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7/24/2019 A Computational Model of the Mechanical Behavior Within Reconstructed LixCoO2 Li-ion Battery Cathode Particles
1/11
Electrochimica Acta 130 (2014) 707717
Contents lists available at ScienceDirect
Electrochimica Acta
journal homepage: www.elsevier .com/ locate /e lectacta
A Computational Model ofthe Mechanical Behavior withinReconstructed LixCoO2Li-ion Battery Cathode Particles
Veruska Malav,J.R. Berger, Huayang Zhu, RobertJ. Kee
Department of Mechanical Engineering, Colorado School ofMines, Golden, CO 80401, USA
a r t i c l e i n f o
Article history:
Received 31 December 2013
Received in revised form 18 March 2014Accepted 24 March 2014
Available online 1 April 2014
Keywords:
Lithium ion battery
Diffusion induced stress
Intercalation process
Phase transformations
Anisotropic stress-strain
Discharge rate
Particle morphology
a b s t r a c t
A coupled electrochemical-mechanical model is developed and applied to predict transient three-
dimensional stress fields within reconstructed Lix
CoO2
cathode particles from commercial Li-ion
batteries. The reconstructed particle geometries are derived from focused-ion-beamscanning-electron-
microscopy (FIB-SEM) experiments. The study uses three individual particles, representing typical sizes
and shapes. The mechanical model incorporates measured anisotropic strain within the LixCoO2 lattice
and includes strains due to phase transformations. The stresses are generally found to be compressive in
the particle interiors and tensile near the surfaces. Small-scale surface morphology, high Li concentration
gradients, and phase transformations are found to have a major influence on the stresses, with partic-
ularly high tensile stresses near small protuberances and concave notch-like features on the electrode
surfaces. The study considers 1C and 5C discharge rates. The qualitative behaviors are similar at different
discharge rates, but the stress magnitudes are higher at higher discharge rates.
2014 Elsevier Ltd. All rights reserved.
1. Introduction
This paper reports the development of a micro-scale three-
dimensional (3D) finite element (FE) linear elastic approach to
predict the mechanical behavior within reconstructed LixCoO2Li-ion battery (LIB) cathode particles during discharge. The
mechanical model, which is coupled directly to an electrochem-
istry model, includes the effects of crystal anisotropy and phase
transformations. The study is particularly concerned with predict-
ing the influence of particle size and surface morphology, as well
as discharge rates. Results show that tensile stresses, especially on
the particle surfaces, can be sufficiently high as to suggest particle
fracture.
Representative cathode particles are reconstructed from a
commercial battery using focused-ion-beamscanning-electron-
microscopy (FIB-SEM) [1]. The FIB-SEM experiments typicallyproduce reconstructions for an assembly of particles within a 3D
rectangular domain measuring a few tens of microns on a side [2].
Individual particles can be extractedfrom the assembly of particles,
and the present study uses three individual reconstructed particles
with different sizes and shapes.
The particle mechanical behavior is closely coupled with
the transient Li-concentration field within the cathode particles.
Corresponding author. Tel.: 1 303 273 3682; fax:+1 303 2733602.
Thus, the approach depends upon coupling an electrochemical
simulation with the mechanical simulation. The electrochemical
simulation is accomplished in a finite-volume (FV) setting using
extensions of the ANSYS Fluent software [2]. The mechanical sim-
ulation is accomplished in an FE setting using extensions of the
ANSYS Mechanical software.1 At each time step during a transient
discharge simulation, the Li-concentration field within the parti-
cles must be communicated from the electrochemical simulation
to the mechanical simulation.
The mechanical simulations depend upon constitutive relation-
ships between the stress and strain tensors. The present study
uses data published by Reimers, et al. [3], considering the effects
of both isotropic and anisotropic stress-strain relationships. The
LixCoO2 lattice experiences significant volume changes and phase
transformations during the lithiation (discharge) process [3]. The
diffusion-induced stresses (DIS) can be very high, potentiallyexceeding the material strength and leading to electrode degra-
dation and particle fracture. The present results show that stresses
can be particularly high in the vicinity of notch-like features on
the particle surfaces. Results also show that high discharge rates
and phase transformation occurringduring the Li intercalation also
contribute to high stresses.
