, a f ixed amount of energy per unit time is injected to the heat source region by increasing kinetic energy of the atoms in the heat source region. Similarly, energy is removed at the same rate from in heat sink region. As a result, the system remains energy constant. Once steady state is accomp lished, which requires about 125 ps the temperature gradient is determined from temperature averaged over each cross sectional area over along the z direction of the cell over some duration of simulation time (in this case, 1125 ps ). The thermal conductiv ity is then calculated from Fourier s law, where J is heat flux which is identical to injected or removed kinetic energies in the system per area per time, is thermal conductivity tensor, and is the temperature gradient determined by the direct method Considering the weak dependence of thermal conductivity on the cross sectional area of the simulation cell the minimum cross s ectional area of UO 2 super cell on x and y directions is 4x4 unit cells due to the cutoff values of UO 2 potentials R c = 1.98 a used for this potential. Here, 10x10x40 super cell on x, y, and z axis is chosen to describe the dependence of thermal conductivit y on lattice void size Even though the c alculated thermal conductivity, for perfect crystal is 9.78 W/mK is higher than experimental

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124 value, it agrees well with other MD results (Table 6 3 ). Watanabe et al. have shown that this discrepancy between theoretical and experimental values is significantly red uced by compensating for the underestimate of the anharmonicity of UO 2 by the potential 108 6 .4.1 UO 2 with Lattice Voids Molecular d ynamics (MD) s tud y : A l attice void in our UO 2 supercell is modeled as a sphere Void s of radius from 2 to 12 with an interval of 2 were simulated. The size of voids is smaller than that of bubble which are experimentally measured (25 < r <120 ) 32 A s mall FORTRAN code generate s spherical voids at the center of two symmetrically identical 10x10x20 cells (Figure 6 2 ). Specifically, based on the location of sphere center and its radius which are user inputs, the code finds U atoms within the radius of the sphere and removes UO 2 units for which the U atoms lie within the sphere. By doing this, charge neutrality of the sy stem can be maintained. The code can also randomly locate He atoms of specific He density ( No. of He atoms/m 3 ) within an already constructed sphere. The percentage volume void ( ) corresponding to void size varies from 0.01% to 2. 2%. Here, 2.2% is maximum % volume void in the simulation cell that maintaining its stability. This is about half of experimental percentage volume of voids. Figure 6 4 shows the thermal conductivity, of UO 2 as a function of the void radius. The dependence of on void radius, r decrease in is insignificant (~1%) This is not surprising as there is only one UO 2 unit is missing in this void. However, as t he void radius decreases further decreases significantly : a t r=12 which is maximum radius considered in this study, is decreased

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125 by ~25% from the perfect crystal value (there are 531 UO 2 units within the void of r=12 ). Figure 6 5 shows the same d ata of Figure 6 4 but plotted with effective cross sectional area ( ) and relative thermal conductivity ( ) Here, k 0 and k r are thermal conductivities of UO 2 without and with a void having radius r resp ectively and A is cross sectional area of perfect UO 2 without a void. The dotted line shows how the thermal conductivity would change if decreased in proporti on to the reduction of cross sectional area of UO 2 Clearly, thermal conductivity decrease more qu ickly than the reduction of cross sectional area. 6 .4.2 UO 2 with Lattice Voids and He B ubbles As discussed in Chapter 1, bubbles (or small lattice voids) are filled with fission gases. Therefore, the effect of He gas bubble on the thermal conductivity of UO 2 is investigated by MD calculations (Figures 6 6 and 6 7 ). In particular, thermal conductivities of UO 2 having different void size (r=4.0, 8.0 and 12.0 ) with He densities ( 3 ) varying from 0.0 to 2.88x10 28 (this is maxi mum He density available based on the interatomic potentials of UO 2 with He) are calculated. This maximum value is consistent with experimental counterpart (see Chapter 1) 32 For small voids (r=4.0 and 8.0 ), the He atoms do not change the thermal conductivity by a significant amount : changes at mos t 3%. However, for the large void with r=12.0 conductivity is reduced significantly by increasing bubble density. At the maximum He density of r=12.0 case, k is reduced by ~ 12 %. This is surprising

