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NORTHWESTERN UNIVERSITY A Study of Oxides for Solid Oxide Cells A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Materials Science and Engineering By Olivier Comets EVANSTON, ILLINOIS December 2013
Transcript of AStudyofOxidesforSolidOxideCells - Voorhees Research...
NORTHWESTERN UNIVERSITY
A Study of Oxides for Solid Oxide Cells
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHILOSOPHY
Field of Materials Science and Engineering
By
Olivier Comets
EVANSTON, ILLINOIS
December 2013
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c� Copyright by Olivier Comets 2013
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ABSTRACT
A Study of Oxides for Solid Oxide Cells
Olivier Comets
As the world energy consumption increases, it is a question of global health to
increase energy production e�ciency 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
e�ciency 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
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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,
I would like to acknowledge my friend Pierre Garreau for his constant support, 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).
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Contents
ABSTRACT 3
Acknowledgements 5
List of Figures 11
List of Tables 14
Chapter 1. Introduction 15
Chapter 2. Background 17
2.1. Electricity production in the US 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
Mixed Ionic Electronic Conductors 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. New free energy function and Maxwell’s equation 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↵ects of stress on the chemical capacitance 48
3.6. Comparisons and predictions 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
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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. Homogeneous and heterogeneous nucleation 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. Mathematical formulation of the system 131
5.4.1. Governing equation 131
5.4.2. Boundary conditions 131
5.4.3. Particle growth rate 132
5.4.4. Undimensionalizing the equations 133
5.5. Approach 135
5.5.1. Green’s function 135
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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
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List of Figures
2.1 World energy consumption in the world as predicted by the United
States Energy Information Administration in 2011. 18
2.2 Composition of the electricity produced in the US by resources. 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
@V . 33
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
a substrate under thermal stress. 62
3.6 Chemical capacitance versus oxygen partial pressure at T = 873K
as estimated for bulk La0.6Sr0.4CoO3��, as reported in a 1.5µm-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��, as reported for a 45nm-thick
LSC film on YSZ and according to the model. 67
4.1 Sketch of a SOEC under operation. If the current is high enough,
bubbles can form in the electrolyte. 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
under an applied current. 95
4.5 Driving force for nucleation explained in the perspective of the mole
fraction of oxygen vacancy. 109
4.6 Plot of the grand potential of the gas bubble, !gv , the homogeneous
part of the grand potential of the oxide, !ox, and the negative
of the elastic energy, �We, as a function of the oxygen electrode
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 , and oxygen
partial pressure at the oxygen electrode, POO2. 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 Parameters for LSC-82 used in establishing Fig. 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 to establish Fig. 3.6. 65
3.3 Parameters for LSC-82 used to compute Cchem as a function of PO2
to establish Fig. 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,
a solid oxide cell (SOC) is a solid state device that converts directly chemical energy
into electricity, or vice versa. Run in fuel cell mode, a SOC produces electricity by a
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 e�cient 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 e�ciency, 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 e�cient 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) and other byproducts (e.g. heat). Developing
e�cient 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 ine�cient process.
Solid oxide cells are devices that address that problem, capable of e�ciently 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 theUnited States Energy Information Administration in 2011 [1]. The grow-ing demand is mainly driven by the non-OECD countries, who representmore 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 e�ciency
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 ine�cient
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 andtide. Fossil fuels (coal, gas and oil) contribute to 71% of the electricityproduced 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 �C by the end of the century [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 e�cient 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 afunction 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
e�ciency 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
A Solid Oxide Cell (SOC) is a solid state electrochemical device capable of e�ciently
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 e�ciency is limited by the Carnot cycle, the maximum
theoretical e�ciency 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
the cathode to form oxygen ions (O2�). 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:
at the cathode:1
2O2(g) + 2e0 ! O2�(ox)(2.1)
at the anode: H2 +O2�(ox) ! H2O(g) + 2e0(2.2)
and overall:1
2O2(g) + H2 ! H2O(g)(2.3)
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 onhydrogen gas. Air and fuel are fed to the cell. At the cathode, air isreduced to oxygen ions. Those ions are then transported to the anodethrough the electrolyte, where they react with the fuel gas (here H2),forming water, and regenerating electrons. The flow of electrons from theanode 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 isreduced 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 < 10µm.
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 � 50µm is the
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 e�ciency 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 e�ciencies 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 e�cient 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] and Lankhorst
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] are used to derive the equilibrium conditions and the
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 Larche [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
a state under stress finally yields the expression of the chemical potential of oxygen in
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, and other gases that do not react with
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 �2 oxidation state and neutral vacancies are found on
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 Kroger-Vink
notation, these elements are noted OxO, V
··O, La
xA, Sr
0A, V
000A , Co
xB, V
000B . 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,
e
oxv0 , is taken to be a function of the entropy s
oxv0 , the deformation gradient tensor F, the
electric displacement field D and the number densities of the di↵erent elements ⇢0LaxA ,
⇢
0Sr0A
, ⇢0V000A, ⇢0CoxB
, ⇢0V000B, ⇢0Ox
O, ⇢0V··
Oand ⇢0h· :
(3.1) e
oxv0
⇣
s
oxv0 ,F,D, ⇢
0LaxA
, ⇢
0Sr0A
, ⇢
0V000
A, ⇢
0CoxB
, ⇢
0V000
B, ⇢
0Ox
O, ⇢
0V··
O, ⇢
0h·
⌘
A variation of any of these variables induces a change in the internal energy:
�e
oxv0 =T
ox�s
oxv0 + T : �F+ JE · �D+ µLaxA
�⇢
0LaxA
+ µSr0A�⇢
0Sr0A
+ µV000A�⇢
0V000
A
+ µCoxB�⇢
0CoxB
+ µV000B�⇢
0V000
B+ µOx
O�⇢
0Ox
O+ µV··
O�⇢
0V··
O+ µh·
�⇢
0h·(3.2)
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/dv
0),
E is the electric field and µi =@eox
v
0@⇢0
i
is the chemical potential of specie i (i = LaxA, Sr0A,
V000A , Co
xB, V
000B , O
xO, V
··O and h
·). The symbol ”·” represents the classical scalar product
while ”:” represents the tensorial scalar product.
The internal energy density of the gas phase in the actual state e
gv is a function of
the entropy s
gv, the pressure in the bubble P b 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:
(3.3) e
gv
�
s
gv, P
b, ⇢O2 , ⇢N2
�
A variation of any of these variables induces a change in the internal energy of:
(3.4) �e
gv = T
g�s
gv � �P
b + µ
gasO2�⇢O2 + µ
gasN2�⇢N2
where T
g is the temperature of the gas phase and µ
gasO2
and µ
gasN2
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 V of the system containing both phases is first identified, as depicted
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 @V translates into constant number of atoms (O, La, Sr, Co
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 V = Vox +Vg is the sum
of the internal energies of both phases
(3.5) " =
ZZZ
V 0ox
e
oxv0 dv
0 +
ZZZ
Vg
e
gvdv + [surface terms]
where ”[surface terms]” group all the integrals on the surfaces: @V and ⌃. Note that
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:
(3.6) S =
ZZZ
V 0ox
s
oxv0 dv
0 +
ZZZ
Vg
s
gvdv
(2) Constant charge:
(3.7) Q =
ZZZ
V 0ox
⇣
3⇢0LaxA + 2⇢0Sr0A + 3⇢0CoxB+ ⇢
0h· � 2⇢0Ox
O
⌘
dv
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@V . The thermodynamic densities relative to the oxide are referred to astress free state, while those relative to the gas are referred to the actualstate.
(3) Constant number of atoms:
NO =
ZZZ
V 0ox
⇢
0Ox
Odv
0 + 2
ZZZ
Vg
⇢O2dv(3.8)
NLa =
ZZZ
V 0ox
⇢
0LaxA
dv
0(3.9)
NSr =
ZZZ
V 0ox
⇢
0Sr0A
dv
0(3.10)
NCo =
ZZZ
V 0ox
⇢
0CoxB
dv
0(3.11)
NN = 2
ZZZ
Vg
⇢N2dv(3.12)
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:
"
⇤ = "� TcS � �ONO � �LaNLa � �SrNSr � �CoNCo � �NNN(3.13)
where " is the total energy of the system, defined by Eq. (4.10), Tc, �o, and the �is —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) �"
⇤ = �"� Tc�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
�"
⇤ =
ZZZ
V 0ox
⇢
[T ox � Tc]�soxv0 + T : �F+ JE · �D+ [µLaxA
� �La � 3eo�c]�⇢0LaxA
+ [µSr0A� �Sr � 2eo�c]�⇢
0Sr0A
+ µV000A�⇢
0V000
A
+ [µCoxB� �Co � 3eo�c]�⇢
0CoxB
+ µV000B�⇢
0V000
B
+ [µOx
O� �O + 2eo�c]�⇢
0Ox
O+ µV··
O�⇢
0V··
O+ [µh· � eo�c]�⇢
0h·
�
dv
0
+
ZZZ
Vg
⇢
[T g � Tc]�sgv � �P
b
+ [µgasO2
� 2�O]�⇢O2 + [µgasN2
� 2�N]�⇢N2
�
dv
+ [surface terms](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.
