Development of First Principles Capacity Fade Model for Li-Ion Cells

TL;DR: In this paper, a first principles-based model was developed to simulate the capacity fade of Li-ion batteries and the effect of parameters such as end of charge voltage and depth of discharge, the film resistance, the exchange current density, and the over voltage of the parasitic reaction on the battery performance was studied qualitatively.

Abstract: A first principles-based model has been developed to simulate the capacity fade of Li-ion batteries. Incorporation of a continuous occurrence of the solvent reduction reaction during constant current and constant voltage (CC-CV) charging explains the capacity fade of the battery. The effect of parameters such as end of charge voltage and depth of discharge, the film resistance, the exchange current density, and the over voltage of the parasitic reaction on the capacity fade and battery performance were studied qualitatively. The parameters that were updated for every cycle as a result of the side reaction were state-of-charge of the electrode materials and the film resistance, both estimated at the end of CC-CV charging. The effect of rate of solvent reduction reaction and the conductivity of the film formed were also studied. © 2004 The Electrochemical Society. All rights reserved.

Summary

Model Development

• The side reaction of general interest in lithium-ion batteries is passive film formation on the negative electrode.
• Thus to develop the model, the side reaction should be considered as consumption of solvent species and Li ions to form a group of such as: Li-alkyl carbonates, Li 2 CO 3 , etc., based on the composition and concentration of solvent.
• Change in volume leads to stretching of the surface films on the edge planes through which Li ions are inserted into the graphite.
• Thus in order to obtain better predictions for discharge performance both in terms of decrease in capacity as well as increase in cell resistance, the authors assume that a mixture of products of reasonable conductivity would be formed as a result of solvent reduction.
• Interfacial reaction kinetics.-For the semi-empirical model, 4 the Butler-Volmer ͑BV͒ kinetic expression was used to describe the overall charge transfer process occurring across the electrode/ electrolyte interfaces.

Model Equations

• Figure 1 shows a schematic representation of a typical Li-ion cell consisting of three regions namely negative electrode ͑graphite͒, separator ͑poly-propylene͒ and positive electrode (LiCoO 2 ).
• Both the graphite and LiCoO 2 are porous composite insertion electrodes.
• The model equations, initial, and boundary conditions that describe the mass transport, Eq. B-1 to B-17 and are discussed in detail in Ref. 12, 13 and 17.
• For incorporating the solvent reduction reaction, the following additional model equations have been added to the existing Li-ion model.
• The decrease in the charge capacity available from positive electrode (Q p ) is the capacity fade of the battery with cycling.

Results and Discussions

• -The capacity fade model was set to run under normal cycling conditions with constant current charging till the cell voltage reached 4.2 V followed by constant voltage charging until the charging current dropped to 50 mA.
• This includes both constant cur- rent and constant voltage parts of the charge cycle and the capacity was calculated using Eq. 16.
• Because the capacity loss due to the side reactions was assumed to occur only during charging the cell, the capacity fade model was programmed to simulate only the charging performance for every cycle.
• Thus as shown in Fig. 5 , due to the side Reaction 1, the film resistance continuously increased with cycling thereby causing an increased drop in the voltage plateau in the simulated discharge curves.

Case Studies

• The discussions given above were primarily focused on the capacity fade simulations for fixed values of adjustable parameters, which control the capacity loss and the film resistance.
• By increasing i os , the rate of the side reaction increased and hence the capacity lost during charging (Q s ), was higher at higher rates.
• The capacity fade model could be used as a predictive tool for cycling performance of Li-ion cells when charged to different end potentials.
• Figure 11 presents the cycling simulations for different EOCVs.
• Thus cells charged from shallow discharge loose less SOC and hence capacity and provide more cycles and longer life.