1 ANSYS, Inc., Canonsburg, PA 15317; www.ansys.com
http://dx.doi.org/10.1016/j.electacta.2014.03.113
0013-4686/ 2014 Elsevier Ltd. All rightsreserved.
http://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.electacta.2014.03.113http://www.sciencedirect.com/science/journal/00134686http://www.elsevier.com/locate/electactahttp://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.electacta.2014.03.113http://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.electacta.2014.03.113http://crossmark.crossref.org/dialog/?doi=10.1016/j.electacta.2014.03.113&domain=pdfhttp://www.elsevier.com/locate/electactahttp://www.sciencedirect.com/science/journal/00134686http://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.electacta.2014.03.1137/24/2019 A Computational Model of the Mechanical Behavior Within Reconstructed LixCoO2 Li-ion Battery Cathode Particles
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708 V. Malav et al./ Electrochimica Acta 130 (2014) 707717
The study focuses on isolated particles that are not constrained
by neighboring particles (i.e., under free-expansion conditions).
The present study also considers isothermal behavior. Thus the
results reveal behaviors that are solely attributed to internal
diffusion-induced stresses.
The electrode particles from actual batteries can vary signif-
icantly and randomly in size, shape, and surface morphology.
The physical connections between particles within the electrode
assembly and the electrical contacts vary randomly within the
porous electrodeassembly.Moreover, the crystallographicorienta-
tions within particles are random. Thus, although the simulations
presented here are quantitative using actual reconstructed parti-
cles, the results must be understood in a qualitative context. The
broad objective is to glean observations and trends that are gener-
ally applicable.
2. Prior literature
There is significant foregoing research concerning the mechani-
cal behavior of Li-ion battery cathodes.ReconstructedLIB electrode
particles have been extracted from electrodes using a variety of
microscopic techniques [46,2,7]. Lim et al. [5] developed com-
putational simulations to show that lithiation-induced stressesdepend on geometric characteristics, with the stresses being much
higher in reconstructed LiyC6 and LixCoO2 particles than in ideal-
ized, spherical particles. Likewise, Seo et al. [4] and Chung et al. [6]
used reconstructed particles of LiMn2O4compounds in an FE solid
mechanics simulation. Chung et al. [6] reported that the DIS are
much greater in actual particles that in spherical particles. These
investigations [46] were based on elastic, isotropic, single-phase
individual particles and made use of the thermal analogy to com-
pute diffusion strains [8,9]. The present investigation develops an
analogous approach, but additionally considers phase transforma-
tion and crystal anisotropy as well as particle surface morphology.
Although a few recent studies have considered anisotropic Li
diffusion in polycrystalline LixCoO2particles [1012], the effects of
anisotropy on the stress response in geometrically complex elec-trode particles has not been reported. Additionally, most analytical
and numerical investigations of DIS in LIBs have not incorporated
the effects of volumetric and/or lattice strains that result from
phase transformations. However, Park et al. [13] have incorporated
phase-transformation-induced stresses in a 3D numerical model
of spherical LiMn2O4 particles. Their results showed that stresses
associated with phase transformations were greater than those
developed when considering the intercalation process alone. Ana-
lytical methods, such as moving boundary and porous electrode
theory, have been developed to investigate the effects of phase
transition and/or phase coexistence during Li intercalation [1417].
Theseapproaches,however, werelimited by the followingassump-
tions: a) isotropic elastic behavior in smooth, idealistic particles, b)
two phases concentrically coexisting, c) Li transport is decoupledfrom intercalation-induced stress phenomena, and d) phase coex-
istence modeled as Li-poor or Li-rich phases. Regarding the latter,
no published literature suggests that either of the two hexagonal
phases in the phase coexistenceregion of LixCoO2 is richerin Lithan
the other. In addition, smooth and spherical particles are unable
to capture stress concentrations that can develop in local concave
regions of actual cathode particles. Understanding and predicting
the mechanical behavior of electrodes is practically important. For
example, capacity fade can be associated with diffusion-induced
stress [4]. Even under normal operating conditions, particles can
fracture and thus degrade battery performance. Particle fracture
can originate from locally high stresses leading to the formation
and growth of microcracks [18]. Such processes are known to be
intensified at highdischarge rates [19]. If a fractured particle looses
electrical contact with neighboring particles or current collection
foils, it can no longer participate electrochemically and the bat-
tery resistance increases and capacity fades [13,20]. Additionally,
fragmentation exposes fresh electrode surfaces to the electrolyte
solvent, thus promoting the growth of new surface solidelectrolyte
interface (SEI) films [20,4].
Hydrostatic stress gradients are known to influence Li diffu-
sion with electrode particles [21]. Thus, in addition to mechanical
degradation associated with diffusion-induced stress, the stress
state couples back into the electrochemistry problem. This effect
is neglected in the present study, but is the subject of active model
development.
