## Abstract

Macrohomogeneous battery models are widely used to predict battery performance, necessarily relying on effective electrode properties, such as specific surface area, tortuosity, and electrical conductivity. While these properties are typically estimated using ideal effective medium theories, in practice they exhibit highly non-ideal behaviors arising from their complex mesostructures. In this paper, we computationally reconstruct electrodes from X-ray computed tomography of 16 nickel–manganese–cobalt-oxide electrodes, manufactured using various material recipes and calendering pressures. Due to imaging limitations, a synthetic conductive binder domain (CBD) consisting of binder and conductive carbon is added to the reconstructions using a binder bridge algorithm. Reconstructed particle surface areas are significantly smaller than standard approximations predicted, as the majority of the particle surface area is covered by CBD, affecting electrochemical reaction availability. Finite element effective property simulations are performed on 320 large electrode subdomains to analyze trends and heterogeneity across the electrodes. Significant anisotropy of up to 27% in tortuosity and 47% in effective conductivity is observed. Electrical conductivity increases up to 7.5× with particle lithiation. We compare the results to traditional Bruggeman approximations and offer improved alternatives for use in cell-scale modeling, with Bruggeman exponents ranging from 1.62 to 1.72 rather than the theoretical value of 1.5. We also conclude that the CBD phase alone, rather than the entire solid phase, should be used to estimate effective electronic conductivity. This study provides insight into mesoscale transport phenomena and results in improved effective property approximations founded on realistic, image-based morphologies.

## 1 Introduction

In order for lithium-ion battery technology to become ubiquitous in applications with more pressing demands, electrode materials must be designed to maximize material utilization under a variety of conditions. Macrohomogeneous models (i.e., Newman models [1,2]) are often used to optimize cell design and macroscopic (effective) material properties. However, the connection between intrinsic microscale material properties, electrode recipes, and manufacturing conditions with macroscopic properties needed for modeling (e.g., electrochemically active specific surface area, pore phase tortuosity, and electrical conductivity) are often tenuous. Numerous experiments covering a wide parameter space are required to obtain these properties, yet even once the data have been gathered, the connection between changes to the manufacturing conditions and resulting impacts on properties remains out of reach.

In the absence of direct experimental data for the exact materials under study, macrohomogeneous models typically utilize effective medium theories such as the Bruggeman approximation [3,4] to estimate these properties as functions of material porosity. However, recent experimental and theoretical studies have shown that these approximations often fail (sometimes drastically) at predicting realistic property values [49].

Recent advances in high resolution 3D imaging, specifically X-ray computed tomography (XCT) [1017], have enabled new avenues of fundamental material studies at the mesoscale. With mesoscale XCT imaging, material phases are resolved at length scales sufficient to resolve particle morphology and spatial arrangement while having a field of view large enough to reconstruct a representative volume element (RVE). These imaging techniques, when coupled with advances in high performance computing, have enabled mesoscale simulations that not only predict effective macroscale properties for use in macrohomogeneous models [9,1822] but also predict electrochemical electrode performance [2332].

Despite these advances in 3D imaging and high-resolution mesoscale simulation, most imaging techniques are incapable of distinguishing the conductive binder domain (CBD, which is composed of polyvinylidene fluoride (PVDF) binder and conductive carbon black (CB)) from the void/electrolyte phase. Recent studies have begun to make advancements in resolving the CBD [11,14,17,19,3335], yet these data are insufficient for mesoscale simulation, either due to resolution or image size. While some have attempted to physically predict the CBD phase [36,37], modelers typically use stochastic or algorithmic approaches to incorporate CBD into image-based mesoscale simulations [19,20,3842]. This includes work by the current authors, who have proposed a “binder bridge” CBD placement algorithm [43,44] designed to replicate the CBD phase that may arise from solvent drying and precipitation processes shown by Jaiser et al. [14]. While our previous publications [43,44] established the binder bridge CBD algorithm and used it to calculate effective transport properties, it was only demonstrated for a single mesostructure.

In this paper, we study the impact of conductive binder loading and calendering pressure on the effective electrochemically active specific surface area, electrical conductivity, pore-phase tortuosity, and elastic modulus of LiNi1/3Mn1/3Co1/3O2 (NMC) cathodes. Sixteen electrodes imaged by Ebner et al. [12] are used to define the particle mesostructure, while the binder bridge CBD algorithm of Trembacki et al. [43,44] is used to insert the CBD phase.

Each simulation uses a large RVE with mesh resolution approaching the image voxel size in order to ensure statistical significance of the predictions [45]. Additionally, 20 random samples are chosen from each of the 16 image sets in order to characterize the heterogeneity within the electrodes. This large set of simulation data is unprecedented in the literature. It covers a large span of manufacturing conditions in order to prove useful for insertion into macrohomogeneous modeling efforts. Results are compared to the traditional effective medium theories, and improved models are proposed where appropriate. Furthermore, the effects of these improved models on macrohomogeneous model predictions are demonstrated to establish the significance of the improved models.

## 2 Electrode Reconstruction

This study focuses on NMC cathodes. NMC particles are combined with a CBD composed of PVDF binder and conductive CB in an equal weight ratio. Sixteen electrodes were manufactured by Ebner et al. [12] spanning a matrix of four NMC:CBD weight ratios (90, 92, 94, and 96 wt%) and four calendering pressures (0, 300, 600, and 2000 bar).

We begin by presenting the workflow used to convert XCT image data to 3D computational meshes that represent cathode particle bed mesostructures composed of NMC active material particles and a non-active region. We then discuss the addition of a synthetic CBD phase to approximate the non-active material phases in the imaging data. The entire workflow from image data to simulation results is visualized in Fig. 1.

### 2.1 X-Ray Computed Tomography Electrode Images.

XCT images of NMC cathodes are provided by Ebner et al. [12]. This dataset consists of 3D XCT images that are approximately 330 × 330 μm2 in cross-sectional area and encompasses the entire electrode thickness, ranging in thickness from $∼50to100μm$. We chose to use these XCT data due to the large sample volume spanning many thousands (7000–17,000) of individual NMC particles. Image stacks represent each electrode, and the authors provide raw, binarized, and particle-labeled versions of each image stack.

Using the commercially available image processing software avizo (Thermo Fisher Scientific), along with the published image data where individual particles have been identified via grayscale pixel coloring, a 3D volume representation for each NMC particle is generated. A surface mesh is generated for each particle by meshing and then smoothing the surface of each particle. The mesh is then exported in a standard tessellation language (STL) file format. The process was performed for each of the 16 electrode image stacks.

Fig. 1
Fig. 1
Close modal

There are inherent errors introduced when reconstructing smoothed 3D geometries from pixelated image data. Furthermore, there is evidence that significant errors were introduced in the particle labeling process performed by Ebner et al. [12], which we have detailed previously [43]. To mitigate these errors and approximations, we confirm that the final volume fractions of active and non-active regions in our finite element mesh match the volume fractions calculated from the binarized images as reported in Ref. [12]. This is accomplished by extending each particle surface by 330 nm normal to the surface, a value that is consistent with the hypothesized source of the error.

While it is possible to create surface meshes for thousands of NMC particles in a porous electrode, meshing and simulating physical phenomena on such a large domain is computationally infeasible. However, a large computational domain size is critical in order to ensure that it is a RVE that exhibits bulk properties [45]. Therefore, we select 20 randomly located subdomains of 600–1000 particles for each electrode, for a total of 320 subdomains.

