NORTHWESTERN UNIVERSITY

AStudyofOxidesforSolidOxideCells

ADISSERTATION

SUBMITTED TO THE GRADUATE SCHOOL

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for the degree

DOCTOR OF PHILOSOPHY

Field of Materials Science and Engineering

By

Olivier Comets

EVANSTON, ILLINOIS

December 2013 UMI Number: 3605699

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ABSTRACT

AStudyofOxidesforSolidOxideCells

Olivier Comets

As the world energy consumption increases, it is a question of global health to increase energy production eciency and to reduce CO2 emissions. In that respect, solid oxide cells are solid state devices that convert directly fuel into electricity, or vice versa. In fact, when run in fuel cell mode, such devices produce electricity with eciency up to twice that of current natural gas power plants. However, systems equipped with them have only seen limited commercialization owing to issues of cost, durability, and performance.

In this thesis, three di↵erent aspects of solid oxide cells are studied. First, the e↵ects of stress on the properties of mixed ionic electronic conducting oxides are considered.

Such oxides can be used as electrode materials, where they are often subject to large stresses, which can, in turn, a↵ect their performance. Hence, understanding the rela- tionship between stress and properties in such materials is crucial. Non-stoichiometry in strontium substituted lanthanum cobaltite is found to increase under tension and to decrease under compression. 4

Then, degradation taking place when the cell is run in electrolysis mode is discussed.

A high current allows for a high production rate of hydrogen gas. However, this can also lead to oxygen bubble nucleating in the electrolyte and subsequent degradation of the cell. The analysis conducted here shows that such nucleation phenomenon can be avoided by keeping the overpotential at the oxygen electrode below a critical value.

Finally, the growth and coarsening of catalyst nanoparticles at the surface of an oxide is studied. Scientists have developed new oxides for anodes in which a catalyst material is dissolved and exsolves under operating conditions. As the performance of the cell is controlled by the surface area of the catalyst phase, understanding the kinetics of the growth is critical to predict the performance of the cell. An approach is developed to study the growth of one particle, in the limiting case where only bulk transport is allowed. 5

Acknowledgements

As I reflect back at my time in the Department of Materials Science and Engineering at Northwestern, I realize how much I have learned, how many great people I have met, and how many amazing experiences I have lived. Undeniably, this department and the people I have met through it have played a major role in my scientific development, personal fulfillment, and my integration in the US.

First and foremost, I would like to thank my advisor, Peter Voorhees, for his knowl- edge and guidance while confronting me with such exciting and stimulating projects.

Knowing that a graduate school experience is both of academic and human nature, he encouraged me to develop my soft skills through various projects unrelated to work.

Peter, thank you for everything!

I would like to thank Scott Barnett, who played the role of a second advisor given the overlap of my research and his expertise, for insightful discussions and thrilling collaboration work. I would also like to thank my committee members Thomas Mason,

Kenneth Poeppelmeier, and Chris Wolverton for thoughts, suggestions, and insight.

I’m very fortunate to be part of such an amazing and complementary group as the Voorhees Research Group and I would like to individually thank each one of you:

Kuo-An, Tony, Thomas, Begum, Larry, Ian, Megna, Alanna, Eddie, Anthony, John T.,

Tom, John G., Kevin, Quentin, and Ashwin. I leave the group with memories of great scientific discussions, help in dire situations, and with great friendships. I am also very 6 grateful to the Barnett group for their thoughts and insights on Solid Oxide Cells, and namely to: David B., Scott, Kyle, Gareth, Ann, Beth, and David K.

This work would not have been possible without the many challenging and en- lightening discussions with our collaborators: professors Jason Nicholas, Stuart Adler,

Katsuyo Thornton, Dr. Hui-Chia Yu, and T. J. McDonald as well as with Prof. Anil

Virkar and Prof. Junichiro Mizusaki. The English in this thesis wouldn’t have been as good without the help of John, Alex, Ahmed and Kyle.

Finally, I would like to thank my loving family and friends for all their support during the process. I am grateful to my parents for the education they provided me with and to my parents, Aude, and Antoine for their constant encouragements. I am very glad to Dave Herman, Ahmed Issa, Carlos Alvarez and Begum Gulsoy for valuable friendships, great advice and much fun Ive had during grad school. Last but not least,

IwouldliketoacknowledgemyfriendPierreGarreauforhisconstantsupport,an infallible friendship, a lot of fun during grad school and many essential conversations weve had together.

This work was financially supported by the US Department of Energy (DOE) and the National Science Foundation (NSF). 7

Contents

ABSTRACT 3

Acknowledgements 5

List of Figures 11

List of Tables 14

Chapter 1. Introduction 15

Chapter 2. Background 17

2.1. ElectricityproductionintheUS 18

2.2. Solid Oxide Cells 20

2.2.1. Fuel cell mode 21

2.2.2. Electrolysis mode 22

2.2.3. Materials 23

2.2.4. Features 25

Chapter 3. The E↵ects of Stress on the Defect and Electronic Properties of

MixedIonicElectronicConductors 27

3.1. Introduction 27

3.2. Thermodynamics 29

3.2.1. Thermodynamic description of the system 29 8

3.2.2. Equilibrium conditions 31

3.2.3. NewfreeenergyfunctionandMaxwell’sequation 39

3.2.4. Chemical potential of oxygen under stress 41

3.3. E↵ects of stress on the non-stoichiometry 43

3.4. E↵ects of stress on the vacancy formation energy 45

3.5. E↵ectsofstressonthechemicalcapacitance 48

3.6.Comparisonsandpredictions 51

3.6.1. E↵ects of a hydrostatic stress on the properties of La0.8Sr0.2CoO3 52 3.6.2. Thin Films 56

3.7. Discussion 69

3.7.1. LSC thin films 69

3.7.2. Generalization to other mixed conductors 72

3.8. Conclusion and future work 73

Chapter 4. Oxygen Bubble Formation in Solid Oxide Electrolysis Cells 76

4.1. Introduction 76

4.2. Thermodynamics of nucleation 78

4.2.1. Thermodynamic model 82

4.2.2. Internal energies 84

4.2.3. Constraints 86

4.2.4. Equilibrium conditions 92

4.3. Driving force 94

4.3.1. Value of the oxygen potential 94

4.3.2. Expression of the oxygen potential 95 9

4.3.3. Expressions of the grand potentials 99

4.3.4. Change in the grand potential 103

4.3.5. Free energy change of nucleation 107

4.4. Results and discussion 112

4.4.1. Critical radius 112

4.4.2. Homogeneousandheterogeneousnucleation 114

4.4.3. E↵ects of parameters on the nucleation polarization 117

4.4.4. Critical current 118

4.4.5. Vacancy concentration 121

4.5. Conclusion and future work 122

Chapter 5. Growth and Coarsening of Nanoparticles on the Surface of an Oxide 125

5.1. Introduction 125

5.2. Background 127

5.2.1. Coarsening in 3D 128

5.2.2. Coarsening in 2D 129

5.3. Modeling considerations 129

5.4. Mathematicalformulationofthesystem 131

5.4.1.Governingequation 131

5.4.2. Boundary conditions 131

5.4.3. Particle growth rate 132

5.4.4. Undimensionalizingtheequations 133

5.5. Approach 135

5.5.1. Green’s function 135 10

5.5.2. Green’s theorem 140

5.5.3. Solving the equations 143

5.6. Extension of the model and future work 143

5.7. Conclusion 144

Chapter 6. Conclusion 146

References 148 11

List of Figures

2.1 World energy consumption in the world as predicted by the United

StatesEnergyInformationAdministrationin2011. 18

2.2 CompositionoftheelectricityproducedintheUSbyresources. 19

2.3 Projections for added electricity generation capacity as a function of

sources through 2040. 20

2.4 Schematic of a solid oxide cell running in fuel cell mode on hydrogen

gas. 22

2.5 Schematic of a solid oxide cell running in electrolysis mode on water. 23

3.1 System under consideration for the derivation of the equilibrium

conditions: oxide and gas phase delimited by an arbitrary interface

@ .33 V

3.2 Thought experiment to understand the e↵ect of stress on the

non-stoichiometry. 44

3.3 Non-stoichiometry as a function of the trace of the stress in LSC-82. 56

3.4 Schematic of the change in non-stoichiometry in a coherent and

dislocation-free thin film due to lattice mismatch with the substrate. 61 12

3.5 Schematic of the change in non-stoichiometry in a thin film grown on

asubstrateunderthermalstress. 62

3.6 Chemical capacitance versus oxygen partial pressure at T =873K

as estimated for bulk La0.6Sr0.4CoO3 ,asreportedina1.5µm-thick LSC film on GDC and according to the model. 64

3.7 Chemical capacitance versus oxygen partial pressure at T =793K

evaluated for bulk La0.8Sr0.2CoO3 ,asreportedfora45nm-thick LSCfilmonYSZandaccordingtothemodel. 67

4.1 Sketch of a SOEC under operation. If the current is high enough,

bubblescanformintheelectrolyte. 77

4.2 Schematic of oxygen bubble formation in the dense YSZ electrolyte

of a SOEC. 79

4.3 Sketch of the system under study for the derivation of the equilibrium

conditions: perfect YSZ lattice with a spherical bubble of oxygen. 83

4.4 Sketch of the oxygen potential near the oxygen electrode of a SOEC

underanappliedcurrent. 95

4.5 Driving force for nucleation explained in the perspective of the mole

fraction of oxygen vacancy. 109

g 4.6 Plot of the grand potential of the gas bubble, !v ,thehomogeneous

part of the grand potential of the oxide, !ox,andthenegative

of the elastic energy, W ,asafunctionoftheoxygenelectrode e polarization. 111 13

4.7 Critical radius of the nucleus versus electrode polarization for the

exact case as given by Eq. (4.85) and the approximation given by

(4.86). 113

4.8 Reversible work for the formation of a critical nucleus as a function

of the oxygen electrode polarization for the homogeneous nucleation

case (within a grain) and heterogeneous case (at a grain boundary). 117

4.9 Nucleation overpotential as a function of the surface energy of the

oxide. 119

4.10 Critical overpotential as a function of temperature, T ,andoxygen

O partial pressure at the oxygen electrode, PO2 .120

5.1 Schematic of catalyst nanoparticles precipitating at the surface of the

anode. 126

5.2 Schematic of the mechanism for the formation of catalyst particles at

the surface of the oxide. 127

5.3 Schematic of the configuration for the coarsening problem. 130 14

List of Tables

3.1 ParametersforLSC-82usedinestablishingFig.3.3. 57

3.2 Parameters for GDC-91, LSC-64, LSC-55 and LSC-73 used to

compute Cchem as a function of PO2 toestablishFig.3.6. 65

3.3 Parameters for LSC-82 used to compute Cchem as a function of PO2 toestablishFig.3.7. 68

4.1 Values of the parameters for nucleation of oxygen bubbles in 8-mol

% YSZ electrolyte. 110 15

CHAPTER 1

Introduction

The growing needs in energy and the depletion of the oil resources have made man consider new, cleaner and sustainable ways to produce energy. In that respect, asolidoxidecell(SOC)isasolidstatedevicethatconvertsdirectlychemicalenergy into electricity, or vice versa.Runinfuelcellmode,aSOCproduceselectricitybya direct oxidation of the fuel, skipping the conversion steps into thermal and mechanical energy present in the standard fossil fuel power plants. Production of electricity by this process is up to twice as ecient as in standard power plants. Such cells can also be run in electrolysis mode, to regenerate gas. SOCs are one of several di↵erent fuel and electrolysis cells, but are of interest because of higher eciency, low emissions, fuel flexibility and potential long-term stability. Thanks to these advantages, solid oxide fuel cells have a wide range of applications from auxiliary power units in big-rig vehicles to dispersed stationary power generation. However, for such systems to be mass produced the issues of performance, durability, and cost must be addressed.

The goal of this thesis is to use thermodynamics and other mathematical tools to study di↵erent aspects of solid oxide cells. In doing so, we hope to gain better under- standing of the processes taking place in the cell and of the cell intrinsic limitations.

Such basic understanding is the cornerstone of SOC systems commercialization.

Chapter 2 o↵ers some background for the current thesis work. First, SOC Research is motivated by the increase in global energy demand, the need for ecient processes 16

to produce electricity and the increase in CO2 emissions. Then, an overview of the fundamentals of operations and the di↵erent components of a cell are discussed.

Chapter 3 discusses the e↵ects of stress in mixed ionic electronic conducting oxides.

Such oxides are used in a variety of di↵erent applications (e.g. sensors, SOCs) and often are in a state of stress (e.g. thermal, mismatch). As stress can a↵ect their performance, understanding how such oxide behave under stress will allow to better predict their performance in applications.

Chapter 4 presents the degradation of solid oxide electrolysis cells (SOECs) by oxygen bubble formation in their electrolyte. SOECs are used to regenerate gas using electricity. High rates of production are achieved with high currents. However, when the voltage applied to the cell is above a critical value, bubbles start forming in the electrolyte of the cells, leading sometimes to deleterious consequences.

Chapter 5 tackles the growth and coarsening of catalyst nanoparticles at the surface of an oxide. Scientists have developed novel anode materials, where catalyst is dissolved within the oxide and exsolves under operating conditions. Because the performance of the cell is controlled by the surface area of those catalyst particles, understanding the kinetics of the process is crucial to predict the performance of the cell.

Finally, Chapter 6 summarizes the main results of this thesis. For each of the projects, only the most important elements of the future work are recalled. 17

CHAPTER 2

Background

It is no surprise that the world energy demand is growing quickly. In 2008, the

United States Energy Information Administration (EIA) evaluated the world energy demand to grow by 2% yearly [1]. In other words, every 10 years the world adds capacities equivalent to the entire annual energy production of the US. The projections for the energy demand, reported in Fig. 2.1, show that growth is mainly driven by non-OECD countries. This fact can be understood when considering that non-OECD countries represented 80% of the world population but 50% of the energy consumed worldwide in 2008. The development of those countries is synonymous with a dramatic increase in energy demand. This then results in an unprecedented increase in the production of greenhouse gases (e.g. CO2)andotherbyproducts(e.g.heat).Developing ecient and cleaner ways to produce energy is thus crucial.

After considering the energy landscape in the US, it will be shown that electricity production represents a large share of the energy mix and is a highly inecient process.

Solid oxide cells are devices that address that problem, capable of eciently convert- ing chemical energy to electricity and vice versa. Various aspects of those devices: electrochemistry, materials and features will be presented in the second section. 18

Figure 2.1. World energy consumption in the world as predicted by the United States Energy Information Administration in 2011 [1]. The grow- ing demand is mainly driven by the non-OECD countries, who represent more than 80% of the global population.

2.1. Electricity production in the US

Today, electricity generation in the United States accounts for approximately 40% of the energy consumed in the US [2]. Fig. 2.2 represents how the electricity is split among the di↵erent resources. Fossil fuels (coal, gas, oil), which have by far the biggest carbon footprint, represent 71% of all the electricity produced in the US. The eciency for a fossil fuel-based power plant is currently about 33% [3], i.e. two third of the energy used to produce electricity is wasted. Furthermore, electricity generation is responsible for 40% of the total US carbon dioxide production [4], almost exclusively due to fossil fuels. All this proves that electricity generation in the US remains a highly inecient process and responsible for much greenhouse gas emissions. 19

Figure 2.2. Composition of the electricity produced in the US by re- sources [2]. ”Renewable” includes hydro, geothermal, solar, wind and tide. Fossil fuels (coal, gas and oil) contribute to 71% of the electricity produced in the US.

Petit et al. have shown a positive correlation between atmospheric levels of CO2 and the earth temperature [5]. Given the recent rise in CO2 concentration in the atmosphere, the Nobel prize-winning Intergovernmental Panel on Climate Change has predicted a temperature increase of 2 to 6 Cbytheendofthecentury[6]. Such an increase in temperature can in turn lead to a disruption of the earth’s fragile climate.

However, addressing the issue of electricity generation could result in vital progress in reducing CO2 emissions, controlling the atmosphere temperature and protecting the planet’s fragile equilibrium.

With a steady increase in the demand for electricity, the reduction of these emissions must result from the development of a more ecient electricity generation process. The

first step is to use cleaner fuels, e.g. natural gas rather than coal. For that matter, the

US EIA projects a drastic increase in the number of natural gas plants as additional capacities, along with renewable resources, which have little carbon footprint, as shown 20

Figure 2.3. Projections for added electricity generation capacity as a function of sources through 2040. Source: United States EIA [7].

in Fig. 2.3. Solid oxide cells can be operated on natural gas and have nearly twice the eciency of current plants running on such fuels. However, for such systems to be mass produced and integrated into power generators, research is needed to lower the costs and limit degradation.

2.2. Solid Oxide Cells

ASolidOxideCell(SOC)isasolidstateelectrochemicaldevicecapableofeciently converting the chemical energy of a fuel gas to electricity and vice versa. The cell can be run both in the fuel cell mode to produce electricity, and in the electrolysis mode to regenerate the gas. 21

2.2.1. Fuel cell mode

In conventional power plants, gas and oxygen are mixed together and combusted. The heat produced by this reaction is transferred to a fluid which then drives a turbine, activating an alternator to finally generate electricity. Given the number of di↵erent steps in the process and that its eciency is limited by the Carnot cycle, the maximum theoretical eciency of a traditional power plant is 47%. In a Solid Oxide Fuel Cell

(SOFC), no direct combustion take place. The reactants (fuel and air) are, in fact, spatially separated and involved in electrochemical reactions at electrodes, separated by an electrolyte, much like in a battery. However, unlike a battery, a fuel cell does not need to be recharged and will run as long as the reactants are supplied. Fig. 2.4 is a schematic of a SOC running in fuel cell mode on hydrogen gas. Oxygen is reduced on

2 the cathode to form oxygen ions (O ). Those ions are then transferred to the anode via the electrolyte, where they react with hydrogen gas to form water and regenerate electrons. Electrons are thus produced at the anode and consumed at the cathode generating a current. The reactions taking place are:

1 2 at the cathode: O (g) + 2e0 O (ox)(2.1) 2 2 !

2 (2.2) at the anode: H +O (ox) H O(g) + 2e0 2 ! 2 1 and overall: O (g) + H H O(g)(2.3) 2 2 2 ! 2

The overall reaction is a reaction of combustion. Finally, in a SOFC, the reaction of combustion has essentially been split up into it’s reduction and oxidation reactions in order to directly use the flow of electrons. 22

Figure 2.4. Schematic of a solid oxide cell running in fuel cell mode on hydrogen gas. Air and fuel are fed to the cell. At the cathode, air is reduced to oxygen ions. Those ions are then transported to the anode through the electrolyte, where they react with the fuel gas (here H2), forming water, and regenerating electrons. The flow of electrons from the anode to the cathode is then used outside of the cell to power appliances.

2.2.2. Electrolysis mode

The operating principle of a Solid Oxide Electrolysis Cell (SOEC) is the very opposite of that of a SOFC. Fig. 2.5 is a schematic of a solid oxide cell running on water in electrolysis mode. Water and an electric current are fed to the cell. Water molecules 23

Figure 2.5. Schematic of a solid oxide cell running in electrolysis mode. Water and electric power are fed to the cell. At the cathode, water is reduced to hydrogen gas and oxygen ions. Those ions are then trans- ported to the cathode through the electrolyte, where they recombine, regenerating oxygen gas and electrons. react with electrons at the cathode producing oxygen ions and hydrogen gas. Oxy- gen ions are then transported via the electrolyte to the anode where they recombine, regenerating oxygen gas and electrons.

2.2.3. Materials

Because the electrolyte and electrodes serve di↵erent purposes, di↵erent materials and geometries are used. 24

The role of the electrolyte is threefold: to physically separate the fuel and the oxidant, to transport the oxygen ions from one electrode to the other, while preventing the passing of electrons. Thus an electrolyte must:

(1) be fully dense,

(2) exhibit high ionic conductivity,

(3) have low electronic conductivity,

(4) be stable in oxidizing and reducing environment,

(5) be chemically compatible with electrode materials,

(6) have a thermal expansion relatively similar to that of the electrodes.

Typical electrolyte materials are 8 mol% yttria-stabilized zirconium (YSZ), Y2O3-ZrO2, and gadolinium doped ceria, Gd2O3-CeO2 [8,9]. As electrolyte resistance is a function of thickness, electrolytes are made very thin, typically < 10 µm.

The anode of a SOFC provides reaction sites for the oxidation of the fuel. It also supports the transport of the various species to and from those reaction sites: gases, oxygen ions and electrons. As a result, it should be porous to allow for the gases to di↵use, capable to conduct oxygen ions and electrons. Finally, anodes must have the following features:

(1) high porosity

(2) electronic conductivity

(3) ionic conductivity

(4) stability at high temperatures and in reducing environments

(5) mechanical compatibility with electrolyte

(6) chemical compatibility with electrolyte and interconnect 25

(7) catalyst activity toward the oxidation of the fuel

(8) fuel flexibility and resistance to impurities and carbon deposition

Typical anode comprise a mixture of nickel and YSZ. The metal phase (nickel) provides a path for the electrons, while the oxide phase (YSZ) provides the transport of oxygen ions and the pores allows for gas di↵usion. The pores are created from the reduction of nickel oxide to nickel after exposure to the fuel. In this case, the active sites are at the junction of the three phases, known as the triple phase boundaries. Often the anode is fabricated using a dual-layer geometry, where a first layer of thickness .5 1mm provides the support for the cell and a second layer of thickness 10 50 µmisthe functional layer.

The cathode of a SOFC is exposed to air. Like the anode, the cathode needs to be porous, capable of transporting both oxygen ions and electrons, compatible (me- chanically and chemically) with the electrolyte and interconnect and stable at high temperature. However, the anode is required to be stable under an oxidizing environ- ment while having a catalytic activity on the reduction of molecular oxygen. Typical cathodes are made of a mixture of strontium substituted lanthanum manganite and

YSZ.

2.2.4. Features

Various aspects of SOCs have caused them to emerge recently as a serious solution to address the problem of growing electricity demand. Solid oxide cells operate at very high temperature (between 400 and 1000 C), which make expensive catalysts unnec- essary. Furthermore, the high quality of the by-products can be used for cogeneration 26

(in fuel cell mode), boosting the eciency of the process even more. Because the cell is entirely solid state, there are no moving parts, making it silent and easier to run. The electrolyte does not require any management, unlike sulfuric acid fuel cells for exam- ple. As a result, SOCs present a potential long life expectancy of 40,000-80,000 hours of operation. SOFCs can achieve eciencies of 45 to 60%, and up to 90% with heat recovery [9]. Last, a tubular geometry has recently emerged promising much shorter start up times than the classic planar geometry, typically on the order of minutes.