3. Particle reconstruction andcomputational discretization
Fig. 1 illustrates the process used to define the single parti-
cles used in the present study. A commercial cell (here, Lishen2
LR18650AH) is disassembled and a small portion of a cathode is
prepared for FIB-SEM imaging [1]. The raw data from the FIB-SEM
consists of approximately 200 two-dimensional SEM slices, with
each slice being separated by approximately 60 nm. In Fig. 1 the
white areas in the FIB-SEM slices represent the cathode particles
and the black space represents the pore space that would be filled
by electrolyte solvent. As discussedby Wiedemann et al. [2], the3D
cube is reconstructedusing theMimics software.3 The 3D geometry
is represented in STL (STereoLithography) format, which is a com-
putational definition of the particlesurfaces.For the purposes of the
present study, individual particles are extracted from the rectan-
gular assembly of many particles. The individual particle geometry
is also represented in STL format, which is used as the basis for
computational discretization.
Fig. 2 illustrates the three reconstructed cathode particles used
in the present study. The particles, labeled P1, P2, and P3, are ren-
dered at the same scale to show the relative particle sizes. In
addition to the reconstructed particles, a perfectly smooth spheri-
cal particle (labeled Ps) is also modeled. Cathode particles may be
polycrystalline with a few grains or be composed of a single grain
[22]. The present model assumes that each particle is composed ofa single crystal for the anisotropic studies.
To be electrochemically active, the particles must be connected
to other particles and ultimately to current-collection foils in the
battery. The yellow patches on particles (Fig. 2) indicate the sur-
face positions through which electrical current enters the particle
during discharge. Lithium enters the particles via charge-transfer
reactions on the surfaces that are in contact with the electrolyte
solution. The anisotropic LixCoO2crystal grain orientation is illus-
tratedby thexyzaxes whichcorrespond to the abccrystallographic
axes. In the present study, the electrical contact areas (yellow
patches) and crystal orientation are assigned somewhat arbitrarily.
However, in all cases the electrical contacts are essentially aligned
with the c-axis of the LixCoO2lattice.
The electrochemical model, which has been described pre-viously [2], uses an FV mesh that is generated using the
TGRID algorithm. The structural model uses the five degree-of-
freedom (DOF) element SOLID 227 that is implemented in ANSYS
Mechanical v14.5. This element consists of a 3D 10-node tetra-
hedron. The tetrahedral elements were selected for a variety of
reasons. First, coupled-field elements are required in order to cou-
ple the mechanical response to the chemical diffusions. These
particular elements in ANSYS Mechanical are tetrahedral or hex-
ahedral elements. However, the tetrahedral elements provide a
lower error when numerical solutions are compared to analytic
2 Tianjin LishenBattery Co., Ltd., Tianjin, China, http://en.lishen.com.cn3
Materialize, NV; Leuven, Belgium; http://www.materialise.com
http://en.lishen.com.cn/http://www.materialise.com/http://www.materialise.com/http://en.lishen.com.cn/7/24/2019 A Computational Model of the Mechanical Behavior Within Reconstructed LixCoO2 Li-ion Battery Cathode Particles
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V. Malav et al./ Electrochimica Acta 130 (2014) 707717 709
Fig. 1. Individual particles are extracted from FIB-SEM cathode reconstructions.
Fig. 2. Three reconstructed cathode particles and a 2.7-m spherical particle. The
electrical contact areas are shown as yellow patches. As indicated by the Cartesian
axis, thezaxis of the LixCoC2 is essentially normal to theelectrical contact area.
solutions, and provide a good representation of the DIS state for
the reconstructed surfaces based on the original STL image format.
Because twodifferentdiscretizationsare used,the Li-concentration
field from the FV simulation of the electrochemistry must be inter-
polated onto the FE mesh at each time step during the simulation.
Table 1 provides some summary information about each of the
particles.
4. Phase transformation and strain
Fig. 3a illustrates a pseudo phase diagram, which shows the
possibility of three phase transformations as a function of Li inter-
calation [3,18,2326]. Fig. 3b shows that the lattice can experience
Table 1
Particle geometric and mesh characteristics and current densitiesat 1C
Particle Volume Surface Area Size FE Nodes i at1C
(m3) (m2) (m) (A m2 )
P1 4 .10 15.0 2.6 86,015 7.56
P2 16.50 37.1 3.6 106,506 7.92
P3 82.34 116.7 10.2 316,624 7.58
Ps 82.34 91.6 2.7 65,637 9.14
asmuch as a 2.6% contraction upon full lithiation. There is also sig-
nificant discontinuity between the lattice volumes associated with
the two hexagonal phases. Figs. 3c-d show significant anisotropy
between thea and clattice parameters. Thea and b lattice parame-
ters are equal in the hexagonal phases. However, in the monoclinic
Fig. 3. Li/LixCoO2 pseudo phase diagram (adapted from Reimers, et al. [3]). The
monoclinic phase is labeled as M1. Two hexagonal phases are labeled as H1 and
H2.
