Reusable Theory for the In Silico optimization of composite electrode fabrication processes for rechargeable battery

Introduction | 9

In this chapter, the operation principles of a lithium-ion battery (LIB) will be detailed as well as the industrial context and the role the ARTISTIC project 1 plays in this highly dynamic field. Then the scope of my work within the projectand the structure of the manuscript will be presented.

i.1. LIBs in the Environmental Context

The rarefication of fossil fuels calls for an improvement of our renewable sources production, which are currently not able to satisfy the increasing electricity demand. Indeed, for the past three years, the International Energy Agency has been reporting a growth in the generation of electricity from renewable sources (Figure i.1.a). 2 It is noteworthy that due to the global pandemic situation, a slowing down of the power capacity addition is expected because of lockdowns and the collapse of the economy. In the years to come intermittent power sources such as wind turbines or solar panels will have an increasing place in the energy mix. However, relying that much on these technologies at a large scale will enhance even more the issues arising from the intermittence of electricity production. As seen in Figure i.1.b, the percentage of wind and solar electricity generation (which are the main renewable sources) used to answer the demand in France varies drastically from less than 5 % to almost 40 % during the first lockdown in March. Even though the demand can vary significantly too at the scale of a day, around 35 % according to the US Energy Information, the intermittence of the production could not match the need at each moment. 3 Currently, thermic power stations and dams are used to adjust the production of electricity from nuclear plants. The lack of flexibility of intermittent sources is the main obstacle to their development. Stationary electric storage is the solution investigated to overcome this flaw. The principle is simple, storing the energy when the supply exceeds the demand and delivering it back when the demand exceeds the supply. Also, more complex grid management solutions are being developed at a theoretical level and might be relevant in the years to come. 4,5 To this end, electrochemical storage is a promising candidate because it offers a wide panel of applications depending on the chemistry of the battery. Conventionally, a lithium-ion battery with liquid electrolyte is as schematized in Figure i.2 with two electrodes, one positive, and one negative. Inside the two electrodes, the active material (AM) in blue will store the Li, the carbon and binder domains (CBD) in orange will respectively guarantee the electronic percolation and mechanical stability of the cell. A liquid ensures the transport of ions between the two electrodes, i.e. the electrolyte in green, a separator prevents any shortcut between the two, and current collectors are at each electrode to gather the electrons, allowing them to enter the external circuit. During a discharge, at the negative electrode, the inserted Li is extracted from the AM and flows as a Li + ion to the positive electrode through the separator. In the meantime, resulting from the electrochemical reaction electrons are flowing from the negative electrode to the positive electrode, creating a current in the opposite direction. Once the Li + ions have reached the positive electrode, they will react with an electron at the AM/Electrolyte surface. During the discharge, the negative electrode is providing electrons to the system, hence its name the anode, and the positive electrode is the cathode. During the charge, the exact opposite phenomena occur, this time the positive electrode is the anode, and the negative one is the cathode.

i.2. ARTISTIC Project in the Industrial Context

In recent years, the Asian battery market has been on the rise thanks to major companies (see Figure i.3) and solid academic works. Indeed, in 2015, Japan, Korea, and China accounted for 88 % of the lithium-ion battery production worldwide. 6 In the meantime, the American industry has launched a program of gigafactories through the firms Panasonic and TESLA to tackle the price of manufacturing batteries. 7 This endeavor has led TESLA to be one of the main cell manufacturers in the world and the USA to catch up with its Asian competitors in 2018. Europe has still to find its place amid this extremely competitive environment. To overcome its drop behind, an agreement has been found in 2019 between France and Germany to co-found a 5-years project as much as 1 billion euros for Germany and 700 million for France, to develop battery production in Europe leading to a so-called "Airbus for batteries". 9 Also, numerous initiatives have seen the light of day with tremendous funding. Industrially wise, there is the Gigafactory Northvolt project in Sweden with industrial investors and government grants to produce up to 32 GWh/yr for 2023/2024.

Introduction | 12

To provide a sense of scale, the first Gigafactory built in Nevada accounted in 2018 for 35 GWh/yr. In January 2019, the EU Research and Innovation Programme Horizon 2020 called for battery-related projects in 7 different topics with 114 million euros as a budget. [START_REF]Horizon 2020: new Next-Generation Batteries call published. Innovation and Networks Executive Agency[END_REF] Introduction | 13 for academic, industrial, or student people. Given the parameters of manufacturing of the battery, the platform provides the electrochemical performance that should be expected for such fabrication processes. It will also embed a reverse-engineering tool based on machine learning, which will find fabrication parameters that will fit the performances requested by the user. Such a tool allows to sharply cut down the number of experiments necessary to tailor a fabrication process by predicting the impact of the parameters at the mesoscale and at the battery scale. Putting an end to the trial and error methodology when optimizing a process is one of the many challenges shared by the academic and industrial worlds. This way of doing, while being used for more than a hundred years [START_REF] Thorndike | Animal Intelligence: An Experimental Study of the Associative Processes in Animals[END_REF] is still active in the industry and consumes many resources, namely time and funds. Besides, emerging ethical and geopolitical issues for the supply of certain chemical compounds raise even more doubts on this technique. The ARTISTIC tool claims in no way to outsmart the experiments, but rather to give insights on which direction to go to drastically shorten the optimization process or the development of new chemistries. The ARTISTIC project is a pioneer work to give the possibility to the community to link the manufacturing parameters to the output performance of a battery cell in a time-efficient and reliable manner for present and future chemistries. This chapter will introduce the two different AM used in this Ph.D. as well as all the existing approaches to do battery modeling at the mesoscale. The strengths and weaknesses of each method will be discussed, making clear why we went toward 3-D modeling.

Then the different ways to acquire 3-D structures and the challenges of 3-D models will be presented. A focus on the 3-D modeling state-of-the-art will follow through a review of noteworthy or representative articles from literature in addition to a short description of the electrochemical model widely used in the battery community.

Finally, the personal approach undergone in this Ph.D. to tackle the previously identified obstacles will be detailed.

I.1. Active Materials

In this section, the AM used in the computational studies will be presented along with their main advantages and drawbacks.

I.1.1. LiNi1-x-y Mnx Coy O2 (NMC)

The LiNi1-x-y Mnx Coy O2 (NMC) is widely used as a positive active material since it has been first reported in 1999. [START_REF] Liu | Synthesis and characterization of LiNi1-x-yCoxMnyO2 as the cathode materials of secondary lithium batteries[END_REF] NMC was synthesized as an alternative to LiCoO2 (LCO), which suffers from a poor specific capacity (140 mAh.g -1 ) due to instabilities when overcharged, [START_REF] Liu | Understanding electrochemical potentials of cathode materials in rechargeable batteries[END_REF] by co-substituting the Ni and Mn at the Co sites. [START_REF] Zhang | Recent progress in advanced electrode materials, separators and electrolytes for lithium batteries[END_REF] As seen in State-of-the-art | 18

Compared to LCO, NMC exhibits higher voltage (3.65 V), [START_REF] Zhang | Recent progress in advanced electrode materials, separators and electrolytes for lithium batteries[END_REF] , and higher thermal stability at a lower cost. Indeed, it has been identified that Mn favors higher thermal stability and that the increasing amount of Ni improves the specific capacity while Co ensures better capacity retention (see Figure I.2.aFigure ). [START_REF] Li | Synthesis and characterization of LiNi0.6Mn0.4-xCoxO2 as cathode materials for Li-ion batteries[END_REF][START_REF] Noh | Comparison of the structural and electrochemical properties of layered Li[END_REF] It is possible to interchange these elements because all these metals share the same sites in the structure as Among the first NMC reported in 1999 were the LiNi0.6 Mn0.2 Co0.2 O2 (NMC-622) and the LiNi0.5 Mn0.2 Co0.3 O2 (NMC-523). [START_REF] Liu | Synthesis and characterization of LiNi1-x-yCoxMnyO2 as the cathode materials of secondary lithium batteries[END_REF] Despite the previously mentioned improvements, the NMC did not receive much interest until the introduction of the LiNi0. 33 Mn0.33 Co0.33 O2 (NMC-111) in 2001 by Ohzuku et al.. [START_REF] Ohzuku | Layered Lithium Insertion Material of LiCo1/3Ni1/3Mn1/3O2 for Lithium-Ion Batteries[END_REF] Its main feature is its high safety while having a higher specific capacity than LCO (160 mAh.g -1 versus 140 mAh.g -1 ). For NMC-111, the Ni and Co will both react and be reduced during discharge and oxidized during charge. In Li1-x Ni0.33 Mn0.33 Co03.3 O2, in the range 0 < x < 2/3 the Ni will be at the oxidation states Ni 2+ /Ni 4+, and for 2/3 < x < 1 the Co 3+ /Co 4+ couple will be activated.

The Mn 4+ is inactive since it is the highest oxidation state possible under the range of potential for cycling. From its orbital configuration (3d 5 4s 2 ) Mn 5+ and Mn 7+ could be obtained but at much higher potentials. [START_REF] Saint | Synthesis and electrochemistry of Li3MnO4: Mn in the +5 oxidation state[END_REF] Unlike Ni and Co, the oxidation number of Mn will remain the same so its contribution will be to stabilize the electrode structure. [START_REF] Kim | The first cycle characteristics of Li[Ni1/3Co1/3Mn1/3]O2 charged up to 4.7 V[END_REF] Even with the growing consideration toward NMC for its improved safety, this material was still capped at a low specific capacity. The research aimed to go beyond this State-of-the-art | 19 limit and to reach the 200 mAh.g -1 for this promising material (the specific capacity of NMC-111 is 280 mAh.g -1 ). To that effort, Ni-rich NMCs were investigated since Ni will increase the specific capacity of the material. For instance, in 2005 Liao et al. reported a series of NMCs with LiNi0.6 Mn0.25 Co0.15 O2 exhibiting a specific capacity of 167 mAh.g -1 . [START_REF] Liao | Effect of Mn Content on the Microstructure and Electrochemical Performance of LiNi0.75 -x Co0.25Mn x O2 Cathode Materials[END_REF] The breakthrough came the following year in 2006 with Kim et al. who successfully synthesized spherical LiNi0.8 Mn0.1 Co0.1 O2 (NMC-811) with a specific capacity of nearly 200 mAh.g -1 . [START_REF] Kim | Synthesis and electrochemical properties of Li[Ni0.8Co0.1Mn0.1]O2 and Li[Ni0.8Co0.2]O2 via co-precipitation[END_REF] This NMC-811 is still drawing interest today especially for automotive applications where car manufacturers are shifting toward this technology. [START_REF] Campagnol | Basic Materials[END_REF] Nevertheless, the NMCs are not free of drawbacks and they demonstrate poor electronic conductivity. [START_REF] Amin | Characterization of electronic and ionic transport in Li1-xNi0.33Mn0.33Co0.33O2 (NMC333) and Li1-xNi0.50Mn0.20Co0.30O2 (NMC523) as a function of Li content[END_REF] The use of Co is also a drawback of the NMC because of its expensiveness and the geopolitical issues around its extraction. The Ni-rich NMC-811, which has reduced the amount of Co, decreasing its cost, suffers as a consequence of the lack of Co and Mn of poor cyclability and chemical/electrochemical instability. [START_REF] Li | Collapse of LiNi1-x-yCoxMnyO2 Lattice at Deep Charge Irrespective of Nickel Content in Lithium-Ion Batteries[END_REF][START_REF] Schipper | From Surface ZrO2 Coating to Bulk Zr Doping by High Temperature Annealing of Nickel-Rich Lithiated Oxides and Their Enhanced Electrochemical Performance in Lithium Ion Batteries[END_REF] I. 1.2. Graphite Graphite is the most common negative active material used in LIBs thanks to its low (de)lithiation potential and its high theoretical capacity of 372 mAh.g -1 . [START_REF] Nishi | Lithium ion secondary batteries; past 10 years and the future[END_REF] While the market share for positive materials is shared between the LCO, the LFP, the LiNi1-x-yCoxAlyO2

(NCA), and the NMC, the negative electrode market is a monopole of the Graphite with 98 % of market share and only 2 % for the Li4Ti5O12 (LTO) between 2010-2020. [START_REF] Zhang | Graphite as anode materials: Fundamental mechanism, recent progress and advances[END_REF] Graphite has been identified in 1975 as a material suitable for LIBs for its ability to intercalate Li. However, all attempts to achieve a reversible Li intercalation did not succeed because of Graphite exfoliation and instabilities in the electrolyte. In 1983,

Yazami and Touzain [START_REF] Yazami | A reversible graphite-lithium negative electrode for electrochemical generators[END_REF] were successful and reported a reversible cycle using a polymer.

Dahn et al. [START_REF] Fong | Studies of Lithium Intercalation into Carbons Using Nonaqueous Electrochemical Cells[END_REF] were the first to report in 1990 a liquid electrolyte, the ethylene carbonate (EC), compatible with Graphite thanks to the formation of a passivating layer at the surface of Graphite, the solid electrolyte interphase (SEI). An optimized electrolyte formulation was proposed by Guyomard and Tarascon in 1993, to improve the cyclability of Graphite electrodes, namely a dimethyl carbonate (DMC) and EC mixture. [START_REF] Guyomard | Rechargeable Li1 + x Mn2 O 4 / Carbon Cells with a New Electrolyte Composition: Potentiostatic Studies and Application to Practical Cells[END_REF] Even today the formation mechanisms of the SEI is still disputed and of the utmost importance to better understand the capacity fading arising from the SEI. [START_REF] Peled | Review-SEI: Past, Present and Future[END_REF] Like NMC, Graphite is a layered material constituted of an ABAB stacking of graphene layers (see Figure I.3). In the ab plane (basal plane) the carbon atoms form a honeycomb pattern by strong covalent bonds. Whereas in the ac plan (prismatic plane) the different layers of graphene are linked by weak van der Waals interactions. [START_REF] Persson | Thermodynamic and kinetic properties of the Li-graphite system from first-principles calculations[END_REF] State-of-the-art | 20 Most of the time the insertion of Li will be favored via the prismatic plane rather than via the basal plane except in the presence of defects. Indeed the space in between two graphene layers (prismatic plan) is 33.5 nm whereas between two carbon atoms along the basal plane is 14.2 nm. [START_REF] Chung | Review Graphite[END_REF] The Li will have more space to come and go on the prismatic plane unless some defects in the structure would enlarge the distance in-between atoms on the basal plane. The Li-ions will be inserted in-between the graphene layers according to the following electrochemical reaction + + + -→ . Due to this phenomenon, upon insertion of Li a change in the layer spacing is observed with an initial spacing of 33.5 nm when there is no Li to 37.1 nm when the LiC6 configuration is reached. [START_REF] Guerard | Intercalation of lithium into graphite and other carbons[END_REF] It results in a volume expansion when fully lithiated of 13.2 %. [START_REF] Schweidler | Volume Changes of Graphite Anodes Revisited: A Combined Operando X-ray Diffraction and In Situ Pressure Analysis Study[END_REF] During the lithiation process the Graphite undergoes different crystallographic structures, which is highlighted by the electrochemical curve in Figure I.4. In the literature, four different stages are identified, stage 4 whose composition is not well defined, then stage 3 (LiC24) where there are four graphene layers in-between Li. Then the composition shifts toward stage 2 (LiC12) where two graphene layers separate the Li, with a diluted stage 2 intermediate state (LiC18). Finally, the stable composition of LiC6 is achieved at stage 1 at the end of lithiation. [START_REF] Asenbauer | The success story of graphite as a lithium-ion anode materialfundamentals, remaining challenges, and recent developments including silicon (oxide) composites[END_REF] State-of-the-art | 21

I.2. LIBs Modeling

It is necessary to define such a term that will be extensively used in this manuscript.

According to the Cambridge Dictionary, the definition of modeling is "the activity of using mathematical models to make calculations or predict what might happen". Modeling has many applications in our daily life, for instance, the weather forecast or the chess programs that have been able to defeat any human for more than twenty years. [START_REF]Deep Blue vs. Garry Kasparov: 20th Anniversary of Epic Chess Match[END_REF] It is also omnipresent in the battery field, where both experimentalist and modeling people are using them daily. In essence, every formula, every empirical equation used to extract parameters from experimental data are part of modeling. For instance, using xray diffraction data to access the lattice parameters of a given structure is modeling.

There are different modeling approaches to simulate a phenomenon, all of which are still used in the literature because each has its advantages (see Figure I.5). The higher the dimensional level, the more flexible the model and the more accurate the prediction will be. Unfortunately, the computational cost follows the same trend with the 3-D simulations being the most expensive ones. There is the particular case of the Artificial Intelligence/Machine-Learning models (AI/ML) that are different because of their working principle. They rely on a first training step which is extremely expensive in terms of computational power and input data. However, once this step is efficiently done, it allows accurate predictions in a short time. The more thorough the training step is, the higher the accuracy will be. The choice of the appropriate modeling approach is essential, it must be able to capture at the fullest the impact of the mesostructure of an electrode on the electrochemical behavior. Below, all the different modeling approaches in Figure I.5 will be discussed, starting with the empirical models.

I.2.1. Empirical Models

By definition, empirical models are based on observations and experiments rather than on physical laws and theories. They are what we could call 0-D models because they take as input 0-D parameters (i.e. numbers). The main advantage of such models is their capability to predict complex behavior in a time-efficient manner. Indeed they can capture the impact of several phenomena through a simple set of equations fitted on exper- State-of-the-art | 23 imental data. This approach is particularly effective when applied to reproducible phenomena. However, they lack flexibility because they are designed to predict a given variable for a specific system. For example, an empirical model predicting the evolution of the cost of Graphite cannot be applied to other chemistries since the variations will be different. In the battery community, empirical models are mainly used to predict the aging of the cell. [START_REF] Liu | An evaluation study of different modelling techniques for calendar ageing prediction of lithium-ion batteries[END_REF][START_REF] Tian | Capacity attenuation mechanism modeling and health assessment of lithium-ion batteries[END_REF][START_REF] Šeruga | Continuous modelling of cyclic ageing for lithium-ion batteries[END_REF][START_REF] Hosen | Battery lifetime prediction and performance assessment of different modeling approaches[END_REF][START_REF] Messing | Empirical modeling of cathode electrode durability in polymer electrolyte fuel cells[END_REF][START_REF] Rechkemmer | Empirical Li-ion aging model derived from single particle model[END_REF] The capacity fade is particularly complex to predict with models of higher dimensions because it occurs over the span of few hundreds to couple thousand cycles. Simulating such a long operating time would require expensive computational resources and an enormous computation time. The empirical models predicting the capacity fade are of particular interest to battery management systems in which they can easily be embedded thanks to their low computational cost.

I.2.2. 1-D Models

Unlike empirical models, 1-D models are based on physical laws and equations, yet, they serve a similar purpose. Indeed, they are mainly applied to assess observables with little computational effort. Such efficiency is possible thanks to the averaging of all the properties of the cell on one single axis (see Nevertheless, they must be based on physical equations, which entails a deep understanding of the system studied. While empirical models can capture the impact of complex phenomena without a profound knowledge of each mechanism at work, the establishment of 1-D models is more challenging in such cases. They also have been applied

to predict the capacity fade in LIBs, [START_REF] Sankarasubramanian | A capacity fade model for lithium-ion batteries including diffusion and kinetics[END_REF] although here the capacity fade was only due to the SEI formation which was solved for. This leads to a finer interpretation of the output result, i.e. the capacity fade. The authors identified the kinetic of the SEI formation and the solvent diffusivity as the governing parameters for the SEI formation. The capacity fade could be described as a function of the SEI thickness with a linear growth over time.

1-D models can also be used to simulate the discharge curve and average lithiation state across the LIB. [START_REF] Martínez-Rosas | Modeling and simulation of lithium-ion batteries[END_REF][START_REF] Fabre | Charge/Discharge Simulation of an All-Solid-State Thin-Film Battery Using a One-Dimensional Model[END_REF] All these examples require an extensive list of parameters because of the implicit consideration of the geometry of the cell components (e.g. porosity, tortuosity, State-of-the-art | 24 width, height, etc…). This is the main drawback of 1-D models which depends on physical equations using parameters not explicitly taken into account in the model. Increasing the dimension of the system is seen as an expensive but effective approach to decrease the number of input parameters while increasing the precision of the model.

I.2.3. Pseudo-2-D Models

To further reduce the number of effective parameters, the pseudo-2-D model (P-2D) is a popular approach among the community. [START_REF] Chadha | Model Based Analysis of One-Dimensional Oriented Lithium-Ion Battery Electrodes[END_REF][START_REF] Farkhondeh | Full-Range Simulation of a Commercial LiFePO 4 Electrode Accounting for Bulk and Surface Effects: A Comparative Analysis[END_REF][START_REF] Klein | Electrochemical Model Based Observer Design for a Lithium-Ion Battery[END_REF][START_REF] Ramadesigan | Modeling and Simulation of Lithium-Ion Batteries from a Systems Engineering Perspective[END_REF][START_REF] Dao | Simplification and order reduction of lithium-ion battery model based on porous-electrode theory[END_REF][START_REF] Kumaresan | Thermal Model for a Li-Ion Cell[END_REF][START_REF] Smith | Control oriented 1D electrochemical model of lithium ion battery[END_REF][START_REF] Ramadass | Development of First Principles Capacity Fade Model for Li-Ion Cells[END_REF] The P-2D model owns its name to the The main advantage of P-2D over 1-D models is the explicit representation of the AM particles in the system. This simple change allows capturing more accurately a variety of important phenomena in the cell. Namely, the ionic transport through the electrolyte that was taken into account through effective parameters of porosity and tortuosity in 1-D models. Also, the (de)intercalation of Li at the AM/Electrolyte interface will not be driven by an effective surface area (1-D) but more realistically by the 2D representation of spherical particles. The increase in dimension of the model can then assess the impact of the particle radius distribution on the electrochemical performance of the LIBs. [START_REF] Farkhondeh | Full-Range Simulation of a Commercial LiFePO 4 Electrode Accounting for Bulk and Surface Effects: A Comparative Analysis[END_REF] It also gives access to capacity fade predictions with the SEI growing as a layer around the explicit AM particles. [START_REF] Liu | Synthesis and characterization of LiNi1-x-yCoxMnyO2 as the cathode materials of secondary lithium batteries[END_REF][START_REF] Liu | Understanding electrochemical potentials of cathode materials in rechargeable batteries[END_REF] As mentioned earlier, the model has been validated for LIBs in State-of-the-art | 25 several works, establishing its credibility in the battery field. [START_REF] Ramadesigan | Modeling and Simulation of Lithium-Ion Batteries from a Systems Engineering Perspective[END_REF][START_REF] Kumaresan | Thermal Model for a Li-Ion Cell[END_REF][START_REF] Doyle | Comparison of Modeling Predictions with Experimental Data from Plastic Lithium Ion Cells[END_REF] That is why P-2D models are often used as a reference when no experimental data is available. [START_REF] Klein | Electrochemical Model Based Observer Design for a Lithium-Ion Battery[END_REF][START_REF] Dao | Simplification and order reduction of lithium-ion battery model based on porous-electrode theory[END_REF][START_REF] Smith | Control oriented 1D electrochemical model of lithium ion battery[END_REF] Nonetheless, due to the 2-D configuration of the system, accurate mesostructure-focused studies will not be possible with this approach. The P-2D model also completely overlooks two components of an electrode, the carbon additives, and the binder phase. In addition to these limitations, the number of effective parameters and the spherical particle hypothesis is a brake toward higher predictability of the model.