When selecting a subdomain, 15 μm is excluded from the edge of each image stack, as the published labeled image data removed all particles that crossed the edge boundary. The full electrode thickness is captured in each subdomain, with the acknowledgment that edge effects, possibly due to sample preparation, in the electrode thickness direction are reduced by avoiding the first and last ∼10 μm. Subdomain size for each electrode is reported in Table 1, where the subdomain volume across all electrodes is consistent while the dimensions vary for each electrode due to the various electrode thicknesses. It has been shown that subdomains of this size are likely representative of the bulk material when many subdomains are studied [45].

Table 1

Computational subdomain size (x, y, z) in μm for each of the 16 electrodes studied, where z is the electrode thickness

0 bar300 bar600 bar2000 bar
90 wt%82 × 82 × 4878 × 78 × 5279 × 79 × 5174 × 74 × 58
92 wt%82 × 82 × 4863 × 63 × 8073 × 73 × 6071 × 71 × 64
94 wt%66 × 66 × 7363 × 63 × 7966 × 66 × 7469 × 69 × 66
96 wt%56 × 56 × 10158 × 58 × 9557 × 57 × 9961 × 61 × 86
0 bar300 bar600 bar2000 bar
90 wt%82 × 82 × 4878 × 78 × 5279 × 79 × 5174 × 74 × 58
92 wt%82 × 82 × 4863 × 63 × 8073 × 73 × 6071 × 71 × 64
94 wt%66 × 66 × 7363 × 63 × 7966 × 66 × 7469 × 69 × 66
96 wt%56 × 56 × 10158 × 58 × 9557 × 57 × 9961 × 61 × 86

Note: The weight percentage of NMC ranges from 90 to 96 wt% and the calendering pressure ranges from 0 to 2000 bar.

### 2.2 Mesh Generation.

Once a set of STL surface mesh files representing individual NMC particles has been extracted from the data, an interface-conformal computational mesh can be created using the conformal decomposition finite element method (CDFEM) [4547]. The process begins with the selection of a generic tetrahedral mesh that defines the bounds and spatial resolution of the computational domain and is referred to as the background mesh. The automated algorithm begins by converting each particle STL file into a level-set distance field representation ($φ$) on the background mesh. CDFEM uses the level-set field to decompose the background mesh into an interface-conformal mesh. The result is a connected 3D mesh that conforms to the interfaces defined by the original STL surfaces. This algorithm has been previously discussed extensively for this application in a number of publications [28,29,4345,48].

A significant shortcoming in the battery electrode literature is difficult distinguishing between the composite electrode non-active phases of void/electrolyte, PVDF binder, and conductive carbon particles. Trembacki et al. [43,44] introduced and demonstrated a binder bridge algorithm that generates a synthetic CBD within the non-active region of a reconstructed electrode that mimics high-fidelity experimental imaging [14] by preferentially placing composite binder in regions close to two or more particles. An in-depth discussion of this algorithm, as well as our methods for selecting algorithm parameters, is included in the appendix. A single CBD shape parameter is used across all 16 electrodes and CBD loadings, assuming $47%$ (CBD) nanoporosity [19,49]. Figure 2 shows 2D slices of the four corner cases of active material weight percent and calendering pressure to demonstrate the variability on binder morphology and volume as well as overall electrode porosity.

Fig. 2
Fig. 2
Close modal

Using a background mesh element size close to the image voxel size has been shown to limit discretization errors in effective properties to less than 10% relative error for this dataset [45]. To maintain computational feasibility while limiting discretization errors, a background mesh element size of 0.5 μm was selected for this study, yielding computational meshes composed of ∼30 − 35 million tetrahedral elements. We utilize the Galerkin FEM within the SIERRA/Aria multi-physics simulation software [50] to solve the governing equations in the following section. Each set of physics (mechanical, electrical, and ionic) is solved separately using Newton–Raphson nonlinear iterations combined with an algebraic multi-level preconditioned generalized minimal residual (GMRES) iterative linear solver. The mechanics simulation is performed first due to the strain-dependent composite binder conductivity relationship discussed in Sec. 3.

## 3 Models for Effective Properties of Porous Electrodes

The goal of this study is to evaluate macroscale performance properties of electrodes using mesoscale simulations. Here, we present the physical phenomena simulated along with the metrics used to evaluate the particle-bed reconstructions. In many cases, these metrics are effective electrode-scale properties that could be experimentally measured and can be used in macroscale cell-level simulations.

### 3.1 Electrochemically Active Surface Area.

While electrochemical simulations are outside the scope of this study, we know that geometric information such as active particle surface area plays a role in electrochemical performance. Since we are explicitly resolving the CBD phase, several interfacial areas will exist. We assume that the composite binder phase is nanoporous in this study [19,51] and therefore acknowledge that, in addition to particle–electrolyte interface area, particle-CBD interface area may be relevant to electrochemistry [32]. We therefore calculate and report particle–CBD interfacial area as well as particle–electrolyte interfacial area as a part of this study.

Traditionally, the specific active area, a, quantifies the amount of available electrochemically active interface per unit electrode volume as
$a=particle−electrolyteinterfacearea(m2)electrodevolume(m3)$
(1)
where the active interface is the particle–electrolyte interface area and electrode volume is the total electrode volume, rather than just solid volume or active material volume [52]. We also consider a potential secondary source of active electrochemical area, the specific particle–CBD area, defined as
$aNMC−CBD=particle−CBDinterfacearea(m2)electrodevolume(m3)$
(2)
where the particle–CBD interface is the total interfacial area between the particle and CBD phases.

### 3.2 Transport Properties.

Along with electrochemistry, transport phenomena largely govern electrode performance at the mesoscale. A main motivation for reconstructing detailed electrode scans and performing mesoscale simulations is to accurately capture the pathways for ion and electron transport and extract homogenized electrode-scale effective material properties. In this paper, we focus on evaluating two important effective transport properties across the electrode reconstructions: effective electrical conductivity ($σeff$) and pore-space tortuosity ($τ$).

In general, we model conduction through the domain by solving for steady-state conservation
$∇⋅Jk=0$
(3)
where Jk is the generic flux vector. When solving for electrical conduction
$Jelec=−σi∇ϕ$
(4)
where $ϕ$ is electric potential and $σi$ is the electrical conductivity of the constituent phases (i). Similarly, ionic conductivity is considered by solving steady-state conservation (3) where the flux, if interest, is ionic flux,
$Jionic=−κi∇ϕ$
(5)
and $κi$ is ionic conductivity of each phase. As in the remainder of the paper, effects on ionic flux due to lithium-ion concentration gradients that may form during high-rate charge/discharge (i.e., the full Nernst–Planck flux) are neglected, resulting in the simplified ionic flux expression in (5). In both cases, a potential difference is applied across the electrode by applying Dirichlet (uniform potential) boundary conditions on opposing boundaries and applying symmetry (no-flux) boundary conditions on the other four boundaries.
The effective electrical conductivity can be calculated by assuming that the computational domain is a single homogeneous material and applying Ohm’s law:
$σeff=−J¯elecΔϕ/L$
(6)
where $J¯elec$ is the mean current density through a surface where the voltage difference is applied, $Δϕ$ is the voltage difference, and L is the distance across which the voltage difference is applied. The generic mean current density $J¯k$ (either $J¯elec$ or $J¯ionic$) is obtained from
$J¯k=∫Sn⋅JkdS∫SdS$
(7)
where n is the normal vector of one of the Dirichlet boundary surfaces S. $σeff$ is likely to be dominated by the conductive composite binder phase with some contributions from the moderately conductive NMC active particle phase.
The effective ionic conductivity, $κeff$, is calculated similarly to (6) by considering total ion flux at the Dirichlet boundary as
$κeff=−J¯ionicΔϕ/L$
(8)
where $J¯ionic$ is again calculated using (7). Electrolyte tortuosity [4,53] can be calculated with
$τ=κeεκeff$
(9)
where $κe$ is the intrinsic ionic conductivity of the electrolyte, and $ε$ is the electrode porosity. We equate the porosity to the electrolyte phase volume fraction as
$ε=εe=electrolytephasevolumetotalvolume$
(10)