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710 V. Malav et al./ Electrochimica Acta 130 (2014) 707717
phase, where b /= a, the average lattice parameter is expressed as[3]
aM1=1
3aM1 + bM1
Note that there is a very small increase in thea lattice parameter
as the Li fraction increases. However, there is a much more signif-
icant contraction in the caxis as Li fraction increases (i.e., during
battery discharge).
The coincidence of two hexagonal phases (Fig. 3, 0.75
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Table 2
Li/LixCoO2 cathode model parameters used in theFE structural analysis
Name Symbol Value Reference
Density 2328.5kg m3 [2]Youngs modulus E 370.0 GPa Current study
Elastic stiffness C11 596.0GPa [10], [30]
C12 200.0GPa
C13 133.0GPa
C33 375.0GPa
C44 124.0GPaPoissons ratio 0.20 Current study
Li saturation concentration C94%Li
22.37kmolm3 [2]Initial Li concentration C50%
Li 11.95kmolm3 Current study
Li diffusivity DLi 5.3871015 m2 s1 [2]
expressed assumingmechanical equilibrium in the absence of body
forces as
ijxj
= 0, (8)
where xj are spatial coordinates. The total elastic strain tensor, Tij
,
can be decomposed into the mechanical mij
and chemical diffusion
dij
contributions [19],
Tij= mij + dij. (9)
The present model neglects external loads (i.e., the surface of
the electrode particle is traction free). Thus, the cathode particle is
considered to be in a free-expansion, quasi-static, state wherein
equilibrium (Eq. 8) is satisfied everywhere within the particle.
However, the spatially non-uniform Li concentration field creates
non-uniform elastic strain andstress fields within the particle(irre-
spective of external constraints).
As a basis for comparison, the material properties within cath-
ode particles are first assumed to be isotropic. Then, in subsequent
simulations, the LixCoO2 lattice is assumed to be anisotropic. A
thermal analogy is used to compute the strains resulting from Li
diffusion [8,9]. A single stress-strain relationship is used to rep-
resent the strains that are induced by Li intercalation as well as
phase transformations. This approach is facilitated by defining a
concentration-dependent chemical expansion tensor, ij. This ten-sor specifies the changes of the crystal volume in the isotropic
case (Fig. 3b), and the changes in the lattice parameters for the
anisotropic case (Fig. 3c-d). Thus, the diffusion-induced strain is
both structure and composition sensitive.
6.1. Isotropic strains
The elastic strains in the isotropic case, where ij =ij, are
expressed as
Tij=
1
E
[(1
+)ij
kkij]
+ijC, (10)
where E is the Young modulus, is the Poisson ratio, ij is theKronecker delta, ij is the stress tensor, and C is the concentra-tion difference between the current and initial Li composition (i.e.,
C=CC0). In the isotropic case, it is assumed that the particlecontains a sufficientnumberof grains to use effectiveisotropic elas-
tic parameters. Reuss averages are used, with an effective Youngs
modulusEandaneffective Poissonratioevaluatedfrom thehexag-onal elastic-constant data reported by Hart and Bates [30]. To the
authors best knowledge, no data have been reported regarding
the LixCoO2 monoclinic-phase material properties. Therefore, the
anisotropic models use the hexagonal elastic constants. Previous
studies have used a wide range of values for the Youngs modulus
of LixCoO2 [16,5,31,30]. The value used here, E=370 GPa(Table 2),
lies within previously reported ranges.
The value of for the isotropic case can be extracted
from the lattice-volume data shown in Fig. 3b. For a given
volumetric strain (represented in Cartesian coordinates), the vol-
ume at a given intercalation fraction x may be evaluated as
V(x) =( da+da)(db+db)(dc+ dc). Expanding this relationship,
and neglecting higher-order contributions, the normal strain due
to both intercalation and phase-transformation can be represented
as
ij= 13V(x) V0
V0
ij, (11)
daa= dbb= dcc=
1
3
jV0
C= jC, (12)
where
j=1
3
jV0
. (13)
In these expressions jis the slope of the linear function corre-sponding to the specific Li intercalation fraction andV0is the initial
crystal volume (evaluated atx0.5).
6.2. Anisotropic strains
Most previous studies of mechanical response in LIB cathodeshave used isotropic elastic properties, even for LixCoO2. How-
ever, among the commonly used cathode materials, anisotropic
lattice strains are especially significant for LixCoO2 [32]. The
anisotropy is particularly important because individual particles
can be monocrystalline or polycrystalline with only a few grains
[22].