Although this technology has been known for over 160 years, much more research is necessary to optimize the cell, control its long term degradation, and eventually bring the cost down, making it competitive with conventional less ecient technologies. 27

CHAPTER 3

The E↵ects of Stress on the Defect and Electronic Properties

of Mixed Ionic Electronic Conductors

3.1. Introduction

Oxides transporting both ionic and electronic species are called mixed ionic elec- tronic conductors (MIECs). Such materials are of particular interest in applications where simultaneous ion and electron conduction is required, such as at the electrodes of

Solid Oxide Cells [8]. Recent studies have shown that such oxides under stress (e.g. in thin film form) display very di↵erent properties —e.g. non-stoichiometry and kinetics— from that of bulk materials [10,11]. This di↵erence in behavior between the thin film configuration and bulk is often attributed to stresses developing in the film. Such stress can be thermal, originating from di↵erent operating and firing temperatures, or due to a misfit between the lattice parameters of the MIEC and the substrate. As oxides in most applications are in the form of thin films, stress is present and it is critical to understand its e↵ects to predict their behavior under operating conditions. Finally, because oxide lattice parameter, oxygen non-stoichiometry and electrical properties are closely related [12], stress will influence all of these simultaneously.

Two types of MIECs have been reported and classified according to their electronic conduction mechanism. The first has a metallic-like electronic conduction mechanism, 28 mediated by holes present in a partially filled delocalized band. Such behavior is de- scribed by the itinerant electron model developed by Mizusaki et al. [13]andLankhorst et al. [14–17]. La1 xSrxCoO3 (LSC) is a such example [18]. The second type has an electronic conductivity described by an activated electron hopping mechanism, also known as the small polaron model [19]. A small polaron is a defect created when an electronic carrier is trapped at a given site fostering a local distortion of the lattice.

The hopping of such defects (the carrier plus its polarization field) is responsible for the electronic conductivity in such materials as La1xSrxMnO3 (LSM) for x ¡ .2 [18]. Un- fortunately, the defect structure is also much more complex in such material, evidenced by extensive work on the topic, e.g. [20–29].

The purpose of this chapter is to illustrate the e↵ects of stress on the properties of mixed conductors with a perovskite structure in equilibrium with an atmosphere, through the example of LSC. In the first section, results from previous studies on elas- tically stressed crystals [30–34]areusedtoderivetheequilibriumconditionsandthe expression of the chemical potential of oxygen in the oxide under stress. In the follow- ing three sections, the expressions for the non-stoichiometry, the chemical capacitance and the vacancy formation energy are respectively derived for an oxide under stress.

The fifth section is dedicated to predictions of the model. First, changes in the non- stoichiometry of an oxide under a hydrostatic stress are considered. The e↵ects on the hole concentration and on the ionic and electronic conductivities are also discussed.

Second, predictions of the chemical capacitance as a function of oxygen pressure are compared to experimental data for a thin film under mismatch strain and thermal build up [10,11]. Third, calculations of the change in vacancy formation energy due to 29 stress are compared to ab initio results [35]. The qualitative agreement resulting from those comparisons show that stress is not the sole controlling factor for the di↵erence in behavior between thin film and bulk. Possible sources of the quantitative discrepancy are discussed in the last section. Finally, the treatment developed here is shown to extend to other mixed conducting oxides, with more complex defect equilibria.

3.2. Thermodynamics

The approach employed here was initially introduced by Cahn and Larch´e[30], and developed by Johnson and Schmalzried [31, 32], see [33] for a review. Swaminathan et al. used a similar approach to study the di↵usion of charged defects in ionic solids

[34, 36]. First, the equilibrium conditions are derived for an oxide under stress, in equilibrium with oxygen gas. Introducing a new free energy function allows us to derive a Maxwell’s equation. Integrating this equation between a stress free state and astateunderstressfinallyyieldstheexpressionofthechemicalpotentialofoxygenin the oxide.

3.2.1. Thermodynamic description of the system

Under consideration is a dislocation-free slab of oxide perovskite structure of general chemical formula ABO3 in equilibrium with a gas containing molecular oxygen, O2.

The oxide has three distinct sublattices: two for the cations (A and B), one for the oxygen ions (O) and the interstitial sites are all vacant. The species assumed to be present in the gas are molecular oxygen, O2,andothergasesthatdonotreactwith the oxide (e.g. N2). Elements from sublattices A and B are not soluble in the gas and 30 no other phases are assumed to form. The various thermodynamic densities relative to the crystal are referred to a reference or stress-free state, while those relative to the gas are referred to the actual state. Thermodynamic densities expressed per-unit-volume in the reference state are designated with a superscript 0.

The oxide used to derive the results in the rest of the chapter is La1 xSrxCoO3 , where x is the strontium substitution level and is the non-stoichiometry. Considering absolute charges, oxygen with a 2oxidationstateandneutralvacanciesarefoundon the O sublattice. The A sublattice is populated with lanthanum ions (III), strontium ions (II), and neutral vacancies, while the B sublattice is populated with cobalt ions

(III) and neutral vacancies. Considering relative charges and using the Kr¨oger-Vink

x x x notation, these elements are noted OO,VO·· ,LaA,SrA0 ,VA000,CoB,VB000. Any given ion must occupy a site on one of the subblattices. Dislocations, exchange of atoms between the anionic and cationic sublattices and interstitial atoms are not included in the model.

The internal energy density per unit volume of the oxide in the stress-free state, eox,istakentobeafunctionoftheentropysox,thedeformationgradienttensor ,the v0 v0 F electric displacement field D and the number densities of the di↵erent elements ⇢0 x , LaA

⇢0 , ⇢0 , ⇢0 x , ⇢0 , ⇢0 x , ⇢0 and ⇢0 : SrA0 VA000 CoB VB000 OO VO·· h·

ox ox (3.1) e s , , D,⇢0 x ,⇢0 ,⇢0 ,⇢0 x ,⇢0 ,⇢0 x ,⇢0 ,⇢0 v0 v0 F LaA SrA0 VA000 CoB VB000 OO VO·· h· ⇣ ⌘ Avariationofanyofthesevariablesinducesachangeintheinternalenergy:

ox ox ox x x ev =T sv + T : F + JE D + µLa ⇢La0 + µSr0 ⇢Sr0 + µV000 ⇢V0 0 0 · A A A A0 A A000

(3.2) + µCox ⇢0 x + µV ⇢0 + µOx ⇢0 x + µV ⇢0 + µh ⇢0 B CoB B000 VB000 O OO O·· VO·· · h· 31

where T ox is the absolute temperature of the oxide, T is the first Piola-Kirchho↵stress tensor, J =detF is the Jacobian of the transformation (also equal to the ratio of the volume of a cell in its deformed state to that in its non-deformed state J = dv/dv0),

@eox v x E is the electric field and µi = 0 is the chemical potential of specie i (i =LaA,SrA0 , @⇢i0 x x V000,Co ,V000,O ,V·· and h·). The symbol ” ”representstheclassicalscalarproduct A B B O O · while ”:” represents the tensorial scalar product.

g The internal energy density of the gas phase in the actual state ev is a function of

g b the entropy sv,thepressureinthebubbleP and the number density species: ⇢O2 and

⇢N2 —assuming nitrogen is the only other nonreactive gas. The internal energy of the gas phase is of the form:

g g b (3.3) ev sv,P ,⇢O2 ,⇢N2 Avariationofanyofthesevariablesinducesachangeintheinternalenergyof:

g g g b gas gas (3.4) e = T s P + µ ⇢O + µ ⇢N v v O2 2 N2 2

g gas gas where T is the temperature of the gas phase and µO2 and µN2 are the chemical potential of oxygen and nitrogen in the gas phase.

3.2.2. Equilibrium conditions

The thermodynamic equilibrium conditions are obtained using a Gibbsian variational approach, stating that the energy of an isolated system is at a minimum. An arbi- trary volume of the system containing both phases is first identified, as depicted V in Fig. 3.1. This volume is then isolated from the rest of the system and subject to 32 virtual perturbations. In order to do so, global constraints must be taken into account.

The condition of no heat flow translates into constant entropy, the absence of atomic

flux across the interface @ translates into constant number of atoms (O, La, Sr, Co V and N) and constant charge in the system [31]. Other constraints that need to be included are local constraints: electrostatics, lattice site conservation and mechanical considerations.

The total energy of the thermodynamic system defined by = + is the sum V Vox Vg of the internal energies of both phases

ox g (3.5) " = e dv0 + e dv +[surfaceterms] v0 v g ZZZVox0 ZZZV where ”[surface terms]” group all the integrals on the surfaces: @ and ⌃. Note that V for this problem, only the bulk equilibrium conditions are important, thus the surface integrals will not be explicitly treated.

As mentioned above, global thermodynamic constraints on the system must first be taken into account

(1) Constant entropy:

ox g (3.6) = s dv0 + s dv v0 v S g ZZZVox0 ZZZV

(2) Constant charge:

(3.7) = 3⇢La0 x +2⇢Sr0 +3⇢Co0 x + ⇢h0 2⇢O0 x dv Q A A0 B · O ZZZVox0 ⇣ ⌘ 33

(a) (b)

Figure 3.1. System under consideration for the derivation of the equilib- rium conditions: oxide and gas phase delimited by an arbitrary interface @ .Thethermodynamicdensitiesrelativetotheoxidearereferredtoa stressV free state, while those relative to the gas are referred to the actual state.

(3) Constant number of atoms:

(3.8) O = ⇢0 x dv0 +2 ⇢O dv OO 2 N g ZZZVox0 ZZZV

(3.9) La = ⇢La0 x dv0 N A ZZZVox0

(3.10) Sr = ⇢Sr0 dv0 N A0 ZZZVox0

(3.11) Co = ⇢Co0 x dv0 N B ZZZVox0

(3.12) N =2 ⇢N2 dv N g ZZZV

Nitrogen atoms remain in the gas and lanthanum, strontium and cobalt are

not soluble in the gas. 34

Those constraints are accounted for in the Lagrangian of the system:

(3.13) "⇤ = " T cS ONO LaNLa SrNSr CoNCo NNN

where " is the total energy of the system, defined by Eq. (4.10), Tc, o,andtheis—for i =O, La, Sr, Co and N— are the Lagrange multipliers associated with the constraints aforementioned. The first variation of this energy is given by

(3.14) "⇤ = " T c S o Q O NO La NLa Sr NSr Co NCo N NN

Substituting the expression of the internal energies, Eq. (4.5) and (4.9), and using the global constraints, (3.6)-(3.12), in that equation yields

ox ox "⇤ = [T Tc]sv + T : F + JE D +[µLax La 3eoc]⇢La0 x 0 · A A ZZZVox0 ⇢

+[µSr0 Sr 2eoc]⇢Sr0 + µV000 ⇢V0 A A0 A A000

x x +[µCo Co 3eoc]⇢Co0 + µV000 ⇢V0 B B B B000

x x +[µO O +2eoc]⇢O0 + µV·· ⇢V0 +[µh eoc]⇢h0 dv0 O O O O·· · · g g b + [T Tc]sv P g ZZZV ⇢ gas gas +[µ 2O]⇢O +[µ 2N]⇢N dv O2 2 N2 2 +[surfaceterms](3.15)

All the variations appearing in Eq. (3.15) are not independent, they are linked via local constraints. 35

First, every site of each sublattice must be occupied either by an atom or a vacancy, ie.

A (3.16) ⇢0 x + ⇢0 + ⇢0 = ⇢ LaA SrA0 VA000

B (3.17) ⇢0 x + ⇢0 = ⇢ CoB VB000

O (3.18) ⇢0 x + ⇢0 ,⇢0 = ⇢ OO VO·· h· where ⇢A, ⇢B and ⇢O are the number densities of sites on each of the sublattices. The perovskite structure further requires ⇢A = ⇢B = ⇢O/3. This imposes a relationship between the concentrations of the di↵erent elements.

Furthermore, the electric displacement must satisfy Gauss law in the oxide:

(3.19) D =3⇢La0 x +2⇢Sr0 +3⇢Co0 x 2⇢O0 x r· A A0 B O

Noting the electric potential, we can rewrite:

(3.20) E D = D = [ (D) ( D)] = (D)+( D) · r · r· r· r · r·

Using this decomposition, the integral involving the electric displacement in the ex- pression of " simplifies to [31]:

E Ddv0 = eo 3⇢La0 x +2⇢Sr0 +3⇢Co0 x 2⇢O0 x dv · A A0 B O ZZZVox0 ZZZVox0 n o (3.21) + D nda @ ox+⌃ · Z V 0 36

The last transformation involves the elastic term T : F = TjiFij using the Einstein notation. Using the divergence theorem, the integral on the elastic strain energy can be rewritten as

ox0 (3.22) TjiFijdv = Tjinj uida Tji,juidv ⌃ ZZZVox0 ZZ 0 ZZZVox0 where the index after the comma in Tji,j denotes a derivative with respect to the i-th component, nox0 is the normal to the interface pointing outward and u is the displacement vector.

Finally, using those local constraints, Eq. (3.16), (3.17), (3.18), (3.21) and (3.22) in (3.15), the first variation of internal energy of the system under the constraints is rewritten as

ox ox "⇤ = [T Tc]sv +(T )u +[⌘Lax La 3eoc]⇢La0 x 0 ·r A A ZZZVox0 ⇢

x x +[⌘Sr0 Sr 2eoc]⇢Sr0 +[⌘Co Co 3eoc]⇢Co0 A A0 B B

+[⌘Ox O +2eoc]⇢O0 x +[⌘h eoc]⇢h0 dv0 O O · · g g b + [T Tc]sv P g ZZZV ⇢ gas gas + µ 2O ⇢O + µ 2N ⇢N dv O2 2 N2 2 ⇥ ⇤ ⇥ ⇤ +[surfaceterms](3.23) 37 where more surface integrals have been added to the last term and the electrochemical potentials are defined as

(3.24) ⌘Lax = µLax µV +3eo A A A000

(3.25) ⌘Sr = µSr µV 2eo A0 A0 A000

(3.26) ⌘Cox = µCox µV 3eo B B B000

(3.27) ⌘Ox = µOx µV +2eo O O O··

(3.28) ⌘h· = µh· + eo

As all the variations in Eq. (3.23) are now independent, the bulk equilibrium con- ditions are read by setting the terms in brackets to 0:

the thermal equilibrium conditions imposes a uniform and constant tempera- • ture throughout the system:

ox g (3.29) T = T = Tc

the mechanical equilibrium condition imposes that •

(3.30) T =0 ·r

the chemical equilibrium condition states that the chemical potential of each • species is constant in the system, and specifically

x (3.31) ⌘LaA = La +3eoc 38

(3.32) ⌘ = +2e SrA0 Sr o c

x (3.33) ⌘CoB = Co +3eoc

(3.34) ⌘Ox = O 2eoc O

(3.35) ⌘h· = eoc

gas (3.36) µO2 =2O

gas (3.37) µN2 =2N

Because the crystal under study is a mixed ionic electronic conductor, we

can make the assumption that it is locally charge neutral, that is:

(3.38) 3⇢0 x +2⇢0 +3⇢0 x =2⇢0 x LaA SrA0 CoB OO

Using this new conditions in Eq. (3.23) simplifies the chemical equilibrium

condition to:

1 gas (3.39) µO = µ 2 O2

where µO is the chemical potential of oxygen in the oxide

(3.40) µO = µOx µV +2⌘h O O·· ·

Although such assumptions make the electrostatic term disappear from the

expressions, this is not in contradiction with the development of an electric 39

field [16]. This equilibrium is consistent with the reaction [37]

x . 1 (3.41) O +2h V·· + O (gas) O ! O 2 2

3.2.3. New free energy function and Maxwell’s equation

Applying a stress to an oxide changes its energy, which in turn a↵ects its chemical potential. This dependence can be determined by using a Maxwell equation for a free energy function [30–33]. This Maxwell equation is integrated from the initial (stress- free) state to the final (stressed) state yielding the oxygen chemical potential under stress.

We introduce a new free energy function for the oxide

ox ox ox (3.42) g = e Ts ij✏ij v0 v0 v0

where T is the temperature of the system, ✏ij and ij are the Eulerian strain and stress tensors that follow from standard linear elasticity, i, j =1, 2, 3andimplicitsummation over repeated indices from 1 to 3 is assumed. Thus, ✏ijij represents the scalar product of those two tensors.

Using the same description as above, the change in the internal energy of the oxide in the limit of small strain is [31,33]:

ox ox e =Ts + ✏ + ⌘ x ⇢ x + ⌘ ⇢ + ⌘ ⇢ v0 v0 ij ij OO OO VO·· VO·· h· h·

(3.43) + ⌘ x ⇢ x + ⌘ ⇢ + ⌘ ⇢ + ⌘ x ⇢ x + ⌘ ⇢ LaA LaA SrA0 SrA0 VA000 VA000 CoB CoB VB000 VB000 40

where ⌘i = µi + zi is the electrochemical potential of specie i, zi its charge (e.g. -2

x for i =OO)and is the electric potential. Note that the e↵ect of the electric energy has been factored into the electrochemical potentials. As mentioned in the previous paragraph, the variations present in this equation are not all independent, but are coupled via the conservation of sublattice sites, Eq. (3.16), (3.17) and (3.18), and the local charge neutrality, (3.38). Using those conditions in Eq. (3.43), the total derivative of the internal energy of the oxides simplifies to

ox ox (3.44) de = Tds + d✏ + µ d⇢ x + µ d⇢ x + µ d⇢ + µ d⇢ x v0 v0 ij ij O OO La LaA Sr SrA0 Co CoB

where µO is defined by Eq. (3.27). The variations in (3.44) are now all independent.

Using Eq. (3.44), one can evaluate the total derivative of the free energy gox,defined v0 by (3.42),

ox ox (3.45) dg = s dT ✏ijdij + µOd⇢Ox + µLad⇢Lax + µSrd⇢Sr + µCod⇢Cox v0 v0 O A A0 B

Noting that the number density can be linked to the oxygen non-stoichiometry, ,in

O O La1 xSrxCoO3 : ⇢Ox = ⇢ ⇢V = ⇢ (1 /3), the total derivative of the new free O O·· energy rewritten as

ox ox 1 O (3.46) dg = s dT ✏ijdij + ⇢ µOd + µLad⇢Lax + µSrd⇢Sr + µCod⇢Cox v0 v0 3 A A0 B 41

Finally taking the cross derivatives of the second and third terms yields a Maxwell relation involving the oxygen chemical potential:

@µ 3 @✏ (3.47) O = ij @ ⇢O @ ij T,kl=ij ,⇢i T,kl,⇢i ✓ ◆ 6 ✓ ◆

Knowing the constitutive equation for the strain, this equation can be integrated to provide the chemical potential of oxygen as a function of stress.

3.2.4. Chemical potential of oxygen under stress

Strain can result from numerous sources. Here we consider three such sources. One is a change in lattice parameter with temperature, in materials that have a nonzero thermal expansion coecient. Similarly, a change in the non-stoichiometry can induce stress. There can also be mismatch strain that is a result of placing a thin film of one lattice parameter coherently (continuous lattice planes) on a substrate with another lattice parameter. Accounting for all of these sources of strain, the relationship between strain and stress is,

c a T (3.48) ✏ij = Sijklkl + ✏ ()ij + ✏ ij + ✏ (T )ij where S is the compliance tensor, ✏c()=ec[ o]/3, ✏a and ✏T (T )arethemagnitude ijkl of the isotropic compositional, mismatch and thermal strains respectively. ec is the compositional strain coecient and is defined by Chen et al. [12]as

ˆ c 1 1 @ ln V (3.49) e = C = 3 3 @cV O·· !T,P 42 where Vˆ is the specific volume of the oxide and o is the non-stoichiometry at which the mismatch strain is computed. Using Eq. (3.48) in (3.47) yields

@µ ec O = @ ⇢O ij ij T,kl=ij ✓ ◆ 6 assuming the compliance tensor and compositional strain coecient are independent of non-stoichiometry. Integrating this equation between the oxide under a state of

o hydrostatic pressure P and stress, ij,yields

ec (3.50) µ (, )=µ (,P o)+ ( + P o ) O ij O ⇢O kk kk

o where summation over repeated indices from 1 to 3 is assumed, µO (,P )isthe chemical potential of oxygen at non-stoichiometry and under hydrostatic pressure of P o =1atm,whichismeasuredexperimentally. P o =1atmisassumedto | kk| be the case in the rest of the chapter, Eq. (3.50) simplifies to

ec (3.51) µ (, )=µ (, 0) + () O ij O ⇢O kk

where again µO (, 0) designates the bulk chemical potential of oxygen at under no stress. Thus, the chemical potential of oxygen changes linearly with the trace of the stress, with a direction that depends on the change in the volume of the oxide with vacancy concentration. Since the compositional coecient in mixed conducting oxides can be as large as ec 0.10 [12], stress may induce large changes in composition. Note ⇡ that the stress, kk(), is a function of the non-stoichiometry. Solving Eq. (3.48) for 43 the stress,

(3.52) ()=C ✏ ✏c () ✏a ✏T (T ) kl klmn mn mn mn mn highlights that dependence on non-stoichiometry. Cklmn is the sti↵ness tensor, the inverse of the compliance tensor Sijkl.

In order to determine the non-stoichiometry under stress, the dependence of µO on

in the absence of stress, µO(, 0), is needed. Various models for the chemical potential of oxygen in bulk LSC under no stress are available in the litterature [13,15,38]. Given the range of temperatures considered in this chapter, T 1073 K, Mizusaki et al.’s  form will be used [13]:

(3.53) µ (, 0) µ (o, 0) = ( ho (x) a(x)) T so (x)+R ln O O 4 O 4 O 3 ✓  ◆

o o where µO ( , 0) is the chemical potential of oxygen in bulk LSC at PO2 =1atm. ho (x), so (x)anda(x) are parameters (dependent on the substitution level, x) 4 O 4 O that are measured experimentally.

3.3. E↵ects of stress on the non-stoichiometry

In order to understand the origin of the stress-induced composition changes, imagine

o aslabofoxideinequilibriumwithanatmosphereatanoxygenpartialpressurePO2 , cf. Fig. 3.2(a). At equilibrium, the chemical potential of oxygen in the gas is equal to the chemical potential of oxygen in the oxide, thus, giving rise to a non-stoichiometry in the stress-free state, o. Now, applying a stress on the slab deforms it as roughly depicted in Fig. 3.2(b), which in turn changes the chemical potential of oxygen in the 44

(a) Schematic of a slab of oxide in equi- (b) Applying a stress (in this case, librium with oxygen gas at P o in the compressive, but not necessarily hydro- O2 reference state. Under no stress, the static) to the oxide deforms it, com- oxygen non-stoichiometry is o. pared to the initial configuration (light grey rectangle). The oxygen in the crystal then equilibrates with that in the gas giving rise to a new non- stoichiometry, .