### 3.3 Mechanics.

Mechanics within the mesostructure are of interest due to their impact on the transport physics previously discussed [43,44]. In addition, mechanical phenomena likely play a significant role in battery cycle life and electrode degradation [29,32,5456]. In these simulations, mechanical deformation of the composite cathode phases is considered to be quasi-static and is governed by the conservation of linear momentum,
$∇⋅T=0$
(11)
where T is the Cauchy stress tensor. Here, we consider two components of the stress: a linear-elastic response and lithiation-induced chemical expansivity in the NMC phase. We approach the lithiation strain analogously to Vegard’s law [5759] and have detailed this approach previously [28]. We assume deformation in the small strain limit and thus the constitutive response takes the form
$T=Ci:(γ−βΔCLi)$
(12)
where $Ci$ is the fourth-order elasticity tensor, $γ=(1/2)[∇d+(∇d)T]$ is the small strain tensor, d is the displacement vector, and $βΔCLi$ represents the lithiation-induced swelling strains within NMC where ΔCLi is a simple lithium concentration difference from the NMC reference state. The second-order tensor $β$ is a material property defined by $β=γlith/ΔCLi$ where $γlith$ is the lithiation-induced strain tensor. While recent studies suggest that the mechanical behavior may be much more complicated than this [60,61], the current approach allows a first look at the mechanical effects.

Solving the conservation equation yields stress and strain fields throughout the computational domain, allowing for comparison between various mesostructures. Strain values are useful when considering strain-dependent material properties.

In addition to effective transport properties, effective mechanical properties are useful for cell-scale simulations where the particle-bed structure cannot be resolved. To extract effective behavior from an REV, we first choose a stationary boundary and then apply a small uniaxial compressive strain to the REV defined as $γapplied=ΔL/L$, where ΔL is the compressive displacement at the opposite boundary, and L is the distance between the stationary boundary and the opposite boundary. We then solve the conservation of linear momentum while applying a displacement boundary condition of the form $Dapplied=dγapplied$ to all boundary nodes, where d is the distance from the stationary boundary. We can then calculate the mean normal stress at the stationary boundary as
$T¯=∫Sn⋅T⋅ndS∫SdS$
(13)
where n is the normal vector of the stationary boundary surface S. Finally, we can define the effective elastic modulus of the mesoscale reconstruction as
$Eeff=T¯γapplied$
(14)

### 3.4 Intrinsic Material Properties.

Here, we present the material properties included in our study. Along with several constant material properties, all of which are shown in Table 2, we include several lithiation-dependent material properties. The electrical conductivity of NMC is dependent on the state of lithiation [7,43]
$σNMC={0.1885delithiated5.05×10−6lithiatedS/m$
(15)
We also include strain-dependent CBD conductivity ($σb$), also previously detailed in Ref. [43] and based on the work by Grillet et al. [49], which takes the form
$σb=−17039.67γv+15.93S/m$
(16)
where $γv$ is engineering volume strain within the CBD that is caused by the swelling of surrounding NMC particles. The value of $σb$ is capped at the conductive additive intrinsic value of 500 S/m.
Table 2

Summary of intrinsic material properties and their sources

PropertySymbolUnitsNMCCBDElectrolyte
Electrical conductivity$σi$S/mf(CLi) (15) [7,43]$f(γ)$(16) [43,49]
Ionic conductivity$κi$0.05 [19]1.0
Young’s modulusEiGPa139 [62]0.0638 [49]
Poisson’s ratio$νi$0.3 [62]0.3 [62]
PropertySymbolUnitsNMCCBDElectrolyte
Electrical conductivity$σi$S/mf(CLi) (15) [7,43]$f(γ)$(16) [43,49]
Ionic conductivity$κi$0.05 [19]1.0
Young’s modulusEiGPa139 [62]0.0638 [49]
Poisson’s ratio$νi$0.3 [62]0.3 [62]

All relevant material properties are included in Table 2. Since the metric of interest when calculating tortuosity is relative ($κeff/κe$), an arbitrary value of $κe=1$ can be used. Additionally, since we are assuming a nanoporous CBD phase, we acknowledge that it exhibits measurable ionic conductivity due to its absorption of liquid electrolyte [19].

## 4 Effective Property Results and Discussion

We utilize the mesh generation methods and effective property definitions from the previous sections to comprehensively simulate statistically significant effective properties of the 16 electrodes from Ebner et al. [12]. Example 3D voltage profiles resulting from both effective electrical conductivity and tortuosity simulations on a subdomain are presented in Fig. 3. Due to our inclusion of lithiation-dependent material properties and lithiation-induced swelling, we report effective properties both at fully lithiated (x = 1.0 in LixNi1/3Mn1/3Co1/3O2) and fully depleted NMC (x = 0.5) where appropriate. Additionally, the calendering process applied during manufacturing likely causes anisotropy in properties [63], requiring simulation of the physical phenomena in all three coordinate directions. We present both in-plane and out-of-plane values for each effective property. The in-plane direction is presented as the mean of the two coordinate directions parallel to the current collector surface. Out-of-plane is the direction perpendicular to the current collector, which is also the direction of the calendering/compression process. This requires each subdomain to be simulated in three directions. The mean property values discussed throughout this section are tabulated in Table 4, and all best-fit relationships are summarized in Table 5.

Fig. 3
Fig. 3
Close modal

All plotted results follow the same convention where each data point corresponds to the compiled results of 20 random subdomains taken from a single 3D image. There are 16 data points for each property set, representing each of the 16 electrode configurations. The marker represents the mean value of the 20 subdomains while the error bars are two standard deviations around the mean, representing the 95% confidence interval in the data, assuming a normal distribution. We choose to plot all properties against porosity ($ε$) for easy comparison against theoretical effective property relations typically used by the cell-scale battery modeling community (porous electrode theory) [64].

### 4.1 Active Area.

Before discussing effective properties, we first investigate the geometry/morphology of the various electrodes by calculating the specific surface area of each battery (a), presented in Fig. 4(a). As expected, the specific surface area increases as the wt% of NMC increases, since there is less CBD to cover the particles’ surfaces. For all four NMC weight loadings, the lowest porosity value (highest calendering pressure) exhibits significantly less available surface area. There is a clear trend between porosity and specific surface area for each loading case, although the trend becomes less clear at lower CBD weight fractions. Significant variability in porosity is present between the relatively large subdomains, indicating that the electrode morphologies are quite heterogeneous, consistent with previous observations [45]. Significant variability in specific surface area exists for several of the electrodes, although the deviations are smaller at the higher porosity values typical of NMC cathodes.

