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Micromechanisms of Capacity Fade in Silicon Anode for
Lithium-Ion Batteries
S. Pala, S. Damleb, S. Patelc, M. K. Duttaa, P. N. Kumtaa,b,d,
and S. Maitia
a Department of Bioengineering, University of Pittsburgh, PA
15261, USA b Department of Chemical Engineering, University of
Pittsburgh, PA 15261, USA
c Department of Mechanical Engineering, Michigan Technological
University, MI 49931, USA
d Mechanical Engineering and Materials Science , University of
Pittsburgh, PA 15261, USA
Large volume change and associated stress generation is known to
cause failure of the silicon thin film anode used for Lithium-ion
batteries after a few cycles. Experimental observations suggest
that plastic deformation of the underlying Cu substrate and
degradation of the active/inactive interface are the primary
reasons responsible for the capacity fade. The goal of the present
study is to examine the interplay between these mechanisms using a
computational mechanics approach. In the present study, a novel
multi-physics finite element framework has been developed to
simulate the lithiation and de-lithiation induced failure of
amorphous Si (a-Si) thin film on Cu foil. The numerical framework
is based on the finite deformation of the active silicon wherein
diffusion of lithium occurs, plastic deformation of the Cu foil,
and debonding of the active/inactive interface. The effect of
substrate property, interfacial energy and kinetics of interface
degradation has been examined.
Introduction Lithium ion batteries are flagship rechargeable
systems for a variety of portable and consumer electronics
employing graphite as the anode material of choice. In recent years
silicon anodes have been researched extensively as potential
alternative anode material owing to its very high theoretical
capacity (~4200 mAh/g). However, the enormous volume expansion
(~300 %) of silicon during lithiation leads to the mechanical
failure of anode (1). Consequently, the loss of electrical contact
within the active material as well as with the current collector
results in poor cyclic performance (2). Capacity retention over a
large number of cycles coupled with minimization of the first cycle
irreversible loss and improved coulombic efficiency by avoiding the
ensuing mechanical failure is the primary concern preventing the
large-scale deployment of silicon based electrodes in future
rechargeable lithium-ion batteries.
A wide range of design strategies has been adopted for the
manufacture silicon based anode materials with increased cycle
life. All these anode designs, through various modifications of
material composition and/or anode geometry, aim to preserve its
mechanical integrity in face of severe volume expansion (3).
However, an in depth understanding of the complex interplay between
electrochemistry and mechanics that leads to the ultimate failure
of anode is still lacking. A number of theoretical and
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numerical reports based on simple geometries have been published
over the years to study various aspects of this complicated problem
(4, 5, 6, 7). However, though fracture and failure of anode is the
primary reason for performance degradation, this aspect is not
studied in detail. Few recent studies have incorporated the
principal stress based failure law that may predict the nucleation
of cracks in a quasi-brittle material, but not the actual
propagation and its effect on the capacity retention. While they
are useful for qualitative understanding of the mechanisms
responsible for failure, these models are not predictive in nature,
which can guide the battery designers towards the design of optimal
configuration and achieving the desired materials properties.
We present herein a thermodynamically consistent theoretical
framework that couples electro-kinetics with the deformation
failure response of the anode materials system. We postulate that a
damage zone precedes the actual nucleation of a crack. Segregation
of Li atoms in this zone reduces the strength of the material in
the damage zone progressively thus resulting in crack formation. A
multi-physics cohesive zone model has been developed to model the
process of crack nucleation and propagation. Transport of Li in Si
is coupled with the mechanical behavior in a finite deformation
setting. Transport of Lithium from the electrolyte to the anode is
modeled by the well-known Butler-Volmer equation. Our model
includes mechanical behavior of the current collector, and in turn
examines the interaction of its passive response with the active
response of a-Si based anode. The input to our model is the C-rate
of charging and the output is the consequent resulting voltage
capacity curve. Our model is capable of simulating the multiple
electrochemical cycles, and thus can predict the interfacial crack
propagation leading to capacity fade mimicking the experimental
current-voltage response for an amorphous Si anode. We describe the
details of the model along with its experimental validation against
a-Si thin film anode deposited on copper, the established anode
current collector used in experiments in the next section. The
article is concluded with a discussion of simulations results
combined with suggestions for future improvements of the model.
