Prof Marie-Cécile Pera

Prof Friedrich Guy

Associate Prof David

Cécile Tessier

Keywords: Li-ion batteries, aging, inhomogeneity, multi-physics models, P2D model, active material loss, cycle aging

for taking the time to evaluate my thesis and for their input during the defense. I would like to thank Dominique for his patience (with the language barrier) and hospitality in the lab, especially in the very beginning; Remy for his timely experienced input with several issues in the laboratory; Philippe, who has been very helpful in fabricating my experimental setup. At a time when I was desperately looking for alternate cell options for mounting in my setup, Charles had reached out to Prof. Jeff Dahn and he had granted about 300 cells for free and a sincere thanks for his generosity. I would like to thank each and every person who has been a part of the R0731S team in IFPEN, during the past three years. I would also like to thank the R05 direction for their inputs for the several characterization tests that I had to perform on my cell samples; and all the other people I have met in IFPEN, who have turned to be some excellent friends of mine; and last but not the least, the ADIFP association for their several activities. I would like to extend a great thanks to the LRCS team, especially my fellow PhD students, in addition to Arash, Benoit, Sylvie, Gregory and Sebastian. Not to forget, I would like to thank my mom, dad and sister

Résumé (court)

Les batteries Li-ion (LIB) sont utilisées comme dispositifs de stockage d'énergie dans les applications automobiles, mobiles ou stationnaire. Cependant, leur vieillissement conduisant à une dégradation de leur performance reste un problème majeur. Les LIB présentent un comportement non uniforme qui entraîne une utilisation incomplète et un vieillissement non uniforme. L'objectif de ce travail est donc d'identifier les facteurs influençant le comportement inhomogène et d'étudier leur effet sur le vieillissement. Une approche combinée modèle/expérimentation est adoptée. Un dispositif expérimental a été développé pour simuler les dispersions thermiques et de potentiels dans les batteries Li-ion commerciales. Ce dispositif est utilisé pour effectuer des tests en cyclage et le vieillissement inhomogène est évalué par des tests de caractérisation effectués pendant et après le cyclage. Des modèles multi-physiques décrivant le comportement des LIB ont été développés pour représenter le comportement du système expérimental. Deux phénomènes de vieillissement identifiés expérimentalement sont pris en compte, à savoir la formation d'une couche de SEI (Solid Electrolyte Interface) et la perte de matière active d'électrode positive. Le premier est fortement dépendant de la température et le second est plus uniforme. Cette approche combinée a permis de montrer que la dispersion thermique avait plus d'impact que les différences de potentiel sur l'homogénéité du vieillissement.

Mot-clés : Batteries Li-ion, Vieillissement, Inhomogénéité, Modelés multi-physiques, Modelé P2D, Perte de matière active, SEI, Tests en cyclage Equilibrium potential V V i , where i = 1, 2, 3 and 4

Voltage of each cell in parallel network V V j , where j = NMC/C, C/Li and NMC/Li

Voltage of each half and full cells V

x 0 Stoichiometry of negative electrode at 0% SOC x 100 Stoichiometry of negative electrode at 100% SOC y 0

Stoichiometry of positive electrode at 0% SOC y 100 Stoichiometry of positive electrode at 100% SOC z Dimension along electrode thickness m

Greek letters

List of figures

List of tables

General introduction

The technological advancements and industrialization over the past century has led to rising alarms concerning climate change and global warming. One of the main issues concerning this are a continuously increasing world demographics, massive urbanization and economic growth demanding a rapid rise in the mobility sector. The optimal solution to curb the greenhouse gases responsible for these issues is to reduce the dependency on hydrocarbon based fuels and instead use alternative types of energy sources and energy storage devices. The automobile industry that is responsible for a huge portion of the greenhouse gases released into the atmosphere, has decided to go hybrid and/or to electrify the drivetrain. Although this curbs the pollution temporarily and locally, with the use of renewable energy sources, this could be a very viable option. The Li-ion batteries has been at the forefront, dating back to its inception in 1991 by Sony Inc. [1]. They have been embraced by the automobile industry and there is a fleet of battery powered vehicles currently storming the market.

The extent of energy and/or power substitution by Li-ion batteries to the conventional internal combustion engine is varied. Depending on this percentage of substitution, there are several options such as range extender or series hybrid vehicles, parallel hybrid vehicles, series-parallel hybrid vehicles, PHEV (Plugin Hybrid Electric Vehicles) and BEV (Battery Electric Vehicles), etc. In all these types of vehicles, multiple Li-ion cells are grouped together to form battery packs, necessary to meet their energy and power requirements.

Figure 1 presents the different length scales, starting from as small as the atomic scale of the elements used in the cell to the system scale for use in automotive applications, for instance. It may be seen that evaluating the performance of Li-ion batteries might begin as early as the choice of appropriate materials and design at each level to have the required performance. Appropriate composition of the materials is required at the atomic scale, followed by studies and analyses at the particle and electrode scales to evaluate their performance [2]. Based on this, cells are constructed and their performance is studied under practical operating conditions followed by grouping the individual cells into smaller modules for easy packaging and then into larger battery packs, to meet the requirements of various applications. At the final system scale, there are several challenges, namely, cost, safety and lifetime [4]. The issue concerning the lifetime of Li-ion batteries may be considered central to the other challenges. For instance, fast depleting performance of a battery pack decreases the overall lifetime of the vehicle and making the vehicle economically less viable. Concerning safe and optimal operation of Li-ion batteries, battery pack is required to be equipped with additional safety features that add to the costs. Thus, in this work, the factors influencing their diminishing performance and resulting in a low lifetime is studied. Understanding this might, to a great extent, solve the central issues concerning Li-ion batteries and also provide optimal solutions for future energy storage applications.

The diminishing performance of Li-ion batteries is due to the several degradation phenomena occurring in them and it is referred to as aging [5][6][7]. Their overall macroscopic effect on the automobile performance, for instance, may result in a decreased driving range and a decreased acceleration and deceleration (significant for regeneration of energy while braking) [8][9][10]. In addition to this, it is also known that within the scale of single Li-ion batteries, they exhibit non-uniform behavior which eventually results in non-uniform aging [11,12]. The inhomogeneous aging is also likely to result in faster degradation of the cells. Thus, it is of prime importance to have greater insight into this inhomogeneous behavior and thus the objective of this work is defined as identifying the factors influencing this behavior and estimating the effect of these parameters on the overall non-uniform aging.

To achieve these objectives, this work consists of two frameworks, namely an experimental and modeling one, as shown in Figure 2 and these frameworks each are divided into two steps. These frameworks work simultaneously, exchanging information between them in both steps. In the first modeling step, multi-physics models describing specific cell behavior are developed in COMSOL Multiphysics v5.3a software and using the model simulation results, a unique electronic setup is developed to surrogate the gradients occurring in commercial Li-ion batteries in the first experimental step. Completing this first step marks a crucial stage in this work, in achieving the proof of concept. Following this, in the second modeling step, the multi-physics models are appropriately coupled to represent the experimental setup behavior while in the second experimental step, the cells undergo long term cycling under different operating conditions. The cycling test results are used to calibrate the coupled model to obtain information on the aging behavior in general and to identify the effect of the parameters on inhomogeneous aging. The desired final deliverable is a fully coupled electrochemical, electrical and thermal model with a special focus on the inhomogeneous aging behavior in commercial Li-ion batteries. Following this, the degradation phenomena are discussed with their influence on the macroscopic cell performance and factors influencing them. The inhomogeneous behavior and the inhomogeneous aging are also presented and finally, the combined modeling and experimental approach for studying the inhomogeneous aging behavior are described.

In Chapter 3, the description, results and conclusions from the choice of four different types of cells, considered in this work are given. Initially, the cell specifications are given followed by the different tests used to assess their performance to have a repeatable and consistent behavior to be used in the setup for performing the long term cycling tests. The test results are presented and, based on that, an optimum choice is made to proceed with the next step.

In Chapter 4, both the first experimental and modeling stages are discussed. Multi-physics models are developed and using the simulation results, the experimental setup is fabricated. The individual cell electrochemical behavior from experiments is compared with an electrochemical model. A coupled model is also developed that describes the setup behavior and simulation analyses give an estimate on the variations in the setup and cell behavior and an ability to forecast the inhomogeneous aging behavior.

In Chapter 5, the cells are cycled in their respective setup and the intermediate checkup tests are performed to assess the cells aging behavior and further post-mortem tests are done and the effect of the factors influencing the inhomogeneous aging behavior is evaluated. Using the test results, the coupled model is calibrated and using these, insight on the nature of the inhomogeneous aging behavior is obtained. Additional simulation results allow quantifying the effect of inhomogeneous aging on the overall cell performance.

Finally, in Chapter 6, a brief summary of the complete thesis, the conclusions drawn from the work and the future scope of this work are highlighted along with a scope for future work.

State of the art Li-ion batteries

The use of lithium in energy storage has gained interest for about 30 years now. This is because of very low reduction potential of lithium equal to -3.05 V vs SHE (Standard Hydrogen Electrode), making it a very strong reducing agent [4]. Lithium is the lightest alkali metal in the periodic table, the least dense solid element and also has a very high specific capacity of about 3800 Ah/kg [13]. A Li-ion cell is an electrochemical redox (reduction-oxidation) system that works based on the potential difference between its electrodes. Based on the nature of the electrode materials used in Li-ion cells, the redox phenomena is referred to as rocking chair, coined by Scrosati et al. [14].

A major concern for the development of Li-ion batteries in stationary and automotive applications is the phenomena called aging, which has already been widely studied over the last few years [6,7]. These studies have revealed some of the main phenomena involved in performance degradation as well as some heterogeneities occurring during cycle life. In this chapter the state of the art concerning Li-ion batteries, inhomogeneous behavior and aging behavior is presented. The first section deals with the composition and the working principle, the second section presents the different degradation phenomena occurring in Li-ion batteries, the third section highlights the occurrence of inhomogeneous degradation phenomena, the fourth section deals with the general inhomogeneous behavior within single cells and finally the combined modeling and experimental approach that is generally adopted in literature dedicated to study the degradation phenomena. Although the general terminology for a single electrochemical Li-ion unit is called a cell and multiple such cells are connected to form a battery (module or pack), in this work, the terms, cell and battery are used interchangeably to refer to a single electrochemical Li-ion cell.

Composition and working of Li-ion batteries

Li-ion batteries are electrochemical energy storage devices transforming electrical energy into chemical energy and vice versa. In a Li-ion cell there are several components performing different tasks and interacting with each other simultaneously, for instance, the electrochemical components -active materials and electrolyte, electrical components -current collectors, electronically conducting carbon filler and other technical components -separator, binder, outer aluminum casing.

Figure 3 shows a schematic of the Li-ion cell and it consists of the positive and negative electrodes and their respective current collectors made of aluminum and copper, respectively, a separator between the electrodes to avoid electrical contact and an electrolyte enabling ionic contact between the two electrodes through the separator. The ability of the electrodes to store lithium atoms directly corresponds to the nominal capacity of the electrodes and eventually relates to that of the cell. The electrodes contain a significant amount of a binder material that holds the active material particles together and the filler material occupies the pores in the electrode boosting its conductivity. The binder and filler materials occupy a considerable volume of about 2 % of the total electrode volume, in case of positive electrodes (and lesser in graphite electrodes) and are responsible for providing structural strength to the electrodes. There are also conducting materials (carbon black, for instance) in positive electrodes to improve the solid-phase conductivity which is otherwise usually low [15]. Coming to the working of a Li-ion cell, the Li + ions travel through the electrolyte, represented by green arrows in the figure below and the electrons (denoted by red arrows) shuttle between the electrodes through an external circuit, in the presence of a load, in the same direction as the ions. During charge, at the positive electrode, the lithium atoms oxidize emitting Li + ions and electrons that move to the negative electrode, where they recombine/reduce forming lithium atoms again. The vice versa occurs during discharge. This is illustrated in the equations below, at the positive and negative electrodes, in eqs. ( 1) and ( 2), respectively. The oxidation and reduction of lithium atoms and its ions is achieved through the de-intercalation and intercalation from and into the complex crystal structure compounds, respectively, and hence referred to as insertion-based compounds. Their composition is such that the Li atoms occupy the vacant spaces present in the structures [16].

Cell components

The positive and negative electrodes are referred to as active materials because they are directly responsible for the amount of energy stored in a Li-ion cell. Thus it is very important for the appropriate choice of these materials and a number of review articles are available for the same [17][18][19]. The electrolyte, responsible for the transfer of Li + ions, its properties and types are presented. In this subsection the different materials, currently in use are briefly described and the future means of energy storage are discussed in the next subsection.

Positive electrode

There are several positive electrode materials and based on their crystal structure, they fall into the category of lamellar, spinel and olivine type, as shown in Figure 4. . Lastly, olivine type of crystal structure, also a three dimensional structure, is of the form LiMPO4, as in Figure 4 (c) [22]. The common example of this type is LFP (LiFePO 4 ) and it is successful because of its low cost, high-rate capability, high specific capacity, low toxicity and high flat stable potential plateau [23]. Each positive electrode technology may be assessed based on various factors such as energy, power, life span, performance and safety that facilitate the choice of an appropriate technology possible based on their score on these factors. Figure 5 shows a polar chart comparison of the present-day positive electrode materials (LMO, LFP, NMC and NCA), available for use in commercial cells based on factors such as cost, safety, specific energy, specific power, life span and performance at hot and cold temperatures [START_REF]Types of Lithium-ion Batteries -Battery University[END_REF]. It is important to do this comparison because their choice for use in commercial cell is dependent on these factors. For instance, if the application demands operating under high power loads, it would be wise to use to LFP or LMO over other materials. In addition to these existing technologies there is also study on other types of insertion compounds capable of achieving higher operating potentials and specific energies, namely Li-rich [START_REF] Whittingham | [END_REF] and Ni-rich [27] and/or Mn-rich [28] layered oxides.

Negative electrode

Li has the lowest reduction potential and thus Li metal foil would be an ideal choice for the negative electrode, as done in the past [29]. However, they are known to undergo degradation through lithium metal plating, eventually growing to form dendrites causing short-circuits when it comes in contact with the positive electrode, thus posing a serious safety issue. Therefore, insertion-based materials with operating potentials in the same range as that of the reduction potential of lithium were substituted with graphite as an alternative and other alternatives are used like LTO (Li 4 Ti 5 O 12 or lithium titanium oxide), TiO 2 (titanium oxide) and Si (silicon) [17].

The most widely used negative electrode material is graphite because of its abundance in availability, low cost, good electrochemical properties (high electrical conductivity and diffusion coefficient), good theoretical capacity of 372 mAh/g, etc. [18]. The crystallographic structure of graphite is lamellar with two-dimensional layered sheets. One notable property of graphite is that there is a passivation layer formation on its particles that consume cyclable Li + ions eventually protecting the graphite particles from further loss.

Although the nominal reduction potential of LTO is far off from that of lithium (1.6 V vs. Li/Li + ), it is still interesting because it doesn't require a passivation layer to maintain stability and there is no risk of lithium plating [30]. Its flat potential provides the ability to deliver a constant power over a wide operating range. LTO has a spinel configuration similar to that of LMO electrodes.

Finally, Si offers higher energy densities when compared to graphite electrodes, in addition to their benefit of low cost. However their use is still challenging due to their large volumetric expansion during charge and discharge process. As a result, sometimes graphite electrodes are blended with Si for benefits of higher energy densities.

Electrolyte

The electrolyte links the two electrodes internally through the transport of Li + ions in an electrochemical cell and must exhibit high transport properties like ionic conductivity, diffusivity and transference number [30]. There are two main classifications of the electrolyte depending on their state, namely liquid and solid or polymer electrolytes.

Most common electrolytes consist of Li + salts dissolved in organic solvents [START_REF] Bandhauer | [END_REF]. A commonly used liquid electrolyte is a salt of LiPF 6 (lithium hexafluorophosphate), LiBF 4 (lithium borophosphate), LiClO 4 (lithium perchlorate), LiTFSI (lithium bis(trifluoromethanesulfonyl)imide), etc. dissolved in a mixture of polar organic solvents composed of EC (Ethylene Carbonate) and one of the following mixed in appropriate weight percentages, namely DMC (Di-Methyl Carbonate), EMC (Ethyl Methyl Carbonate) or DEC (Di-Ethyl Carbonate) [31]. The choice of electrolytes is driven by their electrochemical stability, which are desired to be wider than the overall operating cell voltage (lowest negative electrode and the highest positive electrode potential values). The instability of operating outside this range, especially at lower potentials, is overcome by the formation of passivation layers on the negative electrode particles which protects the electrode surface by reducing the rate of consumption of more Li + ions. However, there are certain drawbacks in using liquids electrolytes, namely, they cannot be used at low temperatures as they are highly viscous and they may not be able to wet the electrodes and separators sufficiently under these conditions [32]. Another important challenge is that electrolytes are highly volatile and inflammable and usage at high temperatures result in degradation of the electrolyte and possibly leading to thermal runaway.

The effects on the use of solid electrolytes was studied and discussed starting from as early as 2000s [33,34]. One of the main advantages is that it lacks a flammable liquid electrolyte which is a major safety aspect in Li-ion batteries. Also there are other advantages of solid-state batteries such as the use of high voltage positive electrodes that directly correspond to higher specific energy and specific power of the battery, allows to operate the battery at higher temperatures, the use of lithium metal on the negative electrode due to its specific capacity and also highly durable when compared to using liquid electrolytes because the side reactions are reduced. However, there are some basic challenges to the use of solid polymer electrolytes such as low ionic conductivity at ambient temperature conditions [35], electrochemical stability issues [36], etc. High specific energies may be obtained with the use of lithium metal as the negative electrode and iIn spite of these challenges, these are commercially available; for instance, electric vehicles by Bollore [37], solid-state battery by Sakti3 [38].

Classification of Li-ion cells

Electrochemical cells may be classified based on energy or power type and geometrical construction of the cell. Both these classifications arise from the application where the cells are employed and are discussed in this subsection. The energy and power type classification arise from the configuration or specifications of the electrode segments in the cell while the cell geometry is defined by the outer construction and the manner of stacking the cell segments.

Energy and power type

Cells are generally grouped or quantified based on their specific energy/power or energy/power density. Specific energy (power) is defined as the energy (power) per unit mass and is given by Wh/kg (W/kg), respectively. While energy (power) density is defined as the energy (power) per unit volume and is given by Wh/L (W/L), respectively. Energy devices are compared based on their specific energies and powers which are represented in a Ragone plot as shown in Figure 6. The Ragone plot is represented as a function of specific energy vs. specific power, where both axes are plotted on logarithmic scales. It can be seen from the figure that if the devices are energy-rich (higher on the energy axis) then they are power-deficit (lower on power axis) and vice-versa. Today's technology is limited to the operating range of either in the mid energy/power region, high power region and/or high energy region. Hence, in the future, research is underway to have a high-energy device on the right-hand top corner in the graph where it is possible to extract high power on demand. According to the current Li-ion battery technology, high energy applications would require thicker and/or less porous electrodes in order to have the capability to store more lithium atoms [39]. The rate of supply of Li + ions that directly corresponds to the power capability of a cell could be achieved by the use of thinner and/or more porous electrodes which is contrary to obtaining high energy. This is a trade-off one has to deal with, when designing batteries but the choice is primarily driven by the type of application.

Cell geometry

There are three major classifications of commercial cells based on their geometry, namely, cylindrical, prismatic and pouch cells, as shown in Figure 7. Although the outer geometry of both pouch and prismatic type of cells are rectangular, in a pouch cell configuration is such that the rectangular electrode sheets are stacked one upon the other while in a prismatic cell, the electrode segments are spirally wound. In addition to this difference, the prismatic cells have a more rigid outer cover while pouch cells are sealed in a soft casing. The cell geometry and dimensions strongly influence the performance of batteries. For example, if the cell has a small surface to volume ratio, its heat dissipating capacity is low, resulting in a rapid temperature build up [41]. The choice of appropriate geometry is driven by the application, for example, on the available space, packaging difficulty and the environment in which they are deployed. The cylindrical and prismatic cells can withstand a higher stress due to their spirally wound nature and hard outer casing.

Challenges with Li-ion cells

There are several challenges associated to the use of Li-ion cells, namely, safety, cost and lifetime. In addition to this, commercial Li-ion cells have small energy and power ratings and may not be employed for highly demanding applications; for this several similar small cells are grouped and connected in either series or parallel to increase the energy and/or power to attain the required performance. This process is referred to as dimensioning or upscaling which also comes with associated problems. This subsection deals with these challenges.

Safety

Concerning safety of Li-ion batteries, it was shown in Figure 5, using a polar chart that not all cell chemistries have the same level of safety during operations. There are several incidents in the past concerning fire accidents or explosions relating to Li-ion batteries [42]; technically speaking, the thermal runaway phenomena is also likely to occur [START_REF] Petit | Li-Battery Safety. 1st[END_REF]. Temperature rise due to inadequate cooling or heat dissipation is a major safety issue. To avoid such conditions, battery systems often are lined up with safety equipment that increase the weight, add to the existing high costs and decrease the performance of batteries [START_REF] Bandhauer | [END_REF]. Insight on the cell and battery performance under extreme operating conditions, also termed as abuse conditions, can be obtained from both experimental and simulation studies [START_REF] Abada | [END_REF]45].

Cost

Currently the battery packs used in automobile applications cost about $ 250 per (kWh) kilowatt-hour but to be more commercially viable and competing with conventional internal combustion engines, the cost is required to be about 150 per kWh [START_REF] Gaines | Costs of Lithium-Ion Batteries for Vehicles[END_REF]. It was described in [START_REF] Ciez | [END_REF] that the cost of Li-ion batteries can be lowered if the cost of the materials used for battery construction is lowered. And as described earlier, the additional equipment needed to upscale batteries and to maintain certain safety standards is a major contributor to its cost. The term Return on Investment (ROI) is very important in batteries because the extent of its deployment is greatly determined by this number as it requires a huge initial investment [48]. High cost associated to early failure is another challenge all-together and requires that the lifetime of batteries is sufficiently long.

Lifetime

As discussed earlier, ROI determines the extent of deployment of batteries it is preferred that their performance does not degrade with time. There are several degradation phenomena occurring in batteries due to several factors (this will be discussed in the following sections) and as a result, the performance of batteries goes down upon usage and/or with time. Tesla Motors predicts the Model S lifetime to be approximately 8 years or 500,000 miles after which the battery pack capacity goes below 80 % and the battery is considered dead [START_REF]Tesla Model S battery pack data shows very little capacity loss over high mileage, mars[END_REF].

Upscaling

Li-ion unitary cells have low energy and power ratings when compared to their application requirements and for use in high demanding applications, several Li-ion cells with same configuration are grouped to form complex battery systems and this grouping process is called upscaling [START_REF] Pollet | [END_REF]. The first step of upscaling is grouping a few cells in a series and/or parallel arrangement to form a significantly larger unit called a module. One of the purposes of forming modules is the ease of packaging. Several of these modules are grouped together to form a battery pack, whose configuration depends on the specific applications. Upscaling is a main challenge, especially for automotive applications because of space constraints. Furthermore, for safety concern it is necessary to maintain a uniform temperature and SOC (State of Charge) of the cells inside the battery pack, thus requiring an efficient BMS (Battery Management System) comprising of cell balancing and thermal management.

Future of energy storage

In the Active materials subsection, the current search for materials, specific to Li-ion technology, with higher specific energies, lower costs and safer materials were described. In this subsection, future technologies, currently in research stage, are highlighted.

For instance, different types of conversion-based materials are studied. These materials undergo a change in their crystalline structure during lithiation and de-lithiation. For instance, some technologies most studied are Li/S (Lithium Sulfur) and Li/O 2 (Lithium Oxygen) [51]. These technologies pose a completely new array of challenges in spite of the additional specific energies they offer.

Going beyond Li-ion technology, there are also studies conducted on other types of alkali and alkaline earth metals, namely Na (sodium), Mg (magnesium), Ca (calcium), Zn (zinc), etc. [52,53]. Each of these technologies pose different types of challenges and the research on these technologies are still in its nascent stages. Use of Mg, Ca and Zn can be very interesting as for every redox reaction there are two electrons released and is believed to give twice the energy when compared to the standard Li-ion cells.

Aging in Li-ion batteries

The degradation phenomena in Li-ion batteries, referred to as aging, are a major concern as they result in performance deterioration upon usage and over time. In this section, the aging phenomena is analyzed in detail; firstly, it is categorized into three major categories and their macroscopic effect on the cell behavior is provided, followed by specific degradation phenomena identified so far are described. Finally, the factors influencing the aging behavior in Li-ion batteries are detailed.

Effect on cell behavior

The three broad categories are as follows -the change in the overall cell performance namely, capacity or power fade [START_REF] Bandhauer | [END_REF]; the operating conditions during which the cell undergoes degradation namely, calendar and cycle aging [54]; and based on the competing phenomena between the positive and negative electrodes namely, reversible and irreversible aging [55].

Capacity and power fade

Firstly, capacity or energy fade corresponds to a reduction in the total extractable energy from the cell while power fade refers to the decreased performance of the cell due to an increase in its internal resistance [56]. Both capacity and power fade are an inevitable process resulting from numerous degradation phenomena. The term SOH (State of Health) may be defined here, as the ratio of remaining deliverable cell capacity by its initial capacity. In commercial Li-ion batteries, when SOH reaches 80 %, the cell is considered dead and ought to be replaced. Concerning power fade, higher cell resistance results in a loss of performance at higher current rates forcing an early onset of the cell End-of-Discharge (EOD) voltage, thus resulting in an incomplete utilization of the cell energy. In the context of automobile application, energy fade reduces the vehicle driving range while power fade reduces the performance during acceleration and deceleration (braking allows energy regeneration).

An interesting feature about the capacity fade in Li-ion batteries is that it occurs in two stages [57]. Usually, at the start of the lifetime of the battery, the capacity fade as a function of time/charge throughput is slow and almost linear, independent of the severity of operating conditions. However, upon continuous usage, this it reaches a kink and accelerates after that point. The occurrence of this inflexion point may either depend on intrinsic parameters such as the cell geometry or cell chemistry, etc. or on extrinsic parameters such as the severity of the cell operating conditions, quality of the cell integrity (as a result of manufacturing defects), etc.

Calendar and cycle aging

Next is the operating condition during which degradation occurs namely calendar and cycle aging. Calendar aging occurs when there is no current passing through the cell, or in other words, when the cell is at rest while cycle aging, as the name suggests, is the degradation occurring when the cell is under usage, either during charge or discharge. It was reported that Li-ion batteries used in automotive applications, for instance, spend about 95 % of their lifetime in a state of rest and thus it is important to study the effect of calendar aging in commercial batteries [58].

The major difference between calendar and cycle aging is the primary aging phenomena that occur during these conditions and its effect on the macroscopic cell performance. The aging conditions and the degradation phenomena accompanying them are studied in detail from time-to-time and sufficient insight is available.

Reversible and irreversible aging

Finally, depending on the competing nature of the parasitic side reactions occurring at the positive and negative electrodes, the degradation phenomena may be classified as reversible and irreversible [54]. It is known that there is involvement of electrons at the positive and negative electrodes upon reaction with the electrolyte solvent resulting in the internal cell balancing. This is a redox reaction where the electrolyte oxidizes and reduces at the positive and negative electrodes, respectively. Thus there is a resulting net charge where the extent of side reactions at one of the electrodes exceeds the other. It was reported that the side reaction charge consumed at the negative electrode is usually higher than the other [58]. Given this is the case, the charge consumed at the positive electrode is recoverable and is referred to as reversible capacity loss while the remainder of the charge consumed at the negative electrode is referred to as irreversible capacity loss.

Degradation phenomena

The degradation phenomena occurring in Li-ion batteries have been identified and detailed over the years and several review articles are available for the same [5,7,9,59]. Some of the easily quantifiable and commonly studied phenomena are formation of passivation layers on the positive and negative electrode particle surfaces, lithium metal plating on negative electrode particles, transition metal dissolution from the positive electrode, degradation of the electrolyte (possibly yielding passivation layers), loss of electronic and/or ionic conductive paths, etc. There are several other phenomena that occur in Li-ion batteries but are not as easily quantifiable as the ones described earlier, namely, loss of electrical contact paths within the solid electrode matrix and with the current collector, loss of the ionic conductive paths in the complete 3D matrix of the electrode and electrolyte assembly, micro-cracking of the electrode particles, etc. [START_REF] Groot | State of Health estimation of Li-ion batteries: cycle life tests methods[END_REF][START_REF] Birkl | [END_REF]. These degradation phenomena are detailed in Figure 8. The general effect of these degradation phenomena on the cell behavior may be classified based on the macroscopic effect on the cell, namely cyclable lithium loss, active material loss and resistance increase [23]. Cyclable lithium loss occurs due to parasitic side reactions at the electrodes, possibly yielding in passivation layers, that involve consumption of electrons and as a result an equivalent amount of lithium is lost that is involved in the rocking chair process between the electrodes. Active material loss occurs due to numerous phenomena occurring at negative and/or positive electrodes and as a result the sites responsible for storage of extractable lithium in the electrodes decrease and eventually a loss of cell capacity. The resistance increase may arise from several sources namely, flow of ions in the electrolyte and of electrons and atoms in the electrode. The phenomena resulting in this may be loss of conductive paths in the solid and/or liquid media or an increase in the passivation layer thickness.

Passivation layer formation

Firstly the passivation layer formation on the electrode particle surface is discussed. The type of passivation layer is a characteristic of the electrolyte solvent used. Electrolytes are stable only in a particular potential window and become unstable outside this range; they undergo reduction at low potentials (at negative electrode) and oxidation at higher potentials (at positive electrode) [7]. Their most important ideal characteristic is that they are permeable to Li + ions but impermeable to electrons and other electrolyte components. They are considered important because they protect the electrolyte from further reduction and additional degradation under ideal conditions.

In commercial Li-ion batteries where graphite is used as a negative electrode, the passivation layer is normally referred to as SEI (Solid Electrolyte Interphase) and is the most commonly studied phenomena because of its significance, in terms of thickness/contribution when compared to its positive counterpart [10]. The SEI formation is believed to be driven by the processes initially described by Peled et al. in [62].

