Claude Guéry

Afef Mastouri

Mattia Giannini

Carine Davoisne

Jacob Lee

Mariem Belhadj

Kan-Hao Xue

Matias Quiroga

Yinghui Yin

Alexis Rucci

Oscar Xavier

Jose Vivo Torayev

Garima Vilches

Rakes Shukla

Vijay Elango

Ronan Challuri

Laura Le Ruyet

Loaiza Cristina

Omer Ulas Rodriguez

Carlos Eduardo Alarcón Kudu

Emiliano Suesca

Primo

Lithium sulfur (Li-S) batteries, mathematical models, 3D kinetic Monte-Carlo (kMC) model, Multi-scale continuum Li-S batteries discharge model, Li 2 S (s) nucleation and growth model, cyclic voltammetry of dissolved sulfur and polysulfides, potentiostatic and galvanostatic discharge of Li-S batteries.

"Ever tried. Ever failed. No matter. Try again. Fail again. Fail better." -Samuel Beckett (Worstward Ho).

Dedicated to my parents

Chitra and Thangavel

I also like to thank my friends from the MESC Master program, Alagaraj Paulraj, Yasemin Duygu Yücel, Violetta Arszelewska and Daria Iermakova for their continuous support. I am grateful to my friends back in India, Bineesh, Gautam, Shriram, Vignesh Kumar and Srinivasan Viswanathan, for their perpetual support. Lastly, I would like thank my parents, Chitra and Thangavel and my brother, Srinivasan, whose love and moral support helped me survive during this PhD work.

Chapter 1. General Introduction

Energy demand, consumption and production

We now live in a technological world, and energy is more crucial than ever in our day to day lives. Additionally, we like the current technologies such as smart phones, computers, selfdriving cars, etc., to become smarter and faster in the future, which could mean that they may consume more energy. These technological innovations and developments, have aided the economic growth of countries around the world, which in turn increased the demand for energy (Fig. 1 These economic and technological developments have also improved the living conditions, increased life expectancy [START_REF] Roser | Life Expectancy, Our World Data[END_REF], reduced child mortality [START_REF] Roser | Child Mortality, Our World Data[END_REF], etc. in developing countries like India and China. Therefore these improvements have also led to the growth in human Chapter 1. General Introduction population until the year 2100 [START_REF] Kremer | Population Growth and Technological Change: One Million B.C. to[END_REF]. Since the people around the world will continue to strive for the betterment of their lives and living conditions, the global energy consumption is projected to increase in the future due to the population growth (Fig. 1.2) [START_REF] Larcher | Towards greener and more sustainable batteries for electrical energy storage[END_REF].

Figure 1.2. Relationship between annual global energy consumption given in ton of oil

equivalent and human population [START_REF] Larcher | Towards greener and more sustainable batteries for electrical energy storage[END_REF].

For the good of the many, the current trends in the technological and socio-economic progress cannot be stopped. Therefore, it is highly necessary to increase the production of energy to meet the demand. However, the majority of the energy currently being consumed is produced from fossil fuel sources (Fig. 1.3) [1]. This energy consumption trend is not sustainable, since the reserves for fossil fuels such as oil and natural gas will be exhausted within the next fifty years, while coal will be exhausted within the next 115 years [START_REF] Höök | Depletion of fossil fuels and anthropogenic climate change-A review[END_REF]. Further issues with the fossil fuels are the emissions of greenhouse gases (GHG) and particulates matter which cause serious damage to environment, health, global climate, etc. In the year 2018, the emission of GHG like carbon dioxide emission (CO 2 ) is projected to increase by 2.7% which could be correlated to the increase in the consumption of fossil fuels in countries such as India, China and the United States of America (USA) [START_REF] Quéré | Global Carbon Budget[END_REF]. This growth in the global CO 2 emission is the largest in the recent years, since it was 1.6% in the year 2017 which was preceded by a period of three years with very low or no growth [START_REF] Quéré | Global Carbon Budget[END_REF]. The most important crisis related to the emission of GHG like CO 2 , is the phenomenon called global warming, which refers to the increase in the global average sea and surface air temperatures during a period of thirty years. In October of 2018, the Intergovernmental Panel for Climate Change (IPCC), published a report stating that the human activities have currently caused global warming of approximately 1.0 o C (Fig. 1.4) above the temperature level of the pre-industrial era (1850-1900 AD) [START_REF]Global warming of 1.5°C[END_REF]. The report recommends to reduce the net global CO 2 emission (i.e. the difference between the anthropogenic emission and removal of CO 2 ) to zero by the year 2040 (Fig. 1.4), in order to limit the global warming to 1.5 o C [START_REF]Global warming of 1.5°C[END_REF]. Since the risks to natural and human systems are higher for the global warming above 1.5 o C (e.g. 2.0 o C) -it is crucial to drastically reduce the CO 2 emission by the year 2040. The reduction of CO 2 emission could be achieved only by transitioning from a fossil fuels-based energy economy to a renewable one [START_REF] Jaforullah | Does the use of renewable energy sources mitigate CO 2 emissions? A reassessment of the US evidence[END_REF] and failing to do so could lead to catastrophic events around the world.

The non-CO 2 radiative forcing refers to the rise in global temperature due to the GHGs other than CO 2 in the atmosphere, which block of power radiated from the Earth's crust back to space.

Renewable energy technologies

Renewable energy refers to the energy produced from the sources which could be naturally replenished therefore sustainable and produce limited or zero GHG emission. Some of the types of renewable energies include hydropower, wind, solar, bioenergy and geothermal energy. The International Renewable Energy Agency (IRENA) reported that the global renewable power generation capacity accounted to 2179 GW in the year 2017 (Fig. 1.5) [START_REF]Renewable capacity highlights[END_REF].

Hydropower constitutes about 53% of the renewable power generation capacity, whereas wind and solar constitute 23% and 18% respectively [START_REF]Renewable capacity highlights[END_REF]. The costs of all the renewable energy technologies have been in steady decline in the recent year and are projected to become more competitive to those of the fossil fuels based technologies in the near future [START_REF]Renewable power generation costs in 2017: Key findings and executive summary, IRENA[END_REF]. Global renewable power generation capacity in gigawatt (GW) and increase in capacity of different renewable power sources [START_REF]Renewable capacity highlights[END_REF]. Some fundamentals of the different types of renewable energies are as follows,

•

Hydropower -is generated from the energy of flowing water in dams. Recently, there has been a boom in the construction of hydropower dams to increase its global capacity to 1700 GW, which will not sufficient to meet energy demands. Furthermore, the mitigation CO 2 emissions due to them will also be low [START_REF] Zarfl | A global boom in hydropower dam construction[END_REF]. Moreover, hydropower dams negatively impact the social and ecological conditions of the surrounding areas due to relocation of the local population, fragmentation of free-flowing rivers, higher risk to freshwater biodiversity, etc. [START_REF] Zarfl | A global boom in hydropower dam construction[END_REF].

• Wind power is generated by the wind turbines, which convert the free and renewable kinetic energy of the wind into electricity [START_REF]Wind Energy Basics[END_REF]. The wind power generation involves zero CO 2 and other pollutant emissions. However, the ideal locations for the wind power plants have wind speeds above 20 km.h -1 [START_REF]Wind Energy Basics[END_REF], which are unevenly distributed in the world and often in remote areas away from cities [START_REF]Advantages and Challenges of Wind Energy, Energy.Gov[END_REF].

•

The solar power -is generated by the photovoltaic devices which convert the Sun's radiation energy into electricity. The solar power generation also does not involve any GHG and particulate emissions.

• Bioenergy -is generated by the combustion of carbon based fuels which are available in the biosphere such as parts of plants, trees, etc. and the products extracted from them (e.g. ethanol) [START_REF] Edenhofer | Intergovernmental Panel on Climate Change, Working Group 3, Renewable energy sources and climate change mitigation: summary for policymakers and technical summary : special report of the intergovernmental panel on climate change[END_REF]. The bioenergy generation involves the emission of CO 2 gas, which was initially absorbed from the atmosphere by the biosphere [START_REF] Berndes | Bioenergy, Land Use Change and Climate Change Mitigation[END_REF].

Furthermore, the CO 2 emitted during the bioenergy generation could be reabsorbed by the biosphere provided that the land use change (LUC) is managed properly.

• Geothermal energy -refers to the naturally replenishable heat energy which is stored in rocks and trapped steam or liquid water in the Earth's interior [START_REF] Edenhofer | Intergovernmental Panel on Climate Change, Working Group 3, Renewable energy sources and climate change mitigation: summary for policymakers and technical summary : special report of the intergovernmental panel on climate change[END_REF]. This trapped steam or liquid water is extracted through wells and fed into turbines to generate electricity in the geothermal power plants. The geothermal energy generation depends on the properties such as the temperature of the hydrothermal resource in the Earth's interior, etc. [START_REF] Sowa-Watrak | The criteria for suitable location of geothermal power plant[END_REF]. Therefore, the power plants are mostly located in hot spots with volcanic activities, which are unevenly distributed in the world [START_REF] Duffield | Geothermal Energy-Clean Power From the Earth's Heat[END_REF].

Need for energy storage systems

The demand for electricity varies significantly with the time of the day (Fig. 1.6) and season.

Therefore, the conventional electricity generation technologies use a complex energy production and transmission system to level the load on the electric power grid [START_REF] Ibrahim | Energy storage systems-Characteristics and comparisons[END_REF]. This is done with the help of the energy supply and demand predictions. A very small percentage of the electricity produced is stored before it is transmitted. The global electricity storage capacity as of September 2017 was 176 GW, which amounted to less than 2% of the global power generation capacity [20]. Furthermore, the power generated by the fastest growing renewable energy technologies such as solar and wind (Fig. 1.5), fluctuates independently from the electricity demand (Fig.

1.7) [START_REF]eco2mix mix énergétique en[END_REF]. This is due the intermittent nature of their energy sources such as the sunlight and wind speed [START_REF] Gagnon | Mémoire concernant la contribution possible de la production éolienne en réponse à l'accroissement de la demande québécoise d'électricité d'ici 2010[END_REF]. Due to this imbalance between the renewable energy supply and the demand for electricity, it is difficult to stabilize the power grid [START_REF] Ibrahim | Energy storage systems-Characteristics and comparisons[END_REF]. for transitioning from the fossil fuels-based technologies. In order to reduce losses, increase efficiency and security of electricity supply from renewable power sources, a new type of power grid infrastructure called smart grid has emerged [START_REF] Zame | Smart grid and energy storage: Policy recommendations[END_REF]. It uses enhanced sensors, automated controls, advanced communication and computing devices, to optimize the way in which electricity is transmitted, consumed and generate [START_REF] Zame | Smart grid and energy storage: Policy recommendations[END_REF]. Energy storage systems are the core components that stabilize the smart grid, since they store the excess energy generated by the renewable sources and balance the electricity demand and supply [START_REF] Römer | The role of smart metering and decentralized electricity storage for smart grids: The importance of positive externalities[END_REF].

Figure 1.7. Electricity generated using wind and solar energy technologies in France from 1 st to 6 th January, 2019. Data acquired from the website of RTE France [START_REF]eco2mix mix énergétique en[END_REF].

Electricity can be converted and stored in different forms of energies such as mechanical (e.g. pumped hydro storage, flywheels and compressed air storage), electrical (e.g. capacitors and super conducting coil) and electrochemical (e.g. hydrogen and batteries) [START_REF] Ibrahim | Energy storage systems-Characteristics and comparisons[END_REF]20,[START_REF] Römer | The role of smart metering and decentralized electricity storage for smart grids: The importance of positive externalities[END_REF]. Pumped storage hydropower (PSH) accounts for about 98% (172 GW) of the electricity storage capacity of the world [START_REF]Energy storage[END_REF]. PSH involves the conversion of electricity into mechanical potential energy. Where water is pumped using an electric pump from a lower to higher elevation, when the electricity demand is low [START_REF] Ibrahim | Energy storage systems-Characteristics and comparisons[END_REF]. During high electricity demand, the stored water is allowed to flow from the higher to lower elevation via a turbine, which generates electricity [START_REF] Ibrahim | Energy storage systems-Characteristics and comparisons[END_REF]. Large scale pumped storage facilities are geographically centralized and they are not suitable for smart grids which require decentralized energy storage [START_REF] Römer | The role of smart metering and decentralized electricity storage for smart grids: The importance of positive externalities[END_REF]. Therefore, other energy storage technologies are being developed. Among them rechargeable batteries such as the lithium-ion (Li-ion), redox flow, sodium-based batteries (e.g. sodium sulfur), etc.

accounts for about 46% of the available non-PSH storage volume (Fig. 1.8) [START_REF]Energy storage[END_REF]. the world as of 2017 [START_REF]Energy storage[END_REF].

In addition to electricity, transportation is another sector which is dominated by the fossil fuels. Even in the developed countries such as the 28 member states of the European Union (EU), GHG emissions from transportation sector are on the rise [START_REF] Ioannis | Li-ion batteries for mobility and stationary storage applications[END_REF], which increased by 3% in the year 2016 [START_REF]Greenhouse gas emissions from transport, Eur. Environ. Agency[END_REF]. The transportation sector contributed to 27% of the total GHG emissions by the EU, in which 72% of the emissions came from road transport (Fig. 1.9) [START_REF]Greenhouse gas emissions from transport, Eur. Environ. Agency[END_REF].

This is because most of the vehicles on the road have internal combustion engines (ICEs) 2016 [START_REF]Greenhouse gas emissions from transport, Eur. Environ. Agency[END_REF].

Rechargeable batteries

Batteries are electrochemical devices which convert chemical energy to electrical energy.

Rechargeable batteries are also called secondary batteries in which redox reactions of the anodes and cathodes are reversible. Therefore, these batteries store electricity as chemical energy during charge, and they convert it back to electricity during discharge. Rechargeable batteries have been in existence for the last 160 years. The first rechargeable battery called lead-acid battery was invented in the year 1859 by a French physicist named Gaston Planté [START_REF]Handbook of batteries[END_REF]. Lead-acid batteries are still used in conventional automobiles with ICEs for purposes such as starting, lighting and ignition (SLI) [START_REF] May | Lead batteries for utility energy storage: A review[END_REF]. Furthermore, they are also used in invertors and uninterruptible power supply (UPS) [START_REF] Stan | A comparative study of lithium ion to lead acid batteries for use in UPS applications[END_REF]. However, the gravimetric energy density which refers to the amount of energy stored per unit mass of the battery is very low for lead-acid Railways batteries, since lead (Pb) is a heavy metal. This has made the lead-acid batteries heavier and unsuitable for energy storage requirements of today and in the past. Therefore, rechargeable batteries with different chemistries and materials such as nickel-cadmium, nickel-metal hydride, Li-ion, lithium-sulfur (Li-S) etc. have been developed [START_REF] Tarascon | Issues and challenges facing rechargeable lithium batteries[END_REF]. Due to these developments, gravimetric energy density has increase at an annual rate of 5% since 1970 (Fig. 1.10) [START_REF] Winter | Before Li Ion Batteries[END_REF].

Figure 1.10. Energy density evolutions of different battery chemistries [START_REF] Winter | Before Li Ion Batteries[END_REF].

The energy stored in a battery (Wh) is the product of its cell potential (V) and capacity (Ah), both of which depend on the chemistries of its anode and cathode [START_REF] Tarascon | Issues and challenges facing rechargeable lithium batteries[END_REF]. Therefore, the energy density which refers to the energy stored per unit mass (Wh.kg -1 ) or volume (Wh.L -1 ) of the battery, also depends on its chemistries. Lithium (Li) is the lightest metal and it has an extremely low reduction potential (-3.045 V vs standard hydrogen electrode) [START_REF] Lewis | The potential of the lithium electrode[END_REF], which makes it a very attractive anode material with very high gravimetric capacity ( 3860 mAh.g -1 ).

Therefore, in the early 1970s, Li metal batteries were developed, which had metallic Li as the anode and Lithium-ion (Li + ) intercalation materials such as the layered TiS 2 as the cathode [START_REF] Whittingham | History, Evolution, and Future Status of Energy Storage[END_REF]. However, these batteries suffered a serious drawback due to the formation of dendrites during Li plating which lead to short circuiting and fires [START_REF] Tarascon | Issues and challenges facing rechargeable lithium batteries[END_REF][START_REF] Whittingham | History, Evolution, and Future Status of Energy Storage[END_REF]. In order to circumvent the issues related to Li metal anode, it was replaced with Li + intercalation anodes. This led to emergence of Li-ion batteries involving rocking chair mechanism between two intercalation electrodes [START_REF] Murphy | Topochemical reactions of rutile related structures with lithium[END_REF][START_REF] Lazzari | A Cyclable Lithium Organic Electrolyte Cell Based on Two Intercalation Electrodes[END_REF]. Further developments such as the high potential Li + intercalation cathode like LiCoO 2 and stable low potential carbon-based anodes (Fig. 1.11) have led to the commercialization of Li-ion batteries by Sony in June 1991 [START_REF] Nagaura | Lithium ion rechargeable battery[END_REF].

Figure 1.11. Schematic of the working principle of Li-ion batteries [START_REF] Nishi | The development of lithium ion secondary batteries[END_REF].

Today, Li-ion batteries (LIBs) dominate the market share of portable electronics and EVs. The energy densities of LIBs have been improved in the recent year through the development of new cathode [START_REF] Rosenman | Review on Li-Sulfur Battery Systems: an Integral Perspective[END_REF] and anode materials. However, the energy densities of LIBs are primarily limited by the specific capacities (charge stored per unit mass of the active material) of their cathode materials [START_REF] Rosenman | Review on Li-Sulfur Battery Systems: an Integral Perspective[END_REF]. Therefore, further improvements to the energy density of LIBs Wh.kg -1 ) (Fig. 1.12(a)) and the driving of the EVs powered by them are highly limited. In order to increase the driving ranges of EVs, next generation batteries such Li-S, Li-air or Lioxygen (Li-O 2 ) batteries, etc. are being developed [START_REF] Bruce | Li-O2 and Li-S batteries with high energy storage[END_REF]. Although Li-O 2 battery can potentially offer very high energy density (≈900 Wh.kg -1 ), it suffers from issues such as poor rechargeability [START_REF] Luntz | Nonaqueous Li-Air Batteries: A Status Report[END_REF], irreproducibility [START_REF] Griffith | Correlating Li/O 2 Cell Capacity and Product Morphology with Discharge Current[END_REF][START_REF] Torayev | Stochasticity of Pores Interconnectivity in Li-O 2 Batteries and its Impact on the Variations in Electrochemical Performance[END_REF], etc. -which make it unviable for commercialization in the near future. In contrast, Li-S batteries are already available for commercial purchase [START_REF] Uav | Oxis Energy[END_REF] and it is used in some niche applications like the Unmanned Arial Vehicle (UAV) [START_REF] Cleaver | Perspective-Commercializing Lithium Sulfur Batteries: Are We Doing the Right Research?[END_REF][START_REF] Fotouhi | Lithium-Sulfur Battery Technology Readiness and Applications-A Review[END_REF].

Figure 1.12. Regone plot of different energy storage and conversion systems [START_REF] Rosenman | Review on Li-Sulfur Battery Systems: an Integral Perspective[END_REF].

Sulfur (S 8 ) is initial active material of the fully charged Li-S batteries cathode, which is one of the most abundant elements in the world and a by-product of the petroleum and natural gas refining [START_REF] Apodaca | Mineral year book[END_REF]. It is also extremely cheap (≈40 $/ton) [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF]. Furthermore, the overall reaction of a Li-S batteries (eq. 1.1) involves transfer of two electrons and Lithium ions (Li + ) per sulfur atom. Therefore, the complete conversion of S 8 to Li 2 S, theoretically delivers a very high specific capacity of 1675 mAh.g -1 and energy density of 2567 Wh.kg -1 based on the mass of the solid sulfur in the cathode [START_REF] Demir-Cakan | Chapter 1: Introduction to Rechargeable Lithium-Sulfur Batteries[END_REF]. Therefore, Li-S batteries could be cheaper and they can offer 2 to 3 times the energy density of the Li-ion batteries [START_REF] Walus | Lithium/Sulfur batteries : development and understanding of the working mechanisms[END_REF].

1 8 𝑆 8 + 2𝐿𝑖 + + 2𝑒 -⇌ 𝐿𝑖 2 𝑆 (1.1)
Despite the progresses made in the past years, there are still several challenges facing the Li-S batteries which impede the further improvements of their performance.

Objective of my thesis

Li-S batteries has a complex working principle (explained in Chapter 2), which involves multiple of phenomena such as chemical and electrochemical reactions of different species that occur simultaneously. This makes it difficult to assess certain underlying mechanisms and limitations behind the electrochemical operation of Li-S batteries. Therefore, the objective of this thesis work is to develop mathematical models which assist in understanding the phenomena behind electrochemical experimental results of Li-S batteries and the impacts of cathode design parameters on the discharge capacities. This thesis work also aims to provide suggestions to optimize the discharge performance of Li-S batteries through the assessments achieved using our models.

This work is a part of the European Union's Horizon 2020 project called High Energy Lithium

Sulfur batteries and cells (HELiS). The entirety of the work presented in this manuscript except the potentiostatic experimental result in Chapter 6, was carried out at Laboratoire de Réactivité et Chimie des Solides (LRCS), CNRS UMR 7314, Université de Picardie Jules Verne, Hub de l'Energie, Amiens, France.

Manuscript structure

The rest of the manuscript is organized as follows,

• Chapter 2 -discusses the operating principles, challenges and developments of Li-S batteries, along with the detailed review of the state of the art of Li-S battery models;

• Chapter 3 -reports the development of a novel kinetic Monte-Carlo (kMC) which is used to study the impact of discharge rate and sulfur loading on the mesostructural evolution of a carbon/sulfur cathode composite;

• Chapter 4 -reports the theoretical development of a microstructural resolved continuum discharge model, which is used to investigate the impact of cathode design on the performance;

• Chapter 5 -presents a cyclic voltammetry model which is used to investigate the cyclic voltammograms of different polysulfides dissolved in the electrolyte;

• Chapter 6 -presents a nucleation and growth model which is used to investigate the electrodeposition of Li 2 S over carbon surface;

• Chapter 7 -summarizes the experimental work carried out in this thesis work along with few comparative discharge modelling results.

Chapter 2. State of the art of lithium sulfur batteries

Chapter 2. State of the art of lithium sulfur batteries

Introduction to Li-S batteries

The concept of using sulfur and lithium metal or its alloy respectively as the electro-positive and the electro-negative electrodes in storage batteries, was first introduced in the mid-1960s

by Herbert et al. [START_REF] Danuta | Electric dry cells and storage batteries[END_REF]. Electrolyte containing Li-salt dissolved in propyl, butyl or amyl amine was also proposed by the same authors. Therefore, the concept of Li-S batteries has existed for more than 5 decades. However, the inherent issues related to the cathode such as insulating nature of solid sulfur [START_REF] Tarascon | Issues and challenges facing rechargeable lithium batteries[END_REF][START_REF] Demir-Cakan | Chapter 1: Introduction to Rechargeable Lithium-Sulfur Batteries[END_REF], formation of highly soluble polysulfides intermediates which migrate and react with lithium metal anode, etc. have limited the specific capacity and the cyclability of the Li-S batteries [START_REF] Cheon | Rechargeable Lithium Sulfur Battery[END_REF]. Due to these reasons, the development of Li-S batteries was held back for several years. The sulfur loaded porous carbon cathode, which had improved electronic conductivity was first introduce by Peled et al. [START_REF] Peled | Rechargeable lithium sulfur battery (extended abstract)[END_REF]. Their Li-S cell delivered better specific capacity at low discharge rate, however the capacity faded upon cycling due to the loss of active materials from the cathode [START_REF] Peled | Rechargeable lithium sulfur battery (extended abstract)[END_REF]. In the year 2009, Nazar et al. used a mesoporous carbon as the cathode host material, which was impregnated with sulfur at 155 o C [START_REF] Ji | A highly ordered nanostructured carbon-sulphur cathode for lithiumsulphur batteries[END_REF]. Their cells containing the aforementioned sulfur impregnated mesoporous carbon delivered better capacity and had good cyclability. This work by Nazar et al., rejuvenated the interest in Li-S batteries which led to the explosive growth in the amount of Li-S research works carried out in the past decade.

Working principle of Li-S batteries

A conventional Li-S battery consists of a carbon/sulfur (C/S) composite cathode and a lithium metal anode which are electronically separated by a porous polymer membrane (Fig. 2.1). A non-aqueous organic electrolyte exists in the pores of the cathode and the separator which provide ionic conductivity between anode and cathode. The carbon particles in the cathode are bound by a polymer binder to form a porous matrix, which provides electronic wiring and acts as a host for solid sulfur based precipitates such as S 8(s) and Li 2 S (s) [START_REF] Manthiram | Challenges and Prospects of Lithium-Sulfur Batteries[END_REF]. The typical charge and discharge curves of a Li-S batteries consist of low and high potential plateaus (Fig. 2.2) [START_REF] Nazar | Lithium-sulfur batteries[END_REF]. During the high plateau stage of the discharge the solid sulfur (S 8(s) )

initially undergoes dissolution and reduction to produce dissolved sulfur and higher order polysulfides, which subsequently get reduced to medium order polysulfides [START_REF] Kawase | Electrochemical reactions of lithiumsulfur batteries: an analytical study using the organic conversion technique[END_REF][START_REF] Cuisinier | Sulfur Speciation in Li-S Batteries Determined by Operando X-ray Absorption Spectroscopy[END_REF]. Further The typical charge and discharge curves of a Li-S batteries [START_REF] Nazar | Lithium-sulfur batteries[END_REF].

The aforementioned reaction mechanism is the simplest one, since the dissolved polysulfides in the electrolyte are known to undergo reaction such as dissociation, disproportionation, etc. [START_REF] Barchasz | Lithium/Sulfur Cell Discharge Mechanism: An Original Approach for Intermediate Species Identification[END_REF][START_REF] Wujcik | Lithium Polysulfide Radical Anions in Ether-Based Solvents[END_REF]. Furthermore, the reaction mechanism of Li-S batteries can change based on the aforementioned properties polysulfides in the electrolyte [START_REF] He | Unraveling the Correlation between Solvent Properties and Sulfur Redox Behavior in Lithium-Sulfur Batteries[END_REF]. Moreover, Dibden et al.

showed that the characteristics of theoretical discharge based on the proportions of solid and dissolved sulfur based species at different stages of using a ternary S 8 -Li 2 S-electrolyte phase diagram (Fig. 2.3) [START_REF] Dibden | Predicting the composition and formation of solid products in lithium-sulfur batteries by using an experimental phase diagram[END_REF]. The potential profiles which had two plateaus correspond to the discharge trajectories (A and B) consisting of three equilibrium stages namely S 8 -liquid electrolyte, completely liquid electrolyte and Li 2 S-liquid electrolyte (Fig. 2.3).

Challenges in Li-S batteries

The components of Li-S batteries such as the cathode active materials (solid sulfur and Li 2 S), polysulfides dissolvable electrolytes and the lithium metal anode pose several challenges to the improvement of Li-S batteries performance. Some of the challenges are listed below,

• The solid sulfur which is the initial active material of a fully charged cathode is electronically insulating in nature [START_REF] Demir-Cakan | Chapter 1: Introduction to Rechargeable Lithium-Sulfur Batteries[END_REF]. Therefore, a proper contact between the solid sulfur and the electron conducting host material should be maintained to ensure full utilization of the active material during the operation of Li-S batteries [START_REF] Manthiram | Challenges and Prospects of Lithium-Sulfur Batteries[END_REF].

• The sulfur is also soluble up to a certain extent in the conventionally used Li-S batteries electrolytes. The dissolved sulfur will migrate to the lithium metal anode, where it gets reduced to polysulfides [START_REF] Al-Mahmoud | A simple, experimentbased model of the initial self-discharge of lithium-sulphur batteries[END_REF]. This phenomenon results in the self-discharge of the Li-S batteries.

• Furthermore, the reaction of the dissolved sulfur with the lithium metal anode could produce insulating film over its surface, which results in the irreversible of loss of active material from the cathode [START_REF] Demir-Cakan | Chapter 1: Introduction to Rechargeable Lithium-Sulfur Batteries[END_REF].

• During charge, the higher order polysulfides produced in the cathode will migrate to the lithium metal anode, where they get reduced to medium or low order polysulfides, which will then transport back to cathode and get oxidized once again (Fig. 2.4a) [START_REF] Busche | Systematical electrochemical study on the parasitic shuttle-effect in lithium-sulfur-cells at different temperatures and different rates[END_REF][START_REF] Moy | Direct measurement of polysulfide shuttle current: A window into understanding the performance of lithium-sulfur cells[END_REF]. The aforementioned reduction and oxidation processes will make the polysulfides to shuttle between the electrodes depending on the charge current, total polysulfides concentration in the electrolyte and temperature [START_REF] Busche | Systematical electrochemical study on the parasitic shuttle-effect in lithium-sulfur-cells at different temperatures and different rates[END_REF][START_REF] Mikhaylik | Polysulfide Shuttle Study in the Li/S Battery System[END_REF]. Furthermore, this shuttling mechanism will reduce the Coulombic efficiency, since the oxidation of polysulfides during charging will take longer time (Fig. 2.4b). • The Li 2 S precipitates produced during the discharge are also insulating in nature, which could passivate electroactive surface of the cathode [START_REF] Fan | Electrodeposition Kinetics in Li-S Batteries: Effects of Low Electrolyte/Sulfur Ratios and Deposition Surface Composition[END_REF]. Furthermore, the Li 2 S is more voluminous than solid sulfur, therefore it could result in the blocking of cathode pores [START_REF] Cheon | Rechargeable Lithium Sulfur Battery[END_REF]. Additionally, the volume expansion of the cathode due to the precipitation of Li 2 S could also result in the disintegration of porous conductive matrix [START_REF] Demir-Cakan | Targeting the role of lithium sulphide formation for the rapid capacity fading in lithium-sulphur batteries[END_REF].

• Lithium metal anode is highly reactive to the electrolyte used in the Li-S batteries.

Therefore, any damage in the protective Solid Electrolyte Interphase (SEI), will result in the degradation of electrolyte and adverse reactions of the lithium with polysulfides [START_REF] Cheng | Review-Li Metal Anode in Working Lithium-Sulfur Batteries[END_REF].

Developments in the components of Li-S batteries

Positive electrode

In order to mitigate the issues involving the cathode, the majority of the Li-S batteries research was dedicated to the tailoring the architecture of the carbon/sulfur composites. As mentioned before, Nazar et al. used a sulfur impregnated highly order mesoporous composite called CMK-3/S-155 as the cathode material which delivered better capacity and cyclability [START_REF] Ji | A highly ordered nanostructured carbon-sulphur cathode for lithiumsulphur batteries[END_REF]. The better performance was achieved through improved conductivity, since the sulfur was impregnated inside the ordered mesopores (Fig. 2.5a). The performance of the CMK-3/S composite was further improved by coating a thin layer of polyethylene glycol (PEG) over it, which inhibited the transport of polysulfides out of the cathode (Fig. 2.5b). nanofiber [START_REF] Manthiram | Challenges and Prospects of Lithium-Sulfur Batteries[END_REF].

The utilization of the sulfur was improved (1400 mAh.g -1 at 0.1C) by its confinement inside the pores of the porous carbon nanofiber [START_REF] Ji | Porous carbon nanofiber-sulfur composite electrodes for lithium/sulfur cells[END_REF]. It was achieved through high surface area and electronic conductivity of carbon nanofiber. Furthermore, the composite containing sulfur confined inside the hollow carbon nanofiber limited the diffusion of polysulfides, thereby it improved the retention of capacity over 150 cycles [START_REF] Zheng | Hollow Carbon Nanofiber-Encapsulated Sulfur Cathodes for High Specific Capacity Rechargeable Lithium Batteries[END_REF].

Wang et al., confined the sulfur inside the micropores of a microporous-mesoporous carbon and the galvanostatic discharge curves of their cell containing mixed carbonate-based electrolyte consisted of only one plateau similar to those of the solid-state Li-S batteries [START_REF] Wang | A microporousmesoporous carbon with graphitic structure for a high-rate stable sulfur cathode in carbonate solvent-based Li-S batteries[END_REF].

The micropores restricted the intrusion of electrolyte and the dissolution of polysulfides [START_REF] Dominko | Polysulfides Formation in Different Electrolytes from the Perspective of X-ray Absorption Spectroscopy[END_REF],

but allowed reversible lithiation and de-lithiation of sulfur. This type of reaction mechanism is called Quasi-Solid-State Reaction (QSSR). Markevich et al., later showed that the stability of the SEI layer over the microporous carbon containing sulfur within their micropores, was crucial for achieving QSSR mechanism in Li-S batteries (Fig. 2.7) [START_REF] Markevich | The effect of a solid electrolyte interphase on the mechanism of operation of lithium-sulfur batteries[END_REF]. The QSSR type Li-S batteries delivered better cyclability due to the restriction to the dissolution of polysulfides [START_REF] Markevich | Review-On the Mechanism of Quasi-Solid-State Lithiation of Sulfur Encapsulated in Microporous Carbons: Is the Existence of Small Sulfur Molecules Necessary?[END_REF].

Negative electrode

Few strategies have been adopted to mitigate the aforementioned issues related to the lithium (Li) metal anode [START_REF] Cao | Anodes for Rechargeable Lithium-Sulfur Batteries[END_REF]. The widely used strategy is the SEI formation over the Li metal anode using electrolyte additive such as LiNO 3 , which improved the cyclability and Columbic efficiency due to the elimination of polysulfides shuttle [START_REF] Yang | A membrane-free lithium/polysulfide semi-liquid battery for large-scale energy storage[END_REF]. Demir-Cakan et al., showed that SEI produced by coating sulfur over the Li metal anode assisted in capacity retention of Li-S batteries [START_REF] Demir-Cakan | Li-S batteries: simple approaches for superior performance[END_REF].

Another strategy is the use of a passivation layer which physically separates the Li metal and the electrolyte. Several polymer and solid-state electrolytes were explored as the passivation layer which assisted in reducing polysulfide shuttle but had very low Li + conductivities. The solid electrolyte layers also suppressed the dendrite formation over the Li metal anode. The passivation layer containing Nafion membrane which selectively blocks the diffusion of polysulfides, largely enhanced the capacity retention and Coulombic efficiency (Fig. 2.8) [START_REF] Huang | Ionic shield for polysulfides towards highly-stable lithium-sulfur batteries[END_REF]. Furthermore, the Li-Al coating over the Li-anode assisted in the suppression of dendrite formation in Li-S batteries [START_REF] Kim | Enhancing performance of Li-S cells using a Li-Al alloy anode coating[END_REF]. is the most commonly used salt and 1 M LiTFSI in DME:DOL (1:1 v/v) is the widely used electrolyte. Moreover, a class of electrolyte called the solvent-in-salt which contains ultrahigh concentration of LiTFSI (7 mol.l -1 ) in DME:DOL were investigate by Suo et al. [START_REF] Suo | A new class of Solvent-in-Salt electrolyte for highenergy rechargeable metallic lithium batteries[END_REF]. This class of electrolyte had very high transference number and inhibited the dissolution of polysulfides.

Furthermore, they also assisted in supressing the formation of dendrites over the Li metal anode.

In the recent years, sparingly soluble electrolytes containing hydrofluorinated ether (HFE):TEGDME (4:1) and ACN 2 -LiTFSI:HFE (1:1) have been investigate [START_REF] Cheng | Sparingly Solvating Electrolytes for High Energy Density Lithium-Sulfur Batteries[END_REF]. Since, these electrolytes inhibit the dissolution of polysulfides in Li-S batteries and they lower the electrolyte/sulfur (E/S) ratio, thereby they assist in increasing the energy density.

Modelling of Li-S batteries

Due to the complex operating principle of Li-S batteries, the mathematical models are used to assess the phenomena taking place in them [START_REF] Franco | Chapter 8: Computational Modeling of Lithium-Sulfur Batteries: Myths, Facts, and Controversies[END_REF]. Numerous modelling techniques have been used to study the phenomena occurring at different spatial and temporal scales. polysulfides (Fig. 2.9) [START_REF] Wang | A quantum-chemical study on the discharge reaction mechanism of lithium-sulfur batteries[END_REF] and nanoconfined Li 2 S [START_REF] Liu | Revealing reaction mechanisms of nanoconfined Li 2 S: implications for lithium-sulfur batteries[END_REF]. It was also used to investigate the oxidation potentials of electrolytes [START_REF] Liu | Revealing reaction mechanisms of nanoconfined Li 2 S: implications for lithium-sulfur batteries[END_REF]. DFT was widely used to study the interaction of polysulfides with different electrode surfaces such as sulfur terminated Ti 2 C MXene [START_REF] Liu | Anchoring effects of S-terminated Ti2C MXene for lithium-sulfur batteries: A first-principles study[END_REF], 2D layered materials [START_REF] Zhang | Understanding the Anchoring Effect of Two-Dimensional Layered Materials for Lithium-Sulfur Batteries[END_REF], nitrogen-doped graphene [START_REF] Yi | Effect of lithium-trapping on nitrogen-doped graphene as an anchoring material for lithium-sulfur batteries: a density functional theory study[END_REF], functionalized graphene [START_REF] Wasalathilake | Interaction between functionalized graphene and sulfur compounds in a lithium-sulfur battery -a density functional theory investigation[END_REF], silicene [START_REF] Liu | Evaluating silicene as a potential cathode host to immobilize polysulfides in lithium-sulfur batteries[END_REF], heteroatom-doped nanocarbon [START_REF] Hou | Design Principles for Heteroatom-Doped Nanocarbon to Achieve Strong Anchoring of Polysulfides for Lithium-Sulfur Batteries[END_REF], etc. The results of the aforementioned studies will assist in development of new cathode materials for Li-S batteries. dissolved Li + and polysulfides in different electrolyte using molecular dynamics simulations [START_REF] Li | Polysulfide-Blocking Microporous Polymer Membrane Tailored for Hybrid Li-Sulfur Flow Batteries[END_REF]. These results were used to design polysulfides blocking microporous polymer membrane.

Mesoscopic modelling

Mesoscopic modelling of Li-S batteries is carried out mostly using molecular dynamics.

Mesoscopic models are widely used to investigate the interfacial phenomena such as the polysulfides interaction [START_REF] Perez Beltran | Formation of Multilayer Graphene Domains with Strong Sulfur-Carbon Interaction and Enhanced Sulfur Reduction Zones for Lithium-Sulfur Battery Cathodes[END_REF] and Li 2 S over carbon surface [START_REF] Liu | Mesoscale Elucidation of Surface Passivation in the Li-Sulfur Battery Cathode[END_REF], volume expansion of carbon/sulfur composite and impregnation of liquid sulfur into the micropores [START_REF] Pascal | Liquid Sulfur Impregnation of Microporous Carbon Accelerated by Nanoscale Interfacial Effects[END_REF]. Few more mesoscopic modelling works have been discussed in the introduction of chapter 3.

Continuum level modelling

The continuum models are conventionally used to identify the phenomena behind the characteristics of charge/discharge curves and the performance limitations of Li-S batteries.

The first comprehensive one-dimensional (1D) continuum model for Li-S batteries was developed by Kumaresan et al. [92]. This model is based on the porous electrode theory, where the cathode and the separator are considered to be homogenous porous media (Fig.

2.13a

). The transport of dissolved species through these porous media are described using the dilute solution theory. Furthermore, this model considers multiple reduction and precipitation/dissolution reactions of different polysulfides and it is capable of simulating the typical discharge curve of a Li-S battery (Fig. 2.13b). Hofmann et al., developed a similar 1D continuum model which had fewer reaction steps but considered the polysulfides shuttle mechanism [START_REF] Hofmann | Mechanistic modeling of polysulfide shuttle and capacity loss in lithium-sulfur batteries[END_REF]. This model was capable of simulating the galvanostatic discharge and charge curves of Li-S batteries and it was used to investigate the impact current densities on the overcharging due to polysulfides shuttle phenomena (Fig.

2.14a

). This model also predicted the capacity loss upon cycling due to irreversible precipitation of Li 2 S over the Li metal anode (Fig. 2.14b). to investigate the reduction of discharge capacity at high discharge rate (1C) [START_REF] Zhang | Modelling transport-limited discharge capacity of lithium-sulfur cells[END_REF]. They showed that due to the slow transport of Li + , polysulfides were forced to migrate to the separator to maintain charge neutrality which lead to the reduction of discharge capacity at high discharge rate (Fig. 2.15a). They also showed that this capacity loss could be recovered by relaxing the cell for 1 hour (Fig. 2.15b). Therefore, they concluded that the discharge capacity of Li-S is mainly limited by the slow transport of Li + during fast discharge. discharge [START_REF] Zhang | Modelling transport-limited discharge capacity of lithium-sulfur cells[END_REF].