Anisotropic stress analysis is based upon directional lattice
strains instead of overall volumetric strains. The diffusion strain is
determined fromthe anisotropic crystal lattice behavior(Fig. 3c-d).
The anisotropic elastic stress-strain relationships may be repre-
sented as
Tij= Sijkk + ijC, (14)
where ij is the anisotropic expansion coefficient and Sijk is theelastic-compliance tensor. The components ofSijk can be deter-
mined from the elastic stiffness values provided in Table 2. The
concentration-dependent ij is evaluated from the linear fits tothe lattice expansion data (Fig. 3c-d). The present model incorpo-
rates the anisotropic ijtensor into the normal strain relationship.These anisotropic relationships capture the combined effect of
the lattice-contraction and phase-transformation strains that are
induced during battery discharge. The anisotropic strains may be
written as
daa= dbb=a
a0, dcc=
c
c0, (15)
where a0and c0are the correspondinga and clattice-constant val-
ues atx
0.5 (i.e., beginning of the discharge process).The resulting
anisotropic stress-strain relationship is then decomposed as
Tij= Sijkk + dij, (16)
where dij
is the strain tensor expressed in Eq. 15.
6.3. Effect of Li intercalation
Fig. 5 plots the approximate isotropic and anisotropic strain
functions that are derived from the Reimers et al. [3] data and
used in themodel. Fig.5a shows slightly swelling anisotropic strain
in the a and b lattice directions as the Li fraction increases. Con-
versely, Fig. 5b shows significanty stronger anisotropic contraction
in the clattice direction as Li fraction increases. By definition, the
isotropic strain must be the same in all directions. Figs. 5a-b show
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712 V. Malav et al./ Electrochimica Acta 130 (2014) 707717
xin LixCoO2
-8000
-6000
-4000
-2000
0
2000
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
b) caxis
0
ChemicalSt
rain(strains)
a) aand baxes
-2500
-2000
-1500
-1000
-500
500
1000
Isotropic model
Isotropic model
Isotropic model
Anisotropic model
Anisotropic model
Anisotropic model
Fig. 5. Approximate linear strains as functionsof Li intercalation, a) alignedwith a
and b lattice directions, daaand dbb
and b) aligned with clattice direction, dcc. The
isotropic strains are the same in both panels, but plotted using different scales for
the ordinates.
the same isotropic strains, but plotted on different ordinate scales.
The discontinuities atx0.75 is the result of phase transformationbetween the H1 and H2 hexagonal phases.
Fig. 5 shows isotropic and anisotropic strain functions to be
significantly different. Thus, it should be anticipated stress fields
predicted by isotropic models will be quite different from those
that assume the more-realistic anisotropic behavior.
7. Results and discussion
Table 2 lists physical parameters that are used in the present
study. Other parameters and implementation details for the
electrochemical model can be found elsewhere [2]. Theresults pre-
sented here express the DIS as maximum principal stresses, whichare especially significant for predicting the failure of ceramic, non-
ductile, materials.
7.1. Isotropic andanisotropic comparison
Fig. 6 shows stress fields in particle P2 at t= 1 820 s during a
1C discharge, using the isotropic and anisotropic models. In both
cases the stresses are compressive in the particle interior, but ten-
sile in the outer regions near the particle surfaces. The anisotropic
Fig. 6. Diffusion-induced stress fields in particle P2 using a) the isotropic model
and b) the anisotropic model. In both cases the battery is discharging at 1C and the
stress fields are shown at t=1820 s, which is approximately midway through the
discharge. The arrow points to a notched region on the surface with high tensile
stress.
Fig. 7. Diffusion-induced stress fields in fully lithiated particles Ps and P3 at th e
end of a 1C discharge(i.e., t= 3600 s).Both simulations arebasedon theanisotropic
model. These two particleshave comparable volumes (Table 1).
model predicts significantly higher tensile stresses. As indicated by
the arrow in Fig. 6b, concave or notched features on the particle
surface show particularly high tensile stresses. Such high tensile
stresses are likely to cause crack initiation and fracture. Although
different particles with different crystallographic orientations and
different electrical contacts will lead to different stress fields, this
result suggests strongly that using an isotropic model may signifi-
cantly under-predict deleterious tensile stresses.