According to this, the solvent radical ions react with the lithium ions and electrons from the negative electrode to form organic lithium alkyl carbonates that are porous in nature. Some of these then convert to inorganic salts of lithium, most commonly lithium carbonate, etc. As a result, the SEI comprises of two layers, an inner impermeable layer with an inorganic composition and an outer porous layer comprising organic compounds. A schematic representation of the SEI composition with an inner inorganic layer and a thicker outer organic layer is shown in Figure 9. It may also be seen from the figure that the solvent moves through the SEI to react with the electrons and the lithium ions in the electrode. The inner inorganic layer is usually not considered, for instance during modeling, due to its insignificant thickness when compared to the total SEI thickness. The general effect of SEI layer growth is a cyclable lithium loss at the time of intercalation and deintercalation and a resistance increase arising from the resistivity of the continuous expansion of the layer.

Lithium metal plating

One of the commonly occurring issues with Li-ion batteries is the lithium metal deposition on the graphite electrode particles that results in a cyclable lithium loss [7,[START_REF] Groot | State of Health estimation of Li-ion batteries: cycle life tests methods[END_REF]. The occurrence of lithium plating may be observed through visual inspection and has light grey color [64]. Lithium plating is triggered when the negative electrode potential is close to lithium reduction potential, approximately about -3.05 [V] vs SHE (Standard Hydrogen Electrode) or 0 [V] vs. Li + /Li. Lithium plating may occur during one of the following conditions, while overcharging (when the cell is forced to operate at higher voltage and as a result the negative electrode potential may go below 0 [V]) [65], charging the cell at higher current rates (bigger potential gradients may result in certain electrode segments to reach low values and slow solid diffusion rate may result in an accumulation of lithium ions on the graphite particles surface) [66] and operating at low temperatures (slow solid diffusion rate) [67]. Lithium plating may also be the result of the electrode balancing in cells; for instance, as stated in [12], the overhang of the anode over the cathode in all commercial cells is to avoid lithium plating during regular operation. The lithium metal accumulates and grows in size and there is a possibility that the metallic lithium reacts with the electrolyte resulting in cyclable lithium loss. Under most extreme conditions, it is also suspected that if the dendrites grow in size, they may penetrate the separator and may cause a shortcircuit if they get in contact with the positive electrode.

Dissolution of transition metal

The transition metals part of the insertion compounds at the positive electrodes have been identified to be subject to dissolution, which may result in a loss of sites for storage of Li-ions and results in a loss of active material in the cell [6], however it may not be directly quantifiable and sophisticated physical characterization may be needed for the same. The most commonly dissolving transition metal is manganese metal from LMO electrodes with a spinel structure [7,68]. The most accepted mechanism for the dissolution is the John Teller mechanism [69]. Studies have been conducted to study the effect of manganese dissolution in the electrolyte and its contamination on the SEI in Li-ion batteries and it was shown that it results in a greater cyclable lithium loss due to enhanced SEI growth [70]. In addition to manganese, studies show that when iron, in LFP electrodes with an olivine structure, is not pure, the impurities dissolve in the electrolyte and deposit on graphite [23,58,71]. Similar to the manganese case, this also results in a loss of cyclable lithium due to additional SEI growth in addition to the loss of active material loss at the positive electrode. Recently, it has been reported that the NMC electrodes of lamellar arrangement also tend to undergo dissolution but the extent of it is not as much as the spinel electrode types [72][73][74][75].

Electrolyte contamination

Since the Li-ions are carried by the electrolyte, its degradation or contamination affects the cell performance significantly. The electrolyte drying is a common occurring undesirable phenomenon resulting from leaks in the seal or casing of the cells. The presence of water traces in the electrolyte result in the loss of cyclable lithium with the formation of LiF precipitate and HF [76,77]. It was reported in [78] that this degradation is temperature enhanced and presence of HF leads to a faster degradation of the electrolyte. The result of salt decomposition can may also result in LiF formation and a loss of cyclable lithium loss [79].

Effect of degradation phenomena on cell performance

Table 1 below summarizes the different degradation phenomena occurring at different parts of a Li-ion cell and their effect on the cell performance through loss of cyclable lithium, active material loss and resistance increase. The degradation phenomena are segregated depending on where they occur, namely, current collectors, negative and positive electrodes. The phenomena occurring at the current collectors result mainly in a resistance increase but also indirectly results in active material loss because certain electrode segments become inaccessible and hence may be considered as active material loss. At the negative and positive electrodes, the effects may be seen depending on the type of the degradation phenomena. The resistance increase, in general, arises from loss of electrical and/or solid-phase conduction paths and interfacial resistance.

Factors influencing degradation phenomena

The extent of the degradation phenomena are characterized by the influence of the operating conditions on the cell performance and the ones that influence the cell performance are commonly termed as stress factors [55]. It is known that the degradation phenomena are dependent on several parameters. Significant work has been done to identify the most significant contributors to the cell degradation and study their effect on aging. Some of the most commonly identified parameters are time, temperature, charge/discharge current rates, storage voltage or SOC (State of Charge) and DOD (Depth of Discharge) [10]. The factors influencing the calendar and cycle aging phenomena in Li-ion batteries are summarized in Figure 10. The contribution of each of these stress factors may depend on the cell operating conditions. For instance, in [80], Safari and Delacourt reported that for LFP/C cells, the effect of cell storage temperature is much higher on the cell degradation than storage SOC and also it was reported that higher temperature and higher SOC result in a faster degradation rate, during calendar aging conditions. Depending on the degradation phenomena, the contributing parameters may be different as well. For example, in the case of growth of passivation layer, the contributing parameters are high temperatures, very low and very high electrode potentials and high current rates whereas in the case of lithium metal plating, the influencing parameters are low temperatures, very low negative electrode potentials and again high current rates.

Inhomogeneities in Li-ion batteries

In this section, the inhomogeneities occurring in Li-ion batteries are looked into. Firstly, the factors resulting in the inhomogeneous behavior are identified from literature and several studies have been conducted to estimate the extent of these inhomogeneities. Following this, the occurrence of the inhomogeneous aging behavior in Li-ion batteries through sufficient proof from experimental and modeling work is detailed. The factors influencing the inhomogeneous aging are identified to be the same as seen in the previous section.

Inhomogeneous behavior in Li-ion batteries

The nature of the construction of a commercial Li-ion battery is looked into details to understand the sources for the non-uniform behavior. To meet the energy and power requirements of the market today, the size of Li-ion batteries are very different [START_REF]Lithium-ion Batteries | Industrial Devices & Solutions | Panasonic[END_REF]. Depending on their size, non-uniformities within the scale of a single large cell is inevitable [START_REF] Kim | [END_REF]. These non-uniformities may arise from different sources; firstly at the particle scale, the differences arise due to manufacturing inaccuracies, especially that of the electrode which involves coating of the electrode slurry over the current collector foils, calendaring and drying processes [START_REF] Brodd | Advances in lithium-ion batteries[END_REF]. This results in a non-uniform thickness and porosity in the active material slurry coating, difference in contact resistances between the electrode particles, electrode and their respective current collector and within the electrolyte, etc. [START_REF] Bazinski | [END_REF]85]. However, we are limited by the accuracy of the manufacturing processes and are entitled to trust that these differences are negligible; hence, this is considered to be beyond the scope of this work.

Secondly, also at the electrode scale, the transport limitation of Li + ions and electrons in the liquid and solid phase, respectively, result in significant potential and concentration gradients along the electrode thickness [12,86]. For instance, in [87], Paxton et al. showed that the inhomogeneous behavior along the electrode thickness is the result of the conduction of Li + ions in the electrolyte along the electrode thickness. It was reported in [88] that these differences may also occur as a result of the electrode tortuosity, crystallite orientation, etc. However, the inhomogeneities resulting from these gradients are considered to be quite small when compared to those arising from those at the electrode surface scale [86]. Lastly, at the cell scale, the inhomogeneities may arise from the physical parameters like dimension (for example, ratio of height and diameter in a cylindrical cell), weight or volume of the cell, ambient and operating conditions, etc. [START_REF] Kim | [END_REF]87,89]. In addition to this, similar to the electrode scale, the limitations in the transport of Li + ions and electrons in the liquid and solid phases, may also influence the inhomogeneous behavior [START_REF] Kim | [END_REF].

It has been reported that commercial Li-ion batteries, owing to their geometry, size and poor anisotropic thermal conductivity, exhibit significant thermal gradients and there is experimental evidence for the same, using thermal imaging techniques [89,90] or using thermocouples [91]. The surface-to-volume or aspect ratio (also called Biot number) of the cell define the extent of the thermal gradients [92]. In [89,91] it was attempted to develop a model to represent this thermal behavior to get an estimate on the temperature gradients in the cell and in the inner most regions of the cells. In [90], Waldmann et al. reported that temperature differences of about 30°C in cylindrical cells and 10°C in pouch cells, under very high load conditions, may be achievable. Although the physical construction along the radial direction in cylindrical cells and the through-plane direction in pouch cells is similar, the lesser differences in the latter is due to its smaller thickness. In cylindrical cells, the temperature gradients along the axial direction is significantly smaller when compared to the radial direction [90].

The second parameter causing non-uniformities within Li-ion batteries may be directly attributed to the large surface areas of electrodes and their corresponding current collectors and is due to potential inhomogeneities. Depending on the cell geometry, the maximum distance between the various electrode segments and the current collecting tabs may be estimated. In [93,94], the authors experimentally measured different output potentials for similar input current conditions, when the point of reference for measurement changed between the different tabs of a modified cylindrical cell. Further light was shed on this observation by developing a model and simulating this behavior under different conditions. In [95], a pouch cell was modified with multiple measurement points on the positive electrode to experimentally observe the potential gradients in the cell as an effect of distance from the current collecting tabs. In [96], the potential gradients along the surface of pouch cell electrode segments was represented by developing a 2D (two-dimension) electrical model.

It is known that the effect of each of the two factors on the macroscopic cell performance is interrelated. The thermal gradients result in significant non-uniformities in the local current density and SOC (state of charge) or potential gradients across the different segments of the cell, as shown in [97,98]. The vice versa is also true, meaning that the potential gradients along the electrode surfaces result in thermal gradients in the cell [98], thus providing testimony to the first statement.

Inhomogeneous aging in Li-ion batteries

It has also been identified and reported that the degradation phenomena within commercial Li-ion batteries occur non-uniformly. This behavior results in poor cell performance and underutilization of the overall cell energy over time. Hence several studies have been performed in the past to better understand this behavior in order to mitigate it. Some of the first steps towards validating and proving the occurrence of inhomogeneous behavior in commercial Li-ion batteries was using sophisticated imaging techniques such as Neutron imaging [99,100], Raman mapping [101,102], laser scanning [103], etc. or identification through colorimetry of extracted electrodes [78,104]. For example, in [105], Klett et al. used color patterns on aged electrode segments from dismantled cells as one of the identification techniques to indicate the occurrence of non-uniform aging.

To be more precise, modeling and simulation studies have been done to precisely estimate the inhomogeneity of the most commonly occurring degradation phenomena. For instance, to study the SEI growth differences in different parts of the cell, uneven SEI growth was observed across the electrode thickness through post-mortem analyses in [106] and a three dimensional model along the complete electrode surface was developed by Awarke et al. in [86]. Since the effect of temperature was not taken into account in this work, it was concluded that the SEI growth may not be studied in three dimension due to its uniformity across the different segments of the cell. The conclusion wouldn't have been the same if the effect of temperature on SEI growth was considered. In other works, in general, the SEI growth was experimentally seen to be higher at the separator end when compared to the current collector end [106] which was also modeled by Liu et al. in [107]. In [64], Burow et al. concluded from their study that the lithium plating was observed to be occurring non-uniformly on the negative electrode.

Since the inhomogeneous behavior influences the aging behavior as well, the factors reported for the degradation phenomena are also applicable here. Figure 10 may be referred to again, to get an estimate on the stress factors influencing the inhomogeneous aging behavior under different conditions. However, it was identified that certain factors have a greater influence on the inhomogeneous aging than others. Before investigating this further, outlook on the inhomogeneous aging behavior may be slightly changed and it may be regarded as the result of operating the cell inhomogeneously for extended periods of time. This was demonstrated by Hunt et al. in [92], where the effect of thermal gradients on the cell aging was studied and it was observed that when the thermal gradient was smaller, the cell aged more homogeneously and vice versa. Thus it is important to understand precisely the inhomogeneous behavior in addition to the several degradation phenomena occurring in the cells.

Combined modeling and experimental approach

In order to design better battery systems and to improve the performance of existing battery systems, it is important to understand the different phenomena occurring during battery operation. To begin with, numerous established experimental tests for studying the electrochemical behavior in lithium ion cells are available. Depending on the scale of the study (for example, electrode or cell scale), different techniques may be used. In addition to this, there is also a modeling and simulation approach to study the behavior of Li-ion batteries. In many cases, both these studies go hand-in-hand and complement each other, in [93,94] for instance. Analysis of batteries purely using experiments can be very expensive and time consuming, thus modeling and simulation is a key step involved in evaluating their performance and estimating and studying their behavior. Since the degrading phenomena in Li-ion batteries are complex, in most of the cases, the several experimental techniques are carried out simultaneously to appropriately estimate and quantify the different phenomena. In addition to this, an appropriate model is also often used complementary to the experimental work to provide additional insight on the degradation phenomena.

Modeling work

In this subsection, the modeling techniques used in literature to study the inhomogeneous aging behavior are detailed. Of the two different techniques, the physics-based one is used in given special attention. Multi-physics models, describing the electrochemical, electrical and thermal aspects of a Liion battery are detailed. Finally, coupling these multi-physics models to appropriately serve different studies or applications are detailed.

Modeling techniques

The phenomena occurring in Li-ion batteries are complex and multi-physics models are needed to describe their behavior. Two types of modeling techniques are available to develop these models, namely, physics-based and equivalent electrical or Randles circuit models. The first is a very straightforward technique where each physical phenomena occurring in the cell is described using its appropriate equation. Multiple phenomena are known to occur simultaneously in a Li-ion cell and there are usually a set of partial or ordinary (or a combination of both) differential and algebraic equations.

Although considered very accurate, solving these models is very demanding, often requiring high computational power and time. However it is the preferred choice of many research teams and is also suited for either material or electrode design purposes, for instance [108,109]. A relatively simpler way of representing the electrochemical cell phenomena is using an equivalent electrical circuit with a combination of different components [START_REF] Bergveld | Battery management systems: Design by modelling[END_REF]. These are again of two types, where in one the average electrode properties are considered, termed as resistor network model while in the other, the porous nature of the electrodes are considered, referred to as transmission line model [START_REF] Nara | [END_REF]. These models are computationally less intensive to solve and hence are generally employed for use in battery management systems (BMS) in automotive applications. However, the main limitation with this technique is that it may not be accurate enough to represent the complete cell behavior as the phenomena are not directly considered and instead their analogical electrical components are considered. This poses a problem when the model parameters evolve, for instance, due to temperature differences, temporal evolution (because of aging), etc. In this work, focus is given on the physics-based model to obtain information on the intricate details related to the inhomogeneous aging behavior.

Electrochemical model

To study the inhomogeneous aging behavior in Li-ion batteries, a detailed physics-based electrochemical model may be needed to study the evolution of the different cell parameters and to evaluate the changes occurring in the cell performance. The electrochemical phenomena mostly occur at the electrode separator assembly level. The accompanying phenomena here were first quantified, formulated and simulated by Doyle et al. in [112] and Fuller et al. [113], and is referred to as P2D (Pseudo Two-Dimensional) model. Several hypotheses were considered for developing this model and the electrochemical phenomena are modeled in two separate dimensions (one, along the electrode thickness and the ,other along the particle radius) and hence the name.

The particles are considered to be spherical in nature (either single or multiple particle dimensions) uniformly distributed along the electrode. To model the electrode phenomena, they are considered to be porous in nature, meaning the solid and liquid phases are superimposed on one another. Several phenomena are considered and solved simultaneously. For instance, in the electrolyte, the transport of lithium ions between the two electrodes is solved using the concentrated solution theory and also the conduction of Li + ions is described using an extended Ohm's law, along the complete thickness of the electrode/separator assembly in 1D (one dimension). In each electrode, the conduction of electrons in the solid electrode medium is described in 1D. The transport of lithium atoms in each of the solid electrodes is described using the second order Fick's law along the particle radius. The reaction kinetics for the oxidation (de-intercalation) and reduction (intercalation) of the lithium and lithium ions, respectively, is described using a Butler-Volmer relation that links the several phenomena occurring along the two different dimensions. Figure 11 shows a schematic representation of the P2D model; the micromodel describes the phenomena occurring along the particle radius and the macromodel describes the transport phenomena and charge conservation along the electrode thickness. The micromodel, spherical in nature, is solved as many times as the number of nodes in the macromodel. In this model, there are several parameters that require to be given as input and these parameters are either constant or depend on the cell under study. Considering the model to be a black box, the input can either be current or potential to the cell and the output being the opposite. The hypotheses for the model and the equations and parameters, are detailed in the following chapters.

Figure 11 : Schematic representation of P2D model showing the micro and macro models in two separate dimensions coupled through an exchange of variables, spherical configuration of micromodel and multiple nodes of micromodel along electrode thickness

Over the years, this model was used as the basis and either additional features were implemented to it or the model complexity was reduced using appropriate approximations. One such effort to improve the model performance at different temperatures is the inclusion of a temperature dependency of certain physical parameters using an Arrhenius dependency [114] or to couple the model the electrochemical model with a thermal model [115]. In [116], to model the two-phase phenomena in LFP electrodes, a shrinking core model was proposed by Srinivasan and Newman. In [117], Safari et al. studied the effect of contact resistance between the particles of LFP electrodes and also the effect of hysteresis by considering 4 particle size groups in a SP (Single Particle) model, that will be discussed later. In [118], Farkhondeh et al. implemented a concentration dependent solid-phase variable diffusivity in their model for LFP electrode materials to have a good agreement with the experimental data. They also considered a particle size distribution of the electrode particles which was also done by Mao et al. in [119].

Others have tried to reduce the model complexity after appropriate assumptions and approximations.

The most commonly used approximation to the P2D model is the SP (Single Particle) model, where the phenomena or equations corresponding to the transport of lithium ions along the electrode separator assembly and the electron charge conservation in the solid-phase electrode are ignored [120]. In this model, only the transport of lithium atoms along the particle radius of each electrode and the reaction kinetics corresponding to the intercalation and de-intercalation of the lithium atoms are considered [10]. The limitation of using this model is that they are accurate only at low current rates. One other type of approximation is the use of polynomial approximations in the place of the complex time consuming non-linear PDEs in the P2D model, as in [121,122], which retains the exhaustiveness of the P2D model but makes the computation easier. In [123], Subramanian et al. developed approximate solutions to the diffusion along the particle radius that was coupled to their respective governing equations of the equations corresponding to the transport phenomena in the electrolyte medium.

Another approximation to the P2D model is the Average Model that assumes that the electrode potential in the solid-phase along the electrode thickness is negligible, as described in [124][125][126]. Other approximations are ISP (Improved single particle) by Rahimian et al. [127]; SEMP (Simplified Electrochemical Multi Particle) and HP2D (Homogenous Pseudo Two Dimensional) by Mastali et al. in [128], etc. A detailed review of the simplified P2D models is described in [129]. The equations describing the electrochemical model are detailed in Chapter 4.

Modeling the degradation phenomena

The above electrochemical model may also consider the degradation phenomena occurring in the Li-ion cell. Again as described earlier, representation of this may either be done through physics-based modeling or using empirical relations. The physics-based models describe, in detail, the phenomena accompanying the degradation mechanism that include either differential or algebraic equations, linked to the original electrochemical model. The effect on the overall cell performance, for instance capacity fade, internal resistance increase, etc., is a direct result of these phenomena. The empirical models do not consider the degradation mechanisms directly but instead describe the resulting cell behavior due to the degradation mechanisms as a function of either time, charge throughput through the cell, current rate and/or temperature conditions. In this work, the equations are detailed in the following chapters.

Earlier works have included modeling the degradation phenomena, for instance, SEI formation and growth on the negative electrode particle surface [130], lithium metal plating on the negative electrode [131] and transition metal dissolution [132,133]. Figure 9 may be referred to for a schematic representation of the SEI composition and the components involved in its formation. In the case of the SEI, it is assumed that there are two competing phenomena driving its formation and growth. While the first is the diffusion of solvent molecules through the SEI, the second is the reaction kinetics defining its formation rate once it reaches the negative electrode particle surface. The reaction is considered to be irreversible and driven by the electrode potential value. This model is linked to the electrochemical model through the side reaction current that defines the reaction rate.

In [134], the lithium plating phenomena is also considered an electrochemical one, driven by the electrode potential and the reaction rate results in a side reaction current that is also coupled to the electrochemical model along with a SEI growth model. The transition metal dissolution is also modeled in a similar fashion and is used in conjunction with the SEI model and are both coupled with an electrochemical model by Kindermann et al. in [72].

It is to be noted that the physics-based models are usually multimodal, meaning that it is valid under different operating conditions under regular operating conditions of temperature, current rates and/or operating voltage limits. Some other publications on studying the degradation phenomena through physics-based modeling approach are detailed below. In [135], Sikha et al coupled a physics-based aging model with a P2D model and the resulting decrease in porosity helped understand the decrease in cell performance. In [54], Spotnitz et al. presented five different classifications to differentiate the effect on cell performance due to the different degradation phenomena.

In the case of empirical models, the empirical equation may depend on the type of aging phenomena, namely calendar and cycle aging. For instance, in Petit et al., different exponential terms were used during calendar and cycle aging conditions [136]. In another work, Ramadass et al. showed that using empirical relations the negative electrode diffusivity as a limiting factor, the loss of rate capability in the cells may be explained in addition to the capacity fade observed during aging [137]. Earlier works, where only the SEI was considered to be the primary degradation mechanism, it was concluded that the capacity fade rate and resistance increase are proportional to the square root of time [10]. In [138], Dakin's degradation model was used to represent the capacity fade and resistance increase in NMC/LMO and NCA based cathodes in commercial cells using a logarithmic dependency. The degradation phenomena, used in this work, are further detailed in Chapter 5.

Electrical model

The construction of an electrical cell is such that the aluminum and copper current collectors are adjacent to the electrode/separator assembly, as shown in Figure 12. The function of the current collectors is to gather and transfer the electrons from the electrodes to the external circuit through the tabs and vice versa and this may be modeled using an electrical model [START_REF] Kim | [END_REF]128]. Since the thickness of the current collectors is considered to be negligible when compared to its length and breadth and owing to their high conductivity, it is considered that the electron conduction occurs in two dimension in this model. The input to the model may either be current input at the current collector tabs (from or to the external circuit) or a mapping of the potential gradient across the current collector surface (from its corresponding electrode) and the output being a potential gradient on the current collector surface or a current flow through the current collecting tabs (from or to its corresponding electrode), respectively. The model equations are further described in Chapter 4.

Thermal model

The redox reactions in Li-ion batteries are both endothermic and exothermic in nature, depending on the SOC of the cell but, in general, batteries are known to heat up during operation [42]. The heat generation and heat loss phenomena are summarized in Figure 13. There are two sources for heat generation, namely, reversible and irreversible terms while the heat is dissipated in the form of conduction, convection and radiation [91]. The irreversible contributions, as shown in Figure 13, is from overpotential (due to Ohmic loss, change transfer resistance and transport limitations), material phase change resulting in enthalpy heating due to solid-phase diffusion of lithium and finally, heat of mixing due to the formation or relaxation of concentration gradients. The reversible contribution is due to the change in entropy due to the equilibrium voltage of the cell. For describing the thermal behavior at the cell scale, appropriate models are used based on the type of cell geometry, dimensions, application, etc. These may be either physics-based models describing the energy balance of the cell in one of the following dimensions -zero dimension (also called lumped model with one nominal temperature throughout the cell), one dimension (radial temperature gradient in cylindrical cells), or three dimension (in large format commercial cells). The thermal model is either solved independently or is coupled with the equations of the electrochemical model. The heat generated in the thermal component of the model using information from the electrochemical model is used to compute the cell temperature. According to this, the heat generation is both reversible (entropic that depends on the relation between the equilibrium cell potential and SOC) and irreversible (overpotential multiplied by the current). The heat dissipation occurs through either conduction (contact with external mounting objects, for instance), convection (natural or forced) or radiation (depending on the difference with ambient temperature). The equations and the physical parameters for the thermal model are detailed in Chapter 4. The above multi-physics models are developed at different length scales and dimensions and they describe specific behavior in Li-ion batteries. However, to describe the complete cell behavior and to describe the interdependency of different phenomena or the cell parameters, these models need to be appropriately coupled. Kim et al. provided a framework for developing a coupled multi-domain multiphysics model at different length scales for large-format prismatic cells in [START_REF] Kim | [END_REF]. A coupled three dimensional model was also developed in [128], using a Resistor Network Methodology. A three dimensional coupled model taking into account the SEI growth along the complete electrode surface was developed by Awarke et al. [86]. A broader perspective of developing a fully coupled 3D electrochemical-electrical-thermal model is briefed in Figure 14. All the computed internal variables are a function of time and/or space. The three models, in different length scales and dimensions, are solved simultaneously alongside each other through an exchange of information. For instance, the cell and electrode potentials, resulting at the current collector ends of the electrochemical model are given as inputs to the electrical and thermal models and in exchange, the distribution of current and temperature along the current collector surfaces and the overall cell geometry is given as input to the electrochemical model. Similar information exchange occurs with the other two models as well.

Theoretically, the electrochemical model is solved as many times as the number of nodes along the electrical and thermal model for a given time instant. Although computationally intensive, this model, after calibration, may be suitable for any commercial cell available in the market with any configuration.

Experimental work

The physical tests performed on Li-ion cells help to evaluate their performance and this is usually performed complementary to the modeling work [139]. For this work, to study the aging behavior in Liion batteries, the tests may be chronologically classified into two categories, namely, during aging, where the cells are aged (either calendar and/or cycling conditions) and interrupted at regular intervals to evaluate their SOH and post aging which includes electrochemical tests on a different scale and postmortem analyses.

During aging

There are several steps involved in this step that is primarily driven by a detailed aging protocol. The protocol is generally custom designed for each aging study based on several factors, for instance, the requirement from the industry, the phenomena that requires to be studied, etc. The protocol may be one of the following -calendar aging, cycle aging with 100 % DOD and hybrid vehicle and electric vehicle drive cycles under realistic driving conditions where the cell SOC either oscillates around 50 % drive cycle or is completely used [105]. For research purposes, the aging campaigns are mostly accelerated, i.e. the cells are aged under extreme or severe conditions owing to time constraints. The disadvantage of this is that the degradation phenomena are forced to occur, often all competing against each other that makes it difficult to distinguish and identify them individually. One other disadvantage of this accelerated aging protocol is that it is conducted under a controlled environment under very specific conditions and as a result the aging phenomena studied are restrictive to these operating conditions.

As part of these elaborate aging protocols, the cells undergo checkup tests at different stages. Usually the first and last checkups are elaborate to obtain a complete knowledge on the cell SOH. In addition to this, the cells are brought out of the aging protocol conditions to perform a series of intermediate checkup tests to assess their SOH [START_REF] Bernard | Rapport final du projet ANR VTT 2009 -SIMCAL[END_REF]. The checkup tests are custom designed, similar to that of the aging protocol, to assess and measure specific behavior of the cells. Some of the commonly used tests are capacity and internal resistance measurement, Galvanostatic and Potentiostatic Intermittent Titration Techniques (GITT and PITT, respectively), Electrochemical Impedance Spectroscopy (EIS), Galvanostatic charge/discharge or rate capability tests, etc. [START_REF] Shi | [END_REF]. By analyzing the results of these intermediate checkup tests, the temporal evolution of the cell behavior is evaluated and information on the reason for these changes are known. Thus care is taken so that the intermediate tests are not too long and must be carried out carefully because frequent and long interruptions may induce other degradation phenomena and may disrupt the, otherwise natural aging behavior. The tests performed during the checkups and how they may help in identifying the origin or cause of the aging phenomena are described below.

Capacity and resistance measurement

The cell capacity at low current rates during aging, gives a direct estimate of the cell capacity fade when compared to the start of aging. The cell resistance is estimated using different techniques, either by measuring the Ohmic potential drop during pulse charge/discharge tests divided by current causing that drop or performing an EIS (Electrochemical Impedance Spectroscopy) and measuring the cell impedance at different frequencies. While the former technique is a direct indication of the hindrance to the flow of positively charged lithium ions and electrons inside the cell, the second is a measure of the cell impedance due to charge transfer kinetics, double layer capacitance and diffusion, in addition to the flow of charged species [8]. Figure 15 shows an excerpt on a sample cell showing the capacity fade rate during cycling and internal resistance from EIS (Electrochemical Impedance Spectroscopy) at the start (fresh) and end (fully aged) of cycling.

EIS is a technique to measure the impedance of an electrochemical system over a range of frequencies and its response expressed graphically in either a Bode or a Nyquist plot [8]. The impedance measurement of an electrochemical Li-ion cell on a Nyquist plot can be divided into three regions, namely, a high-frequency intercept, a mid-frequency capacitive-resistive loop and a low frequency tail.

The high frequency x-intercept corresponds to the Ohmic resistance drops of the electrolyte and other resistive components; the capacitive-resistive loop corresponds to the combined effect of the insertion/de-insertion reaction kinetic process and the interfacial double layer on the particle surfaces; the low frequency tail corresponds to the diffusion of Li-ions in the solid and liquid phases. EIS performed during intermediate checkup tests allows to identify the following -an increase in the internal cell resistance may cause a shift of the intercept along the positive x-axis (Ohmic resistance drop) and an enlargement of the semicircle at high frequencies (interfacial resistance); the increase in low frequency tail may indicate a reduced diffusivity of Li-ions in the electrode particles [142]. GITT and PITT measurement GITT is a technique where pulse currents are given to the cell for short intervals of time followed by a significantly long rest period. During this rest period, the cell reaches equilibrium and the temporal evolution of the cell potential is monitored. This continues until the cell cut-off potential is reached [143]. Similarly, in PITT, the cell is charged/discharged using a constant potential difference applied to it and the evolution of current is monitored during this period. This is followed by a rest period where the cell reaches an equilibrium condition before this process starts all over again until the cut-off potential is reached.

