Quantifying the influence of charge rate and cathode-particle architectures on degradation of Li-ion cells through 3D continuum-level damage models
Jeffery M. Allena, Peter J. Weddleb, Ankit Vermab, Anudeep Mallarapub, Francois Usseglio-Virettab, Donal P. Fineganb, Andrew M. Colclasureb, Weijie Maib, Volker Schmidtc, Orkun Furatc,
David Diercksd, Tanvir Tanime, Kandler Smithb,∗
aComputational Science Center, National Renewable Energy Laboratory, Golden, CO 80401, USA bCenter for Energy Conversion & Storage Systems, National Renewable Energy Laboratory, Golden, CO 80401, USA
cInstitute of Stochastics, Ulm University, D-89069 Ulm, Germany dMaterials Science Program, Colorado School of Mines, Golden, CO 80401, USA
eIdaho National Laboratory, 2525 N. Fremont, Idaho Falls, ID 83415, USA
Abstract
In this article, we develop a 3D, continuum-level damage model implemented on statistically generated LixNi0.5Mn0.3Co0.2O2
(NMC 532) secondary cathode particles. The primary motivation of the particle-level model is to inform cathode-
particle design through detailed exploration of the influence of secondary and primary particle sizes on the dam-
age predicted during operation, and determine charging profiles that reduce cathode fracture. The model considers
NMC 532 secondary particles containing an agglomeration of anisotropic, randomly oriented grains. These brittle, Ni-
based cathodes are prone to mechanical degradation, which reduces overall battery cycle life. The model predicts that
secondary-particle fracture is primarily due to non-ideal grain interactions and high-rate charge demands. The model
predicts that small secondary-particles with large grains develop significantly less damage than larger secondary parti-
cles with small grains. The model predicts most of the chemo-mechanical damage accumulates in the first few cycles.
The chemo-mechanical model predicts monotonically increasing capacity fade with cycling and rate. Comparing to
experimental results, the model is well suited for capturing initial capacity fade mechanisms, but additional physics is
required to capture long-term capacity fade effects.
Keywords: Continuum damage, Li-ion battery, cathode capacity-loss, NMC 532
1. Introduction
and power density improvements have propelled Li-ion battery adoption into commercial electric vehicles, drones,
and electric-powered flight [1, 2]. These energy and power density gains are due to improvements in active-material
∗Corresponding author. Tel: (303) 275-4423. Email address: kandler.smith@nrel.gov (Kandler Smith)
chemistries and minimizing inactive materials/components. Energy density is related to the battery’s open-circuit po-
tential (OCP) coupled with the battery’s Li insertion capacity, while power density is related to the battery’s internal re-
sistances, reaction kinetics, and solid-state diffusion [3]. The present manuscript focuses on the LixNi0.5Mn0.3Co0.2O2
(NMC 532) cathode chemistry, which exhibits relatively high energy and power densities, but suffers from detri-
mental secondary-particle cracking (and eventual capacity fade) [4]. A physics-based, chemo-mechanical model is
developed to study how cracking is induced in these agglomerated NMC 532 particles. The model, implemented on
realistic secondary particles, provides cathode-design criteria (such as preferred secondary-particle size) to minimize
cathode cracking and improve overall battery cycle-life.
The NMC cathode has a poly-crystalline particle architecture. Here, large (order 10-20 µm) secondary parti-
cles are comprised of sub-micron grains (also referred to as primary particles). Each grain is a single, transversely
isotropic crystal [5, 6]. When conglomerated together to form a secondary particle, the grain’s lattice-plane orien-
tation are usually randomly configured [7, 8]. During cycling, grains undergo anisotropic volume change due to Li
(de)intercalation. Because grains have random lattice-orientations, these anisotropic volume changes can result in
grain-to-grain separation within secondary particles [9]. This grain-to-grain separation is most severe when adjoin-
ing grains have significantly different lattice orientations [10]. At the secondary-particle scale, crack nucleation and
growth can lead to grain isolation and reduce overall cathode capacity [11].
Secondary-particle mechanical degradation manifests as inter-granular fracture at grain boundaries [12] and intra-
granular grain fracture [13, 8] eventually culminating in complete morphological disintegration. A two-step proce-
dure is proposed in the literature for short- and long-term mechanical failure of these NMC particles [8, 14]. Initially,
secondary-particles fracture from coupled diffusion-induced stress, mismatched strains at grain boundaries, and het-
erogeneous electrochemical surface reactions [15, 14]. These initial “break-in” degradation modes are escalated by
the crystal’s transversely anisotropic and concentration-dependent material properties [16, 17, 4, 14]. Second, side
reactions at the solid-electrolyte interface cause further stresses/strains. These side reactions are accelerated by break-
in mechanisms and range from phase-transformations, structural disordering, and transition metal dissolution [18, 8].
For the cathode, oxygen release (forming surface rock-salt and spinel-like structures) are of particular concern because
these side-reactions result in large volume change and reduce the Li diffusivity at the particle surface [11, 19, 20, 21].
These large strains from crystal restructuring can further propagate cracking between grains, but also can result in
intra-granular fracture [11, 13, 8, 22]. In the present model, the initial crack formation due to non-ideal grain in-
teractions is referred to as the break-in mechanism, while the feedback cracking due to crystal reorganization/phase
transformations/oxygen release is referred to as the “fatigue” mechanism.
Recently, continuum-scale modeling was used to investigate the mechanical degradation of NMC cathodes. These
2
continuum-scale models implement a finite-element method (FEM) paired with cohesive-zone models (CZM). A
significant number of published models are 2D models that resolve coupled chemo-mechanical NMC dynamics [9,
23, 14, 8]. Secondary particle-level cohesive-zone models reveal intergranular debonding induced by strain mismatch
at grain boundaries (i.e., the break-in mechanism) [9]. Only recently, has a physically based continuum-level model
been proposed for a fatigue-like mechanism [24]. However, coupling this chemistry-based mechanism with chemo-
mechanics is non-trivial and is yet to be developed.
The present continuum-scale damage model is most similar to that of Bai et al. [25]. They consider an ideal-
ized spherical secondary 3D NMC particle with grain-boundary transport and mixed-mode cohesive-zone mechanical
damage. They show that anisotropic diffusion within grains strongly affects the secondary particle damage and Li
diffusion effects. The present work builds on previous work by Bai et al. [25] by 1) considering realistic 3D secondary
particle morphology for NMC cathodes, 2) considering concentration-dependent material properties, 3) considering
how cycling influences break-in capacity fade and comparing this capacity fade to fast-charge experiments at different
rates over the first 25 cycles, 4) implementing the Butler–Volmer boundary condition rather than constant flux assump-
tions, and 5) simulating secondary particle fracture in a continuous-damage model as opposed to the computationally
expensive CZM implementation.
The primary objective of the present study is to develop a predictive NMC 532 cathode-cracking model that cap-
tures secondary-particle fracture dynamics, relates damage to observed capacity fade, and provides design criteria
for future cathode development. The novel coupled electrochemistry-transport-mechanics model is implemented on
realistic three-dimensional NMC 532 secondary particles. The model is developed on an open-source computational
framework, FEniCS. Realistic secondary particles are generated using statistically representative particle generation
software [6, 26]. The chemo-mechanical model simulates Li (de)intercalation during cycling. The model considers
diffusion-induced stress, mismatch strain, and damage evolution coupled through the spatial and temporal evolution
of geometric, transport, and mechanical properties. In other words, the model is particularly concerned with break-in
effects that cause initial secondary-particle fracture. Contrary to most cathode-cracking models, instead of a cohesive-
zone implementation, a continuum-damage model is used [27]. Because the continuum-damage model approach
is faster computationally, cycle dynamics are tractable (a single charge/discharge cycle of a secondary particle re-
solved with 370,000 electrochemical degrees of freedom and 556,000 mechanical degrees of freedom can be run in
≈1 hr 40 min on 72 processors). Additionally, the model simulates several realistic secondary particle geometries
to study the effect of secondary-particle size and grain size on damage. Such a parameter study (secondary-particle
size, grain size, charge-rate dependence, and cycle dependence), is unique in the literature and provides insights into
optimal cathode operation and design. The model is expected to provide insights into ideal cathode geometries (e.g.,
3
Figure 1: Overview of secondary cathode-particle reconstruction using the Furat et al. [26] algorithm. Figure (a) illustrates features of the synthetic grain-architecture algorithm, while (b) illustrates features of the synthetic secondary particle-architecture algorithm. Both of these algorithms are used to construct a representative 3D secondary particle with grain information.
grain morphology, orientation, and particle sizes) and optimal cathode-sensitive charge-protocol development.