Fig. 4
Fig. 4
Close modal

Figure 4(b) provides the secondary source of active particle area, aNMC-CBD. There is a clear trend of increasing CBD coverage with decreasing porosity. This is expected, as the CBD generation algorithm places binder on particle surfaces that are near another particle surface, forming a “bridge” between the particles. In a low porosity environment, most of each particle surface is likely to be near another particle surface and is therefore covered with binder.

It is important to compare the calculated a values here to the theoretical values typically used in battery cell modeling. Traditionally, NMC cathode active material particles are assumed to be spherical. Assuming that all the particles are disconnected and the entire area for each is available for intercalation, active surface area per unit particle volume is theoretically (4πr2)/(4πr3/3) = 3/r. Equivalently, the theoretical specific active area per unit electrode volume is $at=3εAM/r$, where r is the mean particle radius and $εAM$ is the active material (NMC) volume fraction within the electrode. The mean particle size for each of the 16 electrode image datasets (not subdomains) ranges from 2.7 to 3.1 μm, with smaller mean values corresponding to higher calendering pressures, likely due to particle damage/breakage at high calendering pressures. The overall average particle radius across all electrodes is 2.90 μm, which is the value used for r here to calculate at for each electrode. Values of at range from 0.40 to 0.64 μm−1 across the electrodes due to variability in volume fractions, with a mean value of $a¯t=0.53μm−1$.

The second vertical axis in Fig. 4 plots the ratio between our measured value from the NMC + CBD reconstruction and the average theoretical value. We see that reconstructed a values range from 3% to 20% of the theoretical value, which would correspond to an under-prediction of reaction overpotential when using the theoretical value in a cell-scale simulation. We have previously shown that this low $a/a¯t$ ratio is due to the theoretical calculation neglecting both the particle size distribution as well as the CBD covering part of the NMC particle surface [43]. Figure 4(b) highlights that fact as the NMC-CBD interface ranges from 32% to 55% of the theoretical particle surface value and aNMC-CBD/a ratios range from 1.8 to 16.6 across the electrodes. The total reconstructed particle specific surface area (particle–electrolyte + particle–CBD + particle–particle) across the 16 electrodes ranges from 49% to 60% of $a¯t$, suggesting that for this NMC dataset, roughly half of the theoretical surface area is typically introduced into battery-scale simulations erroneously by neglecting the particle size distribution.

### 4.2 Tortuosity.

The first transport property we discuss is the electrolyte phase tortuosity, $τ$, and results for all reconstructions are shown in Fig. 5. We present both in-plane and out-of-plane tortuosity results here to evaluate anisotropy. In general, tortuosity values are larger in the out-of-plane (calendering) direction when compared to the in-plane values. Significant increases in out-of-plane tortuosity are observed in 13 of the 16 electrodes, exhibiting 5–27% higher values. This is expected as the calendering process exerts an out-of-plane pressure to tightly pack the particle slurry, and the pore network inherits that directional packing, decreasing pore connectivity in that direction. This is also consistent with the values of $∼2%$ to $21%$ previously reported [9, Fig. 5].

Fig. 5
Fig. 5
Close modal

There is a very clear relationship between the electrolyte phase tortuosity and porosity, as expected. No discernible dependence on NMC/CB/PVDF loading weights exists outside of the observation that higher CBD loading reduces porosity, indicating that pore morphology or connectivity is not likely affected by solid phase mixture ratios. This is in contrast to all other properties reported in this paper, where trends are not a function of porosity alone. This is despite the fact that the CBD phase does have a small, but finite, ionic conductivity; this CBD conductivity insignificantly impacts the effective tortuosity. Significant variability in tortuosity is observed, again demonstrating the inhomogeneity within each electrode, especially at low porosity values.

We compare our simulated results to the traditional approach used in cell-scale modeling [64], with tortuosity theoretically defined as $τ=ε1−α$ [3,4], where α is the Bruggeman exponent of 1.5 yielding $τ=ε−0.5$. Figure 5 demonstrates that the simulated mean values from the electrode reconstructions do not follow the theoretical equation, exhibiting ∼25% to 40% higher values across all samples. The Bruggeman exponent α is successfully fit to the simulated data, providing alternative forms for both in-plane ($τ=ε−0.666$, α = 1.666) and out-of-plane ($τ=ε−0.722$, α = 1.722) directions for use in cell-scale modeling. To provide the best representation of typical NMC electrodes, best fits are calculated using only the lowest three calendering pressures (0, 300, and 600 bar), which corresponds to the 12 electrodes with the highest mean porosities (0.32–0.42). Calendering pressures of 2000 bar are omitted from the fit, as these pressures are much higher than are industrially used and cause significant particle damage.

Our results match very well with a study by Zielke et al. [39] on a LiCoO2 electrode reconstruction, where they predict tortuosity values of 2.75–3.75 on an electrode with 55% active material volume fraction, corresponding to the lowest porosity electrodes. Experimentally measured tortuosity values (∼2.75 to 3.5) are larger than the values predicted for typical electrode porosities ranging from 0.3 to 0.5 [4,5,8]. This larger experimental value may be due to a CBD morphology that is not as homogeneous as our CBD generation algorithm assumes, which could create significant pore blockages that dominate the tortuosity measurement [65]. Theoretical values predicted by Usseglio-Viretta et al. [9] are also higher than the values in this work, likely due to their more stochastic (and therefore more tortuous) representation of the CBD phase. We note that the Bruggeman model fits in Ref. [9] are not directly comparable to ours because Usseglio-Viretta et al. additionally fit a prefactor before the porosity, meaning it is not consistent with an expected tortuosity of 1 in the limit of large porosities.

### 4.3 Effective Electrical Conductivity.

Another important material property used to characterize composite electrodes is effective electrical conductivity, $σeff$. The calendering process is again expected to cause in-plane versus out-of-plane anisotropy. Additionally, effective electrical conductivity also exhibits lithiation dependence due to the lithiation-dependent NMC conductivity as well as the strain-dependent CBD conductivity. These two observations require the presentation of results for the four scenarios displayed in Fig. 6.

Fig. 6
Fig. 6
Close modal

This extensive dataset allows us to make several observations. We immediately see that effective conductivity is unsurprisingly a function of material loading as well as porosity. More solid-phase material (lower porosity, higher calendering pressure) increases conductivity, and for a given porosity, a higher percentage of highly conductive CBD (lower NMC wt%) further increases effective conductivity by up to ∼4 ×. Although the intrinsic conductivity of NMC decreases with lithiation, the conductivity of the CBD phase increases with lithiation due to the compressive strain it experiences as NMC particles swell. The increase in CBD conductivity prevails over the decreased NMC conductivity as $σeff$ values are up to 7.5 × higher when the NMC is lithiated versus the delithiated NMC case. Interestingly, effective conductivity is generally reduced in the out-of-plane direction, as the calendering process likely pushes particles closer together in that direction, reducing the amount of conductive CBD between particles relative to the in-plane direction. While several electrodes show negligible anisotropy, 12 of the 16 electrodes show relative decreases in out-of-plane conductivity of 7–21% and 12–47% for the delithiated and lithiated cases, respectively.