In [START_REF] Shi | [END_REF], Shi et al. used GITT to estimate and compare the diffusion coefficient values, before and after cycling at different temperatures, as a means to quantify the aging. Additionally, using PITT and GITT curves, differential capacity (or incremental capacity) and differential voltage curves, respectively, are obtained. On a side note, the differential curves may be obtained using low rate galvanostatic charge/discharge curves. The peaks in these differential curves give information on the transition of the phase-change occurring in the electrodes [144,145]. By analyzing and comparing the differential curves of aged cells with fresh ones, information such as cyclable lithium loss, loss of active material and increase in cell overpotential may be obtained [23,146,[START_REF] Edouard | Vieillissement des batteries Li-ion de traction des mécanismes vers le vieillissement accéléré[END_REF]]. An excerpt from [55], in Figure 16, shows the differential voltage curves obtained from the PITT measurements and comparison in the decrease of different lengths of the three regions help identify appropriate phenomena.

Visual inspection

The hard outer aluminum casing, in cylindrical cells, or the soft outer cover, in pouch cells, is safely cut open and the electrode segments are extracted. This is usually done inside an inert glove box atmosphere or a dry fume hood. Figure 17 shows the dismantling of different cells at various stages with different cell geometries and of different construction. Firstly, a simple visual inspection may be performed to identify the inhomogeneities in the electrode segments of a single cell and maybe compare it with of a fresh cell. Although any sophisticated equipment is not needed, as the inspection is done with the naked human eye, sufficient knowledge of the cell, its working components, material composition and possibly occurring outcomes is required to interpret the results. This technique only gives a brief insight on the nature of the degradation phenomena but to obtain greater details, further tests are required to be performed.

Harvesting electrode samples

The electrodes and the separator are usually washed off with dimethyl carbonate (DMC) and vacuum dried for some time. Commercial electrode segments are coated on both sides of the current collector foils with the active material to increase energy density. This poses a hindrance to perform electrochemical tests using half cells by fabricating coin cells, for instance. Thus, one side of the double side coated electrodes is removed by scraping the active material off the surface using a solution of NMP (1-methyl-2-pyrolidinone), for electrodes built upon PVDF polymer. This step is done with utmost attention to prevent the seepage of the solution to the other side. The electrode segments are cut into circular pieces with appropriate dimensions. Following this, in [START_REF] Bernard | Rapport final du projet ANR VTT 2009 -SIMCAL[END_REF], coin cells are assembled where the working electrode is the harvested electrode segment, lithium foil serves as the counter electrode, Whattman GF/D borosilicate glass fiber is the separator and the electrolyte being LiPF6 (1 mol/L) solvent in a 1:1 EC/DMC solution (LP30 by Merck, for instance).

Electrochemical characterization tests

Electrochemical tests are performed using the assembled coin cells. Some of the tests include a series of formation cycles, absolute capacity measurement using a very low current rate (C/20, for instance), residual capacity measurement for positive electrode segments, rate capability tests at different current rates, etc. These tests help to precisely measure exactly the amount of the remaining active material in each electrode and to estimate the change in electrode balancing causing cyclable lithium loss and active material loss [START_REF] Birkl | [END_REF][START_REF] Delacourt | Physical multiscale modeling and numerical simulation of electrochemical devices for energy conversion and storage[END_REF]. These low rate measurement curves may be used to obtain differential curves and reconstruct the pristine aged cell and compare it with the pristine fresh cell to understand the aging behavior [START_REF] Edouard | Vieillissement des batteries Li-ion de traction des mécanismes vers le vieillissement accéléré[END_REF]. Figure 18 shows an example of the low current rate capacity measurements of the positive and negative electrodes harvested from the fresh and aged cells, where the x-axis and y-axis represent the capacity and potential, respectively. The red and green solid lines correspond to the positive and negative electrodes, respectively; the green dashed line corresponds to the aged condition.

When there is parasitic side reaction occurring at the negative electrode, there is cyclable lithium loss and the green curve shifts to the left. The result of this is that the lower potential regions of the red curve are not utilized anymore and thus the overall cell capacity reduces.

Figure 18 : Graphical representation of a shift in the electrodes electrochemical window upon aging and a consequent reduction in the cell capacity

Physical characterization techniques

Numerous sophisticated post-mortem analyses techniques may be adopted to obtain information on the morphological changes at the particle scale and surface topology (using SEM and TEM, for instance) and compositional changes at the electrode scale (using XRD, XPS, FTIR, ICP-OES, for instance), during aging.

In the case of SEM or TEM, a beam of electrons is directed to a sample and the secondary beam resulting from the secondary or backscattered electrons is detected from that sample as a result of its activation. The result is a high precision two dimensional image (three dimensional image possible by stacking multiple two dimensional images) at the particle scale. In the case of graphite electrodes, in aged electrode samples, the porosity decreases and the particles are seen to be more smooth on the surface when compared to the distinctive flakes in fresh electrode samples [10]. However, the limitation with this technique is that the exact nature and composition of the passivation layer may not be known and thus other tests are needed. Additionally, EDX (Energy Dispersive X-ray spectroscopy) may also be performed using the same platform to obtain information on the composition of the electrode samples (especially the major bulky elements or mostly metals) but is subject to noise and hence the exact composition may not be obtained. Also particle size distribution of the electrode samples may be obtained [116].

In the latter case, the exact composition of the electrode samples may be obtained in weight percentage or ppt (parts-per-thousand), for instance. XPS and XRD also work on the principle of focusing a beam of X-ray on the sample and measuring the kinetic energy and number of electrons emitted. Both provide lattice parameters of the crystalline structure of the samples and thus information on the structural changes may be obtained during the course of aging [77]. ICP-OES and ICP-MS uses the inductively coupled plasma to produce ions and using emission spectroscopy and mass spectroscopy, respectively, the ions are appropriately quantified [70].

Overview

This chapter gives a detailed overview of the state of the art related to the inhomogeneous degradation occurring within single commercial Li-ion batteries. Firstly, the different cell components, currently used and future technologies along with their combined working is presented. Following that the various degradation phenomena, their effect on the macroscopic cell performance and the factors influencing them are detailed. The non-uniform behavior in Li-ion batteries and as a result, the non-uniformly occurring degradation phenomena and the causes for this inhomogeneous behavior is described. The individual effect of the factors influencing the inhomogeneous aging behavior is not completely known and this is the objective of this work. Following this, the experimental and modeling techniques available for studying the inhomogeneous aging behavior are described.

Based on this detailed state of the art of Li-ion batteries, it is decided to adopt a combined experimental and modelling approach in our work, where inhomogeneous behavior is studied experimentally using a custom built experimental device that enables us to decouple the effect of the two factors influencing the inhomogeneous behavior and study their individual effect on aging. In addition to this, appropriate tests are performed on the cells to assess their performance, both during the course of aging and post aging. Following this, a coupled physics-based model is used to both complement and supplement the experimental results obtained, giving more insight on the inhomogeneous aging behavior.

3 Benchmarking of cells -choice for aging

Introduction

The primary objective of this work, as described earlier, is to separate the effect of the two parameters influencing the inhomogeneous behavior and to evaluate their individual effect on the inhomogeneous cell aging behavior. Thus, to overcome the issue of the inherent inhomogeneity present in commercial Li-ion batteries, a setup is custom built. The setup is conceptualized based on the hypothesis that a network of smaller cells is connected in parallel, as shown in Figure 19, such that each of these cells represent different electrode segments of a commercial cell and the setup cumulatively mimics the behavior of the commercial cell. Smaller cells in this setup ensure uniformity in performance within each cell and between the different cells. A study by Zhang et al., in [95], testifies this hypothesis where a pouch cell was modified to resemble numerous unit cells when assembled in an appropriate manner. In our work, it is also hypothesized that four cells, with appropriate shunt resistors, sufficiently surrogate the commercial cell behavior. The reason for using four cells is to ensure that the nuances in the cell behavior from different regions of the cell are sufficiently represented and yet at the same time, the setup remains relatively simple. The shunt resistor represents the conductive resistance experienced by an electron to traverse between an electrode segment and the current collecting tab connected to the cell terminal. The transformation between the full commercial cell behavior and the network shown in the figure below is described in the following chapter.

Figure 19: Schematic representation of the equivalent electronic setup showing 4 electrochemical cells connected in parallel with an appropriate shunt resistor and a feature to regulate the cell temperatures individually, to surrogate the inhomogeneities observed in commercial Li-ion cells

To perform this study, the cells must be homogeneous in nature, with uniform thermal and potential distributions within them and must have minimum cell-to-cell variation under the same operating conditions. Thus this chapter deals with the part of identifying the appropriate cells needed for this study. Different options are considered (four in total), namely, two of them using fabricated coin cells where in one the electrode segments are harvested from commercial Li-ion cells and in another, the electrode segments are procured from a commercial supplier; in the third option, pouch cells are used (initial formation cycles are performed after electrolyte filling) and finally, coin cells directly procured from a commercial supplier. A series of tests are performed on these cells to evaluate the consistency and the repeatable nature of the cell behavior that will be discussed in the sections below. To begin with, the different cells used in this study along with their specifications and the protocols used for their testing are briefed. In the section following that, the test results are presented in detail and based on these results, an appropriate choice of the cells, to perform aging on the setup, is made.

Description of cells

The specifications of the cells, the details of cell fabrication process and the cell test protocols are detailed in this section, for each of the four options described before.

Commercial Li-ion cells

The choice of commercial Li-ion cells is primarily driven by the energy or power type of classification. The power type cells (with thin electrodes) operate at high currents causing high temperature and potential gradients while the energy type cells (with relatively thicker electrodes) and large dimensions also have significant gradients within them. Thus these cells with huge inhomogeneities within them, even at moderate operating conditions, may be used in this study. Two different cells are proposed to be used, LFP/C and NMC/C cells manufactured by A123 and Kokam, respectively. The chosen cells are shown in Figure 20 and their specifications are given in Table 2. For the A123 cell, a cutting tool is used to cut open the hard outer aluminum casing and with the Kokam cell, a scissor is used to cut the soft outer casing. The electrode segments are separately (positive and negative electrodes) rinsed in DMC (dimethyl Carbonate) twice, for about 15 minutes each, and then vacuum dried for about 2 hours. The harvested electrode samples are double side coated with active material, except those on the outer ends of the Kokam pouch cell where only one of its surface is coated with the positive electrode material. For future references, the single-side coated aluminum current collector foil is referred using the acronym ssc while the double side coated electrodes are referred to as dsc. It is suspected that the positive electrode from the different, ssc and dsc sheets portrayed different behaviors because of difference in thickness due to mechanical stability and strength, probably an effect during calendaring at the manufacturing stage, and hence these differences are to be evaluated. In the case of the A123 cell, the current collecting tabs are considered to be the reference positions and different electrode segments are investigated, especially those adjacent to the current collecting tabs, as shown in Figure 21. In addition to the electrode segments adjacent to one another, indifferences between electrodes from different tab positions along the spiral winding are also investigated.

Figure 21 : Illustration of harvesting the electrode segments from different tab locations of the A123 cylindrical cell

One side of the double side coated electrode segments is removed by scraping the electrode surface using a solution of NMP (1-methyl-2-pyrolidinone) because the electrodes are prepared based on a PVDF material. This is to ensure that for the coin cell fabrication one of the surfaces is open to establish contact with the terminals. Following this, coin cells are assembled using the negative and positive electrode segments; full cells with the working and counter electrodes as the pristine cell positive and negative electrodes, respectively, and half cells with the working electrode as the pristine cell positive and negative electrodes and the counter electrode as the lithium foil. The schematic of the exploded view of the coin cell components is shown in Figure 22. The coin cell is assembled in the same order as shown in the figure, from the positive terminal cap below to the negative terminal cap above. The separator and the working and counter electrodes are circular with appropriate dimensions but it is ensured that the working electrode is smaller than the counter electrode. This is due to safety concerns (to avoid lithium plating on the counter electrode) and due to ensure appropriate balancing between the electrode electrochemical windows. The other cell components are Whattman GF/D borosilicate glass fiber as the separator and LiPF 6 salt of 1 M concentration in a 1:1 EC/DEC solution (LP40) as the electrolyte. The coin cell configuration is summarized in Table 3. The coin cells after assembly undergo a series of tests to assess their behavior. To quantify their behavior a checkup protocol is established inspired from the SIMCAL project [START_REF] Bernard | Rapport final du projet ANR VTT 2009 -SIMCAL[END_REF], starting with a few cycles for the stabilization of the passivation layer (also referred to as, SEI reformation cycles), rate capability and impedance measurement tests. The checkup protocol is summarized in Table 4; step 1 is to stabilize the passivation layer on the electrode surfaces; steps 2 is to measure the cell capacity using a CCCV (constant-current constant-voltage) protocol at a low current rate of C/20; steps 3 is to perform an EIS (Electrochemical Impedance Spectroscopy) to evaluate the cell internal resistance at different SOC (State of Charge); step 4 is to assess the rate capability at different current rates starting from C/10 to 5C; and step 5 is low current rate at C/10 to verify the stability of the cell even after cycling at high current rates. These results provide a platform to compare the performance of different coin cells and to also to compare with the pristine cell. The operating potential ranges of the full cells are the same as that of the original pristine cell (as given in Table 2).

Custom Cells electrodes

Following the work on electrode segments harvested from commercial cells, electrode segments are procured directly from a commercial supplier, Custom Cells Itzehoe GmbH. The electrode segments are packed in compartments, as shown in Figure 23, that are sealed in vacuum packets in numbers of six and the electrode specifications are given in Table 5. These electrodes are single side coated with active material and it is believed that the quality of the electrodes is high and their performance is more reliable and consistent compared to harvested electrode segments. Two different cell chemistries are chosen for the work, namely, LFP -High power and Graphite -High power. Half and full coin cells are fabricated as described in the previous subsection. The specifications of the electrode segments and the details of the formation cycles for the half cells (with lithium foil as the counter electrode), as described by the manufacturer, are given in Table 5. The protocol for the formation of full cells with LFP and graphite as the working and counter electrodes, respectively, is the same as stated in the table below.

The voltage limits are considered to be the same as that described in the previous subsection. Following the formation cycles, the cells undergo a checkup as described in the previous subsection. In addition to this, the cells also undergo cycling tests at different temperature, namely, at 25 and 45 o C to again assess the repeatability in the capacity fade rate behavior of the cells, for the same reasons detailed in the subsection before.

Li-fun pouch cells

The pouch cells, used in this work, are manufactured and assembled by Li-Fun Tech and are complimentarily provided by Jeff Dahn Research Group with kind. These cells are only partly fabricated, meaning that the cells are devoid of any electrolyte but stacked with just the electrodes and separator. The cells are required to be filled with electrolyte and appropriately formed to get a stable passivation layer. The specifications of the cell are given in Table 6 [START_REF] Xiong | [END_REF]150]. The cells are prepared using the steps described below  An electrolyte with the following specifications is prepared -1 M of LiPF6 in EC:EMC (3:7 by vol. %) with 2 % VC and each cell requires about 0.9 g (0.75 mL approx.) of the electrolyte  Finally, an EIS is performed before assessing their performance through checkup tests  It is to be noted that with 2 % VC, about 0.3 mL gas is expected to be released during the degassing step Following the cell preparation, the cells undergo a checkup to assess their behavior. The checkup protocol is similar to that described in Table 4.



Varta coin cells

In order to ensure repeatable results from industrial processes, commercially manufactured coin cells are considered. It is brought to our notice that the coin cells manufactured by VARTA Microbattery GmbH are currently used in portable and cordless devices in the electronic industry have a high quality in performance with highly repeatable behavior. Owing to its low thermal mass and small geometry, it is assumed that there is thermal and potential uniformity within the cells, thus also satisfying the primary requirements of this work. An image of Varta coin cell is shown in Figure 24 and the cell specifications are given in Table 7. The cells go through a checkup, similar to that described in the previous subsection, and a series of other cycling tests to assess the different behavioral aspects of the cell. The protocols for these tests are the same as described in the previous subsections and the results obtained are given in the following section.

Tests results

The test results of the above detailed cells coming from the checkup and cycling are detailed in this section. The initial cell capacity, internal resistance and aging (whenever available) are individually presented.

Coin cells fabricated using harvested electrodes from commercial Li-ion cells

This subsection consists of two phases; one with the results from checkup and cycle aging data from both A123 and Kokam cells, while in the second, the statistical dispersion of the multiple full coin cell data is presented, obtained from the checkup of just the A123 cell.

Phase I

In this subsection the checkup data of the full coin cells assembled using the harvested electrode segments from the commercial A123 and Kokam cells are presented. Firstly, the absolute capacity and coulombic efficiency of the cells during the passivation stabilization cycles is presented; followed by the absolute and normalized capacity during the rate capability tests of the cells at current rates ranging from C/20 to 5C is plotted which is also compared with the rate capability of the respective pristine cells. Lastly, the temperature dependent capacity fade of the cells at two temperatures, namely, 25 and 45 o C are also presented for both chemistries.

A total of 10 LFP/C and 18 NMC/C coin cells are assembled following a particular chronology. LFP/C cells 1 and 2 and NMC/C cells 1 and 2 (dsc) are assembled first, followed by NMC/C cells 3 and 4 (ssc). LFP/C cells 3 and 4 and NMC/C cells 5 and 6 (ssc and dsc, respectively) are assembled to assess the repeatability of the cells and are eventually used for cycling at higher temperatures. NMC/C cells 7-10 (two each in ssc and dsc) are fabricated again to study the repeatability nature of the cells. Following this, LFP/C cells 5-10 and NMC/C cells 11-18 (four each in ssc and dsc) are assembled later to study the repeatability aspect of the coin cells. It is to be noted that in phase I, with all assembled coin cells, the electrode segments from A123 and Kokam cells are chosen randomly from across the electrode surface. Also, it is necessary to mention that the positive electrodes of the LFP/C cells 5-10 and NMC/C cells 11-18 are smaller, 12 mm in diameter, when compared to the former cells, 12.7 mm in diameter. In the case of NMC/C cells, cells with 12.7 mm and 12 mm positive electrode diameters are grouped together and plotted separately below.

Passivation layer stabilization cycles

Figure 25 shows the discharge capacity and coulombic efficiency of both LFP/C and NMC/C cells during the formation cycles. In Figure 25 (a) and (b), it is observed that LFP/C cells 1 and 2 have a higher capacity when compared to cells 3 and 4 which may be explained by the chronology in which they are prepared; cells prepared/assembled at the same time are observed to exhibit similar behavior. Although LFP/C cells 6 and 7 have similar capacities as the other cells from the same batch at the start (between 1.2 and 1.3 mAh/cm 2 ), they lose a significant portion of their capacity towards the end of the formation cycles and these are declared as faulty cells.

In the case of NMC/C cells, in Figure 25 (c) and (e), there is a greater dispersion in the discharge capacities of cells when compared to LFP/C cells. Of these, cells 7-10 in particular, show very poor performance when compared to the other NMC/C cells. There might have been some discrepancies at the time of the cell assembly or while performing the formation cycles. Comparing the absolute capacities of cells in Figure 25 (c) and (e), it may be seen that the cells with 12 mm diameter, prepared at a later stage, show a poorer performance, which is also seen in LFP/C cells. However, the rate capability of the cells is measured to further assess their performance. Ideally, in the absence of side reactions at the positive electrode, the coulombic efficiency is expected to give an estimate of the charge consumed for side reactions at the negative electrode; a value close to 100 % is desired which indicates a stable passivation layer. The SEI in the negative electrode of pristine LFP/C cell (on the left) seems to be pretty stable resulting in high coulombic efficiencies from the start.

In the case of NMC/C cells, cells 7-10 show poor coulombic efficiency in the start but recover towards the end, indicating that the passivation layers are stable towards the end of the stabilization cycles; while for a few other cells, the coulombic efficiency is small throughout when compared to the other cells.

Rate capability

Next, the rate capability of both LFP/C and NMC/C cells is studied by charging and discharging them at current rates ranging from C/25 to 5C. Figure 26 shows the absolute and normalized discharge capacities of LFP/C and NMC/C cells. The slight decrease in performance of the cells assembled at a later stage may clearly be seen in the case of LFP/C cells, in Figure 26 (a). This difference may arise from the physical degradation of the electrode material either during handling or storage. Verification by manually computing the capacities of the cells with smaller positive electrodes reveal that the measured capacity is smaller than expected hinting the possibility of material deterioration that may be verified using measurements from half-cells. Referring to the rate capability of the LFP/C cells, the total discharge capacity is almost a constant until 2C current rate but decreases at higher current rates. The cutoff voltage limit is reached sooner either because of limitations in the transport of Li + ions in the electrolyte medium or diffusion limitations inside the solid electrode particles.

In the case of NMC/C cells, in Figure 26 . This is comparable with the normalized capacity of the pristine cell (dashed black line, up to 0.5C) and it can be concluded that the performance of the LFP/C coin cells is in good agreement with the pristine cell. The slight increase in capacity of the pristine cells at higher C-rates is because of increase performance from temperature evolution in the cell. In the case of NMC/C cells, the rate capability of the cells is much lower at higher C-rates because it is an energy type cell and thus recoverable capacity is expected to be lower. NMC/C cells 7-10 are normalized with respect to the capacity at C/10 instead of C/20 and still show poorer performance, clearly being the outliers. A few cells, namely 11, 12, 13 and 18 have a normalized capacity slightly greater than 1, probably owing to the same reason as in the case of cells 7-10 but at a much lower rate. Also, similar to the LFP/C cells, a good agreement between the coin cells and the pristine cell is achieved.

Temperature dependent coin cell aging

Figure 27 shows the capacity loss of the cycled LFP/C and NMC/C coin cells, at two different temperatures, namely 25 and 45°C. A total of about 10 LFP/C and 14 NMC/C cells are cycled. In Figure 27 (a) LFP/C cells 1 and 2, cycled at 25°C, and cells 3 and 4, cycled at 45°C, experience a capacity fade corresponding to their respective temperatures (higher capacity with increase in temperature). The capacity fade of cells 7-10 is not very consistent; while 3 other cells have the same starting capacity, cell 7 has a poor performance from the start. It can be observed that cell 8, cycled at 25°C, experiences a similar capacity fade rate compared to cells 1 and 2. Cells 9 and 10, cycled at 45°C, experience a faster capacity fade, at start of cycling, when compared to cells 3 and 4. Overall, the cells cycled at 25°C, experience a gradual capacity fade, retaining about 80 % of the initial capacity, at the end of 1000 cycles while cells cycled at 45°C, experience a faster capacity fade in the start followed by a more gradual fade, leading to 65 % loss (in the best case scenario, with cells 3 and 4), greater than that at 25°C. Comparing this with data from literature, Safari et al. in [80], it is reported that for an equivalent of 1000 cycles measured at 25 o C and 45 o C, the capacity loss is less than 5 % and about 10 %, respectively. Thus, it may be concluded that the capacity fade is more severe in our case and also the it is optimistic that the cells are at 25 o C and 45 o C show consistent behavior.

In the case of NMC/C cells, in Figure 27 (b) and (c), in addition to the capacity fade of the cells at different temperatures, the effect of the ssc electrode segments when compared to the dsc electrode segments is also studied. Firstly, considering cells 1-6, they exhibit a great variation in the capacity fade rate. Cells 1-4 are cycled at 25°C while cells 5 and 6 are cycled at 45°C. Cells 1-4 do not show any consistent trend in their capacity fade behavior; while cells 3 and 4 retain about 50 % of their initial capacity, cells 1 and 2 retain almost 80 %. This difference may be attributed to the temporal differences of the cell assembly. Also strangely, cells 5 and 6 cycled at 45 °C, experience a capacity fade rate similar to that of cells 1 and 2. With NMC/C cells 11-18, in Figure 27 (c), there is even more non-uniform behavior; three cells showed an initial discharge capacity of about half that of the other cells, while cell 14 is damaged and its capacity drops close to 0 within 150 cycles. Irrespective of the temperature, all cells show a steep capacity fade rate at the start of cycling. The inconsistency can further be substantiated with cell 16, cycled at 45°C that experiences capacity fades similar to that of the cells at 25°C. There is also no observable trend or difference between the ssc and dsc electrodes. Preliminary conclusions from phase I Firstly owing to the several non-uniformities, inconsistencies and non-repeatable behavior in the case of NMC/C cells, it is decided to temporarily suspend further work using Kokam cells. Additionally in this work, the overall exercise of performing cycle aging tests using coin cells may be claimed as a good achievement because cells are assembled with a considerably good repeatable behavior and even cycled for about 1000 times. Referring to the credibility of the results, the manufacturer of the A123 ANR26650M1-B 2.3 Ah cylindrical cells, claims that the pristine cells may be cycled for 500 times with a capacity fade of about 5 %; in our case, after 500 cycles a capacity fade of about 10 % or more is seen. Further experiments and additional analysis showed that there are some differences between reconstructed coin cells and the pristine cell; the ratio of the electrode surface area is not the same in the coin cells and the pristine cells. Furthermore, harvesting the electrodes, handling them and assembling them into coin cells contribute to material defects and small modification in the active materials (loss of active matter, short circuit during assembly, etc.). Besides, the assembly of coin cells using the harvested electrodes changes the electrochemical window due to the loss of cyclable lithium during the formation of initial SEI passivation layers. Nevertheless, it has been possible to obtain repeatable results for the LFP/C cells provided that active material and cells are built at the same time.

This extra care will be taken for the building of our next coin cells. Finally, although there is some exception to the temperature dependence on the aging behavior, the effect of temperature is clearly identifiable with the LFP/C cells.

Phase II

Based on the results conclusions drawn from phase I, several coin cells are assembled using electrode segments harvested from a fresh A123 cell to compare their behavior and to identify differences in the electrode segments obtained from different tab positions, as shown in Figure 21. LFP and graphite electrodes from different locations in the harvested cell are used to assemble coin cells. Half cells, as described earlier, are fabricated to assess their performance and consistent behavior is obtained. But these are not shown here because of their insignificance with the current workflow.

The LFP/C full coin cell constitutes of the LFP as the positive electrode of 12 mm diameter, graphite as the negative electrode of 14 mm diameter and same electrolyte and separator as described earlier.

These new cells show a consistent behavior from the rate capability tests and hence a statistical dispersion of the absolute discharge capacity and the ohmic voltage drop is presented.

Figure 28 shows a histogram of the dispersion of absolute discharge capacities and ohmic voltage drop of the cells. The ohmic voltage drop is a measure of the cell internal resistance and is obtained by dividing the potential drops from zero to non-zero current at 1C discharge during about 1 ms. It can be seen that there is a great dispersion in both the cell capacities and internal resistances. The medians of the capacities and resistances are in the range of around 1.4 mAh and 20 Ω, respectively. There is an overall dispersion of about 30 % and 40 % in the capacity and resistance, respectively. However, the capacities are dispersed by about 15 % on either side of the median while the resistances are dispersed by about 30 % and 10 % towards higher and lower values, respectively. In spite of the wide dispersions, both the capacities and internal resistances are seen to be distributed randomly. It is also seen that the cells did not show any specific correlation or trend with respect to the tab locations, with both capacity and internal resistance. This may be because of either lack of such any original trend or that this trend is no longer present because of the human-related errors while fabricating the coin cells at the lab scale. To meet the objectives of this work, it is expected that the cells have a reputable and comparable behavior with similar capacity and resistance values. This is important because to represent the complete cell behavior using the parallel network of coin cells, the differences should come from external sources, for example, placing the cells at different temperatures, and not from the internal differences arising within the cells. However, it may be seen from Figure 28 that the cells with similar capacities did not have the comparable resistances and vice versa which requires us to search for alternate options. In the sections below, the following investigation is done -coin cells assembled using electrode segments provided by a commercial manufacturer and pouch cells delivered by a commercial manufacturer devoid of electrolyte.

Custom cells electrode segments

The coin cells assembled using the commercial electrode segments from Custom cells, using the same procedure as described in the previous subsection, are assessed in this subsection. The cells undergo an initial set of formation cycles, followed by a rate capability test at different C-rates and cycle aging at two different temperatures. A total of about 16 coin cells are assembled and a statistical dispersion of the cell capacity and internal resistance is performed to assess the repeatability and consistency nature of the cells. All cells are assembled at approximately the same point of time to avoid the discrepancies occurring from the temporal changes, as discussed in the previous subsection.

Passivation layer stabilization cycles

The protocol for the initial formation cycles is as presented in the previous section and the discharge capacities and coulombic efficiency is shown in Figure 29 (a) and (b), respectively. Considering Figure 29 (a), the decrease in discharge capacity is small towards the end of the formation cycles, except for one cell. There is also dispersion in the discharge capacities ranging between 1.1 and 1.25 mAh. In the case of Figure 29 (b), the coulombic efficiency at the start is between 85 and 90 % but quickly converges to about 99 % towards the end for all cells, except one. Thus, it may be concluded from this that the cells have a consistent and repeatable behavior.

Rate capability

Following the initial formation cycles, the cells undergo cycling at different current rates ranging from C/20 to 5C and the rate capability test results are shown in Figure 30. Figure 30 (a) shows the absolute discharge capacities of the cells while Figure 30 (b) shows the normalized discharge capacities (calculated with respect to the capacity at C/20 rate) at the different current rates. It may be seen that the cells have a good rate capability, in other words, the decrease in discharge capacity of the cells is small at high current rates when compared to lower current rates. This is because both the positive and negative electrodes are of high power type and thus are designed to operate under high current rates. It may be seen from Figure 30 (a) that the cells display a considerable dispersion in the discharge capacity, similar to the case of the initial formation cycles but when it comes to the normalized capacity the dispersion is very small. It may be concluded that the differences in the cell performance is a result of the imperfections at the time of assembly of the cells; once the cells are assembled, their rate capability is good and repeatable.

