I dedicate this thesis to my grandparents:

Şamyrat Töräyew

Ogulhan Töräyewa Chapter 0.

Introduction and objectives of the thesis

Global warming is an undeniable scientific fact. The average temperature of the Earth is on a gradual rise as confirmed by independent organizations (Figure 0.1). This is mainly due to emissions of greenhouse gases into the atmosphere as a result of our consumption of fossil fuels. Indeed, fossil fuels are consumed for almost all of our energy demands including transportation, industrial applications, and household heating and cooling. In order to reduce our impact on the environment, there is a pressing need to cut down our consumption of fossil fuels. To deal with this societal challenge, there has been an immense push towards the development and use of renewable energies in recent decades, with a particular focus on solar and wind energy. However, in contrast with fossil fuels, these renewable energy sources are intermittent, i.e. they are not present at all times and fluctuate during the day and across seasons.

The sun shines during day time but its intensity reduces on a cloudy day or fade during the nights. Because of the intermittent nature of renewable energies, they cannot directly replace fossil fuels and we need to develop performant energy storage systems in order to enable their widespread use. The energy storage devices will be employed to store the excess energy when it is produced in large quantities, and release it when the energy is needed. Energy storage systems are also needed for applications such as transportation or mobile devices. There are many types of energy storage systems, including pumped hydro, fly-wheels, hydrogen storage, and electrochemical storage. All of these approaches have their advantages and drawbacks and as such, they are to be used in different applications. For example, the specifications for stationary or mobile storage are different. Here we will focus on electrochemical energy storage systems which are well suited for transportation applications, in particular lithium oxygen (Li-O2) batteries.

Li-O2 batteries are attractive thanks to their high theoretical discharge capacity of 1,168 mAh.gLi2O2 -1 , and the low cost of their constituent materials. They offer 5-10 times higher capacities than typical Li-ion battery cathode materials, in theory. Figure 0.2 shows some of the most common battery types and their specific energy densities. Even though the theoretical capacity of Li-O2 batteries is very high, it is not achieved yet due to several challenges. The Li-air battery concept is about half a century old. The idea of extracting energy from the reaction of alkali metals with water was first realized in 1971. [START_REF] Rowley | Electrochemical Cell. US3791871A[END_REF] The initial efforts were made on a system where an aqueous electrolyte was used and water was involved in the electrochemical reaction. In this initial setup, water was continuously added to the electrolyte to compensate for the water consumption. Thanks to the high reactivity of both lithium and water, the system was able to provide large discharge current densities of around 200 mA.cm - 2 . [START_REF] Blurton | Metal/Air Batteries: Their Status and Potential -a Review[END_REF] Nevertheless the reaction of lithium with water lead to the formation of a passivation layer on the surface of the lithium metal which blocked the charge transfer totally when reaching a certain thickness. [START_REF] Rowley | Electrochemical Cell. US3791871A[END_REF][START_REF] Littauer | Anodic Behavior of Lithium in Aqueous Electrolytes I. Transient Passivation[END_REF][START_REF] Littauer | Anodic Behavior of Lithium in Aqueous Electrolytes II. Mechanical Passivation[END_REF] This first system is thus considered to be electrochemically irreversible and is called a primary battery. An option to make it a secondary battery was to propose "mechanically rechargeable batteries" in which the Li negative electrode (anode) is replaced by a "fresh" one at the end of discharge. [START_REF] Blurton | Metal/Air Batteries: Their Status and Potential -a Review[END_REF] Later on, in 1996, Abraham et al. made a major breakthrough when they showed that using a non-aqueous electrolyte, they could cycle a battery. [START_REF] Abraham | A Polymer Electrolyte-Based Rechargeable Lithium/Oxygen Battery[END_REF] To this end, they used a Li metal anode, a non-aqueous electrolyte (a solution of LiPF6 in a mixture of polyacrylonitrile, ethylene carbonate and propylene carbonate) and a porous carbon positive electrode (cathode) in contact with oxygen gas. Their cells provided a discharge capacity of 1,410 mAh.g -2 at 0.1 mA.cm -2 current density and 600 mAh.g -2 at 2 mA.cm -2 discharge current density. Even though the discharge current was much lower than for aqueous electrolytes, the evidence of rechargeability over three cycles opened the door to a wide range of possibilities. Following this work, the non-aqueous electrolyte lithium-oxygen (Li-O2) battery became a highly investigated system.

The basic schematic representation of a Li-O2 battery is shown in Figure 0.3. It consists in:

• A lithium containing anode (typically just Li metal);

• A separator soaked with a lithium ion (Li + ) conducting but electronically insulating electrolyte;

• A porous electronically conducting cathode (typically a porous carbon) in contact with oxygen gas. During discharge, lithium gets oxidized at the anode (eq. 0.1) and lithium ions migrate to the cathode where they react with oxygen to form solid discharge products, predominantly Li2O2 (eq. 0.2-eq. 0.4). During charge, Li2O2 discharge products decompose and form Li + and O2 species. Lithium ions migrate back to the anode where they are reduced. Oxygen molecules are released back to the oxygen gas source.

𝐿𝑖 ↔ 𝐿𝑖 + + 𝑒 - eq. 0.1 𝐿𝑖 + + 𝑒 -+ 𝑂 2 ↔ 𝐿𝑖𝑂 2 eq. 0.2 2𝐿𝑖𝑂 2 ↔ 𝐿𝑖 2 𝑂 2 + 𝑂 2 eq. 0.3 𝐿𝑖 + + 𝑒 -+ 𝐿𝑖𝑂 2 ↔ 𝐿𝑖 2 𝑂 2 eq. 0.4

The predominant discharge product in a Li-O2 battery is Li2O2 which is an electronic insulator. [START_REF] Viswanathan | Electrical Conductivity in Li 2O 2 and Its Role in Determining Capacity Limitations in Non-Aqueous Li-O 2 Batteries[END_REF] The discharge product growth can follow two processes. If the intermediate specie in eq. 0.2 (LiO2: lithium superoxide) is insoluble in the electrolyte, then it will get reduced on the cathode surface and form a film of Li2O2. The thickness of this film is limited by the electron tunneling through Li2O2 leading to very thin films of about 10 nm. [START_REF] Luntz | Tunneling and Polaron Charge Transport through Li2O2 in Li-O2 Batteries[END_REF] If the solubility of LiO2 in the electrolyte is high, it can follow a disproportionation reaction (eq. 0.

3) and form large discharge products, which have been mostly observed in toroidal shape. [START_REF] Mitchell | Mechanisms of Morphological Evolution of Li2O2 Particles during Electrochemical Growth[END_REF][START_REF] Zhai | Disproportionation in Li-O2 Batteries Based on a Large Surface Area Carbon Cathode[END_REF] In a practical cell, even though the real active cathode is oxygen, we have to use an additional conductive material to provide electronic conductivity and mechanical support to host the Li2O2 discharge products. The support structure is typically a porous carbon and is also designated as the cathode. The choice of the cathode material and its structure has a large impact on the performance of a cell. A number of experimental and theoretical works have aimed at understanding the relation between the cathode structure and the cell performances, sometimes leading to conflicting results. [START_REF] Meini | Effect of Carbon Surface Area on First Discharge Capacity of Li-O2 Cathodes and Cycle-Life Behavior in Ether-Based Electrolytes[END_REF][START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2batteries[END_REF][START_REF] Kuboki | Lithium-Air Batteries Using Hydrophobic Room Temperature Ionic Liquid Electrolyte[END_REF] The objective of this thesis is to study how the cathode mesostructure affects the discharge capacity of a cell and the transport of oxygen in the porous media, as well as to optimize the electrode mesostructure for better performances. To this end, we have developed specifically designed models which we have then compared with a collection of existing modeling and experimental tools. This text focuses on non-aqueous Li-O2 batteries, thus all Li-O2 battery mentions refer to non-aqueous Li-O2 batteries. The remainder of this manuscript is structured as follows. In chapter 1, we give a summary of the state of the art on Li-O2 batteries. In chapter 2, we provide details of the experimental techniques used for our studies. In chapter 3, we describe the methodology developed for the pore network and the kinetic Monte Carlo models.

In chapter 4, we assess the quality of the pore network extraction and compare results from the pore network model with the ones from a continuum model. In chapter 5, we show the evolution of the tortuosity along discharge measured by pulsed field gradient (PFG) nuclear magnetic resonance (NMR) experiments and calculation results from the kinetic Monte Carlo model used to study the effect of pore size on diffusion. In chapter 6, we investigate experimentally the effect of applying a magnetic field to improve oxygen transport and discharge capacity. Finally, in chapter 7, we provide a number of conclusions and perspectives based on our study.

Chapter

Chapter 1. State of the art

There has been a significant amount of work done on non-aqueous Li-O2 batteries. It is sometimes difficult to get a complete overview of a scientific field and here, we propose to approach it using bibliometrics. We first use classical tools before moving on to the results obtained with a text mining script which we developed for the current study. This bibliometric approach allows us to extract a number of important concepts relevant for the rest of this thesis.

An overview of the literature on Li-O2 batteries using bibliometrics

As a first test, a simple search on Scopus was done using the keywords "Li-O2 battery", "Liair battery", "lithium air battery" and "lithium oxygen battery". Any article or document containing at least one of these strings is extracted from the database. Figure 1.1 shows the increasing number of publications year-by-year, totaling 7323 documents (search done on October 18, 2018). The same search on the Web of Science database shows a number of documents close to 10,000. This first simple search confirms that the research field on Li-air is growing and that the work of Abraham et al. [START_REF] Abraham | A Polymer Electrolyte-Based Rechargeable Lithium/Oxygen Battery[END_REF] in 1996 was probably a milestone. This type of searches is used to get a rough idea of the number of publications in a field but have many drawbacks. Indeed, when reading some of the search results, we can realize easily that there are many documents which are not on lithium oxygen batteries. To get a better understanding on the actual number of non-aqueous Li-O2 battery publications, the whole search result was filtered manually by reading the title and abstract if needed. This filtered group includes all the derivatives of non-aqueous Li-O2 batteries such as Li-ion oxygen battery, Li-air battery, Li-O2 flow battery, Li-O2 batteries with trace amounts of water as an additive but not aqueous Li-O2 batteries. Figure 1.2 shows the number of articles after the manual filtering, totaling 1803 articles. There has been a boom in the number of papers since 2010. These studies investigate various aspects and shortcomings of a Li-O2 battery. Stability [START_REF] Tan | Advances and Challenges in Lithium-Air Batteries[END_REF][START_REF] Feng | Mapping a Stable Solvent Structure Landscape for Aprotic Li-Air Battery Organic Electrolytes[END_REF] , cyclability [START_REF] Zhang | Functional and Stability Orientation Synthesis of Materials and Structures in Aprotic Li-O2batteries[END_REF][START_REF] Geaney | Tailoring Asymmetric Discharge-Charge Rates and Capacity Limits to Extend Li-O2Battery Cycle Life[END_REF] , low practical capacity [START_REF] Liu | Capacity-Limiting Mechanisms in Li/O2batteries[END_REF] , rate capability [START_REF] Lu | The Discharge Rate Capability of Rechargeable Li-O 2 Batteries[END_REF] , discharge and charge overpotentials [START_REF] Højberg | An Electrochemical Impedance Spectroscopy Investigation of the Overpotentials in Li-O2 Batteries[END_REF] are major problems with Li-O2 batteries. To get a clearer idea of the topics studied in the literature, the articles were categorized according to the aspect of a battery they study (Figure 1.3). This classification is done using a text mining program written in Python developed specifically for this study. All the filtered articles, shown in Figure 1.2, are downloaded and converted into a text file using a program written in R. The text mining program searches for the number of occurrences of some keywords in an article (for instance, for "stability" it looks for the use of words such as stability, stabilized, unstable and its synonyms & antonyms such as parasitic reaction). If these keywords are used in an article more than 10 times, then it is considered that the article studies that topic. Needless to say, an article can belong to several categories. For example if authors study lithium ion and oxygen transport in porous electrodes through modeling, [START_REF] Torayev | Stochasticity of Pores Interconnectivity in Li-O2Batteries and Its Impact on the Variations in Electrochemical Performance[END_REF] the article is counted in both modeling and transport categories. Such a classification gives a good idea of the main challenges faced in a specific field of research, and it also makes literature searching and reading much easier since one can automatically filter out documents studying a particular aspect of Li-O2 batteries. In the following, we focus on some of the challenging aspects of Li-O2 batteries.

Stability and cyclability

For any type of secondary battery, stability is a key issue. To be used in a commercial application, a battery should be stable for at least several hundreds of discharge-charge cycles.

Achieving this with Li-O2 batteries is very challenging.

In most systems, a Li anode is used to fully benefit from the high capacity of the air cathode.

However, Li metal can react with the electrolyte and oxygen coming from the cathode side.

Several researchers tried to design solid-state Li-O2 batteries to tackle this issue, [START_REF] Li | The Pursuit of Rechargeable Solid-State Li-Air Batteries[END_REF][START_REF] Yang | Rechargeable Solid-State Li-Air Batteries: A Status Report[END_REF] but the solid state electrolyte brought more problems than it solved. Low ionic conductivity, poor interface wettability and large resistance are some of the problems encountered with solid-state Li-O2 batteries. In addition, there are still parasitic reactions occurring in the cathode.

Li metal protection is another approach for increasing the stability. Kim et al. coated the Li anode with a conductive polymer and AlF3 particles and achieved an improved stability in line with a suppressed growth of dendrites on the anode. [START_REF] Kim | Improved Cycling Performance of Lithium-Oxygen Cells by Use of a Lithium Electrode Protected with Conductive Polymer and Aluminum Fluoride[END_REF] As dendrites lead to short circuiting and end of the battery life, suppressing them is a good strategy to promote the stability of Li-O2 batteries. Nevertheless, Kim et al. still observe degradations of the electrolyte and cathode.

Such degradations are difficult to avoid as the intermediate species in a Li-O2 battery are quite reactive. The LiO2* radical can attack the electrolyte and the carbon cathode which leads to the formation of lithium carbonates. This was demonstrated by Leskes et al. who studied the end of discharge products of a Li-O2 battery using [START_REF] Zhang | Functional and Stability Orientation Synthesis of Materials and Structures in Aprotic Li-O2batteries[END_REF] O and [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2batteries[END_REF] C solid state NMR spectroscopy. [START_REF] Leskes | Direct Detection of Discharge Products in Lithium-Oxygen Batteries by Solid-State NMR Spectroscopy[END_REF] They showed the formation of Li2CO3 and small amounts of LiOH and HCO2Li side products.

McCloskey et al. also worked on this topic. They prepared a 99% [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2batteries[END_REF] C labelled cathode and analyzed the CO2 gas release during the battery operation by differential electrochemical mass spectrometry (DEMS). They could show that the electrolyte and carbon cathode both contribute in similar amounts to the CO2 evolution. [START_REF] Mccloskey | Twin Problems of Interfacial Carbonate Formation in Nonaqueous Li-O 2 Batteries[END_REF] There has also been some work on using more stable electrolytes and cathodes. Some carbon-free cathodes have been synthesized and tested, 28 e.g.

Cui et al. grew Co3O4 nanorods on

Ni foam and reported discharge capacities above 4,000 mAh.g -1 , with a good capacity retention. [START_REF] Cui | A Free-Standing-Type Design for Cathodes of Rechargeable Li-O 2 Batteries[END_REF] While the carbon cathodes are not the best in terms of stability, the scientific community seems to still prefer this alternative. This is mainly because of their low cost and easy production.

Along the years, researchers have explored various carbon materials going from carbon blacks in the early times to nano-structured cathodes such as graphene based materials [START_REF] Xiao | Hierarchically Porous Graphene as a Lithium-Air Battery Electrode[END_REF][START_REF] Li | Superior Energy Capacity of Graphene Nanosheets for a Nonaqueous Lithium-Oxygen Battery[END_REF] or templated electrodes [START_REF] Yang | Research Progresses on Materials and Electrode Design towards Key Challenges of Li-Air Batteries[END_REF] in more recent works.

Unlike the limited changes on the carbon cathode side, the Li-O2 field moved to much more stable electrolytes. In the early years of Li-O2 batteries, most of the works were carried out on carbonate electrolytes. Recently, more stable ether and amide based electrolytes have been proposed, and ionic liquids are also used and studied.

Discharge capacity

Another problem of Li-O2 batteries is the low practical capacity compared to the theoretical one. This low capacity is the result of two main aspects. The first one is the inefficient use of the pore volume, and the second one is the fact that the reaction rate is too slow for applying high discharge currents. Before going into details about these two capacity limiting factors, it is worth having a look at the capacities reported in the literature, these are shown in Figure 1.5.

This graph clearly shows the progress in achieving higher and higher discharge capacities, starting from 1,410 mAh.g -1 for Chevron acetylene black carbon electrodes with polyacrylonitrile-based polymer electrolyte discharged at 0.1 mA.cm -2 in the work of Abraham et al. [START_REF] Abraham | A Polymer Electrolyte-Based Rechargeable Lithium/Oxygen Battery[END_REF] in 1996, to 150,000 mAh.g -1 for reduced graphene oxide (rGO) electrodes with dimethoxyethane (DME) electrolyte with 10 mM of 2,5-di-tert-butyl-1,4-benzoquinone and 20,000 ppm of H2O electrolyte additives discharged at 0.1 mA.cm -2 in the work of Liu et al. [START_REF] Liu | The Effect of Water on Quinone Redox Mediators in Nonaqueous Li-O2Batteries[END_REF] in 2018.

Figure 1.5. Reported discharge capacities for Li-O2 batteries.

There are about 11,000 data points in this plot which were obtained using an in-house text mining program. The three radical improvements of capacities labelled are Wang et al. [START_REF] Wang | A Lithium-Air Battery with a Potential to Continuously Reduce O2 from Air for Delivering Energy[END_REF] , Zhu et al. [START_REF] Zhu | A High-Rate and Long Cycle Life Solid-State Lithium-Air Battery[END_REF] and Liu et al. [START_REF] Liu | The Effect of Water on Quinone Redox Mediators in Nonaqueous Li-O2Batteries[END_REF] . *Note that the capacity value of 101,376 mAh.g -1 seems to be wrongly reported in Liu et al. [START_REF] Liu | Self-Assembled 3D Foam-Like NiCo2O4as Efficient Catalyst for Lithium Oxygen Batteries[END_REF] , it is actually equal to 10,137 mAh g -1 . The capacity values are per weight of cathode or carbon loading whichever is used in the respective publications.

These capacity values might seem extremely large compared to the typical capacities obtained in Li-ion battery cathodes (150-400 mAh.g -1 ) and the 1,168 mAh.g -1 theoretical capacity mentioned at the beginning of this manuscript. In fact, one should pay attention to the common way in which the capacity in Li-O2 battery is normalized in the literature. Typically, the gravimetric discharge capacity of a Li-ion battery is normalized by the mass of active material and any inactive components in the cathode such as the binder and the conductive carbon.

However, for Li-O2 batteries, the active material is oxygen which is eventually expected to come from air, even though at the development stage most researchers work with pure oxygen gas. Therefore the gravimetric capacities are normalized by the mass of the pristine cathode (mass of the carbon in the cathode + mass of the binder if used) or sometimes just the mass of carbon in the cathode, which is the inactive part of the cell. In principle, the values should be normalized by the mass of the cathode at the end of discharge in order to compare them with Li-ion batteries capacities. In practice, this normalization is rarely adopted because of the difficulty to measure the mass of the cathode after discharge (presence of residual electrolyte, difficulty of detaching the cathode and the separator, etc.…). Because of the normalization approach preferred by many, it is quite common to see reports of very large discharge capacities in the order of several tens of thousands mAh.g -1 whereas the theoretical capacity for Li-O2

batteries is 1,168 mAh.g -1 Li2O2.

Using adequate metrics for measuring performances is important in order to optimize the systems, we thus would like to detail this a bit more here. Actually, when we assume that the only discharge product is Li2O2, there is a direct way to renormalize the capacity. Figure 1.6

shows the relation between the capacities normalized by the mass of the pristine cathode, which is the typical way used to report capacities, and the capacity normalized by the mass of the cathode at the end of discharge. If Li2O2 is the only discharge product, then its mass can be inferred from the current measured through the electrochemical setup. With this renormalization, we see that, indeed, the capacity normalized by the mass at the end of discharge does not exceed the theoretical capacity. We also notice that, interestingly, above 20,000 mAh.g -1 for the typical normalization, the capacities normalized by the mass at the end of discharge are not changing much. Finally, one should keep in mind that it is the values normalized by the mass of the cathode at the end of discharge which should be compared to Li-ion batteries. As such, the typical capacities obtained for Li-O2 batteries are not orders of magnitude larger than the capacities of Li-ion batteries.

Figure 1.6. Different ways to normalize the discharge capacity. (a) Definitions and relationship between discharge capacities when the only discharge product is Li2O2. (b) Correlation between discharge capacities normalized by the pristine cathode mass and the total cathode mass (end of discharge cathode mass).

Coming back to the capacity limiting phenomena, the inefficient use of the porous volume is related to the fact that Li2O2 discharge products are not filling up all the pore volume in the electrode. This phenomenon is strongly linked to the electrode mesostructure and the size of the Li2O2 discharge products. The insulating nature of Li2O2 implies that electrochemical reactions cannot occur far from the cathode surface.

Correlations between the structural properties of carbon cathodes and their discharge capacities have been examined in a number of works. Here, we briefly outline some of the experimental results published on the topic. 1. [START_REF] Meini | Effect of Carbon Surface Area on First Discharge Capacity of Li-O2 Cathodes and Cycle-Life Behavior in Ether-Based Electrolytes[END_REF] Ding et al. measured the discharge capacities of various carbon blacks, mesoporous carbons, multi-walled carbon nanotubes and reduced graphene oxide. [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2batteries[END_REF] Their study showed that the discharge capacity is positively correlated with the pore size but they did not establish any clear correlation between the discharge capacity and the CSA or the pore volume (Table 2). In the same study, Ding et al.

synthesized several electrodes with different pore sizes via silica templating to confirm that the observed differences are not due to different precursors or carbonization temperatures. The silica templated carbons also showed a pore size dependence of the discharge capacities.

Comparable results were obtained by Zeng et al., using two electrodes with similar morphology and wettability but different average pore sizes. [START_REF] Zeng | Synthesis of Mesoporous Carbons and Reduced Graphene Oxide and Their Influence on the Cycling Performance of Rechargeable Li-O2batteries[END_REF] However, Kuboki et al. observed a stronger correlation between the discharge capacity and the electrode pore volume. [START_REF] Kuboki | Lithium-Air Batteries Using Hydrophobic Room Temperature Ionic Liquid Electrolyte[END_REF] In their study, they compare four different carbon electrodes, two made of activated carbons and two made of conductive carbons, and show that a large mesopore volume is a vital feature for increasing the cell capacity (Table 3). Numerous alternative electrode architectures and designs have also been investigated to boost the cell performance such as carbon aerogels, [START_REF] Ma | Effect of Porosity on Electrochemical Properties of Carbon Materials as Cathode for Lithium-Oxygen Battery[END_REF] graphene based electrodes [START_REF] Zeng | Synthesis of Mesoporous Carbons and Reduced Graphene Oxide and Their Influence on the Cycling Performance of Rechargeable Li-O2batteries[END_REF][START_REF] Sun | Porous Graphene Nanoarchitectures: An Efficient Catalyst for Low Charge-Overpotential, Long Life, and High Capacity Lithium-Oxygen Batteries[END_REF] and carbon nanotubes [START_REF] Lim | A New Catalyst-Embedded Hierarchical Air Electrode for High-Performance Li-O2 Batteries[END_REF] . Following these experimental studies, it appears that an ideal cathode requires i) A large CSA for electrochemical reactions to occur and discharge products to deposit on;

ii) Large pores to enhance the transport of active species and prevent pore clogging;

iii) A large porosity to allow for the growth of discharge products.