The traditional approach for calculating effective electrical conductivity for use in cell-scale models is to use a Bruggeman formulation similar to tortuosity $σeff=σs(1−ε)α$, where $σs$ represents solid phase conductivity, which is typically derived from experimental measurements. The theoretical effective conductivity of a multi-material domain can be calculated using many theories [66], and Bruggeman’s approach takes the form
$σeff=σCBD(εCBD)α+σNMC(εNMC)α$
(17)
where $σCBD$ and $σNMC$ represent intrinsic conductivities of the conductive phases, $εCBD$ and $εNMC$ represent volume fractions of those phases, and α = 1.5. Since $σCBD$ is several orders of magnitude larger than $σNMC$, we neglect the second term in (17), resulting in
$σeff=σCBD(εCBD)α$
(18)

Evaluation of (18) for each case is included in Fig. 6, where there are four curves per plot due to the four binder loadings and $σCBD=15.93S/m$ is the unstrained CBD value from (16). The theory matches very well with the simulated results for the delithiated case, indicating that using a typical Bruggeman exponent value of 1.5 is quite accurate for real NMC electrodes, even though the conductive CBD is not necessarily continuous across the domain. However, this conclusion requires the acknowledgment that the CBD phase is the dominant conductive material and requires knowledge of CBD properties and volume fraction. This approach is not commonly taken in cell-level modeling. The theoretical value does not match well in the lithiated cases, likely due to the non-uniform CBD conductivity due to its strain-dependence.

In addition to the theoretical $σeff$ relationship, we also include a non-linear least squares best fit relationship for each of the four subplots, which is again determined by considering only the 12 electrodes with mean porosities greater than 0.3 to increase relevance of the curve fit. The resulting fit is shown in the legend of each plot in Fig. 6.

First, we fit the exponent α for each of the delithiated cases. Again, the best-fit exponents are not significantly different than the exponent value of 1.5 suggested by Bruggeman theory, nor do they significantly deviate from the α values suggested by the tortuosity predictions in Sec. 4.2. The deviation from theory is likely due to a non-ideal morphology as compared to the assumptions made by Bruggeman.

The exponents fit to the delithiated cases are then used as-is for the lithiated cases, as the morphology has not changed between the delithiated and lithiated cases, only the CBD conductivity. The values of $σCBD$ are then fit to the lithiated cases, with the resulting fit parameters shown in the legends of Fig. 6. The pre-factor resides between the allowable range of [15.93, 500] S/m for $σCBD$ and can be considered as an effective $σCBD$ value in the fully lithiated NMC case.

The results presented here support two major conclusions. First, using Bruggeman’s equation is quite accurate for effective conductivity prediction, provided that CBD attributes are known. Second, NMC lithiation effects on $σeff$ can be captured using an appropriate effective value for $σCBD$. In fact, recent mesoscale electrochemical simulation efforts predict that $σeff$ increases almost linearly with NMC lithiation [32], an effect that could be captured in cell-scale modeling using the α exponent values determined here while linearly scaling the intrinsic $σCBD$ value from 15.93 to 42.5 S/m as the NMC is lithiated.

We finally compare $σeff$ values predicted here to others in the literature, although it is difficult to make a direct comparison due to the wide variety of electrode recipes. Peterson and Wheeler [6] measured LiCoO2 composite cathode effective conductivity values ranging from 5 to 25 S/m. Lanterman et al. [67] measured similar values of 10–25 S/m for an NMC532 electrode. NMC532 has been directly shown to be orders of magnitude more conductive than the NMC333 material we studied here [7], which may account for the lower $σeff$ predicted. They also report HE5050 electrode $σeff$ measured values of 1.5–2.0 S/m, which is more aligned with our predicted values. Liu et al. [68] measured values less than ∼1 to 10 S/m at active material (LiNi0.8Co0.15Al0.05O2) weight fractions above $∼94%$. Lastly, Mistry et al. [20] performed simulations with similar binder/PVDF loadings that predict values of ∼5 S/m or less at NMC wt% values above 90%.

### 4.4 Effective Elastic Modulus.

The effective elastic modulus (Eeff) for each electrode in the fully delithiated case is shown in Fig. 7. Poisson’s ratios for each direction were also extracted from the effective modulus simulations by comparing mean transverse strain to the applied compressive axial strain and are included in Table 4 for completeness. Because of our linear elastic and small strain formulations, the modulus of the fully lithiated case will not change. Generally, the effective modulus increases with decreasing porosity in the in-plane direction. This is expected, as a lower pore volume should increase NMC volume and therefore overall material stiffness. This trend is not as clear in the out-of-plane direction, where the 96 wt% case appears to exhibit the opposite trend and values are in lower in general, with 5 of 16 electrodes showing minimal anisotropy while the other 11 exhibit 24–105% higher in-plane values. This may be caused by particle misalignment in the calendering (out-of-plane) direction, which in turn causes indirect load paths, yielding a lower effective modulus. In contrast, there is likely no mechanism to prohibit straight particle chains (i.e., direct load paths) in the in-plane directions.

Fig. 7
Fig. 7
Close modal

There is, however, a trend of increasing modulus with increasing NMC wt%. This is, of course, expected, as the NMC particles have the highest modulus, and including more of them in the electrode for a fixed porosity will result in a higher modulus.

The values predicted here capture mechanical behavior and the associated anisotropy into a material property that can be used as a first-order approach in cell-scale models to predict mechanical performance in both normal and abuse (external compression) scenarios.

### 4.5 Impact on Macrohomogeneous Cell-Scale Modeling.

To evaluate the impact of our mesoscale effective property results on reduced-order macrohomogeneous modeling, we employ a standard Newman pseudo-2D (P2D) electrochemical model [1,69], which we briefly describe and parameterize in Appendix  C. We use a sufficiently sized and highly porous anode to ensure minimal anode effects on battery capacity and transport limitations, and we focus on the cathode half-cell discharge curve to compare various cathode properties. In general, we see that several of the properties extracted from the mesoscale reconstructions and simulations have significant impact on macroscale cell behavior.

Figure 8 demonstrates how the mesoscale effective property/transport properties predicted in the previous sections affect the P2D model. Due to increasing interest in faster charge/discharge rates for applications like electric vehicles [70], we simulate both 1C and 6C discharge rates. We note that for the battery electrode configuration/size used here, altering the effective electrical conductivity ($σeff$) between nominal/Bruggeman values and the mesoscale best-fit values had negligible effects on the discharge performance, even at 6C rates. While this indicates that these battery geometries are likely not limited by electrical conduction, the general trends/fits demonstrated by the mesoscale simulations are still valuable.

Fig. 8
Fig. 8
Close modal

In the top two plots, we consider the effect of increased tortuosity corresponding to Fig. 5(b) for two porosity values ($ε=0.29,0.40$) that represent the low and high range of the NMC electrode samples used throughout this study (excluding the 2000-bar extremely low porosity electrodes). Figure 8(a) shows that battery performance at 1C is largely unaffected for the high porosity cathode; however, the increased tortuosity does noticeably affect the 6C discharge performance. The lower porosity electrode discharge curves in Fig. 8(b) are degraded at both discharge rates, although only slightly at the 1C rate. This is expected as lower porosity significantly increases tortuosity and ionic transport limitations. For the high energy density (low porosity) and high power density (6C) case, realizable capacity decreases by nearly 20% when considering our calculated tortuosity values.