Cell capacity and internal resistance

The nominal capacity of the cell may be calculated from the values given in Table 5. It may be seen that the positive electrode is the limiting one with a capacity of 1.03 mAh/cm 2 when compared to 1.07 mAh/cm 2 for the negative electrode. The absolute nominal capacity of the positive electrode is calculated to be 1.32 mAh but it is known that there is a shift in the electrochemical window of the electrodes; assuming that only 90 % of the positive electrode is used, the nominal cell capacity is calculated to be approximately 1.2 mAh [START_REF] Birkl | [END_REF].

The cell discharge capacities and internal resistances measured from checkup of the assembled cells are shown in Figure 32 (a) and (b), respectively. The cell capacity is at a median value of 1.15 mAh while the cell resistance is at 5 Ω. Comparing the measured median cell capacity of the assembled cells with that of the estimated value above, it may be said that the cells are well fabricated and are of good quality. The overall dispersion in the measured cell capacity, from Figure 32 (a), is about 26 % including one outlier cell; suppose that cell is not taken into account, then the dispersion reduces by half to 13 %.

Considering Figure 32 (b), the dispersion is much greater, about 160 % of the median value on both sides. It may be seen that there are only 1 cell each in between 3 & 4 Ω and 8 & 11 Ω; by excluding these extreme values, the dispersion from the median is still about 80 %, half of the original value. Although no significant conclusion may be drawn from this, it may be seen that there is no clear correlation between the discharge capacities of the cells and their internal resistance values and may be concluded that similar to the previous case, the spread is random due to the human-related inaccuracies at the of fabricating the cells in the laboratory.

Figure 32 : Histogram showing dispersion of (a) absolute discharge capacity and (b) ohmic resistance drops of the high power LFP/C Custom cells electrode segments

Based on the analyses performed using coin cells assembled with commercial electrode segments from Custom cells, it may be concluded that the quality of the assembled cells are better compared to harvested electrode based cells but there are still differences in the behavior occurring from the errors or inaccuracies while assembling the cells. The dispersion in the cell capacities and internal resistances is quite small when compared to the previous case where coin cells are assembled using electrode segments from harvested electrode segments from commercial cells. But the fact that the cell capacity and internal resistance is randomly distributed, these cells may not be a valid choice for assembling them in the parallel network to meet the objective of this work. To overcome this discrepancy, multiple cells may be prepared to pick the best suitable cells but the limitation with this is that the Custom cells electrodes are expensive. Thus, further options are explored below.

Li-fun pouch cells

The Li-fun pouch cells are appropriately prepared using the protocol described in the previous section. The cells undergo an initial set of formation cycles, followed by a rate capability test at different C-rates and cycle aging at two different temperatures. Similar to the work using coin cells, a statistical dispersion of about 34 cells is performed to assess the repeatability and consistency of the cell behavior.

Passivation layer stabilization cycles

The formation cycles is carried out in two steps -first step involves charging the cells to 3.5 V at C/20 current rate; this is followed by a degassing process and step 2 involves a C/20 charge to 4.2 V and a consequent discharge at C/20. The charge and discharge capacity is measured during the formation cycles and a coulombic efficiency is computed, which is shown in Figure 33. The median of the coulombic efficiency of the cells is around 89 %, similar to the case of coin cells fabricated the electrodes from Custom cells, but recovers to about 99 % or even higher for all cells. The dispersion for the different cells is small, less than 1 % in each side which indicates that the cells display a consistent behavior.

Figure 33 : Coulombic efficiency during the passivation layer formation cycles of Li-fun pouch cells

Rate capability

Following the formation cycles, the cells undergo a rate capability test at current rates ranging from C/20 to 1C, except the initial set of 8 cells, which are cycled until 5C rate. Following the poor performance at high current rates and the damage caused to some cells, as seen in Figure 34, it is identified that these cells are high energy cells and do not sustain well under high current rates. Thus for the following set of cells, the rate capability tests are limited to 1C rate. It can be seen that the dispersion of the cells at low C-rates is small while it is higher at C-rates. Further analysis on assessing the absolute cell capacity and internal resistance is done in the following subsection, to obtain more information on the consistency and repeatability of the cell behavior.

Cell capacity and internal resistance

The absolute discharge capacities and the internal resistance (or ohmic voltage drop) of the cells measured during the rate capability tests are shown in Figure 35 and Figure 36, respectively. In Figure 35, the dispersion in the cell capacities at C/10 current rate before (blue) and after (red) the rate capability tests are shown. The nominal capacity of the pouch cells, as specified by the manufacturer is 200 mAh. Before the rate capability tests, the cells, represented in blue, are less disperse, mostly lying between 240 and 260 mAh and the median is approximately at 250 mAh. However after the rate capability tests, the cell capacities are much more disperse with the median still being in the range of 250 mAh but with a smaller frequency. Even if the one outlier cell with the capacity in 180-200 mAh range is disregarded, the dispersion is greater than the former case. The drastic decrease in the nominal overall cell performance is not clearly known. Initially, it is thought that this difference is the result of cycling the cells at current rates up to 5C but the result turned out to be the same even when the maximum current rate is limited to 1C. Thus, the consistency and repeatability in the cell behavior is questioned here. The cell internal resistances are plotted in Figure 36 and it can be seen that the median lies between 0 and 0.5 Ω with a mean close to 0.3 Ω. There is clearly one outlier cell in the lot whose resistance is 10 times the average value which is the result of the cycling at higher current rates. Unlike the cell capacities, the internal resistance values are less dispersed around the median. This may be considered as a subjective argument, owing to the magnitude of the nominal resistance at 0.3 Ω, a dispersion of about 400 % may be considered for the cell whose resistance lies between 1 and 1.5 Ω. However, similar to the previous coin cell studies, it is also observed here that the cell capacity and the internal resistance values are distributed randomly. cycles, the cell's capacity drops down to zero. This breakdown may be the result of a leakage in the cell pouch causing the electrolyte to evaporate and dry up eventually. With the other three cells, the data is not sufficient to draw any valid conclusions from them. However, there is a slight decrease in cell capacity of cell 1 at 25 o C after about 200 cycles and a slight increase in the cell capacity of cells 2 and 3 at 25 and 45 o C, respectively, after about 120 cycles. Thus, no clear conclusions may be drawn from this result.

Figure 37 : Capacity fade of the Li-fun pouch cells cycling at 1C current rate at different temperatures

Although these pouch cells are procured from a commercial supplier and are of high quality, there are several other factors that undermine their usage to achieve the objectives of this work. Firstly, there is a decrease in the cell capacity after performing the rate capability tests and also a huge dispersion in the resulting cell capacity values and the reason for this being small discrepancies occurring at the laboratory scale at the time of fabricating the cells, for instance, improper wetting of the electrodes with the electrolyte or possible leaks in the pouch. However, multiple cells were made at the Jeff Dahn Research Group, based on the same type of cells, and different electrolyte additives were studied and the cells are very reliable [151,152]. It is unfortunate that they didn't work well in our work. The random spread in the cell capacity and internal resistance and the huge dispersion is also another factor.

Varta coin cells

The Varta coin cells are procured and a statistical dispersion of cell capacity and internal resistance is measured for fresh cells. In addition to this, a couple of cells undergo cycle aging at 25 o C for about 1500 cycles and the repeatability and consistency in the cell behavior is assessed and reported in this subsection.

Cell capacity and internal resistance

To check the repeatable and consistency nature of the coin cells, their performance is studied by comparing two metrics. The first parameter is the cell capacity and it is measured using a low discharge current at C/20 rate while the second parameter is the cell internal resistance, estimated using the data from the EIS measurement. The EIS data gives the cell impedance at different frequencies that may be represented as a Nyquist plot. The width of the semicircular region in the mid frequency range at 50 % SOC is used as an estimate of the cell internal resistance. Figure 38 (a) and (b) presents the statistical dispersion of the cell capacity and the internal resistance, respectively, of a total of about 24 cells.

Figure 38 : Histograms showing the statistical dispersion of (a) discharge capacity and (b) internal resistance of the coin cells used in this study

It is observed that the dispersion in the discharge capacity is about 3 mAh or 5 % of the average cell capacity while it peaks at about 61 and 62.5 mAh. In the case of the cell resistance, the absolute dispersion is greater, almost about 20 % of the average resistance while it peaks at about 0.26 and 0.285 Ω. Although the absolute numbers may seem quite small, these differences may be considered significantly high to cause variations when the cells are connected in parallel in the setup. But based on the performance of cells from the previous subsections, it may be considered less dispersive. It is also observed that the differences in dispersion of the discharge capacity and the internal cell resistance may be accounted for the differences pertaining to manufacturing inaccuracies. Thus these cells are chosen to be assembled in the setup to perform long term cycling tests and owing to randomness in the distribution of cell capacity and internal resistance.

Aging

A couple of cells are chosen and cycled using the same protocol for about 1500 cycles at 25 o C, in seven different stages, and their capacity fade rate is shown in Figure 39. The capacity fade rate is almost uniform for both cells, except that the rate is slightly faster for cell 1 when compared to cell 2. These differences may be the result of manufacturing differences between the cells. It is to be noted that these tests are not performed continuously but are interrupted multiple times for several practical reasons and the cycling is resumed within a day or two. The capacity gain occurs during the first few cycles of this resumption and this may be because of the relaxation of potential and concentration gradients within the cell [127]. Thus it may be concluded that the cells have a very repeatable capacity fade behavior with a slight dispersion owing to manufacturing differences, as stated in the case of the initial cell capacity and internal resistance.

Conclusion

In this chapter, the behavior of different cell options are assessed that are required to meet the objectives of this work to obtain small sized cells that have a consistent and repeatable behavior. To begin with, the use of coin cells is proposed owing to their small thermal mass, nominal capacity and geometry. Four options are chosen, namely, assembling coin cells using harvested electrode segments from dismantled commercial cells; secondly, assembled coin cells using electrode segments procured from a commercial supplier; thirdly, using commercial pouch cells devoid of electrolyte that need to be prepared; and finally, using coin cells procured directly from a commercial supplier.

In the first case, two different commercial cells are used in this study, of which only the harvested electrode segments from A123 cell is further proceeded with, as the cells fabricated using harvested electrodes from Kokam cell did not give consistent and repeatable data. With the electrode segments from A123 cell, although the quality of the fabricated cells are good, there is a huge dispersion in the cell behavior occurring either due to human errors during the assembly process or from the degradation of the electrode materials. Thus, the second option using electrodes from a commercial supplier, Custom cells is explored. It is observed that the quality of the electrode segments and as a result, the cells is good but there is a huge dispersion in the data that is random and does not follow any particular trend, similar to the previous case. Following this, pouch cells from a commercial supplier, Li-fun is procured that is devoid of any electrolyte. The cells are carefully assembled and based on the statistical dispersion of the cell behavior, it is concluded that these cells show good repeatability for most cells but experience capacity loss after the rate capability tests, probably because the test conditions are too harsh for the cell specifications. In addition to this, like the previous cell types, these cells also display a huge dispersion in the cell capacity and the internal resistance. Although this wasn't the case in the work done in Jeff Dahn Research group . In the case of Varta coin cells, the dispersion in cell capacity, internal resistance and the capacity fade rate is much smaller when compared to the previous choices. Although these may be the result of manufacturing differences, the cells display a good repeatable and consistent behavior. Table 8 summarizes the results and conclusions from the analyses.

Table 8 : Summary of cells used in this work and their performance

Cell property Coin cells using A123 electrodes

Coin cells using

Custom cells electrodes Li-fun pouch cells Varta coin cells

Formation cycles

Moderate repeatability

Good repeatability Good repeatability Not applicable

Rate capability tests

Good for most cells Good repeatability

Good repeatability for most cells but capacity loss after rate capability tests

Good repeatability for all cells

Aging performance

Good temperature dependence but not good repeatability

Insufficient cycling data

Insufficient cycling data

Good repeatability at 25 o C

Dispersion in capacity and resistance

Big dispersion in both

Big dispersion in both

Big dispersion in cell capacity Smaller dispersion

Following the failure of the first three cell types and the good repeatability and consistency along with low dispersion, the Varta coin cells is used in this study. In the following chapter, further tests are performed using these cells and are eventually aged in the parallel network and the effect of inhomogeneity in commercial Li-ion batteries is studied.

Decoupling the effect of parameters influencing inhomogeneity 4.1 Introduction

The factors influencing the inhomogeneous behavior in large-format commercial Li-ion batteries have already been identified to be thermal and potential gradients, as presented in Chapter 2. Following this, studies have also been conducted to assess the effect of each of these parameters on the degradation phenomena. For example, in [153], two cells at different temperatures are connected in parallel and their non-uniform degradation was studied and it was reported that a bigger difference in temperature between the cells causes a greater extent of non-uniform degradation. However, the shortcoming here is that an appropriate quantification of each of these factors, their combined effect, and their interdependency have not been understood well, thus this is the objective of this work. To achieve this, a unique setup is developed to decouple the effect of these two parameters. The terms T-dep and V-dep are used to represent the conditions or setups depicting the thermal and potential inhomogeneities, respectively. In this chapter, only the work done until the start of the cycle aging tests are presented while the further are reported in Chapter 5.

In this chapter, Materials and Methods section, presents the details of the V-dep and T-dep setup along with the detailed experimental setup with its auxiliary components and a brief recap of the cycling and checkup protocol. In the Model Development section, the 1D (one dimension) electrochemical model used in this work and its several hypotheses are detailed, along with the model equations and physical parameters; this model also has a thermal dependent behavior described using an Arrhenius relation.

The electrochemical model performance is compared with the cell behavior obtained from the previous checkup tests. In this section, a 2D (two dimension) electrical and a 3D (three dimension) thermal model are developed and the simulation results using these models are used to obtain the temperature and shunt resistance values for the V-dep and T-dep setup. The electrochemical models are coupled to represent the V-dep and T-dep setup behavior and the variations in the current distribution in the model are compared with the experimental one. In the Discussion section, additional simulations are performed to provide additional understanding about the setup using simulations from the model to estimate the differences in the cell temperature and charge throughput. And finally, the conclusions from this chapter are presented and this serves as a good platform to perform the cycling tests in Chapter 5.

Materials and methods

After a detailed benchmarking of the different possibilities of cells, presented in Chapter 3, it is decided to use commercial coin cells supplied by VARTA Microbattery GmbH. The reason for considering these coin cells is that they are of high quality with a highly repeatable behavior, low thermal mass and small geometry and ensures minimal potential and thermal variations between the different cells, as described earlier. Before mounting these coin cells in the setup, they are characterized using a protocol and the performance of these cells is evaluated by measuring its capacity and internal resistance. It is seen in Chapter 3 that these cells exhibit a small dispersion that is identified to be distributed randomly.

A schematic of the setup is again shown in Figure 40 with Varta coin cells to be mounted. The setup is physically destined to accommodate repeats of the cycling tests under the same operating conditions. Thus, in V-dep setup, one additional repeat is performed and in T-dep setup, two additional repeats are performed, adding to a total of eight and twelve cells in V-dep and T-dep setup, respectively. It is to be noted that the cells are chosen are arbitrarily for the reasons pertaining to the random spread in the cell capacity and internal resistance, as described in Chapter 3.

Figure 40 : Schematic representation of the setup developed in this work showing 4 electrochemical cells connected in parallel, with a feature to regulate cell temperatures and shunt resistors, separately, to surrogate the inhomogeneities in large-format commercial Li-ion cells

There are two variables in this setup, namely, cell temperatures and shunt resistances; assigning appropriate values to each of these variables, different aspects of the large-format commercial cell behavior may be mimicked/surrogated. For instance, by placing the Varta coin cells at different temperatures, the thermal gradients occurring in large-format commercial cells may be surrogated where the higher temperature coin cells represent hotter inner regions whereas the lower temperature ones represent the colder outer regions, for instance. But by keeping the coin cells at the same temperature and using different shunt resistors for each coin cell in the network, the potential gradients may be surrogated; the coin cell with a higher shunt resistance represents an electrode segment that is farther away from the current collecting tab than the one with a lower shunt resistance. The shunt resistors also serve a dual purpose by providing a means to estimate the current in each cell by measuring the voltage across them. They ought to be small when compared to the cell internal resistance, to have as little influence as possible on the overall setup behavior.

To maintain the cells at different temperatures, a separate thermal chamber is necessary for each cell and the rectangular box around each cell, in Figure 40 may be regarded as a local temperature control chamber. These are built using a Leapfrog 3D printer, made using PLA (Poly-Lactic Acid) material that is a good thermal insulator. The heating, required to maintain the elevated temperatures inside the chambers, is provided using thermal mats, procured from Flexelec and their power is controlled using an automated custom-made temperature regulator developed in-house at IFPEN.

To connect the coin cells in the setup and to get an estimate the voltage and current of each cell, a data logging system is custom built, comprising a set of ADAM 4017 analog input modules and the cell voltage and current is recorded in-situ. The analog data is converted into digital and is then fed to a desktop computer through their respective serial communication channels. The system is calibrated to estimate additional contact losses and offsets in the measuring devices by performing an EIS after shortcircuiting each channel individually and by measuring static ohmic resistance values using a multimeter. After assembly, the cells are cycled to undergo aging using the same protocol as described by the manufacturer/supplier. The protocol is a CCCV charge at a constant current rate of C/2 until the voltage reaches the maximum value followed by a CV part until the current drops to C/50 and a constant current discharge at a rate of 1C over 100 % DOD; a 10-minute rest after each charge and discharge is done.

Model development

In this section, the different phenomena of the cell are modeled; firstly, an electrochemical model describes the various transport, conduction and reaction kinetics phenomena in 1D, occurring in the electrode separator assembly. This is followed by a subsection where the electrochemical model is appropriately calibrated and the model performance is compared with experimental cell data. Secondly, an electrical model in 2D describes the conduction phenomena occurring across the electrode surface area and a thermal model in 3D describes the heat conduction phenomena. Simulation results of these models using parameters of a commercial cell are used to obtain appropriate values of temperature and shunt resistors for the V-dep and T-dep setup. Finally, the electrochemical cell models are appropriate coupled to surrogate the V-dep and T-dep setup and the model performance is again compared with the experimentally obtained data.

Electrochemical cell model

A schematic representation of an electrochemical Li-ion cell is shown in Figure 42 (a). There are several techniques to represent this model and the most commonly used ones are physics-based models and equivalent circuit models. In this thesis work, the former modeling technique is used that is inspired from Newman's P2D (pseudo two-dimensional) model [112,113], as shown in Figure 42 (b). It is a physical model of the anode/separator/cathode sandwich. The name P2D stands for the two separate dimensions, namely one along the sandwich and the other in a radial direction along the electrode particles that are assumed to be spherical. The P2D model is based on the porous-electrode theory, which relies on several hypotheses. The electrodes and the separator are taken to be a superposition of two continua, a porous solid matrix and a liquid electrolyte that fills these pores, that are each characterized by its volume fraction. The physical properties describing the continua are averaged over a volume element that is considered small when compared to the electrode layers but large when compared to the pore dimensions. The electrolyte forms an ionic bridge between the two electrodes through the separator. The transport of Li + ions between the electrodes occurs via the liquid electrolyte medium using a combination of diffusion and migration that is described using the concentrated solution theory. The transport of electrons occurs through the percolating network of the conductive carbon filler. It is considered that there are multiple particles along the thickness of each electrode and these particles may either be of uniform or different radius. The transport of lithium occurs through diffusion along their radius. The rate determining step is the lithiation and de-lithiation of lithium and is described using the Butler-Volmer equation. The P2D model consists of the equations, presented in Table 9.

The model used in this thesis further simplifies the P2D model, as shown in Figure 42 (c). This simplification is based on the hypothesis that the solid-phase lithium concentration c s is uniform along the electrode thickness, as described in the work of Prada et al. [125] and Edouard et al. [124]. Then according to this, c s is given by a representative single particle in which the lithium concentration assumes a mean value along the electrode thickness. This implies that the interfacial current density j f for this representative single particle is equal to a mean value along the electrode thickness. However, the liquid phase Li + concentration c e gradient is considered to be significant and is computed along the cell sandwich assembly. Thus, the exchange current density i o , that is a function of both solid phase surface concentration c s,surf and liquid phase concentration c e , as shown in eq. ( 8), varies along the cell sandwich and an average exchange current density i o,avg is computed as shown in eq. ( 16).

𝑖 𝑜,𝑎𝑣𝑔 = ∫ 𝑖 𝑜 (𝑥) 𝛿 𝑜 𝑑𝑥 (16) where δ is each individual electrode thickness. Using this, the Butler-Volmer relation is solved for the kinetic overpotential value η k,avg , as shown in eq. ( 17), under conditions of equal charge transfer coefficients for both anodic α a and cathodic α c reactions, equal to 0.5.

𝜂 𝑘,𝑎𝑣𝑔 = 𝑅𝑇 𝛼𝐹 ln (𝜉 + √𝜉 2 + 1) (17)
where ξ is given by eq. ( 18)

𝜉 = 𝑅 𝑠 𝐼 6𝑖 𝑜,𝑎𝑣𝑔 𝜀 𝑠 𝐴𝛿 ( 18
)
where I is the input current, A is the electrode surface area, R s is the particle radius and ε s is active material volume fraction. The average kinetic overpotential value is used to compute an average solidphase electrode potential Φ s,avg , as given by eq. ( 20).

𝛷 𝑠,𝑎𝑣𝑔 = 𝜂 𝑘,𝑎𝑣𝑔 -𝜙 𝑒 -𝑈 𝑟𝑒𝑓 (19) The above simplifications are further backed by our necessity to have a simplified yet detailed model that describes the cell behavior sufficiently but reduces the computation time and effort while performing long-term cycle aging simulations. The physical parameters of the model are listed in Table 10. The model is developed in COMSOL Multiphysics v5.3a using the Mathematics module. The differential and algebraic equations in the electrochemical model are solved using the Coefficient Form PDE and Domain ODEs and DAEs physics solvers, respectively.

Thermal dependency of electrochemical model

It is known that the performance of a Li-ion cell is highly dependent on temperature. To describe this behavior using the electrochemical cell model, it is considered that certain physical parameters are temperature dependent. This is achieved by considering that these parameters follow an Arrhenius behavior, as described in eq. ( 20). There are two constants in this equation, namely, a reference value ψ ref , measured at a reference temperature T ref of 25°C and an activation energy E a (ψ). The parameters assumed to have a thermal dependency are solid phase diffusivity D s , charge transfer kinetics k o , liquid phase diffusivity D e and conductivity κ.

𝜓 = 𝜓 𝑟𝑒𝑓 𝑒𝑥𝑝 ( 𝐸 𝑎 (𝜓) 𝑅 ( 1 𝑇 𝑟𝑒𝑓 - 1 𝑇 )) (20)

Electrochemical cell model calibration

In this subsection, the model calibration is presented for the electrochemical cell model and its thermal dependency part. The geometrical parameters for the coin cell such as the electrode thickness L and the separator surface area A el are computed using X-ray micro-tomography images of the cell, shown in Figure 43. It may be seen that the electrode separator assembly is spirally wound inside the aluminum casing. It is assumed that the spirally wound electrodes follow a Archimedean spiral law, as reported in [154]. The maximum Li-ion concentration c s,max is theoretically computed using the specific capacity values for both electrodes [155]. Using the information about the electrode geometric parameters, the maximum lithium concentrations in each electrode and the nominal cell capacity, the operating stoichiometry windows x and y at 0% and 100% SOC in each of the electrodes are appropriately estimated along with the values of the volume fractions of active material in the electrodes. For these parameters, an appropriate range is obtained from literature and they are tuned accordingly. An average radius of the positive and negative electrode particles R p are obtained from SEM (Scanning electron microscope) images of the dismantled cell electrode segments, presented in Chapter 5. All curves correspond to model simulations and the black solid line corresponds the parameters that are already fitted with the experimental data. It is to be mentioned here that the sensitivity of the reaction rate constant is not done but rather is considered to be a constant. Looking at Figure 44 (a), it may be seen that, the influence of a smaller negative electrode diffusivity D s,neg is higher at lower SOC regions, thus drastically reduces the capacity of the cell whereas the effect of positive electrode diffusivity D s,pos is almost negligible. In Figure 44 (b), it may be seen that a decrease in liquid-phase conductivity κ, imparts a huge overpotential on the cell. The effect of the transference number and electrolyte diffusivity is lower as a result of the average model, used in this thesis. From this analyses, while κ is adopted from literature, D s,neg is tuned with different combinations of x 0 , y 0 , x 100 and y 100 and an appropriate fit is obtained. The positive and negative electrode potentials as a function of their respective stoichiometry are given in Table 10 [156,157]. The other parameters required for the model are adopted from literature [65,125]. . Of these, it is identified that the biggest significant contribution comes from the activation energy corresponding to D s,neg followed by a smaller contribution from κ, k o,pos and k o,neg . Thus, the activation energy for D s,neg and κ are slightly increased when compared to values given in the literature. The other activation energies are adopted directly from literature [65] and the values are reported in Table 10. To verify the fitted activation energies, an average activation energy value is computed from the internal resistance of the cell measured at different temperatures. The internal resistance, evaluated by measuring the high frequency intercept values on the x-axis is obtained from the Nyquist plots at 10 different temperatures ranging from -20°C to 60°C. It is known that the cell resistance has an exponential dependency with respect to the temperature, as seen in eq. ( 21). In other words, the logarithm of the cell resistance has a linear dependency on the temperature, thus a linear fit may be obtained. The logarithm of the resistance is plotted as a function of the inverse of cell temperature, as shown in Figure 47. It is observed that this linear fit obtained is repeatable between the different cells. Two data points are chosen from this linear fit to estimate the average activation energy of the cell from the relation below. The resulting values for the activation energy and the reference resistance value are 42678 J/mol and 1.58 Ω, respectively; the activation energy value obtained is in good agreement with the values reported in the literature [65,125]. [125]; ¤ from [156]; § from [157]; ~ from [65]

𝑅 Ω = 𝑅 Ω,𝑟𝑒𝑓 𝑒𝑥𝑝 ( 𝐸 𝑎 (𝑅 Ω ) 𝑅 ( 1 𝑇 - 1 𝑇 𝑟𝑒𝑓 )) (21)

Electrical model

The aluminum and copper current collectors, on the extreme ends of the electrodes have very high electrical conductivity and negligible thickness and hence are considered to be in 2D, as described in Chapter 2. The function of the current collectors is to collect electrons resulting from deintercalation of lithium and transfer them to the external circuit and vice versa. For instance, during discharge, the electrons arrive at the positive current collector tabs and are then conducted to the other portions. The electrons are involved in the reduction of host structure at the positive electrode which is then released at the negative electrode during oxidation. These released electrons are accumulated at the negative current collector where electrons converge towards to their nearest current collector tabs. It is to be noted that the transfer of Li + ions through the electrode/separator assembly and the transfer of electrons through the external circuit is conserved.

The total input current at one of the current collector tabs is I, and the input current density, per unit surface area of the tab i +,tab , that is assumed to be equally distributed in each tab, is given by eq. ( 22).

𝑖 𝑘,𝑡𝑎𝑏 = 𝐼 𝑁 𝑘 𝐴 𝑘,𝑡𝑎𝑏 (22) where N + is the number of current collector tabs and A +,tab is the surface area of tabs. Whatever current that is input to the current collector surface exits it and hence the gradient of current ∇i in the x, y and z direction is equal to zero shown in eq. ( 23).

𝜕𝑖 𝑥 𝜕𝑥 + 𝜕𝑖 𝑦 𝜕𝑦 + 𝜕𝑖 𝑧 𝜕𝑧 = 0 (23)
Using Ohm's law, the voltage across the current collector as a function of current density may be obtained as is given in eq. ( 24).

𝑖 = -𝜎. 𝛻𝑉 (24)

Applying the Ohm's law to the current conservation and assuming that the current collector thickness is small when compared its length and width, the voltage distribution across the current collector surface is obtained in 2D, as described in [96], for the current density per unit surface area of each of the current collectors i k,CC , perpendicular to its surface is as given in eq. ( 25).

( 𝜎 𝑥,𝑘 𝜕 2 𝑉 𝑘 (𝑥, 𝑦) 𝜕𝑥 2 + 𝜎 𝑦,𝑘 𝜕 2 𝑉 𝑘 (𝑥, 𝑦) 𝜕𝑦 2 ) = 𝑖 𝑘,𝐶𝐶 𝛿 𝑘 [START_REF]Types of Lithium-ion Batteries -Battery University[END_REF] where subscript k is for either the positive or negative electrodes, σ x,k and σ y,k is conductivity of current collectors in x and y direction, respectively, V k (x,y) is the potential distribution across the current collectors and δ k is the thickness of the current collectors. The input to the model may be either be a potential gradient across the current collector surface or a current at the tabs while the result is a corresponding current profile on these surfaces or a potential gradient, respectively. The model is developed in COMSOL Multiphysics v5.3a using the Electric Currents module.

Thermal model

To estimate the temperature evolution in Li-ion batteries, a precise thermal model is required. Conventionally, a physics-based thermal model is used and is coupled to an electrochemical model with a variable heat generation term or a constant heat term in the absence of a coupled electrochemical model [65,89,97,121,158]. In this work, a physics-based heat balance differential equation, in 3D given in eq. ( 26), is used that is coupled to the above described electrochemical model. The heat generation is the result of both irreversible heating from the cell overpotential (the difference between terminal voltage and equilibrium potential) and from reversible entropic heating, given by eq. ( 27). Heat loss is due to both convective and radiative cooling as shown in eq. ( 28). The equations are tabulated in Table 11. The model is developed in COMSOL Multiphysics v5.3a using the Heat Transfer in Solids module.