Model description
Transport of Li atoms from electrolyte to anode During
galvanostatic electrochemical cycling of thin film anode, if the
anode with maximum theoretical Li alloying capacity C mAh/g is to
be completely lithiated in n h, then the Li flux across the
electrolyte-anode surface JLi (mole m
-2 s-1) is given by , where A is the surface area per unit
volume of the active
material, F as the Faraday constant and m is mass of the active
material. The overpotential s at the a-Si anode- electrolyte
interface can be estimated through the Butler-Volmer equation
as
[1]
with an estimated exchange current density . Exchange current
density at the anode surface depends on the maximum concentration
of the anode material, cmax , the surface concentration, cs, and
the concentration of the electrolyte, cewith reaction rate constant
k . Values of k depend on the type of reaction
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(lithiation/delithiation) occurring at the anode and are
selected to match the experimentally observed voltage profile of
the anode half-cell. Delamination of the thin film from the current
collector increases the contact resistance leading to additional
voltage drop across the interface. The half-cell voltage Vis thus
given by
, where, UOCP and Rc denoted open circuit potential and
contact
resistance.
To account for the non-ideality in the charge/discharge process,
the Uocp determined experimentally as a function of state of charge
(SOC) by Galvanostatic Intermittent Titration Technique (GITT) has
been used in all the simulations (8). Transport of Li in the anode
and the attendant deformation of the anode material Diffusion of Li
atoms in a-Si anode induces stress due to volume expansion, and
this in turn influences the transport of lithium. Thus, these two
phenomena are strongly coupled. We have developed a
thermodynamically consistent model for these interacting phenomena.
The salient features of this model are discussed in the following.
Kinematics of diffusion induced expansion and balance laws To
account for the large deformation of the silicon anode due to Li
transport, we introduce a deformation map , which maps a material
point X in the reference configuration 0 to the spatial point x in
spatial configuration t at any given time t as
. Therefore, the displacement field u(X,t) can be obtained from
x = u + X . We assume a multiplicative decomposition of the
deformation gradient of the form F = FeF with . Deformation
gradient solely due to the insertion of lithium in the anode
material at zero stress is given as F = (1+c)I , with I being the
identity tensor, c(X, t) is the concentration of the lithium atom
in the material configuration, and the expansion coefficient
assuming isotropic volume expansion. The elastic distortion of the
lattice is characterized by Fe component of the total deformation
gradient. To find the concentration field of lithium in the anode
during alloying, we consider the mass conservation of lithium atoms
in the reference configuration given as t c + X >J = 0 [2] where
J is the flux of Lithium at a material point X(t). Initial and
boundary conditions of alloying of lithium in the anode can be
described as
c(X, t) = c0 (X) on 0c , and J >N=J0 (X, t) on0J [3] with c0
(X)as the initial concentration of lithium in the anode, J0 as the
applied time varying flux of lithium atom at point X on the
boundary 0with normal N . Atomic diffusion occurs at a much slower
time scale; hence, mechanical equilibrium is assumed to be already
attained. Therefore, the balance of linear momentum can be
expressed as
X >P = 0 with P = FS [4] Where, Pand S are the first and
second Piola-Kirchhoff stress tensor, respectively. The boundary
conditions for mechanical equilibrium are given as PN = t on 0t ,
and
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u = u0 on 0u . Thermodynamical considerations The free energy
functional of the anode material can be additively decomposed as
(F,c) = 1(Fe ) +2 (c) [5] with 1(Fe )the elastic free energy
density and 2 (c) the chemical energy density. The elastic free
energy depends on the overall deformation gradient (through F) as
well as the concentration of the lithium atom. However, chemical
energy density of the anode material depends only on the
concentration of lithium. The material time derivative of the free
energy can thus be written as
= 2F-1 Ce
F-T :
C2
Me :L +c
c [6] The second Piola-Kirchhoff stress tensor Se and Mandel
stress tensor Me at the intermediate configuration is expressed as
and Me = CeSe , respectively, with as the right Cauchy-Green
tensor. From the second term, we define p as a pressure like
quantity at the reference configuration, with In the limit of small
deformation, this term reduces to where, is the Cauchy stress