All these criteria have to be met while maintaining a good electronic conductivity and a good mechanical stability which usually decrease when increasing the porosity.

Along with testing new electrode designs, there is another approach to use the pore volume more efficiently, which is preferentially growing large discharge particles. The growth of discharge products follows several steps. After initial nucleation of Li2O2 particle, the particle can grow through both electrochemical and chemical reactions as shown in eq. 0.2 to eq. 0. After the particle grows up to a certain size the insulating nature of Li2O2 will hinder the electrochemical growth and it will follow the chemical growth route (also designated as the solution mechanism). O2 and Li + will react and form LiO2 close to the cathode surface where charge transfer can occur. The lithium superoxide will then dissolve in the electrolyte and consequently get adsorbed on Li2O2 particles. The adsorbed LiO2 can disproportionate and form Li2O2, and the particle can grow further though this mechanism.

The relative importance of this solution growth mechanism with respect to film growth or end of discharge strongly depends on the solubility and stability of LiO2 in the electrolyte. Laoire introduced this concept by using a 1 mM concentration of ethyl viologen ditriflate in their electrolyte and showing the viologen-promoted formation of Li2O2. [START_REF] Lacey | A Redox Shuttle to Facilitate Oxygen Reduction in the Lithium Air Battery[END_REF] Since 2012 there has been a growing interest in using redox mediators to promote formation of large discharge products (Figure 1.3). Several mediators have been tested and demonstrated to promote the solution mechanism. For example, Lim et al. analyzed several redox mediators including tetrathiafulvalene, N,N,N',N'-tetramethyl-p-phenylenediamine and lithium iodide. [START_REF] Lim | Rational Design of Redox Mediators for Advanced Li-O2batteries[END_REF] Gao et al.

showed that using 2,5-di-tert-butyl-1,4-benzoquinone, the discharge capacity of gas diffusion layer electrodes can increase by one to two orders of magnitude. [START_REF] Gao | Promoting Solution Phase Discharge in Li-O2batteries Containing Weakly Solvating Electrolyte Solutions[END_REF] reported that introducing trace amounts of H2O in the electrolyte results in larger Li2O2 particle sizes and hence higher discharge capacities can be achieved. [START_REF] Liu | The Effect of Water on Quinone Redox Mediators in Nonaqueous Li-O2Batteries[END_REF][START_REF] Aetukuri | Solvating Additives Drive Solution-Mediated Electrochemistry and Enhance Toroid Growth in Non-Aqueous Li-O2batteries[END_REF]

Rate capability

The rate capability limitation, the second reason for low practical capacity, occurs when the reaction rate cannot keep up with the discharge current. To understand this limitation, it is useful to write down the net current equation in its general form.

𝐽 = 𝑛𝐹𝐴𝑘 [𝐶 𝑂𝑥 𝑒 -𝛽𝑛𝐹 (𝑈-𝑈 0 ) 𝑅𝑇 -𝐶 𝑅𝑒𝑑 𝑒 (1-𝛽)𝑛𝐹 (𝑈-𝑈 0 ) 𝑅𝑇 ] eq. 1.1
where 𝐽 is the net current, 𝑛 is the number of electrons exchanged in the electrochemical reaction, 𝐹 is the Faraday constant, 𝐴 is the active cathode surface area, 𝑘 is the reaction rate constant, 𝐶 𝑂𝑥 and 𝐶 𝑅𝑒𝑑 are the concentrations of the oxidized and reduced species respectively, 𝛽 is the charge transfer coefficient, 𝑈 is the electrostatic potential of the electrode, 𝑈 0 is the standard potential of the reaction, 𝑅 is the gas constant and 𝑇 is the temperature. From eq. 1.1, it is clear that three parameters can affect the current: the active surface area, the reaction rate constant and the concentration of reactants.

The role of the active surface area on the rate capability can be quite significant. Along the discharge of a Li-O2 cell, the active cathode surface area can be progressively reduced by the formation of solid discharge products on the cathode surface. This effect is not significant at low current densities because at slow discharge the discharge products grow as large particles.

As such, they do not cover large areas of the cathode surface. However, when the discharge current density is increased, there is a preferential formation of Li2O2 as small discharge particles or as a thin film. This leads to a coverage of larger amounts of the cathode surface area per volume of discharge product. The relation between the discharge product size and the current density has been studied in the work of Adams et al. [START_REF] Adams | Current Density Dependence of Peroxide Formation in the Li-O2 Battery and Its Effect on Charge[END_REF] In Figure 1.9, we see that increasing the discharge current leads to a decrease in the discharge product size. Discharging at high currents gives less time for LiO2 intermediates to dissolve in the electrolyte and contribute to the growth of large particles. The higher overpotential occurring at higher currents also leads to the fast reduction of LiO2 into Li2O2 and hence the formation of smaller discharge products as shown in Figure 1.8. The effect of the active surface area on the rate capability can be reduced by using electrolytes that favor large discharge product formation such as electrolytes with additives and mediators as explained previously.

Figure 1.9. Field-emission SEM images of cathodes before discharge (a) and at the end of discharge with discharge currents of (b) 5 µA.cm -2 , (c) 10 µA.cm -2 , (d) 25 µA.cm -2 , (e) 50 µA.cm -2 and (f) 100 µA.cm -2 . Insets show the corresponding discharge curves. The scale bar is 400 nm. Reproduced from Adams et al. [START_REF] Adams | Current Density Dependence of Peroxide Formation in the Li-O2 Battery and Its Effect on Charge[END_REF] Apart from acting on the active surface area or promoting the solution phase mechanism, one can try to increase the reaction rate constant in order to be able to apply larger currents. The reaction rate constant is a function of the activation energy, [START_REF] Bard | Kinetics of Electrode Reactions[END_REF] which can be lowered using catalysts of the oxygen reduction reaction. Since the early years of Li-O2 batteries, there has been an avalanche of articles on catalysts (Figure 1.3). Wu et al. compared discharge capacities of electrodes: Ketjen Black (KB) electrode without a catalyst, carbon electrode with platinum catalyst, KB electrode with pyridinic nitrogen -cobalt catalysts and multi-walled-nanotubes electrode with pyridinic nitrogencobalt catalysts. [START_REF] Wu | Nitrogen-Doped Graphene-Rich Catalysts Derived from Heteroatom Polymers for Oxygen Reduction in Nonaqueous Lithium-O2battery Cathodes[END_REF] They observed that the multi-wallednanotubes electrode containing the pyridinic nitrogencobalt catalysts gave the highest discharge capacity, ~3,700 mAh.g -1 among all. Rate capability tests with a discharge current increasing from 50 mA.g -1 to 800 mA.g -1 also showed an improved rate capability. In particular, the discharge capacity of this electrode at 800 mA.g -1 provides a discharge capacity similar to the KB electrode at 50 mA.g -1 . Lim et al. reported that adding a V2O5/Al2O3 catalyst into the KB electrode increases the discharge capacity of the cell by three times. [START_REF] Lim | Electrochemical and Catalytic Properties of V2O5/Al2O3in Rechargeable Li-O2batteries[END_REF] While catalysts are shown to improve the discharge capacities, there is a risk that they also catalyse parasitic reactions. McCloskey et al. measured O2 and CO2 gas emissions with DEMS on charge for cells with carbon electrodes without any catalyst, and with gold, MnO2 and platinum catalysts. [START_REF] Mccloskey | On the Efficacy of Electrocatalysis in Nonaqueous Li-O 2 Batteries[END_REF] They observed that the electrodes with catalysts released less oxygen and much more carbon dioxide than the electrode without any catalyst. This work clearly shows that catalysts foster parasitic reactions, probably involving the decomposition of some cell components, when present.

The last factor contributing to poor rate capability is the lack of reactants. The two primary reactants in a Li-O2 battery are Li + and O2. The concentration of Li + ions can be modified easily by changing the amount of lithium salt added into the electrolyte. The typical range of lithium salt concentration is between 0.1 M and 1 M. The concentration of oxygen is much more difficult to change and strongly depends on the electrolyte solvent. Typical oxygen solubilities in commonly used electrolytes are just a few milli-molars, e.g. 2.10 mM, 4.43 mM and 9.57 mM for DMSO, tetraglyme and DME respectively. [START_REF] Laoire | Influence of Nonaqueous Solvents on the Electrochemistry of Oxygen in the Rechargeable Lithium-Air Battery[END_REF] Because of this low oxygen concentration, about two orders of magnitude lower than the lithium ion concentration, oxygen is the limiting reactant. There are two ways of tackling this issue.

The first way is to increase the solubility of oxygen in the electrolyte by using electrolytes that have a high oxygen solubility. For instance triglyme has a superior oxygen solubility compared to tetraglyme, and Wu et al. have observed a better performance for the triglyme based electrolyte. [START_REF] Read | Oxygen Transport Properties of Organic Electrolytes and Performance of Lithium/Oxygen Battery[END_REF][START_REF] Wu | Enhanced Electrochemical Performance of Non-Aqueous Li-O 2 Batteries with Triethylene Glycol Dimethyl Ether-Based Electrolyte[END_REF] Chen et al. used octamethylcyclotetrasiloxane as an additive to tetraglymebased electrolyte. [START_REF] Chen | A New Type of Cyclic Silicone Additive for Improving the Energy Density and Power Density of Li-O2batteries[END_REF] They showed that by increasing the additive content from 0 to 10 volume percent, the oxygen solubility, the discharge capacity and the capacity retention improved.

However, this way of improvement is quite challenging since changing the electrolyte can generate stability and reaction rate issues.

The second way is to increase oxygen diffusion. This can be done by moving to electrolytes that have high oxygen diffusion coefficients or by choosing electrode mesostructures with lower tortuosities. Modifying the electrode mesostructure is probably a less risky approach since it does not change the whole electrochemistry of the system. When using carbon materials, we can prepare cathodes with a very large variety of mesostructures. The number of papers working on this approach can be seen in the transport category of Figure 1.3. As described previously, electrodes can be tailored towards achieving some target property, e.g. a high surface area [START_REF] Etacheri | Hierarchical Activated Carbon Microfiber (ACM) Electrodes for Rechargeable Li-O2 Batteries[END_REF][START_REF] Zeng | Enhanced Li-O2battery Performance, Using Graphene-like Nori-Derived Carbon as the Cathode and Adding LiI in the Electrolyte as a Promoter[END_REF] or large pores, [START_REF] Wang | A Dual Pore Carbon Aerogel Based Air Cathode for a Highly Rechargeable Lithium-Air Battery[END_REF] or synthesized in a more controlled fashion to obtain templated electrodes with specific mesostructures [START_REF] Zeng | Synthesis of Mesoporous Carbons and Reduced Graphene Oxide and Their Influence on the Cycling Performance of Rechargeable Li-O2batteries[END_REF][START_REF] Nie | Nitrogen Enriched Mesoporous Carbon as a High Capacity Cathode in Lithium-Oxygen Batteries[END_REF] . It is important to note that the electrode mesostructure can influence the discharge capacity, as already detailed in section 2.3.

Modeling work

Modeling techniques have proven to be helpful in studying electrode performances, downfalls and proposing better designs to enhance cell performance. [START_REF] Tan | Advances in Modeling and Simulation of Li-Air Batteries[END_REF][START_REF] Yoo | A Review of Lithium-Air Battery Modeling Studies[END_REF][START_REF] Franco | Carbon-Based Electrodes for Lithium Air Batteries: Scientific and Technological Challenges from a Modeling Perspective[END_REF] Cell level continuum models (CM) are especially attractive since they allow for the calculation of observables similar to those measured in electrochemical and microscopy experiments. For instance, Albertus et al.

developed a cell level CM which includes oxygen transport, discharge product growth and its resistance in porous electrodes to identify the capacity limiting parameters. [START_REF] Albertus | Identifying Capacity Limitations in the Li/Oxygen Battery Using Experiments and Modeling[END_REF] The authors showed that high current densities lead to the emergence of oxygen concentration and discharge product gradients along the electrode thickness. Their simulations suggest that without CSA passivation, discharge capacities can be increased significantly, from around 300 mAh.g -1 to above 7,000 mAh.g -1 . Indeed, when a thin film of about 7-10 nm of Li2O2 is formed on the carbon surface, it becomes passivated, i.e. the insulating layer blocks electron transfer from the conductive carbon to the electrochemical reaction sites. [START_REF] Luntz | Tunneling and Polaron Charge Transport through Li2O2 in Li-O2 Batteries[END_REF] Using a different approach, Xue et al.

simulated cell voltage profiles for Super P and Ketjen Black electrodes and compared two different resistance scenarios for the passivation process, namely a tunneling limited model, where the Li2O2 layer is assumed to be perfect insulator and electron conduction is only possible via quantum-tunneling process, and a linear resistance model, where resistance of the Li2O2 layer increases linearly with the deposit thickness. [START_REF] Xue | Impact of the Cathode Microstructure on the Discharge Performance of Lithium Air Batteries: A Multiscale Model[END_REF] The simulations for the two passivation processes predict that for the tunneling limited process, the cell voltage shows a long plateau followed by a sharp decline, whereas the linear resistance model induces a gradual decrease of the cell voltage. We note that these potential profile evolutions as a function of discharge product resistance have also been reported in prior works. [START_REF] Viswanathan | Electrical Conductivity in Li 2O 2 and Its Role in Determining Capacity Limitations in Non-Aqueous Li-O 2 Batteries[END_REF][START_REF] Albertus | Identifying Capacity Limitations in the Li/Oxygen Battery Using Experiments and Modeling[END_REF] In their model, oxygen transport, Li2O2 film growth and electrode pore size distributions are captured. The authors show superior performance for Ketjen Black compared to Super P electrode owing to its high CSA. Later, Xue et al. extended their model to consider the influence of the electrolyte donor number on the discharge performance. [START_REF] Xue | A Comprehensive Model for Non-Aqueous Lithium Air Batteries Involving Different Reaction Mechanisms[END_REF] The improved model assumes a competition between the Li2O2 thin film formation mechanism and the formation of large Li2O2 particles in a hall (pore size of several micrometers) driven by a solution phase mechanism. A control parameter, called escape function in that work, allows for the adjustment of the relative weight of each mechanism. The model thus allows to capture the fact that the use of high donor number solvents, stabilizing LiO2, mitigates the CSA passivation and improves the electrochemical performance in good correlation with experiments. [START_REF] Johnson | The Role of LiO 2 Solubility in O 2 Reduction in Aprotic Solvents and Its Consequences for Li-O 2 Batteries[END_REF]

Cell components

Anode -A lithium metal anode is used for all the cells. The lithium anode is cut in disk shape with a 0.73 mm thickness and a 9 mm diameter. Considering that the density of lithium metal is 0.534 g/cm 3 , this negative electrode is sufficient to provide a capacity of about 95.8 mAh per cell. This allows us to be sure that the anode is not a capacity limiting parameter as none of the cells we discharge reaches this high discharge capacity.

Separator -Two layers of Whatman® glass fiber with a diameter of 14 mm are used in each cell. The total thickness of these two layers is about 1 mm. Two layers are used to avoid any short circuit in the cell assembly.

Electrolyte

Electrochemical tests

For the electrochemical experiments, Swagelok cells are used (Figure 2.1.a). A hole is drilled through the plunger at the cathode side for oxygen contact. A lithium metal anode, two layers of separators and a cathode are placed into the cell; then 250 µL of electrolyte is added. The cell is assembled, in a glovebox and put into an airtight glass chamber (Figure 2.1.b-c). Once the glass chamber is sealed, it is taken out of the glovebox and flushed with oxygen gas for 30 minutes to replace the argon gas with oxygen. After flushing and before the electrochemical tests, the cell is rested for 6 hours to give sufficient time for oxygen molecules to dissolve in the electrolyte. The electrochemical tests have been carried out on a Bio-Logic VMP3 multichannel potentiostat. We mainly used the galvanostatic discharge, chronoamperometry and cyclic voltammetry techniques. The galvanostatic discharge consists in doing a discharge at constant current and gives access to the cell voltage along the depth of discharge and the discharge capacity. The chronoamperometry experiment is made by applying a constant voltage to the cell and measuring its current response. This experiment provides information about kinetics of electrochemical reactions, and diffusion processes. The cyclic voltammetry is conducted by applying a potential changing linearly between two potential values. It is used for the same purpose as the chronoamperometry but provides additional information such as the onset potential for reduction and oxidation reactions.

Scanning Electron Microscopy

SEM uses thermionically emitted electrons to reach very high magnifications. The range of magnifications we used are from 1,000x to 120,000x. The sample is placed in a chamber under vacuum to avoid electron beam colliding with air molecules. We use this imaging technique to see the cathode structure and discharge products after discharge. Particles as small as 20-30 nm can be visualized using this technique. However, it is difficult to see the formation of the thin Li2O2 films since their thickness is typically below 10 nm and since they form on the cathode surface without any distinct shape.

SEM images of pristine electrodes are carried out on the cathodes dried at 120°C under vacuum overnight. The cathodes are put on an SEM sample holder in a glovebox and then transported to the SEM room in a sealed argon filled box to avoid air exposure. However, there are a few seconds of air exposure while transferring the sample from the sealed box to the SEM chamber before the vacuum is established.

For the discharged or half discharged cathodes, the cells are disassembled in a glovebox, and the cathode is washed with pure DME to remove traces of salt and electrolyte. Then, it is dried in the pre-chamber of the glovebox for a few minutes under vacuum to remove the DME solvent. The dried cathode is attached onto a SEM sample holder and then transported to the SEM room in an airtight box. Again, there are a few seconds of air exposure while transferring the sample from the sealed box to the SEM chamber before the vacuum is established.

Tomographic imaging

The tomography imaging was carried out by our collaborators: Arnaud Demortière from LRCS and Vincent De Andrade from Argonne National Laboratory. We provided them with the electrodes and they conducted the tomography imaging experiment, reconstruction and segmentation. What we have received from them is a 3D grey scale image with pores and solid regions assigned according to the grey scale value.

For the nano-CT (Computed Tomography) measurements, a series of TXM (Transmission Xray Microscopy) images at different angles are acquired on the beamline 32-ID of the Advanced Photon Source (APS) synchrotron at Argonne National Laboratory (ANL). The data set is acquired at an energy of 8 keV with 1 s of exposure time per projection in a "flying scan" mode, i.e. in continuous rotation. The X-ray objective lens is similar to the Fresnel zone plate (FZP) and the FZP numerical aperture matches the beam shaping condenser, which is a grating with 1.32 mm diameter and 60 nm spacing for the outermost grating. Zernike phase contrast is used to enhance the contrast of light element (here carbon) using a phase ring localized after the FZP. 1,500 projections were acquired along 180˚. The pixel width is 20 nm but the true spatial resolution is 60 nm (60 nm features with at least 3 pixels can be seen). Reconstruction is made using Tomopy tools [START_REF] Gürsoy | TomoPy: A Framework for the Analysis of Synchrotron Tomographic Data[END_REF][START_REF] Miqueles | Generalized Titarenko's Algorithm for Ring Artefacts Reduction[END_REF][START_REF] De Carlo | Scientific Data Exchange: A Schema for HDF5-Based Storage of Raw and Analyzed Data[END_REF] using the algorithm from the ASTRA toolbox [START_REF] Pelt | A Method for Locally Approximating Regularized Iterative Tomographic Reconstruction Methods[END_REF] and accurately approximating algebraic tomographic reconstruction by filtered back projection. As done in previous works, [START_REF] Eustache | High Areal Energy 3D-Interdigitated Micro-Supercapacitors in Aqueous and Ionic Liquid Electrolytes[END_REF][START_REF] Létiche | Atomic Layer Deposition of Functional Layers for on Chip 3D Li-Ion All Solid State Microbattery[END_REF] the final 3D reconstructed object is segmented using the FIJI and AMIRA

Pulsed Field Gradient NMR

Nuclear magnetic resonance spectroscopy is a powerful and versatile tool for identifying species, quantifying them, and characterizing local environments, interatomic distances and dynamics. [START_REF] Pecher | Methods: NMR in Battery Research[END_REF] Atomic nuclei have isotope dependent spin I and, associated with this, a magnetic dipole µ = γ√𝐼(𝐼 + 1). When an ensemble of atomic nucleus with spin ½, like 1 H or [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2batteries[END_REF] C, is put in an external magnetic field (𝑩 𝟎 ), the nuclei are distributed over two quantum states with either a specific positive or negative projection z on the magnetic field axis defining the z axis, as shown in Figure 2.3. The energy gap between the magnetic moment "up" and "down" state is

∆𝐸 = 𝛾ℏ𝐵 0 eq. 2.1
where 𝛾 is the gyromagnetic ratio of the nucleus, ℏ is the reduced Plank's constant and 𝐵 0 is the magnitude of the external magnetic field. Electromagnetic radiation with a photon having energies matching the energy difference causes transitions between the nuclear orientation states. For typical magnetic fields of 1-20 T the required frequencies for most NMR visible isotopes are in the radiofrequency (rf) range (10 -1000 MHz).

Figure 2.3. Splitting of the energy levels under an external magnetic field for a nucleus with spin ½.

The main principle of NMR is shown in Figure 2.4. In thermal equilibrium the splitting of the energy levels will lead to a preferential alignment of nuclear spins along the external magnetic field according to the Boltzmann distribution, which will generate a net magnetization in the B0 direction. In a simple NMR experiment, a short rf pulse at the resonance frequency is applied to rotate the overall nuclear magnetic moment of the nuclear ensemble away from the z axis.

The magnetic moment will then precess about the z axis. The non-zero rotating component in the xy plane, induces a current in the NMR detection coil. The resulting signal is called the free induction decay (FID). By taking the Fourier transform of the FID, we obtain an NMR spectrum. The key aspect of this experiment is the dependence of the Larmor frequency (i.e.

the precession frequency) of the nuclei on the net magnetic field

𝜔 0 = 𝛾𝐵 0 eq. 2.2
The high chemical resolution of NMR lies in the fact that the local magnetic field experienced by the nuclear spins depends on the local electron density. The electronic environment of a nucleus can shield it from the external magnetic field, and the local magnetic field a nucleus experiences will be altered

𝜔 = 𝛾(𝐵 0 -𝐵 𝑒 ) eq. 2.3
where 𝐵 𝑒 is the magnetic field generated by the electronic environment around the nucleus.