Determining parameters for V-dep and T-dep setup

To obtain appropriate values of shunt resistances and temperatures for the V-dep and T-dep setups, the 2D electrical model and 3D thermal model simulations are used. A commercial A123 high power cell is considered for this exercise. Although, not a large-format type of cell, it is known that these cells under high loading conditions exhibit significant thermal and potential gradients [91,93]. The model parameters for the A123 cell pertaining to the electrical model is taken from [94] while some physical parameters pertaining to the 3D thermal model is taken from [125]. While the other parameters are either measured, calculated or estimated and are provided in Table 12. [125]; ~ from [94]

Shunt resistance for V-dep setup

Figure 48 (a) and (b) shows potential distributions on the negative and positive current collector surfaces, respectively. An initial potential for the negative and positive current collectors are taken to be about 0 and 3.2 V, respectively. This model is independent of the electrochemical model and since we are interested only in the potential distribution across the current collectors, the simulation is performed under stationary conditions. An input current equivalent to 2C-rate of the nominal cell capacity (2.3 Ah) is considered and the simulations are performed. When the tabs are considered as the point of reference, the regions in between the tabs exhibit the greatest potential difference; a maximum value of about 9 mV is measured in both the positive and negative current collectors, calculated using eq. ( 29). Thus the A123 cell exhibits a total of 18 mV potential difference at the current collectors for the current input of 2C-rate. Now to obtain the resistance to be used in the setup that is representative of the behavior seen in the simulations, this value is divided by the nominal current value (60 mA) to be used in a Varta coin cell, as shown in eq. ( 30), and a resistance value of 0.3 Ω is obtained. This resistance value obtained from simulations is very small for use in the V-dep setup, from a practical point of view, to spread it uniformly between four cells and yet detectable, at the same, by an inexpensive data logging system. Thus it is multiplied by a factor of 2 and the setup is fabricated considering the maximum difference in resistance being 0.6 Ω, with a minimum value of 0.1 Ω. The others three cells are chosen to be increasing in steps of 0.2 Ω with a maximum value of 0.7 Ω. The resistance values are given in Table 13 and the temperatures of these cells are chosen to be a moderate room temperature of 25 o C. The temperature values for both the T-dep and V-dep setup are set using the simulation results from the thermal model. Figure 49 shows the simulation results of thermal gradients occurring in a cylindrical A123 cell for a constant heat generation term. This value is reported in Table 12 and is obtained by a current equal to C/4 rate (nominal capacity is 2.3 Ah) multiplied by the nominal cell voltage of 3.2 V and also divided by the volume of the cell. In these simulations, the radiative heat loss contribution is considered small and is neglected. The different regions of the cells with different temperatures are identified as shown in Figure 49 and the temperature values are directly used in the setup. The computed values used for the setup is given in Table 13.

𝛥𝜙 𝑗 = 𝜙 𝑗,𝑡𝑎𝑏 -𝜙 𝑗,(𝑥,𝑦) (29)
𝑅 𝑒𝑙𝑒𝑐 = 𝛥𝜙 𝑝𝑜𝑠 + 𝛥𝜙 𝑛𝑒𝑔 𝑖 𝑐𝑜𝑖𝑛 (30)

Coupling

Finally to represent the V-dep and T-dep setup behavior in a model, the electrochemical models need to be appropriately coupled. Necessary information is exchanged between the individual cells in the model, represented as Components in COMSOL Multiphysics v5.3a. Four electrochemical models, representing the four cells in the network, are coupled to one another using Kirchoff's circuit laws. In reference to Figure 40, there are three different closed voltage loops and the resulting equations are given in eq. (31). By conserving the current inflow and outflow in the circuit, the resulting equation is given in eq. (32).

𝑉 𝑗 + 𝑅 𝑗 𝐼 𝑗 -(𝑉 𝑗-1 + 𝑅 𝑗-1 𝐼 𝑗-1 ) = 0, where j = 2, 3 and 4 (31)

𝐼 = 𝐼 1 + 𝐼 2 + 𝐼 3 + 𝐼 4 (32)
where V 1 to V 4 are the cell voltages across each branch, R 1 to R 4 are their respective shunt resistors and T 1 to T 4 are the temperatures of each cell. The two different coupled models are simulated and the model performance is compared with the experimentally obtained data below. Figure 50 and Figure 51 draw a comparison between the experiment and model of an excerpt of the current distribution in each of the cells during the initial stages of the cycling tests, for the T-dep and V-dep setup, respectively. In both figures, on the left is the data from the experimental setup while on the right is from the model. The cell network is charged at a constant current rate of C/2 followed by a constant voltage part where the total current drops to C/50 and is discharged at a constant current rate of 1C. It is known that higher temperatures enhance the transport phenomena and reaction kinetics and as a result cell at a higher temperature has a lower internal resistance. Hence at the start of both charge and discharge cycles, cell 4 at 40°C has the highest current while cell 1 at 25°C has the smallest current flowing through it which may be seen from both experiments and the simulation results. This trend reverses towards the end of both charge and discharge cycles but is more evident during discharge. This is because of higher overpotential values with the use of 1C rate during discharge when compared to C/2 used during charge. Also at 100 % DOD the steep region of graphite electrode is reached which results in big voltage differences between cells for small SOC differences. Cells 1 and 4 at the extreme temperatures have the highest dispersion from the average values while cells 2 and 3 are closer to the mean value. This clearly indicates that the highest level of inhomogeneity occurs in the current profiles of cells 1 and 4. A comparison of the current flow pattern in the cells between the experimental data and the model simulation result shows that the experimental setup is very well represented by the model during both charge and discharge. The transition in the current distribution profiles occurs precisely at the same time although the magnitude of the peaks is different. It is to be noted here that cell-to-cell variations, described earlier, play a significant role during experiments which is absent in the model and the slight differences between the experiments and the model may be explained by this. The current distribution pattern in the cells, seen in Figure 50 (c) and (d), with the same NMC/C chemistry, is comparable to that described by Klein and Park in [159]. According to this, during 100 % DOD usage, there are five different regions corresponding to the five phase transitions occurring in the cells. The offset in the attainment of the phase change and as a result the overpotential between the different cells is the reason for the current dispersion in the networks. It is also seen that the highest dispersion in current distribution profiles is in the small SOC ranges, especially at high and low SOC during both charge and discharge. Thus it may be interesting to see the effect of extended cycling in small SOC ranges when there isn't enough time for the potential and concentration gradients to attain equilibrium.

Figure 51 shows the comparison of current distribution profiles of the cells in V-dep setup obtained from experiments and model simulations, where Figure 51 (a) and (b) show a complete charge and discharge cycle and Figure 51 (c) and (d) concentrate on just the discharge cycle. From our understanding of basic electrical systems, a higher resistance in a channel results in a lower current flowing through it. Using this reasoning, it may be seen that cell 4 with the highest shunt resistance has the least current flowing through it and vice versa for cell 1, at the start of both charge and discharge. This trend reverses towards the end of both charge and discharge. Similar to the previous case, while cells 1 and 4 have the highest current distribution in the network, cells 2 and 3 are closer to the average current in the setup, indicating that the inhomogeneity in the latter set of cells is lesser when compared to the former. Again a comparison of the experimental data with the data from model simulations reveals a very good qualitative agreement between the two. The transition in the current profiles between the cells occur precisely at the same time with a slight difference in magnitude.

Supplementary model simulations

In this section, additional simulation results are presented to get a better insight on the setup behavior and to understand the intricacies in the setup, if any. According to this, charge throughput in the cells and the individual cell temperature evolution are presented. The charge throughput helps identify the cells that are utilized the most when compared to the others and estimate the aging pattern in the cells.

Evaluating the cell temperature evolution helps to estimate the temperature increase in the cells during operation and to ensure that it doesn't influence the temperature of the setup significantly.

Charge throughput

Charge throughput may be defined as the cumulative capacity passing through the cell. Figure 52 (a) and (b) shows the differences in the charge throughput of the cells during the initial stages of cycling in the V-dep and T-dep setups, respectively. Since the charge throughput in cells are in very close proximity upon a complete charge and discharge cycle, a difference is computed by considering the most utilized cell as the reference. In Figure 52 (a) and (b), cell 1 with 0.1 Ω shunt resistor and cell 4 at 40°C have the highest utilization, respectively. The capacities of other cells are subtracted from these; the blue curves correspond to the cells that are closest to the reference cell while the yellow curves correspond to the most dispersive from the reference cell. It may be seen that at the end of CCCV charge, the charge throughput in all the cells, in both setups, is the same and equal to 0, owing to the CV part where there is sufficient time for the relaxation of potential and concentration gradients within the cell. However, this is not the case during discharge, where the charge throughput is non-zero towards the end. It may be seen that the peaks during discharge are higher in magnitude when compared to charging conditions, for both setups. However, the maximum absolute difference at the discharge peaks of the two plots are

Cell temperature evolution

The temperature evolution in the cells is estimated by integrating a thermal model to the existing coupled electrochemical model. Owing to the small size of the coin cells in this work, it is assumed that the thermal conductivity along the radial and longitudinal directions are insignificant and the resulting heat balance equation is given by eq. ( 33), also referred to as a 0D or lumped model. The heat generation is also simplified to just consider the irreversible heating term, as shown in eq. ( 34), that is obtained from the experiments performed. The heat loss is also simplified to just consider convective heat loss term, as shown in eq. ( 35). The cell properties such as its dimensions, R cell and H cell , and mass m cell are measured directly. The conductive heat transfer coefficient h con and specific heat capacity C p are fitted with galvanostatic charge and discharge curves and also by measuring the cell temperature at the same time, using an optimization algorithm by solving eq. ( 33). These parameters are given in Table 14. 53 (a) shows that cell 4 has the highest rise in cell temperature of about 3.25°C while in cell 1, this value goes up only by about 2.8°C. The maximum difference in the temperatures between cells 1 and 4 is about 0.45°C. Figure 53 (b) reveals that cell 1 at 25 o C has the highest absolute rise in cell temperature of about 3.3°C, occurring towards the end of discharge, while in cell 4, this value goes up only by about 1.75°C. This is because, cell 4 is at a higher ambient temperature and with an increase in temperature, the cell internal resistance is lower due to enhanced reaction kinetics and transport properties and thus lower is the heat generation. It may also be seen that the increase in individual cell temperature is concurrent with its ambient temperature conditions and not exceeding the temperature of the other cells. This indicates that the temperature differences between cells are big enough that the individual cell temperature evolution does not influence the setup temperature conditions. After analyzing the patterns in current distribution profiles in V-dep and T-dep cells from Figure 50 and Figure 51, respectively, charge throughput profiles from Figure 52 and temperature profiles from Figure 53, it may be concluded that the temperature differences between the T-dep cells is a bigger contributor to the inhomogeneities between cells when compared to the potential gradients. Also, applying this to the context of a commercial Li-ion battery, it may be stated that the temperature differences in the cell, as a result of just the potential gradients occurring across the electrode surfaces, may be estimated to be minimal. A similar effect may be forecast when cycling the cells in the V-dep and T-dep setup. However, this statement requires further validation and thus, in Chapter 5, the cells are aged for about 1000 cycles and the aging pattern is studied using both experimental and modeling study.

𝑚 𝑐𝑜𝑖𝑛 𝐶 𝑝 𝑑𝑇 𝑑𝑡 = 𝑄 𝑔𝑒𝑛 -𝑄 𝑙𝑜𝑠𝑠 (33)
𝑄 𝑔𝑒𝑛 = 𝐼(𝑉 -𝑈 𝑂𝐶𝑉 ) (34)
𝑄 𝑙𝑜𝑠𝑠 = ℎ 𝑐𝑜𝑛 𝐴 𝑐𝑜𝑖𝑛 (𝑇 -𝑇 𝑎𝑚𝑏 ) (35)

Conclusion

In this chapter, the steps leading up to the fabrication of the setup is detailed. A 2D electrical and a 3D thermal model are developed and simulation results are used to obtain appropriate values of temperature and shunt resistors for the V-dep and T-dep setup, respectively. A 1D electrochemical model is developed and these models are appropriately coupled to represent the V-dep and T-dep setup. The coin cells undergo initial checkup for performance assessment and there is a good agreement with the electrochemical model. The coin cells are assembled in the setup and the cells are cycled shortly to assess the setup behavior. There is a good qualitative agreement between the coupled model and the experimental cell network behavior. Analyses using additional simulation results with the coupled model are done to obtain the network cell temperature evolution and charge throughput. These results give an early indication that the thermal differences between the cells in the T-dep setup may have a bigger influence on the inhomogeneous behavior in large-format commercial Li-ion batteries when compared to the potential gradients, under the conditions considered in this work.

It is noted while discussing the current distribution profiles that the highest dispersion between cells occur within small SOC ranges when there is no time for potential and concentration gradients to attain equilibrium. Thus it may be interesting to see the effect of cycling the cells in V-dep and T-dep conditions in narrow SOC ranges, which is likely to occur under realistic driving conditions in automobile applications.

Overall, the combined experimental and modeling work provide the basis to perform long term cycling tests using the coin cells in the V-dep and T-dep setup. The multi-physics coupled model is updated to incorporate an aging component and the experimental results are compared with the model performance. This analysis allows to identify the primary sources of aging and also to quantify the effect of each of these parameters on the inhomogeneous aging behavior. Further results from long term aging tests and analyses using model simulations are done in Chapter 5.

Cycling tests and inhomogeneous aging behavior

Introduction

In the previous chapter, the steps preceding the long-term aging tests are detailed. A combined modeling and experimental work is used; in the experimental framework, a setup using a parallel network of smaller cells, namely Potential-dependency (V-dep) and Temperature-dependency (T-dep) is developed to mimic the thermal and potential gradients in commercial cells, respectively. In the modeling part, multi-physics models, appropriately coupled to represent the experimental setup behavior, are developed and analyses using simulations substantiate the proof of concept and provide additional insight on the setup behavior. As an extension to the previous chapter, a combined modelling and experimental approach is again used to study the aging behavior. The experimental part consists of using the V-dep and T-dep devices to assess the effect of the factors influencing the inhomogeneity, on the non-uniform aging behavior. Following this, the multi-physics model, from the previous chapter, is additionally equipped with an aging component and the experimental results are compared with modeling simulations to better explain the behavior observed during the long term cycling tests.

Below, the Materials and methods section provides the experimental test details of the individual coin cells to estimate the baseline aging behavior. The steps involved in performing the post-mortem tests and the electrochemical analyses on the harvested electrode segments from dismantled aged cells, are detailed. In the Model description section, the hypotheses and the equations governing the aging behavior with SEI (Solid Electrolyte Interface) growth and PAML (Positive electrode Active Material Loss) models are detailed. Also, appropriate parameters from these aging models are calibrated using the baseline aging behavior of individual coin cells. Following this, in the Results and discussion section, the capacity fade and internal resistance increase results from the intermediate checkup tests performed during the V-dep and T-dep cycle aging tests, drawing a comparison with their corresponding model simulations. Also, the post-mortem test results comprising of the physical characterization and electrochemical tests are highlighted. Using complementary model simulations, the estimates of charge consumed at the positive and negative electrodes and the resulting cyclable lithium and active material loss to the total capacity fade are also presented. Supplementary model simulations give the current distribution in the cells, charge throughput in the cells and the temperature evolution when the cells are fully aged. Finally, the Conclusion section summarizes the results obtained and conclusions drawn from this work along with providing the scope for future work.

Materials and methods

In this section, the steps taken to assess the baseline aging behavior of the Varta coin cells are provided. The aging protocol for the cycling tests that include intermediate checkup tests to assess the SOH (State of Health) of the cells is presented. After the cycling tests, the steps towards dismantling the cells are provided, followed by the post-mortem tests and the electrochemical tests using the harvested electrode segments from the dismantled cells are detailed. Model analyses and additional simulation results are provided wherever necessary.

As a brief recap, it is decided to use commercial coin cells by VARTA Microbattery GmbH because these coin cells are of high quality with a highly repeatable behavior. Also, owing to their low thermal mass and small geometry, there is thermal and potential uniformity within each cell. Before using the cells in the V-dep and T-dep setup for the cycling tests, an initial checkup is performed and the results are presented in Chapter 3.

Aging protocol

To begin with, some preliminary aging tests are done on the cells individually to assess their baseline aging behavior. Three different tests are performed, where, in the first test, the cells are stored in a state of rest at different temperatures, at 100 % SOC (State of Charge), for about 30 days to undergo calendar aging. The elevated temperatures used are the same used to perform the cycle tests, at 30, 35 and 40°C. This provides information to estimate the reversible and irreversible capacity losses in the cell that helps in justifying the hypotheses considered in this work. In the second test, cycle aging of two pristine coin cells at 25°C is done for about 1200 cycles, as described in Chapter 3, a CCCV charge (constant current charge rate of C/2 followed by a constant voltage charge until the current reaches C/50) and a constant current discharge rate of 1C. This enables to assess the baseline aging behavior of the cell and also enables to calibrate the model that is presented in the Model development section. In the final test, the cells are cycled at different temperatures, namely, 25, 30, 35 and 40°C, for about 200 cycles using the same protocol as of the second test. This information is used to estimate the basic thermal dependency of the aging component of the cell model that is also described again in the Model development section.

Similar to the cycling of individual cells, the cells in the V-dep and T-dep setup undergo a CCCV charge (constant current charge rate of C/2 followed by a constant voltage charge until the current reaches C/50) and a constant current discharge rate of 1C. As part of the aging protocol, checkups are performed before the start and at the end of cycling (protocol is detailed in Chapter 3). In addition to this, the cycling tests are interrupted, approximately after every 100-150 cycles, to perform intermediate checkup tests. These are necessary to assess the SOH of the cells by measuring the cell capacity, impedance, etc. The intermediate checkup tests ought to be done swiftly, with little interruption time, to not influence the aging behavior through calendar aging, for instance. Thus, the C/20 charge/discharge step is removed and all constant current charge regimes are replaced by a CCCV charge at a constant current rate of C/2 and a constant voltage cut-off current of C/50 and other steps remain the same. All the checkup tests are done under controlled environment inside a Binder climate chamber at 25°C using a VMP3 potentiostat from Biologic. Post-mortem analyses are performed on the harvested electrodes from both pristine and aged cells. SEM (Scanning Electron Microscopy) analyses for studying the morphological changes of the aged electrode samples are done using a FEG Zeiss Supra 40 microscope from 2007. The elemental composition is studied by performing an EDS (energy dispersive spectrometry) using a SDD detector from Brucker. XRD (X-ray Diffraction) analysis is performed to identify the main constituents of the positive electrode and to quantify their molar ratio using a PANalytical X'Pert Pro diffractometer in the 2θ range 5-80° with a source of Cu Κα, where λ = 1.5406 Å. The results also provide a means (through SEM images) to compare the differences between the pristine and aged electrode samples.

Post-mortem analyses

Model development

The electrochemical cell model used in this work, along with the thermal dependency of certain variable parameters and the coupling of models to represent the V-dep and T-dep setup behavior is described in the previous chapter. In this chapter, an aging component is integrated into the coupled multi-physics model. There are two degradation phenomena considered in this work, namely, SEI growth on the negative electrode particles and active material loss on the positive electrode. These are modeled as separate components, in COMSOL Multiphysics v 5.3a and integrated into the coupled model through an exchange of appropriate information.

SEI growth model

The aging model is integrated into the electrochemical model described in Chapter 4. It is hypothesized in this work that the formation and growth of SEI on the negative electrode particle surface is responsible (either directly or indirectly) for the cell internal resistance increase and one of two major contributors to the cell capacity fade (especially cyclable lithium loss). In other words, it may be said that the SEI growth cumulatively accounts for all the interfacial phenomena (electron tunneling, charge transfer kinetics and ) occurring on the surface of electrode particle. An aging model may be represented in several ways; the most simple and straightforward way is using an empirical relation for the capacity fade, as described in [136,153]. In other works, a physics-based aging model was used to represent this behavior [63]. According to this, it is considered that EC or ethylene carbonate ((CH 2 O) 2 CO) in the electrolyte reduces to form a radical compound and finally, in the end, results in ethylene dicarbonate ((CH 2 OCO 2 Li) 2 ), as shown in eq. ( 36) and it may be seen that the for every molecule of the end product, there are two electrons involved.

2(𝐶𝐻 2 𝑂) 2 𝐶𝑂 + 2𝐿𝑖 + + 2𝑒 -→ (𝐶𝐻 2 𝑂𝐶𝑂 2 𝐿𝑖) 2 (36)
The hypotheses for developing this model are as follows, there is only a single uniform organic porous layer surrounding the particles while the inner inorganic layer is completely neglected assuming that electron tunneling occurs in this phase due to its relatively insignificant thickness. The solvent molecules travel through the SEI to reach the particle surface followed by an irreversible reaction to form the SEI compound. In earlier works while considering the diffusion of solvent molecules through the SEI, an additional convective term corresponding to the growth rate was considered [160], as shown in eq. (37).

Since it was established that the contribution of this term is negligible, it is not considered in this work.

Following this, the result is a non-linear differential equation with appropriate boundary conditions. An additional hypothesis is considered here such that the rate of change of the concentration gradient is considered to negligibly small within the SEI layer. Thus, the concentration gradient is established almost instantly or, in other words, is considered to be almost a steady-state condition. The non-linear differential equation is solved numerically, similar to work of Yang et al. [134], to obtain the relation given by eq. ( 38), in Table 15.

𝜕𝑐 𝐸𝐶 𝜕𝑡 = 𝐷 𝐸𝐶 𝜕 2 𝑐 𝐸𝐶 𝜕𝑟 2 - 𝑑𝛿 𝑆𝐸𝐼 𝑑𝑡 𝜕𝑐 𝐸𝐶 𝜕𝑟 (37)
After the diffusion of the solvent molecules across the SEI, the solvent molecules reach the particle surface where its reduction occurs. This is an irreversible reaction and its rate kinetics is described using Tafel's relation, as given in eq. ( 39). The growth rate of SEI is estimated by the side reaction current density and the relation is given in eq. ( 40). The number two on the right of the equal to sign, in the denominator, is the result of eq. ( 36), where two moles are exchanged per reaction. Finally there is a current balance relation where the total current input at the negative electrode is a result of the computed side reaction current i s and the intercalation or de-intercalation current from the Butler-Volmer relation of the electrochemical model i int , given by eq. ( 41). The increase in cell resistance due to the evolution of SEI thickness is given by eq. ( 42). The solvent diffusivity D EC and reaction rate kinetics constant k f,s are taken to be temperature dependent and thus follow Arrhenius law, as given in eq. ( 43). )) (43)

PAML (Positive electrode Active Material Loss) model

In addition to the SEI growth model, a model to address the active material loss in the positive electrode is developed. In is shown later, in the Model calibration subsection, that there is a side reaction occurring at the positive electrode consuming electrons for the oxidation of the electrolyte. In addition to this, it is also suspected that there is active material loss at the positive electrode. The mechanism for this occurrence was described in [132] by Dai et al., for spinel based cathodes. According to this, the solvent oxidizes and gives out protons, among other reactions, that cause the Mn 3+ to undergo a disproportionation reaction. The effect of this was described to be cyclable lithium loss at the positive electrode, formation of passivation layers on the positive electrode particles and reduction in its volume fraction [133].

In the case of NMC electrodes, similar acid attacks are reported to occur at the positive electrodes [73][74][75]. In this work, it is considered that solvent oxidation occurs at the positive electrode and as a result of this, the acid formed by reaction with the LiPF 6 salt in the electrolyte corrodes an equivalent amount of the transition metal and this is said to occur above 4 [V] [68]. In other words, every electron produced due to the electrolyte oxidation is consumed for the corrosion of an equivalent molecule at the positive electrode. The solvent oxidation is driven by anodic irreversible reaction kinetics (Tafel's expression) with a charge transfer coefficient γ equal to 0.5, as given in eq. ( 44) in Table 16. As given in eq. ( 45), the overpotential driving this phenomenon is given by the difference of solid-phase potential, liquid-phase potential and the potential above which the acid attack is triggered. The solid and liquid phase potentials are obtained from the electrochemical model. The decrease in active material volume fraction is calculated from the computed capacity fade rate using eq. ( 46).

Model calibration

As described in the previous section, three different tests are done on individual coin cells to assess their baseline aging behavior. The first is calendar aging of coin cells at different elevated temperatures. A checkup each at the start and end of these tests gives an estimate of the irreversible and reversible capacity fade values occurring in the cell which corresponds to the parasitic side reactions occurring at the negative and positive electrodes, respectively, resulting in a net cyclable lithium loss. The charge consumed at the negative electrode due to electrolyte reduction, for example, is partially counterbalanced by the charge consumed by electrolyte oxidation on the positive electrode [55]. The counterbalanced capacity is commonly referred to as reversible capacity while the unrecoverable capacity due to excess of charge consumed at one electrode is called irreversible capacity.

Estimation procedure for the reversible and irreversible capacity losses are described by Kassem et al. in [55]; this helps in justifying the hypothesis considered in this work that the electrolyte decomposition at the negative electrode is more prevalent than at the positive electrode. According to this, the cell discharge capacity at the start over 100 % DOD is given by Q 0 . The cell is then charged to 100 % SOC followed by calendar aging for about 30 days. The discharge capacity of the cell measured soon after calendar aging over 100 % DOD is measured as Q 1 , followed by a charge to 100 % SOC and a subsequent discharge over 100 % DOD with a capacity of Q 2 . The total capacity loss is obtained by subtracting Q 1 from Q 0 and the irreversible capacity loss by subtracting Q 2 from Q 0 . The reversible capacity loss is obtained by the difference of Q 1 and Q 2 . The measured discharge capacities and reversible and irreversible losses are given in Table 17. It is observed that the overall contribution from the positive electrode is less (about 5 % of the total capacity fade) except at relatively higher temperatures, 35 and 40°C, which is about 18 and 21 %, respectively. This clearly shows that the parasitic side reactions occurring at the negative electrode are predominant over those at the positive electrode.

The second test is cycling two cells for about 1200 cycles at 25oC and their capacity fade as markers is plotted in Figure 55. The capacity fade rate is almost uniform for both cells, except that the rate is slightly faster for cell 1 when compared to cell 2, which may be the result of manufacturing differences between the cells. It is believed that SEI is a kinetically limited process in the early stages of its lifetime and as the thickness of this passivation layer increases, the limitation due to solvent diffusion takes over in the later stages [63]. The result the long-time SEI growth is known to have a quadratic behavior (with a proportionality to the square root of time) and thus its rate of formation, as a whole, decreases [59,161]. However, with respect to the solvent oxidation at the positive electrode and the consequently resulting decrease in its volume fraction, the trend is reversed; it is low at the early stages of cycling and the rate increases at later stages. Thus, Figure 55 may be split into two parts to fit parameters from both SEI growth and PAML models. This is done by considering that in the 250 cycles, the contribution from PAML model to the total capacity fade is smaller; hence the solvent diffusivity D EC and electrolyte reduction rate constant k f,s are fitted to this data. The fitted curve is shown as a dashed line in Figure 55.

As described earlier, at about 1100 cycles, when the SEI contribution to the total capacity fade decreases, the exchange current density i o,PAML pertaining to the solvent oxidation at the positive electrode is fitted appropriately and the cumulative total capacity fade due to both SEI growth and PAML models is shown as a solid line. The fitted parameters are given in Table 19; the other SEI growth model parameters are taken from [63] and PAML model parameters are taken from [72]. 18. These values are used to calibrate the activation energy values in the SEI growth model pertaining to the solvent diffusivity D EC and the side reaction rate constant k f,s and the obtained values are 5.5x10 4 J/mol for both E a,D,EC and E a,kf,s . In this calibration, it is taken that the effect of PAML on the total capacity fade is negligibly small, due to the same reasons stated for the previous set of experiments. Looking into the V-dep setup experimental data in Figure 57 (c) it may be seen that there is no clear trend in the evolution of resistance between the cells, unlike their respective capacity fade. However, for the T-dep setup from Figure 57 (d), the evolution of cell resistance is in the exact same order of cell temperatures, with cell 4 having the highest resistance increase while cell 1 being the slowest. It may be observed from the experimental data that the resistance evolution is small until about 400 to 500 cycles and the rate of increase is greater after this point, in both V-dep and T-dep setup cells. This may be explained by taking a look again at the Nyquist plots; it may be noticed that for the fully aged cells, there are two distinguishable semi-circles that is not as clear with the fresh cells. Thus, the sudden growth in the internal resistance increase is a result of the emergence of this second semi-circle in the Nyquist plots. The resistance at the start of cycling for all cells is in the range of 0.275 Ω or 1 Ωm 2 (obtained by multiplying with the electrode surface area and this is comparable with the values given in literature [63]). This initial resistance corresponds to the resistance due to interfacial phenomena for the fresh cell.

Figure 57 (c) and (d) also shows the estimated cell resistance obtained from model simulations (solid lines) for the V-dep and T-dep setup cells, respectively. The cell internal resistance is a direct measure of the SEI thickness increase and is computed, as described earlier, using eq. (42). It is to be noted that the κ SEI used for computing this resistance, is at 25 o C for all the cells (especially T-dep setup) and unlike the capacity fade curves. Comparing the cell resistance obtained from model simulations with the experimental data, a strong dependence of temperature on the SEI growth is seen. This is evident in In the case of the V-dep setup cells, the simulations show that the resistance increase follow a similar trend with all the cells which may be attributed to the same temperature conditions. This is different from what is observed from experiments and thus requires further analysis of the individual capacity fade contribution from each of the electrodes as well. The resistance increase after 1100 cycles, from simulations, is comparable with that of cell 4 from the experiments. But the dispersion in the resistance increase between the V-dep setup cells from experiments is not captured by the model. This may be because the dispersion is caused by the onset of another phenomenon that is not represented in the model. It may be seen that the general trend in the experimental data, the cell resistance increase occurs in two stages, the first until 400-500 cycles where the increase is linear, followed by a kink after which the resistance rapidly increases. This may be the result of an accelerated increase in the cell internal resistance after 400-500 cycles and the contribution could be either from the positive or the negative electrode [162]. This could be verified by fitting the impedance curves with a Randel's circuit and identifying the different contributions appropriately. The solution to this is to update the model to take this additional resistance into account.

Also, in the current model, it can be clearly seen that the solvent diffusivity D EC is the limiting case as it reaches a quadratic behavior with the current number of cycles. To avoid this, the solvent diffusivity value may be slightly increased along with the decrease in the slight decrease in the side reaction rate constant (due to overestimation by the model at the start of cycling). But overall, from the complete nature of aging over 1100 cycles, it may be concluded that the macroscopic model performance is represented well when compared to the experimental data.