2. Secondary-particle meshing and reconstruction
Five representative secondary-particle geometries are studied in the present manuscript. These particles vary in
both secondary-particle radius and number of constitutive grains. The particles are reconstructed using statistically
representative generation software [26]. The particle-generation algorithm is described briefly here for completeness,
but the reader is directed to Furat et al. [26] for more formal derivation. Figure 1 illustrates a general process-flow
diagram describing how these 3D secondary particles are generated. Figure 1a illustrates features of the synthetic
grain-architecture algorithm. The grain-architecture algorithm uses focused-ion beam electron backscatter diffraction
(FIB-EBSD) data to extract grain-level properties (e.g., grain size and sphericity). Figure 1b illustrates features for
the synthetic secondary particle-level algorithm. Here, particle-level features are extracted from the X-ray nano-
computed tomography (nano-CT) data. From these data sets, machine-learning algorithms are trained to generate
representative grain and secondary-particle architectures, respectively. These two algorithms are then combined to
produce a secondary cathode particle with resolved granular features. The combined synthetic algorithm outputs 3D
voxel data.
The 3D voxel data is loaded into Matlab to detect edges and identify unique grains from the image-quality maps.
Distinct grains are segmented and given unique grain IDs [26]. A grain-size filter is then applied to remove small
grains (≤ 1000 voxels) to de-noise the image and facilitate mesh creation. Removed voxels (≈ 0.003% of the total
4
Figure 2: Five reconstructed secondary particle geometries modeled. Figures (a), (c),and (e) illustrate the original particles generated from Furat el al. [26]. Figures (b) and (d) illustrate particles with different grain sizes, but scaled to have the same secondary particle volume as the “Baseline” particle. The secondary particles are named “Small particle”, “Large grains”, “Baseline”, “Small grains”, and “Large particle” for Figures (a)-(e), respectively.
Table 1: Secondary-particle labels and geometry characteristics Secondary-particle
name Figure
Average grain
volume (µm3)
Standard deviation
grain volume (µm3) Small particle Figure 2a 80.8 6.23 152 5.32E−1 3.99E−1 Large grains Figure 2b 487 11.34 152 3.20E+0 2.40E+0
Baseline Figure 2c 487 10 916 5.32E−1 4.11E−1 Small grains Figure 2d 487 11.25 6095 7.99E−2 6.13E−2
Large particle Figure 2e 3240 21.16 6095 5.32E−1 4.08E−1
volume) are re-assigned to nearest un-removed grain using a Euclidean-distance map. A standard morphology opening
(an erosion step followed by a dilatation step [28]) is used per grain to reduce surface complexity. The filtered tiff-
stack is converted into a 3D tetrahedral mesh using Iso2Mesh [29]. Then, a triangle tessellation of each iso-surface
is created using cgalsurf and subsequently converted to a tetrahedral mesh using cgalmesh. This process creates a
single contiguous mesh that 1) accounts for the grain boundaries, 2) produces relatively smooth internal surfaces, and
3) creates a marker function that properly identifies each grain by a unique domain ID.
Figure 2 illustrates the reconstructed secondary particles studied. As illustrated, the particles in the section la-
belled “Variable secondary size study” (Figures 2a, c, e) are directly reconstructed from synthetic-particle generation
algorithm [26]. The secondary particles vary in diameter, but have nominally the same grain size. To clearly see the
3D grains, a cube is cut from the reconstructed picture. The particles illustrated in the box labelled “Variable grain size
study” have the same nominal secondary-particle volume as the Baseline particle (487 µm3), but have different grain
volumes. The Baseline particle radius and grain sizes are representative of reconstructed NMC 532 particles (cf.,
Figure S1) [26]. The purpose of studying these two simulation sets is to study the chemo-mechanical influences of
5
Figure 3: Reconstructed secondary particle with a single-grain call-out. Overlaid on the single-grain call-out are layers illustrating the crystal-lattice orientation. The y-axis is in the same nominal direction as the c-lattice direction, and the x-axis represents one of the a-lattice directions.
particle size and grain size in realistic NMC particle architectures. Table 1 provides the particle names and geometry
characteristics.
Each secondary-particle is comprised of multiple grains. These grains are empirically assigned random, trans-
versely isotropic crystal orientations. These layered crystals have corresponding a- (in-plane) and c- (out-of-plane)
lattice orientations [30, 16, 31]. Figure 3 illustrates a single grain call-out from a composite secondary particle. Over-
laid on this grain is an x and y axis. The x-axis correlates to an a-lattice orientation and the y-axis correlates to the
c-lattice orientation. Randomly assigned crystal orientations can result in mis-oriented grain-to-grain interfaces where
the crystal orientations do not line up. The worse-case mis-orientation scenario is when the c-axis is perpendicular
between two adjoining grains. The grain-orientation also determines Li-diffusion dynamics, where in-plane diffusion
is simulated as 100 times faster than through-plane difffusion (cf., Supplemental Material).
3. Numerical implementation
The continuous-damage model resolves the intercalated lithium concentration [Li], solid-phase potential ΦNMC,
and chemo-mechanically induced displacement ui, at each time step. First, the [Li] and ΦNMC field variables are
resolved using Equations S1 and S9. Second, the displacement due to chemo-mechanics ui is resolved using Equa-
tions S3 and S8. Third, the elastic stresses (cf., Equation S8) advance the damage factor field variable (Equation S23).
Both the electrochemical and mechanical system are solved using a conjugate-gradient method preconditioned with
successive over-relaxation. The governing field variables (i.e., [Li], ΦNMC, and ui) are all approximated using first-
order continuous Galerkin finite elements. Finally, the solution state is advanced using a backward Euler time-stepping
scheme. The simulation is written in Python using the FEniCS project’s finite-element framework [32, 33]. Full model
6
Figure 4: Model validation using the Baseline particle using half-cell (Li/NMC 532) data from Tanim et al [35]. Figure (a) illustrates the C/20 discharge capacity for half-cells alongside the calibrated continuum-level damage model using the Baseline particle geometry after 25 cycles. Figure (b) illustrates the experimental half-cell C/20 discharge capacity after 600 cycles.
documentation, material parameters, and model assumptions are provided in the Supplemental Material.
The model boundary conditions dynamically change to simulate a CC-CV charge/discharge cycle. The cycle starts
by charging the particle from the low-voltage cut-off (cf., Table S1) using the constant-current boundary condition
(Equation S14). Once the potential reaches the upper cut-off voltage, ΦCV,max, the simulation switches to the constant-
voltage boundary conditions (Equation S15). After a set amount of charge time, the simulation stops charging and
starts to discharge. The discharge step is complete when the potential reaches the lower voltage cut-off, ΦCV,min. After
the discharge, the cycle is complete. Subsequently, another cycle can be modeled.
To mimic laboratory experiments, the model simulates a formation cycle. During the formation cycle, the particle
is initialized at a discharged (lithiated) state. The formation cycle consists of 3 charge/discharge cycles at C/10
followed by 3 cycles at C/2 [34, 35]. The distributions (e.g., damage D and Li distribution [Li]) after formation are
then saved and used as the initial conditions for further simulations.
4. Results
Five reconstructed secondary particles are simulated using the continuum-level chemo-mechanical damage model.
The particles are initialized from the formation cycle results and subsequently cycled an additional 25 times using the
constant-current–constant-voltage (CC-CV) charge and constant-current discharge scheme described in Section 3.
During these cycles, the particles are charged using a CC-CV protocol at either 1 C, 4 C, 6 C, or 9 C for 3600, 900,
600, or 600 seconds respectively. (Note: The 9 C CC-CV charge is for 10 min and not 6.66 min to compare to
experimental results). The upper-cutoff voltage is 4.2 V. For the CC-CV protocol, as the charge rate increases, the
more percentage of the charge is in CV mode [35]. The particles are then discharged at C/2 until the minimum voltage
cut-off is reached, ΦCV,min. During cycling, the particles expand and contract due to Li (de)intercaltion. Such strains
can induce damage, which adversely effects Li diffusion and decreases the material stiffness.
7
4.1. Model calibration
The continuum-damage model requires two parameters that can not be easily measured: the initiate damage strain
ki and the loss of integrity strain kf . To estimate these parameters, the model is calibrated to half-cell experiments.
The details of these experiments are in Tanim et al. [35]. In the experiments, graphite/NMC 532 pouch cells undergo
a formation cycle and then the charge/discharge demand described in Section 3 (full-cell voltage bounds of 3-4.1 V).
After 25, 225, and 600 cycles, several pouch cells are disassembled and half-cells are formed from the harvested
cathodes. These half-cells are then discharged at C/20 to measure the cathode-specific capacity loss due to cycling.
