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Research Papers

# Insights Into Lithium-Ion Battery Degradation and Safety Mechanisms From Mesoscale Simulations Using Experimentally Reconstructed MesostructuresOPEN ACCESS

[+] Author and Article Information
Scott A. Roberts

Thermal/Fluid Component Sciences Department,
Sandia National Laboratories,
Albuquerque, NM 87185
e-mail: sarober@sandia.gov

Hector Mendoza

Thermal/Fluid Component Sciences Department,
Sandia National Laboratories,
Albuquerque, NM 87185
e-mail: hmendo@sandia.gov

Victor E. Brunini

Thermal/Fluid Science and Engineering Department,
Sandia National Laboratories,
Livermore, CA 94550
e-mail: vebruni@sandia.gov

Thermal/Fluid Component Sciences Department,
Sandia National Laboratories,
Albuquerque, NM 87185
e-mail: btremba@sandia.gov

David R. Noble

Fluid and Reactive Processes Department,
Sandia National Laboratories,
Albuquerque, NM 87185
e-mail: drnoble@sandia.gov

Anne M. Grillet

Thermal/Fluid Component Sciences Department,
Sandia National Laboratories,
Albuquerque, NM 87185
e-mail: amgrill@sandia.gov

Manuscript received May 16, 2016; final manuscript received July 6, 2016; published online October 20, 2016. Assoc. Editor: Partha Mukherjee.The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Electrochem. En. Conv. Stor. 13(3), 031005 (Oct 20, 2016) (10 pages) Paper No: JEECS-16-1068; doi: 10.1115/1.4034410 History: Received May 16, 2016; Revised July 06, 2016

## Abstract

Battery performance, while observed at the macroscale, is primarily governed by the bicontinuous mesoscale network of the active particles and a polymeric conductive binder in its electrodes. Manufacturing processes affect this mesostructure, and therefore battery performance, in ways that are not always clear outside of empirical relationships. Directly studying the role of the mesostructure is difficult due to the small particle sizes (a few microns) and large mesoscale structures. Mesoscale simulation, however, is an emerging technique that allows the investigation into how particle-scale phenomena affect electrode behavior. In this manuscript, we discuss our computational approach for modeling electrochemical, mechanical, and thermal phenomena of lithium-ion batteries at the mesoscale. We review our recent and ongoing simulation investigations and discuss a path forward for additional simulation insights.

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## Introduction

Lithium-ion batteries (Li-ion batteries or LIB) are electrochemical storage devices consisting of an anode, separator, and cathode. The anode and cathode are typically multicomponent composite materials comprised of an active material, a polymeric conductive binder, and void space which is filled with electrolyte. The active material in cathodes is often lithium cobalt oxide (LiCoO2 or LCO), which is most commonly found in consumer devices such as phones and laptops, or nickel manganese cobalt oxide (LiNiMnCoO2 or NMC), most often used in electric vehicles. The polymeric binder, typically polyvinylidene fluoride (PVDF), is present to mechanically hold the active material together, but is also often impregnated with carbon black (CB) to provide the binder with electrical conductivity.

Because of the ubiquity of LIBs in consumer devices and because of the potential for wide use in electric vehicles, there is significant interest in understanding and improving their behavior and performance, both in normal operation (predicting electrochemical performance under a variety of operating conditions) and in abnormal scenarios (thermal or mechanical abuse/insult) [1]. The cathode is often the key contributor to battery performance, limiting the energy and power density of the battery. Additionally, the cathode performance often degrades with cycling in a process known as capacity fade, further reducing the usable energy density and requiring larger batteries to meet long-life requirements [26].

While many of these observations are macroscopic (cell-level or larger), they are actually driven by particle-scale or mesoscale (network of particles) phenomena. At this scale, the complex-shaped active material particles are bound together by the polymeric conductive binder into a bicontinuous (particle/binder and void/electrolyte phases) percolated network [710]. Intercalation of lithium into the particles results in swelling (volume change) of the particles [11,12], generating significant mechanical stress, potentially rearranging or damaging the network, and affecting the electrochemical performance [5,1317]. However, the scientific community generally has a lack of understanding of exactly what happens at the mesoscale. This is primarily because it is difficult to experimentally observe phenomena at this length scale, particularly in situ.

Mesoscale simulations are a promising approach for answering these many questions. The idea of mesoscale simulations is not new, with many researchers attempting to look at particle-scale and mesoscale phenomena in various manners. Early research involved simple approaches for studying single particles [1822] and later networks of simple particles [2328]. However, the complex shape of the active material particles casts doubt on some of the mechanics findings. Recent advances in experimental tomographic capabilities have enabled the visualization and study of the morphology of these complex mesostructures [7,8,10,2935]. This has enabled researchers to model the complex shape of single particles [36,37]. While some researchers have successfully performed electrochemical calculations or simple image-based analyses on computational reconstructions of these tomographic images [3844], the capability to look at fully coupled phenomena involving all the three constituent phases has been lacking.

In this manuscript, we review our approach for performing fully coupled electrochemical, mechanical, and thermal mesoscale simulations on experimentally reconstructed lithium-ion battery mesostructures involving particle, binder, and void phases. A similar effort for molten salt batteries has also been performed [45]. The mathematical equations that govern these physics are first presented in Sec. 2. Our approach for obtaining computational representations/reconstructions of the complex mesostructures is described in sec. 3, and our novel approach for performing finite-element simulations on these mesostructures is discussed in Sec. 4. Highlights of results from the recent studies [46,47] using this approach are shown in Sec. 5, and in Sec. 6, we conclude by discussing our path forward for this technique.

## Mathematical Model

A mesoscale (many-particle scale) domain is considered in this work and is illustrated in Fig. 1. Three single-crystal active particles are shown in various shades of green, each with their own coordinate system aligned with the crystallographic orientation. Each particle is coated in a uniform coating of binder, shown in blue, although we are also considering other binder morphologies, such as those discussed by Rahani and Shenoy [27] and those reconstructed from images [8]. The electrolyte phase is shown in gray. The domain is bounded by a separator (shown in red at the bottom) and current collector (shown in purple at the top). While this illustration is in 2D and only includes three particles, the simulations are 3D and include hundreds of particles.