ICA (Incremental Capacity Analysis)

Capacity fade contribution

Complementary model simulation analyses is done using model simulations involve quantifying the contribution of capacity fade from each of the different mechanisms to the total fade. This is shown in Figure 59, where, in (a) and (b), the charge corresponding to the electrons consumed at each electrode as a result of solvent oxidation and reduction is shown and in (c) and (d), the contribution of cyclable lithium loss and active material loss to the total capacity fade is shown. It may be seen that in Figure 59 (a), with V-dep setup cells, the charge consumed at the negative electrode (blue) is almost the same for all cells, owing to the same temperature while the charge consumed at the positive electrode (red) decreases from cell 1 to cell 4 with an increase in the shunt resistance values. In Figure 59 (b), it may be seen that with an increase in cell temperature between cells 1 and 4, the charge consumed at the negative electrode is dependent on the cell temperature whereas the charge consumed at the positive electrode is almost the same. Using the values of charge consumed for side reactions at both the electrodes, a net cyclable lithium loss is computed. This is because, the charge consumed at the positive and negative electrodes compensate each other, due to the electrode balancing, as seen in the Model calibration section earlier. Overall it may be seen that the effect of cyclable lithium loss on all the 4 cells in V-dep setup is small; but by comparing the blue regions, it may be seen that it is higher for cell 1 when compared to cell 4. In the case of T-dep setup, the contribution of cyclable lithium loss to the total capacity fade is much higher when compared to V-dep setup cells and corresponds to its temperature values (higher the temperature, higher is the cyclable lithium loss) whereas the active material loss is almost a constant. It may be seen from Figure 59 (b) that the cyclable lithium loss contribution is higher because a higher amount of charge is consumed for solvent reduction at the negative electrode.

To complement the above results, Figure 60 In the Tdep setup cells, the higher SOC of cell 4 (due to its higher temperature) and thus a higher overpotential, is compensated when divided by its corresponding temperature. Thus, for the temperature conditions chosen in this study, there is no difference in η PAML /T between the different cells. In the case of V-dep It can be seen that with V-dep setup cells, the volume fraction ε s,pos is slightly different among the cells but the stoichiometry at 0 % SOC y 0 is the same. This is because, as discussed earlier, the cyclable lithium loss is small when compared to the active material loss and, as a result, the major contribution to the total capacity fade is the active material loss at the positive electrode. In the case of T-dep setup cells, the positive electrode volume fraction ε s,pos for all cells is taken to be the same, from the simulation results, equal to 0.47. Values for y 0 are obtained by fitting the reference curve with the pristine aged cells (obtained before dismantling them). It is seen that y 0 is in the decreasing order of the increasing temperature of the cells where cell 1 at 25 o C experiences the smallest shift and cell 4 at 40 o C the largest.

Comparing the trend seen here, in Table 21, with the values reported from simulations, in Table 20, the claims made earlier can be ascertained. These are as follows, firstly, the contribution from cyclable lithium loss is small and almost the same in V-dep setup cells but in T-dep setup cells, it increases with an increase in cell temperature. The active material loss is almost the same for all cells if the shunt resistance values are the same, irrespective of the temperature but if the shunt resistances are different, then the active material loss increases with a decrease in shunt resistance value.

In addition to the above fitting exercise, Figure 63 shows the lithiation curves of NMC/Li (above) and delithiation curves for C/Li (below), mimicking a charge on the full-cell at a C-rate of C/10, for three cells is shown. These are, the fresh cell (blue dotted line), V-dep aged cell 1 with 0.1 Ω shunt resistor and T-dep aged cell 4 at 40 o C. The fresh cell is directly obtained from half-cell measurements, while the other curves are the fitted using the values reported in Table 21. The x-axis shows the absolute cell capacities of the pristine cells before dismantling them. Looking closer at Figure 63, it may be seen that the positive electrode potential shifts up when compared to the fresh cell. This may be because of two reasons, namely, the decreasing positive electrode stoichiometry at 0 % SOC y 0 accounting for the cyclable lithium loss or the other reason being the reduction in volume fraction ε s,pos which may not be ascertained. One limitation of this analysis is that any increase in individual electrode overpotential may not be observed.

Supplementary model simulations

In this part, the model simulation results, at the end of the cycling tests, of the current distribution patterns between the cells in V-dep and T-dep setup is shown. This allows to compare the differences in the cell behavior at the end of aging with those at the start of the cycling tests. In addition to this, the charge throughput in the cells and the temperature increase in the cells is also shown. Overall, comparing the current distribution profiles of the fully aged cells and the fresh ones from Chapter 4, one major difference seen is the emergence of a significant peak towards the end of discharge. In addition to this, the dispersion in current distribution profiles between the T-dep cells is always higher than V-dep cells. This spike in magnitude at the end of discharge is the result of SOC differences between the cells. It is seen earlier that the positive electrode stoichiometry at 0% SOC y 0 shifts to lower values as the cells age and as a result, the steeper potential regions of the positive electrode is not accessible. As a result, the steeper potential region of the graphite electrode significantly drives the cell more quickly to the end of discharge voltage and thus the big differences in current distribution between the cells. It may be seen that the spread in current between the cells during discharge is greater between T-dep setup cells than in V-dep setup cells. 64, where the T-dep cells had a greater current dispersion during discharge. However when comparing this to the pattern observed in fresh cells, firstly, the magnitude in dispersion in charge throughput is comparable to that of the fresh cells. But the V-dep aged cells follow a reverse trend when compared to the fresh cells as well. This is noteworthy because a higher dispersion between the cells during charging conditions, especially at high SOC conditions, results in a greater risk of lithium plating on the negative electrode. Similar to the fresh cells, the aged cells also In Figure 66 (a), the temperature evolution is slightly higher for the aged cells when compared to the fresh cells scenario; the maximum value for cell 4 is 4.3 o C higher than the nominal temperature of 25 o C with a difference from cell 1 of about 0.6 o C. In the case of T-dep cells, although this difference is higher, the temperature of the fresh cells never exceeds their consequently higher temperature cells. The same trend may be seen in Figure 66 (b); however, due to aged nature of the cells, the maximum temperature evolution of the cells is slightly higher than the fresh cells. In terms of absolute temperature evolution, at the highest point, cell 1 has a 3.7 o C and cell 4 has a 2.1 o C. Overall it may be concluded that the cells with a higher resistance to current flow has a higher temperature evolution and the temperature evolution between fresh and aged cells is not high. A coupled electrochemical model is developed to represent the V-dep and T-dep setup behavior, as described in the previous chapter. In this work, an aging component is integrated to the coupled model. A set of initial tests are performed to appropriately parameterize the model. The capacity fade in the cells is a result of the contribution of both the cyclable lithium loss and active material loss. It is hypothesized in this thesis that the resistance increase in the cells is considered to strictly be the result of an increase in the SEI layer thickness. However, based on the resistance evolution pattern from the experiments, a significant contribution from the charge transfer resistance may be suspected after about 400 cycles during cycling. Also based on the model behavior when compared to the experimental data, it may be concluded that the model is to be appropriately calibrated. In addition to this, the differential curve analyses reveal a loss of either active material loss or inhomogeneity within the scale of the coin cell and an overpotential increase in the cells.

Current distribution

Following the cycling tests, the electrode segments are harvested from the aged cells and electrochemical tests are performed. Based on the analyses of parameter fitting using half-cell measurements, appropriate values of the positive electrode volume fraction ε s,pos and the positive electrode stoichiometry at 0 % SOC y 0 are estimated. Also using the model simulations the contributions of the two degradation mechanisms to the total capacity fade is appropriately estimated. Overall, it is seen that the active material loss is the greatest contributor to the cell capacity fade in both V-dep and T-dep setups. In the V-dep setup case, the charge consumed at both electrodes is almost the same with minor differences. The total capacity fade is also within a comparable range and the cyclable lithium loss is very small. The cell with the lowest resistance path is observed to have undergone the highest positive electrode volume fraction ε s,pos loss. In the T-dep setup case, large differences in capacity fade come from cyclable lithium loss as a direct effect of the temperature difference between the cells but the active material loss follows is almost a constant. The volume fraction ε s,pos is almost a constant between the cells.

It has also been identified that, under the conditions used here, temperature is a greater cause for the degradation in the Li-ion batteries and also for the inhomogeneity occurring in them. Also, utilization of higher voltages regions of the positive electrode, negatively impact the performance of Li-ion batteries due to the positive electrode active material loss. Reversible capacity fade is insignificant at lower temperatures but is considerable at higher temperatures. It is also observed that during the time frame of the cycling, the point of inflexion or kink in the capacity fade is not reached. The temperature increase in the aged cells is not very high when compared to the fresh cells. The utilization of T-dep cells is more disperse during discharge conditions, as seen in the case of fresh cells but for V-dep cells, this trend is reversed.

As part of these post-mortem analyses in this chapter, fully aged cells are dismantled and electrode segments are harvested and electrochemical tests are performed. There are two parameters, namely ε s,pos and y 0 , each pertaining to one type of capacity fade, namely, cyclable lithium loss and active material loss at the positive electrode. ε s,pos values are taken from literature and y 0 is fitted with the data and a trend similar to that reported from simulation results is obtained.

In this study, although the thermal dependency of the cell parameters are considered, the temperature rise during operation is not considered. Also, to identify the source of PAML and for verification with the half-cell analyses, further analyses using ICP-MS is required. As described in Chapter 4, the cells may be cycled within small SOC ranges, where there is a bigger dispersion between the cells and its effect on short term and long term cycling may be investigated. As part of the future scope of this work, a fully coupled electrochemical, electrical and thermal model to enable the modeling of a complete commercial cell behavior with any geometry and any cell chemistry, to study the effect of the two factors influencing the inhomogeneous aging behavior may be considered. Also, the aging model integrated to the electrochemical model may also include additional degradation phenomena, lithium plating for instance, that is also observed to occur under normal operating temperatures.

Conclusions and scope for future work

The objective of this work is to estimate the effect of the factors influencing the inhomogeneous aging behavior observed within large format commercial Li-ion batteries. The literature on state of the art of Li-ion batteries is humungous as well as, on the degradation phenomena. The inhomogeneous aging behavior and the parameters influencing them are surveyed. It has been identified that the thermal and potential gradients inherently present in large format commercial Li-ion batteries are the prime factors influencing the inhomogeneous aging behavior. A study of the different modeling and experimental techniques to assess the cell performance during aging is done.

Based on all the above literature study, it is decided to focus on a combined modeling and experimental approach in this work. According to this, a setup is to be designed and fabricated for decoupling the effect of the two parameters influencing the inhomogeneous aging behavior. The setup representing the potential and thermal inhomogeneities is referred to as V-dep and T-dep setups, respectively. For this setup to work effectively, small cells are to be used with minimal cell-to-cell variations. The setup also has the provision to perform repeats under same conditions simultaneously.

A benchmarking study is done on the appropriate choice of cells for use in the experimental setup to perform the long-term cycling tests. Three different types of cells, are fabricated manually in the laboratory, and it is observed that these cells have a huge dispersion due to human-related errors. Thus it is decided to opt for commercial coin cells, manufactured with a greater level of accuracy, from Varta Microbattery. These cells are observed to have a low mass, low specific heat capacity and minimal cellto-cell variations that satisfy the requirements of this work are used.

In order to parameterize the devices appropriately, multi-physics models, describing the electrical and thermal aspects of a commercial cell is to be developed. For this, it is decided to choose an A123 cell for developing the multi-physics models, as there is abundant information available in the literature, for the same. Using simulation results, the temperature and shunt resistance values are deduced. Following this, the Varta cells are assembled in the setup. The coin cells undergo checkup tests to assess their general behavior before cycling the cells. From these tests, charge and discharge curves are obtained and are used to calibrate the electrochemical cell model and, in the end, a good agreement is achieved. The electrochemical cell models are appropriately coupled to represent the experimental setup conditions. Using the simulation results, additional insight on the overall setup behavior is obtained. It is foreseen that the thermal non-uniformity may result in a more inhomogeneous aging behavior based on the obtained results and thus the setup is used for performing the long term cycling tests. It is also seen that the dispersion between the cells is highest in small SOC ranges; thus it may be interesting to investigate the effect of cycling the cells in small SOC ranges on the inhomogeneous aging behavior.

The aging tests are performed and the aging behavior of the cells is obtained from intermediate checkup tests performed at regular intervals by interrupting the cycling tests. From these tests information on the capacity fade, internal resistance increase and differential curves from the charge and discharge curves are presented. It is seen that cells with the least resistance path, namely V-dep setup cell 1 with 0.1 Ω shunt resistor and T-dep setup cell 4 at 40 o C, undergo the highest capacity fade. Following the end of the cycling tests, post-mortem tests are also performed through physical characterization and electrochemical tests on harvested electrode samples from dismantled coin cells. The main degradation phenomena contributing to the loss of cell performance have been identified to be SEI (Solid Electrolyte Interphase) growth and PAML (Positive electrode Active Material Loss). It is also identified that there is a significant contribution to the cell internal resistance due to additional interfacial phenomena at the positive electrode after about 400 cycles that is not accounted for in the current model.

The coupled electrochemical model is integrated with an aging component. It is calibrated using the baseline aging behavior, obtained from the individual coin cells. This coupled model is then compared with the experimental results and a good correlation is obtained between the two. However, it is observed that the model performance could further be improved by appropriately calibrating the model and by considering the effect of additional internal resistance from the positive electrode. Simulation results using the current model reveal appropriate contributions from cyclable lithium loss and active material loss at the positive electrode to the total capacity fade. It can be concluded that the effect of temperature on the inhomogeneous aging behavior is higher than the potential non-uniformities. Thus, the cells cycling at higher temperatures have a bigger contribution from cyclable lithium loss (and thus a higher total capacity fade) while the contribution from active material loss at the positive electrode is the same. Among V-dep setup cells, it is identified that the cells with smaller shunt resistance undergo a bigger capacity fade. Additional simulation results show that the dispersion in the current distribution of the cells is higher at low SOC (State of Charge) conditions but the difference in cell capacity utilization is almost the same between the cells in both setups. The difference in temperature evolution of the fully aged cells is comparable to that of the fresh cells.

It is seen that there is a huge dispersion in current distribution pattern of the cells in small SOC ranges, and simulation analyses may be required to understand its consequences on the inhomogeneous aging behavior. To begin with, the coupled model is used to cycle the cells within a 10 % SOC range, at 0 % and 100 % SOC conditions, where the dispersion is the highest between the cells. The current distribution pattern is shown in Figure 67 for both V-dep and T-dep setup conditions. Figure 67 From this it may be foreseen that if the cells are cycled over long-term under low SOC conditions, the non-uniform aging behavior is likely to be more existent. However, cycling at high SOC conditions may also be interesting because there is still a significant dispersion in the current distribution of the cells and other types of degradation phenomena are most active in these potential ranges, for instance lithium plating at the negative electrodes. Integrating a thermal component to this analysis may be distribution between the cells. This may be useful for developing appropriate BMS (Battery Management System) for the automobile. It may also be applicable to other large-format electrochemical energy storage systems. For instance, in the stationary energy storage applications, the sources for inhomogeneity are similar to that reported in this work and may be mitigated, thus increasing their lifetime and making them economically viable. The technique may also be applicable for other types of energy storage devices such as fuel cells and redox flow batteries.

Résumé (long)

Le travail dans cette thèse fait partie de l'effort de recherche global visant à la diffusion de véhicules propres (électriques et hybrides) dans lesquels l'énergie électrique est stockée sur un principe électrochimique. Les accumulateurs électrochimiques, en particulier dans le cas des véhicules électriques << purs >>, représentent une part importante du coût et leur comportement dans le temps reste difficile à prévoir. La question de la durée de vie des batteries Li-ion peut être considérée comme essentielle pour les autres défis. La diminution des performances des batteries Li-ion est due à plusieurs phénomènes de dégradation qui s'y produisent et on parle de vieillissement. Dans les batteries commerciales au lithium-ion, lorsque la capacité de la batterie atteint 80% de sa capacité initiale, la cellule est considérée comme morte et doit être remplacée, ce qui augmente encore le coût. En plus de cela, il est également connu que dans l'échelle des batteries Li-ion simples, elles présentent un comportement non uniforme, ce qui conduit finalement à un vieillissement non uniforme. Le vieillissement inhomogène est également susceptible d'entraîner une dégradation plus rapide des cellules. Il est donc primordial de mieux comprendre ce comportement inhomogène. L'objectif de ce travail est donc d'identifier les facteurs qui influencent ce comportement et d'estimer l'effet de ces paramètres sur le vieillissement global non uniforme.

Pour atteindre ces objectifs, ce travail consiste en deux cadres, à savoir un expérimental et un modélisateur, comme illustré à la figure 1, qui sont chacun divisés en deux étapes. Ces cadres fonctionnent simultanément, échangeant des informations entre eux au cours des deux étapes. Dans la première étape de modélisation, des modèles multi-physiques décrivant le comportement de cellules spécifiques sont développés dans le logiciel COMSOL Multiphysics v5.3a et, à l'aide des résultats de simulation de modèles, une configuration électronique unique est développée pour remplacer les gradients apparaissant dans les batteries Li-ion du commerce étape expérimentale. L'achèvement de cette première étape marque une étape cruciale dans ce travail, dans la réalisation de la preuve de concept. Ensuite, dans la deuxième étape de modélisation, les modèles multi-physiques sont correctement couplés pour représenter le comportement de la configuration expérimentale, tandis que dans la deuxième étape expérimentale, les cellules subissent un cycle à long terme dans différentes conditions de fonctionnement. Les résultats du test de cyclage sont utilisés pour calibrer le modèle couplé afin d'obtenir des informations sur le comportement au vieillissement en général et d'identifier l'effet des paramètres sur le vieillissement non homogène. Le produit final souhaité est un modèle électrochimique, électrique et thermique entièrement couplé, qui met l'accent sur le comportement de vieillissement inhomogène des batteries Li-ion du commerce. Les différents mécanismes de dégradation qui entraînent tous le vieillissement des batteries Li-ion sont identifiés. Les plus étudiés dans la littérature sont la formation de couche de passivation, le placage de lithium métallique, la dissolution de métaux de transition et la contamination électrolytique. Bien que les phénomènes soient nombreux, leurs effets principaux sont la perte de lithium cyclable, la perte de matière active et l'augmentation de la résistance. Les facteurs suivants influent sur les phénomènes de dégradation. Il existe dans la littérature plusieurs facteurs tels que la température, la vitesse du courant, le nombre de temps / cycle, le potentiel / l'état de charge (SOC) et la profondeur de décharge (DOD). Des simulations supplémentaires sont effectuées à l'aide du modèle couplé. Cela aide à comparer les différences entre les cellules de la configuration dans des conditions fraîches et vieillies. Les profils de distribution de courant, de débit de charge et d'évolution de la température sont analysés. On peut donc en conclure que les différences thermiques entre les cellules ont une influence plus grande sur l'inhomogénéité globale par rapport aux différences de potentiel.

Les comportements inhomogènes des batteries

Enfin, au chapitre 6, un bref résumé et les conclusions tirées des travaux sont décrits en détail. L'objectif principal de ce travail est d'estimer l'effet des facteurs qui influencent le comportement de vieillissement inhomogène observé dans les batteries Li-ion grand format du commerce. Comme décrit précédemment, les différences thermiques ont une plus grande influence sur le comportement inhomogène de la cellule par rapport aux différences de potentiel. Une partie de la portée future de ce travail est que des tests supplémentaires doivent être effectués pour attester / quantifier le matériau actif des électrodes positives au niveau des électrodes positives et négatives après le démontage et la récolte des électrodes. Le protocole de vieillissement dans ce travail consistait à cycler les cellules à 100% de DOD (profondeur de décharge), si les cellules avaient été cyclées dans des plages étroites de SOC, à 0% et à 100% de SOC, où la dispersion entre les cellules est la plus élevée. , l'inhomogénéité globale de la configuration aurait pu être plus significative. Ceci est illustré ci-dessous à la figure 7, où les modèles couplés T-dep et V-dep sont utilisés pour simuler de petites plages de SOC à différentes conditions de SOC moyennes. De plus, la méthodologie utilisée dans ce travail peut être utilisée pour étudier les inhomogénéités dans d'autres dispositifs de stockage électrochimiques tels que les piles à combustible ou les applications de stockage stationnaire, etc., où les inhomogénéités sont plus prononcées.

Figure 1 : 20 Figure 2 : 21 Figure 3 : 24 Figure 4 : 25 Figure 5 : 26 Figure 6 : 29 Figure 7 : 30 Figure 8 : 34 Figure 9 : 35 Figure 10 :Figure 11 : 43 Figure 12 : 46 Figure 13 : 47 Figure 14 : 48 Figure 15 : 50 Figure 16 : 51 Figure 17 : 52 Figure 18 : 53 Figure 19 : 55 Figure 20 : 56 Figure 21 : 57 Figure 22 : 58 Figure 23 : 60 Figure 24 : 62 Figure 25 : 64 Figure 26 : 66 Figure 27 : 68 Figure 28 : 71 Figure 31 : 72 Figure 32 : 73 Figure 33 : 74 Figure 34 : 75 Figure 35 : 76 Figure 36 : 76 Figure 37 : 78 Figure 39 : 79 Figure 40 : 82 Figure 41 : 83 Figure 42 : 88 Figure 44 : 89 Figure 45 : 90 Figure 46 : 90 Figure 47 : 91 Figure 48 : 96 Figure 49 : 97 Figure 50 : 99 Figure 51 : 100 Figure 52 : 101 Figure 53 : 103 Figure 54 : 107 Figure 55 : 112 Figure 56 : 114 Figure 57 : 115 Figure 58 : 118 Figure 59 : 119 Figure 60 : 120 Figure 61 : 121 Figure 62 : 123 Figure 63 : 126 Figure 64 : 127 Figure 65 : 128 Figure 66 : 129 Figure 67 :

 120221324425526629730834935101143124613471448155016511752185319552056215722582360246225642666276828713172327333743475357636763778397940824183428844894590469047914896499750995110052101531035410755112561145711558118591196012061121621236312664127651286612967 Figure 1 : Representation of battery systems at different length scales starting at the atomic scale on the left to system scale on the right [3] ............................................................................................................ 20 Figure 2 : Global scientific approach of the work focusing on the aim, combined modeling and experimental approach and final deliverable of the work ......................................................................... 21 Figure 3 : Schematic representation of the composition of an electrochemical Li-ion cell ....................... 24 Figure 4 : Types of insertion compound crystal structures used in batteries (a) lamellar, (b) spinel and (c) olivine [24] .................................................................................................................................................. 25 Figure 5 : Polar chart comparing the aspects of different commercial positive electrode materials (a) LMO, (b) LFP, (c) NMC and (d) NCA [25] ..................................................................................................... 26 Figure 6 : Ragone plot showing a comparison of power and energy ratings for different energy-storage devices [40] ................................................................................................................................................. 29 Figure 7 : Commercial cell formats -(a) cylindrical, (b) prismatic and (c) pouch cell composition [20] .... 30 Figure 8 : Degradation phenomena occurring in a Li-ion cell [61] .............................................................. 34 Figure 9 : Schematic representation of the SEI composition with an inner inorganic layer and a thicker outer organic layer [63] .............................................................................................................................. 35 Figure 10 : Factors influencing the calendar and cycle aging phenomena in Li-ion batteries .................... 38 Figure 11 : Schematic representation of P2D model showing the micro and macro models in two separate dimensions coupled through an exchange of variables, spherical configuration of micromodel and multiple nodes of micromodel along electrode thickness .................................................................. 43 Figure 12 : Schematic representation of the current flow through an unwound cylindrical cell; in this case the current enters at the positive current collector tabs, traverses through the electrode/separator assembly and goes out through the negative current collector tabs ......................................................... 46 Figure 13 : Thermal phenomena occurring in batteries -sources for heat generation and dissipation during operation [42] .................................................................................................................................. 47 Figure 14 : Coupled 3D model where the electrochemical, electrical and thermal models are solved simultaneously and information is exchanged between them .................................................................. 48 Figure 15 : An excerpt on a sample cell showing (a) capacity fade rate during cycling and (b) internal resistance from impedance spectroscopy at the start (fresh) and end (aged) of cycling .......................... 50 Figure 16 : An excerpt of differential voltage curves obtained from PITT measurements, comparing (a) fresh cell with (b), (c) and (d) aged cells (calendar aged at SOC and different temperature) [55] ............ 51 Figure 17 : Pictures showing dismantling of commercial cells at various stages with different cell geometries and of different construction, namely, large pouch cell, small pouch cell and small cylindrical/coin cell (clockwise starting from top left) ................................................................................ 52 Figure 18 : Graphical representation of a shift in the electrodes electrochemical window upon aging and a consequent reduction in the cell capacity ............................................................................................... 53 Figure 19: Schematic representation of the equivalent electronic setup showing 4 electrochemical cells connected in parallel with an appropriate shunt resistor and a feature to regulate the cell temperatures individually, to surrogate the inhomogeneities observed in commercial Li-ion cells ................................ 55 Figure 20 : Commercial cells chosen in this work -(a) A123 cylindrical and (b) Kokam pouch cells ......... 56 Figure 21 : Illustration of harvesting the electrode segments from different tab locations of the A123 cylindrical cell .............................................................................................................................................. 57 Figure 22 : An exploded view of the coin cell components in the order of their assembly ....................... 58 Figure 23 : Custom cells electrode segments ............................................................................................. 60 Figure 24 : Image depicting a Varta coin cell .............................................................................................. 62

Figure 1 :

 1 Figure 1 : Representation of battery systems at different length scales starting at the atomic scale on the left to system scale on the right [3]

Figure 2 :

 2 Figure 2 : Global scientific approach of the work focusing on the aim, combined modeling and experimental approach and final deliverable of the work This manuscript consists of five other chapters following this one. Firstly, in Chapter 2, the state of the art Li-ion battery technology is presented starting with the composition and working principle of Li-ion batteries and the present-day materials currently in use and future technologies under research.Following this, the degradation phenomena are discussed with their influence on the macroscopic cell performance and factors influencing them. The inhomogeneous behavior and the inhomogeneous aging are also presented and finally, the combined modeling and experimental approach for studying the inhomogeneous aging behavior are described.

1 )Figure 3 :

 13 Figure 3 : Schematic representation of the composition of an electrochemical Li-ion cell 1

Figure 4 (

 4 a) shows the lamellar structure which is a two-dimensional layered sheet and is of the general form LiMO 2 , where M is transition metal, usually a combination of any two or three metals, namely, Ni (nickel), Co (cobalt) or Mn (manganese). The first known Li-ion batteries for Sony electronic devices used a positive electrode of this type, LCO (LiCoO 2 )[1]. However since cobalt is expensive and due to safety issues, it was partially substituted by other metals such as nickel, aluminum and manganese and as a result other technologies emerged, namely NCA (LiNi 0.8 Co 0.15 Al 0.05 O 2 ), NMC's (LiNi x Mn y Co 1-x-y O 2 , with multiple mole fraction possibilities like LiNi 0.8 Mn 0.1 Co 0.1 O 2 , LiNi 0.6 Mn 0.2 Co 0.2 O 2 , LiNi 0.5 Mn 0.3 Co 0.2 O 2 and LiNi 0.33 Mn 033 Co 0.33 O 2 ), etc.[START_REF] Badin | Les véhicules hybrides: Des composants au système[END_REF]. In parallel, the spinel configuration is a three dimensional electrode type which is of the form LiM 2 O 4 , as in Figure4(b) and the most popular electrode of this type is LMO (LiMn 2 O 4 )[START_REF] Bandhauer | [END_REF]

Figure 4 :

 4 Figure 4 : Types of insertion compound crystal structures used in batteries (a) lamellar, (b) spinel and (c) olivine [24]

Figure 5 :

 5 Figure 5 : Polar chart comparing the aspects of different commercial positive electrode materials (a) LMO, (b) LFP, (c) NMC and (d) NCA [25]

Figure 6 :

 6 Figure 6 : Ragone plot showing a comparison of power and energy ratings for different energy-storage devices [40]

Figure 7 :

 7 Figure 7 : Commercial cell formats -(a) cylindrical, (b) prismatic and (c) pouch cell composition [20]

Figure 8 :

 8 Figure 8 : Degradation phenomena occurring in a Li-ion cell [61]

Figure 9 :

 9 Figure 9 : Schematic representation of the SEI composition with an inner inorganic layer and a thicker outer organic layer [63]

Figure 10 :

 10 Figure 10 : Factors influencing the calendar and cycle aging phenomena in Li-ion batteries

Figure 12 :

 12 Figure 12 : Schematic representation of the current flow through an unwound cylindrical cell; in this case the current enters at the positive current collector tabs, traverses through the electrode/separator assembly and goes out through the negative current collector tabs

Figure 13 :

 13 Figure 13 : Thermal phenomena occurring in batteries -sources for heat generation and dissipation during operation [42] 2.4.1.5 Coupled models

Figure 14 :

 14 Figure 14 : Coupled 3D model where the electrochemical, electrical and thermal models are solved simultaneously and information is exchanged between them

Figure 15 :

 15 Figure 15 : An excerpt on a sample cell showing (a) capacity fade rate during cycling and (b) internal resistance from impedance spectroscopy at the start (fresh) and end (aged) of cycling

Figure 16 :

 16 Figure 16 : An excerpt of differential voltage curves obtained from PITT measurements, comparing (a) fresh cell with (b), (c) and (d) aged cells (calendar aged at SOC and different temperature) [55]

Figure 17 :

 17 Figure 17 : Pictures showing dismantling of commercial cells at various stages with different cell geometries and of different construction, namely, large pouch cell, small pouch cell and small cylindrical/coin cell (clockwise starting from top left)

Figure 20 :

 20 Figure 20 : Commercial cells chosen in this work -(a) A123 cylindrical and (b) Kokam pouch cells

Figure 22 :

 22 Figure 22 : An exploded view of the coin cell components in the order of their assembly

Figure 23 :

 23 Figure 23 : Custom cells electrode segments

  Cells are vacuum dried at 100 o C overnight in a vacuum chamber before filling with the electrolyte to remove any small traces of moisture present in the cells. To ensure that the cells do not come in contact with ambient atmospheric conditions, this is done in the dry room facility at LRCS which is maintained at about -45 o to -50 o C dew point temperature)  After drying, the cells are cut open and filled with electrolyte inside the dry room before vacuum sealing them with a residual pressure of about 80 -90 mbar at 160 o C for about 6 seconds  The cells are taken out of the dry room to undergo formation at C/20 (with respect to the nominal capacity) at 40 o C to 3.5 V in a VMP3 potentiostat from Biologic  The cells are brought back to the dry room for degasification, meaning that, they are cut open for the gas formed during formation to be vented out and then vacuum sealed under the same conditions as above  The cells are again taken out to the potentiostat to charge the cell at C/20 to 4.2 V followed by a discharge to 3.8 V

Figure 24 : 6

 246 Figure 24 : Image depicting a Varta coin cell

Figure 25 :

 25 Figure 25 : Discharge capacity and coulombic efficiency of (a) and (b) LFP/C cells, (c) and (d) NMC/C cells with 12.7 mm positive electrode and (e) and (f) NMC/C cells with 12 mm positive electrode diameter, respectively, during SEI reformation cycles Next, the coulombic efficiency of the LFP/C and NMC/C cells during the formation cycles is computed by taking the ratio of discharged capacity and the charged capacity and is shown in Figure 25 (b), (d) and (f).Ideally, in the absence of side reactions at the positive electrode, the coulombic efficiency is expected to give an estimate of the charge consumed for side reactions at the negative electrode; a value close to 100 % is desired which indicates a stable passivation layer. The SEI in the negative electrode of pristine LFP/C cell (on the left) seems to be pretty stable resulting in high coulombic efficiencies from the start.In the case of NMC/C cells, cells 7-10 show poor coulombic efficiency in the start but recover towards the end, indicating that the passivation layers are stable towards the end of the stabilization cycles; while for a few other cells, the coulombic efficiency is small throughout when compared to the other cells.