Conclusions

• The capacity fade model developed and discussed in this paper could be used as a basis for predicting the cycle life and analyzing the discharge characteristics of Li-ion cells after any cycle number.
• The effect of parameters ͑EOCV and DOD, the film resistance, the exchange current density and the overvoltage of the parasitic reac-tion͒ was studied qualitatively.
• The next step involves estimation of these time-dependent parameters based on the initial cycling data obtained experimentally.
• More than one mechanism could also be incorporated in the model to explain the capacity loss.
• The model developed assumes that the entire capacity loss was due to the side reaction over the surface of negative electrode during CC-CV charg-ing.

Figures

Figure 11 presents the cycling simulations for different EOCVs. Based on simulation results for 10 cycles, it was found for all cases that the capacity continued to decrease with cycling. However capacity decrease was found to be higher for 4.2 V when compared with the other EOC potentials.

Figure 6. Variation of cell capacity (QP), capacity loss per cycle (Qs), and SOC loss per cycle (u l) with cycle number. Figure 7. Effect of i os over capacity loss with charging.

Figure 11 presents the cycling simulations for different EOCVs. Based on simulation results for 10 cycles, it was found for all cases that the capacity continued to decrease with cycling. However capacity decrease was found to be higher for 4.2 V when compared with the other EOC potentials.

Figure 8. Variation of current during CC-CV charging for the cells charged to different EOCV.

Figure 10. Comparison of capacity loss (Qs) with charging and variation of overpotential for side reaction (hs) for Li-ion cells charged to different EOCV.

Figure 9. Variation of charge capacity after the 10th cycle when the cells were charged to different EOCV.

Figure 1. Schematic of a typical Li-ion cell sandwich.

Figure 1. Schematic of a typical Li-ion cell sandwich.

Figure 14. Variation of capacity loss during charging for the cells charged from different DOD.

Figure 12. Charge curves Li-ions for the first cycle when charged from different DOD.

Figure 13. Variation of SOC of LiCoO2 with cycle number for the cells cycled from different DOD.

Figure 2a and b presents simulations of the variation of cell voltage and current during CC-CV charging with cycle numbers, respectively. The cell voltage shown in Fig. 2a is the difference in the solid phase potentials (f1) between the positive (x 5 L) and negative ends (x 5 0) of the Li-ion cell sandwich. Becausef1 was set to zero atx 5 0, the solid phase potential at the positive end (f1ux5L) is the cell voltage. The applied current during both constant current and constant voltage charging was estimated using Ohm’s law given by

Figure 4. Discharge characteristics of Li-ion cells for various cycles.

Figure 5. Variation of film resistance during charging for~solid line! cycle 40 and~dotted lines! variation of film resistance with cycle number.

Figure 3. Charge curves of Li-ion cells for various cycles.

Figure 2a and b presents simulations of the variation of cell voltage and current during CC-CV charging with cycle numbers, respectively. The cell voltage shown in Fig. 2a is the difference in the solid phase potentials (f1) between the positive (x 5 L) and negative ends (x 5 0) of the Li-ion cell sandwich. Becausef1 was set to zero atx 5 0, the solid phase potential at the positive end (f1ux5L) is the cell voltage. The applied current during both constant current and constant voltage charging was estimated using Ohm’s law given by

