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Mechanical stresses which develops during lithiation of crystalline silicon particles in lithium silicon battery causes fracture and limits the life of silicon based lithium batteries. We formulated an elasto-plastic stress formulation for a two-phase silicon model and investigated the influence of different mechanical properties of lithiated silicon on the fracture of nanoparticles during first cycle charging. A chemo-mechanical model was developed to determine lithium distribution and associated stress states during first cycle lithiation. The concentration gradient of lithium and an elastic perfectly plastic material behavior for silicon were considered to evaluate stress distribution formulation and determine stress field in the particle. The stress profile was used to perform a crack growth analysis. The stress distribution formulation was validated by evaluating stress field for different elastic modulus value for lithiated silicon and comparing our inference against observations from prior experiments. The results showed lower modulus of lithiated silicon yielded results like experimental observations for nanoparticles. The size dependent fracture behavior was also observed in lower elastic modulus of lithiated silicon. We conclude that accurate mechanical characterization of lithiated silicon nanoparticle is necessary to model the failure of silicon particle and improving the mechanical properties may suppress crack growth in silicon nanoparticles during charging.
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E3606 Journal of The Electrochemical Society,164 (11) E3606-E3612 (2017)
JES FOCUS ISSUE ON MATHEMATICAL MODELING OF ELECTROCHEMICAL SYSTEMS AT MULTIPLE SCALES IN HONOR OF JOHN NEWMAN
Fracture Modeling of Lithium-Silicon Battery Based on Variable
Elastic Moduli
Abhishek Sarkar, Pranav Shrotriya,zand Abhijit Chandra
Department of Mechanical Engineering, Iowa State University, Ames, Iowa 50010, USA
Mechanical stresses which develops during lithiation of crystalline silicon particles in lithium silicon battery causes fracture and
limits the life of silicon based lithium batteries. We formulated an elasto-plastic stress formulation for a two-phase silicon model
and investigated the influence of different mechanical properties of lithiated silicon on the fracture of nanoparticles during first cycle
charging. A chemo-mechanical model was developed to determine lithium distribution and associated stress states during first cycle
lithiation. The concentration gradient of lithium and an elastic perfectly plastic material behavior for silicon were considered to
evaluate stress distribution formulation and determine stress field in the particle. The stress profile was used to perform a crack growth
analysis. The stress distribution formulation was validated by evaluating stress field for different elastic modulus value for lithiated
silicon and comparing our inference against observations from prior experiments. The results showed lower modulus of lithiated
silicon yielded results like experimental observations for nanoparticles. The size dependent fracture behavior was also observed in
lower elastic modulus of lithiated silicon. We conclude that accurate mechanical characterization of lithiated silicon nanoparticle
is necessary to model the failure of silicon particle and improving the mechanical properties may suppress crack growth in silicon
nanoparticles during charging.
© The Author(s) 2017. Published by ECS. This is an open access article distributed under the terms of the Creative Commons
Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any
medium, provided the original work is properly cited. [DOI: 10.1149/2.0631711jes] All rights reserved.
Manuscript submitted April 3, 2017; revised manuscript received July 11, 2017. Published July 25, 2017. This paper is part of the
JES Focus Issue on Mathematical Modeling of Electrochemical Systems at Multiple Scales in Honor of John Newman.
Lithium ion Batteries (LIBs) are the leading source of energy
storage in the electronic devices and electrical vehicles.1Since the
development of first commercial LIBs by Sony in 1991, there has been
a paramount research and development related to this battery sector.
The batteries started with a graphite anode and a lithium oxide of a
transition metal (specifically cobalt) as cathode.2Recent experimental
and theoretical research has focused on identification and utilization
of next generation electrode materials such as silicon in order to
achieve higher energy density, longer cycle life and safer operation.3
Silicon nanosphere and nanowire have shown extreme energy storage
capacity (around 4200 mAh/g for Li15Si4). This high-energy storage
capacity of silicon occurs due to its ability to bind up to four lithium
atoms for each silicon atom. The accommodation of the large volume
expansion associated with lithiation results in significant stresses in
the electrode particle causing its capacity to fade over a few cycles.4
A lithium silicon battery system schematically shown in Figure
1a stores and discharges electrical energy through the exchange of
lithium ions between the electrodes. The cell charges with lithium ions
moving out of the cathode (metal oxide of lithium) and reacting with
crystalline silicon at the anode to form amorphous lithiated silicon.
