Self-Organization of Electroactive Suspensions in Discharging Slurry

Research Article www.acsami.org

Self-Organization of Electroactive Suspensions in Discharging Slurry Batteries: A Mesoscale Modeling Investigation Garima Shukla,†,‡ Diego del Olmo Diaz,†,‡ Vigneshwaran Thangavel,†,‡ and Alejandro A. Franco*,†,‡,§,∥ †

Laboratoire de Réactivité et Chimie des Solides (LRCS), CNRS UMR 7314, Université de Picardie Jules Verne, 33 Rue Saint Leu, 80039 Amiens Cedex, France ‡ Réseau sur le Stockage Electrochimique de l’Energie (RS2E), FR CNRS 3459, 80039 Amiens Cedex, France § ALISTORE-ERI, European Research Institute, FR CNRS 3104, 80039 Amiens Cedex, France ∥ Institut Universitaire de France, 103, Boulevard Saint Michel, 75005 Paris, France S Supporting Information *

ABSTRACT: We report a comprehensive modeling-based study of electroactive suspensions in slurry redox flow batteries undergoing discharge. A three-dimensional kinetic Monte Carlo model based on the variable step size method is used to describe the electrochemical discharge of a silicon/carbon slurry electrode in static mode (i.e., no fluid flow conditions). The model accounts for Brownian motion of particles, volume expansion of silicon upon lithium insertion, and formation and destruction of conducting carbon networks. Coupled to an electrochemical model, this study explores the impact of carbon fraction in the slurry and applied c-rate on the specific capacity. The trends obtained are analyzed by following the behavior of parameters such as number of contacts between electroactive particles and the percentage of electroactive silicon particles. Furthermore, instead of studying the bulk behavior of the slurry, here the focus is given to the slurry/current collector interface in order to illustrate its importance. Hereby, it is demonstrated how this modeling tool can lead to deeper understanding and optimization of electroactive particle suspensions in redox flow batteries. KEYWORDS: redox flow batteries, slurry electrodes, electroactive suspensions, silicon, carbon, mesoscale, computational modeling, kinetic Monte Carlo

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different types of materials.2−4 Silicon is one such material that offers high gravimetric capacity by forming a lithium rich alloy (Li3.75Si) at the end of discharge, at room temperature. However, the volume of Li3.75Si is about 175% larger than that of pristine Si.4 This results in significant loss of contacts between silicon and carbon particles in solid state electrodes leading to poor cyclability; thus the use of binders and polymeric additives with modified silicon surfaces has been studied to retain capacity over cycling.4,5 An elegant alternative approach to overcome this challenge consists of using silicon in the form of a slurry electrode, without the use of any additive or binder, as demonstrated by Hamelet et al.4 In this case, the stable suspension of silicon and carbon in an electrolyte offers a surprisingly appealing electrochemical performance on cycling in the absence of flow conditions hereinafter referred to as the static mode. The growth of SEI on silicon due to large volume fraction of electrolyte is more than in solid state electrodes; however the experiments show that 80% of the theoretical capacity can be

INTRODUCTION Developing energy storage devices for smart grids is critical to meet both consumer and industrial electricity demands and to inhibit further ecological damage by providing sustainable means to incorporate renewable energy sources. Electrochemical energy storage systems offer a wealth of redox couples that can be tailored to satisfy requirements of a large variety of scales. Furthermore, out of all the stationary technologies, redox flow batteries (RFBs) provide the highest technical flexibility by decoupling energy and power. Although vanadium-based RFBs have dominated stationary applications for decades,1 the concept of slurry redox flow batteries (SRFBs) introduced by Chiang et al.2 offers a new approach for incorporation of a variety of families of materials that were previously only studied in the context of solid state electrodes for nomad applications. SRFBs are fueled by semisolid suspensions of high energy density lithium storage compounds that are electrically wired by dilute percolating networks of nanoscale conductor particles. Furthermore, a slurry electrode system can help overcome disadvantages of higher energy density materials that are cheap but fail as solid state electrodes. Recent efforts allowed significant progress in the understanding of their electrochemical characterization with © 2017 American Chemical Society