I.2.4. 3-D Models

Despite an early example of thermal modeling applied to a simplified structure in 2005, [START_REF] Chen | Thermal analysis of lithium-ion batteries[END_REF] the emergence of the 3-D models arose in the 2010s. This development was intrinsically tied to the improvement of computational power at lower costs [START_REF] Bell | Effect of Technology on Near Term Computer Structures[END_REF] and the development of the tomography imaging technique. [START_REF] Scharf | Bridging Nano and Micro-scale X-ray Tomography for Battery Research by Leveraging Artificial Intelligence[END_REF] The benefit of running computation- LiCoO2 images in 2012, [START_REF] Yan | Three Dimensional Simulation of Galvanostatic Discharge of LiCoO 2 Cathode Based on X-ray Nano-CT Images[END_REF] numerous simulations were based on tomography images to have realistic mesostructure for their studies. [START_REF] Yan | Simulation of heat generation in a reconstructed LiCoO2 cathode during galvanostatic discharge[END_REF][START_REF] Lim | Simulation of diffusion-induced stress using reconstructed electrodes particle structures generated by micro/nano-CT[END_REF][START_REF] Lu | 3D microstructure design of lithium-ion battery electrodes assisted by Xray nano-computed tomography and modelling[END_REF][START_REF] Danner | Thick electrodes for Li-ion batteries: A model based analysis[END_REF][START_REF] Hein | Stochastic microstructure modeling and electrochemical simulation of lithium-ion cell anodes in 3D[END_REF][START_REF] Ferraro | Electrode Mesoscale as a Collection of Particles: Coupled Electrochemical and Mechanical Analysis of NMC Cathodes[END_REF][START_REF] Trembacki | Mesoscale Effective Property Simulations Incorporating Conductive Binder[END_REF][START_REF] Roberts | A framework for threedimensional mesoscale modeling of anisotropic swelling and mechanical deformation in lithium-ion electrodes[END_REF][START_REF] Kotaka | Microstructural analysis of mass transport phenomena in gas diffusion media for high current density operation in PEM fuel cells[END_REF][START_REF] Kashkooli | Multiscale modeling of lithium-ion battery electrodes based on nano-scale X-ray computed tomography[END_REF][START_REF] Xu | Heterogeneous damage in Li-ion batteries: Experimental analysis and theoretical modeling[END_REF] Thanks to their fast development, 3-D models could be applied to complex cases such as coupled electrochemical-mechanical models. [START_REF] Roberts | A framework for threedimensional mesoscale modeling of anisotropic swelling and mechanical deformation in lithium-ion electrodes[END_REF][START_REF] Mai | A particle-resolved 3D finite element model to study the effect of cathode microstructure on the behavior of lithium ion batteries[END_REF][START_REF] Jia | Effective thermo-electro-mechanical modeling framework of lithium-ion batteries based on a representative volume element approach[END_REF][START_REF] Fathiannasab | Chemo-mechanical modeling of stress evolution in all-solid-state lithium-ion batteries using synchrotron transmission X-ray microscopy tomography[END_REF][START_REF] Jeong | A fully coupled diffusionalmechanical finite element modeling for tin oxide-coated copper anode system in lithium-ion batteries[END_REF][START_REF] Zhao | A review on modeling of electro-chemo-mechanics in lithium-ion batteries[END_REF][START_REF] Bai | A chemo-mechanical grain boundary model and its application to understand the damage of Li-ion battery materials[END_REF][START_REF] Singh | Coupled chemo-mechanical modeling of fracture in polycrystalline cathode for lithium-ion battery[END_REF][START_REF] Xu | Stochastic 3D microstructure reconstruction and mechanical modeling of anisotropic battery separators[END_REF] However, one of the challenges to achieving the next generation of electrochemical-mechanical models is to be able to simulate dynamically the swelling of CBD and the volumetric expansion of the AM upon lithiation. In that way, the mechanical properties will have a direct impact on the electrochemical behavior of the system. For instance, the volumetric expansion of AM will increase the diffusion path length for both solid Li and Li + . For the former the volume of the AM will be larger, hence the increase of the distance Li(s) will have to cross. The tortuosity in the electrolyte would increase due to the same phenomenon, hindering the ionic transport. Also, with the appearance of cracks, the model will capture the dynamic evolution of the active surface area where the electrochemical reaction takes place.

ally
State-of-the-art | 26 the binder, the binder, the LiCoO2, and the pores respectively from dark to light grey. [START_REF] Hutzenlaub | Three-dimensional reconstruction of a LiCoO 2 Li-ion battery cathode[END_REF] It appears that 3-D models are the most suited to investigate the impact of the mesostructure on the electrochemical performance of a battery. Indeed, their ability to drastically reduce the number of implicit parameters thanks to a 3-D representation is essential to highlight heterogeneities due to the location of each phase.

I.2.5. Artificial Intelligence/Machine Learning

The modeling approaches introduced previously have been the most popular since the introduction of the Newman model in 1993. [START_REF] Doyle | Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell[END_REF] Yet in recent years, a new methodology has been attracting much interest in the battery field and is now the most dynamic domain of computational research, the Artificial Intelligence/Machine Learning (AI/ML) models. [START_REF] Xu | A novel data-driven method for predicting the circulating capacity of lithium-ion battery under random variable current[END_REF][START_REF] Li | Lithium-ion battery capacity estimation -A pruned convolutional neural network approach assisted with transfer learning[END_REF][START_REF] Yravedra | A complete machine learning approach for predicting lithiumion cell combustion[END_REF][START_REF] Khaleghi | Online health diagnosis of lithium-ion batteries based on nonlinear autoregressive neural network[END_REF][START_REF] Gao | Designed high-performance lithium-ion battery electrodes using a novel hybrid model-data driven approach[END_REF][START_REF] Xu | Improving the state of charge estimation of reused lithium-ion batteries by abating hysteresis using machine learning technique[END_REF][START_REF] Song | Intelligent state of health estimation for lithium-ion battery pack based on big data analysis[END_REF][START_REF] Fan | A novel deep learning framework for state of health estimation of lithium-ion battery[END_REF][START_REF] Yang | Extreme Learning Machine Based Thermal Model for Lithium-ion Batteries of Electric Vehicles under External Short Circuit[END_REF][START_REF] Yang | Lifespan prediction of lithium-ion batteries based on various extracted features and gradient boosting regression tree model[END_REF][START_REF] Ma | The capacity estimation and cycle life prediction of lithium-ion batteries using a new broad extreme learning machine approach[END_REF][START_REF] Liu | Mass load prediction for lithium-ion battery electrode clean production: A machine learning approach[END_REF][START_REF] Hong | Towards the swift prediction of the remaining useful life of lithium-ion batteries with end-to-end deep learning[END_REF][START_REF] Kilic | Assessment of critical materials and cell design factors for high performance lithium-sulfur batteries using machine learning[END_REF][START_REF] Shen | Deep convolutional neural networks with ensemble learning and transfer learning for capacity estimation of lithium-ion batteries[END_REF][START_REF] Shen | A deep learning method for online capacity estimation of lithium-ion batteries[END_REF] The strength of these models is the ability to predict in a short time and accurately complex phenomena. In that sense, they are similar to empirical models since they both rely on experimental input data. Indeed, the AI/ML algorithms require a training step where usually 80 % of the input dataset is provided to the machine. The training is the most computationally expensive step because the amount of data needed to achieve a high fidelity model is significant. Then 20 % of the remaining data will be used to ascertain the accuracy of the model during the validation step. If the accuracy is

State-of-the-art | 27 too low to be acceptable, then the training step will have to be repeated with more datasets, or the structure of the model will have to be tuned.

For instance, a popular approach is the data-driven models that are based on the analysis of the data to unravel relationships between the observables and the input parameters of a given system. [START_REF] Zhu | Data-driven analysis on thermal effects and temperature changes of lithium-ion battery[END_REF][100][101][102][103][104][105][106][107][108][109][110] As stated previously their main application is for the prediction of the state of life of the battery due to their low computational cost when the training is done. Nonetheless, they are occurrences of data-driven models applied to the manufacturing of batteries. [START_REF] Liu | Mass load prediction for lithium-ion battery electrode clean production: A machine learning approach[END_REF]111,112 Among them, Cunha et al. 111 Firstly, the methods to generate structures suitable for models with implicit CBD will be detailed, and secondly, the approaches to generate structures with explicit CBD will be addressed. CT images due to the coarse spatial resolution. Despite this limitation, it allows capturing the impact of the real morphology of the AM on the electrochemical behavior. [START_REF] Yan | Three Dimensional Simulation of Galvanostatic Discharge of LiCoO 2 Cathode Based on X-ray Nano-CT Images[END_REF][START_REF] Lim | Simulation of diffusion-induced stress using reconstructed electrodes particle structures generated by micro/nano-CT[END_REF] As an example, thanks to its wide field of view, micro-CT has been used in the redox-flow battery field to capture a representative volume that is larger than for LIBs. [114][115][116] However the impact of the arrangement of the AM and CBD cannot be investigated through this approach.

To achieve more accurate 3-D models, the CBD has to be resolved and thus must be distinguishable in the 3-D structures. To do so the same stochastic generation workflow can be used except with an additional step for the stochastic generation of CBD. It introduces new parameters for the electrodes, the amount of CBD, and its morphology.

Indeed, the arrangement of the CBD as a film around the AM or as clusters can be controlled. [START_REF] Trembacki | Mesoscale Effective Property Simulations Incorporating Conductive Binder[END_REF][117][118][119] The commercial software GeoDict ® is widely used in the community for electrode generation. 120 This methodology gives access to very large volumes of electrodes because it is not restricted by experimental limitations. It is noteworthy that AM reconstructed from micro-CT has been used with the stochastic addition of CBD to capture both the impact of the latter and the influence of the AM morphology. [START_REF] Danner | Thick electrodes for Li-ion batteries: A model based analysis[END_REF][START_REF] Trembacki | Mesoscale Effective Property Simulations Incorporating Conductive Binder[END_REF] State-of-the-art | 29 To fully capture the mesostructure of an electrode from tomography data, the nano-CT is the most used technique because from the images the AM, the CBD, and the pores can be distinguished. [START_REF] Scharf | Bridging Nano and Micro-scale X-ray Tomography for Battery Research by Leveraging Artificial Intelligence[END_REF] The structures arising from this workflow can directly be used in 3-D models with spatially resolved CBD. Nevertheless, the high contrast between the phases comes at the cost of a narrower field of view, resulting in volumes the size of a cube of 30 µm of length. This entails a problem about the representativeness of the volume and has already been discussed in the literature. 121,122 Furthermore, in the LIBs modeling field the number of input geometrical parameters is identified as a brake toward higher precision of the model. Indeed, average effective parameters cannot capture the heterogeneities in the cell which is the aim of this work. Even with nano-CT data, most models need to implement an effective porosity and/or tortuosity to the CBD to account for its micro-porosity which is not captured in the images. Yet, in 2020 the group of Shearing was able to develop a COMSOL model without effective parameters for the inactive phase thanks to the microstructure of the CBD explicitly resolved. [START_REF] Lu | 3D microstructure design of lithium-ion battery electrodes assisted by Xray nano-computed tomography and modelling[END_REF]123 To tackle the number of effective parameters, the explicit representation of the mesostructure is of the utmost importance to predict the heterogeneities in a battery. It will be even more

State-of-the-art | 30

important with the new rising technology of all-solid-state batteries with the introduction of a significant volume of void in the structure.

As illustrated by the ARTISTIC project, an alternative computational method to generate realistic 3-D structures with spatially resolved CBD is the manufacturing simulations. [START_REF] Ngandjong | Multiscale Simulation Platform Linking Lithium Ion Battery Electrode Fabrication Process with Performance at the Cell Level[END_REF]124,125 This workflow models each step of the fabrication process, from the slurry formulation to the calendering process. Thanks to experimental observables (Young modulus, slurry viscosity, the porosity of the calendered electrode, etc...) the models are validated at each step. This methodology is very unique and has a pioneering role in linking the modeling and the experiments with satisfactory results.

Finally, the last source of 3-D structures is the rising field of machine learning. In 2020, The benefit from their work is the ability to generate structures with periodic boundary conditions, meaning that the volume of a particle that is cut out due to the spatial boundary of the system will be integrated at the opposite boundary. It provides a -of-the-art | 31 representative volume of an electrode with a much smaller dimension than one without periodic boundary conditions.

State

I.4. 3-D LIBs Models

In this section, the modeling aspect will be addressed, through the numerical method and mesh. Then the models of interest regarding LIBs simulation will be discussed and divided into two groups based on their implicit or explicit consideration of CBD.

I.4.1. Finite Element Method

Most of the 3-D models for LIBs rely on one of the most widespread techniques in which is the finite element method (FEM). This method has started to arise interest in the 1940s when Richard Courant recognized its usefulness and was later applied to many engineering fields. 129 In essence, this method consists in break down a large system into several smaller sub-volumes, called finite elements. These finite elements are defined through the discretization of the space by the mesh of the system. The mesh generation is a critical process since the representation of the geometry relies on it and will be discussed later in this section.

For the electrochemical modeling of a LIB in 3-D, the FEM will have to solve a set of partial differential equations (PDEs). These PDEs will be locally approximated in each finite element by a trial function whose error will be quantified through an observable called the residual. Overall, the locally linearized PDEs will become a set of algebraic equations in steady-state systems and a set of ordinary differential equations for timedependent systems. These equations will be solved at each node of the mesh and not in the entire system. Then to obtain continuous values of the variables, the values will be interpolated between the nodes thanks to polynomial functions. Each finite element will have its matrix describing the set of approximated PDEs, yet they cannot be solved independently since the boundary conditions are set for the whole system and not for individual elements.

For a given problem -∆ + = (I.1

) with f a continuous function, and u the solution of this PDE, the matrix equivalent notation is as follows

= (I.2 )
State-of-the-art | 32

With A the matrix with the sets of approximated equations, U the matrix with the approximated solutions, and B the matrix accounting for the values of the function f in each node. The FEM workflow is then:

• Determine the matrix A,

• Compute the matrix B based on the function f,

• Solve Equation I. Now that the FEM has been discussed, how to discretize a system through the generation of a mesh should be detailed. As mentioned previously, the mesh will be a geometrical object composed of nodes that are "anchor points" for higher dimension elements.

The most widely used geometries for the finite elements is the tetrahedron, where the surface is defined by triangles (each formed by 3 nodes) and the volume is defined by In the LIB modeling literature, most studies rely on commercial software to perform

the meshing or open-source ones which are not suitable to incorporate CBD into the mesh. [START_REF] Lu | 3D microstructure design of lithium-ion battery electrodes assisted by Xray nano-computed tomography and modelling[END_REF]130 Roberts developed his own meshing algorithm which allows powerful simulations such as coupled electrochemical-mechanical in his workflow, but the algorithm is not open-access. [START_REF] Roberts | A framework for threedimensional mesoscale modeling of anisotropic swelling and mechanical deformation in lithium-ion electrodes[END_REF]119 Similar to the choice of the numerical methods, the meshing workflow and the resolution of the mesh will have a large influence on the results. This is an important step of the FEM that should be optimized.

State-of-the-art | 33 To further study the impact of the morphology of the particles, Nelson et al. 130 used tomography data of NMC-111 electrodes to assess their AM geometrical features. From

Table I.1 it transpires that ball milling and calendering processes tend to decrease the mean radius of the secondary AM particles due to cracking (higher number of particles).

In turn, the mean specific surface area will be greater, inducing a higher surface of contact between the active material and the electrolyte. To investigate the influence of the morphology on the electrochemical behavior, they analyzed a single particle from tomography data in a 3-D model (reported in Figure I.14.a). From there, they played with the diffusion coefficient value and followed the concentration gradient in the particle at a high rate during charge (Figure I.14.b). It was shown that the diffusion model could capture the shift from diffusion-limited toward surface change transfer controlled regime. For the lower diffusion coefficient, at each time illustrated, the distribution of Li intercalated in the NMC-111 is inhomogeneous where as for the higher diffusion coefficient the concentration is uniform at all times. From the results

of the simulations of one AM particle of each electrode, they were able to highlight that the rate capability is improved when the particle size is reduced because it will mitigate the concentration gradients. Then, the optimal case with the carbon black layer covering all the surface of the AM was compared to an inhomogeneous case with different sub-volumes. Figure I.17 Indeed, during the discharge, the electrons are flowing to the positive electrode from the current collector. If the electronic percolation network is disconnected close to the current collector region, all the electrodes will suffer from a lack of electrons. Hence, a charge transfer limited system where the region close to the current collector will be lithiated first. A full cell configuration was also reported with stochastic Graphite as negative AM.

A charge at 1C illustrated the ionic transport limitation, probably due to larger Graphite particles, which are more likely to obstruct the Li + path.

In 2019, our group published a study investigating the physics that should be applied c. The difference is of several orders of magnitude due to hotspots where the conductivity of the CBD reaches more than 400 S.cm -1 . The cause of these local maximums is the stress under which the CBD is upon discharging because of mechanical deformations or lithiation-induced swelling of the AM. From a careful individual particle analysis, they concluded that CBD is essential and has a positive impact on performance even if it hinders ionic transport and reduces the available active surface area.

The very same year, Lu et al. [START_REF] Lu | 3D microstructure design of lithium-ion battery electrodes assisted by Xray nano-computed tomography and modelling[END_REF] published the most precise 3-D LIB study in terms of spatial resolution of the structure in the literature. The last brake towards a model free of implicit parameters was the microstructure of the CBD. Indeed, the porosity and tortuosity of the CBD were input parameters in 3-D models to account for the pores inside this phase that was too narrow to be explicitly considered. However, thanks to a novel technique called the Dual Scan Superimposition (DSS), the group of Shearing could report for the first time the micro-pores of the CBD spatially resolved. In essence, the method consists in performing a first nano-CT scan to assess the location of each phase in an NMC-111 electrode, i.e. the AM, the CBD, and the pores. Then, a second scan was done with a sample solely composed of carbon and binder, allowing a better contrast between the pores and the CBD. From there, the microstructure of the CBD could be only as an electronic conductive medium. This impressive spatial resolution was possible due to the significant number of elements in the mesh (ca. 30 million), which requires extremely expensive computational resources to solve.

After a thorough analysis of the geometrical features of the electrodes, several simulations were simulated at different C-rates. The state of lithiation was plotted versus the distance from the separator at different depths of discharge in Figure I.19.(c,d). It transpires that there is a gradient of lithiation throughout the thickness of the electrode, with the region near the separator being more lithiated than the region closer to the current collector. This trend is characteristic of an ionic transport limited system. A relationship can also be established between the size of the particles and their lithiation state. Indeed, the smaller the particles are, the higher the state of lithiation will be. This study reached the same conclusions related to the link between particle size and rate capability as the work led by Nelson et al. 130 It is noteworthy that both trends will be more pronounced with higher C-rates and depth of discharge. In fact, with increasing rates of discharge the system will be more kinetically driven and the impact of the ionic transport limitation even greater.

State-of-the-art | 41 In essence, this approach is quite similar to the work undergone by Danner et al. [START_REF] Danner | Thick electrodes for Li-ion batteries: A model based analysis[END_REF] previously detailed. In both cases, they monitored the impact of modifying the percolation network of Li + (Lu et al. [START_REF] Lu | 3D microstructure design of lithium-ion battery electrodes assisted by Xray nano-computed tomography and modelling[END_REF] ) or the electrons (Danner et al. 67 ) close to the origin of their flux, the separator, and the current collector respectively.

State-of-the-art | 42 Discharge simulations were performed at 3.75 C to investigate the impact under a high C-rate. The graded electrode exhibits a smoother Li + concentration gradient than the asprepared one in the graded region, and a distribution similar for the rest of the electrode (Figure I.20.c). The authors attributed this phenomenon to the particle geometry that decreased the ionic transport resistance. In other words, the smaller size of the graded particles has reduced the tortuosity, i.e. the pathways of the ions to reach the AM particles. Also, as highlighted previously smaller particles will result in a higher rate capability, which is demonstrated here for the grade electrode in lithiation for the graded electrode is higher in the graded region thanks to a smaller solid diffusion path and higher active surface area. This higher utilization of the graded region is also contributing to the softening of the Li + concentration gradient since more Li + will be intercalated, hence consumed.

While this study significantly pushed forward the state-of-the-art, it also stood out thanks to its thorough geometrical and modeling observables analyses. More generally the need of having each AM particle identified individually for post-processing has been evidenced by the two previous studies. It will help in the understanding of the complex interactions and phenomena inside LIBs.

I.5. Newman-based Model

This section will shortly describe the physical model used in the simulations reported in this manuscript. The set of equations are originated from Newman's model that was firstly reported in 1993 to describe the behavior of a half-cell in P-2D. [START_REF] Doyle | Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell[END_REF] As the model used in this Ph.D. is in 3-D, there will be some minor modifications to the original Newman's model.

First of all, mass transport in the AM is accounted for by fickian diffusion, where migration is neglected. The Li + will move under a potential or concentration gradient in the electrolyte and will react at the interface with the AM following Buttler Volmer law.

The preexponential term is accounting for the number of available sites in the AM for the intercalation of Li + . Ohm's law characterizes the evolution of the potential of the electrolyte and solid (AM+CBD) phases. Both the electronic conductivities and diffusion coefficients are corrected by the porosity to access their effective value following the Bruggeman relation. The work strategy of this Ph.D. will be explicated in the following section, as well as the approaches to tackle each obstacle identified in this chapter.

I.6.1. Modeling Strategy

The ARTISTIC project aims to be a pioneering tool to optimize and accelerate the current and next LIBs generation. In that sense, the modeling work was firstly focused on the NMC under a half-cell configuration (Li foil as the negative electrode), one of the most used positive AM. Once the model was operational, to demonstrate the flexibility of the platform, the study shifted toward Graphite which is also one of the most used AM, only this time at the negative electrode. Finally, the full cell configuration was achieved once the model was capable of predicting the behavior of both positive and negative electrodes.

From the descriptions in Section I.2, it transpires that the 3-D modeling methodology is the most suited to shed light on the link between the mesostructure and the behavior of an electrode and to highlight heterogeneities due to the location of each phase. This approach will be used throughout this manuscript to predict the electrochemical performance of any given electrode. As it has been stated, such simulations are computationally expensive and require solving a significant number of partial derivative equations. The commercial software COMSOL Multiphysics ® (referred to as COMSOL in the following manuscript) was used to solve these equations. COMSOL is a modeling software based on solving complex sets of partial differential equations (PDEs) in complex geometries.

I.6.2. Carbon-Binder Domains

As evidenced earlier, the explicit consideration of the CBD is a cornerstone toward the improvement of the state-of-the-art LIBs models. In 2018, at the time this study started, CBD drew very little interest in the modeling community. It was then a challenge to implement this phase into the model with its own physical properties. However, there are challenges to tackle, the meshing and the acquiring of 3-D structures with the AM and CBD distinguished. Regarding the first point, COMSOL is based on the FEM introduced previously and requires a mesh to compute. That is why a new meshing algorithm was developed in the course of this Ph.D. to satisfy the needs of the ARTISTIC project with high flexibility. This algorithm relying on voxelization will be introduced in the following chapter. Regarding the second, since the explicit use of CBD was not popular,

State-of-the-art | 46 finding 3-D structures with CBD was an issue that will be discussed in the following section.

I.6.3. Structure Generation

In the ARTISTIC workflow, the structure of the electrodes arose from simulations modeling each step of the manufacturing process. The strength of the project is the ability to link the electrochemical behavior to the fabrication parameters. That is why most of the NMC structures used in that work come from the manufacturing simulations.

Since the CBDs were explicitly considered throughout all the fabrication models, it was not a problem to distinguish between AM and CBD. However, this model is predictive and cannot generate structures with desired observables such as porosity, CBD morphology, etc…. Moreover, this methodology must be optimized for each new AM, and for most of the duration of this work, only the NMC model was experimentally validated.

Stochastic structure generation was used in the course of these studies to provide electrodes the manufacturing simulations were not yet ready to generate. The main source of stochastic structures was an in-house MATLAB-based script I developed and on one occurrence the commercial software Geodict ® . The last source of structures was tomography data performed on electrodes from this project. Even though it might not be directly in the scope of this project, using tomography data allows demonstrating the efficiency of a model and its capability to predict results based on real mesostructures. In this chapter, all the computational tools developed during the course of this Ph.D.

will be presented. They were all developed in Matlab to help either to automatize or enable new approaches for the ARTISTIC project team. To increase their reach, all these algorithms were compiled into a Graphic User Interface (GUI) freely available as a Web

App through the ARTISTIC website. 133 The first tool was a meshing algorithm that allows having an indefinite number of different phases in the mesh, which tackles the issue related to the explicit consideration of CBD. In Section II.1, the workflfow of the algorithm will be explicited with an article published by us in 2019. 134 The importance of the mesh will also be discussed in detail accompanied with suggestions on how to design a modeling study having the meshing step in mind.