The mathematical model for these simulations consists of three distinct sets of physics: electrochemical transport (species and current) and reaction, stress generation and mechanical deformation, and heat generation and transport. These equations have been derived and described in great detail elsewhere [46,47], but will be summarized here for completeness. A summary of the key parameters and their values is shown in Table 1.

###### Electrochemical Model.

Two sets of volumetric equations must be solved on the entire computational domain, including the particle, binder, and electrolyte regions. First, lithium transport is governed by a species conservation equation Display Formula

(1)$∂Ci∂t+∇·J=0$

where Ci is the concentration of lithium (i = Li in the particles, i = Li+ and $i=PF6−$ in the electrolyte and binder), and J is the species flux. In the particle phase, the flux is given by Display Formula

(2)$JLi=−MCLi∇(μLichem+μListress)$

which was derived in detail by Mendoza et al. [47]. This formulation relies on nonideal solution theory and allows for the mechanical stress to affect the species transport. In Eq. (2), $μLichem=−Fϕeq, μListress=βtr(σ)/3$, M is the mobility of Li using the Nernst-Einstein relation, F is the Faraday's constant, $ϕeq$ is the equilibrium potential, β is a volumetric expansion coefficient, and $tr(σ)/3$ is the hydrostatic pressure. In the electrolyte and binder phases, the fluxes are calculated using the Nernst–Planck model Display Formula

(3)$Ji=−Di(ziCiFRT∇ϕl+∇Ci)$

where i represents either Li+ or $PF6−$, Di is the diffusivity, zi is the species valence, R is the gas constant, T is the temperature, and $ϕl$ is the liquid-phase potential. Finally, we assume electroneutrality to determine the anion concentration, $CPF6-=CLi+$, and the $PF6−$ diffusivity comes from the transference number [48,53]. While we expect that the diffusivities will be lower in the binder phase, we have no direct experimental data and therefore use the pure electrolyte values in our simulations.

Next, current (ionic and electrical) is conserved following a quasi-static assumption, where Ohm's law applies in the particle and binder phases Display Formula

(4)$∇·(κ∇ϕi)=0$

and ionic conservation applies in the electrolyte and binder phases Display Formula

(5)$∇·[F(JLi+−JPF6-)]=0$

Here, κ is the electrical conductivity, and $ϕi$ is either the particle-phase ($ϕs$) or binder-phase ($ϕb$) potential.

The volumetric equations in the particle and electrolyte/binder domains are coupled by the intercalation reaction at the particle–electrolyte interface, modeled using the Butler–Volmer equation [54]. The computed intercalation reaction rate, r, is used to apply normal flux boundary conditions to each of the volumetric conservation equations.

###### Mechanical Model.

Our treatment of mechanical deformation in the particle and binder phases is based on the assumption of quasi-static equilibrium, governed by the balance of linear momentum Display Formula

(6)$∇·σ=0$

where $σ$ is the Cauchy stress tensor (incorporating the Young's modulus and Poisson's ratio). For our previous work with LCO, we have treated motion in the small deformation limit with a linear elastic strain constitutive model. However, our implementation is general enough to handle finite deformation with a variety of nonlinear hyperelastic or hypoelastic constitutive equations.

For an elastic material, the constitutive model is given generally as Display Formula

(7)$σ=Ci:(ε−βΔCLi)$

where $Ci$ is the fourth-rank elasticity tensor, $ε$ is the total small strain tensor, and $βΔCLi$ represents the swelling strains that develop due to lithium intercalation. The swelling of individual particles of active material is captured in a second-rank tensor, $β$, which is a material parameter defined such that $β=ε/ΔCLi$ for a single crystal under stress-free boundary conditions, allowing for anisotropic (namely, transversely isotropic) deformation. This β parameter is calibrated from experimental data [11], and ΔCLi is simply the concentration difference from the fully lithiated state. For LCO, 1.6% deformation in the c-axis direction is expected. As each particle may be arbitrarily oriented in the simulation (as denoted by the n and t coordinate frame in Fig. 1), $β$ must be rotated into the laboratory reference frame for solution, allowing each particle to have a different principal direction. For an isotropic behavior, $β$ reduces to a scalar β.

###### Thermal Model.

Heat generation and transport is governed by the standard advection–diffusion equation for temperature Display Formula

(8)$∂T∂t+v·∇T=∇·(α∇T)+Qj+Qr=0$

Here, v is the fluid velocity (in the electrolyte), α is the thermal diffusivity, and the Qi are the source terms. Two source terms are considered. Joule (or resistive) heating arises from electrical current flow, $Qj=κ−1J·J$. Second, the intercalation reactions at the particle surface can release heat, Qr = ΔHrr, where ΔHr is the heat of reaction. In current simulations, the reaction heat sources are neglected.

## Mesostructure Reconstruction

The key for performing mesoscale simulations is a high-resolution, experimentally derived reconstruction of the material's mesostructure. These 3D images can be obtained by a variety of methods, but most frequently scanning electron microscopy (SEM) of focused-ion beam (FIB) cross sections or micro- or nanocomputed tomography (micro-CT or nano-CT) techniques are used.

In our previous work, we have focused on LCO, which has very irregular, nonspherical shape, as shown in Fig. 2(a) [7]. We have been recently shifting focus to NMC, which has a much more spherical particle morphology, as shown in Fig. 2(b) [15]. In both cases, the particles diameters are in the range of 5–20 μm.

The 3D stack of images is reconstructed and processed using Avizo 9 (FEI, Hillsboro, OR). The 2D images are first binarized to identify the particle, electrolyte, and potentially binder phases. Each individual particle is then identified and separated using a watershed algorithm. Finally, a surface mesh is created on each individual particle with minimal smoothing (only to remove pixelation), and the resulting surface mesh is exported as a standard tessellation language (STL) file for analysis. This process has been described in detail elsewhere [46,47]. An image of a surface-meshed LCO mesostructure [7] is shown in Fig. 3.

## Numerical Method

As many researchers have noticed that creating a high-quality, interface-conformal mesh that includes both particle and electrolyte phases on an experimentally reconstructed mesostructure is nearly impossible with available tools, including commercial and open-source software specifically designed for this purpose. This, we believe, is why many researchers focus their studies on either simplified, artificially generated mesostructures, single particles, networks of particles avoiding the electrolyte or binder phases, or mesostructures with nonconformal meshes [7,8,1844].