7.2. Surface morphology
Fig.7 compares predictedstressfields in the ideal spherical par-
ticle Ps with the reconstructed particle P3 that has a comparable
volume (Table 1). Both cases use the anisotropic model, but the
spherical particle has a smooth convex surface compared to the
relative rough features of the reconstructed particle. At the end
of a 1C discharge, the maximum predicted tensile stress in Ps is
approximately 53.8 MPa. By comparison, the maximum predicted
tensilestressinP3 is approximately79.9 MPa.In bothcases,stresses
in the core of the particles are compressive, with tensile stresses
in the outer regions. The non-smooth surface morphology of the
actual particle leads to much higher local stress variations than arepredicted in the smooth-surface spherical particle.
The stress field withinthe sphericalparticle tends to be concen-
tric, which is the result of an essentially concentric Li concentration
field. However, because the electrical contact occupies a portion
of the particle surface, the fields are not exactly concentric. In
any case, the shapes of actual particles are usually very different
fromspheres. Consequently,although thepredicted fields maylook
qualitatively concentric (i.e., more compressive toward the inte-
rior), the actual fieldscan bequitedifferent from those ina spherical
particle.
These results demonstrate that surface features dominate the
mechanical response of the cathode particle. Local small-scale pro-
trusions in thesurface topology createlocal areas that lithiate more
quickly than the surrounding areas. Consequently, these locationsundergo phase transformations more rapidly than the surrounding
material, leading to high, localized, stresses. Such behavior is not
captured by simulations using smooth spherical particles.
7.3. Particle shape and size
Figs. 810 show predicted maximum-principal-stress fields in
the three reconstructedparticles at t= 1650s intoa 1Cdischarge. In
all cases, the stress patterns are qualitatively similar. That is, com-
pressive in the interior and tensile near the outside edges. Concave
features on the surfaces tend to be regions of relatively high ten-
sile stress. Particles P1 and P2, which are comparable in particle
size, show comparable stress distributions. Particle P3 is larger and
shows a wider range of DIS. However, as discussed subsequently,
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V. Malav et al./ Electrochimica Acta 130 (2014) 707717 713
Fig. 8. Maximum principal stress field in P1 at t=1650 s under a 1C discharge. a)
stresscontours on thesurface and b) stress contours on a cross-section cut.
Fig. 9. Maximum principal stress field in P2 at t=1650 s under a 1C discharge. a)
stresscontours on thesurface and b) stress contours on a cross-section cut.
the particle size itself may not be the dominant factor in governing
the maximum stress.Although not shown in Figs. 810, the model reveal that the
magnitude of the principal stresses depend upon the magnitude of
theLi-concentration gradients. In P3 themaximumLi concentration
gradients are on the order of 108 kmol m4, while inP1and P2themaximum Li concentrationgradients wereon the orderof 106 kmol
m4.Althoughthe predictedresults are basedupon quantitative sim-
ulations, they should be understood in terms of qualitative trends.
Individual particle shapes and sizesare random within the full elec-
trode. The electrical contacts and crystallographic orientations are
also random. Thus,any particularsimulation cannot produce a fully
general result.
Fig. 10. Maximum principal stress field in P3 at t=1650 s under a 1C discharge. a)
stresscontours on thesurface and b) stress contours on a cross-section cut.
x
Discharge time (s)
100 600 1100 1600 2100 2600 3100 3600
Max.
Prin.
Stress(MPa)
0
100
200
300
400
500
600
700
a) Variation in Li
b) Maximum DIS
1C discharge
P3
P2P1
P3
P2
P1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Fig. 11. a) Variationof intercalatedLi fraction withinthe three reconstructed cath-
ode particles during 1C discharge. b) Maximum principal tensile stress within the
reconstructed particles throughout a 1C discharge.
7.4. Maximum principal stress
Fig. 11a shows the maximum variation in Li fraction x withinthe three reconstructed particles. The relatively small particles (P1and P2) show relatively small variations that are essentially con-
stant atx0.02. The variations in thelarger particle(P3)aremuchgreater, reaching a peak at x0.14 about midway through thedischarge. Fig. 11b shows the transient maximum principal tensile
stresses during a 1C discharge. The maximum-stress profiles are
qualitatively similar for all the particles, but the magnitudes are
greater for P3. The stress peaks at approximately midway through
the discharge are caused by the phase transformations between the
two hexagonal phases (H1 to H1+H2) that occur atx0.75 (Figs. 3and 5). These sharp increasesin tensile stress, whichtendto be con-
centrated around features such as small protuberances or notches
on the particles surfaces,can serve to damage, crack, or fracturethe
cathode particles.