Thus the differences in chemical environment in molecules and materials will lead to slightly different precession frequencies. This causes small NMR frequency differences between chemically inequivalent nuclei of the same isotope. The corresponding chemical shift  is expressed in relative frequency units (parts per million; ppm) with respect to the NMR frequency of a mutually agreed reference compound for the observed NMR isotope, such as tetramethylsilane (TMS) for 1 H, [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2batteries[END_REF] C and [START_REF] Cui | A Free-Standing-Type Design for Cathodes of Rechargeable Li-O 2 Batteries[END_REF] Si:

𝜈 = 𝛾𝐵 0 2𝜋 eq. 2.4 𝛿 = 𝜈 𝑠𝑎𝑚𝑝𝑙𝑒 -𝜈 𝑟𝑒𝑓 𝜈 𝑟𝑒𝑓 eq. 2.5
where 𝜈 𝑠𝑎𝑚𝑝𝑙𝑒 is the NMR frequency of the NMR distinct nucleus in the studied molecule and 𝜈 𝑟𝑒𝑓 is the frequency of the selected reference compound. The sensitivity of the NMR frequency on the local magnetic field can be exploited to study many different aspects of a system: local atomic environments, interatomic distances and diffusion. For the diffusion studies the position of a nucleus is encoded by applying a spatially dependent magnetic field superimposed on the static homogeneous magnetic field B0 (Figure 2.5). For a linear magnetic field gradient gz along the magnetic field axis, the total magnetic field experienced by the observed NMR nuclei is:

𝐵(𝑧) = 𝐵 0 + 𝑔 𝑧 𝑧 eq. 2.6
After the first 90° rf pulse, a gradient of magnetic field is applied to encode the locations of the molecules in the system. Then a certain time is given for molecules to diffuse before a second pulse, a 180°, followed by another gradient pulse is applied to decode the location tags. The rf pulses combined with gradient pulses produce a so-called echo signal. In another version designed for slow diffusion, the time between the encoding and decoding stages is extended by replacing the 180° by two 90° pulses separated by a waiting time  (90° - -90°), which results in a stimulated echo (Fig. 2.5). The idea behind this is that if molecules do not move at all, the encoding and decoding magnetic fields will be the same and the echo intensity of the (stimulated) echo signal for these molecules will be high. On the contrary, if molecules diffuse, then they will experience different magnetic encoding and decoding fields. As a consequence the intensity of the echo signal will be smaller than in the non-diffusing case. In this way, one can determine the effective diffusion coefficients (𝐷 𝑒𝑓𝑓 ) by applying a range of gradients and fitting the signal intensity as a function of gradient strength gz to the Stejskal-Tanner equation.

For the pulsed field gradient (PFG) stimulated echo experiment with the pulse sequence depicted in Figure 2.5 the equation is:

𝐼′ = 𝐼′ 0 𝑒𝑥𝑝[-𝐷 𝑒𝑓𝑓 𝛾 2 𝑔 𝑧 2 𝐿𝐷 2 (∆ -𝐿𝐷 3 ⁄ )] eq. 2.7
where 𝐼′ is the intensity of the signal, LD is the time the gradient is applied, and ∆ is the diffusion time between two gradient pulses. [START_REF] Stejskal | Spin Diffusion Measurements: Spin Echoes in the Presence of a Time-Dependent Field Gradient[END_REF][START_REF] Price | NMR Studies of Translational Motion[END_REF] The PFG-NMR experiment is done without exposing the samples to air. The cells are disassembled in a glovebox, and then each cathode is put in a vial with pure DME to remove the salt and electrolyte. We used DME to wash out the salt because of its low viscosity and because it evaporates easily. Then the cathode is dried in the pre-chamber of the glovebox under vacuum. The dried electrode is put in another vial with pure tetraglyme. The vial is put under vacuum for 3 minutes to allow for the solvent molecules to go into small pores, and the electrode is kept in the solvent for one hour to allow for soaking by the solvent. After that, the electrode surface is gently dried on a lab paper-towel and put into a sealed plastic bag (Figure lead to more collisions and slower diffusion but we wanted to get a more quantitative view on this effect for the pore sizes we are interested in. The kMC model is adapted from a kMC model previously developed in LRCS which simulated the discharge process of Li-O2 batteries. [START_REF] Blanquer | Modeling Investigation of the Local Electrochemistry in Lithium-O 2 Batteries: A Kinetic Monte Carlo Approach[END_REF] To simulate diffusion in a pore we used a cylindrical pore geometry (Figure 3.1). The pore volume is sliced into voxels with a side length of 0.5 nm, which is assumed to be the hydrodynamic size of a dissolved O2 molecule. The hopping frequency (𝛾 𝑖 ), also called the event or process frequency, is calculated as

𝛾 𝑖 = 2𝐷 𝑂 2 𝑠 2 eq. 3.1
where 𝐷 𝑂 2 is the bulk diffusion coefficient of O2, and 𝑠 is the voxel size. In the model, the O2 molecules are able to jump to 6 face sharing voxels and 12 edge sharing voxels. In total, an O2 molecule sitting on a voxel can jump to 18 sites, provided that those neighboring voxels are empty.

We used periodic boundary conditions in the direction of the pore length. In this kMC model, we assumed that there is only translational motion and no reaction. We use the variable step size [START_REF] Blanquer | Modeling Investigation of the Local Electrochemistry in Lithium-O 2 Batteries: A Kinetic Monte Carlo Approach[END_REF] and the time step is calculated as

∆𝑡 = - ln (𝜌 1 ) 𝛾 𝑡𝑜𝑡 ⁄ eq. 3.2 𝛾 𝑡𝑜𝑡 = ∑ 𝛾 𝑖 𝑁 𝑖=1 eq. 3.3
where 𝜌 1 is a random number in the interval (0,1], 𝛾 𝑡𝑜𝑡 is the sum of all event frequencies, and 𝑁 is the total number of possible events. The event occurring at a given step is chosen according to the weighted probability of each event. The event 𝑗 is selected according to the following rule

∑ 𝛾 𝑖 𝑗 𝑖=1 < 𝜌 2 𝛾 𝑡𝑜𝑡 ≤ ∑ 𝛾 𝑖 𝑗+1 𝑖=1 eq. 3.4
where 𝜌 2 is a random number in the interval (0,1]. This process allows one to simulate a system with various event frequencies, including a mixture of fast and slow processes. In our case, we only have events related to the transport of a single type of molecule (O2), however, the script is written in a general form to be able to work with many event types if necessary. It is worth noting that in this work, we use an approach slightly different from the ones previously developed at LRCS [START_REF] Blanquer | Modeling Investigation of the Local Electrochemistry in Lithium-O 2 Batteries: A Kinetic Monte Carlo Approach[END_REF] and reported in the literature [START_REF] Reuter | First-Principles Kinetic Monte Carlo Simulations for Heterogeneous Catalysis: Concepts, Status, and Frontiers[END_REF] . A simplified scheme of the algorithm is given in Figure 3.2.a. The key difference between our model and previous ones is that we determine all possible processes at the beginning of the calculation, and store them in an easy-to-update data structure. In the main loop, we then only update the processes which are in the vicinity of the executed process. The number of processes updated at each timestep is around 18 (depending on the process type) against several thousands of processes, when the full ensemble of possible processes is scanned. As a consequence, this procedure is much more efficient and calculations are much faster. In fact, we wrote two programs, one for each approach, and showed that our new procedure is about 2000 times faster than the one reported in the literature.

Extraction of diffusion coefficients from mean square displacements

To extract the diffusion coefficients from the particles' trajectories, we use the mean square displacement (MSD) method. The MSD function is defined as

𝑀𝑆𝐷 = 〈|𝑑 𝑖,𝑡 -𝑑 𝑖,𝑡 0 | 2 〉 eq.

An illustration of the determination of a diffusion coefficient from the MSD curve. Two vertical lines show the time interval where the MSD curve is linear with time.

The red line is the fit to the MSD curve in the selected region.

nano-computed tomography with phase contrast imaging using synchrotron X-ray source (APS-ANL). The 3D reconstructed structure allows us to generate porous structures with realistic porosities and pore interconnectivities. For studying the effect of pore size, tomography like images are generated in a computerized fashion.

Generating tomography like images

To generate a tomography like image, we wrote a program (using the C programming language) which uses four input parameters: (1) the box size, (2) the voxel size, (3) the porosity and ( 4) the pore size. Using these quantities we build a 3D matrix according to the box size and the voxel size, and we set all the values of the matrix to 1 (solid). Then, we randomly place a pore with the chosen pore size in the matrix and set all the voxels within its radius to 0 (pore).

We continue this process till the given porosity is achieved. In the program, we allow for overlapping of pores.

Tomography image converter

Before extracting a pore network, the tomography image is converted into a binary format. For this, a python script was written, together with Alexis Rucci, a postdoctoral researcher in LRCS. Each image is divided into pixels characterized by a gray scale value. Conventionally, in an electrode image stack, a zero gray value represents a pore. Accordingly, the algorithm developed takes the gray value of each pixel image by image and assigns a 0 or 1 value to it. If the gray value is different from zero, it will be represented by '1' in the binary text file, which represents a carbon electrode site (solid). If it is zero, it will be represented by '0' in the binary text file and will correspond to a void (pore). The final file is then a text sequence with a succession of 0 and 1, where each number corresponds to one pixel.

Pore network extraction

For the pore network extraction, an open source code developed by Dong et al. is utilized and adapted to the current system. [START_REF] Dong | Pore-Network Extraction from Micro-Computerized-Tomography Images[END_REF] To extract the pore network, a pore is assigned to each void voxel of the tomography data and expanded till a carbon site is reached. After filling up the pore space of the tomography data, an iterative process is carried out to identify the clusters of overlapping pores, connected by bottlenecks. To do that (step 1), the largest pore is taken from the list and assigned as a parent pore and all the pores overlapping with that pore are assigned as child pores. Then the pores overlapping with the child pores are also included as child pores of the parent pore unless they have a bigger size than the parent pore. This process continues till there are no new overlapping pores or the expansion is truncated by larger overlapping pores. This cluster of pores is designated as pore family i, where i is the index of this pore family. All of these pores are removed from the initial list of pores and the process returns to step 1. The process is repeated till all the pores are assigned to a pore family.

After identifying pore families (two are shown in Figure 3.5.a), only two parameters need to be defined: an effective pore radius for each pore family and the length of the throats which connect these effective pores. To assign these parameters we use a different approach from the one reported by Dong et al. [START_REF] Dong | Pore-Network Extraction from Micro-Computerized-Tomography Images[END_REF] They use a parameter called the pore-throat segmentation coefficient (𝛼′) to determine the effective pore radius and throat length. To calculate the radii of the effective pores, the distances between the center of the parent pores and the bottleneck pore (𝑙 𝑖 𝑡 and 𝑙 𝑗 𝑡 ) are calculated (Figure 3.5.b) and the radii of the effective pores are given by:

𝑙 𝑖 = 𝑙 𝑖 𝑡 (1 -𝛼′ 𝑟 𝑡 𝑟 𝑖 ) eq. 3.7 𝑙 𝑗 = 𝑙 𝑗 𝑡 (1 -𝛼′ 𝑟 𝑡 𝑟 𝑗 ) eq. 3.8
𝑙 𝑡 = 𝑙 𝑖𝑗 -𝑙 𝑖 -𝑙 𝑗 eq. 3.9

where 𝑙 𝑖 , 𝑙 𝑗 , 𝑙 𝑡 are radii of the effective pore for pore family 𝑖, 𝑗 and the length of the throat, (Figure 3.5.d), 𝑟 𝑡 is the radius of the throat; 𝑟 𝑖 is the radius of the parent pore for pore family 𝑖;

𝑙 𝑖𝑗 is the distance between centers of parent pores 𝑖 and 𝑗. This method has two caveats; first, it uses a fitting parameter 𝛼 and second, it can assign more than one effective radius for a pore family depending on the number of throats it has. We amended this method to assign the effective pore radius (𝑙 𝑖 ) so that the volume of the effective pore is equal to the total volume of the pore family. (Figure 3.5.c). A 3D pore network is then obtained (Figure 3.5.e). growth in our model: particle and thin film formation.

Pore network model

The transport equations for the PNM are adapted from the polymer electrolyte membrane fuel cell work of Fazeli et al. [START_REF] Fazeli | Incorporating Embedded Microporous Layers into Topologically Equivalent Pore Network Models for Oxygen Diffusivity Calculations in Polymer Electrolyte Membrane Fuel Cell Gas Diffusion Layers[END_REF] The evolution of the concentration in each pore is calculated by solving the balance eq. 3.

𝐼 𝑡𝑜𝑡 = ∑ 𝑆𝐴 𝑖 𝑛𝐹𝜗 𝐴𝑙𝑙 𝑝𝑜𝑟𝑒𝑠 𝑖=1 eq. 3.20
where 𝑆𝐴 𝑖 is the surface area of pore 𝑖, which can be either a spherical or a cylindrical pore with the surface area considered accordingly.

The discharge products growth can follow two types of pathways: the formation of large particles and the deposition of a thin film. Experimentally, the Li2O2 formation path depends on several parameters, such as the electrolyte used, [START_REF] Johnson | The Role of LiO 2 Solubility in O 2 Reduction in Aprotic Solvents and Its Consequences for Li-O 2 Batteries[END_REF] the current density [START_REF] Adams | Current Density Dependence of Peroxide Formation in the Li-O2 Battery and Its Effect on Charge[END_REF] and the presence of impurities (e.g. the H2O content in the electrolyte enhancing the LiO2 solubility). [START_REF] Aetukuri | Solvating Additives Drive Solution-Mediated Electrochemistry and Enhance Toroid Growth in Non-Aqueous Li-O2batteries[END_REF] In our model, both the formation of large particles and the deposition of a thin film are considered using the escape factor concept reported in a previous work. [START_REF] Xue | A Comprehensive Model for Non-Aqueous Lithium Air Batteries Involving Different Reaction Mechanisms[END_REF] The escape factor is the probability of LiO2 to escape or dissolve in the electrolyte once it is formed close to the carbon surface. The dissolved LiO2 is assumed to follow a solution phase mechanism leading to the formation of large Li2O2 particles, whereas the LiO2 molecules adsorbed on the carbon surface lead to thin film formation. The discharge reaction including the escape factor (χ) can be represented by the eq. 3.18-eq. 3.24, and the amounts of 𝐿𝑖 2 𝑂 2 particles and thin films (eq.

3.25) can be calculated by combining these four equations.

𝐿𝑖 + + 𝑂 2 + 𝑒 -→
2𝐿𝑖 + + 𝑂 2 + 2𝑒 -→ 2 -2𝜒 2 -𝜒 𝐿𝑖 2 𝑂 2,𝑓𝑖𝑙𝑚 + 𝜒 2 -𝜒 𝐿𝑖 2 𝑂 2,𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 eq. 3.25
Since the escape factor represents the discharge product size and how much carbon surface area it covers, it depends on many parameters. In simple terms any parameter that affects the discharge product size will also affect the value of the escape factor. High donor number electrolytes foster formation of large discharge particles thus, for example, DMSO will have a higher escape factor than tetraglyme. [START_REF] Laoire | Influence of Nonaqueous Solvents on the Electrochemistry of Oxygen in the Rechargeable Lithium-Air Battery[END_REF] The presence of mediators and trace amount of water will also facilitate the formation of large particles. [START_REF] Lim | Rational Design of Redox Mediators for Advanced Li-O2batteries[END_REF][START_REF] Aetukuri | Solvating Additives Drive Solution-Mediated Electrochemistry and Enhance Toroid Growth in Non-Aqueous Li-O2batteries[END_REF] This means that an electrolyte with mediators or water (intentionally added or present as an impurity due to improper drying) has a higher escape factor than an electrolyte consisting in a well-dried salt dissolved in a pure solvent. Additionally, the discharge rate also has an effect on the particle size, and a system which is discharged at lower current will have a higher escape factor than the same system discharged at higher rates. To quantify all these effects into a number is a daunting task, therefore in this thesis the escape factor is more used as a qualitative descriptor than a welldefined value for a specific system.

For the PNM calculations, we need to specify the operating conditions (e.g. discharge current density, temperature and lithium ion concentration). Specific operating conditions will be mentioned in all chapters in which the PNM is used. Here, we specify the parameters which are always the same, except otherwise mentioned. The pore network simulated in this work contains about 8,000 pores and corresponds to a volume of 5 μm (electrode thickness) x 4 μm x 4 μm. For the PNM, an electrolyte composition of 1M Li + ion concentration in tetraglyme solvent is considered. A constant oxygen concentration (solubility of O2 in tetraglyme) and zero oxygen flux boundary conditions are applied at the oxygen gas inlet side and at the separator side respectively. For the Li + ions, boundary conditions of a constant 1 M concentration and zero flux are used at the separator side and at the gas inlet end respectively.

Periodic boundary conditions are used in the two directions perpendicular to the electrode thickness. The initial condition for the model is uniform concentrations of O2 and Li + ions.

Over the course of a PNM calculation, we simulate an electrochemical reaction leading to the formation of solid Li2O2 in both spherical and cylindrical pores. This solid deposit can lead to pore blockage which is considered in the model. A pore is considered to be blocked when it is filled with discharge products, i.e. the thickness of the thin film added to the particle size is equal to the radius of the pore. In this case there can be no reaction in that pore anymore and no transport through it. Additionally, a pore is surface passivated when the thickness of the thin film reaches 10 nm. This is a consequence of the electronic insulator character of the Li2O2.

There can be no electrochemical reaction in a surface passivated pore but it can still contribute to the transport of species. The list of parameters and their values used for PNM is shown in Table 4.

The code developed is written in C and parallelized using the Open Multi-Processing (OMP) library. Calculations are carried out on the Odyssey compute cluster owned by the Grey group.

Two nodes with 8 cores each and a 2.

Image processing: Artificial film growth

In addition to the relatively complex PNM approach, we use an image processing method to analyze crudely the effect of a film growth on the three dimensionally-resolved electrode mesostructures. Starting with experimentally measured or artificially generated mesostructures, we expand the solid region of the image uniformly, to imitate a homogeneous thin Li2O2 film growth along discharge (Figure 3.7). The expansion of the solid region is done in iterative steps. First, a random walker is placed in pore voxels sitting within 10% of an edge in each direction. This walker randomly tries to move to any of the six neighboring voxels provided that they are void/pore. All the pore voxels reached by the random walker which have at least one solid neighboring voxel are converted to solid voxels. This random walk approach is adopted to account for the accessibility of the different regions to Li + and O2. In other words, a monolayer of film with thickness of a voxel size is deposited on the solid region accessible from the edges of the structure. This same process is repeated till all of the pore voxels reachable by the active species are converted to solid voxels. After each deposition step, the percentage of isolated pores is calculated in a similar manner: a random walker is placed in voxels sitting within 10% of an edge in each direction, the walker randomly tries to move to any of the six neighboring voxels provided that they are pores and all pore voxels which cannot be reached by any of these random walkers are considered isolated.

Modified continuum model

We compared PNM results to CM for several structures. For the continuum approach, we used a slightly modified version of a model previously developed by Xue et al. [START_REF] Xue | A Comprehensive Model for Non-Aqueous Lithium Air Batteries Involving Different Reaction Mechanisms[END_REF] The faradaic reduction current is calculated following

𝑗 𝐹𝑎𝑟 = 𝑛𝐹𝑘𝑐 𝑂 2 {𝑒𝑥𝑝 ( -𝛽𝐹(𝑈 -𝑈 0 ) 𝑅𝑇 ) -𝑒𝑥𝑝 ( (1 -𝛽)𝐹(𝑈 -𝑈 0 ) 𝑅𝑇 )} (4)
where 𝑐 𝑂 2 is the oxygen concentration. The oxygen concentration in the cell is given by

𝜕(𝜀𝑐 𝑂 2 ) 𝜕𝑡 = 𝜕 𝜕𝑥 [𝜀 1.5 𝐷 𝑂 2 ( 𝜕𝑐 𝑂 2 𝜕𝑥 )] - 𝐴 • 𝑗 𝐹𝑎𝑟 𝑛𝐹 ( 5
)
where 𝜀 is the porosity. The modification made in this work is related to the growth of large particles. As discussed in Chapter 1 of this thesis, the original model assumed that large particles grow in one single "hall" (i.e. pore) with a size of about tens of micrometers. Here,

we have modified the model so that all the LiO2 intermediate species dissolved in the electrolyte stay in the pore in which they appear and contribute to the growth of large Li2O2 particle in the center of the concerned pore. This change implies that pore clogging through large particle formation is now possible.

With all of these tools we can explore the role of the electrode mesostructure on the cell performance, on the processes occurring at the pore level and interpret some experimental observations more accurately. In the following chapters, these tools will be used and the results will be analyzed. The list of parameters and their values used for CM is shown in Table 4. To analyze the mapping accuracy in a more numerical way, we characterize the porous features of the structure obtained via PNE and the original structure. The pore size distribution (PSD), the total porosity and the surface area of the original structure are calculated using the GeoDict commercial software 86 . The same properties are easily determined for the extracted pore network since all the pores are well defined. On Figure 4.2.a, showing the pore size distributions, we can see that the position and shape of the main peak are similar but the PNE structure gives a bigger tail towards the large pores. This effect, already observed by Dong et al. [START_REF] Dong | Pore-Network Extraction from Micro-Computerized-Tomography Images[END_REF] , arises from the assignment of a large effective pore size to a cluster of overlapping pores (see Figure 3.5.c for details). The porosity obtained by PNE is very similar to the value calculated using GeoDict for the original structure. On the contrary, the carbon surface area calculated for the PNE structure is about 27% smaller than the value determined for the original structure. The pores in the tomographic image of the Super P electrode have rough surfaces so by assigning spherical or cylindrical shapes to these pores, we smoothen the surface of the porous media. That is the main reason why we have less carbon surface area in the pore network extracted by PNE. For this computer generated structure, the surface area mismatch is just 3.4%, compared to the 27% difference for the Super P structure. The smaller difference in the carbon surface area for the computer generated structure reinforces our confidence in the idea that surface smoothening leads to a reduction in surface area. Indeed, the computer generated structure has a relatively smoother pore surface than the Super P image.

From this study, we can conclude that the PNE allows one to obtain a usable and realistic pore network which is a good representation of the original porous structure while reducing its complexity. In particular, the total porosity is very well maintained while the PNE tends to broaden a bit the PSD features and reduces the surface area. The accuracy of the PNE seems good enough to compare results for different electrode structures.

Important checks for input parameters and information provided by the PNM

In this section we report on some tests done to validate important input parameters and provide examples of the typical information we can get using the PNM approach. All the calculations The system size convergence is checked using the following scheme: the electrode thickness is kept constant (5 µm) while the electrode dimensions in the two directions perpendicular to the electrode thickness are changed. While periodic boundary conditions are applied in these two directions, artefacts can happen if the lateral sizes are not large enough. Three system sizes are tested: 0.5 µm x 0.5 µm x 5 µm, 0.7 µm x 0.7 µm x 5 µm, and 1.0 µm x 1.0 µm x 5 µm. A structure is generated for each system size, all structures being generated with a pore size of 60 nm. The calculated discharge capacities are very similar which means that the system sizes chosen are big enough (Figure 4.4.c). Based on these calculations, a system size of 0.7 µm x 0.7 µm x 5 µm is selected as the appropriate size. As such, the calculation results shown in the remainder of this section are done with an electrode size of 0.7 µm x 0.7 µm x 5 µm and a 1 10 -4 s time-step.

We now explore the possibilities offered by the PNM approach in terms of calculating various quantities. Rate capability tests can be carried out using the model. The reason why the oxygen transport becomes a significant issue can be analyzed from other outputs of the PNM. 1. Clogged: A pore is clogged when filled with Li2O2 discharge products.

2. O2 depleted: A pore is O2 depleted when its O2 concentration is below 0.1 mM. This can happen for two main reasons: i) if neighboring pores are clogged and isolate the pore from the O2 source (O2 inlet side) ii) if the transport of O2 decreases significantly due to changes in the electrode mesostructure, and it cannot catch up with the O2 consumption rate.

3. Passivated: A pore is passivated when the thickness of the thin Li2O2 film on its surface reaches the tunneling distance limit (10 nm). These passivated pores contribute to the ionic transport, but are inactive for electrochemistry.