tensor. For an irreversible process, the second law of
thermodynamic states that:
with s as the entropy density, J as the diffusion flux and as
the chemical potential. Incorporating Helmholtz free energy
function and mass conservation of lithium in the anode, the
Clausius-Duhem inequality for anode with isothermal condition can
be written as
D = S :C2
+c J 0 [7] Substituting the expression for free energy
functional [6], the above inequality can be expressed as
[8]
From the first part of the above inequality, we derive the
thermodynamically consistent definition of second Piola-Kirchhoff
stress tensor S in the reference configuration as S = F
-1SeF-T, The second part of the inequality offers the expression
of the chemical
potential in the reference configuration as Furthermore, the
inequality above can be ensured only when the third component is
negative semi-definite, which provides the definition of the
lithium flux of anode in the reference configuration as
where the mobility is a symmetric and positive definite tensor.
In the present study, we assume isotropic mobility tensor such that
. Thus, the atomic flux can be recast as with D = MRT / c as the
diffusivity of the lithium atom in the anode material. Therefore,
the equation for the transport of lithium in the anode material can
be expressed as
[9]
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This equation suggests that the change of concentration of
lithium in the anode is driven by the concentration gradient as
well as the gradient of pressure. Continuum modeling of debonding
and interface degradation kinetics To model the crack initiation
and propagation, we resorted to a popular method called the
cohesive element technique. In its classical form, this method is
formulated only for the deformation field. It relates the
displacement jump ahead of a crack tip to the traction T the
material exerts to resist fracture through a cohesive failure law.
However, segregation of Li atoms in the interfacial zone can cause
embrittlement and thus enhance failure. Moreover, Maranchi et. al.,
(9) have reported the presence of lithium atoms in the interfacial
region after failure. We extend the classical formulation to
account for this effect from a thermodynamic viewpoint (10). In the
spirit of Gibbs isotherm, the excess of the energy per unit area
can be written as [10] assuming isothermal conditions. In this
expression, is the surface energy of each separating surface, and
can be related to the work of separation as in the
absence of any segregation. The last term of the above equation
stems from the segregation of Li atoms, with is the chemical
potential and as the excess of these atoms on the interface. Note
that, in the absence of this term, our formulation coincides with
the classical displacement based cohesive technique. When
segregation is present, is a function of both and . We assume a
separable form as follows: [11]
where, is a parameter varying with the fraction of absorbed
atoms cint (equivalent to ), and degradation constant Sb . To
estimate adsorption of Li atoms from the bulk to the interface, the
Langmuir-McLean model is assumed to be valid:
[12]
where Sm is the segregation factor and cb is the surface
concentration of Li on the bulk. Finally, the cohesive traction can
be found as
[13]
In the thin film anode system, the delamination of the active
material from the inactive current collector is one of the primary
reasons contributing to capacity fade and increase in the
electrical resistance between the active/ inactive interfaces.
Therefore, an additional voltage drop occurs at the interface,
which reduces charge discharge capacity.
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Continuum modeling of substrate The anode is attached to the
current collector to provide an electrical pathway. During alloying
of the anode, the current collector undergoes severe deformation
often in the inelastic regime. Such inelastic deformations may
contribute to delamination of the active material as discussed
previously. The present analysis accounts for such inelastic
deformation considering finite deformation kinematics through
multiplicative decomposition of the total deformation gradient as F
= FeFp . To determine the plastic part
of the deformation gradient Fp , a flow rule Fp Fp1 = Np is
considered with the flow direction Np = f / Me where f is the yield
function and is the plastic parameter. In the present formulation,
the yield function with isotropic linearly hardening is assumed and
is of following type
f Me , p( ) = 32 Med :Med y + Hp = 0 [] with Me
d is the deviatoric part of Mandel stress tensor Me . The yield
stress and hardening moduli are represented as y and H ,
respectively.