Figure 3.2. Thought experiment to understand the e↵ect of stress on the non-stoichiometry.

oxide to some new value. The pressure of the gas is also changed to a di↵erent value

PO2 , and at equilibrium a new non-stoichiometry, ,results. At equilibrium, the chemical potential of oxygen in the oxide is equal to the chemical potential of molecular oxygen in the gas. Since the oxide is in equilibrium in both the stress-free and stressed states, Eq. (3.39) applies in both cases,

o 1 gas o (3.54) µO ( , 0) = µ P 2 O2 O2

1 gas (3 . 5 5 ) µO (,ij)= µ (PO )( 2 O2 2 45

Using Eq. (3.51) in (3.55) yields:

c 1 gas e (3.56) µ (PO )=µO (, 0) + kk() 2 O2 2 ⇢O

Subtracting Eq. (3.54) from this last equation results in

c 1 gas 1 gas o o e (3.57) µ (PO ) µ P = µO (, 0) µO ( , 0) + kk() 2 O2 2 2 O2 O2 ⇢O 1 µgas Assuming an ideal gas and using the expression of 2 O2 from Eq. (3.53) in (3.57) finally yields

RT P ln O2 =( ho (x) a(x)) 2 P o 4 O ✓ O2 ◆ 2ec (3 . 5 8 ) T so (x)+R ln + ()( 4 O 3 ⇢O kk ✓  ◆

This equation shows that the non-stoichiometry is a function of both the oxygen pres- sure and the stress. In most experiments, the composition-independent part of the stress, e.g. the mismatch strain, remains constant and the non-stoichiometry is re- ported as a function of the oxygen pressure.

3.4. E↵ects of stress on the vacancy formation energy

One valuable way to consider the e↵ects of stress on the nonstoichiometry of an oxide is to consider the oxygen vacancy formation energy. This is frequently computed using first-principles methods wherein a block of oxide is stressed and the change in energy on adding an oxygen vacancy is considered. This energy change can be related 46 to the chemical potential discussed above and can be determined using first-principles calculations [29,35,39–41].

The energy of formation of a vacancy is given by [40]:

1 (3.59) Eo = G + G G f,vac crystal+vac 2 O2 crystal

where Gcrystal and Gcrystal+vac are the Gibbs free energies of the block of oxide — at a given non-stoichiometry— with no extra vacancy and with one extra oxygen

vacancy —and two extra holes— respectively, while GO2 is the Gibbs free energy of an oxygen molecule. The factor 1/2 accomodates for the fact that the Gibbs free energies correspond to the exchange of one oxygen ion, while a molecule of oxygen is composed of 2 oxygen atoms. The total Gibbs free energy of the block of oxide under no stress can be expressed as the sum of chemical potentials of its constitutive elements [42]

x N N x N NLaA SrA0 VA000 NCoB VB000 G =µ x + µ + µ + µ x + µ crystal LaA SrA0 VA000 CoB VB000 NA NA NA NA NA x NOO NVO·· Nh· x + µOO + µVO·· + µh· NA NA NA where the N s are the number of atoms of each species and is Avogadro’s number. i NA Using the conservation of lattice sites on each sublattice

A N x + N + N = N LaA SrA0 VA000

A N x + N = N CoB CoB000

A x NOO + NOO·· =3N 47 where N A is the number of A sites in the crystal and noting that there are 3 oxygen sites per unit cell, and charge neutrality: 3N x +2N +3N x +2N x = N in this LaA SrA0 CoB OO h· equation yields

x N x NLaA SrA0 NCoB Gcrystal =µLa + µSr + µCo NA NA NA A N NOx (3.60) + 3µ + µ + µ + µ (, 0) O VO·· VA000 VB000 O NA NA ⇥ ⇤ where µO(, 0) is the chemical potential of oxygen in the bulk under no stress, defined by Eq. (3.27) and µLa, µSr and µCo are defined by (3.24), (3.25) and (3.26). Assuming that adding one extra vacancy to the crystal has a negligible impact on the non- stoichiometry, the Gibbs energy of the oxide with one extra vacant site reads:

x N x NLaA SrA0 NCoB Gcrystal =µLa + µSr + µCo NA NA NA A N (NOx 1) (3.61) + 3µ + µ + µ + µ (, 0) O VO·· VA000 VB000 O NA NA ⇥ ⇤ Using Eq. (3.60) and (3.61) in (3.59) finally yields the vacancy formation energy for a given composition under no stress:

1 1 (3.62) Eo = µ µ (, 0) f,vac 2 O2 O NA NA

If the gas was taken such that it gave rise to the non-stoichiometry ,thisvaluewould

be 0 —cf. Eq. (3.39). The formation energy of a vacancy under stress, Ef,vac,canbe computed in a similar manner. Taking the crystal to be at the same non-stoichiometry,

,andkeepingtheappliedstressconstantbetweentheconfigurationswithandwithout 48 an extra vacancy, =cst,theelasticenergycancelsoutbetweenthoseconfigurations and

1 1 (3.63) E = µ µ (,) f,vac 2 O2 O NA NA

Assuming a constant oxygen gas pressure, the change in the vacancy formation energy between a state under stress and a stress-free state while keeping the nonstoichiometry constant is given by the di↵erence between Eq. (3.62) and (3.63):

1 1 ec (3.64) E = E Eo = µ (,)+ µ (, 0) = f,vac f,vac f,vac O O ⇢O kk NA NA NA

Under such assumptions, the change in the vacancy formation energy is of the same sign as the stress. If an increase in vacancy concentration increases the lattice parameter, as in many oxides [12], a compressive stress —kk < 0— yields an increase in the vacancy formation energy, which then results in a smaller equilibrium concentration of vacan- cies. In addition, within the assumptions used above of a stoichiometry-independent solute expansion coecient, and elastic constants, the change in vacancy formation energy is an odd function of stress and can be computed using only thermodynamic information from the stress-free state and the trace of the stress.

3.5. E↵ects of stress on the chemical capacitance

Electrochemical impedance spectroscopy (EIS) is an experimental method of char- acterization, during which the impedance of a system is recorded over a range of fre- quencies. Such method is used to characterize e.g. fuel cells, thin films or batteries. 49

Processes taking place in those devices can be modeled with equivalent circuits. Fit- ting this impedance data allows to quantify the underlying processes and to compare them among di↵erent devices. In the case of a thin film where the ionic and elec- tronic bulk resistances (due to thickness) are small compared to the surface reaction resistance, such device can be represented by a chemical capacitance in parallel with aresistance[10, 43]. This capacitance, Cchem,ischaracteristicofthechargeinthe

film, which is due to oxygen non-stoichiometry. It can then be used to compute the non-stoichiometry of the oxide, ,e.g.[10]. Using the results above it is possible to express Cchem,aquantitythatisdirectlymeasuredexperimentally,asafunctionofthe stoichiometry and stress.

Unless perfectly lattice-matched, thin films deposited on a substrate are usually in a state of lattice-mismatch induced stress. Consider a thin film on a substrate that is under biaxial stress, ⇤,andstoichiometry⇤ in equilibrium with oxygen of

partial pressure PO2 .Thisistheinitialconfiguration.EISconsistsofapplyinga sinusoidal voltage between the oxide and the substrate and measuring the resulting time-dependent current flowing through the sample. Comparing those two signals yields the impedance as a function of frequency. Applying an electric potential to the oxide drives oxygen ions into or out of the oxide, displacing momentarily the non- stoichiometry, ,awayfromtheinitialone,⇤. This change in composition changes the lattice parameter and thus the stress in the film. The chemical equilibrium condition,

Eq. (3.39), requires the non-stoichiometry to return to its initial value by an exchange of oxygen atoms with the surrounding atmosphere. The kinetics of the return to the initial configuration, ⇤,isdictatedbythatimpedance. ! 50

In [43], Adler derives the chemical capacitance of a thin film in the case of surface- limited kinetics, common to many SOFC anodes and cathodes. His approach is followed to determine the stress-dependence of the chemical capacitance. The driving force for incorporation of oxygen at the surface of the oxide is the change in a free energy function under the constraints of constant entropy, pressure, and number of oxygen atoms in the system. Furthermore, the composition and stress field are taken to be uniform in the oxide thin film, since the rate limiting step is assumed to be the incorporation of oxygen at the surface. The film is bonded to the substrate and cannot slide along the interface. Since the lattice parameters of the film and substrate are di↵erent, the film

is under biaxial stress.The film is in contact with a gas at pressure PO2 that induces strains in the substrate that are very small compared to the lattice mismatch strain.

Oxygen incorporation results in a change in the lattice parameter. Since the lattice can only expand normal to the substrate and the pressure of the gas is low (1 atm or below), this expansion does no work. The driving force is simply the displacement of the chemical potential of oxygen in the oxide under stress from equilibrium with the gas

1 gas 1 1 (3.65) D = µO (,) µ = µO (,) µO (⇤,⇤) 2 O2 2 2

where ⇤ is the stress in the film at a non-stoichiometry ⇤. Following Adler’s approach

[43]withthisnewdrivingforce,thechemicalcapacitanceis

4F 2L⇢O (3.66) C = chem 3RTf 51 where L is the thickness of the film and f is

1 @µ (3.67) f = O RT @ =⇤ Again, the expression of the chemical capacitance derived above is valid for a compo- sitionially uniform system in a homogeneous stress field. Using the expression for the stress-dependent chemical potential, Eq. (3.51) in (3.67) and (3.66) yields:

2 O c 1 4F L⇢ 3RT e @kk (3.68) Cchem = a(x)+ O 3 ⇤(3 ⇤) ⇢ @ ✓ =⇤ ◆ We note that ⇤ is the non-stoichiometry of the oxide under stress in equilibrium with

oxygen at PO2 . The chemical capacitance is a function of stress through two e↵ects. The non-stoichiometry of the oxide under stress can be di↵erent from that in the absence of stress. This a↵ects the chemical capacitance through the presence of the ⇤ terms. The chemical capacitance will also vary with stress explicitly since the lattice parameter of the oxide varies with composition —that is if the compositional expansion coecient ec is nonzero. The term involving the derivative of the trace of the stress with respect to non-stoichiometry captures the energy change required to add an atom in a distorted lattice.

3.6. Comparisons and predictions

Applications of the model are considered in this section. First, the e↵ects of a hydrostatic stress on the non-stoichiometry and on the conductivity are examined.

Then, a thin film configuration is considered: the types of stresses developed in such 52 configurations are briefly presented before comparing predictions given by the model to experimental measurements and ab initio calculations.

3.6.1. E↵ects of a hydrostatic stress on the properties of La0.8Sr0.2CoO3

La0.8Sr0.2CoO3 (LSC-82) is considered as an example in here. Using Eq. (3.53) with the appropriate coecients —c.f. Table (3.1), the non-stoichiometry of LSC-82 in the

o o stress-free state at T =1073KandPO2 = 1 atm is computed to be =0.0059. The oxygen pressure is further assumed to be equal in the initial and final states,

o ie. PO2 = PO2 .Thee↵ectsofahydrostaticstressareconsideredhere. 3.6.1.1. Small changes in non-stoichiometry. Aqualitativeideaofthee↵ectsof stress on the non-stoichiometry can be obtained by considering small changes in from the stress-free value and in the simple hydrostatic stress case, where 11 = 22 = 33.

o Assuming that the stress is applied at constant gas pressure, setting PO2 = PO2 in Eq. (3.57) yields the equation governing the non-stoichiometry as a function of stress:

ec (3.69) µ () µ (o)= () O O ⇢O kk

Using Eq. (3.52), the stress can be written as the sum of a stoichiometry-independent

o kl( )andastoichiometry-dependentterms,

ec ()=C ✏ ✏a ✏T (T ) ( o) C kl klmn mn mn mn 3 klmn mn ec (3.70) = (o) ( o) C kl 3 klnn 53

The stoichiometry dependence of the stress, and thus the right hand side of Eq. (3.69), is now explicit. (o)canalsobeenseenasthestressappliedtothereferencestate.

The chemical potential is a nonlinear function of ,see(3.58),thustosolve(3.69)for

,weneedtoexpandthechemicalpotentialstofirstorderin o:

c c 2 1 o e @µO (e ) E o (3.71) = O + O kk ( ) ⇢ @ o ⇢ 3(1 2⌫)  = where E is Young’s modulus and ⌫ is Poisson’s ratio [44]. In most cases, @µO/@ <

0andincreasingvacancyconcentrationexpandsthelattice,ec > 0. Thus, non- stoichiometry, ,decreasesunderacompressivestress(kk < 0) and conversely, increases under a tensile stress. Furthermore, using parameters from Table 3.1 and a temperature of T =1073K,onecanestimatetheratioofthetwotermsinthebrackets of Eq. (3.71) as c 2 O (e ) E/(3(1 2⌫)⇢ ) 2 10 @µO ⇡ @ =o As mentioned earlier, a change in the non-stoichiometry results in both a change in the chemical potential of oxygen in the oxide as well as a change in the stress of the oxide

—via the compositional strain. Such a small ratio means that the latter e↵ect is small compared to the change in the chemical potential with ,forthisparticularoxideand non-stoichiometry. However, this may not be true for other oxides in which the change in the oxygen chemical potential of the oxide with non-stoichiometry is smaller. As a result, to a good approximation, Eq. (3.71) can be further simplified:

c 1 e @µO o 13 o (3.72) o = O kk ( )=6.5 10 kk( ) ⇢ @ o ⇥  = 54

o o where kk ( ) is expressed in Pa. Hence, a compressive stress of 100MPa, kk ( )=

5 0.1GPa, induces a change in non-stoichiometry of = 5 10 ,corresponding o ⇥ to a 1% relative change. The non-stoichiometry is directly proportional to the mole fraction of oxygen va-

cancies, via the lattice constraint cVO·· = /3, and is linked to the mole fraction of holes in the system, via local charge neutrality. Expressed in terms of the relative charge of each site, this charge neutrality relation is ⇢ +3⇢ +3⇢ =2⇢ + ⇢ where the SrA0 VA000 VB000 VO·· h·

⇢is denote the number density of the various species. Given the crystal structure, the lattice imposes 3 times as many oxygen sites as A or B sites. Dividing by the number density of sites on the A sublattice and neglecting the vacancy concentration on both cation sublattices, local charge neutrality further simplifies to

(3.73) c = x 2 h·

for La1 xSrxCo3 O, where ch· is the fraction of holes per B sublattice sites. Hence, both the vacancy concentration change, cV ,andtheelectroniccarriersconcentration 4 O·· change, c ,canbeevaluatedforagivenstress. 4 h· Assuming the ionic and electronic mobilities don’t change significantly with the stress, and assuming the ionic and electronic conductivity to be proportional to the concentration oxygen vacancies and holes, respectively, the relative change in conduc- tivity —between the stressed and the stress-free states— is equal to the relative change in carriers, that is

c 1 cV e @µ ion 4 O·· O o 4 o = o = O o kk ( )= 1%(3.74) ion cV ⇢ @ =o O··  55

1 c elec ch· 2e @µO o 4 o = 4o = O o kk ( )=.2%(3.75) c ⇢ (x 2 ) @ o elec h·  = for a stress (o)= 0.1GPa. Becausetheconductivitiesareintrinsicallysohigh, kk such e↵ect is likely negligible in most SOFC applications.

3.6.1.2. Larger changes in non-stoichiometry. If pressures are too large, the deviation from equilibrium can be significant and the linearization made above does

o not hold. One must then solve the equations numerically. Setting PO2 = PO2 in Eq. (3.58) yields a nonlinear equation for the non-stoichiometry in LSC-82

ec (3.76) ( ho a) T so + R ln = () 4 O 4 O 3 ⇢O kk ✓  ◆ where the coecients ho , so ,anda are evaluated for a substitution level of x = 0.2. 4 O 4 O Conducting the same analysis as in the previous paragraph, the composition-dependent part of the stress is shown to be negligible compared to the other terms in the equation

o above. The stress tensor becomes independent of composition and kk()=kk( )in

Eq. (3.76).

o The variations of with kk ( ) are plotted in Fig. 3.3 for both the linearized —

Eq. (3.72)— and the exact form —Eq. (3.76)— using the values found in Table 3.1 for a temperature T =1073K.Sinceceramicsarenotpronetofractureincompression,the

o calculations extends much more with compression (kk ( ) < 0) than with tension. The graph shows that a change in is dictated by the sign of the stress: decreases under compression and increases under tension. Note the amplitude of the variations: a large compressive stress, = 5GPa,decreasesthenon-stoichiometrybyapproximatelya kk factor 2. Furthermore, the change in is roughly exponential in the trace of the stress, 56

0.01

−o −

exact linearized 0.001 −10 −5 0 o σkk (− ) (GPa)

Figure 3.3. Log of the non-stoichiometry of LSC-82 as a function of the o trace of the stress in the reference state, kk( ), in the hydrostatic case o at T =1073K,underconstantoxygenpartialpressurePO2 = PO2 = o 1atm. Negative values of kk( )correspondtocompressivestresses while positive values correspond to tensile stresses. since the curves are nearly straight lines near zero stress. As a result, non-stoichiometry would increase by roughly a factor 2 for a tensile stress of kk =5GPa.Comparing the linearized solution to the exact one show a good agreement for stresses less than

3GPainabsolutevalueanda largediscrepancyforstresseslargerthanthatvalue. ⇡

3.6.2. Thin Films

Unlike the case of a hydrostatic stress, stressed thin films experience nonzero biaxial stress. It is assumed that the chemical expansion coecient is purely dilational and as aresulttheoxygenchemicalpotentialcouplesonlytothetraceofthestress.Thefilm is taken to be suciently thin that the substrate is infinite. Stress can be present in the oxide thin film as a result of multiple sources of strain, such as the lattice parameter 57

Table 3.1. Parameters for LSC-82 used in Eq. (3.76) and (3.68) to com- pute the change in non-stoichiometry due to stress in LSC-82: lattice constant, compositional coecient, Young’s modulus, Poisson’s ratio and parameters used in the itinerant electron model.

Parameter Value 1 aLSC (nm) .3833 [45] ec .129 [12] E (GPa) 160 [35] ⌫ .25 [11] o hO (kJ/mol) -146 [13] 4 o sO (J/mol) -86.6 [13] a4(kJ/mol) 418 [13]

1The number density of oxygen lattice sites is ⇢O =3/( (a )3), where is Avogadro’s constant. NA LSC NA

di↵erence between the crystals, a change in the composition of the film and thermal expansion. These sources of stress each can a↵ect the chemical capacitance and degree of non-stoichiometry of the film.

3.6.2.1. Sources of Strain. As mentioned above, possible sources of stress consid- ered here are thermal, compositional and lattice mismatch between the film and sub- strate. For the sake of simplicity, linear isotropic elasticity is assumed to hold.

The strain in the film can be a result of:

(1) lattice mismatch strain. This arises when the lattice parameters of the oxide

and substrate are di↵erent. This is the strain experienced by the oxide to make

the lattice parameter of the oxide match that of the substrate. Assuming that

both the film and substrate are cubic:

a aox as a (3.77) ✏ij = ij = ✏ ij as 58

where aox and as are the lattice parameters of the oxide film and of the sub-

strate respectively. In many cases the lattice parameters of the two phases can

change with the degree of non-stoichiometry. Thus, the lattice parameters are

o taken to be those at a temperature To and non-stoichometry .

(2) compositional strain. The lattice parameter of oxides depend strongly on their

oxygen content, this strain arises when the oxygen composition of the oxide

is di↵erent from that in the reference state. Assuming that only the lattice

parameter of the film varies with ,thatthelattticeparameterofthesubstrate

remains unchanged, and a cubic crystal, the strain in the film varies with as,

ec (3.78) ✏c = ( o) = ✏c() ij 3 ij ij

where ec is defined by Eq. (3.49).

(3) thermal strain. When the coecients of thermal expansion (CTE) of the oxide

and substrate are di↵erent, changing the temperature generates strain in the

film. Assuming a cubic crystal for both film and substrate, the strain in the

oxide is due to the di↵erence in thermal expansions of the two materials, i.e.

(3.79) ✏T =(↵ ↵ )(T T ) = ✏T (T ) ij ox s o ij ij

where ↵ox and ↵s are the CTE of the oxide film and the substrate respectively,

T T is the change in temperature from that at which the lattice parameter o in Eq. (3.77) is defined. 59

3.6.2.2. Possible configurations. In thin films, the strains mentioned above com- bine to induce a stress that, in turn, a↵ects the non-stoichiometry. It will be assumed that oxides deposited as thin films on a substrate have a displacement imposed by the substrate in the plane of the film, yielding a film under biaxial strain. The surface of the film is stress-free since it is in contact with a gas at very low pressure. It can be shown (e.g. [46]) that in the thin film configuration, assuming the crystal to be elastically isotropic and no slip at the interface with the substrate, the trace of the stress in the film is

(3.80) (T,)= 2Y ✏c ()+✏a + ✏T (T ) kk where ✏c ✏a and ✏T are the amplitudes of the compositional, mismatch and thermal strains, as defined by Eq. (3.77)-(3.79), and Y is an e↵ective modulus, defined as

E (3.81) Y = 1 ⌫

When the lattice parameters of the oxide and substrate are close but not equal, the thin film can grow coherently and dislocation-free. Once grown, the thin film will be (biaxially) strained in the in-plane directions and stress-free in the normal direction. This is particularly true for very thin films, as dislocations will start forming in thicker films. Fig. 3.4 illustrates the di↵erent steps to achieve such configuration in an isothermal system.

(a) In the stress-free state, the lattice parameter of the oxide, aox,isdi↵erentfrom

o that of the substrate, as.Theoxygennon-stoichiometryinthisstateis . 60

(b) The oxide film is isotropically stressed to fit the substrate. The lattice of the

film now matches that of the substrate.

(c) Assuming the substrate to be infinite, it applies the in-plane stress to make

the lattice parameters match. However, the oxide is free to move in the normal

direction, it relaxes.

(d) Oxygen equilibrates between the film and the atmosphere, the non-stoichiometry

changes, which, in turn, changes the stress applied by the substrate yielding

the final non-stoichiometry ⇤.

The reference state for the stress and strain is as,thelatticeparameterofthesubstrate.