The discussions about the few other continuum models are presented in the introductions of chapters 3, 4, 5 and 6.

Conclusions

Although the concept of Li-S batteries exists for several decades, the improvements to its performance such as the increase of its discharge capacity, coulombic efficiency and cyclability were achieved only in the last decade. Most of the improvements were achieved through the tailoring of the architecture of the carbon/sulfur composites. However, they were also improved by inhibiting the adverse reactions of the polysulfides with the Li metal anode either through SEI formation using electrolyte additives or by using protective coatings or layers. Even after all these developments, the assessment of Li-S batteries phenomena and optimization of its performance is not trivial, due to its complex operating principle.

Therefore, numerous models were also developed to investigate the phenomena behind the The carbon/sulfur (C/S) composite cathodes of lithium sulfur batteries undergo mesostructural evolutions during discharge due to the dissolution/precipitation reactions of solid sulfur (S 8(s) ) and Li 2 S [START_REF] Manthiram | Challenges and Prospects of Lithium-Sulfur Batteries[END_REF][START_REF] Yermukhambetova | Exploring 3D microstructural evolution in Li-Sulfur battery electrodes using in-situ X-ray tomography[END_REF]. Where, the morphology of the Li 2 S precipitates depends on the operation and cathode designs such as discharge rate, sulfur loading, etc. [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF][START_REF] Fan | Electrodeposition Kinetics in Li-S Batteries: Effects of Low Electrolyte/Sulfur Ratios and Deposition Surface Composition[END_REF].

Since these Li 2 S precipitates are insulating, they impact the discharge performance due to phenomena such as surface passivation [START_REF] Andrei | Theoretical and experimental analysis of precipitation and solubility effects in lithium-sulfur batteries[END_REF] and pore-clogging [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF]. Therefore, the interest to investigate the impact of discharge performance on the mesostructural properties of the Li 2 S precipitate and vice-versa using mathematical models have increased recently [START_REF] Andrei | Theoretical and experimental analysis of precipitation and solubility effects in lithium-sulfur batteries[END_REF][START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF].

However, most of the continuum models used for the investigation of charge and discharge performances of Li-S batteries, only consider the effective cathode structural properties [START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF][START_REF] Hofmann | Mechanistic modeling of polysulfide shuttle and capacity loss in lithium-sulfur batteries[END_REF][START_REF] Zhang | Modelling transport-limited discharge capacity of lithium-sulfur cells[END_REF][START_REF] Zhang | Modeling the voltage loss mechanisms in lithium-sulfur cells: the importance of electrolyte resistance and precipitation kinetics[END_REF][START_REF] Zhang | What Limits the Rate Capability of Li-S Batteries during Discharge: Charge Transfer or Mass Transfer?[END_REF]. Thereby, they overlook the three-dimensional (3D) nature of the C/S mesostructure and the Li 2 S deposits.

Recently, Mistry et al. developed a modelling framework, where the precipitates are grown randomly (i.e. without explicitly describing the reaction mechanisms) in 3D porous carbon microstructures based on deposition energy and a morphology parameter [START_REF] Mistry | Precipitation-Microstructure Interactions in the Li-Sulfur Battery Electrode[END_REF]. They also calculated the effective cathode structural evolutions of those microstructures and incorporated them in their 1D continuum discharge model to predict the impacts of Li over a flat carbon substrate [START_REF] Liu | Mesoscale Elucidation of Surface Passivation in the Li-Sulfur Battery Cathode[END_REF]. This model is capable of predicting the impact of temperature, S 2-concentration, etc. on the mesoscale properties such as thickness and coverage Li 2 S (s) deposits [START_REF] Liu | Mesoscale Elucidation of Surface Passivation in the Li-Sulfur Battery Cathode[END_REF]. However, this model cannot be used to study the direct impact of the discharge on the deposition, since it does not consider any electrochemical reduction process [START_REF] Liu | Mesoscale Elucidation of Surface Passivation in the Li-Sulfur Battery Cathode[END_REF]. The side length of each voxel was set to 5 Å which is close to the S-S bond length (3.1 Å) in an isothermally stabilized graphene/S microstructure [START_REF] Perez Beltran | Formation of Multilayer Graphene Domains with Strong Sulfur-Carbon Interaction and Enhanced Sulfur Reduction Zones for Lithium-Sulfur Battery Cathodes[END_REF] which was simulated using a classical reactive molecular dynamics model. Therefore, this distance was set as the resolution between the coarse-grained atoms which are in contact with each other. The code uses the grey scale values of these images to create the simulation box with voxels containing carbon atoms. Locations of the carbon atoms in the simulation box were exported to location file (*.xyz) along with those of the solid sulfur (𝑆 8(𝑠) ) particles (Fig. 3.1). The visualizations of the simulation box were done using the open source software Ovito [START_REF] Stukowski | Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool[END_REF].

The locations of each coarse-grained carbon (𝐶) and sulfur (𝑆) atoms in 𝑆 8(𝑠) were identified using the integer numbers 1 and 2. Each 𝑆 8(𝑠) particle consists of eight coarse-grained 𝑆 atoms that are in contact with each other (Fig. 3.2). The 𝑆 8(𝑠) particles were randomly distributed next to the coarse-grained 𝐶 atoms at the carbon particle surface since the impregnated 𝑆 8(𝑠) sticks to the surface of the carbon particles [START_REF] Ji | A highly ordered nanostructured carbon-sulphur cathode for lithiumsulphur batteries[END_REF]. Furthermore, the mass ratio between 𝐶 and 𝑆 atoms was set to 1:0.27. We have used this low sulfur loading in our simulations to reduce the computational cost.

Finally, the resulting location file is read by our in house developed kMC python code, which reconstructs the simulation box and utilizes it as the initial C/S mesostructure. It should be noted that our in silico C/S mesostructure creation method and kMC code are not specific to the aforementioned dimensions, structural and geometric parameters. In fact, we can customize the initial C/S mesostructure by changing the parameters such as of the shape and size of the carbon particles, mesostructure porosity, sulfur loading, etc. In the future, this in silico method will also be used to transform the tomographic images of a real C/S composite electrode into a simulation box which will then be used as the initial C/S mesostructure of our kMC code.

Development of the 3D kMC-VSSM model and its assumptions

In the past, an on-lattice kMC algorithm called Variable Step Size Method (VSSM) was developed and used by us to describe the reaction and diffusion events in Fuel cells [START_REF] Quiroga | A Multi-Paradigm Computational Model of Materials Electrochemical Reactivity for Energy Conversion and Storage[END_REF] and Li-O 2 batteries [START_REF] Blanquer | Modeling Investigation of the Local Electrochemistry in Lithium-O 2 Batteries: A Kinetic Monte Carlo Approach[END_REF][START_REF] Yin | Compactness of the Lithium Peroxide Thin Film Formed in Li-O 2 Batteries and Its Link to the Charge Transport Mechanism: Insights from Stochastic Simulations[END_REF] and Brownian motion of suspended particles in slurry redox flow batteries [START_REF] Shukla | Self-Organization of Electroactive Suspensions in Discharging Slurry Batteries: A Mesoscale Modeling Investigation[END_REF][START_REF] Shukla | Handling Complexity of Semisolid Redox Flow Battery Operation Principles through Mechanistic Simulations[END_REF]. Here, we have adopted a similar method to select and execute the reaction and diffusion events (Fig. 3.2) during the discharge simulation of an in silico created C/S mesostructure (Fig. 3.1).

Although there could be several reactions involving multiple dissolved polysulfide species that occur during the operation of Li-S batteries, a reduced set of reaction steps (Eqs. 3.1-3.4) was considered in our kMC model. The aforementioned simplification was done in order to limit the computational costs of our simulations and complexities arising through multiple unknown parameters. This approximation is common in many previously reported Li-S batteries models [START_REF] Hofmann | Mechanistic modeling of polysulfide shuttle and capacity loss in lithium-sulfur batteries[END_REF][START_REF] Marinescu | A zero dimensional model of lithium-sulfur batteries during charge and discharge[END_REF][START_REF] Danner | Modeling of nano-structured cathodes for improved lithium-sulfur batteries[END_REF][START_REF] Dysart | Towards next generation lithium-sulfur batteries: non-conventional carbon compartments/sulfur electrodes and multi-scale analysis[END_REF][START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF].

The reaction events considered in the 3D kMC model are as follows,

𝑆 8(𝑠) → 𝑆 8(𝑙) (3.1) 𝑆 8(𝑙) + 4𝑒 -→ 2𝑆 4(𝑙) 2- (3.2) 𝑆 4(𝑙) 2-+ 2𝑒 -→ 2𝑆 2(𝑙) 2- (3.3) 𝑆 2(𝑙) 2-+ 4𝐿𝑖 + + 2𝑒 -→ 2𝐿𝑖 2 𝑆 (𝑠) (3.4)
where, Eq. particles are also identified using unique set of integer numbers namely 3, 4, 5 and 6

respectively. The coarse-grained structures of the different sulfur based particles along with the schematic representation of the reaction events between them and the directions in which the dissolved particles can diffuse are shown in Fig. 3.2. Each of the 𝑆 8(𝑙) , 𝑆 4(𝑙) 2-and 𝑆 2(𝑙)

2-

particles have 8, 4 and 2 coarse-grained 𝑆 atoms, respectively (Fig. 3.2). The distance between centers of the atoms which are within a sulfur-based particle is equal to the mesh size (5 Å).

Our kMC model does not include 𝐿𝑖 atoms explicitly, thus each 𝐿𝑖 2 𝑆 (𝑠) particle is simulated using a single coarse-grained atom. Among the reactions considered in our model (Eq. 3.1-3.4), only the Li 2 S electrodeposition involves dissolved 𝐿𝑖 (𝑙) + in the electrolyte. Moreover, the 𝐿𝑖 (𝑙) + concentration in Li-S batteries electrolyte is much larger (≈1000-5000mM) than that of 𝑆 2(𝑙) 2-(≈10 mM) [START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF], therefore the kinetics of Li 2 S electrodeposition will primarily depend on the latter. Since the dissolved 𝐿𝑖 (𝑙) + and anion of Li salt are highly concentrated in the electrolyte, the probilities of our kMC model selecting their diffusion events are much higher.

This will ultimately increase the simulation cost. Therefore, the Li salt containing supporting electrolyte is not explicitly considered in our model, and we assume that they are uniformly distributed in the void volume of the simulation box. the six directions in which the dissolved particles (𝑆 8(𝑙) , 𝑆 4(𝑙) 2-and 𝑆 2(𝑙) 2-) can diffuse.

Equations for rate constants of different types of events

According to the VSSM algorithm used in our 3D KMC model presented here, an event is selected and executed in a given time step or an iteration based on the weighted probabilities of all the possible events which depend on their corresponding rate constants. Therefore, it is important to determine the rate constants for different types of events.

The rate constants of the electrochemical reactions in our model (Eqs. 3.2-3.4) could be calculated using Butler Volmer type equations [START_REF] Blanquer | Modeling Investigation of the Local Electrochemistry in Lithium-O 2 Batteries: A Kinetic Monte Carlo Approach[END_REF][START_REF] Yin | Compactness of the Lithium Peroxide Thin Film Formed in Li-O 2 Batteries and Its Link to the Charge Transport Mechanism: Insights from Stochastic Simulations[END_REF]. However, the overpotentials in the Butler Volmer equations vary a lot during the discharge of Li-S batteries [START_REF] Zhang | Modeling the voltage loss mechanisms in lithium-sulfur cells: the importance of electrolyte resistance and precipitation kinetics[END_REF]. Since we intend to simulate the discharge of our in-silico created C/S mesostructures under a galvanostatic condition, we have derived the rate constants of the electrochemical reactions based on the discharge current (𝐼) which remains constant at any given time. Therefore, the kinetic rate constant (𝐾 𝑗 𝑒𝑙𝑒 ) of an electrochemical reaction (𝑗) is given by a Faraday's law type equation,

𝐾 𝑗 𝑒𝑙𝑒 = 𝐼 𝑛 𝑗 𝑞 𝑒 Θ(𝛿 𝑒 ) (3.5)
The discharge current 𝐼 is determined from the discharge C-rate and initial mass of 𝑆 8(𝑠)

present inside simulation box. 𝑛 𝑗 and 𝑞 𝑒 in eq. 3.5 are the number of electrons transferred in an electrochemical reaction (𝑗) and the charge of the electron respectively. According to eq. 5, the applied current is equal to the Faradaic current at each iteration. We have neglected the double layer phenomenon [START_REF] Neidhardt | A Flexible Framework for Modeling Multiple Solid, Liquid and Gaseous Phases in Batteries and Fuel Cells[END_REF] in our kMC model, since its impact on the simulated results and mechanisms in the Li-S batteries are still unclear. Furthermore, implementation of double layer dynamics would increase the computational cost of our model, since it would require us to simulate the supporting electrolyte explicitly or coupling the KMC model with a physical double layer model.

In order for the electrochemical reactions to occur, the dissolved polysulfides should be present within the electron tunnelling distance (𝛿 𝑒 ) from the carbon surface and the electron tunnelling probability (Θ(𝛿 𝑒 )) is given by a simple function [START_REF] Blanquer | Modeling Investigation of the Local Electrochemistry in Lithium-O 2 Batteries: A Kinetic Monte Carlo Approach[END_REF],

Θ(𝛿 𝑒 ) = { 1, 0 ≤ 𝛿 𝑒 ≤ 10 nm 0,
𝛿 𝑒 > 10 nm

(3.6)
Furthermore, the kinetic rate constant of the 𝐿𝑖 2 𝑆 (𝑠) deposition reaction is considered only when the 𝑆 2(𝑙) 2-particles are present next to either a carbon atom or a 𝐿𝑖 2 𝑆 (𝑠) particle. This condition mimics the nucleation and growth processes of 𝐿𝑖 2 𝑆 (𝑠) observed in Li-S batteries.

In few chronoamperometric investigations of 𝐿𝑖 2 𝑆 (𝑠) electrodeposition [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF][START_REF] Li | Solvent-Mediated Li 2 S Electrodeposition: A Critical Manipulator in Lithium-Sulfur Batteries[END_REF], Beweick,, Fleischman, and Thirsk (BFT) model [START_REF] Bewick | Kinetics of the electrocrystallization of thin films of calomel[END_REF][START_REF] Jafarian | Effect of potential on the early stage of nucleation and growth during aluminum electrocrystallization from molten salt (AlCl3-NaCl-KCl)[END_REF] and Scharifker-Hills (SH) Model were used fit the dimensionless current signals. These models assume that the charge transfer step of the electrodeposition is fast and the growth of the existing Li 2 S nuclei is controlled by mass transport of dissolved species to the electrode surface. However the model developed by Ren et al., uses a modified Tafel equation to calculate current for the growth of Li 2 S nuclei, [START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF] which is the rate of charge transfer step. Therefore, we have considered the charge transfer steps along with the diffusion 𝑆 2(𝑙) 2-for the 𝐿𝑖 2 𝑆 (𝑠) nucleation and growth reactions.

The transport rates of electrons through solid 𝑆 8(𝑠) and 𝐿𝑖 2 𝑆 (𝑠) are quite low [START_REF] Park | Adiabatic and Nonadiabatic Charge Transport in Li-S Batteries[END_REF]. Therefore, we have currently neglected the electronic conductivities of solid 𝑆 8(𝑠) and 𝐿𝑖 2 𝑆 (𝑠) in our kMC model. However, the inclusion of these conductivities could be a further improvement of our model and it could be carried out in the future.

As mentioned before, our model also considers the diffusion of dissolved particles such as 𝑆 8(𝑙) , 𝑆 4(𝑙) 2-and 𝑆 2(𝑙) 2-along six directions (Fig. 3.2). The diffusion rate constant of a dissolved particle (𝑖) is given by the Stokes-Einstein's equation,

𝐾 𝑖 𝑑𝑖𝑓 = 𝜅 𝐵 𝑇 6𝜋𝜇𝑟 𝑖 𝑧 2 (3.7)
where 𝜇 and 𝑟 𝑖 are the viscosity of the electrolyte and radius of gyration of a dissolved particle (𝑖) respectively. 𝑧 is the distance displaced by the dissolved particle along a given direction.

The parameters used in the kMC model is listed in Table 3.1.

Since our model simulates the redox reaction of the dissolved particles in the electrolyte phase near the electrode surface instead of solid-state-like reactions [START_REF] Markevich | Review-On the Mechanism of Quasi-Solid-State Lithiation of Sulfur Encapsulated in Microporous Carbons: Is the Existence of Small Sulfur Molecules Necessary?[END_REF], the ionic transport events through 𝑆 8(𝑠) and 𝐿𝑖 2 𝑆 (𝑠) deposits [START_REF] Park | Adiabatic and Nonadiabatic Charge Transport in Li-S Batteries[END_REF] are neglected. Standard potential adopted from Ref. [START_REF] Zhang | Modeling the voltage loss mechanisms in lithium-sulfur cells: the importance of electrolyte resistance and precipitation kinetics[END_REF].

Working principle of the 3D kMC-VSSM code

Reconstruction of the simulation box containing our in silico C/S mesostructure, is the initial step of our model, after which the kMC code enters into an iterative loop to execute the reaction and diffusion events during discharge simulation (Fig. 3.3). In any given iterative cycle, the entire simulation box is initially scanned in order to find all the different types of particles and the possible events which could be performed by them. A list containing all the possible events is then stored in the computer memory, along with their corresponding particle types, individual and cumulative sums of rate constants, current and final locations inside the simulation box. After this step, the sum of all the possible events (𝐾 𝑇 ) is calculated as follows,

𝐾 𝑇 = ∑ 𝐾 𝑛 𝑁 𝑛=1 (3.8)
where 𝑁 is the total number of all the possible events in a given iterative cycle and 𝐾 𝑛 is the rate constant of 𝑛𝑡ℎ event in the aforementioned list. 𝐾 𝑛 could be either a rate constant of a diffusion (𝑘 𝑖 𝑑𝑖𝑓 ) or a chemical (𝑘 𝑘 𝑐ℎ𝑒 ) or an electrochemical reaction event (𝑘 𝑗 𝑒𝑙𝑒 ).

After calculating 𝐾 𝑇 , a random number (𝜌 1 ) ∈ (0,1) is generated. According to the conditional algorithm of our kMC-VSSM model -an event is selected based on the cumulative sums of the rate constants of all the possible events and the product of 𝜌 1 and 𝐾 𝑇 as follows,

∑ 𝐾 𝑛 𝑚 𝑛=1 ≥ 𝜌 1 𝐾 𝑇 ≥ ∑ 𝐾 𝑛 𝑚-1 𝑛=1 (3.9)
where 𝑚 is the number of the selected event in the list.

According to Eq. 3.9, the product of the first random number and the sum of the rate constants of all the events (𝜌 1 𝐾 𝑇 ) -is within the range between the partial cumulative sums of the rate constants of (𝑚 -1) th and 𝑚 th event in the list. Therefore, according to the condition Eq. 3.9, in any given iterative cycle, the events with large rate constants have larger probabilities to be selected. Furthermore, an event type has a large probability of being selected, when there is a large number of events of that type present in the list of possible events.

Since diffusion rate constants are normally larger than electrochemical rate constants, our kMC code has to go through a large number of iterative cycles which select diffusion events before an electrochemical reaction event is selected. Therefore, in each iterative cycle, we only consider the diffusion rates of dissolved particles which could be displaced to a particular distance along anyone of the six directions within the simulation box. This criterion will increase the frequency in which electrochemical events are selected, and thereby it aids the discharge simulation to progress faster.

Following the event selection process of an iterative cycle, its corresponding time step (∆𝑡) is calculated as follows,

∆𝑡 = - ln 𝜌 2 𝐾 𝑇 (3.10)
where 𝜌 2 is the second random number ∈ (0,1). Since ∆𝑡 is inversely proportional to 𝐾 𝑇 , it varies from one iterative cycle to another and it also makes the calculated time (𝑡) after each iteration to be low. However, since we are interested in comparing the simulated results of two different discharge rates, it is much more relevant to represent them as functions of specific capacity (𝑄). Therefore, we also calculate the specific capacity gained (∆𝑄) during each iteration using the following equation,

∆𝑄 = { 𝑛 𝑗 𝑞 𝑒 𝑚 𝑆 8(𝑠) , if K m is a K j ele 0, else (3.11)
where the product of 𝑛 𝑗 and 𝑞 𝑒 in Eq. 3.11 is the charge transferred during the selected electrochemical event (𝑗) and 𝑚 𝑆 8(𝑠) is the initial mass of 𝑆 8(𝑠) present in the simulation box.

The final step in the iterative loop our kMC code is the execution of the selected event and the evolution of the C/S mesostructure inside the simulation box. This evolved C/S mesostructure, once again goes through another subsequent cycle consisting of steps such as scanning to find all possible events, selection and execution of an event and evolution C/S mesostructure. After a selected amount of cycles of this iterative process, the details of the

Impact of discharge rate

In this section, we have presented the discharge simulation results for two different C-rates namely C/2 and 2C. The same initial C/S mesostructure created using our in silico method (presented in subsection 3.2.1) was used for both the simulations. The C/2 and 2C discharge simulations were concluded at 1230 and 1400 mAh. g S 8(s) -1 , respectively. Around these capacities diffusion events dominated over reaction events, resulting in very few changes in the types of sulfur-based particles inside the simulation box. However, the results produced using these simulations were still used to compare the impact of C-rates on the evolutions inside the C/S mesostructure. Due to the stochastic nature of our kMC model, each of these simulations were carried out three times to determine the confidence region which are presented as shaded regions or error bars.

General effective evolutions of C/S mesostructure

As mentioned in the subsection 3.2.4, the locations of the different types of sulfur-based particles are saved to a location file after a selected number of cycles during the discharge simulation. This file can be used to track and visualize the evolution of the simulation. The visualization of the simulation box could be done even when the simulation is performing. When the simulation box is visualized, each type of sulfur-based particle is assigned a unique colour (Fig. 3.4). Therefore, it is possible to visually get an idea about the DoD of the simulation just from the colours of the particles inside the simulation box (Fig. 3.4).

Furthermore, this colour coding visually aids us to see the different types of events taking place inside the simulation box at different DoDs. In addition to visualization, the quantification of the different types of particles in the simulation box can be used to predict the effective properties such as concentrations of dissolved particles, porosity of the mesostructure, etc.

The concentration of a dissolved particle (𝑐 𝑖 ) inside the porous volume of C/S mesostructure is determined using the equation,

𝑐 𝑖 = 𝑛 𝑖 𝑁 𝐴 𝜀𝑉 (3.12)
where 𝑁 𝐴 and 𝑛 𝑖 respectively are the Avogadro's number and the total number of dissolved particles of type 𝑖 (i.e. 𝑆 8(𝑙) or 𝑆 4(𝑙)

2-or 𝑆 2(𝑙) 2-). 𝑉 is the total volume of the simulation box and 𝜀 is the porosity of the C/S mesostructure, which is determined from the fraction between the number of voxels which are unoccupied by the atoms of solid particles (such as carbon, 𝑆 8

and 𝐿𝑖 2 𝑆 particles) and the total number of voxels in the simulation box. Since the rate constants of the electrochemical reactions increase with the C-rates, the dissolved particles during the fast discharge are consumed faster. Therefore, the concentrations of all the dissolved particles during 2C discharge simulation are always lower than those of the C/2 (Fig. 3.5). The evolutions of concentrations of different dissolved particles (Fig. 3.5) and numbers of 𝑆 8(𝑆) and 𝐿𝑖 2 𝑆 (𝑠) particles in the simulation box (Fig. 3.8), assist in providing insights into the reactions taking place at different stages of discharge simulations. A typical discharge curve of a conventional Li-S battery, consists of a high and a low potential plateaus, with an intermediate slopy stage, during which the cell potential decreases [START_REF] Nazar | Lithium-sulfur batteries[END_REF][START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF]. The cell potentials in the continuum models are derived from the current balance equation [START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF][START_REF] Hofmann | Mechanistic modeling of polysulfide shuttle and capacity loss in lithium-sulfur batteries[END_REF][START_REF] Zhang | Modelling transport-limited discharge capacity of lithium-sulfur cells[END_REF]. However, our kMC model does not have equations that directly relate current and potential. Therefore, here we have predicted the approximate discharge curves (Fig. 3.6) from the concentrations of dissolved particles such as 𝑆 4(𝑙) 2-and 𝑆 2(𝑙) 2-using Nernst's equation [START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF],

𝑈 = 𝑈 𝑜 + 𝑅𝑇 2𝐹 (ln 𝑐 𝑆 4(𝑙) 2- 1000 -ln ( 𝑐 𝑆 2(𝑙) 2- 1000 ) 2 ) (3.13)
where 𝑈 is the approximate discharge potential and 𝑈 0 is the standard potential for the 𝑆 4(𝑙) 2-/𝑆 2(𝑙) 2-electrochemical reaction. A similar Nernst's equation was used to predict equilibrium potentials during the discharge in a published Li-S batteries model [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF]. The details about this galvanostatic discharge experiment, such as electrode preparation, electrolyte loading, etc. are given in Chapter 7. As mentioned before, the calculated discharge curves are only used to correlate the stages of our simulations with the experiments and we did not make any attempt to predict discharge curves that quantitatively match the experimental results. Since we have used the Nernst's equation, discharge curves correspond to theoretical equilibrium potentials, which is why the variation between them are smaller in comparison with the experiments.

Initially, during the first slopy discharge stage, the concentrations of the dissolved 𝑆 8(𝑙) and 𝑆 4(𝑙) 2-particles (Fig. 3.5) increase due to the chemical dissolution of 𝑆 8(𝑠) particles and subsequent reduction of some 𝑆 8(𝑙) particles to 𝑆 4(𝑙) 2-, respectively. This could be understood from the decrease in the number of 𝑆 8(𝑠) during this initial stage in both C/2 and 2C discharge simulations (Fig. 3.8). However, in the middle of the first slopy discharge stage (≈100 mAh. g S 8(s) -1 ), the 𝑆 8(𝑙) concentrations for both C/2 and 2C simulations start to decrease (Fig. -1 (Fig.

3.5)

, which suggest that the electrodeposition of 𝐿𝑖 2 𝑆 (𝑠) particles is the most dominant reaction from this specific capacity. Since, the S 8(s) dissolves during the slopy first stage of the predict discharge curve, it corresponds to the high potential plateau and slopy intermediate stage of a typical Li-S battery discharge curve [START_REF] Waluś | Lithium/Sulfur Batteries Upon Cycling: Structural Modifications and Species Quantification by In Situ and Operando X-Ray Diffraction Spectroscopy[END_REF]. Whereas, the relatively flat second stage corresponds to the low potential plateau of a typical discharge curve [START_REF] Waluś | Lithium/Sulfur Batteries Upon Cycling: Structural Modifications and Species Quantification by In Situ and Operando X-Ray Diffraction Spectroscopy[END_REF]. These trends are consistent with the numerous continuum scale simulation results.

Since the rate constant of the 𝐿𝑖 2 𝑆 (𝑠) electrodeposition reaction increases with the C-rate, the number of 𝐿𝑖 2 𝑆 (𝑠) particles increase slightly faster during 2C than C/2. This impact of the discharge rate on the precipitation rate is consistent with the continuum simulation results [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF]. However, an opposite trend is observed for the decrease in the number of 𝑆 8(𝑠) particles (Fig. electrodeposition, the mesostructure porosity of 2C is always lower than that of the C/2 (Fig.

3.8)

. However, the mesostructure porosities of both the simulations, increase during the first slopy discharge stage and decrease during the second relatively flat one, which is consistent with many of the previously reported modelling results [START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF][START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF][START_REF] Dysart | Towards next generation lithium-sulfur batteries: non-conventional carbon compartments/sulfur electrodes and multi-scale analysis[END_REF]. Finally, the decrease of discharge potential from around 1200 mAh. g S 8(s) -1 (Fig. 3.6) is due to the significant depletion in the concentrations of all the dissolved sulfur based particles (Fig. 3.5).

Mesoscale evolutions of Li 2 S deposits over carbon

Since the deposition of 𝐿𝑖 2 𝑆 (𝑠) over the carbon surface impacts the discharge performance due to the surface passivation, it is important to understand the impact of the C-rates on the evolutions of mesoscale properties. particles are formed, which then grow bigger along with the formation of newer clusters occur during the subsequent stages of the discharge simulations. A similar type of 𝐿𝑖 2 𝑆 (𝑠) deposition process over the carbon surface was experimentally observed by Fan et al. [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF]. However, it is difficult to make conclusions about the impact of C-rates on the mesoscale properties of 𝐿𝑖 2 𝑆 (𝑠) deposits over the carbon surface just from the visualizations. Therefore, in the following, we have presented the analysis of the post-processed results of the 𝐿𝑖 2 𝑆 (𝑆) deposits produced using computational tools such as radial distribution function and cluster recognition algorithm.

Distribution of Li 2 S particles from the carbon surface

Due to the discrete nature of our model, we decided to approximate the radial distribution function (RDF) as a histogram of the distances between the particles and the carbon surface.

Therefore, RDF assist in determining the distribution of the number of 𝐿𝑖 2 𝑆 (𝑠) particles at different distances from the the carbon particles (Fig. 3.11). At first, we used the results of the RDF to predict the coverage of 𝐿𝑖 2 𝑆 (𝑠) particles over carbon surface (𝜃 𝐿𝑖 2 𝑆 (𝑠) ), which quantifies the fraction of carbon surface that is directly blocked by 𝐿𝑖 2 𝑆 (𝑠) and it is calculated using the following equation,

𝜃 𝐿𝑖 2 𝑆 (𝑠) = 𝑁 𝑐 𝐿𝑖 2 𝑆 (𝑠) 𝑁 𝑐 𝑇 (3.14)
where, 𝑁 𝑐 𝐿𝑖 2 𝑆 (𝑠) and 𝑁 𝑐 𝑇 respectively are the number of surface 𝐶 atoms covered by 𝐿𝑖 2 𝑆 (𝑠)

particles and the total number of surface 𝐶 atoms in simulation box. It should be noted that

𝑁 𝑐 𝐿𝑖 2 𝑆 (𝑠)
is also the number of 𝐿𝑖 2 𝑆 (𝑠) particles present at 5 Å from the surface of the carbon particles, since this distance refers to the voxels that are present right next to the 𝐶 atoms. The increase of 𝐿𝑖 2 𝑆 (𝑠) coverage over carbon surface during 2C discharge simulation is faster than during C/2 (Fig. 3.10). This trend is consistent with the previously observed modelling results of Andrei et al. [START_REF] Andrei | Theoretical and experimental analysis of precipitation and solubility effects in lithium-sulfur batteries[END_REF]. Furthermore, the number of 𝐿𝑖 2 𝑆 (𝑠) particles at different distances from the carbon surface (Fig. 3.11), show that the some of the 𝐿𝑖 2 𝑆 (𝑠) deposits, produced during both simulations are away from the carbon surface (3D growth). This is the reason behind the low coverages of 𝐿𝑖 2 𝑆 (𝑠) over carbon surface. Although there is only a slight difference between the average numbers of 𝐿𝑖 2 𝑆 (𝑠) particles produced at around 1229 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 during 2C and C/2 discharge simulations (Fig. 3.8), the difference between the average coverages of 𝐿𝑖 2 𝑆 (𝑠) over the carbon surface is relatively larger at this specific capacity (Fig. 3.11). This suggests that the 𝐿𝑖 2 𝑆 (𝑠) deposits, produced at 1229 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠)

-1

during C/2 discharge simulation, have slightly more 𝐿𝑖 2 𝑆 (𝑠) particles that are away from the surface than those produced during 2C. This can also be understood from the distributions of The impact of C-rates on the nucleation and growth dynamics of 𝐿𝑖 2 𝑆 (𝑠) deposition during discharge simulations, could be understood from the comparisons of the evolutions of the 𝐿𝑖 2 𝑆 (𝑠) distributions on the carbon surface (Fig. -1 , the average numbers of 𝐿𝑖 2 𝑆 (𝑠) particles beyond 2.5 nm for C/2 discharge simulation are slightly larger than those of the 2C (Fig. 3.11). These evolutions show that 3D growth of 𝐿𝑖 2 𝑆 (𝑠) particles over carbon during C/2 discharge is slightly faster than during 2C, which is also consistent with the modelling predictions of Ren et al. [START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF].

Since the rate constant of the 𝐿𝑖 2 𝑆 (𝑠) electrodeposition reaction increases with the C-rate, 𝑆 2 2-particles tend to react more when they come closer to the carbon surface during 2C than C/2 (Fig. 3.12). Whereas, they tend to diffuse more during C/2 than during 2C. Therefore, this competition between reaction and diffusion events of 𝑆 2 2-particles, could be the reason why the 3D growth of 𝐿𝑖 2 𝑆 (𝑠) deposits of C/2 discharge simulation is slightly faster than those of the 2C.

Size distribution of Li 2 S (s) clusters

As mentioned in subsection 3.3.2, clusters of 𝐿𝑖 2 𝑆 (𝑠) particles are formed during discharge simulations (Fig. 3.9). The sizes of these clusters provide us details about the proximity of the 𝐿𝑖 2 𝑆 (𝑠) deposits with each other. Here, the size of a cluster refers to the number of 𝐿𝑖 2 𝑆 (𝑠) particles in that cluster and it assists in providing insights about the local passivation of carbon surface. Since a large 𝐿𝑖 2 𝑆 (𝑠) cluster could cover a large area of the carbon surface, locally the surface passivation by that cluster will be higher than if a small cluster is formed at that same area. This local passivation of carbon surfaces could have an impact on the electrochemical performance towards the end of discharge when the concentrations of dissolved polysulfides are low and unevenly distributed. Therefore, we have estimated the size distributions of 𝐿𝑖 2 𝑆 (𝑠) clusters formed at 1229𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 (Fig. 3.13 and 3.14), using a cluster recognition algorithm called Density-based Spatial clustering of Applications with Noise (DBSCAN) [START_REF] Ester | CiteSeerX -A density-based algorithm for discovering clusters in large spatial databases with noise[END_REF].

DBSCAN does not require prior knowledge about the shapes and amounts of clusters, which is its main advantage over other cluster recognition algorithms. Here, we have implemented the DBSCAN algorithm in a python code and we specially adjusted it to analyse the 𝐿𝑖 2 𝑆 (𝑠)

particles data which were produced using our kMC code. The input parameters that DBSCAN requires are -minimum number of particles required for a region to be considered as a cluster (MinPts) and minimum distance between the particles to be considered belonging to the same cluster (𝜀). In this chapter, MinPts and 𝜀 were set to 5 particles and 1.5 voxel sides (7.5

Å) respectively, in order to reduce the background noise.

The visualizations and size distributions of 𝐿𝑖 2 𝑆 (𝑠) clusters formed at 1229 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 during the both discharge simulations are shown in Fig. 3.13 and 3.14. The cluster sizes vary from 4

to 367, which is too wide a range to visualize using a single image and to represent its size distribution in a single histogram. Therefore, we have classified clusters based on their cluster size classes such as 4-8, 9-13, 14-18 and so on until 39-43 𝐿𝑖 2 𝑆 (𝑠) particles per cluster (Fig.

3.13).

There are a large number of small clusters (<19) formed during both C/2 and 2C simulations. Therefore, to clearly highlight the differences between the cluster sizes formed 𝐿𝑖 2 𝑆 (𝑠) clusters with big sizes represent the 𝐿𝑖 2 𝑆 (𝑠) deposits which are closer to each other.

Therefore, the 𝐿𝑖 2 𝑆 (𝑠) deposits produced during 2C are relatively closer than those produced during C/2. This means the local passivation of carbon surfaces are relatively high for 2C. This effect could also be due to the competition between reaction and diffusion events of 𝑆 2(𝑙)

2-

particles during 2C and C/2 discharge simulations. Since 𝑆 2(𝑙) 2-particles diffuse more during C/2 discharge (Fig. 3.11), they get separated more from each other, which could result in isolated deposits of 𝐿𝑖 2 𝑆 (𝑠) .

Impact of sulfur loading

This subsection presents the 2C discharge simulation results of a C/S mesostructure with 1:0.54 C/S mass ratio. The structural parameters such as the shape and size of the carbon particles, and the carbon porosity of this mesostructure are same as the one with 1:0.27 C/S mass ratio (Fig. 3.1). Therefore, the discharge simulation results of the aforementioned C/S mesostructures (i.e. 1:0.27 and 1:0.54 C/S mass ratios) are compared to assess the impact of the sulfur loading on the overall and 𝐿𝑖 2 𝑆 (𝑠) mesostructural evolutions.

The number of 𝑆 particles in the mesostructure with 1:0.54 C/S mass ratio (high-𝑆 loaded mesostructure) is twice that in the low-𝑆 loaded one (i.e. 1:0.27 C/S mass ratio). Therefore, to compare the evolution rates of 𝑆 8(𝑠) and 𝐿𝑖 2 𝑆 (𝑠) particles in high and low-𝑆 loaded mesostructures, we have normalized their amounts with the corresponding maximum possible number of particles (Fig. 3.15). During discharge, the porosity of the high-𝑆 loaded mesostructure is significantly lower than that of the low-𝑆 loaded one (Fig. 3.15). This is because the absolute total number of 𝑆 8(𝑠) and 𝐿𝑖 2 𝑆 (𝑠) particles in the high-𝑆 loaded mesostructure is always larger than that of the low-𝑆 loaded one. The rate of dissolution of 𝑆 8(𝑠) in high-𝑆 loaded mesostructure is slower than that in the low-𝑆 one (Fig. 3.15). Since the discharge current increases with the 𝑆 loading, the absolute rates of all the electrochemical reactions will also increase with it. Therefore, around 200 mAh. g S 8(s) -1 , the normalized amount of 𝐿𝑖 2 𝑆 (𝑠) particles in the high-𝑆 loaded mesostructure starts to increase significantly, even when there is a large amount undissolved 𝑆 8(𝑠) in it.

Initially, the increase in the normalized amount of 𝐿𝑖 2 𝑆 (𝑠) particles in high-𝑆 loaded mesostructure is relative slower than that in the low-𝑆 loaded one, which is due to the slow dissolution 𝑆 8(𝑠) particles. However, beyond 1000 mAh. g S 8(s) -1 much of the 𝑆 8(𝑠) particles are dissolved, therefore the rate of increase in the normalized 𝐿𝑖 2 𝑆 (𝑠) particles in high-𝑆 loaded mesostructure becomes relatively faster. The visual comparison of the 𝐿𝑖 2 𝑆 (𝑠) deposits over the carbon surface during discharge in the mesostructures with different 𝑆-loading show that the absolute number 𝐿𝑖 2 𝑆 (𝑠) particles are always higher in the high-𝑆 mesostructure (Fig. At the initial stage of discharge, the coverage of 𝐿𝑖 2 𝑆 (𝑠) over the carbon surface in the high-𝑆 loaded mesostructure increases at faster rate than that in the low-S loaded mesostructure (Fig. 3.17). This effect is due to the increase in the discharge current with the 𝑆 loading.

3.16).

However, at the mid stage (≈800 mAh. g S 8(s) -1 ), the increase in the 𝐿𝑖 2 𝑆 (𝑠) coverage in the high-𝑆 loaded mesostructure starts to slow down (Fig. 3.17). This is due to the low availability of free carbon surface, since a significant amount of it is covered by the previously deposited 𝐿𝑖 2 𝑆 (𝑠) and undissolved 𝑆 8(𝑠) particles (Fig. 3.15). Since, beyond 1000 mAh. g S 8(s) -1 the 𝐿𝑖 2 𝑆 (𝑠)

distributions at different distances from carbon surface in the high-𝑆 loaded mesostructure are significantly broader than those in the low-𝑆 loaded one (Fig. 3.18), they limit the transport of 𝑆 2 2-to the carbon surface. Therefore, the 𝐿𝑖 2 𝑆 (𝑠) coverage in the high-𝑆 loaded mesostructure remain relative lower even when the majority of 𝑆 8(𝑠) particles have dissolved beyond this specific capacity. Ultimately, the 𝐿𝑖 2 𝑆 (𝑠) coverage in the high-𝑆 mesostructure at the end discharge is relatively lower than the that of the low-𝑆 loaded one (Fig. 3.17). During discharge, the peaks of the 𝐿𝑖 2 𝑆 (𝑠) distributions over carbon surface in the high-𝑆 loaded mesostructure shift from 1.0 to 1.5 nm (Fig. 3.18). This indicates that the growth dynamics of 𝐿𝑖 2 𝑆 (𝑠) electrodeposition is more dominant than the nucleation dynamics in the high-𝑆 loaded mesostructure. This is also due to the low availability of free carbon surface in the high-𝑆 loaded mesostructure during discharge, since most of it is covered by previously deposited 𝐿𝑖 2 𝑆 (𝑠) and undissolved 𝑆 8(𝑠) particles. Furthermore, the slow nucleation dynamics in the high-𝑆 loaded mesostructure is also due to the thick 𝐿𝑖 2 𝑆 (𝑠) deposits over its carbon surface which limit the diffusion of 𝑆 2 2-.