The present continuum-level model damage strain parameters (ki and kf) are calibrated to the half-cell 25-cycle data.
Figure 4 illustrates the experimental half-cell data from Tanim et al. [34] and the Baseline particle capacity loss
with calibrated ki and kf (cf., Table S1). Figure 4a illustrates the half-cell capacity loss after 25 cycles and Figure 4b
illustrates the half-cell capacity loss after 600 cycles. For each C-rate (1, 4, 6, and 9), three half-cells capacities are
measured (Note: the 6 C cases have two essentially overlapping points). The procedure to produce the model results
(points connected by the line in Figure 4a), is
1) Estimate unknown damage parameters (ki and kf)
2) Simulate a formation cycle
3) Cycle for 25 CC-CV cycles
4) Discharge at C/20 discharge cycle
5) Compare the C/20 capacity to that after formation
After calibration, the initiation damage strain is found to be ki = 0.0125 and the loss of integrity strain is found to be
kf = 0.0150.
There are at least two important insights made from the calibration study. First, a significant amount of damage
is accumulated during formation (nominally 21.5% volume percent for the baseline case with calibrated parameters).
These after-formation damage percentage capture the particle-strain inherently due to the non-ideal, anisotropic grain
orientations. In other words, damage is induced during the slow formation that is predominately Li-distribution and
C-rate independent, but occur simply because the grains expand/contract in different 3D directions.
The second important insight from the calibration study is that the C/20 capacity loss increases monotonically
with C-rate. During high C-rate charging, Li-gradients are most severe and these gradients can induce more damage
than less demanding charge demands (see line labelled “Model” in Figure 4a). This response is expected. However,
experimental data in Figure 4b clearly shows that capacity loss decreases with C-rate after 600 cycles. The chemo-
mechanical continuum-damage model will not predict the behavior in Figure 4b. Instead, a feed-back fatigue-like
mechanism (cf., Section 1) is required to predict such effects. For example, suppose a detrimental solid/electrolyte
8
chemical reaction is preferred at low Li intercalation fractions (e.g., rock-salt formation, oxygen release, metal disso-
lution). The cathode cycled at 1 C spends more time at low intercalation fractions than the cathode cycled at 9 C. Thus,
the chemical reaction may proceed further in the 1 C case as compared to the 9 C case. Some of these chemical reac-
tions, such as rock-salt formation, change the cathode crystal structure [11, 19, 20, 21]. Thus, this chemical reaction
may induce stresses that cause additional crack growth. Additional crack growth would then increase the solid-
electrolyte interface area that could cause more non-ideal chemical reactions. The present model does not consider
these long-term chemical reactions. Adding this physics is a topic of future research. Thus, the continuum-damage
model is best used to interpret break-in damage mechanisms during early cycling.
4.2. Baseline particle results
Figure 5 illustrates a significant amount of simulation results for the Baseline particle. Figure 5a illustrates the
pristine Baseline particle surface (blue indicates damage is zero D = 0). Figure 5b illustrates (i) the particle dis-
placement and surface damage and (ii) a cut-slice of the damage variable after formation. For all surface images,
the particle displacement u is magnified ten times. As illustrated there is significant damage after formation. These
damaged areas occur primarily at grain interfaces, but do significantly propagate into some grains. By significantly
propagating into grains, the model is essentially predicting micro-cracks form inside poorly situated grains. The re-
sults from the “After formation” step are the initial conditions for the cycling steps (Figure 5c-f). This initialization is
illustrated as black arrows on the left side.
Figures 5e-f illustrate the internal parameters (displacement u, damage D, and lithium intercalation fraction x) after
the last charge after 25 CC-CV cycles. Figures 5e-f illustrate the 1 C, 4 C, 6 C, and 9 C results, respectively. Each of
these figures contain sub-figures (indicated as Roman numerals i-iv). Sub-figures i) illustrates the particle surface with
10x displacement u and is colored based on damage D. Sub-figures ii) illustrate continuum-level damage on a cut-
plane. Sub-figures iii) illustrate the current damage less the damage after formation (Fig, 5b,ii). This essentially shows
what damage is induced due to fast-charge cycling. Sub-figures iv) illustrates the intercalation fraction distribution.
Comparing the cycle responses, after the last charge the particles have shrunk from their initial state (compare
Figures 5a-f,i) due to reduced c- and a- lattice parameters at these concentrations (cf., Figure S2d). Additionally, the
surface damage (i) is higher in crevasses where grain-grain interfaces intersect. Comparing the damage on cut planes
(sub figures ii) is difficult because a significant amount of damage occurs during formation (cf., Figure 5b,ii). Thus,
sub-figures iii) are shown to illustrate the damage induced due to cycling. Comparing Figures 5e-f,iii, 1C cycling does
not significantly introduce any additional damage (Figure 5e,iii). However, as the rate increases, additional “cracks”
are formed where 9 C has at least three additional cracks in the cut-plane as compared to the “after formation”
response.
9
Figure 5: Continuum-level damage model results for the Baseline particle. Figure (a) illustrates the pristine particle before cycling. Figure (b) illustrates the particle after formation. Figures (c)-(f) illustrate the particle after 25 cycles using a 1 C, 4 C, 6 C, and 9 C charging rate, respectively. Figures (b)-(f) contain sub figures where (i) illustrates the surface damage D and displacement u with magnification of 10, (ii) illustrates a cut-slice of damage, (iii) illustrates the change in damage from the formation cycle, and (iv) illustrates the intercalation fraction distribution. Figure (g) illustrates the percent volume damaged with respect to time for each charge rate.
10
Figure 6: Half-cell capacity after 25 cycles loss comparing (a) different secondary particle radii at C/20, (b) different relative grain sizes at C/20, (c) different secondary particle radii at C/2, and (d) different relative grain sizes at C/2 with respect to cycling rate.
The intercalation fraction distributions (Figures 5e-f,iv) illustrate Li concentration gradients due to fast charging,
and “hot-spots” due to Li being “trapped” in highly damaged areas. For example, in the 1C case, there is essentially
no visible Li gradients, but there are hot-spots where Li appears to be trapped. This trapped Li occurs in regions
of high damage (cf., Figure 5e,ii) where the diffusion coefficient is hindered by the mechanically induced damage
(Equation S24). As the C-rate increases, additional hot-spots are created in high-damage areas and the Li-gradients
become more pronounced due to the fast-charging rate. The one exception is the 9 C case (compare Figures 5e-f,iv).
The 9 C case shows less Li-gradients than the 6 C case. This is because the 9 C and 6 C cases are both charged for
10 min (this is consistent with the experiment the model is validated against [35]). After 10 min, the 9 C case is under
a constant-voltage hold longer than the 6 C case and thus Li gradients are less pronounced in the 9 C case as compared
to the 6 C case.
Figure 5g illustrates the volume-averged integral of damage with respect to time. As illustrated, damage within
the secondary particles tend to increase quickly within the first couple of cycles. This initial increase is non-linear and
monotonic with respect to C-rate. After the initial damage increase, damage tends to level-out and saturate.
4.3. Comparing particle responses
Figure 6 illustrates half-cell capacity loss after 25 cycles with respect to charge rate for the five different secondary
particles. Figure 6a compares the particles with the same nominal secondary particle volume. Figure 6b compares
the particles with the same nominal grain volumes. These responses are extracted after mimicing the experimental
cycling profiles (formation, 25 CC-CV cycles, C/20 discharge) used to validate the Baseline particle.
11
Figure 6a shows a clear correlation between C-rate and secondary particle volume on capacity fade. The capacity
fade is exacerbated with increased secondary particle size (Large particle > Baseline > Small particle) and current
rate (9C > 6C > 4C > 1C). The secondary particle size increases from 6.23 µm to 10 µm to 21.6 µm for the Small,
Baseline, and Large particle, respectively. Such a dramatic dependence on secondary particle size is due to diffusion
distance and surface lithium flux. As the secondary particles become larger, more Li flux is required for a given
C-rate (the surface area does not increase as fast as volume). Additionally, the diffusion length from the secondary
particle surface to center increases. Both of these effects can increase Li gradients and lead to additional chemo-
mechanical stress/strains. Elevated stress results in increased damage, which further increases diffusion lengths and
hinders Li transport. As the particle is cycled, these coupled damage-diffusion effects cause a feed-back loop. An
interesting observation is the Small particle experiences miniscule capacity fade (< 0.25%) with high-current cycling.
The nominal 2 percentage point spread in capacity fade loss between particle sizes is on the same scale as the spread
in experimental data (cf., Figure 4a). A major outcome in this study is the suggestion that small secondary particles
are expected to dramatically reduce cycle losses. Although small secondary-particles are predicted to have preferred
chemo-mechanics, there are practical trade-offs with this geometry. For example, small particles result in low tap
density and have increased surface area for detrimental electrode/electrolyte interface reactions.