The stochastic generation algorithm used in this project will also be described in Section II.2 to demonstrate its capability to have fine control over the output 3-D electrode.

The GUI that embeds all the previously presented functionalities will be detailled in Section II.3 through an article that was published in 2021 by us. 135

II.1. Meshing

As evidenced in Chapter 2, the electrode structure generation and meshing were a brake to the early development of 3-D battery modeling. In the ARTISTIC project, a variety of approaches are used to generate 3-D structures, namely stochastic generation, tomography data, and spheres coordinates from manufacturing simulations. There was not an existing tool that allowed to import all these different formats and to generate the associated mesh. Especially when having in mind the requirements of the project, i.e.

distinguishing between the AM and the CBD to achieve the next generation of battery models. Since the meshing appeared to be the most challenging blocking point, it was the first investigation of this Ph.D..

In late 2018, the only reported meshing tools dedicated to the battery were not openaccess hence could not be used in the project. [START_REF] Danner | Thick electrodes for Li-ion batteries: A model based analysis[END_REF]119 However, in the ceramic modeling field, to capture the impact of different grain boundaries, the issue of a multiphasic volumetric had been tackled in 2014 by Müller et al.. 136 They claimed that through the use of an easy to implement meshing algorithm developed in the 90s, 137 they could obtain a mesh with all the different grain boundaries distinguishable during their simulations.

An example of the meshing algorithm in one given configuration and a rendering of a mesh resulting from this algorithm are displayed respectively in Nonetheless, in order to reduce the fabrication costs and to further optimize batteries in regards of their integration in the transportation sector, the development of predictive mechanistic models is critically needed. [138][139][140] Among the most used mechanistic modeling approaches, the so-called pseudo-2D (P2D) models dominates since many years the continuum simulation studies of LIBs since its introduction by Newman and co-workers. [START_REF] Doyle | Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell[END_REF][START_REF] Doyle | Comparison of Modeling Predictions with Experimental Data from Plastic Lithium Ion Cells[END_REF] However, P2D models cannot capture the impact of the 3D electrode mesostructure and associated anisotropy on the LIB cell behavior. The electrode mesostructure, characterized by its pore network, the active

Figure II.2. Schematic of a lithium-ion battery highlighting its complexity

In the last 10 years there has been a sharp rise of 3D-resolved electrochemical models in the scientific community on multiple chemistries. [141][142][143][144] Used electrode mesostructures are either computer-generated by using random location algorithms (e.g. GeoDict software) or obtained by computer-tomography. However, reported 3D-resolved models rely on oversimplifications, such as an implicit representation of the CBD through the use of effective parameters for porosity and tortuosity [START_REF] Ngandjong | Multiscale Simulation Platform Linking Lithium Ion Battery Electrode Fabrication Process with Performance at the Cell Level[END_REF][START_REF] Hein | Stochastic microstructure modeling and electrochemical simulation of lithium-ion cell anodes in 3D[END_REF] or by merging CBD with the Active Material (AM) as a single solid phase. 145 Because of the disparate spatial scales, meshing an explicit representation of CBD and AM would be like trying to paint with the same paintbrush a rock and a mountain next to each other. Either the paintbrush is too wide or is too fine. In the first case, the time needed to paint the mountain will be short but the rock geometry will appear as deformed.

In the second case, the rock will retain its apparent geometry but the time needed to paint the mountain will be extremely long.

In the case of an adaptive mesh, the complexity relies on assembling a coarser and a finer mesh together. Where the AM and the CBD are in contact, the elements of the mesh of the AM should be finer to match the ones of the CBD. This task is challenging to do at each interface between AM and CBD phases.

This work reports a computationally efficient and versatile algorithm capable of meshing multiple phases by using voxelization. In this paper, INNOV is illustrated through its application in LIB composite electrodes predicted by using CGMD using LAMMPS software. 147 Such composite electrodes are made of AM particles with multiple sizes (radius 1.4-9.4 µm) and CBD particles (radius 0.65 µm). This is useful in order to implement multiscale modeling workflows able to predict the impact of the manufacturing process on the battery performance, as carried out in the scope of the ARTISTIC project. 133 However, the algorithm can be applied to a wider spectrum of scientific fields.

II.1.1.2. Results and Discussion

In this section we discuss the results of INNOV applied to a three-phased LiNiMnCo The voxelization technique is known for its staircase shape so the algorithm embeds a smoothing option, which relies on Kroon's function. 149,150 It is noteworthy that the smoothing algorithm tends to shrink the volume of the elements, resulting in a slight increase of the porosity (for the example here, 49.69 % before and 50.07 % after smoothing).

Observables

The core of this work is to increase the current level of precision in the modeling of battery cell electrodes by separating active and inactive materials. In doing so, one must not sacrifice the integrity of the mesostructure geometry. To ensure this, INNOV provides a number of observables, which can be compared to experimental numbers (e.g.

arising from tomography characterizations). Among them, two distinct values of porosity are displayed by INNOV: the porosity of the mesh and the porosity of the stack of images. In the example provided in this section, the porosity of the stack is 50.1 % and the porosity of the resulting mesh is 49.7 %.

Computational Tools | 54

Furthermore, the volume ratio of each phase in the mesh is compared to its value in the stack of images in Table II.1. This can render proof of the overall volume conservation of each phase; however, these are average values and cannot highlight local deformations within the mesh. Here, the meshing step introduces an error below to 2% on the volume of each phase, which is considered satisfactory in our study. Another useful insight is the surface coverage between different phases. For example, the surface area of the interfaces between Active Material and CBD (36.2 %) or the Active Material and Electrolyte (63.8 %) are provided once the mesh is done. Application of this algorithm to a composite electrode will be presented later in this work.

Inputs of INNOV

The algorithm takes as input binary stack of images. Such a stack of images can result from different processes. Firstly, it can be generated from a various selection of imaging tools, such as the segmentation of tomography data. Secondly, binary images can be obtained through the slicing of 3D object files. Finally, the origin can be a file with the coordinates of particles centers and radii. This algorithm has been designed to receive as input the coordinates arising from CGMD simulations of the LIB electrode fabrication process. A specific function converts this input into binary images to feed the algorithm with and to reconstruct the electrodes.

Node generation

In the voxelization technique, each node is related to a pixel in the segmented images so each node is coded with a particle type depending its phase (AM, CBD, electrolyte, separator, etc…). The node generation is the computational-time limiting step for the next to come simulation due to the number of faces and elements which increases with the number of nodes. Since the number of nodes equals the number of pixels, aiming for high-resolution images can be a double edge sword. To address this specific issue, the function embeds an anisotropic downsizing tool in the form of the "precision" parameter.

This can be a scalar (isotropic) or a vector (anisotropic) indicating the downsizing factor along each axis i.e. one node out of how many will be saved for the node generation.

While slightly losing in precision, the gain in time can be tremendous. Taking a given electrode structure with 3 phases (AM, CBD, Electrolyte) with input stack of images (201×200×179 pixels), the computation time for the generation of the mesh is 177.20 s without downsizing, and 4.48 s with a precision = [4 4 3].

Going further, a more relevant observation would be that the simulation time (which can be up to several days in some cases) of the downsized mesh will also decrease

Computational Tools | 56

compared to the original one. Indeed, the downsized mesh has more than 1.5 million elements when the mesh without downsizing has more than 48 million elements. An In a 3D mesh, elements can be of different shape; in the algorithm they are tetrahedrons which are a classical choice due to their efficiency (4 nodes/elements). The cubic elements

Computational Tools | 57

need to be divided into tetrahedrons, the function offers two different decompositions: into 5 or 6 tetrahedrons. On the one hand, the decomposition into 6 is more precise and easier to implement since the decomposition into 5 requests for some additional operations as discussed in the original paper of Nielson and Franke. 137 On the other hand, the computational cost will be higher as it will result in more elements.

Algorithm

The voxelization addresses the issues at the interface by meshing every phase all at once. In that way, the surface mesh at the interfaces are by design the same across a given interface between two different phases. To do so, it has to distinguish between four different cases depending on the phases of the nodes of a given element. A trivial one is when all nodes belong to the same phase, no decomposition is needed and the element is retained. The three others occur when the nodes belong to different phases and demand for a decomposition into several more tetrahedrons and creation of triangular faces to effectively separates the phases. Which means the more intricate the phases are, the more surface area there is between them, the more the number of elements and faces will be high.

Importation

A non-trivial step in the modeling process is the importation of a mesh into an outside software. Whether due to the formalism of the format or its lack of documentation, one must tread carefully when converting a mesh. The algorithm presented here performs a successful importation of the mesh in the native format mphtxt of COMSOL Multiphysics software.

INNOV takes full advantage of the new "Selections" feature in COMSOL 5.4 146 which allows the user to create automatically selections of faces and elements in the text file.

In this case, manual selections would have been prohibitory since they are typically thousands of domains and boundaries in the imported mesh in COMSOL. "Selections" provide access to every domains or boundary of interest to apply partial differential equations representing physical mechanisms in COMSOL.

Computational Tools | 58 II.1.2. Why Does the Mesh Matter?

One could wonder why such efforts are focused on the mesh while this step is the furthest away from electrochemistry and battery components. Indeed, the meshing process inherently does not have any physics, but it can impact in a significant manner the electrochemical simulations since the mesh will serve as an intermediate between the 3-D battery structure and the physical model.

To understand how the mesh can propagate uncertainties through the simulation, elements under one minute. Then, for both cases, a significant improvement can be observed from the coarser mesh (light blue) to the finer mesh (red). The morphology of the AM particles as well as the AM/CBD interfaces are more precisely defined with higher numbers of elements. Nonetheless, it seems that a certain number of elements threshold can be reached where no real improvement can be made to the mesh since there is no significant change between the green and red meshes.

Then, is this threshold an optimal number of elements in a mesh to have a perfectly defined geometry? Unfortunately, it does not exist an optimal number suited for all the different cases. The optimal number depends on several geometrical parameters, but most importantly on the computational resources available, which will most likely limit the precision in the mesh.

Putting aside the limitation of the computational resource, let us investigate the geometrical features that are important to have in mind while generating a mesh. To do so, a systematic study was led where stochastic cubic electrodes of various lengths were generated with different constant particles' size radii. To quantify the deformation of the AM, the standard deviation of the ratio between the volume of AM in the 3-D electrode and the actual volume of AM in the mesh was plotted in Figure II.9.a versus the dimensionless parameter Γ which is the ratio between the AM radius and the length of the electrode. The electrodes generated were 94:6 (AM:CBD) with 35 % of porosity, and the mesh was kept at 2M of elements (+/-5 %).

The expected trend is obtained, where the smaller Γ is, meaning the larger is the electrode and the smaller is the AM, the higher is the deviation of volume in the mesh from the initial electrode. Since with decreasing Γ the AM particles will be more and more deformed, one could be tempted to increase the value of Γ by either increasing the particle size (when possible)

or by decreasing the size of the system. However, by adopting this approach the overall number of particles inside the system will decrease, leading to worse representability of the electrode. To claim the representativeness of a sub-volume, in addition to the same volume ratio and tortuosity, the sub-volume should also capture a large enough sample of the overall volume, with as many AM particles as possible. From this short study, it transpires that the threshold for the optimal number of elements in the mesh would be a compromise between a high Γ and the representability of the sub-volume. Nevertheless, in most cases, the AM does not have a constant radius but rather a distribution of particle size. To capture the impact of a heterogeneous AM size, a 40 x

Computational Tools | 61 40 x 40 µm 3 electrode was stochastically generated with a particle size distribution (PSD) from NMC-111. To visualize the size effect, the ratio between the volume in the electrode and the mesh was plotted in Figure II.10 for each particle with markers proportional to the size of the particle in the electrode. Three different cases are compared, based on the number of elements in the mesh.

As expected based on the previous results, with a higher number of elements in the mesh, the particles green are closer to the line Vimage / Vmesh = 1 compared to the blue and red particles. It is noteworthy that larger particles tend to be less deformed, while smaller ones are the most impacted by the meshing step. Indeed, the highest and lowest values of Vimage / Vmesh are observed for the smallest particles, since a slight variation will result in a huge relative change on the individual volume.

From this section, it transpires that the meshing step is crucial for the integrity of the simulation results and should be carefully performed, especially to distinguish between AM and CBD in the model. It was the major blocking point for the ARTISTIC project and the community to achieve the next generation of battery modeling.

That obstacle has been tackled through INNOV which allows to mesh an infinite number of phases and to smoothly import the mesh into COMSOL. All the simulations that will be presented in this manuscript were performed using meshes from INNOV.

The impact an ill-designed mesh can have through the deformation of battery components has also been highlighted in this section. The ratio between the characteristic length of interest and the size of the system as well as the representability of the sample should be closely monitored to achieve an optimal mesh.

II.2. Structure Generation

Below will be detailed the two different approaches explored in this Ph.D. and the ARTISTIC project to acquire in-silico, i.e. computationally generated, 3-D structures of electrodes. Namely, the manufacturing simulations to capture through physical modeling the fabrication process and the stochastic generation which is faster and more flexible.

II.2.1. From Manufacturing Simulations

In the Artistic project, the goal is to model the whole manufacturing process and to use the resulting mesostructure in a performance model to evaluate how the fabrication parameters will impact the electrochemical performance. During the manufacturing simulations, the particles of AM and CBD are considered spherical, so the output of those simulations is a list of coordinates and radii of all the particles.

The first objective of this Ph.D. was to automatize the processing of the coordinates list to be able to use the meshing algorithm we developed. Since INNOV takes as input images, an intermediate step was required to use the output from manufacturing simulations. A simple script was written to generate a stack of images given the coordinates and radii of the spheres and the resolution of the images, i.e. the amount of µm per pixel.

With fine resolutions, the geometry of the spheres will be retained while going toward coarser resolutions will inherently deform the spherical shape of the particles (see Figure

II.11.(a,b)).
Then, the code was refined to embed the so-called periodic boundaries condition (PBC).

PBC's principle is to achieve a higher degree of representativeness for the sub-volume by introducing each time a particle is intersected by one spatial boundary of the electrode the cut-out volume at the opposite boundary (see Figure II.11.c). In the manufacturing simulation case, the PBC was taken into account during the simulation but the coordinates of the additional sphere were not in the output list. That is why it was mandatory in the Artistic project to implement the PBC to the slicing algorithm to ensure the proper volume ratio between the different phases in the stack of images.

Computational Tools | 63 Due to the CGMD approach, the CBD phase was by essence a collection of spheres, hence discontinuous. Moreover, to avoid having too much overlapping during the CGMD simulations, the CBD particles were most of the time close to the surface of AM particles but not in contact. When using a fine resolution for the slicing, the CBD was then a discontinuous phase with little contact with the AM. This was not a problem when using the images to generate a mesh because the loss of resolution introduced by the meshing step was smoothing the CBD, resulting in a more continuous phase that had better contact with the AM. However, when performing simulations directly on the images of the structure (tortuosity, electrolyte impregnation), the CBD morphology was an issue.

To have a more continuous phase with a higher area of contact with the AM, a "dilate and erode" algorithm was developed. The workflow is the following, the dilatation consists in identifying the neighbors of the pixels in the outer layer of the CBD, and the ones that are electrolyte will be converted into CBD. This operation will be repeated the desired amount of iterations then the erode step will follow. During the latter, the exact amount of CBD pixels that have been added will be converted back to electrolyte to keep Computational Tools | 66 Computational Tools | 67 The very same methodology is used to generate the structure as with the spherical particles, i.e. the particles are added using the distance and overlapping of the particles as a condition to accept a particle but instead of using the radius and the center, an

Computational Tools | 69

effective radius and the centroid were used. Only the periodic boundaries condition algorithm will be different since it is no more perfectly spherical particles. It is noteworthy that before being added to the simulation box, particles must be rescaled to fit the desired resolution of the user as the particles arise from micro-CT with a fixed resolution of 642 nm.

II.3. Web App INNOV

All the previously mentioned functionalities (meshing, stochastic generation, slicing) are written as Matlab codes, meaning the user should know how to program in this language and have a valid software license to execute them. These restrictions drastically limit the number of users and the impact INNOV can have on the battery research community.

To address this point, INNOV was chosen to be offered to the battery field as a Web App, freely accessible through the Artistic Website. 1 The benefits of this approach are that through a Graphical User Interface (GUI) the user's experience can be highly improved by facilitating access to the functionalities, furthermore, no software license is required, which makes it entirely free to use, overall the user only needs a proper internet connection.

The following article goes through the functionalities of the Web App and the different built-in examples to help the user to understand the workflow of the application. 135 From the meshing algorithm reported in 2019, 134 on Web of Science with the keywords "battery" and "battery modeling"). This trend has been persistent despite improvements and promising computational modeling achievements in recent years (e.g. higher resolution, coupled electrochemical-mechanical model). [START_REF] Lu | 3D microstructure design of lithium-ion battery electrodes assisted by Xray nano-computed tomography and modelling[END_REF][START_REF] Danner | Thick electrodes for Li-ion batteries: A model based analysis[END_REF][START_REF] Ferraro | Electrode Mesoscale as a Collection of Particles: Coupled Electrochemical and Mechanical Analysis of NMC Cathodes[END_REF]126,138 Computational models can be very useful, especially time dependent

Computational Tools | 70 ones resolving processes in three Cartesian spatial dimensions (3D), for assisting in the design of components (e.g. electrodes) thanks to their capabilities to assess heterogeneities upon battery cell operation. [START_REF] Lu | 3D microstructure design of lithium-ion battery electrodes assisted by Xray nano-computed tomography and modelling[END_REF]154 However, there are numerous obstacles to the development of the modeling field, and most of them are related to the need of expertise outside of the electrochemist's conventional skillset. Firstly, the acquisition of a component (e.g. electrode, separator) mesostructure spatially resolved in 3D can be challenging if access to a tomogram is not available. The mesostructure will have to be computationally generated, typically using a stochastic algorithm. 117,153,155 "Stochastic" means here that the electrode generation will be performed through a random process that can be more or less controlled. For instance, spherical particles can be placed randomly in a

given volume representing the electrode, and these random positions can be chosen in a way to reach a given surface area between the particles and the electrolyte or a given amount of overlap between the particles. It means that launching the algorithm several times with the same input parameters will result in different outputs.

Then, a mesh that connects the component mesostructure to the computational model will be needed before proceeding with the numerical simulation through the solving of partial differential equations describing electrochemical and transport processes. Meshing is the way to divide the component mesostructure into finite elements (usually tetrahedrons) without deforming the original geometry. This step might be the most obscure to unexperienced users but must be well understood to avoid to avoid introducing errors resulting from complications in the computational representation of the mesostructure used in battery cell simulations. In a first effort to ease this step, we reported in 2019 an efficient meshing algorithm called "INdefinite Number Of phases meshing through

Voxelization" (INNOV) capable of importing both the active material (AM), pores and the carbon and binder domain (CBD) present in Lithium Ion Battery (LIB) electrodes into computational modeling software. 134 Since then, this algorithm has been deeply improved with several new features that have been implemented into a graphical user interface (GUI) to make it more accessible, which are presented in the present Concept.

GUIs allow users to control the software with interactive elements (e.g. push button, slider, etc…) rather than editing the code directly, which significantly reduces the required expertise.

With INNOV, the objective is to enable more participation to the battery modeling field thanks to a free and user-friendly tool that significantly accelerates the computational modeling process involving 3D-resolved components. It is noteworthy that very recently, efforts in a similar direction start to emerge ("Matbox" 156 ). Still, such efforts are scarce due to the complexity for scientists unfamiliar with both programming and modeling to undertake such initiatives.

Computational Tools | 71

II.3.1.2. INNOV Purpose
INNOV is a Web App freely accessible through our ARTISTIC project website 1 that offers a user-friendly tool for both new and advanced modeling researchers. All the computation will be performed in the project servers, which means that only a proper internet connection will be required. For the importation, the user can have the data as a stack of images, as a matrix with the native Matlab format, or, for the case of electrodes for instance, as a list of the coordinates and radii of AM and CBD spheres, with the CBD having a constant radius.

For the latter, an additional slicing step will be required (see Section II.2.1).

For the stochastic generation of electrodes, a built-in tool allows tailoring the ). An option to generate an additional "void" phase, particularly relevant for all-solid-state batteries, is also available.

Computational Tools | 74 Finally, users can add a separator that can be homogeneous, based on tomography data (e.g. Celgard PP andd Targuay PE 157,158 ), stochastically generated with the same algorithm used for the CBD generation, or as filaments (see Figure II.20). For models that use explicit current collectors or Li foil as counter-electrode (half-cell configuration),

i.e. 3-D resolved, homogeneous ones can be added to the final structure. The whole structure can be saved as a .tif file to be used for analyses in other software (e.g.

TauFactor 159 or GeoDict ®120 to calculate tortuosity factors).

Computational Tools | 75 The mesh 3-D rendering as well as several other functions were reused from Iso2mesh. 160 The mesh can then be exported in a variety of formats depending on the users' needs.

The mesh can be exported as a surface mesh (.ply or .stl) or as a volume (.bdf, .mphtxt, .mat). INNOV facilitates the importation of the mesh to COMSOL Multiphysics ® by using its native format .mphtxt into which built-in selections are embedded. However, the use is not restricted to COMSOL Multiphysics ® since the .bdf (Nastran) format is accepted by most software and the .mat format allows the user to convert it to its desired format. Computational Tools | 77

In the first example, the focus is on the positive electrode so a half-cell configuration with an implicit Li foil (i.e. no thickness considered) as negative electrode was chosen.

To simulate a calendered electrode, a porosity of The precision of the mesh was slightly refined to reach a mesh composed of ca. 650 000 tetrahedrons. The precision should always be adapted to the size of the system, the characteristic length of the system, and to the computational power available to perform the electrochemistry/transport simulations afterwards. Then the mesh was imported into COMSOL Multiphysics ® to perform the electrochemical simulations. Using a half-cell model similar to those already reported by our group, 124,154,161 INNOV is freely accessible through the Computational Portal in the ARTISTIC project webpage. To use it, the interested person needs to register in the ARTISTIC Computational Platform with her/his e-mail address and will receive a password. To extend the App capabilities, several new features such as the generation of non-spherical particles, CBD with PSD, or adding a separate app to perform basic image segmentation will be considered in the future.

To conclude, INNOV gathers expertise developed for several years within our group for a wide battery field audience to raise the interest in 3D-resolved battery computational modeling.

Computational Tools | 79 II.4.

Chapter Conclusion

In this chapter, the applications and algorithms developed to serve the ARTISTIC project have been detailed. They enable to tackle all the previously mentioned obstacles during a modeling study, i.e. the generation of 3-D structures (either stochastic or by processing the CGMD output) and the meshing of those electrodes with the AM and the CBD distinguished.

All these functions have then been embedded into a single graphical user interface available as a Web App in the ARTISTIC project website. This choice was motivated by the need to provide to the community user-friendly tools that do not require expensive computational resources. Indeed, with INNOV Web App, all the computation will be done in the ARTISTIC servers, meaning the user would only require a proper internet connection to use this tool.

The algorithms presented in this chapter will be used in all the studies undergone in the next chapters, and the development of these scripts was one of the biggest challenges of this Ph.D.. Now that the workflow to obtain 3-D structures, and to generate and import their meshes is established, the last impediment to simulations with the CBD explicitly con-

sidered is what will be the physics applied to this phase. Indeed, only one example of 3-D modeling with explicit CBD was reported in the literature by Danner et al. in 2016. 67 In their case, they considered implicitly the carbon as a surface film around AM, and the binder explicitly but they were not providing the equations applied to these phases.