Conclusions

In this chapter, we presented a novel 3D kMC model which is capable of simulating the evolutions inside a C/S mesostructure during discharge. Our model can predict effective evolutions inside the mesostructure such as concentrations of dissolved particle, numbers of solid sulfur based particles and mesostructure porosity. Furthermore, the approximate discharge curves calculated from the results of our kMC model, assist in assessing the phenomena taking place at different stages of discharge. The evolutions of mesostructure porosities, such as their increase during the first slopy discharge stage and decrease during the second relative flat discharge stage are consistent with the previously reported continuum modelling results [START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF][START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF][START_REF] Dysart | Towards next generation lithium-sulfur batteries: non-conventional carbon compartments/sulfur electrodes and multi-scale analysis[END_REF]. Furthermore, the reduction of long chain (S 8 ) to medium chain polysulfides (S 4 2-) during the first slopy stage and subsequent reduction of S 4 2- to S 2 2-and precipitation of Li 2 S (s) during the relatively flat stage are consistent with the continuum simulation [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF][START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF] and experimental results [START_REF] Kawase | Electrochemical reactions of lithiumsulfur batteries: an analytical study using the organic conversion technique[END_REF][START_REF] Cuisinier | Sulfur Speciation in Li-S Batteries Determined by Operando X-ray Absorption Spectroscopy[END_REF].

The post-processed results of 𝐿𝑖 2 𝑆 (𝑠) data produced by the simulations our kMC model, assist in assessing the impact of C-rates on the mesoscale properties of 𝐿𝑖 [118] and Ren et al. [START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF]. Furthermore, comparison of the size distributions of 𝐿𝑖 2 𝑆 (𝑠) clusters of C/2 and 2C discharge simulations, show that relatively small number of big deposits are produced during slow discharge. This could be due to the competition between reaction and diffusion 𝑆 2 2-. Due to the insulating nature of 𝐿𝑖 2 𝑆 (𝑠) , the mesoscale properties of its deposits over carbon will impact the surface passivation [START_REF] Andrei | Theoretical and experimental analysis of precipitation and solubility effects in lithium-sulfur batteries[END_REF][START_REF] Mistry | Precipitation-Microstructure Interactions in the Li-Sulfur Battery Electrode[END_REF]. Since the 𝐿𝑖 2 𝑆 (𝑠) coverage over carbon increases faster during fast discharge (2C), we believe that the overall surface passivation is also faster during fast discharge. Furthermore, since a relatively large number of bigger clusters are formed during fast discharge, local passivation of carbon surface is also larger during the fast discharge. These passivation effects could explain, at least partially, why the discharge capacity decreases when the C-rate increases.

The results of the radial distribution function show that the 𝐿𝑖 2 𝑆 (𝑠) deposits over the carbon surface in high-𝑆 loaded mesostructure (i.e. 1:0.54 C/S mass ratio) are relative thicker than those in low-𝑆 loaded mesostructure. This is also evident from the relatively low 𝐿𝑖 2 𝑆 (𝑠)

coverage over carbon surface in high-𝑆 loaded mesostructure at the end of the 2C discharge simulation. Therefore, the growth process of the 𝐿𝑖 2 𝑆 (𝑠) electrodeposition in high-𝑆 loaded mesostructure is more dominant than the 𝐿𝑖 2 𝑆 (𝑠) nucleation process. These phenomena are primarily due to the lack of free carbon surface in high-𝑆 loaded mesostructure, since most of the surface is covered by previously deposited 𝐿𝑖 2 𝑆 (𝑠) and undissolved 𝑆 8(𝑠) . Due to the combined effect of thick 𝐿𝑖 2 𝑆 (𝑠) deposits and coverage of undissolved 𝑆 8(𝑠) in high-𝑆 loaded mesostructure, surface passivation increases significantly with the 𝑆 loading. This conclusion about the impact of 𝑆 loading on the surface passivation is consistent with the experimental results reported by Fan et. al. [START_REF] Fan | Electrodeposition Kinetics in Li-S Batteries: Effects of Low Electrolyte/Sulfur Ratios and Deposition Surface Composition[END_REF]. Furthermore, thick 𝐿𝑖 2 𝑆 (𝑠) deposits will also limit the transport of solvate species towards the electrode surface, which could impede the electrochemical reactions. Finally, our kMC results show that porosity of the high-𝑆 loaded mesostructure throughout 2C discharge simulation is lower than that of the low-𝑆 loaded mesostructure. This suggest that the possibility of pore blocking during discharge could increase with the 𝑆 loading. Therefore, the increase in the surface passivation rate and possibility of pore blocking can explain, why the discharge capacity of Li-S batteries decreases when the 𝑆 loading in the cathode is increased. As mentioned in Chapter 2, mesoporous carbon/sulfur composites for Li-S batteries were developed to improve the electronic conductivity of the cathode, to confine polysulfides and to accommodate excess volume of the Li 2 S precipitates in the cathode [START_REF] Ji | A highly ordered nanostructured carbon-sulphur cathode for lithiumsulphur batteries[END_REF][START_REF] Zhang | Enhancement of long stability of sulfur cathode by encapsulating sulfur into micropores of carbon spheres[END_REF][START_REF] Schuster | Spherical Ordered Mesoporous Carbon Nanoparticles with High Porosity for Lithium-Sulfur Batteries[END_REF]. A large variety of mesoporous carbon exists, therefore assessing the impact of their architecture on the discharge performance could be time consuming. Alternatively, mathematical models could be used to the accelerate the assessment of the mesoporous carbon/sulfur composite design on the discharge performance. The continuum models applied to Li-S batteries have been successful in simulating various battery operation phenomena [START_REF] Kumaresan | A Mathematical Model for a Lithium -Sulfur Cell[END_REF][START_REF] Neidhardt | A Flexible Framework for Modeling Multiple Solid , Liquid and Gaseous Phases in Batteries and Fuel Cells[END_REF][START_REF] Fronczek | Author ' s personal copy Insight into lithium e sulfur batteries : Elementary kinetic modeling and impedance simulation Author[END_REF][START_REF] Hofmann | Mechanistic modeling of polysul fi de shuttle and capacity loss in lithium e sulfur batteries[END_REF][START_REF] Ghaznavi | Sensitivity analysis of a mathematical model of lithium e sulfur cells : Part II : Precipitation reaction kinetics and sulfur content[END_REF][START_REF] Ghaznavi | Sensitivity analysis of a mathematical model of lithium e sulfur cells part I : Applied discharge current and cathode conductivity[END_REF][START_REF] Ghaznavi | Electrochimica Acta Analysis of a Mathematical Model of Lithium-Sulfur Cells Part III : Electrochemical Reaction Kinetics , Transport Properties and Charging[END_REF]. However, most of them, model the cathode as a homogenous porous medium, described by an effective porosity, not accounting for the microstructure, thus not enabling any study of the impact of different architectures. In contrast, Danner et al. developed a full cycle model using a mesoporous cathode containing microporous carbon particles [START_REF] Danner | Electrochimica Acta Modeling of nano-structured cathodes for improved lithium-sulfur batteries[END_REF]. However, this model assumes the sulfur-based species to be electrochemically active only as long as they remain confined in the micropores within the carbon particles, hence ignoring the transport of these species along the cell and their electrochemistry over the external surface of the carbon particles. Dysart et al. reported a multi-scale analysis of Li-S batteries using a stochastic model to reconstruct the cathode microstructure [START_REF] Dysart | Towards Next Generation Lithium-Sulfur Batteries : Non-Conventional Carbon Compartments / Sulfur Electrodes and Multi-Scale Analysis[END_REF] and calculated the associated effective transport properties, but without evolving the microstructure upon discharge.

In this chapter, we report a multi-scale model devoted to the simulation of the discharging Li-S cells using a cathode made up of mesoporous carbon particles with inter-particular pores in-between. Our model brings novel features such as the consideration of the mass exchange of all the electrolyte solutes between mesopores within carbon particles and inter-particular pores, of the chemical and electrochemical reactions in both types of pores, and of the dynamical evolution of species diffusion coefficients. The novel features in our model permit the exploration of the impact on the discharge performance by the properties such as the inter-particular porosity between the carbon particles, the mesoporosity within the carbon particles, the particle and the mesopore sizes, and the sulfur loading at two different pore scales. These are features either impossible to study or simply not studied in previously reported models. In the following, we present the physical and geometrical assumptions and demonstrate how it can be utilized to study the effects of the initial cathode microstructure and sulfur repartition within on the overall discharge cell performance. Although our model has limitations of experimentally non-validated assumptions and assumed parameters, it can still qualitatively predict some experimentally observed discharge trends and provide insights on the limiting reasons behind them.

The contents of this chapter were reported in a research paper titled, 'A microstructurally resolved model for Li-S batteries assessing the impact of the cathode design on the discharge performance', which was published in the Journal of the Electrochemical Society [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF]. This The assumptions on which our model (Fig. 4.1) is based are the following:

Methodology: overall assumptions

• The separator and cathode components are represented in one-dimension;

• The cathode consists of mesoporous spherical carbon particles with a uniform size distribution;

• Each carbon particle contains spherical mesopores with a uniform pore size distribution;

• The impregnated sulfur initially exists both in mesopores within the carbon particles and in the inter-particular pores between them;

• The active surface area of the mesopores within each carbon particle is assumed to remain unchanged after loading 𝑆 8(solid) inside them;

• Transport of all the electrolyte solutes, such as sulfur, polysulfides, and lithium ions, is considered along the cell, as well as their mass exchange between the mesopores within the carbon particles and inter-particular pores;

• The transport of the electrolyte solutes is assumed to be diffusive and thus described by using Fick's laws;

• The diffusion coefficients of the solutes are assumed to depend on the electrolyte viscosity, calculated on-the-fly as dependent on the concentration of long chain polysulfides;

• Due to precipitation/dissolution reactions the porous volume of the cathode evolves along the discharge which is assumed to be completely filled with electrolyte;

• The concentration of the dissolved species in the electrolyte evolve due to transport, electrochemical and precipitation/dissolution reactions and their resulting evolution of porous/ electrolyte volume.

• The equilibrium potentials of all the electrochemical reactions follow Nernst's equations;

• The porous carbon matrix is perfectly percolated allowing proper electron wiring for all the particles;

• The carbon is assumed to be a perfect electronic conductor;

• The cathode surface area and porosity losses due to the presence of binder are neglected.

• The electrochemical reaction kinetics are assumed to follow the Butler-Volmer equations;

• The capacitive current due to the electrochemical double layer effect is neglected;

• The parasitic chemical and electrochemical reactions involving the polysulfide species reduction at the anode are neglected.

For the discharge reaction scheme at the anode we simply assume that lithium metal oxidation takes place, while at the cathode the reaction mechanism is assumed to be much more complex. Here, a part of the solid sulfur (𝑆 8(solid) ) contained in the cathode dissolves into the electrolyte to produce uncharged dissolved sulfur (𝑆 8(soln) ). The dissolved sulfur subsequently undergoes a series of reduction reactions to 𝑆 (soln) 2-. The reduction reactions take place in the electrolyte by accepting electrons from the surface of the carbon particles.

The cathode reduction reactions and the anode side oxidation reactions are listed in Table 4.1. The cathode reactions can take place both in the mesopores of the carbon particles and in the inter-particular pores. For the sake of clarity all the electrochemical reactions were indexed with angular brackets "<>". Since, the equations of our model are solved using numerically the initial concentrations and the volume fraction of the species cannot be zero. Therefore, we assumed certain values for the concentrations and volume fractions of the dissolved and solid species, respectively. The dissolved lithium salt concentration in the electrolyte is assumed to be 1000 mM (1 M), however in order to ascertain electroneutrality, the initial lithium ion concentration (Table 4.1) is determined by the total negative charge due to anion of lithium salt and polysulfides ions.

Assumed parameters are calibrated by qualitatively matching the calculated discharge profile trends with those experimentally observed for different current densities or C-rate [START_REF] Jayaprakash | Porous hollow carbon@sulfur composites for high-power lithium-sulfur batteries[END_REF][START_REF] Shim | The Lithium Õ Sulfur Rechargeable Cell Effects of Electrode Composition and Solvent on Cell Performance[END_REF][START_REF] Cheon | Rechargeable Lithium Sulfur Battery[END_REF].

The dissolved polysulfides (𝑆 𝑦(soln)

2-

) may react with lithium ions (𝐿𝑖 (soln)

+

) and produce insoluble lithium polysulfide that precipitates (𝐿𝑖 2 𝑆 𝑦(solid) ). These chemical reactions are reversible, hence termed as precipitation/dissolution reactions (Table 4.2), and indexed by curly brackets "{}". All parameters and expressions corresponding to inter-particular pores are henceforth assigned the subscript 1 and those of the mesopores within the carbon particles subscript 2.

The mathematical flow chart of our model is shown in Fig. 4.2, where the global input is the applied current density and the global output is the electrode potential. The inter-particular porosity and the carbon mesoporosity before sulfur impregnation are given by Eqs. 4.1 and 4.2 respectively

𝜀 1 𝑚𝑎𝑥 = 1 -𝜌 𝑐 ( 4𝜋𝑅 𝑝 3 3 ) (4.1) 𝜀 1 𝑚𝑎𝑥 = 𝑁 𝑝 𝜌 ( 4𝜋𝑟 𝑝 3 3 ) (4.2)
Where 𝜌 𝑐 and 𝑅 𝑝 are the number density and radius of the carbon particles. 𝑁 𝑝 and 𝑟 𝑝 are the number and radius of mesopores within each carbon particle.

After 𝑆 8(𝑠𝑜𝑙𝑖𝑑) impregnation the initial porosities are given by Eqs. 4.3 and 4.4 respectively

𝜀 1 𝑖𝑛𝑖𝑡 = 𝜀 1 𝑚𝑎𝑥 -(𝜖 1 ) 𝑠 8 (4.3) 𝜀 2 𝑖𝑛𝑖𝑡 = 𝜀 2 𝑚𝑎𝑥 -(𝜖 2 ) 𝑠 8 (4.4)
where (𝜖 1 ) 𝑆 8 and (𝜖 2 ) 𝑆 8 , are the volume fractions of the 𝑆 8(𝑠𝑜𝑙𝑖𝑑) impregnated.

The maximum carbon surface areas of the inter-particular pores and the mesopores are given by Eqs. 4.5 and 4.6, respectively

𝑎 1 𝑚𝑎𝑥 = 𝜌 𝑐 (4𝜋𝑅 𝑝 2 ) (4.5) 𝑎 2 𝑚𝑎𝑥 = 𝜌 𝑐 (4𝜋𝑟 𝑝 2 ) (4.6)
Due to their smaller size, the mesopores have a very high specific surface area and the passivation effect by 𝑆 8(𝑠𝑜𝑙𝑖𝑑) is considered to be negligible, therefore the initial surface area is given by

𝑎 2 𝑖𝑛𝑖𝑡 = 𝑎 2 𝑚𝑎𝑥 (4.7)
However, the carbon active surface area of the inter-particular pores is reduced, given by Bruggeman relation [START_REF] Thangavel | HELIS project D6.2 deliverable report -databases for results produced using interfacial models[END_REF],

𝑎 1 𝑖𝑛𝑖𝑡 = 𝑎 1 𝑚𝑎𝑥 ( 𝜀 1 𝑖𝑛𝑖𝑡 𝜖 1 𝑚𝑎𝑥 ) 𝛽 (4.8)
where 𝛽 Bruggeman coefficient.

Electrochemical reaction kinetics

The rate of each electrochemical reaction reported in where 𝑘 𝐵 is the Boltzmann constant, 𝑇 is the temperature, ℎ is the Planck constant, 𝜅 is the frequency factor, 𝐴 is an area factor, 𝑐 𝑟𝑒𝑓 is the reference concentration, and 𝐸 𝐹 and 𝐸 𝐵 are the forward and backward activation energies, respectively. The subscript 𝑖 in the Eqs. 4.10 and 4.11 corresponds to the species index (Table 4.1). 𝑃 𝑖𝑗 and 𝑄 𝑖𝑗 are the absolute values of the stoichiometric coefficients of the oxidized and reduced species, respectively which are involved in the electrochemical reactions listed in Table 4.1.

The Eqs. 4.9-4.11, hold for both the electrochemical reactions in the inter-particular pores and in the mesopores, but all concentration variables must be substituted accordingly.

Chemical reaction rates

As in previously reported models [START_REF] Kumaresan | A Mathematical Model for a Lithium -Sulfur Cell[END_REF][START_REF] Ghaznavi | Sensitivity analysis of a mathematical model of lithium e sulfur cells : Part II : Precipitation reaction kinetics and sulfur content[END_REF][START_REF] Ghaznavi | Sensitivity analysis of a mathematical model of lithium e sulfur cells part I : Applied discharge current and cathode conductivity[END_REF][START_REF] Ghaznavi | Electrochimica Acta Analysis of a Mathematical Model of Lithium-Sulfur Cells Part III : Electrochemical Reaction Kinetics , Transport Properties and Charging[END_REF], the chemical reaction rates of the dissolution/precipitation reactions (Table 4.2) depend on the concentration of dissolved species in the electrolyte 𝑐 𝑖 , and the volume fraction of the solid precipitates in the cathode 𝜖 𝑘 . Since the physics are the same for both inter-particular pores and mesopores, the rate equation is given without the subscripts. The rate of a dissolution/precipitation reaction 𝑅 𝑘 is given by

𝑅 𝑘 = 𝑘 𝑘 𝑝𝑒𝑟𝑐 𝜖 𝑘 (∏(𝑐 𝑖 ) 𝛾 𝑖𝑘 𝑖 -𝐾 𝑘 𝑠𝑝 ) (4.12)
where 𝑘 𝑘 𝑝𝑒𝑟𝑐 is the chemical rate constant, 𝐾 𝑘 𝑠𝑝 is the solubility product and 𝛾 𝑖𝑘 is the coefficient of the species involved in the chemical precipitation/dissolution reaction 𝑘.

Transport in the electrolyte

Generally, the electrolytes used in Li-S batteries are highly viscous and the dissolved species concentrations will reach very high values along discharge, thus for the sake of simplicity we assume transport of species in the electrolyte to be diffusive. Apart from the anion of the dissolved lithium salt, the evolution of the concentrations of all the other solutes in the interparticular pores is given by the mass conservation expression

𝜕(𝑐 1 𝜀 1 ) 𝜕𝑡 = 𝜕 𝜕𝑥 (𝐷𝜀 1 𝛽 𝜕𝑐 1 𝜕𝑥 ) + 𝑔 1 -𝐺 1 + 𝑓 (4.13)
where 𝜀 1 is the inter-particular porosity, 𝑐 1 is the dissolved species concentration, 𝐷 is the diffusion coefficient, 𝑔 1 and 𝐺 1 are the rates of the electrochemical and chemical reactions.

The final term 𝑓 is called the "intra-flux"; accounting for the exchange of dissolved species between inter-particular pores and mesopores. The factor 𝜀 𝛽 represents the classical Bruggeman correction to the diffusion coefficient in porous media [START_REF] Bruggeman | Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen[END_REF], [START_REF] Fuller | Simulation and Optimization of the Dual Lithium Ion Insertion Cell[END_REF].

A similar equation is used to describe the concentration evolution inside the mesopores, but the global diffusion flux term is removed:

𝜕(𝑐 2 𝜀 2 ) 𝜕𝑡 = 𝑔 2 -𝐺 2 -𝑓 (4.14)
The electrochemical reaction source term (𝑔 1 /𝑔 2 ) for a specific species 𝑖 is given by

𝑔 𝑖 = -𝑎 ∑ 𝑆 𝑖𝑗 𝑖 𝑗 𝑛 𝑗 𝐹 𝑗 (4.15)
where 𝑎 is the active surface area and 𝑆 𝑖𝑗 the coefficients of the species and number of electrons involved, respectively (Table I).

The chemical reaction source term (𝐺 1 or 𝐺 2 ) for a species 𝑖 is given by

𝐺 𝑖 = ∑ 𝛾 𝑖𝑘 𝑘 𝑅 𝑘 (4.16)
For the separator the evolution of the concentration is also given by the mass conservation Eq. 4.13. However, the chemical, electrochemical and intra-flux source terms are not applicable as the separator is assumed to be a homogenous porous medium without any chemical or electrochemical reactions occurring inside.

Intra-flux

The intra-flux term 𝑓 used in Eqs. 4.13 and 4.14 is a key concept in our multi-scale model. It is featured as "intra" rather than "inter-" as it describes the exchange of species between the inter-particular pores and mesopores only within the same control volume of the discretized cathode (Fig. 4.3). The intra-flux of a given species within a control volume is given by

𝑓 = 𝐾 𝑓𝑙𝑢𝑥 𝐷(𝑐 2 -𝑐 1 )Ξ(𝛿 1 ) (4.17)
Where 𝐾 𝑓𝑙𝑢𝑥 is a rate constant and the "choking function", Ξ(𝛿 1 ) describes the resistance to mass exchange between the inter-particular pores and the mesopores,

Ξ(𝛿 1 ) = 1 -𝑒𝑟𝑓 ( 𝛿 1 -𝛿 𝑐ℎ𝑜𝑘 𝐿 𝑐ℎ𝑜𝑘 ) 2 (4.18)
In Eq. (4.18), 𝑒𝑟𝑓 is the error function, 𝛿 1 is the thickness of the 𝐿𝑖 2 𝑆 (𝑠𝑜𝑙𝑖𝑑) thin film covering the external surface of the carbon particles, 𝛿 𝑐ℎ𝑜𝑘 is a characteristic 𝐿𝑖 2 𝑆 (𝑠𝑜𝑙𝑖𝑑) thin film thickness of halved probability of mass exchange, and 𝐿 𝑐ℎ𝑜𝑐𝑘 is a scaling factor. When Ξ = 0 and the mesopores are fully choked, there will be no intra-flux within that finite volume element.

Dynamic viscosity evolution

The diffusion coefficient D in Eq. (4.13) is assumed to be affected by the long chain polysulfides generated during discharge, increasing the viscosity of the electrolyte. The diffusion coefficient of a spherical species in a viscous solution is expressed through the Stokes-Einstein relation as

𝐷 = 𝑘 𝐵 𝑇 6𝜋𝜇𝑟 𝑖 (4.19)
Where 𝜇 is the viscosity and 𝑟 𝑖 is the radius of gyration of the species 𝑖. For non-spherical species, a correction factor will appear in the denominator, but the relation between diffusion coefficient and viscosity remains as

𝐷 ∝ 1 𝜇 (4.20)
Since 𝑆 8(soln) and 𝑆 8(soln)

2-

are larger than the solvent molecules and other solutes in the electrolyte, we account for them as suspended particles. The relative viscosity is hence calculated using Einstein's formula for a monodispersed suspension [164],

𝜇 𝜇 0 = 1 + 0.5𝜑 (1 -𝜑) 4 (4.21)
where 𝜇 0 is the viscosity of the pure electrolyte and 𝜇 is the viscosity with suspended particles, whose volume fraction is 𝜑. We assume that, concentration can be very large and contribute significantly to the electrolyte viscosity. There is lack of experimental data on the evolution of electrolyte viscosity during discharge and the contributions of different solutes, but this approach suggests a viable path towards a better understanding.

Active surface area and porosity

The loss of inter-particular active surface area due to solid sulfur loading is calculated based on a phenomenological expression also adopted by previous models [START_REF] Kumaresan | A Mathematical Model for a Lithium -Sulfur Cell[END_REF][START_REF] Ghaznavi | Sensitivity analysis of a mathematical model of lithium e sulfur cells : Part II : Precipitation reaction kinetics and sulfur content[END_REF][START_REF] Ghaznavi | Sensitivity analysis of a mathematical model of lithium e sulfur cells part I : Applied discharge current and cathode conductivity[END_REF][START_REF] Ghaznavi | Electrochimica Acta Analysis of a Mathematical Model of Lithium-Sulfur Cells Part III : Electrochemical Reaction Kinetics , Transport Properties and Charging[END_REF]. Since we assume that the 𝐿𝑖 2 𝑆 (solid) film passivates the carbon particle surface, its active surface is calculated based on the film thickness using an electron tunnelling probability function modified from a model for Li-O 2 batteries [START_REF] Xue | Impact of the Cathode Microstructure on the Discharge Performance of Lithium Air Batteries : A Multiscale Model P + Q Air inlet[END_REF] by some of us

𝑎 1 = 𝑎 1 𝑚𝑎𝑥 (1 - (𝜖 1 ) 𝑆 8 𝜀 1 𝑚𝑎𝑥 ) 𝛽 Θ(𝛿 1 ) (4.23)
where Θ(𝛿 1 ) is the electron tunneling probability function,

Θ(𝛿 1 ) = 1 -𝑒𝑟𝑓 ( 𝛿 1 -𝛿 𝑡𝑢𝑛 𝐿 𝑡𝑢𝑛 ) 2 (4.24)
𝛿 1 comes from Eq. (4.18), 𝛿 𝑡𝑢𝑛 is a threshold thickness at halved electron tunnelling probability, and L tun is a scaling factor.

The passivation effect due to the loaded 𝑆 8(solid) is neglected, but the losses due to the passivation by the 𝐿𝑖 2 𝑆 (solid) film on the internal mesoporous surface and the choking of some mesopores entrances by the film formed over the external particle surface are both included.

The fraction of mesopores that are not choked are computed using the choking function (Eq 4.18).

𝑎 2 = 𝑎 2 𝑚𝑎𝑥 ( 𝑅 𝑝 -𝛿 2 𝑅 𝑝 ) 2 Θ(𝛿 2 )Ξ(𝛿 1 ) (4.25)
where the term

( 𝑅 𝑝 -𝛿 2 𝑅 𝑝 ) 2
characterizes the decrease of the mesoporous surface area (Fig. 4.4).

Similar to previous models [START_REF] Kumaresan | A Mathematical Model for a Lithium -Sulfur Cell[END_REF][START_REF] Ghaznavi | Sensitivity analysis of a mathematical model of lithium e sulfur cells : Part II : Precipitation reaction kinetics and sulfur content[END_REF][START_REF] Ghaznavi | Sensitivity analysis of a mathematical model of lithium e sulfur cells part I : Applied discharge current and cathode conductivity[END_REF][START_REF] Ghaznavi | Electrochimica Acta Analysis of a Mathematical Model of Lithium-Sulfur Cells Part III : Electrochemical Reaction Kinetics , Transport Properties and Charging[END_REF], the inter-particular porosity and the mesoporosity are evaluated based on the volume fraction of the solid precipitates within the pores. The precipitation rate of a solid precipitate 𝑘 in the inter-particular pores is,

𝜕(𝜖 1 ) 𝑘 𝜕𝑡 = ( 𝑀 𝑘 𝜌 𝑘 ) 𝜀 1 (𝑅 1 ) 𝑘 (4.26)
Similarly, the precipitation rate of a solid precipitate 𝑘 in the mesopores is,

𝜕(𝜖 2 ) 𝑘 𝜕𝑡 = ( 𝑀 𝑘 𝜌 𝑘 ) 𝜀 2 (𝑅 2 ) 𝑘 (4.27)
The evolution of the inter-particular porosity and the mesoporosity of the cathode, respectively, are calculated as follows, is the primary solid precipitate and the precipitation in the inter-particular pores may occur either through a solution phase route in the bulk electrolyte or a surface-limited thin film route (Fig. 4.4). The volume fraction of the 𝐿𝑖 2 𝑆 (solid) thin film in the inter-particular pores is given by

𝜀 1 = 𝜀 1 𝑚𝑎𝑥 -∑(
(𝜔 1 ) 𝐿𝑖 2 𝑆 = (𝜖 1 ) 𝐿𝑖 2 𝑆 (1 -𝜒) (4.30)
where 𝜒 is the escape function [START_REF] Xue | A Comprehensive Model for Non-Aqueous Lithium Air Batteries Involving Different Reaction Mechanisms[END_REF], the fraction of 𝐿𝑖 2 𝑆 (solid) precipitated as particles in the inter-particular pores. Assuming the film to cover the external surface of the carbon particle as a hollow sphere, its thickness is given by

(𝛿 1 ) 𝐿𝑖 2 𝑆 = ( 3((𝜔 1 ) 𝐿𝑖 2 𝑆 -𝜀 1 𝑚𝑎𝑥 + 1) 4𝜋𝜌 𝑐 ) 1 3
-𝑅 𝑝 The total volume fraction of the 𝐿𝑖 2 𝑆 (solid) produced in the mesopore is simply the volume fraction of the 𝐿𝑖 2 𝑆 (solid) film formed over its internal carbon surface:

(𝜔 2 ) 𝐿𝑖 2 𝑆 = (𝜖 1 ) 𝐿𝑖 2 𝑆 (4.32)
Similarly, the thickness of the 𝐿𝑖 2 𝑆 (𝑠𝑜𝑙𝑖𝑑) film formed inside the mesopore is given by

(𝛿 2 ) 𝐿𝑖 2 𝑆 = 𝑟 𝑝 -( 3(𝜀 1 𝑚𝑎𝑥 -(𝜔 1 ) 𝐿𝑖 2 𝑆 ) 4𝜋𝜌 𝑐 𝑁 𝑝 ) 1 3 (4.33)

Current balance

The sum of the Faradaic current densities of all the electrochemical reactions is set equal to the applied current density, I G (we neglect the double layer effect) [START_REF] Quiroga | A Multiscale Model of Electrochemical Double Layers in Energy Conversion and Storage Devices[END_REF] 𝐼 𝐺 = -∑ ∫(𝑎 1 (𝑖 1 ) 𝑗 + 𝑎 2 (𝑖 2 ) 𝑗 ) 𝐶 𝐵 𝑗 𝑑𝑥 (4.34) where the minus sign emerges as the anodic direction is positive in Eq. (4.9).

Cell voltage

The general form of the equilibrium potential expression for any electrochemical reaction is

𝑈 𝑗 = 𝑈 𝑗 0 - 𝑅𝑇 𝑛 𝑗 𝐹 ∑ 𝑆 𝑖𝑗 𝑖 ln((𝑐 1 ) 𝑖 ̅̅̅̅̅̅ ) (4.35)
where 𝑈 𝑗 0 is the standard potential of a electrochemical reaction 𝑗, (𝑐 1 ) 𝑖 ̅̅̅̅̅̅ is the avearge concentration of a species 𝑖.

As the concentration of 𝐿𝑖 (soln) + evolves significantly in Li-S batteries the anode equilibrium potential is hence determined using Nernst's equation for lithium metal oxidation

𝑈 1 = 𝑅𝑇 𝐹 ln((𝑐 1 ) 𝐿𝑖 ̅̅̅̅̅̅̅ ) (4.36)
Similarly, the cathode equilibrium potential is calculated based on the 𝑆 2(soln) (4.37)

The cell voltage is the difference between the cathode and anode equilibrium potentials, plus the cathode overpotential

𝑈 𝑐𝑒𝑙𝑙 = 𝑈 6 -𝑈 1 + 𝜂 (4.38)
The cell potential drop due to electrolyte resistance during the high plateau and the intermediate stage between the discharge plateaus has been found to be more significant than the activation overpotential [START_REF] Zhang | Modeling the voltage loss mechanisms in lithium-sulfur cells: the importance of electrolyte resistance and precipitation kinetics[END_REF]. However, EIS studies carried out by Deng et al. [START_REF] Deng | Electrochemical Impedance Spectroscopy Study of a Lithium/Sulfur Battery: Modeling and Analysis of Capacity Fading[END_REF],

showed that the charge transfer resistance, corresponding to the activation overpotential, and the surface film resistance, are much larger than electrolyte resistance throughout the discharge event. This suggests that the cell potential can be limited by charge transfer and surface resistance even when the electrolyte conductivity is adequate.

Therefore, it is reasonable to assume that the cell potential drop is dictated by the cathode overpotential and as the latter is determined by inverting the overall current balance equation it also includes the mass transport and surface passivation effects, via our model's strong coupling of these phenomena.

Computational implementation

The model was implemented in MATLAB. The coupled partial/ordinary differential equations were solved using the finite volume method (cf. Appendix A.4).

All simulations were carried out on a PC equipped with four 3.30 GHz processors Intel ® core™ i5-4590 with a typical simulation time of 1.5 to 2 days for a full polarization curve.

The discharge simulations are stopped when either one of these conditions is satisfied:

(i) all the dissolved sulfur species get completely reduced to 𝑆 (soln)

2-

in which case the discharge capacity equals the theoretical capacity;

(ii) a clogging of pores due to 𝐿𝑖 2 𝑆 (solid) precipitation prevents the transport of active species through the cathode and cause the loss of active surface area;

(iii) a complete passivation by the 𝐿𝑖 2 𝑆 (solid) thin film, defined as a thickness beyond the electron tunneling threshold.

Physical model

A cathode with 70% porosity is assumed as a reference case where the areal loading of sulfur is 2.48 mg.cm -2 and the inter-particular porosity constitutes 25% of the volume, whereas the mesoporosity accounts for 45%. The C/S volume ratio is 1:1. 70% of the total volume of the impregnated 𝑆 8(𝑠𝑜𝑙𝑖𝑑) exists inside the mesopores. a Assumed parameters. b Taken from the TEM image in Ref. [START_REF] Jayaprakash | Porous hollow carbon@sulfur composites for high-power lithium-sulfur batteries[END_REF]. c Calculated based on the assumed inter-particular porosity and mesoporosity.

Results and discussion

A systematic study of the effects of the battery operation and C/S microstructural properties on the discharge is presented in this section. Only those parameters whose effects are being investigated are changed for the simulations reported in each subsection. This way it is possible to track the unique roles of several experimentally modifiable parameters singlehandedly, paving the way for a rational design of more performant Li-S cells in general and C/S cathodes in particular.

Rate capability

The rate capability of the reference cathode microstructure is investigated by simulating discharge using three different current densities: 0.5, 1.0 and 1.5 mA.cm -2 (Fig. 4.5). • The stage IV corresponds to loss of active surface area and porosity in the cathode;

• Finally, the discharge simulation stops when the inter-particular porosity falls to zero (Fig. 4.7).

These calculated concentration profiles are almost consistent to those observed in the Ref. [START_REF] Kumaresan | A Mathematical Model for a Lithium -Sulfur Cell[END_REF]. The predicted solid species evolution during discharge is also consistent with the crystalline in situ XRD study [START_REF] Ca??as | In-situ X-ray diffraction studies of lithiumsulfur batteries[END_REF], which suggest that reaction mechanisms in our model are reasonable. During discharge, the total concentration of polysulfides will reach a very high value (≈ 4.8 M), but the lithium ion concentration will be more than twice this value. Since the production and consumption rate of electrons in the anode and the cathode, respectively, are identical, the amount of positive and negative charges produced in the cell should also be identical.

Even though our model does not have any constraints for electroneutrality, the charge difference between the lithium ions and polysulfide ions minus the initial lithium salt anions is found to be negligible for the complete cell.

At the end of discharge the 𝐿𝑖 2 𝑆 (solid) film thickness still allows for electron tunneling, but the inter-particular porosity of the cathode bin closest to the separator falls to zero due to clogging by 𝐿𝑖 2 𝑆 (solid) (Fig. 4.7(a)).

Impact of porosity

The discharge dependence on porosity is simulated by using different C/S cathodes. The C/S volume ratio is 1:1 for all three configurations, while the sulfur loading is reduced as the porosity is increased (Table 4.4). The mesoporosity is increased by increasing the number of mesopores per particle, while the inter-particular porosity is increased by decreasing the particle number density and in all the cathodes 70% of the total impregnated sulfur volume exists in the mesopores. reduction reaction is faster when the activation energy is low ( 𝐸 <𝑆 2 2-/𝑆 2-> =3.0 kJ.mol -1 ). Therefore, the discharge capacities of all the cathodes decrease when 𝐸 <𝑆 2 2-/𝑆 2-> is decreased (Fig. 4.9), due to faster precipitation of 𝐿𝑖 2 𝑆 (solid) and earlier clogging of inter-particular pores closest to the separator (Fig. 4.10(a)).

In general, the cathode surface area increases as the mesoporosity increases, which facilitates better active material utilization and hence, as expected, the discharge capacity increases.

However, surprisingly, the increase in discharge capacity is slightly pronounced for the cathode with 35% maximum inter-particular porosity despite its surface area being the lowest among three due its low particle number density (Table 4.4). The slight increase in the discharge capacity is achieved through its large transport path for the soluble active species, thereby facilitating better access to the carbon surface throughout the cathode. This effect is more pronounced for the discharge curve simulated using high activation energy

( 𝐸 <𝑆 2 2-/𝑆 2-> =3.6 kJ.mol -1 ). Under such slow 𝑆 2(soln) 2- /𝑆 (soln) 2-
reduction kinetics, the active species possess ample time to get transported across the cathode. Therefore, unlike the other two cathodes, with 25% maximum inter-particular porosity, the electrochemical and precipitation reactions take place isotropically in this cathode. This way the inter-particular porous network does not get clogged (Figs. 10a andb), facilitating the active species access to the entire carbon surface, resulting in a complete utilization. Due to the low particle number density, the inter-particular surface area is low for the cathode with larger particles (Table 4.5) and the 𝐿𝑖 2 𝑆 (solid) film thickness grows at a faster rate (Fig. 4.12), producing an earlier choking of the mesopores with some unutilized active species trapped inside and hence a slightly lowered discharge capacity. However, for the case 𝜒 = 0.3, 70% of the 𝐿𝑖 2 𝑆 (solid) precipitates as a thin film, causing a rapid growth of its thickness in the cathode with larger particles, producing a much earlier choking of mesopores and a substantial decrease in the discharge capacity. The potential at the latter part of the discharge decreases for the cathode with large particles which is caused due to the larger passivation effect produce by the thick 𝐿𝑖 2 𝑆 (solid) film. We simulate the discharge profiles of the cathodes with different C/S volume ratios (Table

Conclusions and perspectives

Our comprehensive multi-scale model allows to investigate the effects of the practical and experimentally modifiable operation and C/S microstructure properties on the discharge performance. The analysis of the calculated results reveals the physical reasons that limit the discharge capacities. The results qualitatively agree with the experimentally observed trends for example, the discharge capacities decrease when applied current density [START_REF] Jayaprakash | Porous hollow carbon@sulfur composites for high-power lithium-sulfur batteries[END_REF][START_REF] Shim | The Lithium Õ Sulfur Rechargeable Cell Effects of Electrode Composition and Solvent on Cell Performance[END_REF][START_REF] Cheon | Rechargeable Lithium Sulfur Battery[END_REF] and 𝑆 8(solid) volume fraction are increased [START_REF] Ding | Key parameters in design of lithium sulfur batteries[END_REF]. With the qualitatively validated results, our model is used to perform a prospective study of the roles of various microstructural design parameters, such as the inter-particular porosity, the mesoporosity, the particle and pore sizes, etc. on the final cell performance. A sensitivity analysis of the activation energy of the 𝑆 2(soln)

2- /𝑆 (soln) 2-
reduction reaction shows that the discharge capacity decreases as the activation energy is increased. However, the predicted discharge capacity trends and the capacity limiting mechanisms for different cathode designs remain the same. The discharge capacity of the cathode with the larger particles decreases substantially as the escape function is decreased, since its capacity is mainly limited by the choking of mesopores by the 𝐿𝑖 2 𝑆 (solid) film formed over the carbon particles. In all the other cathode designs, the discharge capacities are shown to be affected by clogging of inter-particular pores, caused by the slow transport of dissolved species.

The cathode with 35% maximum inter-particular porosity, 45% mesoporosity and 1:1 C/S ratio provided the highest capacity. This result suggest that the discharge capacity and rate capability can be improved by increasing the inter-particular porosity. This is because, the dissolved species transport through the cathode and the accommodation of the excess volume of the 𝐿𝑖 2 𝑆 (solid) precipitates are improved by increasing the inter-particular porosity.