In this state of absence of displacement along the interface, the stress is proportional to the sum of the lattice mismatch ✏a (at zero stress) and the compositional strain ✏c:

a c (3.82) (⇤)= 2Y (✏ + ✏ (⇤)) kk

Another possible configuration for the thin film is to be in a stress-free state at the firing temperature and the change of temperature to the experimental conditions induces a biaxial strain on the oxide. This is often assumed to be the case when depositing a layer on top of a substrate where the lattice parameter of film and substrate are very di↵erent. The di↵erent steps to determine the stress in the film are depicted

Fig. 3.5.

(a) At the firing temperature Tf ,theoxideandsubstrateareintheirstress-free

state. The oxygen non-stoichiometry in this state is o.

(b) The oxide thin film is grown on the substrate such that the film is fully relaxed. 61

Figure 3.4. Schematic of the change in non-stoichiometry in a coherent and dislocation-free thin film grown on a substrate with a lattice mis- match. (a) Stress-free state: the lattice parameters of the oxide and the substrate are di↵erent, aox = as.(b)Thefilmisisotropicallystrainedto make the lattice parameters6 equal. (c) The oxide in-plane displacement is set by by the substrate, but it relaxes in the normal direction. (d) Oxygen equilibrates with the atmosphere resulting in ⇤ and modifying the stress applied by the substrate.

(c) The temperature is changed to T

di↵erent rates. Assuming the substrate to be infinite, it imposes the in-plane

displacement of the film, while the top surface of the film is free to relax. In

the present example, the oxide was assumed to have a larger CTE than the

substrate which results in a tensile strain in the oxide.

(d) Oxygen equilibration with the atmosphere induces a change in that then

decreases the stress applied by the substrate. The final non-stoichiometry of

the film is ⇤. 62

Figure 3.5. Schematic of the change in non-stoichiometry in a thin film grown on a substrate under thermal stress. (a) At the firing temperature Tf , the oxide and substrate are stress-free. (b) The thin film is grown on the substrate, stress-free. (c) Changing the temperature, the two materials shrink, but at di↵erent rates. The substrate imposes the in- plane displacement of the film, while top surface relaxes. (d) Oxygen equilibrates with the atmosphere resulting in ⇤ and modifying the stress applied by the substrate.

Assuming no sliding at the interface and an infinite substrate, the stress in the film is

T proportional to the sum of the thermal strain ✏ and the compositional strain ✏c and

T (3.83) (T,⇤)= 2Y (✏ (T )+✏(⇤)) kk

Finally, ⇤ is the non-stoichiometry of the film in equilibrium with oxygen at pres-

sure PO2 under biaxial stress, given by one of the configurations just described. Using the oxygen pressure in the final state along with the expression of the trace of the stress, Eq. (3.82) or (3.83), into (3.58) for = ⇤,yieldsthatnon-stoichiometry⇤. 63

3.6.2.3. Chemical capacitance of LSC films. Kawada et al. [10] have measured the chemical capacitance as a function of oxygen partial pressure of a 1.5µmLa0.6Sr0.4CoO3

(LSC-64) thin film grown on a Ce0.9Gd0.1O1.95 (GDC-91) substrate. Given the absence of literature on bulk LSC-64, Kawada et al. interpolated the parameters —a, ho 4 O and so —fromMizusakiet al. [13]forcomparison.Thosecoecientswereusedin 4 O Eq. (3.53) to obtain the non-stoichioimetry as a function of oxygen pressure. Using

Alder’s work [43], the chemical capacitance for stress-free LSC was then estimated as a function of oxygen pressure. The estimations for bulk LSC-64 and the thin film’s measurements are reported in Fig. 3.6 for a temperature T =873K.Thechemical capacitance in the film is significantly lower than that computed for the stress-free

LSC-64, by factors of 2 to 10 depending on the oxygen pressure. It has been suggested that this large di↵erence in chemical capacitance is due to the stress in the film [10].

Assuming the thin film to be coherent and dislocation-free, the trace of the stress is given by Eq. (3.82). Using this in (3.68) simplifies the expression of the chemical capacitance

1 4F 2L⇢O 3RT 2Y (ec)2 (3.84) C = a(x)+ + chem 3 (3 ) 3⇢O ✓ ⇤ ⇤ ◆ where ⇤,thenon-stoichiometryinthefilm,isnumericallyevaluatedusingEq.(3.82)

in (3.58) for a given PO2 . All the parameters used for the calculations are collected in Table 3.2. Under such conditions, the lattice mismatch strain at T =873Kis

✏a =2.23%. The predictions of the chemical capacitance under stress are plotted in

Fig. 3.6 using a solid green line. The sensitivity of the predictions with a, ho and 4 O 64

100 ) 2

-1

(F/cm 10 chem C LSC-64 bulk LSC-64 thin film LSC-64 under stress LSC-73 under stress LSC-55 under stress 10-2 -6 -4 -2 0 log P (atm) O2

Figure 3.6. Chemical capacitance versus oxygen partial pressure at T = 873 K estimated for bulk La0.6Sr0.4CoO3 [10, 13](solidblueline),as reported by Kawada et al. in a 1.5µm-thick LSC film on GDC [10](red symbols) and that in presence of stress (solid green line). The dotted lines o o represent the predictions when using coecients —a, hO or sO—cor- responding to the thermodynamics of LSC-73 and LSC-554 respectively.4 The composition of the bulk material is assumed to be well characterized and thus the coecients are fixed at the measured values.

so is also reported in Fig. 3.6 using doted lines, where the coecients correspond to 4 O the thermodynamics of La0.7Sr0.3CoO3 (LSC-73) and La0.5Sr0.5CoO3 (LSC-55). The plot shows that a positive lattice mismatch decreases the chemical capacitance of the oxide, in agreement with experimental measurements. At a given oxygen partial pressure, the chemical capacitance of the oxide under stress is predicted to be smaller

than that of the bulk, with a decrease that is larger for high PO2 .Infact,atlow oxygen pressure, the left hand side of Eq. (3.58) can be of the same order of magnitude as the stress term making ,andthereforeCchem,muchmoresensitivetostressat 65

Table 3.2. Parameters used to compute Cchem as a function of PO2 — Eq. (3.58) and (3.68)— to compare to Kawada et al. measurements [10], Fig. 3.6: lattice parameter temperature of the measurement ,coe- cient of thermal expansion, compositional{ coecient, Young’s modulus,} Poisson’s ratio and parameters used in the itinerant electron model.

Parameter GDC-91 LSC-64 LSC-55 LSC-73 2 12 abulk (nm) .5407 [47] .3874 [48] idem idem TinK 1673 900 { } { 6}{6} CTE 12 10 [49]20 10 [12]idemidem c 1 ⇥ ⇥ e (K ) 0.129[12]idemidem E (GPa) -160[35]idemidem ⌫ -.30idemidem o 3 hO (kJ/mol) --85.8-70.7 [13]-112[13] 4 o 3 sO (J/mol) --69.4-64.4 [13]-112[13] 4a (kJ/mol) -2893 222 [13]385[13]

2The [110] direction of LSC aligns with the [100] direction of GDC, to give a lattice mismatch strain a p2a a of ✏ = LSC GDC =2.23% at T = 873 K. aGDC 3Interpolated by Kawada et al. [10] oxygen pressures close to 1atm. However, the magnitude of the change in the chem- ical capacitance from its stress-free value is smaller than in the experiments. Lastly, predictions are sensitive to the values of the bulk coecients: a, ho and so ,as 4 O 4 O illustrated by the magnitude of the change in the chemical capacitance with variations in those coecients and a good agreement is found between experiment and theory for a LSC-55 film, instead of that assumed in the experiment of LSC-64. Reasons for such discrepancies are discussed in the next section.

AsimilarcomparisoncanbemadewiththeresultsofLaO’et al. [11, 50]. They measured the chemical capacitance of La0.8Sr0.2CoO3 (LSC-82) films of thicknesses 20, 45 and 130nm grown on a 8 mol% yttria-stabilized zirconia (YSZ) substrate for atemperatureT =793K.ThevaluesforbulkLSC-82arecomputedasdescribed above, where values for the parameters are available in the literature [13]. Results 66 are plotted in Fig. 3.7. The capacitance of their thin film is significantly increased compared to the bulk. La O’ et al. also report the in-plane lattice parameter in the

film at room temperature, that is di↵erent from the bulk one, confirming that the

film is under stress. Comparing this value to that in the relaxed bulk yields a lat- tice strain at room temperature ✏a =(a a )/a . Accounting for the thermal bulk film film strain arising from the high temperature of the experiments, the mismatch strain is

✏a(T =793K)=✏a(T =298K)+(↵ ↵ ) T = 1.01%, 1.64% and 0.30% LSC YSZ 4 for the 20, 45 and 130 nm films respectively, proving that the film is not coherent with the substrate, but has been partially relaxed by the formation of interfacial dislocations or other defects. As the thermal strain has already been taken into account, the trace of the stress is given by Eq. (3.82). Using the same procedure as above with a mis- match strain of ✏a = 1.64%, we can numerically evaluate the capacitance of LSC-82 under stress as function of oxygen pressure for the 45nm film. Such results are given in Fig. 3.7. The parameters used for calculations are shown in Table 3.3. It should be noted that the curves of Cchem versus oxygen pressure in the present case appear much straighter than in the case of Kawada et al. [10]whichdisplaytworegions(highand

low PO2 ). Such e↵ect is due to the di↵erent Strontium concentrations and its e↵ect on the coecients in the expression of the chemical potential. This figure demonstrates that the predicted chemical capacitance of an oxide under a negative mismatch is in- creased compared to the bulk, in agreement with the experimental results. However, the quantitative agreement is poor. To illustrate the high sensitivity of the predictions 67

104

103 ) 3

2

(F/cm 10 chem C

1 10 LSC-82 bulk LSC-82 film LSC-82 under stress LSC-73 under stress 100 -4 -3 -2 -1 0 log P (atm) O2

Figure 3.7. Chemical capacitance versus oxygen partial pressure at T = 793 K evaluated for bulk La0.8Sr0.2CoO3 [13](solidblueline),asre- ported by la O’ et al. in a 45nm-thick LSC film on YSZ [11](redsymbols) and the predicted values using a lattice strain of -1.64% (solid green line) The dotted green line is the predictions for a LSC-73 film.

on the values of the coecients of the chemical potential, di↵erent strontium substitu- tion were considered, to bracket LSC-82. The result for LSC-73 is represented by the dotted green line in Fig. 3.7, while the result for LSC-91 is below the graphic window.

3.6.2.4. Vacancy formation energy. Using ab initio calculations, Kushima et al. [40] studied the oxygen vacancy formation energy in LaCoO3 as a function of biaxial strain while Donner et al. [35] investigated the change in the vacancy formation energy due to strain in La0.875Sr0.125CoO3 for an epitaxial configuration. They both find that a tensile strain decreases the formation energy. Using the relationship between the trace

app of the stress and the applied strain in a biaxial configuration, kk =2Y✏ [44], in

Eq. (3.64), the change in the energy of formation relative to a stress-free state can be 68

Table 3.3. Parameters used to compute Cchem as a function of PO2 — Eq. (3.58) and (3.68)— for comparisons with La O’ et al. work [11]: lattice parameter in the bulk at room tempetarure, coecient of thermal expansion, compositional coecient, Young’s modulus, Poisson’s ratio and parameters used in the itinerant electron model.

Parameter YSZ LSC-82 arelax (nm) -.3837[45] 1 6 6 CTE (K ) 10 10 [51]17 10 [12] ec ⇥ 0.112[⇥ 12] E (GPa) -160[35] ⌫ -.25[11] o hO (kJ/mol) --146[13] 4 o sO (J/mol) --86.6[13] 4a (kJ/mol) -418[13]

computed as a function of strain and is

2ecY (3.85) E (eV) = = 0.059 ✏app(%) f,vac ⇢O q kk ⇥ NA

c using a chemical expansion coecient e = .112 (corresponding to La0.8Sr0.2CoO3 in [12]), a number density of oxygen sites ⇢O =84 103 mol/m3 (corresponding to a ⇥ lattice parameter of a =0.39 nm [39]), an e↵ective modulus Y =213GPa(correspond- ing to a Young’s modulus E =160GPa[35]andaPoisson’sratioof⌫ = .25 [11]) and

19 the conversion factor q =1.6 10 J/eV. As a result, a 4% biaxial tensile strain, ⇥ ✏app =4%,inducesachangeinthevacancyformationenergyofE = 236 meV, f,vac consistent with the 300 meV and 500 meV changes reported for such strain by Don- ner et al. [35]andKushimaet al. [40] respectively. As stated earlier, this means that atensilestrainfacilitatestheformationofvacancies,henceincreasingtheirconcen- tration. Note however, that the model does not account for the strong jump in the 69 vacancy formation energy seen in [40]forsmallertensilestrains,anditassumeslinear elasticity to hold.

3.7. Discussion

3.7.1. LSC thin films

As illustrated in the previous section, the model does a good job in qualitatively capturing the shift in chemical capacitance due to stress in LSC. Because the non- stoichiometry and chemical capacitance vary similarly with stress, the comparison with experiments validates a comment made earlier: the change in non-stoichiometry is of the same sign as the stress (or equivalently the sign of the strain). For Kawada et al.’s experiments, the film is under positive lattice mismatch strain or correspondingly under negative stress (compression) and the non-stoichiometry decreases, which trans- lates in a decreased chemical capacitance; in the case of La O’ et al., the film is under negative misfit strain or correspondingly under positive stress (tension) and the non- stoichiometry is increased, which translates into an increase in chemical capacitance.

It is clear, however, that there is a quantitative disagreement between predictions and experimental results.

Furthermore, when comparing predictions for the chemical capacitance of itinerant electron oxide thin films to experimental data in section 3.6.2.3, the oxide was assumed to be coherent with the substrate. In reality, this is likely not the case and there may well be many defects, such as dislocations, at that interface. The presence of those defects lowers the elastic energy. The number of dislocations increases with

film thickness so as to minimize the total energy. It is possible to estimate the critical 70 thickness above which dislocations appear in the film. Using People and Bean’s theory, h 500 Aforathinfilmwitha2%strain[˚ 52]. Because Kawada’s film (1.5 µm) is c ⇡ one and a half orders of magnitude thicker than that critical length, dislocations must be present in the film. As a consequence, the strain and the absolute change in non- stoichiometry in the film should be smaller than that shown in Fig. (3.6), in contrast to that observed in the experiments. It is thus reasonable to conclude that some other e↵ect is largely responsible for the shift in the chemical capitance seen in the experiments.

One possibility is the inaccuracy of the coecients used in the thermodynamic model of the oxide. Such coecients were calculated from thermogravimetric data [13].

In this case, the sample nonstoichiometry is evaluated with respect to a reference state, assumed to be at zero non-stoichiometry. As a result, errorneously defining that reference state can have a large e↵ect on the calculated absolute non-stoichiometry, and subsequently on the coecients a, ho and so . However, there is no reason to 4 O 4 O believe that this is the case in these experiments.

Another more likely reason is a nonuniform distribution of lanthanum and strontium cations throughout the film, that occurs during thin film growth [53]. Such nonunifor- mity can certainly result in a change in the chemical capacitance of the oxide. If there is an enhanced concentration in the film or at the film surface, and since the oxygen incorporation process is surface limited, the value of Cchem can be di↵erent from that predicted. In fact, Fig. 3.6 shows good agreement between Kawada et al.’s LSC-64 film measurements and predictions for LSC-73 under compressive stress, and Fig. 3.7 shows reasonable agreement between la O’ et al.’s LSC-82 film and predictions for LSC-73 71 under tensile stress. Given that strontium ions have a larger radius than lanthanum ions, rSr2+ =1.32 A˚ >rLa3+ =1.172 A[˚ 54], it is possible that a tensile stress will favor larger atoms, thus fostering a higher strontium concentration in the film and segregat- ing lanthanum ions to the surface, while a compressive stress will favor smaller atoms, thus fostering a higher lanthanum concentration in the film, keeping strontium ions at the surface. Such changes are consistent with the changes in strontium concentrations needed to bring the theory close to the experiment. However, as the predictions of the chemical capacitance are exclusively valid for a uniform strontium concentration throughout the film, further work is needed to properly account for such enhanced surface concentration compared to the bulk.

The discrepancy between predictions and experimental results can also originate from nonlinear e↵ects in the film such as non constant compositional coecients or vacancy ordering in particular directions, as recently reported by Donner et al. [35] and by Kim et al. [55]. There is no doubt that such e↵ects combining with cation segregation can result in a nonuniform chemical capacitance, very di↵erent from the one computed assuming uniform concentration and stress fields.

Alastpossibleexplanationforthedi↵erencebetweenpredictionsandexperiments is the change in the electronic structure induced by stress. As explained by Kushima et al. [40], stress induces a low-spin / internediate-spin transition in LaCoO3 char- acterized by a change in symmetry of the electronic density around the Co atom from cubic-like to spherical. Such change in symmetry could even result in an alteration of its electronic conduction mechanism, in turn changing the thermodynamics of the system. 72

3.7.2. Generalization to other mixed conductors

The results derived above for LSC can be extended to other mixed conductors. LSC was considered so far in the paper, as the dominant defect mechanism is a simple equilibrium between oxygen in the gas and oxygen vacancies on the oxygen sublattice, as given by

Eq. (3.41). Other mixed conductors can have much more complex defect equilibria involving multiple reactions, such as vacancies formation on the A and B sublattices, exchange of cations between the subattices or charge disproportionation [29]. However, the equilibrium of all those reactions can be determined by one variable: the oxygen content of the oxide. As a result, for every mixed conductor, however complex the defect mechanism is, one can define a chemical potential of oxygen in the oxide, µO, which value is set by that in the gas, according to Eq. (3.39). Furthermore, the expression of the variation of the new free energy function, Eq. (3.42), remains valid as the energy to add one unit of oxygen to the system is dictated by the chemical potential of oxygen in that system. As a result, for every mixed conductor, whatever the defect mechanism is, the chemical potential of oxygen under stress is linear in the trace of the stress, as shown by Eq. (3.51). As di↵erent species will coexist in the system, the expression of the charge neutrality is expected to be di↵erent from Eq. (3.73) and the stress is expected to have a more complex e↵ect on concentrations and conductivities. However, the results derived for LSC can be easily modified to describe any mixed conductor by using the appropriate form of the chemical potential of oxygen in the oxide. 73

3.8. Conclusion and future work

Using the example of strontium doped lanthanum cobaltite, the e↵ects of stress on the properties of mixed conducting oxides have been addressed. The chemical po- tential of oxygen in the oxide was shown to be linear in the trace of the stress. As a result, the change in the non-stoichiometry and in the chemical capacitance was found to be of the same sign as the applied stress in LSC. Furthermore, the change in in

LSC-64 was shown to be exponential in the trace of the stress. The comparison of the model predictions to experimental data for the case of thin films has shown that oxides under stress are only qualitatively described by the presented model. Plausible reasons accounting for the quantitative inaccuracy of the model have also been discussed. Fi- nally, the results derived for LSC were shown to be easily transposable to other mixed conducting oxides, by using the appropriate form of the chemical potential of oxygen under no stress.

Clearly, further work is needed to increase our understanding of stress on the prop- erties of mixed conductors and only a coordinated e↵ort from both the experimental and theoretical sides will lead to success.

As mentioned in the discussion section, it is possible that mismatch stress a↵ects the film growth resulting in cation segregation in the film. A first experiment of great interest would be to measure the chemical capacitance of a non-substituted perovskite

—e.g. LaCoO3—inaconfigurationunderstress.Ifthethermodynamicsoftheper- ovskite under no stress are known, a direct comparison between experimental results and predictions from the model would be possible. Because cation segregation would have no e↵ect, such experiment would allow us to conclude on the presence of such 74 e↵ect in a film configuration. If such experiment is not possible, it would be of great interest to accurately measure the concentration of strontium in LSC films. Further- more, as stress alters the electronic structure, this could result in an alteration of the conduction mechanism of the film and more profoundly in a change of the thermody- namics of the system. In a thin film configuration, an itinerant electron oxide could hypothetically turn into an electron hoping oxide. In general, to measure the thermo- dynamic of a system, scientists use thermogravimetry, which cannot be used on thin

films. As a result, developing a method to accurately measure the thermodynamics of oxide films —and not just the chemical capacitance— would be a critical advance for our understanding.

More work can also be conducted on the theory side. First, as the concentration of strontium is very likely not uniform in the thin film, it would be interesting to expand the present model to compute the chemical capacitance of films with nonuniform cation concentrations in order to compare those predictions to experiments. Because actual electrodes are porous, non-uniform stresses develop. Expanding the model to such configurations would allow us to gain insight on both the local and global behavior of the electrode under stress. Nonuniform stresses induce regions of varying oxygen non- stoichiometry and varying conductivity. One would thus be able to observe preferred paths for oxygen migration and correlate those with the microstructure. Adding all those microscopic contributions over the entire electrode would allow to predict its macroscopic performance under stress, e.g. by computing the chemical capacitance.

Finally, stress has been shown to lead in certain cases to vacancy ordering [35]anda modification of the electronic structure around the cobalt atom [40]. Such e↵ect are not 75 accounted for in the presented model and further refinement of the thermodynamics of the oxide would be necessary. 76

CHAPTER 4

Oxygen Bubble Formation in Solid Oxide Electrolysis Cells

4.1. Introduction

As mentioned in Chapter 2, solid oxide cells are very promising devices to eciently convert electricity into chemical energy (electrolysis mode). In order to produce gas at high rates, these devices have to be stable under high current densities, which requires an understanding of possible degradation processes. Sources of degradation in such cells have been mostly studied at the electrode level and include local heating [56], passivation [57], incompatibility between materials [58]anddefectformationatthe interface with the electrolyte [59]butrecentstudieshaveshownthatdeterioration can also occur inside the dense yttrium-stabilized zirconia (YSZ) electrolyte [60–62].

Oxygen bubble formation has been observed under the oxygen electrode if the current density is above a critical value, experimentally determined to lie between 1.0 A/cm2 and 1.5 A/cm2 [61].