Fig. 12, which shows contours of maximum principal stress,
strain, and Li concentration on the surface of P3at t= 1820 s during
a 1C discharge, helps to explain the profiles shown in Fig. 11. The
dominantbehaviors arelocalized around a small protruding surface
feature that is also shown at an expanded scale. The peak compres-
sive and tensile stresses are highly localized around this surface
feature (Fig. 12a). The strain is relatively small on the tip of the
feature, but increases greatly near the base of the feature where it
joins the bulk of the particle (Fig. 12b). Because thefeature is small,
the Li intercalation fraction is highest near the tip of the feature(Fig. 12c). Because the small feature lithiates so rapidly relative
to its nearby surroundings, a highly localized phase transforma-
tion (H1 to H1+H2) contributes to locally high stresses. The local
phase transformation results in a relaxation of the local diffusion
strains.
7.5. Discharge rate
Increasing the discharge rate increase the stresses, but the
general behaviors remain qualitatively similar. The present study
compares mechanical behaviors at 1C and 5C. At the higher dis-
charge rate, the resulting Li concentration field is found to be more
non-uniformly distributed spatially. As a consequence, there is an
increase in the concentration gradients of at least one order of
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714 V. Malav et al./ Electrochimica Acta 130 (2014) 707717
Fig. 12. Localized fields within P3 at t= 1820 s during a 1C discharge. a) Maximum principal stress; b) equivalent diffusion strain; c) local Li fractionx.
magnitude. The increases in maximum stresses are approximately
proportional to the increases in Li concentration gradients. In large
measure, the stresses increase at higher discharge ratesbecausethe
Li-concentration variations within the particle are greater at high
rates.
Figs. 13, 14, and 15 show maximum principal stress maps forforP1, P2, and P3, respectively. As is the case at lower discharge rates,
stresses are generallycompressive in the particle interiors andten-
sile near the surfaces. Also, locally high stresses are evident around
small protuberances and concave features on the surfaces. These
observations are consistent with the previously reported results
[19,5].
Fig. 16a shows that after an initial transient, the maximum vari-
ations in intercalated Li are approximately steady at x0.07
Fig.13. DIS fieldin P1at t= 335s duringa 5C dischargecondition.a) Particlesurface
and b) Particle cross section.
Fig.14. DIS fieldin P2at t= 335s duringa 5C dischargecondition.a) Particle surface
and b) Particle cross section.
Fig.15. DIS fieldin P3at t= 335s duringa 5C dischargecondition.a) Particle surface
and b) Particle cross section.
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V. Malav et al./ Electrochimica Acta 130 (2014) 707717 715
x
0.05
0.10
0.15
0.20
0.25
Max.
Prin.
Stress(M
Pa)
20 120
220
320
420
520
620
720
Discharge time (s)
0
200
400
600
800
0
a) Normalized Li
b) Maximum DIS
P1
P2
P3
P3
P2
P1
5C discharge
Fig.16. a) Maximum difference in intercalation fractionxwithin particles during
a 5C discharge. b) Maximum principalstress historiesfor threeparticles duringa 5C
discharge.
for P1 and x
0.10 P2. The Li variation is greater for P3, with
larger variations during the course of the lithiation. The peakvariation x0.23 occurs during the H2-to-(H2+H1) phase tran-sition (t= 413 s) and gradually declines for the remainder of the
lithiation process. As was the case in the 1C discharge, the small
feature illustrated in Fig. 12 plays a significant role.
Fig. 16b shows the history of the DIS for the particles during a
5C discharge. Comparison with Fig. 11 shows that maximum prin-
cipal stresses in the M1 and H1 phases are generally greater than
they are during a 1C discharge. The H2-to-(H2+H1) phase transi-
tion occurs at t400 s, leading to the highest stresses. Becausethe intercalation-induced strains decline in the later stages of
the discharge where the H1 and H2 phases coexist, the stress
levels decrease. The stress state depends upon the Li intercala-
tion fraction, x, its variation, x, and the concentration gradient
throughout the particle. This agrees withthe observations reportedin [33]. Both of these factors are highly dependent on the dis-
charge rate as well as particle size and morphology. The stresses
are generally higher at higher discharge rates because x is
greater.
Fig. 17 illustrates maximum tensile stress histories at four
selected points on the surface of P3with discharge rates of 1C and
5C. The particular locations are chosen near topological features
that would tend to create high local stresses. To assist compari-
son between 1C and 5C the abscissa is a normalized time, t = t/tD,where tD is the time required for full discharge (e.g., tD =3600
s at 1C). At most points the stress histories are similar but the
magnitudes are greater at high C rates. The most striking behav-
ior occurs at t0.5, where the H2-to-(H2+H1) phase transitiontakes place at pronounced concave and convex locations. Thesesites are accommodating regions under stress peaks and decays.
Enteringthe coexistence region, a comparable stressevolution pat-
tern occurs at all sites evaluated in the particle throughout the
remainder of the lithiation.