We have pioneered a novel approach for generating interface-conformal meshes of particle, binder, and electrolyte phases from reconstructed mesostructures that is suitable for finite-element simulations [46,47]. This method uses the conformal decomposition finite-element method (CDFEM) [55,56] to automatically decompose an arbitrary tetrahedral mesh into phase-conformal tetrahedral elements. A 2D illustration of this process on a reconstructed LCO mesostructure is shown in Fig. 4. Figure 4(a) shows the uniform tetrahedral background mesh in gray. Overlaid on that mesh are curves that represent the intersections of the particle phases (represented by their STL description) with the mesh surface. At this point, the particles are not a part of the computational mesh, but are shown on the same image for reference. The resulting mesh from the CDFEM process is shown in Fig. 4(b), where the mesh is now conformal to the original particle surfaces, with the elements that spanned the surface being cut to conform to that surface. A 3D visualization of the computational domain, with the particle and electrolyte phases visualized separately, is shown in Fig. 5. Note that the computational mesh shown here is relatively coarse for ease of visualization. Simulations are run on a more refined mesh, as described later.

The mathematical model in Sec. 2 is solved on the reconstruction-based computational mesh using the Galerkin finite-element method (FEM), implemented in SIERRA/Aria [57]. The numerical model is solved on this conformal, tetrahedral element mesh using linear (Q1) basis functions. Each set of physics (electrochemical, mechanical, and thermal) is solved in a segregated, operator-split method using a Newton–Raphson nonlinear iteration scheme for each equation system. Finally, the systems are integrated in time using a standard second-order backward differentiation formula (BDF2).

A challenge with mesoscale simulations is that large 3D simulations can be quite computationally expensive, with millions of computational elements. One must simulate a large enough domain with enough particles to obtain a representative volume element (RVE) while also having a fine enough mesh resolution to resolve the physics that you are trying to study. Kanit et al. [58] proposed a methodology for assessing the RVE that is required. However, assessing the requisite mesh resolution requires solution verification (mesh refinement studies) on the domain of interest.

Solution verification for an LCO mesostructure using CDFEM is shown in Fig. 6. In this problem, we investigate five quantities of interest (QOIs). The first three are related to the ability of the CDFEM method to capture the geometry of the mesostructure, namely, the particle volume (curve with square markers), the external surface area of the particles (curve with circular markers), and the particle-to-particle contact area (curve with triangle markers). Capturing the volume is necessary for getting the electrochemical capacity and porosity of the electrode. The electrochemical reactions take place on the particle surfaces, so capturing the surface area is critical for this phenomena. Additionally, we expect that the mechanical stresses can be dominated by stress concentrators on the surface, which will be more accurately captured as the surface area converges. Finally, electronic transport through the particle network will be dominated by the contact area between particles, making this QOI relevant. We would expect that these geometric quantities would converge at second-order, and this is exactly what is observed for the volume (2.17) and surface area (2.18), based on a Richardson extrapolation. The volume gets captured very well with very coarse meshes, while more refined meshes are required for converged surface and contact areas, with the contact area not yet approaching quadratic convergence at the most refined mesh (1.54). Note that the most refined mesh, which has an element edge length of 0.0625 μm, which is on the order of the experimental data voxel size, is comprised of over 100 × 106 computational elements, requiring significant computational resources to solve.

Arguably more important than the geometric QOIs, however, is a measure of the physical transport mechanisms. A simple way to study these physics is to perform steady-state effective property simulations. Convergence of the effective electrical conductivity (curve with pentagon markers) and effective Young's modulus (curve with octagon markers) is also shown in Fig. 6. The expected convergence of these quantities is not as clear as the geometric quantities, however, as they can demonstrate singular behavior near sharp contacts between particles. For example, when two large particles come into contact at a small point, all of the electrical current must bend to flow through that contact, and much of this current will become concentrated at the edge of the contact. This phenomenon is shown in Fig. 7, where the current streamlines bend around the sharp edge of the contact, and the magnitude of the current density forms a singularity. Nonetheless, both of these QOIs show reasonable convergence with mesh resolution (1.76), albeit at a lower rate than the geometric parameters. A more in-depth look at the applicability and verification of CDFEM for mesoscale simulations, along with the use of adaptive mesh refinement and a study of the role of binder, is under preparation for separate publication [59].

## Highlights of Recent Results

In this section, we highlight some physical insights into the mesoscale behavior of LCO that have been obtained from our coupled electrochemical–mechanical mesoscale simulations. In Sec. 5.1, we discuss coupled electrochemical–mechanical simulations that were first presented by Mendoza et al. [47]. We conclude by showing some new results on the effective electrical conductivity of mesostructures with conductive polymeric binder coatings in Sec. 5.2.

###### Intercalation-Induced Mechanical Deformation.

In Ref. [47], we performed coupled electrochemical–mechanical simulations on an LCO cathode that was manufactured and imaged in-house. This analysis did not include the presence of the binder phase. While many insights were gained from this study, one of the key findings is summarily shown in Fig. 8.

Lithium concentration, in terms of the x in LixCoO2, is shown in Fig. 8(a). Since this snapshot is taken when the cathode is fully charged, Lix = 0.5 nearly everywhere in the domain. The exceptions to this are the two particles shown in red, which are still fully lithiated. These particles were isolated from the rest of the LCO network, not touching a neighboring particle. While lithium ions were able to access the surface of these particles through the electrolyte, electrons were not able to transport in from the current collector to complete the electrochemical circuit. This highlights the importance of the bicontinuous percolated network and shows that if particles can rearrange their network structure during cycling, additional particles could become isolated from the network, leading to degradation of the cathode's capacity.

While the intercalation-induced swelling of a spherical LCO particle would be radially symmetric and stress free, Fig. 8(b) shows that complex-shaped particles arranged into a constrained network exhibit very asymmetric displacements. In the middle of this domain, the particles are shifting to the left as a whole due to the particle–particle contact constraints and boundary conditions applied to them. While this behavior is clearly influenced by the size of the domain and the exact sample of material imaged, it is clear that these materials will deform in a very complex manner.

This complex deformation is due to the intercalation-induced stress shown in Fig. 8(c). High stresses are generally observed (1) near stress concentrators (sharp concave features) in the particles and (2) at particle–particle contacts. The stresses observed here are much higher than in previous studies that investigated single particles [22,36,37,60], primarily due to the network confinement effects. While we recognize that the polymeric binder present in the system will mitigate these stresses [27,47], the network effect will lead to more complicated and generally higher stresses than will be observed by isolated particles, further motivating these mesoscale simulation approaches.