To In the process of changing the paradigm on how human civilization uses and transforms the energy, Li-ion batteries (LIBs) play nowadays a predominant role, in part thanks to several decades of research and industrial development. 162,163 All the properties of these devices such as performance, cycle life and safety depend on the electrode mesostructure reached at the end of its manufacturing process. Because the electrode mesostructure plays a crucial role in the battery performance, [START_REF] Ramadesigan | Modeling and Simulation of Lithium-Ion Batteries from a Systems Engineering Perspective[END_REF]138,148 many experimental studies have been carried out in order to unravel the link between the manufacturing parameters and the cell electrochemical response. [START_REF] Yan | Three Dimensional Simulation of Galvanostatic Discharge of LiCoO 2 Cathode Based on X-ray Nano-CT Images[END_REF]114,164,165 Nonetheless, despite very significant progresses in the development of advanced 3D electrode imaging techniques, there is still a lack of complete understanding of the electrode mesostructural features. 138,139 Their correlation with the electrochemical behavior remains, in part, unclear due to the complexity of the

The Impact of CBD Location | 84 electrode texture and the disparate length scales involved (active material -AM-particles of ~1-10 µm size and carbon-binder domains -CBD-of ~50-100 nm size containing carbon additive particles and binder). Due to these challenges, the optimization of the LIB cell performance through its manufacturing process is mainly supported on a time-consuming trial-and-error approach.

Computational modeling gives the promise to overcome this flaw thanks to its predictive capabilities. 138 Based on physics, it has the potential to suggest ways to enhance the performance while reducing the R&D cost and time. Several modeling works have been published in order to study the influence of the electrode mesostructure on the electrochemical performance. [START_REF] Trembacki | Mesoscale Effective Property Simulations Incorporating Conductive Binder[END_REF]153,166 Such models have been revealed at some extent useful at understanding the discharge/charge behavior of the LIB electrodes. Nevertheless, due to the technical difficulties related to the consideration of the AM and CBD in the electrodes as separated phases, some simplifications are common in the models reported in the literature. These models consider either an electrode mesostructure generated from a stochastic distribution of both phases based on average properties (e.g. weight ratio between AM and CBD), either a hybrid approach with AM reconstructed from imaging techniques and a stochastically-generated CBD. [START_REF] Danner | Thick electrodes for Li-ion batteries: A model based analysis[END_REF]153,167,168 Such electrode mesostructures are used either to extract effective parameters (such as porosity and tortuosity) injected them in 1D or pseudo-2D models, either in 3D models solving electrochemistry and transport by considering AM and CBD as a single solid phase. The physics related to an explicit representation of CBD, as charge balance and flux between its interfaces with AM and pores is not fully addressed in the literature. Additionally, the impact of CBD properties on the overall performance is not analyzed in depth and no one considers both AM and CBD as explicitly separated phases both originated from the simulation of the electrode manufacturing process. In our previous publications, we have reported a first version of a multiscale computational platform able to predict the influence of some manufacturing parameters on the final electrode performance. Such a platform combines Coarse Grained Molecular Dynamics (CGMD) for the simulation of the slurry, coating and drying, with a 4D-resolved (three spatial dimensions + time) performance model describing electrochemistry and transport mechanisms. [START_REF] Ngandjong | Multiscale Simulation Platform Linking Lithium Ion Battery Electrode Fabrication Process with Performance at the Cell Level[END_REF]148 Despite the implicit consideration of CBD, the 4D-resolved model was able to capture its influence on the discharge performance through effective parameters (such as the electronic conductivity of an effective solid medium merging CBD with AM) [START_REF] Ngandjong | Multiscale Simulation Platform Linking Lithium Ion Battery Electrode Fabrication Process with Performance at the Cell Level[END_REF]148 Section III.1.2 presents a significant advance of that 4D-resolved performance model, where the three-dimensional spatial locations of both AM and CBD within the electrode, as predicted from CGMD simulations, are considered explicitly as separated phases. Such a new feature of the performance model permits the study of the impact of the CBD morphological properties on the overall electrode and cell performance, an aspect which is very difficult to assess based only on electrochemical experiments. The first approach

The Impact of CBD Location | 85 studied here is to consider the CBD as a blocking media towards Li + , i.e. to avoid its transport inside CBD. Although it might seem extreme, the primarily aim of the CBD is to provide an electronic conductive network, not to transport Li + . [START_REF] Trembacki | Mesoscale Effective Property Simulations Incorporating Conductive Binder[END_REF]118,169 However, such an assumption could result in underestimating the performance of the cell by increasing the tortuosity towards Li + transport in the electrolyte. The second approach consists in the representation of the CBD as a fully open medium for Li + , i.e. Li + is able to move in the CBD as freely as in the electrolyte. The Li + intercalation is assumed to occur at both AM/electrolyte and AM/CBD interfaces. This second approach might overestimate the performance of the cell by strongly favoring the Li + arrival to the AM. The last considered approach is an intermediate case between the first two, where Li + can diffuse through the CBD at a lower rate than in the fully open case, mimicking the influence of the CBD microporosity and/or of its swelling driven by the electrolyte. In this intermediate case, the Li + intercalation is assumed to occur only at the AM/electrolyte interface. The chosen value for the diffusion coefficient in the partially open case results from an effective consideration of the CBD micro-porosity and allows a transition from thermodynamic to kinetic driven system as the C-rate increases. The effective Li + diffusion coefficient in the electrolyte is a function of the porosity and of the ratio between the constrictivity and the tortuosity following Equation III. 1. 170,171 The constrictivity reflects the hindering of the mass transport when the size of the pores is close to the mean path length of the ions moving in the electrolyte. It is a dimensionless constant ranging between 0 (the pores are smaller than the solvated ion size) and 1 (the pores are much larger than the solvated ion size). In the case of the electrolyte in the bulk, δ is equal to 1.

, =

(III.1 )

However, for the partially open hypothesis, Li + can diffuse through the CBD micro-pores.

In this case, it is appropriate to choose a constrictivity smaller than 1 to take into account the small size of the pores. The ratio between the constrictivity and the tortuosity is set at 5×10 -3 .

III.1.2.2. Methodology

In order to achieve this study, the use of a meshing technique constitutes the first step for the importation of the electrode mesostructures into the 4D-resolved electrochemistry We then use our 4D-resolved performance model of a Li-foil/separator/NMC electrode cell to first evaluate how the differences in the NMC electrode mesostructural parameters arise in differences in the electrochemical response. Equations, parameters and other computational details are provided in the Appendix A2. Figure III.4. Comparison between discharge curves at different C-rates for the same 96:4 structure with a coarser mesh (7 Mo), a regular mesh (10 Mo) and a finer mesh (16 Mo).

The Impact of CBD Location | 89 III.1.2.3.

Results and Discussions

What it follows from the geometrical features is that a lower content of CBD (4% vs. 6%) leads to a larger active surface area for the AM (74% vs. 70%). The slight differences of these two electrodes is emphasized by the simulated discharge curves at 0.8 C. Here CBD is considered blocking, i.e. Li + cannot diffuse through this phase. In fact, in Figure 2b they show rather similar specific capacities with regard to the dry electrode mass (mass of AM plus CBD). Since the difference between the two compositions is small, an important gap between capacities is not to be expected. However, Figure III.3.c displays a slight difference in specific capacities in regards to the AM mass. Indeed, the higher AM content electrode has a poorer specific capacity compared to the 94:6 one because its electronic percolation network is less effective than the one of a higher CBD content electrode. [173][174][175][176] This evolution of the specific capacity per mass of AM with the CBD content is also reported at the experimental level. 148,174 This demonstrates that the meshing part can shed light on simple trends and that the 4D-resolved performance model is able to capture the expected behavior in regards to the cell performance. From an experimental point of view, the importance of the carbon additives 177-179 and binder 180 mesostructures in rechargeable batteries has already been acknowledged at the experimental level. The small size of the conductive additive and binder, compared to the active material, provides microporosity where lithium can diffuse/migrate through The Impact of CBD Location | 90 and that can modify the effective electronic properties of the CBD. [START_REF] Trembacki | Mesoscale Effective Property Simulations Incorporating Conductive Binder[END_REF] Our model is not able yet to reflect explicitly microporosities, so an effective transport coefficients are used to capture their impact on Li + diffusion/migration through the CBD (see Table A2 The partially open approach provides an intermediate scenario where features under both hypotheses can be observed. Indeed, Li + concentration in CBD is obviously higher than in the case of blocking CBD however there are also some sharp Li(s) concentration gradients, e.g., in the lower edge of Figure III.6. Therefore, the model is able to capture the expected trends regarding the different hypotheses: it predicts heterogeneous intercalation of lithium in the AM when the CBD is blocking and the homogenization when the CBD is fully open. As far as we know, this is the first time that a 4D model predicting explicitly the CBD spatial location impact on the electrochemical response is reported. Further investigations are planned, including performing simulations with electrode mesostructures with other compositions ranges, using higher C-rates or with mesostructures from tomography imaging: for the latter, a proof-of-concept simulation result is presented and discussed in the Supporting Information. This model paves the way to more sophisticated studies, for instance considering the swelling of CBD depending on the electrolyte salt concentration using a dynamic mesh and focusing on the impact of solid/cathode electrolyte interphases on cell aging.

III.2. Case Study: Diffusion Coefficient Extraction

During the previous study, when choosing the diffusion coefficient of inserted Li in the NMC, we noticed a huge discrepancy in the values reported in the state-of-the-art. Confronted to this curiosity, we decided to investigate through a combined experimental and modeling approach the reasons for the difference between the reported diffusion coefficient values.

The objectives were to acquire the diffusion coefficient as a function of the state of lithiation of the NMC used in the project and also to unravel the cause of the discrepancy in values reported.

For the first time to our knowledge, the galvanostatic intermittent titration technique was modeled in 3-D, allowing us to provide valuable modeling insights on the limitations of the formula used to extract the diffusion coefficient.

For the comfort of the readers, the relevant paragraphs and figures from the supplementary information have been added to the article. 161

III.2.1.1. Introduction

In the energy materials field, there is an increasing urge to develop new active materials (AMs) for lithium-ion batteries (LIBs) using elements not linked to geopolitical issues or limited supplies. During the screening of AM candidates, the comparisons are most of the time based on their capacity, working voltage and diffusion coefficient of lithium in the AM (DLi). The easiest experimental ways to obtain the latter is through potentiostatic or galvanostatic intermittent titration techniques (PITT and GITT, respectively) applied to half cells (i.e. the electrode made of the AM under investigation vs. a lithium foil counter/reference electrode). PITT and GITT consist in applying a pulse of potential or current respectively, followed by a rest period long enough to reach the equilibrium before the next pulse. In PITT and in GITT, DLi is usually extracted by applying an equation derived from the combination of the first and the second Fick's laws and the Butler-Volmer equation to the output current or potential. 184,185 For the case of GITT, chosen here as case study, the equation depends on the geometry assumed for AM particles. For instance, for spherical AM particles such equation is given by

= 4 9 ∆ ∆ ( . )
where R is the AM particle radius, τ is the pulse time, ∆Es is the change of steady state voltage and ∆Et is the total change of cell voltage during a current pulse neglecting the ohmic drop (Figure III.7.a). This equation has been used in the LIB field since its derivation, five decades ago. 184 The first assumption of this expression is that it is valid for one single AM spherical particle. It is extended to many particles systems by assuming that the ratio between the surface area and volume of AM is constant. The other assumptions are that the Li diffusion in the AM is one-dimensional (radial) and that the electrochemical double layer charging (at the interface between the AM and the electrolyte) and the phases transformations (within the AM) are neglected. Furthermore, the equation assumes that the charge transfer process throughout the AM and the carbon binder domains (CBDs) is neglected. In addition, the applied current has to be small enough so the variation of potential at each time step is also small. The calculated GITT diffusion coefficient should respect the criteria

≪ = ( . )
where τdiff represents the characteristic diffusion time. 186 The Impact of CBD Location | 94

However, in the literature there is a tremendous variability in the values reported for a given AM, as it has been discussed by Bazant et al. for the specific case of the multiphase AM Graphite. 187 This is also true for solid solution materials (for which solid phase separation does not occur): for instance, DLi for Li Ni1/3 Mn1/3 Co1/3 O2 (NMC-111) ranges from 4 ×10 -18 to 1.8 ×10 -13 m 2 .s -1 in various modeling and experimental papers [START_REF] Noh | Comparison of the structural and electrochemical properties of layered Li[END_REF][START_REF] Amin | Characterization of electronic and ionic transport in Li1-xNi0.33Mn0.33Co0.33O2 (NMC333) and Li1-xNi0.50Mn0.20Co0.30O2 (NMC523) as a function of Li content[END_REF][START_REF] Roberts | A framework for threedimensional mesoscale modeling of anisotropic swelling and mechanical deformation in lithium-ion electrodes[END_REF][188][189][190][191] as it is evidenced in Figure III.7.b. There is a lack of consensus about the origin of those discrepancies: researchers have speculated that the infinite charge transfer hypothesis might be the cause of the variability, but without any experimental proof. Section III.2.1 aims to investigate the origins of these differences in the literature and evaluate the limitations of Equation III.2 through an original study combining GITT data coming both from experimental and 3D modeled NMC-111 positive electrodes. NMC-111 is adopted here as a case study because it is one of the most popular spherical AM for LIB positive electrodes. Moreover it does not manifest solid phase separation upon (de-) lithiation which makes easier the study and analysis reported in the following than in the case of materials such as Graphite. 187 Our study workflow is organized as follows: firstly, experimental GITT data and Equation III. Models simulating GITT have been already reported, based on the single particle model 192,193 and the pseudo-2D electrochemical Newman model. 188,192 The former considers a system of a single sphere which is not representative of a real electrode mesostructure but it is closer to the assumptions made to obtain Equation III. ). This polarization is mainly due to the steep drop in electronic conductivity of fully lithiated NMC-111. [START_REF] Amin | Characterization of electronic and ionic transport in Li1-xNi0.33Mn0.33Co0.33O2 (NMC333) and Li1-xNi0.50Mn0.20Co0.30O2 (NMC523) as a function of Li content[END_REF] As a consequence, the DLi extracted from the charge was used as an input for the simulation.

The Impact of CBD Location | 96 To create the digital twin of this experiment, a 96:4 AM:CBD (weight ratio) electrode was generated through an in house CGMD simulation that we have already reported. [START_REF] Ngandjong | Multiscale Simulation Platform Linking Lithium Ion Battery Electrode Fabrication Process with Performance at the Cell Level[END_REF]154 In short, CGMD constitutes a physical model which simulates the slurry preparation and inactive phase (first row) and one free of CBD (second row). In this case, blocking CBD condition is analyzed in order to emphasize the effect. Before the pulse they have roughly the same lithiation, yet during the pulse the distribution evolves differently for the two.

We observe steeper gradients for the particle covered with CBD than for the one free of inactive phase. This demonstrates the impact of CBD location on the dynamic evolution of solid Li concentration during GITT pulse. This effect has not been studied so far in previous works and points out, on one side, the importance of the inactive phase mesostructure, and on the other, reinforces the fact that the DLi directly obtained from a GITT measurement using Equation III.2 is an apparent property.

Moreover, Equation III.2 assumes a unique radius for the spherical AM particles, yet in our experimental and modeling system, AM radii can range from 0.65 μm up to 8 μm 11). It is evident that due to transport limitation in bigger particles, the gradient of lithiation will be steeper. Furthermore, Equation III.2 is valid for noninterconnected particles because it assumes that all the surface of the AM is in contact

(Figure III.
The Impact of CBD Location | 100 with the electrolyte. In the model/experimental case, interparticle solid Li diffusion takes place as a consequence of inhomogeneous gradient of concentration through the NMC phase. All these contributions lead to higher polarization, introducing a dispersion of concentration gradients that depends both on the AM radii distribution and the electrode mesostructure.

It is noteworthy that the DLi extracted from the simulated electrode is different from the experimental one. Even though the flaws of Equation III.2 have been highlighted, the simulated electrode shares the same particle size distribution as the experimental one and uses DLi taken from the latter as input for the physical model. Moreover, the simulated electrode arose from a physical model (CGMD) of the manufacturing process, so the geometrical properties should be similar between model and experiment. Thus, the contributions of the particle size and inactive phase location should affect in the same manner the two electrodes, resulting in the same (or very similar) DLi. Nonetheless, the magnitude of the simulation-derived DLi is 5 times smaller than the experimental one.

The only difference between the two systems is the volume of the electrode, 6.64×10 -9 m 3 for the experiment while 4.83×10 -14 m 3 for the simulation. Due to the smaller size of the model system, the heterogeneity and anisotropy (mesoscale) contributions unaccounted by Eq. ( 1) would be even more significant in the model. However, regardless the initial DLi values assumed in the model physical equations, the DLi values extracted by applying Equation III.2 on the simulated GITT will be different from the initial DLi values due to these mesoscale effects.

Normalized Distribution

The Impact of CBD Location | 101 III.2.1.3. Conclusion Section III.2.1 aims to identify the origins of the observed differences in value of diffusion coefficients through an original approach. The main contributions of the deviation from the assumptions linked to Equation III.2 were highlighted thanks to the insight provided by 3D modeling. To the authors' knowledge, this is the first time a 3D GITT model including CBD is reported in the field. The benefit of such a model is to provide a tool to have access to more observables or to generate reliable results in a shorter time. In this letter, it is provided for the first time physical insights on the limitations associated with the most widely used equation to extract DLi from GITT experiments. In addition, the workflow presented in this work could be implemented also in order to evaluate the error on the diffusion coefficient value for a given electrode based on its composition and fabrication process parameters.

III.3. Chapter Conclusion

In this chapter, the importance of the CBD location was highlighted firstly through the behavior of this phase in the 3-D model by investigating three different hypotheses.

It was shown that the lithiation process will be highly impacted by the physics applied to the CBD and will entail heterogeneities based on the CBD location. From this study, an intermediate hypothesis was chosen for the CBD, where the Li + can diffuse through it but not intercalate at the AM/CBD interface.

Then, the same methodology was applied to model an electrochemical technique used to extract the OCV curve, the diffusion coefficient of Li in the AM, or the kinetic coefficient, the GITT. This modeling endeavor demonstrated that the environment of the AM particles, i.e. the contact with the CBD or other AM particles, can impact the electrochemical response, hence an impact also on the parameters that can be extracted. This phenomenon, challenging to observe experimentally, could explain the discrepancy of reported diffusion coefficient values for Li in a given AM (here NMC-111). Indeed, the experimental conditions are never strictly identical between different groups and these differences can lead to a different environment for the AM particles.

The Impact of the Mesostructure In this chapter, the link between mesostructural parameters and the electrochemical behavior of an electrode will be investigated. The influence of the calendering (Section IV.1), the morphology of AM upon Solid Electrolyte Interphase formation (Section IV.2 through an article published in 2021) 195 , and the effect of a heterogeneous CEI (Section IV.3) will be studied below. The focus on the calendering will demonstrate the capability of the ARTISTIC workflow to link the manufacturing parameters to the performance of a battery through the addition of the calendering modeling.

IV.1. Impact of Calendering

Calendering is a crucial step of the manufacturing process that takes place after the coating/drying process and before the cell assembly/electrolyte impregnation. It consists in applying pressure with two rotating rolls with the electrode in the middle in order to decrease the amount of pores and to improve the contact between the solid phases. It is an important step during the electrode fabrication, hence it must be modelled within the ARTSTIC chain.

The impact of the calendering on the mesostructural parameters and the electrochemical performance will be described in this section.

IV.1.1. Prediction of Mesostructural Observables

The first approach aimed to predict features such as the tortuosity or the porosity for a given pressure applied during calendering. It would greatly benefit experimentalists in their optimization process, allowing them better control and understanding over the manufacturing step. The following endeavor led to an article by Duquesnoy et al. in 2020. 117 IV. 1.1.1. Methodology The workflow applied during this study is represented in This is both the strength and the weakness of this study, as it is more realistic since it is based on experimental data, but also dependent on experimental results that will require to do many measurements, hence not a significant decrease in the number of experiments compared to a naïve trial and error approach. The data were provided by a team member in charge of the experimental aspect, Dr. Emiliano Primo.

The Impact of the Mesostructure | 106

The second step B is the stochastic generation of 3-D structures, exploring the space made out of three parameters, i.e. the initial porosity, the AM content, and the calendering pressure (the initial thickness was kept constant at 100 µm). The initial porosity ranged from 40% to 50%, the AM content from 93% to 96%, while the calendering pressure from 20 MPa to 170 MPa. Based on these structures, relevant observables could be extracted during step C in an automatized manner, allowing to have a dataset of more than 4400 structures for 440 data points (10 repetitions per data point). The selected output observables were the tortuosity for both solid and liquid phases (with TauFactor 159 ), the contact between the current collector and the AM and CBD, and lastly the active surface area. They all have a significant impact on the electrochemical response of an electrode under cycling and are all likely to change upon calendering.

Then, the observables fed a machine-learning algorithm (step D) in order to predict the observables for cases that are not in the dataset and to build an optimization map with the contribution of each input on each output (step E). Both steps D and E were performed by Marc Duquesnoy, so to have further details on the machine-learning process, the reader should read the original article as it is not the author's area of expertise. 117

IV.1.1.

Results

Thanks to the ML algorithm, the output observables can be interpolated between values from the initial dataset, allowing to plot graphs in Figure IV.3 for a given initial porosity of 48%. As expected, with increasing calendering pressure, the tortuosity of the

The Impact of the Mesostructure | 108 electrolyte increases while the tortuosity of the solid phase decreases because of the decrease in porosity. Also, the active surface area is directly proportional to the AM content and the calendering pressure, as with less CBD and more pores, the percentage of active surface area will increase. Regarding the percentage of the surface of the current collector in contact with CBD, it will obviously increase with a higher CBD content, hence a lower AM content, and with the calendering pressure that will improve the solid phase contact. The color green indicates that the relation between the input and output is direct and the color red is inverse. For instance with increasing porosity after calendering, the tortuosity of the liquid phase will decrease, which means a faster ionic transport, hence a red circle.

The Impact of the Mesostructure | 109

Conclusion

Although all these interdependencies could have been guessed by a battery researcher, the strength of this methodology resides in its ability to be applied to predict more complex input/output relations. With this approach, one can quantify the contributions of each input on the relevant observables and have a clear idea of how to optimize its system.

Nonetheless, to uncover the impact of calendering upon discharge, 3-D physical modeling will be needed to access dynamically the observables. The following section will present discharge simulations on half-cells with different degrees of calendering for the positive electrode.

IV.1.2.

3-D Electrochemical Modeling

In this section, the 3-D model presented before will be applied to different positive electrodes to which different calendering pressures were applied. The structures arose from CGMD simulations and the modeling results reported here were part of an article on the simulation of the calendering process by Ngandjong et al. in our group. 124 The Impact of the Mesostructure | 110 A displays the meshes used in this work, with the full thickness of the positive electrode considered to capture not only the mesostructural changes but also the variation of thickness of the electrodes. Upon discharge at 1 C, a difference in specific capacity is observed for the three electrodes in Figure IV.6.B, i.e. a higher capacity with higher calendering pressures (lower porosities). It is noteworthy that the uncalendered electrode appears to have the least ohmic drop during most of the discharge, while the two calendered electrodes follow the expected trend. An opposite behavior could have been expected for the uncalendered since the calendering step should improve the contact between the solid phase, hence a higher electronic conductivity. In this case, this inconsistency is probably due to the calendering simulations that may result in structures

The Impact of the Mesostructure | 111 having poorer contact with the current collector compared to the initial uncalendered ones. It arises from the CGMD (drying process) model having periodic boundary conditions along the z-axis (thickness of the electrode) while the calendering simulation does not. Work is undergone to develop a drying model without periodic boundary conditions on the z-axis to match with the calendering model. Thanks to the 3-D simulations, we can access many local observables that could explain the differences in specific capacity with increasing calendering pressure. Upon calendering, one could expect to have a change in the electrolyte transport as the porosity is reduced, a change in the electronic percolation as the particles are brought closer to one another, and the surface area of AM in contact will also evolve, as seen previously in Section IV.1.1.

The average Li + concentration gradient from bottom to top was extracted to account for the efficiency of the electrolyte transport. The inverse of this gradient was reported in Figure IV.6.C so that the higher will be the gradient, the lower the electrolyte transport observable will be. A decline is observed with lower porosities as the Li + will have fewer pathways to reach the bottom of the electrode.