⇢
0LaxA
+ ⇢
0Sr0A
+ ⇢
0V000
A= ⇢
A(3.16)
⇢
0CoxB
+ ⇢
0V000
B= ⇢
B(3.17)
⇢
0Ox
O+ ⇢
0V··
O, ⇢
0h· = ⇢
O(3.18)
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) r ·D = 3⇢0LaxA + 2⇢0Sr0A + 3⇢0CoxB� 2⇢0Ox
O
Noting � the electric potential, we can rewrite:
(3.20) E · �D = �r� · �D = �[r · (��D)� �(r · �D)] = �r · (��D) + ��(r ·D)
Using this decomposition, the integral involving the electric displacement in the ex-
pression of �" simplifies to [31]:
ZZZ
V 0ox
E · �Ddv
0 =
ZZZ
V 0ox
eo�
n
3⇢0LaxA + 2⇢0Sr0A + 3⇢0CoxB� 2⇢0Ox
O
o
dv
+
Z
@Vox+⌃0��D · nda(3.21)
36
The last transformation involves the elastic term T : �F = Tji�Fij using the Einstein
notation. Using the divergence theorem, the integral on the elastic strain energy can
be rewritten as
(3.22)
ZZZ
V 0ox
Tji�Fijdv =
ZZ
⌃0Tjin
ox0
j �uida�ZZZ
V 0ox
Tji,j�uidv
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
�"
⇤ =
ZZZ
V 0ox
⇢
[T ox � Tc]�soxv0 + (T ·r)�u+ [⌘LaxA � �La � 3eo�c]�⇢
0LaxA
+ [⌘Sr0A � �Sr � 2eo�c]�⇢0Sr0A
+ [⌘CoxB� �Co � 3eo�c]�⇢
0CoxB
+ [⌘Ox
O� �O + 2eo�c]�⇢
0Ox
O+ [⌘h· � eo�c]�⇢
0h·
�
dv
0
+
ZZZ
Vg
⇢
[T g � Tc]�sgv � �P
b
+⇥
µ
gasO2
� 2�O⇤
�⇢O2 +⇥
µ
gasN2
� 2�N⇤
�⇢N2
�
dv
+ [surface terms](3.23)
37
where more surface integrals have been added to the last term and the electrochemical
potentials are defined as
⌘LaxA= µLaxA
� µV000A+ 3eo�(3.24)
⌘Sr0A= µSr0A
� µV000A� 2eo�(3.25)
⌘CoxB= µCoxB
� µV000B� 3eo�(3.26)
⌘Ox
O= µOx
O� µV··
O+ 2eo�(3.27)
⌘h· = µh· + eo�(3.28)
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:
(3.29) T
ox = T
g = Tc
• the mechanical equilibrium condition imposes that
(3.30) T ·r = 0
• the chemical equilibrium condition states that the chemical potential of each
species is constant in the system, and specifically
⌘LaxA= �La + 3eo�c(3.31)
38
⌘Sr0A= �Sr + 2eo�c(3.32)
⌘CoxB= �Co + 3eo�c(3.33)
⌘Ox
O= �O � 2eo�c(3.34)
⌘h· = eo�c(3.35)
µ
gasO2
= 2�O(3.36)
µ
gasN2
= 2�N(3.37)
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⇢0LaxA + 2⇢0Sr0A + 3⇢0CoxB= 2⇢0Ox
O
Using this new conditions in Eq. (3.23) simplifies the chemical equilibrium
condition to:
(3.39) µO =1
2µ
gasO2
where µO is the chemical potential of oxygen in the oxide
(3.40) µO = µOx
O� µV··
O+ 2⌘h·
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]
(3.41) OxO + 2h. ! V··
O +1
2O2(gas)
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
(3.42) g
oxv0 = e
oxv0 � Ts
oxv0 � �ij✏ij
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, 3 and implicit summation
over repeated indices from 1 to 3 is assumed. Thus, ✏ij�ij 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]:
�e
oxv0 =T �s
oxv0 + �ij�✏ij + ⌘Ox
O�⇢Ox
O+ ⌘V··
O�⇢V··
O+ ⌘h·
�⇢h·
+ ⌘LaxA�⇢LaxA
+ ⌘Sr0A�⇢Sr0A
+ ⌘V000A�⇢V000
A+ ⌘CoxB
�⇢CoxB+ ⌘V000
B�⇢V000
B(3.43)
40
where ⌘i = µi + zi� is the electrochemical potential of specie i, zi its charge (e.g. -2
for i = OxO) 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
(3.44) de
oxv0 = Tds
oxv0 + �ijd✏ij + µOd⇢Ox
O+ µLad⇢LaxA
+ µSrd⇢Sr0A+ µCod⇢CoxB
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 g
oxv0 , defined
by (3.42),
(3.45) dg
oxv0 = �s
oxv0 dT � ✏ijd�ij + µOd⇢Ox
O+ µLad⇢LaxA
+ µSrd⇢Sr0A+ µCod⇢CoxB
Noting that the number density can be linked to the oxygen non-stoichiometry, �, in
La1�xSrxCoO3��: ⇢Ox
O= ⇢
O � ⇢V··O= ⇢
O(1 � �/3), the total derivative of the new free
energy rewritten as
(3.46) dg
oxv0 = �s
oxv0 dT � ✏ijd�ij +
1
3⇢
OµOd� + µLad⇢LaxA
+ µSrd⇢Sr0A+ µCod⇢CoxB
41
Finally taking the cross derivatives of the second and third terms yields a Maxwell
relation involving the oxygen chemical potential:
(3.47)
✓
@µO
@�ij
◆
T,�kl 6=ij
,⇢i
=3
⇢
O
✓
@✏ij
@�
◆
T,�kl
,⇢i
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 coe�cient. 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,
(3.48) ✏ij = Sijkl�kl + ✏
c(�)�ij + ✏
a�ij + ✏
T (T )�ij
where Sijkl is the compliance tensor, ✏c(�) = e
c[���o]/3, ✏a and ✏T (T ) are the magnitude
of the isotropic compositional, mismatch and thermal strains respectively. e
c is the
compositional strain coe�cient and is defined by Chen et al. [12] as
(3.49) e
c =1
3�C =
1
3
@ ln V
@cV··O
!
T,P
42
where V 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
✓
@µO
@�ij
◆
T,�kl 6=ij
=e
c
⇢
O�ij
assuming the compliance tensor and compositional strain coe�cient are independent
of non-stoichiometry. Integrating this equation between the oxide under a state of
hydrostatic pressure P
o and stress, �ij, yields
(3.50) µO (�, �ij) = µO (�, P o) +e
c
⇢
O(�kk + P
o�kk)
where summation over repeated indices from 1 to 3 is assumed, µO (�, P o) is the
chemical potential of oxygen at non-stoichiometry � and under hydrostatic pressure
of P o = 1atm, which is measured experimentally. |�kk| � P
o = 1atm is assumed to
be the case in the rest of the chapter, Eq. (3.50) simplifies to
(3.51) µO (�, �ij) = µO (�, 0) +e
c
⇢
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 coe�cient 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) �kl (�) = Cklmn
�
✏mn � ✏
c (�) �mn � ✏
a�mn � ✏
T (T )�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 1073K, Mizusaki et al.’s
form will be used [13]:
(3.53) µO (�, 0)� µO (�o, 0) = (4h
oO(x)� a(x)�)� T
✓
4s
oO(x) +R ln
�
3� �
�◆
where µO (�o, 0) is the chemical potential of oxygen in bulk LSC at P
oO2
= 1atm.
4h
oO(x), 4s
oO(x) and a(x) are parameters (dependent on the substitution level, x)
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
a slab of oxide in equilibrium with an atmosphere at an oxygen partial pressure P
oO2,
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-
librium with oxygen gas at P oO2
in the
reference state. Under no stress, the
oxygen non-stoichiometry is �o.
(b) Applying a stress (in this case,
compressive, but not necessarily hydro-
static) to the oxide deforms it, com-
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 thenon-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,
µO (�o, 0) =1
2µ
gasO2
�
P
oO2
�
(3.54)
µO (�, �ij) =1
2µ
gasO2
(PO2)(3.55)
45
Using Eq. (3.51) in (3.55) yields:
(3.56)1
2µ
gasO2
(PO2) = µO (�, 0) +e
c
⇢
O�kk(�)
Subtracting Eq. (3.54) from this last equation results in
(3.57)1
2µ
gasO2
(PO2)�1
2µ
gasO2
�
P
oO2
�
= µO (�, 0)� µO (�o, 0) +e
c
⇢
O�kk(�)
Assuming an ideal gas and using the expression of 12µ
gasO2
from Eq. (3.53) in (3.57) finally
yields
RT
2ln
✓
PO2
P
oO2
◆
=(4h
oO(x)� a(x)�)
� T
✓
4s
oO(x) +R ln
�
3� �
�◆
+2ec
⇢
O�kk(�)(3.58)
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]:
(3.59) E
of,vac = Gcrystal+vac +
1
2GO2 �Gcrystal
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]
Gcrystal =µLaxA
NLaxA
NA
+ µSr0A
NSr0A
NA
+ µV000A
NV000A
NA
+ µCoxB
NCoxB
NA
+ µV000B
NV000B
NA
+ µOx
O
NOx
O
NA
+ µV··O
NV··O
NA
+ µh·Nh·
NA
where the Nis are the number of atoms of each species and NA is Avogadro’s number.