LiMO2
Cathode
−−Li1xMO
2+xLi++xe[1]
xLi++xe+Si Anode
−−LixSi [2]
The lithiation of nanoparticles in the silicon anode is schemati-
cally represented in Figure 1b. During the first cycle, lithiation process
causes the crystalline silicon nanoparticle to convert into amorphous
lithiated silicon (Li15Si4). Lithiation initiates with the formation of
lithiated silicon amorphous shell on the crystalline silicon nanopar-
ticle. The diffusion of lithium is very slow through the crystalline
silicon, hence the phase boundary separating the crystalline core from
amorphous shell is extremely sharp (1 nm).5As the lithium diffuses
into the particle, the reaction front propagates forward. The surface is
assumed to have the highest concentration of lithiated silicon (Li4.4Si)
while the reaction front has the lowest in the lithiated domain (mini-
mum of Li3.75Si). The lithium concentration drops sharply in the reac-
tion zone which is negligibly thick with no lithium in the crystalline
zE-mail: shrotriy@iastate.edu
silicon core.6Lithiation of the particle proceeds through progression
of the sharp front along the radial direction toward the center and
conversion of the nanoparticle from crystalline silicon to amorphous
lithiated silicon.
Over the past decade, several theoretical models have been devel-
oped to predict the critical particle size for silicon.7One of the earliest
work on fracture analysis of silicon bilayer was done by Huggins and
Nix8in which they have established the critical dimensions for sili-
con bilayer based on linear-elastic mechanics. Two significant work
by Zhao et al.9,10 model lithiation of silicon considering a single and
Figure 1. a) Schematic of lithium-silicon battery. b) Diffusion-Reaction
Mechanism of lithium during charging in silicon nanoparticle.
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Journal of The Electrochemical Society,164 (11) E3606-E3612 (2017) E3607
two-phase model. In their two-phase model, they have considered a
plastic deformation formulation during lithiation assuming a higher
elastic modulus (35 GPa) and ignoring the elastic domain due to its
negligible thickness.10 Previous approaches10,11 toward mechanical
analysis of silicon lithiation have either assumed a perfectly plastic
material or elastic perfectly plastic material behavior for amorphous
lithiated silicon in order to analyze the stresses due to massive volume
expansion during lithiation. Ryu et al.12 and Yao et al.13 performed
experimental and mathematical analysis based on elastic deforma-
tion to infer the critical size of silicon nanoparticle between 300–400
nm. Several different mechanical properties have been considered to
mathematically model lithiation of silicon and have led to contra-
dicting inferences. Xie et al.14 and Zhao et al.9have modeled the
mechanical stresses in lithiated silicon with a modulus assumption of
80 GPa and flow stress of 1 GPa for silicon nanowires. Such high mod-
ulus assumption for silicon might have been based on brittle nature
of crystalline silicon. Sethuraman et al.15 andZhaoetal.
10 reported
35 GPa as the biaxial elastic modulus for amorphous silicon. The
consideration of high elastic modulus and perfectly plastic material
works well for brittle materials. However, the mechanical behavior of
amorphous lithiated silicon is not well understood. Recent nanoinden-
tation tests on amorphous silicon nanowires have reported an elastic
modulus within 8–12 GPa.16 While experiments based on tensile test-
ing reveal the flow stress between 500–750 MPa.17 Such low elastic
modulus values represent a softer, polymer-like, elasto-plastic behav-
ior of silicon when it gets lithiated. As a step toward designing of
advanced energy systems, it is important to develop a mathematical
formulation for stress analysis in lithiated silicon nanoparticles which
encompasses a three-layer (two-phase) model. The model needs to
incorporate the crystalline silicon core which has no lithium diffusion
and an amorphous shell of lithiated silicon with elastic-plastic behav-
ior. It is necessary to understand how the mechanical properties of
amorphous phase can influence the stress fields and fracture behavior
of silicon nanoparticles during lithiation.
In this paper, we formulated the stress field for a two-phase silicon
particle based on elasto-plastic chemo-mechanical model. Wereported
the influence of mechanical properties on stress and fracture analysis
of silicon nanoparticles during lithiation. Stress field associated with
lithiation of silicon nanoparticle was modeled using reaction driven
diffusion equation. Calculated stress field was utilized to determine the
driving forces for an initial radial crack in the nanoparticle. Mechanical
behavior of crystalline silicon was modeled as elastic while amorphous
lithiated silicon was modeled as an elastic perfectly plastic material.