Furthermore, regarding the CBD transport parameter, it was defined as the inverse of the average standard deviation of local current density at the CBD surface. Meaning that if there is a poor electronic percolation network in the electrode entailing high fluxes of current in some local regions, the standard deviation of the current flowing through the CBD will be high, and the CBD transport (i.e. the inverse of that standard deviation) will be low. A heterogeneous distribution of the current will lead to a CBD transport value close to 0, as it is the case for the uncalendered electrode's value which is almost 3 times lower than for the 27.2% electrode.

Finally, the evolution of the active surface area is consistent with the results from the previous approach in Figure IV.4 with a decline with increasing calendering pressure.

From the analysis of these observables, it transpires that the most significant difference between the three electrodes is the CBD transport as it is where the gap between the uncalendered and most calendered electrodes is the largest. From The Impact of the Mesostructure | 113 IV. 1.2.3. Conclusion This work sheds light on the aptitude of the coupled manufacturing and electrochemical simulations to capture the impact of the calendering process on the electrochemical performance of an electrode. The trends highlighted in the previous study in Section IV.1.1 were validated through this work with the active surface area, the CBD and electrolyte transport. Also, the improvement of the electronic percolation upon calendering was proved to be essential to prevent strong SoL gradients in the electrode.

IV.2. Impact of the AM Morphology: SEI Formation

Another important component of the mesostructure is the morphology of the AM particles since it will directly impact the active surface area, the diffusion of inserted Li, and the arrangement of the other phases around the AM. In the meantime, the project had to demonstrate flexibility in terms of the chemistries modeled so the chosen system for this study was a Graphite electrode.

To investigate the effect of AM morphology, two different shapes were selected, spherical and ellipsoidal particles. Then, the Solid Electrolyte Interphase (SEI) formation was simulated, followed by a charge at a higher rate to analyze the differences due to the Graphite morphology. For the comfort of the readers, the relevant paragraphs and figures from the supplementary information have been added to the article. 195 IV.2.1. Article 5

IV.2.1.1. Introduction

Nowadays Graphite is one of the most used negative electrode materials in lithium-ion batteries (LIBs). 196 Its ability to intercalate Li has been known since 1975 [START_REF] Guerard | Intercalation of lithium into graphite and other carbons[END_REF] but all endeavors to achieve reversible intercalation were unsuccessful because of Graphite exfoliation and instabilities in the electrolyte. 197,198 The first reported reversible cycle with Graphite using a polymer electrolyte was done in 1983 by Yazami and Touzain. [START_REF] Yazami | A reversible graphite-lithium negative electrode for electrochemical generators[END_REF] Later, in 1990, [START_REF] Fong | Studies of Lithium Intercalation into Carbons Using Nonaqueous Electrochemical Cells[END_REF] Dahn et al. highlighted the possible use of ethylene carbonate (EC) as a liquid electrolyte in order to form a passivating layer around the Graphite, now known as solid electrolyte interphase (SEI). Further investigation on the electrolyte formulation led Guyomard and Tarascon to identify, in 1993, a dimethyl carbonate (DMC) and EC mixture as an excellent candidate to improve the cyclability of Graphite-based electrodes.

31

Despite tremendous efforts to better understand and control the SEI formation its mechanism has been subject to debate for 40 years. [START_REF] Peled | Review-SEI: Past, Present and Future[END_REF] For instance, it is disputed whether the electrolyte reduction occurs at the Graphite/electrolyte interface, or at the SEI/electrolyte interface, with the electrons transferring from the Graphite to the SEI

The Impact of the Mesostructure | 114 by tunneling effect. 199 Numerous first principles and mathematical models have been reported to shed light on the SEI growth process, [199][200][201][202][203] but still at a very local scale (electrode/electrolyte interface) or by oversimplifying the actual porous electrode mesostructure, described either through effective parameters (e.g. porosity and tortuosity factor like in the so-called pseudo-2D model), either through a 1D representation of the system. It is noteworthy that these type of models have been also used in the battery context to simulate lithium plating. 145

Among the latter approaches, we can refer to the work of Pinson and Bazant 204 in which the rate of growth of the SEI iSEI was described with a Butler-Volmer law in the same manner as the Li intercalation rate

= - - (IV.1)
where kSEI is a kinetic constant; ce and cs are the Li + concentration and Li concentration at the SEI interface, respectively, and ηSEI is the overpotential taking into account the resistance due to the SEI. Following a similar approach, Kolzenberg et al. 205 defined the rate of growth of the SEI as directly proportional to the intercalation rate iint

= + -̃ (IV.2)
where c ο is the standard concentration, DLi is the diffusion coefficient of Li in the SEI, σLi+,SEI is the ionic conductivity of Li + across the SEI layer, and , is the dimensionless overpotential due to the Li formation. Despite these approaches provide very useful formulas to correlate the SEI growth with experimental observables, 2-D approaches are incapable, by construction, to capture the impact of the porous electrode threedimensional features, such as active material and carbon-binder spatial heterogeneities on the growth process and its impact on the electrode performance. The explicit representation of such mesostructure features is of paramount importance since the SEI formation may initiate at the Graphite/electrolyte interface, so an implicit consideration of the carbon and binder domain (CBD) would not capture the actual active surface area loss and the electron transport heterogeneities due to the CBD percolation network. Section IV.2.1 aims to propose a methodology to investigate the seldom-discussed topic of heterogeneous SEI, for the first time to the authors' knowledge through a 3Dresolved continuum modeling approach with CBD explicitly considered at the mesoscale.

For demonstration purposes, we focus here on different half-cells with Graphite as the positive electrode's active material (AM) with two different morphologies. The concerned Graphite electrodes are 35 % porous and were stochastically generated using the software resistance due to solid-state diffusion when compared to spherical Graphite. 206 Then, 8 % of volume of CBD was added, using an in-house MATLAB-based algorithm already reported by us. 117 A 6 µm thick homogeneous separator was added on top of the Graphite electrodes (75×75×75 µm 3 ), the positive electrode's current collector and the Li foil were considered implicitly. The electrode mesostructures were then imported into software COMSOL Multiphysics ® as meshed volumes using our INNOV algorithm, 134 where the classical Newman model [START_REF] Lu | 3D microstructure design of lithium-ion battery electrodes assisted by Xray nano-computed tomography and modelling[END_REF]154,161,207 was implemented in three dimensions. Then, an interfacial SEI domain is added only at the Graphite/electrolyte interface, the SEI being considered implicitly due to its nanometric thickness. Based on Ref. 205 we assume here that the Li + can either be inserted in the Graphite or in the SEI where it will later react with the solvent to form SEI as it follows

+ → (IV.3a) + → (IV.3b)
Similarly to equation (IV.1) ,the SEI growth current is assumed in our model to be controlled by

= - - (IV.4)
As a consequence, the SEI will grow faster where Li + concentration is high and where the overpotential is low. Since the system under study is a half-cell (infinite reservoir of Li at the negative electrode) and the aging is out of the scope of this study, the first discharge will trigger an accelerated SEI formation. Therefore, it is assumed that the SEI layer will not change during the following charge, so no Li + will be consumed for the SEI formation during the charge.

More information regarding the electrode mesostructures generation and the physical model can be found in the Appendix A4.

The Impact of the Mesostructure | 116 The Impact of the Mesostructure | 117

IV.2.1.2.

Results and Discussion

The simulations consist in a discharge at C/20 vs lithium (equivalent to a charge if the Graphite is used as the negative electrode) during which the SEI will be formed and a charge at 1C. During charge, the SEI thickness is assumed to remain constant and to impact the electrode only as a resistive layer for Li deintercalation. The results of this cycle are then compared to a cycle of the same half-cell but without any SEI contribution.

The simulated cycling curves for the spherical and ellipsoidal Graphite are reported in Figure IV.8a. For both spherical and ellipsoidal Graphite, the discharges with and without SEI are almost identical. As expected, the charge with the SEI displays a higher ohmic loss than the one without SEI for both spherical and ellipsoidal electrodes.

However, for the ellipsoidal case, the ohmic loss is much higher than the spherical one, on average 0.147 V vs. 0.033 V respectively when compared between the charge without SEI. This trend would suggest that the impact of the heterogeneous SEI formation differs according to the active material particle morphology. In addition, the electrode with ellipsoidal particles demonstrates a specific capacity much lower than the spherical one (27 vs. 280 mAh.g -1 respectively for the charge with SEI). Nonetheless, this dramatic difference does not arise from the SEI but rather from a limitation at a high rate of the electrode morphology itself (as detailed later) since poor yield is also observed for the electrode with ellipsoidal particles after the charge without SEI (103 mAh.g -1 ).

To quantify the SEI heterogeneity, the calculated distribution of its thickness within the electrode volume is represented in Figure IV.8b for both ellipsoidal and spherical Graphite at the end of discharge. The variation of values of the SEI thickness is higher for the ellipsoidal electrode (standard deviation σSTD = 21.1 nm) than for the spherical Graphite (σSTD = 11.6 nm). It is noteworthy that the average value of SEI thickness is higher for the spheres than for the ellipsoids (45.6 nm and 39.3 nm respectively) because the active surface area is higher for the ellipsoid 206 (7.79×10 -8 m versus 7.29×10 -8 m for the spherical) and the time of discharge is lower for the ellipsoid (54 629 s versus 63 272 s for the spherical). The histogram used for the analysis proves the capability of this modeling approach to form heterogeneous SEI layer dependent on the Graphite electrode mesostructure.

To further assess the SEI heterogeneity, the anisotropy of the SEI thickness has been Indeed, during the charge, Li is deinserted from the positive active material and travels through the electrode and the separator to be deposited into the negative active material.

Since we consider high kinetics for the deposition of Li on the Li foil, the process is limited by the transport of Li + , hence this gradient of concentration. Interestingly the gradient is much more pronounced for the ellipsoidal electrode with the electrolyte concentration being almost 10 times higher near the current collector than near the separator. This steep concentration gradient entails the poor specific capacity and the higher ohmic loss during charge reported earlier for the ellipsoidal electrode displayed in becomes more favorable for the lithium to be deinserted at Sthick where the lithium content is the highest, even though the SEI is the thickest. Indeed, the rate of delithiation increases by 60 % after 1500 s at Sthick while only by 10 % at Sthin, hence a homogenization of the system and a decrease of the difference in average lithiation between the two subsurfaces is observed. Regarding the electrode with ellipsoidal particles, the same increase of the difference in average lithiation is reported but since the charge could not last long because of the ionic transport limitation, the following decrease could not be observed.

It is noteworthy that the trend is reversed for the ellipsoidal compared to the spherical case, indeed the charge without SEI has a higher heterogeneity in terms of lithiation than the one with SEI. As seen previously, part of the electrode with ellipsoidal particles will be highly favored for the deinsertion because of a poorly connected electrolyte phase. So during the charge, the lithium will not have as much freedom as in the spherical Graphite to be deintercalated where the SEI is the thinnest: it will mainly be constrained to react in the upper part of the electrode, near the separator. It entails a higher deintercalation rates at Sthick (5.50 × 10 -4 s -1 ) than at Sthin (4.26 × 10 -4 s -1 ) for the charge with SEI, which is also a cause of the high ohmic loss reported previously. Since the SEI layer is the thickest in the upper part of the electrode, the SEI will still favor a deinsertion at the bottom of the electrode, where it was the least favorable in the case of the charge without SEI. It will tend to reduce the heterogeneity in the system, hence the observed trend between the two ellipsoidal case curves in Figure IV.10c. However, this homogenization of the system is not beneficial to the overall performance of the electrode, since it occurs

The Impact of the Mesostructure | 121 at the prize of a significant ohmic loss, and the poor utilization of the cell remains due to a poorly connected electrolyte phase. As explicited earlier, this is due to the SEI being thinner where it is the least favorable for the Li to be deinserted in the case without SEI. For instance, in Figure IV.10(c,d), in this slice near the bottom half of the electrode with ellipsoidal particles (37 µm < z < 40 µm), the average SOD is lower in the case with SEI than compared to the case without SEI. It was shown that due to ionic limitation, this region of the electrode with ellipsoidal particles was poorly utilized. During the charge with SEI, it will then be more favorable (thanks to the thinner SEI) for the Li to be deinserted in this region where it was the least favorable before, entailing a homogeneization of the SOD. Indeed, the model highlights the influence of the electrode's mesostructure in the SEI formation process, and its impact on the electrochemical performance. In both cases, it results in an ohmic loss but the SEI will have a higher influence on the ellipsoidal Graphite since the system does not have the time to equilibrate along the charge because Sthick never becomes more favorable for the deintercalation of lithium.

The Impact of the Mesostructure | 123

IV.2.1.3. Conclusion

To conclude, this work presents a novel 4D-resolved computational approach to assess heterogeneous SEI growth impact on Graphite porous electrode performance. However, there is still room for improvement for the description of the SEI formation step in the model, for instance by introducing several formation cycles. Nevertheless, the model has proven to be capable to form heterogeneous SEI layers and predict their spatial distribution as function of the Graphite type (spherical or ellipsoidal particles). It was found that the thickness of the SEI becomes more homogeneous in the spherical electrode where the ionic transport limitation is the mildest, which leads to a better electrochemical performance. We believe that this work provides an insightful tool to assist experimental SEI analysis and, we hope, to stimulate future advanced characterizations of this complex and fascinating interphase.

The Impact of the Mesostructure | 124 IV.3. Impact of a Heterogeneous CEI

Once the formation of the layer arising from the decomposition of the electrolyte has been studied for the Graphite (negative AM), one should also investigate its impact on the positive AM of interest in this manuscript, i.e. the NMC.

Modeling-wise, few articles have investigated the influence of the Cathode Electrolyte Interphase (CEI) that forms at the surface of the NMC on the electrochemical behavior of an electrode. The main approaches used are the density functional theory or the molecular dynamics to study the composition of the CEI locally at the atomic scale. 208,209 More recent endeavors built a transmission line model in order to quantify the additional resistive term induced by the CEI, thanks to FIB-SEM images. 210 Nonetheless, there is no report of continuum models accounting for the formation of the CEI because of the lack of theory and complexity of this phenomenon. It is noteworthy that the formation of the SEI (as seen in the previous section) has been attracting more interest in the modeling community with continuum models at different scales. 195,[199][200][201][202][203] This endeavor led to an article that is in the writing process and that will be referenced here by the time of the final version of the manuscript.

IV.3.1. Methodology

Unlike the SEI in Section IV.2, the CEI layer formation will not be dynamically resolved but rather a heterogeneous distribution will be assumed and its impact along different hypotheses will be investigated. A discharge at a rate of C/2 will be simulated Four different hypotheses will be investigated and compared to a pristine case without CEI to capture its influence on the electrochemical behavior of the electrode.

The first two are the "CEI" and "Thick CEI" cases, in the latter the CEI distribution is the exact same as in Figure IV.13, except the thickness of the CEI was multiplied by a factor 2. The heterogeneous CEI distribution has been built based on TEM images after 15 cycles, so the "Thick CEI" scenario simulates an electrode after several hundreds of cycles.

The third hypothesis is that since electrons are needed in the electrolyte decomposition reaction, it might be favorable at the surface where the electronic conductivity is the highest, i.e. at the carbon surface. The carbon is several orders of magnitude more conductive than NMC and is taken into account in the model through the CBD. It is reported in the literature that this merger of carbon and binder has nanometric pores 211 that allow the Li + to diffuse through them, and reach the surface of the NMC where it will react. If the CBD is indeed the most favorable place for the electrolyte decomposition to happen, then the CEI layer could clog the pores of the CBD, hindering the ionic transport of this phase. The "CBD clogging" hypothesis adds to the CEI heterogeneous layer a blocking behavior of the CBD toward the Li + .

The last and most extreme hypothesis is that the CEI could also clog some small pores or thin throats between particles at the electrode scale. This would result in a rise of the

The Impact of the Mesostructure | 127 tortuosity in the electrolyte phase, entailing a hindering of the ionic transport in the bulk of the electrolyte. In the "Full clogging" hypothesis, in addition to the "CBD clogging", an effective diffusion coefficient has been assigned to the bulk electrolyte to account for an increase of the tortuosity by a factor 2.

IV.3.2. Results

The obtained CEI thickness for all the cases except the "Pristine" case and the "Thick CEI" hypothesis is represented in Figure IV.14.a. As it was observed in the TEM images, the CEI thickness varies both at the electrode and particle levels.

The simulated discharge curves at C/2 are reported in Figure IV.14.b for each case, with a cut-off voltage at 3.5 V. The case that performs the best in terms of specific capacity is the "Pristine" one (120 mAh.g -1 ) and the one that demonstrates the lowest capacity is the "Thick CEI" hypothesis with . Other than the specific capacity, an ohmic drop can be seen when comparing the cases with CEI to the pristine case.

Moreover, the average state of lithium in a cross-sectional plane (xy) is reported in Figure IV.14.c as a function of the distance from the current collector at the end of discharge. It transpires that the milder gradient of concentration is attributed to the "Pristine" case, while the steeper one belongs to the "Thick CEI" hypothesis, whose state of lithiation is significantly lower in the upper half of the electrode (closer to the separator). As it was evidenced in Section IV.2, this observation can be attributed to the heterogeneous CEI thickness, which will favor intercalation where the CEI is the thinnest, i.e. at the bottom of the electrode. This phenomenon is particularly obvious in the case of the "Thick CEI" for two reasons, firstly this is the hypothesis where the CEI will be the thickest, and secondly because of the resulting ohmic drop that will be greater than for the other cases. Indeed, the "Thick CEI" case will reach the cut-off voltage faster than the others, thus the system will never reach the point where it is more favorable to The Impact of the Mesostructure | 129

Nevertheless, the two clogging hypotheses do not seem to have a noticeable impact on both the discharge curve and the distribution of Li concentration. It can be explained by the nature of these hypotheses that will mainly impact ionic transport. Such an impact on the distribution of lithiation or specific capacity can be too small at the scale of one discharge to be noticeable. However, when the concentration in Li + is plotted against the distance from the current collector in

IV.3.3. Conclusion

A noteworthy conclusion that can be drawn from these results is that even after one discharge (starting from an ideal fully charged state) with a CEI thickness corresponding to 15 cycles, a loss of capacity was reported. Furthermore, with further cycling, the CEI will keep growing, and the loss in capacity with increase as illustrated through the "Thick CEI" hypothesis. The CEI will impact the electrode through an overpotential during the (de)lithiation and a higher heterogeneity of the Li concentration. The regions that will

The Impact of the Mesostructure | 131 be more favored for the (de)intercalation will be used the most and could be more prone to the formation of cracks.

Also, the clogging of the pores will impact the cyclability of the cell, which explains why little difference in the specific capacity could be seen after one discharge. Indeed, despite using a homogeneous (in terms of phase distribution) sub-volume in this modeling endeavor, a strong gradient of Li + concentration appeared in the electrolyte after one discharge, and will only increase with the number of cycles, entailing a higher polarization.

Since the clogging of the pores of the CBD can result in a loss of active surface area (and capacity), it comes that the size of its pores will play a role in the cyclability of an electrode. So, if the CBD is indeed favored for the formation of the CEI layer, the speed at which the pores are clogged will have an impact on its aging. Then, an optimal architecture could be designed in terms of the morphology of the carbon and binder to get wider pores.

IV.4. Chapter Conclusion

In this chapter, a machine learning algorithm has been developed based on observables extracted from mesostructures stochastically generated with the tools presented in Section II.2.2. It allows optimizing certain key descriptors based on the manufacturing parameters to improve the electrochemical performance of the electrodes. In particular, the degree of calendering was investigated, with experimental data to support the stochastic generation of the structures.

However, to have a deeper understanding of the impact of calendering on the electrochemical behavior of the cell under discharge, the NMC model presented in

Chapter III was applied to the same structures arising from CGMD with different degrees of calendering. The coupling of these two modeling approaches was successful in capturing the improvement of the electronic conductivity upon calendering, leading to a milder gradient of the state of lithiation along the thickness of the electrode.

Once the impact of the calendering on the mesostructure has been highlighted, the influence of the mesostructure on the SEI was studied to show the flexibility of the project by moving to the modeling of a negative active material (Graphite). The formation of the SEI was dynamically resolved and led to a heterogeneous layer which was different depending on the morphology of the AM, i.e. spherical or ellipsoidal. Overall, it was found that the electrode with spherical particles was more homogeneous in terms of the state of lithiation and SEI thickness compared to the ellipsoidal case because of the better ionic transport.

Lastly, the layer arising from the decomposition of the electrolyte on positive active material (NMC) was the focus of the last section. Relying on experimental insights and data, an NMC-111 electrode was reconstructed from tomography images and different

The Impact of the Mesostructure | 132 extreme hypotheses were investigated to have a better grasp on the influence of the CEI on the cycling capability of NMC electrodes.

Full-cell Simulation In this chapter, all the functionalities developed during this Ph.D. will be embedded in a single model to demonstrate the capability to perform state-of-the-art 3-D battery simulation using the Web App INNOV. To do so, a full cell model will be presented here, with an NMC positive electrode and Graphite negative electrode, and all the particles individually identified.

V.1.

From Half to Full: Challenges

In the battery 3-D modeling literature, few studies investigate full cells with the CBD explicitly considered, the only one that did so (to the author's knowledge) is the work of Danner et al. [START_REF] Danner | Thick electrodes for Li-ion batteries: A model based analysis[END_REF] presented in Chapter I. Several reasons might explain this lack of full cell models in the state-of-the-art, indeed, a half-cell configuration allows to study and isolate one electrode, which is more suitable when one wants to investigate one given AM.

Moreover, it is easier at an experimental level to gather data for a half-cell than for a full cell, because there are fewer parameters to obtain, and only one electrode has to be fabricated/optimized.

However, there also exist technical modeling challenges to the development of full cell models. Indeed, as detailed in Section II.1.2, the precision of the mesh is crucial during a finite element methods simulation, and can greatly impact the results. By resolving in 3-D an additional electrode, the number of elements will have to drastically increase in order to ensure the integrity of the structure after the meshing step. Even though some of the early simulations in this manuscript used a resolved Li foil, the number of elements in the mesh dedicated to this electrode is not comparable with having a full cell. Indeed, the Li foil was thin (ca. 5µm thick), and homogeneous, meaning there was no complex arrangement of several phases within the electrode, which increases the number of elements needed.

Another issue is the difference in size between the two AM at the positive and negative electrodes. Each electrode has to have enough particles inside to have a system worthy of simulation. Nonetheless, if there is a significant difference in size between the two AM, the number of particles of the larger AM will drive the dimension of the system and may lead to having a large number of particles of the smaller AM. The parameter Γ introduced in Section II.1.2 (radius of AM/length of the cube) will then sharply decrease while we have seen one should try to have Γ as high as possible to ensure the precision of the mesh. It is the case for the full cell under study in this chapter, as the NMC used as positive AM is quite smaller than the spherical Graphite used at the negative electrode, as highlighted in Figure V.1. Then, the NMC electrode will have to have a cross-section area wider than ca. 25 25µm as it was the case in the simulations to capture the calendering effect (Section IV. 1.2.3). Such a narrow cross-section area would not allow having enough particles, regardless of the thickness chosen.

Full-cell Simulation | 136 As established in the above paragraphs, the number of particles for a full cell configuration might be larger for a given electrode compared to a half-cell configuration if there exists a difference in the size of the positive and negative AMs. Adding to that number of particles the ones 3-D resolved for the second electrode entails several hundreds of different particles in the system. However, the importance and benefits of having the particles individually identified have been detailed in Chapter I, and the identification of several hundreds of particles in a mesh can be challenging. Indeed, it is noteworthy that tools relying on Iso2Mesh 160 to perform the mesh (for instance MATBOX [ref]) don't allow to have more than 256 different domains in the mesh since the inputs are greyscale images with indexes from 0 to 255.

In this chapter, through the use of the tools developed during this Ph.D., the previously presented challenges will be addressed through a basic example of a discharge modeling of a full cell.