Received: February 21, 2017 Accepted: May 11, 2017 Published: May 11, 2017 17882

DOI: 10.1021/acsami.7b02567 ACS Appl. Mater. Interfaces 2017, 9, 17882−17889

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ACS Applied Materials & Interfaces

Figure 1. (a) Experimental setup of the static cell in our lab; (b) schematic representation of the cell assembly containing the silicon−carbon slurry in the electrolyte; (c) mesoscopic representation of the interface between the silicon (blue)−carbon (red) slurry and the current collector (yellow) simulated by the model (here the empty space in slurry represents the electrolyte).

Figure 2. Simulation workflow describing the three main modules: (a) channel initialization setting up the three-dimensional slurry based on size and composition requirements (module only used once at the beginning of the simulation); (b) module to simulate the Brownian motion using VSSM under the kMC framework; (c) electrochemical phenomena of silicon discharge. Modules b and c are implemented until the potential calculated in part c reaches a cutoff potential.

achieved with very low polarization and reasonable capacity retention. This is an interesting system from a multidisciplinary modeling perspective as particle suspension dynamics, particle growth dynamics, and electrochemistry can be studied simultaneously.

Despite the promise and progress achieved experimentally, a holistic picture of parameters and operation principles that influence the electrochemical behavior of SRFBs is yet to be explored.2,6−12 Indeed, very few theoretical studies have been attempted so far on SRFBs.13,14 Computational SRFB models 17883


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electron conducting carbon is to assemble itself as a network that wires silicon particles in the slurry. When a carbon network has contact with silicon particles that are not yet fully discharged, this network and the corresponding silicon particles are considered to be electroactive. Silicon particles are assumed to be electron sinks only, thus making them incapable of participating within a carbon network to conduct electrons. Once a silicon particle is fully discharged, it is no longer considered electroactive and the carbon network it was connected to becomes inactive. Hence our model can track the durability of these carbon networks under different operating conditions and slurry compositions. While Brownian motion drives suspension stability, a single particle motion can potentially break essential carbon networks. In the model we consider only Brownian motion, while other destructive forces for carbon networks such as sedimentation and flocculation, which can induce suspension instability, are ignored. Interaction potentials such as van der Waals and electrostatic forces were ignored in this first modeling attempt due to the incomplete fundamental understanding of how potentials arising from atomic interactions manifest at the mesoscale for particles and how the solid−liquid interface transforms during the electrochemical cycling of particles experiencing electron flow while aggregating with other particles and being solvated from the electrolyte. In our model, kMC-VSSM is used to choose from a weighted list of all possible particle displacements occurring within Brownian motion (Figure 2b). The weight attributed to any displacement is based on the diffusion rate. A particle with a high diffusion rate is most probably small or has available space around it; thus VSSM is most likely to select this particle to move. All particles in this model are given a basal diffusion property arising from the macroscopic feature of viscosity of the slurry, modified by their respective sizes. For this purpose, the Stokes− Einstein equation (eq 1) expresses the diffusion coefficient (Di,j) as a function of viscosity of the slurry (ηsl) and particle radius (ri,j),24 where subscript “i” refers to particle type (Si or C) and “j” refers to the identity:

available in the literature consider the slurry as a continuum medium in which the impact of dynamic mesostructural selforganization properties of suspended particles on the overall electrochemical response has been overlooked. Additionally, carbon and silicon/carbon suspensions in bulk have been modeled with discrete coarse-grained particles using threedimensional Boltzmann dynamics approaches.15−18 The latter works reported the effects of temperature and mass ratios of carbon and silicon particles on the number of contacts between them within the percolation networks. However, since the aforementioned studies do not simulate electrochemical reactions in the slurry, they address only a part of the challenge in understanding SRFBs. Valuable insights can come from a model that couples particle suspension dynamics with electrochemical mechanisms. Therefore, the primary objective of this paper is to report an innovative model designed to capture the mesoscopic self-organization of suspensions in an anodic slurry, comprising silicon and carbon particles in an organic electrolyte, when simulating a galvanostatic discharge versus metallic lithium in static mode (Figure 1). We focus here on the interface between the current collector and the suspension, within the experimental setup configuration of Hamelet et al.4 In the following section, we present the theoretical background of our model. Then, we illustrate its capabilities by reporting simulated sensitivity analysis of electrochemical performance vs carbon fraction and c-rate. Finally, we conclude and indicate future directions of this work.