Numerical inferences from our model for high and low elastic modulus
were compared and validated with TEM imaging results reported by
Liu et al.18 and FEM analysis by Lee et al.19 Model predictions were
used to identify the critical particle radius for fracture and to identify
a mechanism for improvement of the mechanical performance of the
battery.
Mathematical Modeling
Lithium transport in the silicon nanoparticle was modeled based
on the following assumptions: 1) Steady state diffusion in amorphous
shell, 2) Negligible diffusion in the crystalline core, 3) Sharp con-
centration change at the reaction front thickness, 4) Linear decay
of surface lithium concentration as the reaction front penetrates the
particle,14 5) Traction free particle expansion on the surface. The dif-
fusion of lithium in the amorphous shell is governed by Fick’s Second
Law of diffusion.
D
r2
rr2c
r=0[3]
Where, D is the diffusivity of lithium in amorphous silicon shell, c
is the lithium concentration in the amorphous shell. To solve for steady
state, the surface of silicon was initially considered to be at maximum
concentration followed by a linear decay of the surface concentration
as the lithium reaction front penetrated the crystalline silicon particle.
The following two boundary conditions were considered to solve
Figure 2. Schematic of stress domain in partialy lithiated silicon particle
battery.
Equation 3.
Dc
rr=R=Jb[4]
c|r=R=cb=cmax y(cmax c0)[5]
y=Rr0
R[6]
Where, Jbis the surface flux of lithium ion, y is the relative lo-
cation of the reaction front with respect to the particle surface which
changes from 0 to 1, R is the particle radius, r0is the crystalline silicon
radius and cmax is the maximum lithium concentration possible during
lithiation of a silicon particle. The reaction is carried forward by the
reaction of lithium with crystalline silicon at the reaction front. A
mass balance was satisfied across the boundary of the reaction front.
Dc
rr=r0=k(c|r=r0c0)[7]
Where, k is the rate of reaction, c|r0 is the concentration of lithium
at the reaction front and c0is the minimum possible concentration of
lithium in amorphous silicon (at the reaction front).
The concentration profile of lithium in silicon nanoparticle and
the flux was obtained by solving Equation 3using the boundary con-
ditions (Equations 47). We formulated a piecewise function for the
concentration that was dependent on the location of the reaction front.
c=0; rr0
cbJbR
D1r
r;r>r0[8a]
Jb=k(Cmax C0)(
1y)
1
(1y)2+kR
D
y
1y
[8b]
An elastic perfectly plastic stress model was solved to determine
the stress field generated due to the lithium concentration gradient in
the particle. Figure 2represents the stress domain representation of
the silicon nanoparticle. The particle was divided into three zones, i.e.
crystalline core (under no equivalent stress), elastic lithiated silicon
and plastic lithiated silicon. The elastic part was driven by expansion
stresses due to lithium diffusion in amorphous silicon. When the yield
criterion was attained, perfect plasticity was assumed to account for
the plastic deformation of the electrode particle. Stress field in the
particle satisfied the stress equilibrium equation.
σr
r+2σrσθ
r=0[9]
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E3608 Journal of The Electrochemical Society,164 (11) E3606-E3612 (2017)
Where σrand σθare the radial and hoop stress in the spherical
particle. In the elastic domain, the radial and hoop strain produced
were calculated based on the volumetric expansion due to lithium
diffusion.
εr|el =1
E[σr2νσθ]+c
3[10]
εθ|el =1
E[(1ν)σθνσr]+c
3[11]
Where, cis the concentration difference with respect to lithium
concentration. The yield criterion was used to establish the plastic
domain based upon the yield stress of lithiated silicon (σy).
|σθσr|σy[12]
The three domains within the silicon particle (Figure 2), required
two boundary conditions to maintain continuum and a free expansion
condition for the particle. The following boundary conditions were
used to merge the free expanding core to the elastic (el) domain and
the latter with the plastic (pl) domain.