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126 since He conducts heat (~0.3 W/mK at 800 K ) 109 ; thus filling the empty bubble with gas will be expected to increase the thermal conductivity, not decrease i t. To understand th is decrease, the distribution of He atoms in the voids is analyzed. Figure 6 8 shows cross sections of He bubble (r=8.0 and =1.44x10 28 atoms/m 3 ) He atoms are distributed near bubble/UO 2 interface. Figure 6 9 shows radial distribution of U, O and He atoms within simulation cells (r=12.0 ) with different He densities. In low He density He atoms are located out of r=6.3 As He density is increased, due to corresponding increase in gas pressure, He atoms distribute out of r=4.0 Even though portion of He atoms which are existed as interstitials in UO 2 lattice does not show significant change as a function of He density (ave rage is 0.37 with standard deviation of 0.08)., the number of He atoms penetrated into the lattice is increased as He density is increased From the reported evidence, it is possible to make a reasonable proposal concerning the significant decrease in the rmal conductivity by He bubble. Due to high pressure of He bubble, He atoms in the bubble can penetrate into the UO 2 lattice. These He atoms behave as point defects in the matrix and reduce the thermal conductivity of UO 2 locally. This trend becomes signif icant as the density of He interstitials in the lattice increases. Similarly, it is well known that thermal conductivity of UO 2 decreases with oxygen interstitials and this trend is strengthened by increasing degree of off stoichiometry 110 112 The contribution of He interstitials to the reduction of thermal conductivity becomes significant as the size of void is increased. Assuming that penetration depth, d of He atoms is constant ( e.g. d varies from 16 to 18 in the UO 2 supercell with r=12.0 void

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127 (Figure 6 9 ) ) the volume of the defective UO 2 region is increased by in which is the volume of He bubble Thus, s imple calculation shows that the volume of defecti ve region for the bubble with r=4.0 (~1206 2 ) is 1/9 with respect to that with r=12.0 (~10852 2 ). Considering that the portion of the region with He interstitials to the UO 2 supercell is 0.37% for the bubble with r=4.0 and 3.31% for that with r=12. 0 respectively, the penetration of He atoms into the UO 2 lattice will be main source of additional reduction in thermal conductivity by He bubble.

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128 Table 6 1 Parameters of interatomic potentials (a) Buckingham A ij [eV] ij [] C ij [eV 6 ] O O 108 0.38 56.06 U O 2494.2 0.341 40.16 U U 18600 0.275 32.64 (b) Lennard Jones He O 0.0153864 2.400015 He U 0.027106 2.019993 He He 0.00219 2.559 Table 6 2 Lattice constants of UO 2 unitcell Calculated DFT Experimental Lattice Constant () 5.463 5.49 5.478 Table 6 3 Thermal conductivities of UO 2 Calculated Potential 1 113 Potential 2 114 Experimental Thermal (W/mK) 9.78 12.49 11.2 0 4.33

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129 Figure 6 1. Normalized lattice parameter as a function of temperature from the Grimes potential and experiment. Figure 6 2. Schematic of 10x10x40 simulation cell of UO 2 with lattice void which contains He atoms

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130 Figure 6 3. Simulation setup of the non equilibrium molecular dynamics for calculating thermal conductivity. The heat source and heat sink are located at a quarter of the cell length away from the center of the system (gray regions). The same amount of kinetic energy, is added to the atoms in the heat source, and removed from those in the heat sink. This setup provides two equivalent heat currents J in opposite directions along the z axis. Figure 6 4. Calculated therm al conductivity as a function of a radius of spherical void.