V.2. Full Cell Configuration

Unlike the half-cell modeling, here both electrodes will be 3-D resolved with a positive electrode of NMC-111, and a negative electrode with Graphite. The thickness of the two electrodes will be adjusted so they have the same amount of capacity, and all the particles will be individually identified. At the time this chapter was written, there was no report for the Graphite particles compared to the NMC due to their bigger size while they share a diffusion coefficient for Li of the same order of magnitude for 0.5 < SoL < 1. Thus, the Full-cell Simulation | 139 (de)lithiation process is limited by the solid diffusion of Li rather than the kinetics of the reaction. Indeed, if the kinetics of the reaction was limiting, the sense of the gradient would be the opposite. For instance, for the Graphite particles, there would be an accumulation of Li at the surface of the particle, "waiting" to react while the core would be almost empty. [START_REF] Lu | 3D microstructure design of lithium-ion battery electrodes assisted by Xray nano-computed tomography and modelling[END_REF] where the state of lithiation for each particle is represented as a function of the distance to the current collector at a depth of discharge equal to 66%. The size of the marker is proportional to the size of the particles, and the lighter the color, the less there is a loss of active surface area due to CBD contact.

It appears that most particles are at a state of lithiation higher than 0.4, except for some small, probably poorly connected particles. Furthermore, a trend highlighted by the blue shape can be observed, where the particles closer to the separator tend to be more lithiated than the ones closer to the current collector. It would induce a slightly ionic transport limited positive electrode where the Li + cannot easily diffuse through the thickness of the electrode, entailing a higher concentration of Li + (and utilization) near the separator, i.e. from where the ions come from. This statement is consistent with the low porosity of the NMC electrode (25%) that would lead to a proper solid connection but a more tortuous path for the Li + .

V.4. Chapter Conclusion

In this chapter, the challenges to model a full cell have been detailed, followed by an example of an NMC/Graphite discharge simulation. This study combines all the tools and models developed in the course of this Ph.D. to perform a full cell model.

It is additional proof that with INNOV, state-of-the-art modeling studies are possible as no full cell modeling approach (with particles individually identified) has been reported in the literature to our knowledge.

With the COMSOL-Matlab Livelink, observables can be extracted automatically in a time-efficient manner to gain further insights into the behavior of the cell. As highlighted in Chapter I, the use of individually identified particles is extremely valuable and the author believes that its use will be more and more widespread in the battery modeling field in the future.

The results demonstrated that the positive electrode utilization is limited by the poor ionic transport due to its low porosity and that the Graphite particles suffer from a steep concentration gradient due to their size. Moreover, the benefit of having a 3-D model with all the components of the cell accounted for has been evidenced through the impact of the environment of a particle on its (de)lithiation process.

General Conclusion & Perspectives | 143

General Conclusion & Perspectives The scope of this Ph.D. project was to couple experiments and simulations to obtain further insights on the influence of the mesostructure of an electrode on its electrochemical behavior. The strength and uniqueness of this approach was its capability to link a given mesostructure to a set of manufacturing parameters thanks to the workflow of the ARTISTIC project. To achieve that pioneering goal, several challenges needed to be addressed as highlighted in the review of the state-of-the-art of battery modeling.

The choice of a 3-D approach was the most suited for linking the arrangement of the different phases inside an electrode to its performance, yet it introduces additional obstacles. Indeed, most battery 3-D models at the time this project started were using an implicit representation of the inactive phase due to the complexity and the computational cost to resolve an additional phase (an order of magnitude smaller than the rest) in 3-D. Nevertheless, in the 3 years that followed, increasing interest in the simulation of an electrode at the mesoscale has been observed in the modeling community.

With this growing attention, several groups achieved to include the CBD explicitly in their workflow, increasing significantly the quality of the state-of-the-art. The improvement and the cost drop of computational resources also had an impact on the development of such models that are more computationally expensive.

The simulation endeavor described in this manuscript focused on two different active materials, the NMC for the positive electrode and the Graphite for the negative electrode)

whose properties were introduced in Chapter I. Demonstrating the flexibility of our approach was one of the requirements of the project, that is why both positive and negative active materials were modeled individually and then combined into a full cell simulation.

To address the previously identified obstacles, scripts and applications were developed, not only for the scope of this Ph.D. but also for the ARTISTIC project as a whole.

Regarding the electrode structures, it was made possible to convert the CGMD/DEM produced structures into a stack of images for them to be used throughout the ARTISTIC chain. Moreover, a stochastic generation algorithm was coded to fit the needs of the modeling investigations. This tool enables fine-tuning of the features of an electrode for instance through the particle size distribution and the AM-AM overlap. Periodic boundaries conditions were also applied to the particles and the morphology of the inactive phase could be controlled from aggregate-like to a film-like arrangement. In addition, a meshing algorithm has been coded in Matlab to generate a multiphase volumetric mesh

General Conclusion & Perspectives | 144 with an indefinite number of domains inside, making it possible to identify individually each particle of AM.

The integrality of these scripts as well as other functions has been integrated to a Web App INNOV, in the ARTISTIC project website, that only requires an internet connection to be used. It embeds all several years of work of our battery modeling group into a single application accessible freely in an internet browser.

Once the obstacles and the tools to overcome them were presented, the first electrochemical investigation was led on the impact of the CBD location in an electrode. At first, the physics applied to this phase was chosen after the exploration of three hypotheses for the behavior of the CBD. An intermediate behavior was decided, where the Li + can diffuse through the inactive phase with an effective diffusion coefficient that depends on the microporosity but they cannot react at the AM/CBD interface.

The benefits of explicitly considering the inactive phase were further demonstrated through the GITT modeling that followed. The objectives were to gather the OCV curve and the diffusion coefficient as a function of the state of lithiation for the NMC used in our project, and also to provide modeling insights on the limitations of the formula used to extract the diffusion coefficient. It was shown that the environment of the AM particles is one of the causes of the discrepancy in the reported value of diffusion coefficient in the literature for a given material.

Then, the impact of the mesostructure on the performance has been assessed firstly through the influence of calendering. Based on experimental inputs and observables automatically extracted out of thousands of electrode structures generated stochastically, a machine learning algorithm was created by a team member to assist in the optimization of the fabrication process. This approach was complemented by a 3-D modeling endeavor that simulated the discharge of a given NMC electrode for different degrees of calendering.

It was shown that the calendering helps to mitigate the steep gradient of state of lithiation along the thickness of the electrode (lower near the separator), which entails that the calendering improved the electronic percolation through better solid/solid contact.

To demonstrate the flexibility of this modeling framework to unravel the link between mesoscopic features and performance, we shifted from NMC to Graphite and studied the growth of the AM morphology on the SEI growth and impact. The heterogeneous layer dynamically formed was different for the spherical and ellipsoidal particles, and led to different behaviors. The spherical case had a more homogeneous behavior than the ellipsoidal one, entailing a higher specific capacity, particularly due to better ionic transport.

The layer formed out of the decomposition of the electrolyte was also studied for the NMC, especially the effect of a heterogeneous CEI along different hypotheses with a combined experimental/modeling approach. the same AM skeleton were to accompany this experimental approach to provide insights on why a given arrangement performs better than the other.

• The investigation of the heterogeneous design of an electrode to optimize its performance would also be quite interesting. One could play for instance with an inhomogeneous distribution of porosity, phase distribution (AM, CBD, Pores), and particle size.

• One of the most challenging perspectives would be to establish a modeling framework where a dynamic mesh can be used to capture the changes in the • Regarding the tools developed in this work, the optimization of the GUI of INNOV and the addition of new functionalities would be beneficial to the user's experience. Also, it would be welcome to add a smoothing algorithm to the meshing in order to avoid the "staircase" shape when the resolution of the mesh is finer. For the stochastic generation, the algorithms can be improved to be more efficient for larger systems and lower porosities. Moreover, it could be extended to different shapes of AM particles and allow the user to import and use its own library of particles.

• Lastly, given high enough computational resources, it would be interesting to study the impact of the mesostructure at a higher scale. Indeed, while it can have an effect on 40x40x40 µm 3 systems, will it also be the case on systems ten times larger? Maybe the key features identified at the mesoscale will be different This Ph.D. investigations aim to unravel the link between structural observables and the performance of an electrode. The novelty of this study relies on the explicit consideration of the inactive phase of the electrode, as well as the investigations on the interphase that forms at the interface between the active material and the electrolyte.

Keywords: GITT, 3-D mesh, SEI, CEI, Stochastic Generation.

Introduction i. 1 .

 1 LIBs in the environmental context ............................................. i.2. ARTISTIC project in the industrial context ............................. i.3. Scope of my Ph.D ..................................................................... i.4. Manuscript structure ................................................................

Figure i. 1 .

 1 Figure i.1. (a) Annual growth for renewable electricity generation 2018-2020. (b) Wind and solar share of hourly electricity demand in 2020 in France. Both replotted from data of the Global Energy Review 2020.2

Figure i. 2 .

 2 Figure i.2. Schematic of a LIB under charge or discharge, with the active material in blue, the carbon and binder in orange, and the liquid electrolyte in green.

Figure i. 3 .

 3 Figure i.3. Worldwide LIB cell market company market share in 2018. Replotted from data of AVICENNE ENERGY Analyses 2019. 8

Figure i. 4 .

 4 Figure i.4. ARTISTIC project logo.

  Figure I.1.a, the NMC is a layered material such as the LCO or the LiFePO4 (LFP), meaning the material structure is a stacking of layers into which the Li will be free to(de)intercalate.

Figure I. 1 .

 1 Figure I.1. (a) Structure of the LiNi0.33Mn0.33Co0.33O2 (space group R-3m) with (b) zoom on the co-shared Ni, Co, Mn sites.

  evidenced by Figure I.1.b where the sites are shared equally between the Ni, Co, and Mn. The ability to play with the proportions of Ni, Mn, and Co to tailor its characteristics provides the NMC huge flexibility.

Figure I. 2 .

 2 Figure I.2. Diagram of the relationship between the metal content and the AM properties for NMCs.

Figure I. 3 .

 3 Figure I.3. Structure of the LiC6 (space group P6/mmm)

Figure I. 4 .

 4 Figure I.4. The open-circuit voltage of a Graphite electrode with the intermediate compositions' structures.

Figure I. 5 .

 5 Figure I.5. Overview of the different modeling approaches and their predictability as a function of the computational cost.

  Figure I.6).

Figure I. 6 .

 6 Figure I.6. Schematic of LIB system in 1-D models.

  Figure I.7. Schematic of LIB system in P2-D models.

  expensive 3-D simulations is to capture the geometrical features of the electrode explicitly (see Figure I.8.a). Having tomography data as input to perform the simulation has considerably accelerated the development of this approach. In 2010, the first tomography images on a negative electrode (Graphite) were accomplished by Shearing et al. 61 followed by the first tomography images on a positive electrode with the binder as an individual phase (see Figure I.8.b) in 2011. 62 Starting with Yan et al. using the latter

Figure I. 8 .

 8 Figure I.8. (a) Schematic of LIB system in 3-D models. In blue is the positive AM, in red is the negative one, in yellow is the CBD, in black is the separator, and transparent green is the electrolyte. (b) Reconstructed structure of a LiCoO2 positive electrode with

  from our group built an experimental dataset of the porosity and mass loading under different manufacturing conditions for NMC electrodes (see Figure I.9). This allowed to propose for the first time a machine learning model able to automatically classify into three categories, i.e. high, medium, and low, the porosity and mass loading observables, and to predict them as a function of the viscosity and solid to liquid ratio of the slurry.

Figure I. 9 . 28 I

 928 Figure I.9. Workflow of a data-driven model for LIBs manufacturing. Figure adapted from the work of Cunha et al. 111

  There are two pathways to obtain 3-D structures; the first is through tomography imaging and the second is through computational generation (seeFigure I.10). Regarding the case of implicit CBD, only the AM is of importance (if the CBD is merged with the electrolyte) in the structure. Then, for stochastic generation, several parameters are needed, namely the particle size distribution, the porosity, the amount of AM, and the dimensions of the electrode. The active material (most of the time spheres) is randomly generated inside the simulation box until the amount of AM is reached. To further control the mesostructure, for instance, the packing of the AM, the overlapped volume of AM, and the active surface area can be controlled throughout the process. The electrodes are then ready to be fed to a 3-D model with an implicit consideration of CBD.113 Nonetheless, stochastic structures suffer from the randomness inherent to their generation. That is why in recent years, tomography images have been used to improve the reliability of 3-D structures. Positive electrodes arising from micro-CT imaging have been used to reconstruct the AM domains In fact, the CBD cannot be extracted from micro-

Figure I. 10 .

 10 Figure I.10. Summary of the methods to generate 3-D structure adequate for 3-D models with implicit or explicit CBD. Figure adapted from Scharf et al.. 60

(

  GANs) model applied to the battery field. This method that has already been used in rock physics and material analysis in 2018 128 , enabled them to generate under a short computation time large realistic 3-D structures. They have been successful in applying this approach to lithium-ion battery positive electrodes (see Figure I.11) and solid oxide fuel cells.

Figure I. 11 .

 11 Figure I.11. Images of real lithium-ion positive electrodes used as a training set and computationally generated images through GANs. 126 In black are the pores, in grey is the AM, and in white is the CBD.

  tetrahedrons (each formed by 4 nodes) see Figure I.12.a. For 2-D meshes, the geometry will be described with nodes and triangles as in Figure I.12.b, 3-D meshes will have also tetrahedrons as illustrated in Figure I.12.c.

Figure I. 12

 12 Figure I.12. (a) Representation of the different elements in a mesh. Example of (b) 2-D mesh and (c) 3-D mesh from Team Geometrica 131 .

Figure I. 14 .

 14 Figure I.14. (a) Representation of the NMC-111 particle under study from the electrode 1. (b) Results of the 3-D diffusion model at different values of diffusion coefficient, adapted from Nelson et al. 130

Figure I. 15 .

 15 Figure I.15. (a) Discharge curves as a function of the capacity normalized by the electrode mass at 1C. (b) Slice view of the distribution of the normalized Li concentration in the AM particles at different stages of discharge from Ngandjong et al.. 12

Figure I. 16 .

 16 Figure I.16. (a) NMC reconstructed from tomography with stochastically added CBD. (b) The distribution of current density at C/10 discharge rate for homogeneous and inhomogeneous carbon location. Figure adapted from Danner et al..67

  Figure I.17. Discharge curves at different C-rates for different sub-volumes of the thick electrode, investigating the (a-b) cross-section width and (c-d) the location of the carbon layer heterogeneities effect. Figure adapted from Danner et al..67

  to the CBD explicitly considered for the first time in 3-D in an NMC-111 electrode, as well as the first report in the supplementary information of a 3-D discharge model accounting explicitly for the CBD, based on tomography data (Section III.1). In the following years, other simulation studies have been led on LIBs, for instance, or the work of Ferraro et al. 69 in 2020. It is a coupled electrochemical-mechanical model on an NMC-111 electrode from the same open-access tomography data 132 as the previously introduced work of Danner et al. 67 A 50×50×50 µm 3 was extracted and the CBD was stochastically State-of-the-art | 39 added to reach a composition of 92 % weight percent of AM and 8 % CBD. In their model, the CBD acts as an electronic conductive medium whose electronic conductivity depends on stress. CBD has a micro-porosity of ca. 50 % that allows the Li + to diffuse and react at the AM surface area, whether AM/Electrolyte or AM/CBD. However, the current density at the AM/CBD interface was corrected by the CBD porosity to account for the loss of active surface area. A discharge at 5C was simulated on this positive electrode, and some of the reported observables are displayed in Figure I.18.

Figure I. 18 .

 18 Figure I.18. Snapshots along the discharge at 5C of the distribution of (a) the concentration (b) the electrical current, (c) and the electronic conductivity. When not colored, the electrolyte and AM regions are in light grey while the CBD region is in dark grey.

Figure

  Figure adapted from Ferraro et al.. 69

  extracted and was used in the first scan where the CBD was located. One of the resulting images is shown in Figure I.19.a and the reconstructed 3D volume is illustrated in Figure I.19.b. Thanks to the explicit representation of the micro-pores in the CBD, this model did not have additional assumptions regarding this phase, and the CBD was considered

Figure I. 19 . 1

 191 Figure I.19.1 Representation of (a) the output microstructure from DSS and (b) its 3-D reconstructed volume. State of lithiation as a function of the distance to the separator for each AM particle at 50 and 70 % of the depth of discharge at (c) 1.25 C and (d) 2.5C. The color of the symbols corresponds to the particle size (blue to red: large to small).

Figure

  Figure adapted from Lu et al.. 66

Figure I. 20 .

 20 Figure I.20. Distribution of the AM particles diameter in the graded region for the (a) as-prepared and (b) graded electrode. Distribution at 50 % depth of discharge at 3.75 C of the (c) mean electrolyte concentration and (d) state of lithiation as a function of the distance to the separator for the graded and as-prepared electrodes. Figure adapted from Lu et al.. 66

  Figure I.20.d. The state of State-of-the-art | 43

. 1 .

 1 Meshing ...................................................................................... II.1.1. Article 1 .............................................................................................. II.1.2. Why does the mesh matter? ................................................................

  Figure II.1.a and Figure II.1.b. Introduction Despite of the rapid success of lithium ion batteries (LIBs), most of the technical progresses reached during the last years have arisen from trial-and-error approaches without deep theoretical understanding of the detailed operation mechanisms behind.

Computational

  ) and the carbon-binder domain (CBD) particles location and resulting pore network, drives the electrochemical performance of the cell (see Figure II.2).

  electrode (AM + CBD + Void) generated from CGMD calculations using LAMMPS software, as explained in the article of Ngandjong et al. 12 and Rucci et al.. 148 In this case, the input is a text file with the coordinates of the centers and the radius of each of the particles (see Figure II.3).

Figure II. 3 .

 3 Figure II.3. Process of meshing from CGMD structure to the volumetric mesh.

Figure II. 4 .

 4 Figure II.4. Mesh of an NMC electrode generated without smoothing (a) and with smoothing (b). The spherical active material particles are in red and the carbon binder domains are in yellow.

  a flexible MATLAB-based algorithm (INNOV) to tackle the meshing process and importation of multiphase objects in computational frameworks for multiphysics simulations based on sets of partial differential equations. From a selection of input format, INNOV generates a volumetric mesh able to contain an unlimited number of phases. This mesh can then be converted into the desired format to suit a variety of commercial and open-source software. INNOV embeds a set of observables to ensure the geometric integrity of the structure after the meshing step. While designed for the battery field, this algorithm can be applied to a wide spectrum of scientific fields. It offers a time-efficient tool to perform meshing without requiring substantial computational resources. II.1.1.4. Methods The INNOV algorithm embeds as its segmentation method the mathematical approach of Nielson and Franke 137 (Figure II.5) and a function of the open source program Iso2Mesh 151 to remove duplicated nodes. This segmentation method has been translated and optimized for MATLAB language and modified to suit the COMSOL Multiphysics meshing importation process. For instance, this new algorithm could lead to considering explicitly the complex CBD morphology in an electrode performance model, Computational Tools | 55 which has not yet been reported in the battery field to the knowledge of the authors.

Figure II. 5 .

 5 Figure II.5. Algorithmic workflow of INNOV with the inputs of each step.

  example of the impact of downsizing is shown in Figure II.6. Further in Section II.1.1.4 we present INNOV calculated observables to account for the integrity of the structure geometry after the meshing. Finally, a scale factor parameter (ratio m/pixel) ensures the geometric dimensions of the structure.

Figure II. 6 .

 6 Figure II.6. Slices of a mesh without downsizing (precision = 1) on the left and with downsizing (precision = 5) on the right. In green AM, in yellow CBD and in purple void.

Figure II. 7 .

 7 Figure II.7. INNOV process to generate elements from stack of images.

  Figure X reports meshes for different numbers of elements (tetrahedrons) for tomography data (Figure II.8.a) and a stochastic electrode (Figure II.8.b). First of all, it can be seen that the algorithm is well optimized because INNOV can generate a mesh of ca. 20 M

  Figure II.9.b and Figure II.9.c illustrate the extreme cases with respectively AM particles of 3 µm of radius in a 75 µm cubic electrode and AM particles of 7 µm of radius on a 25 µm cubic electrode.

Figure II. 8 .

 8 Figure II.8. The number of tetrahedrons in the mesh as a function of the computational time for (a) tomography data and (b) a stochastic electrode. In both cases, a coarse mesh (light blue), a medium mesh (green), and a fine mesh (red) are displayed.

Figure

  Figure II.9. (a) The standard deviation of the ratio between the volume of AM in the initial electrode and the mesh as a function of the ratio Γ between the AM radius and the size of the electrode. 3-D rendering of AM particles meshes in the extreme cases (b) rAM = 3 µm, Lcube = 75 µm and (c) rAM = 7 µm, Lcube = 25 µm.

Figure II. 10 .

 10 Figure II.10. Relative variation of the individual volume of each AM particle during the meshing process at different degrees of precision.

Figure II. 11 .

 11 Figure II.11. Images from the slicing algorithm on the same manufacturing simulation output with (a) a coarse resolution and (b) a fine resolution. (c) Illustrated case with particles impacted by the PBC highlighted. In purple is the electrolyte, in green is the AM, and yellow is the CBD.

Figure II. 12 .

 12 Figure II.12. Example of the "dilate and erode" algorithm applied to a CGMD structure with an increasing number of iterations.

Figure II. 13

 13 Figure II.13. (a) Illustration of the 6 and 26 connectivity for a pixel. (b) The output of the dilate and erode algorithm with different connectivity for each step.

Figure II. 14 .

 14 Figure II.14. Flowchart of the stochastic generation algorithm in INNOV.

  many new features have been embedded into INNOV, namely the importation and exportation of data in different formats, and the stochastic generation of electrode mesostructure (half and full cell) with more parameters to play with. The User Guide of INNOV which is accessible through the Web App is reported in this manuscript in the Annex. For the comfort of the readers, the relevant paragraphs and figures from the supplementary information have been added to the article. II.3.1. Article 2 II.3.1.1. Introduction When compared with the experimental community, the battery modeling community is significantly smaller. Indeed, during the last thirty years less than 20 % of published articles in the battery field were related to modeling (query on the 19th of April, 2021,

Figure

  Figure II.16. (a) Workflow of INNOV with the different options to import or generate 3-D structures to perform the mesh.(b) Graphical user interface of INNOV with the highlighted panels related to the importation (red) or generation of electrode (blue) and meshing (green).

  Figure II.17. Panels in INNOV GUI for (a) the generation of the electrode and (b) the advanced particle size distribution settings.

Figure II. 18 .

 18 FigureII.18. Example with (a) an accepted sphere based on the distance from the boundaries and (b) a case where the sphere is rejected because it is too close to the boundary.

Figure II. 19 .

 19 Figure II.19. Slice view available in INNOV GUI of stochastically generated NMC electrodes with in green the AM, in yellow the CBD, and in purple the electrolyte. Comparison between (a) film-like and (b) cluster-like morphology for the CBD and between (c) uncalendered and (d) calendered electrodes.

Figure II. 20 .

 20 Figure II.20. Popup menu for the different types of separator available in INNOV Web App.

Figure II. 21 . 3 -

 213 Figure II.21. 3-D rendering of the mesh of a LIB positive electrode in INNOV GUI with the NMC in red and the CBD in yellow.

  30 % was set with 20 % (in volume) of overlap between AM particles. A volume of 35 x 35 x 35 µm 3 was selected with a realistic AM weight percentage (94 %) and an experimental PSD for NMC-111. CBD was added as a film covering the NMC particles. To finalize the structure, a 5 µm thick homogeneous separator was added. The pores in the separator were not resolved since the impact of the separator is out of the scope of this example. A slice view of the generated structure and a 3-D rendering of the output mesh are shown in Figure II.22.(a,b), respectively.

  three different discharges were simulated at C/10, 1C, and 2C. The discharge curves for each case are represented in Figure II.22.c displaying a higher specific capacity at lower discharge rates. The profile of lithiation at the end of discharge at 2C is also reported in Figure II.22.d where the solid diffusion can be identified as the limiting phenomenon because of the steep Li concentration gradient in the NMC. A second built-in example provided with the software generates a full cell battery with a uniform PSD for the positive electrode's particles, a homogeneous separator and a custom Gaussian PSD for the AM at the negative electrode. This example illustrates two important features of the App, namely the full cell generation and the use of a custom Gaussian distribution which is more realistic than a uniform distribution. The last built-in example sheds light on the possibility with INNOV to generate structures suited for all-solid-state batteries modeling. It consists of a generated half-cell with NMC, CBD, solid electrolyte and 10 % of void. Then 20 µm in thickness of solid electrolyte is automatically added to act as the separator with 15 % of void. With the improvement of simulations accuracy, modeling should assist in the development of this promising new chemistry that all-solid state is. Thanks to these provided examples, users can understand the overall procedure behind a component structure generation and then interact with the variety of parameters available to study afterwards the desired phenomena in half-cell or full cell computer simulations.