In the final plot (Fig. 8(c)), we investigate the effect of specific surface area by comparing two cases to the nominal value. First, we select the highest particle–electrolyte area demonstrated in Fig. 4(a), which is $∼20%$ of the nominal at value. Due to recent evidence that particle–CBD interface area also contributes to electrochemistry [32], we also consider the highest demonstrated total particle surface area represented by the combination of particle–electrolyte (Fig. 4(a)) and particle–CBD (Fig. 4(b)) areas, which is ∼60% of the nominal at value. These decreases in available area and have a significant impact on the discharge curves, although the effect is much more pronounced for the 0.2at case and at the higher 6C discharge rate. It should be noted that in the P2D model, the value of the Butler–Volmer reaction rate constant (k) can be adjusted to compensate for deficiencies in a value that would otherwise cause significant disagreement with experimental data.

## 5 Conclusions

In this paper, we predict bulk effective properties of 3D imaged NMC electrodes across both active material loading fractions and calendering pressures. Novel to our approach is the consistent representation of nanoporous CBD into the particle bed mesostructure. Specific surface area, tortuosity, electrical conductivity, and elastic modulus are all studied. Results are presented statistically, with computations performed on 20 large subdomains (representative volume elements) from each of the 16 electrodes. Simulation of effective properties across the 320 subdomains allows us to provide detailed analysis on property trends across the reconstructed electrodes and enables comparison with traditionally used theoretical relationships based on Bruggeman theory [3]. Three thousand eight hundred and fourty finite element simulations of ∼30 million elements each are presented in this work. We highlight trends with porosity, in-plane versus out-of-plane anisotropy, and the effect of NMC lithiation/delithiation.

Our approach is unique largely due to the variety of real electrodes studied as well as the number and size of reconstructed subdomains selected from each electrode. This approach provides significant confidence that predictions are not highly sensitive to a single electrode/subdomain and allows observation of trends across electrodes that are impossible to see when studying a small electrode sample size. Selecting 20 subdomains per electrode also allows quantification of the significant heterogeneity across an individual electrode, which is largely ignored in cell-scale modeling. Statistical results are provided that will enable cell-scale modelers to not only perform deterministic simulations but also to quantify uncertainty or study spatial heterogeneities.

Additionally, we include lithiation-induced particle expansion, which ultimately affects localized CBD electrical conductivity, a phenomena rarely considered in mesoscale simulations. While our binder bridge nanoporous CBD generation algorithm is unique, recent comparisons demonstrate good agreement with other CBD generation algorithms [44].

We report specific surface area (electrochemically available area) across the electrodes and note the significant deviation from the traditional analytical calculation based off of a single particle radius. Particle–electrolyte interface areas fall below 20% of theoretical values across all electrodes. We observe that CBD coverage of particle surfaces is quite significant, as expected, and that using a single particle radius for these NMC electrodes causes particle surface area overprediction by a factor of ∼2. Quantifying values of particle–electrolyte and particle–CBD interface areas allow for more accurate cell-scale modeling, and these results represent a significant improvement over the theoretical approach typically used.

Tortuosity values across the electrodes follow a very clear trend with porosity, as predicted by theory. We demonstrate that the typically used Bruggeman exponent (α) of 1.5 is not adequate for these reconstructions, as all electrodes exhibit 25–40% higher tortuosity than predicted by theory. Out-of-plane tortuosity values are observed to be consistently higher than in-plane values for the same electrode, with increases ranging from 5% to 27%. We therefore provide exponents of 1.666 and 1.722 for in-plane and out-of-plane tortuosity, respectively, to accurately represent these mesostructures. Comparisons to experimental data suggest that tortuosities of real electrodes are likely higher than predicted, perhaps due to strongly heterogeneous CBD localization dominating pore-transport resistance.

Trends in effective electrical conductivity ($σeff$) are also clear. Conductivity increases with decreases in porosity. Bruggeman theory sufficiently captures conductivity trends when using only CBD volume fraction and CBD conductivity, while ignoring effects from the relatively non-conductive NMC particles. This is an important finding, as traditional approaches typically use solid-phase volume fraction (NMC + CBD or NMC only) when determining Bruggeman exponents. A Bruggeman exponent of 1.5 generally slightly underpredicts $σeff$. Best-fit Bruggeman exponents of 1.625 and 1.654 for both in-plane and out-of-plane morphologies, respectively, improve the fit.

We also consider the effective electrical conductivity of fully lithiated NMC, where NMC intrinsic conductivity is significantly reduced and NMC particle swelling compressively strains the CBD phase, locally increasing CBD conductivity. Trends with porosity are similar to the delithiated case, with up to 7.5 × increase in conductivity compared to delithiated electrodes. The effective CBD conductivity in the Bruggeman formula is fit to the lithiated data, arriving at a value of 42.5 S/m that is interestingly consistent between both in-plane and out-of-plane directions. This analysis suggests that Bruggeman theory is appropriate when predicting $σeff$, provided that CBD properties are known, and also suggests that lithiation effects on $σeff$ can be captured using a constant anisotropic exponent while scaling intrinsic CBD conductivity appropriately.

This rigorous study utilizes electrode reconstructions to provide a foundation for improved effective property correlations used in traditional macrohomogeneous cell-scale porous electrode theory models. We demonstrate the significant cell-scale model effects predicted by our mesoscale reconstructions. Depending on electrode constituent phases and mesoscale geometry, specific surface area, effective electrical conductivity, and tortuosity can all heavily influence model predictions, and the results presented here allow those model parameters to be founded on reconstructions of real battery morphologies rather than idealized conditions.

## Acknowledgment

We appreciate Vanessa Wood’s group at ETH Zurich for publishing their experimental mesostructure image stacks online. We also appreciate a peer review by Lincoln Collins prior to submission and the anonymous peer reviewers for their insightful comments that improved the quality of this manuscript.

This work was supported by the U.S. Department of Energy Vehicle Technologies Office’s CAEBAT III program (Brian Cunningham, technology manager) and the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.

### Appendix A: Binder Bridge Algorithm

Here, we detail the binder bridge level set algorithm as well as the methods used to determine consistent parameters across the various electrodes considered in this study. In the level-set framework utilized by the CDFEM method, the composite binder domain (CBD) level-set function is defined as
$φB=min[(φP,i+O)*(φP,j+O)−S,…,(φP,N+O)*(φP,N+O)−S],i≠j$
(A1)
where $φP,i$ represents each individual particle level-set, particle number indexes i and j independently range from 1 to particle count N, O denotes a level-set offset, and S is a non-physical size parameter that dictates the size/volume of the binder bridge created between particles. A large O value places more binder near the particle contacts, exposing more particle surface area to electrolyte as visualized in Fig. 9. For a given O value, there is only one S value that will yield the correct volume fraction of CBD. The correct NMC/CBD volume ratio can be calculated using the weight fractions of NMC, PVDF, and carbon used to manufacture the electrode while accounting for the assumed 47% nanoporosity of the CBD phase [19,44].
Fig. 9
Fig. 9
Close modal

We note that a shortcoming of this model is that there are an infinite number of (O, S) parameter pairs that yield the correct CBD volume fraction. The O parameter dictates shape of the CBD, while the S parameter dictates the CBD phase volume. Therefore, we have developed a method to determine a single O parameter for all electrodes in this dataset to ensure that all CBD phases exhibit similar morphology across electrodes, significantly improving the model. Once a single O value has been selected, then the appropriate S value for each of the 16 electrodes must also be determined.