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131 Figure 6 5. The same thermal conductivity dat a as in Figure 6 2 in relative thermal conductivity ( ) and effective cross sectional area ( ) perpendicular to heat flux direction. k 0 and k r are thermal conductivities of UO 2 without and with a void having radius r respectively and A is cross sectional area of perfect UO 2 without a void. If decrease in thermal conductivity is proportional to th e reduction of cross sectional area of UO 2 relative thermal conductivity would follow dotted line.

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132 Figure 6 6. Calculated thermal conductivities of sphere voids with different radii as a function of He bubble density. Numbers on each data point are the number of He atoms corresponding to each bubble density. Figure 6 7 Change in thermal conductivity [ ] as a function of bubble density. 0 is thermal conductivity at each void without He bubble.

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133 Figure 6 8. Cross sections of He bubble (r=8.0 A with =1.44x10 28 atoms/m 3 ). Number of He atoms in the bubble, n He is 31. He atoms are distributed near the interface betwee n the bubble and UO 2 lattice. Figure 6 9 Radial distributions of U, O, and He atoms from the center of the void within UO 2 simulation cell (r=12.0 ).

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134 Figure 6 10 No. of He atoms penetrated into UO 2 lattice, and the ratio of He atoms penetrate d into UO 2 lattice over total No. of He atoms at UO 2 with voids of r=12.0 Numbers on each data point are the No. of He atoms corresponding to each He density.

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135 CHAPTER 7 CONCLUSIONS LaFeO 3 : I have dissected the stabilization of LFO ( 010) surfaces in terms of three factors: charge redistribution at the surface the contribution of sublayers to charge compensation, and the contribution of atomic relaxation to charge compensation. All these factors are interrelated, but the results indic ate that in the case of LaO type surfaces, atomic relaxation is the most important factor because it facilitates charge redistribution of LaO type surfaces In the case of FeO 2 type surfaces, the ability of the Fe ion to modulate its valence makes charge r edistribution at the surface more important. The proposed stabilization mechanisms of LFO (010) sur faces are summarized in Figure 7 1 The mechanisms identified allow two important predictions for the surface compositio n of the LFO (010) surface. First, th e change of La charge can be more difficult than th at of Fe charge and oxygen reduction should occur more readily on the FeO 2 surface termination than on the LaO surface termination T his may be one of the reasons that in ABO 3 perovskite cathode materials of SOFC B site cations are catalytically more ac tive than A site cations for oxygen reduction reaction s 115 So, it may be critical to control the concentration of surface oxygen vacancies on the LaO type surface terminat ions to prom ot e oxygen reduction on the LFO (010) surface. Second, strong charge transfer to O low atoms within the LaO type surface termination implies that oxid ation of this surface may be energetically more favorable than oxidation of the FeO 2 type surface terminati on

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1 36 It is likely that the proposed mechanisms for surface charge compensation can provide routes to systematically predict the most stable surface compositions and corresponding electronic structures of other base materials for SOFC cathodes P hase diagra m of L FO (010) surfaces for varying oxygen stoichiometr ies is developed b y combining density functional theory and thermodynamics T h e diagram has shown that the contribution of the chemical potentials is significant in determination of surface stability. Also, the calculated surface phase diagram is consistent with results from TPD and TPR experiments. The investigation of LaO type terminations in LFO (010) is justified in comparison with LaMnO 3 surfaces. The surface phase diagrams lead us to conclude that oxidized surface terminations must be considered to understand oxygen reduction reaction s especially for the research of cathode materials for intermediate temperature SOFCs. Also, the instability of the reduced surfaces with respect to stoichiometric te rminations at SOFC operating conditions emphasize s the role of extrinsic defects including Sr and Co in increasing the concentration of surface oxygen vacancies and the improving catalytic properties of LFO surfaces. The higher prev a l e nce of LaO type surfa ces in LFO (010) than in LaMnO 3 may provide an important clue to understand ing the difference between LFO and LaMnO 3 as base materials for SOFC cathode. Also, considering the stabilizing effect of Sr dopant s 116 on LaO surface, it may be important to investigate oxygen reduction reaction s on both LaO and FeO 2 type surface terminations rather than focusing only on FeO 2 type surface terminations I believe that the calculated surface phase diagram