Furthermore, our modelling results show that the accumulation of 𝐿𝑖 2 𝑆 (solid) precipitates in the inter-particular pores, could be reduced by increasing the mesoporous surface area of the carbon particles. Therefore, carbon particles with high mesoporous surface area, can inhibit the early clogging inter-particular pores in the cathode, thereby resulting in the increase of its capacity. The mesoporous surface area can be increased by reducing the mesopore size and by increasing the number of mesopores.

Our discharge modelling results show that the discharge capacity, can also be improved by reducing the particle size and by increasing the number of carbon particles. Since, these aforementioned design modifications will result in the increase of the inter-particular surface area and produce thin 𝐿𝑖 2 𝑆 (solid) layer over the carbon surface which do not choke the mesopores.

Finally, our modelling result show that the discharge capacity decreases when the sulfur loading of the cathode is increased. The reduction in the discharge capacities are due to the clogging of inter-particular pores and chocking mesopores. Therefore, an appropriate cathode design solution such as increasing of inter-particular and decreasing particle size, could assist in improving the discharge capacity of the cathodes with high 𝑆 8(solid) loading.

A.4. Appendix: Numerical Schemes

The discretized form of the Eq. (4.13) is derived by applying a second order finite volume method as follows

(𝐼𝑑 - ∆𝑡 2(∆𝑥) 2 𝑀 𝑛+1 ) (𝑐 1 𝜀 1 ) 𝑛+1 = (𝐼𝑑 + ∆𝑡 2(∆𝑥) 2 𝑀 𝑛+1 ) (𝑐 1 𝜀 1 ) 𝑛 + ∆𝑡(𝑔 1 𝑛 -𝐺 1 𝑛 + 𝑓 𝑛 ) (A.4.1)
where

( ) ( ) ( ) ( ) ( ) ( ) ( ) . 2 2 2 2 1 5 . 1 1 , 1 5 . 1 , 1 1 5 . 1 1 , 1 5 . 1 , 1 1 5 . 1 1 , 1 1 5 . 1 1 , 1 5 . 1 , 1                             +         + + -         + = + + + + - - - -        n i n i n i n i n i n i n i n i n i n i n i n i n i n i n D D D D D D D M       
Here 𝐼𝑑 denotes the identity matrix, 𝑖 and 𝑛 respectively are the spatial and temporal indices, ∆𝑥 is the time step and ∆𝑥 is the length of the control volume element.

At the anode/separator interface (at 𝑥 = 0 in Fig. 4.1) the fluxes of all the dissolved species is zero, except for 𝐿𝑖 (𝑠𝑜𝑙𝑛) + which is given by

(𝑁 1 𝑛 ) 𝑥=0 = 𝑖 1 𝐹 (A.4.2)
where is the Faradaic current density of the lithium oxidation reaction at the anode (Table 4.1).

The fluxes for all the dissolved species are continuous at the separator/cathode (at 𝑥 = 𝐿 𝑠𝑒𝑝 ) interface is continuous, therefore fluxes of the species entering the cathode are equal to the those leaving the separator, which is given as

(𝑁 𝑖 𝑛 ) 𝑠𝑒𝑝 = (𝑁 𝑖 𝑛 ) 𝑐𝑎𝑡 = (𝜀 𝑃+ 1 2 𝑛 ) 1.5 𝐷 𝑃+ 1 2 𝑛 ( 𝜕𝑐 𝜕𝑥 ) 𝑃+ 1 2 𝑛 𝑎𝑡 𝑥 = 0 (A.4.3)
where 𝑃 + 1 is the cathode bin closest to the separator (Fig. 4.1).

Finally, at the cathode/current collector interface (𝑥 = 𝐿 𝑠𝑒𝑝 + 𝐿 𝑐𝑎𝑡 ) the fluxes of all the dissolved species is equal to zero.

Chapter 5. Cyclic voltammetry model

Background and motivation

In our discharge model presented in chapter 4, an arbitrary electrolyte was assumed to fill the pores of the cathode and the separator. Therefore, a conventionally used reaction mechanism was considered for our discharge [START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF][START_REF] Zhang | Modelling transport-limited discharge capacity of lithium-sulfur cells[END_REF][START_REF] Ghaznavi | Sensitivity analysis of a mathematical model of lithium-sulfur cells part I: Applied discharge current and cathode conductivity[END_REF][START_REF] Zhang | What Limits the Rate Capability of Li-S Batteries during Discharge: Charge Transfer or Mass Transfer?[END_REF][START_REF] Ghaznavi | Analysis of a Mathematical Model of Lithium-Sulfur Cells Part III: Electrochemical Reaction Kinetics, Transport Properties and Charging[END_REF]. However, hence forth we have used 1 M LiTFSI in TEGDME:DOL as the electrolyte in our experiments. Therefore, in this chapter we have investigated the validity of the conventional reaction involving solid and dissolved sulfur based species in our experimental electrolyte.

The Cyclic voltammetry is an electrochemical potential sweep technique and it has been extensively used to analyse dissolved polysulfides in nonaqueous solvents for several decades [START_REF] Merritt | Electrochemical reduction of elemental sulfur in aprotic solvents. Formation of a stable S8-species[END_REF][START_REF] Xie | Studies on electroreduction mechanism of sulfur in aprotic medium II: Researches of the second redox step[END_REF][START_REF] Xie | Studies on electroreduction mechanism of sulfur in aprotic medium I: Researches of the first redox step[END_REF][START_REF] Kim | In situ spectroelectrochemical studies on the reduction of sulfur in dimethyl sulfoxide solutions[END_REF][START_REF] Han | Time-Resolved In Situ Spectroelectrochemical Study on Reduction of Sulfur in N,N[sup ʹ]-Dimethylformamide[END_REF][START_REF] Gaillard | Visible time-resolved spectroelectrochemistry: application to study of the reduction of sulfur (S8) in dimethylformamide[END_REF][START_REF] Zou | Solvent-Dictated Lithium Sulfur Redox Reactions: An Operando UV-vis Spectroscopic Study[END_REF]. Cyclic voltammograms (CVs) of dissolved sulfur (S 8 ) in nonaqueous organic solvents, have long been known to possess two distinctive reduction peaks sometimes with a small intermediate peak termed prewave in between them which is observed in some electrolytes [START_REF] Merritt | Electrochemical reduction of elemental sulfur in aprotic solvents. Formation of a stable S8-species[END_REF][START_REF] Kim | In situ spectroelectrochemical studies on the reduction of sulfur in dimethyl sulfoxide solutions[END_REF][START_REF] Han | Time-Resolved In Situ Spectroelectrochemical Study on Reduction of Sulfur in N,N[sup ʹ]-Dimethylformamide[END_REF][START_REF] Jung | Effect of organic solvents and electrode materials on electrochemical reduction of sulfur[END_REF][START_REF] Manan | Electrochemistry of Sulfur and Polysulfides in Ionic Liquids[END_REF]. However, CV of S 8 in 1M LiTFSI in Diox:DME consists only one reduction peak (Fig. 5.1) [START_REF] He | Unraveling the Correlation between Solvent Properties and Sulfur Redox Behavior in Lithium-Sulfur Batteries[END_REF]. While, the number of oxidation peaks vary from one to three based on the solvents used in the electrolyte solutions (Fig. 5.1 and Table 5.1) [START_REF] He | Unraveling the Correlation between Solvent Properties and Sulfur Redox Behavior in Lithium-Sulfur Batteries[END_REF]. Due to these significant characteristics variations, CVs of dissolved S 8 in different electrolytes were used to compare the impact of solvent properties such as donor number [START_REF] Zou | Solvent-Dictated Lithium Sulfur Redox Reactions: An Operando UV-vis Spectroscopic Study[END_REF], acceptor number and dielectric constant, on the stabilities and reaction mechanisms of polysulfides [START_REF] He | Unraveling the Correlation between Solvent Properties and Sulfur Redox Behavior in Lithium-Sulfur Batteries[END_REF]. The changes in the characteristics of CVs in different electrolyte can also be correlated to the changes in the galvanostatic curves such as the separation of discharge plateaus, charge/discharge polarization, etc. [START_REF] He | Unraveling the Correlation between Solvent Properties and Sulfur Redox Behavior in Lithium-Sulfur Batteries[END_REF][START_REF] Zou | Solvent-Dictated Lithium Sulfur Redox Reactions: An Operando UV-vis Spectroscopic Study[END_REF]. Furthermore, characteristics variations of CVs with the operation conditions in some individual electrolytes were also used to deduce the reaction mechanisms of S 8 and polysulfides. the second reduction peak using Spectro electrochemical measurements and proposed a cyclic voltammetry model which considers S 4 *-and S 3 *-as the only reducible polysulfides that produce S 4 2-and S 3 2-respectively, which could then be deoxidized [START_REF] Gaillard | Polysulfides in dimethylformamide: Only the radical anions S3-and S4-are reducible[END_REF]. Gaillard et al., also studied the CV of Li 2 S 6 in NH 3 , in which two reductions peaks were observed, one of which increases with the temperature and decreases with the scan rate while the other exhibited the opposite trend [START_REF] Levillain | Reduction of S-3 and S2-6 polysulfide ions in liquid ammonia[END_REF]. The CV simulations using their proposed reaction mechanism have shown that the former peak corresponds the S 3 *-/S 3 2-redox reaction, where S 3 *-is produced by the dissociation S 6 2-at high temperatures, while the latter peak was assigned to the S 6 2- /S 6 3-redox reaction [START_REF] Levillain | Reduction of S-3 and S2-6 polysulfide ions in liquid ammonia[END_REF]. particularly attractive for commercial use due to its safety. DME has a very low flash point [START_REF] Hess | Flammability of Li-Ion Battery Electrolytes: Flash Point and Self-Extinguishing Time Measurements[END_REF], therefore the electrolyte becomes highly flammable when it is used [START_REF] Kaiser | Structure-Property Relationships of Organic Electrolytes and Their Effects on Li/S Battery Performance[END_REF]. On the contrary, TEGDME has a very high flash point [START_REF] Di Lecce | Rechargeable lithium battery using nonflammable electrolyte based on tetraethylene glycol dimethyl ether and olivine cathodes[END_REF], therefore it is much safer to use. A mathematical model is used to simulate and interpret the experimental CVs, using a same set of reaction steps and parameters. The objective of this chapter is to assess the reaction steps between the dissolved polysulfides that correspond to certain characteristics of CVs and their variation with operating conditions such as the dissolved polysulfides speciation and scan rates. Through this procedure, we have attempted to determine reaction mechanism of the dissolved polysulfides in 1 M LiTFSI in TEGDME:DOL.

It
The contents of this chapter will soon be submitted as a research paper to a peer-reviewed journal. The co-authors for the research paper in preparation are V. Thangavel, A. Mastouri, C. Guéry, M. Morcrette and A. A. Franco.

Experimental Section

Cell setup

A relatively simple three electrode cell setup was used to carry out cyclic voltammetry measurements of sulfur and polysulfides in 1 M LiTFSI in TEGDME:DOL (1:1) contained in a glass beaker (Fig. 5.2). A glassy carbon electrode with diameter 1.6 mm was used as the working electrode and a Pt grid was used as the counter electrode. [START_REF] Thangavel | HELIS project D6.2 deliverable report -databases for results produced using interfacial models[END_REF] It should be known that most of the previously reported cyclic voltammetry studies of dissolved S 8 and polysulfides were carried out using planar working electrode like glassy carbon [START_REF] He | Unraveling the Correlation between Solvent Properties and Sulfur Redox Behavior in Lithium-Sulfur Batteries[END_REF][START_REF] Han | Time-Resolved In Situ Spectroelectrochemical Study on Reduction of Sulfur in N,N[sup ʹ]-Dimethylformamide[END_REF][START_REF] Jung | Effect of organic solvents and electrode materials on electrochemical reduction of sulfur[END_REF][START_REF] Lu | Probing the Lithium-Sulfur Redox Reactions: A Rotating-Ring Disk Electrode Study[END_REF][START_REF] Zheng | Reduction mechanism of sulfur in lithium-sulfur battery: From elemental sulfur to polysulfide[END_REF][START_REF] Evans | The mechanism for the cathodic reduction of sulphur in dimethylformamide: low temperature voltammetry[END_REF], since it is considered to be a well behaved electrode [START_REF] Dibden | Quantitative Galvanostatic Intermittent Titration Technique for the Analysis of a Model System with Applications in Lithium-Sulfur Batteries[END_REF].

Furthermore, the porous carbon working electrode was not used in this study -since it introduces complexities such as anisotropic distribution of species concentrations, additional physics and parameters for modelling, etc. which impede the proper understanding of the polysulfides reaction mechanism in the electrolyte. Ag/AgNO 3 electrode was used as the reference electrode, which consists of an Ag wire immersed in a solution containing 10 mM of AgNO 3 and 0.1 M of TBAP in acetonitrile contained in a glass tube fitted with a frit at its extremity. Since the reference and working electrolytes are different in our experiment, a liquid junction potential difference (LJP) could exist between them. However, this LJP and the overall potential of our Ag/AgNO 3 reference electrode could be assumed to remain stable, since the frit restricts the diffusion of species across it and prevents leakage of reference electrolyte into the experimental one.

Cyclic voltammetry measurements

The entire cell setup was assembled inside a glove box under Ar atmosphere and sealed using a rubber lid. Cyclic voltammetry measurements were done outside the glove box using Biologic VMP3 with the temperature maintained at 25 o C. The potential between the working and reference electrodes was swiped, starting from and ending at an open circuit potential (OCV) after reaching a minimum value during the scan towards negative direction and a maximum valude for the scan towards positive direction, thereby constituting a cycle.

Theoretical Methodology

Simulated domain and governing equations

Since, the redox reactions in the experiment occur at the working electrode/electrolyte interface, the simulation domain consists only the diffusion layer (DL) at the vicinity of the working electrode (Fig. 5.3) whose thickness is given by [START_REF] Compton | Understanding voltammetry: simulation of electrode processes[END_REF],

𝛿 𝐷𝐿 = 6√𝐷 𝑚𝑎𝑥 𝑡 𝑚𝑎𝑥 (5.1)
where 𝐷 𝑚𝑎𝑥 is the diffusion coefficient of the species which has the fastest diffusivity (Table 5.1) and 𝑡 𝑚𝑎𝑥 is the total duration of the experimental cyclic voltammetry measurement. Since the working glassy carbon electrode is flat and planar, the diffusion fields of different species in the DL will have points that are equivalent at a given distance (𝑥) from the electrode surface [START_REF] Compton | Understanding voltammetry: simulation of electrode processes[END_REF]. Therefore, the net flux of species will occur perpendicular to the electrode surface along the thickness of the DL and the distribution of a species (𝑖) concentration can be suitably described by a one-dimensional mass conservation equation, as follows

𝜕𝑐 𝑖 𝜕𝑡 = 𝜕 𝜕𝑥 (𝐷 𝑖 𝜕𝑐 𝑖 𝜕𝑥 ) -𝑠 𝑖 -𝐺 𝑖 (5.2)
Where 𝑐 𝑖 and 𝐷 𝑖 are the concentration and diffusion coefficient of species 𝑖, while 𝑠 𝑖 and 𝐺 𝑖 respectively are its sink/source terms, brought about by some homogeneous chemical and heterogeneous dissolution/precipitation reactions.

𝑠 𝑖 = ∑ 𝜗 𝑖𝑗 𝑘 𝑗 𝑓 (∏((𝑐 𝑖 ) 𝜗 𝑖𝑗 ) 𝜗 𝑖𝑗>0 𝑖 -𝐾 𝑗 𝑒𝑞 ∏((𝑐 𝑖 ) -𝜗 𝑖𝑗 ) 𝜗 𝑖𝑗<0 𝑖 ) 𝑗 (5.3)
In the equation above, 𝜗 𝑖𝑗 is the stoichiometric coefficient for the species 𝑖 involved in the homogenous chemical reaction 𝑗 (Eq. 5.23 or 5.25). 𝑘 𝑗 𝑓 and 𝑘 𝑗 𝑟 are forward and reverse kinetic constants. 𝐾 𝑗 𝑒𝑞 is the equilibrium constant of the chemical reaction 𝑗, which is given by

𝐾 𝑗 𝑒𝑞 = 𝑘 𝑗 𝑟 𝑘 𝑗 𝑓 (5.4)
The bulk concentrations of all the species (𝑐 𝑖 * ) are assumed to remain constant and the concentrations of species at the boundary 𝑥 = 𝛿 𝐷𝐿 are equal to those of the bulk (Eq. 5.5).

(𝑐 𝑖 ) 𝑥=𝛿 𝐷𝐿 = 𝑐 𝑖 *

(5.5)

while the flux of a species (𝑁 𝑖 ) at the electrode surface (𝑥 = 0) is given by the Faraday's law where 𝑎 0 and Θ(𝛿) are the initial specific surface area of the electrode and the electron tunnelling probability function [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF],

(
Θ(𝛿) = 1 -𝑒𝑟𝑓 ( 𝛿 -𝛿 𝑡𝑢𝑛 𝐿 𝑡𝑢𝑛 ) 2 (5.14)
In Eq. 5.14, 𝛿 𝑡𝑢𝑛 and 𝐿 𝑡𝑢𝑛 respectively are threshold thickness at halved electron tunnelling and scaling factor. 𝛿 is the total thickness of sulfur based solid deposits on the working electrode,

𝛿 = ∑ 𝜖 𝑘 𝑘 𝑎 𝑚𝑎𝑥
(5.15)

Computational implementation

The model was implemented in the MATLAB software and the coupled partial differential equations were solved spatially using the Finite Volume Method and temporally using the Crank-Nicholson method (see Appendix -A3). As mentioned in the introduction, only a single set of parameter values were used for all the simulations (Table 5.1). The CV characteristics are very sensitive to kinetic and diffusion parameters, however well-defined experimental values are not available for parameters such as exchange current densities, charge transfer coefficients of different polysulfides reactions in 1 M LiTFSI in TEGDME:DOL, etc. Therefore, the initial values of these parameters were determined using an automatized MATLAB program which matches the simulated first reduction curves of different polysulfides with their corresponding experimental results using a Nonlinear least-squares solver called lsqnonlin. Only two or three parameters whose initial values were determined after each run of this automatized MATLAB program. These initial values of parameters were further adjusted until we were able to match all the simulated results as close as possible with the experimental ones. The parameters chosen for the cyclic voltammetry model are given in Table 5.1. Assumed parameters taken from Refs. [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF], [START_REF] Jung | Effect of organic solvents and electrode materials on electrochemical reduction of sulfur[END_REF], [START_REF] Gaillard | Polysulfides in dimethylformamide: Only the radical anions S3-and S4-are reducible[END_REF] and [START_REF] Zhang | Modelling transport-limited discharge capacity of lithium-sulfur cells[END_REF] respectively. fExperimenal parameters taken from Ref. [START_REF] Drvarič Talian | Reactivity and Diffusivity of Li Polysulfides: A Fundamental Study Using Impedance Spectroscopy[END_REF]. m Measured parameter.

The concentration of the species in the diffusion layer are determined explicitly by solving a set of non-linear equations using Newton-Raphson method. If the initial concentrations of the dissolved species are zero, the solution of the non-linear equations will be infinity. Therefore, in our model we assume trace values for the concentrations of polysulfides which should not be present in the experimental electrolyte solution (Table A-5.1).

Determining reaction mechanism for the model

The experimental CV of 6 mM 𝑆 8 in 1 M LiTFSI in TEGDME:DOL recorded at 5 mV.s -1 , consists of two reduction (R 1 and R 2 ) and two oxidation (O 1 and O 2 ) peaks (Fig. 5.3). Initially, the simulation of the CV of dissolved 𝑆 8 at 5 mV.s -1 (Fig. 5.3), was done using the following set of reaction steps (Eq. 5. [START_REF] Berndes | Bioenergy, Land Use Change and Climate Change Mitigation[END_REF] The initial simulated CV of S 8 (Fig. 5.4), produced using the conventional reaction mechanism consists of two reduction peaks R 1 and R 2 and one oxidation peak O 1 . However, the oxidation peak O 2 is absent in it (Fig. 5.4). Furthermore, the experimental peak intensity of O 2 does not increase with the scan rate as much as the intensities of other peaks namely R 1 , R 2 and O 1 (Fig. 5.5). This suggests that the underlying reaction mechanism behind O 2 should implicate some electrochemical reactions which are coupled to other chemical reactions. Therefore, these aforementioned characteristics trends of O 2 cannot be reproduced by the cyclic voltammetry model using only the reaction steps (Eq. 5.16 to 5.22) considered in the conventional reaction mechanism. This conclusion was also arrived after an extensive parameter analysis. Various studies in the literature involving the spectroscopic analysis of dissolved polysulfides in the electrolyte, have shown that the stability of the polysulfide ions and radicals primarily depend on the properties of the solvents such as the donor numbers and dielectric constants [START_REF] Wujcik | Lithium Polysulfide Radical Anions in Ether-Based Solvents[END_REF][START_REF] Cuisinier | Radical or Not Radical: Revisiting Lithium-Sulfur Electrochemistry in Nonaqueous Electrolytes[END_REF]. More specifically, in-situ and operando Raman spectroscopy studies have proven that the polysulfide radicals such as 𝑆 4 * -and 𝑆 3 * -are stable in LiTFSI in TEGDME:DOL [START_REF] Hannauer | The Quest for Polysulfides in Lithium-Sulfur Battery Electrolytes: An Operando Confocal Raman Spectroscopy Study[END_REF][START_REF] Wu | In Situ Raman Spectroscopy of Sulfur Speciation in Lithium-Sulfur Batteries[END_REF]. Therefore, the assumed reaction mechanism of the cyclic voltammetry model was replaced with a comprehensive set of reaction steps (Eq. 5.16 to 5. After reversing the potential scanning direction and during the period leading up to the oxidation peak O 2 (-1.5 to -1.15 V) -the concentrations of S 2 2-and S 4 2-decrease (Fig. 5.8a) due to their oxidation which produce S 3 2-and S 6 2-respectively. These reactions are the reasons behind the observed increase in S 6 2-concentration. Whereas, the increase of S 3 *- concentration is due to the oxidation of S 3 2--which is evident from the S 3 *-/S 3 2-current peak at -1.15 V (Fig. 5.7a). At this potential (-1.15 V) -S 3 *-concentration start to decrease significantly (Fig. 5.8a). During the potential window between -1.5 to -0.9 V, the concentrations of S 8 2-and S 4 *-increase (Fig. 5.8a) due to the oxidation of S 6 2-and simultaneous dissociation S 8 2-. Beyond -0.9 V, concentrations of all the polysulfides ions and radicals decrease, while the S 8 concentration increases (Fig. 5.8a). This indicates that all the polysulfides simultaneously undergo oxidation -to produce S 8 . This is also evident from the individual current peaks of S 4 *-/S 4 2-, S 6 2-/S 4 2-, S 8 2-/S 6 2-and S 8 /S 8 2-redox reactions around -0.6 V (Fig. 5.7a), which form the basis for the oxidation peak O 1 (Fig. 5.6).

In summary, oxidations of S 2 2-, S 3 2-and S 4 2-ions occur during O 2 , whereas S 4 *-, S 6 2-and S 8 2-in addition to all the aforementioned polysulfides undergo oxidation during O 1 . After switching the potential scanning direction once again, reduction of S 8 takes place which is evident from the decrease in their concentration and increase in the concentrations of all the other polysulfides (Fig. 5.8b).

Throughout the reduction and oxidation scans of the cyclic voltammetry simulation, the Li + concentration and volume fractions of S 8(s) and Li 2 S (S) at the electrode surface do not change in any significant way (Fig. 5.9). This suggests that the depositions of S 8(s) and Li 2 S (S) over the working electrode surface are negligible. It is important to note that the visibility of O 2 around -1.15 V in the simulated of CV of S 8 produced at 5 mV.s -1 (Fig. 5.6), is primarily due to the intense peak of S 3 *-/S 3 2-reaction current at that potential (Fig. 5.7a). Since the intensity of S 3 *-/S 3 2-current peak is kinetically controlled by the dissociation of S 6 2-, it does not increase when the potential scan rate is increased (Fig. solution (potential scan rate of 500 mV.s -1 ).

CV of dissolved Li 2 S 8

In order to simulate the CVs of Li 2 S 8 -the initial concentration of S 8 2-was set to 50 mM (Table 5.2). Furthermore, the characteristics of the simulated CVs (Fig. 5.12b) -qualitatively resemble the experimental ones (Fig. 5.12a). This suggests that the characteristics of the CV primarily depend on the type of the polysulfide species present in the solution.

In general, the intensities of all the distinctive peaks (R 1 , R 2 and O 1 ) in the CVs of Li 2 S 8 are higher (Fig. 5.12a) than those of S 8 (Fig. 5.5 and 5.10). This is due to the increase of polysulfides concentrations in the Li 2 S 8 solution. Furthermore, the characteristics of the first CV of Li 2 S 8are also different from those of S 8 , since the intensity of its reduction peak R 1 is significantly lower than that of the R 2 (Fig. 5.12a and b). The O 2 peak is not visible in the CV of dissolved Li 2 S 8 , due to the fast scan rate of 100 mV.s -1 . During the first oxidation scan after the reduction concentrations of S 2 2-and S 3 2-decrease due to their oxidations between -1.70 to -1.10 V (Fig. 5.13b). Furthermore, the S 3 *-/S 3 2-reaction current form an intense peak around -1.10 V (Fig. 5.14a), which is when the concentration of S 3 2-starts to decrease significantly due its oxidation (Fig. 5.13b). Whereas, S 4 2-, S 6 2-, S 8 2-, etc.

simultaneously undergo oxidations around -0.6 V -to produce S 8 . Due to these reactions the concentrations of these polysulfides decrease significantly around -0.6 V, while that of the S 8 increases (Fig. 5.13b). This can also be understood from the positive peaks of S 4 *-/S 4 2-, S 6 2-/S 4 2- , S 8 2-/S 6 2-and S 8 /S 8 2-reaction currents around -0.6 V (Fig.

Conclusions

CV investigations of different polysulfides solutions, using our relatively simple cell setup have shown that the complex characteristics of the cyclic voltammogram depend primarily on the type and concentration of the dissolved polysulfide in the electrolyte. The initial polysulfides speciation in the electrolyte, will influence the characteristics of CV, even if the Pt counter electrode is replaced with Li, which has been previously observed experimentally by Zheng et al. [START_REF] Zheng | Reduction mechanism of sulfur in lithium-sulfur battery: From elemental sulfur to polysulfide[END_REF]. The characteristics also depend on the scan rate and cycle number. The combined experimental and modelling investigations, have also assisted in determining the reactions steps that are required to explain the reasons behind the changes in the characteristics of the CV under wide range of experimental conditions (e.g. initial solvate polysulfide speciation and scan rate). There are various analytical spectroscopy tools such as the UV-vis, Raman, etc.

which assist in the measurement of the evolutions of different of polysulfides in the electrolyte during the operation of Li-S batteries [START_REF] Barchasz | Lithium/Sulfur Cell Discharge Mechanism: An Original Approach for Intermediate Species Identification[END_REF][START_REF] Zou | Solvent-Dictated Lithium Sulfur Redox Reactions: An Operando UV-vis Spectroscopic Study[END_REF][START_REF] Gorlin | Understanding the Charging Mechanism of Lithium-Sulfur Batteries Using Spatially Resolved Operando X-Ray Absorption Spectroscopy[END_REF][START_REF] Conder | Direct observation of lithium polysulfides in lithium-sulfur batteries using operando X-ray diffraction[END_REF]. However, the studies using these tools hypothesize reaction mechanism with large number of reaction steps. Therefore, modelling the CVs of dissolved sulfur and polysulfides can assist in verifying the possibility of these reaction steps.

The cyclic voltammetry simulation results have revealed that the visibility of the oxidation peak O 2 depends on the intensity of the S 3 *-/S 3 2-reaction current peak, thus it shows the presence of S 3 *-radical in 1 M LiTFSI in TEGDME:DIOX. This indicates that the modelling of the CV curves not only helps in identifying the initial speciation of dissolved polysulfide in the electrolyte, but it also assists in identifying the presence of some polysulfide intermediates as well. Simulation results have also shown, that the increase in intensity and appearance of CV peaks during the second cycle as in the case Li 2 S 8 and Li 2 S 6 solution respectively, are due to occurrence of additional electrochemical reactions and change in the polysulfide speciation at the end of the first cycle. This effect should be considered during the characterization of dissolved polysulfide using continuous cycling as in the study of effectiveness of polysulfide crossover blocking by an ion-selective membrane [START_REF] Li | Polysulfide-Blocking Microporous Polymer Membrane Tailored for Hybrid Li-Sulfur Flow Batteries[END_REF][START_REF] Doris | Understanding and controlling the chemical evolution and polysulfide-blocking ability of lithium-sulfur battery membranes cast from polymers of intrinsic microporosity[END_REF].

A.5 Appendix: Numerical Schemes

A.5.1. Finite volume method

The temporal unit mesh or time step is giving by

∆𝑡 = 𝑡 𝑛+1 -𝑡 𝑛 𝑡 𝑚𝑎𝑥 [A-5.1]
where 𝑡 𝑛+1 and 𝑡 𝑛 are the time at (𝑛 + 1) 𝑡ℎ and 𝑛 𝑡ℎ iteration respectively. 𝑡 𝑚𝑎𝑥 is the total simulation time.

The spatial unit mesh is given by

∆𝑥 = 𝑥 𝑘+1 -𝑥 𝑘 [A-5.2]
where 𝑘 refers to the nodal points (Fig At the nodal point 𝑘 = 1,

(𝑐 𝑖 ) 1 𝑛+1 -(𝑐 𝑖 ) 1 𝑛 ∆𝑡 = 𝐷 𝑖 (∆𝑥) 2 ( 4 3 (𝑐 𝑖 ) 2 -4(𝑐 𝑖 ) 1 + 8 3 (𝑐 𝑖 ) 0 ) + (𝑆 𝑖 ) 1 [A-5.6] (𝑆 𝑖 ) 1 = (𝑠 𝑖 ) 1 [A-5.7]
For nodal points from 𝑘 = 2 to 𝑁 𝑏𝑖𝑛 -1 The Crank-Nicholson method is used calculate the concentration of the species at (𝑛 + 1) 𝑡ℎ time step which is half implicit and is given by

(𝐼𝑑 𝑘×𝑘 - 𝐷 𝑖 ∆𝑡 2(∆𝑥) 2 𝑀 𝑘×𝑘 ) (𝑐 𝑖 ) 𝑘×1 𝑛+1 = (𝐼𝑑 𝑘×𝑘 + 𝐷 𝑖 ∆𝑡 2(∆𝑥) 2 𝑀 𝑘×𝑘 ) (𝑐 𝑖 ) 𝑘×1 𝑛 + ∆𝑡(𝑆 𝑖 ) 𝑘×1 𝑛+1 [A- 5.12]
Where,

𝑀 𝑘×𝑘 = ( -8 8 0 ⋯ ⋯ ⋯ ⋯ 0 8/3 -4 4/3 0 ⋮ ⋮ ⋮ ⋮ 0 1 -2 1 0 ⋮ ⋮ ⋮ ⋮ 0 1 -2 1 ⋱ ⋮ ⋮ ⋮ ⋮ 0 ⋱ ⋱ ⋱ 0 ⋮ ⋮ ⋮ ⋮ ⋱ 1 -2 1 0 ⋮ ⋮ ⋮ ⋮ 0 1 -2 1 0 ⋯ ⋯ ⋯ ⋯ 0 1 -2 ) [A-5.13] 𝐼𝑑 𝑘×𝑘 = ( 1 0 ⋯ ⋯ ⋯ ⋯ ⋯ 0 0 1 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 1 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 1 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 1 ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋱ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 1 0 0 ⋯ ⋯ ⋯ ⋯ ⋯ 0 1 ) [A-5.14]
The electrochemical reaction coefficients of final reaction steps are given by

𝑆 𝑖𝑗 = ( 0 0 0 0 0 0 0 0 -1/2 0 0 0 0 0 0 0 1/2 -1/2 0 0 0 0 0 0 0 2 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 3/2 1 -1/2 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 3/2 1/2 0 0 0 0 0 0 0 1 ) [A-5.15]
The homogeneous chemical reaction coefficients of final reaction steps (Eqs. 5. 23 and 5.25) are given by The seminal investigation of the Li 2 S precipitation phenomenon in Li-S batteries was carried out by Fan et al. [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF]. In this work, the authors performed potentiostatic discharge experiments using Li-S cells containing porous cathodes made up of carbon fibers and electrolyte solutions based on dissolved Li 2 S 8 . Typically, at a stage during the potentiostatic discharge, the current density of the cell increases to a peak before it decreases (Fig. 6.1a).

𝜗 𝑖𝑗 = ( 0 0 0 0 1 0 0 1 -2 0 0 0 0 -2 0 0 0 0 0 0 ) [A-5.
The SEM images show that the nucleation and growth of the Li 2 S deposits over the carbon surface occur during this stage (Fig. 6.1b to 6.1d). Fan et al. also observed that the Li 2 S nuclei and the current density peak do not appear if the constant discharge potential was above certain critical value (> 2.05 V) [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF]. Furthermore, they showed that the combined nucleation and growth rate constant depend on the constant discharge potential and the solvent type used in the electrolyte. Fleishman and Thirsk (BFT) models [START_REF] Bewick | Kinetics of the electrocrystallization of thin films of calomel[END_REF] and 3D instantaneous (3DI) and progressive (3DP)

Scharifker-Hills (SH) models [START_REF] Jafarian | Effect of potential on the early stage of nucleation and growth during aluminum electrocrystallization from molten salt (AlCl3-NaCl-KCl)[END_REF][START_REF] Scharifker | Theoretical and experimental studies of multiple nucleation[END_REF]. The experimental curves of the ether-based solvents matched well with the simulated results of the 2DI BFT model, while those of DMF and DMA matched well with the results of the 3DP SH model [START_REF] Li | Solvent-Mediated Li 2 S Electrodeposition: A Critical Manipulator in Lithium-Sulfur Batteries[END_REF].

It is important to note that the BFT and SH models can only simulate the current corresponding to the electrochemical growths of the existing nuclei [START_REF] Bewick | Kinetics of the electrocrystallization of thin films of calomel[END_REF][START_REF] Jafarian | Effect of potential on the early stage of nucleation and growth during aluminum electrocrystallization from molten salt (AlCl3-NaCl-KCl)[END_REF][START_REF] Scharifker | Theoretical and experimental studies of multiple nucleation[END_REF]. These models also assume that the electrochemical growths of the existing nuclei have fast charge transfer steps and are controlled by the mass transport of ions to the electrode surface.

Furthermore, BFT and SH models can be applied only to the potentiostatic discharge stage where the cell current density form a peak [START_REF] Bewick | Kinetics of the electrocrystallization of thin films of calomel[END_REF][START_REF] Jafarian | Effect of potential on the early stage of nucleation and growth during aluminum electrocrystallization from molten salt (AlCl3-NaCl-KCl)[END_REF][START_REF] Scharifker | Theoretical and experimental studies of multiple nucleation[END_REF]. These models have limited capability to investigate the precipitation of Li 2 S in Li-S batteries, since they cannot predict the reaction mechanism behind the nucleation and growth processes, particle sizes of the deposits, their coverage over carbon surface, etc. charge and discharge. However, both these aforementioned models assume that the nucleation of Li 2 S is driven by supersaturation of S 2-instead of overpotential. Therefore, they completely contradict the experimental observations of Fan et al. [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF] and Li et al. [START_REF] Li | Solvent-Mediated Li 2 S Electrodeposition: A Critical Manipulator in Lithium-Sulfur Batteries[END_REF].

Furthermore, none of the aforementioned nucleation and growth models have been applied

to simulate the potentiostatic discharge experiments.

Therefore, in this chapter we propose a different 1D model to simulate the potentiostatic discharge of a simple Li-S cell. Our model considers the electrochemical nucleation and growth of Li 2 S deposits over the carbon surface. The Li 2 S nucleation of our model was described using the classical electrochemical nucleation theory [START_REF] Pletcher | 9 -Electrocrystallisation[END_REF], which is driven by the overpotential and is capable of predicting the critical sizes of the nuclei.

Experimental Setup

The experimental work presented in this chapter was carried by Dr. Sara Drvarič Talian at National Institute of Chemistry (NIC) in Ljubljana, Slovenia. A two-electrode coffee bag cell was used for the electrochemical measurement (Fig. 6.2). A planar glassy carbon electrode with surface area 2 cm 2 was used as the cathode and a lithium metal foil was used as the anode. These two electrodes were separated using two glass fiber separators and an Ohara membrane, which was placed in middle (Fig. 6.2). The glass fiber separator on the anode side was wetted with 60 µL of the pure 1 M LiTFSI in TEGDME:DOL electrolyte. The glass fiber separator in the cathode side was wetted with 60 µL of the 100 mM electrolyte solution of Li 2 S 6 in 1 M LiTFSI in TEGDME:DOL. The Ohara membrane is an ion selective one which permits Li + to pass through it while blocking the crossover of polysulfides from the cathode to anode compartment. Therefore, this setup prevents the reactions of polysulfides with the lithium metal anode.

Theoretical methodology

Thermodynamics and kinetics of Li 2 S nucleation

As proposed by the experimental works of Fan et al. [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF] and Li et al. [START_REF] Li | Solvent-Mediated Li 2 S Electrodeposition: A Critical Manipulator in Lithium-Sulfur Batteries[END_REF] and modelling work of Ren et al. [START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF], the following Li 2 S nucleation reaction was assumed in our model,

8𝐿𝑖 + + 𝑆 4 2-+ 6𝑒 -→ 4𝐿𝑖 2 𝑆 (6.1)
According to the classical electrochemical nucleation theory, the overall homogenous free energy for the formation of Li 2 S nuclei (∆𝐺 ℎ𝑜𝑚𝑜 ) has two components [START_REF] Pletcher | 9 -Electrocrystallisation[END_REF],

∆𝐺 ℎ𝑜𝑚𝑜 = ∆𝐺 𝑏𝑢𝑙𝑘 + ∆𝐺 𝑠𝑢𝑓𝑎𝑐𝑒 (6.2)
where, ∆𝐺 𝑏𝑢𝑙𝑘 and ∆𝐺 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 are the bulk and surface free energies, respectively. Assuming that the Li 2 S nuclei are cubic, the bulk and surface energies are given by the Eqs. 6.3 and 6.4

∆𝐺 𝑏𝑢𝑙𝑘 = 𝑟 3 𝑛 𝑗 𝐹𝜂 𝑗 𝜌 𝑘 𝑀 𝑘 (6.3)
where, 𝑟 is the side length of the Li 2 S nucleus. 𝜌 𝑘 and 𝑀 𝑘 are the density and molar mass of the Li 2 S, respectively. 𝑛 𝑗 and 𝜂 𝑗 are the number of electrons and overpotential of the Li 2 S nucleation reaction (Eq. 6.1), respectively. Finally, 𝐹 is the Faraday's constant.

Δ𝐺 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 = 6𝑟 2 𝛾 (6.4)
Where, 𝛾 is the surface energy of the Li 2 S nucleus.

According to the Eqs. 6.2 to 6.3, Δ𝐺 ℎ𝑜𝑚𝑜 depends on the particle size of the nuclei (Fig. 6.3). Furthermore, nucleation of Li 2 S occurs when the particle size reaches a critical value, where the total homogenous free energy for the formation of nuclei is maximum (Eq. 6.5):

𝜕(Δ𝐺 ℎ𝑜𝑚𝑜 ) 𝜕𝑟 = 0 (6.5) Therefore, the critical particle size (𝑟 𝑐𝑟𝑖𝑡 ) and the critical homogenous free energy for the formation of the nuclei (Δ𝐺 𝑐𝑟𝑖𝑡 ℎ𝑜𝑚𝑜 ) can be derived by differentiating Eq. 6.2 with respect to 𝑟 and applying Eq. 6.5 to the differential equation.

𝑟 𝑐𝑟𝑖𝑡 = - 4𝑀 𝑘 𝑛 𝑗 𝐹𝜂 𝑗 (6.6) Δ𝐺 𝑐𝑟𝑖𝑡 ℎ𝑜𝑚𝑜 = 32𝑀 𝑘 2 𝛾 3 𝑛 𝑗 2 𝐹 2 𝜂 𝑗 2 𝜌 𝑘 (6.7)
Since the nucleation of Li 2 S occurs over the carbon electrode surface, its energy barrier is calculated using the heterogenous free energy (Δ𝐺 𝑐𝑟𝑖𝑡 ℎ𝑒𝑡𝑟𝑜 ) which is given as follows [START_REF] Yin | Linking the Performances of Li-O 2 Batteries to Discharge Rate and Electrode and Electrolyte Properties through the Nucleation Mechanism of Li 2 O 2[END_REF],

Δ𝐺 𝑐𝑟𝑖𝑡 ℎ𝑒𝑡𝑒𝑟𝑜 = Δ𝐺 𝑐𝑟𝑖𝑡 ℎ𝑜𝑚𝑜 (1 + 𝐸 𝑏𝑖𝑛𝑑 2𝛾𝑟 𝑐𝑟𝑖𝑡 2 ) (6.8)
where, 𝐸 𝑏𝑖𝑛𝑑 is the binding energy of the Li 2 S nuclei to the carbon surface.