Another key finding first presented by Mendoza et al. [47] is the role that macroscopic mechanical constraints (such as the constraint of the cell within a battery or pack) have on the mesoscale deformation. While it is well-known that putting external pressure on a pouch cell affects the battery life [5], the role that mesostructure plays in that phenomenon is not well studied. Figure 9 explores how the mechanical boundary conditions applied to the reconstructed cathode affect (1) the total amount of intercalation-induced volume change and (2) the partitioning of that volume change between decreasing the porosity of the mesostructure and macroscopic swelling (or “breathing”) of the electrode. When the cathode is under stiff mechanical constraints, the overall volume change (curve with square markers) is lower than when the cathode is allowed to expand freely. With these stiff conditions, all of the volume change goes into decreasing the porosity (curve with circular markers) rather than swelling the cathode (curve with triangular markers). Conversely, when the cathode is allowed to expand nearly all of the volume change goes into swelling, while only a small amount goes into reducing the porosity. This analysis is quite similar to the work of Garrick et al. [61], who studied the “case compressibility,” which is a more 3D effect than these 1D-constrained results. This study highlights the ability of mesoscale simulations to connect macroscale phenomena to electrochemical transport phenomena at the mesoscale.

###### Electrical Conductivity Through LCO-Binder Networks.

LCO cathodes are comprised of LCO active particles and a PVDF/CB binder. Grillet et al. [50] reported a range of values for the electrical conductivity of the binder, depending on its state of stress and number of mechanical or electrochemical cycles it has been subjected too. What they made clear is that it is likely that the binder is the key conductor of electrical current due to its electrical conductivity often being much higher than that of the LCO particles and because it is located at/in particle–particle contacts. This behavior is illustrated in Fig. 10, where two cases of electrical conductivity are shown. The left image shows the distribution of current density when the binder's conductivity is 10× that of LCO, consistent with an uncycled cathode, and nearly all of the current travels through the binder. Even when the conductivities are equal, as shown in the right image, a significant portion of the current density is found in the binder due to its location at or near particle–particle contacts. This work highlights the ability of mesoscale simulations to elucidate electronic transport mechanisms to help tailor the electrical conductivity of binder materials and their stress and cycling response.

Motivated by the large range of electrical conductivity values for both LCO and binder, we performed a number of simulations of the effective electrical conductivity of the cathode for three thicknesses of a uniform binder coating and a very wide range of LCO and binder conductivities, consistent with the experimentally reported values [62,50], and the results of these simulations are shown in Fig. 11. The first three images (a–c) represent the effective conductivity for each of the three binder thicknesses, while the last three images (d–f) show the voltage drop associated with this effective conductivity across a 100 μm cathode at a 1C (dis)charge rate.

During battery charging, the conductivity of both the LCO and the binder increases, with LCO's increase due to its inherent properties [62] and the binder's increase attributed to the increase in mechanical stress pressing the CB particles closer together [50]. Therefore, as the battery charges the effective conductivity increases significantly and the resulting voltage drop decreases. In the fully charged state, the voltage drop is likely as low as 0.1–0.01 V. As the battery discharges, however, the effective conductivity sharply decreases during the last 5% of the theoretical capacity of the cathode, resulting in a voltage drop across the cathode that may reach the 1–10 V range. This behavior may represent a mechanism for the capacity loss that is often observed after the first cycle; after the battery is charged, the high voltage drop required to bring the material back to a state of Li1CoO2 may be prohibitive, making the last 5% of the capacity inaccessible. This effect only gets worse with electrochemical cycling, as the binder conductivity degrades with cycling [50].

This effect is even more pronounced for lower binder loadings (smaller thicknesses), where the binder conductivity needs to be much higher to obtain the same effective conductivity. For reference, the standard loading of 3 wt.% PVDF and 3 wt.% CB results in the 32 nm binder thickness, as shown in Figs. 11(b) and 11(e). It is worth noting that the reconstruction used for these simulations places a uniform coating of binder on the outside of all of the particles, but not necessarily between particle contacts because the particles were not separated prior to simulation. These nuances may have an effect on the results and are the focus of ongoing simulation efforts.

## Ongoing Efforts

While the discussion in Sec. 5 represents significant insight into the role of mesostructures on battery behavior and the ability of mesoscale simulations to provide that insight, many additional simulation efforts are ongoing to further address unanswered questions about the mesostructure.

Our most recent activities have been focused on properly representing the binder in mesoscale simulations. While tomographic data exist that show binder locations [810], there is still significant uncertainty as to where exactly the binder is located within the mesostructure. Rahani and Shenoy [27] suggested that the placement of binder between particle–particle contacts will significantly mitigate stress generation within the particles. However, our previous simulations only placed a uniform coating of binder on the outside of the particles as we had not separated the particles in the reconstructions. Simulations are ongoing to explore the behavioral differences between uniform coatings of binder [63], binder located just near particle–particle contacts (which is suggested from knowledge of the coating process) [9], and from reconstructions of the binder phase from experiment [8,10].

The data of Grillet et al. [50] show that the binder has a much higher electrical conductivity when under mechanical stress, which forces the CB particles into more intimate contact. This suggests that while there may be binder coating all of the particles and throughout the cathode domain, only that binder which experiences significant stress states, such as at particle–particle contacts, will exhibit increased conductivity. To explore this, we are developing constitutive relationships for the binder that captures the stress-dependence of the electrical conductivity. We are performing simulations where the particles experience intercalation-induced swelling and therefore only activate the enhanced electrical conductivity of the binder where it is stressed. These simulations are coupled with thermal calculations to additionally capture temperature-dependent behaviors.

LCO is known to swell anisotropically [11,12], with a preferential swelling direction normal to its c-axis. Experimental data suggest that most of the LCO particles within a cathode are single crystal and that the crystal plane is oriented along the principal c-axis of the particle [64]. Having separate representations for each particle will allow us to analyze the particle orientation and apply our anisotropic constitutive models to these simulations. A round of fully coupled electrochemical–mechanical simulations with anisotropic particle swelling and appropriate binder representation is planned. We also plan to investigate the role of manufacturing conditions (calendaring pressure, etc.) on performance.