σel
r
rr0=0 [13]
σel
r=σpl
rrp[14]
σpl
rR=0 [15]
At the reaction front (r0), the equivalent stress (from Equation 12)
was considered zero because the crystalline core expanded only due to
the stress state in amorphous shell. Incorporating the zero-equivalent
stress from Equation 12 into the stress equilibrium (Equation 10), we
obtained the gradient of the radial stress ( σr
r) at the crystalline core
surface to be zero. The radial stress continuity was maintained across
the elastic-plastic interface at r =rp. We developed the stress formu-
lation by substituting the lithium concentration profile (Equation 8)
into the elastic stress formulation (Equations 911) and solved the
combined elasto-plastic differential equation based on the boundary
conditions (Equations 1315), which yielded:
σr=
E
(1+ν)(12ν)(1+ν)C12(12ν)C2
r032E
3(1ν)
1
r03
r0
0
cr2dr;0r<r0
E
(1+ν)(12ν)(1+ν)C12(12ν)C2
r32E
3(1ν)
1
r3
r
0
cr2dr;r0r<rp
2σyloger
R;rprR
[16]
σθ=
E
(1+ν)(12ν)(1+ν)C1+(12ν)C2
r03+E
3(1ν)1
r03
r0
0
cr2dr c;0r<r0
E
(1+ν)(12ν)(1+ν)C1+(12ν)C2
r3+E
3(1ν)1
r3
r
0
cr2dr c;r0r<rp
σy12loger
R;rprR
[17]
The constants C1 and C2 were found using the radial stress boundary conditions.
C1=1
9(1+2ν)
2cbDrp3+JbRRro23Rr p2+2rp3
Drp3(1+ν)+
18σylog R
rp
E
[18]
C2=JbR2ro2(1+ν)
18D(1ν)[19]
Computed stress fields were used to determine the crack driving
force for radial cracks. The stress intensity factor (KI) was calculated
using the weight function theory for a semi-elliptic edge crack de-
veloped by Newman and Raju.20 This model compared the spherical
electrode particle with an equivalent cuboid having a semi-circular
crack and under the same stress conditions as the particle.2022
KI=E
Kref (1 ν2)
a
0
σθ(x)m(x,a)dx [20]
m(x,a)=
aσθmax(1 ν2)
E24Fa
laax
+Ga
l(ax)3
2
a [21]
Where, Kref is the reference stress intensity, E is the Young’s Mod-
ulus, υis the Poisson’s Ratio and m is the weight function. The stress
intensity factors were compared to fracture toughness (K1C)ofamor-
phous lithiated silicon.
K1C= 2Eγ[22]
Where γis the surface energy of the material. The governing
equations were normalized to simplify the calculations.
σn=σ
σy
[23a]
rn=r
R[23b]
cn=c
cmax
[23c]
xn=1rn[23d]
Kn=KI
K1C
[23e]
The non-dimensionalized equations were solved for a sil-
icon nanoparticle using Mathematica.30 The mechanical and
electrochemical properties listed in Table Iwere used to describe
the crystalline silicon mechanical and lithiation response. In addition,
the stress and fracture behavior were computed for three different
elastic moduli of lithiated silicon as listed in Table II. First modulus
corresponds to 80 GPa which has been extensively used in earlier
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Journal of The Electrochemical Society,164 (11) E3606-E3612 (2017) E3609
Table I. Silicon Properties.
Property Value Reference
Diffusivity, D (m2/s) 1016 23
Reaction Rate, k (m/s) 2.54 ×10924
Maximum Concentration, cmax (mol/m3)0.36×10625
Molar Volume, (m3/mol) 1.20 ×10526
Poisson’s Ratio, υ0.30 27
Surface Energy, γ(J/m2)0.8528
Young’s Modulus of crystalline silicon (GPa) 190.00 29
Yield Strength of lithiated silicon, σy(MPa) 720.00 16
Table II. Amorphous Lithiated Silicon Mechanical Properties.
Young’s Modulus, E(GPa)
1 80.009
2 24.5010,15
3 12.0016
literature. Lithiated silicon had been considered to have a brittle na-
ture. Second modulus value was considered based on inferences from
prior Density Functional Theory calculations and experimental obser-
vations for biaxial modulus. The biaxial modulus was converted to
Young’s modulus using the Poisson’s ratio (E=Ebia xi al (1 ν)). The
final modulus was taken as 12 GPa based on tensile and creep test
observations for silicon nanowires.
Results and Discussion
The concentration profiles of lithium in the amorphous silicon shell
computed for different penetration depths of the reaction front in a
200 nm particle is represented by Figure 3. The propagation of the
reaction front depended on the reaction rate. The particle started with
zero bulk lithium concentration at t =0 sec. As the lithium started to
react with the crystalline silicon, it moved the reaction front forward
by converting it in to amorphous lithiated silicon. The crystalline and
amorphous layers were separated by a sharp reaction front. The lithium
diffusing into the amorphous phase from the electrolyte pushed the
reaction front into the particle. As the amorphous phase expanded, it
provided the saturated lithium in the outer shell more space to diffuse.