A significant number of pores get clogged along the discharge, and the number of connections between non-clogged pores decreases which impedes transport of mobile species in the electrode. The pore sizes decrease as well with a significant number of pores getting clogged (radius goes to zero). Because of the slow oxygen transport, an oxygen concentration gradient emerges leading to a gradient of discharge products along the electrode thickness. Figure 4.5.ef show the evolution of the PSD and the amount of discharge products formed in the cathode during discharge. Overall, the end of discharge in this system is due to a depletion of O2 which renders the electrochemical reactions impossible.

Using the above example, we have demonstrated that the PNM provides us with global as well as local properties which allow us to predict capacities and assess the reasons for the end of discharge. This luxury of information comes with the price of a larger computational cost in comparison with other models. We will now compare this PNM approach with more common and faster models.

PNM vs. continuum model

In this section, we compare our PNM with a CM but before doing so, we use a simple image analysis to evaluate the consequences of ignoring the spatial resolution of the electrode mesostructure.

Thin film and isolated pores from image processing

To see the likelihood of forming isolated pores and pore regions by shrinkage of the porous media along discharge, a simple tomographic image processing is carried out (see chapter 3 for details). Three 3D mesostructures are used: an actual tomography image of a Super P electrode (porosity = 36%) and two computer generated mesostructures obtained by randomly placing 40 nm and 100 nm pores in a solid medium till a 70% target porosity is reached. Following the homogeneous growth of a thin film, small bottlenecks connecting large pores becomes blocked and the porous media loses interconnectivity. Figure 4.6.a shows the evolution of the percentage of isolated pores with the porosity. For porosities larger than 30%, there is a weak dependence of the percentage of isolated pores with the porosity. On the contrary, when the porosity drops below 30%, the percentage of isolated regions increases drastically. In Figure 4.6.b, we represent schematically the underlying mechanism. We consider a simple case where two different pore sizes are randomly distributed in a 3D lattice. Since it is a random distribution, there will be cases where some large pores are connected to each other through smaller pores. Then when the small pores (blue spheres) get clogged, some of the interconnectivity of large pores (red spheres) will be lost and even form isolated pores (orange sphere) that do not add to the electrode performance since they are not connected to any O2 or Li + sources. A more likely phenomenon before the formation of isolated pores is the path becoming more tortuous. We expect both the pore clogging and the tortuosity increase to be better described in the pore network model than in the continuum model.

(a) Fraction of isolated pores calculated by applying the image processing approach (see chapter 3 for details) on the experimentally measured Super P tomography image and two computer generated mesostructures. (b) An example of randomly

distributed large and small pores in a three-dimensional lattice.

Calculated discharge capacities and end of discharge for various escape factors and mesostructures

To compare the PNM and CM, we generate four structures with average pore sizes of 40 nm, 60 nm, 80 nm and 100 nm. We chose to use these structures because of two reasons. First, a prior experimental work has been reported using silico-templated structures with similar pore sizes, so we can compare our simulation results to these experiments. [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2 Batteries[END_REF] Secondly, these are relatively simple structures on which we can carry out fundamental studies, e.g. on the role of the surface area and pore sizes, before studying more complex structures with different pore size distributions such as a mix of micro, meso and macro pores. [START_REF] Gaya | Investigation of Bi-Porous Electrodes for Lithium Oxygen Batteries[END_REF] In order to compare the CM and the PNM methods, they need to share a number of similarities.

We remind here that in both models an overall electrochemical reaction is used, and intermediate steps and side reactions are neglected. In both approaches, the calculation focuses on the processes happening at the cathode and the anode and separator only give rise to specific boundary conditions. To compare the two models, we analyze their behavior for different mechanisms of discharge products formation. To do this, we consider three escape factor values, 𝜒 = 0 (formation of only thin film), 𝜒 = 0.5 (mixed formation of thin film and large particles, i.e intermediate size discharge particles) and 𝜒 = 1 (formation of only large particles). By using three different escape factors, we will be able to see how the electrode mesostructure affects the cell performance changes according to the type of discharge products formed.

Spatially, the formation of discharge products in a pore is considered as shown in The thin film forms on the entire pore surface and grows uniformly while large particles, corresponding to the solution mechanism, grow from the center of a pore. The assumption of a particle growth from the center of a pore allows us to neglect all contacts between the particle and the carbon surface, and any surface coverage of the carbon by Li2O2 through this growth mechanism. The carbon surface is assumed to be a perfect electronic conductor when not passivated and a perfect insulator when the Li2O2 film thickness reaches 10 nm. Regarding the electrode mesostructures, a 5 µm electrode thickness and a porosity of 70% are considered. This relatively small thickness was chosen because of the restrictive computational cost of the PNM method. The pore size distributions considered in this work are given in Figure 4.8 shows the calculated discharge capacities of these cathodes for the three different particle growth mechanisms considered (only thin film formation, mixed growth, only solution phase particle growth). All the discharge capacities calculated using the CM method are larger than the ones from the PNM method. This difference comes mainly from the slower transport of O2 in PNM compared to CM. Even though we used the same diffusion coefficient for both models, we use the one dimensional Fick's second law in the CM while we solve a threedimensional network of fluxes (Fick's first law) in the PNM. This results in more important transport limitations in the PNM model, which affects the absolute values of the discharge capacities. We note that both models show higher capacities for cases where the Li2O2 growth occurs at least partially through particle growth mechanism. This is in agreement with previously reported data. [START_REF] Aetukuri | Solvating Additives Drive Solution-Mediated Electrochemistry and Enhance Toroid Growth in Non-Aqueous Li-O2batteries[END_REF][START_REF] Gao | Promoting Solution Phase Discharge in Li-O2 Batteries Containing Weakly Solvating Electrolyte Solutions[END_REF] The results of the PNM and CM methods are similar for the case where only thin film formation occurs. We observe that the discharge capacity is high for mesostructures with high surface area (Figure 4.8.a-c) and that pores with radius larger than the tunneling limitation distance do not get clogged. When the escape factor is 0.5 or 1, then significant differences are observed between the two models. In particular, when the growth happens only in solution, the CM gives the same capacity value for all the mesostructures. To understand the differences observed between the CM and PNM methods, we now focus on the end of discharge for all the simulated systems. The end of discharge can be either due to a transport limitation or to a lack of active carbon surface area. The carbon surface is considered active when it is both accessible to the reactants (O2 and Li + ) and not passivated. As discussed previously, in the CM, the transport depends on the porosity and on the tortuosity which is written as a function of porosity through the Bruggeman relation. Due to the computational cost of carrying out simulations of large systems with the PNM approach, an electrode with a relatively small 5 µm thickness has been considered for both pore network and continuum models. We check here that the discharge capacity trends observed with the CM with 5 µm thick electrodes are not due to the small electrode thickness, but to the actual limitations arising from the representation of the 3D phenomena into a 1D model. We calculated the discharge capacities for the four electrodes with a 100 µm electrode thickness, which is within the typical range used in experiments. We can see that the calculated discharge capacity trends are similar for the 5 µm (Figure 4.8.a) and 100 µm (Figure 4.11) cases. For both electrode thicknesses, the continuum model is not able to capture the pore size effect when the solution phase mechanism is dominant. Yet, the calculated capacities for the mesostructures with these two electrode thicknesses differ. Thick electrodes yield lower capacities than thin ones, this is due to the O2 transport limitation in thick electrodes. We now compare our modeling results with experimental data reported in the literature. As discussed above, Ding et al. carried out an experiment in which they synthesized electrode mesostructures with 20 nm, 40 nm, 60 nm, 80 nm and 100 nm average pore sizes using a silica templating method. [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2 Batteries[END_REF] The discharge capacity trend they observed is similar to the calculated one with our PNM with 𝜒 = 1, i.e. the discharge capacity is higher for electrodes with larger pores. This result cannot be captured by the CM: an explicit 3D representation of the electrode and pore interconnectivity is necessary. The capacities calculated by the PNM also help us to understand various seemingly conflicting trends reported in the literature. For instance, Meini et al. reported a discharge capacity increase as the CSA increases. [START_REF] Meini | Effect of Carbon Surface Area on First Discharge Capacity of Li-O2 Cathodes and Cycle-Life Behavior in Ether-Based Electrolytes[END_REF] As previously mentioned, Ding et al. demonstrated that the cell capacity increases as the average pore size increases. [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2 Batteries[END_REF] Yang et al. synthesized mesocellular carbon foam electrodes with an average pore size around 30 nm and compared their performance with electrodes made of activated carbon (pore size ~2 nm), carbon nanotubes (pore size ~10 nm) and Super P (pore size ~50 nm). [START_REF] Yang | Preparation of Mesocellular Carbon Foam and Its Application for Lithium/Oxygen Battery[END_REF] They observed neither a capacity dependence on the CSA nor on the pore size but the highest capacity was obtained for the electrode with a 30 nm average pore size. These experimental variations can be attributed to possible differences in the synthesis conditions, such as temperature treatment or precursors used, but it can also be due to different particle formation mechanisms, corresponding to different escape factor values. When we plot the PNM calculated discharge capacities for the exclusive thin film formation case (𝜒 = 0) as a function of the mesostructure surface area (Figure 4.8.d), we obtain a similar trend as the one reported by Meini et al. [START_REF] Meini | Effect of Carbon Surface Area on First Discharge Capacity of Li-O2 Cathodes and Cycle-Life Behavior in Ether-Based Electrolytes[END_REF] . The escape factor depends on several parameters, such as the solvent, the salt, the additives, the discharge current density and even the temperature. In fact, the reason why Ding et al. [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2 Batteries[END_REF] and Meini et al. [START_REF] Meini | Effect of Carbon Surface Area on First Discharge Capacity of Li-O2 Cathodes and Cycle-Life Behavior in Ether-Based Electrolytes[END_REF] observed different correlations between the discharge capacity and the electrode properties, can be due to different Li2O2 growth mechanisms.

To get an idea of how different the growth mechanisms are for these two studies, we estimate an equivalent film thickness (EFT). This quantity is calculated as the volume of discharge products divided by the BET surface area of the pristine electrode. The average Li2O2 film thickness per electrode surface area is calculated as

𝛿 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 = 𝑄 𝑔 𝐴 𝑔 • 𝑀 𝐿𝑖 2 𝑂 2 • 1 𝐹 eq. 4.1
𝑄 𝑔 is the gravimetric discharge capacity (mAh.g -1 ), 𝐴 𝑔 is the electrode surface area (m 2 .g -1 ), 𝑀 𝐿𝑖 2 𝑂 2 is the molar volume of Li2O2 (m 3 .mol -1 ), and 𝐹 is Faraday's constant (C.mol -1 ). This calculation is a coarse approach to get an idea of the relative importance of thin film and particle growth mechanisms. It assumes that all the discharge current is due to the formation of Li2O2, therefore neglecting parasitic reactions arising for example from electrolyte decomposition.

While this is a very rough estimate for a thickness of Li2O2 deposit, large values will suggest a dominant particle growth mechanism. This is especially true for EFT above 10 nm as such film thicknesses are not reachable due to the surface passivation. Figure 4.12 shows the EFT for the experiments reported by Ding et al. [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2 Batteries[END_REF] and Meini et al. [START_REF] Meini | Effect of Carbon Surface Area on First Discharge Capacity of Li-O2 Cathodes and Cycle-Life Behavior in Ether-Based Electrolytes[END_REF] . Most of the values for Meini et al. [START_REF] Meini | Effect of Carbon Surface Area on First Discharge Capacity of Li-O2 Cathodes and Cycle-Life Behavior in Ether-Based Electrolytes[END_REF] are very low while values for Ding et al. [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2 Batteries[END_REF] vary a lot more from around 1 nm to 46 nm. This suggests that the lower current density and the different electrolytes used by Ding et al. [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2 Batteries[END_REF] in their experiments would foster large particle formation, in agreement with previous studies. [START_REF] Adams | Current Density Dependence of Peroxide Formation in the Li-O2 Battery and Its Effect on Charge[END_REF][START_REF] Horstmann | Rate-Dependent Morphology of Li2 O2 Growth in Li-O2 Batteries[END_REF] Differences in the Li2O2 growth mechanisms could explain the apparently contradicting conclusions these groups of authors made as we have shown using the PNM that the evolution of the discharge capacity with the average pore size depends quite strongly on the growth mechanism considered.

Evolution of the tortuosity along discharge

The calculated evolution of the tortuosity along discharge is given in Figure 4.13. The tortuosity values for the PNM are calculated by exporting the 3D mesostructures snapshots at 4 different depths of discharge to external image files which are then analyzed using the Matlab-developed TauFactor code. [START_REF] Cooper | An Open-Source Application for Calculating Tortuosity Factors from Tomographic Data[END_REF] For the CM, the tortuosity evolves in the same way for the four electrode mesostructures, i.e. following the Bruggeman relation. However, the PNM results show that the tortuosity evolutions are not the same for the various electrode mesostructures under investigation and confirm that the CM tends to underestimate the tortuosity. According to the PNM, the electrode mesostructure with the smallest average pore size, 40 nm, becomes much more tortuous along discharge. This is due to the large amount of small pores which makes pore clogging and loss of pore interconnectivity more likely. The second most tortuous structure is the 60 nm one. For pore sizes above approximately 60 nm, the tortuosity evolution seems to follow a similar trend. The calculated tortuosities confirm that the lower discharge capacities observed for the PNM (

Conclusions

In this chapter, we presented the accuracy of the PNE and typical results the PNM provides.

We have compared the pore network and continuum modeling techniques in their abilities to capture the pore size and discharge product growth mechanisms effects in Li-O2 batteries. For the surface growth mechanism both models show similar results and can capture the effect of the electrode mesostructures properly. For the solution phase mechanism however, the continuum model is not able to capture the pore size effect properly while the PNM calculation results show pore size dependent discharge capacities in agreement with experiments reported in the literature. [START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2 Batteries[END_REF] Moreover, the calculated discharge capacities extracted from the PNM for different electrode mesostructures with three different discharge product growth scenarios allow us to propose an explanation for conflicting trends previously reported. In particular, the current densities and the nature of the solvent can affect the Li2O2 growth mechanisms and potentially lead to different ends of discharge for a given mesostructure. As a result, depending on the Li2O2 growth mechanism, the discharge capacity was suggested to be correlated with various properties of the electrode mesostructures (e.g. carbon surface area, pore volume, pore size). For a given set of parameters, one can also identify the existence of an optimum pore size offering the highest capacity. These results highlight that one should be cautious when using a continuum model to simulate the electrochemical performance of Li-O2 batteries, particularly when the solution growth mechanism is considered: the pores interconnectivities in three dimensions play a major role in determining the overall cell performance.

Chapter

Chapter 5. Diffusion coefficients: results from PFG-NMR experiments and kMC model

As we have now demonstrated in many ways, that the porous media and its mesostructure play an important role in Li-O2 batteries. They not only provide the space needed for the solid discharge products to form but also enable the transport of reactants (Li + and O2) involved in the discharge reactions. As electrochemical reactions occur, the porous media evolves: some pores shrink and some pores get clogged because of the formation of discharge products. These mesostructural changes of the electrode modify its transport properties: tortuosity, porosity and effective diffusion coefficients are affected. In this section we quantify these properties and how they change upon discharge. This will help us to understand some of the performance limiting factors and potentially improve them.

Evolution of tortuosity along the depth of discharge: PFG-NMR experiments

The effective diffusion coefficient (𝐷 𝑒𝑓𝑓 ) in a porous media is related to the bulk diffusion coefficient (𝐷 0 ) as [START_REF] Chung | Validity of the Bruggeman Relation for Porous Electrodes[END_REF] 𝐷 𝑒𝑓𝑓 = 𝜀 𝜏 ⁄ 𝐷 0 eq. 5.

1
where 𝜏 is the tortuosity. By knowing 𝐷 0 and 𝐷 𝑒𝑓𝑓 , one can calculate 𝜀 𝜏 ⁄ . In this work, the porosity and tortuosity are considered structural properties. That is, they mainly depend on the porous media and its mesostructure, but not on the diffusion of the species measured, provided that the diffusing specie does not have a specific interaction with the carbon mesostructure. In principle, a molecule A and a molecule B will not necessarily have the same porosity and tortuosity in a given media if they sample different space. If molecule B is twice as big as molecule A, some parts of the porous electrode might not be accessible to B, and as a result, a smaller effective porosity and a larger tortuosity will be determined when using B compared to the case using A. The porosity and tortuosity might also depend on the rigidity/flexibility of molecules A and B. In practice however, differences between relatively similar molecules, e.g.

the different solvent used in Li-O2 batteries electrolytes, are very limited and the porosity and tortuosity are considered to be structural properties. The oxygen molecules are dissolved in the electrolyte and we assume oxygen molecules diffusion media is similar to the solvent molecules in the electrolyte. Following the hypothesis that the tortuosity depends only on the electrode mesostructure, and as we require a molecule with the appropriate diffusion coefficient and longer relaxation time for it to sample the different pores and move a sufficient distance within the timescale of the PFG-NMR experiment, we measured tetraglyme solvent diffusion by tracking motion of 1 H nuclei by NMR.

PFG-NMR experiments are used by the Grey group to measure diffusion in porous media for applications in batteries and supercapacitors. [START_REF] Forse | Direct Observation of Ion Dynamics in Supercapacitor Electrodes Using in Situ Diffusion NMR Spectroscopy[END_REF][START_REF] Engelke | Three Dimensional Pulsed Field Gradient NMR Measurements of Self-Diffusion in Anisotropic Materials for Energy Storage Applications[END_REF] Forse et al. studied the effect of the pore size distribution on the self-diffusion of ions, and measured diffusion in and out of nanopores. [START_REF] Forse | Direct Observation of Ion Dynamics in Supercapacitor Electrodes Using in Situ Diffusion NMR Spectroscopy[END_REF] Engelke et al. showed that by measuring effective diffusion coefficients with incremental diffusion times, mean square displacements can be calculated, and from there one can extract information about pore dimensions. [START_REF] Engelke | Three Dimensional Pulsed Field Gradient NMR Measurements of Self-Diffusion in Anisotropic Materials for Energy Storage Applications[END_REF] In our case, we focus on long range diffusion, and extracting tortuosity values from that.

Measuring bulk diffusion coefficients

PFG-NMR measurements were conducted as described in Chapter 2. A diffusion time, the time between two gradient pulses, is selected. Then the NMR signal is recorded for 16 different gradient strengths. The signal recorded for pure tetraglyme is shown in Figure 5.1. The signal should follow the Stejskal-Tanner relation which describes its intensity as a function of the gradient strength and the diffusion coefficient (eq. 2.7). The same experiment can be conducted with a porous electrode soaked with tetraglyme. In this case the obtained diffusion coefficient will be the effective diffusion coefficient of tetraglyme molecules in the porous media.

For the pure tetraglyme measurement, the diffusion time  selected in the PFG-NMR experiment does not affect the diffusion coefficient, provided that ∆ ≫ LD and ∆ < 𝑇 1 , where 𝑇 1 is the longitudinal relaxation time which has been independently measured to be 1.42 seconds in bulk tetraglyme. For the diffusion in porous media the diffusion time plays an important role. If the diffusion time is too short, then the diffusing molecules will not encounter any pore walls, and the calculated diffusion coefficient will be similar to the bulk diffusion coefficient. By increasing ∆, the mobile molecules will diffuse further and encounter more pore walls (Figure 5.

Ensuring the removal of excess solvent on the electrode surface

To measure the diffusion of tetraglyme in the porous structure of a Super P electrode, it is important to have all the solvent inside the porous structure and no molecules on the surface of the electrode. If some molecules are on the surface of the carbon, their effective diffusion coefficients will be faster than the confined ones and affect the average measured diffusion coefficient. To ensure that there is no significant amount of tetraglyme molecules on the surface, we tested two different drying approaches. For the first one, the soaked electrode is gently touched and dragged on both sides using a lab paper-towel, this method will be called gently-dried. For the second drying approach, the same procedure is done but followed by an extra step. The gently-dried electrode is put on a lab paper-towel and another piece of papertowel is put on top of it. Then a slight pressure is applied by hand to soak away any possibly remaining liquid on the surface, this second drying approach is called excess-dried. The measurements on electrodes dried with the two techniques are compared with two other systems: bulk tetraglyme, and an electrode flooded with an excess of tetraglyme solvent as shown in Figure 5.3.a. The diffusion coefficient of the bulk tetraglyme does not decrease when increasing the diffusion time, as expected for free diffusion. The other three samples show the signature of confined diffusion. The similar diffusion coefficients obtained for the gently-dried and excessdried samples give us confidence in the fact that we have removed all the excess liquid from the surface of the electrodes via both approaches. If that was not the case, the diffusion coefficients for the gently-dried case would be higher than the excess-dried case as demonstrated by the results for the flooded sample. For the following part, we used the gentlydried approach.

Tortuosity evolution along discharge

To After discharge, the cells are disassembled and the cathodes are washed with pure DME to remove the salt, dried, and soaked with pure tetraglyme as described in details in Chapter 2.

The 1 H diffusion coefficients are measured for all of these electrodes and shown in Figure 5.5.

The diffusion time is set to 1.2 seconds for these measurements in agreement with the convergence observed in Figure 5.3. The root mean square displacement <r 2 > 1/2 shows that the displacement of tetraglyme molecules covers around 50 µm in this time

(<r 2 > 1/2 =(6•D•t) 1/2 =(6•D•Δ) 1/2 =(6•3.42•10 -10 m 2 /s•1.2 s) 1/2 =49.6 µm
). This ensures that the measured diffusion coefficients are not local (i.e. short range) motions but correspond to those of long range diffusion.

The overall trend observed for both solvents is that the effective diffusion coefficient decreases as the cell is discharged further. This is what we expect as the growth of discharge products leads to a reduction of porosity and a clogging of some pores. As a result, the electrode becomes more tortuous. We measure higher effective diffusion coefficients for the electrodes discharged in the tetraglyme based electrolyte than for the electrodes discharged in the DME based electrolyte. This could be due to larger particles formed and more pore clogging happening in DME. Indeed, DME has a higher donor number than tetraglyme; and we expect the corresponding cells to show larger Li2O2 particles than the tetraglyme ones. [START_REF] Laoire | Influence of Nonaqueous Solvents on the Electrochemistry of Oxygen in the Rechargeable Lithium-Air Battery[END_REF][START_REF] Johnson | The Role of LiO2 Solubility in O2 Reduction in Aprotic Solvents and Its Consequences for Li-O2 Batteries[END_REF] Formation of larger particles would lead to more pore clogging and higher tortuosities and because of that lower effective diffusion coefficient values. this assumption widely used for continuum models [START_REF] Yin | Linking the Performances of Li-O2Batteries to Discharge Rate and Electrode and Electrolyte Properties through the Nucleation Mechanism of Li2O2[END_REF][START_REF] Xue | A Comprehensive Model for Non-Aqueous Lithium Air Batteries Involving Different Reaction Mechanisms[END_REF] is not adapted to predict electrochemical performances accurately. The results are in agreement with what was shown in Chapter 4.