Full text

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Development of First Principles Capacity Fade Model
for Li-Ion Cells
P. Ramadass,
*
Bala Haran,
**
Parthasarathy M. Gomadam,
*
Ralph White,
***
and Branko N. Popov
**
,z
Department of Chemical Engineering, University of South Carolina, Columbia, South Carolina 29208, USA
A first principles-based model has been developed to simulate the capacity fade of Li-ion batteries. Incorporation of a continuous
occurrence of the solvent reduction reaction during constant current and constant voltage 共CC-CV兲 charging explains the capacity
fade of the battery. The effect of parameters such as end of charge voltage and depth of discharge, the film resistance, the exchange
current density, and the over voltage of the parasitic reaction on the capacity fade and battery performance were studied qualita-
tively. The parameters that were updated for every cycle as a result of the side reaction were state-of-charge of the electrode
materials and the film resistance, both estimated at the end of CC-CV charging. The effect of rate of solvent reduction reaction and
the conductivity of the film formed were also studied.
© 2004 The Electrochemical Society. 关DOI: 10.1149/1.1634273兴 All rights reserved.
Manuscript submitted May 4, 2003; revised manuscript received July 30, 2003. Available electronically January 8, 2004.
In this study, an attempt was made to develop a first principles
capacity fade model for Li-ion batteries. Darling and Newman
1
made a first attempt to model the parasitic reactions in lithium bat-
teries by incorporating a solvent oxidation side reaction into a
lithium-ion battery model. The model explains the self-discharge
process occurring in Li-ion cells. Recently, Spotnitz
2
developed
polynomial expressions for estimation of irreversible and reversible
capacity losses due to solid electrolyte interphase 共SEI兲 growth and
dissolution. According to the author the expressions were difficult to
use in conjunction with time temperature superposition. Also, the
model requires extensive experimental cycling data to resolve the
model parameters.
Side reactions and degradation processes in lithium-ion batteries
may cause a number of undesirable effects leading to capacity loss.
3
If the cyclable lithium in the cell is reduced due to side reactions of
any type, the capacity balance is changed irreversibly and the degree
of lithium insertion in both electrodes during cell cycling is
changed. The objective of this paper was to develop a capacity fade
model through incorporation of side reactions with the existing Li-
ion intercalation model.
Model Development
The side reaction of general interest in lithium-ion batteries is
passive film formation on the negative electrode. The reduction re-
actions taking place which lead to the deposition of solid products
are less understood, large in number, and varied in their nature de-
pending on the composition of the electrolyte solution.
3
Thus to
develop the model, the side reaction should be considered as con-
sumption of solvent species and Li ions to form a group of such as:
Li-alkyl carbonates, Li
2
CO
3
, etc., based on the composition and
concentration of solvent. Similar to semiempirical capacity fade
models developed earlier,
4
only the negative electrode was consid-
ered for developing a simplified first principles capacity fade model.
The solvent diffusion model developed for Li-ion cells under
storage
5
explains the aging mechanism and helps to predict the cal-
endar life. The model was based upon diffusion of the organic sol-
vent present in the battery electrolyte followed by reduction near the