This caused the slope of the concentration to gradually flatten with
the increment of the reaction front. The steady state diffusion process
meant that the concentration profile does not grow with time. So, the
flux of lithium was not constant and reduced as the front propagated
toward the center of the spherical particle.
The dimensionless stress (radial and hoop) profile was computed
for the nanoparticle (radius of 200 nm) at 50% reaction corresponding
to the different values of mechanical properties in Table II. The stress
profiles are shown in Figures 4a,4b and 4c for particle with 80 GPa,
Figure 3. Normalized concentration vs normalized radius variation at differ-
ent stages of penetration of reaction front into silicon particle.
Figure 4. Normalized stress (radial and hoop) vs normalized radius (200 nm)
for different elastic moduli a) 80 GPa b) 24.5 GPa c) 12 GPa.
24.5 GPa and 12 GPa, respectively. The surface of the nanosphere
was considered traction free, i.e. the particle was allowed to expand
freely. Due to the low diffusivity of lithium in crystalline silicon, the
inner crystalline core has minimal lithium concentration. Therefore,
the core has no stress gradient as it got stretched (or compressed)
by the amorphous layers expanding (or contracting) during lithiation
(or delithiation). In the material near the reaction front, the lithium
concentration jumped and rapidly built up. The expansion due to
the formation of lithiated silicon from crystalline silicon was analo-
gous to thermal expansion and the concentration gradient across the
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E3610 Journal of The Electrochemical Society,164 (11) E3606-E3612 (2017)
reaction front led to variation in the stress in the amorphous material.
The elastic stresses were found compressive where the lithium con-
centration was higher and the stress state moved toward tension as the
concentration decreased. This occurs because the higher concentra-
tion domain wanted to expand but it’s free expansion was restricted by
the zones with lower concentration. The particle expanded elastically
until the difference between the radial and hoop stress exceeded the
yield strength of the material causing plastic deformation. After the
onset of yield, the material expanded through plastic deformation. As
amorphous silicon was modeled as perfectly plastic, the equilibrium
and yield criteria were used to determine the stress distribution in the
plastically deformed zone. The stress profiles generated in Figures
4a,4b were validated against Xie et al.14 and Zhao et al.10 Zhao et
al.10 have neglected the elastic domain due to its negligible thickness
and only considered plastic deformation. However, the FEM analysis
performed by Xie et al.14 closely resembles the results from our formu-
lation in Figure 4a. As shown in Figure 4, the thickness of the elastic
zone increased from (a) to (c) as lower magnitude of moduli were
used in the analysis. More interestingly, lower magnitude of modu-
lus resulted in a fascinating outcome. The core silicon with higher
modulus for lithiated silicon (shown in Figure 4a) was subjected to
compressive stresses because the elastic domain was thin. However,
for a much lower value of lithiated silicon modulus (shown in Figure
4c), the stress distribution resulted in a larger elastic domain. This led
to tensile hoop stresses in both the core and the plastic domain. The
development of tensile stress near the surface and core would cause
any surface or internal flaws to open causing the particle to fail during
charging. The crack propagation corresponding to different stress dis-
tributions was computed to better understanding the fracture of silicon
nanoparticles.
An initial semi-circular flaw was considered in the particle and the
computed stress intensity factors based on the stress profile in Figure
4for different radial crack length are represented in Figures 5a,5b
and 5c. The stress intensity factors were computed for an initial radial
crack emanating from outer surface and growing toward the center
(based on analysis reported by Woodford et al.21). The tensile hoop
stress in the particle resulted in opening and propagation of radial
cracks. The domains having positive (or tensile) hoop stress are prone
to cracking while compressive hoop stress causes crack closure and
arrests the crack propagation. For Figures 5a,5b the short cracks on
the outer surface had a positive stress intensity because of positive
hoop stress on the particle surface. This result was inferred by Zhao
et al.10 However, these results do not yield with the TEM results
observed by Liu et al.18 In Figures 5a,5b, the crack grew toward the
center having compressive hoop stress which causes crack closure.