Using the conservation of lattice sites on each sublattice
NLaxA+NSr0A
+NV000A
= N
A
NCoxB+NCo000B
= N
A
NOx
O+NO··
O= 3NA
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: 3NLaxA+2NSr0A
+3NCoxB+2NOx
O= Nh· in this
equation yields
Gcrystal =µLa
NLaxA
NA
+ µSr
NSr0A
NA
+ µCo
NCoxB
NA
+⇥
3µV··O+ µV000
A+ µV000
B
⇤
N
A
NA
+ µO(�, 0)NOx
O
NA
(3.60)
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:
Gcrystal =µLa
NLaxA
NA
+ µSr
NSr0A
NA
+ µCo
NCoxB
NA
+⇥
3µV··O+ µV000
A+ µV000
B
⇤
N
A
NA
+ µO(�, 0)(NOx
O� 1)
NA
(3.61)
Using Eq. (3.60) and (3.61) in (3.59) finally yields the vacancy formation energy for a
given composition under no stress:
(3.62) E
of,vac =
1
2NA
µO2 �1
NA
µO(�, 0)
If the gas was taken such that it gave rise to the non-stoichiometry �, this value would
be 0 —cf. Eq. (3.39). The formation energy of a vacancy under stress, E�f,vac, can be
computed in a similar manner. Taking the crystal to be at the same non-stoichiometry,
�, and keeping the applied stress constant between the configurations with and without
48
an extra vacancy, � = cst, the elastic energy cancels out between those configurations
and
(3.63) E
�f,vac =
1
2NA
µO2 �1
NA
µO(�, �)
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):
(3.64) �Ef,vac = E
�f,vac � E
of,vac = � 1
NA
µO(�, �) +1
NA
µO(�, 0) = � e
c
⇢
ONA
�kk
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 coe�cient, 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
a resistance [10, 43]. This capacitance, Cchem, is characteristic of the charge in the
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, a quantity that is directly measured experimentally, as a function of the
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, �⇤, and stoichiometry �
⇤ in equilibrium with oxygen of
partial pressure PO2 . This is the initial configuration. EIS consists of applying a
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, �, away from the initial one, �⇤. 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, � ! �
⇤, is dictated by that impedance.
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
(3.65) D = µO (�, �)� 1
2µ
gasO2
=1
2µO (�, �)� 1
2µO (�⇤, �⇤)
where �⇤ is the stress in the film at a non-stoichiometry �⇤. Following Adler’s approach
[43] with this new driving force, the chemical capacitance is
(3.66) Cchem = �4F 2L⇢
O
3RTf
51
where L is the thickness of the film and f is
(3.67) f = � 1
RT
@µO
@�
�
�
�
�
�=�⇤
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:
(3.68) Cchem =4F 2
L⇢
O
3
✓
a(x) +3RT
�
⇤(3� �
⇤)� e
c
⇢
O
@�kk
@�
�
�
�
�
�=�⇤
◆�1
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
coe�cient e
c 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 coe�cients —c.f. Table (3.1), the non-stoichiometry of LSC-82 in the
stress-free state at T = 1073K and P
oO2
= 1 atm is computed to be �o = 0.0059.
The oxygen pressure is further assumed to be equal in the initial and final states,
ie. PO2 = P
oO2. The e↵ects of a hydrostatic stress are considered here.
3.6.1.1. Small changes in non-stoichiometry. A qualitative idea of the e↵ects of
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.
Assuming that the stress is applied at constant gas pressure, setting PO2 = P
oO2
in
Eq. (3.57) yields the equation governing the non-stoichiometry as a function of stress:
(3.69) µO (�)� µO (�o) = � e
c
⇢
O�kk (�)
Using Eq. (3.52), the stress can be written as the sum of a stoichiometry-independent
�kl(�o) and a stoichiometry-dependent terms,
�kl (�) = Cklmn
�
✏mn � ✏
a�mn � ✏
T (T )�mn
�
� e
c
3(� � �
o)Cklmn�mn
= �kl (�o)� e
c
3(� � �
o)Cklnn(3.70)
53
The stoichiometry dependence of the stress, and thus the right hand side of Eq. (3.69),
is now explicit. �(�o) can also been seen as the stress applied to the reference state.
The chemical potential is a nonlinear function of �, see (3.58), thus to solve (3.69) for
�, we need to expand the chemical potentials to first order in � � �
o:
(3.71) � � �
o =e
c
⇢
O
�@µO
@�
�
�
�
�
�=�o+
(ec)2
⇢
O
E
3(1� 2⌫)
��1
�kk (�o)
where E is Young’s modulus and ⌫ is Poisson’s ratio [44]. In most cases, @µO/@� <
0 and increasing vacancy concentration expands the lattice, e
c> 0. Thus, non-
stoichiometry, �, decreases under a compressive stress (�kk < 0) and conversely, �
increases under a tensile stress. Furthermore, using parameters from Table 3.1 and a
temperature of T = 1073K, one can estimate the ratio of the two terms in the brackets
of Eq. (3.71) as
(ec)2E/(3(1� 2⌫)⇢O)
�@µO
@�
�
�
�=�o
⇡ 10�2
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 �, for this particular oxide and
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:
(3.72) � � �o =e
c
⇢
O
�@µO
@�
�
�
�
�
�=�o
��1
�kk (�o) = 6.5⇥ 10�13
�kk(�o)
54
where �kk (�o) is expressed in Pa. Hence, a compressive stress of 100MPa, �kk (�o) =
�0.1GPa, induces a change in non-stoichiometry of �� �o = �5⇥ 10�5, corresponding
to a �1% relative change.
The non-stoichiometry is directly proportional to the mole fraction of oxygen va-
cancies, via the lattice constraint cV··O= �/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 ⇢Sr0A + 3⇢V000A+ 3⇢V000
B= 2⇢V··
O+ ⇢h· where the
⇢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) ch· = x� 2�
for La1�xSrxCo3��O, where ch· is the fraction of holes per B sublattice sites. Hence,
both the vacancy concentration change, 4cV··O, and the electronic carriers concentration
change, 4ch· , can be evaluated for a given stress.
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
4�ion�
oion
=4cV··
O
c
oV··
O
= � e
c
⇢
O�
o
@µO
@�
�
�
�
�
�=�o
��1
�kk (�o) = �1%(3.74)
55
4�elec�
oelec
=4ch·
c
oh·
=2ec
⇢
O(x� 2�o)
@µO
@�
�
�
�
�
�=�o
��1
�kk (�o) = .2%(3.75)
for a stress �kk (�o) = �0.1GPa. Because the conductivities are intrinsically so high,
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
not hold. One must then solve the equations numerically. Setting PO2 = P
oO2
in
Eq. (3.58) yields a nonlinear equation for the non-stoichiometry in LSC-82
(3.76) (4h
oO � a�)� T
✓
4s
oO +R ln
�
3� �
�◆
= � e
c
⇢
O�kk(�)
where the coe�cients4h
oO, 4s
oO, and a are evaluated for a substitution level of x = 0.2.
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
above. The stress tensor becomes independent of composition and �kk(�) = �kk(�o) in
Eq. (3.76).
The variations of � with �kk (�o) 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. Since ceramics are not prone to fracture in compression, the
calculations extends much more with compression (�kk (�o) < 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, �kk = �5GPa, decreases the non-stoichiometry by approximately a
factor 2. Furthermore, the change in � is roughly exponential in the trace of the stress,
56
0.001
0.01
!10 !5 0
!
"kk (!o) (GPa)
!o
exactlinearized
Figure 3.3. Log of the non-stoichiometry of LSC-82 as a function of thetrace of the stress in the reference state, �kk(�o), in the hydrostatic caseat T = 1073K, under constant oxygen partial pressure PO2 = P
oO2
=1 atm. Negative values of �kk(�o) correspond to compressive stresseswhile 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
⇡ 3GPa in absolute value and a large discrepancy for stresses larger than that value.
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 coe�cient is purely dilational and as
a result the oxygen chemical potential couples only to the trace of the stress. The film
is taken to be su�ciently 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: latticeconstant, compositional coe�cient, Young’s modulus, Poisson’s ratio andparameters used in the itinerant electron model.
Parameter ValueaLSC (nm) .3833 [45] 1
e
c .129 [12]E (GPa) 160 [35]⌫ .25 [11]4h
oO (kJ/mol) -146 [13]
4s
oO (J/mol) -86.6 [13]
a (kJ/mol) 418 [13]
1The number density of oxygen lattice sites is ⇢O = 3/(NA(aLSC)3), where NA is Avogadro’s constant.
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:
(3.77) ✏
aij =
aox � as
as
�ij = ✏
a�ij
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
taken to be those at a temperature To and non-stoichometry �o.
(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 �, that the latttice parameter of the substrate
remains unchanged, and a cubic crystal, the strain in the film varies with � as,
(3.78) ✏
cij =
e
c
3(� � �
o) �ij = ✏
c(�)�ij
where e
c is defined by Eq. (3.49).
(3) thermal strain. When the coe�cients 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) ✏
Tij = (↵ox � ↵s) (T � To)�ij = ✏
T (T )�ij
where ↵ox and ↵s are the CTE of the oxide film and the substrate respectively,
T � To is the change in temperature from that at which the lattice parameter
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) �kk(T, �) = �2Y�
✏
c (�) + ✏
a + ✏
T (T )�
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
(3.81) Y =E
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, is di↵erent from
that of the substrate, as. The oxygen non-stoichiometry in this state is �o.
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, the lattice parameter of the substrate.
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:
(3.82) �kk(�⇤) = �2Y (✏a + ✏
c(�⇤))
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 , the oxide and substrate are in their stress-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 coherentand dislocation-free thin film grown on a substrate with a lattice mis-match. (a) Stress-free state: the lattice parameters of the oxide and thesubstrate are di↵erent, aox 6= as. (b) The film is isotropically strained tomake the lattice parameters equal. (c) The oxide in-plane displacementis set by by the substrate, but it relaxes in the normal direction. (d)Oxygen equilibrates with the atmosphere resulting in �⇤ and modifyingthe stress applied by the substrate.
(c) The temperature is changed to T < Tf and the two materials shrink, but at
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 filmgrown on a substrate under thermal stress. (a) At the firing temperatureTf , the oxide and substrate are stress-free. (b) The thin film is grownon the substrate, stress-free. (c) Changing the temperature, the twomaterials shrink, but at di↵erent rates. The substrate imposes the in-plane displacement of the film, while top surface relaxes. (d) Oxygenequilibrates with the atmosphere resulting in �⇤ and modifying the stressapplied by the substrate.