Figure II. 22 .

 22 Figure II.22. Complete workflow of the NMC half-cell example in INNOV from the (a) electrode generation and (b) meshing, to the electrochemical simulations in COMSOL Multiphysics ® with (c) the discharge curves and (d) the state of lithiation at the end of the discharge at 2C with the CBD in green.

  highlight the importance of the behavior and the location of the CBD, different hypotheses were investigated in the following paper. It allowed us to choose the equations to apply to the CBD phase in the rest of the study. For the comfort of the readers, the relevant paragraphs and figures from the supplementary information have been added

6 (

 6 simulator. The meshes have been generated with our in house algorithm INNOV, recently published by us, 134 which relies on a voxelization technique to create multi-phase volumetric meshes. Meshed electrode mesostructures for compositions 96:4 and 94:AM:CBD) weight ratios, are provided in Figure III.1. It is assumed that all the void space in the mesostructures is filled by the electrolyte. 96:4 and 94:6 compositions have been chosen for this study because they are close to the compositions used in the scientificThe Impact of CBD Location | 86 community.132,169 However, intrinsically, the applicability of the meshing algorithm has no limitation in terms of composition. INNOV algorithm is used in this work just as a tool to import the CGMD/tomography structures into our electrochemical simulator: the present paper novelty comes from the numerical simulation of the electrochemical behavior of these electrode mesostructures and the analysis of the CBD impact (with a spatial location explicitly considered for the first time).

Figure III. 1 . 87 Figure III. 2 .

 1872 Figure III.1. INNOV rendering of the meshes of the AM (red) and CBD (yellow) for the compositions (a) 96:4 and (b) 94:6 (weight ratio AM:CBD).

Figure III. 3 .

 3 Figure III.3. For both electrodes with 96:4 and 94:6 compositions (a) observables from INNOV, discharge curves at 0.8 C with specific capacity per gram of (b) dry electrode and (c) AM. The simulated discharge curves are extrapolated from 3.4 V to 3.3 V.

Figure III. 5 .

 5 Figure III.5. Discharge curves of the 96:4 electrode along the blocking, partially open and fully open assumptions at (a) 0.8 C, (b) C/5, (c) C/10, (d) C/20 and (e) the histogram of the specific capacities at the end of discharge (3.3 V) for each C-rate. The simulated discharges curves that stopped at 3.4 V are extrapolated in order to reach 3.3 V.

  .1 in the Annex). On the other hand, changing the Li + diffusion coefficient values through CBD can mimic the use of different types of carbon additives and binders. Simulations at different C-rates were carried out for each of the assumptions reported in Figure III.5, namely blocking, partially open and fully open CBD. The C-rates do not exceed 0.8 C because of the current model limitation. The origin comes from the high current density at the current collector arising from low CBD surface area in contact with the solid. Considering that the CGMD simulation used to generated the electrode mesostructures does not take into account explicitly yet the calendering process, it is believed that the explicit consideration of this step would improve the contact between the current collector and the solid phases. Firstly, the loss of specific capacity while increasing Crate is captured by the model, i.e. the capacity drops from ca. 201.5 mAh/g at C/20 to 144.1 mAh/g at 0.8 C for the blocking case. Furthermore, the decay of the discharge curves during the voltage drop in Figure III.5.a is smoother than the one at lower Crates which is in agreement with the experimental results. 181 Another trend arising from the Figure III.5 is that the specific capacity at a given C-rate is higher for the fully open hypothesis than for the partially open and blocking ones. Such behavior is expected because in the case of the fully open CBD, by assuming a diffusion coefficient higherthan it should be, the limitations due to diffusion are underestimated, hence a better performance. On the contrary, in the case of the blocking CBD, avoiding the access of CBD to Li + leads to an increase of the tortuosity of its transport paths, which results in a decrease of the specific capacity, mostly at high C-rate.182,183 Obviously, the partially open scenario is in between the previous two assumptions in terms of specific capacity.Interestingly, the difference of specific capacity between the assumptions tends to shrink by decreasing the C-rate. Especially, this is the case for the fully open and partially open cases where the discharges curves are almost similar at low C-rate(Figure III.5.d). This emphasizes the shifting from a kinetics-regulated system at high C-rate to a thermodynamics driven one at lower C-rate.The different approaches regarding CBD can also be characterized by quantifying the lithium (Li + and Li(s)) inside the CBD and the AM. Figure III.6 illustrates how each assumption impacts the distribution of lithium in the system at 0.8 C. It is evident that in the case of blocking CBD the concentration inside the CBD will not change during the discharge because Li + cannot diffuse through it. Since CBD behaves like an insulating wall towards Li + , the latter tends to be accumulated in some regions in the active material leading to steeper concentration gradients, hence a higher polarization resulting in poorer specific capacities. Assuming fully open CBD facilitates the diffusion and the homogenization of Li(s) concentration in the AM. There are smoother concentrationThe Impact of CBD Location | 91 gradients thanks to these new diffusion pathways which implies a better cell performance.

Figure III. 6 .

 6 Figure III.6. Snapshots at different depths of discharge of the lithium concentration inside the solid phase for a 96:4 electrode for (a) Blocking CBD, (b) partially open CBD and (c) Fully open CBD hypothesis at 0.8 C.

Figure III. 7 .

 7 Figure III.7. (a) Schematic of the GITT pulse current in dashed red line and the output potential in solid blue. (b) Dispersion of reported value of DLi,NMC111. The error bars represent the range of value when DLi,NMC111 is reported as a function of the lithiation state x in LixNMC-111.

  2 are used to extract DLi and the Open-Circuit Voltage (OCV) as a function of the lithiation/delithiation state and the maximal Li concentration in the AM for a NMC-111-based electrode made of spherical particles. Those parameters are used as inputs for a 3D electrochemical model, which accounts for a detailed description of the positive electrode mesostructure, spatially resolving the location of AM, the inactive phases made of CBD and pores. Then, the 3D model is used to simulate the GITT experiment by assuming different Li + transport properties for the CBD and comparing their implications on the results. Thereafter, Equation III.2 is used again to extract DLi (a) (b) The Impact of CBD Location | 95 from the simulated GITT curve. Finally, DLi arising from experiments and simulation are compared and discussed.

2 .

 2 The latter assumes a single particle size and effective transport parameters (in terms of the electrode tortuosity and porosity) along only the electrode thickness. Both models are not able to capture the heterogeneous and anisotropic spatial distribution of AM, CBD and pores in the electrode. It is evident that a 3D representation of the electrode at the mesoscale will allow a more realistic consideration of the AM environment and its influence on the behavior of the half-cell (partial coverage of AM particles by CBD and therefore partial exposure of the AM particles to pores). Moreover, the explicit spatially-resolved consideration of CBD is of the highest importance to capture the impact of the AM environment. In that sense, we have already reported a first model spatially resolving AM and CBD for the simulation of LIB discharge.154 This model, adapted here for the simulation of GITT, constitutes the pillar of the present study.III.2.1.2. Results and DiscussionThe experimental GITT curve was obtained with NMC111:carbon black:PVdF cathode with a composition of 96:2:2, in weight ratio. After three formation cycles at C/10, the GITT was performed by stepping the current (between 3.0 and 4.4 V) at C/10 for 30 min followed by a resting period of 6 h in both charge and discharge. The current pulse and the output potential are plotted in Figure III.8.a. DLi can be deduced from Equation III.2 at the different lithiation states x. The average value for the radius (3.94 μm) was used. Particle size distribution for our NMC-111 and additional information can be found in the Experimental Methods section, in the Appendix A3.1. DLi values calculated during the experimental GITT charge and discharge are represented in Figure III.8.b. On the interval 0 < x < 0.5, the coefficients from charge and discharge are well in agreement and almost constant at approximatively 1.6 ×10 -15 m 2 .s -1 . This value is within the lower range of reported value in the literature. Indeed, it is of the same order of magnitude as the DLi reported for a positive electrode of NMC-111:carbon Super P:PVdF in the work of Cabelguen. 190 Furthermore, there is an important polarization occurring in the two last discharge current pulses, resulting in a dramatic drop of DLi (in the region x > 0.6 in Figure III.8.b

Figure III. 8 .

 8 Figure III.8. (a) Potential as a function of time during the GITT experiment (with the extracted OCV curve, in red). (b) Experimental DLi obtained through Eq. (1) during charge and discharge.

Figure

  Figure III.9. (a) Mesh of the electrode used in the simulation. In green is the Li foil, in magenta the separator, in red the AM, in yellow the CBD and in black the electrolyte which fills all the void. (b) Charge potential as a function of time during the GITT simulations, (c) the extracted OCV curves for each CBD hypothesis compared to the input one and (d) the comparison of the experimental and simulation diffusion coefficients calculated during charge.

Figure III. 10 .

 10 Figure III.10. (a) Mean lithiation state x along electrode thickness (z direction on Figure 3a) in terms of the experiment time. Cross-sectional 2D plots of lithiation state x along the AM NMC111 phase (b) under the three different CBD physics assumptions: fully open, partially open and blocking from z = 24.5 μm at a simulation time equal to 35 000 s (after the seventh pulse); (c) on two AM particles of same size within the composite electrode mesostructure at three different times during the seventh pulse. The grey areas correspond to the CBD and the white ones, to the electrolyte.

Figure

  Figure III.11. (a) AM image and (b) particle size distribution used in the CGMD simulations.

  Figure IV.1, and is a combination of experimental data, stochastic generation, and machine learning. The first step A is the evaluation of the function predicting the porosity after calendering as a function of the initial porosity, the initial thickness, the AM content, and the calendering pressure. To do so, systematic experiments were performed to construct an experimental database from which the function could be extracted (see Figure IV.2).

Figure IV. 1 .

 1 Figure IV.1. Workflow from the experimental data to the machine-learning predictions used in this work. Figure from Duquesnoy et al.. 117

Figure

  Figure fromDuquesnoy et al.. 117

Figure IV. 3 .

 3 Figure IV.3. Evolution as a function of the AM content and calendering pressure at initial porosity equals 48% of (A) the tortuosity of the liquid phase, (B) the tortuosity of the solid phase, (C) the percentage of the surface of AM in contact with the electrolyte, and (D) the contact between the current collector and the CBD. The cross markers indicate the value of the porosity after calendering. Modified from Duquesnoy et al.. 117

Figure IV. 4 .

 4 Figure IV.4. Interdependencies based on the ML algorithm between the inputs and the relevant observables. The size of the circle marker is proportional to the degree of correlation, and the color green indicates that the output is directly proportional while red indicates an inversely proportional relation. Figure from Duquesnoy et al.. 117

  , in the ARTISTIC project the electrode is modeled from the manufacturing process to the galvanostatic cycling. Here, Dr. Alain Ngandjong and Teo Lombardo modeled the solvent evaporation and the calendering as illustrated in Figure IV.5, for an NMC electrode with realistic particles size provided by measurements on SEM images. Then the electrodes were meshed and imported into COMSOL using IN-NOV 135 to simulate a single discharge at 1C for three half-cells with different porosities corresponding to different calendering pressures, namely 41.6% (uncalendered), 31.5%, and 27.2%.

Figure IV. 5 .

 5 Figure IV.5. Workflow of the modeling of the whole fabrication step through CGMD and the electrochemical discharge with finite element methods. Figure from Ngandjong et al.. 124

  Figure IV.6. (A) 3-D rendering of the three half-cells with the AM in red, the CBD in yellow, the separator in green, and the Li foil in purple. (B) Discharge curves at 1 C as a function of the specific capacity. (C) Electrolyte transport stands for the inverse of the average concentration gradient of Li + along the z-axis, CBD transport refers to the inverse of the average standard deviation of local current density at the CBD surface, and Active surface area is defined as the surface of AM in contact with electrolyte. (D) Distribution of the state of lithiation at a discharge capacity of 110 mAh.g -1 . Only the AM and the CBD (in grey) are represented. Figures from Ngandjong et al.. 124

  Geodict ® . One electrode contains spherical particles (Figure IV.7.a) and the other one The Impact of the Mesostructure | 115 contains ellipsoidal particles (Figure IV.7.b). Ellipsoidal particles, despite forming electrodes with high tortuosity factor (Figure IV.7.c), have been identified as a promising morphology through impedance measurements simulations because of their lower

Figure IV. 7 .

 7 Figure IV.7. 3D rendering of the (a) spherical and (b) ellipsoidal particles constituting the simulated electrodes, the separator and the current collector are respectively located at the top and the bottom of the electrodes; (c) the associated tortuosity factors calculated using TauFactor 159 software and the specific active surfaces which are the active surface areas divided by the volumes of Graphite. The CBD location can be visualized in Appendix A4.1. ; (d) Scheme of the simulated system under charge and discharge with the Graphite in red, the CBD in orange, and the electrolyte in green.

  displayed in FigureIV.8(c,d), which represents the average value of the SEI thickness within the electrode volume as a function of the distance from the (0,0,0) point (origin of the coordinate system) indicated in Figure IV.7. While the spherical Graphite has a rather isotropic distribution of the SEI thickness (ca. 15 % of variation), the SEI thickness for the ellipsoidal Graphite is strongly anisotropic. Indeed, the z-axis trend shows that the SEI near the separator is almost three times thicker than near the current collector, while along the x and y axes variations of the same magnitude as the spherical case are observed. To summarize, in the ellipsoidal case the SEI is the thinnest at the bottom of The Impact of the Mesostructure | 118 the electrode and gets thicker as we get closer to the negative electrode side. Such a gradient along the z-direction during discharge in a positive electrode is usually due to ionic transport limitation. Figure IV.9 reports the lithiation state at the end of charge for both electrodes, and highlights this ionic transport limitation for the ellipsoidal case.

Figure IV. 8 .

 8 Figure IV.8. (a) Simulated cycling curves of both structures with and without SEI. (b) The distribution of the SEI thickness at the end of discharge for both cases. The spatial evolution of the SEI thickness along the three axes for (c) ellipsoidal and (d) spherical Graphite.

Figure

  Figure IV.8a. Moreover, it can also explain the anisotropy of the SEI thickness for the ellipsoidal Graphite. Indeed, the location of the thicker SEI from Figure IV.8c stems from the limited ionic transport. During discharge, the Li + flows from the negative electrode through the separator to the positive electrode where it deposits. As it has been evidenced in the ellipsoidal positive electrode, the region near the current collector is poorly connected to the upper part of the electrode. During charge, it leads to an accumulation of ions near the current collector (Figure IV.10a), but during discharge, it leads to an accumulation of ions near the separator, causing a higher utilization of the upper part of the positive electrode. This results in a thicker SEI near the separator and a thinner SEI close to the current collector. The calculated anisotropy of the SEI due to the electrode mesostructure further proves the capability of the model to capture the impact of the electrode mesostructure. Once the discharge and the anisotropy of the SEI have been explained through the structural properties of the electrodes, one should analyze the impact of the heterogeneous SEI layer on the observables along charge. The active surface area has

Figure IV. 10 .

 10 Figure IV.10. 3D representation of the relative electrolyte concentration( [Li + ] / [Li + ]max) at time = 270 s for the charge simulation with SEI for (a) the spherical particles; (b) the ellipsoidal particles as calculated by our electrochemical model, the separator and the current collector are located at the top and at the bottom of the electrodes, respectively; (c) Evolution of the difference of the average lithiation state between the surface Sthick (half of the active surface where the SEI is thicker) and Sthin (half of the active surface where the SEI is thinner) during the charge simulation with SEI; (d) The average rate of delithiation at Sthick and Sthin along time during the charge simulation with SEI for the spherical particles (in orange) and the ellipsoidal ones (in blue).

Figure IV. 11 .

 11 Figure IV.11. 3D representations of the relative State of Discharge (SOD) for a time of charge = 1500 s and 67 µm < z < 70 µm for the electrode with spherical particles (a) without and (b) with SEI, and for a time of charge = 270 s and 37 µm < z < 40 µm for the electrode with ellipsoidal particles (c) without and (d) with SEI. The CBD is represented in green, the SEI thickness as wireframe in thermal scale. The separator and the current collector are respectively located at the top and the bottom of the electrode slices.

  Figure IV.12. (a) Reconstructed FIB-SEM images for the entirety of the segmented volume with the NMC in red, the CBD in dark blue, the pores in light blue and the cracks in green. The yellow dash line delimits the sub-volume used for the simulation approach and is represented in 3-D in (b). (c) The phase distribution through the thickness of the electrode is represented as a function of the distance to the current collector.

Figure IV. 13 .

 13 Figure IV.13. The description of the methodology to build the heterogeneous CEI thickness distribution.

intercalate

  Figure IV.14. (a) The CEI thickness distribution on the sub-volume used for the simulation. (b) The simulated discharge curves with cut-off voltage at 3.5 V and (c) the distribution of the state of lithiation as a function of the distance to the current collector for different cases. (d,e) 3D distribution of the state of lithiation at the end for the pristine (without CEI) and thick CEI cases, respectively.

  Figure IV.15. (a) Distribution of the Li + concentration along the thickness of the electrode at the end of discharge. 3-D distribution of the relative Li + concentration ([Li + ]/[ Li + ]max) at the end of discharge without CEI consideration (b), with CEI and CBD clogging (c), and with CEI and full pores clogging (d).

Figure V. 1 .

 1 Figure V.1. Particle size distribution used for the generation of the full cell's electrodes for (a) NMC and (b) spherical Graphite.

Full

  of 3-D modeling of a full-cell with resolved CBD and individually identified particles. This demonstrates that with the tools presented in this manuscript, one can perform state-of-the-art battery simulations. The positive electrode consists of a cube of 40 µm of length with NMC-111 particles with an experimental particle size distribution, the weight ratio between NMC and CBD is 94:6 with a porosity of 25 %. The negative electrode consists of a 40 4048.66 µm 3 volume (to balance the capacity of the two electrodes) with spherical Graphite spherical particles. The porosity is 40 % and the weight ratio between AM and CBD is 95:5. The 3-D rendering of the mesh of the two electrodes and the 10 µm thick homogeneous separator is available in Figure V.2. The mesh is ca. 4M elements large and all the particles are individually identified.

Figure V. 2 Full 1 Figure V. 3 .

 213 Figure V.2. 3-D rendering of the mesh used for the full cell, with the NMC particles in cyan, the Graphite particles in red, the CBD in yellow, and the homogeneous separator in green.

Figure V. 4 .

 4 Figure V.4. Distribution of the state of lithiation at the end of discharge with the negative electrode (Graphite) at the top, the positive one (NMC) at the bottom, and the CBD in grey.

Figure V. 6 .Figure V. 7 .

 67 Figure V.6. State of Lithiation for each NMC particle at depth of discharge = 66% with the size of the marker that indicates the size of the particle and the color of the relative amount of active surface area available despite the loss due to contacts with CBD. The region highlighted in blue provides the overall trend of the distribution.

Finally

  , the NMC and Graphite models were combined into one full cell model by overcoming the obstacles to build such a model. This work uses most of the algorithms General Conclusion & Perspectives | 145 and scripts developed in the course of this Ph.D. to achieve a state-of-the-art model with several hundreds of particles individually identified. This work will be soon valorized through a scientific publication by us.With a Ph.D. project scope as wide as mine, there exist obviously many exciting perspectives and studies to be undergone beyond what has been presented in this manuscript:• The first one is to include more to my investigations, and be more involved in experimental studies, by taking advantage of the experimental expertise within the ARTISTIC project.. An ongoing work by us that aims to investigate the impact of the CBD morphology (film or aggregate) at both the experimental and modeling levels is not included in this manuscript because it is not mature enough. It consists in playing with the solid content (SC), i.e. solid-to-liquid weight ratio in the slurry preparation and with the temperature of drying during the coating to achieve the desired morphology. Some preliminary characterizations on the morphology of the CBD were performed through Energy Dispersive X-ray spectroscopy (EDX) to map the distribution of carbon and binder in different electrodes. The difference in the manufacturing process also appears during the cycling or for the electronic conductivity of the electrodes as reported in Figure C.1. It transpires that the solid content does not seem to be driving the trend, but rather the temperature of drying with the two electrodes dried at the highest temperature displaying the highest electronic conductivities. The modeling of an NMC electrode with the CBD as a film and as aggregate with

Figure C. 1 .

 1 Figure C.1. Effective electronic conductivity for the different electrodes extracted from chronoamperometry.

  ressources fossiles requiert l'amélioration des filières de productions d'énergie renouvelables qui ne sont actuellement pas suffisantes pour satisfaire la demande croissante en électricité. Néanmoins ces sources d'énergies telles que l'énergie solaire ou éolienne sont connues pour avoir une production instable intermittente, c'est-àdire que la quantité d'électricité générée peut varier significativement à l'échelle d'une journée, d'une semaine et d'un mois. Cette production intermittente sera un frein dans l'optique de l'incorporation plus importante des sources renouvelables dans le mix énergétique, car l'offre ne pourra pas satisfaire la demande à tout instant. C'est pourquoi des recherches sont entreprises dans le stockage de l'électricité afin de pouvoir stocker l'énergie produite quand elle est en surplus par rapport à la demande et la restituer quand la production n'est pas au niveau de la demande. Le stockage électrochimique apparait comme un candidat prometteur, cependant ces systèmes sont complexes et même la technologie Li-ion commercialisée pour la première fois il y 30 ans, souffre de verrous technologiques. En effet, de nombreux problèmes restent à élucider avant de pouvoir obtenir des batteries Li-ion aux standards exigés par l'industrie automobile par exemple. Parmi ces freins, on compte entre autres les matériaux actifs utilisés qui ne garantissent pas une puissance et autonomie suffisantes. Ceux considérés comme les matériaux du futur souffrent quant à eux d'instabilités qui résultent en une faible utilisation de la capacité du matériau ou bien une durée de vie faible. Ces problèmes peuvent être solutionnés par une optimisation du processus de fabrication, composé de plusieurs étapes successives où chacune aura un impact sur les performances finales des batteries. Au sein du projet ARTISTIC, l'objectif est d'assister l'optimisation des matériaux actifs actuellement commercialisés ainsi que d'accompagner la transition vers des matériaux plus prometteurs. Pour ce faire, une plateforme de simulation numérique est mise en place avec chaque étape de fabrication modélisé via des modèles physiques. Ce travail s'inscrit à la fin de cette chaîne de modélisation, avec la prédiction du comportement de la batterie pendant son fonctionnement en charge/décharge. L'apport de ces travaux vis-à-vis de l'état-de l'art est la considération explicite de la phase inactive dans un modèle électrochimique 3-D, ainsi que la mise à disposition de scripts et d'algorithmes développés durant cette thèse afin d'encourager la communauté à utiliser la modélisation pour assister les études expérimentales. Tout d'abord, des outils Matlab ont été créés afin de pouvoir répondre au cahier des charges du projet, à savoir inclure la phase inactive composé de Carbone et de Binder (CBD) au sein du modèle 3-D afin de pouvoir assigner à cette phase une physique qui Résumé Long | 212 lui est propre. Ainsi, un algorithme de maillage 3-D qui autorise un nombre indéfini de phase a été codé en Matlab, permettant l'importation de n'importe quelle électrode dans Comsol Multiphysics, le logiciel qui résoudra les équations différentielles. De plus, une fonctionnalité permettant la génération stochastique de structure 3-D d'électrodes a été mise en place. Cet outil permet de contrôler un grand nombre d'observables et également d'utiliser des particules issues d'une librairie établie grâce à des données tomographiques sur une électrode de Li1-x Ni0.33 Mn0.33 Co03.3 O2 (NMC-111). L'ensemble de ces fonctions a été implémenté au sein d'une unique interface graphique d'une application web, accessible gratuitement via le portail du site du projet ARTISTIC (voir Figure R.1).

