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:
[email protected] (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
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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