Fig.18 illustratesamatrixofcross-sectionalimagesforP 3,show-
ing principal stresses, strain, and lithium intercalation fractions at
three times during a 5C discharge. The row at t= 85 s corresponds
to the M1-to-H2 phase transition. The row at t= 413 s corresponds
to the H2-to-(H2+H1) phase transition. The row at t= 7 20 s cor-
responds to the end of the 5C discharge. Fig. 18a shows surface
tensile stresses around 550MPa. Fig. 18d shows that during the H2-
to-(H2+H1) phase transition the surface tensile stresses are even
higher, in the range of 670 MPa. Even within the particle interior,
tensile stresses can exceed 300 MPa. At t= 4 13 s, Fig. 18d show
Fig. 17. Maximum principal stress histories at selected sites on the surface of P3when discharged at 1C and5C. Thetime is normalized between 0 t 1 toaccom-modate thedifferent C rates (t= 1 corresponds to full discharge).
a nominally concentric ring of stresses that are relatively more
compressive than the surrounding regions outside the ring. The
ring corresponds to a chemical strain misfit at x0.75 as the Liintercalation proceeds inward (Fig. 18e). Fig. 18g shows that at
the end of the lithiation process (t= 720 s) the stress magnitude
decreases, which is caused by the increasingly uniform Li interca-
lation. Although the stress magnitudes decrease near the end of
discharge, the general pattern of being tensile near the surface and
compressive in the interior is preserved.
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716 V. Malav et al./ Electrochimica Acta 130 (2014) 707717
Fig. 18. Mechanical and chemical response of particle P3 at three instants in time during a 5C discharge.
8. Summary and conclusions
The present study is based upon the computational simula-
tion of reconstructed LixCoO2 cathode particles during discharge.
Threeindividual particles were selected fromFIB-SEM reconstruct-
ions of a commercial battery, with the particles being selected to
represent a range of sizes and shapes. The approach couples three-
dimensional finite-volume simulations of the electrochemistry and
Li transport with three-dimensional finite-element simulations of
the mechanical response. The mechanical models compare both
isotropic and anisotropic behavior, considering phase transitions
(monoclinic and two hexagonal phases) associated with the extent
of lithiation in the LixCoO2lattice.
A number of general observations can be drawn from the study.
During discharge, the stresses are generallycompressive in the par-
ticle interiors and tensile near the particles surfaces. By comparingthe behavior of a smooth spherical particle with the reconstructed
particles, it is evident that the surface morphology plays a strong
role on the maximum tensile stresses. Especially high stresses are
found near concave, notch-like, features on the surfaces. Surface
features such as small protuberances that lithiate rapidly relative
to the surrounding surface also promote high stresses.
The anisotropic strain characteristics of the LixCoO2 lattice
are important. Comparison with equivalent isotropic stress-strain
models show that that stresses are significantly higher when
the anisotropic models are used. Thus, predictions based on
isotropic assumptions are likely to under-estimate the propensity
for electrode degradation, crack formation, and particle fracture.
Additionally, a major factor contributing to the stress state in
the individual Lix
CoO2
particles was the phase transformation the
crystal structure undergoes after half lithiation. This phase trans-
formation, combined with large concentration gradients, allowdramatic changes in the stress that evolve in the particle.
The mechanical behavior depends significantly upon discharge
rate. The present study compared 1C and 5C discharges. Although
the qualitative behaviors are similar at differentdischarge rates, the
stresses are significantly higher at higher discharge rates. Larger
Li-concentration gradients at high discharge rates contributes to
the higher stresses. Spatially varying lithiation rates, especially
near small-scale surface features, also contributes to high stresses.
Assuming that one is concerned with mechanical damage and par-
ticle fracture, the study suggest some considerations for electrode
design and operation. As much as practical, nominal spherical par-
ticles with smooth surface morphologies are desirable. Because
crystal anisotropy is important, it may be desirable to fabricate
polycrystalline particles with numerous randomly oriented grains,leading to overall more isotropic behavior. Limiting high discharge
rates is also desirable from the perspective of limiting mechanical
degradation.
Acknowledgements
We are grateful to Prof. Scott Barnett (Northwestern Univer-
sity), who provided datafrom FIB-SEM experiments and to Andreas
Wiedemann who assisted with the generation of the model geom-
etry. The authors also gratefully acknowledge the assistance of Drs.
Graham Goldin and Bill Bulat (ANSYS, Inc.) in the development of
the computational models. This effort was supported by the Office
of Naval Research via Grant N00014-08-1-0539.
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V. Malav et al./ Electrochimica Acta 130 (2014) 707717 717
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