Applying a load on a Solid Oxide Electrolysis Cell (SOEC) shifts the oxygen chem- ical potential away from its equilibrium value, driving oxygen ions from the hydrogen to the oxygen electrode, as sketched in Fig. 4.1. The applied current in the presence of apolarizationresistanceduetotheoxygenelectrodeisresponsibleforahighoxygen potential under this electrode [63–68]. Assuming local equilibrium between oxygen ions, electrons and oxygen gas in the electrolyte, such a potential can be interpreted as 77

Figure 4.1. Sketch of a SOEC under operation. Applying a load drives oxygen from the hydrogen to the oxygen side. The applied current in the presence of an oxygen electrode polarization resistance is responsible for a high oxygen potential under the O electrode and, if the current is above a critical current, the formation of oxygen bubbles in the electrolyte. ahighoxygenpressure(incertaincasesmuchhigherthan1atm)[37]. It was concluded that this high pressure was responsible for the pressurization and growth of pores found in the electrolyte and consequent degradation of the SOEC [65–68]. Using mechanical stability arguments, Virkar and Lim provided an estimate of the critical value of the pore pressure above which delamination can take place [66–68].

Virkar et al. also studied the pressurization of existing pores in the electrolyte due to spatial variations of the conductivity [69]. They showed that the constant influx of gas in a pore leads to its pressurization and that this pressure can be very large. Once the pressure exceeds a critical value, delamination of the electrode and electrolyte takes place, bringing about failure of the SOEC.

While all the aforementioned articles consider degradation originating from the pressurization of pores in the electrolyte, bubbles are observed to form in the dense 78 part of the electrolyte —be it within grains [60] or at grain boundaries [60, 61]. As a result, nucleation of oxygen gas from the dense electrolyte must be considered. In addition, the large pressures that are expected to exist within the nucleus will generate stress in the electrolyte. This stress leads to an inhomogeneous solid that renders the results of classical nucleation theory invalid and can potentially alter the driving force for nucleation and the critical radius of the nucleus.

In this chapter, the homogeneous and heterogeneous nucleation of oxygen gas bub- bles in the electrolyte of SOECs are addressed. In the first section, the thermodynamic model is developed and the equilibrium conditions are derived. Next, the governing equations are set up, highlighting the driving force for nucleation. A critical oxygen electrode polarization above which nucleation is possible naturally arises from this analysis. In the second part, the e↵ects of the di↵erent parameters on this critical polarization are systematically analyzed and the possible role of grain boundaries is addressed. Finally, a comparison with experiments is given that shows that the theory and experiment are in very good agreement.

4.2. Thermodynamics of nucleation

By definition, nucleation is the local clustering of atoms which display characteris- tics of a di↵erent phase. In the present case, the initial phase is the dense solid oxide, e.g. YSZ, composed of cations, oxygen ions, vacancies and electronic species. As the focus of this chapter is on the high oxygen pressure side of the electrolyzer cell, elec- tron holes is the main electronic specie [65,70]. The nucleus that forms is assumed to be a gas of molecular oxygen. At high temperature and under an applied potential, 79

(a) (b)

Figure 4.2. Schematic of oxygen bubble formation in the dense oxide electrolyte phase. The electrolyte is assumed to be YSZ, composed of two sublattices: one for the cations and one for the oxygen ions. (a) The large green spheres represent cations while the medium-sized blue sphere represents a cation vacancy and the small red ones represent oxygen ions. Albeit present in the system, electron holes and oxygen vacancies are not represented. (b) Under certain conditions, oxygen ions can react with holes around a cation vacancy, forming a molecule of oxygen and simul- taneously destructing a unit cell. This local destruction of the lattice gives way to a bubble (purple ellipse) containing molecular oxygen (grey small spheres). cations, oxygen ions, vacancies and electron holes move in the crystal, c.f. Fig. 4.2(a).

Under appropriate conditions, oxygen ions and holes can react next to a cation va- cancy to form a molecule of oxygen. As those elements react, a unit cell of the solid is simultaneously destroyed, creating a bubble filled with molecular oxygen as depicted in Fig. 4.2(b).

YSZ has a cubic fluorite crystal structure [71], with 2 distinct sublattices: one for the cations and one for the anions, as depicted in Fig. 4.2. Any given ion must occupy a site on one of the sublattices. On the cation sublattice, two di↵erent ions

x 0 are present, zirconium ions, ZrZr,andyttriumions,YZr,withrealabsolutechargesof

0000 +4 and +3 respectively. Cation vacancies, VZr , are also present on this sublattice, in 80

x small concentration. On the anion sublattice, the only element present is oxygen, OO, bearing a 2realcharge,andvacancies,V·· .Last,electronholes,h·,areassumedto O be present in the system, located on the zirconium sites [71,72].

As shown by Matus et al. [60]andKnibbeetal.[61]oxygenbubblesnucleatefrom ahomogeneoussolidphase.Spacemustthusbecreatedinthelatticetoallowforthe oxygen gas molecule to form. This is only possible if lattice sites are destroyed. This can happen by the elimination of Schottky defects, consuming cation and oxygen sites in stoichiometric ratio. Cation vacancies have been shown to be present in YSZ [73] and cations to di↵use under an electric field [74]. Given those two facts, elimination of Schottky defects is possible without the creation of an extra phase. The reaction to form a bubble can be summarized as:

x (4.1) V0000 +2O +4h· O (bubble) Zr O ! 2 where the word ”bubble” designates the implicit elimination of the lattice at the reac- tion site. This reaction is in fact the sum of the two following reactions:

V0000 +2V·· nil(4.2) Zr O !

x (4.3) 2O +4h· O +2V·· O ! 2 O

The first reaction is the inverse of the reaction of formation of Schottky defects while the second one is the oxygen equilibrium reaction between the gas and the oxide. At high oxygen pressure, consuming Schottky defects is not favorable. However, as it will be seen below, the energy gained by creating oxygen gas from an oxygen-rich oxide is 81 larger than the cost to destroy Schottky defects, making the overall equilibrium (4.1) favorable. While this bubble formation process requires mobile vacancies, it is not a simple creep process since there is a chemical reaction, or phase formation, process at the solid-vapor interface.

Because the initial phase is solid, the properties are not uniform throughout the material upon nucleation. Indeed, when a bubble of oxygen gas forms in the electrolyte, the oxygen pressure is uniform in the bubble. However, the stress field in the oxide, which is due to a combination of the hydrostatic pressure from that bubble and the surface stress —of that newly created surface— depends on the position [46]. As shown below, such stresses can only be withstood by a material with a non-zero shear modulus, i.e. by a solid. For example, the trace of the stress (pressure) in the oxide surrounding a bubble is zero, if the crystal is elastically isotropic.

Those two aspects of nucleation of oxygen bubbles in the electrolyte of a SOEC

—destruction of the lattice and non uniform stress field— are the major di↵erences with nucleation in fluids (e.g. condensation of a liquid from a vapor). Thus, results for the present case are expected to di↵er from the classical homogeneous nucleation case.

The critical nucleus is defined as the bubble that neither grows nor shrinks and is thus in unstable equilibrium with the metastable, electrolyte phase [42]. Bubbles greater than the critical size will grow while bubbles smaller than the critical size will shrink. Since a bubble of the critical radius is in equilibrium with the electrolyte, its radius and the reversible work for the formation of the bubble can be determined using the conditions for thermodynamic equilibrium in a stressed oxide in contact with a spherical bubble of oxygen gas. In a system without stress, the equilibrium conditions 82 are given by Gibbs [42]. Equilibrium conditions for a stressed solid have been developed by Leo and Sekerka [75] and by Johnson and Schmalzried [31, 32], then reviewed by

Voorhees and Johnson [33]. Their approach is followed and modified accordingly for the nucleation of an oxygen bubble in stressed YSZ.

The method used here is in fact very similar to the one used in Chapter 3. In the

first part, we describe the system, composed of a perfect crystal and a spherical bubble separated by an interface. In this case, the interface is very important as the nucleation process is driven by the gain in energy by creating a new phase with extra surface. In the next part, the energies of the di↵erent components are subject to virtual variations.

However those variations are not all independent and constraints should be taken into account. Conditions for equilibrium are then obtained as those minimizing the energy of the system.

4.2.1. Thermodynamic model

The system is composed of a perfect crystal and a spherical bubble of oxygen, sep- arated by an interface, denoted ⌃, as depicted in Fig. 4.3. The crystal lattice can be distorted because of the stresses and interrupted because of the bubble, but it is continuous elsewhere. The various thermodynamic densities relative to the crystal and to the interface are referred to a reference state of an undeformed YSZ lattice under ahydrostaticpressureP 1 (determined by the operating conditions, as it will be seen below), while those relative to the gas are referred to the actual or deformed state.

Thermodynamic densities expressed per-unit-volume in the reference state are desig- nated with a superscript 0.Thee↵ectsofdislocations,possibleexchangeofatoms 83

Figure 4.3. Sketch of the system under study for the derivation of the equilibrium conditions: perfect YSZ lattice with a spherical bubble of oxygen. The larger green spheres represent the cation sublattice (pop- x 0 0000 ulated with ZrZr,YZr and VZr ) and the smaller red ones represent the x oxygen sublattice (populated with OO and VO·· ) in the electrolyte. Al- though not represented here, holes are also present in the oxide. The smaller grey spheres inside the spherical bubble represent oxygen gas. The two phases are separated by an interface, ⌃. between the anionic and cationic sublattices and interstitial atoms are not considered here. The only specie assumed to be present in the bubble is molecular oxygen, O2.

Yttrium and zirconium are assumed to be insoluble in the gas and no other phase forms upon nucleation of the oxygen bubble. As it will be seen later, the growth of the bubble is due to the elimination of unit cells, which requires cations to migrate and thus suciently high temperatures is required. While there may be gradients in the composition of the species or potentials, it will be assumed that, on the scale of the critical radius for nucleation of a bubble, the system is in thermodynamic equilibrium. 84

4.2.2. Internal energies

The internal energy density per unit volume of the oxide in the stress-free state, eox, v0 is taken to be a function of the entropy sox,thedeformationgradienttensor ,the v0 F electric displacement field D and the number densities of the di↵erent elements ⇢0 x , ZrZr

⇢0 , ⇢0 , ⇢0 x , ⇢0 and ⇢0 . YZr0 VZr0000 OO VO·· h·

ox ox (4.4) e s , , D,⇢0 x ,⇢0 ,⇢0 ,⇢0 x ,⇢0 ,⇢0 v0 v0 F OO VO·· h· ZrZr YZr0 VZr0000 ⇣ ⌘ Exchange of atoms between the cation and anion sublattices is not allowed. A variation of any of these variables induces a change in the internal energy:

ox ox ox x x ev =T sv + T : F + JE D + µO ⇢O0 + µV·· ⇢V0 + µh ⇢h0 0 0 · O O O O·· · ·

(4.5) + µZrx ⇢0 x + µY ⇢0 + µV ⇢0 Zr ZrZr Zr0 YZr0 Zr0000 VZr0000 where T ox is the absolute temperature of the oxide, T is the first Piola-Kirchho↵stress tensor, J =detF is the Jacobian of the transformation (also equal to the ratio of the volume of a cell in its deformed state to that in its non-deformed state J = dv/dv0), E

x x the electric field and µi the chemical potential of specie i (i =OO,VO·· , h·,ZrZr,YZr0 or

V0000). The operator ” ”representstheclassicalscalarproductwhile”:”representsthe Zr · tensorial scalar product.

In a similar fashion, the energy density per unit area of the interface in the stress- free state e⌃ is taken to be a function of the entropy s⌃ , the deformation gradient tensor a0 a0 ˆ of the interface F,themeancurvatureatthesurface0, and the surface densities of the di↵erent elements 0 x ,0 ,0 ,0 x ,0 and 0 . Here, the bubble is assumed to ZrZr YZr0 VZr0000 OO VO·· h· 85

be spherical. In the present configuration, the mean curvature, 0,isnegativeasthegas bubble is inside the solid phase. Note that even though the electric displacement field

D can have important e↵ects on the interface energy e⌃ ,itisnottakenintoaccount. a0 The energy of the interface is of the form:

⌃ ⌃ (4.6) e s , ˆ,0, 0 x , 0 , 0 , 0 x , 0 , 0 a0 a0 F OO VO·· h· ZrZr YZr0 VZr0000 ⇣ ⌘ Avariationofanyofthesevariablesinducesachangeintheinternalenergyof:

⌃ ⌃ ⌃ e =T s + ˆ : ˆ + K0 + Ox 0 x + V 0 + h 0 a0 a0 T F O OO O·· VO·· · h·

(4.7) + µZrx 0 x + µY 0 + µV 0 Zr ZrZr Zr0 YZr0 Zr0000 VZr0000 where T ⌃ is the temperature of the interface, Tˆ is the surface stress tensor, K is a coecient linking a change in curvature to a change in energy and i is the interfacial

x x chemical potential of specie i (i =OO,VO·· , h·,ZrZr,YZr0 or VZr0000). Assuming the radius of curvature to be small compared to the thickness of the interface, the term K can be neglected [33].

g The internal energy density of the gas phase in the present state ev is a function of

g b the entropy sv,thepressureinthebubbleP and the number density species. Assuming no other phase forms and that yttrium and zirconium are not soluble in the gas, the gas is only composed of oxygen and its density per unit volume of the current state is

noted ⇢O2 .Theinternalenergyofthegasphaseisoftheform:

g g b (4.8) ev sv,P ,⇢O2 86

Avariationofanyofthesevariablesinducesachangeintheinternalenergyof:

(4.9) eg = T gsg P b + µb ⇢ v v O2 O2

g b where T is the temperature of the gas phase and µO2 the chemical potential of oxygen in the bubble.

The total internal energy of the system is the sum of these three contributions:

ox g ⌃ (4.10) " = e dv0 + e dv + e da0 v0 v a0 g ⌃ ZVox0 ZV Z 0 and its first variation reads:

ox g ⌃ " = e dv + e dv + e da0 v0 v a0 g ⌃ ZVox0 ZV Z 0 ox ox g g ⌃ ox (4.11) + e y da0 + e y da + e 20y da0 v0 v a0 Z⌃0 Z⌃ Z⌃0 where yi represents the variation due to accretion of phase i (solid in the reference state or gas in the actual state).

Not all the variables that come into play in Eq. (4.11) —through the variations of the energies and accretions— are independent. They are linked via constraints expressed in the following section.

4.2.3. Constraints

The system is assumed to be isolated from the rest of the universe by a virtual surface in the solid, far from the bubble. To depict the physical situation, a certain number of constraints must be applied to this system. These constraints are of three kinds: 87 global, local and continuity. The global thermodynamic constraints imposed on the system are:

constant entropy, , • S

ox ⌃ g (4.12) = s dv0 + s da0 + s dv v0 a0 v S ⌃ g ZVox0 Z 0 ZV

constant number of oxygen atoms across the oxide, the interface and the • NO gas

(4.13) O = ⇢0 x dv0 + Ox da0 +2 ⇢O dv OO O 2 N ⌃ g ZVox0 Z 0 ZV

constant number of zirconium and yttrium atoms, and ,acrossthe • NZr NY oxide and the interface given their non solubility in the gas

(4.14) Zr = ⇢0 x dv0 + 0 x da0 ZrZr ZrZr N ⌃ ZVox0 Z 0

(4.15) Y = ⇢0 dv0 + 0 da0 YZr0 YZr0 N ⌃ ZVox0 Z 0

These constraints are taken into account in the Lagrangian of the system:

(4.16) "⇤ = " T cS ONO ZrNZr YNY

where " is the total energy of the system, defined by Eq. (4.10), Tc, O, Zr and Y are the Lagrange multipliers associated with the aforementioned constraints. The first variation of this energy reads:

(4.17) "⇤ = " T c S O NO Zr NZr Y NY 88

In addition to the global thermodynamic constraints, there are local constraints:

The lattice network constraint that stipulates that density of sites on the anion • and cation sublattices are constant and in the ratio 2 to 1:

O (4.18) ⇢0 x + ⇢0 = ⇢ OO VO··

1 O (4.19) ⇢Zr0 x + ⇢Y0 + ⇢V0 = ⇢ Zr Zr0 Zr0000 2

O (4.20) 0 x +0 = OO VO··

1 O (4.21) Zr0 x +Y0 +V0 = Zr Zr0 Zr0000 2

where ⇢O and O are the densities of oxygen sites in the bulk and on the

surface respectively. Taking the first variation of those constraints yields

⇢0 x + ⇢0 =0(4.22) OO VO··

⇢0 x + ⇢0 + ⇢0 =0(4.23) ZrZr YZr0 VZr0000

0 x + 0 =0(4.24) OO VO··

0 x + 0 + 0 =0(4.25) ZrZr YZr0 VZr0000

local charge neutrality in the oxide and at the surface, •

(4.26) 4⇢0 + ⇢0 = ⇢0 +2⇢0 VZr0000 YZr0 h· VO··

(4.27) 40 +0 =0 +20 VZr0000 YZr0 h· VO·· 89

Taking the first variation of those equations yields

(4.28) 4⇢0 + ⇢0 = ⇢0 +2⇢0 VZr0000 YZr0 h· VO··

(4.29) 40 + 0 = 0 +20 VZr0000 YZr0 h· VO··

The electric displacement must satisfy Gauss’s law at all points inside the • crystal:

(4.30) D = eo 4⇢Zrx +3⇢Y + ⇢h 2⇢Ox =0 r· Zr Zr0 · O assuming local charge neutrality. Note that local charge neutrality is not

incompatible with the development of an electric field in the bulk of the oxide

[16]. Hence, the two sides of the solid oxide cell can be at di↵erent potentials

even if local charge neutrality is assumed inside the electrolyte.

The last constraint links the variations of the interface due to accretions of the oxide and gas phases. The surface integrals that appeared in Eq. (4.11) are due to virtual variations that permit the transformation of material from one of the phases into the other. Because the two phases remain in contact during the transformation

(no vacuum or fault is created between the two phases), the accretion yox and yg are linked to the elastic deformation u by the relation [33,75]:

(4.31) yg = u + nox0 Fyox nox0 · · ⇣ ⌘ where nox0 is the unit vector normal to the interface, pointing into the gas, in the reference state. 90

Three transformations of surface and volume integrals are required. Using the divergence theorem, the integral of the elastic strain energy can be rewritten as

ox0 (4.32) T : Fdv0 = T n uda0 (T R0 ) udv0 ⌃ · · ·r · ZVox0 Z 0 ⇣ ⌘ ZVox0 where denotes the gradient with respect to the initial state in the volume 0 . Using rR0 Vox results from Leo and Sekerka [75]assuminganisotropicsurfacestress,theintegralof the surface stress energy on the spherical closed surface ⌃0 is rewritten

ˆ ˆ ˆ (4.33) T : Fda0 = T ⌃0 uda0 ⌃ ⌃ ·r Z 0 Z 0 ⇣ ⌘ where is the gradient acting on the interface coordinates in the reference state. r⌃0 Such isotropic in-plane stress at the interface has the following form

(4.34) Tˆ = fˆI where ˆI is the unit matrix that acts on the surface coordinates and f is the surface stress in the reference state. In the case of a spherical bubble, integral (4.33) simplifies to:

2f (4.35) Tˆ : Fˆda0 = uda0 Ro Z⌃0 Z⌃ where we have noted R = 1/0 the radius of the bubble in the reference state. o The divergence theorem applied to the integral of the electric energy yields:

ox0 JE Ddv0 = eo 4⇢0 x +3⇢0 + ⇢0 2⇢0 x dv0 + D n da0 ZrZr YZr0 h· OO · ⌃ · ZVox0 ZVox0 n o Z 0 91

ox (4.36) = D n 0 da0 · Z⌃0 using Eq. (4.30). is the electric potential.

Substituting Eq. (4.12-4.15), (4.31), (4.35-4.36) in the expression of the first varia- tion of "⇤,Eq.(4.17),andusing(4.22-4.29)thefirstvariationofthatenergyreads:

ox ox 1 ox "⇤ = [T Tc] s ( R ) u + µ O ⇢0 x v0 T 0 O2 OO ox ·r · 2 ZV 0 ⇢ 

+[µZr Zr] ⇢Zr0 x +[µY Y] ⇢Y0 dv0 Zr Zr0 g g b + [T Tc] sv + µO2 2O ⇢O2 dv g ZV ⇥ ⇤ ⌃ ⌃ 1 ⌃ + T Tc sa + µO O O0 x 0 2 2 O Z⌃0 ⇢  ⇥ ⇤ +[µZr Zr] Zr0 x +[µY Y] Y0 da0 Zr Zr0 ox g 2 ox + ! J! y da0 v0 v Ro Z⌃0  2f ox0 g ox0 1 + n T J!v n F uda0 · · Ro Z⌃0  ox (4.37) + [] D n 0 da0 · Z⌃0 where

ox (4.38) µ =2µOx +4µV 2µh O2 O O·· · is the chemical potential of oxygen in the oxide, defined as that of oxygen gas in

x equilibrium with the oxide at that given composition according to the reaction 2OO +

⌃ 4h· O2(gas) + 2V·· , µ =2Ox +4V 2h is the chemical potential of oxygen at ! O O2 O O·· · 92

b the interface defined in a similar fashion, µO2 is the chemical potential of oxygen gas in the bubble. µZr = µZrx µV 4µh and µY = µY µV 3µh are the chemical Zr Zr0000 · Zr0 Zr0000 · potentials of yttrium and zirconium in the oxide, defined in a similar way as oxygen

⌃ ⌃ and µ = Zrx V 4h and µ = Y V 3h are their surface chemical Zr Zr Zr0000 · Y Zr0 Zr0000 · potentials.

ox ox ox (4.39) !v = ev Tcsv O⇢O0 x Zr⇢Zr0 x Y⇢Y0 0 0 0 O Zr Zr0 is the grand potential of the oxide,

(4.40) !g = eg T sg 2 ⇢ v v c v O O2

⌃ ⌃ is the grand potential of the gas and 0 = ea Tcsa OO0 x ZrZr0 x Y0 Y0 is 0 0 O Zr Zr the interfacial energy. Note the absence of the electric field in the expressions of the chemical potentials as the charges cancel due to the constraint of charge neutrality.

Note that Eq. (4.37) is comparable to the one obtained by Leo and Sekerka [75] and by Voorhees and Johnson in [33].