  (c) and (e), the absolute discharge capacities of most of the cells have a big dispersion in the values and may be seen that cells assembled at a later stage have a capacity. Cells 7-10, also pointed out earlier, belonging to one batch of fabrication, have a poor discharge capacity at C/25 rate. From earlier, the rate coulombic efficiency recovered at the end of reformation cycles and thus may indicate that the cells undergo self-discharge. This suggests the possibility of a discrepancy at the time of fabrication where there might have been a slight short circuit. Unlike LFP/C cells, the NMC/C cells show a constant discharge capacity only until 1C or, in some cases, even earlier, C/2, because they are energy type cells and have thicker electrodes.

Figure 26 :

 26 Figure 26 : Absolute and normalized capacities of (a) and (b) LFP/C cells, (c) and (d) NMC/C cells with 12.7 mm positive electrode and (e) and (f) NMC/C cells with 12 mm positive electrode diameter, respectively, during rate capability tests at different C-rates ranging from C/25 to 5C and their respective comparison with its respective pristine commercial cell Figure 26 (b), (d) and (e) shows the normalized discharge capacities of LFP/C and NMC/C cells. With LFP/C cells, as seen in Figure 26 (a), except cells 6 and 7, other cells have a constant capacity at current rates of up to 2C. This is comparable with the normalized capacity of the pristine cell (dashed black line, up to 0.5C) and it can be concluded that the performance of the LFP/C coin cells is in good agreement with the pristine cell. The slight increase in capacity of the pristine cells at higher C-rates is because of increase performance from temperature evolution in the cell. In the case of NMC/C cells, the rate capability of the cells is much lower at higher C-rates because it is an energy type cell and thus recoverable capacity is expected to be lower. NMC/C cells 7-10 are normalized with respect to the capacity at C/10 instead of C/20 and still show poorer performance, clearly being the outliers. A few cells, namely 11, 12, 13 and 18 have a normalized capacity slightly greater than 1, probably owing to the same reason as in the case of cells 7-10 but at a much lower rate. Also, similar to the LFP/C cells, a good agreement between the coin cells and the pristine cell is achieved.

Figure 27 :

 27 Figure 27 : Capacity fade rate of (a) LFP/C cells, (b) NMC/C cells with 12.7 mm positive electrode diameter and (c) NMC/C cells with 12.7 mm positive electrode diameter during 1C charge/discharge cycling at 25°C and 45°C

Figure 28 :

 28 Figure 28 : Histogram showing dispersion of (a) absolute discharge capacities and (b) ohmic resistance drops of the LFP/C coin cells assembled using electrode segments obtained from different tab locations

Figure 29 :

 29 Figure 29 : (a) Absolute discharge capacity and (b) coulombic efficiency during the SEI reformation cycles of the high power LFP/C Custom cells electrode segments

Figure 30 : 3 . 3 . 2 . 3 Aging

 303323 Figure 30 : (a) Absolute and (b) normalized capacity during the rate capability at different C-rates ranging from C/25 to 5C of the high power LFP/C Custom cells electrode segments

Figure 31 :

 31 Figure 31 : Capacity fade rate during 1C charge/discharge cycling at 25°C and 45°C of (a) LFP/C and (b) NMC/C coin cells

Figure 34 :

 34 Figure 34 : Absolute discharge capacity during the rate capability at different C-rates ranging from C/25 to 5C of the Li-fun pouch cells

Figure 35 :

 35 Figure 35 : Histogram showing the statistical dispersion of the absolute discharge capacity at C/10 current rate before (blue) and after (red) the rate capability test of the Li-fun pouch cells

Figure 36 :-fun pouch cells 3 . 3 . 3 . 4 Aging

 363334 Figure 36 : Histogram showing the statistical dispersion of the resistance due to ohmic voltage drop of the Li-fun pouch cells 3.3.3.4 Aging Four cells from the initial set of 8 cells that are intact and with comparable capacities and internal resistances are chosen for cycling them using a 1C charge and discharge constant current rate at 25 and 45 o C. The cells undergo cycling for about 200 times and the capacity fade rate is shown in Figure 37. The behavior of cell 4 is quite strange, suddenly after about 70 cycles the cell degrades and in another 30

Figure 39 :

 39 Figure 39 : Capacity fade rate of two Varta coin cells during cycling for about 1500 cycles

Figure 41

 41 Figure41shows a schematic representation of the complete experimental setup. According to this, there are four main components, namely the parallel connected cell network, a temperature regulator to maintain the cell temperatures within their dedicated compartments at predetermined values, a data logging system to store the voltages of both the cell and the shunt resistor and a VMP3 potentiostat from Biologic connected to the cell network terminals. Although not shown here, the cell network has the provision to duplicate the V-dep and T-dep setup conditions. Thus, a total of about two and three tests of the V-dep and T-dep setup, respectively, is performed simultaneously under the same conditions but only the best results are shown in the following sections.

Figure 41 :

 41 Figure 41 : Schematic representation of the complete experimental setup showing the four main components, namely the experimental setup, Biologic VMP3 potentiostat, temperature regulator and data logging system

Figure 42 :

 42 Figure 42 : Schematic representation of (a) electrochemical Li-ion cell with (left to right), negative copper current collector, negative electrode, separator, positive aluminum current collector and positive electrode; (b) Newman's P2D (pseudo two-dimension) model approximation with spherical particles uniformly distributed along its thickness in each electrode; (c) average model approximation with a single solid particle in each electrode assuming negligible solid-phase electrode potential gradient [43]

Figure 43 :

 43 Figure 43 : X-ray micro tomography images of Varta coin cell (a) top view and (b) front view, showing the spiral winding within

Figure 44 :

 44 Figure 44 : Parameter sensitivity analyses of the electrochemical model at C/10 current rate of different physical parameters from (a) solid-phase and (b) liquid-phase The charge and discharge curves at different C-rates (C/20 to 2C) obtained from galvanostatic tests, is compared with simulations from the model. The markers in Figure 45 (a) and (b) show discharge and charge curves, respectively, obtained from experiments. The solid lines correspond to the electrochemical cell model, after calibration using the experimental data. It can be observed that the model captures the experimental cell behavior very well, during both discharge and charge conditions at different C-rates albeit at 2C rate during charge conditions.

Figure 45 :

 45 Figure 45 : Comparison of galvanostatic (a) discharge and (b) charge curves of Varta coin cell between the electrochemical model (solid lines) and experimentally obtained data (markers) at different Crates at 25 o C Moving on to the thermal dependency of the electrochemical model, a parameter sensitivity is done for the activation energies corresponding to the solid-phase diffusivities D s,pos and D s,neg , liquid phase diffusivity D e and conductivity κ and reaction rate constants k o,pos and k o,neg. Of these, it is identified that the biggest significant contribution comes from the activation energy corresponding to D s,neg followed by a smaller contribution from κ, k o,pos and k o,neg . Thus, the activation energy for D s,neg and κ are slightly increased when compared to values given in the literature. The other activation energies are adopted directly from literature[65] and the values are reported in Table10. Figure46(a) shows a comparison of the discharge curves from experiments (markers) and model simulations (solid lines) at a wide range of temperatures, starting from -20°C to 40°C at a low C-rate of C/10, after appropriately tuning the activation energies. Figure46(b) shows the discharge curves at a high current rate of 2C-rate but just between 25°C and 40°C. Overall it may be concluded that the fit may be considered good at all temperatures.

  Figure 46 (a) shows a comparison of the discharge curves from experiments (markers) and model simulations (solid lines) at a wide range of temperatures, starting from -20°C to 40°C at a low C-rate of C/10, after appropriately tuning the activation energies. Figure46 (b)shows the discharge curves at a high current rate of 2C-rate but just between 25°C and 40°C. Overall it may be concluded that the fit may be considered good at all temperatures.

Figure 46 :

 46 Figure 46 : Comparison of galvanostatic discharge curves at (a) C/10 and (b) 2C current rates of the cell model (solid lines), at different temperatures, with the experimentally obtained data (markers)

Figure 47 :

 47 Figure 47 : Logarithm of cell resistance, obtained from the Nyquist plot, plotted as a function of the inverse of temperature

Figure 48 : 4 . 3 . 6 . 2

 484362 Figure 48 : Simulations showing potential distributions along (a) negative and (b) positive current collector surfaces in a cylindrical A123 cell, for a 2C-rate input current, under static conditions 4.3.6.2 Temperature for T-dep setup

Figure 49 :

 49 Figure 49 : Simulations showing temperature distribution in a cylindrical A123 cell along the radial and axial directions for an C/4 input current rate, after the cell has reached saturation where heat generated inside the cell is equal to heat lost from the cell; cells 1 to 4 represent the different regions chosen for assembly in the coin cell network Table 13 : T-dep and V-dep setup temperature and shunt resistance specifications

Figure 50 shows

 50 Figure 50 shows the comparison of current distribution profiles of the cells in T-dep setup obtained from experiments and model simulation. While Figure 50 (a) and (b) show a complete charge and discharge cycle, Figure 50 (c) and (d) concentrate on just the discharge cycle.It is known that higher temperatures enhance the transport phenomena and reaction kinetics and as a result cell at a higher temperature has a lower internal resistance. Hence at the start of both charge and discharge cycles, cell 4 at 40°C has the highest current while cell 1 at 25°C has the smallest current flowing through it which may be seen from both experiments and the simulation results. This trend reverses towards the end of both charge and discharge cycles but is more evident during discharge. This is because of higher overpotential values with the use of 1C rate during discharge when compared to C/2 used during charge. Also at 100 % DOD the steep region of graphite electrode is reached which results in big voltage differences between cells for small SOC differences. Cells 1 and 4 at the extreme temperatures have the highest dispersion from the average values while cells 2 and 3 are closer to the mean value. This clearly indicates that the highest level of inhomogeneity occurs in the current profiles of cells 1 and 4. A comparison of the current flow pattern in the cells between the experimental data and the model simulation result shows that the experimental setup is very well represented by the model during both charge and discharge. The transition in the current distribution profiles occurs precisely at the same time although the magnitude of the peaks is different. It is to be noted here that cell-to-cell variations, described earlier, play a significant role during experiments which is absent in the model and the slight differences between the experiments and the model may be explained by this.

Figure 50 :

 50 Figure 50 : Comparison of current distribution in T-dep cells from (a) experiment and (b) model simulation in 1C discharge and C/2 CCCV charge current; (c) and (d) represent the zoomed discharge regions from (a) and (b), respectively

Figure 51 :

 51 Figure 51 : Comparison of current distribution in each cell in the V-dep network of (a) experiment and (b) model simulation in 1C discharge and C/2 CCCV charge current; (c) and (d) represent the zoomed discharge regions from (a) and (b), respectively

  3 and 5 mAh, for V-dep and T-dep setup, respectively. It may also be seen that the occurrence of the peak in the T-dep setup occurs close to 100 % DOD, unlike the V-dep setup. It may be forecasted here that the cells with the highest utilization (cell 1 in V-dep setup and cell 4 in T-dep setup) are expected to have the highest capacity fade.

Figure 52 :

 52 Figure 52 : Simulation data showing the difference in charge throughput between the cells in (a) T-dep (with respect to cell 4) and (b) V-dep (with respect to cell 1) setup using 1C discharge and C/2 CCCV charge current

Figure 53 (

 53 Figure53(a) and (b) shows the temperature evolution in the cells during the first stages of cycling from the V-dep and T-dep setups, respectively. Figure53(a) shows that cell 4 has the highest rise in cell temperature of about 3.25°C while in cell 1, this value goes up only by about 2.8°C. The maximum difference in the temperatures between cells 1 and 4 is about 0.45°C. Figure53(b) reveals that cell 1 at 25 o C has the highest absolute rise in cell temperature of about 3.3°C, occurring towards the end of discharge, while in cell 4, this value goes up only by about 1.75°C. This is because, cell 4 is at a higher ambient temperature and with an increase in temperature, the cell internal resistance is lower due to enhanced reaction kinetics and transport properties and thus lower is the heat generation. It may also be seen that the increase in individual cell temperature is concurrent with its ambient temperature conditions and not exceeding the temperature of the other cells. This indicates that the temperature differences between cells are big enough that the individual cell temperature evolution does not influence the setup temperature conditions.

Figure

  Figure53(a) and (b) shows the temperature evolution in the cells during the first stages of cycling from the V-dep and T-dep setups, respectively. Figure53(a) shows that cell 4 has the highest rise in cell temperature of about 3.25°C while in cell 1, this value goes up only by about 2.8°C. The maximum difference in the temperatures between cells 1 and 4 is about 0.45°C. Figure53(b) reveals that cell 1 at 25 o C has the highest absolute rise in cell temperature of about 3.3°C, occurring towards the end of discharge, while in cell 4, this value goes up only by about 1.75°C. This is because, cell 4 is at a higher ambient temperature and with an increase in temperature, the cell internal resistance is lower due to enhanced reaction kinetics and transport properties and thus lower is the heat generation. It may also be seen that the increase in individual cell temperature is concurrent with its ambient temperature conditions and not exceeding the temperature of the other cells. This indicates that the temperature differences between cells are big enough that the individual cell temperature evolution does not influence the setup temperature conditions.

Figure 53 :

 53 Figure 53 : Simulation showing the temperature evolution of the individual cells in the (a) V-dep and (b) T-dep setup (after coupling each of them to a 0D thermal model) using 1C discharge and C/2 CCCV charge current

  After the end of cycling aging tests in the V-dep and T-dep setup and the final checkup, one of the two repeats in the V-dep setup and one of the three repeats, that is considered the best representative of the whole behavior is identified and chosen for further steps. The extreme cells from each of two setups, namely cell 1 at 25°C and cell 4 at 40°C from the T-dep setup and cell 1 with 0.1 Ω and cell 4 with 0.7 Ω shunt resistors from the V-dep setup are dismantled. The electrode samples are unwound and separately rinsed in dimethyl carbonate twice, for about 15 minutes each time and then vacuum dried for approximately 2 hours.The harvested electrode samples are double side coated with active material. One side of the double side coated negative electrode is removed by scraping the electrode surface using a solution of NMP (1methyl-2-pyrolidinone). Coin cells are assembled with graphite electrode segments and for NMC electrodes, 3 electrode Swagelok cells are assembled. This is because of the difficulty in removing the positive electrode material on one side of the double side coated electrodes and the electrodes may directly be used in the Swagelok cells. All the above steps (dismantling, rinsing, preparing and assembling) are carried out in an inert environment, in an Argon filled glovebox. The schematic for coin cell and 3 electrode Swagelok cell fixtures are shown in Figure54. Since the width of the electrodes is only about 4mm, rectangular electrode segments are used in half-cells, unlike the usual circular ones. The length of the electrode segments are not the same in all the cells but attention is paid so that they are smaller than the lithium foil used as the counter electrode. The electrolyte used is 1M LiPF6 in 1:1 EC:DMC with 2 % VC and while the separator is the same as described in Chapter 3. The cells undergo a set of three initial formation cycles (charge/discharge at C/10 current rate) and the cell capacity is estimated during the last cycle.

Figure 54 :

 54 Figure 54 : Schematic representation of the components and the order of assembly of (a) coin cell and (b) 3 electrodes Swagelok cell

Figure 55 :

 55 Figure 55 : Cell capacity of two cells during intermediate checkups of cycling tests at 25 o C and comparison with model simulations with and without PAML (Positive electrode Active Material Loss) component The final test is cycling the cells for about 200 cycles at different temperatures and observing the capacity fade behavior. A check at the start and end of cycling gives an estimate of the overall capacity fade, presented in Table18. These values are used to calibrate the activation energy values in the SEI growth model pertaining to the solvent diffusivity D EC and the side reaction rate constant k f,s and the obtained values are 5.5x10 4 J/mol for both E a,D,EC and E a,kf,s . In this calibration, it is taken that the effect of PAML on the total capacity fade is negligibly small, due to the same reasons stated for the previous set of experiments.

  (a) and (b), respectively, as markers. The error bars in the capacity fade values are obtained from the two repeats performed on the setup. The respective capacity fade values from simulations of the coupled model for about 1100 cycles is also shown in Figure56(a) and (b) as solid lines. The cycling protocol for the simulations is the same as that of the experimental tests, except that there is a C/10 charge and discharge cycle at approximatively every 85 cycles. It may be seen that the capacity fade rate of the cells in the T-dep setup is faster, especially for the cells at higher temperatures whereas it is almost similar in the case of V-dep setup cells. The capacity fade between the extreme cells 1 and 4, in the V-dep and T-dep setup is 1.84 and 7.56 mAh, respectively. This dispersion in the magnitudes of capacity fade from both the setups indicates a great influence of temperature on the capacity fade rate of the cells. This observation validates the forecast from Chapter 4 that temperature gradients may have a greater influence on the inhomogeneous aging behavior when compared to the potential gradients. Comparing the model performance with the experimental data, the trend observed from the experiments that in V-dep and T-dep setup cells, cell 1 with 0.1 Ω shunt resistor and cell 4 at 40 o C, respectively, lose the capacity fastest, is well reproduced by the model. From model simulations, it may be seen that in the V-dep setup from Figure 56 (a), the cells undergo almost the same amount of capacity fade, similar to the experiments. With T-dep setup cells in Figure 56 (b), there is a significant difference between the model simulations and the experimental data, especially at higher temperatures because the checkups from simulations are performed at their respective temperatures (cell 1 at 25 o C, cell 2 at 30 o C, etc.) whereas the intermediate checkup tests are all performed at 25 o C. Hence there is a good agreement at lower temperatures and at higher temperatures, the model underestimates the capacity fade.

Figure 56 : 5 . 4 . 2

 56542 Figure 56 : Comparison of evolution of capacity fade between model simulations (solid lines) and experiment (markers) of (a) V-dep and (b) T-dep setup cells from intermediate checkups measured at C/10 current rate, at 25 o C, during cycling for 1100 cycles

Figure 57 :

 57 Figure 57 : Nyquist plots of pristine (blue) and fully aged (red) cells of (a) V-dep and (b) T-dep setup obtained from EIS measurements performed during intermediate checkups, at 25 o C; comparison of evolution of cell internal resistance between model (solid lines), computed as a direct measure of SEI thickness, and experiment (markers) of (c) V-dep and (d) T-dep setup cells from intermediate checkups, at 25 o C, during cycling for 1100 cycles

Figure 57 (

 57 Figure57 (c) and (d) also shows the estimated cell resistance obtained from model simulations (solid lines) for the V-dep and T-dep setup cells, respectively. The cell internal resistance is a direct measure of the SEI thickness increase and is computed, as described earlier, using eq. (42). It is to be noted that the κ SEI used for computing this resistance, is at 25 o C for all the cells (especially T-dep setup) and unlike the capacity fade curves. Comparing the cell resistance obtained from model simulations with the experimental data, a strong dependence of temperature on the SEI growth is seen. This is evident in Figure57 (d), where cell 4 at 40 o C has the highest resistance increase and cell 1 at 25 o C the lowest. It may be seen that the model represents the resistance increase in T-dep setup towards the end of 1100 cycles very well, although during the initial stages it overestimates the internal resistance greatly.

Figure 58 (

 58 Figure 58 (a) and (b) shows the evolution of the charge and discharge curves of the fresh (before the start of cycling in blue) and fully aged (at the end of cycling in red) cells and their respective ICA or differential capacity curves for the four V-dep cells, respectively, obtained from the intermediate checkup tests performed at 25 o C. The corresponding ICA curves for the T-dep cells are simultaneously shown in Figure 58 (c) and (d).It is to be noted that the charge and discharge curves are at a C-rate of C/2 and C/10, respectively. Firstly, taking a look at the curves corresponding to the fresh cells from both the setups, it may be observed that the dispersion in the cells is very little. This indicates that the cells have a repeatable and comparable behavior at the start of the cycling tests. Next looking at the aged curves in Figure58(a) and (b), the dispersion in the capacity fade of the cells in both setups is in the same order of magnitude as that observed in the Capacity fade section earlier. It may be seen that the overpotential in the aged cells cause an upwards and downward shift from their respective charge and discharge curves, respectively. This shift is greater in magnitude for the charge curves when compared to the discharge curves because the discharge curves are obtained using a C/10 current while the charge curves are obtained using a C/2 current.

Figure 58 :

 58 Figure 58 : Galvanostatic C/10 discharge and C/2 charge curves and their respective incremental capacity curves of pristine (blue) and fully aged (red) cells of (a) & (b) V-dep and (c) & (d) T-dep setup, respectively, obtained from intermediate checkups performed at 25 o C

Figure 59 :

 59 Figure 59 : Charge consumed at negative and positive electrode during solvent reduction and oxidation, respectively, in (a) V-dep and (b) T-dep setup cells; Cyclable lithium loss and LAM (Los of Active Material) contribution to the total capacity fade for (c) V-dep and (d) T-dep setup cells, obtained from model simulations after about 1100 cycles

  net cyclable lithium loss to the total capacity fade is shown in red in Figure59 (c) and (d) for V-dep and T-dep setup cells, respectively, while the LAM (Loss Active Material) contribution is shown in blue.

  (a) and (b) show the positive electrode volume fraction ε s,pos of V-dep and T-dep setup cells, respectively. It may be seen that in V-dep setup, cell 1 has the greatest decrease in ε s,pos and this value decreases with an increase in the shunt resistance values used for the cells. However, in T-dep setup, all the cells experience almost the same amount of positive electrode volume fraction decrease. The trend seen in the active material loss in both setups is in-line with the evolution of ε s,pos over time; smaller the absolute value of ε s,pos , the active material loss is greater.

Figure 60 :

 60 Figure 60 : Comparison of evolution of positive electrode active material loss between model (solid lines) and experiment (markers) of (a) V-dep and (b) T-dep setup cells from intermediate checkups, at 25 o C, during cycling for 1100 cycles

Figure 62 :

 62 Figure 62 : SEM images of negative graphite electrode of samples from (a) fresh pristine cell, (b) cell 4 with 0.7 Ω shunt resistor from V-dep setup and (c) cell 4 at 40 o C from T-dep setup; (e) EDS analysis data from two points, namely spectrums 7 and 8, performed on (d), a negative electrode sample from cell 4 at 40 o C

Figure 63 :

 63 Figure 63 : Comparison of capacity Q vs voltage V curves of NMC/Li (blue lines) and C/Li (red lines) curves obtained, by fitting ε s,pos and y 0 of the fresh reference half-cell, against their respective pristine aged cells from cell 4 of T-dep aged cell 4 at 40 o C (solid line), V-dep aged cell 1 with 0.1 Ω shunt resistor (dashed line) and pristine fresh cell (dotted line)

  Figure 64 (a) and (b) show the current distribution profiles from simulation results of the fully aged cells, for the V-dep and T-dep setup, respectively and Figure 64 (c) and (d) show just the discharge parts from Figure64(a) and (b), respectively. Overall, comparing the current distribution profiles of the fully aged cells and the fresh ones from Chapter 4, one major difference seen is the emergence of a significant peak towards the end of discharge. In addition to this, the dispersion in current distribution profiles between the T-dep cells is always higher than V-dep cells. This spike in magnitude at the end of discharge is the result of SOC differences between the cells. It is seen earlier that the positive electrode stoichiometry at 0% SOC y 0 shifts to lower values as the cells age and as a result, the steeper potential regions of the positive electrode is not accessible. As a result, the steeper potential region of the graphite electrode significantly drives the cell more quickly to the end of discharge voltage and thus the big differences in current distribution between the cells. It may be seen that the spread in current between the cells during discharge is greater between T-dep setup cells than in V-dep setup cells.

Figure 64 :

 64 Figure 64 : Simulation results showing the comparison of current distribution of (a) V-dep and (b) Tdep cells at the end of cycling tests in 1C discharge and C/2 charge current; (c) and (d) represent the zoomed discharge regions from (a) and (b), respectively

  Figure 65 (a) and (b) show the charge throughput from simulation results of V-dep and T-dep setup cells, respectively. In Figure 65 (a), cell 1 with 0.1 Ω shunt resistor has the highest utilization or the most capacity flowing through it, thus cell 1 is taken as the reference and the capacities of other cells are subtracted from this. Similarly, in Figure 65 (b), cell 4 at 40°C has the highest utilization and the capacities of other cells are subtracted from this. Comparing the charge throughput pattern between the V-dep and T-dep cells, it may be seen that in V-dep setup, the greater difference in the charge throughput between cells occur during charge conditions while in T-dep, it occurs during discharge. This is synonymous to the current flow pattern seen in Figure64, where the T-dep cells had a greater current dispersion during discharge. However when comparing this to the pattern observed in fresh cells, firstly, the magnitude in dispersion in charge throughput is comparable to that of the fresh cells. But the V-dep aged cells follow a reverse trend when compared to the fresh cells as well. This is noteworthy because a higher dispersion between the cells during charging conditions, especially at high SOC conditions, results in a greater risk of lithium plating on the negative electrode. Similar to the fresh cells, the aged cells also show a maximum peak at mid-SOC regions in V-dep setup and at low SOC regions in T-dep setup. One

Figure 65 :Figure 66

 6566 Figure 65 : Simulation data showing the difference in charge throughput between the fully aged cells from (a) T-dep (with respect to cell 4) and (b) V-dep (with respect to cell 1) setup using 1C discharge and C/2 charge current

Figure 66 :

 66 Figure 66 : Simulation results showing the comparison of temperature of (a) and (b) fresh and (c) and (d) aged cells, for the V-dep and T-dep setup, respectively, using 1C discharge and C/2 CCCV charge current

  (a) and (b) show the current distribution in V-dep and T-dep setup cells, respectively, at 0 % SOC whereas Figure67 (c) and (d)show the current distribution at 100 % SOC. The cells are cycled continuously using a constant current charge and discharge rate of 1C each. It may be seen that dispersion in the cells at low SOC conditions, in Figure67(a) and (b), is the highest, as the current profiles do not converge within the 10 % SOC range used. Whereas at high SOC conditions, Figure67 (c) and (d), the current profiles although widely spread at the start of both charge and discharge, the profiles almost converge towards the end.

Figure 1 :

 1 Figure 1 : Approche scientifique globale des travaux axée sur l'objectif, modélisation combinée et approche expérimentale et résultat final des travaux Le manuscrit comprend six chapitres au total et chacun de ces chapitres sera brièvement décrit cidessous. Le chapitre 1 est une introduction au contexte des batteries Li-ion. Le stockage de l'énergie est considéré comme une option viable pour surmonter l'intermittence des sources d'énergie renouvelables. Ensuite, la raison pour comprendre le comportement de vieillissement non uniforme est présentée.

Figure 2 :

 2 Figure 2 : Représentation schématique de la composition d'une cellule électrochimique au Li-ionEnsuite, on comprend le vieillissement des batteries Li-ion. La classification basée sur leur effet sur les performances globales des cellules macroscopiques est réalisée en trois catégories, à savoir la décroissance de la puissance / puissance, le vieillissement calendaire / cycle et le vieillissement réversible / irréversible. La diminution de la capacité ou de l'énergie correspond à une réduction de l'énergie totale extractible de la cellule, tandis que la diminution de la puissance correspond à la diminution des performances de la cellule en raison de l'augmentation de sa résistance interne. Le vieillissement du calendrier se produit lorsqu'il n'y a pas de courant traversant la cellule, ou en d'autres termes, lorsque la cellule est au repos pendant le cycle, comme son nom l'indique, est la dégradation qui se produit lorsque la cellule est utilisée, pendant la charge ou la décharge. Pour définir la perte de capacité réversible et irréversible, il faut considérer les réactions secondaires parasites aux électrodes positives et négatives. Dans ce cas, en règle générale, la charge consommée dans l'électrode positive est considérée comme étant réversible et la différence entre la charge consommée au niveau des électrodes positives et négatives correspond à la perte de capacité irréversible.