Indeed, the tortuosity evolution calculated using the PNM also shows larger changes along discharge than what is calculated using the continuum model (Figure 4.13). The better match between the tortuosity evolutions measured and calculated using the PNM, than tortuosity evolutions calculated using CM, shows the importance of incorporating explicit 3D electrode structure in modelling approaches. To check if indeed the formation of larger discharge particles lead to more pore clogging as we see slower effective diffusion for electrodes discharged with DME than tetraglyme, we use the PNM approach to plot (Figure 5.7) the number of clogged pores along discharge for the structure with a 60 nm average pore size and with three escape factors. The number of clogged pores are higher for higher escape factors, even at the same depth of discharge. This calculation shows a good plausible interpretation of why the effective diffusion coefficients for electrodes discharged in DME based electrolyte decreases faster along discharge. Plots are shown for three escape factors (χ =0, 0.5 and 1).

Measuring Li + self-diffusion in tetraglyme based electrolytes

The diffusion coefficient of Li + used in modeling works is mostly assumed, [START_REF] Yin | Linking the Performances of Li-O2Batteries to Discharge Rate and Electrode and Electrolyte Properties through the Nucleation Mechanism of Li2O2[END_REF][START_REF] Gaya | Investigation of Bi-Porous Electrodes for Lithium Oxygen Batteries[END_REF][START_REF] Torayev | Stochasticity of Pores Interconnectivity in Li-O2 Batteries and Its Impact on the Variations in Electrochemical Performance[END_REF] as few experimental measurements of this property have been reported. Most of the reports refer to the ionic conductivity of the electrolyte. [START_REF] Laoire | Influence of Nonaqueous Solvents on the Electrochemistry of Oxygen in the Rechargeable Lithium-Air Battery[END_REF] Therefore, we decided to measure Li + self-diffusion coefficients in tetraglyme based electrolytes with 0.1, 0.25, 0.5, 0.75 and 1 M concentrations of LiTFSI. In these series of PFG-NMR experiments, a shorter diffusion time of 30 ms is selected, because [START_REF] Abraham | A Polymer Electrolyte-Based Rechargeable Lithium/Oxygen Battery[END_REF] Li T1 relaxation is much faster than for 1 H in the electrolyte. The measured diffusion coefficients are shown in Figure 5.8. In this section, we measured the effective diffusion coefficients and tortuosity values along the depth of discharge for self-standing super P electrodes discharged in two electrolytes. The obtained results show an increasing tortuosity when an electrode is discharged further; and the results show that the tortuosity evolution depends on the electrolyte. Even though we measure the diffusion of solvent molecules, we expect the transport of both oxygen molecules and Li + to be affected in a similar way. We calculate the MSD in all three directions, x, y and z. The y direction is along the pore length and x and z are the two directions perpendicular to the pore length. Figure 5.10 shows those MSD plots. The MSD curves for the x and z directions reach a plateau as expected for confined diffusion. [START_REF] Bickel | A Note on Confined Diffusion[END_REF] On the other hand, the MSD along y direction does not converge and reaches a linear regime since there are no restrictions along this direction. The diffusion coefficients are obtained from the fitting of the MSD values in the y direction which is the long direction with periodic boundary condition along the cylindrical pore length.

Monte Carlo simulations are stochastic in nature and we need to do several simulations to ensure a sufficient statistical accuracy. For each structure, the kMC calculations are repeated three times; and we report the average of those three repetitions. The diffusion coefficient as a function of the pore diameter is shown in Figure 5.11. The effective diffusion coefficient does not vary much for pores with diameters larger than 25 nm. Below 25 nm, we start seeing a pore size effect on the diffusion. The effective diffusion coefficient decreases sharply for pores with diameters below 4-5 nm. These trends are consistent with PFG-NMR measurements conducted by Forse et al. [START_REF] Forse | Direct Observation of Ion Dynamics in Supercapacitor Electrodes Using in Situ Diffusion NMR Spectroscopy[END_REF] for self-diffusion coefficients of ions in activated carbons for supercapacitors.

They measured effective diffusion coefficients about two orders of magnitude lower for ions in nanopores compared to the bulk values.

Conclusions

In this Chapter we measured diffusion in porous electrodes using PFG-NMR experiments. We discharged several Super P electrodes to various depths of discharge and measured the effective diffusion coefficients of tetraglyme molecules in these materials. This allowed us to extract the tortuosity evolution along discharge for Li-O2 batteries. We showed that the tortuosity increases when discharging the electrodes further. We studied the effect of the solvent on the tortuosity evolution of Super P electrodes. When a DME based electrolyte is used, the tortuosity increases faster than when tetraglyme is used as the electrolyte solvent. We attribute this difference to the difference in the size of discharge products in these two electrolytes. We show from PNM calculations that electrolytes with higher donor number (larger escape factor) lead to more pore clogging. That is a plausible reason why the tortuosity of the electrodes discharged by DME increases faster than tetraglyme.

In the second part of this Chapter, we investigated the pore size effect on diffusion using our kMC model. We showed that there is indeed a pore size effect, but it manifests itself only for very small pores with diameters below 5 nm. The effective O2 diffusion coefficient for pores with a 1 nm diameter is calculated to be 5.5 times lower than for pores with a 100 nm diameter.

In the future, these results could be integrated in the PNM model to improve its accuracy by including the pore size effect on diffusion, the PNM would capture in a better way the influence of the pore shrinkage along discharge.

Chapter 6. Discharge capacity: variability and potential ways of improvement

In this chapter we will discuss the possible sources of dispersion in the discharge capacities measured experimentally for Li-O2 batteries and propose a way to improve the discharge capacity by accelerating the oxygen transport along the electrode thickness. For all of the electrochemical tests in this Chapter, the airtight glass chamber shown in Figure 2.1.b is used.

Possible sources of dispersion in the calculations and measurement of discharge capacities

Measurements of Li-O2 batteries' performances are quite sensitive to operating conditions, cell components and impurity if any present. That is, when the same experiment is repeated several times, the discharge capacities show a non-negligible dispersion. It is often assumed to result from contact between the electrodes and separator [START_REF] Wang | High Capacity Pouch-Type Li-Air Batteries[END_REF] , the uncertainty in the material quantities (e.g., mass loading, electrolyte amount 100 ), handling issues or variations from one experiment to another. To explore the possibility of a lack of consistency in the experimental conditions,

Griffith et al. reported reproducibility tests for 60 cells at 5 different current densities done

with well controlled experimental procedures. They still observed a large variation in the measured discharge capacities. 1 Such a dispersion of results is observed in many experiments but insufficient attention is paid to the origin of this phenomenon. As such, the variations in experimental conditions seem insufficient to explain the extent of the dispersion observed. Here we contribute to this investigation by selecting several parameters and examining how they affect dispersion. This section is part of a collaborative work with a PhD student at LRCS, on 7 of those 11 factors. To avoid electrode macro-heterogeneity, electrode casting nonuniformity and formation of large cracks while drying, we used commercial GDL electrodes for all of the experiments in this section. The effect of water impurity, operating temperature and its control, and electrode surface defects and removing them has been investigated by Gaya 101 and is not discussed here. To briefly summarize what she found, we can say that increasing the H2O ppm level increases the discharge capacity and also increases the dispersion. Controlling the temperature helps reducing the dispersion, but discharging a cell at elevated temperatures does not have a monotonous effect. It can either increase or reduce the dispersion depending on the discharge current density and electrolyte used. Removing electrode surface defects by heat treatment improves reproducibility significantly, but reduces the discharge capacity as well.

Here, we focus on the 4 possible sources colored in red in Figure 6.1. We conducted our study using both experimental and modelling techniques as will be described in the next sections.

Electrode size and position inside the cell

Initially we checked if some cell preparation methods affect the dispersion of discharge capacities. The first one is the cathode size. For this study, we used commercial GDL electrodes, since they are quite uniform in thickness and loading. A triglyme solvent with 0.5 M LiTFSI is used. The Swagelok cell used for the experiments is shown in Figure 6.2.a. The size of the cylinder in the Swagelok body is 0.5 inches (12.7 mm). If a cathode with a size smaller than 12.7 mm is used, it may not necessarily sit at the center of the cell, there can be a mismatch between the cathode and the plunger locations as shown in Figure 6.2.b, the plunger can cover anywhere from 5% to 20% of the electrode surface depending on how they are placed. If the cathode is not located at the exact same place every time, which is hard to ensure in practice, the area the plunger covers on the cathode will vary from one experiment to another.

On the contrary, if the cathode size is exactly 12.7 mm, it is much easier to ensure that the plunger will cover the same area in every experiment. We tested the effect of a mismatch between the plunger and the cathode in our experiments. We selected two cathode sizes with diameters equal to 11 mm and 12.7 mm on which we ran some galvanostatic discharges. Each experiment was repeated 4 times. The discharge currents were applied according to the geometrical surface area of the electrodes, meaning that a higher discharge current was applied to the cell with a larger cathode size. This is to balance the mass loading difference. Since both electrode sizes have the same thickness, then the electrode with larger diameter (size) will have higher mass. In the literature the discharge current densities are normalized either to mass of the electrode or geometrical surface area of the electrode. In our case, since the electrode thickness is constant, the larger electrode has 33% higher mass loading and geometrical surface area. The discharge capacities obtained for the two cathode sizes are shown in Figure 6.3.a. The dispersions in the discharge capacities were not significantly different for these two cathode sizes. It suggests that the electrode size and position in the Swagelok cell does not affect the dispersion to any noticeable extent. The discharge capacity is slightly lower for the cell with the large electrode. The apparent improvement in cell performance when decreasing the mass loading in the cathode is discussed in the work of Noked, Schroeder and coauthors. 102 The reduction of discharge capacity when increasing the mass loading (electrode size) is most probably caused by the fixed volume of electrolyte in both cells, and having less dissolved oxygen per electrode volume when the electrode size is increased.

Salt concentration in the electrolyte

With the same experimental setup, we also tested the possible effect of the salt concentration on the dispersion of discharge capacities. The idea behind this test was the possible existence of a Li + concentration gradient along the cell during discharge. This gradient will potentially influence the electrochemical reactions and will be much more pronounced when the salt concentration is low. For this test, two salt concentrations were selected, 0.5 M and 1.0 M LiTFSI in triglyme solvent. Based on the discharge capacities, it seems that the salt concentration does not to affect the dispersion (Figure 6.3.b). The average discharge capacity increased marginally by increasing the LiTFSI salt concentration from 0.5 M to 1.0 M, yet the dispersion of the capacities stays the same. As a conclusion, in these conditions quite typical of Li-O2 experiments reported in the literature, no influence of the salt concentration on the dispersion can be supported. In fact, these results are consistent with the PNM results of Chapter 4 which have shown that the Li + concentration is not a capacity limiting parameter. In particular, Figure 4.5.a shows that there is no significant Li + concentration gradient along the electrode.

Electrode compression effect

Another source of dispersion could come from the assembly of the Swagelok cell during which we need to press the two plungers by hand to close and tighten the cell. Since this process is done manually, the compression strength may be different from cell to cell. Thus, the cathode compression can and does vary. 103 As it is hard to evaluate the force applied to the electrodes during the cell tightening, we studied the contribution of the varying cathode compression on the discharge capacity dispersion by a modelling approach. Initially a GDL electrode structure is generated from the gas diffusion layer model in the FiberGeo module of the GeoDict software (Figure 6.4) by Alexis Rucci, a postdoctoral researcher in LRCS. This in silico structure is then compressed along the cathode thickness direction to obtain 3 cathode structures with different compressions (0%, 10%, and 20%). We applied our PNM approach to 123 these three structures to study the effect of compression on the discharge capacity. In Chapter 4, we showed that the PNM captures the role of the mesostructure on the discharge performance of a cell more accurately than a continuum approach and as such, it should be well suited for The compression of the electrode alters the electrode mesostructure and the pore interconnectivity. This effect, known as calendering, has already been studied for Li-ion batteries. 104 However, we expect it to be much more pronounced for Li-O2 batteries. Unlike in The calculated discharge capacities show that the compression of the cathode impacts the discharge capacity. In fact, the discharge capacity decreases as the cathode is compressed (Figure 6.5.d). Compressing an electrode by 20% can lead to a change of discharge capacity of about 15%. Based on these calculation results, the variation in compression of the cell can be a source of dispersion of the discharge capacities.

Role of micro-heterogeneity on dispersion

We now turn to the study of another possible source of dispersion which is the microheterogeneity in the mesostructure of the cathodes and investigate its effect on the dispersion in electrochemical results. As opposed to the previous parameters (electrode position, salt concentration, electrode compression), which could vary mainly by differences in handling and can in principle be controlled, the heterogeneity or homogeneity of the mesostructure of the carbon cannot be controlled finely. As a result, cathodes presenting identical macroscopic quantities such as porosity and pore size distribution could be locally different. To analyze this aspect further, we use two different electrode structures: GDL and Super P electrodes.

We start with the calculation for a GDL electrode. We generated a GDL electrode with 1 mm x 1 mm x 0.23 mm dimensions and divided it into four equally sized zones, with a 0.5 mm x 0.5 mm x 0.23 mm size. Then a pore network is extracted from these four regions. The resulting porosity and surface area values are given in Table 5. The variation in porosity is about 1.8% and the change in surface area is about 2.7%. The discharge capacities are calculated using the PNM for all of these structures. We see some dispersion in the calculated discharge capacity, about 5.7% (Figure 6.6), but it is very limited.

However, since the pores are in the micrometer range and the porosity is very high, we do not observe much pore clogging, and hence a limited effect on the dispersion in the data. We then follow the same approach for Super P electrodes. To study the effect of local heterogeneity, we select four random zones, with equal sizes of 4 µm x 4 µm x 5 µm, from a single tomography image acquired for a Super P carbon electrode (Figure 6.7.a) and apply the PNM model to these distinct regions. A 1 M Li + concentration and a 4.43 mM O2 solubility are used in these calculations. We check that the active surface areas and porosity values are very close to each other (Table 6). There is only a difference of about 7% for the porosity and 8%

for the CSA. The PSDs are also superimposed (Figure 6.7.b). While the four selected regions have similar CSAs, porosities and PSDs, the discharge curves of these four slices show a significant dispersion of capacities for the three current densities explored (Figure 6.7.c). The relative differences between the discharge capacities are 29% for 400 mA.g -1 , 23% for 100 mA.g -1 and 16% for 20 mA.g -1 . These variations in capacities are much larger than the ones for porosity and CSA. To make sure these differences are not due to slight differences in porosities and CSAs, the four electrodes are flipped; the air inlet and separator sides of the cathode are exchanged. In this case the macroscopic properties are identical. Yet, the discharge profiles again show variations (Figure 6.7.d). The main reasons of these large dispersions are inherent to the stochastic nature of the pores interconnectivity and the dynamic change of the porous mesostructure. For the four selected zones, while the macroscopic properties are similar, the mesoscopic arrangement of pores is not identical and in particular the pores are not connected in the same fashion. In Li-O2 batteries, the transport of O2 and Li + in the electrolyte crucially impacts the cell capacity, and it is expected that the differences in interconnectivity and pore arrangements will generate a significant dispersion. Besides, unlike in Li-ion batteries, discharge products of Li-O2 batteries are solid particles that fill up the porous volume. As such, the porous structure dramatically evolves along discharge. This is illustrated in Figure 6.8 which shows the evolution of the pore size distribution along the discharge for zone 2 and the number of inactive pores as a function of time for the same zone in normal and reversed configurations. Once a pore gets clogged, it does not only affect itself but also its neighboring pores. This blocks the transport of O2 and Li + to the connected pores through this clogged pore. This can impede long range transport and also form isolated regions in the pore network which do not contribute to the cell capacity. From the calculation on Super P and GDL electrodes, we can say that the heterogeneity and local microscopic differences between the structures affect dispersion. When the electrode has micro or meso-pores and a low porosity, the effect of the pore interconnectivity and heterogeneity on the dispersion will be much bigger. On the other hand, if the electrode has a high porosity and macro-pores, then the impact on the dispersion will be much smaller.

A strategy to increase the discharge capacity

In Li-O2 batteries one well known reason of premature battery death is the clogging of pores at the O2 inlet of the positive electrode. This can be easily observed by comparing SEM characterizations of opposite sides of the positive electrode at the end of discharge. [START_REF] Bickel | A Note on Confined Diffusion[END_REF] Due to the limited solubility of oxygen, its concentration in Li-O2 battery electrolytes is much lower than the Li + concentration. On top of that, the sluggish diffusion of O2 within the electrolyte increases the importance of this problem. Before discharge the electrolyte is saturated with O2, but during discharge the O2 consumption rate exceeds the rate at which O2 is transported from the air inlet side through the electrolyte to the bulk of the porous electrode. This generates a gradient of O2 concentration along the electrode and the electrochemical reaction happen more rapidly at the O2 inlet side of the positive electrode. This phenomenon facilitates the production of much larger amounts of Li2O2 at the gas inlet side that fills up the pore space leading to a complete blocking of the O2 transport. This problem is much more significant at high discharge rates for which oxygen is consumed faster and its slow transport becomes even more problematic. As seen in Figure 6.10, 105 when increasing the discharge current density from 0.05 mA.cm -2 to 1.0 mA.cm -2 , the amount of discharge products decreases in the center of the electrode. Particularly for electrodes discharged at 1.0 mA.cm -2 current density, there is almost no discharge product in the inner part of the electrode. However, at the air inlet side there is a significant amount of discharge product for all cases. Thus it is evident that higher current densities produce increasingly inhomogeneous distributions of discharge products. We can see the formation of significant oxygen concentration and Li2O2 gradients along the electrode in Figure 4.5. Similar results have been shown in the modeling work of Albertus et al. [START_REF] Albertus | Identifying Capacity Limitations in the Li/Oxygen Battery Using Experiments and Modeling[END_REF] in which the authors developed a continuum model to identify capacity limiting parameters in Li-O2 batteries. Figure 6.11 shows the concentration of oxygen and the amount of discharge products along the positive electrode thickness for two different current densities. When the current density is low there is no significant O2 concentration gradient.

However, for high current densities, the O2 consumption rate increases, resulting in a nanofoams with four different pore size distributions. In their study they observed higher discharge capacity as they increased the pore volume or the average pore size. However, this approach does not affect the spatial inhomogeneous distribution of the discharge products, it only delays the gas inlet clogging time since it takes longer for particles to grow and fill up the pore volume. Furthermore, the pore volume can only be increased up to a certain value as after a given threshold the electrical conductivity is compromised.

One other concept proposed to solve gas inlet clogging is the use of electrode structures with a porosity gradient. 107 That is, an electrode can be designed to have large pores at the gas inlet side, and low porosity on the separator side. However experimentally achieving such an electrode design without compromising the electronic conductivity can be cumbersome.

Besides modifications of the electrode structure, mitigation strategies using different electrolytes have also been proposed. Some electrolytes offering enhanced O2 diffusion can be employed to hinder this issue. Zheng et al. 108 proposed acetonitrile mixed with propylene carbonate as an electrolyte solvent. This electrolyte has a low viscosity, fast O2 diffusion and a higher oxygen solubility than common electrolytes but it still does not offer an optimal solution to the clogging problem.

Electrolytes like DME, with low donor number resulting in poor LiO2 solubility can also be used. In this case the discharge products form an electronically insulating film on the surface of the electrode, [START_REF] Johnson | The Role of LiO2 Solubility in O2 Reduction in Aprotic Solvents and Its Consequences for Li-O2 Batteries[END_REF] the thickness of which is restricted to a few nanometers due to limited electron tunneling distance. Thus pores with radii larger than the electron tunneling distance are not clogged. However, this approach offers low discharge capacity because discharge products form only a thin film on the surface of electrode and passivate all available surface area available for electrochemical reactions.

Here we propose an original approach to try and reduce the gradient of O2 along the electrode thickness without modifying other experimental parameters such as the electrolyte or the porous structure.

Using a magnetic field gradient to improve oxygen transport

Li-O2 batteries are similar to polymer electrolyte membrane fuel cells (PEMFC) in many ways.

They both consume oxygen coming from the gas state and they both have a porous cathode for electronic conduction and mechanical support. Moreover, in both systems there is an oxygen transport issue. For PEMFCs, an interesting concept to increase oxygen transport has been employed. It has been observed that applying a magnetic field gradient improved oxygen mass transport and hence the performance of PEMFC. [109][110][111][112] We applied the same concept to Li-O2 batteries. This involves the introduction of an external physical force to accelerate the diffusion of oxygen within the electrolyte without replacing any battery material or making significant changes to the battery architecture. The active material for lithium air batteries, namely oxygen and the intermediate discharge product

LiO2 113 are paramagnetic in nature. Indeed, the molecular orbital description of molecular oxygen shows the presence of two unpaired electrons which may orient themselves in the direction of a magnetic field. Due to this paramagnetic nature, the oxygen and LiO2 molecules would be attracted towards a magnet as they would be experiencing a magnetic force, called Kelvin force, described by eq. 6.1.

𝐹 𝑚 ⃗⃗⃗⃗ = 𝑐𝜒 𝑚 2𝜇 𝑜 ∇𝐵 ⃗ 2 eq. 6.1

Here, the force experienced by paramagnetic molecules is given by 𝐹 𝑚 ⃗⃗⃗⃗ which is calculated using 𝜒 𝑚 , the molar susceptibility of a species, the magnetic permeability 𝜇 𝑜 of vacuum, the concentration of paramagnetic species (𝑐) and the gradient of the magnetic field 𝐵 ⃗ applied by the magnet.

Therefore, O2 and LiO2 could drift away from the air inlet side and enter the central and far end of the cathode under the influence of a magnet placed behind the separator and anode. This could produce a more homogenous distribution of solid Li2O2 discharge products along the porous electrode because Li2O2 is formed from reduction of the soluble and paramagnetic LiO2 species. Any paramagnetic radicals existing in the system would also experience the Kelvin force. However organic electrolyte molecules which are diamagnetic would be experiencing a weak diamagnetic force and would be repelled. Other potentially present molecules, such as the diamagnetic nitrogen gas molecules introduced from the air inlet side would be repelled as well. This can prove to be useful when the lithium air battery technology moves from using pure O2 to an air like mixture of O2 and N2 wherein N2 would compete with O2 for dissolution in the electrolyte and decrease the saturation concentration of O2. A magnetic field can potentially reduce the ability of N2 to dissolve even if the solubility of N2 in the electrolyte is relatively high. The magnetic behavior of the ions, radicals and molecules in the system depends on the materials used, and a greater understanding can be developed by performing detailed studies in the future.

Overall, this idea has two main benefits. First, by introducing a Kelvin force in the battery, pore clogging at the inlet side can potentially be drastically reduced, allowing the battery to undergo deeper discharge and hence offer higher specific capacity. Second, it enhances the transport of paramagnetic species, namely O2 and LiO2, which in-return elevates the rate capability of metal air batteries.

Cell design

A scheme of the Swagelok incorporating the magnet is shown in Figure 6.12. The main difference with the original setup described in Figure 6.2 is that a permanent magnet (NdFeB) with a 0.5 inch diameter and a 0.5 inch height is placed behind the Li metal anode. This specific size is chosen to match the size of the Swagelok plunger. The magnet is coated with a Ni-Cu-Ni conductive layer and has a magnetic field strength of 1.48 T.

The gradient of magnetization across the total distance occupied by the thickness of the lithium metal, separator and porous cathode is vital for the paramagnetic oxygen entering from the air inlet side to experience a noticeable Kelvin force and thereby influence the discharge behavior.