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COMPUTATIONAL MODEL Kinetic Monte Carlo (kMC) models based on the variable step size method (VSSM) have been conventionally used for atomistic-type modeling.19 Recently, they have been adapted by us to simulate, with three-dimensional resolution, mesoscopic transport and electrochemical mechanisms taking place in energy devices such as fuel cells and lithium O2 batteries.20−23 In this work, we adopt this approach to simulate the behavior of the silicon−carbon particle suspensions upon the slurry electrochemical discharge. In our model (Figure 2), the particle suspension is confined within a cubic grid with unit cells of edge length 100 nm (Figure 2a (ii)), a parameter that can be tuned for obtaining the desired resolution. It is assumed that the top face of the grid is the surface of current collector and the rest of the faces are boundaries that contain a part of the slurry considered as the interfacial system under study. The relative impact of parameters on the calculated specific capacities represents only the simulated part of the slurry electrode and does not correspond to that of the entire slurry electrode. The slurry itself is composed of carbon particles of 100 nm diameter, silicon particles of about 237 nm effective diameter (Figure 2a (i)), and organic electrolyte in the empty spaces around the particles. Initially, carbon and silicon particles are considered to occupy one unit cell and seven unit cells, respectively, in order to generate a particle size difference which is computationally easy to handle while allowing experimental viability. Additionally, the volume occupied by each subunit for all theoretical considerations is taken as a cube of the matrix but represented as spherical subunits for the purpose of visualization. The initial configuration of the slurry is generated by placing silicon and carbon particles randomly in the grid (Figure 2a (v)). An ideal slurry electrode would be one that is capable of fully discharging all silicon particles by providing them with sufficient electrons and lithium ions, thus offering the highest possible capacity. The role of the purely

Di , j =

kBT 6πηsl ri , j

(1)

The viscosity of the slurry (ηsl) is a product of pure liquid viscosity (ηl) and relative solid viscosity (ηr), wherein the latter can be estimated on-the-fly based on solid volume fraction (ϕsol) using Thomas’ expression:25,26 ηr = 1 + 2.5 + ϕsol + 10.05ϕsol 2

(2)

The diffusion rate is obtained by multiplying the diffusion coefficient with the degrees of freedom of the particle, which are evaluated in the model (Figure 2b (i)) by checking how many immediate neighboring locations can be occupied by the particle in question. Once the diffusion rates of all particles (nC + nSi) have been calculated, a weighted cumulative list of particle displacements can be obtained, upon which VSSM can be implemented (Figure 2b (ii)). The total diffusion rate (KT) is taken as a sum of individual diffusion rates (ki,j) of all particles: j = nC

KT =

j = nSi

∑ (k C,j) + ∑ j=1

j=1

(k Si, j) (3)

In order to choose a random particle for motion, a pseudo random number (ρ1) is multiplied by the total rate. Then a search 17884


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ACS Applied Materials & Interfaces algorithm produces a pair of rates (km, km+1) between which the randomly generated rate lies (eq 4) (Figure 2b (iii)): n=m

ions are only consumed by the silicon insertion reaction. Any other contribution to the potential drop arising from capacitive effects of carbon, resistance across carbon particle contacts in the network, parasitic effects such as electrolyte degradation or lithium uptake by carbon, and cracking of silicon particles is ignored but can be easily introduced in the future. Resistance arising from lithium ion transport in the electrolyte is neglected under the low c-rate conditions in which batteries containing silicon are usually operated. It is assumed that lithium ions are readily available at the silicon−electrolyte interface and do not limit the insertion reaction (eq 7). The activation overpotential is derived from the Butler− Volmer kinetic equation

n=m+1

∑ (kn) ≤ ρ1KT < ∑

(kn)

n=1

n=1

(4)