Assuming no sliding at the interface and an infinite substrate, the stress in the film is
proportional to the sum of the thermal strain ✏T and the compositional strain ✏c and
(3.83) �kk(T, �⇤) = �2Y (✏T (T ) + ✏(�⇤))
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 � = �
⇤, yields that non-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 1.5µm La0.6Sr0.4CoO3��
(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, 4h
oO
and 4s
oO— from Mizusaki et al. [13] for comparison. Those coe�cients were used in
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. The chemical
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
(3.84) Cchem =4F 2
L⇢
O
3
✓
a(x) +3RT
�
⇤(3� �
⇤)+
2Y (ec)2
3⇢O
◆�1
where �⇤, the non-stoichiometry in the film, is numerically evaluated using Eq. (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 = 873K is
✏
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, 4h
oO and
64
10-2
10-1
100
-6 -4 -2 0
Cch
em (
F/c
m2)
log PO2 (atm)
LSC-55 under stressLSC-73 under stressLSC-64 under stressLSC-64 thin filmLSC-64 bulk
Figure 3.6. Chemical capacitance versus oxygen partial pressure at T =873K estimated for bulk La0.6Sr0.4CoO3�� [10, 13] (solid blue line), asreported by Kawada et al. in a 1.5µm-thick LSC film on GDC [10] (redsymbols) and that in presence of stress (solid green line). The dotted linesrepresent the predictions when using coe�cients —a, 4h
oO or4s
oO— cor-
responding to the thermodynamics of LSC-73 and LSC-55 respectively.The composition of the bulk material is assumed to be well characterizedand thus the coe�cients are fixed at the measured values.
4s
oO is also reported in Fig. 3.6 using doted lines, where the coe�cients correspond to
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 . In fact, at low
oxygen pressure, the left hand side of Eq. (3.58) can be of the same order of magnitude
as the stress term making �, and therefore Cchem, much more sensitive to stress at
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 coe�cient, Young’s modulus,Poisson’s ratio and parameters used in the itinerant electron model.
Parameter GDC-91 LSC-64 LSC-55 LSC-73abulk (nm) .5407 [47] 2 .3874 1 2 [48] idem idem{T in K} {1673} {900}CTE 12⇥ 10�6 [49] 20⇥ 10�6 [12] idem idem
e
c (K�1) 0 .129 [12] idem idemE (GPa) - 160 [35] idem idem
⌫ - .30 idem idem4h
oO (kJ/mol) - -85.8 3 -70.7 [13] -112 [13]
4s
oO (J/mol) - -69.4 3 -64.4 [13] -112 [13]
a (kJ/mol) - 289 3 222 [13] 385 [13]
2The [110] direction of LSC aligns with the [100] direction of GDC, to give a lattice mismatch strain
of ✏a =
p2aLSC�aGDC
aGDC= 2.23% at T = 873K.
3Interpolated by Kawada et al. [10]
oxygen pressures close to 1 atm. 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 coe�cients: a, 4h
oO and 4s
oO, as
illustrated by the magnitude of the change in the chemical capacitance with variations
in those coe�cients 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.
A similar comparison can be made with the results of La O’ et al. [11,50]. They
measured the chemical capacitance of La0.8Sr0.2CoO3�� (LSC-82) films of thicknesses
20, 45 and 130 nm grown on a 8 mol% yttria-stabilized zirconia (YSZ) substrate for
a temperature T = 793K. The values for bulk LSC-82 are computed as described
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 = (abulk � afilm)/afilm. Accounting for the thermal
strain arising from the high temperature of the experiments, the mismatch strain is
✏
a(T = 793K) = ✏
a(T = 298K) + (↵LSC � ↵YSZ)4T = �1.01%, �1.64% and �0.30%
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 45 nm 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] which display two regions (high and
low PO2). Such e↵ect is due to the di↵erent Strontium concentrations and its e↵ect on
the coe�cients 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
100
101
102
103
104
-4 -3 -2 -1 0
Cch
em (
F/c
m3)
log PO2 (atm)
LSC-73 under stressLSC-82 under stressLSC-82 filmLSC-82 bulk
Figure 3.7. Chemical capacitance versus oxygen partial pressure at T =793K evaluated for bulk La0.8Sr0.2CoO3�� [13] (solid blue line), as re-ported by la O’ et al. in a 45nm-thick LSC film on YSZ [11] (red symbols)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 coe�cients 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
of the stress and the applied strain in a biaxial configuration, �kk = 2Y ✏app [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, coe�cient of thermalexpansion, compositional coe�cient, Young’s modulus, Poisson’s ratioand parameters used in the itinerant electron model.
Parameter YSZ LSC-82arelax (nm) - .3837 [45]CTE (K�1) 10⇥ 10�6 [51] 17⇥ 10�6 [12]
e
c 0 .112 [12]E (GPa) - 160 [35]
⌫ - .25 [11]4h
oO (kJ/mol) - -146 [13]
4s
oO (J/mol) - -86.6 [13]
a (kJ/mol) - 418 [13]
computed as a function of strain and is
(3.85) �Ef,vac(eV) = � 2ecY
⇢
ONAq�kk = �0.059⇥ ✏
app(%)
using a chemical expansion coe�cient e
c = .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] and a Poisson’s ratio of ⌫ = .25 [11]) and
the conversion factor q = 1.6⇥ 10�19 J/eV. As a result, a 4% biaxial tensile strain,
✏
app = 4%, induces a change in the vacancy formation energy of �Ef,vac = �236meV,
consistent with the �300meV and �500meV changes reported for such strain by Don-
ner et al. [35] and Kushima et al. [40] respectively. As stated earlier, this means that
a tensile strain facilitates the formation of vacancies, hence increasing their concen-
tration. Note however, that the model does not account for the strong jump in the
69
vacancy formation energy seen in [40] for smaller tensile strains, and it assumes linear
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,
hc ⇡ 500 A for a thin film with a 2% strain [52]. Because Kawada’s film (1.5 µm) is
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 coe�cients used in the thermodynamic
model of the oxide. Such coe�cients 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 coe�cients a, 4h
oO and 4s
oO. However, there is no reason to
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 coe�cients 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.
A last possible explanation for the di↵erence between predictions and experiments
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— in a configuration under stress. If the thermodynamics of the per-
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] and a
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 e�ciently
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] and defect formation at the
interface with the electrolyte [59] but recent studies have shown that deterioration
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 1.0A/cm2
and 1.5A/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
a polarization resistance due to the oxygen electrode is responsible for a high oxygen
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 drivesoxygen from the hydrogen to the oxygen side. The applied current in thepresence of an oxygen electrode polarization resistance is responsible fora high oxygen potential under the O electrode and, if the current is abovea critical current, the formation of oxygen bubbles in the electrolyte.
a high oxygen pressure (in certain cases much higher than 1 atm) [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 oxideelectrolyte phase. The electrolyte is assumed to be YSZ, composed oftwo sublattices: one for the cations and one for the oxygen ions. (a) Thelarge green spheres represent cations while the medium-sized blue sphererepresents a cation vacancy and the small red ones represent oxygen ions.Albeit present in the system, electron holes and oxygen vacancies are notrepresented. (b) Under certain conditions, oxygen ions can react withholes around a cation vacancy, forming a molecule of oxygen and simul-taneously destructing a unit cell. This local destruction of the latticegives way to a bubble (purple ellipse) containing molecular oxygen (greysmall 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
are present, zirconium ions, ZrxZr, and yttrium ions, Y0Zr, with real absolute charges of
+4 and +3 respectively. Cation vacancies, V0000Zr , are also present on this sublattice, in
80
small concentration. On the anion sublattice, the only element present is oxygen, OxO,
bearing a �2 real charge, and vacancies, V··O. Last, electron holes, h·, are assumed to
be present in the system, located on the zirconium sites [71,72].
As shown by Matus et al. [60] and Knibbe et al. [61] oxygen bubbles nucleate from
a homogeneous solid phase. Space must thus be created in the lattice to allow for the
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:
(4.1) V0000
Zr + 2OxO + 4h· ! O2(bubble)
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
Zr + 2V··O ! nil(4.2)
2OxO + 4h· ! O2 + 2V··
O(4.3)
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
a hydrostatic pressure P
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. The e↵ects of dislocations, possible exchange of atoms
83
Figure 4.3. Sketch of the system under study for the derivation of theequilibrium conditions: perfect YSZ lattice with a spherical bubble ofoxygen. The larger green spheres represent the cation sublattice (pop-ulated with ZrxZr, Y
0Zr and V
0000Zr) and the smaller red ones represent the
oxygen sublattice (populated with OxO and V··
O) in the electrolyte. Al-though not represented here, holes are also present in the oxide. Thesmaller 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 su�ciently 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, eoxv0 ,
is taken to be a function of the entropy s
oxv0 , the deformation gradient tensor F, the
electric displacement field D and the number densities of the di↵erent elements ⇢0ZrxZr ,
⇢
0Y0
Zr, ⇢0V0000
Zr, ⇢0Ox
O, ⇢0V··
Oand ⇢0h· .
(4.4) e
oxv0
⇣
s
oxv0 ,F,D, ⇢
0Ox
O, ⇢
0V··
O, ⇢
0h· , ⇢
0ZrxZr
, ⇢
0Y0
Zr, ⇢
0V0000
Zr
⌘
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:
�e
oxv0 =T
ox�s
oxv0 + T : �F+ JE · �D+ µOx
O�⇢
0Ox
O+ µV··
O�⇢
0V··
O+ µh·
�⇢
0h·
+ µZrxZr�⇢
0ZrxZr
+ µY0Zr�⇢
0Y0
Zr+ µV0000
Zr�⇢
0V0000
Zr(4.5)
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/dv
0), E
the electric field and µi the chemical potential of specie i (i = OxO, V
··O, h
·, ZrxZr, Y0Zr or
V0000Zr). The operator ”·” represents the classical scalar product while ”:” represents the
tensorial scalar product.