For a proof of concept demonstration, a commercial neodymium magnet with N52 grade is used. Depending on the thickness of the electrode, the magnetic field gradient along the positive electrode thickness, from the separator side end to the air inlet end of the electrode, can be as high as 146 T.m -1 . This design can potentially be extended using a coil wrapped around the body of the Swagelok to introduce a temporal gradient of magnetization. By varying the current in the coil, transport properties of oxygen experienced by the Kelvin force can be studied and fine-tuned for optimum performance.

argon are very small compared to the cell under oxygen environment. From these results we can say that the presence of the magnet does not create any new side reactions. Another stability test with Li vs. Li symmetric cells is conducted, this is to see if there is any parasitic reaction occurring between the electrolyte and a part of the cell other than the cathode.

We use the same electrolyte as in the CV test but change the GDL cathode to Li metal. A typical Swagelok cell with two complete plungers (no gas inlet), is used. We applied 25 µA charge and discharge currents with 2 hours periods. The resulting curve is shown in Figure 6.14. For both cells the initial cell voltage values are small, meaning less resistancethat is because the Li metal does not have a resistive surface film. Then the voltages increases, most probably due to Li metal reacting with the electrolyte and forming a slightly passivating layer.

Then after some time the voltage values decrease and stabilize, indicating the stabilization of the Li metal surface layer. The main difference between the two cells is that the cell with the magnet has higher voltage values than the reference cell, even after several cycles. The voltage difference between two cells at cycle 38, the last cycle in Figure 6.14, is 9.4 mV. Knowing the applied current we can calculate the extra resistance value in the magnetic cell, which is 376 ohms. Considering all of the components are the same in these two cells except the permanent magnet, the extra resistance is most likely due to the contact resistance between the Li metal and the magnet; but also considering the difference is quite small, it might be because of a slight variation in the compression or cell assembly. From this study on a symmetric cell we conclude that the magnet does not have an intrusive effect of the system: the cell voltage stabilizes after around 20 cycles, similar to the reference cell case.

Tests on self-standing gas diffusion layer electrodes

We start the electrochemical tests with GDL electrodes and a 0.5 M LiTFSI in triglyme electrolyte.

The advantages of using GDL electrodes are:

• It is a widely available, commercial material.

• It allows us to obtain a uniform loading and thickness.

• There is no need for a binder, so we do not have to worry about the dissolution and stability of the binder.

Using GDL also has disadvantages:

• The pores are large pores so no pore clogging is expected.

• The porous structure is not very tortuous, so there are less transport limitations.

We ran several galvanostatic tests using GDL electrodes. Figure 6.15 shows these tests. The variation of discharge capacities is quite wide, that is why several repetitions are performed to get a more accurate average capacity for each cell configuration. The magnetic cells provided a 31% higher discharge capacity on average. Since the dispersion of discharge capacities is huge, the improvement using a magnet seems less convincing. A 31% improvement is a small improvement. In fact, we do not expect the magnetic field to have huge role when a GDL electrode is used. As listed above, pores in GDL are in the micrometer range and they do not get clogged by the discharge products. The end of discharge is usually due to surface passivation. We can see from SEM images (Figure 6.16) that there is no visible gradient of discharge products along the electrode. The amount and size of discharge particles at the oxygen inlet side of the cathode are similar to those at the separator side. The differences between the capacities of the magnetic and reference cells are larger at higher discharge currents (Figure 6.18). This reinforces our belief that the magnetic field plays a beneficial role. However, the variations in the achieved discharge capacities are huge. In fact, some of the cells with discharge capacities less than 0.1 mAh have most likely failed due to some assembly, missing contacts or some other unclear reasons. In this study we observed improved performance for the magnetic cells but it is still hard to conclude on a systematic gain. There are two issues about the tests on dip-coated electrodes.

First, the mass loading and coating uniformity are hard to control. Secondly, even though a In short, the suggested design helps to improve the performance of Li-O2 batteries by enhancing the transport of oxygen along the positive electrode. This effect is less visible when a small current density is applied on an electrode with macropores. The magnetic effect is much clearer when a thick electrode with mesopores is used. As a perspective, this study should be completed by making SEM images of the discharged electrodes with and without a magnet.

Such an experiment would allow us to confirm the change in discharge products gradient with the application of a magnetic field.

times from 100 nm pore to 1 nm pore. In the future, these results could be included in PNM to enhance its accuracy.

Then we moved to the study of the possible origins of the large dispersions observed when measuring discharge capacities in Li-O2 batteries. We studied the effect of the electrode size and position in the Swagelok cell, the salt concentration, the compression of the cathode and the micro-heterogeneity in the cathode mesostructure. Two different GDL electrode sizes were tested with 11 mm and 12.7 mm disk diameters. The discharge capacities obtained from galvanostatic discharge experiments do not show any noticeable role of the electrode size or position on the dispersion. Our experiments with 0.5 M and 1.0 M solutions of LiTFSI in triglyme also suggest that the variation in discharge capacities does not depend on the salt concentration. However, we have limited our study to two commonly used concentrations.

Lower salt concentrations could lead to different results although the lithium concentration is always much higher than the oxygen concentration and as such is not a limiting factor for the capacity. We tested the compression effect using the PNM approach. We used a GDL model from the GeoDict software to generate a structure and compress it by 10% and 20% along the electrode thickness direction. We observed that the discharge capacity decreases by 15% when the electrode is compressed by 20%. This shows that the variation in the compression strength during cell assembly can impact the cell discharge capacity. We tested the role of the microheterogeneity for two electrodes: GDL and Super P. For both electrode types we select 4 different zones from a large structure and calculate the discharge capacities using the PNM.

The macroscopic properties (PSD, surface area, porosity) for these four zones are almost the same. However, the calculated discharge capacities show some variation. This effect is much more significant for the Super P electrode than for the GDL one. The variations in discharge capacities for Super P are as high as 29%. The smaller pore sizes and lower porosity values for the Super P structure compared to the GDL are the main reasons for this larger effect of the gradient on the Li-O2 battery performance is needed. In future studies magnetic nano and micro particles could be embedded in the cathode to generate extremely high magnetic field gradients.

Résumé développé de la thèse en Français (Summary in

French)

La Le produit de décharge prédominant dans une batterie Li-O2 est le Li2O2, qui est un isolant électronique. La croissance du produit de décharge peut suivre deux voies. Si l'espèce intermédiaire de eq. 2 (LiO2: superoxyde de lithium) est insoluble dans l'électrolyte, elle sera réduite à la surface de la cathode et formera un film de Li2O2. L'épaisseur de ce film est limitée par l'effet tunnel à travers Li2O2, ce qui conduit à des films très minces d'environ 10 nm. [START_REF] Luntz | Tunneling and Polaron Charge Transport through Li2O2 in Li-O2 Batteries[END_REF] Si la solubilité de l'espèce LiO2 dans l'électrolyte est élevée, elle peut suivre une réaction de dismutation (eq. 3) et former un produit de décharge volumineux Ces produits de décharge ont été principalement observés sous forme toroïdale. [START_REF] Mitchell | Mechanisms of Morphological Evolution of Li2O2 Particles during Electrochemical Growth[END_REF][START_REF] Zhai | Disproportionation in Li-O2 Batteries Based on a Large Surface Area Carbon Cathode[END_REF] Dans la pratique, le réactif réel au niveau de la cathode étant l'oxygène, nous devons utiliser un matériau conducteur supplémentaire pour fournir une conductivité électronique et un support mécanique pour permettre le dépôt de Li2O2. La structure support est typiquement un carbone poreux. Le choix du matériau de cathode et de sa structure a un impact important sur les performances de la cellule électrochimique. Un certain nombre de travaux expérimentaux et théoriques sont dédiés à la compréhension de la relation entre la structure de la cathode et les performances de la cellule, conduisant parfois à des résultats contradictoires. [START_REF] Meini | Effect of Carbon Surface Area on First Discharge Capacity of Li-O2 Cathodes and Cycle-Life Behavior in Ether-Based Electrolytes[END_REF][START_REF] Ding | Influence of Carbon Pore Size on the Discharge Capacity of Li-O2batteries[END_REF][START_REF] Kuboki | Lithium-Air Batteries Using Hydrophobic Room Temperature Ionic Liquid Electrolyte[END_REF] Les batteries Li-O2 sont intéressantes grâce à leur capacité de décharge théorique élevée de l'ordre de 1168 mAh/g et au faible coût des matériaux qui les constituent. Cependant, ces systèmes font face à plusieurs défis tels que leur stabilité, leur faible capacité pratique, leur vitesse de cyclage limitée et leur durée de vie. Dans le cadre de ma thèse, nous nous sommes concentrés sur l'étude des mécanismes de transport à l'intérieur des électrodes poreuses des batteries Li-O2. Le transport de l'oxygène joue un rôle crucial sur la performance de la batterie et en l'optimisant on peut en principe améliorer la capacité de décharge et la vitesse de cyclage. Plusieurs modèles ont été développés durant ma thèse. Un modèle cinétique de type Monte Carlo a été développé pour étudier l'effet de la taille des pores sur la diffusion. Un modèle continu a été utilisé afin d'étudier l'impact de la taille des pores sur la capacité de décharge, et la précision de ce modèle a été évaluée. Un modèle de réseau de pores a été développé pour décrire en détail la mésostructure des électrodes et pour prédire leurs performances de manière plus précise. Nous avons montré que le modèle de réseau de pores est plus précis que le modèle continu pour décrire l'effet de la taille des pores sur la performance des batteries. Des expériences de PFG-NMR ont été menées pour mesurer l'évolution de la tortuosité de l'électrode durant la décharge. D'après nos résultats, la tortuosité diminue au cours de la décharge et cette diminution dépend de l'électrolyte. Enfin, nous proposons une stratégie pour améliorer le transport de l'oxygène en utilisant son caractère paramagnétique. En particulier, nous introduisons un aimant dans la batterie. Mots-Clés : Li-O2 batteries, électrodes poreuses, transport de l'oxygène, PFG-NMR

Abstract in English

Li-O2 batteries are attractive thanks to their high theoretical discharge capacity of 1,168 mAh.g - 1 , and the low cost of their constituent materials. However, Li-O2 batteries face several challenges: stability, low practical capacity, rate capability and cycle life. In my PhD research, we focused on studying transport processes in porous Li-O2 battery electrodes. Oxygen transport plays a crucial role in defining the battery performance and by improving it, we can in principle tackle two of the four aforementioned challenges, namely the discharge capacity and the rate capability. In this PhD, we developed several models. A kinetic Monte Carlo model was developed to study the pore size effect on diffusion. A continuum model is used to study the pore size effect on the discharge capacity of Li-O2 batteries, and the accuracy of this model is assessed. A pore network model is developed to represent the electrode mesostructure in detail and to predict the electrode performance in a more accurate way. We show that the pore network model is more accurate in describing the pore size effect on the battery performance than the continuum model. We also conducted Pulsed Field Gradient NMR experiments to measure the evolution of the electrode tortuosity along discharge; and showed that the tortuosity decreases along depth of discharge and the rate of decay depends on the electrolyte. Finally, we propose a way to improve oxygen transport using its paramagnetic character. In particular we introduce magnet in the setup. Keywords: Li-O2 batteries, porous electrode, oxygen transport, PFG-NMR

Figure 0

 0 Figure 0.1. Annual temperature anomalies of Earth relative to the average temperature for the same month during 1951-1980. Reproduced from NASA's web page. 1

Figure 0 . 2 .

 02 Figure 0.2. Practical discharge capacity of several rechargeable batteries with estimated driving distance and pack price per energy unit. The darker shaded area shows the practical energy density and the lighter shaded area shows the theoretical range. Reproduced from Bruce et al. 2

Figure 0 . 3 .

 03 Figure 0.3. Schematic representation of a Li-O2 battery.

  State of the art ............................................................................................ An overview of the literature on Li-O 2 batteries using bibliometrics.... Stability and cyclability ....................................................................... Discharge capacity ............................................................................... Rate capability ..................................................................................... Modeling works ...................................................................................

Figure 1 . 1 .

 11 Figure 1.1. Number of documents mentioning any of the phrases "Li-O2 battery", "Li-air battery", "Lithium air battery" or "Lithium oxygen battery" on Scopus. Search done on October 18, 2018.

Figure 1 . 2 .

 12 Figure 1.2. Number of articles published on non-aqueous Li-O2 batteries. The numbers are obtained after manual filtering of the Scopus search result on Li-O2 batteries.

Figure 1 . 3 .

 13 Figure 1.3. Articles from the literature categorized according to the topic(s) studied. These categories cover 86% of all articles. This figure was obtained using an in-house text mining program.

  Figure 1.4 shows a classification of the text mined articles according to the electrolytes investigated. It is considered that the article studies an electrolyte if the electrolyte name appears more than 4 times in the text. It confirms that the interest for carbonate electrolytes seems to be on the decline while a variety of other electrolytes are explored.

Figure 1 . 4 .

 14 Figure 1.4. Use of different electrolytes in recent years. Some examples of these electrolytes are carbonates: EC, PC, DEC, DMC, FEC; amides: DMA,TMA; glymes: DME, diglyme, triglyme, tetraglyme… These electrolytes cover 50% of all articles. This figure was obtained using an in-house text mining program.

4 and

 4 Figure 1.7.

Figure 1 . 7 .

 17 Figure 1.7. Illustration of the discharge process of a Li-O2 battery. Reproduced from Yin et al. 41

  et al. showed for the first time the role of the solvent properties, particularly the donor number, and its stabilization effect on dissolved lithium superoxide.[START_REF] Laoire | Influence of Nonaqueous Solvents on the Electrochemistry of Oxygen in the Rechargeable Lithium-Air Battery[END_REF] High donor number (DN) solvents stabilize the lithium superoxide molecules in the electrolyte and facilitate the growth of large particles. Johnson et al. investigated four solvents, 1-methylimidazole (Me-Im), dimethyl sulfoxide (DMSO), DME, acetonitrile, with different donor numbers and proposed a model mechanism for the discharge particle growth (Figure1.8).[START_REF] Johnson | The Role of LiO 2 Solubility in O 2 Reduction in Aprotic Solvents and Its Consequences for Li-O 2 Batteries[END_REF] They showed that high donor number solvents (Me-Im and DMSO) give much higher capacities than the low donor number solvents (DME, acetonitrile). The solution mechanism occurring in high-DN solvents would delay the surface passivation mechanism.

Figure 1 . 8 .

 18 Figure 1.8. (a) Schematic representation of surface and solution mechanisms and (b) the effect of donor number and potential on these two mechanisms. Reproduced from Johnson et al. 43

Chapter

  Experimental techniques ......................................................................... Cell components .................................................................................. Electrochemical tests ........................................................................... Scanning Electron Microscopy ............................................................ Tomographic imaging .......................................................................... Pulsed Field Gradient NMR ................................................................. Chapter 2. Experimental techniques Several experiments have been carried out to study the structural evolution of the mesoporous cathode during discharge, the electrochemical performances of different cells and provide input and validation parameters for the models we developed. The main experimental techniques used in this work are electrochemical tests, pulsed field gradient nuclear magnetic resonance spectroscopy (PFG-NMR), scanning electron microscopy (SEM) and nano-computed tomography (nano-CT) measurements.

Figure 2 .

 2 Figure 2.1. (a) Scheme of the Swagelok cell used for the electrochemical experiments. (b) (c) Glass chambers used to keep the Swagelok cells under oxygen. (b) Glass chamber used for the experiments carried out at the Université de Picardie Jules Verne which can take 2 Swagelok cells at a time. (c) Glass chamber used for the experiments done at the University of Cambridge, which has space for only one Swagelok cell.

  software. A part of tomographic image is shown in Figure 2.2.

Figure 2 . 2 .

 22 Figure 2.2. Slice of a tomographic image used in this work.

Figure 2 . 4 .

 24 Figure 2.4. Basic NMR principle. RF stands for the radiofrequency pulse.

Figure 2 . 5 .

 25 Figure 2.5. Pulse program used for the stimulated echo pulsed field gradient NMR experiments.

2. 6 )

 6 . A 300 MHz Avance III Bruker NMR spectrometer equipped with gradient amplifier and 10 mm PFG probe was used and 1 H NMR was conducted to track the motion of tetraglyme molecules within the electrode pores.

Figure 2 . 6 .

 26 Figure 2.6. Sample holders for PFG-NMR.

Figure 3 .

 3 Figure 3.1. Cylindrical pore geometry considered in the kMC model.

Figure 3 . 2 .

 32 Figure 3.2. kMC model flow chart used (a) in this work and (b) literature by Blanquer et al. 77 and Reuter 78 .

3 . 5 whereFigure 3 .

 353 Figure 3.3. An illustration of the determination of a diffusion coefficient from the MSD curve. Two vertical lines show the time interval where the MSD curve is linear with time.The red line is the fit to the MSD curve in the selected region.

Figure 3 .

 3 Figure 3.5. (a)-(d) Schematic illustrations of the pore network extraction. (a) Two pore families connected via a bottleneck. (b) Dong et al. approach to define effective pore radii and throat lengths. 79 (c) Approach used in this work to assign effective pore radii (d) Effective pores and throat obtained after assigning effective pore radiuses and throat lengths either by approaches (b) or (c). (e) Final pore network obtained using the pore network extraction. (f) Illustration of the two mechanisms considered for Li2O2 (in red) growth in our model: particle and thin film formation.

7

 7 GHz processor speed are used with a RAM of 64 Gb per node. The parallelization speedup of the program is shown in Figure 3.6. The program runs about 5 times faster with 16 cores. However, the speedup decreases when more than 32 cores are used. This is mainly because the PNM code uses shared memory in parallelization. So probably data transfers between different nodes are the limiting factor.

Figure 3 . 6 .

 36 Figure 3.6. Parallelization speedup of the PNM code. Calculated as the walltime it takes for completing a calculation with a single processor (T1) divided by the time it takes for completing the same calculation with n processors (Tn).

Figure 3 . 7 .

 37 Figure 3.7. Schematic illustration of the image processing method used here to identify the formation and fraction of isolated pores.

Ref 85 ChapterChapter 4 .

 854 Pore network model: typical results and comparison with a continuum model ............................................................................. Pore network extraction accuracy ........................................................ Important checks for input parameters and information provided by the PNM ...................................................................................................... PNM vs. continuum model .................................................................. 4.3.1 Thin film and isolated pores from image processing ...................... 4.3.2 Calculated discharge capacities and end of discharge for various escape factors and mesostructures ............................................................. 4.3.3 Evolution of the tortuosity along discharge .................................... Conclusions ......................................................................................... Pore network model: typical results and comparison with a continuum model Numerous carbon-based cathode mesostructures have been studied experimentally and computationally aiming at higher capacities. In this context, one-dimensional continuum models are widely used to predict discharge capacities of electrode mesostructures. However, continuum models use a mean field approach which lacks the ability to capture pore clogging and 3D electrode structure evolution. Because of that we have developed a pore network model which allows us to take these effects into account. The technical and physical details of the PNM model are described in the computational methods chapter. Here we focus on the results of such a model applied to various systems. First, the accuracy of the pore network extraction from a tomographic image is discussed. Secondly, we present discharge profiles, concentration evolution and microstructural changes in the electrode calculated by the PNM. Lastly, the PNM and CM results are compared to identify which model can capture the effect of the electrode mesostructure on the cell performance more accurately. Pore network extraction accuracy Before detailing PNM results, we question the pore network extraction (PNE) accuracy and assess if the extracted 3D network provides a good representation of the original structure. We start the demonstration with two simple structures for which we can easily visualize the pore network and judge the accuracy of the pore network mapping. A structure with two unconnected pores (Figure 4.1.a.1) and another structure with three overlapping pores (Figure 4.1.b.1) are generated on a computer. These structures are converted into a binary format with pore (green) and solid (red) voxels (Figure 4.1.a.2 and Figure 4.1.b.2) and given as inputs to the pore network extraction program. As described in Chapter 3, the program maps the porous media with spherical and cylindrical pores. The outcome of the PNE is drawn together with the input structure to visualize the mapping accuracy (Figure 4.1.a.3 and Figure 4.1.b.3). The program gives two unconnected spherical pores as the resulting pore network for the first structure (Figure 4.1.a.1). This is what we expect as the two pores in the input structure do not overlap. For the second structure (Figure 4.1.b.1), the PNE results in two spherical pores connected by a cylindrical throat. The program thus identifies correctly that two large pores are connected by a bottleneck (i.e. a pore smaller than the two pores it connects). As can be seen from these two simple examples, the PNE is able to accurately map the porous media of simple structures. In Figure 4.1.c, we show the same type of data for an actual tomographic image of a Super P electrode and verify that the pores of the resulting pore network (blue spheres and cylinders) are also well located in the porous areas (green zones) of the input structure.

Figure 4 .

 4 Figure 4.1. Examples of pore network extraction outcomes for three structures: a simple structure with two unconnected pores (a), three connected pores (b) and a tomographic image of a Super P electrode (c). Row (1) is the initial structure; (2) is the binary format of the input structure, green is pore and red is solid region; (3) is the superposed visualization of input structure and extracted pore network (blue spheres and cylinders).

Figure 4 .

 4 Figure 4.2. (a) Pore size distributions, (b) surface areas and porosities obtained for the PNE structure (direct calculation from the enumerated pores) and the original structure (calculation with the GeoDict software) for the Super P tomography image.

Figure 4 . 3 .

 43 Figure 4.3. In silico generated structure with a 70% porosity achieved by randomly placing 20 nm porous spheres in a solid matrix. (a) 3D and (b) 2D slices of the structure plotted using the Fiji/ImageJ software; (c) pore size distributions and (d) specific surface area and porosities obtained for the PNE structure (direct calculation from the enumerated pores) and the original structure (calculation with the GeoDict software).

Figure 4 . 4 .

 44 Figure 4.4. PNM results for a computer generated mesostructure with an average pore size of 60 nm. (a) Pore size distribution obtained by PNE; (b) time-step sensitivity of the calculated discharge capacity (discharge current: 400 mA.gc -1 ); (c) system size effect on the discharge profile, each system size calculation is repeated three times with 1x10 -4 s time-step and a discharge current of 400 mA.gc -1 ; (d) Rate capability calculations with a 1x10 -4 s time-step and a 0.7 µm x 0.7 µm x 5 µm system size.

Figure 4 .

 4 4.d showsdischarge curves for 3 different current densities. As expected, when increasing the discharge current density, the discharge capacity decreases and the overpotential increases. With the PNM, we can now move beyond this macroscopic property and analyze the local properties leading to the determined capacities. We can for example visualize the evolution of the oxygen and lithium concentrations along the electrode thickness with the depth of discharge (Figure4.5.a-b). We note that the lithium concentration does not change significantly while the oxygen concentration is clearly depleted in some regions, and particularly close to the separator side of the cathode. This indicates that the oxygen transport can be a limiting process for reaching large discharge capacities.

Figure 4 . 5 .

 45 Figure 4.5. Example of discharge calculation analysis for a discharge current of 400 mA.gc -1 . (a) Lithium ion and (b) O2 concentrations evolution along the electrode thickness; (c) Loss of active surface area with depth of discharge; (d) Evolution of the number of inactive pores; (e) Evolution of the PSD with depth of discharge; (f) Amount of Li2O2 formed along the electrode thickness.

Figure 4 . 5 .

 45 c shows the evolution of the active surface area during discharge. The active surface area decreases by 40% along discharge but 60% of the initial surface remains active. We can conclude that the electrode surface passivation is not the capacity limiting parameter in this case.The number of inactive pores along the depth of discharge is shown in Figure4.5.d. Inactive pores are categorized into three types: clogged, O2 depleted, passivated.

Figure 4 . 6 .