Figure 6b illustrates the importance of relative grain size within secondary particles and C-rate effects on capacity
fade. All of these secondary particle have the same volume of 487 µm3, but have grain volumes that span two orders of
magnitude (cf., Table 1). An interesting outcome is that the variance between the three particles is rather small for the
range of grain sizes simulated. These results suggest that the grain-size has a less important effect than the secondary
particle radii. Additionally, all cases have similar C-rate dependence (i.e., the same slope). The results suggest that
the damage-diffusion feed-back loop that dominated the secondary particle volume study (Figure 6a) is less prevalent
in the grain study.
The reduced feedback-loop for the grain-size study may be the reason for a minor non-monotonic response that
causes the baseline case to have the most capacity loss. For the small-grain case, there are more chances for non-
ideal lattice orientations between grains. These non-ideal grain-to-grain interactions increase the damage (the volume
percent damage for the small grains case is 28% after formation as opposed to 13.8% for the large grains case after
formation). However, even though more damage occurs in the small-grain case, there are a multitude of Li-diffusion
pathways available. Even if Li needs to go through a damaged grain to penetrate to the secondary particle center, these
grain distances are short and thus not as costly. On the other hand, the large grains case has the least damage after
formation (13.8% as compared to the baseline of 21.5% volume damaged). This reduced damage allows for easier
Li-diffusion to the secondary particle center even though fewer Li-diffusion pathways are available.
12
100
120
140
160
180
C V
c ap
ac it
y (m
A h/
Small grains Baseline
Large grainsSmall particle
Figure 7: Fast-charge rate capacity for each secondary particle on the 25th cycles.
Figures 6c-d illustrate the modeled capacity loss if the particles are discharged at C/2 instead of C/20 after 25
cycles. Figure 6c illustrates that as the secondary particle radius increases, the capacity loss increases dramatically.
This is the same response as in Figure 6a, but the issues are exacerbated at a higher discharge rate (C/2 as opposed
to C/20). Figure 6d illustrates the capacity loss between the different grain studies. Again, the particles in Figure 6d
all have the same secondary particle volume, but have different number of grains. Here, the small grains case shows
more capacity loss than either the Baseline or Large-grains particle. It is expected that the increased damaged volume
in the Small-grains particle becomes more important at higher rates (C/2 as opposed to C/20) as the tortuous paths
required to lithiate the cathode are more hindering. For Figures 6c-d the reader may note that the 9 C case has less
capacity loss than the 6 C case. This is because the particles are discharged directly after the last charge of the 25
cycles. Recall that both the 9 C and 6 C cases are both charged for 10 min (as opposed to 6.66 min and 10 min,
respectively). Thus, the 9 C case is charged at the constant-voltage constraint longer and has less Li gradients than the
6 C case (cf., Figures 5e-f,iv). Comparing the grain-density responses, the large-grain case is more preferable because
its observable capacity is less rate-dependent and the large-grain case has less continuum-level damage.
Figure 7 illustrates the capacity reached in the CC-CV charge for each secondary particle on the 25th charge cycle.
As illustrated, the secondary-particle diameter is a dominate parameter, where the Small particle has the best capacity
performance and the Large particle has the worst. The grain density (compare Large grains, Baseline, and Small
grains) has a less dominate effect, but the Large-grains particle has better performance than the Small-grains particle.
The rate-capacity response is a realization of transport and kinetic resistances. The responses indicate that increased
diffusion lengths (whether due to larger secondary particles or more tortuous paths due increased grains) leads to
increased resistances that dominate the fast-charge capability of the secondary particle. The fast-charge capacity has
the same trends as the C/2 capacity results (Figures 6c-d). Thus, not only are small secondary particles with fewer
grains expected to have less chemo-mechanical damage, they are expected to have better power performance.
13
5. Summary and conclusions
A continuum-level damage model is developed and implemented on realistic 3D NMC 532 secondary cathode par-
ticles. The model resolves chemo-mechanically induced stress/strains that promote mechanical failure and eventual
capacity fade. The model considers many (order 1000s) grains (primary particles) that comprise a single secondary
cathode particle. These realistic reconstructions are generated using synthetic, statistically representative architec-
tures [26]. Five secondary particles (three unique geometries) are simulated to study how secondary particle size and
how grain size influence cathode capacity fade after cycling. The model is calibrated using half-cell capacity loss
measurements.
The model predicts cathode chemo-mechanical damage due to fast-charging protocols. These chemo-mechanics
failure modes are realized during high-rate cycling and are due to non-ideal grain-to-grain interactions. The model pre-
dicts that chemo-mechanic failure modes increase with increased charge rates. The model predicts that the most ideal
particle geometries for fast charging are small secondary particles (primary effect) with few grains (secondary effect).
Additionally, the model predicts that long-term cycling effects is not due to classical single-particle chemo-mechanics.
Instead, either electrode-level physics (e.g., multi-particle phase separation [36], secondary-particle isolation) or sur-
face chemistry effects (e.g., chemically induced rock-salt formation) are required as a feedback mechanism to capture
long-term cycle fade.
1) Using a computationally efficient method, secondary-particle damage is simulated on realistic 3D geometries.
2) Continuum-level damage is induced by mis-oriented grains and Li-gradients.
3) Early-cycle break-in damage is simulated to occur in formation and during the first 25 fast-charge cycles.
4) Additional physics (referred to as fatigue-like mechanisms), other than chemo-mechanics, is required to predict
long-term cathode capacity loss.
5) Cycling and charge-rate is shown to influence chemo-mechanically induced capacity loss.
6) Smaller secondary particles are predicted to have significantly less chemo-mechanically induced capacity loss
than larger secondary particles.
7) Secondary particles comprised of larger grains (primary particles) are predicted to have slightly less capacity
loss than secondary particles with small grains.
6. Acknowledgements
We gratefully acknowledge insightful discussions and suggestions from Kasra Taghikhani (Colorado School of
Mines). This work is authored in part by the National Renewable Energy Laboratory, operated by Alliance for Sus-
14
tainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308, and in
part by Idaho National Lab, operated by Battelle Energy Alliance for the U.S. Department of Energy under contract
DE-AC07-05ID14517. Funding is provided by the U.S. DOE Office of Vehicle Technology Energy Storage Pro-
gram, eXtreme Fast Charge and Cell Evaluation of Lithium-Ion Batteris (XCEL) Program, program manager Samuel
Gillard. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government.
The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S.
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 U.S. Government purposes.
Nomenclature
a0 Initial in-plane strain free length m
a Change in in-plane lengths m
Ci jkl Fourth-order stiffness tensor Pa
c0 Initial ou-of-plane strain free length m
c Change in out-of-plane lengths m
D Damage factor −
Di j Anisotropic diffusion coeff. tensor m2 s−1
DD i j Damaged anisotropic diffusion coeff. tensor m2 s−1
DLi In-plane diffusion coeff. m2 s−1
E Young’s modulus Pa
Eeq Equilibirum potential V
f Damage loading function −
i0 Exchange current density A m−2
Ji Lithium flux in direction i mol m−2 s−1
k Scalar-history variable −
ki Initiate damage strain −
[Li]max Maximum lithium concentration mol m−3
[Li] Change in lithium concentration mol m−3
[Li+]el Li-ion concentration mol m−3
nk Unit normal in direction k -
R Universal gas constant J mol−1 K−1
t Time s
T Temperature K
x Intercalation fraction −
αa Anodic transfer coefficient −
αc Cathodic transfer coefficient −
βi j Anisotropic chemical-expansion coeff. tensor m3 mol−1
εi j Total strain tensor −
εe eq Equivalent strain −
εe i Principle elastic strain −
η Kinetic overpotential V
µ Mean −
σ Standard deviation −
ΦCV Constant voltage potential constraint V
ΦCV,max Constant voltage potential constraint (max) V
ΦCV,min Constant voltage potential constraint (min) V
Φel Electrolyte potential V
ΦNMC Electrode potential V
References
[1] E.J. Cairns and P. Albertus. Batteries for electric and hybrid-electric vehicles. Annu. Rev. Chem. Biomol. Eng., 1:299–320, 2010.
[2] M. Tariq, A.I. Maswood, C.J. Gajanayake, and A.K. Gupta. Aircraft batteries: current trend towards more electric aircraft. IET Electrical
Systems in Transportation, 7:93–103, 2016.