4.2.4. Equilibrium conditions

As all the variations in Eq. (4.37) are now independent, the equilibrium conditions can be read straightforwardly by setting all the terms in brackets to 0:

ox g ⌃ (4.41) T = T = T = Tc

ox b ⌃ (4.42) µO2 = µO2 = µO2 =2O 93

⌃ (4.43) µZr = µZr = Zr

⌃ (4.44) µY = µY = Y

(4.45) T R = 0 ·r 0 2 !ox J!g 0 =0(4.46) v0 v Ro 2f ox0 g ox0 1 ox0 (4.47) n T J!v n F n = 0 · · Ro

=0(4.48) ⌃0 All those conditions are well known. The first equation above states that, at equi- librium, the temperature is uniform and constant throughout the system, a condition assumed to hold in the rest of the chapter. This temperature is noted as T .The next three equations, Eq. (4.42)-(4.44), are the chemical conditions for equilibrium: the chemical potentials of oxygen, yttrium and zirconium are uniform and constant throughout the system. Because yttrium and zirconium are not soluble in the gas, that constant cannot be defined. As it will be seen in the next paragraph, the value of that constant for oxygen is set by the applied electric potential. Eq. (4.45) is the condition for mechanical equilibrium. The next two equations are conditions represent- ing respectively an energy and a force balance at the interface, specifically Eq. (4.46) is associated with the addition or removal of lattice sites at the bubble-oxide inter- face. The last equation ensures that there is no jump in the electric potential between the solid phase and the gas phase, assuming the potential in the bubble to be 0 and non-accumulation of charges at the solid/bubble interface. 94

4.3. Driving force

In this section, we derive the driving force for nucleation. The first two parts are devoted to the oxygen chemical potential: to compute its value as a function of overpotential and to derive its expression. Next, the expressions of the grand potentials are derived, allowing to explicitly derived the driving force for nucleation as the change in the grand potential between the phases. Last, the driving force is examined as a function of electric overpotential.

4.3.1. Value of the oxygen potential

Knibbe et al. [61] and Virkar [68]haveshownthatthepointofhighestoxygenpotential in a SOEC under an applied current is located in the electrolyte at the interface with the oxygen electrode, where the bubbles are represented in Fig. 4.1. The Nernst equation, applied between the oxygen electrode and that electrolyte, links the potential of oxygen

O in the oxide to the chemical potential of oxygen at the oxygen electrode (µO2 )andto the oxygen electrode overpotential —or electrical bias from the open circuit— (⌘), as sketched Fig. 4.4.

ox,m O (4.49) µO2 (cVO·· ,ij)=µO2 +4F⌘

ox,m where µO2 is maximum value of the oxygen electrochemical potential in the elec- trolyte and F is Faraday’s constant. As shown in the following paragraph, due to the crystalline lattice and the condition of charge neutrality in the limit of a negligible

ox,m cation vacancy concentration, µO2 is a function of only one of the concentrations in 95

Figure 4.4. Sketch of the oxygen potential near the oxygen electrode of a SOEC under an applied current. ⌘ is the oxygen electrode polar- ization. The maximum value of the oxygen chemical potential in the ox,m system, µO2 ,islocatedintheelectrolyteattheinterfacewiththeoxy- gen electrode [61, 68]. The critical polarization ⌘c is the polarization above which nucleation takes place, as defined in subsection 4.4.2, such O ox,m that µO2 +4F⌘c is chosen to be larger than µO2 in this figure. the system, which is taken to be the oxygen vacancy mole fraction on the oxygen sub-

lattice, cVO·· .Theoxygenpotentialcanalsobeafunctionofthestressintheoxide,ij. However, given the current spherical geometry, the hydrostatic stress in the matrix is independent of the pressure in the bubble [76], which in turn has no e↵ect on the potential of oxygen [77]. As it will be seen later, ⌘ can be computed knowing the applied current using an estimate of the oxygen electrode polarization resistance.

4.3.2. Expression of the oxygen potential

In the rest of the paper, the solid is treated as an ideal solution and the gas is assumed to follow the ideal gas law. Thus, the chemical potential of oxygen vacancies in the oxide is:

o (4.50) µV (cV )=µ + RT ln cV O·· O·· VO·· O·· 96

where cVO·· is the mole fraction of vacancies on the oxygen sublattice. Since the molar volume of solids is so small, the e↵ect of pressure on the value of the chemical potential is small, and thus in expressions of the chemical potentials, the di↵erence between

o the standard state pressure, P ,andthepressureappliedtotheSOEC,P 1,will

5 3 be neglected. Given the molar volume of YSZ, V =2.1 10 m /mol [78], and m ⇥ ahydrostaticpressureP 1 =10atm,theenergydi↵erencebetweenthetwostatesis

o 3 V (P P 1) = 19J/mol, which is indeed negligible compared to RT =8.3 10 J/mol m ⇥ at 1000 K.

Using Eq. (4.50) in (4.38) and a similar ideal solution model for the oxygen ions and holes, defines the chemical potential of oxygen in the oxide under zero stress,

ox ox,o (4.51) µO2 cVO·· = µO2 +2RT ln h(cVO·· ) ox,o where µO2 is the collection of the standard state terms and

2 1 x h(cVO·· )=cOO (ch· ) cVO·· However, the concentrations of the di↵erent species are coupled to one another via site conservation on both the oxygen and cation sublattices, and local charge neutrality, see Eq. (4.22), (4.23) and (4.26) in the Appendix. Such equations, expressed as mole fraction per lattice sites, read

x cOO + cVO·· =1(4.52)

c x + c + c =1(4.53) ZrZr YZr0 VZr0000 97

(4.54) 4c + c = c +4c VZr0000 YZr0 h· VO··

In the limit of a negligible cation vacancy concentration, c ,andassumingyttrium VZr0000 and zirconium to be homogeneously distributed in the oxide, those conditions can be rewritten:

(4.55) cOx =1 cV O O··

(4.56) cZrx =1 y Zr

(4.57) ch = y 4cV · O·· where the mole fraction of yttrium atoms per cation sites has been noted c = y and YZr0 is a function of z,themolepercentofyttria,Y2O3,inzirconia,ZrO2,

N 3+ 2z (4.58) y = Y = N A 1+z

The factor 2 comes from the stoichiometry of yttria involving 2 cations. As a result, y =0.148 for 8 mol% YSZ. Using Eq. (4.55) and (4.57) in (4.51) shows that the chemical potential of oxygen is a function of one concentration only and

ox ox,o (4.59) µO2 cVO·· = µO2 +2RT ln h(cVO·· ) where

2 1 (4.60) h(cV )= 1 cV y 4cV cV O·· O·· O·· O·· 98

As mentioned in the previous paragraph, the chemical potential of oxygen in the oxide is independent of stress, ij,forthisparticularsphericalgeometry.

The bubble will preferably form in the region where the potential is at its maximum value, which has been shown to be at the oxide / electrode interface [68]andisgiven

ox,m by µO2 , Eq. (4.49). Using (4.59) in (4.49) yields:

ox,o O (4.61) µO2 +2RT ln h(cVO·· )=µO2 +4F⌘

where cVO·· is the vacancy concentration at the bubble / oxide interface for a bubble located in a position with the maximum value of the oxygen chemical potential.

To eliminate the standard state value in Eq. (4.61), consider the case where the oxide

O is in equilibrium with oxygen gas at pressure PO2 in the absence of an overpotential. Using Eq. (4.49) in the limit ⌘ =0gives

(4.62) µox,o +2RT ln h(co )=µO O2 VO·· O2 where co is the equilibrium vacancy concentration at the same T and P O as in (4.61). VO·· O2 Using Eq. (4.62) in (4.61) finally yields

h(cV ) (4.63) RT ln O·· =2F⌘ h(co ) " VO·· #

This equation gives the oxygen vacancy concentration as a function of electrode polar- ization. The vacancy concentration under no overpotential is largely controlled extrin- sically by the dopant (yttria) concentration but the function h requires the departure from that dopant concentration, hence demanding the hole concentration. This is, of 99 course, small but nonzero. Taking YSZ to have a unit cell of volume V =67.92 A˚3 for

2 formula units of YSZ [78], the hole concentration per oxygen lattice site, under a

O pressure PO2 ,isgivenby[71]:

o 2 0.62 eV O 1/4 c =5.84 10 exp P h· ⇥ kT O2 ✓ ◆ O where PO2 is in units of atm. Using the local charge neutrality condition, Eq. (4.57), the oxygen vacancy concentration at ⌘ = 0 for YSZ reads

o y 2 0.62 eV O 1/4 (4.64) cV = 1.46 10 exp PO O·· 4 ⇥ kT 2 ✓ ◆ Note that at P O =0.21 atm and T =1073K,co = .03702 while y = .03704, which O2 VO·· 4 supports the fact that the vacancy concentration is mostly controlled extrinsically.

However, this small nonzero change from the extrinsic value is central to allowing nucleation of oxygen bubbles.

4.3.3. Expressions of the grand potentials

The grand potential of the gas is defined by Eq. 4.40 and that of the oxide by Eq. (4.39).

Using Eq. 4.41-(4.44) to define Tc, O, Zr and Y gives

(4.65) !g =egas Tsg µb ⇢ v v v O2 O2

ox ox ox 1 ox ! =e cV ,cZrx ,cY ,ij Ts µ cV ,ij ⇢Ox v0 v0 O·· Zr Zr0 v 2 O2 O·· O (4.66) µZr(cZrx ,ij)⇢Zrx µY(cY ,ij)⇢Y Zr Zr Zr0 Zr0 100

In order to complete these calculations, it is necessary to express the internal energies of the gas and of the crystal as a function of its variables.

4.3.3.1. Grand potential of the bubble. Recalling the description of the gas made in 4.2.2, the internal energy of the gas phase is a homogeneous function of degree 1 in entropy, volume and the number of oxygen molecules:

Eg = TSg P bV b + µ N O2 O2

Dividing the total energy of the phase by its volume yields

eg = Tsg P b + µb ⇢ v v O2 O2 and the grand potential density is the negative of the pressure:

!g = P b v

4.3.3.2. Grand potential of the oxide. The energy density of the solid phase can be expressed as the sum of the homogeneous energy density under hydrostatic pressure

P 1 —the pressure in the reference state— and the elastic energy density [32]. ij Assuming small strains, this is

ox ox 1 e T,cV ,cZrx ,cY ,ij = e T,cV ,cZrx ,cY , P 1ij + ✏ijij v0 O·· Zr Zr0 v0 O·· Zr Zr0 2 where the elastic strain energy density ✏ijij/2iscomputedfromthereferencestate

—under hydrostatic pressure P —totheactualstate. 1 101

Following the same treatment as for the gas phase, the internal energy of a hydro- statically stressed crystal is a homogeneous function of degree 1 in entropy, volume and the number of oxygen ions, cations, vacancies and holes:

ox ox E = TS P 1V + µOx NOx + µV NV + µh Nh + µZrx NZrx + µY NY + µV NV O O O·· O·· · · Zr Zr Zr0 Zr0 Zr0000 Zr0000

Dividing by the volume in the reference state v0

ox ox x x x x ev = Tsv P 1 + µO ⇢O0 + µV·· ⇢V0 + µh· ⇢h0 + µZr ⇢Zr0 + µY0 ⇢Y0 + µV0000 ⇢V0 0 0 O O O O·· · Zr Zr Zr Zr0 Zr Zr0000 makes it more obvious that the number density of the various elements are not inde- pendent. In fact those number densities are related to one another by Eq. (4.18) and

(4.19). Using those equations, the internal energy simplifies to:

ox ox 1 ox 1 O e = Ts P 1 + µ ⇢Ox + µV + µV ⇢ + µZr⇢Zrx + µY⇢Y v0 v0 2 O2 O O·· 2 Zr0000 Zr Zr0  where all the chemical potentials are evaluated at a hydrostatic pressure P 1. Using this in Eq. (4.66) yields:

ox ox 1 ox ox !v =Tsv P 1 + µO µO (ij) ⇢O0 x +[µZr µZr(ij)] ⇢Zr0 x 0 0 2 2 2 O Zr ⇥ ⇤ 1 O +[µY µY(ij)] ⇢Y0 + µV·· + µV0000 ⇢ Zr0 O 2 Zr 

As shown in Chapter 3 [77], the chemical potential of oxygen under stress is linear in the trace of the stress in an isotropic crystal:

ox ox µO2 (ij)=µO2 + Akk 102

where A is a constant and kk is the trace of the stress. Using the method described in [76]andincludingthestressinthereferencestateP 1,thestrainandstresstensors are computed:

3 ˆ 1 Rc b 2f 1 ✏rr = ox 3 P P 2G r " Rc # 3 ˆ (4.67) 1 Rc b 2f 1 ✏✓✓ = = ox 3 P P 4G r " Rc #

✏i=j =0 6 and

3 ˆ Rc b 2f 1 rr = 3 P P r " Rc # 3 ˆ (4.68) 1 Rc b 2f 1 ✓✓ = = 3 P P 2 r " Rc #

i=j =0 6 with i, j = r, ✓,, Gox is the shear modulus of the oxide, r is the position of the point in the oxide at which those stresses are evaluated, Rc is the radius of the bubble and f is the surface stress. As a result, the trace of the stress tensor for this particular configuration is 0 and µox µox ( ) = 0. Assuming this result to also hold for the O2 O2 ij chemical potential of yttrium and zirconium, the grand potential of the oxide reads

ox 1 1 O ! T,cV ,ij = P 1 + ✏ijij + µo⇢ v0 O·· 2 2 103

where µo is the energy to add an extra unit cell at the surface, or equivalently to create aSchottkydefect,definedas

(4.69) µ =2µ + µ o VO·· VZr0000

Interfaces such as grain boundaries act as sources of cation vacancies [73]. Assuming

YSZ to be formed of fine grains, the chemical potential of cation vacancies is taken constant through the oxide. Using an ideal solution model for the oxygen vacancies,

Eq. (4.50), the energy to add an extra vacant lattice site rewrites

o (4.70) µo = µo +2RT ln cVO·· ⇥ ⇤ o where µo is the sum of the standard state of oxygen vacancies and the chemical potential of cation vacancies.

4.3.4. Change in the grand potential

As we have just shown, the grand potentials can be expressed in terms of other ther- modynamic quantities:

(4.71) !g = P b v

ox (4.72) !v = !ox + We

b where P is the oxygen pressure in the bubble, !ox is the composition dependent portion of the grand potential of the oxide, and We is the elastic energy density. !ox is defined 104 as

1 O (4.73) ! = P 1 + ⇢ µ ox 2 o

P 1 is the hydrostatic pressure on the system due to the gas in the electrode (di↵erent from P o,thestandardpressureunderwhichchemicalpotentialsaremeasured),⇢O is the number density of oxygen sites in the oxide and µo is the energy required to add an extra empty unit cell to the crystal, as defined by Eq. (4.70). P 1 is the pressure applied to the SOEC, which can be greater than atmospheric pressure in some cases [79]. We is the elastic strain energy density given by

1 (4.74) W = ✏ e 2 ij ij

where ✏ij and ij are the Eulerian strain and stress tensors that follow from standard linear elasticity in the small strain approximation.

Assuming the bubble forms at the point of highest potential in the cell, Eq. (4.49) sets the value of the oxygen potential in the oxide at the point the bubble forms. Since the bubble is in equilibrium with the matrix, this in turn sets the value of the chemical potential of oxygen gas in the bubble via Eq. (4.42). Using an ideal gas model for the oxygen gas, the pressure in the bubble is:

4F (4.75) P b = P O exp ⌘ O2 RT ✓ ◆ 105

O where PO2 is the oxygen partial pressure at the oxygen electrode. The grand potential of the gas phase is given by substituting P b in Eq. (4.71):

4F (4.76) !g = P O exp ⌘ v O2 RT ✓ ◆

Eq. (4.70) defines the energy µo:

o (4.77) µo = µo +2RT ln cVO·· ⇥ ⇤ To determine the value of the standard state chemical potential, consider the system to be open and the vacancies are at equilibrium at ⌘ =0[42]:

o µo c =0(4.78) VO·· ⇣ ⌘ where co is the equilibrium vacancy concentration of the oxide under a temperature VO·· O T and an oxygen partial pressure of PO2 , defined by Eq. (4.64). Using this in Eq. (4.70)

o allows µo to be determined and thus,

cVO·· (4.79) µo cV =2RT ln O·· co VO·· !

The strains and stresses in the YSZ for an isolated bubble of radius Rc under hydrostatic pressure P b inside an elastically isotropic oxide under a hydrostatic pressure

P 1 are given by Eq. (4.67) and (4.68) [76]. As discussed in [76], the e↵ects of the surface stress are usually negligible: for a pressure of P b =1 104 atm, surface stress ⇥ a↵ect only bubbles with a radius Rc < 1.5 A.˚ For this reason, the surface stress term will be neglected in the rest of the paper. Finally, the elastic energy density of the 106 oxide reads:

6 3 Rc b 2 (4.80) W = P P 1 e 8Gox r ✓ ◆ ⇥ ⇤ where r is the position in the oxide from the center of the bubble and Gox the shear modulus of the oxide. As the presence of only the shear modulus in Eq. (4.80) implies, the gas pressure in the bubble only induces a state of pure shear in the oxide and thus the hydrostatic stress in the oxide is only given by the applied pressure on the SOEC,

P 1.Thisemphasizestheimportanceoftreatingtheoxideasasolidandnotusing standard thermodynamic treatments of fluids that can only account for the e↵ects of hydrostatic pressure. The energy density of interest in the rest of the paper is the elastic energy density evaluated at the surface of the bubble, found by setting r = Rc in (4.80):

3 b 2 (4.81) W = P P 1 e 8Gox ⇥ ⇤ Using Eq. (4.72), (4.79) and (4.81) in (4.72) yields the grand energy density of the oxide at the interface with the gas bubble

ox O cVO·· 3 b 2 (4.82) ! = P 1 + RT⇢ ln + P P 1 v co 8Gox VO·· ! ⇥ ⇤ The determinant of the deformation gradient (ratio of the volume in the actual state to that in the initial state) is J 1+✏ .Sincethetraceofthestrainiszero ⇡ jj ✏ =0[76], the energy associated with the bubble in Eq.(4.46) is J!g !g. jj v ⇡ v 107

4.3.5. Free energy change of nucleation

The reversible work for the formation of a bubble depends on both the surface energy and bulk free energy change on the formation of the vapor phase. In this section, we examine the bulk free energy change.

4.3.5.1. Stability of the phases . The bulk free energy change is associated with the destruction of lattice sites from a planar solid-vapor interface and the simultaneous transformation of oxygen ions from the oxide into oxygen gas. The solid-vapor interface is in contact with a vapor at pressure given by the pressure inside the bubble. The oxide is at the same overpotential, under a hydrostatic stress set by the gas at the electrode and a strain energy set by the pressure in the bubble. Since we are neglecting the surface energy of the bubble in this section, the conditions under which it is energetically favorable to form a bubble are necessary, but not sucient, conditions for nucleation.

In particular, for nucleation of a bubble to occur, it is first necessary for the di↵erence in the bulk free energies, !g !ox,tobenegative,thatisfortheremovalofcationand v v anion lattice sites from the interface and a simultaneous conversion of oxygen from the lattice to the gas to be favorable.

Setting R to in the equilibrium conditions Eq. (4.42)-(4.46) results in two equa- c 1 tions to compute the vacancy concentration as a function of overpotential. The first equation is the bulk equilibrium condition involving the oxygen potential —Eq. (4.63)— referred to as the“µ-condition”,

h(cV ) (4.83) RT ln O·· =2F⌘ h(co ) " VO·· # 108

This is the equation commonly used to describe the conditions on oxygen incorporation into the anion sublattice. If we allow for the di↵usive motion of cations, a solid-vapor interface can form. This leads to a second energy balance at the interface —Eq. (4.46)— referred to as the ”!-condition”,

O cVO·· 3 b 2 O 4F (4.84) P 1 RT⇢ ln P P 1 = P exp ⌘ o ox O2 cV 8G RT O·· ! ✓ ◆ ⇥ ⇤

Fig. 4.5 is the plot of the mole fraction of oxygen vacancies, cVO·· ,asafunctionof overpotential, ⌘,asgivenbythe”µ-condition” and the ”!-condition” for a 8-mol%

YSZ where the oxygen electrode is exposed to pure oxygen at T =1123K.Those two equations give two di↵erent vacancy concentrations, thus defining two regions. In region I, ⌘<⌘s = 24mV, the vacancy concentration in the bulk is lower than the the one at the surface. The surface wants to create extra lattice sites empty of oxygen to accommodate for a higher vacancy concentration constraint, as depicted by the left schematic in Fig. 4.5. This is done via the creation of Schottky defects. The bulk acts as a reservoir of oxygen ions and populates those newly created sites, thus resulting in a motion of the interface towards the gas. The bubble is bound to vanish. In region

II, where ⌘>⌘s,thevacancyconcentrationimposedbytheinterfaceissmallerthan in the bulk. The interface wants to destroy empty lattice sites, provided cations can di↵use away from the interface, resulting in an expansion of the bubble, as depicted by the right schematic in Fig. 4.5. The bubble will grow.

It will become even more clear in the next paragraph but the overpotential ⌘s divides the range of polarization into two regions depending on the stability of an oxide interface against oxygen gas. Note that in the case of a perfectly planar interface, this 109

Figure 4.5. Mole fraction of oxygen vacancy cVO·· as a function of overpo- tential ⌘ as given by the ”µ-condition” —Eq. (4.83)— (red continuous line) and as given by the ”!-condition” —Eq. (4.84)— (blue dashed line) for a 8-mol% YSZ where the oxygen electrode is exposed to pure oxygen at T = 1123K. There are two regions, depending on how the vacancy concentrations computed from both conditions compare. In region I, ⌘<⌘s = 24mV, the vacancy concentration in the bulk is lower than the the one at the surface. The surface wants to create extra empty lattice sites (grey squares) to accommodate for the constraint, resulting in a mo- tion of the interface towards the bubble. The bubble is bound to vanish. In region II, where ⌘>⌘s,thevacancyconcentrationimposedbythe interface is smaller than that imposed by the bulk. The interface tends to destruct empty lattice sites (grey squares), resulting in an expansion of the bubble. The bubble can grow. approach would not be correct as the pressure in the bubble would have to be the equal to the hydrostatic pressure in the oxide [76].

4.3.5.2. Energetics. As mentioned earlier, the driving force for nucleation is the di↵erence between the grand potentials of the gas and the solid. Eq. (4.76) sets the

value of the grand potential of the gas, while solving (4.63) for cVO·· and using that in (4.82) sets the value of the grand potential in the electrolyte. Plotting the values of the 110

Table 4.1. Values of the parameters for nucleation of oxygen bubbles in 8-mol % YSZ electrolyte.

parameter value T 1123 K x 8% ⇢O 97780 mol/m3 4 5 P 1 1atm O 5 PO2 1atm Gox 69 GPa 6 f 0 J/m2 [76] 2 0 1.45 J/m [81] 2 gb 0.813 J/m [81]

4Computed using a volume V = 67.92 A˚3 for 2 units formula of YSZ [78]. 5When not specified otherwise. 6Computed using a Young’s modulus of Eox = 180 GPa [80] and a Poisson’s ratio of ⌫ =0.3via Gox = Eox/(2(1 + ⌫)) [44]. grand potentials as a function of overpotential will provide insight on the sign of the driving force, and will thus allow for a complementary interpretation of the nucleation condition.

g Fig. 4.6 is a plot of the grand potential of the gas, !v —Eq. (4.76), the composition- dependent part of the grand potential of the oxide, !ox —Eq. (4.73), and the negative of the elastic energy in the oxide, W —Eq. (4.81)— as a function of the electrode e polarization ⌘ for a 8-mol% YSZ electrolyte, where the oxygen electrode is exposed to pure oxygen at T =1123K.Theenergiesvarybyordersofmagnitude,hencetheuse of a log-scale for the y-axis. The values of the parameters used in the construction of

Fig. 4.6 are specified in Table 4.1.