 46 Figure 4.6. (a) Fraction of isolated pores calculated by applying the image processing approach (see chapter 3 for details) on the experimentally measured Super P tomography image and two computer generated mesostructures. (b) An example of randomly distributed large and small pores in a three-dimensional lattice.

Figure 4 .

 4 7.a.

Figure 4 . 7 .

 47 Figure 4.7. (a) Scheme of the continuum and pore network models used in this work including a representation of the Li2O2 formation mechanisms in a pore, (b) pore size distributions used in the two types of models.

Figure 4 .

 4 Figure 4.7.b. Transport equations are solved only for the oxygen species. The lithium

Figure 4 . 8 .

 48 Figure 4.8. Discharge capacities calculated by (a) CM and (b) PNM, for 3 escape factors and for 4 mesostructures. The electrode thickness is 5 µm, the discharge current density is 400 µA.gc -1 (i.e. 1.16 A.m -2 ); (c) Carbon surface areas of the pristine mesostructures; (d) Discharge capacities as a function of the pristine carbon surface areas calculated using the PNM approach.

Figure 4 . 9 shows

 49 the calculated evolution of the active carbon surface area and the overall oxygen concentration in the electrodes along discharge. The capacity limiting process when 𝜒 = 0 is the surface passivation for both CM and PNM. The active surface area drops to zero at the end of discharge (Figure4.9.a-b). In the case of 𝜒 = 1, the end of discharge is due to transport limitation. The overall oxygen concentration drops to around 3.5 mM by the end of discharge for the CM while it decreases much more drastically in the case of the PNM (Figure4.9.c-d).

Figure 4 . 9 .

 49 Figure 4.9. Active carbon surface area evolution along discharge for (a) continuum and (b) pore network models. O2 concentration evolution along discharge for (c) continuum and (d) pore network models. Legend and color coding are shown in (c).

Figure 4 . 10 .

 410 Figure 4.10. Evolution of the porosity as a function of the depth of discharge. It evolves in the same way for all mesostructures when the same discharge current is applied. This plot is the same for both continuum and pore network models.

Figure 4 . 11 .

 411 Figure 4.11. Discharge capacities calculated with the continuum model for four electrode mesostructures assuming an electrode thickness of 100 µm and three escape factors (0, 0.5 and 1).

Figure 4 . 12 .

 412 Figure 4.12. Equivalent film thicknesses for the experimental data reported in Meini et al. 12 and Ding et al., 87 estimated by dividing the volume of discharge products by the BET surface area of the pristine carbons. The labels are the ones used in those papers.

Figure 4 .

 4 8.a-b) are due to the higher tortuosity values in the 3D pore network (Figure 4.13).

Figure 4 . 13 .

 413 Figure 4.13. Calculated evolution of the tortuosity for the PNM and CM along discharge for the case with 𝜒 = 1. The tortuosity for the CM is independent of the starting pore size distribution. For all of the calculations 400 µA.gc -1 current density is used.

  Diffusion coefficients: results from PFG-NMR experiments and kMC model .................................................................................................. Evolution of tortuosity along the depth of discharge: PFG-NMR experiments .................................................................................................. 5.1.1 Measuring bulk diffusion coefficients............................................ 5.1.2 Ensuring the removal of excess solvent on the electrode surface . 5.1.3 Tortuosity evolution along discharge ........................................... 5.1.4 Measuring Li + self-diffusion in tetraglyme based electrolytes...... Pore size effect on diffusion: kMC model .......................................... Conclusions .......................................................................................

Figure 5 .

 5 Figure 5.1. PFG-NMR signal intensities recorded for various gradient strengths for pure tetraglyme. ∆= 1200 𝑚𝑠; LD=1 ms.

  2.a). Hence the effective diffusion coefficient will decrease (Figure 5.2.b). After a sufficiently large diffusion time, the confined diffusion coefficient will converge to a value corresponding to long range diffusion, which is what we are interested in to extract the tortuosity.

Figure 5 . 2 .

 52 Figure 5.2. (a) Illustration of the area covered by a diffusing molecule with time. (b) Schematic representation of free diffusion and confined diffusion as a function of diffusion time.

Figure 5 . 3 .

 53 Figure 5.3. (a) The four samples tested. (b) Obtained diffusion coefficients as a function of diffusion time.

Figure 5 . 4 .

 54 Figure 5.4. For the DME solvent, four electrodes are discharged and the discharge processes

Figure 5 . 4 .

 54 Figure 5.4. The discharge profile of self-standing Super P electrodes with 0.5 M LiTFSItetraglyme electrolyte, discharged with 50 µA current.

Figure 5 . 5 .

 55 Figure 5.5. Effective diffusion coefficients of tetraglyme molecules measured by 1 H PFG-NMR as a function of discharge capacity for self-standing Super P electrodes. The solvent used for discharge is indicated in the plot legend.

Figure 5 . 6 .

 56 Figure 5.6. Evolution of tortuosity in pristine and fully or partially discharged selfstanding Super P electrodes extracted from PFG-NMR measurements. The solvent used for discharge is indicated in the plot legend. The Bruggeman relation is calculated by assuming all the discharge products are Li2O2. Dashed lines for experimental data are added as a guide to the eye.

Figure 5 . 7 .

 57 Figure 5.7. Number of clogged pores for the computer generated electrode structure with an average pore size of 60 nm. The discharge current density is 400 µA.g -1 . Plots are shown for three escape factors (χ =0, 0.5 and 1).

Figure 5 . 8 .

 58 Figure 5.8. Diffusion coefficients, measured by PFG-NMR, of 7 Li + in tetraglyme based electrolytes with different salt concentrations. The diffusion time between two gradient pulses is 30 ms.

Figure 5 . 9 .

 59 Figure 5.9. MSD plots obtained from kMC calculations for several pore sizes.

Figure 5 .

 5 Figure 5.10. MSD curves calculated from kMC trajectories in x, y and z directions for pores with (a) 100 nm, (b) 50 nm, (c) 5 nm and (d) 1 nm diameters.

Figure 5 .

 5 Figure 5.11. O2 diffusion coefficients calculated using the kMC model for various pore diameters.

Figure 6 .

 6 Figure 6.1. Possible sources of non-reproducibility. The red colored items are studied by us and discussed here. The blue colored parameters were studied by C. Gaya 101 .

Figure 6 .

 6 Figure 6.2. (a) The Swagelok cell used for the experiments. (b) The illustration of the cathode location for two cathode sizes.

Figure 6 .

 6 Figure 6.3. (a) Discharge capacities obtained for two cathode sizes (diameters of 11 mm and 12.7 mm) with 4 repetitions of each galvanostatic discharge. The current density is 52.6 µA.cm -2 . (b) The dispersion observed for 0.5 and 1.0 molar LiTFSI concentrations in the electrolytes with 4 repetitions. The discharge current is 25 µA.

  this investigation. The dimensions of the input structures along the electrode thickness are 230 µm, 207 µm, 184 µm. The geometrical electrode surface area is 1 mm x 1 mm. The large electrode size is possible in this calculation since the pore sizes in GDL are in the micrometer range. A 1 M Li + concentration and a 4.43 mM O2 solubility in the electrolyte are used along with an escape factor of 0.95, corresponding to a predominant formation of large particles.

Figure 6 . 4 .

 64 Figure 6.4. The GDL electrode structure used in the calculations, with schematics of compression.

Figure 6 . 5 .

 65 Figure 6.5. Evolutions of the electrode properties as the cathode is compressed: (a) porosity; (b) carbon surface area; (c) pore size distribution; (d) discharge capacities.

Figure 6 . 6 .

 66 Figure 6.6. The calculated discharge capacities for four zones selected in a GDL electrode. Discharge current density is 25 µA.cm -2 .

Figure 6 . 7 .

 67 Figure 6.7. (a) Illustration of the 4 zones taken from the tomography data. (b) PSDs corresponding to the four slices. (c) Calculated discharge curves for the four slices at three different applied current densities. (d) Calculated discharge curves for the flipped electrodes (A: separator side, B: air inlet side).

Figure 6 . 8 .

 68 Figure 6.8. (a) Calculated evolution of PSDs and (b) calculated number of inactive pores along discharge for Zone 2 at 100 mA.g -1 , in normal and reversed configurations. Pores are considered inactive if they are i) clogged, ii) passivated by the Li2O2 deposit or iii) depleted in O2.

Figure 6 .

 6 Figure 6.9 shows snapshots of the oxygen concentration at two different depths of discharge

Figure 6 . 9 .

 69 Figure 6.9. Calculated oxygen concentration color maps at two different depths of discharge, (a) 55 mAh.g -1 and (b) 97 mAh.g -1 , for two zones at 20 mA.g -1 discharge current density. Throats (cylindrical pores) are removed for clarity.

Figure 6 .

 6 Figure 6.10. SEM image of PTFE/Super P positive electrode (Left: gas inlet side, Right: center of electrode). a) Before discharge. Discharged at current densities of b) 0.05 mA.cm -2 , c) 0.2 mA.cm -2 , and d) 1.0 mA.cm -2 . Image reproduced from reference 105 .

Figure 6 . 13 .

 613 Figure 6.13. Cyclic voltammetry experiments run under argon and oxygen for reference and magnetic cells (scan rate 5 mV.s -1 , potential range 1.5 V -4.5 V).

Figure 6 . 14 .

 614 Figure 6.14. Galvanostatic cycling tests for Li vs. Li symmetric cell with and without a magnet in the cell. Applied current is 25 µA, and the length of time it is applied is 2 hours.

Figure 6 . 15 .

 615 Figure 6.15. Galvanostatic discharge experiments using GDL electrodes in cells with and without a magnet.

Figure 6 .

 6 Figure 6.16. SEM images of GDL electrode before and after full discharge without a magnet. Discharge current is 25 µA and discharge capacity 0.986 mAh.

Figure 6 .

 6 Figure 6.17. (a)-(e) Galvanostatic discharge and (f) chronoamperometry experiments carried out for reference cells without a magnet and magnetic cells containing a magnet. Discharge current density, mass loading and applied voltage for the chronoamperometry test are given on the figure. A 0.1 M LiClO4 in DMSO electrolyte is used for (a) to (e); and a solution of 0.5 M LiTFSI in triglyme is used for (f).

Figure 6 . 18 .

 618 Figure 6.18. Discharge capacity ratios for magnetic and reference cells shown in Figure 17.

Figure 6 . 19 .

 619 Figure 6.19. Galvanostatic discharge profiles of cells with self-standing Super P cathode. Applied current is 200 µA.

  Par ailleurs, plusieurs défis sont à relever pour les batteries Li-O2: faibles stabilité, capacité effective, efficacité coulombique et cyclabilité.Durant cette thèse, nous nous sommes concentrés sur l'étude de l'influence du transport ionique dans les électrodes poreuses de batterie Li-O2. En particulier, le transport de l'oxygène joue un rôle crucial dans les performances de la batterie et en l'améliorant, nous pouvons en principe répondre à deux des quatre défis mentionnés précédemment, à savoir la capacité délivrée lors de la décharge et l'efficacité coulombique.Plusieurs techniques de modélisation sont utilisées dans cette thèse pour étudier divers aspects d'une batterie Li-O2. Un modèle cinétique de type Monte Carlo (kMC) est développé pour étudier et quantifier l'effet de la taille des pores sur la diffusion. La technique des déplacements quadratiques moyens (MSD) est utilisée pour extraire les valeurs de coefficient de diffusion à partir des trajectoires des molécules mobiles obtenues en kMC. Un modèle de type "réseau de pores" (pore network model, PNM) est développé pour étudier le rôle de la mésostructure des électrodes sur les performances des cellules électrochimiques. Ce modèle est constitué de : (1) un code facultatif qui génère des images tomographiques in silico si l'on n'a pas accès à ce genre de données (2) un script python pour convertir les images tomographiques en images binaires (3) un programme d'extraction d'un réseau de pores 3D à partir d'un tomogramme(4) le modèle de réseau de pores qui est utilisé pour simuler le transport et les réactions électrochimiques dans le réseau de pores extrait et pour déterminer les performances de la cellule. De plus, un programme de traitement d'images est développé pour étudier la formation de régions poreuses isolées. Par ailleurs, un modèle continu est modifié et utilisé pour calculer les capacités en décharge de plusieurs mésostructures d'électrodes afin de les comparer aux résultats expérimentaux publiés dans la littérature et aux résultats du modèle de réseau de pores.Dans la thèse, nous avons présenté la précision de la méthode d'extraction du réseau de pores à partir des tomogrammes et les résultats typiques fournis par l'approche PNM. Nous avons comparé les techniques de modélisation de type réseau de pores et de type modèle continu et en particulier leur capacité à capturer les effets de la taille des pores et des mécanismes de croissance du produit de décharge dans les batteries Li-O2. En effet, le produit de décharge peut être généré sous forme de film passivant ou sous forme de particules de différentes tailles. Pour le mécanisme de croissance de type film, les deux modèles montrent des résultats similaires et représentent correctement l'effet de la mésostructure des électrodes. Pour le mécanisme de type particule, le modèle continu ne permet pas de représenter correctement l'effet de la taille des pores. En revanche, les résultats du PNM montrent des capacités de décharge dépendantes de la taille des pores. Celui-ci est en accord avec les expériences publiées dans la littérature.[START_REF] Lu | Recent Research Progress on Non-Aqueous Lithium-Air Batteries from Argonne National Laboratory[END_REF] De plus, l'utilisation du modèle PNM pour différentes mésostructures avec trois scénarios de croissance de produit de décharge différents permet de suggèrer une interprétation aux tendances contradictoires rapportées précédemment dans la littérature. En particulier, les densités de courant et la nature du solvant peuvent affecter les mécanismes de croissance de Li2O2 et potentiellement conduire à différentes fins de décharge ("morts" de la batterie) pour une mésostructure donnée. En conséquence, en fonction du mécanisme de croissance de Li2O2, il a été suggéré que la capacité de décharge était corrélée à diverses propriétés des mésostructures d'électrodes (par exemple, la surface de carbone, le volume des pores, la taille des pores). Ces résultats montrent qu'il faut faire preuve de prudence lorsqu'on utilise un modèle continu pour simuler les performances électrochimiques des batteries Li-O2, en particulier lorsque le mécanisme de croissance sous forme de particules est à prendre en compte : la connectivité des pores en trois dimensions joue un rôle majeur dans la détermination des performances globales des cellules électrochimiques.Au cours de la thèse, nous avons aussi mesuré l'évolution de la tortuosité au cours de la décharge. Le coefficient de diffusion effectif (𝐷 𝑒𝑓𝑓 ) dans un milieu poreux est lié au coefficient de diffusion (𝐷 0 ) suivant𝐷 𝑒𝑓𝑓 =𝜀 𝜏 ⁄ 𝐷 0 eq. 5 où ε est la porosité et τ est la tortuosité. Connaissant 𝐷 0 et 𝐷 𝑒𝑓𝑓 , on peut calculer 𝜀 𝜏 ⁄ . La porosité et la tortuosité sont des propriétés structurales. En d'autres termes, elles dépendent largement du matériau poreux et de sa mésostructure, et peu des espèces mobiles, à condition que les espèces diffusant n'interagissent pas de manière spécifique avec la mésostructure de carbone. Suivant ces considérations, nous avons mesuré la diffusion uniquement pour le solvant tétraglyme en suivant le mouvement des noyaux 1 H par RMN (Résonance Magnétique Nucléaire). Plusieurs cellules électrochimiques avec deux solvants différents (DME et tétraglyme) ont été préparées puis déchargées à différents états de décharge. Après la décharge, les cellules sont démontées et les cathodes sont lavées au DME pur pour éliminer le sel, puis séchées et imbibées de tétraglyme pur. Les coefficients de diffusion sont détérminés pour toutes ces électrodes. La tendance générale observée pour les deux solvants est que le coefficient de diffusion effectif diminue à mesure que la cellule est déchargée. Cette tendance est due à la réduction de la porosité et à l'obstruction de certains pores. L'électrode devient alors plus tortueuse. Les coefficients de diffusion effectifs sont plus élevés pour les électrodes déchargées avec le solvant tétraglyme que pour celles déchargées avec le solvant DME. Le DME a un caractère donneur plus important que le tétraglyme. C'est pourquoi, en accord avec des résultats de la littérature, on s'attend à ce que les cellules déchargées avec le DME aient des particules de Li2O2 plus grosses que celles obtenues dans le tétraglyme. 42,96 La formation des plus grosses particules entraînerait une obstruction des pores plus importante et des tortuosités plus élevées, ce qui conduiraient à des valeurs de coefficient de diffusion effectif plus faibles. Dans cette thèse, nous abordons aussi le problème de la dispersion des mesures de capacités de décharge et proposons une nouvelle façon d'améliorer les performances des batteries Li-O2. Initialement, nous avons vérifié si certaines méthodes de préparation des cellules affectaient la dispersion des capacités de décharge. La première est la taille de la cathode et un éventuel défaut d'adaptation entre le piston et la cathode. Pour cette étude, nous avons utilisé des électrodes à base de GDL commercial, car leur épaisseur et leur charge sont assez uniformes. Un solvant triglyme avec 0,5 M de LiTFSI a été utilisé. Nous avons sélectionné deux tailles de cathodes de diamètre égal à 11 mm et 12,7 mm sur lesquelles nous avons effectué des décharges galvanostatiques. Les dispersions de capacités de décharge n'étaient pas significativement différentes pour ces deux tailles de cathodes. Cela montre que la taille et la position de l'électrode dans la cellule de Swagelok n'affectent pas la dispersion de manière significative. Nous avons également testé l'effet possible de la concentration en sel sur la dispersion des capacités de décharge. L'idée derrière ce test était l'existence possible d'un gradient de concentration en Li + le long de la cellule pendant la décharge. Ce gradient, s'il existe, aura potentiellement une influence sur les réactions électrochimiques et sera beaucoup plus on peut dire que l'hétérogénéité et les différences microscopiques locales entre les structures affectent la dispersion. Lorsque l'électrode présente des micro ou mésopores et une faible porosité, l'effet de l'interconnectivité et de l'hétérogénéité des pores sur la dispersion sera beaucoup plus important. D'autre part, si l'électrode a une porosité élevée et des macropores, l'impact sur la dispersion sera beaucoup plus petit. Enfin, pour améliorer la capacité de décharge, nous proposons une nouvelle approche. Dans les batteries Li-O2, l'origine bien connue de la 'mort prématurée' est l'obstruction des pores du côté de l'arrivée d'oxygène à l'électrode positive. Notre méthode implique l'introduction d'une force magnétique externe pour accélérer la diffusion de l'oxygène sans remplacer le matériau de la batterie ni apporter de modifications importantes à l'architecture de la batterie. En raison de leur nature paramagnétique, les molécules d'oxygène et de LiO2 seraient attirées vers un aimant, car elles subiraient une force magnétique appelée force de Kelvin. Nous montrons l'effet de la force magnétique avec trois électrodes différentes : de électrodes épaisses de type GDL, des électrodes de type Super P préparées par la méthode "dip-coating" et des électrodes épaisses de type Super P. Dans tous les cas, notre étude expérimentale a montré que les cellules avec aimant ont fourni des valeurs de capacité de décharge plus élevées que celles des cellules non magnétiques. Cependant, les résultats sont plus marqués avec les électrodes de types Super P pour lesquels les micro et méso pores sont plus affectés par l'amélioration du transport que les électrodes macroporeuses de type GDL. in the interval (0,1] 𝛾 𝑡𝑜𝑡 Sum of all event frequencies 𝑁 Total number of possible events 𝑑 𝑖,𝑡 The position of molecule 𝑖 at time 𝑡 𝑑 𝑖,𝑡 0 The position of molecule 𝑖 at time 𝑡 = 0 𝐷 Diffusion coefficient 𝑡 Time 𝑙 𝑖 𝑡 The distances between the center of a parent pore and the bottleneck pore 𝛼′ Pore-throat segmentation coefficient 𝑙 𝑡 Length of a throat or cylindrical pore 𝑙 𝑖 Radius of pore 𝑖 𝑟 𝑡 Radius of a throat 𝑙 𝑖𝑗 Distance between pore centers of 𝑖 and 𝑗 𝑟 𝑖 Radius of a pore in pore family 𝑖 𝑐 𝑖,𝑥 Concentration of 𝑥 in pore 𝑖 𝑘 𝑖𝑗,𝑥 Transfer parameter of species 𝑥 between pore 𝑖 and 𝑗

  Meini et al. compared discharge capacities of several carbon

	black type electrodes (Timcal Super C65, Vulcan XC72, Ketjenblack EC600JD, Black Pearls
	2000) and obtained larger discharge capacities for the electrodes with larger carbon surface
	areas (CSA). Their results are reported in Table

Table Discharge

 Discharge

capacities and BET surface areas (the surface area obtained by Brunauer-Emmett-Teller method) of the carbon materials reported by

Meini et al.

  10, written in terms of the fluxes between neighboring pores 𝑥 is the concentration of mobile species 𝑥 (Li + or O2) in the pore 𝑖 and 𝑐 𝑗,𝑥 is the concentration in the pores connected to 𝑖. 𝑘 𝑖𝑗,𝑥 is the transfer parameter between pores 𝑖 and 𝑗, 𝑡 𝑓,𝑖 and 𝑡 𝑝,𝑖 are the thicknesses of the Li2O2 thin film and particle formed in pore 𝑖 considered 𝑘 𝑏 are the forward and backward reaction rate constants; 𝑎 𝐿𝑖 + , 𝑎 𝑂 2 and 𝑎 𝐿𝑖 2 𝑂 2 are the activities of 𝐿𝑖 + , 𝑂 2 and 𝐿𝑖 2 𝑂 2 respectively. The cell potential 𝑈 is calculated by equating the sum of all currents from all the pores in the network to the input current density or discharge current (𝐼 𝑡𝑜𝑡 )

	The cross-sectional areas are calculated by: where 𝑘 𝑓 ,
		𝐴 𝑖 = 𝜋 [(𝑙 𝑖 -𝑡 𝑓,𝑖 ) 2 -𝑡 𝑝,𝑖	2 ]	eq. 3.15
		𝐴 𝑗 = 𝜋 [(𝑙 𝑗 -𝑡 𝑓,𝑗 ) 2 -𝑡 𝑝,𝑗	2 ]	eq. 3.16
	𝑑𝑐 𝑖,𝑥 𝑑𝑡	𝑛𝑒𝑖𝑔ℎ𝑏𝑜𝑟𝑠 = ∑ 𝑘 𝑖𝑗,𝑥 (𝑐 𝑗,𝑥 -𝑐 𝑖,𝑥 ) 𝑗 𝐴 𝑡 = 𝜋 [(𝑟 𝑡 -𝑡 𝑓,𝑡 ) 2 -𝑡 𝑝,𝑡 2 ]	+ 𝑠 𝑖,𝑥	eq. 3.10 eq. 3.17
	𝑂 2 + 𝑒 -→ 𝑂 2 -𝐿𝑖 + + 𝑂 2 -→ 𝐿𝑖𝑂 2 where 𝑐 𝑖,𝑘 𝑖𝑗,𝑥 = ( 𝐿𝑖 + + 𝑒 -+ 𝐿𝑖𝑂 2 → 𝐿𝑖 2 𝑂 2 𝑘 𝑖,𝑥 1 2𝐿𝑖𝑂 2 → 𝐿𝑖 2 𝑂 2 + 𝑂 2	+	1 𝑘 𝑗,𝑥	+	1 𝑘 𝑡,𝑥	)	Reduction of oxygen Formation of lithium superoxide -1 eq. 3.11 Formation of lithium peroxide Disproportionation reaction
	𝑙 𝑖 𝐴 𝑖 𝐷 0,𝑥 In this work, for simplicity reasons, an overall reaction is considered 𝑘 𝑖,𝑥 =	eq. 3.12
	𝐴 𝑗 𝐷 0,𝑥 2𝐿𝑖 + + 𝑂 2 + 2𝑒 -→ 𝐿𝑖 2 𝑂 2 𝑘 𝑗,𝑥 = 𝑙 𝑗 The associated electrochemical reaction rate is formulated as	eq. 3.18 eq. 3.13
	𝐴 𝑡 𝐷 0,𝑥 𝑙 𝑡 -𝛽𝑛𝐹(𝑈 -𝑈 0 ) 𝑘 𝑡,𝑥 = 2 𝑎 𝑂 2 𝑒𝑥𝑝 ( 𝜗 = 𝑘 𝑓 𝑎 𝐿𝑖 + 𝑅𝑇	)	eq. 3.14 eq. 3.19
		-𝑘 𝑏 𝑎 𝐿𝑖 2 𝑂 2 𝑒𝑥𝑝 (	(1 -𝛽)𝑛𝐹(𝑈 -𝑈 0 ) 𝑅𝑇	)

and 𝑠 𝑖,𝑥 is the sink term for species 𝑥 which is calculated assuming a set of electrochemical reactions taking place in pore 𝑖. 𝑘 𝑖𝑗,𝑥 is a function of the bulk diffusion coefficient of 𝑥 (𝐷 0,𝑥 ), the distance between the centers of pores 𝑖 and 𝑗, and the cross-sectional areas of pores 𝑖, 𝑗 and the connecting throat, which are 𝐴 𝑖 , 𝐴 𝑗 and 𝐴 𝑡 respectively, and can be formulated as (Figure

3

.5.f).