Assuming first order kinetics, the kinetic rate of the Li 2 S nucleation reaction (𝑘 𝑛𝑢𝑐 ) is determined as follows,

𝑘 𝑛𝑢𝑐 = 𝑘 𝑛𝑢𝑐 0 𝑒𝑥𝑝 (- Δ𝐺 𝑐𝑟𝑖𝑡 ℎ𝑒𝑡𝑒𝑟𝑜 𝑘 𝐵 𝑇 ) (6.9)
where, 𝑘 𝑛𝑢𝑐 0 is the rate constant of the Li 2 S nucleation reaction. 𝑘 𝐵 and 𝑇 are the Boltzmann constant and the temperature, respectively.

Furthermore, the current of the Li 2 S nucleation reaction is calculated as follows,

𝑖 𝑛𝑢𝑐 = { -𝑛 𝑗 𝐹𝑘 𝑛𝑢𝑐 , 𝑟 𝑐𝑟𝑖𝑡 ≥ 𝑟 𝑚𝑖𝑛 0, 𝑟 𝑐𝑟𝑖𝑡 < 𝑟 𝑚𝑖𝑛 (6.10)
According to Eq. 6.10, the nucleation current is zero if the critical particle size is below a certain minimum positive value (𝑟 𝑚𝑖𝑛 ). In the classical electrochemical nucleation theory, the nucleation is driven by the overpotential (Eqs. 6.6 to 6.9). Furthermore, the overpotential should reach a certain critical value to produce stable nuclei due to the high energy barrier of the Li 2 S nucleation reaction. Since, the particle size depends on the overpotential (Eq. 6.6), 𝑟 𝑚𝑖𝑛 is considered to be the minimum size at which the Li 2 S nuclei are stable. The overpotential of the Li 2 S nucleation reaction (Eq. 6.1) is calculated as follows,

𝜂 𝑗 = 𝐸 𝑐 -
where, the number 4 is the stoichiometric coefficient of Li 2 S in the nucleation reaction (Eq.

6.1). 𝑎 𝑓𝑟𝑒𝑒 is the specific surface area of the free electrode surface available for the nucleation of Li 2 S. The parameters used in the classical nucleation theory of Li 2 S are given in Table 6.1. The dissolved Li 2 S 6 is the initial polysulfide species present in the electrolyte of the cathode compartment. Therefore, the following reaction steps (Eqs. 6.13 to 6.19) were considered to take place at the cathode surface.

𝑆 6 2-+ 𝑒 -⇌ 3 2 𝑆 4(𝑙) 2- (6.13) 𝑆 6 2-⇌ 2𝑆 3 * - (6.14) 1 2 𝑆 4(𝑙) 2-+ 𝑒 -⇌ 𝑆 2(𝑙) 2- (6.15)
𝑆 3 * -+ 𝑒 -⇌ 𝑆 3 2- (6.16) 𝑆 3 2-+ 𝑒 -⇌ 3 2 𝑆 2 2- (6.17) 1 2 𝑆 2(𝑙) 2-+ 𝑒 -⇌ 𝑆 (𝑙) 2- (6.18)
2𝐿𝑖 (𝑙) + + 𝑆 (𝑙) 2-⇌ 𝐿𝑖 2 𝑆 (𝑠) (6.19)
These aforementioned reaction steps (Eqs. 6.13 to 6.19) were determined using the cyclic voltammetry model (Chapter 3). Furthermore, we also consider the electrochemical growth of the existing Li 2 S nuclei in our model (Eq. 6.20).

8𝐿𝑖 + + 𝑆 4 2-+ 6𝑒 -→ 4𝐿𝑖 2 𝑆 (6.20)
Since, a two-electrode cell was used to in experiment, we also consider the reaction at the lithium metal anode (6.21).

𝐿𝑖 + + 𝑒 -⇌ 𝐿𝑖 (6.21)
An electrochemical reaction (Eqs. 6.13, 6.15-6.18 and 6.21), can be written in a generic form as follows,

∑ 𝑆 𝑖𝑗 𝑀 𝑖 𝑧 𝑖 = 𝑛 𝑗 𝑒 - 𝑖 (6.22)
Where 𝑀 𝑖 𝑧 𝑖 and 𝑒 -respectively are the symbols that represent a species 𝑖 and an electron. 𝑧 𝑖 is the charge of the species 𝑖. Furthermore, the current density of an electrochemical reaction is determined using the Butler-Volmer equation,

𝑖 𝑗 = 𝑖 𝑗 0 (∏ ((𝑐 𝑖 ) 𝑥=0 𝑆 𝑖𝑗 ) 𝑆 𝑖𝑗 >0 𝑖 𝑒 ( (1-𝛼 𝑗 )𝑛 𝑗 𝐹(𝐸-𝑈 𝑗 0 ) 𝑅𝑇 ) -∏ ((𝑐 𝑖 ) 𝑥=0 -𝑆 𝑖𝑗 ) 𝑆 𝑖𝑗 <0 𝑖 𝑒 ( -𝛼 𝑗 𝑛 𝑗 𝐹(𝐸-𝑈 𝑗 0 ) 𝑅𝑇 ) ) (6.23)
where, 𝑈 𝑗 0 and 𝑖 𝑗 0 respectively are the standard potential and exchange current density of an electrochemical reaction 𝑗 and 𝐸 is the electrode potential. 𝑐 𝑖 is the concentration of dissolved species in the electrolyte.

Eq. 6.13 is a homogenous chemical dissociation reaction of 𝑆 6 2-, whose kinetic equation is as follows,

𝑘 𝑗 𝑐ℎ𝑒𝑚 = 𝑘 𝑗 𝑓 (∏((𝑐 𝑖 ) 𝜗 𝑖𝑗 ) 𝜗 𝑖𝑗 𝑖 -𝐾 𝑗 𝑒𝑞 ∏((𝑐 𝑖 ) -𝜗 𝑖𝑗 ) 𝜗 𝑖𝑗<0 𝑖 ) (6.24)
where, 𝜗 𝑖𝑗 is the stoichiometric coefficient for the species 𝑖 involved in the homogenous chemical reaction 𝑗 (Eq. 6.14 where, 𝑁 𝑟 is the number density of Li 2 S deposits with size 𝑟 and 𝑎 𝑟 𝑔𝑟𝑜𝑤 is electroactive specific surface area around an individual deposit where the electrochemical growth reaction occurs.

Active surface areas and porosity

In our model the electroactive specific surface area available for an electrochemical reaction will vary based on the reactiontype. As mentioned before, the nucleation of Li 2 S is assumed to occur only in the free electrode surface which is uncovered by the existing Li 2 S deposits.

Therefore, the free active specific surface area of the Li 2 S nucleation reaction is determined as follows,

𝑎 𝑓𝑟𝑒𝑒 = 𝑎 0 -∑ 𝑁 𝑟 𝑟 2 (6.30)
where 𝑎 0 is the initial specific surface area of the deposit free electrode and 𝑟 2 is the basal surface area of an individual 𝐿𝑖 2 𝑆 deposit with size 𝑟.

The redox reaction in the solution phase can occur on the free electrode surface and on the surface of the Li 2 S deposits which permit tunnelling of electron through them. Therefore, the electroactive surface area of the solution phase redox reactions is calculated as follows,

𝑎 𝑗 = 𝑎 0 -∑ 𝑁 𝑟 𝑟 2 + ∑ 𝑁 𝑟 𝑟 2 Θ(𝑟) (6.31)
where, Θ(𝑟) is the electron tunnelling probability function,

Θ(𝑟) = 1 -𝑒𝑟𝑓 ( 𝑟 -𝛿 𝑡𝑢𝑛 𝐿 𝑡𝑢𝑛 ) 2 (6.32)
In Eq. 6.32, 𝛿 𝑡𝑢𝑛 and 𝐿 𝑡𝑢𝑛 are the threshold thickness at halved electron tunnelling and scaling factor, respectively.

As proposed by Fan et al. [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF], our model considers that the electrochemical Li 2 S growth reaction occurs at the triple phase boundary between the electrolyte, the deposit and the electrode surface. Furthermore, due to the finite nature of the electrode surface, some of the phase boundaries surrounding the existing deposits could be covered by the neighbouring deposits. Therefore, the electroactive specific surface area for the electrochemical Li 2 S growth reaction is determined using the following equation,

𝑎 𝑔𝑟𝑜𝑤 = (∑ 𝑁 𝑟 Δ𝑟 2 ) (1 -𝜃) (6.33)
where, Δ𝑟 2 is the change in the basal surface area of an individual Li 2 S deposit due to its growth. 𝜃 is the fraction of electrode surface which is covered by the existing Li 2 S deposits, which is calculated using Eq. 6.33:

𝜃 = ∑ 𝑁 𝑟 𝑟 2 𝑎 0 (6.34)
Due to the precipitation of Li 2 S over the cathode surface, the porosity of the glass fiber separator close to the cathode surface decreases. Therefore, this porosity (is given by,

𝜀 𝑥=0 = 𝜀 𝑥=0 0 -∑ 𝜖 𝑟 (6.35)
where 𝜀 𝑥=0 0 and 𝜖 𝑟 are the initial porosity of the glass fiber separator and volume fraction of the Li 2 S deposits with size 𝑟.

Mass balance and boundary conditions

The mass balance of a dissolved species in the cathode compartment is determined using the following equation,

𝜕(𝑐 𝑖 𝜀) 𝜕𝑡 = 𝜕 𝜕𝑥 (𝐷 𝑖 𝜀 𝛽 𝜕𝑐 𝑖 𝜕𝑥 ) -𝑠 𝑖 -𝐺 𝑖 (6.36)
Since 𝐿𝑖 + can pass through the Ohara membrane, its mass balance equation (Eq. 6.35) applies to the entire cell. In the Eq. 6.35, 𝐷 𝑖 is the diffusion coefficient of a species 𝑖 and 𝛽 is the The flux at the glassy carbon surface is given by the Faraday's law,

(𝑁 𝑖 ) 𝑥=0 = (𝐷 𝑖 𝜀 𝛽 𝜕𝑐 𝑖 𝜕𝑥 ) 𝑥=0 = 1 𝐴 0 ∑ 𝑆 𝑖𝑗 𝑖 𝑗 𝑛 𝑗 𝐹 𝑗 𝐴 𝑗 (6.39)
where 𝐴 𝑗 is a general symbol for the electroactive surface area for an electrochemical reaction 𝑗, which is determined by the product of the volume of the discretized bin closest to the electrode and the specific surface area of the reaction 𝑗 (Eqs. 6.30, 6.31 and 6.33).

Except for 𝐿𝑖 + , the flux of all the dissolved species at the separator/Ohara membrane interface (𝑥 = 𝐿 𝑐 ) at the cathode compartment is zero (Eq. 6.40).

(𝑁 𝑖 ) 𝑥=𝐿 𝑐 = (𝐷 𝑖 𝜀 𝛽 𝜕𝑐 𝑖 𝜕𝑥 )

𝑥=𝐿 𝑐 = 0 (6.40)

The flux of the 𝐿𝑖 + at the lithium anode surface (𝑥 = 𝐿 𝑐 + 𝐿 𝑎 ) is given by,

(𝑁 𝑖 ) 𝑥=𝐿 𝑐 +𝐿 𝑎 = (𝐷 𝑖 𝜀 𝛽 𝜕𝑐 𝑖 𝜕𝑥 ) 𝑥=𝐿 𝑐 +𝐿 𝑎 = 𝑖 𝐿𝑖 + 𝐿𝑖 ⁄ 𝐹 (6.41)
where 𝑖 𝐿𝑖 + 𝐿𝑖 ⁄ the current of the lithium oxidation reaction (Eq. 6.21). This current is equal to the total cathode current but has an opposite sign.

Finally, the total cathode current is calculated using the following equation,

𝐼 = ∑ 𝑖 𝑗 𝑗 𝐴 𝑗 (6.41)
𝑉 𝑘 (m -3 .mol) 2 2𝐿𝑖 + + 𝑆 2- ⇌ 𝐿𝑖 2 𝑆 (𝑠) (619) 5 ×10 -7d 10 d 𝐿𝑖 2 𝑆 (𝑠) 2.768×10 -4d
a Fitted parameters. b, c and d Assumed parameters taken from Refs. [START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF], [START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF] and [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF], respectively. e Experimental parameter taken from Ref. [START_REF] Drvarič Talian | Reactivity and Diffusivity of Li Polysulfides: A Fundamental Study Using Impedance Spectroscopy[END_REF].

The electrochemical measurement

Due to the presence of the dissolved Li 2 S 6 in the catholyte, the open circuit potential (OCV) of the coffee bag cell was around 2.30 V. Therefore, the cell was discharged galvanostatically to 2.0 V at C/20, in order to trigger the nucleation of Li 2 S. Immediately after this galvanostatic step, the cell was discharged potentiostatically at a constant potential of 2.0 V.

Results and discussion

6.5.1 The experimental vs the simulated electrochemical signals

The simulated current and potential vs. time curves produced using our nucleation and growth model have excellent semi-quantitative agreement with the experimental results (Fig. 6.4). During the galvanostatic stage the cell potential decreases to 2.0 V (Fig. 6.4). Subsequently the overall reduction current decreases monotonically during the initial stage of the potentiostatic discharge. After this stage, the overall reduction increases to a peak before it starts to decrease once again. Finally, the overall reduction current levels off during the later stage of the potentiostatic discharge.

The simulated dissolved species concentrations and the individual reaction currents

During the Galvanostatic stage, the 𝑆 6 2-and 𝑆 3 * -concentrations decrease (Fig. 6.5), while, the concentrations of all the other polysulfides increase. These effects indicate that the 𝑆 6

2-

reduction and dissociation reactions occur during this stage along with the subsequent reduction reactions of the resulting lower order polysulfides. Furthermore, 𝑆 6 2-/𝑆 4 2-is the most dominant reaction during the galvanostatic stage, since its current is almost equal to the constant discharge current (Fig. 6.6a). Additionally, the 𝑆 4 2-/𝑆 2 2-(Fig. 6.6a) and 𝑆 3 * -/𝑆 3 2-(Fig. 6.6b) reactions also occur during the same time due to the increase in their currents.

Towards the end of the galvanostatic stage, the nucleation current starts to increase when the cell potential is slightly below 2.02 V (Fig. 6.6a). As mentioned before the nucleation process in our model is driven by its overpotential due to the high energy barrier required for the formation of stable nuclei. Therefore below 2.02 V, the overpotential of the nucleation reaction facilitates the formation of the stable Li 2 S nuclei. Following this stage, the cell potential quickly drops to 2.0 V and the potentiostatic discharge starts. Following the continuous nucleation of the 𝐿𝑖 2 𝑆 deposits, the current of their electrochemical growth starts to increase (Fig. 6.6a). The increase of electrochemical 𝐿𝑖 2 𝑆 growth current is due to the increase of its electroactive specific surface area (Fig. 6.8), which correspond to the newly formed triple phase boundary surrounding the existing 𝐿𝑖 2 𝑆 deposits. The overall reduction current also increases during the same time. Therefore, the 𝑆 4 2-concentration decreases throughout the potentiostatic stage due to the nucleation and growth of the 𝐿𝑖 2 𝑆 (Fig. 6.5).

The nucleation current decreases after some time due to the significant decrease of the free carbon surface (Fig. 6.6a). In the other side, the current of the electrochemical growth of the 𝐿𝑖 2 𝑆 continues to increase and reaches a peak around the same time when the overall reduction current forms a peak (Fig. 6.4). Following this stage, the 𝑆 2-concentration starts to decrease (Fig. 6.5), which indicates that the chemical growth of the existing 𝐿𝑖 2 𝑆 nuclei also occurs simultaneously. During the same time, the overall and the individual reduction reaction currents start to decrease and they level off during the later stage of the potentiostatic discharge (Fig. 6.6).

Simulated evolutions of the Li 2 S deposits

During the potentiostatic stage, the total volume fraction of the 𝐿𝑖 2 𝑆 deposits starts to increase significantly at the same time when the overall reduction current starts to increase (Figs. 6.5 and 6.7a). However, the initial increase of the 𝐿𝑖 2 𝑆 (𝑠) volume fraction is relatively slow. It is because this stage primarily involves the formation of the 𝐿𝑖 2 𝑆 nuclei, which is evident from the increase in the peak intensities of the 𝐿𝑖 2 𝑆 particle size distribution (Fig. Following the decrease of the nucleation current (Fig. 6.6a), the 𝐿𝑖 2 𝑆 volume fraction increases at a relatively faster rate (Fig. 6.7a). This is due to the chemical and electrochemical growth of the existing nuclei which occur simultaneously along with the formation of the new nuclei. Therefore, the peak intensities and their positions of the 𝐿𝑖 2 𝑆 particle size distribution also increase significantly during this stage (Fig 6 .7b). The growth of the existing 𝐿𝑖 2 𝑆 deposits is much faster when the current of the electrochemical growth reaction increases (Figs. 6.6a and 6.7a). This is due to the increase in the electroactive specific surface area surrounding the newly formed Li 2 S deposits, over which the electrochemical growth reaction of the existing deposits occurs (Fig. 6.8). Furthermore, the peak positions of the Li 2 S particle size distributions increase significantly during this stage.

However, the current and the electroactive specific surface area of the electrochemical growth reaction start to decrease (Figs. [START_REF] Höök | Depletion of fossil fuels and anthropogenic climate change-A review[END_REF].6a and 6.8), when the Li 2 S coverage over the carbon surface increases over 40 % (Fig. 6.8). Finally, the increase of the Li 2 S volume fraction and the peak positions of the particle size distributions slow down significantly (Fig. 6.7), when the 𝐿𝑖 2 𝑆 coverage over the carbon surface increases over 80%. Beyond this point, the overall and the individual reaction currents level off (Fig. 6.6) due to the very low availability of electroactive surface area for the electrochemical reactions. growth and the coverage of the 𝐿𝑖 2 𝑆 deposits over the cathode during the galvanostatic and the potentiostatic discharge stages.

Conclusion and perspectives

In this chapter, we have proposed a comprehensive nucleation and growth model to simulate and investigate the 𝐿𝑖 2 𝑆 electrodeposition during the potentiostatic discharge of a Li-S cell.

The simulated results of our model have good qualitative agreements with the experimental results. Furthermore, the results of our model show that the 𝐿𝑖 2 𝑆 nucleation occurs below 2.02 V, during which its overpotential facilitates the formation of the stable nuclei. The overall current peak of the potentiostatic discharge stage is primarily dominated by the current of the electrochemical growth of the existing 𝐿𝑖 2 𝑆 deposits. This is consistent with the assumptions of the BFT and SH models [START_REF] Bewick | Kinetics of the electrocrystallization of thin films of calomel[END_REF][START_REF] Jafarian | Effect of potential on the early stage of nucleation and growth during aluminum electrocrystallization from molten salt (AlCl3-NaCl-KCl)[END_REF][START_REF] Scharifker | Theoretical and experimental studies of multiple nucleation[END_REF]. However, our model shows that the electrochemical growth of the 𝐿𝑖 2 𝑆 deposits is primarily limited by the electroactive cathode surface area rather than the transport of the dissolved species. This conclusion is consistent with the experimental observations of Fan et al. [START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF]. Note that this surface limitation to the 𝐿𝑖 2 𝑆 growth may not be true in the case of conventional Li-S cells with porous electrodes which have very large surface area. The growth of the 𝐿𝑖 2 𝑆 deposits are considered to be progressive, since our modelling results show that the nucleation and growth reactions occur simultaneously.

In the future, our nucleation and growth model could be incorporated into our discharge model to investigate the electrodeposition of the 𝐿𝑖 2 𝑆 in the practical Li-S batteries.

Furthermore, since the heterogenous free energy for the formation of 𝐿𝑖 2 𝑆 in our model depends on the binding and surface energies, it could be utilized to investigate the electrodeposition of the 𝐿𝑖 2 𝑆 from different electrolytes and over the surfaces of different cathode host materials.

A.6 Appendix: Numerical Schemes

A.6.1. Discretization of the population balance equation

The population balance of the Li 2 S deposits was solved using a discretization method adopted from the work previously carried out in our lab by Yin et al. [START_REF] Yin | Linking the Performances of Li-O 2 Batteries to Discharge Rate and Electrode and Electrolyte Properties through the Nucleation Mechanism of Li 2 O 2[END_REF]. In this method, the possible particle size range of the Li 2 S deposits were discretized into 𝐿 equal sized bins (Fig. A-6.1). The bin size is determined as follows (Fig. A-6.1),

∆𝑟 = 𝑟 𝑙+1 -𝑟 𝑙 = 𝑟 𝐿 -𝑟 1 𝐿 (A-6.1)
where, 𝑟 𝑙+1 and 𝑟 𝑙 are the particle sizes of the deposits in (𝑙 + 1) 𝑡ℎ and 𝑙 𝑡ℎ bin. Furthermore, 𝑟 1 and 𝑟 𝐿 are the smallest and the largest particle sizes, respectively in the possible assumed particles size range.

Furthermore, the time step is calculated as follows,

∆𝑡 = 𝑡 𝑛+1 -𝑡 𝑛 (A-6.2)
where, 𝑡 𝑛+1 and 𝑡 𝑛 are simulation time at (𝑛 + 1) 𝑡ℎ and 𝑛 𝑡ℎ time step.

In this method, the population balance equation is discretized in terms of number density (𝑁 𝑟 ) instead of number density per unit volume (𝑓(𝑣 𝑟 )) of the Li 2 S deposits of size 𝑟. Since 𝑁 𝑟 could be used directly in our model. Therefore, the discretized form of the population balance of the deposits in 𝑙 𝑡ℎ particle size bin is given as follows,

𝑁 𝑙 𝑛+1 -𝑁 𝑙 𝑛 ∆𝑡 = 𝑁 𝑙 ́+ ∆𝑁 𝑙 𝑡𝑜𝑡𝑎𝑙 ∆𝑡 (A-6.3)
where, 𝑁 𝑙 𝑛+1 and 𝑁 𝑙 𝑛 are the number densities of the deposits in the 𝑙 𝑡ℎ bin at the (𝑛 + 1) 𝑡ℎ and 𝑛 𝑡ℎ time step. 𝑁 𝑙 ́ is the nucleation source term for the particle number density of the deposits in 𝑙 𝑡ℎ bin. Furthermore, ∆𝑁 𝑙 𝑡𝑜𝑡𝑎𝑙 is the total change in the particle number density of the deposits in 𝑙 𝑡ℎ bin brought about by the Li 2 S growth and shrinkage reactions.

The change in the particle number density in the 𝑙 𝑡ℎ bin due to the growth of some of its deposits, whose size change from 𝑟 𝑙 to 𝑟 𝑙+1 , is computed as follows,

Δ𝑁 𝑙 + = { - ∆𝜖 𝑙 𝑟 𝑙 3 -𝑟 𝑙+1 3 , Δ𝜖 𝑙 > 0 0, Δ𝜖 𝑙 ≤ 0 (A-6.4)
where ∆𝜖 𝑙 is the change in the volume fraction of the deposits in the 𝑙 𝑡ℎ bin.

The change in the particle number density in the 𝑙 𝑡ℎ bin due to the shrinkage of some its deposits, whose size change from 𝑟 𝑙 to 𝑟 𝑙-1 , is determined using the Eq. A-6.5.

Δ𝑁 𝑙 -= { ∆𝜖 𝑙 𝑟 𝑙-1 3 -𝑟 𝑙 3 , Δ𝜖 𝑙 < 0 0, Δ𝜖 𝑙 ≥ 0 (A-6.5)

Note that the particle number density in the 𝑙 𝑡ℎ bin will also change due to the growth and shrinkage of the deposits in the (𝑙 -1) 𝑡ℎ and (𝑙 + 1) 𝑡ℎ bins, respectively. Therefore, the total change in the particle number density of the deposits in 𝑙 𝑡ℎ bin is computed as follows,

∆𝑁

Materials used

A graphitized mesoporous carbon nano powder from Sigma Aldrich (699624) was used as the carbon host material in the cathodes of our Li-S cells. A 99.5% pure sulfur powder from Alfa Aesar was used for impregnating the carbon host material of the cathode. In the carbon/sulfur cathode films, the carbon super P from Timcal and the polyvinylidene fluoride (PVDF) respectively were the conductive additive and the binder. Finally, 1 M LiTFSI in TEGDME:DOL

(1:1 v/v) was the electrolyte used and it was acquired from Solvionic.

Surface area and porosimetry analyses of carbon nano powder

In order to compare the experimental and modelling results, the details about the average mesostructural properties of the cathode are needed. Therefore, the pristine and the sulfur impregnated mesoporous carbon nano powders were analysed using the Micrometrics ASAP 2020 surface area and porosity analyser. Initially, about 1 g of the pristine carbon nano powder was degassed under vacuum at 300 o C for 12 hours. The N 2 adsorption and desorption isotherms of the degassed carbon nano powder were measured at 77 K (Fig. 7.1). The hysteresis between the adsorption and desorption isotherms indicates that the carbon nano powder was porous (Fig. 7.1). Therefore, the pore size distribution was estimated using the Barret-Joyner-Halenda (BJH) model. The pore size distribution shows that the mesoporous carbon nano powder actually had micropores with diameter around 2.5 nm (Fig. 7.2). The microporosity and total porosity of the pristine carbon nano powder was estimated using the following equation,

𝜀 = 𝑉 𝑝𝑜𝑟𝑒 𝑉 𝑝𝑜𝑟𝑒 + 1 𝜌 𝑐 7.1
where 𝑉 𝑝𝑜𝑟𝑒 is either micropore volume or total pore volume (Table 7.1) and 𝜌 𝑐 is the density of the carbon powder (= 1.88 𝑔. 𝑐𝑚 -3 ) estimated using the pycnometer. The scanning electron microscopy (SEM) micrographs show that the pristine carbon nano powder is an agglomerate of carbon nanoparticles with average diameter around 50nm (Fig. 7.3). Assuming spherical carbon particles, the external specific surface area (𝐴 𝑐 ) could be calculated (Eq. 7.2) using particle radius (𝑟 𝑐 ).

𝐴 𝑐 = 3 𝑟 𝑐 𝜌 𝑐 × 1000 7.2 The calculated specific surface area is 65.5𝑚 2 . 𝑔 -1 , which is very close to the external surface area of the carbon nano powder (64𝑚 2 . 𝑔 -1 ) determined using the t-plot method (Table 7.1).

However, the BET surface area of the carbon nano powder is 80.5𝑚 2 . 𝑔 -1 , where the additional surface corresponds to the those inside the micropores. This is because the difference between the BET and the calculated surface areas (= 15𝑚 2 . 𝑔 -1 ) is very close to the t-plot micropore area (Table 7.1). This analysis reveals that there are no mesopores (𝑑 𝑝𝑜𝑟𝑒 > 4𝑛𝑚) within the carbon particles themselves, since the external and micropore surface areas completely account for its total surface area. Therefore, most of the cumulative pore volume of the carbon nano powder correspond to inter-particular void volume, since the micropore volume is very low (Table 7.1).

Sulfur impregnation

Prior to sulfur impregnation, the mesoporous carbon nano powder was heated overnight at 300 o C under secondary vacuum of about 10 -4 mbar to remove the adsorbed moisture form its surface. The precipitated sulfur powder was dried overnight at 60 o C under vacuum using Buchi oven. The dried carbon and sulfur powders were then taken inside the glove box and mixed thoroughly using a mortar and a pestle. The well mixed carbon/sulfur mixture was then sealed inside an autoclave bomb parr under argon atmosphere. The autoclave bomb parr was placed inside an oven and heated at rate of 5 o C.min -1 up to 155 o C which was maintained for 6 hours. Subsequently, the autoclave was allowed to cool down to room temperature. Finally, the autoclave was opened under argon atmosphere inside a glove box and the sulfur impregnated carbon nano powder was stored in a bottle for future use.

The S 8(s) impregnated carbon nano (C/S) powder was degassed at 60 o C for 6 hours and analysed using the Micrometrics ASAP 2020 surface area and porosity analyser. The hysteresis in the N 2 adsorption/desorption isotherms of the C/S powder was smaller than that of the pristine carbon nano powder. This is because the cumulative pore volume of the interparticular mesopores has reduced to (Table 7.2), due to the presence of solid sulfur in them (Fig. 7.4). The BET surface area of the C/S powder was also lower than that of the pristine carbon powder (Tables 7.1 and 7.2). The was due to the presence large solid sulfur precipitates in the inter-particular pores of the C/S powder (Fig. 7.4). Note that the BET and the t-plot external surface areas of the C/S powder are similar (Table 7.2). This indicates that the micropores in the carbon nano powder are blocked by the solid sulfur precipitates. Furthermore, the SEM image of the C/S powder with higher magnification, shows deposits of sulfur over the surface of some of the carbon particles (Fig. 7.5). These sulfur deposits will directly block the micropores of the carbon nanoparticles. Furthermore, the relatively very low t-plot micropore volume of the C/S powder proves that the micropores of the carbon nanoparticles are blocked by the solid sulfur precipitates. The negative sign of the t-plot micropore volume of the C/S powder could be due to the sublimation of some of the solid sulfur precipitates during the porosimetry analysis. After the analysis, we found some yellow deposits on the wall of the glass sample holder of the ASAP porosimetry analyser. Therefore, we carried out Thermal Gravimetric Analyses (TGA) of the C/S powder before and after porosimetry analysis to find out the amount of solid sulfur lost from the C/S powder due to sublimation.

During TGA, the C/S powders were heated from 25 to 500 o C at a rate of 10 K.min -1 , under argon atmosphere. As expected, the weight loss of the C/S powder before the porosimetry or BET analysis was 49.99%, which corresponds to the weight percentage of the solid sulfur (Fig. 7.6). Whereas, the solid sulfur weight percentage of the C/S powder after the porosimetry analysis was found to be 48.36%. Therefore, during the porosimetry analysis around 1.6% of the solid sulfur precipitates sublimized during the porosimetry analysis. All the solid powder ingredients of the slurry (Table 7.3) were dried and taken inside the glove box. The solid powder ingredients were then mixed properly using a mortar and a pestle inside the glove box. The well mixed solid powder was sealed inside a bottle. NMP solvent was added to the bottle containing the well mixed solid ingredients and the whole mixture was stirred well using a magnetic stirrer for 12 hours. The masses of the solid powder mixture and the liquid NMP solvent are listed in Table 7.4. The C/S electrode slurry was coated over an aluminium (Al) current collector using a doctor blade and an electrode coating machine. The thickness of the doctor blade was adjusted to 40 µm. The C/S electrode slurry was very viscous; therefore, it was taken out of the bottle using a spatula or a syringe while applying it to the doctor blade. The freshly coated electrode film was dried overnight at 40 o C. After drying, the electrodes for making coin cells were punched using a 11 mm puncher. Finally, the punched-out electrodes were dried at 50 o C for 6 hours using the Bushi oven under vacuum to remove the moisture.

Coin cell assembly

The galvanostatic measurements were carried using coin cells (Fig. 7.7), which were assembled inside the glove box under Ar atmosphere. As mentioned before, a C/S electrode film with diameter 11 mm was used as the cathode and a similar sized Li metal foil was used as the anode. Prior to the cell assembly, the surfaces of the Li metal foil were gently scratched using a spatula to remove the oxide layers. A celgard separator (SK innovation) was used to separate the anode and the cathode. The celgard separator was wetted with pure 1 M LiTFSI TEGDME:DOL (1:1 v/v) electrolyte. The electrolyte to solid sulfur (E/S) ratio was 20 𝜇𝐿. 𝑚𝑔 𝑆 8(𝑠) -1 . Due to its high viscosity, the electrolyte does not wet the separator properly.

Therefore, the electrolyte was spread out evenly over the separator using the tip of the micropipette. After assembling the coin cell, it was tightly sealed inside the glove box using a pneumatic press.

Galvanostatic measurement

In order to prevent the voltage loss due to self-discharge, the galvanostatic cycling of the coin cells were started immediately after their assembly. The galvanostatic measurements of the coin cells were carried out using the VMP Biologic rectifier, outside the glove box at room temperature. Measurements at three different C-rates namely C/20, C/10 and C/5 were carried. The information of the coin cells used for different C-rates are listed in the Table 7.5.

Theoretical methodology

We slightly modified our discharge model (Chapter 4), to well represent the experiments. The microporosity of our carbon nano powder was only 1.2% (Table 7.1) and their micropores got blocked after sulfur impregnation. Furthermore, the microporosity of our cathodes were very low even after the dissolution, since the volume fraction carbon used in our cathodes were very low (≈ 0.25). Therefore, we neglected the microporosity of the carbon particles in our model. In our modified discharge model, the intra-flux term in the mass conservation equation of the species within the inter-particular pores (Eq. 4.13) was not considered.

Additionally, we also neglected the mass conservation equation of the species within the micropores of the carbon particles (Eq. 4.13). The cathode potential was calculated directly from the current balance equation,

𝐼 𝐺 = -∑ ∫(𝑎 1 (𝑖 1 ) 𝑗 )
𝐶 𝐵 𝑗 𝑑𝑥 (7.3) where, 𝐼 𝐺 and 𝑎 1 are the applied current density and specific surface area of the cathode.

(𝑖 1 ) 𝑗 is the current density of the electrochemical reaction 𝑗, which is determined using the Butler-Volmer equation.

Since 1 M LiTFSI in TEGDME:DOL was used in our Li-S coin cells, we have considered a comprehensive set of cathode reaction steps (Eqs. 7. [START_REF] Kremer | Population Growth and Technological Change: One Million B.C. to[END_REF] Assumed parameters taken from Refs. [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF], [START_REF] Jung | Effect of organic solvents and electrode materials on electrochemical reduction of sulfur[END_REF], [START_REF] Gaillard | Polysulfides in dimethylformamide: Only the radical anions S3-and S4-are reducible[END_REF] and [START_REF] Zhang | Modelling transport-limited discharge capacity of lithium-sulfur cells[END_REF] respectively. f Experimenal parameters taken from Ref. [START_REF] Drvarič Talian | Reactivity and Diffusivity of Li Polysulfides: A Fundamental Study Using Impedance Spectroscopy[END_REF]. m Measured parameter.

Since, the sulfur mass loadings in our Li-S coin cells are different, the initial cathode design parameters were calculated and listed in the table 7.7. The main objective of this chapter is to compare the simulated results of our discharge model with the experimental ones. Therefore, the galvanostatic cycling of our Li-S coins cells were continued only for few cycles (Fig. 7.8). The highest initial discharge capacity was delivered by the cell with the lowest C-rate (C/20). Whereas, the lowest initial discharge capacity was delivered by the one with the highest C-rate (C/5). However, the overcharging of our Li-S coin cells increased when the C-rate was decreased. The overcharging is due to the polysulfides shuttle between the electrodes of our coin cells. This is because we have used 20 𝜇𝐿. 𝑚𝑔 𝑆 8(𝑠)

-1

of pure 1 M LiTFSI in TEGDME:DOL in our cells without any additive or ion selective membrane between the electrodes. The overcharging due the polysulfides shuttle could be potentially reduced by decreasing the E/S ratio. However, lowering E/S ratio increases the concentration of the polysulfides in the electrolyte and its viscosity. Therefore, the decrease of E/S ratio could lead to the reduction of discharge capacities, due to the increase of the electrolyte and the ionic transport resistances in highly viscous electrolyte. These aforementioned resistance effects are not considered in our discharge model.

The experimental vs the simulated discharge curves

The simulated discharge curves have certain qualitative resemblance with the experimental results (Fig. 7.9). The perfect matching of the simulated results with the experimental ones is difficult due to the multiple parameters corresponding to the reaction steps and the species considered in our modified discharge model. However, the trends such as the decrease of discharge capacity and the increase of polarization with the increase of C-rate are consistent with experiments. The increase of polarization with the discharge rate is due to increase of the activation overpotentials of electrochemical reactions. However, the separation between high plateau potentials is relatively large in the simulated resulted. This discrepancy could be due to the Bruggeman relation for calculating active surface (Eq. 7.17), which only gives an approximate estimation of the surface passivation by solid sulfur precipitates. Therefore, the The cathodes used in our Li-S cells were highly porous, therefore their discharge capacities were not limited by the transport of dissolved species in the electrolyte. Furthermore, our simulation results show that around 68 to 77% of the porous volume of the cathodes remain unoccupied by the 𝐿𝑖 2 𝑆 (𝑠) precipitates at the end of discharge (Fig. 7.10b). However, due to their high porosities, our cathodes had very low specific surface areas. Since, the specific surface area depends on the amount of carbon per unit volume of the electrode, which is very low in our cathodes. Therefore, the low surface area could have led to the early cell potential drop could be due to increase in the overpotential of the sluggish lower order polysulfides.

reduction reactions.

Conclusions

In this chapter, we have presented some of the experimental results produced during this thesis work. Furthermore, we also compared the simulated results of our discharge model with the experimental results of our Li-S coin cells. The surface area and porosimetry analysis of the carbon nano powder used in the cathodes of our Li-S coin cells, showed that they are agglomerates of 50 nm carbon particles which have micropores of diameter 2.5 nm. However, the microporosity of the carbon nano powder was very low and their micropores get blocked after sulfur impregnation. Our Li-S coin cells overcharged during galvanostatic cycling due to the polysulfides shuttle between their electrodes. This could be eliminated either by adding additives such as LiNO 3 to our electrolyte or by placing an ion-selective membrane between the electrodes.

We modified our discharge model by considering a comprehensive set of reaction steps determined using our cyclic voltammetry model, which is presented in Chapter 5. The simulated results of our modified discharge model have qualitative agreement with the experiments. However, the exact matching of the simulated results with the experiments was not achieved. This is mainly due to the complications arising from the multiple parameters of to the reaction steps and species considered in our model. However, simulation results of our model have assisted in assessing the phenomena behind the characteristics of the experimental discharge curves. The simulated results revealed that the 𝑆 8(𝑠) dissolution rate decreases with the discharge rate which is consisted with our kMC simulation results.

Furthermore, the simulated results also show the 𝐿𝑖 2 𝑆 (𝑠) precipitation start before the complete of dissolution 𝑆 8(𝑠) during discharge at C/10 and C/5. This could be the reason behind the invisibility of the potential dips in their corresponding experimental discharge curves. Overall, the reduction of higher order polysulfides dominate during the high and slopy stages of discharge. Whereas, the reductions of medium and low order polysulfides dominate the low plateau stage. These results are consistent with the numerous previously reported experimental results [START_REF] Kawase | Electrochemical reactions of lithiumsulfur batteries: an analytical study using the organic conversion technique[END_REF][START_REF] Cuisinier | Sulfur Speciation in Li-S Batteries Determined by Operando X-ray Absorption Spectroscopy[END_REF]. Furthermore, our simulations results show that radicals such as 𝑆 3 * -are produced during the high plateau stage of the discharge, which get consumed during the subsequent slopy and low plateau stages. This is also consistent with the previously reported experimental results [START_REF] Hannauer | The Quest for Polysulfides in Lithium-Sulfur Battery Electrolytes: An Operando Confocal Raman Spectroscopy Study[END_REF].

Finally, the simulation results reveal that the discharge capacities are not reduced due to the transport limitations of species in this electrolyte. This is because the cathodes used in our Li-S cells had very high porosity. However, the surface areas of our cathodes were very low. Our simulations show that there were some lower order polysulfides which remain unreduced in the electrolytes of the cathode and the separator. This suggests that the discharge capacities are kinetically limited by the sluggish reduction reactions of these lower order polysulfides.

Therefore, the discharge capacity could be increased by using cathodes with higher surface areas.

Chapter 8. General conclusions and perspectives

Li-S batteries are the most attractive among the next generation of batteries, primarily due to its viability. Since, Li-S batteries are being used in some commercial applications. Numerous developments have been done in the past decade in terms of carbon/sulfur composite architectures, protective coating over Li metal anode, etc. which have improved the discharge capacities and cyclability of Li-S batteries. However, due the complex nature of its operating principle, the assessment of the phenomena that occur during the operation of Li-S batteries is not trivial. Moreover, the phenomena behind certain performance limitations and electrochemical characteristics are still under debate in Li-S batteries. A better understanding of these phenomena could assist in the further improvement of the Li-S batteries performance. Therefore, we have developed multiple models that could be applied to investigate certain phenomena that occur at different spatial scales of the Li-S batteries.