Additionally, we have recently been funded through the U.S. Department of Energy's Vehicle Technology Program to advance these techniques and apply them toward vehicle battery scenarios. Most vehicle batteries use NMC cathodes, and we are beginning to obtain experimental reconstructions and develop the necessary constitutive models to study those materials. A major theme for vehicle batteries is safety. One use for mesoscale simulation techniques is to predict material response in abnormal scenarios. Such scenarios may include external heating, nail penetration, and crush. The material's response to these scenarios may differ depending on the state of charge and any residual stress states from the battery operation. Being able to simulate the electrochemical, mechanical, and thermal states at the moment, the abnormal insult occurs and then simulating the insult itself could provide a vast array of insight not available through volume-averaged models and experiment.

A new area of interest in battery modeling is multiscale models [43,44,65]. Volume-averaged models, such as the virtual integrated battery environment (VIBE) [66], are required for cell-, module-, and pack-scale simulations. However, the parameters required for these models often require experimental data for exactly the configuration that is being simulated, which may prevent their use for battery design outside of current experience. In this scenario, multiscale modeling may become useful, with mesoscale simulations performed throughout the macroscopic domain to obtain the material properties for the volume-averaged simulation. As part of the consortium for advanced battery simulation (CABS), we are working with the VIBE team to create such a capability.

Finally, this manuscript has been focused almost entirely on lithium-ion battery cathodes. Yet the anode can play an equally important role. In some ways, the anode is much more complicated, as the common materials used in anodes can swell significantly more (up to a 400% volume increase for silicon [67]) than the cathode materials being considered. This type of volume change will require advanced numerical methods that are beyond the scope of what we present here. However, we are in the process of creating such a capability and hope to apply it to anode materials in the future.

## Acknowledgements

The authors would like to acknowledge the entire Lithium-Ion Battery Degradation LDRD team and the Consortium for Advanced Battery Simulation team for many helpful discussions. We gratefully acknowledge Likun Zhu, Cheolwoong Lim, Robert Kee, Scott Barnett, and Tobias Hutzenlaub for sharing their LCO experimental data with us. This work was partially funded as part of Sandia's Laboratory Directed Research and Development Program. We also acknowledge funding from the U.S. Department of Energy's Vehicle Technologies Office under DE-FOA-0001201 and under the Lab Call as part of the Consortium for Advanced Battery Simulation. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.