Materials and Methods As mentioned earlier, prediction of the
solid phase potential requires an accurate estimation of the
overpotential and the state of charge dependent open circuit
voltage Uocp through experiments. However, the actual experimental
results depend on the anode configuration. We have performed GITT
experiment for estimating the Uocp for 250 nm thick a-Si thin films
deposited on Cu substrate (11). The 250 nm thick film of a-Si was
prepared by radio frequency magnetron sputtering. The details of
the deposition conditions and the fabrication of the 2016 coin
cells for electrochemical testing can be found in the literature
(12). Prior to GITT, the half-cell was cycled at C/4 charge rate
for 5 charge/discharge cycles to ascertain the formation of the SEI
layer. For GITT, galvanostatic lithiation of fully discharged a-Si
anode was carried out in repeated segments of 1 h at C/25 current
followed by relaxation period of 10 h to ensure equilibration. The
cut-off Voltage was set to 0.02 V vs. Li/Li+ electrode. Similar
process for discharge of the fully lithiatiated anode was carried
out and the cut off voltage was set to 1.2 V vs. Li/Li+ electrode.
The Uocp data obtained from GITT experiments was fitted as a
function of state of charge (SOC) using cubic splines and is used
for all the simulations executed in this study. As we have the open
circuit voltage experimental data only for a-Si thin film anode,
computational simulations reported in this article were performed
on this set-up only. A finite element framework has been developed
to study the coupled transport and stress generation problem in a
domain. Variational forms of the coupled equilibrium and transport
equations were obtained to perform finite element
semi-discretization in space. A Newton-Raphson scheme based
linearization procedure was used to solve the resulting algebraic
equations in an iterative manner. A backward Euler implicit time
stepping algorithm was employed to solve the resulting temporal
ordinary differential equations numerically which is presented
here.
s
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Problem description
The numerical model presented in the previous section has been
used to simulate the thin film anode combined with the base and the
interface between them during the lithiation/delithiation cycles
and the ensuing chemo-mechanical response has been studied. After
the first cycle of charging and discharging, the a-Si film
undergoes vertical fracture and forms islands attached to the
current collector (12). During further cycling, the stresses
generated in the film are not sufficient to generate additional
vertical cracks and the island structure is therefore preserved
during further electrochemical cycling (13). Thus, in this study we
will simulate the mechanical response of a single a-Si island along
with the elasto-plastic deformation of the substrate attached to
it. To study the anode structure, we have considered a
three-dimensional domain as shown in Figure 1(c), where the bottom
substrate represents the current collector and the top island
represents the Si thin film having a cohesive layer in between. The
separation between the vertical cracks formed in the thin film (9)
structure is observed to be in the micron range. The a-Si thin film
thickness was considered as 250 nm with 1 m x 1 m sized island to
mimic the experimental conditions, and 3 times larger dimensions
were considered for the substrate to sustain the volume expansion
of the island having a rigid support at the bottom surface.
Selection of domain for simulation is explained in Figure 1. As we
are considering galvanostatic charging and discharging, a constant
Li-ion flux is applied through the top surface of the a-Si island
as shown in Figure 1.
Figure 1: (a) Schematics of a-Si thin film anode half cell. (b)
a-Si thin film anode after 1st electrochemical cycle. (c) Domain
selected for simulation.
In this numerical study, material parameters for the a-Si island
are given as Diffusion coefficient (7), expansion co-efficient (8),
Youngs
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modulus E = 80 GPa and Poisson ratio (9). Taking advantage of
symmetry we have considered only 1/4th of the domain presented in
Figure 1(c) to minimize the computational time. The finite element
discretization of the domain is shown in Figure 4(a). Appropriate
symmetric boundary conditions have been applied on the surfaces of
the symmetry.