The deviation of the crack growth behavior for the 24.5 GPa case
could be accounted for the difference in geometry. Sethuraman et al.15
measured the biaxial modulus for a planar stress field, while the TEM
results are for spherical nanowires. For the lower elastic modulus
system shown in Figure 4c, the particle had tensile hoop stress both
on the surface and in the core with a domain of compressive stress in
the middle. This was reflected by the positive stress intensity factor
on the surface (Figure 5c). The stress intensity factor became negative
in a domain between the surface and core, as the compressive hoop
stress caused crack closure and was positive due to the tension in the
particle core. Although none of these particle cases would not fail as
the KIwaslessthatK
1C in Figures 5a,5b,5c, but Figure 5c yielded
inferences similar to experimental observations of charging induced
particle fracture in in-situ TEM studies.18 Generally, the positive
stress intensity on the surface suggests that surface flaws would
propagate to cause fracture. But higher compressive hoop stress in
the core for higher elastic moduli would lead to surface cracks getting
arrested. Experiments by Liu et al.18 and Lee et al.19 have shown that
silicon particles above 250 nm and 350 nm, respectively, undergo
complete fracture when charged. This observation suggested presence
of tensile stresses both near the surface and in the core. The fracture of
the particle was found to be dependent on the particle size. In addition,
our results suggest that improving the moduli or hardness of the
lithiated silicon may improve the fracture response during charging.
Figure 5. Normalized stress intensity factor vs normalized radius (200 nm)
for different elastic moduli a) 80 GPa b) 24.5 GPa c) 12 GPa.
In order to the investigate the size dependent fracture nature of
silicon nanoparticles observed by Liu et al.,18 stress distribution and
crack driving forces were computed for a particle with radius of 450
nm with a semi-circular flaw on the surface. Figure 6a shows the
normalized stress state of the particle with lithiated silicon modulus
of 12 GPa during charging when the reaction front had propagated half
way through the material. On comparison with Figure 4c, the tensile
hoop stress in the core had higher magnitude for the 450 nm particle.
Figure 6b shows the normalized stress intensity for the same particle
corresponding to elastic modulus of 12 GPa. The crack driving force
was found greater than the fracture toughness for cracks near the core
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Journal of The Electrochemical Society,164 (11) E3606-E3612 (2017) E3611
Figure 6. a) Normalized stress (radial and hoop) vs normalized radius (450 nm) for 12 GPa b) Normalized stress intensity factor vs normalized radius (450 nm)
for 12 GPa.
(0.00 R 0.30 R) as well as near the surface (0.85 R 0.95 R).
Presence of flaws in these zones would propagate and fail the particle.
These results explained the size-dependent failure behavior of the
silicon particles during charging. We conclude that particles smaller
than or equal to 200 nm have lower stress distribution and thus the
crack driving forces are smaller than the fracture toughness of the
particle. As the particle size increases, the magnitude of charging
induced stresses as well as the crack driving forces increases and thus
may result in failure of larger particle (near or above 450 nm).
These results clearly showed that accurate characterization of lithi-
ated silicon moduli is important to accurately model the stress devel-
opment and cracking response of silicon nanoparticles. We suggest
that increasing the modulus or hardening of lithiated silicon would
improve the fracture response of silicon nanoparticles and in turn lead
to designing of better energy storage materials. To test our hypothe-
sis, we plotted the stress and stress intensity profile for the 450 nm
particle having Young’s Modulus of 24 GPa, i.e. twice as predicted
by experiments.16 Figure 7a shows that the elastic domain narrowed
as compared to Figure 6a due to the influence of the increased elastic
modulus. The narrowing of the elastic domain caused the crystalline
core to have a compressive stress state. This got reflected in the stress
intensity as shown in Figure 7b. The stress intensity factor due to a
semi-circular crack is positive and just below the critical stress inten-
sity near the surface (which had tensile hoop stress); and negative at the
core due to compressive hoop stress. From this analysis, we conclude
that an increment in the elastic modulus of the particle would allow
the viable use of bigger sized silicon nanoparticles, without failure.
Recent work in the field of combining silicon particles with graphene
sheets allowed a mechanically and chemically stable electrode.31 We
speculate that by incorporating graphene in silicon, the volumetric ex-
pansion could be reduced leading to lower expansion stresses and the
elastic modulus could be improved for better mechanical performance.
Conclusions
An elasto-plastic stress model was developed for a two-phase sil-
icon nanoparticle during lithiation. The influence of elastic modulus
and yield strength of amorphous silicon on the stress and fracture
behavior of the nanoparticle was investigated. The computed stress
field was utilized to understand the fracture behavior and determine
the crack driving force for the growth of radial cracks present in the
particle. The stress model was validated based upon the compari-
son among three different elastic moduli of lithiated silicon found
in the literature. The numerical results indicated that predictions
corresponding to lower elastic moduli (as observed from nanoin-
dentation experiments) were able to yield with the experimentally-
observed size dependence fracture of silicon particles. Our model
was able to satisfy the fracture inferences drawn for high elastic
Figure 7. a) Normalized stress (radial and hoop) vs normalized radius (450 nm) for 24 GPa b) Normalized stress intensity factor vs normalized radius (450 nm)
for 24 GPa.