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Diercks, D. R. , Musselman, M. , Morgenstern, A. , Wilson, T. , Kumar, M. , Smith, K. , Kawase, M. , Gorman, B. P. , Eberhart, M. , and Packard, C. E. , 2014, “ Evidence for Anisotropic Mechanical Behavior and Nanoscale Chemical Heterogeneity in Cycled LiCoO2,” J. Electrochem. Soc., 161(11), pp. 3039–3045.
Seo, J. H. , Chung, M. , Park, M. , Han, S. W. , Zhang, X. , and Sastry, A. M. , 2011, “ Generation of Realistic Particle Structures and Simulations of Internal Stress: A Numerical/AFM Study of LiMn2O4 Particles,” J. Electrochem. Soc., 158(4), pp. 434–442.
Miller, D. J. , Proff, C. , Wen, J. G. , Abraham, D. P. , and Bareo, J. , 2013, “ Observation of Microstructural Evolution in Li Battery Cathode Oxide Particles by In Situ Electron Microscopy,” Adv. Energy Mater., 3(8), pp. 1098–1103.
Ebner, M. , Marone, F. , Stampanoni, M. , and Wood, V. , 2013, “ Visualization and Quantification of Electrochemical and Mechanical Degradation in Li-Ion Batteries,” Science, 342(6159), pp. 716–720. [PubMed]
Takahashi, K. , Higa, K. , Mair, S. , Chintapalli, M. , Balsara, N. , and Srinivasan, V. , 2016, “ Mechanical Degradation of Graphite/PVDF Composite Electrodes: A Model-Experimental Study,” J. Electrochem. Soc., 163(3), pp. 385–395.
Chen, C.-F. , Barai, P. , Smith, K. , and Mukherjee, P. P. , 2016, “ Scaling Relations for Intercalation Induced Damage in Electrodes,” Electrochim. Acta, 204, pp. 31–49.
Zhang, X. , Shyy, W. , and Sastry, A. M. , 2007, “ Numerical Simulation of Intercalation-Induced Stress in Li-Ion Battery Electrode Particles,” J. Electrochem. Soc., 154(10), pp. 910–916.
Zhang, X. , Sastry, A. M. , and Shyy, W. , 2008, “ Intercalation-Induced Stress and Heat Generation Within Single Lithium-Ion Battery Cathode Particles,” J. Electrochem. Soc., 155(7), pp. A542–A552.
Woodford, W. H. , Chiang, Y.-M. , and Carter, W. C. , 2010, “ “Electrochemical shock” of Intercalation Electrodes: A Fracture Mechanics Analysis,” J. Electrochem. Soc., 157(10), pp. 1052–1059.
Park, J. , Lu, W. , and Sastry, A. M. , 2011, “ Numerical Simulation of Stress Evolution in Lithium Manganese Dioxide Particles Due to Coupled Phase Transition and Intercalation,” J. Electrochem. Soc., 158(2), pp. 201–206.
Miehe, C. , Dal, H. , Schänzel, L.-M. , and Raina, A. , 2016, “ A Phase-Field Model for Chemo-Mechanical Induced Fracture in Lithium-Ion Battery Electrode Particles,” Int. J. Numer. Methods Eng., 106(9), pp. 683–711.
Renganathan, S. , Sikha, G. , Santhanagopalan, S. , and White, R. E. , 2010, “ Theoretical Analysis of Stresses in a Lithium Ion Cell,” J. Electrochem. Soc., 157(2), pp. 155–163.
Goldin, G. M. , Colclasure, A. M. , Wiedemann, A. H. , and Kee, R. J. , 2012, “ Three-Dimensional Particle-Resolved Models of Li-Ion Batteries to Assist the Evaluation of Empirical Parameters in One-Dimensional Models,” Electrochim. Acta, 64, pp. 118–129.
Purkayastha, R. , and McMeeking, R. , 2012, “ An Integrated 2D Model of a Lithium Ion Battery: The Effect of Material Parameters and Morphology on Storage Particle Stress,” Comput. Mech., 50(2), pp. 209–227.
Bower, A. F. , and Guduru, P. R. , 2012, “ A Simple Finite Element Model of Diffusion, Finite Deformation, Plasticity and Fracture in Lithium Ion Insertion Electrode Materials,” Modell. Simul. Mater. Sci. Eng., 20(4), p. 045004.
Rahani, E. K. , and Shenoy, V. B. , 2013, “ Role of Plastic Deformation of Binder on Stress Evolution During Charging and Discharging in Lithium-Ion Battery Negative Electrodes,” J. Electrochem. Soc., 160(8), pp. 1153–1162.
Stershic, A. , Simunovic, S. , and Nanda, J. , 2015, “ Modeling the Evolution of Lithium-Ion Particle Contact Distributions Using a Fabric Tensor Approach,” J. Power Sources, 297, pp. 540–550.
Less, G. B. , Seo, J. H. , Han, S. , Sastry, A. M. , Zausch, J. , Latz, A. , Schmidt, S. , Wieser, C. , Kehrwald, D. , and Fell, S. , 2012, “ Micro-Scale Modeling of Li-Ion Batteries: Parameterization and Validation,” J. Electrochem. Soc., 159(6), pp. 697–704.
Lim, C. , Yan, B. , Yin, L. , and Zhu, L. , 2012, “ Simulation of Diffusion-Induced Stress Using Reconstructed Electrodes Particle Structures Generated by Micro/Nano-CT,” Electrochim. Acta, 75, pp. 279–287.
Ebner, M. , Geldmacher, F. , Marone, F. , Stampanoni, M. , and Wood, V. , 2013, “ X-ray Tomography of Porous, Transition Metal Oxide Based Lithium Ion Battery Electrodes,” Adv. Energy Mater., 3(7), pp. 845–850.
Gross, T. , and Hess, C. , 2014, “ Spatially-Resolved In Situ Raman Analysis of LiCoO2 Electrodes,” ECS Trans., 61(12), pp. 1–9.
Babu, S. K. , Mohamed, A. I. , Whitacre, J. F. , and Litster, S. , 2015, “ Multiple Imaging Mode X-ray Computed Tomography for Distinguishing Active and Inactive Phases in Lithium-Ion Battery Cathodes,” J. Power Sources, 283, pp. 314–319.
Liu, H. , Foster, J. M. , Gully, A. , Krachkovskiy, S. , Jiang, M. , Wu, Y. , Yang, X. , Protas, B. , Goward, G. R. , and Botton, G. A. , 2016, “ Three-Dimensional Investigation of Cycling-Induced Microstructural Changes in Lithium-Ion Battery Cathodes Using Focused Ion Beam/Scanning Electron Microscopy,” J. Power Sources, 306, pp. 300–308.
Lagadec, M. F. , Ebner, M. , Zahn, R. , and Wood, V. , 2016, “ Technique for Visualization and Quantification of Lithium-Ion Battery Separator Microstructure,” J. Electrochem. Soc., 163(6), pp. 992–994.
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Malavé, V. , Berger, J. R. , and Kee, R. J. , 2014, “ The Influence of Crystallographic Orientation on the Chemo-Elastic Response of Reconstructed LixCoO2 Cathode Particles,” J. Electrochem. Soc., 161(11), pp. 3156–3163.
Trembacki, B. , Vadakkepatt, A. , Mathur, S. R. , and Murthy, J. Y. , 2013, “ A Coupled Finite Volume Method for Particle Scale Electrochemical Modeling of Lithium-Ion Batteries,” ASME Paper No. IMECE2013-65774.
Wiedemann, A. H. , Goldin, G. M. , Barnett, S. A. , Zhu, H. , and Kee, R. J. , 2013, “ Effects of Three-Dimensional Cathode Microstructure on the Performance of Lithium-Ion Battery Cathodes,” Electrochim. Acta, 88, pp. 580–588.
Hutzenlaub, T. , Thiele, S. , Paust, N. , Spotnitz, R. , Zengerle, R. , and Walchshofer, C. , 2014, “ Three-Dimensional Electrochemical Li-Ion Battery Modelling Featuring a Focused Ion-Beam/Scanning Electron Microscopy Based Three-Phase Reconstruction of a LiCoO2 Cathode,” Electrochim. Acta, 115, pp. 131–139.
Cooper, S. , Eastwood, D. , Gelb, J. , Damblanc, G. , Brett, D. , Bradley, R. , Withers, P. , Lee, P. , Marquis, A. , Brandon, N. , and Shearing, P. , 2014, “ Image Based Modelling of Microstructural Heterogeneity in LiFePO4 Electrodes for Li-Ion Batteries,” J. Power Sources, 247, pp. 1033–1039.
Chen-Wiegart, Y. K. , DeMike, R. , Erdonmez, C. , Thornton, K. , Barnett, S. A. , and Wang, J. , 2014, “ Tortuosity Characterization of 3D Microstructure at Nano-Scale for Energy Storage and Conversion Materials,” J. Power Sources, 249, pp. 349–356.
Lee, S. , Sastry, A. M. , and Park, J. , 2016, “ Study on Microstructures of Electrodes in Lithium-Ion Batteries Using Variational Multi-Scale Enrichment,” J. Power Sources, 315, pp. 96–110.
Kashkooli, A. G. , Farhad, S. , Lee, D. U. , Feng, K. , Litster, S. , Babu, S. K. , Zhu, L. , and Chen, Z. , 2016, “ Multiscale Modeling of Lithium-Ion Battery Electrodes Based on Nano-Scale X-ray Computed Tomography,” J. Power Sources, 307, pp. 496–509.