The electrochemical discharge reaction generally does not happen in a single step but can involve several intermediate steps such as the formation of superoxides (LiO2)

[START_REF] Yin | Linking the Performances of Li-O2Batteries to Discharge Rate and Electrode and Electrolyte Properties through the Nucleation Mechanism of Li2O2[END_REF][START_REF] Belova | Mechanism of Oxygen Reduction in Aprotic Li-Air Batteries: The Role of Carbon Electrode Surface Structure[END_REF]

The intermediate reactions that are involved in the overall reaction shown in eq. 3.18 are

Table 4 .

 4 List of parameters used for PNM and CM models

	Parameters	Values	Units	Sources
		Physical constants
	Gas constant	8.31	J.K -1 .mol -1	universal value
	Faraday constant	96485	C.mol -1	universal value
		Cell parameters
	Cathode thickness (Z direction)
		510 -6	m	Defined
	electrode thickness
	Li + initial concentration	110 3	mol.m -3	Assumed
	O2 initial concentration	4.43	mol.m -3	Ref 82
	Li2O2 molar volume	1.9810 -5	m 3 .mol -1	Ref 83
	O2 diffusion coefficient	2.1710 -9	m².s -1	Ref 82
	Temperature	298	K	Defined
	Electrochemical parameters
	Reaction equilibrium potential	2.96	V	Ref 84
	Charge transfer coefficient	0.5	/	Assumed
		Kinetic parameters
	Forward reaction rate constant	110 -11	PNM: mol.s -1 .m -²	Fitted
			CM: m. s -1
	Backward reaction rate constant	110 -11	PNM: mol.s -1 .m -² CM: m. s -1	Fitted
	Tunneling distance	10	nm

Table 5 .

 5 Porosities and specific surface areas for the four zones of the GDL electrode.

	Zone number	Porosity	Surface area
	1	0.797	5.76 x 10 4 m 2 /m 3
	2	0.786	5.81 x 10 4 m 2 /m 3
	3	0.801	5.84 x 10 4 m 2 /m 3
	4	0.794	5.92 x 10 4 m 2 /m 3

Table 6 .

 6 Porosities and specific surface areas for the four zones

	Zone number	Porosity	Surface area
	1	0.356	7.97 x 10 6 m 2 /m 3
	2	0.352	7.68 x 10 6 m 2 /m 3
	3	0.382	7.67 x 10 6 m 2 /m 3
	4	0.363	7.27 x 10 6 m 2 /m 3

  température moyenne de la Terre augmente progressivement, comme l'ont confirmé certaines organisations scientifiques. Ceci est principalement dû aux émissions de gaz à effet de serre dans l'atmosphère résultant de notre consommation de combustibles fossiles. En effet, les combustibles fossiles sont utilisés pour la quasi-totalité de nos besoins en énergie, y compris les transports, les applications industrielles et le chauffage et la climatisation domestiques. Afin de réduire notre impact sur l'environnement, il est urgent de réduire notre consommation de combustibles fossiles. Pour faire face à ce défi sociétal, le développement et l'utilisation des Toutes ces approches ont leurs avantages et leurs inconvénients et doivent donc être utilisées dans différentes applications. Par exemple, les applications stationnaire ou mobile ont des critères différents. Nous nous intéresserons ici aux systèmes de stockage d'énergie électrochimiques potentiellement adaptés aux applications de transport, et en particulier aux batteries lithium-oxygène (Li-O2).

	En raison de cette nature intermittente des énergies renouvelables, celles-ci ne peuvent remplacer directement les combustibles fossiles et nous devons développer des systèmes de stockage d'énergie performants afin de généraliser leur utilisation. Les dispositifs de stockage d'énergie doivent stocker l'excès d'énergie lorsqu'il qu'il y a du soleil et du vent, et l'utiliser lorsque c'est nécessaire. Il y a plusieurs méthodes pour stocker l'énergie telles que l'hydroélectricité pompée, les volants d'inertie, le stockage d'hydrogène et le stockage migrent vers la cathode où ils réagissent avec l'oxygène pour former des produits de décharge solides, principalement Li2O2 (eq. 2-4). Pendant la charge, Li2O2 se décompose et forme les espèces Li + et O2. Les ions lithium retournent à l'anode où ils sont réduits et les molécules d'oxygène sont relâchées. 𝐿𝑖 ↔ 𝐿𝑖 + + 𝑒 -eq. 1 𝐿𝑖 + + 𝑒 -+ 𝑂 2 ↔ 𝐿𝑖𝑂 2 eq. 2 2𝐿𝑖𝑂 2 ↔ 𝐿𝑖 2 𝑂 2 + 𝑂 2 eq. 3 électrochimique. Lors de la décharge d'une batterie Li-O2, le lithium s'oxyde à l'anode (eq. 1) et les ions lithium 𝐿𝑖 + + 𝑒 -+ 𝐿𝑖𝑂 2 ↔ 𝐿𝑖 2 𝑂 2 eq. 4

énergies renouvelables ont pris une énorme ampleur ces dernières décennies, notamment avec les énergies solaire et éolienne. Cependant, contrairement aux combustibles fossiles, ces sources d'énergie renouvelables sont intermittentes, c'est-à-dire qu'elles ne sont pas toujours présentes et l'intensité de celles-ci fluctue au cours de la journée et d'une saison à l'autre. Il y a du soleil pendant la journée mais son intensité diminue par temps nuageux ou s'estompe pendant la nuit.

prononcé lorsque la concentration en sel est faible. Pour cet essai, deux concentrations en LiTFSI en solution dans le triglyme ont été sélectionnées : 0,5 M et 1,0 M. Sur la base des capacités de décharge, il semble que la concentration en sel n'affecte pas la dispersion. La capacité de décharge moyenne augmente quand on augmente la concentration en sel, mais la dispersion des capacités reste la même.Le fait d'exercer une certaine pression au moment de l'assemblage de la cellule Swagelok pourrait être une autre source de dispersion. En effet, nous exerçons une pression avec les mains sur les deux pistons pour fermer et resserrer la cellule. Ce processus étant effectué manuellement, la force de compression peut varier d'une cellule à l'autre. Ainsi, la compression de la cathode peut varier. Comme il est difficile d'évaluer la force appliquée aux électrodes lors du serrage de la cellule, nous avons étudié la contribution de la compression variable de la cathode à la dispersion par une approche de modélisation. Une structure d'électrode de type GDL a été générée à l'aide du logiciel GeoDict. Cette structure in silico est ensuite comprimée dans le sens de l'épaisseur de la cathode pour obtenir 3 structures de cathode avec des compressions différentes (0%, 10% et 20%). Nous avons appliqué notre approche PNM à ces trois structures pour étudier l'effet de la compression sur la capacité de décharge. Les capacités de décharge calculées montrent que la compression de la cathode a un impact sur la capacité de décharge. En fait, la capacité de décharge diminue quand la cathode est comprimée. Une compression de 20% peut entraîner une modification de la capacité de décharge d'environ 15%.Sur la base de ces calculs, nous concluons que la variation de compression de la cellule peut être une source de dispersion des capacités de décharge.Nous nous tournons ensuite vers l'étude d'une autre source possible de dispersion, à savoir l'hétérogénéité intrinsèque de la mésostructure des cathodes, et examinons son effet sur la dispersion des résultats électrochimiques. D'après le calcul sur les électrodes Super P et GDL,

Acknowledgements

First and foremost, I would like to thank my supervisors Alejandro A. Franco, Clare P. Grey, Céline Merlet and Pieter C. M. M. Magusin for arranging this collaborative PhD position, and

Chapter 3. Developed programs and models

Several modelling techniques are used in this thesis to study various aspects of a Li-O2 battery.

A kinetic Monte Carlo (kMC) model is developed to investigate and quantify the effect of pore size on diffusion; the mean square displacement (MSD) technique is used to extract diffusion coefficients from the trajectories of mobile molecules in kMC; and a pore network model package is created to study the role of the electrode mesostructure on a cell performance. This package includes: (1) an optional tomographic image generation code if the actual experimental tomographic structure is missing, we call these computer generated structures as tomography like images to distinguish them from the actual tomographic images; (2) a python script to convert tomographic images into binary format files; (3) a pore network extraction program to create a 3D pore network based on a given tomographic image; (4) the actual pore network model which simulates transport and reactions in the input pore network and determines the cell performance. We also developed an image processing program to study the formation of isolated porous regions. Finally, a continuum model is modified and used to calculate discharge capacities of several electrode mesostructures in order to compare them with experimental reports from literature and pore network model results. In this chapter we will detail all of these programs and models developed throughout the PhD.

Kinetic Monte Carlo model

To study the effect of the pore size on diffusion, and particularly the importance of the Knudsen diffusion effect in the systems we explored, we developed a kinetic Monte Carlo model. The Knudsen effect describes the diffusion in narrow pores where the diffusion coefficient is affected by collisions of the mobile species with the pore wall. Intuitively, smaller pores will

Pore network model package

To investigate the effect of the electrode mesostructure and pore interconnectivity on the electrochemical performances of carbon electrodes, we developed a pore network model (PNM) which enables us to calculate transport properties and simulate electrochemical reactions in three dimensional (3D) porous structures. 3D structures of cathodes can be obtained either by tomographic imaging or by generating them in silico which allows us to study new designs and compare performances of various mesostructures.

Figure 3.4. Workflow of our PNM approach

The calculation of model discharge curves is made following several steps as shown in Figure 3.4. The initial step is synthesizing the electrode and obtaining its 3D image, or alternatively generating it in silico. For studying the sensitivity of the discharge capacity of a cell on the cathode mesostructure and pore interconnectivity, we used an actual tomographic image of a Super P electrode. In this case, the 3D structure of a Super P type carbon is obtained from TXM for this part were done based on a computer generated structure with an average pore size of 60 nm. The pore size distribution is given in Figure 4.4.a. For the following calculations, both the formation of thin film and large particles are considered. This is achieved by using an escape factor of 0.5.

When starting to use our newly developed PNM program, we need to run a number of tests to check that we are using suitable input parameters. We for example check what are the appropriate time-step and system size.

For the time-step determination, we calculate the full discharge capacities with various timesteps. It is expected that the calculated discharge capacities might be inaccurate when the timestep is too large. Indeed, with too large a time-step, the electrochemical reactions and diffusion processes cannot follow realistic timescales. The calculation results corresponding to this study are shown in Figure 4.4.b. From this plot, we can see clearly that the capacity value converges when the time-step is small enough. More specifically, for time-steps below 2 10 -4 seconds, the discharge capacity does not change, so for the following calculations we select a time-step of 1 10 -4 s.

Pore size effect on diffusion: kMC model

In this section, we investigate the effect of the pore size on diffusion. For this, we use the kMC model described in Chapter 3. The bulk diffusion coefficient given as an input is 5 10 -9 m 2 .s - 1 , which is within the typical range of oxygen diffusion in Li-O2 battery electrolytes. [START_REF] Laoire | Influence of Nonaqueous Solvents on the Electrochemistry of Oxygen in the Rechargeable Lithium-Air Battery[END_REF] We tested 9 different pore sizes, corresponding to cylindrical pores with diameters of 1 nm, 2 nm, 3 nm, 4 nm, 5 nm, 10 nm, 25 nm, 50 nm and 100 nm. If the cylinder length was kept the same for all pore sizes, we would have a very significant variation in the number of oxygen molecules in the pores as cylinders with large diameters would contain many more molecules than cylinders with small diameters. This would affect the fitting accuracy of the MSD which we would like to avoid. To have the same amount of oxygen molecules in all the calculations we keep the porous volume constant, and equal to 783,398 nm 3 (i.e. 6,267,184 voxels), which corresponds to the pore volume of a cylinder with a diameter of 100 nm and a length of 100 nm. This means that cylinders with smaller pore diameters will have larger pore lengths. We set the amount of oxygen in each calculation to 1256 molecules. The total simulation time is 2 µs.

The obtained MSD curves are shown in Figure 5.9. The MSD values increase non-linearly with time at the beginning of simulation but become linear after a certain time (above 1 µs). This is particularly visible for 100 nm and 50 nm pore sizes. This is the typical trend for MSD in a porous structure. The short range motion is nonlinear but after some time, the linear regime (or long range diffusive regime) is reached. The pore size distribution, porosity and carbon surface area evolutions as the electrode is compressed are shown in Figure 6.5. With the compression, the porosity of the electrode decreases, its pore size distribution changes, the pores get smaller on average, and the available surface area decreases as a consequence of fibers making more contacts with each other. We might expect an increase in the electronic conduction when more fibers make contact with each other but this effect is not included in our model since it is assumed that the electronic conduction is much faster than the ionic conduction and as such is not a limiting factor. substantial O2 concentration gradient along the electrode thickness thus, an inhomogeneous Li2O2 distribution is obtained.

Avoiding pore clogging -Literature

Several solutions to the problems mentioned hereinabove have been discussed in the literature.

An electrode structure with large pores is the most commonly suggested solution. Simply put, if an electrode has very large pores, it will not be blocked by discharge products easily. Chervin et al. 106 have showed a pore volume effect on the discharge capacity in their study using carbon

Experimental results with and without a magnet

Stability tests under argon

The permanent magnet added is in contact with the Li metal anode and the electrolyte. To ensure no new side reactions appear due to the magnet, we carry out CV tests under argon. We use GDL as the cathode and a 0.5 M solution of LiTFSI in triglyme as the electrolyte. A 5 mV/s scan rate and a potential range of 1.5 V to 4.5 V are used. Two cells are run under argon gas: a reference cell and a magnetic cell. Another reference cell is run under oxygen. When there is no oxygen present in the glass chamber, we should not observe any current unless there is a parasitic reaction. Figure 6.13 shows these results. The current response from both cells under

Tests on dip-coated Super P electrodes

We now move from GDL electrodes to Super P electrodes which have much smaller pores sizes than GDLs, in the 10-100 nm range. The transport limitations are expected to be much more pronounced here. The experiments are carried out with dip-coated Super P electrodes (90% Super P and 10% PVdF, weight percentages). Dip-coating is an easy method to prepare electrodes and that is why we start with this process to test the magnetic field effect in Super P based systems. We used 0.1 M LiClO4 in DMSO as an electrolyte. For these experiments, we run 5 pairs of tests. Each pair of tests consists in two electrochemical setups kept in the same glass chamber so that they experience the same environmental conditions such as the same temperature and oxygen pressure. The discharge profiles are shown in Figure 6.17. We note that controlling the mass loading is quite difficult with the dip-coating approach. Therefore, we make about 20 electrodes in a batch and select pairs whose mass loadings are very close to each other for comparison. The test on Figure 6.17.d is very likely to have some contact or preparation problem. Indeed, the open circuit voltages for both cells were much higher than the normal range we see for this cell type, which is around 2.9-3.2 V. As such we think this test is invalid, all of the other comparisons show that the magnetic cells performed better than the reference cells.

PVdF binder is commonly used with DMSO as a solvent in Li-O2 batteries, 114,115 PVdF is reported to dissolve in DMSO. 116 The stability should not be a big problem here, since the tests are done on a single discharge, but the dispersion is a big issue.

Tests on self-standing Super P electrodes

After testing dip-coated Super P electrodes, which have mass loading control problem, and GDL electrodes, which have only macropores and in which the end of discharge is not limited by oxygen transport, we decided to make self-standing Super P electrodes with PTFE binder (70-30 wt%). The main features of this type of electrodes are listed below.

• These are thick electrodes (300 µm thickness).

• These are mesoporous with pore sizes in the 10-100 nm range.

• In these electrodes, due to the electrode thickness and pore sizes, we expect a dominant pore clogging and critical transport limitations.

The improvement of transport should be much clearer with these electrodes. We run galvanostatic discharge experiments using a 0.5 M LiTFSI in triglyme electrolyte, on several magnetic and reference cells (Figure 6.19). The three cells, two magnetic and one reference cells, with discharge capacities below 0.5 mAh are most likely failed experiments. For the tests run properly, the magnetic cells performed 2-3 times better than the reference cases.

Conclusions

In this chapter we studied several parameters that could affect the discharge capacity variations.

Electrode size and salt concentrations does not have any noticeable effect on the observed discharge capacity dispersion. However, we show using the PNM model that the variation in electrode compression during cell assembly can affect the dispersion. The micro-heterogeneity, local variation of pores and their interconnectivities, contribute to the dispersion. This effect is less significant when electrodes with large pores and high porosities are used, but more visible when electrodes with mesopores and lower porosities are used.

We also propose a new approach to enhance the oxygen transport in Li-O2 batteries and thus improve discharge capacities. We demonstrated that this effect is small when an electrode showing no or low amounts of pore clogging is used, such as GDLs. When using Super P based electrodes, which have mesopores and are more prone to pore clogging, the discharge capacities of the magnetic cells increase very significantly. Based on our experimental results, we see a clear positive effect of a magnetic field gradient in the cell performance but we also acknowledge the big dispersion in experimental results. Hence, even though the effect of magnetic field gradient is accepted and used, 110,117-120 it would be a good confirmation if some other researchers can reproduce this effect on Li-O2 batteries. Finally, in the very close future, we plan to complete a mathematical model we started to develop, which will be able to describe the O2 transport enhancement with the magnetic field gradients, and explaining its interplay with diffusion and electrochemical processes.

Chapter 7. General Conclusions

In this PhD project, we studied Li-O2 batteries and in particular the transport of dissolved oxygen molecules in porous electrodes. We investigated the interplay between the electrode mesostructure, the discharge capacity and the discharge products sizes. We used a combination of modeling and experimental methods to accomplish these goals.

We We developed a pore network model where explicit 3D electrode structures with pores and connections between them are included. We showed in Chapter 4 that the pore network extraction from tomographic images is accurate. The pore size distributions, surface areas and porosity values of the extracted pore networks are in good agreement with the ones of the initial structures. Then we showed a comparison between results from the PNM and a continuum model for four electrode structures with different average pore sizes. These comparisons demonstrated the better accuracy of the PNM results compared to the CM results and showed the importance of incorporating explicit porous structures in modeling works. The PNM results allowed us to explain various trends reported in the literature on the correlation between electrode properties and the discharge capacity. The reported surface area, pore size dependence of the discharge capacity and the existence of an optimum pore size to achieve high discharge capacities can be explained by including discharge product sizes and a 3D electrode structure in models. Depending on the pore sizes present in the electrode and the size of the discharge particles, the number of clogged pores does vary and affects the transport properties in the cell and consequently the discharge capacity. The PNM model can be used in the future for designing and systematically studying high performance electrode structures. It can also be adapted to other types of battery applications, particularly lithium-sulfur batteries, in which the electrode porous mesostructure changes significantly during cycling.

In chapter 5, we measured the evolution of the tortuosity along discharge for self-standing Super P electrodes. To do that, we use PFG-NMR measurements and track the effective selfdiffusion of 1 H atoms of tetraglyme molecules in porous electrodes discharged at different states. Two solvents were used for discharging the cells: tetraglyme and DME. The measurements showed that the tortuosity evolves much faster for the electrodes discharged with DME compared to those discharged with tetraglyme. We explain this difference by the different discharge product sizes. DME has a higher donor number than tetraglyme, thus the size of the Li2O2 particles are expected to be larger for DME. We confirm this by calculating the number of clogged pores using the PNM and a computer generated structure with an average pore size of 60 nm. The PNM results show that a larger proportion of particle formation (as opposed to thin film formation) leads to more clogging, even at the same depth of discharge.

In Chapter 5, we also investigated the pore size effect on diffusion coefficients using a kinetic Monte Carlo model. We simulated transport in cylindrical pores with pore diameters of 1 nm, 2 nm, 3 nm, 4 nm, 5 nm, 10 nm, 25 nm, 50 nm and 100 nm. The calculation results showed no significant pore size effect for pores with diameters larger than 25 nm. The pore size effect starts to be noticeable for pores below 10 nm diameters; and the effective diffusion coefficients follow a sharp decay for pores below 4 nm. The effective diffusion coefficient decreases by 5.5 micro-heterogeneity on dispersion. Indeed, the pores are more prone to clogging in the Super P electrode.

Finally, we proposed a novel approach to improve the oxygen transport in Li-O2 batteries. The idea is to apply a magnetic field gradient along the electrode thickness to facilitate the transport of the paramagnetic oxygen from low magnetic field regions to high magnetic field regions.

Such an idea has been already used in polymer electrolyte membrane fuel cells for similar purposes. First, we carried out some cyclic voltammetry and galvanostatic tests under argon, to see the stability of the cell with a permanent magnet inside. The experimental results confirm the stability of the system with a magnet. Then we carried out galvanostatic discharge experiments for three different electrode structures: GDL self-standing electrodes, Super P dipcoated electrodes and Super P self-standing electrodes. We compare the discharge capacities obtained with the cells with a magnet inside (magnetic cells) to the cells without a magnet (references cells). The discharge capacities measured for GDL electrodes showed an improvement of about 30% for the cells with a magnet. Nevertheless, this improvement is hard to distinguish from the dispersion of the discharge capacities. Therefore, we moved towards Super P electrodes which have much smaller pore sizes. The discharge experiments of the initially prepared dip-coated Super P electrodes showed a much larger improvement than the GDL electrodes. However, mass loading control and thickness uniformity of the electrode are hard to achieve using the dip-coating approach leading to huge variations in discharge capacities. Finally, we prepared self-standing Super P electrodes with a 300 µm thickness. This allowed us to better control the mass loading and the electrode thickness uniformity. While three of the eight (four magnetic and four reference cells) assembled cells failed, the five cells which worked fine showed a clear improvement of the discharge capacity for the cells with a magnet. This approach of improving transport is promising but more experimental characterizations and modeling studies to understand in detail the effects of a magnetic field