Furthermore, the main focus of this thesis work was given to the assessment of the underlying phenomena and limitations behind the electrochemical performance of Li-S batteries. It was achieved through the thorough analysis of the simulated results produced using our comprehensive models which consider multiple species, reaction steps and spatial scales. Less focus was given to the exact matching of the simulation results with the experiments.

Numerous atomistic/molecular and continuum level models have been reported in the literature of Li-S batteries. However, very few mesoscopic models have been reported in the literature, which are either too small or less comprehensive. Therefore, to bridge the gap between the atomistic and continuum levels, we have developed a novel 3D kinetic Monte-Carlo (kMC) model, which explicitly simulates the discharge of in silico created carbon/sulfur (C/S) mesostructures at the mesoscopic level. This is also the first kMC model applied to Li-S batteries, which considers multiple reaction and diffusion events, that occur from the start to the end of discharge. However, the number of considered events were simplified to reduce the computational cost. The correlation of the simulated effective evolutions with the calculated approximate discharge curves, shows that the 𝑆 8(𝑠) dissolution and the reduction of higher order polysulfides occur during the high plateau and slopy stages of the discharge.

Whereas, the reduction of higher order polysulfides and the electrodeposition of the 𝐿𝑖 2 𝑆 (𝑠)

of occur during the low plateau stage of the discharge. Due to the 𝑆 8 dissolution and the 𝐿𝑖 2 𝑆 (𝑠) electrodeposition, the cathode porosity initially increases and then decreases during discharge. These effective evolutions during the discharge are consistent with the numerous previous reported modelling [START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF][START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF] and experimental results [START_REF] Kawase | Electrochemical reactions of lithiumsulfur batteries: an analytical study using the organic conversion technique[END_REF][START_REF] Cuisinier | Sulfur Speciation in Li-S Batteries Determined by Operando X-ray Absorption Spectroscopy[END_REF].

Furthermore, the impacts of the discharge rate and the solid sulfur loading on the mesoscale evolutions of the 𝐿𝑖 2 𝑆 (𝑠) deposits over the carbon surface, were also investigated. It was done by analysing the kMC simulation results using the radial distribution function and the cluster recognition algorithm. The results of these analysis, show that the 𝐿𝑖 2 𝑆 (𝑠) coverage over the carbon surface increases faster during fast discharge. Furthermore, relatively more big clusters of 𝐿𝑖 2 𝑆 (𝑠) deposits are produced during fast discharge. Moreover, during slow discharge some of the 𝐿𝑖 2 𝑆 (𝑠) deposits are formed slightly away from the surface. These variations in the mesoscale evolutions of the 𝐿𝑖 2 𝑆 (𝑠) deposits for two different discharge rates are mainly due to the competition between the diffusion and reaction events of 𝑆 2 2-.

Additionally, these results suggest that the passivation of the carbon surface by the 𝐿𝑖 2 𝑆 (𝑠) deposits, increases faster during fast discharge. Furthermore, the analysis of our kMC results, show that relatively thick 𝐿𝑖 2 𝑆 (𝑠) deposits are produced in the C/S mesostructure which has high sulfur loading (high-𝑆 loaded mesostructure). This is mainly due to the low availability of free carbon surface which remains uncovered during the majority of the discharge due to the slow dissolution of 𝑆 8(𝑠) . Furthermore, the thick 𝐿𝑖 2 𝑆 (𝑠) deposits also impede the transport of the dissolved species to the electrode surface. Our kMC results also show that the electrodeposition of 𝐿𝑖 2 𝑆 (𝑠) in the high-𝑆 loaded mesostructure, starts before the complete dissolution of 𝑆 8(𝑠) and its corresponding effective mesostructure porosity remains low throughout the discharge. These aforementioned effects suggest that the passivation of the carbon surface and the blocking of mesopores by the 𝑆 8(𝑠) and 𝐿𝑖 2 𝑆 (𝑠) precipitates, could increase with the sulfur mass loading. Finally, the effects of the discharge rate and the sulfur mass loading on the 𝐿𝑖 2 𝑆 (𝑠) electrodeposition, could partly explain why the discharge capacities reduce, when these aforementioned parameters are increased.

Our kMC model is very generic, and it is not limited to the geometrical and effective structural properties of the in silico created C/S mesostructures reported in this thesis work. It can also be applied to investigate the discharge of mesostructures, which are produced from the experimental tomographic images.

As mentioned before, our kMC model simulates the discharge of the C/S cathode composites at the mesoscopic level. Therefore, it does not simulate the macroscopic phenomena that limit the discharge capacities of Li-S batteries. Consequently, we have developed a continuum model to investigate the limitations to the discharge of Li-S batteries, caused by its design and discharge parameters. However, unlike most of the continuum models applied to Li-S batteries, our model considers a multi-scale description of the cathode. In our model, we consider that the cathode is made up of mesoporous carbon particles which have interparticular pores between them. The chemical and electrochemical reactions are considered to take place in both mesopores and inter-particular pores. Furthermore, we also consider the mass exchange between these pores with different scales. Our simulation results, show that the discharge capacity is mainly limited by the transport of dissolved species in the cathode, which is made up of highly mesoporous carbon particles (45%), which have very low inter-particular porosity between them (25%). Furthermore, the discharge capacity was improved by increasing the inter-particular porosity (35%). These results indicate that the inter-particular porosity of the cathode is a crucial parameter, since the inter-particular pores permit the transport of dissolved species in the cathode and accommodate the volume expansion of the 𝐿𝑖 2 𝑆 precipitates. Therefore, these results show that the rate capability of Li-S batteries could be improved by increasing the inter-particular porosity. Furthermore, the discharge capacity could also be improved by increasing the mesoporosity (55%). Since, the surface area of the carbon particles increases when the mesoporosity is increased, which facilitates more reactions within the mesopores. Additionally, the increase of mesoporosity will also increase the accommodation of 𝐿𝑖 2 𝑆 (𝑠) within the mesopores of the carbon particle.

Therefore, comparatively less amount 𝐿𝑖 2 𝑆 (𝑠) gets accumulated in the inter-particular pores of the cathode containing carbon particles with very high mesoporosity (55%). Less 𝐿𝑖 2 𝑆 (𝑠)

accumulation in the inter-particular pores will prevent their early clogging during discharge, thereby assisting in the increase of its capacity.

The inter-particular surface area increases when the size of the carbon particles is decreased.

It also increases when the number or amount of carbon particles in the cathode is increased.

Therefore, the inter-particular surface area is low for the cathode with large carbon particles.

Our simulations, results show that the thickness of the 𝐿𝑖 2 𝑆 (𝑠) layer formed over the external surface of the mesoporous carbon increases faster during discharge in the cathode with large carbon particles. This is because, the aforementioned cathode does not have enough surface to distribute and form a thin layer of 𝐿𝑖 2 𝑆 (𝑠) over the external surface of the carbon particles.

Consequently, the mesopores get chocked by the aforementioned thick 𝐿𝑖 2 𝑆 (𝑠) layer.

Furthermore, some of the unutilized polysulfides get trapped inside them. This will reduce the discharge capacity of the cathode. In conclusion, our discharge model results suggest that the discharge capacity of the cathode can be improved through,

• The increase of the inter-particular porosity of the cathode;

• The increase of mesoporosity of the carbon particles;

• The increase of the inter-particular surface area of the cathode, by decreasing the size of the carbon particles and by increasing the amount of carbon particles;

• The increase of the mesoporous surface area of the carbon particles, by decreasing the mesopore size and by increasing the number of mesopores.

The discharge model presented in the chapter 4, considers the existence of an arbitrary electrolyte in the pores of the cathode and the separator. Therefore, we have assumed a reaction mechanism which is conventionally used in many continuum models applied to Li-S batteries. However, in the subsequent chapters, we have used 1 M LiTFSI in TEGDME:DOL in our experiments. Therefore, in order to determine reaction mechanism of the polysulfides in 1 M LiTFSI in TEGDME:DOL, we carried out cyclic voltammetry experiments using the electrolyte solutions containing dissolved S 8 , Li 2 S 8 and Li 2 S 6 . We also developed a mathematical model to simulate and interpret the reaction steps behind the characteristics of the experimental cyclic voltammograms (CVs). The CV of the dissolved S 8 measured at 5

mV.s -1 , consists of two reduction peaks (R 1 and R 2 ) and two oxidation peaks (O 1 and O 2 ).

However, the simulated CV produced using the conventional reaction mechanism did not reproduce the O 2 peak. Furthermore, the experimental intensity of the O 2 peak does not increases when the scan rate is increased, which suggest that its underlying reaction mechanism must contain some electrochemical steps that are coupled to chemical reaction steps. Therefore, these results prove that the conventional reaction mechanism used in many In chapter 7, we have presented the results of our experimental works, which were carried to validate our discharge model. The galvanostatic cycling of the Li-S coins were carried out for this purpose. Each of these Li-S coin cells, consists of a C/S cathode, a Li metal anode and a celgard separator which is wetted with the pure 1 M LiTFSI in TEGDME:DOL electrolyte. The porosimetry and surface area analysis of the carbon powder used in our C/S cathodes, are agglomerates of 50 nm carbon nanoparticles which have micropores of size 2.5 nm. However, the microporosity of the carbon powder was very low and its micropores got blocked after sulfur impregnation. Therefore, we modified our discharge model by neglecting the microporosity within carbon particles. Furthermore, we also considered the comprehensive reaction mechanism, which was determined using the cyclic voltammetry model. The simulated results of our modified discharge model semi-quantitative match with the experiments. The exact matching of the simulation results with the experiment was not achieved due to the complications arising from the numerous parameters of the reaction steps and species considered in our model. The simulation results of our model show the discharge capacities are kinetically limited by the sluggish reactions of the lower order polysulfides rather than transport of the dissolved species in the electrolyte within the cathode. This is mainly because, our C/S cathodes have very high porosities, which assists with the better transportation of the species within the cathode. Due to their high porosities, the volume percentages of the carbon particles in our cathodes are low. Furthermore, the electroactive specific surface areas of our cathodes are also low, since they have less amount carbon particles per unit volume of the electrode. Therefore, due to the low electroactive surface area in our cathode, the activation overpotential of the sluggish lower polysulfides reduction reactions, could have led to the potential towards the end of discharge. This conclusion suggests that the discharge capacity can be improved by increasing the electroactive surface area. This could be achieved either by increasing the volume percentage of the carbon particles or by using highly mesoporous carbon particles in the cathode.

In conclusion, we have developed multiple models to address some of the assessment and optimization issues related to Li-S batteries. Our models also assist in analysing the results of different electrochemical experiments such as the galvanostatic discharge, potentiostatic discharge and cyclic voltammetry. The most important perspective of this PhD work, would be the determination of parameter values using specially dedicated experiments or through first principles calculations. Finally, we did not study the charge process of the Li-S batteries, using our models. Therefore, in the future our discharge model could be adapted to simulate the charge process of Li-S batteries, by including phenomena such as nucleation and growth 𝑆 8(𝑠) , polysulfides shuttle, etc. De plus, l'augmentation de la mésoporosité des particules de carbone améliorera également la capacité de décharge, puisqu'elle augmentera la surface spécifique des mésopores ce qui facilitera l'utilisation correcte des polysulfures électroactifs. De plus, il augmentera les dépôts de Li 2 S (s) dans les mésopores. Par conséquent, l'augmentation de la mésoporosité diminuera la quantité de précipités Li 2 S (s) qui sont pris en compte dans les pores inter-particulaires.

Résumé développé de la thèse en Français (Summary in French)

Comme moins de précipités de Li 2 S (s) sont logés dans les pores interparticulaires, leur colmatage précipité pendant la décharge est inhibé dans la cathode avec des particules de carbone ayant une mésoporosité très élevée (55%).

Les résultats de notre modèle de décharge montrent que l'épaisseur de la couverture de Li 2 S (s)

formée sur la surface externe des particules de carbone, dépendra de l'aire de la surface interparticulaire. Dans la cathode à faible surface interparticulaire, une couverture épaisse de

  .1) [1].

Figure 1 . 1 .

 11 Figure 1.1. Rates of change in Gross Domestic Products (GDP), primary energy demand and energy productivity, in the world and in the countries within and outside the Organisation for Economic Co-operation and Development (OECD) [1].

Figure 1 . 3 .

 13 Figure 1.3. World consumption of energy in million tonnes oil equivalent produced using different energy sources [1].

Figure 1 . 4 .

 14 Figure 1.4. Global warming relative to pre-industrial era (1850-1900 AD) temperature level and projections for the future with three different scenarios, where the blue, grey and pink curves respectively correspond to global CO 2 emissions reaching net zero in the years 2040, 2055 with and without reduction of non-CO 2 radiative forcing [8].

Figure 1 . 5 .

 15 Figure 1.5. Global renewable power generation capacity in gigawatt (GW) and increase in

Figure 1 . 6 .

 16 Figure 1.6. Actual and forecasted (D: made on the day and D-1: made on the day ahead) electricity demand in France on January 4 th , 2019. Data acquired from the website of Réseau de Transport d'Électricité France (RTE France) [21].

Figure 1 . 8 .

 18 Figure 1.8. Shares of different non-PSH storage volumes (15300 MWh) which are available in

  by fossil fuels such as diesel and gasoline. Therefore, to reduce the GHG emissions, the ICE vehicles on the road must be switched with electric vehicles (EVs), which are powered by electrochemical energy storage and conversion devices such as rechargeable batteries and fuel cells.

Figure 1 . 9 .

 19 Figure 1.9. Share of GHG emissions from different transportation sectors in the EU in the year

Contents 2 . 1 . 2 . 2 . 2 . 3 . 2 . 4 . 2 . 5 . 2 . 6 .

 212223242526 Introduction to Li-S batteries .............................................................................................. Working principle of Li-S batteries ...................................................................................... Challenges in Li-S batteries ................................................................................................. Developments in the components of Li-S batteries ............................................................. 2.4.1. Positive electrode ............................................................................................................... 2.4.2. Negative electrode .............................................................................................................. 2.4.3. Electrolyte ........................................................................................................................... Modelling of Li-S batteries .................................................................................................. 2.5.1. Atomistic and Molecular level modelling ........................................................................... 2.5.2. Mesoscopic modelling ........................................................................................................ 2.5.3. Continuum level modelling ................................................................................................. Conclusions ........................................................................................................................

Figure 2 . 1 .

 21 Figure 2.1. Schematic representation of a conventional Li-S cell.

  reductions of the medium to low order polysulfides and the precipitation of Li 2 S take place during the low potential plateau stage of the discharge. During charge, the reversal of the aforementioned reactions occurs.

Figure 2 . 2 .

 22 Figure 2.2. The typical charge and discharge curves of a Li-S batteries [57].

Figure 2 . 3 .

 23 Figure 2.3. (a) The experimental phase diagram of the S 8(s) , Li 2 S (s) and 1 M LiTFSI in 1,3 dioxolane system [63]. The lines A, B and C in the phase diagram are discharge trajectories in which have different proportions of solid and dissolved precipitate at different depths of discharge (𝑥 𝐿𝑖 2 𝑆 ). (b) Theoretical discharge profiles of three different discharge trajectories [63].

Figure 2 . 4 .

 24 Figure 2.4. (a) Schematic representation of polysulfides shuttle mechanism and (b) the influence of polysulfides shuttle on charge and discharge curves measured at different discharge rates and temperatures [65].

Figure 2 . 5 .

 25 Figure 2.5. Schematic representation of CMK-3/S-155 composite and (b) discharge capacities during cycling of pure (in red) and PEG coated CMK-3/S composites (in black) [55].

Figure 2 . 6 .

 26 Figure 2.6. Schematics and SEM images of porous carbon/sulfur composites containing (a) microporous carbon, (b) spherically ordered mesoporous carbon, (c) mesoporous hollow carbon, (d) graphene oxide sheets (e) porous carbon nanofiber and (c) hollow carbon

Figure 2 . 7 .

 27 Figure 2.7. The schematics, cyclic voltammograms and galvanostatic discharge/charge curves sulfur impregnated microporous carbon (a) ether-based and (b) FSI-based ionic liquid electrolytes [80].

Figure 2 . 8 .

 28 Figure 2.8. The SEM images of routine (a) PP/PE/PP, and (b) ion selective Nafion-PP/PE/PP layers of Li metal and (c) their corresponding discharge capacities and Coulombic efficiencies for different cycles [85].

  However, modelling works on Li-S batteries are quite recent. The very first Li-S battery model was developed by Mikhaylik et al. in 2004 and it was used to study the polysulfides shuttle [67]. In 2008, Kumaresan et al. developed the first comprehensive 1D continuum model to simulate the discharge of Li-S batteries [92]. 2.5.1. Atomistic and Molecular level modelling Density functional theory (DFT) is an atomistic modelling technique and it has been used to determine the properties of the materials used in the Li-S batteries. It was used to explore reaction mechanism of Li-S by predicting the structures and reduction potentials of

Figure 2 . 9 .

 29 Figure 2.9. Structure and discharge plateau potentials of different polysulfides.

Figure 2 . 10 .

 210 Figure 2.10. Schematic representations of possible charge and ionic transport mechanisms in in α-S and Li 2 S [103].

Figure 2 . 11 .

 211 Figure 2.11. Discharge curves measured using a modified pouch cell with X-ray transparent window and (b) theoretical and experimental XAS spectra at different stage of discharge [107].

Figure 2 . 12 .

 212 Figure 2.12. Structures and energies of different Li-S nanoparticles.

Figure 2 . 13 .

 213 Figure 2.13. (a) Schematic representation of the Li-S cell design considered in the Kumaresan et al.'s model and (b) the simulated average open circuit potential curves of different reactions and the full discharge curves [92].

Figure 2 . 14 .

 214 Figure 2.14. (a) Simulated galvanostatic charge and discharge curves for different current density and (b) simulated charge and discharge capacities for 10 cycles calculated using Hofmann et al.'s model [113].

Figure 2 . 15 .

 215 Figure 2.15. Simulated Li + and S 4 2-concentrations along the Li-S cell at the end of 0.2C and 1C discharge and (b) experimental and simulated capacities after relaxation following the 1C

3 . 2

 32 experimental results and observations of the Li-S batteries. The models applied to Li-S batteries are based on different approaches and the phenomena that they are used to assess span different spatial and temporal scales. Majority of the Li-S batteries models are atomistic/molecular level models which are used to estimate the properties of the materials used such as solid sulfur, Li 2 S and polysulfides, etc. Atomistic/molecular level models also assist in simulating and analysing the experimental spectroscopic results. Continuum models are used to simulate and analyse the electrochemical experiments of Li-S batteries. They also help in the identification of certain phenomena and limitations during the operation of Li-S batteries. However, most of the continuum models do not consider the detailed architecture of the carbon/sulfur composites used in the cathode. Therefore, they do not provide insights into the impacts of the carbon/sulfur composite design on the performance of Li-S batteries.Furthermore, there are very limited mesoscopic models were developed to investigate the phenomena occurring at the mesoscopic level in Li-S batteries. In this PhD, we have attempted to address the issues such as evolution of carbon/sulfur cathode mesostructures during discharge, impacts of cathode design parameters on discharge and understanding the reaction steps involving dissolved polysulfides and electrodeposition of Li 2 S. Background and Motivation ................................................................................................ Theoretical methodology .....................................................................................................

3. 4 . 3 . 5 .

 435 Impact of sulfur loading ...................................................................................................... Conclusions ........................................................................................................................ Chapter 3. A 3D kinetic Monte-Carlo model for Li-S batteries 3.1 Background and Motivation

Figure 3 . 1 .

 31 Figure 3.1. Schematics of the in silico method to create the initial C/S mesostructure.

Figure 3 . 2 .

 32 Figure 3.2. Schematic representations, of (a) the reaction events considered in our model along with the coarse-grained structures of the different types sulfur based particles and (b)

Figure 3 . 3 .

 33 Figure 3.3. Workflow of the 3D kMC-VSSM code.

Fig. 3 . 4 ,

 34 Fig. 3.4, shows the visual evolution of the mesostructure inside the simulation box at different depths of discharge (DoDs) or specific capacities, during the 2C discharge simulation.

Figure 3 . 4 .

 34 Figure 3.4. Visualization of the simulation box at 208, 647, 1050 and 1473 mAh. g S 8(s) -1 during the 2C discharge simulation. The colour map in the image shows the colour coding assigned to different types of sulfur based particle types, where 𝑆 8(𝑠) , 𝑆 8(𝑙) 2-, 𝑆 4(𝑙) 2-, 𝑆 2(𝑙) 2-and 𝐿𝑖 2 𝑆 (𝑆) particles are shown in navy blue, sky blue, green, yellow and red respectively.

Figure 3 . 5 .

 35 Figure 3.5. Evolutions of the concentrations of 𝑆 8(𝑙) (royal blue lines), 𝑆 4(𝑙) 2-(green lines) and 𝑆 2(𝑙) 2-(orange lines) during C/2 (solid lines) and 2C (dashed lines) discharge simulations.

Figure 3 . 6 .

 36 Figure 3.6. Approximate C/2 and 2C discharge curves. Calculated using Nernst's equation for 𝑆 4(𝑙) 2-/𝑆 2(𝑙) 2-electrochemical reaction.

Figure 3 . 7 .

 37 Figure 3.7. Experimental discharge curve of a Li-S coin cell with 1.85 mg.cm -2 of sulfur in the cathode.

  3.8). The rate constant for the chemical dissolution of 𝑆 8(𝑠) particles does not depend on the C-rate of the discharge simulation. Since the 𝑆 8(𝑠) particles have more time to dissolve during the C/2 discharge simulation, the number of 𝑆 8(𝑠) particles decreases faster with the specific capacity. Whereas, the number of 𝑆 8(𝑠) particles decreases very slowly with the specific capacity during 2C discharge simulation, since 𝑆 8(𝑠) particles have less time to dissolve. Due to the combined effect of slow 𝑆 8(𝑠) dissolution and fast 𝐿𝑖 2 𝑆 (𝑠)

Figure 3 . 9 .

 39 Figure 3.9. Visualizations of 𝐿𝑖 2 𝑆 (𝑠) deposits (in red) over the surface of the carbon particles at 208, 648, 1050 and 1229 mAh. g S 8(s) -1 during (a) C/2 and (b) 2C discharge simulations.

Figure 3 . 10 .

 310 Figure 3.10. The evolutions of coverages of 𝐿𝑖 2 𝑆 (𝑠) over carbon surface for both C/2 (black line) and 2C (red line) discharge simulations.

𝐿𝑖 2

 2 𝑆 (𝑠) particles on the carbon surface at 1229 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 during C/2 and 2C discharge simulations (Fig.3.11). At this capacity, the average number of Li 2 S particles present at distances beyond 2.5 nm from the carbon surface during C/2, are slightly larger than those of during 2C.

Figure 3 . 11 .

 311 Figure 3.11. The Number of 𝐿𝑖 2 𝑆 (𝑠) particles at different distances from the carbon surface at 208 (grey lines), 648 (red lines), 1050 (blue lines) and 1229 (green lines) 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 during C/2 (lines with squares) and 2C (lines with open circles) discharge simulations. The error bars in the plot are shown in black.

Figure 3 . 12 .

 312 Figure 3.12. Number of diffusion and reactions events selected during C/2 and 2C discharge simulations.

during C/ 2

 2 and 2C discharge simulations, the visualizations and size distributions of the clusters with sizes above 43, are shown in Fig. 3.14. Here, we classified the clusters based on the cluster size such as 44-79, 80-115, 116-151 and so on until 332-367 𝐿𝑖 2 𝑆 (𝑠) particles per cluster.

Figure 3 . 13 .

 313 Figure 3.13. Visualizations of 𝐿𝑖 2 𝑆 (𝑠) clusters of belonging to size classes of 4-8, 9-13, 14-18 and so on until 39-43 formed at 1229 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 during (a) C/2 and (b) 2C discharge simulations and (c) their corresponding cluster size distributions, where the cluster size distributions of C/2 and 2C are given in grey and pink bars respectively. The error bars over the histograms are shown in black. The colour map in the image shows the colour coding assigned to different cluster size classes (e.g. 4-8 and 39-43 are shown in navy blue and red, respectively) during visualization.

Figure 3 . 14 .

 314 Figure 3.14. Visualizations of 𝐿𝑖 2 𝑆 (𝑠) clusters of belonging to size classes of 44-79, 80-115, 116-151 and so on until 332-367 formed at 1229 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 during (a) C/2 and (b) 2C discharge simulations and (c) their corresponding cluster size distributions, where the cluster size distributions of C/2 and 2C are given in grey and pink bars respectively. The error bars over the histograms are shown in black. The colour map in the image shows the colour coding assigned to different cluster size classes (e.g. 64-94 and 95-125 are shown in navy blue and red, respectively) during visualization.

Figure 3 . 15 .

 315 Figure 3.15. Evolutions of the normalized amounts of 𝑆 8(𝑠) (navy blue lines) and 𝐿𝑖 2 𝑆 (red lines) particles and the porosities of the C/S mesostructures with 1:0.27 (solid lines) and 1:0.54 (dashed lines) C/S mass ratios during 2C discharge simulation.

Figure 3 . 16 .

 316 Figure 3.16. Visualizations of 𝐿𝑖 2 𝑆 (𝑠) deposits (in red) over the surface of the carbon particles in the mesostructures with (a) 1:0.27 and (b) 1:0.54 C/S mass ratios and at different DoDs during 2C discharge simulations.

Figure 3 . 17 .

 317 Figure 3.17. The evolutions of coverages of 𝐿𝑖 2 𝑆 (𝑠) over carbon surface in mesostructures with 1:0.27 (red lines) and 1:0.54 (blue lines) C/S mass ratios, during 2C discharge simulations.

Figure 3 . 18 .

 318 Figure 3.18. The distributions of 𝐿𝑖 2 𝑆 (𝑠) particles at different distances from the carbon surface in the mesostructures with 1:0.27 (lines with squares) and 1:0.54 C/S (lines with open circles) mass ratio and at 208 (grey lines), 647 (red lines), 1050 (blue lines) and 1370 (green lines) 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 during 2C discharge simulations. The error bars are shown in black.

4 . 2 .

 42 Introduction ....................................................................................................................... Methodology: overall assumptions ..................................................................................... 4.3. Methodology: Overall construction and governing equations .............................................

4. 6 .

 6 Conclusions and perspectives ........................................................................................... A.4. Appendix: Numerical Schemes ......................................................................................... Chapter 4. Microstructurally resolved continuum discharge model 4.1. Introduction

  research paper was co-authored by V. Thangavel, k.-H Xue, Y. Mammeri, M. Quiroga, A. Mastouri, C. Guéry, P. Johansson, M. Morcrette and A. A. Franco.

Figure 4 . 1 .

 41 Figure 4.1. Schematic representation of our Li-S cell model at various resolutions.

Figure 4 . 2 . 4 . 3 .

 4243 Figure 4.2. Flow chart of our model.

Figure 4 . 3 .

 43 Figure 4.3. Schematic of the exchange of mass between mesopores and inter-particular pores.

  molar volumes of 𝑆 8(soln) and 𝑆 8(soln) 2-, respectively. While indeed, Eq. 4.21 neglects the effects of other solutes, the long chain polysulfides can potentially affect the transport of all the solutes in the confined environment of the porous media, especially as the 𝑆 8(soln) 2-

(4. 31 )Figure 4 . 4 .

 3144 Figure 4.4. Schematic representation of the 𝐿𝑖 2 𝑆 (𝑠) precipitation routes.

Figure 4 . 5 .• 2 -

 452 Figure 4.5. Calculated discharge profiles at current densities 0.5, 1.0 and 1.5 mA.cm -2 .

Figure 4 . 6 .

 46 Figure 4.6. Evolution of concentration of species (solid lines) and volume fractions of the solid precipitates (dashed lines) for a discharge at 1.0 mA.cm -2 in: (a) the inter-particular pores, and (b) the mesopores. Faradaic current densities for different electrochemical reactions in: (c) the inter-particular pores, and (d) the mesopores.

Figure 4 . 7 .

 47 Figure 4.7. Evolution of porosities and volume fractions of the solid precipitate in: (a) the inter-particular pores, and (b) the mesopores, for the cathode bin closest to the separator.

Figure 4 . 8 .

 48 Figure 4.8. Evolution of the fractions of all the sulfur-based species, precipitates and dissolved, residing at different porous regions for a discharge at 1.0 mA.cm -2 .

Table 4 . 4 .

 44 Sulfur loadings for the three configurations with different carbon porosities. calculated discharge capacities are shown to be limited by the clogging of inter-particular pores closest to the separator by 𝐿𝑖 2 𝑆 (solid) . Therefore to check the reproducibility of the predicted trends, discharge simulation of different cathodes are repeated for two different activation energies corresponding to 𝑆 2(soln) reduction reaction (𝐸 <𝑆 2 2-/𝑆 2-> ).

Figure 4 . 9 .

 49 Figure 4.9. Simulated discharge profiles at 1.0 mA.cm -2 for the three cathode configurations with 𝐸 <𝑆 2 2-/𝑆 2-> =3.6 kJ.mol -1 (solid lines) and 𝐸 <𝑆 2 2-/𝑆 2-> =3.0 kJ.mol -1 (dashed lines).

Figure 4 . 10 .Table 4 . 5 .Figure 4 . 11 .

 41045411 Figure 4.10. (a) The evolution of the inter-particular porosities and the volume fractions of 𝑆 8(solid) and 𝐿𝑖 2 𝑆 (solid) in the inter-particular pores, solid lines correspond to 𝐸 <𝑆 2 2-/𝑆 2-> =3.6 kJ.mol -1 and dashed lines correspond to 𝐸 <𝑆 2 2-/𝑆 2-> =3.0 kJ.mol -1 and (b) the distribution of inter-particular porosity along the cathode thickness at the end of discharge, solid points correspond to 𝐸 <𝑆 2 2-/𝑆 2-> =3.6 kJ.mol -1 and hollow points correspond to 𝐸 <𝑆 2 2-/𝑆 2-> =3.0 kJ.mol -1 .

Figure 4 . 12 .

 412 Figure 4.12. The growth of the 𝐿𝑖 2 𝑆 (solid) film thickness in the inter-particular pores for χ = 0.5 (solid lines) and 𝜒 = 0.3 (dashed lines).

Figure 4 . 13 .

 413 Figure 4.13. Calculated evolution of the porosities and the volume fractions of 𝑆 8(solid) and 𝐿𝑖 2 𝑆 (solid) in the (a) inter-particular pores, and (b) mesopores of the carbon particles, both for the cathode bin closest to the separator. The solid lines correspond to χ = 0.5 and dashed lines correspond to χ = 0.3.

4. 6 )

 6 using the same current density, why the C-rate "automatically" increases as the volume fraction 𝑆 8(solid) is decreased, causing the potential in the latter part of the discharge curve to decrease due to activation polarization (Fig.4.14). The sensitivity of the 𝐸 <𝑆 2 2-/𝑆 2-> (= 3.6 and 3.0 kJ.mole -1 ) is also tested for the cathodes with different 𝑆 8(solid) loadings. Although, the capacities of all the cathodes decrease when 𝐸 <𝑆 2 2-/𝑆 2-> is increased, the predicted discharge capacity trends for different 𝑆 8(𝑠𝑜𝑙𝑖𝑑) loadings remain unchanged.

Figure 4 . 14 .

 414 Figure 4.14. Simulated discharge profiles for cathodes with different C/S volume ratio discharged at 1.0 mA.cm -2 with 𝐸 <𝑆 2 2-/𝑆 2-> =3.6 kJ.mol -1 (solid lines) and 𝐸 <𝑆 2 2-/𝑆 2-> =3.0 kJ.mol -1 (dashed lines).

Figure 4 . 15 .

 415 Figure 4.15. Calculated evolution of the porosities and the volume fractions of 𝑆 8(solid) and 𝐿𝑖 2 𝑆 (𝑠𝑜𝑙𝑖𝑑) in the (a) inter-particular pores, and (b) mesopores, both for the cathode bin closest to the separator. The solid lines correspond to 𝐸 <𝑆 2 2-/𝑆 2-> =3.6 kJ.mol -1 and dashed lines correspond to 𝐸 <𝑆 2 2-/𝑆 2-> =3.0 kJ.mol -1 .

Contents 5 . 1 . 5 . 2 .

 5152 Background and motivation.............................................................................................. Experimental Section ........................................................................................................ 5.

5. 4 . 5 . 4 . 2 . 5 . 5 . 5 . 6 . 5 . 7 .A. 5 Chapter 5 .

 454255565755 CVs of dissolved S 8 ........................................................................................................... 5.4.1. Determining reaction mechanism for the model ............................................................. Impact of scan rate on the characteristics of the CV of S 8 ................................................ CV of dissolved Li 2 S 8 .......................................................................................................... CV of dissolved Li 2 S 6 .......................................................................................................... Conclusions ...................................................................................................................... Appendix: Numerical Schemes .......................................................................................... A.5.1. Finite volume method ................................................................................................... Cyclic voltammetry model

Figure 5 . 1 .

 51 Figure 5.1. CVs recorded at 50 mV.s -1 -of S 8 dissolved in 1 M LiTFSI supporting electrolytes with different solvents namely: (a) DMSO, (b) DMF, (c) DMA, (d) DME, (e) DOL:DME, (f) TEGDME, (g) ACN, (h) TMS and (i) Diox:DME [62].

Figure 5 . 2 .

 52 Figure 5.2. Three electrode cell setup used in this comparative cyclic voltammetry study.

Figure 5 . 3 .

 53 Figure 5.3. Schematic representation of the simulated domain.

Figure 5 . 4 .

 54 Figure 5.4. Experimental CV (Expt.) of 6 mM of 𝑆 8 in 1 M LiTFSI in TEGDME:DOL measured at 5 mV.s -1 and the corresponding simulated CV (Simu.-Conv.) produced using the conventional reaction mechanism of Li-S batteries models (Eq. 5.16 to 5.22).

Figure 5 . 5 .

 55 Figure 5.5. Experimental CVs (Expt.) of 6 mM of 𝑆 8 in 1 M LiTFSI in TEGDME:DOL recorded with different scan rates.

  27) which include the following chemical and electrochemical reactions of 𝑆 4(𝑙) * -and 𝑆 3to 5.26 have been proposed in a previously reported cyclic voltammetry model which was used to simulate the CVs of dissolved Li 2 S 6 in DMF [184]. Whereas 𝑆 3(𝑙) 2-/𝑆 2(𝑙) 2-redox reaction was proposed by Barschasz et al. [60].

Figure 5 . 6 .

 56 Figure 5.6. Experimental CV (Expt.) of 6 mM of 𝑆 8 in 1 M LiTFSI in TEGDME:DOL measured at 5 mV.s -1 and the corresponding simulated CV (Simu.-Comp.) -producing using the comprehensive mechanism (Eq. 5.16 to 5.27).

Figure 5 . 7 .

 57 Figure 5.7. Simulated (a) individual currents of redox reactions and (b) species concentrations at the electrode surface ((𝑐 𝑖 ) 𝑥=0 ) during reduction potential scans at a rate of 5 mV.s -1 , for the 6 mM S 8 solution.

Figure 5 . 8 .

 58 Figure 5.8. Species concentrations at the electrode surface ((𝑐 𝑖 ) 𝑥=0 ) during (a) oxidation and (b) return to OCV potential scans at a rate of 5 mV.s -1 , for the 6 mM S 8 solution.

Figure 5 . 9 .

 59 Figure 5.9. Simulated evolutions of Li + concentration and volume fractions of S 8(s) and Li 2 S (s) at the electrode surface (𝑥 = 0) during (a) reduction and (b) oxidations potential scans at a rate of 5 mV.s -1 , for the 6 mM S 8 solution.

Figure 5 . 10 .

 510 Figure 5.10. Simulated CVs (Simu.-Comp.) of 6 mM of 𝑆 8 in 1 M LiTFSI in TEGDME:DOL with different scan rates -produced using the comprehensive reaction mechanism (Eq. 5.16 to 5.27).

5. 11 )

 11 , therefore the peak O 2 becomes visibly indistinctive during fast scan rates (Fig.5.10).

Figure 5 . 11 .

 511 Figure 5.11. Simulated individual currents of redox reactions corresponding CV of 6 mM S 8

Figure 5 . 12 . 2 -

 5122 Figure 5.12. (a) Experimental and (b) simulated CV of 50 mM Li 2 S 8 solution for a potential scan rate of 100 mV.s -1 .

Figure 5 . 13 .

 513 Figure 5.13. Simulated concentration evolutions of species at the electrode surface during, the first (a) reduction and (b) oxidation scans of 50 mM Li 2 S 8 solution, with a scan rate of 100 mV.s -1 .

  5.14a), which form the underlying basis for O 1 (Figs. 5.12a and b). Due to the aforementioned reactions, S 8 becomes the most concentrated species near the electrode surface instead of S 8 2-at the end of the first oxidation scan (Fig. 5.13b). Therefore, during the second reduction scan -both S 8 /S 8 2-and S 8 2-/S 6 2-reaction currents form peaks around -1.10 V (Fig. 5.14b). This is why the intensity of R 1 increases during the second reduction scan (Fig.s 5.12a and b). However, the rest of the CV characteristics of the second cycle are similar to those of the first cycle.

Figure 5 . 14 . 5 . 6 . 6

 514566 Figure 5.14. Simulated individual currents for electrochemical reactions during the (a) first and (b) second CV cycles 50 mM Li 2 S 8 solution, with a scan rate of 100 mV.s -1 .

Figure 5 . 15 .

 515 Figure 5.15. (a) Experimental and (b) simulated CVs of 50 mM Li 2 S 6 solution produced using potential scan rate 100 mV.s -1 .

Figure 5 . 16 .

 516 Figure 5.16. Simulated individual currents for electrochemical reactions during the (a) first and (b) second CV cycles 50 mM Li 2 S 6 solution, with a scan rate of 100 mV.s -1 .

Figure A- 5 . 1 .

 51 Figure A-5.1. Schematic representation of the adopted 1D meshing for the simulation domain

16 ] 6 .Contents 6 . 1 6 . 2 . 6 . 3 . 6 . 4 . 6 . 5 . 6 . 6 .A. 6 Chapter 6 .

 16661626364656666 Chapter Nucleation and growth of Li 2 S Introduction ...................................................................................................................... Experimental Setup .......................................................................................................... Theoretical methodology .................................................................................................. 6.3.1. Thermodynamics and kinetics of Li 2 S nucleation ............................................................. 6.3.2. Other reaction steps and their kinetic equations ............................................................. 6.3.3. Population balance of Li 2 S particles .................................................................................. The electrochemical measurement ................................................................................... Results and discussion ...................................................................................................... 6.5.1 The experimental vs the simulated electrochemical signals ............................................. 6.5.2. The simulated dissolved species concentrations and the individual reaction currents ... 6.5.3 Simulated evolutions of the Li 2 S deposits .......................................................................... Conclusion and perspectives ............................................................................................. Appendix: Numerical Schemes .......................................................................................... A.6.1. Discretization of the population balance equation .......................................................... Nucleation and growth of Li 2 S 6.1 Introduction Experimental and modelling results have shown that the precipitation of Li 2 S occurs during the low potential plateau stage of the conventional Li-S battery discharge[START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF][START_REF] Neidhardt | A Flexible Framework for Modeling Multiple Solid, Liquid and Gaseous Phases in Batteries and Fuel Cells[END_REF][START_REF] Yu | Direct visualization of sulfur cathodes: new insights into Li-S batteries via operando X-ray based methods[END_REF]. The Li 2 S deposits which form over the conductive carbon surface of the cathode have very low electronic conductivity[START_REF] Park | Adiabatic and Nonadiabatic Charge Transport in Li-S Batteries[END_REF][START_REF] Yang | High-Capacity Micrometer-Sized Li 2 S Particles as Cathode Materials for Advanced Rechargeable Lithium-Ion Batteries[END_REF]. Therefore, they will impact the electrochemical performance due to the passivation of the electroactive surface of the cathode. However, the kinetics and the reaction mechanism behind the precipitation of Li 2 S in Li-S batteries are still under debate. A better understanding of the Li 2 S precipitation phenomenon will assist in improving the discharge performance of the Li-S battery.

Figure 6 .