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E3612 Journal of The Electrochemical Society,164 (11) E3606-E3612 (2017)
modulus as well as capture the size-dependent particle failure ob-
served in low elastic modulus experiments. The results showed that
lower elastic modulus of amorphous silicon led to development of
tensile hoop stresses in both surface and core regions of the sil-
icon particles and thus enhanced the crack driving forces leading
to their fracture. Beyond a certain critical radius (about 450 nm)
a large portion of the particle was affected by stress intensity above
critical value predicting failure. These results emphasized the im-
portance of accurate characterization of the mechanical properties of
amorphous lithiated silicon. Furthermore, we suggest that hardening
of particle, to increase its elastic modulus, would lead to higher critical
stress intensity and lower tendency to have a tensile core. This would
prevent crack propagation in larger silicon nanoparticles.
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... The misfit dilatational strain could reach, for ex-studies conducted with the aim to unravel the underlying mechanisms behind capacity loss in phase changing lithium intercalation materials have pointed particularly to the presence of fracture and fatigue as an active degradation mechanism, e.g. in LiCoO 2 [23] , LiFePO 4 [ 21 , 24 , 25 ], as well as in other lithium compounds [26][27][28] . As a result, crack growth in two-phase electrode particles has been subject of several theoretical studies in the past, with the result of predicting a critical size below which a pre-existing crack irrespective of its initial size remains arrested subject to the diffusion induced stresses [29][30][31][32][33] . These studies rely on the calculation of a stress intensity factor which by comparison with a critical threshold predict whether or not growth of a pre-existing crack is energetically favorable according to the Griffith-Irwin fracture criterion [34] . ...
... Eqs. (32) and (33) however further reveal that the maximum opening displacement 0 falls below c for ∞ > 0.625 c . The analysis above therefore suggests that crack formation remains suppressed on the right hand side of the solid blue line in Fig. 3 (a) as the maximum opening could not reach the critical separation c . ...
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... The fracture energy of pure silicon was measured to be around 8 J m −2 and essentially independent of the degree of lithiation [6]. Theoretical models at particle level have been developed to account for the plasticity [7] and elasto-plasticity [8] of silicon, where it was demonstrated that fracture can be averted for small particle sizes and yield strengths, by increasing the elastic modulus. Atomistic simulations for fracture of pure silicon reveal a transition from intrinsic nanoscale cavitation to extension shear banding ahead of the crack tip [9]. ...
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A phase-field model, accounting for large elasto-plastic deformation, is developed to study the evolution of phase, morphology and stress in crystalline silicon (Si) electrodes upon lithium (Li) insertion. The Li concentration profiles and deformation geometries are co-evolved by solving a set of coupled phase-field and mechanics equations using the finite element method. The present phase-field model is validated in comparison with a non-linear concentration-dependent diffusion model of lithiation in Si electrodes. It is shown that as the lithiation proceeds, the hoop stress changes from the initial compression to tension in the surface layer of the Si electrode, which may explain the surface cracking observed in experiments. The present phase-field model is generally applicable to high-capacity electrode systems undergoing both phase change and large elasto-plastic deformation.
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In this paper, a kinetic model is proposed that combines lithium ion diffusion through a lithiated phase with chemical reaction at the interface between lithiated amorphous and crystalline silicon. It is found out that a dimensionless parameter, relating the concentration distribution of lithium ions to the movement velocity of phase interface, can be used to describe the lithiation process. Based on the stress distributions and lithium ion diffusion profiles calculated by an elastic and perfectly plastic model, it is shown that, as lithiation proceeds, the hoop stress that changes from initial compression to tension in the surface layer of silicon particles may lead to surface cracking.