[3] B. Li, J. Zheng, H. Zhang, L. Jin, D. Yang, H. Lv, C Shen, A. Shellikeri, Y. Zheng, R. Gong, J.P. Zheng, and C. Zhang. Electrode materials,
electrolytes, and challenges in nonaqueous lithium-ion capacitors. Adv. Mater., 30:1705670, 2016.
[4] R. Xu, L.S. de Vasconcelos, and K. Zhao. Mechanical and structural degradation of LiNixMnyCozO2 cathode in Li-ion batteries: An
experimental study. J. Electrochem. Soc., 164:A333–A3341, 2017.
[5] A. Quinn, H. Moutinho, F. Usseglio-viretta, A. Verma, K. Smith, M. Keyser, and D.P. Finegan. Electron backscatter diffraction for investi-
gating lithium-ion electrode particle architectures. Cell Rep., 1:100137, 2020.
[6] O. Furat, D.P. Finegan, D. Diercks, F. Usseglio-Viretta, K. Smith, and V. Schmidt. Mapping the architecture of single lithium ion electrode
particles in 3D, using electron backscatter diffraction and machine learning segmentation. J. Power Sources, 483:229148, 2021.
[7] A. Quinn, H. Moutinho, F. Usseglio-Viretta, A. Verma, K. Smith, M. Keyser, and D.P. Finegan. Electron backscatter diffraction for investi-
gating lithium-ion electrode particle architectures. Cell Rep. Phy. Sci., 1:100137, 2020.
[8] L. Mu, R. Lin, R. Xu, L. Han, S. Xia, D. Sokaras, J.D. Steiner, T-C. Wend, D. Nordlund, M.M. Doeff, Y. Liu, K. Zhao, H.L. Xin, and F. Lin.
Oxygen release induced chemomechanical breakdown of layered cathode materials. Nano Lett., 18:3241–3249, 2018.
[9] R. Xu and K. Zhao. Corrosive fracture of electrodes in Li-ion batteries. J. Mech. Phys. Solids, 121:258–280, 2018.
16
[10] J. Kim, H. Lee, H Cha, M. Yoon, M. Park, and J. Cho. Prospect and reality of Ni-rich cathode for commercialization. Adv. Energy Mat.,
8:1702028, 2018.
[11] H.H. Ryu, K.J. Park, C.S. Yoon, and Y.K. Sun. Capacity fading of Ni-rich Li[NixCoyMn1−x−y]O2 (0.6 ≤ x ≤ 0.95) cathodes for high-energy-
density lithium-ion batteries: Bulk or surface degradation? Chem. Mater., 30:1155–1163, 2018.
[12] H. Liu, M. Wolf, K. Karki, Y.S. Yu, E.A. Stach, J. Cabana, K.W. Chapman, and P.J. Chupas. Intergranular cracking as a major cause of
long-term capacity fading of layered cathodes. Nano Lett., 17:3452–3457, 2017.
[13] G. Qian, Y. Zhang, L. Li, R. Zhang, J. Xu, Z. Cheng, S. Xie, H. Wang, Q. Rao, Y. He, Y. Shen, L. Chen, M. Tang, and Z-F. Ma. Single-
crystal nickel-rich layered-oxide battery cathode materials: synthesis, electrochemistry, and intra-granular fracture. Energy Storage Mater.,
27:140–149, 2020.
[14] Y. Mao, X. Wang, S. Xia, K. Zhang, C. Wei, S. Bak, Z. Shadike, X. Liu, Y. Yang, R. Xu, P. Pianetta, S. Ermon, E. Stavitski, K. Zhao, Z. Xu,
F. Lin, X-Q. Yang, E. Hu, and Y. Liu. High-voltage charging-induced strain, heterogeneity, and micro-cracks in secondary particles of a
nickel-rich layered cathode material. Adv. Func. Mat., 9:1900674, 2019.
[15] T. Li, X.Z. Yuan, L. Zhang, D. Song, and K. Shi. Degradation mechanisms and mitigation strategies of nickel-rich NMC-based lithium-ion
batteries. Energy Storage Rev., 3:43–80, 2020.
[16] J.-M. Lim, T. Hwang, D. Kim, M.S. Park, K. Cho, and M. Cho. Intrinsic origins of crack generation in Ni-rich LiNi0.8Co0.1Mn0.1O2 layered
oxide cathode material. Sci. Rep., 7:39669, 2017.
[17] L.S. de Vasconcelos, R. Xu, J. Li, and K. Zhao. Grid indentation of mechanical properties of composite electrodes in Li-ion batteries. Extreme
Mech. Lett., 9:495–502, 2016.
[18] Y. Li, X. Cheng, Y. Zhang, and K. Zhao. Recent advance in understanding the electro-chemo-mechanical behavior of lithium-ion batteries by
electron microscopy. Materials Today Nano, 7:100040, 2019.
[19] A. Tornheim, S. Sharifi-Asi, J.C. Garcia, J. Bareno, H. Iddir, R. Shahbazian-Yassar, and Z. Zhang. Effect of electrolyte composition on rock
salt surface degredation in NMC cathodes during high-voltage potential holds. Nano Energy, 55:216–225, 2019.
[20] H. Zheng, P. Xu, M. Gu, J. Xiao, N.D. Browning, P. Yan, C. Wang, and J.-G. Zhang. Structural and chemical evolution of Li- and Mn-rich
layered cathode material. Chem. Mater., 27:1381–1390, 2015.
[21] T. Li, X.-Z. Yuan, L. Zhang, D. Song, K. Shi, and C. Bock. Degradation mechanisms and mitigation strategies of nickel-rich NMC-based
lithium-ion batteries. EER, 3:43–80, 2019.
[22] C. Wei, Y. Zhang, S-J. Lee, L. Mu, J. Liu, C. Wang, Y. Yang, M. Doeff, P/ Pianetta, D. Nordlund, X-W. Du, Y. Tian, K. Zhao, J-S. Lee,
F. Lin, and Y. Liu. Thermally driven mesoscale chemomechanical interplay in Li0.5Ni0.6Mn0.2Co0.2O2 cathode materials. J. Mater. Chem. A,
6:23055–23061, 2018.
[23] R. Xu, L.S. de Vasconcelos, J. Shi, J. Li, and K. Zhao. Disintegration of meatball electrodes for LiNixMnyCozO2 cathode materials. Exp.
Mech., 58:549–559, 2018.
[24] A. Ghosh, J.M. Foster, G. Offer, and M. Marinescu. A shrinking-core model for the degredation of high-nickel cathodes (NMC811) in Li-ion
batteries: Passivation layer growth and oxygen evolution. J. Electrochem. Soc., 168:020509, 2021.
[25] Y. Bai, K. Zhao, Y. Liu, P. Stein, and B-X. Xu. A chemo-mechanical grain boundary model and its application to understand the damage of
Li-ion battery materials. Scripta Mater., 183:45–49, 2020.
[26] O. Furat, L. Petrich, D.P. Finegan, D. Diercks, F. Usseglio-Viretta, K. Smith, and V. Schmidt. Artificial generation of representative single
Li-ion electrode particle architectures from microscopy data. npj Comput. Mat., 7:105, 2021.
[27] J. Lemaitre. A Course on Damage Mechanics. Springer, 2012.
[28] P. Soille. Morphological Image Analysis: Principles and Applications. Springer, 2013.
17
[29] A. Phong Tran, S. Yan, and Q. Fang. Improving model-based f NIRS analysis using mesh-based anatomical and light-transport models.
Neurophotonics, 7(1):015008, 2020.
[30] J.-M. Lim, H. Kim, K. Cho, and M. Cho. Fundamental mechanisms of fracture and its suppression in Ni-rich layered cathodes: Mechanics-
based multiscale approaches. Extreme Mech. Lett., 22:98–105, 2018.
[31] M.A. Slawinski. Waves and Rays in Elastic Continua: 3rd Edition. World Scientific, 2015.
[32] M.S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M.E. Rognes, and G.N. Wells. The fenics project
version 1.5. Arch. Num. Soft., 3(100), 2015.
[33] A. Logg, K.-A. Mardal, and G.N. Wells (eds). Automated Solution of Differential Equations by the Finite Element Method. Springer, 2012.
[34] T.R. Tanim, E.J. Dufek, M. Evans, C. Dickerson, A.N. Jansen, B.J. Polzin, A.R. Dunlop, S.E. Trask, R. Jackman, and I. Bloom. Extreme fast
charge challenges for lithium-ion battery: Variability and positive electrode issues. J. Electrochem. Soc., 166:A1926, 2019.