In a similar fashion, the energy density per unit area of the interface in the stress-
free state e⌃a0 is taken to be a function of the entropy s⌃a0 , the deformation gradient tensor
of the interface F, the mean curvature at the surface 0, and the surface densities of the
di↵erent elements �0ZrxZr
, �0Y0
Zr, �0
V0000Zr, �0
Ox
O, �0
V··Oand �0
h· . Here, the bubble is assumed to
85
be spherical. In the present configuration, the mean curvature, 0, is negative as the gas
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
⌃a0 , it is not taken into account.
The energy of the interface is of the form:
(4.6) e
⌃a0
⇣
s
⌃a0 , F,0,�0
Ox
O,�0
V··O,�0
h· ,�0ZrxZr
,�0Y0
Zr,�0
V0000Zr
⌘
A variation of any of these variables induces a change in the internal energy of:
�e
⌃a0 =T
⌃�s
⌃a0 + T : �F+K�
0 + �Ox
O��0
Ox
O+ �V··
O��0
V··O+ �h·
��0h·
+ µZrxZr��0
ZrxZr+ µY0
Zr��0
Y0Zr+ µV0000
Zr��0
V0000Zr
(4.7)
where T
⌃ is the temperature of the interface, T is the surface stress tensor, K is a
coe�cient linking a change in curvature to a change in energy and �i is the interfacial
chemical potential of specie i (i = OxO, V
··O, h
·, ZrxZr, Y0Zr or V
0000Zr). Assuming the radius
of curvature to be small compared to the thickness of the interface, the term K� can
be neglected [33].
The internal energy density of the gas phase in the present state egv is a function of
the entropy sgv, the pressure in the bubble P b 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 . The internal energy of the gas phase is of the form:
(4.8) e
gv
�
s
gv, P
b, ⇢O2
�
86
A variation of any of these variables induces a change in the internal energy of:
(4.9) �e
gv = T
g�s
gv � �P
b + µ
bO2�⇢O2
where T g is the temperature of the gas phase and µ
bO2
the chemical potential of oxygen
in the bubble.
The total internal energy of the system is the sum of these three contributions:
(4.10) " =
Z
V 0ox
e
oxv0 dv
0 +
Z
Vg
e
gvdv +
Z
⌃0e
⌃a0da
0
and its first variation reads:
�" =
Z
V 0ox
�e
oxv0 dv +
Z
Vg
�e
gvdv +
Z
⌃0�e
⌃a0da
0
+
Z
⌃0e
oxv0 �y
oxda
0 +
Z
⌃
e
gv�y
gda+
Z
⌃0e
⌃a02
0�y
oxda
0(4.11)
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,
(4.12) S =
Z
V 0ox
s
oxv0 dv
0 +
Z
⌃0s
⌃a0da
0 +
Z
Vg
s
gvdv
• constant number of oxygen atoms NO across the oxide, the interface and the
gas
(4.13) NO =
Z
V 0ox
⇢
0Ox
Odv
0 +
Z
⌃0�Ox
Oda
0 + 2
Z
Vg
⇢O2dv
• constant number of zirconium and yttrium atoms, NZr and NY, across the
oxide and the interface given their non solubility in the gas
NZr =
Z
V 0ox
⇢
0ZrxZr
dv
0 +
Z
⌃0�0ZrxZr
da
0(4.14)
NY =
Z
V 0ox
⇢
0Y0
Zrdv
0 +
Z
⌃0�0Y0
Zrda
0(4.15)
These constraints are taken into account in the Lagrangian of the system:
"
⇤ = "� TcS � �ONO � �ZrNZr � �YNY(4.16)
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) �"
⇤ = �"� Tc�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:
⇢
0Ox
O+ ⇢
0V··
O= ⇢
O(4.18)
⇢
0ZrxZr
+ ⇢
0Y0
Zr+ ⇢
0V0000
Zr=
1
2⇢
O(4.19)
�0Ox
O+ �0
V··O= �O(4.20)
�0ZrxZr
+ �0Y0
Zr+ �0
V0000Zr
=1
2�O(4.21)
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
�⇢
0Ox
O+ �⇢
0V··
O= 0(4.22)
�⇢
0ZrxZr
+ �⇢
0Y0
Zr+ �⇢
0V0000
Zr= 0(4.23)
��0Ox
O+ ��0
V··O= 0(4.24)
��0ZrxZr
+ ��0Y0
Zr+ ��0
V0000Zr
= 0(4.25)
• local charge neutrality in the oxide and at the surface,
4⇢0V0000Zr
+ ⇢
0Y0
Zr= ⇢
0h· + 2⇢0V··
O(4.26)
4�0V0000
Zr+ �0
Y0Zr= �0
h· + 2�0V··
O(4.27)
89
Taking the first variation of those equations yields
4�⇢0V0000Zr
+ �⇢
0Y0
Zr= �⇢
0h· + 2�⇢0V··
O(4.28)
4��0V0000
Zr+ ��0
Y0Zr= ��0
h· + 2��0V··
O(4.29)
• The electric displacement must satisfy Gauss’s law at all points inside the
crystal:
(4.30) r ·D = eo
�
4⇢ZrxZr + 3⇢Y0Zr+ ⇢h· � 2⇢Ox
O
= 0
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) � �y
g =⇣
�u+ nox0 · F�yox⌘
· 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
(4.32)
Z
V 0ox
T : �Fdv0 =Z
⌃0
⇣
T · nox0⌘
· �uda0 �Z
V 0ox
(T ·rR0) · �udv0
whererR0 denotes the gradient with respect to the initial state in the volume V 0ox. Using
results from Leo and Sekerka [75] assuming an isotropic surface stress, the integral of
the surface stress energy on the spherical closed surface ⌃0 is rewritten
(4.33)
Z
⌃0T : �Fda0 = �
Z
⌃0
⇣
T ·r⌃0
⌘
�uda0
where r⌃0 is the gradient acting on the interface coordinates in the reference state.
Such isotropic in-plane stress at the interface has the following form
(4.34) T = f I
where I 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:
(4.35)
Z
⌃0T : �Fda0 = �
Z
⌃
2f
Ro
�uda0
where we have noted Ro = �1/0 the radius of the bubble in the reference state.
The divergence theorem applied to the integral of the electric energy yields:
Z
V 0ox
JE · �Ddv
0 =
Z
V 0ox
eo�
n
4⇢0ZrxZr + 3⇢0Y0Zr+ ⇢
0h· � 2⇢0Ox
O
o
dv
0 +
Z
⌃0��D · nox0
da
0
91
=
Z
⌃0��D · nox0
da
0(4.36)
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), and using (4.22-4.29) the first variation of that energy reads:
�"
⇤ =
Z
Vox0
⇢
[T ox � Tc] �soxv0 � (T ·rR0) · �u+
1
2µ
oxO2
� �O
�
�⇢
0Ox
O
+ [µZr � �Zr] �⇢0ZrxZr
+ [µY � �Y] �⇢0Y0
Zr
�
dv
0
+
Z
Vg
�
[T g � Tc] �sgv +
⇥
µ
bO2
� 2�O⇤
�⇢O2
dv
+
Z
⌃0
⇢
⇥
T
⌃ � Tc
⇤
�s
⌃a0 +
1
2µ
⌃O2
� �O
�
��0Ox
O
+ [µZr � �Zr] ��0ZrxZr
+ [µY � �Y] ��0Y0
Zr
�
da
0
+
Z
⌃0
!
oxv0 � J!
gv �
2�
Ro
�
�y
oxda
0
+
Z
⌃0
nox0 · T� J!
gvn
ox0 · F�1 � 2f
Ro
�
�uda0
+
Z
⌃0[�] �D · nox0
da
0(4.37)
where
(4.38) µ
oxO2
= 2µOx
O+ 4µV··
O� 2µh·
is the chemical potential of oxygen in the oxide, defined as that of oxygen gas in
equilibrium with the oxide at that given composition according to the reaction 2OxO +
4h· ! O2(gas)+ 2V··O, µ
⌃O2
= 2�Ox
O+4�V··
O� 2�h· is the chemical potential of oxygen at
92
the interface defined in a similar fashion, µbO2
is the chemical potential of oxygen gas
in the bubble. µZr = µZrxZr� µV0000
Zr� 4µh· and µY = µY0
Zr� µV0000
Zr� 3µh· are the chemical
potentials of yttrium and zirconium in the oxide, defined in a similar way as oxygen
and µ
⌃Zr = �ZrxZr
� �V0000Zr
� 4�h· and µ
⌃Y = �Y0
Zr� �V0000
Zr� 3�h· are their surface chemical
potentials.
(4.39) !
oxv0 = e
oxv0 � Tcs
oxv0 � �O⇢
0Ox
O� �Zr⇢
0ZrxZr
� �Y⇢0Y0
Zr
is the grand potential of the oxide,
(4.40) !
gv = e
gv � Tcs
gv � 2�O⇢O2
is the grand potential of the gas and �0 = e
⌃a0 � Tcs
⌃a0 � �O�0
Ox
O� �Zr�0
ZrxZr� �
0Y�
0YZr
is
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:
T
ox = T
g = T
⌃ = Tc(4.41)
µ
oxO2
= µ
bO2
= µ
⌃O2
= 2�O(4.42)
93
µZr = µ
⌃Zr = �Zr(4.43)
µY = µ
⌃Y = �Y(4.44)
T ·rR0 = 0(4.45)
!
oxv0 � J!
gv �
2�0
Ro
= 0(4.46)
nox0 · T� J!
gvn
ox0 · F�1 � 2f
Ro
nox0 = 0(4.47)
�
�
�
⌃0 = 0(4.48)
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] have shown that the point of highest oxygen potential
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
in the oxide to the chemical potential of oxygen at the oxygen electrode (µOO2) and to
the oxygen electrode overpotential —or electrical bias from the open circuit— (⌘), as
sketched Fig. 4.4.