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•Provides vital packaging technologies and process knowledge for silicon direct bonding, anodic bonding, glass frit bonding, and related techniques •Shows how to protect devices from the environment and decrease package size for a dramatic reduction in packaging costs •Discusses properties, preparation, and growth of silicon crystals and wafers •Explains the many properties (mechanical, electrostatic, optical, etc.), manufacturing, processing, measuring (including focused beam techniques), and multiscale modeling methods of MEMS structures •Geared towards practical applications rather than theory
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The time-independent and time-dependent mechanical behavior of electrochemically lithiated silicon was studied with nanoindentation. As indentation was performed with continuous stiffness measurements during loading and load-hold, new insight into the deformation behavior of lithiated silicon is furnished. Supporting other research, Young's modulus and the hardness of lithiated silicon are found to decline with increasing lithium content. However, the results of this study indicate that Young's modulus of the fully lithiated phase, at 41 GPa, is in fact somewhat larger than reported in some other studies. Nanoindentation creep experiments demonstrate that lithiated silicon creeps readily, with the observed viscoplastic flow governed by power law creep with large stress exponents (>20). Flow is thought to occur via local, shear-driven rearrangement at the scale of the Li15Si4 molecular unit volume. This research emphasizes the importance of incorporating viscoplasticity into lithiation/delithiation models. Additionally, more broadly, the work offers insight into nanoindentation creep methodology.
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Practical applications of high-capacity silicon anodes face significant challenges due to the huge volume change during the lithiation/delithiation process and the relatively low intrinsic electrical conductivity. J. Chen and co-workers report on page 758 a multilayered Si/reduced graphene oxide anode that offers superior reversible specific capacity, rate capability, and cycling performance.
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A model is developed to study the stress generation in a spherical particle subjected to lithium insertion. The model accounts for both the plastic deformation and the coexistence of lithium-poor and lithium-rich phases with a sharp and curved phase boundary. Such two-phase and inelastic deformation characteristics often arise during lithiation of crystalline particles with high capacity. A flexible sigmoid function is used to create the lithium profile with a step-like change in lithium concentration, mimicking a sharp phase boundary that separates a pristine core and a lithiated shell in the particle. The mechanics results, obtained by an analytic formulation and finite difference calculations, show the development of tensile hoop stress in the surface layer of the lithiated shell. This hoop tension provides the driving force of surface cracking, as observed by in situ transmission electron microscopy. The two-phase lithiation model is further compared with the single-phase one, which assumes a gradual and smooth variation in radial lithium distributions, and thus predicts only hoop compression in the surface layer of the particle. Furthermore, the effect of dilatational vs. unidirectional lithiation strains in the two-phase model is studied, thereby underscoring the critical role of anisotropy of lithiation strain in controlling stress generation in high-capacity electrodes for lithium ion batteries.
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Solid state diffusion in a binary system, such as lithiation into crystalline silicon, often involves two symbiotic processes, namely, interfacial chemical reaction and bulk diffusion. Building upon our earlier work (Cui et al., 2012b, J. Mech. Phys. Solids, 60 (7), 1280–1295), we develop a mathematical framework in this study to investigate the interaction between bulk diffusion and interfacial chemical reaction in binary systems. The new model accounts for finite deformation kinematics and stress–diffusion interaction. It is applicable to arbitrary shape of the phase interface. As an example, the model is used to study the lithiation of a spherical silicon particle. It is found that a dimensionless parameter β=kfeVmBR0/D0 plays a significant role in determining the kinetics of the lithiation process. This parameter, analogous to the Biot number in heat transfer, represents the ratio of the rate of interfacial chemical reaction and the rate of bulk diffusion. Smaller ββ means slower interfacial reaction, which would result in higher and more uniform concentration of lithium in the lithiated region. Furthermore, for a given ββ, the lithiation process is always controlled by the interfacial chemical reaction initially, until sufficient silicon has been lithiated so that the diffusion distance for lithium reaches a threshold value, beyond which bulk diffusion becomes the slower process and controls the overall lithiation kinetics.
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Alloying anodes such as silicon are promising electrode materials for next-generation high energy density lithium-ion batteries because of their ability to reversibly incorporate a high concentration of Li atoms. However, alloying anodes usually exhibit a short cycle life due to the extreme volumetric and structural changes that occur during lithium insertion/extraction; these transformations cause mechanical fracture and exacerbate side reactions. To solve these problems, there has recently been significant attention devoted to creating silicon nanostructures that can accommodate the lithiation-induced strain and thus exhibit high Coulombic efficiency and long cycle life. In parallel, many experiments and simulations have been conducted in an effort to understand the details of volumetric expansion, fracture, mechanical stress evolution, and structural changes in silicon nanostructures. The fundamental materials knowledge gained from these studies has provided guidance for designing optimized Si electrode structures and has also shed light on the factors that control large-volume change solid-state reactions. In this paper, we review various fundamental studies that have been conducted to understand structural and volumetric changes, stress evolution, mechanical properties, and fracture behavior of nanostructured Si anodes for lithium-ion batteries and compare the reaction process of Si to other novel anode materials.