To select the O shape parameter, we focus on the four corner cases rather than study the (O, S) parameter space across all 16 electrodes with the assumption that the corner case extremes in both weight percentage and calendering pressure bound the morphologies exhibited within these extremes. A CBD that entirely coats the particle surface is likely not physical [43], so we attempt to maximize particle–electrolyte interface surface area while maintaining the correct CBD volume fraction. To study the effect of O value on particle–electrolyte surface area, we select 20 randomly located 50 × 50 × 50 μm3 subdomains from each of the four corner cases (90 wt% NMC, 0 bar; 90 wt% NMC, 2000 bar; 96 wt% NMC, 0 bar; 96 wt% NMC, 2000 bar). For each subdomain, we vary O from $3to50μm$ and determine the correct S value. We evaluate the particle–electrolyte specific surface area as a function of O value for each subdomain, as shown in Fig. 10. We observe that there is no clear maximum surface area value but that each subdomain appears to approach an asymptote as O value increases. Therefore, to consistently compare the subdomains, we define a single slope threshold value (a/O = 2 × 10−4μm−2) where the surface area appears to be sufficiently insensitive to O value, represented by the black vertical lines in Fig. 10. We see from this study that an O value of 30 μm corresponds to sufficiently maximized surface area for all subdomains on all corner cases, prompting us to use this one O value for the entirety of this paper.

Fig. 10
Fig. 10
Close modal

While the previous study provides us with a single shape parameter (O value), there is still uncertainty in how to choose an S value for each of the 16 electrodes. We know that each electrode should exhibit a specific active material to CBD volume ratio from the original manufacturing parameters [12]. However, by choosing to represent each electrode with 20 subdomains, we create the possibility for two possible approaches. The simpler path would be to ensure that each subdomain exhibits the correct NMC/CBD volume ratio, which is not likely in reality due to inhomogeneities within the electrode. We opt instead to choose a single S value for the electrode that yields the correct overall volume fraction for the entire electrode image stack, but may result in individual subdomains having more or less CBD than expected. While there are multiple methods to determine a single S value for each electrode, we choose to find a single S value that results in the correct overall NMC/CBD volume ratio across 100 48 × 48 × 48 μm3 subdomains. The S value for each electrode is displayed in Table 3. We note that there is some dependence on calendering pressure and a significant dependence on NMC/CBD wt% ratios as expected. For completeness, for each electrode we also found an S value for each of the 100 subdomains that yielded the correct NMC/CBD volume ratio for the individual subdomain. The arithmetic mean of the 100 S values for each electrode was very similar in value to those in Table 3 ($<0.1%$ relative difference). The O and S values described by Table 3 are applied to each of the 20 large subdomains for each electrode (see Table 1) to determine effective properties for each electrode.

Table 3

Values of the S (size) parameter for each electrode in μm2 corresponding to an O (shape) parameter value of 30 μm

0 bar300 bar600 bar2000 bar
90 wt%1000.501001.93998.72991.73
92 wt%987.99982.79982.41976.95
94 wt%972.20970.86967.04966.23
96 wt%957.72957.07954.53954.80
0 bar300 bar600 bar2000 bar
90 wt%1000.501001.93998.72991.73
92 wt%987.99982.79982.41976.95
94 wt%972.20970.86967.04966.23
96 wt%957.72957.07954.53954.80

### Appendix B: Tabulated Results

Here, we present the numerical values corresponding to plots throughout the paper in Table 4. Values represent means across all subdomains and both in-plane and out-out-plane values are included where appropriate. Table 5 provides a summary of curve fits of the relationship between material/void volume fractions and effective properties.

Table 4

Resulting values of various mesoscale metrics across the 16 NMC electrodes considered

NMC/CB/PVDF (wt%)Pressure (bar)Porosity ($ε$)NMC-pore specific area (a) (μm−1)CBD-pore specific area (μm−1)$τ$$σeff$, delithiated NMC (S/m)$σeff$, lithiated NMC (S/m)Eeff (GPa)$νeff$
90/5/500.3600.0380.1922.04/1.991.27/1.363.13/3.656.36/5.110.23/0.31
90/5/53000.4180.0450.1681.81/1.760.97/0.962.06/2.085.00/3.530.20/0.26
90/5/56000.3180.0350.2062.18/2.301.57/1.535.00/4.937.31/5.240.22/0.34
90/5/520000.1560.0180.2913.19/3.952.77/2.4521.41/17.1613.75/6.720.20/0.38
92/4/400.3780.0540.1831.92/1.930.89/0.911.89/2.076.82/5.410.23/0.33
92/4/43000.3390.0480.2102.08/2.301.18/1.023.33/2.738.44/8.810.27/0.28
92/4/46000.3400.0540.2072.06/2.231.11/0.983.24/2.887.48/5.730.24/0.35
92/4/420000.1830.0320.2843.06/3.681.98/1.7915.14/11.7116.59/8.130.20/0.37
94/3/300.3700.0730.1861.96/2.060.67/0.611.58/1.388.96/8.690.27/0.27
94/3/33000.3760.0770.1881.91/2.070.64/0.571.42/1.168.34/8.660.27/0.26
94/3/36000.3280.0770.2072.06/2.280.76/0.662.25/1.7610.43/7.220.23/0.33
94/3/320000.2260.0550.2482.85/3.161.19/1.077.36/6.2819.90/10.940.20/0.38
96/2/200.3480.0940.1792.01/2.170.46/0.401.24/0.8414.80/15.340.28/0.25
96/2/23000.3530.1000.1822.01/2.100.44/0.411.03/0.8613.97/14.350.27/0.24
96/2/26000.3170.1040.1912.03/2.270.47/0.401.06/0.7214.75/11.440.24/0.29
96/2/220000.2580.0890.2202.28/2.900.69/0.572.59/2.0720.77/10.840.22/0.31
NMC/CB/PVDF (wt%)Pressure (bar)Porosity ($ε$)NMC-pore specific area (a) (μm−1)CBD-pore specific area (μm−1)$τ$$σeff$, delithiated NMC (S/m)$σeff$, lithiated NMC (S/m)Eeff (GPa)$νeff$
90/5/500.3600.0380.1922.04/1.991.27/1.363.13/3.656.36/5.110.23/0.31
90/5/53000.4180.0450.1681.81/1.760.97/0.962.06/2.085.00/3.530.20/0.26
90/5/56000.3180.0350.2062.18/2.301.57/1.535.00/4.937.31/5.240.22/0.34
90/5/520000.1560.0180.2913.19/3.952.77/2.4521.41/17.1613.75/6.720.20/0.38
92/4/400.3780.0540.1831.92/1.930.89/0.911.89/2.076.82/5.410.23/0.33
92/4/43000.3390.0480.2102.08/2.301.18/1.023.33/2.738.44/8.810.27/0.28
92/4/46000.3400.0540.2072.06/2.231.11/0.983.24/2.887.48/5.730.24/0.35
92/4/420000.1830.0320.2843.06/3.681.98/1.7915.14/11.7116.59/8.130.20/0.37
94/3/300.3700.0730.1861.96/2.060.67/0.611.58/1.388.96/8.690.27/0.27
94/3/33000.3760.0770.1881.91/2.070.64/0.571.42/1.168.34/8.660.27/0.26
94/3/36000.3280.0770.2072.06/2.280.76/0.662.25/1.7610.43/7.220.23/0.33
94/3/320000.2260.0550.2482.85/3.161.19/1.077.36/6.2819.90/10.940.20/0.38
96/2/200.3480.0940.1792.01/2.170.46/0.401.24/0.8414.80/15.340.28/0.25
96/2/23000.3530.1000.1822.01/2.100.44/0.411.03/0.8613.97/14.350.27/0.24
96/2/26000.3170.1040.1912.03/2.270.47/0.401.06/0.7214.75/11.440.24/0.29
96/2/220000.2580.0890.2202.28/2.900.69/0.572.59/2.0720.77/10.840.22/0.31

Note: Values are means across 20 subdomains for each electrode and anisotropy is presented in an in-plane/out-of-plane format.