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137 will serve as a platform for gaining more insight into oxygen reduction reaction at SOFCs cathode materials. Based on developed phase diagram of LFO (010) surfaces, ORR of oxidized FeO 2 +0.5O low and stoichi ometric FeO 2 surfaces are investigated. The behavior of surface oxygen vacancy is important to determine rate limiting step of ORR on both FeO 2 and FeO 2 +0.5O low surfaces. In high concentration of surface oxygen vacancy rate limiting step is predicted to b e elementary step with the highest activation energy. On the contrary, in low concentration of surface oxygen vacancy ORR is limited by site availability for incorporation reaction (so called entropy barrier). The r esults of the ORR study provide directio n for further research on the surface oxygen vacancy In particular, it will be critical to understand how the formation of surface oxygen vacancy is changed by surface stoichiometry. Considering that electronic effect in surface stabilization, it is highl y probable that the preference of oxygen vacancy formation will be influenced by surface stoichiometry. This will enable the interpretation of how entropy barrier is changed by surface stoichiometry. By the same token the dependence of formation of surfac e oxygen vacancy on Sr and Co should be investigated with surface stoichiometry as one variable. Even though it is we ll known that the formation of surface oxygen vacancy in bulk LFO is promoted by Sr and Co dopants 117, 118 it is still unclear how the efficiency of Sr and Co in increasing the concentration of surface oxygen vacancy is affected by surface stoichiometry. UO 2 : The effect of lattice void and He bubble on thermal conductivity of UO 2 has been investigat ed by non equilibrium MD The relationship between the size of lattice

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138 void and thermal conductivity is established, and it was found that penetration of He atoms into UO 2 matrix is the source of additional reduction in thermal conductivity. Considering t hat t he lowest energy planes of UO 2 are {111 } {011}, and {001} 119, 120 t he boundary of the voids in UO 2 lattice consists of these lowest energy plan es The penetration behavior of He atoms in different crystallog raphic planes may not be similar and investigating the behavior of He atoms on different planes will be critical to understand the contribution of He atoms on thermal conductivity of UO 2 Similarly, interaction of He atoms with various defects in UO 2 will also be important to understand the effect of He atoms on thermal conductivity.

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139 Table 7 1 Summary of stabilization mechanism s of L a F e O 3 (010) surface LaO type FeO 2 type Reduced Negative charge variations of surface cations Negat ive charge variations of surface cations Stoichiometric Negative charge variations of surface cations Positive charge variations of surface cations Oxidized Charge transfer from surface region to O low atoms + Strong atomic relaxation at surface region Ch arge transfer from surface region to O low atoms

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147 BIOGRAPHICAL SKETCH Chan Woo Lee was born in Seoul the capital of Republic of Korea at 1976 Inspired by his math tutor who was attending college of engineering at Hanyang University ( HYU, Hanyang is old name of Seoul), he entered HYU and majored Metallurgical E ngineering. After his Bachelor, he had his master under the guidance of Prof. Changhee Lee, with majoring experiment and simulation of welding metallurgy. As his master work wa process, he got curiosity about ab initio approach and decided to study atomic level calculations. So, after serving Republic of Korea Army, he joined Computational Materials Science and Engineering La boratory in Department of Ceramic Engineering, HYU as visiting scientist and learned electronic structure calculations and atomistic simulations under the guidance of Prof. Yong Chae Chung. In 2005 fall, h e joined the Department of Materials Science and En gineering at University of Florida to pursue his Ph.D. with Prof. Susan B. Sinnott. Chan Woo is expecting his Ph.D. in the summer of 2010.