The mth particle is chosen to move within a time step that varies with the total diffusion rate calculated at that instant, modulated by another pseudo random number (ρ2) (eq 5) (Figure 2b (iv)): dt = −

ln(ρ2 ) KT

(5)

During the time step taken for any particle to move, we consider that simultaneously, electrochemical discharge occurs (Figure 2c). Within the next part of the model a search algorithm is implemented to identify carbon networks formed at that time step, identify electroactive silicon particles, and provide shortest carbon pathway to those silicon particles (Figure 2c (i)). The redox couple in the system is the lithium metal oxidation at the anode +

Li ⇄ Li + e

−

⎛ ⎛ αFη ⎞ ⎛ (1 − α)Fη ⎞⎞ c c ⎟⎟⎟ ig = i0⎜⎜exp⎜ ⎟ − exp⎜ − ⎝ ⎠ RT RT ⎠⎠ ⎝ ⎝

where the transfer coefficient (α) is taken as 0.5 such that the expression is simplified to ηc = −

(6)

and lithium insertion into silicon at the cathode +

−

Si 0.26 + Li + e ⇄ LiSi 0.26

(7)

nactSi

Sact =

(11)

⎛ Sf, j + k1S bSi, j + k 2S bC, j ⎞ ⎟⎟4πrj 2 ⎝ Sf, j + S bSi, j + S bC, j ⎠

∑ ⎜⎜ j=0

(12)

The prefactor indicates the fraction of faces of the silicon particle j that are not blocked by other particles and are available for lithium ion uptake from the electrolyte. Sf,j is the number of free facets of a silicon particle, which are only in contact with electrolyte. SbSi,j and SbC,j are the number of facets blocked by another silicon particle and carbon particle, respectively. Constants (k1 = 0.2) and (k2 = 0.5) are assumed to describe the extent of interaction between particle surfaces such that silicon particles do not tend to stick together as much as they would stick to carbon aggregates. The exchange current density (i0) in eq 11 assumes concentration of lithium ion in electrolyte (cLi,el) as 1 mM and is given by

(8)

Here, Ea is set to zero and the kinetics for the lithium oxidation reaction are assumed to be fast enough to consider having negligible contribution in the activation overpotential. Ec is calculated based on a standard Nernst potential wherein E00.26 is the standard potential of silicon and y is the state of charge of silicon: Ec = ESi0 0.26 +

⎛ ig ⎞ RT sinh−1⎜ ⎟ αnF ⎝ 2i0 ⎠

The current density applied (ig) is calculated as the galvanostatic current divided by the surface area of electroactive silicon (Sact) within that time step. This surface area is obtained as the addition of the approximate spherical surface area of each silicon particle j and further modified by a prefactor that accounts for other particles in its environment (eq 12):

where although conventionally the discharged silicon host is expressed as Li15Si4, for simplicity we assume LiSi0.26 (i.e., LiSi4/15) such that stoichiometry of lithium (y in LiySi0.26) at any point of discharge is between 0 and 1.4 On the basis of the value of time step, it is straightforward to estimate how much electronic charge can be delivered to the electroactive silicon particles under galvanostatic conditions (Figure 2c (ii)). The silicon particles are capable of undergoing volume expansion by taking up electrons and lithium ions, which is done by adding 12 extra discrete particle subunits to the existing partially discharged particle (Figure 2a (i)). Thus, the model allows charge accumulation on silicon particles until they possess enough electrons to grow a new subunit (Figure 2c (iii)). The model estimates the state of charge of the cell by tracking the potential difference between the electroactive silicon particles as cathode (Ec) and the lithium metal anode (Ea) (eq 8) (Figure 2c (iv)):

Ecell = Ec − Ea

(10)

i0 = FKc Li,el α cSi α c LixSi1 − α

(13)

It further incorporates volume expansion effects based on the state of charge y, a volume expansion factor of 1.75 times, and the initial concentration of carbon (cSi,in) to represent the concentration of silicon (cSi) and lithium inserted silicon (cLixSi):