(4.49) µ
ox,mO2
(cV··O, �ij) = µ
OO2
+ 4F⌘
where µ
ox,mO2
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
cation vacancy concentration, µox,mO2
is a function of only one of the concentrations in
95
Figure 4.4. Sketch of the oxygen potential near the oxygen electrodeof a SOEC under an applied current. ⌘ is the oxygen electrode polar-ization. The maximum value of the oxygen chemical potential in thesystem, µox,m
O2, is located in the electrolyte at the interface with the oxy-
gen electrode [61, 68]. The critical polarization ⌘c is the polarizationabove which nucleation takes place, as defined in subsection 4.4.2, suchthat µO
O2+ 4F⌘c is chosen to be larger than µ
ox,mO2
in this figure.
the system, which is taken to be the oxygen vacancy mole fraction on the oxygen sub-
lattice, cV··O. The oxygen potential can also be a function of the stress in the oxide, �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:
(4.50) µV··O(cV··
O) = µ
oV··
O+RT ln cV··
O
96
where cV··Ois 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
the standard state pressure, P
o, and the pressure applied to the SOEC, P1, will
be neglected. Given the molar volume of YSZ, Vm = 2.1⇥ 10�5 m3/mol [78], and
a hydrostatic pressure P
1 = 10 atm, the energy di↵erence between the two states is
Vm(P o�P
1) = 19 J/mol, which is indeed negligible compared to RT = 8.3⇥ 103 J/mol
at 1000K.
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,
(4.51) µ
oxO2
�
cV··O
�
= µ
ox,oO2
+ 2RT lnh(cV··O)
where µ
ox,oO2
is the collection of the standard state terms and
h(cV··O) = cOx
O(ch·)2
�
cV··O
��1
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
cOx
O+ cV··
O= 1(4.52)
cZrxZr+ cY0
Zr+ cV0000
Zr= 1(4.53)
97
4cV0000Zr
+ cY0Zr= ch· + 4cV··
O(4.54)
In the limit of a negligible cation vacancy concentration, cV0000Zr, and assuming yttrium
and zirconium to be homogeneously distributed in the oxide, those conditions can be
rewritten:
cOx
O= 1� cV··
O(4.55)
cZrxZr= 1� y(4.56)
ch· = y � 4cV··O
(4.57)
where the mole fraction of yttrium atoms per cation sites has been noted cY0Zr= y and
is a function of z, the mole percent of yttria, Y2O3, in zirconia, ZrO2,
(4.58) y =NY3+
N
A=
2z
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
(4.59) µ
oxO2
�
cV··O
�
= µ
ox,oO2
+ 2RT lnh(cV··O)
where
(4.60) h(cV··O) =
�
1� cV··O
� �
y � 4cV··O
�2 �cV··
O
��1
98
As mentioned in the previous paragraph, the chemical potential of oxygen in the oxide
is independent of stress, �ij, for this particular spherical geometry.
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] and is given
by µ
ox,mO2
, Eq. (4.49). Using (4.59) in (4.49) yields:
(4.61) µ
ox,oO2
+ 2RT lnh(cV··O) = µ
OO2
+ 4F⌘
where cV··Ois 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
is in equilibrium with oxygen gas at pressure P
OO2
in the absence of an overpotential.
Using Eq. (4.49) in the limit ⌘ = 0 gives
(4.62) µ
ox,oO2
+ 2RT lnh(coV··O) = µ
OO2
where coV··Ois the equilibrium vacancy concentration at the same T and P
OO2
as in (4.61).
Using Eq. (4.62) in (4.61) finally yields
(4.63) RT ln
"
h(cV··O)
h(coV··O)
#
= 2F⌘
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 A3 for
2 formula units of YSZ [78], the hole concentration per oxygen lattice site, under a
pressure P
OO2, is given by [71]:
c
oh· = 5.84⇥ 10�2 exp
✓
�0.62 eV
kT
◆
�
P
OO2
�1/4
where P
OO2
is in units of atm. Using the local charge neutrality condition, Eq. (4.57),
the oxygen vacancy concentration at ⌘ = 0 for YSZ reads
(4.64) c
oV··
O=
y
4� 1.46⇥ 10�2 exp
✓
�0.62 eV
kT
◆
�
P
OO2
�1/4
Note that at POO2
= 0.21 atm and T = 1073K, coV··O= .03702 while y
4= .03704, which
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
!
gv =e
gasv � Ts
gv � µ
bO2⇢O2(4.65)
!
oxv0 =e
oxv0�
cV··O, cZrxZr
, cY0Zr, �ij
�
� Ts
oxv � 1
2µ
oxO2
�
cV··O, �ij
�
⇢Ox
O
� µZr(cZrxZr , �ij)⇢ZrxZr � µY(cY0Zr, �ij)⇢Y0
Zr(4.66)
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:
E
g = TS
g � P
bV
b + µO2NO2
Dividing the total energy of the phase by its volume yields
e
gv = Ts
gv � P
b + µ
bO2⇢O2
and the grand potential density is the negative of the pressure:
!
gv = �P
b
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�ij —the pressure in the reference state— and the elastic energy density [32].
Assuming small strains, this is
e
oxv0�
T, cV··O, cZrxZr
, cY0Zr, �ij
�
= e
oxv0�
T, cV··O, cZrxZr
, cY0Zr,�P
1�ij
�
+1
2✏ij�ij
where the elastic strain energy density ✏ij�ij/2 is computed from the reference state
—under hydrostatic pressure P1— to the actual state.
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:
E
ox = TS
ox � P
1V + µOx
ONOx
O+ µV··
ONV··
O+ µh·
Nh· + µZrxZrNZrxZr
+ µY0ZrNY0
Zr+ µV0000
ZrNV0000
Zr
Dividing by the volume in the reference state v
0
e
oxv0 = Ts
oxv0 � P
1 + µOx
O⇢
0Ox
O+ µV··
O⇢
0V··
O+ µh·
⇢
0h· + µZrxZr
⇢
0ZrxZr
+ µY0Zr⇢
0Y0
Zr+ µV0000
Zr⇢
0V0000
Zr
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:
e
oxv0 = Ts
oxv0 � P
1 +1
2µ
oxO2⇢Ox
O+
µV··O+
1
2µV0000
Zr
�
⇢
O + µZr⇢ZrxZr+ µY⇢Y0
Zr
where all the chemical potentials are evaluated at a hydrostatic pressure �P
1. Using
this in Eq. (4.66) yields:
!
oxv0 =Ts
oxv0 � P
1 +1
2
⇥
µ
oxO2
� µ
oxO2(�ij)
⇤
⇢
0Ox
O+ [µZr � µZr(�ij)] ⇢
0ZrxZr
+ [µY � µY(�ij)] ⇢0Y0
Zr+
µV··O+
1
2µV0000
Zr
�
⇢
O
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:
µ
oxO2(�ij) = µ
oxO2
+ A�kk
102
where A is a constant and �kk is the trace of the stress. Using the method described
in [76] and including the stress in the reference state P1, the strain and stress tensors
are computed:
(4.67)
✏rr = � 1
2Gox
R
3c
r
3
"
P
b � P
1 � 2f
Rc
#
✏✓✓ = ��� =1
4Gox
R
3c
r
3
"
P
b � P
1 � 2f
Rc
#
✏i 6=j = 0
and
(4.68)
�rr = �R
3c
r
3
"
P
b � P
1 � 2f
Rc
#
�✓✓ = ��� =1
2
R
3c
r
3
"
P
b � P
1 � 2f
Rc
#
�i 6=j = 0
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 µ
oxO2
� µ
oxO2(�ij) = 0. Assuming this result to also hold for the
chemical potential of yttrium and zirconium, the grand potential of the oxide reads
!
oxv0�
T, cV··O, �ij
�
= �P
1 +1
2✏ij�ij +
1
2µo⇢
O
103
where µo is the energy to add an extra unit cell at the surface, or equivalently to create
a Schottky defect, defined as
(4.69) µo = 2µV··O+ µV0000
Zr
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
(4.70) µo = µ
oo + 2RT ln
⇥
cV··O
⇤
where µoo 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:
!
gv = �P
b(4.71)
!
oxv = !ox +We(4.72)
where P b 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
(4.73) !ox = �P
1 +1
2⇢
Oµo
P
1 is the hydrostatic pressure on the system due to the gas in the electrode (di↵erent
from P
o, the standard pressure under which chemical potentials are measured), ⇢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). P1 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
(4.74) We =1
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:
(4.75) P
b = P
OO2
exp
✓
4F
RT
⌘
◆
105
where POO2
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):
(4.76) !
gv = �P
OO2
exp
✓
4F
RT
⌘
◆
Eq. (4.70) defines the energy µo:
(4.77) µo = µ
oo + 2RT ln
⇥
cV··O
⇤
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]:
µo
⇣
c
oV··
O
⌘
= 0(4.78)
where c
oV··
Ois the equilibrium vacancy concentration of the oxide under a temperature
T and an oxygen partial pressure of POO2, defined by Eq. (4.64). Using this in Eq. (4.70)
allows µoo to be determined and thus,
µo
�
cV··O
�
= 2RT ln
cV··O
c
oV··
O
!
(4.79)
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:
(4.80) We =3
8Gox
✓
Rc
r
◆6⇥
P
b � P
1⇤2
where r is the position in the oxide from the center of the bubble and G
ox 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. This emphasizes the importance of treating the oxide as a solid and not using
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):
(4.81) We =3
8Gox
⇥
P
b � P
1⇤2
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
(4.82) !
oxv = �P
1 +RT⇢
O ln
cV··O
c
oV··
O
!
+3
8Gox
⇥
P
b � P
1⇤2
The determinant of the deformation gradient (ratio of the volume in the actual
state to that in the initial state) is J ⇡ 1 + ✏jj. Since the trace of the strain is zero
✏jj = 0 [76], the energy associated with the bubble in Eq.(4.46) is J!gv ⇡ !
gv .