Results Simulation of voltage-capacity curves and experimental
validation Figure 2 shows a comparison of the experimental and
simulated Voltage-Capacity plots for electrochemical cycling of 250
nm thin film of a-Si anode at a C/2.5 charge/discharge cycling
rate. Since the formation of SEI layer and the irreversible
capacity loss during the first electrochemical cycle is neglected
in the present study, the simulation result shown is only for the
2nd charge/discharge cycle. The electrochemical parameters used for
the simulation are given in Table I. It should be noted that the
cell potential predicted by the simulation are in excellent
agreement with the experimental values except at the onset of
lithiation.
Figure 2. Comparison of Simulated and Experimental
Voltage-Capacity plot for C/2.5
charge/discharge rate
Table I: Parameters and model properties for the intercalation
model
F (Faraday constant) 96485.34 C.mol-1 R (Universal gas constant)
8.314 m2.kg.s-2.K-1.mol-1 T (Room temperature) 398 K a,c (Symmetry
factor) 0.5 ce (Li concentration in electrolyte)
1000 mol.m-3
cmax (Max. Li concentration in a-Si)
3.651 105 mol.m-3 (corresponding to 4200 mAh/g)
k (Reaction rate constant) m2.5.s-1.mol-0.5 (Lithiation)
m2.5.s-1.mol-0.5 (Delithiation)
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Simulation of crack propagation Figure 3 shows the simulation of
the interfacial crack propagation at the a-Si island - current
collector interface. The simulation was performed employing the
mechanical properties of the current collector similar to that of
Copper with E =100 GPa, ,
and (14) for 40 consecutive electrochemical cycles. The study
was conducted with the interface degrading due to Li segregation at
various rates ( corresponding to no interface degradation). Sm was
kept constant at 0.05. It can be seen that when there is no effect
of lithium segregation on the interface strength, crack propagation
stopped after the 4th cycle. However, for all other cases where
interfacial segregation of Li has been taken into account, crack
propagates slowly in an intermittent manner progressively degrading
the electrical contact between a-Si and Cu. This slow delamination
of the interface is a major contributing factor towards the gradual
capacity fade observed in Si-thin film anodes.
Figure 3: Propagation of interfacial crack with electrochemical
cycles. An interfacial strength of 2 GPa and a fracture toughness
of 25 J/m2 have been assumed. Crack propagation is presented as the
percentage of delamination of the original area. Effect of current
collector material property Datta et. al., (15) recently showed
that the presence of an amorphous carbon intermediate layer between
the a-Si thin film and the current collector can significantly
improve the cycling performance of the anode. High Columbic
efficiency and low capacity fade during electrochemical cycling
indicates that the interfacial crack propagation is suppressed due
to presence of the soft intermediate layer. This leads to the
conclusion that modifying the mechanical properties of the
substrate can significantly alter the anode stability during
electrochemical cycling. To study the effect of mechanical
properties of the current collector on the stability of a-Si
island, we simulate the electrochemical cycling of a-Si island on
elastic substrates (current collectors) with different stiffness
mismatch (ESubstrate/ESi) between the active material (silicon) and
the substrate (Figure 5). For elastic substrates, due to absence of
plastic flow in the current collector, crack
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initiates and propagates when the stress at the middle of the
island exceeds the interfacial strength (see Figure 4(b)). As the
maximum tensile stress at the interface occurs at the completion of
lithiation, if the interface survives the stress state at complete
lithiation (~280% volume expansion), the island structure will
survive the consecutive electrochemical cycles (assuming that
fatigue and any other mechanism does not degrade the interfacial
strength appreciably). Figure 5 shows the normalized interfacial
crack initiation time for anode configurations with elastic current
collector for different ratios of ESubstrate/ESi. Active material
is cycled at C/2.5 charge rate and the crack initiation time is
normalized with the theoretical time required for complete
lithiation of 250nm a-Si thin film; =0 indicating start of
lithiation and =1 indicating completion of lithiation or start of
delithiation. The interfacial strength is taken as 1 GPa and values
of fracture toughness are varied between 15 to 50 J/m2.