  Li-ion sont classés en gradients de température et de potentiel. Un comportement inhomogène continu conduit à un vieillissement inhomogène qui entraîne une utilisation incomplète de l'énergie de la cellule. Enfin, les différentes techniques de modélisation et expérimentales et leur approche combinée utilisées pour étudier le comportement de vieillissement non homogène sont décrites. La technique de modélisation utilisée dans ce travail est appelée modèle multiphysique basé sur la physique, dans laquelle les phénomènes physiques se produisant dans la cellule sont décrits à l'aide d'une combinaison d'équations différentielles partielles et algébriques. En fonction du type de phénomène et du lieu où il se produit, trois modèles différents sont classés: modèle électrochimique, modèle électrique et modèle thermique. Le modèle électrochimique utilisé dans ce travail est basé sur le modèle pseudo à deux dimensions (P2D) de Newman et un schéma est présenté à la figure 3. Ce modèle prend en compte les phénomènes de transport se produisant dans la section transversale de l'ensemble électrodes et séparateur. Ce modèle est également étendu à un phénomène de dégradation qui y est associé.

Figure 3 :

 3 Figure 3 : Représentation schématique du modèle P2D montrant les modèles micro et macro dans deux dimensions distinctes couplées par un échange de variables, la configuration sphérique du micro-modèle et les noeuds multiples du micro-modèle le long de l'épaisseur de l'électrodeIl existe un modèle électrique permettant de représenter les potentiels d'électrodes sur leur surface des collecteurs de courant en raison du courant qui les traverse ou inversement. En fonction de la surface des collecteurs de courant et de la distance de segments de collecteur de courant spécifiques par rapport à leurs languettes de collecte de courant, le potentiel électrique varie. Ce modèle est en deux dimensions (2D) contrairement au modèle électrochimique à une dimension (1D). Enfin, il existe un modèle thermique en trois dimensions (3D) représentant le comportement thermique de la cellule à la

Figure 4 :

 4 Figure 4 : Représentation schématique de la configuration électronique équivalente montrant 4 cellules électrochimiques connectées en parallèle avec une résistance shunt appropriée et une fonction permettant de réguler les températures de la cellule individuellement, afin de remplacer les inhomogénéités observées dans les cellules Li-ion du commerce Ce chapitre consiste en une technique d'étalonnage permettant de choisir de manière appropriée quatre types de cellules. Différentes options sont envisagées (quatre au total), à savoir deux d'entre elles utilisant des piles à pièces fabriquées dans lesquelles, dans l'une, les segments d'électrodes sont récoltés à partir de cellules Li-ion commerciales et dans une autre, les segments d'électrodes sont achetés auprès d'un fournisseur commercial; dans la troisième option, les cellules de la poche sont utilisées (les cycles de formation initiaux sont effectués après le remplissage de l'électrolyte) et enfin, les cellules numismatiques sont directement achetées auprès d'un fournisseur commercial. Une série de tests est effectuée sur ces cellules pour évaluer la cohérence et le caractère répétable du comportement de la cellule. Ces tests incluent la capacité de la cellule et la mesure de la résistance interne, la performance pendant la capacité de débit et le vieillissement de la pile bouton en fonction de la température. Sur les quatre choix différents, il a été conclu que les piles bouton commerciales Varta étaient les mieux adaptées à cette application avec le comportement le plus cohérent / reproductible et la moindre dispersion des données.

Figure 5 :

 5 Figure 5 : Images micro-tomographiques à X-rayons de la pile de monnaie Varta (a) vue de dessus et (b) vue de face, montrant la spirale s'enroulant Un modèle couplé est développé en couplant quatre modèles de cellules électrochimiques d'une manière similaire à celle du montage expérimental. La figure 6 montre une comparaison de la distribution actuelle entre le modèle et la configuration expérimentale pour la condition T-dep. On peut constater que le modèle représente très bien le comportement de la configuration expérimentale. L'inhomogénéité entre les cellules est bien représentée par la configuration. Un comportement similaire a été observé dans le cas des cellules V-dep. Des simulations de modèles couplés supplémentaires

Figure 7 :

 7 Figure 7 : Résultats de la simulation montrant les profils de distribution actuels dans les cellules V-dep à (a) 0% et (c) 100% de SOC et dans les cellules T-dep à (b) 0% et (d) 100% de SOC, en utilisant une charge 1C et 1C taux de décharge actuel

Table of contents

 of Nomenclature ...............................................................................................................................................

	L		Electrode/separator assembly thickness	m
	m coin m cell	Varta cell mass Nomenclature A123 cell mass	g kg
	M SEI		SEI molar mass	kg/mol
	English letters N k q gen q loss	Number of tabs Heat input per unit volume Heat dissipated per unit volume	-W/m 3 W/m 3
	Symbol Q gen a s A cell Q loss Q PAML A coin r A sep R A k,tab R Ω brugg R cell c e R coin R i , where i = 1, 2, 3 and 4 c s R s c s,max R SEI c s,surf R el c EC t 𝑐 𝐸𝐶 𝑏 t + 𝑐 𝐸𝐶 𝑠 T C p,cell T amb C p,coin T ref D e U OCV D EC D s 𝐷 𝑒 𝑒𝑓𝑓 U ref	Parameter Total heat input Interfacial surface area per unit volume A123 cell external surface area Total heat dissipated Charge consumed for solvent oxidation Varta coin cell external surface area Dimension in radial direction of electrode particles Separator/electrode surface area Universal gas constant Current collector tab surface area Ohmic resistance drop Bruggman coefficient A123 cell radius Varta coin cell radius Electrolyte concentration along electrode/separator sandwich Shunt resistance in each cell of parallel network Solid-phase lithium concentration Solid-phase particle radius Maximum solid lithium concentration Resistance contribution from SEI Surface solid lithium concentration Resistance on current collector surface Solvent concentration across SEI time Bulk solvent concentration across SEI Transference number Solvent concentration at electrode/SEI interface Temperature A123 cell specific heat capacity Ambient temperature Varta coin cell specific heat capacity Reference temperature Liquid phase Li + ion diffusivity Open circuit voltage Solvent diffusivity Solid-phase lithium diffusivity Effective liquid phase Li + ion diffusivity	W Unit 1/m W As/m 2 m 2 m mm 2 J/mol/K m 2 Ω m 2 mm -m Ω mol/m 3 m mol/m 3 Ω mol/m 3 Ω mol/m 3 s mol/m 3 -mol/m 3 K mol/m 3 K J/kg/K K J/kg/K V m 2 /s m 2 /s m 2 /s m 2 /s
	E a_ct		Activation energy of charge transfer reaction	J/mol
	E a_diff,s	Activation energy of solid phase diffusivity	J/mol
	E a_diff,e	Activation energy of liquid phase diffusivity	J/mol
	E a_cond,e	Activation energy of liquid phase conductivity	J/mol
	E a_D,EC	Activation energy of solvent diffusivity	J/mol
	E a_kf,s		Activation energy of solvent reduction charge transfer reaction	J/mol
	E ref		Electrode equilibrium potential as a function of its stoichiometry	V
	f ±		Thermodynamic factor	-
	F		Faraday's constant	As/mol
	h con		Convective heat transfer coefficient	W/m 2 /K
	H cell		A123 cell axial height	m
	H coin		Varta coin cell axial height	mm
	i coin i int		Varta coin cell current input (De)Intercalation current density	mA A/m 2
	i k,tab i o i o,avg i o,PAML i PAML i s i t	Current density in current collector tab Solvent reduction exchange current density Solvent reduction average exchange current density Solvent oxidation exchange current density Solvent oxidation current density Solvent reduction current density Total current involved at the negative electrode	A A/m 2 A/m 2 A/m 2 A/m 2 A/m 2 A/m 2
	I i , where i = 1, 2, 3 and 4 j f	Current in each cell of parallel network Interfacial current density	A A/m 2
	k f,s k o		Reaction rate constant of solvent reduction Charge-transfer reaction rate constant	m/s A m 2.5 /mol 1.5

Table 1 :

 1 Degradation phenomena and their effect on cell performance .................................................. 37

Table 2 :

 2 Specifications of the pristine commercial cells ............................................................................ 56

Table 3 :

 3 Coin cell configuration .................................................................................................................. 58

Table 4 :

 4 Check-up protocol for coin cells ................................................................................................... 59

Table 5 :

 5 Specifications of the electrode segments from Custom cell ........................................................ 60

Table 6 :

 6 Specifications of Li-fun pouch cells .............................................................................................. 61

Table 7 :

 7 Varta Microbattery coin cell specification ................................................................................... 62

Table 8 :

 8 Summary of cells used in this work and their performance ........................................................ 80

Table 9 :

 9 Equations representing the electrochemical cell model .............................................................. 85

Table 10 :

 10 Model parameters of electrochemical coin cell model ............................................................. 91

Table 11 :

 11

Equations representing thermal behavior in a cell .................................................................... 94

Table 12 : A123 cell parameters suitable for the 2D electrical and 3D thermal model ............................. 94 Table 13 : T-dep and V-dep setup temperature and shunt resistance specifications ................................ 97 Table 14 : Varta coin cell thermal model parameters .............................................................................. 102 Table 15 : Equations representing SEI growth model on the surface of negative electrode particles..... 109 Table 16 : Equations

representing the phenomena corresponding to the active material loss at the positive electrode ..................................................................................................................................... 110 Table 17 : Discharge capacities at the start and end of calendar aging, reversible and irreversible capacities of the cells and percentage of reversible capacity loss to the total loss, at different elevated temperature conditions ............................................................................................................................ 111 Table 18 : Discharge capacities at the start and end of cycle aging and total capacity fade of cells, cycled at different temperatures after about 200 cycles each ............................................................................ 112 Table 19 : Model parameters of SEI growth and PAML model ................................................................. 113 Table 20 : Comparison of positive electrode stoichiometry at 0% SOC y 0 of fresh and fully aged cells obtained at the end of a galvanostatic C/10 discharge performed during the checkup for about 1100 cycles ......................................................................................................................................................... 121 Table 21 : Estimates of positive electrode stoichiometry at 0% SOC y 0 and positive electrode active material volume fraction ε s,pos (taken from simulation result) for the V-dep and T-dep setup cells obtained by fitting with half-cell data....................................................................................................... 125

Table 1 : Degradation phenomena and their effect on cell performance Mechanism Cyclable lithium loss Active material loss Resistance increase At current collector

 1

	Copper cracking and dissolution	Oxidation of Cu may result in cyclable lithium loss	Inaccessibility of electrode particles	Loss of electrical paths
	Aluminum surface cracking	-	Inaccessibility of electrode particles	Loss of electrical paths
	Binder decomposition and contact loss	Loss of cyclable lithium if the electrode is lithiated	Inaccessibility of electrode particles	Loss of electrical paths
	At negative electrode
	Solvent co-intercalation and graphite exfoliation	Loss of cyclable lithium if the electrode is lithiated	Loss of graphite particles	Loss of solid-phase conduction paths
	SEI decomposition and precipitation	More cyclable lithium consumed to replenish the decomposed SEI	-	Probable decrease in internal resistance
	Particle cracking, SEI formation and buildup	Cyclable lithium consumed for more SEI formation	-	Additional interfacial resistance
	Li plating and dendrite formation	Cyclable lithium intercalation process consumed during	-	-
	At positive electrode
	Transition metal dissolution	-	Loss of transition metal sites	Loss of solid-phase conduction paths
		Cyclable lithium
	Cathode surface film	consumed for passivation layer	-	Additional interfacial resistance
		formation
	Structural disordering	-	Loss of active material sites	Influence on solid-phase diffusion
	Particle cracking	-	Loss of active material sites	Loss of solid-phase conduction paths

Table 2 : Specifications of the pristine commercial cells Cell property A123 cell Kokam cell Cell geometry

 2

	Cylindrical	Rectangular or pouch

Both cells are discharged completely and then cut open in an inert environment, Argon filled glovebox.

Table 3 : Coin cell configuration Cell component Material

 3

	Negative electrode	Graphite or lithium foil
	Positive electrode	LFP, NMC or graphite
	Separator	Whattman GF/D borosilicate glass fiber
	Electrolyte	1M LiPF6 in 1:1 EC:DEC (LP40)

Table 4 : Check-up protocol for coin cells

 4 Wherever possible aging tests are performed on the fabricated cells to estimate the nuances in the electrochemical and temperature dependent aging behavior on cells fabricated using the electrode segments harvested from different parts of the cell, as shown in Figure21. As part of the analyses, the cells undergo a constant current rate cycling at 1C, spanning over 100 % DOD at different temperatures, namely, at 25 and 45 o C to also assess the repeatability in the capacity fade rate behavior of the cells.

	Steps	Electrochemical test	Measurement	Comments
	1	Passivation layer reformation cycles	C/5	Discharge and charge 5 times in total
	2	Capacity measurement	C/20	CCCV (Constant current and constant voltage) charge discharge
	3	EIS (Electrochemical Impedance Spectroscopy)	1x10 -2 -2x10 5 [Hz]	0, 50 and 100 % SOC (State of Charge)
	4	Rate capability tests	C/10 -5C	CCCV (Constant current and constant voltage) charge and CCCV discharge
	5	Capacity verification	C/10	After rate capability test

Table 5 : Specifications of the electrode segments from Custom cell Parameter LFP electrode Graphite electrode Diameter [mm] 13 13 Specific capacity of active material [mAh/g] 150 350

 5

	Areal capacity [mAh/cm 2 ]	1.03	1.07
	Nominal voltage [V vs. Li/Li + ]	3.4	0.1
	Maximum voltage [V vs. Li/Li + ]	3.8	1.5
	Minimum voltage [V vs. Li/Li + ]	2.5	0.01
		1. 2 x C/10	1. 2 x C/10
		2. 2 x C/5	2. 2 x C/5
		3. 1 x 1C	3. 1 x 1C
	Formation cycles protocol	Charge -CCCV (C/10 cutoff current)	Charge -CCCV (C/10 cutoff current)
		Discharge -CCCV (C/10 cutoff	Discharge -CCCV (C/10 cutoff
		current)	current)
		30 min rest after each cycle	30 min rest after each cycle

Table 6 : Specifications of Li-fun pouch cells Cell specification Value/Detail Positive electrode

 6 LiNi 0.5 Mn 0.3 Co 0.2 O 2 -lithium nickel-manganese-cobalt oxide coated with Al 2 O 3

	Negative electrode	Graphite
	Nominal capacity [mAh]	200
	Dimensions [mm]	40 x 20 x 3.5

Table 9 : Equations representing the electrochemical cell model

 9

	Electrochemical phenomena		Governing equations	Boundary conditions	Eq.
	Solid-phase Li-ion mass conservation	𝜕𝑐 𝑠 𝜕𝑡	-	1 𝑟 2	𝜕 𝜕𝑟	(𝑟 2 𝐷 𝑠	𝜕 𝜕𝑟	𝑐 𝑠 ) = 0	𝐷 𝑠 -𝐷 𝑠 𝜕𝑟 𝜕 𝜕𝑟 𝜕 𝑐 𝑠 | 𝑐 𝑠 | 𝑟=𝑅 𝑠 𝑟=0	= 0 𝑎 𝑠 𝐹 = 𝑗 𝑓	(3)
	Solid-phase Li-ion charge conservation	𝜕 𝜕𝑧	(𝜎 𝑒𝑓𝑓 𝜕𝜙 𝑠 𝜕𝑧	) -𝑗 𝑓 = 0	-𝜎 𝑒𝑓𝑓 𝜕𝜙 𝑠 𝜕𝑧 = -𝜎 𝑒𝑓𝑓 𝜕𝜙 𝑠 | 𝑧=0 𝜕𝑧 | 𝑧=𝐿	=	𝐴 𝐼	(4)

Table 10 : Model parameters of electrochemical coin cell model

 10

					Values
	Parameters	Symbol	Unit	Positive electrode	Separator	Negative
						electrode
	Electrochemical model
	Electrode thickness*	L	m	5×10 -5	2.5×10 -5	5.5×10 -5
	Particle radius*	R p	m	5×10 -6	-	5×10 -6
	Separator surface area*	A el	m 2	2.77×10 -3	2.77×10 -3	2.77×10 -3

*measured; **calculated; ***fitted; ¹ estimated; º from

Table 11 : Equations representing thermal behavior in a cell

 11

	Thermal phenomena		Governing equations	Eq.
	Heat balance equation	𝜌𝐶 𝑝	𝜕𝑇 𝜕𝑡	= 𝛻(𝑘𝛻𝑇) + 𝑞 𝑔𝑒𝑛 -𝑞 𝑙𝑜𝑠𝑠	(26)
	Heat generation	𝑞 𝑔𝑒𝑛 = 𝐼(𝑉 -𝑈 𝑂𝐶𝑉 ) + 𝐼𝑇	𝜕𝑈 𝑂𝐶𝑉 𝜕𝑇	(27)
	Heat loss	𝑞 𝑙𝑜𝑠𝑠 = ℎ(𝑇 -𝑇 𝑎𝑚𝑏 ) + 𝜎𝜖(𝑇 4 -𝑇 𝑎𝑚𝑏 4 ) (28)

Table 12 : A123 cell parameters suitable for the 2D electrical and 3D thermal model

 12

	Parameters	Symbol	Unit	Values Positive electrode Negative electrode
	Electrical model
	Number of tabs~	N k	-	4	4
	Tab surface area~	A k,tab	m 2	2x10 -5	2x10 -5
	Current collector surface area~	A k,CC	m 2	0.18	0.18
	Current collector thickness~	δ k	m	2x10 -5	1.3x10 -5

Table 14 : Varta coin cell thermal model parameters

 14

	Parameters	Symbol	Unit	Values
	Convective heat coefficient***	h con	W/m 2 /K	50
	Cell mass*	m coin	g	1.68
	Specific heat capacity***	C p	J/kg/K	1500
	Cell radius*	R coin	mm	6
	Cell height*	H coin	mm	5.4
	Cell surface area*	A coin	mm 2	203.57
	Ambient temperature~	T amb	K	298
	Reference temperature~	T ref	K	298
	*measured; **calculated; ***fitted; ~ estimated

Table 15 : Equations representing SEI growth model on the surface of negative electrode particles

 15

	Phenomena		Governing equations	Eq.
	Solvent diffusion	𝐷 𝐸𝐶	𝜖 𝑆𝐸𝐼 𝑐 𝐸𝐶 𝑏 -𝑐 𝐸𝐶 𝑠 𝛿 𝑆𝐸𝐼	= -	𝑖 𝑠 𝐹	(38)
	Solvent reduction reaction	𝑖 𝑠 = -𝐹𝑘 𝑓,𝑠 𝑐 𝐸𝐶 * 𝑒𝑥𝑝 [	-𝛽𝐹 𝑅𝑇	(𝜙 𝑠,𝑛 -	𝛿 𝑆𝐸𝐼 𝜅 𝑆𝐸𝐼	𝑖 𝑡 )]	(39)
	SEI growth rate		𝑑𝛿 𝑆𝐸𝐼 𝑑𝑡	= -	𝑖 𝑠 𝑀 𝑆𝐸𝐼 2𝐹𝜌 𝑆𝐸𝐼	(40)
	Current balance			𝑖 𝑡 = 𝑖 𝑖𝑛𝑡 + 𝑖 𝑠	(41)
	Resistance due to SEI	𝑅 𝑆𝐸𝐼 =	𝛿 𝑆𝐸𝐼 𝜅 𝑆𝐸𝐼 𝑆 𝑛	, where 𝑆 𝑛 =	3𝜖 𝑠,𝑛 𝛿 𝑛 𝐴 𝑠𝑒𝑝 𝑅 𝑝,𝑛	(42)
	Arrhenius dependency of solvent diffusivity and reaction rate kinetics	𝜓 = 𝜓 𝑟𝑒𝑓 𝑒𝑥𝑝 ( 𝐸 𝑎 (𝜓) 𝑅	(	1 𝑇	-	1 𝑇 𝑟𝑒𝑓

Table 16 : Equations representing the phenomena corresponding to the active material loss at the positive electrode

 16

	Phenomena		Governing equations	Eq.
	Rate of capacity fade due to dissolution	𝑑𝑄 𝑃𝐴𝑀𝐿 𝑑𝑡	= 𝑖 𝑃𝐴𝑀𝐿 = 𝑖 𝑜,𝑃𝐴𝑀𝐿 𝑒𝑥𝑝 (	𝐹 𝑅𝑇	𝛾𝜂 𝑃𝐴𝑀𝐿 ) (44)
	Dissolution overpotential		𝜂 𝑃𝐴𝑀𝐿 = 𝜙 𝑠,𝑝 -𝜙 𝑒 -𝑈 𝑃𝐴𝑀𝐿	(45)
	Active material volume fraction		𝜖 𝑠,𝑝 = 𝜖 𝑠,𝑝,𝑜 -	𝑄 𝑃𝐴𝑀𝐿 𝑐 𝑠,𝑚𝑎𝑥,𝑝 𝛿 𝑝 𝐹	(46)

Table 17 : Discharge capacities at the start and end of calendar aging, reversible and irreversible capacities of the cells and percentage of reversible capacity loss to the total loss, at different elevated temperature conditions Measured cell capacities [mAh] Cell at 30°C Cell at 35°C Cell at 40°C

 17

	Q 0 [mAh]	59.92	59.98	60.06
	Q 1 [mAh]	54.25	53.27	52.16
	Q 2 [mAh]	54.54	54.47	53.85
	ΔQ tot = Q 0 -Q 1 [mAh]	5.67	6.71	7.90
	ΔQ irr = Q 0 -Q 2 [mAh]	5.38	5.51	6.21
	ΔQ rev = ΔQ tot -ΔQ irr [mAh]	0.29	1.20	1.69
	% ΔQ rev = ΔQ rev /ΔQ tot [-]	5.11	17.88	21.39

Table 18 : Discharge capacities at the start and end of cycle aging and total capacity fade of cells, cycled at different temperatures after about 200 cycles each Capacity measurements Cell at 25°C Cell at 30°C Cell at 35°C Cell at 40°C

 18 The capacity fade of the cells from the V-dep and T-dep setup, obtained from the intermediate checkups measured with a C/10 current rate, is shown in Figure56

	Before start of cycling Q c,o [mAh]	57.96	59.16	60.22	60.88
	End of cycling Q c,f [mAh]	55.58	55.85	56.47	56.97
	Capacity fade ΔQ c [mAh]	2.38	3.31	3.75	3.91
	Percent capacity fade [%]	4	5.6	6.2	6.4

  The

			V-dep		(a)
		16
	[mAh]	11
	consumed	1 6
	Charge	-4
		-9	Cell#1	Cell#2	Cell#3	Cell#4		Cell#1	Cell#2	Cell#3	Cell#4
			(0.1Ω)	(0.3Ω)	(0.5Ω)	(0.7Ω)		(25°C)	(30°C)	(35°C)	(40°C)
		Cathode -6.74	-6.32	-5.93	-5.62	Cathode	-7	-7.13	-7.1	-7.02
		Anode	5.72	5.82	5.84	6.01	Anode	5.74	7.78	10.12 12.98

Table 21 : Estimates of positive electrode stoichiometry at 0% SOC y 0 and positive electrode active material volume fraction ε s,pos (taken from simulation result) for the V-dep and T-dep setup cells obtained by fitting with half-cell data Positive electrode stoichiometry at 0% SOC y 0 Positive electrode active material volume fraction ε s,pos

 21

		Fresh cell	0.96	0.56
		Cell #1 (0.1Ω)	0.96	0.465
	V-dep	Cell #2 (0.3Ω) Cell #3 (0.5Ω)	0.96 0.96	0.473 0.48
		Cell #4 (0.7Ω)	0.96	0.49
		Cell #1 (25 o C)	0.96	0.47
	T-dep	Cell #2 (30 o C) Cell #3 (35 o C)	0.95 0.93	0.47 0.47
		Cell #4 (40 o C)	0.91	0.47

Source : Lecture slides of the European Master Course, MESC (Materials for Energy Storage and Conversion), given by Dr. Charles Delacourt at LRCS, Amiens

Acknowledgments

Solvent oxidation exchange current density* i o,PAML

A/m 2 3.5x10 -4 *fitted; ** from [63]; *** from [130]; ~ estimated; º calculated; ¤ measured

Results and discussion

The baseline aging behavior has been investigated till now. In this section, the cycle aging tests with the coin cells assembled in V-dep and T-dep network is presented. Firstly, experimental results during cycling, from the intermediate checkup tests is presented followed by the results from post mortem tests. Complementary to this, wherever possible, the experimental results are compared with the model simulation results. Additionally, supplementary simulation results from the coupled model are presented. setup, cell 1 has a higher overpotential (due to its low shunt resistance value) and thus a higher rate of reduction of ε s,pos . Following the evolution of ε s,pos during cycling, obtained from simulations, a comparison of the values of positive electrode stoichiometry at 0 % SOC y 0 of fresh and fully aged cells is done. This is obtained at the end of a galvanostatic C/10 discharge, performed as part of the checkup tests during the simulations for about 1100 cycles, is presented in Table 20. The reported values are in accordance with what is seen in Figure 59. In the V-dep cells, the contribution of cyclable lithium loss to the total capacity fade is almost negligible; it is seen in Table 20 as well that there is no change in y 0 values as well. In T-dep setup, the contribution of cyclable lithium loss in Figure 59 (d) increases with an increase in temperature. A similar trend is seen in Table 20, where a reduction in y 0 indicates a shift in the electrochemical window of the electrodes and thus a greater cyclable lithium loss. 62 (a), it can be seen that the potato shaped graphite particle flakes are masked in the aged cells resulting in a smaller porosity, in both V-dep and T-dep cells (greater in the latter) when compared to the pristine cell. This is usually expected to be seen in aged Li-ion cells [58].

Physical characterization

In addition to this, it can also be seen that there are white spots in the aged electrodes which are usually deposits of substances with higher atomic number. In this case, it is suspected that these may be deposits from the positive electrodes on the graphite electrode, as it has been reported in literature that NMC electrodes undergo transition metal loss. It was reported earlier that NMC electrodes tend to undergo transition metal dissolution upon usage [74,75]. An EDS is performed on the graphite electrode sample from cell 4 at 40 o C of T-dep setup, shown in Figure 62 (d) and the result of the analysis is shown in Figure 62 (e). Two different regions from Figure 62 (d) are tested where Spectrum 8 is on the white spot and Spectrum 7 elsewhere. The result, shown in Figure 62 (e), reveals that the white patch indicates a high concentration of transition metals, namely manganese, nickel and cobalt, in the decreasing order. However using EDS analyses results, the deposits cannot be precisely quantified and is also subject to a lot of measurement noise and the information obtained may not be very accurate. Thus additional physical chemistry tests to quantify this loss from the positive electrode and deposition on the negative electrode may be needed, such as ICP-OES (Inductively Coupled Plasma Optical Emission Spectroscopy) or ICP-MS (Inductively Coupled Plasma Mass Spectroscopy). Although not shown here, similar to the graphite electrode segments, SEM images on the positive electrodes also indicate a masking on the particles with an overall decrease in the porosity.

In addition to SEM, XRD is also performed on both pristine and aged positive and negative electrode samples. The analyses did not show any differences in the detected species between the fresh and the aged cells. However, using the XRD pattern, it is identified that the precise configuration of the NMC electrode is 6, 2 and 2 by mole fraction of nickel, manganese and cobalt, respectively.

Electrochemical analyses

Using the harvested electrode samples, half cells are fabricated (coin cells with graphite electrodes and Swagelok cells with NMC electrodes) as described in the Materials and Methods section. After assembly, three formation cycles are performed followed by measuring the capacity of each half cell. The capacity Q vs voltage V curves of the reference fresh cell during lithiation and delithiation of the negative and positive electrodes, respectively, mimicking a charge scenario on a full cell, are subtracted to get an estimate of an equivalent full cell Q vs V curve. This is then fitted with the data obtained from the fresh pristine full cell which is summarized by eq. ( 47). This fitted curve from the fresh cell is used as a reference to fit the Q vs V curves from the pristine aged cells by calibrating the parameters corresponding to the degradation phenomena. These are the positive electrode stoichiometry at 0% SOC y 0 accounting to the cyclable lithium loss (cumulative value for both positive and negative electrode shift) and the positive electrode volume fraction ε s,pos resulting due to PAML. It is to be noted that there is a difference in the value of y 0 , reported in this section, equal to 0.96 and the value obtained from simulation results, equal to 0.98. This difference probably arises from minor errors while performing the electrochemical tests, detailed below.

To eliminate the arbitrary nature of the fitting exercise, one of the two values ε s,pos and y 0 may be obtained from an appropriate source. To obtain values for y 0 , half-cell measurement data can be used, obtained from the harvested electrode segments from aged cells. Although it was tried to estimate the residual capacity values from NMC/Li half-cells, unfortunately this data is not very reliable and is used in the fitting process. This is because of many discrepancies while working with the harvested electrodes.

To name a few, the presence of a big over-potential in the half-cells with either the use of glass fiber separator or improper electric contact between the electrode and the coin cell or Swagelok cell components, minor errors in precisely estimating the electrode surface area and also clumsiness in handling the extremely tiny electrode segments with a width of 4mm. Thus, values for ε s,pos are adopted from simulation results, shown in Figure 60, and y 0 is fitted appropriately. These values are reported in Table 21.

interesting as well; as the cells are cycled, surrogating real-time driving conditions, the temperature of the cells are more inhomogeneously distributed. In addition to the work done in this thesis, there is still a huge scope for development, with respect to both modeling and experiments. In the experimental part, quantitative elemental analyses will be done on the harvested electrode segments to identify the appropriate source of active material loss in the positive electrode. With respect to the modeling work, the current coupled model may be appropriately calibrated. In addition to this, the multi-physics models and coupling techniques may be used to develop a fully coupled electrochemical, electrical and thermal model to represent the commercial cell behavior completely with a focus on the inhomogeneous aging behavior. In addition to the two degradation phenomena considered in this work, lithium metal plating may also be considered as it is reported in the literature to be frequently occurring.

Beyond the scope of Li-ion batteries, the methodology and techniques, developed and used in this thesis, to identify the inhomogeneous aging behavior, may be well applicable to a wide range of applications. Firstly, the non-uniformities occurring at the scale of battery packs in electric and hybrid vehicles may be studied using the technique developed in this work of discretizing the temperature révèlent que les différences de température dans les cellules peuvent avoir une influence plus grande sur le comportement inhomogène global par rapport aux différences de potentiel.