[35] T.R. Tanim, Z. Yang, A.M. Colclasure, P.R. Chinnam, P. Gasper, S.-B. Son, P.J. Weddle, E.J. Dufek, I. Bloom, K. Smith, C.C. Dickerson,
M.C. Evans, A.R. Dunlop, S.E. Trask, B.J. Polzin, and A.N. Jansen. Extended cycle life implications of fast charging for lithium-ion battery
cathode. Energy Storage Mater., 41:656–666, 2021.
[36] J. Park, H. Zhao, S.D. Kang, K. Lim, C.-C. Chen, Y.-S. Yu, R.D. Braatz, D.A. Shapiro, J. Hong, M.F. Toney, M.Z. Bazant, and W.C. Chueh.
Fictitious phase separation in Li layered oxides driven by electro-autocatalysis. Nat. Mater., 2021.
[37] F.C. Larche and J.W. Cahn. The interactions of composition and stress in crystalline solids. J. Res. Natl. Bur. Stand., 89:467, 1984.
[38] K. Taghikhani, P.J. Weddle, J.R. Berger, and R.J. Kee. Chemo-mechanical behavior of highly anisotropic and isotropic polycrystalline
graphite particles during lithium intercalation. J. Electrochem. Soc., 167:110554, 2020.
[39] R. Xu, L.S. de Vasconcelos, and K. Zhao. Computational analysis of chemo-mechanical behaviors of composite electrodes in Li-ion batteries.
J. Mater. Res., 31:2715–2727, 2016.
[40] A.F. Bower, P.R. Guduru, and V.A. Sethuraman. A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a
lithium-ion half-cell. J. Mech. Phys. Solids, 59:804–828, 2011.
[41] A. Jelvehpour. Development of a Transient Gradient Enhanced Non Local Continuum Damage Mechanics Model for Masonry. PhD thesis,
Queensland University of Technology, 2016.
[42] G.H. Sun, T. Sui, B.H. Song, H. Zheng, L. Lu, and A.M. Korsunsky. On the fragmentation of active material secondary particles in lithium
ion battery cathodes induced by charge cycling. Extreme Mech. Lett., 9:449–458, 2016.
[43] A.M. Colclasure, A.R. Dunlop, S.E. Trask, B.J. Polzin, A.N. Jansen, and K. Smith. Requirements for enabling extreme fast charging of high
energy density Li-ion cells while avoiding lithium plating. J. Electrochem. Soc., 166:A1412–A1424, 2019.
[44] M. Guo, G. Sikha, and R.E. White. Single-particle model for a lithium-ion cell: Thermal behavior. J. Electrochem. Soc., 158:A122–A132,
2011.
[45] J.M. Reniers, G. Mulder, and D.A. Howey. Review and performance comparison of mechanical-chemical degredation models for lithium-ion
batteries. J. Electrochem. Soc., 166:A189–A3200, 2019.
[46] R. Amin and Y.-M. Chiang. Characterization of electronic and ionic transport of Li1−xNi0.33Mn0.33Co0.33O2 (NMC333) and
Li1−xNi0.50Mn0.20Co0.30O2 (NMC523) as a function of Li content. J. Electrochem. Soc., 163:A1512–A1517, 2016.
[47] A. Dubois, K. Taghikhani, J.R. Berger, H. Zhu, R.P. O’Hayre, R.J. Braun, R.J. Kee, and S. Ricote. Chemo-thermo-mechanical coupling in
protonic ceramic fuel cells from fabrication to operation. J. Electrochem. Soc., 166:F1007–F1015, 2019.
[48] K. Vyshenska. How to provide structural stability in thermal expansion simulations (e.g., COMSOL blog). 2018.
[49] K. Smith and C.-Y. Wang. Power and thermal characterization of a lithium-ion battery pack for hybrid-electric vehicles. J. Power Sources,
160:662–673, 2006.
18
[50] A. Verma, K. Smith, S. Santhanagopalan, D. Abraham, K.P. Yao, and P.P. Mukherjee. Galvanostatic intermittent titration and performance
based analysis of LiNi0.5Co0.2Mn0.3O2 cathode. J. Electrochem. Soc., 164:A3380–A3392, 2017.
[51] O. Dolotko, A. Senyshyn, M.J. Muhlbaurer, K. Nikolowski, and H. Ehrenberg. Understanding structural changes in NMC Li-cells by in situ
neutron diffraction. J. Power Sources, 255:197–203, 2014.
[52] X. Zhu, Y. Chen, H. Chen, and W. Laun. The diffusion induced stress and cracking behavior of primary particle for Li-ion battery electrode.
Int. J. Mech. Sci., 178:105608, 2020.
[53] A. Colclasure and R.J. Kee. Thermodynamically consistent modeling of elementary electrochemistry in lithium-ion batteries. Electrochem.
Acta, 55:8960–8973, 2010.
[54] A.M. Colclasure, T.R. Tanim, A.N. Jansen, S.E. Trask, A.R. Dunlop, B.J. Polzin, D. Robertson, L. Flores, M. Evans, E.J. Dufek, and K. Smith.
Electrode scale and electrolyte transport effects on extreme fast charging of lithium-ion cells. Electrochem. Acta, 337:135854, 2020.
[55] E.J. McShane, A.M. Colclasure, D.E. Brown, Z.M. Konz, K. Smith, and B.D. McCloskey. Quantification of inactive lithium and solid-
electrolyte interface species on graphite electrodes after fast charging. ACS Energy Lett., 5:2045–2051, 2020.
[56] F.M. de Sciarra. A nonlocal model with strain-based damage. Int. J. Solids and Struct., 46:4107–4122, 2009.
[57] R. de Borst. Fracture in quasi-brittle materials: a review of continuum damage-based approaches. Eng. Fract. Mech., 69:95–112, 2002.
[58] A. Ansell and T. Gasch. Cracking in quasi-brittle materials using isotropic damage mechanics. Proc. Comsol Conf., 2016.
[59] Z. Gao, L. Zhang, and W. Yu. A nonlocal continuum damage model for brittle fracture. Eng. Fract. Mech., 189:481–500, 2018.
[60] R.H. Peerlings, R. de Borst, W.M. Brekelmans, and M.G.D. Geers. Localisation issues in local and nonlocal continuum approaches to
fracture. Eur. J. Mech. A-Solid., 21:175–189, 2002.
[61] Z.P. Bazant and G. Pijaudier-Cabot. Nonlocal continuum damage, localization instability and convergence. J. Appl. Mech., 559:287–293,
1988.
[62] R.H. Peerlings, R. de Borst, W.M. Brekelmans, and J.H.P. de Vree. Gradient enhanced damage for quasi-brittle materials. Int. J. Numer.
Meth. Eng., 39:3391–3403, 1996.
[63] E. Kuhl, E. Ramm, and R. de Borst. An anisotropic gradient damage model for quasi-brittle materials. Comput. Methods Appl. Mech. Engrg.,
183:87–103, 2000.
[64] A.H.M.A. Gafoor and D. Dinkler. A macroscopic gradient-enhanced damage model for deformation behavior of concrete under cyclic
loadings. Arch. Appl. Mech., 90:1179–1199, 2020.
19
50
100
Figure S1: Grain-volume distributions for the five secondary particles. Distribution statistics are given in Table 1.
Figure S1 illustrates histograms describing the grain volume distributions. Figures S1a and b are the same dis-
tribution because these particles are just scaled versions of each other (this is also true for Figure S1d and e). The
1
grain-volume distribution for the Baseline, Small grains, and Large particle follows a fairly smooth distribution (Fig-
ure S1c-e). Conversely, the Small particle and Large grains secondary particles have more erratic grain-volume distri-
butions. This erratic distribution is because these secondary particles have very few grains (152 grains) as compared
to the more grain-dense particles.
S2. Governing equations
centration [Li], 2) chemo-mechanically induced displacement ui, and 3) solid-phase potential ΦNMC. These indepen-
dent variables are resolved in a reconstructed secondary particle consisting of an agglomeration of grains. The grains
are assumed to have anisotropic properties with randomly assigned lattice orientations. The secondary particle is as-
sumed to be electronically connected to the electrode matrix and surrounded by a electrolyte bath (both not explicitly
modeled). The surrounding electrolyte is assumed to have constant concentrations and potential.
S2.1. Lithium transport
∂[Li] ∂t
∂xi = 0, (S1)
where Ji is the Li flux in the i direction, xi is the ith spatial coordinate, and t is time. Einstein notation is used to
express equations compactly (i.e., ∂Ji/∂xi = ∑
i ∂Ji/∂xi). Intercalated lithium flux Ji is assume to follow Fickian
diffusion, which can be expressed as
Ji = −Di j ∂[Li] ∂x j
, (S2)
where Di j is the anisotropic diffusion coefficient tensor. Because the Li flux only depends on lithium concentrations
(and not hydrostatic stress [37, 38]), the present formulation can be considered one-way coupled. For these NMC
cathodes, the coupled diffusion/stress effect is not expected to be significant when considering anisotropic effects [38]
and such small volume expansion [39] (≈2%). However, there is an implicit coupling effect between stress and lithium
flux. The elastic strain/stress can cause internal damage, which adversely effects the solid-phase diffusion tensor, Di j
(cf., Section S3).