Figure 4: (a) Finite element discretization of an intact a-Si
island and current collector. (b) Interfacial crack (at middle)
between a-Si island and the elastic substrate ( ). (c) Simulated
interfacial crack (at corner) between a-Si island and the Cu
substrate after undergoing severe plastic deformation.
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Figure 5: Effect of elastic substrate stiffness on normalized
crack initiation time ( ). Interfacial strength is taken as 1 GPa.
Simulations indicate that having a compliant elastic substrate can
improve mechanical stability of the a-Si island during
electrochemical cycling. For the cases where Esub / Esi was less
than 0.18, there was no interfacial crack propagation. Also, having
higher interfacial energy can delay the interfacial crack
initiation to some extent.
Discussion and conclusions
In this paper, we report a novel multiphysics model taking into
account the coupled effect of lithium transport in the silicon
anode and the associated stress generation, combined with the
ultimate failure of the anode. We have validated our model against
experimental results obtained from half-cell experiments conducted
on anodes of similar configuration. The simulated voltage-capacity
curve is compared against the experimental results in Figure 2. A
sudden drop in voltage at the onset of lithiation in the
experimental data indicates that a higher overpotential is
encountered. We have experimentally determined the open circuit
potential to minimize this error. However, in our theory we have
not considered the stress-potential coupling that is present in the
physical configuration. We also have not considered the presence of
SEI layer on top of the thin film and the side reactions known to
occur during Li intercalation. These facts may explain the
discrepancy of our simulation results with the experiments at the
early stage of lithiation. Apart from this, the model prediction is
in close agreement with the experimental results.
While prediction of single cycle performance demonstrates the
capability of a theoretical model, batteries are typically
subjected to multiple charge/discharge cycles under practical
service condition. Cycling performance and capacity fade combined
with first cycle irreversible loss and coulombic efficiency are the
overall primary design concerns for Li-ion batteries. To show the
predictive capability of our model over many
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cycles, we have chosen two particular experimental findings. One
of them is concerned with the stability of the Si thin film of Cu
Substrate. During lithiation, plastic flow of the base occurred at
the corner of the island that reduced the tensile stress at the
center as shown in Figure 4(c). However, during delithiation or
de-alloying, the permanent set due to plasticity of the substrate
attempts to pull back on the corner of the island that was trying
to relax to its original configuration. This mechanism produced
very high tensile stress at the interface and caused the nucleation
of primarily mode I crack at the corner. However, after a few
cycles the plastic shake down occurred; the stress at the crack tip
fell just below the interfacial strength, and the crack was
arrested. But subsequent segregation of Li in the interfacial
region, as experimentally observed in (9), weakens the interface
gradually causing an intermittent motion of the crack (and
attendant loss in capacity). Finally the crack propagates all
through the interfacial region from the corner towards the center
causing loss of contact and drastic fade in capacity. For the
second case, the low modulus of carbon enabled very large
deformation of the substrate. During lithiation as shown in Figure
4(b), the middle of the island arched upward along with the
substrate material thus preventing the build up of high stress. The
maximum principal stress remained far below the interfacial
strength. During delithiation, the whole system comes back to its
original configuration elastically. Interfacial stress was low
enough such that strength reduction due to segregation was not
enough to result in failure of the interface over many cycles.
These simulations thus reveal that the stiffness of the base is an
important design parameter that influences long term performance of
silicon anode. It is hoped that these computational mechanics based
models will help the experimentalists design improved silicon based
anode systems.
Acknowledgments
The authors gratefully acknowledge support to this work by the
Centre for Complex Engineered Multifunctional Materials (CCEMM),
University of Pittsburgh. PNK acknowledges support of the DOE-BATT
program and the Edward R. Weidlein Chair Professorship funds for
partial support of this research.
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