 6 Figure 6.1. (a) Current density vs. time curve during potentiostatic discharge at 2.02 V. SEM micrographs showing Li 2 S deposits over a carbon fiber at (a) 2.5, (b) 4 and (c) 6 hours [203].

Figure 6 . 2 .

 62 Figure 6.2. Schematic representation of the two-electrode coffee bag cell.

Figure 6 . 3 .

 63 Figure 6.3. The total homogenous free energy vs. the particle size of the nuclei [206].

  Bruggeman coefficient. The 𝑠 𝑖 and 𝑅 𝑖 are the homogeneous chemical reactions and heterogeneous dissolution/precipitation reactions which are determined using the following equations, 𝐺 𝑖 = ∑ 𝛾 𝑖𝑗 (𝑘 𝑝𝑟𝑒𝑐 ∑(𝜀 𝑟 ) (∏(𝑐 𝑖 )

Figure 6 . 4 .

 64 Figure 6.4. (a) Experimental and (b) simulated potential and current vs. time curves.

Figure 6 . 5 .

 65 Figure 6.5. Simulated evolutions of dissolved species concentrations at the surface of the cathode during both galvanostatic and potentiostatic stages of the cell discharge.

Figure 6 . 6 .

 66 Figure 6.6. Simulated individual reaction currents during the galvanostatic and the potentiostatic stages of the cell discharge.

  6.7b).

Figure 6 . 7 .

 67 Figure 6.7. (a) Simulated evolution of the Li 2 S volume fraction on the cathode during the galvanostatic and the potentiostatic stages, and (b) simulated particle size distributions of Li 2 S deposits at different times.

Figure 6 . 8 .

 68 Figure 6.8. The simulated evolutions of the electroactive specific surface area for the Li 2 S

Figure A- 6 . 1 .

 61 Figure A-6.1. The schematic representation of the discretization of the particle sizes of Li 2 S deposits within the range 𝑟 1 ≤ 𝑟 𝑙 ≤ 𝑟 𝐿 .

6 ) 7 .

 67 𝑙 𝑡𝑜𝑡𝑎𝑙 = -Δ𝑁 𝑙 + -Δ𝑁 𝑙 -+ Δ𝑁 𝑙-1 + + Δ𝑁 𝑙+1 -(A-6.Chapter Experimental work and discharge model validation 7.1. Introduction In this chapter the experimental works such as the surface area and the porosimetry analyses of carbon particles, sulfur impregnation, carbon/sulfur (C/S) composite electrode fabrication and galvanostatic cycling of Li-S coin cells are presented. These experiments were carried out to validate our discharge model. Therefore, the experimental discharge curves are presented in comparison to the simulated ones.

Figure 7 .

 7 Figure 7.1. (a) The N 2 adsorption/desorption isotherm and (b) pore size distribution of pristine mesoporous carbon nano powder 699624.

Figure 7 . 2 .

 72 Figure 7.2. (a) Secondary electron and (b) back scattered electron SEM micrographs of pristine carbon nano powder 699624.

Figure 7 . 3 .

 73 Figure 7.3. (a) The N 2 adsorption/desorption isotherm and (b) pore size distribution of S 8(s) impregnated carbon nano powder.

Figure 7 . 4 .

 74 Figure 7.4. SEM micrographs of sulfur impregnated carbon (C/S) nano powder where (a), (c) and (e) are secondary electron images and (b), (d) and (f) are their respective back scattered electron images. The yellow arrows in the images show the locations of solid sulfur.

Figure 7 . 5 .

 75 Figure 7.5. SEM back scattered electrons micrographs of: (a) pristine and (b) sulfur impregnated carbon nano powders.

Figure 7 . 6 .

 76 Figure 7.6. TGA results of the C/S powder before and after porosimetry analysis.

Figure 7 . 7 .

 77 Figure 7.7. Coin cell set-up.

Figure 7 . 8 .

 78 Figure 7.8. The experimental charge and discharge curves of our Li-S coin cells recorded at (a) C/20, (b) C/10 and (c) C/5.

Figure 7 . 9 .

 79 Figure 7.9. (a) Experimental and (b) simulated initial discharge curves produced at different c-rates.

Figure 7 . 10 . 2 - 2 - 2 -

 710222 Figure 7.10. (a) The simulated evolutions of the normalized S 8(s) and Li 2 S (s) volume fractions and cathode porosities, and (b) the calculated normalised porosities at the end of discharge along the cathode thickness.

Figure 7 . 11 .

 711 Figure 7.11. Simulated average concentrations of the dissolved species within the pores of the (a) cathode and (b) separator of the Li-S coin cell discharged at C/10.

  simulations. Les évolutions effectives au sein de la mésostructure C/S, lors de la simulation de décharge kMC, peuvent être quantifiées et visualisées à l'aide de tout logiciel de visualisation.De plus, des courbes de décharge approximatives peuvent être calculées à partir des concentrations de S 4 2-et S 2 2-. La corrélation entre les évolutions effectives simulées et les courbes de décharge approximatives calculées montre que la réaction de dissolution de S 8(s) et la réduction des polysulfures avec des longues chaînes se produisent pendant le plateau à haut potentiel et au cours de l'étape intermédiaire (pente entre les deux plateaux) de la décharge. Alors que la réduction des polysulfures avec de chaînes courtes et l'électrodéposition du ou des Li 2 S (s) se produisent pendant l'étape du plateau à bas potentiel. De plus, nos résultats sur les kMCs montrent que la porosité mésostructurelle diminue au cours des stades de plateau à haut potentiel et de l'étape intermédiaire de la décharge en raison de la dissolution des S 8(s) . Tandis qu'il diminue en raison de la précipitation de Li 2 S (s) pendant la phase du plateau à bas potentiel de la décharge. Ces évolutions effectives au cours de la décharge sont cohérentes avec les résultats expérimentaux précédemment rapportés[START_REF] Kawase | Electrochemical reactions of lithiumsulfur batteries: an analytical study using the organic conversion technique[END_REF][START_REF] Cuisinier | Sulfur Speciation in Li-S Batteries Determined by Operando X-ray Absorption Spectroscopy[END_REF].Les impacts du taux de C et de la charge massique S 8(s) -sur les propriétés mésoéchelle des Li 2 S (s) déposés sur la surface du carbone -ont été étudiés en analysant les emplacements Li 2 S (s) à l'aide de la fonction de distribution radiale (RDF) et de l'algorithme de reconnaissance des clusters. Les RDF aident à déterminer la couverture du carbone par Li 2 S (s) et le nombre de dépôts formés à différentes distances de la surface du carbone. Les résultats du RDF montrent que la couverture du carbone par Li 2 S (s) augmente plus rapidement lors d'une décharge rapide. De plus, les résultats montrent également que les dépôts de Li 2 S (s) se forment légèrement à l'écart de la surface du carbone lors d'une décharge lente. Ces propriétés mésoéchelle des dépôts de Li 2 S (s) indiquent que la réaction de nucléation est plus rapide lors d'une décharge rapide, alors que la croissance des dépôts existants est plus rapide lors d'une décharge lente. Ces variations dans la dynamique de nucléation et de croissance avec le taux C, sont dus à la concurrence entre les phénomènes de diffusion et de réaction du S 2 2-. L'analyse utilisant l'algorithme de reconnaissance des clusters montre qu'un nombre relativement plus important de grands clusters Li 2 S (s) sont produits pendant la décharge rapide. Par contre, un nombre relativement plus restreint de grands clusters de Li 2 S (s) sont produits pendant la décharge lente. En vue que les particules S 2 2-sont plus mobiles pendant la décharge lente, elles sont donc séparées les unes des autres avant d'être réduites pour produire du Li 2 S (s) . En raison de l'augmentation rapide de la couverture Li 2 S (s) et de la formation de grands amas, la passivation de la surface du carbone augmente plus rapidement pendant la décharge rapide. Cela pourrait être en partie la raison de la réduction de la capacité de décharge lorsque le taux C est augmenté.La dissolution de S 8(s) est relativement lente dans la mésostructure C/S qui a une charge massique élevée en soufre (mésostructure fortement chargée en S). En raison de cette lente dissolution, la surface de carbone disponible est faible pendant la majeure partie de la décharge. Par conséquent, la croissance du Li 2 S (s) sur les dépôts existants est beaucoup plus rapide que la formation de cristaux de Li 2 S (s) dans la mésostructure contenant une quantité importante de S. En raison de cette croissance rapide, les dépôts de Li 2 S (s) sur le carbone dans la mésostructure avec une quantité élevée de S sont très épais. De plus, les dépôts épais de Li 2 S (s) empêchent également le transport des espèces solvatées vers la surface de l'électrode.Par conséquent, en raison de l'effet combiné de la dissolution lente de S 8(s) et des dépôts épais de Li 2 S (s) , la passivation de la surface aura lieu toujours dans la mésostructure ayant une quantité élevée de S.Nos résultats kMC montrent également que la précipitation de Li 2 S (s) commence avant la dissolution complète de S 8(s) dans la mésostructure contenant une quantité importante de S, donc la porosité de la mésostructure est faible pendant toute la simulation de décharge. Ce résultat suggère que les pores entre les particules peuvent être obstrués lorsque la quantité de soufre augmente. Enfin, l'augmentation de la passivation et la possibilité d'obstruction des pores pourraient expliquer en partie pourquoi les capacités de décharge diminuent lorsque la quantité de soufre augmente. Notre modèle kMC ne se limite pas aux propriétés structurales géométriques et effectives supposées de nos mésostructures C/S créées in silico.Il peut également être utilisé pour étudier la décharge des mésostructures, qui sont produites à partir des images tomographiques expérimentales.Bien que notre modèle kMC soit capable d'étudier les phénomènes méso-échelle qui se produisent dans les composites cathodiques C/S, il ne peut décrire les phénomènes macroscopiques qui limitent les capacités de décharge des batteries Li-S. Par conséquent, nous avons développé un modèle de continuum pour étudier les impacts des architectures des composites C/S sur la décharge. Puisque la plupart des améliorations apportées à la performance des batteries Li-S ont été réalisées grâce à l'adaptation de l'architecture du composite C/S, notre modèle de décharge du continuum a considéré une description multiéchelle de la cathode[START_REF] Thangavel | A Microstructurally Resolved Model for Li-S Batteries Assessing the Impact of the Cathode Design on the Discharge Performance[END_REF].Dans notre modèle, nous avons considéré que la cathode est constituée de particules de carbone mésoporeuses qui ont des pores inter-particulaires entre elles. Les réactions chimiques et électrochimiques sont considérées comme ayant lieu à la fois dans les mésopores et les pores inter-particulaires. De plus, les diffusions d'espèces solvatées à travers les pores inter-particulaires (entre les particules de carbone) ont été considérées. Nous considérons également un phénomène appelé intra-flux, qui décrit l'échange de masse entre ces pores à différentes échelles. La capacité de décharge de la cathode contenant des particules de carbone très mésoporeuses (45%) est principalement limitée par l'inhibition du transport des espèces solvatées due à l'obstruction des pores inter-particulaires. De plus, la capacité de décharge a été améliorée en augmentant la porosité inter-particulaire (35%). Ceci montre que la porosité inter-particulaire de la cathode est un paramètre crucial, car elle permet le transport d'espèces solvatées dans la cathode et permet l'expansion en volume des précipités Li 2 S. Par conséquent, ces résultats montrent que la capacité de décharge des batteries Li-S pourrait être améliorée en augmentant la porosité inter-particulaire.

  et l'interprétation des réactions à l'origine des caractéristiques des voltammogrammes cycliques expérimentaux (CV) ont été effectuées à l'aide d'un modèle mathématique. Le CV expérimental de la S 8 solvatée mesuré à 5 mV.s -1 , se compose de deux pics de réduction (R 1 et R 2 ) et de deux pics d'oxydation (O 1 et O 2 ). De plus, les CV expérimentaux de S 8 mesurés à différentes vitesses de balayage montrent que l'intensité du pic d'O 2 n'augmente pas lorsque la vitesse de balayage augmente. Cela suggère que les réactions électrochimiques qui sont couplées à des réactions chimiques, forment la base du pic d'O 2 . Cependant, le mécanisme de réaction conventionnel comprend de telles étapes de réaction. Par conséquent, le CV simulé produit à l'aide du mécanisme de réaction conventionnel n'a pas reproduit le pic d'O 2 . Ces résultats prouvent que le mécanisme de réaction conventionnel utilisé dans de nombreux modèles de batteries Li-S n'est pas suffisant pour simuler les caractéristiques complètes du CV expérimental du S 8 solvaté dans 1 M LiTFSI dans TEGDME:DOL.Des études spectroscopiques Raman in situ et opérationnelles ont montré que les radicaux S 3 *-et S 4 *-sont produits dans des batteries Li-S contenant 1 M LiTFSI en TEGDME:DOL[START_REF] Hannauer | The Quest for Polysulfides in Lithium-Sulfur Battery Electrolytes: An Operando Confocal Raman Spectroscopy Study[END_REF][START_REF] Wu | In Situ Raman Spectroscopy of Sulfur Speciation in Lithium-Sulfur Batteries[END_REF]. Par conséquent, nous avons mis à jour le mécanisme de réaction de notre modèle de voltampérométrie cyclique en incluant quelques étapes supplémentaires dans la réaction qui impliquent des radicaux S 3 *-et S 4 *-. ont déjà été rapportées par Gaillard et al. et elles ont été déterminées en utilisant les CV du Li 2 S 6 solvaté dans le DMF[START_REF] Gaillard | Polysulfides in dimethylformamide: Only the radical anions S3-and S4-are reducible[END_REF]. Alors que la réaction d'oxydoréduction S 3 *-/S 3 2-a été proposée par Barschasz et al.[START_REF] Barchasz | Lithium/Sulfur Cell Discharge Mechanism: An Original Approach for Intermediate Species Identification[END_REF].Le CV simulé de S 8 solvaté, produit à l'aide du mécanisme de réaction mis à jour, non seulement reproduisait l'O 2 , mais ses intensités simulées n'augmentaient pas avec la vitesse de balayage. Les évolutions simulées des courants de réaction individuels montrent que le pic O 2 correspond à l'oxydation de S 3 2-à S 3 *-. La production de S 3 *-et sa réduction ultérieure en S 3 2-, sont cinétiquement contrôlées par la dissociation chimique de S 6 2-. Puisque la vitesse de la réaction de dissociation S 6 2-ne dépend pas de la vitesse de balayage, les productions de S 3 *- et S 3 2-n'augmentent pas avec la vitesse de balayage. Par conséquent, l'intensité du pic d'O 2 n'augmente pas avec la vitesse de balayage. L'intensité expérimentale du pic R 1 était faible dans le premier CV du Li 2 S 8 solvaté. Alors que le pic R 1 était totalement invisible dans celui du Li 2 S 6 solvaté. Les évolutions du courant de réaction individuel montrent que le pic R 1 correspond aux réactions de réduction S 8 /S 8 2-et S 8 2-/S 6 2-. Initialement, la solution solvatée de Li 2 S 8 et Li 2 S 6 ne contient pas de S 8 . Par conséquent, le pic R 1 était moins intense dans le premier CV du Li 2 S 8 et il est complètement invisible dans celui du Li 2 S 6 . Ces deux effets sont dus à l'absence de réaction S 8 /S 8 2-. Ces résultats confirment en outre que le mécanisme de réaction complet mis à jour représente bien les réactions réelles des polysulfures dans 1 M LiTFSI en TEGDME:DOL. Par conséquent, nous avons examiné un ensemble complet d'étapes de réaction dans le modèle présenté dans les chapitres 6 et 7.Le mécanisme de réaction et la limitation des précipitations de Li 2 S (s) dans les batteries Li-S sont encore en discussion. C'est pourquoi, au chapitre 6, nous avons développé un modèle complet de continuum 1D basé sur la théorie classique de la nucléation électrochimique pour étudier la précipitation du Li 2 S (s) dans une pouch cell simplifiée. Notre modèle, considère que la nucléation et la croissancede Li 2 S (s) se produisent en raison d'un type de réaction similaire qui est mentionné ci-dessous, 8𝐿𝑖 + + 𝑆 4 2-+ 6𝑒 -→ 4𝐿𝑖 2 𝑆 (Eq.14) De plus, notre modèle est capable de prédire la distribution granulométrique et la couverture Li 2 S (s), ce qui est obtenu par l'incorporation de l'équation bilan de la population des particules. Une électrode plane en carbone vitreux a été utilisée comme cathode pour la « pouch cell » et une feuille de Li métallique a été utilisée comme anode. Les électrodes étaient séparées par deux séparateurs en fibre de verre et une membrane Ohara sélective d'ions. Le catholyte était 100 mM Li 2 S 6 dans 1 M LiTFSI dans TEGDME:DOL et 1 M LiTFSI pur dans TEGDME:DOL était utilisé comme anolyte. La pouch cell a été déchargée galvanostatiquement pour réduire son potentiel à 2V et déclencher la nucléation de Li 2 S (s) , qui a été suivie par une étape de décharge potentiostatique. Pendant la phase de décharge potentiostatique, on observe l'apparition d'un pic. Les résultats simulés de notre modèle de nucléation et de croissance ont un bon accord semiquantitatif avec les résultats expérimentaux. Les résultats de la simulation montrent que la réduction de S 6 2-à S 4 2-domine l'étape de décharge galvanostatique. De plus, les résultats montrent également que la nucléation de Li 2 S (s) commence lorsque le potentiel cellulaire tombe légèrement en dessous de 2,02 V. En dessous de 2,02 V, le sur-potentiel de la réaction de nucléation est favorable à la production de noyaux stables. Nos résultats de simulation montrent également que le courant de la croissance électrochimique des dépôts existants de Li 2 S (s) domine le pic de courant global formé pendant l'étape de décharge potentiostatique. Dans notre modèle, nous avons supposé que la réaction de croissance électrochimique de Li 2 S (s) se produit à la limite de la triple phase qui entoure les dépôts de Li 2 S (s) existants. Initialement, en raison de la nucléation de Li 2 S (s) , l'aire de surface de la limite triphasée augmente, donc le courant de la réaction de croissance électrochimique de Li 2 S (s) augmente également pendant le même temps. Cependant, la surface de la limite triphasée diminue lorsque la couverture de Li 2 S (s) dépasse 40 %. Par conséquent, le courant de la réaction de croissance électrochimique de Li 2 S (s) et le courant total diminuent en raison de l'augmentation de la couverture de Li 2 S (s) sur la surface de l'électrode. De plus, le courant total et les courants de toutes les réactions électrochimiques se stabilisent lorsque la couverture de Li 2 S (s) augmente de plus de 80%. Les résultats de nos simulations montrent que la nucléation de Li 2 S (s) est pilotée par son sur potentiel et que la réaction de croissance électrochimique de Li 2 S (s) est limitée par la surface électroactive. Ces conclusions concordent avec les observations expérimentales de Fan et al. [212]. Cependant, la limitation de la surface de la croissance électrochimique de Li 2 S (s) peut ne pas se produire dans les cathodes poreuses qui ont une très grande surface spécifique. À l'avenir, la réaction de nucléation et de croissance du Li 2 S (s) pourrait être incorporée dans notre modèle de décharge pour étudier l'impact de la décharge galvanostatique sur le Li 2 S (s) dans les batteries Li-S pratiques. De plus, l'énergie libre pour la formation de noyaux hétérogènes dans notre modèle, dépend des énergies de liaison et de surface. Par conséquent, notre modèle de nucléation et de croissance pourrait être utilisé pour étudier l'électrodéposition de Li 2 S (s) sur différentes surfaces d'électrodes et en utilisant différents électrolytes. Au chapitre 7, nous avons présenté les résultats de nos travaux expérimentaux qui ont été réalisés pour valider notre modèle de rejets. Le cycle galvanostatique de nos cathodes(composites C/S) a été réalisé à l'aide de piles bouton Li-S. Chacune de ces piles, se compose d'une cathode C/S, d'une anode en Li métallique et d'un séparateur celgard qui est mouillé avec 1 M LiTFSI pur dans un électrolyte TEGDME:DOL. La porosimétrie et l'analyse de surface de la poudre de carbone utilisée dans nos cathodes C/S, montrent que le précurseur carboné est formé par des agglomérats de nanoparticules de carbone de 50 nm qui ont des micropores de 2,5 nm. Cependant, la microporosité de la poudre de carbone était très faible et ses micropores ont été bloqués après imprégnation au soufre. Nous avons donc modifié notre modèle de décharge en négligeant la microporosité des particules de carbone. Cela a été fait en négligeant le terme intra-flux dans l'équation de conservation de masse de l'espèce dans les pores inter-particulaires. De plus, nous avons également ignoré la conservation de masse de l'espèce dans les mésopores. En outre, nous avons également examiné le mécanisme de réaction global, qui a été déterminé à l'aide du modèle de voltampérométrie cyclique. Les résultats simulés de notre modèle de décharge modifié correspondent de façon semiquantitative aux expériences. La correspondance exacte des résultats de la simulation avec l'expérience n'a pas été obtenue en raison des complications découlant des nombreux paramètres des étapes de réaction et des espèces considérées dans notre modèle. Les résultats de simulation de notre modèle montrent que les capacités de décharge sont cinétiquement limitées par la lenteur des réactions des polysulfures d'ordre inférieur plutôt que par le transport des espèces solvatées dans l'électrolyte de la cathode. Ceci est principalement dû au fait que nos cathodes C/S ont des porosités très élevées, ce qui permet un meilleur transport de l'espèce à l'intérieur de la cathode. En raison de leur porosité élevée, les pourcentages en volume des particules de carbone dans nos cathodes sont faibles. De plus, les surfaces spécifiques électroactives de nos cathodes sont également faibles, car elles contiennent moins de particules de carbone par unité de volume de la cathode. Par conséquent, en raison de la faible surface électroactive de notre cathode, le surpotentiel d'activation des réactions lentes de réduction des polysulfures inférieurs pourrait avoir entraîné une baisse potentielle vers la fin de la décharge. Cette conclusion suggère que la capacité de décharge peut être améliorée en augmentant la surface électroactive. On pourrait y parvenir soit en augmentant le pourcentage en volume des particules de carbone, soit en utilisant des particules de carbone très mésoporeuses dans la cathode. En conclusion, nous avons développé de multiples modèles pour évaluer et optimiser les phénomènes de décharge des batteries Li-S. Nos modèles aident également à analyser les résultats de différentes expériences électrochimiques telles que la décharge galvanostatique, la décharge potentiostatique et la voltampérométrie cyclique. Les perspectives de ce travail de doctorat comprennent la détermination des paramètres du modèle à l'aide des expériences dédiées et l'incorporation de mécanismes à notre modèle de décharge pour le transformer en un modèle pour le cycle complet.

  3.2.1 Creation of initial C/S mesostructure................................................................................... 3.2.2. Development of the 3D kMC-VSSM model and its assumptions ........................................

3.2.3. Equations for rate constants of different types of events .................................................. 3.2.4. Working principle of the 3D kMC-VSSM code .................................................................... 3.3. Impact of discharge rate ..................................................................................................... 3.3.1. General effective evolutions of C/S mesostructure ............................................................ 3.3.2. Mesoscale evolutions of Li 2 S deposits over carbon ............................................................ 3.3.2.1. Distribution of Li 2 S particles from the carbon surface ................................................. 3.3.2.2. Size distribution of Li 2 S (s) clusters .................................................................................

  Therefore, we have developed a novel 3D kMC model which explicitly simulates structural evolutions of C/S mesostructure and Li 2 S precipitation during discharge. Our model includes phenomena such as dissolution reaction of S 8(s) , diffusions and reduction reactions of dissolved sulfur and polysulfides and electrodeposition of Li 2 S. The main objective of this AB) carbon nanoparticles in the cathode whose surface area is 123.6 m 2 .g -1[START_REF] Zheng | Revisit Carbon/Sulfur Composite for Li-S Batteries[END_REF], which is closer to that of our mesostructure (133.3 m 2 .g -1 ).

	representing the porous carbon mesostructure was created using a commercial software
	called Geodict. The side length of this cubic box and the diameter of the carbon particles were
	set to 50 and 25 nm respectively. The porosity of the entire mesostructure was set to 67%.
	The cubic box was then meshed along each side into 100 cubic volumetric elements called
	voxels. Although the carbon mesostructure presented here is not tied to a direct experimental
	measurement, its continuum-level descriptors are relevant with the previously reported
	carbon host materials. Ma et al., utilized cauliflower like carbon/sulfur composite cathode
	material, in which the size of the carbon particles is 25 nm [130]. The volume percentage of
	the pores in their cathode material with sizes above 20 nm is 63% which is closer to the
	porosity of the mesostructure (67%) presented in our manuscript. Zheng et al., used
	Acetylene Black (
	cathode activation overpotentials [127], transport overpotential [114,128], etc. Since, our
	kMC model simulates the discharge of C/S composite cathodes at mesoscopic level, the
	aforementioned macroscopic phenomena cannot be described by it. The impact of the
	cathode design and the C-rate on the discharge performance is discussed in the Chapter 4.
	The contents of this chapter were reported in a research paper titled, 'A three dimensional
	kinetic Monte Carlo model for simulating the carbon/sulfur mesostructural evolutions of
	discharging lithium sulfur batteries', which was published in the journal called Energy Storage
	Material [129]. This research paper was co-authored by V. Thangavel, O. X. Guerrero, M.
	Quiroga, A. M. Mikala, A. Rucci and A. A. Franco.
	3.2 Theoretical methodology
	3.2.1 Creation of initial C/S mesostructure
	An in silico method was used to create a 3D simulation box based on the structural and
	geometrical properties of the desired initial C/S mesostructure used in our kMC model (Fig.
	3.1). Initially, a cubic box containing randomly distributed spherical carbon particles

chapter is to present the development our kMC model as a methodology to understand the mesoscale evolutions of C/S composite cathodes and Li 2 S deposits during discharge. Note that the impacts of discharge C-rate on the performance such as the capacity limitation and polarization of the discharge curves are due to certain macroscopic phenomena. Some of these phenomena are the overpotential due to electrolyte resistance

[START_REF] Zhang | Modeling the voltage loss mechanisms in lithium-sulfur cells: the importance of electrolyte resistance and precipitation kinetics[END_REF]

, anode and

Table 3 .1. Parameters

 3 used in our Li-S kMC model Assumed parameters which are chosen to speed up the discharge simulation.

	Parameter	Name	Value	unit
	𝑟 𝑆 8(𝑙)	Radius of gyration of 𝑆 8(𝑙)	2.0 × 10 -9b	𝑚
	𝑟 𝑆 4(𝑙) 2-	Radius of gyration of 𝑆 4(𝑙) 2-	3.0 × 10 -9 b	𝑚
	𝑟 𝑆 2(𝑙) 2-	Radius of gyration of 𝑆 2(𝑙) 2-	2.0 × 10 -9b	𝑚
	𝜂	Viscosity of the electrolyte	2.5 a	𝑘𝑔. 𝑚 -1 . 𝑠 -1
	𝑧	Distance for diffusion	24 × 10 -9a	𝑚
	𝐾 𝑆 8(𝑠)→ 𝑆 8(𝑙) 𝑐ℎ𝑒	Rate constant for chemical	10 a	𝑠 -1
		dissolution of 𝑆 8(𝑠)
	𝑈 0	Standard potential for	2.1 c	𝑉
		𝑆 4(𝑙) 2-/𝑆 2(𝑙) 2-electrochemical reaction
	b Radius of gyration values adopted from Ref. [109].

a c

Figure 3.8. Evolutions

  of number of 𝑆 8(𝑠) (navy blue lines) and 𝐿𝑖 2 𝑆 (red lines) particles and 𝐿𝑖 2 𝑆 (𝑠) reduction reactions start to become dominant at around 250 mAh. g S 8(s) -1 in both the simulations. The simulated discharge curves remain relatively flat from 250 to 1200 mAh. g S 8(s)-1 : this stage in the discharge simulations correspond to the second low discharge plateau seen in experiments (Fig.3.7). The number of 𝐿𝑖 2 𝑆 (𝑠) particles continue to increase during this relatively flat discharge stage and until the end of the simulations (Fig.3.8), whereas the concentrations 𝑆 2(𝑙) 2-particles start to decrease at around 800 mAh. g S 8(s)

	3.5), while the concentrations of 𝑆 4(𝑙) 2-continue to increase and those of the 𝑆 2(𝑙) 2-start to 𝑆 2(𝑙) 2-/
	increase. This indicates that the 𝑆 4(𝑙) 2-/𝑆 2(𝑙) 2-reduction reaction starts at this stage in both the
	discharge simulations. Furthermore, the electrodeposition of 𝐿𝑖 2 𝑆 (𝑠) also starts at this stage
	(≈100 mAh. g S 8(s) -1 ), since the numbers of 𝐿𝑖 2 𝑆 (𝑆) particles for both C/2 and 2C simulations
	start to increase (Fig. 3.8).
	the porosity of the C/S mesostructure during the C/2 (solid lines) and 2C (dashed lines)
	discharge simulations.
	At around 250 mAh. g S 8(s) -1 when the approximate discharge curves start to become relatively
	flat -the 𝑆 4(𝑙) 2-concentrations for both simulations start to decrease, whereas the numbers of
	𝐿𝑖 2 𝑆 (𝑠) particles start to increase at a faster rate. This suggest that the 𝑆 4(𝑙) 2-/𝑆 2(𝑙) 2-and

  3.11). The peaks of all the 𝐿𝑖 2 𝑆 (𝑠) distributions for the both discharge simulations -are situated at 1 nm from the carbon surface. This indicates that the 𝐿𝑖 2 𝑆 (𝑠) electrodepositions during C/2 and 2C discharge simulations are dominated by the nucleation of 𝐿𝑖 2 𝑆 (𝑠) deposits since they are very close to the carbon surface (Fig.3.11). Initially, at 208 and 648 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 , the 𝐿𝑖 2 𝑆 (𝑠) distributions of C/2 discharge simulation are all lower than those of 2C. Furthermore, at 1050 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 𝐿𝑖 2 𝑆 (𝑠) distributions for both C/2 and 2C discharge simulations are relatively close to each other.

Moreover, the numbers of 𝐿𝑖 2 𝑆 (𝑠) particles beyond 2 nm for both C/2 and 2C discharge simulations start to overlap each other at 1050 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠) -1 (Fig.

3.11)

. Finally, at 1229 𝑚𝐴ℎ. 𝑔 𝑆 8(𝑠)

  2 𝑆 (𝑠) . The results produced using the radial distribution function, show that 𝐿𝑖 2 𝑆 (𝑠) coverage over carbon increases with the C-rate, while the 𝐿𝑖 2 𝑆 (𝑠) deposits formed during slow C-rate (C/2) have

relative large number of 𝐿𝑖 2 𝑆 (𝑠) particles away from the carbon surface. These effects are due to nucleation and growth dynamics of 𝐿𝑖 2 𝑆 (𝑠) electrodepositions, where the evolutions of the 𝐿𝑖 2 𝑆 (𝑠) distributions over the carbon surface indicate that growth process of 𝐿𝑖 2 𝑆 (𝑠) is slightly faster during slow C-rate (C/2). These aforementioned conclusions made from our kMC model are consistent with previously reported modelling and experimental results of Andrei et al.

Table 4 .

 4

	Species	Specie	D	(𝑪 𝒓𝒆𝒇 ) 𝒊	U	  j	
	Index	s	(m 2 .s -1 )	mM c	V
	(i)	formul
		a

1. Electrochemical reactions and physical parameters.

Electrochemical reactions 𝑬 𝑩 /𝑬 𝑭 a (J. mol -1 ) b Assumed parameters taken from reference

[START_REF] Kumaresan | A Mathematical Model for a Lithium -Sulfur Cell[END_REF]

. c Calculated based on the assumed densities.

Table 4 .

 4

	React.	Precipitation/			𝒌 𝒌 𝒑𝒓𝒆𝒄 a	Species	𝑲 𝒌 𝒔𝒑	Molar
	Index	dissolution reactions	(mol 2 .m -6 .s -1 )	formula	(mol 3 .m -9 )	volume
								(m -3 .mol)
	{1}	𝑆 8(solid)  𝑆 8(soln)			35 (s -1 )	𝑆 8(solid)	19	1.239× 10 -4b
								(mol.m -3 ) b
	{2}	2 𝐿𝑖 (soln) 2+	+ 𝑆 8(soln) 2-		10 -20	𝐿𝑖 2 𝑆 8(solid) 500 a	1.5× 10 -4c
		𝐿𝑖 2 𝑆 8(solid)
	{3}	2 𝐿𝑖 (soln) 2+	+ 𝑆 6(soln) 2-		10 -20	𝐿𝑖 2 𝑆 6(solid) 500 a	1.1× 10 -4c
		𝐿𝑖 2 𝑆 6(solid)
	{4}	2 𝐿𝑖 (soln) 2+	+ 𝑆 4(soln) 2-		10 -20	𝐿𝑖 2 𝑆 4(solid) 500 a	7.5× 10 -5c
		𝐿𝑖 2 𝑆 4(solid)
	{5}	2 𝐿𝑖 (soln) 2+	+ 𝑆 2(soln) 2-		10 -9	𝐿𝑖 2 𝑆 2(solid) 30 a	4.317× 10 -5b
		𝐿𝑖 2 𝑆 2(solid)
	{6}	2 𝐿𝑖 (soln) 2+	+ 𝑆 (soln) 2-		5×10 -7	𝐿𝑖 2 𝑆 (solid) 10 a	2.768× 10 -4b
		𝐿𝑖 2 𝑆 (solid)

2. Precipitation/dissolution reactions and physical parameters. b Assumed parameters taken from reference

[START_REF] Kumaresan | A Mathematical Model for a Lithium -Sulfur Cell[END_REF]

. c Calculated based on the assumed densities.

Table 1

 1

	is given by the Butler-Volmer

𝑖

(4.11)

  𝜑 = (𝑐 1 ) 𝑆 8 𝑉 𝑆 8 + (𝑐 1 ) 𝑆 8 2-𝑉 𝑆8 2-

Table 4 . 3 .

 43 General parameters of the simulated Li-S cell.

	Parameter name	Parameter	Value	Unit
		symbol
	Cathode thickness	𝐿 𝑐𝑎𝑡	40 a	μm
	Separator thickness	𝐿 𝑠𝑒𝑝	20 a	μm
	Separator porosity	𝜀 𝑠𝑒𝑝	50 a	%
	Particle radius	𝑅 𝑝	100 b	nm
	Pore radius	𝑟 𝑝	3 b	nm
	Particle number density	𝜌 𝑐	1.79× 10 20c	m -3
	Number pores per particle	𝑁 𝑝	22222 c	-
	Number of cathode bins	𝑃	10 a	-
	Number of separator bins	𝑄	5 a	-
	Escape function	𝜒	0.5 a	-
	Choking thickness	𝛿 𝑐ℎ𝑜𝑘	3 a	nm
	Choking extension	𝐿 𝑐ℎ𝑜𝑘	4 a	nm
	Tunneling threshold	𝛿 𝑡𝑢𝑛	5 a	nm
	Scaling factor tunneling	𝐿 𝑡𝑢𝑛	2.5 a	nm
	function

Table 4 . 6 .

 46 Areal sulfur loadings of the cathodes with different C/S composite structure

	C/S ratio	Areal 𝑺 𝟖(𝐬𝐨𝐥𝐢𝐝) loading
	1:1.33	3.30 mg.cm -2
	1:1 (Reference)	2.48 mg.cm -2
	1:0.67	1.66 mg.cm -2

  3.1. Cell setup ........................................................................................................................... 5.2.2. Preparation of polysulfides containing electrolyte solutions ........................................... 5.2.3. Cyclic voltammetry measurements ..................................................................................

5.3. Theoretical Methodology ................................................................................................. 5.3.1. Simulated domain and governing equations .................................................................... 5.3.2. Computational implementation .......................................................................................

Table 5 . 1 .

 51 Properties of the solvents in different 1 M LiTFSI supporting electrolyte and visible cyclic voltammetry peaks measure at 50 o C [62].

	Solvents	Dielectric	Donor	Accept	Reduction peaks		Oxidation peaks
		constant (𝜺)	No. (DN)	or No. (AN)	R 1	[V]	R 2	O 2	[V] O 1	O i (inter-
										mediate)
	DMSO	46.5	29.8	19.3	-1.05		-1.55	-1.50 -0.65	-
	DMF	39.7	26.6	16	-1.10		-1.60	-1.55 -0.80	-1.20
	DMA	37.8	27.8	13.6	-1.10		-1.80	-1.70 -0.85	-1.30
	DME	7.1	24/20	10.2	-1.30		-1.60	-	-0.70	-
	DOL:DME 7.1 (DOL)	18	-	-1.15		-1.40	-	-0.60	-
	TEGDME	7.5	16.6	10.5	-1.10		-1.50	-1.30 -0.75	-
	ACN	35.9	14.1	18.9	-1.00		-1.20	-1.10 -0.70	-
	TMS	43.3	14.8	19.2	-1.00		-1.30	-1.10 -0.60	-
	Diox:DME 2.2 (Diox)	14.8	10.8	-1.40		-	-	-0.50	-
		7.1 (DME)	20	10.2
	Few of the previous cyclic voltammetry studies of dissolved S 8 and lithium polysulfides have
	utilized mathematical models to simulate the experimental CVs and to interpret the
	underlying reaction mechanisms behind them. Levillain et al. and Jung et al. have proposed
	mathematical models to simulate the first reduction and oxidation peaks of S 8 dissolved in

Yamin et. al. studied the cyclic voltammetry of different polysulfides dissolved in 1M LiClO 4 in THF, where they compared impact of chain lengths, concentration and scan rate on the peak currents in order to deduce the kinetics and plausible reaction mechanisms

[START_REF] Yamin | Oxidation/Reduction Mechanismsof Polysulfidesin THF Solutions[END_REF]

. Cyclic voltammetry was also used to quantify the crossover of polysulfides through ion-selective membranes, in order to assess their effectiveness in limiting polysulfide shuttle

[START_REF] Li | Polysulfide-Blocking Microporous Polymer Membrane Tailored for Hybrid Li-Sulfur Flow Batteries[END_REF][START_REF] Doris | Understanding and controlling the chemical evolution and polysulfide-blocking ability of lithium-sulfur battery membranes cast from polymers of intrinsic microporosity[END_REF]

. Dimethylformamide (DMF), which are based on partially similar reaction mechanisms involving two one-electron electrochemical reactions namely S 8 /S 8 *-and S 8 /S 8 2-[179,183]. Gaillard et al., investigated the CV of Li 2 S 6 in DMF to identify the reaction mechanism behind

  is clear from the above examples, that there are several reported studies devoted to the investigation of the impacts of different electrolytes on the characteristics of CVs of dissolved S

8 . However, the comparative cyclic voltammetry studies of different types of lithium polysulfides and S 8 dissolved in a single electrolyte are very limited. Therefore, this chapter presents a comparative cyclic voltammetry study of different polysulfides solutions such as S 8 , Li 2 S 8 and Li 2 S 6 dissolved in 1 M LiTFSI in TEGDME:DOL, which is one of the widely used Li-S battery electrolyte. Although DME:DOL is the most widely used electrolyte, TEGDME:DOL is

  𝐴 and 𝐴 0 are in electro-active and geometrical surface area, respectively of the working electrode. 𝑆 𝑖𝑗 , 𝑛 𝑗 and 𝑖 𝑗 are respectively the stoichiometric coefficient of the species 𝑖 (See Appendix -Eq. A-5.15), number of electrons and current density of an electrochemical reaction 𝑗 of form, It should be noted that the 𝐺 𝑖 is zero in the finite volume bins which are not closest to the electrode surface. In the equation above, 𝛾 𝑖𝑘 is the stoichiometric coefficient of the species 𝑖 involved in the dissolution/precipitation of the sulfur based solid species 𝑘 (See Apendix -Eq. 𝑉 𝑘 is the molar volume of the sulfur based solid species. Since sulfur based solid species such as 𝑆 8(𝑠) and 𝐿𝑖 2 𝑆 (𝑠) are insulating, their precipitation over the surface of the working electrode reduces the active surface (𝑎) due to surface passivation.

	where, 𝑈 𝑗 0 and 𝑖 𝑗 0 respectivel are the standard potential and exchange current density of an
	electrochemical reaction 𝑗 and 𝐸 is the working electrode potential.
	The total electrode current (𝐼 𝑡 ) is calculated as follows,
	𝑁 𝑖 ) 𝑥=0 = (𝐷 𝑖 𝐼 𝑡 = 𝐴 ∑ 𝑖 𝑗 𝜕𝑐 𝑖 𝜕𝑥 ) 𝑥=0 = 𝑗 𝑎 = 𝑎 0 Θ(𝛿)	𝐴 𝐴 0 ∑ 𝑗	𝑛 𝑗 𝐹 𝑆 𝑖𝑗 𝑖 𝑗	(5.6) (5.9) (5.13)
	𝑧 𝑖 = 𝑛 𝑗 𝑒 -where 𝐴 is the active surface area of the working electrode, which is the product of the (5.7) volume of the finite volume bin closest to the electrode (𝑉 𝑥=0 𝑏𝑖𝑛 ) and the active specific surface area (𝑎), In the equation above, ∑ 𝑆 𝑖𝑗 𝑀 𝑖 𝐴 = 𝑎𝑉 𝑥=0 𝑏𝑖𝑛 (5.10)
		𝑖
	Where 𝑀 𝑖 𝑧 The current density of an individual electrochemical reaction is given by the Butler Volmer 𝑘 𝐺 𝑖 = ∑ 𝛾 𝑖𝑘 𝑘 𝑘 𝑝𝑟𝑒𝑐 𝜖 𝑘 ((𝑐 𝑖 ) 𝛾 𝑖𝑘 -𝐾 𝑘 𝑠𝑝 ) (5.11)
	equation,
	𝑖 𝑗 = 𝑖 𝑗 0 (∏ ((𝑐 𝑖 ) 𝑥=0 𝑆 𝑖𝑗 ) 𝑠𝑝 and 𝜖 𝑘 respectively are solubility product and the volume fraction of the sulfur ( (1-𝛼 𝑗 )𝑛 𝑗 𝐹(𝐸-𝑈 𝑗 0 ) 𝑅𝑇 ) (5.8) A-5.17). 𝐾 𝑘 𝑒 𝑆 𝑖𝑗 >0 𝑖 based solid species. 𝑘 𝑘 𝑝𝑟𝑒𝑐 rate constant of precipitation/dissolution reaction 𝑘.
	𝑑𝜖 𝑘 𝑑𝑡	-∏ ((𝑐 𝑖 ) 𝑥=0 -𝑆 𝑖𝑗 ) 𝑖 = 𝑉 𝑘 𝜖 𝑘 𝑘 𝑘 𝑝𝑟𝑒𝑐 ((𝑐 𝑖 ) 𝛾 𝑖𝑗 -𝐾 𝑘 𝑒 ( -𝛼 𝑗 𝑛 𝑗 𝐹(𝐸-𝑈 𝑗 0 ) ) 𝑅𝑇 𝑆 𝑖𝑗 <0 𝑠𝑝 )	)	(5.12)

𝑖 and 𝑒 -respectively are the symbols that represent a species 𝑖 and an electron. 𝑧 𝑖 is the charge of the species 𝑖.