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Diercks, D. R. , Musselman, M. , Morgenstern, A. , Wilson, T. , Kumar, M. , Smith, K. , Kawase, M. , Gorman, B. P. , Eberhart, M. , and Packard, C. E. , 2014, “ Evidence for Anisotropic Mechanical Behavior and Nanoscale Chemical Heterogeneity in Cycled LiCoO2,” J. Electrochem. Soc., 161(11), pp. 3039–3045.
Seo, J. H. , Chung, M. , Park, M. , Han, S. W. , Zhang, X. , and Sastry, A. M. , 2011, “ Generation of Realistic Particle Structures and Simulations of Internal Stress: A Numerical/AFM Study of LiMn2O4 Particles,” J. Electrochem. Soc., 158(4), pp. 434–442.
Miller, D. J. , Proff, C. , Wen, J. G. , Abraham, D. P. , and Bareo, J. , 2013, “ Observation of Microstructural Evolution in Li Battery Cathode Oxide Particles by In Situ Electron Microscopy,” Adv. Energy Mater., 3(8), pp. 1098–1103.
Ebner, M. , Marone, F. , Stampanoni, M. , and Wood, V. , 2013, “ Visualization and Quantification of Electrochemical and Mechanical Degradation in Li-Ion Batteries,” Science, 342(6159), pp. 716–720. [PubMed]
Takahashi, K. , Higa, K. , Mair, S. , Chintapalli, M. , Balsara, N. , and Srinivasan, V. , 2016, “ Mechanical Degradation of Graphite/PVDF Composite Electrodes: A Model-Experimental Study,” J. Electrochem. Soc., 163(3), pp. 385–395.
Chen, C.-F. , Barai, P. , Smith, K. , and Mukherjee, P. P. , 2016, “ Scaling Relations for Intercalation Induced Damage in Electrodes,” Electrochim. Acta, 204, pp. 31–49.
Zhang, X. , Shyy, W. , and Sastry, A. M. , 2007, “ Numerical Simulation of Intercalation-Induced Stress in Li-Ion Battery Electrode Particles,” J. Electrochem. Soc., 154(10), pp. 910–916.
Zhang, X. , Sastry, A. M. , and Shyy, W. , 2008, “ Intercalation-Induced Stress and Heat Generation Within Single Lithium-Ion Battery Cathode Particles,” J. Electrochem. Soc., 155(7), pp. A542–A552.
Woodford, W. H. , Chiang, Y.-M. , and Carter, W. C. , 2010, “ “Electrochemical shock” of Intercalation Electrodes: A Fracture Mechanics Analysis,” J. Electrochem. Soc., 157(10), pp. 1052–1059.
Park, J. , Lu, W. , and Sastry, A. M. , 2011, “ Numerical Simulation of Stress Evolution in Lithium Manganese Dioxide Particles Due to Coupled Phase Transition and Intercalation,” J. Electrochem. Soc., 158(2), pp. 201–206.
Miehe, C. , Dal, H. , Schänzel, L.-M. , and Raina, A. , 2016, “ A Phase-Field Model for Chemo-Mechanical Induced Fracture in Lithium-Ion Battery Electrode Particles,” Int. J. Numer. Methods Eng., 106(9), pp. 683–711.
Renganathan, S. , Sikha, G. , Santhanagopalan, S. , and White, R. E. , 2010, “ Theoretical Analysis of Stresses in a Lithium Ion Cell,” J. Electrochem. Soc., 157(2), pp. 155–163.
Goldin, G. M. , Colclasure, A. M. , Wiedemann, A. H. , and Kee, R. J. , 2012, “ Three-Dimensional Particle-Resolved Models of Li-Ion Batteries to Assist the Evaluation of Empirical Parameters in One-Dimensional Models,” Electrochim. Acta, 64, pp. 118–129.
Purkayastha, R. , and McMeeking, R. , 2012, “ An Integrated 2D Model of a Lithium Ion Battery: The Effect of Material Parameters and Morphology on Storage Particle Stress,” Comput. Mech., 50(2), pp. 209–227.
Bower, A. F. , and Guduru, P. R. , 2012, “ A Simple Finite Element Model of Diffusion, Finite Deformation, Plasticity and Fracture in Lithium Ion Insertion Electrode Materials,” Modell. Simul. Mater. Sci. Eng., 20(4), p. 045004.
Rahani, E. K. , and Shenoy, V. B. , 2013, “ Role of Plastic Deformation of Binder on Stress Evolution During Charging and Discharging in Lithium-Ion Battery Negative Electrodes,” J. Electrochem. Soc., 160(8), pp. 1153–1162.
Stershic, A. , Simunovic, S. , and Nanda, J. , 2015, “ Modeling the Evolution of Lithium-Ion Particle Contact Distributions Using a Fabric Tensor Approach,” J. Power Sources, 297, pp. 540–550.
Less, G. B. , Seo, J. H. , Han, S. , Sastry, A. M. , Zausch, J. , Latz, A. , Schmidt, S. , Wieser, C. , Kehrwald, D. , and Fell, S. , 2012, “ Micro-Scale Modeling of Li-Ion Batteries: Parameterization and Validation,” J. Electrochem. Soc., 159(6), pp. 697–704.
Lim, C. , Yan, B. , Yin, L. , and Zhu, L. , 2012, “ Simulation of Diffusion-Induced Stress Using Reconstructed Electrodes Particle Structures Generated by Micro/Nano-CT,” Electrochim. Acta, 75, pp. 279–287.
Ebner, M. , Geldmacher, F. , Marone, F. , Stampanoni, M. , and Wood, V. , 2013, “ X-ray Tomography of Porous, Transition Metal Oxide Based Lithium Ion Battery Electrodes,” Adv. Energy Mater., 3(7), pp. 845–850.
Gross, T. , and Hess, C. , 2014, “ Spatially-Resolved In Situ Raman Analysis of LiCoO2 Electrodes,” ECS Trans., 61(12), pp. 1–9.
Babu, S. K. , Mohamed, A. I. , Whitacre, J. F. , and Litster, S. , 2015, “ Multiple Imaging Mode X-ray Computed Tomography for Distinguishing Active and Inactive Phases in Lithium-Ion Battery Cathodes,” J. Power Sources, 283, pp. 314–319.
Liu, H. , Foster, J. M. , Gully, A. , Krachkovskiy, S. , Jiang, M. , Wu, Y. , Yang, X. , Protas, B. , Goward, G. R. , and Botton, G. A. , 2016, “ Three-Dimensional Investigation of Cycling-Induced Microstructural Changes in Lithium-Ion Battery Cathodes Using Focused Ion Beam/Scanning Electron Microscopy,” J. Power Sources, 306, pp. 300–308.
Lagadec, M. F. , Ebner, M. , Zahn, R. , and Wood, V. , 2016, “ Technique for Visualization and Quantification of Lithium-Ion Battery Separator Microstructure,” J. Electrochem. Soc., 163(6), pp. 992–994.
Malavé, V. , Berger, J. R. , Zhu, H. , and Kee, R. J. , 2014, “ A Computational Model of the Mechanical Behavior Within Reconstructed LixCoO2 Li-Ion Battery Cathode Particles,” Electrochim. Acta, 130, pp. 707–717.
Malavé, V. , Berger, J. R. , and Kee, R. J. , 2014, “ The Influence of Crystallographic Orientation on the Chemo-Elastic Response of Reconstructed LixCoO2 Cathode Particles,” J. Electrochem. Soc., 161(11), pp. 3156–3163.
Trembacki, B. , Vadakkepatt, A. , Mathur, S. R. , and Murthy, J. Y. , 2013, “ A Coupled Finite Volume Method for Particle Scale Electrochemical Modeling of Lithium-Ion Batteries,” ASME Paper No. IMECE2013-65774.
Wiedemann, A. H. , Goldin, G. M. , Barnett, S. A. , Zhu, H. , and Kee, R. J. , 2013, “ Effects of Three-Dimensional Cathode Microstructure on the Performance of Lithium-Ion Battery Cathodes,” Electrochim. Acta, 88, pp. 580–588.
Hutzenlaub, T. , Thiele, S. , Paust, N. , Spotnitz, R. , Zengerle, R. , and Walchshofer, C. , 2014, “ Three-Dimensional Electrochemical Li-Ion Battery Modelling Featuring a Focused Ion-Beam/Scanning Electron Microscopy Based Three-Phase Reconstruction of a LiCoO2 Cathode,” Electrochim. Acta, 115, pp. 131–139.
Cooper, S. , Eastwood, D. , Gelb, J. , Damblanc, G. , Brett, D. , Bradley, R. , Withers, P. , Lee, P. , Marquis, A. , Brandon, N. , and Shearing, P. , 2014, “ Image Based Modelling of Microstructural Heterogeneity in LiFePO4 Electrodes for Li-Ion Batteries,” J. Power Sources, 247, pp. 1033–1039.
Chen-Wiegart, Y. K. , DeMike, R. , Erdonmez, C. , Thornton, K. , Barnett, S. A. , and Wang, J. , 2014, “ Tortuosity Characterization of 3D Microstructure at Nano-Scale for Energy Storage and Conversion Materials,” J. Power Sources, 249, pp. 349–356.
Lee, S. , Sastry, A. M. , and Park, J. , 2016, “ Study on Microstructures of Electrodes in Lithium-Ion Batteries Using Variational Multi-Scale Enrichment,” J. Power Sources, 315, pp. 96–110.
Kashkooli, A. G. , Farhad, S. , Lee, D. U. , Feng, K. , Litster, S. , Babu, S. K. , Zhu, L. , and Chen, Z. , 2016, “ Multiscale Modeling of Lithium-Ion Battery Electrodes Based on Nano-Scale X-ray Computed Tomography,” J. Power Sources, 307, pp. 496–509.
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## Figures