For small values of the electrode polarization, the oxide phase has an energy lower than the gas phase, !ox

−1

) 5 3 −10

10

Energy (J/m −10 σ g σv ox − We 0 100 200 300 400 − (mV)

g Figure 4.6. Plot of the grand potential of the gas bubble, !v ,thehomo- geneous part of the grand potential of the oxide, !ox,andthenegativeof the elastic energy, We,asafunctionoftheoxygenelectrodepolariza- tion. The parameters used in the evaluation of those energies are found in Table 4.1. The shaded area delimits the range of electrode polariza- tion for which the oxide phase is more stable than the gas phase and thus nucleation is not possible. The situation is reversed above the critical polarization ⌘s =24mVandnucleationispossible.

!ox. Provided cations can re-equilibrate quickly, an oxygen bubble present in YSZ v0 would tend to disappear by the mechanism of formation of vacant unit cells or by the

Schottky reaction, corroborating the vision developed in the previous section. Above the stability overpotential, ⌘s,thesituationisreversed:thegasphaseispreferableto the oxide phase, nucleation is possible around cation vacancies, and the critical radius in Eq. (4.46) exists. Under these conditions it is possible for the gas phase to form in the solid electrolyte, however, as it will be seen below, the radius and reversible work for the formation of this bubble is much too high to allow for its formation. This 112

stability polarization ⌘s depends on the electrolyte material, the temperature and the oxygen partial pressure at the electrode.

Below ⌘ 300mV, the elastic energy is negligible compared to the grand potential ⇡ of the gas. Above this polarization, the gas pressure in the bubble becomes suciently large that the elastic energy density is of the same order of magnitude or bigger than the grand potential energy density of the oxide phase. The elastic energy is of the opposite sign compared to the grand potential of the fluid, see Eq. (4.71) and (4.81), which contributes to making the gas phase even more stable than the oxide. This corresponds to replacing a bit of oxide with an elastic energy density proportional to

(P b)2 by the corresponding amount of material in gas phase under a pressure of P b, which for suciently large pressures reduces the energy of the system. However, the pressures in the bubble may yield stresses that are beyond the fracture strength of the oxide.

4.4. Results and discussion

In this section, various aspects of nucleation in a 8-mol% YSZ electrolyte are ex- amined. However, the results can be adapted to other electrolyte materials by an appropriate selection of materials parameters.

4.4.1. Critical radius

Solving for Rc in Eq. (4.46) yields

2 2 (4.85) Rc = = ox g ! ! J!v v v 113

100 exact approx. 80

60 (nm) c

R 40

20

0 0 100 200 300 400 − (mV)

Figure 4.7. Critical radius of the nucleus versus electrode polarization for the exact case as given by Eq. (4.85) and the approximation given by (4.86) for a 8-mol% YSZ electrolyte, where the oxygen electrode is exposed to pure oxygen at T = 1123K. The shaded area, defined by ⌘ 24mV, delimits the range of polarizations for which nucleation is not possible.

The di↵erence between two grand potentials is the driving force for nucleation, as in the case of fluids [42]. However the expressions of those potentials di↵er between the two cases —cf. Eq. (4.76) and (4.82).

Using the values of the grand potentials, Eq. (4.76) and (4.82), in (4.85), the critical radius is computed as a function of electrode polarization in Fig. 4.7. The values of the parameters can be found in Table 4.1. Once again, it is assumed that oxygen ions, vacancies and holes form an ideal solution and that cation vacancies —albeit present and necessary— have a negligible concentration in the system.

For reasons mentioned previously, the radius is negative for ⌘<⌘s —non physical— and decays exponentially from infinity to 0 as polarization grows larger than ⌘ss.At 114

⌘ = ⌘s the critical radius is infinite, since the driving force for nucleation is zero. This implies that it will be very dicult to nucleate bubbles for polarizations near the critical value. Fig. 4.7 further shows that the critical radius attains values in the nanometer range for polarizations around 200mV. Homogeneous nucleation is expected to take place around such polarizations.

As illustrated by Fig. 4.6, for overpotentials between ⌘ and 350 mV, the grand s ⇡ potential of the gas is much larger than the homogeneous part of the grand potential of the oxide or the elastic energy. As a result, the critical radius for this range of polarizations can be approximated by

2 (4.86) Rc = O (4F/RT)⌘ PO2 e

This approximate expression of Rc is also plotted on Fig. 4.7 and is accurate to within

12% for 120 mV <⌘<300 mV.

4.4.2. Homogeneous and heterogeneous nucleation

Nucleation can also be looked at from an energy prospective. Two forces are in direct competition in the formation of a gas bubble: the energy due to the phase change of the cluster of material and the energy associated with the creation of an extra interface,

ox g (4.87) W ⇤ = V (! ! )+S R v v where V and S are the volume and the surface of the spherical bubble respectively [42].

Substituting Eq. (4.85) for Rc in the equation above yields the reversible work for the 115 formation of a critical nucleus in the homogeneous case

4 2 (4.88) W ⇤ = ⇡ (R ) 3 c

Nucleation will take place when that energy of formation is as small as 1 to 100 times the thermal energy, kBT [82].

Eq. (4.88) is the energy of forming a nucleus in a defect-free mother phase, e.g. within grains, but nucleation can also take place at other sites, such as grain boundaries —this is heterogeneous nucleation. In fact, Kingery [83]andothershavereportedahigher concentration of cation vacancies at grain boundaries in oxides, characteristic of a lower formation energy for those vacancies at such sites. Given the mechanism proposed here for the formation of oxygen bubbles, illustrated by Eq. (4.1), such grain boundaries would be preferential nucleation sites. This is heterogeneous nucleation and the critical work of formation is given by [84]

2 3cos✓ +cos3 ✓ (4.89) W ⇤ = W ⇤ het 4

where W ⇤ is the energy defined above and the contact angle ✓ is defined as:

(4.90) cos ✓ = gb 20

with gb the surface energy of grain boundaries that is taken to be di↵erent from that of the solid-vapor surface 0.

2 Using gb =0.813 J/m [81], the reversible work for the formation of the nucleus in both the homogeneous and heterogeneous cases is given in Fig. 4.8. The horizontal 116 hatched area represents the range of energies of 1 100 k T ,inwhichnucleation ⇥ B can be expected. The curve of the critical work of formation for the homogeneous case falls into that zone for an overpotential ⌘c =265mVandthatfortheheterogeneous cases for ⌘c = 252mV, that is much above the stability polarization ⌘s computed in the previous section. As mentioned earlier, it is necessary for the overpotential to be above the critical electrode overpotential but it is clearly not sucient to be a few tens of mV above for nucleation to actually take place. For a given electrode overpotential, the work of formation of a nucleus in the heterogeneous case is lower than that in the homogeneous case. Due to the strongly exponential dependence of the nucleation rate on W ⇤,afactorof2changeinW ⇤ can yield a change in the nucleation rate of almost 1020 [82]. Hence, the nucleation rate of oxygen bubbles will be much higher at grain boundaries than within the grains, and bubbles will most likely form at those grain boundaries. This is consistent with the experiments of Knibbe et al. [61]and

Laguna-Bercero et al. [62].

Because W ⇤ 100 k T for ⌘ 260mV, Eq. (4.86) is a good approximation for ⇡ ⇥ B ⇡ the critical radius. This also means that as a first approximation, the driving force for nucleation is the grand potential of the gas. Using the form of Rc given by Eq. (4.86) in the expression of W ⇤,(4.88),yieldsanapproximateexpressionforthereversiblework of formation of a bubble as a function of the overpotential. Setting W ⇤ =100kBT and solving for the polarization yields the critical polarization above which nucleation happens

RT 4⇡3 (4.91) ⌘ = ln ln T 2lnP O c 8F 75k O2  ✓ B ◆ 117

103

102 T B 1

/ k 10 * W

100 homogeneous heterogeneous 10−1 200 250 300 350 − (mV)

Figure 4.8. Reversible work for the formation of a critical nucleus as a function of the oxygen electrode polarization for the homogeneous nucle- ation case (within a grain) and heterogeneous case (at a grain boundary) for a 8-mol% YSZ electrolyte, where the oxygen electrode is exposed to pure oxygen at T =1123K.Thehorizontalhatchedarearepresentsthe range of energies of 1 100 k T , at which nucleation can be expected. ⇥ B

Because the driving force is dictated by the grand potential of the bubble, it is normal for the nucleation polarization to not depend on the bulk characteristics of the oxide.

The expression of the polarization derived above needs to be modified for the hetero- geneous nucleation by introducing a factor (2 3cos✓ +cos3 ✓)/4atthedenominator. However, using Eq. (4.91) to compute the polarization for the heterogeneous case is accurate to within 5 %.

4.4.3. E↵ects of parameters on the nucleation polarization

The critical polarization, ⌘c,givenbyEq.(4.91),isafunctionofthreeparameters.The

first parameter is a characteristic of the oxide: 0,thesurfaceenergyoftheoxide.The 118

O two other parameters are set experimentally: PO2 ,theoxygenpartialpressureatthe oxygen electrode and T , the temperature of the electrolyte. While other parameters come into play in the expressions of the grand potentials: the applied pressure P 1,the number density of oxygen sites ⇢O and the shear modulus Gox of the oxide, their e↵ect on the critical overpotential (defined as that for which W ⇤ =100kBT )isnegligible given how large the thermodynamic driving force is.

First, attention will be focused on the influence of the type of material. Fig. 4.9 is the plot of the critical polarization as function of the surface energy of the crystal,

2 2 where ranges from 0.9 J/m (corresponding to Al2O3 at 1850 C[85]) to 1.45 J/m

(corresponding to YSZ at 50 C[81]). The critical overpotential depends on the log- arithm of the interfacial energy and a 60% increase in the surface energy results in a

17mV-increase in the polarization, i.e. 7%.

Parameters chosen through the experimental conditions are now considered. Fig. 4.10 shows the nucleation overpotential versus temperature T and log of the oxygen partial

O pressure at the oxygen electrode PO2 .Thecriticalpolarizationspansfrom120mV to 400 mV when varying the temperature from 600 to 1000 Candtheoxygenpartial

2 2 pressure from 10 to 10 atm. The critical polarization decreases with decreasing tem- perature and increasing oxygen partial pressure. Thus, it is easier to nucleate bubbles at lower temperatures and higher oxygen partial pressures at the oxygen electrode.

4.4.4. Critical current

The critical parameter that naturally arises is the overpotential ⌘c at the oxygen elec- trode, which cannot be directly measured experimentally. One can apply a bias on 119

270

265 YSZ

260

255 (mV) n − 250

245

240 0.8 1 1.2 1.4 1.6 σ (J/m2)

Figure 4.9. Overpotential above which nucleation takes place, ⌘c,asa function of the surface energy of the oxide. The left most point of the curve is the critical polarization for a surface energy of 0.9 J/m2 (same as Al2O3). the SOEC and measure the resulting polarization across the entire cell. Unfortunately, the local polarization due solely to the oxygen electrode cannot be easily extracted.

However, as the di↵erent components of the cell —electrodes and electrolyte— are in series, the current running through the whole cell is also the local current. Extracting the local resistance then allows to compute the local overpotential.

Knibbe et al. [61]reporttheinitialelectrodepolarizationresistancesoftheirSOECs showing degradation of R 0.27 0.30 ⌦cm2. Hauch et al. [86]haveshownthatthe p ⇡ polarization resistance for such cells can be split into 3 roughly equal contributions: one of which corresponds to the hydrogen electrode and two of which correspond to the oxygen electrode —high and low frequency processes. As a result, the oxygen electrode polarization resistance makes up for approximately 2/3 of the value reported 120

⌘c (mV) 2 400

360 1 200mV 320

280 (atm)

2 0 + O O

P 240 log 200 1 300mV 160

2 120 600 700 800 900 1000 T (C)

Figure 4.10. Electrode polarization above which nucleation takes place, ⌘c,asafunctionoftemperature,T , and oxygen partial pressure at the O oxygen electrode, PO2 for a 8-mol % YSZ electrolyte. The ”+” sign on the figure denotes the conditions under which the critical polarization O was evaluated earlier —⌘c =265mVforPO2 =1atmandT =1123K.

by Knibbe et al. that is:

R 0.20 ⌦cm2 O ⇡

O Given the testing conditions: T =850CandPO2 =1atm,thecriticalpolarizationis

⌘c = 265mV. As a result, the critical current is estimated to be

⌘c 2 (4.92) Jc 1.3 A/cm ⇡ RO ⇡

While currents exceeding this critical value lead to thermodynamic favorable condi- tions for nucleation, the formation of bubbles could be hindered by slow kinetics, e.g. slow di↵usion of cations in the oxide. However, the critical current just calculated 121 is in very good agreement with what Knibbe et al. have experimentally obtained [61]:

2 2 1.0 A/cm

The critical overpotential is the natural parameter in the problem of oxygen bubble nucleating in the electrolyte, as ⌘c depends solely on the electrolyte material and op-

O erating conditions —T and PO2 —anddonotdependonotherparametersrelativeto the oxygen electrode. However, to convert this critical overpotential into a current, the oxygen electrode polarization resistance is necessary, which depends on the electrode material as well as on those operating conditions [87, 88]. As a result, extracting the variations of the critical current with temperature and oxygen pressure may not be

O easy. In fact, increasing the oxygen pressure at the oxygen electrode, PO2 ,decreases both the critical polarization —cf. Fig. 4.10— and the polarization resistance for LSM /

YSZ electrodes [87,88], which do not allow any conclusions on how the critical current changes as a function of oxygen pressure, using Eq. (4.92). As shown in the previous paragraph, decreasing the temperature, T ,decreasesthecriticaloverpotentialandin- creases the polarization resistance [88]. Those e↵ects combine in Eq. (4.92) to yield a decrease in the critical current with decreasing temperature. However, decreasing the temperature leads to a lower cation vacancy mobility, which can potentially impede the nucleation of oxygen bubbles via the mechanism proposed here.

4.4.5. Vacancy concentration

As mentioned earlier, Eq. (4.83) yields the vacancy concentration in the bulk sur- rounding the bubble. The red continuous curve in Fig. 4.5 is the plot of that vacancy 122 concentration as a function of the overpotential for a YSZ electrolyte where the oxygen electrode is exposed to pure oxygen at T =1123K.

From Fig. 4.5 it is clear that the vacancy concentration decreases with increasing electrode polarization. This result is qualitatively consistent with Laguna-Bercero et al. [62]whoreportanincreaseoftheoxygenatomiccontent(i.e. adecreaseinthe oxygen vacancy concentration) in the YSZ near the oxygen electrode after running the cell in electrolysis mode. This graph also supports the fact that the vacancy

concentration in YSZ, cVO·· ,is(mainly)dictatedbythefractionofyttriumcations,y, and that only a large electrode overpotential (or equivalently, an exponentially large oxygen pressure) can make it depart from that value.

4.5. Conclusion and future work

Conditions for the nucleation of oxygen bubbles in the electrolyte of SOECs have been derived. Such bubbles form in the solid oxide electrolyte under the oxygen elec- trode by destruction of units cell of the solid. As soon as those bubbles are stably formed, a high oxygen equilibrium pressure sets in and further growth is likely to take place via a creep mechanism, sometimes leading to delamination [61,68].

This work led us to define: the stability polarization, ⌘s,belowwhichtheoxide phase is more stable than the gas phase rendering nucleation thermodynamically im- possible, and the critical overpotential, ⌘c,atwhichtheworkofformingacritical nucleus of gas is equal to 100kBT ,acknowledgedtobelowenoughfornucleationto occur. The critical overpotential was shown to be much larger than the stability over- potential. Albeit the complexity of the nucleation process involving both destruction 123 of unit cells and a chemical reaction, the driving force for nucleation was shown to be dictated by the grand potential of the gas in the bubble, as it is for the classical nucleation case. An analytical expression for the critical overpotential was derived and ⌘c increases with increasing surface energy, increasing temperature and decreasing oxygen pressure at the oxygen electrode. Thus, SOECs can be run below a critical current without degradation occurring due to bubble formation. This critical polar- ization yields an equivalent current, for a 8-mol% YSZ electrolyte at 850 Cwherethe oxygen electrode is exposed to pure oxygen, of J 1.3 A/cm2,whichisintherange c ⇡ of critical currents measured experimentally. Last, it has been shown that nucleation is much more likely to take place at grain boundaries rather than within the grains, consistent with that seen experimentally.

Here again, future work includes both an experimental and a theoretical aspect.

So far, scientists have only reported indirect proof that those bubbles were filled with oxygen: high oxygen concentration in the oxide around the bubble and delamination of the electrode. As a first sanity check, measurements of the content of the bubbles should be made, before they are opened or leak in the atmosphere. This is extremely challenging as the bubbles are as small as a few tens of nanometers [61]. Furthermore, measuring the e↵ects of the parameters on the critical overpotential is essential to confronting the predictions. Two easily accessible parameters are the oxygen pressure

O at the electrode PO2 and the temperature T ;theire↵ectsonthecriticaloverpotential are being investigated by the Barnett group at Northwestern University. As pointed out previously, scientists do not have access to the overpotential due solely to the oxygen electrode and can only measure the current throughout the whole cell. As a result, in 124 such experiments, multiple cells should be run under di↵erent loads and identification of cells which have a very high degradation rate (via cell voltage) will show us which cells are prone to nucleation and which are not. A post-mortem examination would also have to be run to correlate this with the absence or the existence of bubbles in the electrolyte. Extracting the electrode polarization resistance from e.g. EIS, the current can be converted back to into a critical overpotential.

On the theory side, expanding the model to include interstitial oxygen should be the first step. In fact, a quick calculation on the data reported by Laguna-Bercero et al. [62] indicates that the increase in the atomic oxygen weight percent under the oxygen electrode of their degraded cell is larger than the number of substitutional vacancies. Last, by looking at the kinetics of the nucleation process, it would be interesting to see whether there is a time below which the bubbles can redissolve by switching back to fuel cell mode, without any significant irreversible consequences. 125

CHAPTER 5

Growth and Coarsening of Nanoparticles on the Surface of an

Oxide

5.1. Introduction

Ideally, SOFCs should work on a variety of di↵erent fuels such as reformed hy- drocarbons or gasified coal, requiring the anode to operate in mixtures of CO, CO2,

H2,H2OandCH4,containingsometimesimpuritiessuchasH2S. However, the current standard anode, Ni – 8 mol% yttria-stabilized zirconia is susceptible to coking, poi- soning by fuel impurities [89]andhasbeenshowntodegradeuponredoxcycling[90].

E↵orts have been made to develop new materials to address these problems. Ru- and

Pd-substituted (La,Sr)CrO3 are conducting-oxide anode materials where a catalyst material, formerly dissolved in the oxide, precipitates into nano-scale particles at its surface under operating conditions [91–93], as represented in Fig. 5.1. Such materials have been shown to out-perform NiYSZ without catalysts in every aspect of perfor- mance. The presence of such particles on the surface of the anode clearly enhances per- formance compared to an unsubstituted anode. Two regimes have been observed: 1) rapid nucleation of the particles, followed by 2) coarsening. Because the performance of the cell is controlled by the surface area of the catalyst, understanding the kinetics of growth —past nucleation— of those particles is crucial to predict the performance of the cell. 126

Figure 5.1. Schematic of catalyst nanoparticles precipitating at the sur- face of the anode. The green phase represent YSZ while the grey phase represent the oxide and the black dots correspond to the nanoparticles.

At very early times, these particles grow because of a flux of solute coming from the oxide, as depicted in Fig. 5.2(a). In an oxidizing environment, the solute, initially dissolved in the oxide, wants to phase separate, thus precipitating into particles at the surface. At later times, coarsening has been reported to take place [92]; catalyst material flows from smaller particles to larger particles, as depicted in Fig. 5.2(b).

Because, in many systems, di↵usion is quicker at the surface than in bulk, it is assumed that matter is flowing mainly via the surface between particles at those later times.

However, for intermediate times, both 2D and 3D transport compete. Both fluxes will have to be considered in the modeling approach. The purpose of this work is to set up amodeltoinvestigatethegrowthandcoarseningofsuchparticlesand,eventually,to gain insight on the e↵ects of the various parameters on the process. 127

(a) Growth (b) Coarsening

Figure 5.2. Schematic of the mechanism for the formation of catalyst particles at the surface of the oxide. (a) At early times, particles grow with solute flowing from the oxide. (b) At later times, surface transport is dominant and particles coarsen. Catalyst flow from smaller particles to larger particles.

In the first part of the chapter, past coarsening theories are studied. Theories for nucleation of particles in a 3D matrix (e.g. precipitates in solids) as well as nucleation of particles nucleating on a surface (e.g. coarsening of adatoms) have been developed.

Then, various aspects of the modeling are discussed. The issue of growth from solute dissolved in the bulk is addressed and is shown to be modeled by adding a bulk ex- traction term to the di↵usion equation. After discussing the complexity of the ideal configuration, a simpler geometry is presented, involving a single particle on the top of a box. In the last part, we derive the governing equations for such a configuration, in the limiting case, where transport of the catalyst material is done exclusively via the bulk and surface di↵usion is not allowed.

5.2. Background

Coarsening, or Ostwald ripening, is the late stage of a first order transition. At this point, the system has separated into two phases and the evolution of the precipitate is driven by the minimization of the total interfacial energy, resulting in fluxes from 128 smaller to bigger particles. As the particles coarsen, the total volume fraction of the second phase remains constant while the average radius of the particles increases.

Theories have been developed for both particles coarsening in a volume (3D), or on a surface (2D).