Finally, 𝐺 𝑖 is the souce/sink term related to the heterogeneous dissolution/precipitation of sulfur based solid species over the electrode

[START_REF] Kumaresan | A Mathematical Model for a Lithium-Sulfur Cell[END_REF]

.

Table 5 .

 5 1. Parameters of different reactions (Eq. 5.16-5.27) and diffusion coefficients of dissolved species considered in the cyclic voltammetry model.

	i.	Electrochemical reaction parameters			iv.	Dissolved
	React.	Reaction		Eq. No. 𝑖 0	𝑈 0		𝛼	species
	No. (𝑗)				(A)	(V)			parameters
	1 2	𝑆 8 /𝑆 8 2-𝑆 8 2-𝑆 6 2-⁄	(5.17) (5.18)	0.150 a 3×10 -3a	-0.8281 a -0.8681 a	0.460 c Species 𝑧 𝑖 𝑀 𝑖 0.500 b	𝐷 𝑖 (m.s -1 )
	3	𝑆 6 2-𝑆 4 2-⁄	(5.19)	3×10 -4a	-0.9400 a	0.420 a	𝐿𝑖 +	1.6335×10 -10a
	4	𝑆 4 * -𝑆 4 2-⁄	(5.24)	2×10 -4a	-0.9400 a	0.500 d	𝑆 8	1.6335×10 -10a
	5	𝑆 4 2-𝑆 2 2-⁄	(5.20)	2.96×10 -6a -1.0544 a	0.200 a	𝑆 8 2-	8.0000×10 -11a
	6	𝑆 3 * -𝑆 3 2-⁄	(5.26)	7.4×10 -2a	-1.1644 a	0.420 a	𝑆 6 2-	8.0000×10 -11b
	7	𝑆 3 2-𝑆 2 2-⁄	(5.27)	1.48×10 -7a -1.2044 a	0.350 a	𝑆 4 * -	3.5000×10 -11f
	8	𝑆 2 2-𝑆 2-⁄	(5.21)	3×10 -9a	-1.4000 a	0.658 a	𝑆 4 2-	3.5000×10 -11f
		ii.	Homogeneous chemical reaction parameters	𝑆 3 * -	3.5000×10 -11f
	Reacti on No. (𝑗)	Reaction		𝑓 Eq. No. 𝑘 𝑗 (s -1 )	(mol -1 )	𝑘 𝑗 𝑒𝑞		𝑆 3 2-𝑆 2 2-	3.5000×10 -11f 3.5000×10 -11f
	1	𝑆 8 2-⇌ 2𝑆 4 * -	(5.23)	8.4 a	4.2088 a			𝑆 2-	3.5000×10 -11f
	2	𝑆 6 2-⇌ 2𝑆 3 * -	(5.25)	9.5 a	12.6263 a
		iii.	Heterogeneous precipitation/dissolution reaction parameters
	Reacti on No. (𝑘)	Reaction		𝑝𝑟𝑒𝑐 (mol 2 Eq. No. 𝑘 𝑘 .m -6 .s -1 )	𝐾 𝑘 𝑠𝑝 (mol 3 .m -9 )		Solid species	𝑉 𝑘 (m -3 .mol)
	1	𝑆 8 2-⇌ 𝑆 8(𝑠) (5.16)	5 (s -1 ) e	6 (mol.m -3 ) m		𝑆 8(𝑠)	1.239×10 -4b
	2	2𝐿𝑖 + + 𝑆 2-	(5.22)	5 ×10 -7b	10 b			𝐿𝑖 2 𝑆 (𝑠) 2.768×10 -4b
		⇌ 𝐿𝑖 2 𝑆 (𝑠)
	a Fitted parameters.

b, c, d and e

Table A -5.1.

 A Initial concentrations of dissolved polysulfides and volume fractions of solid species.

		i.	Initial concentrations of dissolved species (mM)
	Species	Species	In 6 mM S 8	In 50 mM Li 2 S 8	In 50 mM
	No. (𝑖)	𝑀 𝑖 𝑧 𝑖	solution	solution	Li 2 S 6 solution
	1	𝐿𝑖 +	1000 c	1100 c	1100 c
	2	𝑆 8	6 e	1×10 -3a	6×10 -5a
	3	𝑆 8 2-	1.78×10 -7a	50 e	1.78×10 -6a
	4	𝑆 6 2-	5.0×10 -7a	5.0×10 -5a	50 e
	5	𝑆 4 * -	2.015×10 -8a	2.015×10 -8a	2.015×10 -8a
	6	𝑆 4 2-	2.015×10 -8a	2.015×10 -8a	2.015×10 -8a
	7	𝑆 3 * -	1.990×10 -8a	1.990×10 -8a	1.990×10 -6a
	8	𝑆 3 2-	1×10 -7a	1×10 -6a	1×10 -7a
	9	𝑆 2 2-	5.2×10 -7a	5.2×10 -7a	5.2×10 -7a
	10	𝑆 2-	8.2×10 -7a	8.2×10 -10a	8.2×10 -7a

ii. Initial solid species volume fractions (no units)

	5.4. CVs of dissolved S 8
	Species	Solid	In 6 mM S 8	In 50 mM Li 2 S 8	In 50 mM
	No. (𝑘)	species	solution	solution	Li 2 S 6 solution
	1	𝑆 8(𝑠)	1×10 -10a	1×10 -10a	1×10 -10a
	2	𝐿𝑖 2 𝑆 (𝑠)	1×10 -7b	1×10 -7b	1×10 -7b
	a Assumed parameters.

b Assumed Parameters taken from Ref. [114]. c Calculated parameters based on charge conservation. e Experimental parameters.

  Ren et al. developed a comprehensive 1D continuum model which incorporates electrochemical nucleation and growth of Li 2 S deposits[START_REF] Ren | Modeling of lithium-sulfur batteries incorporating the effect of Li2S precipitation[END_REF]. This model predicts the impact of the galvanostatic discharge rates on the particle size distributions of the Li 2 S deposits. The Li 2 S nucleation kinetics of this model was described using the atomistic theory of nucleation which cannot predict the critical sizes of the nuclei and the initial nucleation densities were determined through fitting. Andrei et al.[START_REF] Andrei | Theoretical and experimental analysis of precipitation and solubility effects in lithium-sulfur batteries[END_REF] and Danner et al.[START_REF] Danner | On the influence of nucleation and growth of S8 and Li2S in lithium-sulfur batteries[END_REF] developed similar 1D models which incorporated the chemical nucleation and growth of S 8 and Li 2 S deposits. The model by Andrei et al. predicts the coverages of S 8 and Li 2 S deposits over carbon surface. The model by Danner et al. predicts the particle size distributions of S 8 and Li 2 S deposits during

Table 6 .

 6 1. Parameters of our Li 2 S electrochemical nucleation model.

	S. No Parameters Values	Units	References
	1	𝜌 𝑘	1.63	g.cm -3	[208]
	2	𝑀 𝑘	45.95	g.mol -1	Constant
	3	𝛾	0.3364	J.m -2	[209]
	4	𝐸 𝑏𝑖𝑛𝑑	-0.8811×10 -	J	[111]
			19
	5	𝑘 𝑛𝑢𝑐 0	2.6118×10 124 mol. s -1	Fitted
	6	𝑈 𝑛𝑢𝑐 0	2.086	V	Fitted
	7	𝑇	298	K	Measured
	8	𝑘 𝐵	1.38×10 -23	J.K -1	Universal
					constant
	6.3.2. Other reaction steps and their kinetic equations

  ). 𝑘 𝑗 𝑓 and 𝑘 𝑗 𝑟 are forward and reverse kinetic constants. 𝐾 𝑗 𝑒𝑞 is the equilibrium constant of the chemical reaction 𝑗, which is given by, 𝑣 𝑟 is the volume of a nucleus with size 𝑟. 𝑓(𝑣 𝑟 ) is the number density per unit volume of the nuclei with size 𝑟. The numerical solution of the population balance equation is given in the Appendix subsection A.6.1. Since, we consider both chemical (Eq. 6.19) and electrochemical growth (Eq. 6.20) of the existing nuclei in our model, the rate of change of volume fraction of the nuclei with size 𝑟 has two components (Eq. 6.27). 𝑘 𝑝𝑟𝑒𝑐 and 𝐾 𝑠𝑝 are the rate constant and solubility product of chemical precipitation of Li 2 S (6.19).

				𝜕𝑓(𝑣 𝑟 ) 𝜕𝑡	=	𝑁 ∂𝑣 𝑟	-	𝜕 𝜕𝑣 𝑟	(𝑓(𝑣 𝑟 )	𝑑𝜖 𝑟 𝑑𝑡	)	(6.26)
	where, 𝑑𝜖 𝑟 𝑑𝑡	= (	𝑑𝜖 𝑟 𝑑𝑡	)	𝑐ℎ𝑒𝑚	+ (	𝑑𝜖 𝑟 𝑑𝑡	)	𝑒𝑙𝑒𝑐𝑡𝑟𝑜	(6.27)
	The rate change of volume fraction of the nuclei with size 𝑟 due to chemical reaction (6.19) is
	given by,
	(	𝑑𝜖 𝑟 𝑑𝑡	)	𝑐ℎ𝑒𝑚	= (	𝑀 𝑘 𝜌 𝑘	𝑖 ) 𝑘 𝑝𝑟𝑒𝑐 𝜖 𝑟 (∏(𝑐 𝑖 ) 𝛾 𝑖𝑗	-𝐾 𝑠𝑝 )	(6.28)
	𝐾 𝑗 𝑒𝑞 = 6.3.3. Population balance of Li 2 S particles where The rate of change of volume fraction of the nuclei with size 𝑟 due to electrochemical reaction 𝑘 𝑗 𝑟 𝑘 𝑗 𝑓 (6.25)
	(6.20) is given by
	Since, the nucleation and growth of the Li 2 S deposits are described by separate reaction steps,
	the particle size distribution of the deposits can be determined by the following population balance equation [207], ( 𝑑𝜖 𝑟 𝑑𝑡 ) 𝑒𝑙𝑒𝑐𝑡𝑟𝑜 = -𝑁 𝑟 4 𝑛 𝑗 ( 𝑀 𝑘 𝜌 𝑘 ) 𝑖 𝑗 𝐹 𝑎 𝑟 𝑔𝑟𝑜𝑤 (6.29)

Table 6 . 2 .

 62 Kinetic parameters of the different reactions and the diffusion coefficient of the dissolved species taken into account in our model.

	ii.	Electrochemical reaction parameters		v.	Dissolved
	Reactio	Reaction	Eq.	𝑖 0	𝑈 0	𝛼		species
	n No. (𝑗)		No.	(A)	(V)			parameters
	1 2	𝑆 6 2-𝑆 4 2-⁄ 𝑆 4 2-𝑆 2 2-⁄	(6.13) 1.90×10 -2b 2.20 a 0.500 b Species 𝑀 𝑖 𝑧 𝑖 (6.15) 1.97×10 -5a 2.17 a 0.500 b	𝐷 𝑖 (m.s -1 )
	3	𝑆 3 * -𝑆 3 2-⁄	(6.16) 9.73×10 -4a 2.17 a 0.420 a	𝐿𝑖 +	1.6335×10 -10a
	4	𝑆 3 2-𝑆 2 2-⁄	(6.17) 1.97×10 -8a 2.13 a 0.350 a	𝑆 6 2-	5.6000×10 -10a
	5	𝑆 2 2-𝑆 2-⁄	(6.18) 3.94×10 -10a 2.11 a 0.658 a	𝑆 4 2-	6.0000×10 -10a
	6	𝑆 4 2-𝐿𝑖 2 𝑆 ⁄	(6.20) 7.2 ×10 -5a	2.086 a 0.500 c	𝑆 3 * -	3.5000×10 -10e
	ii. Homogeneous chemical reaction parameters		𝑆 3 2-	3.5000×10 -10e
	Reactio n No.	Reaction	Eq. No.	𝑘 𝑗 𝑓 (s -1 )	𝑘 𝑗 𝑒𝑞 (mol -1 )	𝑆 2 2-	3.5000×10 -10e
	(𝑗)
	2	𝑆 6 2-⇌ 2𝑆 3 * -(6.12) 9.5 a	12.6263 a	𝑆 2-	3.5000×10 -10e
		iii. Heterogeneous precipitation/dissolution reaction parameters
	Reactio	Reaction	Eq.	𝑘 𝑘	𝐾 𝑘 𝑠𝑝		Solid
	n No. (𝑘)		No.	1 ) (mol 2 .m -6 .s -	(mol 3 .m -9 )	species

Table 7

 7

	.1. Mesostructural properties of pristine carbon powder
	i.	Porosimetry parameters
	Type of pore volumes	Values	Porosity
	a. BJH adsorption cumulative pore	0.27 cm 3 . g -1 0.33
	volume (1.7nm < 𝑑 𝑝𝑜𝑟𝑒 < 300nm)
	b. t-plot micropore volume	0.0066	0.012
			cm 3 . g -1
	ii.	Surface area parameters
	Types of surface area	Values
	a. BET surface area	80.54 m 2 . g -1
	b. t-plot external surface area	64 m 2 . g -1
	c. t-plot micropore area	16.5 m 2 . g -1
	d. Calculated surface area	65.5 m 2 . g -1

Table 7 . 2 .

 72 Mesostructural properties of sulfur impregnated carbon nano powder.

	iii.	Porosimetry parameters
	Type of pore volumes	Values
	a. BJH adsorption cumulative pore volume	0.0746 𝑐𝑚 3 . 𝑔 -1
	(1.7nm < 𝑑 𝑝𝑜𝑟𝑒 < 300nm)
	b. t-plot micropore volume	-0.000232 𝑐𝑚 3 . 𝑔 -1
	iv.	Surface area parameters
	Types of surface area	Values
	e. BET surface area	16.5 𝑚 2 . 𝑔 -1
	f. t-plot external surface area	16.6 𝑚 2 . 𝑔 -1
	g. t-plot micropore area	Not available

Table 7 . 3 .

 73 The breakdown of solid ingredients used in the slurry.

	S.No	Materials	Mass	Weight %
	1	C/S (mass ratio=1:1)	0.4425 g	75%
	2	PVdF (Binder)	0.0885 g	15%
	3	Carbon super P (Csp) (Conductive additive) 0.059 g	10%
	4	Total solid mixture (C/S + PvDF + Csp)	0.590 g	100%

Table 7 . 4 .

 74 The masses and the weight fractions of the solid powder mixture and the liquid NMP solvent used in the slurry.

	S.No Materials	Mass	Weight %
	1	Total solid mixture (C/S + PVdF + Csp)	0.590 g	16.4%
	2	NMP solvent	3.000 g	83.6%
	3	Total slurry weight (C/S + PVdF + Csp + NMP)	3.590 g	100%
	7.5.2. Coating of electrode film

Table 7 . 5 .

 75 Details of the coin cells used for galvanostatic cycling at different C-rates.

	C-rate	Sulfur mass loading	Electrolyte volume	Current	density
		𝒎𝒈. 𝒄𝒎 -𝟐	𝝁𝑳	𝒎𝑨. 𝒄𝒎 -𝟐
	C/20	1.46	27.7	6.12
	C/10	1.85	35.0	15.53
	C/5	1.70	32.3	25.2

Table 7 . 6 .

 76 The parameters of our modified discharge model are listed in the Table 7.6. Most of the parameter values used in our discharge model are same as those used in the cyclic voltammetry model. However, in our Li-S coin cells we have used Li metal foil as the anode.Furthermore, there are uncertainties in the conversion factors for converting the standard potentials from vs Ag/Ag + to vs Li/Li + , due to the LJP in Ag/AgNO 3 electrode. Therefore, the standard potentials reported in the Table7.6 were fitted by qualitatively matching the simulated results with the experiments. Additionally, we have used different form Butler-Parameters of different reactions and diffusion coefficients of dissolved species considered in our modified discharge model.

	to 7.12) in our modified discharge

Table 7 . 7 .

 77 The initial cathode design parameters of our Li-S coin cells cycled which were at different C-rates.

	Parameters	Symbols Units	C/20	C/10	C/5
	Number density	𝜌	𝑚 -3	3.79×10 21 4.81×10 21 4.42×10 21
	of particles
	Volume fraction	𝜖 𝑆 8(𝑠)	-	0.1763	0.2234	0.2052
	of S 8(s)
	Volume fraction	𝜖 𝑃𝑉𝐷𝐹	-	0.08156	0.1034	0.0950
	of PVDF
	Maximum	𝜀 𝑚𝑎𝑥	-	0.6702	0.5821	0.6157
	cathode porosity
	Initial	cathode	𝜀 𝑖𝑛𝑖𝑡	-	0.4932	0.3581	0.4099
	porosity
	7.9. Results and discussion
	7.9.1 Galvanostatic cycling

  Bruggeman relation could have led to the over estimation of polarization during high plateau stage of the discharge.In the Eq. 7.17, (𝜖 1 ) 𝑆 8 and (𝜖 1 ) 𝐿𝑖 2 𝑆 are 𝑆 8(𝑠) and 𝐿𝑖 2 𝑆 (𝑠) volume fractions, respectively. 𝜒 is the escape function, which is the fraction 𝐿𝑖 2 𝑆 (𝑠) which do not form a thin layer over carbon surface. 𝛽 is

	𝑎 1 = 𝑎 1 𝑚𝑎𝑥 (1 -	(𝜖 1 ) 𝑆 8 + 𝜒(𝜖 1 ) 𝐿𝑖 2 𝑆 𝜀 1 𝑚𝑎𝑥	) 𝛽	Θ(𝛿 1 )	(7.17)
	Bruggeman coefficient.

  Li-S batteries models are not sufficient to simulate the complete characteristics of the experimental CV of the dissolved S 8 in TEGDME:DOL. current peak formed during the potentiostatic discharge stage. The current of the electrochemical 𝐿𝑖 2 𝑆 (𝑠) growth reaction initially increases due to increase of the electroactive surface area corresponding to the triple phase boundaries which surrounds the newly formed 𝐿𝑖 2 𝑆 (𝑠) deposits. However, this electroactive surface decreases after some time due to the growth and increase of the 𝐿𝑖 2 𝑆 (𝑠) coverage over the carbon surface, which also led to the decrease of the electrochemical 𝐿𝑖 2 𝑆 (𝑠) growth current. Furthermore, the overall current and electrochemical 𝐿𝑖 2 𝑆 (𝑠) growth current, level off when the 𝐿𝑖 2 𝑆 (𝑠) coverage increases over 80%. Our simulations results show that the nucleation of 𝐿𝑖 2 𝑆 (𝑠) are driven by its overpotential and the electrochemical 𝐿𝑖 2 𝑆 (𝑠) growth reaction is limited by the electroactive surface area. These conclusions are consistent with the experimental observations of Fan et al.[START_REF] Fan | Mechanism and Kinetics of Li 2 S Precipitation in Lithium-Sulfur Batteries[END_REF]. In the future, the 𝐿𝑖 2 𝑆 (𝑠) nucleation and growth reaction could be incorporated in our discharge model to investigate the impact of galvanostatic discharge on the 𝐿𝑖 2 𝑆 (𝑠) in the practical Li-S batteries. Furthermore, the free energy for the formation heterogeneous nuclei, in our model depend on the binding and surface energies. Therefore, our nucleation and growth model could be used to investigate the electrodeposition of 𝐿𝑖 2 𝑆 (𝑠) over different electrode surfaces and using different electrolytes.

  L'énergie est cruciale pour notre vie dans ce monde technologique moderne. En outre, le progrès technologique de notre époque a contribué à l'amélioration des conditions socioéconomiques dans le monde entier. Il est donc indispensable de produire l'énergie nécessaire afin de répondre à la demande créée par ces développements technologiques et par l'amélioration des conditions de vie du monde entier. Cependant, la majorité de l'énergie actuellement consommée est produite à partir des ressources limités tels que les combustibles fossiles. Lorsqu'ils brûlent, les combustibles fossiles émettent des gaz à effet de serre qui sont les principales causes de l'augmentation de la température mondiale et du changement climatique. C'est le problème le plus existentiel de notre époque, car Les batteries lithium-ion (LIB) sont actuellement utilisées dans les VE. Les cathodes des LIBs sont principalement des oxydes de transition métalliques, qui ont une masse molaire élevée. Une meilleure compréhension de ces phénomènes pourrait aider à améliorer les performances de ces batteries. Pour cette raison, au cours de mon doctorat, nous avons développé de multiples modèles mathématiques qui pourraient être appliqués à l'étude des phénomènes qui se produisent à différentes échelles spatiales des batteries Li-S.Dans la littérature, très peu de modèles mésoscopiques ont été rapportés. Ces modèles pourraient être utilisés pour étudier les phénomènes des batteries Li-S qui se produisent à l'échelle mésoscopique. Pour cette raison nous avons développé un nouveau modèle de type Monte-Carlo cinétique 3D (kMC), qui simule explicitement la réaction et la diffusion de différentes espèces à base de soufre dans les mésostructures carbone/soufre (C/S) créées in silico[START_REF] Thangavel | A three dimensional kinetic Monte Carlo model for simulating the carbon/sulfur mesostructural evolutions of discharging lithium sulfur batteries[END_REF]. Notre modèle kMC est utilisé pour étudier la décharge des cathodes C/S à l'échelle mésoscopique. Les différentes étapes de réaction et de diffusion considérées dans notre modèle kMC, sont celles qui se produisent du début à la fin de la décharge. Cependant, le nombre d'étapes considérées a été simplifié afin de réduire les coûts informatiques de nos

	Par conséquent, les cathodes limitent les capacités spécifiques et les densités énergétiques
	des LIBs [210]. Ainsi, afin d'augmenter l'énergie pouvant être stockée dans les batteries, la
	prochaine génération de batteries, telles que le lithium soufre (Li-S), le lithium oxygène (Li-
	O 2 ), etc. est en développement. Les batteries Li-S sont les plus prometteuses parmi la
	prochaine génération de batteries en raison de leur matière active cathodique : le soufre
	solide. La réaction cathodique globale des batteries Li-S est la suivante,
	1 8	𝑆 8 + 2𝐿𝑖 + + 2𝑒 -⇌ 𝐿𝑖 2 𝑆	(Eq. 1)
	météorologiques. Par conséquent, l'énergie produite à partir des sources intermittentes doit derrière eux sont assez complexes. En effet, différentes réactions chimiques,
	être stockée lorsque la demande est faible. Les batteries sont considérées comme des électrochimiques et de précipitation/dissolution se produisent pendant le fonctionnement
	candidats prometteurs pour stocker l'énergie produite par les sources intermittentes. Ceci est des batteries Li-S. Ces réactions impliquent de multiples espèces telles que le soufre solvaté,
	principalement dû à l'avènement des réseaux intelligents, qui nécessitent des systèmes de des polysulfures et des espèces solides à base de soufre.
	stockage d'énergie distribués. En outre, la plupart des gaz à effet de serre sont émis par le De nombreux développements ont été réalisés pour améliorer la capacité de décharge,
	transport routier dans les pays développés. Par conséquent, la seule façon d'atténuer ces l'efficacité coulombique et la cyclabilité des batteries Li-S. Cependant, en raison de la
	émissions est de passer des véhicules à moteur à combustion interne (ICE) classiques aux
	véhicules électriques (VE).

l'augmentation de la température de plus de 1,5°C peut provoquer des événements catastrophiques dans le monde entier, tels que la fonte des calottes glaciaires, la montée du niveau de la mer, des conditions météorologiques extrêmes, etc. Par conséquent, pour limiter l'augmentation de la température mondiale en dessous de 1,5 o C, nous devons passer d'une économie énergétique basée sur les combustibles fossiles à une économie énergétique renouvelable. L'éolien et le solaire sont parmi les technologies d'énergie renouvelable les moins chères et celles qui ont le moins d'impact sur l'environnement. Cependant, les sources d'énergie de ces technologies sont intermittentes. En effet, le soleil ne brille pas toute la journée et le vent ne souffle pas tout le temps à la même vitesse. De plus, la demande d'énergie dans un endroit donné varie en fonction de l'heure de la journée et des conditions La masse molaire de soufre est faible (32 g.mol -1 ) et c'est l'un des matériaux les plus abondants sur terre. Comme chaque mole de soufre accepte deux moles de Li + , les batteries Li-S ont une densité énergétique supérieure à 2567 Wh.kg -1 [210]. Le soufre solide est isolant, donc les cathodes des batteries Li-S sont des composites carbone/soufre (C/S) poreux. Le carbone dans la cathode aide à la conductivité électronique et agit également comme matériau hôte pour retenir les précipités solides tels que les précipités S 8(s) et Li 2 S (s) . Des feuillards de Li métallique sont utilisées comme anode dans les batteries Li-S, qui sont séparées de la cathode par un séparateur polymère poreux. Des électrolytes organiques non aqueux à base d'éther remplissent les pores du séparateur et de la cathode. Les courbes galvanostatiques typiques des batteries Li-S ont deux plateaux, et les mécanismes de réaction complexité de son principe de fonctionnement, l'évaluation de certains phénomènes et l'identification des limites des batteries Li-S est difficile.

  Li 2 S (s) est produite sur les particules de carbone. En effet, la cathode à faible surface interparticulaire n'a pas assez de surface pour distribuer et former une couverture fine de Li 2 S (s) sur les particules de carbone. Les mésopores sont bloqués, quand la couverture de Li 2 S (s) devient trop épaisse. De plus, ces mésopores obstrués peuvent piéger certains des polysulfures inutilisés qui s'y trouvent. Elles entraînent donc une réduction de la capacité de décharge de la cathode. Par conséquent, la capacité de décharge diminue lorsque la surface inter-particulaire est réduite. L'aire de surface interparticulaire peut être augmentée en augmentant le nombre de particules de carbone et en diminuant la taille des particules de carbone. En conclusion, la capacité de décharge d'une cathode peut être augmentée soit en augmentant la porosité et la mésoporosité inter-particulaires, soit en diminuant la taille des particules de carbone ou en augmentant le nombre de particules de carbone. Dans notre modèle de décharge multi-échelle présenté dans le chapitre 4, nous avons supposé que le volume vide des pores de la cathode et du séparateur est rempli par un électrolyte arbitraire.Par conséquent, nous avons supposé un mécanisme de réaction conventionnellement utilisé dont les étapes de réaction sont données ci-dessous, Cependant, on sait que la stabilité et le mécanisme de réaction des polysulfures changent en fonction de l'électrolyte. De plus, 1 M LiTFSI dans TEGDME:DOL a été utilisé dans les expériences rapportées dans les chapitres 5 à 7. C'est pourquoi, au chapitre 5, nous avons étudié la validité du mécanisme de réaction conventionnel utilisé en effectuant des mesures de voltampérométrie cyclique à l'aide de solutions électrolytiques contenant S 8 , Li 2 S 8 et Li 2 S 6

	𝑆 6(𝑙) 2-+ 𝑒 -⇌	3 2	𝑆 4(𝑙) 2-	(Eq. 5)
	1 2	𝑆 4(𝑙) 2-+ 𝑒 -⇌ 𝑆 2(𝑙) 2-	(Eq. 6)
	1 2	𝑆 2(𝑙) 2-+ 𝑒 -⇌ 𝑆 (𝑙) 2-	(Eq. 7)
	2𝐿𝑖 (𝑙) + + 𝑆 (𝑙) 2-⇌ 𝐿𝑖 2 𝑆 (𝑠)	(Eq. 8)
			𝑆 8(𝑠) ⇌ 𝑆 8(𝑙)	(Eq. 2)
	1 8	𝑆 8(𝑙) + 𝑒 -⇌	1 8	𝑆 8(𝑙) 2-	(Eq. 3)
	3 2	𝑆 8(𝑙) 2-+ 𝑒 -⇌ 2𝑆 6(𝑙) 2-	(Eq. 4)

  Volume fraction of 𝐿𝑖 2 𝑆 (𝑠) which form as thin layer over carbon particle Surface energy of 𝐿𝑖 2 𝑆 (𝑠) deposit Δ𝐺 𝑏𝑢𝑙𝑘 Bulk free energy for the formation 𝐿𝑖 2 𝑆 (𝑠) nuclei Δ𝐺 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 Surface free energy for the formation 𝐿𝑖 2 𝑆 (𝑠) nuclei Δ𝐺 ℎ𝑜𝑚𝑜 Homogenous free energy for formation 𝐿𝑖 2 𝑆 (𝑠) nuclei Δ𝐺 ℎ𝑒𝑡𝑒𝑟𝑜 Heterogenous free energy for formation 𝐿𝑖 2 𝑆 (𝑠) nuclei 𝐸 𝑏𝑖𝑛𝑑 Binding energy of 𝐿𝑖 2 𝑆 (𝑠) deposit over carbon surface 𝑘 𝑛𝑢𝑐 Rate of the 𝐿𝑖 2 𝑆 (𝑠) nucleation reaction Rate constant of the 𝐿𝑖 2 𝑆 (𝑠) nucleation reaction 𝑖 𝑛𝑢𝑐 Current density of the 𝐿𝑖 2 𝑆 (𝑠) nucleation reaction

	G2 𝜔 𝐿𝑖 2 𝑆 (𝑠) 𝜌 𝑐	Diglyme Number density of carbon particles
	G3	𝑅 𝑝 𝛿 𝐷𝐿	Triglyme Radius of carbon particle Diffusion layer
	G4		𝜀 1 𝑡	Tetraglyme Inter-particular porosity Time
	GDP 𝜀 2 𝑠 𝑖	Gross Domestic Product Mesoporosity Source/sink term of species 𝑖 brought about homogenous chemical reactions
	GHG 𝑎 1 𝑘 𝑗 𝑓	Greenhouse Gas Inter-particular specific surface area Forward rate constant of homogenous chemical reaction 𝑗
	HELiS HFE 𝑎 2 𝑒𝑞 𝐾 𝑗 ICEs 𝛽 𝜗 𝑖𝑗 IPCC 𝜂 𝐴 IRENA 𝑖 𝐴 0 kMC 𝐸 𝑗 𝑎 LIBs 𝑃 𝑖𝑗 𝑁 𝑖 Li-O 2 𝑄 𝑖𝑗 𝑐 𝑖 * Li 𝐼 𝐺 𝑆 𝑖𝑗 Lithium-ion LJP 𝑅 𝑘 𝑘 𝑘 𝑖 𝑗 𝑝𝑟𝑒𝑐 𝐸 LRCS ℎ 𝛼 𝑗 lsqnonlin 𝜅 𝐼 𝑡 LUC 𝐷 Δ𝑥	High Energy Lithium Sulphur batteries and cells Mesoporous specific surface area Equilibrium constant of chemical reaction 𝑗 Hydrofluorinated ether Bruggeman coefficient Stoichiometric coefficient of species 𝑖 of chemical reaction 𝑗 Internal Combustion Engines Overpotential Electroactive surface area of the electrode Intergovernmental Panel for Climate Change Current density Initial surface area of the electrode International Renewable Energy Agency Activation energy of electrochemical reaction 𝑗 Specific surface area of the electrode kinetic Monte-Carlo Stoichiometric coefficient of oxidized species 𝑖 in electrochemical reaction 𝑗 Flux of species 𝑖 Lithium-ion Batteries Stoichiometric coefficient of reduced species 𝑖 in electrochemical reaction 𝑗 Bulk concentration of species 𝑖 Lithium-oxygen Applied current Stoichiometric coefficient Lithium Lithium-ion Rate of the precipitation/dissolution reaction 𝑘 Current density of reaction 𝑗 Rate constant of the precipitation/dissolution reaction 𝑘 Electrode potential Liquid Junction Potential Planck constant Transfer coefficient of electrochemical reaction 𝑗 Laboratoire de Réactivité et Chimie des Solides Frequency factor Total current Nonlinear least-squares Land Use Change Diffusion coefficient Bin size of the control volume
	NIC	𝑔 𝑟	National Institute of Chemistry Source/sink term for electrochemical reaction Particle size of 𝐿𝑖 2 𝑆 (𝑠) deposit
	NMP OCP 𝐺 𝑓 𝜂 𝑗 𝛾	N-Methyl-2-Pyrrolidone Open Circuit Potential Source/sink term for precipitation/dissolution reaction 𝑘 Overpotential of electrochemical reaction 𝑗 Intra-flux
	OECD 𝛾 𝑖𝑘 PEG 𝐾 𝑘 𝑠𝑝	Stoichiometric coefficient of species 𝑖 in precipitation/dissolution reaction 𝑘 Organization for Economic Co-operation and Development Polyethylene Glycol Solubility product of precipitation/dissolution reaction 𝑘
	PEGDME 𝐾 𝑓𝑙𝑢𝑥	Polyethylene glycol dimethyl ether Rate constant of Intra-flux
	PSH 𝛿 1	Pumped Storage Hydropower Thickness of Li 2 S (s) layer over mesoporous carbon particle
	PVDF 𝜑	Polyvinylidene fluoride Volume fraction of the suspended particles
	QSSR 𝑉	Quasi-Solid-State Reaction Partial molar volume
	RDF SEI 𝑀 𝑘 𝜌 𝑘 𝑘 𝑛𝑢𝑐 0	Radial Distribution Function Solid Electrolyte Interphase Molar mass of solid species 𝑘 Density of solid species 𝑘
	SEM 𝜖 𝑘	Scanning Electron Microscopy Volume fraction of solid species 𝑘

Acknowledgements

NIC, Monica Marinescu of Imperial college, Claude Guéry of LRCS and Christian Jordy of SAFT batteries, for accepting our invitation to evaluate this PhD work. My PhD work is a part of the European Union (EU)'s Horizon 2020 project called High Energy Lithium Sulphur cells and batteries (HELiS). Therefore, I would like thank the EU and CNRS for funding this work. I deeply appreciate the useful discussions with all the HELiS project partners, especially Robert Dominko and Patrik Johansson. I also specially thank Sara Drvarič

Talian for her collaboration and for providing the potentiostatic experimental result, which is presented in the chapter 6.

The homogeneous chemical reaction coefficients of final reaction steps (Eqs. [START_REF] Larcher | Towards greener and more sustainable batteries for electrical energy storage[END_REF]. 16 and 5.22) are given by, 𝛾 𝑖𝑘 = ( 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ) [A-5.17] Finally, the particle number density in the 𝑙 𝑡ℎ bin at the (𝑛 + 1) 𝑡ℎ time step is given by the Eq. A-6.7.

𝑁 𝑙 𝑛+1 = 𝑁 𝑙 𝑛 + ∆𝑡𝑁 𝑙 ́-Δ𝑁 𝑙 + -Δ𝑁 𝑙 -+ Δ𝑁 𝑙-1 + + Δ𝑁 𝑙+1 -(A-6.7)

The matricized form of the Eq. A-6.7, is used to calculate the particle number density of the deposits in all the bins at the (𝑛 + 1) 𝑡ℎ time step as follows, 𝑁 𝑛+1 = 𝑁 𝑛 + ∆𝑡𝑁 ́-𝑀 𝑙 (∆𝑁 + + ∆𝑁 -) + 𝑀 𝑙-1 ∆𝑁 + + 𝑀 𝑙+1 ∆𝑁 - (A-6.8)

where, 𝑁 𝑛+1 and 𝑁 𝑛 are the 𝐿 × 1 vectors which contain the lists of particle number densities of the deposits in all the bins at the (𝑛 + 1) 𝑡ℎ and 𝑛 𝑡ℎ time steps, respectively. 𝑁 ́ is the 𝐿 × 1

vector which contains a list of nucleation source terms of all the bins.

In the equation Eq. A-6.8, ∆𝑁 + and ∆𝑁 -respectively are the 𝐿 × 1 vectors which contain the changes in the particle number densities due to growth and shrinkage some deposits in all the bins. The ∆𝑁 + and ∆𝑁 -vectors are shown in Eq. A-6.9 and A-6.10, respectively.

In the equation Eq. A-6.8, the 𝑀 𝑙 , 𝑀 𝑙-1 and 𝑀 𝑙+1 are 𝐿 × 𝐿 matrices which are shown in Eq. A-6.11, A-6.12 and A-6.13, respectively.

Chapter 8. General conclusions and perspectives

In situ and operando Raman spectroscopic studies have shown that 𝑆 3 * -and 𝑆 4 * -radicals are produced in Li-S batteries containing 1 M LiTFSI in TEGDME:DOL [START_REF] Hannauer | The Quest for Polysulfides in Lithium-Sulfur Battery Electrolytes: An Operando Confocal Raman Spectroscopy Study[END_REF][START_REF] Wu | In Situ Raman Spectroscopy of Sulfur Speciation in Lithium-Sulfur Batteries[END_REF]. Therefore, we updated the reaction mechanism of our cyclic voltammetry model by adding dissociation reactions of 𝑆 8

Thesis abstract in English

Li-S batteries have attracted a lot of attention in the past decade due to their very high theoretical energy density of 2567 Wh.kg -1 . However, the assessment of some of Li-S batteries phenomena and limitations still remain challenging. In my PhD work, we have developed multiple mathematical models, which assist with assessment and optimization of some the Li-S batteries phenomena. We developed a novel 3D kinetic Monte-Carlo (kMC) model, which simulates the discharge of carbon/sulfur (C/S) composites at the mesoscopic level. Our kMC model also provides insights into the impacts of the discharge rate and sulfur loading on the mesoscale properties of the 𝐿𝑖 2 𝑆 (𝑠) deposits over the carbon surface. We also developed a multi-scale continuum model to investigate the impacts of the C/S cathode design parameters on the discharge of Li-S batteries. In order to determine the reaction mechanism of the dissolved polysulfides in our experimental electrolyte (1 M LiTFSI in TEGDME:DOL (v/v=1:1)), we carried out cyclic voltammetry measurements of different electrolyte solutions containing dissolved sulfur and polysulfides. A mathematical model was used to interpret the reactions behind the characteristics of experimental cyclic voltammograms. Furthermore, we also developed a nucleation and growth model to understand the phenomena behind the 𝐿𝑖 2 𝑆 (𝑠) electrodeposition in a simplified Li-S cell. Finally, we carried out some galvanostatic experiments using Li-S coin cells to validate our discharge model.