Particle displacement ui is represented using infinitely small-strain theory. Small-strain theory is appropriate for
modeling electrodes that expand less than ≈ 10% volume expansion during cycling [31, 38, 39]. Assuming that lithium
2
transport is much slower than elastic-wave deformation and that there is no external body forces on the particle, linear-
elastic body forces can be represented as ∂σi j
∂x j = 0, (S3)
where σi j is the Cauchy stress tensor. The stress-strain relationship relates the Cauchy stress tensor to the elastic strain
tensor εe i j as
σi j = Ci jk` ε e k`, (S4)
where Ci jk` is the stiffness tensor. The elastic strain, εe, is related to the total strain, εi j, and the inelastic diffusion-
induced strain, εD i j as
εe i j = εi j − ε
D i j . (S5)
The total strain εi j is related to the displacement field ui as
εi j = 1 2
εD i j = βi j [Li], (S7)
where βi j is the anisotropic chemical-expansion coefficient tensor and [Li] is the difference in lithium concentration
from the stress-free state [38, 37]. Combining these equations, the Cauchy stress can be expressed as a function of the
independent variables (displacement and Li concentration) as
σi j = Ci jk`
] . (S8)
This formulation assumes that the quasi-brittle NMC material does not exhibit plastic strain [40, 41, 42].
S2.3. Solid-phase potential
∂
3
Table S1: Constant model properties Property Variable Value Unit Young’s modulus (isotropic) [17, 23] E 138 GPa Maximum Li concentration [43] [Li]max 49.6 kmol m−3
Density [43] ρ 4700 kg m−3
Li-ion concentration [43] [Li+]el 1.2 kmol m−3
Temperature [35] T 273.15 K Anodic & cathodic transference numbers [43] αa, αc 0.5 −
Loss of integrity strain i) kf 0.0150 −
Initiate damage strain i) ki 0.0125 −
Effective electronic conductivity ii) κ 10 S m−1
Poisson’s ratio (isotropic) [39, 17] ν 0.3 −
Electrolyte potential (reference) ii) Φel 0 V Cut-off voltage (min) [35] ΦCV,min 3.4 V Cut-off voltage (max) [35] ΦCV,max 4.2 V
i) fit, ii) assumed
where κ is the solid-phase conductivity. It is standard-practice for single-particle models to assume that the solid-
phase potential only evolves in time (not space) [44, 45]. However, such a hard constraint coupled tightly to the
(de)intercalation reaction (cf., Section S2.4) imposed on a complex reconstructed geometry is numerically unstable.
To stay consistent with other single-particle models, a high solid-phase conductivity κ is chosen (Table S1). The
high solid-phase conductivity causes the solid-phase potential to be essentially uniform across the secondary particle.
However, cathode NMC particles do not have intrinsically high solid-phase conductivity κ [46]. This somewhat
contradictory condition is rationalized by assuming the surrounding carbon-binder layer (not modeled) removes any
significant gradients in solid-phase potential resulting in a higher apparent solid-phase conductivity than that of bulk
NMC 532. By assuming the surrounding binder-layer electronically conducts to all grains in the secondary particle,
grain electrical isolation is not captured. In other words, even if a grain is predicted to damage in such a way that it can
no longer diffuse lithium to an adjacent grain, the isolated grain can still intercalate/ deintercolate from the electrolyte.
Capturing grain electrical isolation, is a subject of future work and will be incorporated in electrode-scale studies.
S2.4. Boundary and initial conditions
The model is initialized at a uniform lithium concentration (xinit[Li]max) and a stress-free initial state. The stress-
free initial state assumes no residual stresses remain from calendaring, applied cell pressure, or thermal strains from
the sintering temperature [47]. Rigid-motion suppression boundary conditions are specified for conservation of mo-
mentum (Equation S3) [48]. The lithium flux (Equation S1) and electrostatic potential (Equation S9) boundary con-
ditions at the secondary-particle surface are coupled and expressed using a Butler–Volmer formulation. The current
density due to electrochemical reactions at the particle surface can be expressed as
i = i0 [ exp
where i0 is the concentration-dependent exchange current density (cf., Figure S2b), R is the universal gas constant, F
is the Faraday constant, T is temperature, and η is the overpotential. The overpotential, η, is expressed as
η = ΦNMC − Φel − Eeq(x), (S11)
where Eeq is the equilibrium potential (cf., Figure S2a). The current i at a surface is related to the Li-flux as
Dk j ∂[Li] ∂x j
nk = i F , (S12)
where nk is the outward-pointing normal. The current at the surface is also related to the solid-phase current as
i = −κ ∂ΦNMC
∂xk nk. (S13)
Because the electrolyte potential and concentration is assumed constant and uniform, and the solid-phase electric
potential ΦNMC is resolved spatially, a solid-phase potential boundary condition is required. If a current demand is
specified, the constant-current boundary condition can be expressed as
idemand = − ∑
k
∂xk Ak, (S14)
where idemand is the current demand (in Amps), Ak is the projected area normal to direction k, and the summation is
over all surfaces. A constant-voltage boundary condition requires that boundary is a set potential
ΦNMC = ΦCV, (S15)
where ΦCV is the specified constant-voltage potential. In the present study, the solid-phase conductivity, κ, is set to a
high value such that the potential is essentially uniform across the entire secondary particle (cf., Table S1).
The loading applied to the modeled particle follows a standard constant-current–constant-voltage (CC-CV) charg-
ing profile. During the constant-current demand, the total current is constant, but the current at the particle surface
is allowed to vary locally. Previous models [49], predict that at electrode-scale the current density is not uniform
along the electrode thickness due to concentration and potential gradients. These gradients are increased during high-
current demand. In the present work, current variation due to electrode-scale effects is not considered. Thus, the
single-particle model can reasonably approximate a full-electrode response for relatively thin electrodes, or relatively
small current demands, or at a specific depth in the electrode. In future work, the team plans on addressing the uni-
5
0
2
4
6
a la
tt ic
e pa
ra m
et er
)
Figure S2: Concentration dependent LixNi0.5Mn0.3Co0.2O2 parameters: (a) equilibrium voltage [43], (b) exchange current density [43], (c) in-plane Li diffusion coefficient [50], (d) lattice parameters [51].
form electrode-scale assumption by dynamically changing the applied current based on current-demands predicted by
a macroscale model.
S2.5. Model parameters
The chemo-mechanical model requires several material properties. Table S1 lists constant material properties used
in the model. The model assumes that the Young’s modulus and the Poisson’s ratio are constant and isotropic, which is
standard practice in modeling these cathode materials (i.e., Ci jkl is assumed isotropic) [42, 52]. Additionally, the sur-
rounding electrolyte concentration and potential are assumed constant. Assuming constant electrolyte concentrations
and potential neglects electrode-thickness variations that develop during cycling.
Figure S2 illustrates the concentration-dependent properties. The equilibrium potential Eeq and exchange current
density is from Colclasure et al. [43]. The exchange current density includes species activity coefficient contributions
6
Figure S3: Voltage response of the current “reconstructed model” using the Baseline particle and from Colclasure et al. [43, 54, 55] (Macro-model) at 1 C, 4 C, 6 C and 9 C rates.
from both the cathode and the electrolyte (cf., Equation S10) [53]. Figure S2d illustrates the a- and c-lattice param-
eters for NMC 532 from Dolotko et al. [51]. The anisotropic lattice parameters are used to compute the anisotropic
chemical-expansion coefficient tensor represented as [38]
βi j =
The in-plane chemical expansion coefficient βa can be expressed as
βa = a/a0
[Li] , (S17)
where a is the change in the a lattice parameter from the initial state, a0 is the lattice parameter at the initial state
and [Li] is the change in Li concentration from the initial state. Similarly, the out-of-plane chemical expansion
coefficient βc can be expressed as
βc = c/c0
[Li] . (S18)
These coefficients result in a unit cell volume expansion of ≈2% between intercalation fractions of 1 to 0.4, which is
in-line with experimentally measured volume swelling [51].