5.2.1. Coarsening in 3D

Ostwald ripening was initially developed for spherical particles in a 3D volume, with applications in metallurgy. The initial description of this process was proposed by

Lifshitz and Slyozov [94]andWagner[95](LSWtheory)overfiftyyearsago.They showed that the cube of the average radius of particles increases linearly with time, and that the particle size distribution, normalized by the average radius, is independent of time. In deriving such kinetics they considered the second phase to have a zero volume fraction. However, at non-zero volume fraction, the di↵usion field is perturbed by other particles resulting in a dependence of the coarsening rate on the volume fraction.

Many theories accounting for the di↵usion interactions in systems with non-zero volume fraction have since emerged, e.g. [96–98], but the most complete study is from Akaiwa and Voorhees [98]. They developed a solution to the multiparticle di↵u- sion problem considering a non-zero volume fraction under the quasistatic assumption.

They further showed how the coarsening rate of a given system depends on the par- ticles’ size and the spatial distribution. In fact, a given particle surrounded by bigger particles will shrink, while if that same particle is surrounded by smaller particles, it will grow. Their calculations also include dipolar terms, allowing for particle migration 129 due to di↵usional interaction. Their method allowed to accurately predict the evolu- tion of a system by simply solving a set of equations linking radius and position of the particles, instead of solving the di↵usion equation every timestep, keeping track of the concentration at every point of the simulation box. This streamlined approach is com- putationally less expensive and retains much of the accuracy in fitting these di↵usion processes.

5.2.2. Coarsening in 2D

Daddyburjor et al. saw the limits of the mean field approach in treating the coarsening of hemispherical particles nucleating on a 2D surface [99]. It seems even more evident in 2D, coarsening of particles depends on the screening from surrounding particles.

Using a 2D periodic Green’s function, they calculated the di↵usion field of an array of droplet on a surface.

5.3. Modeling considerations

As mentioned in the introduction, the ultimate goal of this project is to set up a model for the coarsening of a large number of nanoparticles on the top surface of an oxide. Given an array of seeds of di↵erent sizes on the top surface of an oxide, we would like to predict the kinetics of their growth and coarsening. At very early times bulk

flux dominates: particles grow due to material flowing out of the oxide. At later times, coarsening takes place: larger particles grow at the expanse of smaller particles. It is believed that such coarsening happens via rapid transport of matter at the surface. As a result, the model should encompass both limiting cases; predominantly bulk fluxes 130 to the surface at early times and mostly 2D transport at later stages. However, at intermediate times, both 2D and 3D di↵usion processes compete.

As it can be seen from the existing literature, no paper allows for growth of particles due to solute present in the matrix. However, such bulk extraction can be easily taken into account by adding a constant uniform term in the di↵usion equation.

In the real system, a large number of particles are nucleating in a porous electrode, c.f. Fig. 5.1. However, given the small size of the particles compare to the curvature of the electrode, the surface can be assumed to be planar. As a result a good ap- proximate configuration to the real system is a large number of particles nucleating on a parallelepiped block of oxide. As a first step to derive the equations for such a system, we will start with a very simple configuration: one particle nucleating on the top surface. We will further assume the transport of matter to happen exclusively via the bulk. Fig. 5.3 is a schematic of the configuration. The following section describes the configuration in details and the mathematical approach.

Figure 5.3. Schematic of the configuration: one particle coarsening at the surface of a slab. The reference of the axes is taken as shown on the figure. 131

5.4. Mathematical formulation of the system

In this section, we consider a simplified system composed of one hemispherical particle of radius R seating on the top surface of a box of dimensions L L L , x ⇥ y ⇥ z as depicted in Fig. 5.3. Solute is dissolved in the box and di↵usion is exclusively three dimensional. The particle is assumed to be in the shape of a hemisphere and the Gibbs-

Thomson boundary equation applies. In the first three parts, we describe the system: di↵usion equation and boundary / continuity conditions. The last part is dedicated to undimensionalize these equations. Those equations are then solved in the next section.

5.4.1. Governing equation

Under the quasistatic assumption, the governing equation for the bulk concentration,

CB,is:

(5.1) D 2C = Br B

3 2 where CB has units of mol/m , DB m /s and is a constant. As it will be seen below, this constant corresponds to the bulk extraction term mentioned above.

5.4.2. Boundary conditions

All the surfaces not in contact with the particle are assumed to be under a zero flux boundary condition:

(5.2) C n =0 r B · 132 where n is the normal to the interface. However this condition does not hold under the particle, as solute is flowing into the particle. Local equilibrium is assumed to be valid at the particle-matrix interface so that the Gibbs-Thomson equation gives the concentration under the particle as a function of its radius R:

l (5.3) C (R)=Ceq 1+ c B m R ✓ ◆

eq where Cm is the equilibrium concentration at a planar interface in the matrix and lc is the capillary length, defined as:

2vm (5.4) lc = eq eq eq Cm (Cp Cm )G m00

eq where is the interfacial energy, vm is the molar volume of the matrix phase, Cp is the concentration at a planar interface in the particle phase and Gm00 is the second derivative of the molar Gibbs energy with respect to composition in the matrix phase.

5.4.3. Particle growth rate

The growth rate of the particle is determined by the mass balance condition at the interface:

dV (5.5) Ceq p = D C nda p dt B r B · ZZSparticle

2 3 where Vp = 3 ⇡R is the volume of the hemispherical particle.

1 (5.6) 2Ceq = D C nda p Vn B ⇡R2 r B · ZZSparticle 133 where = dR is the velocity of the particle surface. Vn dt A couple of comments should be made. First, the boundary conditions are of a mixed type, that is, involving both flux and value of the function on the boundary.

Furthermore, integrating the Laplacian of CB over the entire box and using the diver- gence theorem yields:

2 (5.7) CBdv = CB nda = CB nda V r @V r · r · ZZZ ZZ ZZSparticle given that the flux is non-zero only under the particle and assuming the flux to be uniformly distributed over the surface under the particle. n is defined as pointing outward and is the surface under the particle. Integrating the right hand side Sparticle of Eq. (5.1) and equating it with the integral just above yields:

V = CB nda DB r · ZZSparticle D @C (5.8) i.e. = B B da V @n ZZSparticle

As a result, is linked to the solute flowing in the particle, and that solute is extracted from the bulk, hence its name ”bulk extraction term”.

5.4.4. Undimensionalizing the equations

The problem can be recast in the following dimensionless variables

eq Ro(C Cm ) (5.9) = eq lcCm R (5.10) r = Ro 134

eq Cm DBlc (5.11) ⌧ = eq 3 t Cp Ro where Ro is the initial radius of the particle. In the case of the multiparticle problem, this would be the average initial radius. The dimensionless lengths of the box are now noted A = Lx/lc, B = Ly/lc and C = Lz/lc. The problem now becomes:

(5.12) 2 = ↵ r with the boundary conditions:

1 (5.13a) under the particle: (⇢,✓, z = C)= r 1 @ (5.13b) v = da n 2⇡r2 @n ZZSparticle @ elsewhere on the faces of the box: =0(5.13c) @n

3 Ro dr where ↵ = eq is a constant and vn = is the dimensionless velocity of the lcCm DB d⌧ interface.

The constant ↵ can be reevaluated as a function of using the divergence theorem:

@ ↵V = 2 dv = da r @n ZZZV ZZ@V 1 @ (5.14) i.e. ↵ = da V @n ZZSparticle where V = ABC is the volume of the box. 135

5.5. Approach

The Akaiwa-Voorhees approach assumes that the contribution from the exterior surface is negligible [98]. Unfortunately, the only sink of matter in the present case is located at the surface (particle) which do not allow us to make such assumption.

We thus have to use a modified Greens function for which the flux is 0 on the surface to formulate the problem. It is explained below how such Greens function can be constructed explicitly, as a combination of reflection and 3D periodic sources.

After constructing a Green’s function appropriate for the current problem, Green’s theorem is applied. In the last part, we explain how to solve the equations obtained.

5.5.1. Green’s function

Let’s define the Green’s function, GB(p, q), given a point source at q.Suchfunction should satisfy the following conditions:

1 (5.15a) 2G = (p q)+ for p and q in the box A B C r B V ⇥ ⇥ @G B =0onallthefacesoftheparallelepiped(5.15b) @n

Note that the term 1/V represents the bulk extraction necessary to balance the ex- traction of mass due to the sink term (p q). Without that term, the integration of the equation above over the volume V = A B C and using the zero-flux boundary ⇥ ⇥ condition would raise a contradiction.

Such a function can be explicitly constructed, as explained in [100, 101]. Given a point source, a zero flux along a plane boundary is achieved by placing a source of same 136 intensity, image of the point source with respect to the plane. The fluxes generated by both sources add up to result in a zero flux on the boundary. Generalizing this idea to the box, we superpose sources of same intensity, images of the initial source with respect to the sides of the box. In other words, given a source at (⇠,⌘,µ), we need to place sources at ( ⇠,⌘,µ), (⇠, ⌘, µ), (⇠,⌘, µ), ( ⇠, ⌘, µ)andsoon.Thereare8 di↵erent possibilities, given the 8 adjacent cubes in contact at any given corner of a parallelepiped. Repeating that pattern in the 3 directions of space then ensures the zero flux condition. This thought can be mathematically translated into:

(5.16) G (x, y, z ⇠,⌘,µ)= G(x, y, z ⇠, ⌘, µ) B | |± ± ± X± where designates all the possible combination of + and .Thesumincludes8 ± terms. G is the Green’s function for a periodic array of sources of period 2A,2B and

2C,solutionof:

1 (5.17) 2G = (r r ) r mnp V m,n,p 2 X ✓ ◆ where rmnp =(⇠ +2Am, ⌘ +2Bn,µ +2Cp)denotesthethepositionofthesinksand

V2 =8ABC.ThisGreen’sfunctiontakestheform:

G(x, y, z ⇠,⌘,µ)=G(x ⇠,y ⌘, z µ) | 1 1 (5.18) = 4⇡ 2 2 2 1/2 m,n,p [(x ⇠ +2Am) +(y ⌘ +2Bn) +(z µ +2Cp) ] X This field is similar to the electrostatic field created by a 2A-, 2B- and 2C-periodic array of charges. 137

Unfortunately, this series converges very slowly. The Ewald method splits this sum into two contributions: a real space one and a spectral one [100,101]. The real-space sum gives good convergence for nearby image sources and the spectral sum gives good convergence for the long-range periodic images. It makes use of the identity:

1 2 1 R2s2 (5.19) = e ds R p⇡ Z0

Using this in Eq. (5.18) yields

E 1 R2 s2 1 R2 s2 (5.20) G(X, Y, Z)= e mnp ds + e mnp ds 2⇡p⇡ m,n,p 0 E X ⇢Z Z where X = x ⇠, Y = y ⌘ and Z = z µ and R is defined as mnp

2 2 2 1/2 (5.21) Rmnp = (X +2Am) +(Y +2Bn) +(Z +2Cp) ⇥ ⇤

Let’s define G1(X, Y, Z)asthefirstsumandG2(X, Y, Z)asthesecond:

E 1 R2 s2 (5.22) G (X, Y, Z)= e mnp ds 1 2⇡p⇡ m,n,p 0 X Z 1 1 R2 s2 (5.23) G (X, Y, Z)= e mnp ds 2 2⇡p⇡ m,n,p E X Z and manipulate them to obtain better convergence of the series.

Let’s rewrite the first function as:

1 X Y Z (5.24) G (X, Y, Z)= + m, + n, + p 1 2⇡p⇡ F 2A 2B 2C m,n,p X ✓ ◆ 138 where the function is defined as F

E 4(A2 2+B2 2+C2 2)s2 (5.25) ( , , )= e X Y Z ds F X Y Z Z0

Making use of Poisson’s summation formula:

1 1 2i⇡kx 1 2i⇡kx (5.26) f(x + n)= e f(x0)e 0 dx0 n= k= X1 X1 Z1 in Eq. (5.24) yields:

1 2i⇡(m X +n Y +p Z ) G (X, Y, Z)= e 2A 2B 2C 1 2⇡p⇡ m,n,p X 1 2i⇡(mx+ny+pz) (5.27) (x, y, z)e dxdydz F ZZZ1

We note I (E)thetripleintegralabove,as involves an integral which bound is mnp F E.Swappingtheintegralons in and that on x, y, z yields: F

E (4A2x2s2+2i⇡mx) (4B2y2s2+2i⇡ny) Imnp(E)= e e 0 (x,y,z) Z ZZZ ⇣ ⌘⇣ ⌘ (4C2z2s2+2i⇡pz) (5.28) e dxdydz ⇣ ⌘

⇡m Isolating the integral on x,achangeofvariables,⇠ =2Asx + i 2sA ,isdone:

2 2 2 2 1 (4A2x2s2+2i⇡mx) ⇡ m 1 ⇠2 1 p⇡ ⇡ m (5.29) e dx = e 4A2s2 e d⇠ = e 4A2s2 2As 2As Z1 Z1 139

Performing a similar change of variables and integration with the integrals on y and z,

Imnp rewrites:

2 mnp E 2 ⇡p⇡ e 4s ⇡p⇡ 2 /4E2 (5.30) I (E)= ds = e mnp mnp 8ABC s3 4ABC2 Z0 mnp

2 2 2 2 where mnp = m + n + p . Finally, G1 writes

2 /4E2 2i⇡(m X +n Y +p Z ) 1 e mnp e 2A 2B 2C (5.31) G (X, Y, Z)= 1 ABC 2 (m,n,p)=0 mnp X6

Substituting by t = Rmnps in G2 yields:

1 1 1 t2 G (X, Y, Z)= e dt 2 2⇡p⇡ R m,n,p mnp RmnpE X Z 1 erfc(R E) (5.32) = mnp 4⇡ R m,n,p mnp X where erfc(x)=1-erf(x)isthecomplementaryerrorfunction.

Finally, the 2A-, 2B-and2C-periodic Green’s function writes

2 /4E2 2i⇡(m X +n Y +p Z ) 1 e mnp e 2A 2B 2C G(X, Y, Z)= ABC 2 (m,n,p)=0 mnp X6 1 erfc(R E) (5.33) + mnp 4⇡ R m,n,p mnp X

This series has the best convergence when the parameter E is equal to Eo =

⇡ 1/2 4AC ,whereA and C are respectively the biggest and smallest dimensions of the box⇥ [⇤101]. 140

Keep in mind that the actual Green’s function of interest is the sum of 8 such contributions:

(5.34) G (x, y, z ⇠,⌘,µ)= G (x ⇠,y ⌘, z µ) B | 3D ± ± ± X±

5.5.2. Green’s theorem

Applying Green’s identity to the box yields:

(q) 2G (p, q) 2 (q)G (p, q) dq r B r B ZZZV ⇥ @G @ ⇤ (5.35) = (q) B (p, q) (q)G (p, q) dq @n @n B ZZ@V  where q is an integration point, p is a field point, is the dimensionless bulk concen- tration defined by Eq. (5.9) and GB is the Green’s function defined by (5.34). Using the properties of GB,thefirstpartofthelefthandsideisevaluated:

(5.36) (q) 2G (p, q)dq = (p) r B h i ZZZV where =(1/V ) (q)dq is the average concentration in the box. Note that h i V when p on the surface,RRR the Green’s function associated with the initial source provides half of that contribution, while the Green’s function associated with the reflection of the source with respect to that surface provides the other half. Using the properties of

,thesecondpartofthelefthandsideisevaluated:

2 (q)G (p, q)dq = ↵ G (q)dq r B B ZZZV ZZZV 141

1 (5.37) = (q)dq GB(q)dq V V ZZSparticle ZZZ

Given the simple boundary conditions on GB and ,therighthandsideofEq.(5.35) is easily evaluated:

@G @ (q) B (p, q) (q)G (p, q) dq @n @n B ZZ@V  @ = (q)G (p, q)dq @n B ZZSparticle @ (5.38) = da G @n h Bi ZZSparticle using Eq. (5.14). Here again, G is the average value of G over the box. Note that h Bi B given its construction and periodicity, this value do not depend on the position of the

field point, p,

Gathering all those results together, Eq. (5.35) reads

(5.39) (p) GB (q)dq = (q)GB(p, q)dq h ih i part part ZZS ZZS where (q)= @ (q) is the single layer density. Note that this is valid for q inside the @nq box as well as at the surface, as justified earlier.

Eq. (5.39) is the governing equation for the field .Thatequationalongwiththe boundary conditions given by Eq. (5.13) is enough to uniquely define the concentration

field in the box. The term G (q)dq in (5.39) corresponds to the mass that B part h i S departed from the box, and thusRR that flowed into the particle. In English, (5.39) gives the concentration at a given point in the box as a function of the average concentration, as well as the flux of solute flowing into the material. 142

Taking the derivative of Eq. (5.39) with respect to np yields:

@ (p) @ (5.40) (p)= = (q)GB(p, q)dq @np @np part "ZZS #

When p is on , G (p, q) is singular but its integral converges. Following Jawson Spart B and Symm [102], we derive:

@ @GB 1 (5.41) (q)GB(p, q)dq = (q) (p, q)dq (p) @np part part @np 2 "ZZS # ZZS

As a result the equation above simplifies to:

@G (5.42) (p)=2 (q) B (p, q)dq part @np ZZS

This is the homogeneous Fredholm integral equation of the second kind. A solution of that equation gives a single layer density with a constant potential on the surface of the particle.

One last unknown has to be determined: which is the average value of the h i concentration field in the box. This value can be computed at each time step using global conservation.

2 2 (5.43) V + ⇡r3 = oV + ⇡ h i 3 p 3 p

o where p is the supersaturation in the particle and is the initial supersaturation in the box. 143

5.5.3. Solving the equations

Solving Eq. (5.43) yields the average value of the supersaturation in the box, . As h i mentioned above, Eq. (5.42) can be solved to obtain the single layer density yielding a constant potential on the surface of the particle. Substituting that form for the single layer density in Eq. (5.39) and using the boundary condition (5.13), (p)isdetermined uniquely, and by extension the supersaturation field throughout the entire box. Such methods have been implemented before, e.g. [98,103]. Solving those equations is done by expanding the functions —(p)andGB(p, q)— in a series of harmonics. Akaiwa and Voorhees use spherical harmonics [98], which cannot be used in the current, more complex, geometry. It appears natural to decompose (p)inaseriesofpolarharmonics of the form Jn(r)exp(in✓)andYn(r)exp(in✓)whereJn and Yn are the Bessel function of the first and second kind respectively. However, decomposing GB(p, q)ismore complex as the point p can be in the volume of the box, not at the surface, imposing to take into account a z component into the decomposition.

5.6. Extension of the model and future work

How to solve the governing equations was briefly described above. However, the next first step should be to decompose the functions on the appropriate base and to implement a solver for such system. This will allow for the simulation of the growth of one hemispherical particle driven by solute flowing from the bulk.

Secondly, surface fluxes should be added. Such fluxes allow for matter to be trans- ported faster at the surface. In the situation described above, where only bulk trans- port is allowed, only the bulk near the particles is expected to be depleted. However, 144 if transport at the surface is faster than in the bulk, we expect the depletion zone to extend to the rest of the surface on which the particle is growing. The issue of connecting bulk flux to those at the surface will be important. It can be assumed for example that the flux of atoms between the top layer of the bulk and the surface to be proportional to the jump in concentrations.

After correctly incorporating the surface transport into the model, it will be in- teresting to add more particles to the system. Because the problem is linear, adding more particles should be relatively easy. The homogeneous Fredholm integral equation,

Eq. (5.42), will include the sum of the single layer contributions from the other parti- cles. This equation will have N unique solutions completed by N boundary conditions of the type Gibbs-Thomson, Eq. (5.13a). This will allow us to observe the competitive mechanisms in the system: bulk versus surface flux. Furthermore, this will allow us to investigate both the growth of the particles due to bulk transport and coarsening mainly due to surface flux between particles. At that point, we will finally be able to investigate the e↵ects of the parameters on the whole process.

5.7. Conclusion

In this chapter, growth and coarsening of an ensemble of particles at the surface of an oxide has been discussed. An approach has been developed to study the growth of one particle nucleating at the surface of an oxide, in the limiting case where only transport is allowed. We have shown that the addition of a constant term in the di↵usion equation allows for bulk extraction of solute and thus particle growth. This 145 approach yields a new set of equations to solve: a Fredholm equation along with boundary conditions to determine uniquely the concentration field.

As discussed in the previous paragraph, the amount of work necessary to extend the model presented here to an ensemble of particles, where surface flux is also present is copious. The multiparticle problem, where particles are nucleating at the surface of an oxide, has never been tackled before and is rich in interesting applications. 146

CHAPTER 6

Conclusion

The work presented in this thesis has expanded our understanding of the e↵ects of stress on mixed oxides, oxygen nucleation in the electrolyte of electrolysis cells, and catalyst nanoparticle formation on the surface of oxides.

Using the example of LSC, the e↵ects of stress in mixed ionic electronic conducting oxides have been analyzed. The chemical potential of oxygen in the oxide was shown to be linear in the trace of the stress, which translated into a change in non-stoichiometry and in the chemical capacitance of the same sign as the applied stress in LSC. Com- parisons of the model predictions with experiments for thin film configurations showed only qualitative agreement. Furthermore, results derived for LSC were shown to be easily transposable to other mixed conducting oxides. Future work for this project includes both experimental and theoretical aspects. Our analysis led us to wonder if the cations were uniformly distributed in the film. Measurements of the cation con- centration throughout the film is critical to support or disprove this hypothesis. Many improvements could be made to the model. However, the first one would be to include nonlinear e↵ects, such as vacancy ordering or involving electronic states.

Conditions for the nucleation of oxygen bubbles in the electrolyte of solid oxide electrolysis cells have been developed. Such bubbles form in the solid electrolyte under the oxygen electrode by a destruction of formula units of the solid. Despite the com- plexity of the current nucleation process, the driving force was shown to be similar to 147 that for the classical nucleation case. An oxygen electrode critical overpotential was defined, as that above which bubbles are likely to nucleate. An analytical expression for this critical polarization was derived and was shown to increase with increasing surface energy, increasing temperature and decreasing oxygen pressure at the electrode. Fi- nally, it was shown how to estimate the equivalent critical current. Future work should include gathering more critical currents for di↵erent values of the temperature and oxygen pressure. Furthermore, as there could be interstitial oxygen in YSZ, extending the model to include such sites is essential.

Finally, growth and coarsening of an ensemble of particles at the surface of an oxide has been discussed. Equations for the limiting case of one particle nucleating on an oxide where only bulk fluxes exist were derived. We have shown that the addition of aconstantterminthedi↵usionequationaccountsforbulkextractionofsolute.This approach needs to be extended to more than one particle before being able to compare with experimental data. Furthermore, surface transport should be taken into account in this model, as it plays a dominant role at later times. 148

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