Figure R. 1

 1 Figure R.1 Interface graphique d'INNOV disponible sur le site du projet ARTISTIC avec les régions dédiées à l'importation (rouge) et la génération (bleu) des électrodes, et la génération du maillage 3-D (vert).

Figure R. 2 .

 2 Figure R.2. Images de la concentration en Li de la phase solide à différentes profondeurs de décharge pour les hypothèses Bloquante, Partiellement ouverte et Totalement ouverte à un régime de 0.8 C.

Figure R. 3 .

 3 Figure R.3. Distribution de l'état de lithiation à un état de décharge de 110 mAh.g -1 pour des électrodes avec des degrés de calandrage différents. Seulement le matériau actif et e CBD (gris) sont représentés. La figure est extraite de Ngandjong et al.. 124

Figure R. 4 .

 4 Figure R.4. Représentation 3D de l'éétat de décharge relative pour un temps de charge = 1500 s et 67 µm < z < 70 µm pour l'électrode à particules sphériques (a) sans et (b) avec SEI, et pour un temps de charge = 270 s et 37 µm < z < 40 µm pour l'électrode aux particules ellipsoïdales (c) sans et (d) avec SEI. Le CBD est représenté en vert, l'épaisseur de la SEI est représentée en fil de fer avec une échelle thermique. Le séparateur et le collecteur de courant se situent respectivement en haut et en bas des tranches d'électrodes.

Figure R. 5 .

 5 Figure R.5. Simulation de la courbe de décharge à un régime de 1 C avec la distribution 3-D de l'état de lithiation en milieu et fin et décharge avec l'électrode positive de NMC en bas et l'lectrode negative de Graphite en haut, et le CBD en gris.

   reported for the first time a generative adversarial networks 127

  The choice of the numerical method to solve PDEs is not insignificant in the output results. It is important to keep in mind that different numerical methods will predict different solutions to the same problem as evidenced byTrembacki et al..

	2 by inverting A
	= -	(I.3 )

118 I.4.2. Meshing Tool

Table I

 I

				State-of-the-art | 35
	.1. Geometrical features of four NMC-111 electrodes with differ-
	ent preparation methods adapted from Nelson et al.. 130
	Treatment	Elect. 1	Elect. 2 Elect.3	Elect.4
	Ball Milled	✘	✘	✘	✓
	Drying	Gradual	Rapid	Gradual	Rapid
	Calendering	✘	✘	✓	✘
	Metric
	Particle Count	199	217	960	3532
	Mean Radius (±60nm)	576	597	240	181
	Mean Specific Surface (nm -1 )	0.018	0.016	0.026	0.031

Table I

 I

	.2. Governing equations for the 3-D electrochemical model based on Newman's
	model. 56
	Lithium foil electrode
	Lithium foil current

  .2.2. Stochastic Generation .......................................................................... II.3. Web App INNOV ...................................................................... II.3.1. Article 2 ..............................................................................................

II.2. Structure Generation .................................................................. II.2.1. From Manufacturing Simulations ........................................................ IIII.4. Chapter Conclusion .................................................................... Computational Tools | 49

Table II .

 II 1. Observables provided by INNOV after the meshing of the electrode in Figure II.4.

		Volume percentage Volume conservation
	Void	49.7 %	99.2 %
	AM	30.5 %	101.6 %
	CBD	19.8 %	99.7 %
		AM with CBD	AM with Electrolyte
	Coverage percentage 36.2 %	63.8 %

II.1.1.

  Computational Tools | 64 the same volume ratio. Only the pixels that have at least one pixel of electrolyte as a neighbor can be converted back to electrolyte. An example of this algorithm is displayed in Figure II.12 with different numbers of iterations.

  II.2.2. Stochastic GenerationWhile manufacturing simulations are more physically grounded than stochastic generation, they also lack flexibility when it comes to a study disconnected from the fabrication process parameters. Indeed, you cannot achieve desired observables (active surface area, AM overlapping, CBD morphology) through this physical-based approach and the simulations are much longer to complete. For all these reasons, stochastic generation (and more recently ML algorithms) is still quite popular in the battery modeling community, since it offers a time-efficient tool to investigate specific features. For instance the clogging of the pores in a current collector due to Li deposition at different current densities.

152

Thanks to the slicing algorithm developed to process the manufacturing simulation output, it was possible to convert into a stack of images the stochastically generated AM sphere coordinates. Moreover, having the CBD as spheres is one of the constraints of the manufacturing simulation that the stochastic algorithm does not have to mimic. So, to achieve a more continuous CBD, the phase was added either pixel per pixel or voxel per voxel to speed up the process. Another benefit of stochastic generation is the capacity to control the CBD morphology.

153

The workflow to control the arrangement of the CBD phase and the generation of the AM particles coordinates is reported in Figure II.14.

While the script is quite simple, it offers efficient results and fine control over the electrode mesostructure. It is noteworthy that if the set of parameters deviates from the usual ranges it was designed for, the algorithm can suffer from significant speed reduction or even get stuck in a loop. For instance, if the algorithm had to generate enormous electrodes with a fine resolution, or generate a huge amount of CBD as a film with a limited AM surface.

  The Impact of CBD Location III.1. How to deal with CBD? ............................................................. III.1.1. Context ............................................................................................... III.1.2. Article 3 .............................................................................................. III.2. Case Study: Diffusion Coefficient Extraction ............................. III.2.1. Article 4 .............................................................................................. III.3. Chapter Conclusion ................................................................... The Impact of CBD Location | 83In this chapter, investigations related to the physics applied to the CBD phase will be undergone. Through different hypotheses, the impact of the CBD location on the electrochemical behavior of an electrode in a 3-D model will be illustrated in Section III.1 with an article published in 2019.154 Then, thanks to this established model, a mixed modeling/experimental study was led to provide valuable insights on the limitations of a technique used to extract the diffusion coefficient of Li inside the AM among other parameters. The results and discussions were compiled in an article published in 2020 which is presented in Section III.2. 161 III.1. How to Deal with CBD?

	III.1.1.	Context

  IV.1. Impact of Calendering ......................................................... IV.1.1. Prediction of Mesostructural Observables .......................................... IV.1.2. 3-D Electrochemical Modeling ........................................................... IV.2. Impact of the AM Morphology: SEI Formation ........................ IV.2.1. Article 5 ............................................................................................ IV.3. Impact of a Heterogeneous CEI ................................................ IV.3.1. Article 6 ............................................................................................

IV.4. Chapter Conclusion ................................................................... The Impact of the Mesostructure | 105

  .1. From Half to Full: Challenges ................................................... V.2. Full Cell Configuration ............................................................. V.3. Electrochemical Behavior under Discharge ............................... V.4. Chapter Conclusion ...................................................................

	Full-cell Simulation | 135

V

Research Council (ERC) offers the possibility to individual scientist to apply for generous grants to provide the ressources to lead ambitious and ground-breaking projects. In 2018, Pr. Alejandro A. Franco was granted by the ERC 2 million euros (Consolidator Grant) as the principal investigator of the "Advanced and

Last Update: June 24, 2021 INNOV is released under the CC BY-NC-SA 4.0 license.

Citations to the following papers are requested:

1. ACS Omega, 2019, doi.org/10.1021/acsomega.9b01279 2. Batteries & Supercaps, 2021, doi.org/10.1002/batt.202100096 * Amiens, France. E-mail: mehdi.chouchane@u-picardie.fr alejandro.franco@u-picardie.fr Appendix | The web app INNOV aims to provide a free and user-friendly tool to ease the battery modeling process but can also be used for a variety of applications. Indeed, INNOV embeds a battery components generation function that allows in few clicks to obtain half or full cells with the active material (AM) and carbon binder domain (CBD) explicitly resolved. Nonetheless its meshing tool can be used for any scientific field thanks to the importation of data through images or matrices. INNOV was designed to facilitate the importation of mesh into COMSOL Multiphysics ® but the output meshes can be imported into other modeling software.

Table of Contents

A1.1.2. Graphic User Interface (GUI)

The GUI can be divided into different areas:

• Slice Rendering: You can visualize the structure, change the slice orientation, scroll with the slider or click on "Play" to launch an automatic scroll through the structure.

Appendix | 174

• Importation: Panel for the importation of data (see Section 2).

Appendix | 175

• Electrode Generation: Panel with the generation parameters for the electrodes, separator, Li foil and current collector (see Section 3).

Appendix | 176

• Meshing: Panel with the meshing inputs (see Section 4).

Appendix | 177

• 3-D Rendering: You can visualize the generated mesh, rotate the view with the slider or click on "Play" to launch an automatic 360° rotation.

A1.2. Import Electrode

Appendix | 178 A1. 2.1. From Images To import a stack of images into INNOV, click on the "Load Images" in the "Import Data" panel. Then select all the images you want to load. The images should already be segmented, meaning a given phase should have the same unique Id (or pixel value) in all the stack.

INNOV doesn't support single .tif files containing all the stack of images, the user Then, please refer to Section A1.2.4.

A1.2.2. From Matlab Matrices

To import 3-D matrices into INNOV, click on "Load 3D Matrix" and select or drop your file. You can only import ".mat" files, which is the native format of Matlab ® .

Then, please refer to Section A1.2.4.

A1.2.3. From Spheres' Coordinates

You can import your electrode with AM and CBD defined as a collection of spheres through the "Load xyz file" button. INNOV will then generate your electrode based on the coordinates and diameters of the particles. The CBD should have a constant diameter and be smaller than the AM particles.

The input file should be reporting the coordinates in x, y and z and the diameter in either a .txt or .xyz file. An example of an input file is provided below.

To import such a file, specify the resolution (µm/pixel) in the "Generate Electrode" panel with the drop down element, then click on "Load xyz file" and the importation of your data is complete.

Appendix | 179

Recommendations: If your electrode is in the range of 50*50*50 µm 3 , a resolution of 0.33 µm/pixel is recommended. A finer resolution is possible but this one offers the best precision/time efficiency compromise. To generate an electrode, you can play the following parameters:

If

• Alpha: A particle will be accepted into the electrode if its center is no closer than Alpha * Radius from the boundaries of the box. Meaning Alpha = 0 will lead to AM particles being accepted on the edge of the system, with the part outside of the system being added through periodic boundary conditions. While Alpha = 1 will keep the AM particles to be intersected by the boundaries of the system. (Alpha>0) Recommendations: The closer Alpha will be to 0, the fastest the structure generation will be. It will limit the chances that the generation algorithm gets stuck. We recommend to keep it at 0 ideally, or to have between 0 and 1.

• Overlap: Qualitative parameter to control if the AM particles will be more or less packed. It is the volume of AM that is overlapped, so a high value for Overlap will result in a packed AM structure will a low amount of Overlap will result in a more dispersed AM structure.

Recommendations: We recommend you adjust the Overlap value to the other parameters you are using. For instance if you want to have AM particles of 5 µm of radius in a cube of 20 µm of length, it will obviously be tricky to manage the desired amount of AM without having any overlap. The higher Overlap is, the faster the generation will be. If your generation is taking too long, you can always play with Alpha and this parameter to increase the speed.

• AM weight ration: It is defined as the weight ratio between AM and CBD.

Recommendations: For values > 95 %, the algorithm might experience some issues, especially if the porosity is lower than 25 %.

• AM density: Set the density of your active material. If you want to use the default density for "NMC" for positive electrode, and "Graphite" for negative electrode, please leave the value equal to 0.

Appendix | 181

• Porosity: Volume ratio of pores in the electrode.

• Radius max/min: Minimum and maximum values for a uniform distribution of radius for the AM particles. For more refined options you can check "Advanced PSD", see the following 3.2 section.

Recommendations: We recommend you adjust the size of your AM to the size of the electrode. The AM size should neither be too small nor too big for the set of parameters you are generating.

• Dimensions: Dimensions of the electrode.

Recommendations: The typical size of an electrode generated with INNOV is around 50*50*50 µm 3 . Obviously you can increase the size of the system if you are being careful with the other parameters (resolution/AM size).

• Stochastic CBD: The parameter "w" controls the morphology of the CBD. The closer w will be to 0, the more film-like the CBD will be. The closer w is to 1, the more the CBD will form clusters. You can also control the rate at which the CBD is added, either pixel per pixel with "pixel" or by a cube of 3*3 pixel with "voxel".

Recommendations: If you want to have a high amount of CBD in the electrode while having a film like morphology, if you put w = 0 you might experience some issues due to the CBD having no more AM surface to deposit. In this case we would recommend to decrease the value of w until the generation can be completed.

Regarding the "pixel" or "voxel" method, we recommend for resolutions >0.33 µm to use the "pixel" and "voxel" for the others. You can always start with "voxel" and if the CBD is too coarse refine it using "pixel".

• Resolution: Set the number of µm/pixel in the generated images.

Recommendations: As stated before, we recommend for an electrode of 50*50*50 µm 3 to start with a resolution of 0.33 µm/pixel. For larger electrodes, we recommend to start with 1 µm/pixel.

• ASSB: If you click on the check box, you will be invited to fill a new "void" parameter that will be the percentage of volume of void in your electrode. A new phase will then be generated that will be the actual void in your electrode, whereas the "porosity" parameter controls the amount of solid electrolyte.

A1.3.2. Particle Size Distribution (PSD)

If you click on "Advanced PSD" the following panel should appear below the "Li Foil/Current Collector" panel.

Appendix | 182

It allows you to either use default PSD for NMC or Graphite, either to generate a Gaussian distribution either to import your own.

To import your PSD, you can either import it through a single column file with all the radius listed or through a two columns file as follow:

6.25 0.995

1

These files can either be .mat or any text files that can be read by Matlab ® . To import it, click on the popup menu "Create your PSD", select "Import", click on the button "Import", select your file and once the importation is complete you should see the representation of your PSD on the graph.

To generate a Gaussian distribution, click on the popup menu, select Gaussian.

Choose the mean value, standard deviation and lower/higher limits for the distribution and click on "Generate". Once the generation is complete you should see the representation of your PSD on the graph.

To use default PSD, click on the popup menu and select either "NMC-111" or "Spherical Graphite". Once the importation is done you should see the representation of the PSD on the graph.

A1.3.3. Separator

To add a separator, check that you have specified the resolution in the "Meshing

Options" panel and go to the "Separator" panel and click on the popup menu. You can choose separator from tomography images of "Targuay" and "Celgard"

where you will only have to fill the thickness of the separator and click on "Generate".

Appendix | 183

By choosing "Homogeneous" you will generate a homogeneous separator (no pores)

to which you will be able to assign a porosity in your electrochemical model later on.

You can also generate stochastically a separator with "Stochastic" using the same algorithm than for the CBD generation. We recommend that you keep w = 0.5 and that you use "voxel" if your resolution is lower than 0.33 µm/pixel.

Finally, you have the option to create a filament-like or crossed filaments-like separator. We recommend you keep the default parameters for the "Separator Options" panel.

A1.3.4. Li Foil / Current Collector

To add a Li foil or current collectors, check that you have specified the resolution in the "Meshing Options" panel and go to the "Li foil/Current Collector" panel. Indicate the thickness of both, if you want only one and not the other, keep the other thickness at 0 µm and click on "Generate".

The current collector will be added to the bottom of the electrode, while the Li foil will be added to the top.

In the case of a full cell, the current collectors will be added at the bottom and at the top of the electrode.

A1.3.5. Full Cell

At the top of the GUI, you have the following elements:

The To generate a mesh, you need to fill the following parameters:

• Decomposition: Number of tetrahedrons into which each voxel will be divided.

Recommendations: We recommend to keep it at 5 and to increase it to 6 only if you want to achieve a specific amount of elements you cannot reach with 5. • Resolution: The resolution parameter is automatically assuming the same value as the resolution used to generate stochastic electrode if this parameter is filled.

Otherwise if you imported your own data, you will have to specify the number of µm/pixel in your input electrode.

Appendix | 185

• Precision: Integer superior to 0, specifying 1 out of how many pixel will be kept for the meshing in x,y and z.

Recommendations: This parameter will govern the amount of elements in the mesh. As a rule of thumb, we recommend you start with precision parameters so that Number of pixel / precision = 30 for each axis. To help you, the values of Precision are automatically updated when you generate a new electrode to follow this rule. Then you will be able to adjust the parameters to reach the desired precision in your mesh.

A1.4.2. Plot

If you tick "Plot", once the mesh is generated you will be able to visualize in 3-D the output in the visualization panel at the bottom of the GUI. The 3-D rendering can cause some performance issues, especially with a large number of elements (>10 6 ). It is possible to change the angle of the view with the slider under the figure and to launch a 360° rotation of the structure by clicking on "Play" next to the slider.

A1.4.2. Saving Mesh

Once you have generated a mesh, you can export it in the following formats:

• Mphtxt: The native format of Comsol, it will embed selections with automatized names to facilitate their use in your models.

• Bdf: NASTRAN Free format which should be recognized by most of the mainstream modeling software. Here, the selections are based on the Id of each phase (1 = electrolyte, 2 = AM, 3 = CBD, Current Collector = 6, Li Foil = 16, 21 = Separator, 101 =Void).

• Stl: Surface mesh that is recognized by most of the modeling software.

• Ply: Surface mesh that is recognized by most of the modeling software.

• Mat: Native format of MATLAB ® by clicking on "Browse" then "Download" in the "Save Mesh as Matrice" panel.

A1.5. Guided Examples

Appendix | 186 A1.5.1. Example 1: NMC Half-cell You can click on the button "Example: Half-cell" at the top of the GUI to generate a NMC electrode with a separator and perform the mesh. This button will activate the procedure described below. From there you can play with the "Precision" parameter under "Meshing Options" to refine the mesh, or for instance change the "w" parameter in the "Generate Electrode" panel from 0.05 to 1 and click on "Generate Electrode" to see how the CBD morphology will change.

• Click on "Clear Geometry" at the top of the GUI • Click on "Positive Electrode" at the top of the GUI • Go to the panel "Generate Electrode" 35 35] µm

• Click on the checkbox "Advanced PSD"

• Go to the panel "Advanced PSD"

• Select "NMC-111" in the drop list

• Go back to the panel "Generate Electrode"

• Click on the button "Generate Electrode"

• Once the generation is done, go to the panel "Separator"

• Select "Homogeneous" in the drop list

• *OPTIONAL* Go to the panel "Li Foil/CC" and put Thickness Li Foil = 3 µm and Thickness CC = 3 µm • Go to the panel "Meshing Options"

• Click on the checkbox "Plot"

• From the drop list select "All"

• Put Decomposition = 5,

• Click on "Generate Mesh" Appendix | 187

• Once the mesh is generated you can now visualize it and save it in the suited format by clicking on "Save Mesh" A1.5.2. Example 2: Full Cell You can click on the button "Example: Full Cell" at the top of the GUI to generate a positive electrode with a uniform PSD, a homogeneous separator and a negative electrode with a custom Gaussian PSD and perform the mesh. This button will activate the procedure described below.

• Click on "Clear Geometry" at the top of the GUI • Click on "Positive Electrode" at the top of the GUI • Go to the panel "Generate Electrode" • Click on the checkbox "Advanced PSD"

• Go to the panel "Advanced PSD"

• Select "Gaussian" in the drop list

• Go back to the panel "Generate Electrode"

• Click on the button "Generate Electrode"

• Go to the panel "Meshing Options"

• Click on the checkbox "Plot"

• From the drop list select "All"

• Put Decomposition = 5,

• Click on "Generate Mesh" A1.5.3. Example 3: All Solid-State Half-cell You can click on the button "Example: ASSB" at the top of the GUI to generate a NMC electrode with solid electrolyte and 10 % of void and a separator with 15 % of void. This button will activate the procedure described below. From there you can play with the "Precision" parameter under "Meshing Options" to refine the mesh, or for instance change the "w" parameter in the "Generate Electrode" to see how the CBD morphology will change. You can observe in the 3-D rendering in red the AM, in yellow the CBD, in blue the void in the positive electrode and in green the void in the separator. The solid electrolyte is not displayed for visibility sake.

• Click on "Clear Geometry" at the top of the GUI • Go to the panel "Advanced PSD"

• Select "NMC-111" in the drop list

• Go back to the panel "Generate Electrode"

• Click on the button "Generate Electrode"

• Once the generation is done, go to the panel "Separator"

• Select "Stochastic" in the drop list

• Put Thickness = 20 µm

• Put Porosity = 85 %

• Put w = 0.5

• Select "voxel" in the dropping list • Click on "Generate"

• *OPTIONAL* Go to the panel "Li Foil/CC" and put Thickness Li Foil = 3 µm and Thickness CC = 3 µm • Go to the panel "Meshing Options"

• Click on the checkbox "Plot"

• From the drop list select "All"

• Put Decomposition = 5,

• Click on "Generate Mesh"

• Once the mesh is generated you can now visualize it and save it in the suited format by clicking on "Save Mesh"

A2. Article's 2 Model & Parameters

A2.1. 4D-resolved electrochemical model parameters

The parameters listed here below were obtained from literature, experimental measurements, optimization and sensitivity analysis. [213][214][215] . The effective Li + diffusion coefficient in the electrolyte is a function of the porosity and of the ratio between the constrictivity and the tortuosity following equation 1. 220,221 The constrictivity reflects the hindering of the mass transport when the size of the pores is close to the mean path length of the ions moving in the electrolyte. It is a dimensionless constant ranging between 0 (the pores are smaller than the solvated ion size) and 1 (the pores are much larger than the solvated ion size). In the case of the electrolyte in the bulk, δ is equal to 1.

However, for the partially open hypothesis, Li + can diffuse through the CBD micropores. In this case, it is appropriate to choose a constrictivity smaller than 1 to take into account the small size of the pores. The ratio between the constrictivity and the tortuosity is set at 5×10 -3 . A3.2. Computational Section A3.2.1. GITT current pulse

A3. Article's 4 Experimental & Modeling Sections

The full GITT cycle was modelled in two separate simulations, one for the charge and one for the discharge. As the GITT conditions should allow reaching the thermodynamic equilibrium at each step, having two separate simulations should not impact the final results. The complexity of a 3D GITT simulation with explicit CBD relies on the dramatic changes in current applied to a complex system. To smoothen these current variations, exponential functions were used before (Equation A3.1) and after (Equation A3.2) the pulse to shift gradually the current value from zero to the pulse value. It mitigates the stiff variations while having a minor impact on the pulse shape.

Normalized Distribution

Appendix | 196

In addition, it was observed when the resting time between two pulses was too long that the simulation would at some point ignore the following pulses. In that case, the current was kept constant and equal to zero for the rest of the simulation time. To avoid such behavior, the resting time was decreased from 6 hours (experimental) to 1 hour in the simulation. It is assumed 1 hour of resting time is long enough for the system to reach the equilibrium.

The same issue was encountered when the current during the resting time was equal to zero. A current density of 6 mA.m -2 was applied during rest which corresponds approximatively to a C-rate of C/6600. It is assumed this C-rate is negligible when applied for an hour.

A3.2.2. Ohmic drop correction

A3.2.5. 4D-resolved electrochemical model parameters

The parameters listed here below were obtained from literature, experimental measurements, optimization and sensitivity analysis. The electronic conductivity of 700 S.m-1 for the CBD adopted in this work is in the same order of magnitude compared to what is reported in the literature (760 S.m -1 in References 2 and 3). 2,3 Table A3.1. 4D-resolved electrochemical model parameters. σe (S.m -1 ) 0.00158×Ce×exp(-5.363×10 - 5 ×(Ce/C0) 1.4 Then, the overlaps between the generated objects are removed to approach the specified volume fraction as much as possible. The porosity of the resulting structures without the addition of carbon-binder is 35% for both. From these structures two representative volumes of 75×75×75 µm 3 with the exact same phase ratio are extracted.

Name

The CBD was later added in Matlab using an in-house algorithm already reported by us in a previous work. σe (S.m -1 ) 0.00158×Ce×exp(-5.363×10 - 5 ×(Ce/C0) 1.4 ) Electronic conductivity of