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 su�cient, 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, !gv �!
oxv , to be negative, that is for the removal of cation and
anion lattice sites from the interface and a simultaneous conversion of oxygen from the
lattice to the gas to be favorable.
Setting Rc to 1 in the equilibrium conditions Eq. (4.42)-(4.46) results in two equa-
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”,
(4.83) RT ln
"
h(cV··O)
h(coV··O)
#
= 2F⌘
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”,
(4.84) P
1 �RT⇢
O ln
cV··O
c
oV··
O
!
� 3
8Gox
⇥
P
b � P
1⇤2 = P
OO2
exp
✓
4F
RT
⌘
◆
Fig. 4.5 is the plot of the mole fraction of oxygen vacancies, cV··O, as a function of
overpotential, ⌘, as given by the ”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, the vacancy concentration imposed by the interface is smaller than
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 cV··Oas a function of overpo-
tential ⌘ as given by the ”µ-condition” —Eq. (4.83)— (red continuousline) 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 oxygenat T = 1123K. There are two regions, depending on how the vacancyconcentrations computed from both conditions compare. In region I,⌘ < ⌘s = 24mV, the vacancy concentration in the bulk is lower than thethe one at the surface. The surface wants to create extra empty latticesites (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, the vacancy concentration imposed by theinterface is smaller than that imposed by the bulk. The interface tendsto destruct empty lattice sites (grey squares), resulting in an expansionof 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 cV··Oand 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 in8-mol % YSZ electrolyte.
parameter valueT 1123Kx 8 %⇢
O 97780 mol/m3 4
P
1 1 atm 5
P
OO2
1 atm 5
G
ox 69 GPa 6
f 0 J/m2 [76]�
0 1.45 J/m2 [81]�gb 0.813 J/m2 [81]
4Computed using a volume V = 67.92˚A3
for 2 units formula of YSZ [78].5When not specified otherwise.
6Computed using a Young’s modulus of Eox
= 180GPa [80] and a Poisson’s ratio of ⌫ = 0.3 via
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.
Fig. 4.6 is a plot of the grand potential of the gas, !gv —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, �We —Eq. (4.81)— as a function of the electrode
polarization ⌘ for a 8-mol% YSZ electrolyte, where the oxygen electrode is exposed to
pure oxygen at T = 1123K. The energies vary by orders of magnitude, hence the use
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, !oxv0 < !
gv , the oxide phase is more stable than the gas phase.
Energetically, this means that creating extra lattice sites releases energy, as µo ⇡
111
!1010
!105
!1
0 100 200 300 400
En
erg
y (
J/m
3)
! (mV)
! We
"ox
"v
g
Figure 4.6. Plot of the grand potential of the gas bubble, !gv , the homo-
geneous part of the grand potential of the oxide, !ox, and the negative ofthe elastic energy, �We, as a function of the oxygen electrode polariza-tion. The parameters used in the evaluation of those energies are foundin 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 thusnucleation is not possible. The situation is reversed above the criticalpolarization ⌘s = 24mV and nucleation is possible.
!
oxv0 . Provided cations can re-equilibrate quickly, an oxygen bubble present in YSZ
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, the situation is reversed: the gas phase is preferable to
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 su�ciently
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 su�ciently 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
(4.85) Rc =�2�
�!v
=2�
!
oxv � J!
gv
113
0
20
40
60
80
100
0 100 200 300 400
Rc (
nm
)
! (mV)
exactapprox.
Figure 4.7. Critical radius of the nucleus versus electrode polarizationfor the exact case as given by Eq. (4.85) and the approximation givenby (4.86) for a 8-mol% YSZ electrolyte, where the oxygen electrode isexposed to pure oxygen at T = 1123K. The shaded area, defined by⌘ 24mV, delimits the range of polarizations for which nucleation isnot 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 di�cult 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 ⌘s and ⇡ 350mV, the grand
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
(4.86) Rc =2�
P
OO2e
(4F/RT )⌘
This approximate expression of Rc is also plotted on Fig. 4.7 and is accurate to within
12% for 120mV < ⌘ < 300mV.
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,
(4.87) W
⇤R = V (!ox
v � !
gv) + �S
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
(4.88) W
⇤ =4
3⇡� (Rc)
2
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] and others have reported a higher
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]
(4.89) W
⇤het = W
⇤2� 3 cos ✓ + cos3 ✓
4
where W
⇤ is the energy defined above and the contact angle ✓ is defined as:
(4.90) cos ✓ =�gb
2�0
with �gb the surface energy of grain boundaries that is taken to be di↵erent from that
of the solid-vapor surface �0.
Using �gb = 0.813 J/m2 [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 ⇥ kBT , in which nucleation
can be expected. The curve of the critical work of formation for the homogeneous case
falls into that zone for an overpotential ⌘c = 265mV and that for the heterogeneous
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 su�cient 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
⇤, a factor of 2 change in W
⇤ 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⇥ kBT for ⌘ ⇡ 260mV, Eq. (4.86) is a good approximation for
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), yields an approximate expression for the reversible work
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
(4.91) ⌘c =RT
8F
ln
✓
4⇡�3
75kB
◆
� lnT � 2 lnPOO2
�
117
10!1
100
101
102
103
200 250 300 350
W* /
kBT
! (mV)
homogeneousheterogeneous
Figure 4.8. Reversible work for the formation of a critical nucleus as afunction 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 topure oxygen at T = 1123K. The horizontal hatched area represents therange of energies of 1� 100⇥ kBT , at which nucleation can be expected.
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� 3 cos ✓+ cos3 ✓)/4 at the denominator.
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, given by Eq. (4.91), is a function of three parameters. The
first parameter is a characteristic of the oxide: �0, the surface energy of the oxide. The
118
two other parameters are set experimentally: P
OO2, the oxygen partial pressure at the
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 P1, 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 ) is negligible
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,
where � ranges from 0.9 J/m2 (corresponding to Al2O3 at 1850 �C [85]) to 1.45 J/m2
(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
pressure at the oxygen electrode P
OO2. The critical polarization spans from 120mV
to 400mV when varying the temperature from 600 to 1000 �C and the oxygen partial
pressure from 10�2 to 102 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
240
245
250
255
260
265
270
0.8 1 1.2 1.4 1.6
!n (
mV
)
" (J/m2)
YSZ
Figure 4.9. Overpotential above which nucleation takes place, ⌘c, as afunction of the surface energy of the oxide. The left most point of thecurve is the critical polarization for a surface energy of 0.9 J/m2 (sameas 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] report the initial electrode polarization resistances of their SOECs
showing degradation of Rp ⇡ 0.27� 0.30⌦ cm2. Hauch et al. [86] have shown that the
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
600 700 800 900 1000
T (
�C)
�2
�1
0
1
2
logP
O O2(atm)
⌘c (mV)
+
2
0
0
m
V
3
0
0
m
V
120
160
200
240
280
320
360
400
Figure 4.10. Electrode polarization above which nucleation takes place,⌘c, as a function of temperature, T , and oxygen partial pressure at theoxygen electrode, PO
O2for a 8-mol % YSZ electrolyte. The ”+” sign on
the figure denotes the conditions under which the critical polarizationwas evaluated earlier —⌘c = 265mV for PO
O2= 1atm and T = 1123K.
by Knibbe et al. that is:
RO ⇡ 0.20⌦ cm2
Given the testing conditions: T = 850�C and P
OO2
= 1 atm, the critical polarization is
⌘c = 265mV. As a result, the critical current is estimated to be
(4.92) Jc ⇡⌘c
RO
⇡ 1.3A/cm2
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]:
1.0A/cm2< jn < 1.5A/cm2.
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-
erating conditions —T and P
OO2— and do not depend on other parameters relative to
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
easy. In fact, increasing the oxygen pressure at the oxygen electrode, POO2, 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 , decreases the critical overpotential and in-
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] who report an increase of the oxygen atomic content (i.e. a decrease in the
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, cV··O, is (mainly) dictated by the fraction of yttrium cations, 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, below which the oxide
phase is more stable than the gas phase rendering nucleation thermodynamically im-
possible, and the critical overpotential, ⌘c, at which the work of forming a critical
nucleus of gas is equal to 100kBT , acknowledged to be low enough for nucleation to
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 �C where the
oxygen electrode is exposed to pure oxygen, of Jc ⇡ 1.3A/cm2, which is in the range
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
at the electrode P
OO2
and the temperature T ; their e↵ects on the critical overpotential
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, H2O and CH4, containing sometimes impurities such as H2S. However, the current
standard anode, Ni – 8 mol% yttria-stabilized zirconia is susceptible to coking, poi-
soning by fuel impurities [89] and has been shown to degrade upon redox cycling [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 phaserepresent 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
a model to investigate the growth and coarsening of such particles and, 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 catalystparticles at the surface of the oxide. (a) At early times, particles growwith solute flowing from the oxide. (b) At later times, surface transportis dominant and particles coarsen. Catalyst flow from smaller particlesto 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] and Wagner [95] (LSW theory) over fifty years ago. 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 atthe surface of a slab. The reference of the axes is taken as shown on thefigure.
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 Lx ⇥ Ly ⇥ Lz,
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) �DBr2CB = ��
where CB has units of mol/m3, DB m2/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) rCB · n = 0
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:
(5.3) CB(R) = C
eqm
✓
1 +lc
R
◆
where C
eqm is the equilibrium concentration at a planar interface in the matrix and lc
is the capillary length, defined as:
(5.4) lc =2�vm
C
eqm (Ceq
p � C
eqm )G00
m
where � is the interfacial energy, vm is the molar volume of the matrix phase, Ceqp
is the concentration at a planar interface in the particle phase and G
00m 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:
(5.5) C
eqp
dVp
dt
= DB
ZZ
Sparticle
rCB · nda
where Vp =23⇡R
3 is the volume of the hemispherical particle.