RT ⎛ (1 − y) ⎞ RT ln⎜ A(2y − 1) + ηc ⎟+ nF ⎝ y ⎠ nF

cSi α c LixSi1 − α =

(9)

In the equation herein above, the slope of the Nernst discharge is modified to correspond with experiment using an expression containing the Margules constant (A).4 In addition, only the activation overpotential (ηc) of lithium insertion reaction 7 is considered. In the interest of starting from a simple and elegant modeling framework, it is assumed that no mechanisms for irreversible loss of capacity are present in the system; i.e., electrons and lithium

cSi,in(1 − y)1 − α y α (1 + 1.75y)

(14)

Fully discharged silicon particles (LiSi0.26) obtain an effective diameter of 330 nm by growing new particle subunits when the accumulated charge on the silicon particles is sufficient (Figure 2c (v)). The model control alternates between implementation of Brownian motion of particles and electrochemical discharge of silicon particles until a cutoff potential is reached. During the 17885


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ACS Applied Materials & Interfaces simulation, parameters such as the electric potential, time, state of charge, fraction of electroactive silicon, size of the carbon networks, viscosity, total kinetic rate, particle size distribution, and three-dimensional trajectories of all particles are tracked. These diagnostic tools allow greater insight into the dynamics of the slurry mesostructure during discharge.

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RESULTS AND DISCUSSION For the simulation shown in this paper, the slurry composition was taken as 9 vol % of carbon and 14 vol % of silicon within a channel depth of 1 μm and square current collector of edge 2 μm discharging at a c-rate of C/20.4 For this particular case, the time steps as calculated from VSSM were dominantly within the range of 10−1−10−3 s and the total time of battery discharge was 12 h, for which the time of simulation was 10 min. The simulation of the slurry and particle self-organization along the discharge can be tracked visually using the open-source visualization software OVITO (Figure 3a−c and Supporting Information video).27 The

Figure 4. (a) Calculated impact of c-rate (2C, C, C/5, C/10, and C/20) on specific capacity. (b) Calculated impact of the depth of slurry considered as a part of the interface (3−17 μm in steps of 2 μm), on the specific capacity.

affected only by activation overpotential arising from the c-rate and equilibrium potentials from the state of charge of the silicon particles. The c-rate of C/20 is chosen for the reference simulations as it is slow enough to ensure that the slurry has sufficient relaxation time to respond to effects of silicon volume expansion. The sensitivity of the specific capacity toward the slurry interface depth (Figure 4b) shows that larger depth results in lower capacity. Choosing 1 μm as the interface depth puts the reference system in the middle of this trend. With the aforementioned reference system, a detailed study of the impact of carbon fraction on the specific capacity is presented (Figure 5). The three-dimensional visualization offers an interesting picture of the evolution of carbon networks as the carbon fraction is increased (Figure 5a−h). The behavior of the specific capacity (Figure 5i) shows that there exists an upward trend up to 9 vol % which can be referred to as the percolation region.28 The error bar while going from low carbon fractions to high carbon fractions is gradually decreased owing to the decrease in sensitivity of kMC to the initial particle configuration in slurry, reduced diffusion rate, and the decrease in importance of the role of each carbon particle in the carbon network. However, despite the error bars, the model clearly shows a percolation region within which the conductivity is limited by the number of carbon particles which determine the size of the carbon network. Above 11 vol % the specific capacity plateaus as

Figure 3. Snapshots of three-dimensional visualization of the slurry with silicon (in blue), electroactive carbon (in yellow), inactive carbon (in red), and electrolyte (empty space) at (a) beginning of simulation, (b) half discharge, and (c) full discharge. (d) Average of three simulation runs (in black) and the final discharged curve obtained as a fitted cubic polynomial (in red). The current collector on the top has been removed for clarity.