Fig. 2

Two-dimensional images of cathode mesostructures from nano-CT: (a) LCO, image courtesy Yan et al. [7] and (b)NMC (Reprinted with permission from Ebner et al. [31]. Copyright 2013 by John Wiley and Sons.)

Fig. 3

Three-dimensional visualization of a mesostructure reconstruction of LCO. The shades (color online) represent particles that have been separated and individually surface-meshed.

Fig. 4

Illustration of the CDFEM process for a reconstructed LCO mesostructure, visualized on a 2D plane: (a) uniform background mesh (gray) with the curved outlines representing the intersection of the STL-described particles and (b) decomposed mesh after CDFEM, with the electrolyte phase represented by gray elements and the separate particles phases in distinct shades

Fig. 5

Visualization of the computational mesh resulting from the CDFEM process for a reconstructed LCO mesostructure: (a) particles and (b) electrolyte

Fig. 6

Solution verification of the CDFEM approach to calculate effective electrical conductivity and modulus on an LCO mesostructure

Fig. 7

Concentration of electrical current as it flows through the contact between two spheres. The color field is the magnitude of the current density, forming a singularity near the edge of the particle contact. The curved arrows are streamlines of current.

Fig. 1

Two-dimensional illustration of the particle-scale geometry considered in this work, showing the particle, conductive binder, and electrolyte phases. Each of the three particles is shown to have a different crystallographic orientation.

Fig. 8

Cross section of a coupled electrochemical–mechanical simulation on an LCO mesostructure showing how the interplay between the complex particle shapes and network arrangement affects lithium intercalation (a), strain (b), and stress (c). Simulation is of battery charging at 1C, with this snapshot in the fully charged state. (a) Lithium fraction, (b) von Mises equivalent strain and displacement vectors, and (c) von Mises stress (Reprinted with permission from Mendoza et al. [47]. Copyright 2016 by Elsevier.).

Fig. 9

Role of system flexibility (stiffness) on the partitioning of volume change between porosity and macroscopic swelling. The curve with square markers represents volume change of the particle phase (representing the total volume change), the curve with circular markers shows the volume change that reduces porosity, and the curve with triangle markers the volume change from macroscopic swelling. The dashed curve with left triangle markers (corresponding to the right ordinate axis) is the partition coefficient of the fraction of the total volume change that goes into porosity reduction. The vertical dashed line indicates the effective stiffness of the particle network [47].

Fig. 10

Electrical conduction through an LCO cathode including a 100 nm layer of polymeric binder on the outside of the particles. For an uncycled cathode (left), the binder conductivity is 10× that of the LCO and a vast majority of the current travels through the binder. Even when the conductivities of the two phases are equal, the peak current densities are localized in the binder at the contact points between particles (Reproduced with permission from Grillet et al. [50]. Copyright 2016 by The Electrochemical Society.).

Fig. 11

Contour maps of the effective electrical conductivity (a)–(c) and the cathode voltage drop (d)–(f) for a range of particle and binder conductivity values at three binder thicknesses. The large arrows on (a) and (d) show how the results change with charging and cycling. (a) Conductivity, 10 nm binder, (b) conductivity, 32 nm binder, (c) conductivity, 100 nm binder, (d) voltage drop, 10 nm binder, (e) voltage drop, 32 nm binder, and (f) voltage drop, 100 nm binder.

## Tables

Table 1 Key parameters of the mathematical model of Sec. 2, with adopted values for the LCO system

## Discussions

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