(5.6) 2Ceqp Vn = DB
1
⇡R
2
ZZ
Sparticle
rCB · nda
133
where Vn = dRdt
is the velocity of the particle surface.
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:
(5.7)
ZZZ
V
r2CBdv =
ZZ
@V
rCB · nda =
ZZ
Sparticle
rCB · nda
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 Sparticle is the surface under the particle. Integrating the right hand side
of Eq. (5.1) and equating it with the integral just above yields:
�
DB
V =
ZZ
Sparticle
rCB · nda
i.e. � =DB
V
ZZ
Sparticle
@CB
@nda(5.8)
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
=Ro(C � C
eqm )
lcCeqm
(5.9)
r =R
Ro
(5.10)
134
⌧ =C
eqmDBlc
C
eqp R
3o
t(5.11)
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) r2 = ↵
with the boundary conditions:
under the particle: (⇢, ✓, z = C) =1
r
(5.13a)
vn =1
2⇡r2
ZZ
Sparticle
@
@nda(5.13b)
elsewhere on the faces of the box:@
@n= 0(5.13c)
where ↵ = R3o
�lc
Ceqm
DB
is a constant and vn = drd⌧
is the dimensionless velocity of the
interface.
The constant ↵ can be reevaluated as a function of using the divergence theorem:
↵V =
ZZZ
V
r2 dv =
ZZ
@V
@
@nda
i.e. ↵ =1
V
ZZ
Sparticle
@
@nda(5.14)
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. Such function
should satisfy the following conditions:
r2GB = ��(p� q) +
1
V
for p and q in the box A⇥ B ⇥ C(5.15a)
@GB
@n= 0 on all the faces of the parallelepiped(5.15b)
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) and so on. There are 8
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) GB(x, y, z|⇠, ⌘, µ) =X
±G(x, y, z| ± ⇠,±⌘,±µ)
where ± designates all the possible combination of + and �. The sum includes 8
terms. G is the Green’s function for a periodic array of sources of period 2A, 2B and
2C, solution of:
(5.17) �r2G =
X
m,n,p
✓
�(r � rmnp)�1
V2
◆
where rmnp = (⇠ + 2Am, ⌘ + 2Bn, µ + 2Cp) denotes the the position of the sinks and
V2 = 8ABC. This Green’s function takes the form:
G(x, y, z|⇠, ⌘, µ) = G(x� ⇠, y � ⌘, z � µ)
=1
4⇡
X
m,n,p
1
[(x� ⇠ + 2Am)2 + (y � ⌘ + 2Bn)2 + (z � µ+ 2Cp)2]1/2(5.18)
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:
(5.19)1
R
=2p⇡
Z 1
0
e
�R2s2ds
Using this in Eq. (5.18) yields
(5.20) G(X, Y, Z) =1
2⇡p⇡
X
m,n,p
⇢
Z E
0
e
�R2mnp
s2ds+
Z 1
E
e
�R2mnp
s2ds
�
where X = x� ⇠, Y = y � ⌘ and Z = z � µ and Rmnp is defined as
(5.21) Rmnp =⇥
(X + 2Am)2 + (Y + 2Bn)2 + (Z + 2Cp)2⇤1/2
Let’s define G1(X, Y, Z) as the first sum and G2(X, Y, Z) as the second:
G1(X, Y, Z) =1
2⇡p⇡
X
m,n,p
Z E
0
e
�R2mnp
s2ds(5.22)
G2(X, Y, Z) =1
2⇡p⇡
X
m,n,p
Z 1
E
e
�R2mnp
s2ds(5.23)
and manipulate them to obtain better convergence of the series.
Let’s rewrite the first function as:
(5.24) G1(X, Y, Z) =1
2⇡p⇡
X
m,n,p
F✓
X
2A+m,
Y
2B+ n,
Z
2C+ p
◆
138
where the function F is defined as
(5.25) F (X ,Y ,Z) =
Z E
0
e
�4(A2X 2+B2Y2+C2Z2)s2ds
Making use of Poisson’s summation formula:
(5.26)1X
n=�1f(x+ n) =
1X
k=�1
e
2i⇡kx
Z 1
�1f(x0)e�2i⇡kx0
dx
0
in Eq. (5.24) yields:
G1(X, Y, Z) =1
2⇡p⇡
X
m,n,p
e
2i⇡(m X
2A+n Y
2B+p Z
2C)
ZZZ 1
�1F(x, y, z)e�2i⇡(mx+ny+pz)
dxdydz(5.27)
We note Imnp(E) the triple integral above, as F involves an integral which bound is
E. Swapping the integral on s in F and that on x, y, z yields:
Imnp(E) =
Z E
0
ZZZ
(x,y,z)
⇣
e
�(4A2x2s2+2i⇡mx)⌘⇣
e
�(4B2y2s2+2i⇡ny)⌘
⇣
e
�(4C2z2s2+2i⇡pz)⌘
dxdydz(5.28)
Isolating the integral on x, a change of variables, ⇠ = 2Asx+ i
⇡m2sA
, is done:
(5.29)
Z 1
�1e
�(4A2x2s2+2i⇡mx)dx = e
� ⇡
2m
2
4A2s
2
Z 1
�1e
�⇠2 1
2Asd⇠ =
p⇡
2Ase
� ⇡
2m
2
4A2s
2
139
Performing a similar change of variables and integration with the integrals on y and z,
Imnp rewrites:
(5.30) Imnp(E) =⇡
p⇡
8ABC
Z E
0
e
��
2mnp
4s2
s
3ds =
⇡
p⇡
4ABC�
2mnp
e
��2mnp
/4E2
where �2mnp = m
2 + n
2 + p
2. Finally, G1 writes
(5.31) G1(X, Y, Z) =1
ABC
X
(m,n,p) 6=0
e
��2mnp
/4E2e
2i⇡(m X
2A+n Y
2B+p Z
2C)
�
2mnp
Substituting by t = Rmnps in G2 yields:
G2(X, Y, Z) =1
2⇡p⇡
X
m,n,p
1
Rmnp
Z 1
Rmnp
E
e
�t2dt
=1
4⇡
X
m,n,p
erfc(RmnpE)
Rmnp
(5.32)
where erfc(x) = 1 - erf(x) is the complementary error function.
Finally, the 2A-, 2B- and 2C-periodic Green’s function writes
G(X, Y, Z) =1
ABC
X
(m,n,p) 6=0
e
��2mnp
/4E2e
2i⇡(m X
2A+n Y
2B+p Z
2C)
�
2mnp
+1
4⇡
X
m,n,p
erfc(RmnpE)
Rmnp
(5.33)
This series has the best convergence when the parameter E is equal to Eo =⇥
⇡4AC
⇤1/2, where A 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) GB(x, y, z|⇠, ⌘, µ) =X
±G3D(x± ⇠, y ± ⌘, z ± µ)
5.5.2. Green’s theorem
Applying Green’s identity to the box yields:
ZZZ
V
⇥
(q)r2GB(p,q)�r2
(q)GB(p,q)⇤
dq
=
ZZ
@V
(q)@GB
@n
(p,q)� @
@n
(q)GB(p,q)
�
dq(5.35)
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, the first part of the left hand side is evaluated:
(5.36)
ZZZ
V
(q)r2GB(p,q)dq = (p)� h i
where h i = (1/V )RRR
V (q)dq is the average concentration in the box. Note that
when p on the surface, 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
, the second part of the left hand side is evaluated:
ZZZ
V
r2 (q)GB(p,q)dq = ↵
ZZZ
V
GB(q)dq
141
=1
V
ZZ
Sparticle
�(q)dq
ZZZ
V
GB(q)dq(5.37)
Given the simple boundary conditions onGB and , the right hand side of Eq. (5.35)
is easily evaluated:
ZZ
@V
(q)@GB
@n
(p,q)� @
@n
(q)GB(p,q)
�
dq
= �ZZ
Sparticle
@
@n
(q)GB(p,q)dq
=
ZZ
Sparticle
@
@ndahGBi(5.38)
using Eq. (5.14). Here again, hGBi is the average value of GB over the box. Note that
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
(p)� h i � hGBiZZ
Spart
�(q)dq =
ZZ
Spart
�(q)GB(p,q)dq(5.39)
where �(q) = @ (q)@nq
is the single layer density. Note that this is valid for q inside the
box as well as at the surface, as justified earlier.
Eq. (5.39) is the governing equation for the field . That equation along with the
boundary conditions given by Eq. (5.13) is enough to uniquely define the concentration
field in the box. The term hGBiRR
Spart�(q)dq in (5.39) corresponds to the mass that
departed from the box, and thus 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:
(5.40) �(p) =@ (p)
@np=
@
@np
"
ZZ
Spart
�(q)GB(p,q)dq
#
When p is on Spart, GB(p,q) is singular but its integral converges. Following Jawson
and Symm [102], we derive:
(5.41)@
@np
"
ZZ
Spart
�(q)GB(p,q)dq
#
=
ZZ
Spart
�(q)@GB
@np(p,q)dq� 1
2�(p)
As a result the equation above simplifies to:
(5.42) �(p) = 2
ZZ
Spart
�(q)@GB
@np(p,q)dq
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: h i which is the average value of the
concentration field in the box. This value can be computed at each time step using
global conservation.
(5.43) h iV +2
3⇡r
3 p =
oV +
2
3⇡ p
where p is the supersaturation in the particle and o 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, h i. As
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) is determined
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) and GB(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) in a series of polar harmonics
of the form Jn(r) exp(in✓) and Yn(r) exp(in✓) where Jn and Yn are the Bessel function
of the first and second kind respectively. However, decomposing GB(p,q) is more
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
a constant term in the di↵usion equation accounts for bulk extraction of solute. 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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