trend for the electrochemical potential on discharge (Figure 3d) obtained from the simulation has fluctuations originating from the discrete nature of kMC. In order to produce statistical trends for all parameters, for each case under study three simulations are run, averaged and the resultant discharge curve is fitted either with a cubic polynomial function or a power law to show its mean characteristic. Parameter sensitivity trends are reported with error bars to indicate the corresponding fluctuation of data. The fitted curve (Figure 3d) shows semiqualitative correspondence to in-house and published experiments.4 The specific capacity is observed to decline at faster c-rates (Figure 4a). As resistance due to lithium transport is not considered, the trend of calculated discharge potential profiles is 17886


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Figure 5. 3D visualization of slurries for each carbon fraction represented as percentage by volume wherein the blue particles are silicon, red particles are carbon, and yellow particles are those that compose the carbon network: (a) 3%, (b) 5%, (c) 7%, (d) 9%, (e) 11%, (f) 13%, (g) 15%, and (h) 17%. (i) Behavior of specific capacity with carbon fraction varied between 3 and 17 vol % in steps of 2 vol %. The error bars capture the data fluctuation.

it reaches a high limiting conductivity. Beyond this composition, carbon networks are always available and the specific capacity is no longer a function of the number of carbon particles. The model provides diagnostic tools (Figure 6) to analyze the impact of carbon fraction (Figure 5) in greater detail, two of which are discussed here. The nature of the particle assembly (Figure 5a−h) can be quantified based on the number of interparticle contacts made by the carbon network, offering a means to visualize and analyze the extent of conductivity. By studying the evolution of the number of such electroactive contacts made over normalized time of discharge (Figure 6a; see Supporting Information Figure S1), it can be seen that within 7− 11 vol % there is a significant jump in the number of electroactive contacts, after which it begins to saturate above 11 vol % of carbon. This can be explained by the increase in probability of carbon to form through-going carbon networks across the volume of the slurry due to greater number of carbon particles available. Thus, it can be said that 11 vol % is the threshold for formation of through-going carbon network which goes beyond the interface and is essential to obtain conductivity in the bulk of the slurry, leading to a well-functioning cell. To understand how particle self-organization influences the discharge potential, it is useful to track the electroactivity of silicon particles in terms of percentage along discharge (Figure 6b; see Supporting Information Figure S2). The impact of volume expansion of silicon on the silicon particle size

distribution can easily be captured by the model (see Supporting Information Figure S3). A corresponding transition as seen above (Figure 6a) is noted between 7 and 11 vol % (Figure 6b). For carbon fractions at and below 7 vol %, the slope of the plot or rate of electroactivation plateaus before achieving 100% of silicon, indicating that some silicon particles are out of reach, while for carbon fractions at and above 11 vol %, the rate of electroactivation captured by the slope of the trend indicates that within the first few hours, 100% of silicon electroactivation is achieved. This can be explained by the existence of the threshold for through-going carbon network which lies at 11 vol %, which allows fast access to all silicon particles and consequently higher specific capacity obtained from the cell.

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CONCLUSIONS In this paper we presented an in-house kMC modeling framework in three dimensions devoted to a holistic study of electroactive particle suspensions. Although presented here with simplifications, this framework can accommodate new physical phenomena once the theory behind is defined at the mesoscale. The model can simultaneously predict the evolution of the mesostructural self-organization and electrochemical performance based on physical phenomena such as the Brownian motion of particles and the volume expansion of silicon. Since calculations are done on-the-fly, the model presents a host of diagnostic tools based on tracking descriptors that can 17887



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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsami.7b02567. Description and raw data of trends (number of percolation contacts and percentage of electroactive silicon versus normalized time of discharge) and the variation of particle size on volume expansion of silicon particles (PDF) Video of simulation showing electrochemical discharge of the slurry containing silicon (blue) and carbon (red, electroinactive; yellow, electroactive) (AVI)

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AUTHOR INFORMATION

Corresponding Author

*Phone: 0033 3 22 82 53 36. Fax: 0033 3 22 82 75 90. E-mail: [email protected]. ORCID

Alejandro A. Franco: 0000-0001-7362-7849 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors deeply acknowledge the Région Hauts de France for the financial support through the project “WONDERFUL”. Discussions with Dr. Charles Delacourt and Dr. Matias Quiroga (LRCS) are greatly appreciated.