Figure S2c illustrates the in-plane Li diffusion coefficient. The diffusion coefficient’s functional dependence with
respect to Li concentration is scaled from Verma et al. [50] . Verma et al. [50] estimate the bulk, isotropic diffusion
coefficient from galvanostatic intermittent titration experiments during delithiation. However, NMC 532 crystals are
inherently transversely isotropic (i.e., layered) [51, 11]. To account for NMC 532 anisotropy, the out-of-plane diffusion
coefficient is assumed to be one hundred times smaller (slower) than the in-plane diffusion coefficient. Additionally,
7
the in-plane diffusion coefficient is scaled such that the reconstructed model produces a similar polarization response
as the half-cell macro-model from Colclasure et al. [43, 54, 55].
Figure S3 compares the half-cell voltage response simulated with a macro-homogeneous model from Colclasure
et al. [43, 54, 55] (dots) and the present reconstructed model (lines). For this comparison study, mechanics/damage are
not resolved. The macro-model is simulated using a very thin NMC electrode with minimal effects from electrolyte
transport or electron conduction through electrode. Again, the in-plane (and thus out-of-plane) diffusion coefficient in
the “reconstructed” model are scaled such that the present reconstructed model has a similar polarization response to
Colclasure et al. [43, 54, 55]This is equivalent to a backward homogenization approach, where a macroscale model is
used to set a lower-scale parameter model. The voltage responses predicted by the current reconstructed model and
that of the macromodel from Colclsure et al. [43, 54, 55] are essentially identical at 1 C, 4 C, 6 C and 9 C rates.
S3. Quasi-brittle continuum-level damage
Continuum damage model (CDM) theory is used to determine the spatial and temporal evolution of strain-induced
damage for the quasi-brittle cathode [27, 41]. While the cohesive zone [9] and extended finite element method (X-
FEM) [52] have been used in literature to investigate fracture of polycrystalline NMC battery cathodes, the continuum
damage model provides a balance of including detailed model physics while reducing computational complexity.
The CZM and X-FEM methods are more computationally expensive because they are required to resolve the grain-
boundary separation. This computational expense is one of the reasons a majority of studies are 2D, an approximation
that oversimplifies transport pathways and limits the extension of a model to new polycrystalline geometries. The
continuous-damage model offers a different approach by employing a continuous-damage variable D throughout the
spatial domain. Because the continuous-damage model does not require remeshing, the model has significantly faster
computation times [56]. It is worth noting that the continuous-damage model does not resolve at least three grain-to-
grain separation physics inherently captured in cohesive-zone models. These are
1) Discontinuous lithium concentrations and fluxes
2) Discontinuous stress/strain fields
However, the continuous-damage model can simulate hundreds to thousands grains and charge-discharge cycle dy-
namics, which are intractable for cohesive-zone models.
The model considers isotropic damage evolution for quasi-brittle materials [57, 58, 59]. Here, the Poisson’s
ratio is assumed to be unaffected by damage. Instead, the damage variable adversely effects the material stiffness.
8
Incorporating isotropic damage, the stress σi j takes the form
σi j = max ( (1 − D), 0.1
) Ci jklε
e kl, (S19)
where D is the damage parameter, which varies from 0 to 1 as damage accumulates (compare to Equation S4). Here,
a severely damaged region will have 1/10 of the original stiffness. This minimum stiffness cut-off is enforced for
numerical stability.The stress-strain relation is complemented by a damage loading function, D = f (εe eq,σeq, k), where
εe eq and σeq are scalar-valued functions of elastic strain and stress tensors, respectively, and k is the scalar-history
variable. Discrete Kuhn-Tucker loading-unloading conditions are satisfied by the damage loading function f and the
rate of the history variable such that
f ≥ 0, dk dt ≥ 0, f
dk dt
= 0. (S20)
f (εe eq, k) = εe
eq − k. (S21)
The equivalent elastic strain εe eq is defined based on principle elastic strains such that the material only fractures un-
der tension. Quasi-brittle materials like NMC 532 cathode particles exhibit much greater strength under compression
than tension. Hence, the equivalent strain is defined as
εe eq =
i > )2 , (S22)
where εe i is the principle elastic strain (cf., Equation S5) and < εe
i >= εe i if εe
i > 0 and < εe i >= 0 otherwise. A known
difficulty with continuum-damage models is mesh-dependence of the equivalent strain variable εe eq, which informs
damage. To quantify/resolve this issue a mesh-dependence study is provided in the Supplementary Material.
The history parameter k starts at the damage threshold level ki and is updated by the requirement that f = 0 during
damage growth (Equation S21). Damage growth occurs according to an evolution law
D(k) =
, (S23)
9
where ki is the threshold strain upto which linear elastic behavior is followed and kf is the strain where there is
complete loss of integrity. Beyond the threshold strain ki, the stress-strain curve follows a linear descending profile
until complete fracture at kf .
If a given volume is damaged, that damage decreases the material’s stiffness (cf., Equation S19) and decreases
the Li-diffusion coefficient. Essentially, if a volume is stretched too much, the volume is assumed to contain many
micro-cracks that hinder Li diffusion. Damage modulates the Li-diffusion coefficient as
DD i j = (1 − D)Di j, (S24)
where DD i j is the damaged, anisotropic diffusion coefficient and Di j is the un-damaged Li-diffusion coefficient (cf.,
Figure S2c).
ented grains. The continuum-damage formulation allows for rapid (order hours) cycling simulations with coupled
chemo-mechanical responses on complex, realistic 3D reconstructed particles. While computationally tractable and
applicable to real geometries, a disadvantage for continuum-damage models is intrinsic mesh dependence [60, 57, 61].
The present supplementary material discusses mesh-dependent study results for the chemo-mechanical model. The
mesh-dependent study results quantify the particle damaged volume percentage and capacity loss mesh-dependency
(or lack thereof). These results are shown in Figure 4 and 5 in the main text.
Figure S4: Mesh dependence results using the Baseline particle. The results are for nominally 50, 114, and 371 degrees of freedom (DOF). Figure a illustrates the percent volume damaged after cyling, and b illustrates the predicted capacity loss using a C/20 discharge.
A mesh dependence study is implemented on the Baseline particle (cf., Figure 2c in the main text). The mesh
resolution is reported in degrees of freedom (DOF), where the higher degrees of freedom, the more dense the mesh.
Three meshes are studied with approximately 50, 114, and 371 thousand DOF. The percent volume damaged and the
10
capacity loss (using a C/20 discharge) are compared between the three meshes.
Figure S4 illustrates the volume percent damaged (a) and the capacity loss (b) for the three meshes under different
cycling rates after 25 cycles. Figure S4a shows that as the C-rate increases, the damaged particle volume increases. As
the mesh density increases, the damaged volume decreases. This decrease in damaged volume is because the meshed
volumes near the mis-oriented grain interfaces, which are most-likely to damage, are smaller. Figure S4b illustrates the
capacity loss predicted using a C/20 discharge for the three meshes after 25 cycles under different charging demands
(e.g., 1 C, 4 C). For the 1C case, the all meshes predict the same nominal capacity fade (i.e., mesh independent). For
the 4 C and 6 C cases, the capacity increases with increased mesh density. For the 9 C case, the capacity loss varies
with mesh density, but does not have a clear trend. In other words, for the 9C case, even though the damage volume
decreases with increased mesh, the capacity loss is non-monotonic. Such a non-linear effect means that the location
of the damage is now a dominating contributor to capacity loss. Thus, further increasing mesh density is expected
to continue to decrease the percent volume damaged (Figure S4a), but the capacity loss is not expected to decrease
further (Figure S4b).
In the present study, the local-damage model is considered converged for the 371 thousand DOF case (i.e., the
results in the main article are reported using the 371 thousand DOF density). This mesh density is considered to be
converged because the capacity loss is no longer monotonically mesh dependent. The authors realize that developing
such a highly refined mesh is non-ideal in that the model becomes much more computationally expensive. One method
to reduce the mesh density is to implement a non-local damage approach [62, 63, 64]. These, and other methods will
be explored in future work to further improve the continuum-level damage model.
S5. Time in constant-current mode
Figure S5: Percentage of time in CC mode as compared to CC-CV mode during the first charge. Simulated particle responses are reported for 1 C, 4 C, 6 C, and 9 C.
Figure S5 illustrates the percentage of charge time spent in CC mode for the particles at 1 C, 4 C, 6 C, and 9 C
after formation. As illustrated, as the C-rate increases, less charge time is spent in constant-current (CC) mode. The
11
particles that spend more time in CC mode have less resistance. Like the C/2 capacity results (cf., Fig. 6c-d), the
smaller particles have less diffusion resistance than larger particles. Additionally, particles with larger grains have less
diffusion resistance (less tortuous pathway) than small grains. Note that the 9 C case is charged for 10 min as opposed
to the standard 6.66 min to replicate the experiment procedure [35].
12