Table 5

Summary of curve fit relations of various mesoscale metrics corresponding to the values in Table 4

In-planeOut-of-plane
Tortuosity ($τ$)$τ=ε−0.666$$τ=ε−0.722$
$σeff$, delithiated NMC (S/m)$σeff=σCBD(εCBD)1.625$$σeff=σCBD(εCBD)1.654$
$σeff$, lithiated NMC (S/m)$σeff=42.47(εCBD)1.625$$σeff=42.49(εCBD)1.654$
In-planeOut-of-plane
Tortuosity ($τ$)$τ=ε−0.666$$τ=ε−0.722$
$σeff$, delithiated NMC (S/m)$σeff=σCBD(εCBD)1.625$$σeff=σCBD(εCBD)1.654$
$σeff$, lithiated NMC (S/m)$σeff=42.47(εCBD)1.625$$σeff=42.49(εCBD)1.654$

### Appendix C: Pseudo-Two-Dimensional Macrohomogeneous Model

Here, we present the P2D model used to compare typical effective transport assumptions to the values predicted by our mesoscale reconstructions throughout the paper. We do not formally introduce the P2D model here, but the equations that constitute the model are summarized in Table 6. Details omitted are abundant in the literature [1,64,71]. Our model uses the diffusion length method to account for solid-phase diffusion within spherical electrode particles [72], a common approach used to reduce the numerical cost of the P2D model [71]. We assume constant transport properties and lithium ion transference number (t+). To simulate discharge current density (Iapplied), a flux boundary condition is applied to the $ϕs$ equation ($−σseff∇ϕs=Iapplied$) on the cathode boundary while a constant voltage boundary condition ($ϕs=0$) is applied at the anode boundary. All other equation boundary conditions are zero flux conditions. Finally, relevant NMC cathode properties sourced from Ref. [32] are included in Table 7.

Table 6

Pseudo-Two-Dimensional (P2D) model equation summary [64]

VariableEquation
Separator region
$ϕe$, electrolyte potential$∇⋅(−κe∇ϕe)+∇⋅(−κD∇lnce)=0$
ce, electrolyte Li+ concentration$∂ce∂t+∇⋅(−De∇ce)=0$
Electrode regions
$ϕe$, electrolyte potential$∇⋅(−κeeff∇ϕe)+∇⋅(−κDeff∇lnce)−aj=0$
$ϕs$, solid potential$∇⋅(−σseff∇ϕs)+aj=0$
ce, electrolyte Li+ concentration$ε∂ce∂t+∇⋅(−Deeff∇ce)−(1−t+)ajF=0$
cs, solid Li concentration$(1−ε)∂cs∂t+∇⋅(−Dseff∇cs)+ajF=0$
j, Butler–Volmer current density$j=i0[exp(αaFηRT)−exp(−αcFηRT)]$
$η=ϕs−ϕe−ϕeq$
i0, exchange current density$i0=kF(ce)αa(cs,max−cs¯)αccs¯αc$
$ϕeq$, equilibrium potential [64,73]$ϕeq=f(cs¯/cs,max)$
$cs¯$, solid Li surface concentration [72]$cs¯=cs−jr5FDs$
Effective transport$κeeff=κeετ$, $κDeff=κDετ$, $Deeff=Deετ$, $τ=ε−0.5$
$σseff=σs(1−ε)1.5$, $Dseff=Ds(1−ε)1.5$
VariableEquation
Separator region
$ϕe$, electrolyte potential$∇⋅(−κe∇ϕe)+∇⋅(−κD∇lnce)=0$
ce, electrolyte Li+ concentration$∂ce∂t+∇⋅(−De∇ce)=0$
Electrode regions
$ϕe$, electrolyte potential$∇⋅(−κeeff∇ϕe)+∇⋅(−κDeff∇lnce)−aj=0$
$ϕs$, solid potential$∇⋅(−σseff∇ϕs)+aj=0$
ce, electrolyte Li+ concentration$ε∂ce∂t+∇⋅(−Deeff∇ce)−(1−t+)ajF=0$
cs, solid Li concentration$(1−ε)∂cs∂t+∇⋅(−Dseff∇cs)+ajF=0$
j, Butler–Volmer current density$j=i0[exp(αaFηRT)−exp(−αcFηRT)]$
$η=ϕs−ϕe−ϕeq$
i0, exchange current density$i0=kF(ce)αa(cs,max−cs¯)αccs¯αc$
$ϕeq$, equilibrium potential [64,73]$ϕeq=f(cs¯/cs,max)$
$cs¯$, solid Li surface concentration [72]$cs¯=cs−jr5FDs$
Effective transport$κeeff=κeετ$, $κDeff=κDετ$, $Deeff=Deετ$, $τ=ε−0.5$
$σseff=σs(1−ε)1.5$, $Dseff=Ds(1−ε)1.5$
Table 7

Pseudo-2D (P2D) model cathode region parameters [32]

PropertyValue
Cathode thickness70 μm
Porosity, $ε$0.29 and 0.40
Specific surface area, a$3(1−ε)/r$
Initial electrolyte concentration, $ce0$1200 mol/m3
Initial Li concentration, $cs0$15,162 mol/m3
Maximum Li concentration, cs,max36,100 mol/m3
Reaction rate constant, k4.38 × 10−11 m2.5/s mol0.5
Effective solid-phase electrical conductivity, $σseff$1.5 S/m
Electrolyte ionic conductivity, $κ$1.165 S/m
Solid phase Li diffusivity, Ds2.51 × 10−14 m2/s
Electrolyte phase Li+ diffusivity, De2.6 × 10−10 m2/s
PropertyValue
Cathode thickness70 μm
Porosity, $ε$0.29 and 0.40
Specific surface area, a$3(1−ε)/r$
Initial electrolyte concentration, $ce0$1200 mol/m3
Initial Li concentration, $cs0$15,162 mol/m3
Maximum Li concentration, cs,max36,100 mol/m3
Reaction rate constant, k4.38 × 10−11 m2.5/s mol0.5
Effective solid-phase electrical conductivity, $σseff$1.5 S/m
Electrolyte ionic conductivity, $κ$1.165 S/m
Solid phase Li diffusivity, Ds2.51 × 10−14 m2/s
Electrolyte phase Li+ diffusivity, De2.6 × 10−10 m2/s

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