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Figure 6. (a) Impact of carbon fraction (3−17 vol %) on the evolution of average number of electroactive contacts over normalized time of discharge. (b) Impact of carbon fraction (3−17 vol %) on percentage of electroactive silicon particles along normalized time of discharge.

REFERENCES

(1) Wang, W.; Luo, Q.; Li, B.; Wei, X.; Li, L.; Yang, Z. Recent Progress in Redox Flow Battery Research and Development. Adv. Funct. Mater. 2013, 23 (8), 970−986. (2) Duduta, M.; Ho, B.; Wood, V. C.; Limthongkul, P.; Brunini, V. E.; Carter, W. C.; Chiang, Y.-M. Semi-Solid Lithium Rechargeable Flow Battery. Adv. Funct. Mater. 2011, 1 (4), 511−516. (3) Hamelet, S.; Tzedakis, T.; Leriche, J.-B.; Sailler, S.; Larcher, D.; Taberna, P.-L.; Simon, P.; Tarascon, J.-M. Non-Aqueous Li-Based Redox Flow Batteries. J. Electrochem. Soc. 2012, 159 (8), A1360−A1367. (4) Hamelet, S.; Larcher, D.; Dupont, L.; Tarascon, J.-M. Silicon-Based Non Aqueous Anolyte for Li Redox-Flow Batteries. J. Electrochem. Soc. 2013, 160 (3), A516−A520. (5) Touidjine, A.; Morcrette, M.; Courty, M.; Davoisne, C.; Lejeune, M.; Mariage, N.; Porcher, W.; Larcher, D. Partially Oxidized Silicon Particles for Stable Aqueous Slurries and Practical Large-Scale Making of Si-Based Electrodes. J. Electrochem. Soc. 2015, 162 (8), A1466−A1475. (6) Dennison, C. R.; Beidaghi, M.; Hatzell, K. B.; Campos, J. W.; Gogotsi, Y.; Kumbur, E. C. Effects of Flow Cell Design on Charge Percolation and Storage in the Carbon Slurry Electrodes of Electrochemical Flow Capacitors. J. Power Sources 2014, 247, 489−496. (7) Campos, J. W.; Beidaghi, M.; Hatzell, K. B.; Dennison, C. R.; Musci, B.; Presser, V.; Kumbur, E. C.; Gogotsi, Y. Investigation of Carbon Materials for Use as a Flowable Electrode in Electrochemical Flow Capacitors. Electrochim. Acta 2013, 98, 123−130. (8) Youssry, M.; Madec, L.; Soudan, P.; Cerbelaud, M.; Guyomard, D.; Lestriez, B. Formulation of Flowable Anolyte for Redox Flow Batteries: Rheo-Electrical Study. J. Power Sources 2015, 274, 424−431. (9) Youssry, M.; Madec, L.; Soudan, P.; Cerbelaud, M.; Guyomard, D.; Lestriez, B. Non-Aqueous Carbon Black Suspensions for Lithium-Based Redox Flow Batteries: Rheology and Simultaneous Rheo-Electrical Behavior. Phys. Chem. Chem. Phys. 2013, 15 (34), 14476.

offer valuable insight to understand parameter sensitivity such as the threshold observed for through-going carbon networks. Only above this threshold can the carbon networks penetrate the interface of slurry and current collector and reach the bulk of the slurry. Below this threshold it is likely that the bulk of the slurry will not be electroactive, thus offering specific capacity much lower than what is estimated at the interface. Thereby, it is worthwhile to understand the mechanism of network formation by self-organization at the interface before attempting a computationally expensive task such as simulating the entire slurry. Additionally, once parameter sensitivities of the model are in accordance with experiment, the model targets performing optimization of compositional parameters and operating conditions to aid experimentalists. This generic modeling framework can also be applied to analogous electrochemical devices such as redox flow capacitors as well as simulation of slurries for lithium ion battery fabrication. This work lays a theoretical foundation for studying a highly complex system that encompasses electrochemistry, suspension dynamics, particle dynamics, and eventually fluid dynamics. 17888


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Self-Organization of Electroactive Suspensions in Discharging Slurry

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