negative electrode surface thereby forming unwanted products
which form as a passive film 共SEI兲. Previous studies of the SEI
on lithiated carbon, both theoretical
6-8
and experimental,
9
have
recognized that the film may have a significant porosity. Thus the
mechanism for SEI growth as a result of solvent diffusion through
the SEI seems plausible.
According to Aurbach et al.,
10
Li-ion insertion into graphite par-
ticles during charging causes increase in lattice volume due to an
increase in the space between the graphene planes. Change in vol-
ume leads to stretching of the surface films on the edge planes
through which Li ions are inserted into the graphite. It is well known
that the surface film, usually comprised of a mixture of Li salts 共both
organic and inorganic兲, has a limited flexibility. Accordingly, one
can expect the surface film to break during the Li-ion insertion re-
action due to increase in particle volume, which alters the film pas-
sivity and exposes more of the underlying carbon to the electrolyte.
This phenomenon supports our assumption that continuous
small-scale reactions occur between the lithiated carbon and solvent
species, which increase the surface impedance with cycling. Also,
the same process explains the large increase of the electrode imped-
ance at higher temperatures, which is attributed to the increased rate
of the repeated film formation.
The first principles capacity fade model developed here is based
on a continuous occurrence of a very slow solvent diffusion/
reduction near the surface of the negative electrode in case when the
cell is in charge mode 共both constant current and constant voltage
charging兲. In other words, loss of the active material with continu-
ous cycling was attributed to a continuous film formation over the
surface of the negative electrode.
Choice of side reaction and assumptions.—
1. There are several possible reaction mechanisms between lithi-
ated carbon and the electrolyte solution. The nature of the reaction
depends upon the type of solvent mixture used in the battery elec-
trolyte. Possible contaminants in the system include gases such as:
CO
2
,O
2
, and N
2
. Since most of the Li-ion systems use ethylene
carbonate 共EC兲 as one of the organic solvent for the electrolyte, the
simplest reaction scheme that can be considered for modeling ca-
pacity loss is the reduction of EC. The reaction can be expressed as
S ⫹ 2Li
⫹
⫹ 2e
⫺
→ P 关1兴
where S refers to the solvent and P is the product formed as a result
of side reaction.
2. The solvent reduction reaction occurs only during charging the
Li-ion cell and it occurs during both constant current and constant
voltage charging. Because the ratio of charge to discharge capacity
remains close to 100%, it would be a valid assumption not to con-
sider any side reaction or capacity fade during discharge.
3. The products formed as a result of side reaction 共EC reduc-
tion兲 may be a mixture of organic and inorganic Li-based com-
pounds and not Li
2
CO
3
alone. The reason behind this is that, if we
consider the entire product formed as lithium carbonate, it would
result in overestimation of film resistance with cycling because it is
a very poor conductor. Thus in order to obtain better predictions for
discharge performance both in terms of decrease in capacity as well
*
Electrochemical Society Student Member.
**
Electrochemical Society Active Member.
***
Electrochemical Society Fellow.
z
E-mail: popov@engr.sc.edu
Journal of The Electrochemical Society, 151 共2兲 A196-A203 共2004兲
0013-4651/2004/151共2兲/A196/8/$7.00 © The Electrochemical Society, Inc.
A196
Downloaded 01 Aug 2011 to 129.252.86.83. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp

as increase in cell resistance, we assume that a mixture of products
of reasonable conductivity would be formed as a result of solvent
reduction.
4. The side reaction is assumed to be irreversible and a value of
0.4 V vs. Li/Li
⫹
10,11
has been chosen as the open circuit potential for
the solvent reduction reaction.
5. The initial resistance of the SEI formed during the formation
period was taken as 100 ⍀ cm
2
.
6. To make the model simpler, no overcharging conditions have
been considered thereby the other side reaction, namely lithium
deposition, could be eliminated.
Interfacial reaction kinetics.—For the semi-empirical model,
4
the Butler-Volmer 共BV兲 kinetic expression was used to describe the
overall charge transfer process occurring across the electrode/
electrolyte interfaces. In this model BV kinetics was defined sepa-
rately for Li-ion intercalation reaction
12-15
and for the solvent reduc-
tion reaction. Thus, for the negative electrode, the local volumetric
charge transfer current density was defined as the summation of
intercalation and side reaction current densities which is given by
J ⫽ J
l
⫹ J
s
关2兴
BV kinetics for Li-ion intercalation reaction.—The local volumetric
transfer current density due to Li-ion intercalation occurring across
both electrode/electrolyte interfaces is given by
J
l
⫽ a
j
i
0,j
冋
exp
冉
␣
a,j
F
RT
␩
j
冊
⫺ exp
冉
⫺
␣
c,j
F
RT
␩
j
冊
册
j ⫽ n,p 关3兴
where i
0,j
is the concentration dependent equilibrium exchange cur-
rent density at an interface and is given by
i
0,j
⫽ k
j
共
c
1,j
max
⫺ c
1,j
s
兲
␣
a,j
共
c
1,j
s
兲
␣
c,j
共
c
2
兲
␣
a,j
j ⫽ n,p 关4兴
The overpotential for the Li-ion intercalation reaction was given by
␩
j
⫽ ␾
1
⫺ ␾
2
⫺ U
j,ref
⫺
J
a
n
R
film
j ⫽ n,p 关5兴
The equilibrium potentials (U
j,ref
) of positive and negative electrode
are expressed as functions of state-of-charge 共SOC兲
U
p
ref
→ fn
共
␪
兲
U
n
ref
→ fn
共
␪
兲
关6兴
where ␪ is the SOC of the electrode. The empirical expressions for
equilibrium electrode potentials as functions of SOC are given in
Appendix A, Eq. A1, A2. For the quantitative description of electro-
chemical Li intercalation/deintercalation into Li-insertion electrodes,
Frumkin intercalation isotherm can also be adopted as explained by
Levi et al.
16
However, only empirical expressions were used in this
paper to represent equilibrium potentials of positive and negative
electrodes as a function of SOC. The term R
film
in Eq. 5 represents
the film resistance developed as a result of solvent reduction reac-
tion that takes place during charging of Li-ion cell.
BV kinetics for the solvent reduction reaction.—Similar to Li-ion
intercalation reaction, BV kinetic expression was used to explain the
rate of solvent reduction 共Eq. 1兲 as
J
s
⫽ i
os
a
n
再
冉
C
P
C
P
*
冊
e
共
␣
a
nf␩
s
兲
⫺
冉
C
S
C
S
*
冊
冉
C
Li
⫹
C
Li
⫹
*
冊
2
e
共
⫺␣
c
nf␩
s
兲
冎
关7兴
While including the side reaction along with the intercalation reac-
tion, some approximations were made to simplify the calculations
and hence the model. The kinetic expression 共Eq. 7兲 can be reduced
to either a Tafel or linear approximation depending on the reaction
conditions. The cathodic Tafel approximation could be used if the
solvent reduction reaction is considered to be irreversible. Thus the
rate expression for the side reaction becomes
J
s
⫽⫺i
os
a
n
冉
C
S
C
S
*
冊
冉
C
Li
⫹
C
Li
⫹
*
冊
2
e
共
⫺␣
c
nf␩
s
兲
关8兴
There may not be much variation in the concentration of Li ions in
solution for low to moderate rates of charge and discharge. More-
over, solution phase Li-ion concentration as well as the solvent con-
centration may not be limiting for the side reaction to take place, as
they will be present in excess. Based on these assumptions, the
cathodic Tafel kinetics developed for the side reaction can still be
simplified by not considering the concentration dependencies. Thus
the rate expression can be represented as
J
s
⫽⫺i
os
a
n
e
共
⫺␣
c
nf␩
s
兲
关9兴
where the overpotential term ␩
s
is expressed as
␩
s
⫽ ␾
1
⫺ ␾
2
⫺ Uref
s
⫺
J
a
n
R
film
关10兴
As mentioned earlier in the assumption, Uref
s
was taken as 0.4 V vs.
Li/Li
⫹
. For the first cycle, the film resistance, R
film
, is defined as
R
film
⫽ R
SEI
⫹ R
P
共
t
兲
关11兴
where R
SEI
refers to the resistance of the SEI layer formed initially
during the formation period and R
P
(t) is the resistance of the prod-
ucts formed during charging and is defined by
R
P
共
t
兲
⫽
␦
film
␬
P
关12兴
and the rate at which the film thickness increases is given by
⳵␦
film
⳵t
⫽⫺
J
s
M
P
a
n
␳
P
F
关13兴
Thus for any cycle number N, the film resistance is given by
R
film
兩
N
⫽ R
film
兩
N⫺1
⫹ R
P
共
t
兲
兩
N
关14兴
Model Equations
Figure 1 shows a schematic representation of a typical Li-ion cell
consisting of three regions namely negative electrode 共graphite兲,
separator 共poly-propylene兲 and positive electrode (LiCoO
2
). Both
the graphite and LiCoO
2
are porous composite insertion electrodes.
A Li-ion intercalation model
12
was used as a basis for developing
this capacity fade model. The model equations, initial, and boundary
Figure 1. Schematic of a typical Li-ion cell sandwich.
Journal of The Electrochemical Society, 151 共2兲 A196-A203 共2004兲 A197
Downloaded 01 Aug 2011 to 129.252.86.83. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp

conditions that describe the mass transport, and charge transport of
Li-ions in both solid and solution phases are summarized in Appen-
dix B, Eq. B-1 to B-17 and are discussed in detail in Ref. 12, 13 and
17.
For incorporating the solvent reduction reaction, the following
additional model equations have been added to the existing Li-ion
model.
1. The mass transport of Li-ion inside the particle has been rep-
resented by means of spherical diffusion equation for both positive
and negative electrode. At the surface of the negative electrode, as a
result of side reaction, the boundary condition was modified as
r ⫽ r
n
⫺D
n
s
⳵c
1,n
⳵r
⫽
J
I
a
n
F
关15兴
because a part of applied current (J
s
) is utilized for the solvent
reduction reaction.
2. For calculating the total charge capacity available from the
positive electrode after a complete constant current and constant
voltage 共CC-CV兲 charging for any cycle, the following equation has
been used
Q
P
⫽
冕
0
t⫽T
CC⫹CV
␧
p
␴
p
⳵␾
1
⳵x
冏
x⫽L
P
dt 关16兴
3. For the estimation of capacity lost as a result of side reaction
at the negative electrode surface, the following equation has been
used
Q
s
⫽⫺
冕
0
t⫽T
CC⫹CV
i
s
dt 关17兴
where the term i
s
refers to the current due to the side reaction inte-
grated across the length of the negative electrode
i
s
⫽
冕
0
L
n
J
s
dx 关18兴
4. At the end of every charge cycle, the total capacity lost as a
result of side reaction is estimated based on Eq. 17, which is fol-
lowed by calculation of loss of SOC as follows
␪
l
⫽
Q
s
Q
max
关19兴
In the above equation, Q
max
is the initial rated capacity of the cell.
For simulating the capacity in the next charge cycle, the SOC of the
positive electrode has to be updated and hence the general initial
condition for SOC of cathode for any cycle number 共N兲 is given by
␪
p
o
兩
N
⫽ ␪
p
o
兩
N⫺1
⫺ ␪
l
兩
N⫺1
关20兴
In the above expression, it is assumed that although capacity loss
occurs only at the negative electrode, it causes the capacity of the
positive electrode to diminish by the same magnitude.
As a result of side reaction, the film resistance over the surface of
the negative electrode continues to increase during both constant
current and constant voltage charging. Hence, an average value of
film resistance calculated over the entire length of negative electrode
was chosen as initial condition for the next cycle. The decrease in
the charge capacity available from positive electrode (Q
p
) is the
capacity fade of the battery with cycling.
The design adjustable parameters for positive and negative elec-
trodes are presented in Table I. The parameters for the solvent re-
duction reaction are given in Table II. The set of eight independent
governing equations for eight dependent variables (c
1
, c
2
, ␾
1
, ␾
2
,
J
I
, J
s
, Q
s
, and Q
p
) are solved as a 1D-2D coupled model for the
three domains 共negative/separator/positive兲 using FemLab software.
Results and Discussions
Simulation of charge characteristics.—The capacity fade model
was set to run under normal cycling conditions with constant current
charging till the cell voltage reached 4.2 V followed by constant
voltage charging until the charging current dropped to 50 mA. Thus
the negative electrode potential (␾
2
⫺ ␾
1
) at the current collector
end never reached 0 V or less and hence, a lithium deposition side
reaction was not considered for this model.
Figure 2a and b presents simulations of the variation of cell
voltage and current during CC-CV charging with cycle numbers,
respectively. The cell voltage shown in Fig. 2a is the difference in
the solid phase potentials (␾
1
) between the positive (x ⫽ L) and
negative ends (x ⫽ 0) of the Li-ion cell sandwich. Because ␾
1
was
set to zero at x ⫽ 0, the solid phase potential at the positive end
(␾
1
兩
x⫽L
) is the cell voltage. The applied current during both con-
stant current and constant voltage charging was estimated using
Ohm’s law given by
i
app
⫽ ␧
p
␴
p
⳵␾
1
⳵x
冏
x⫽L
关21兴
As shown in Fig. 2b, the model predicts a decrease in CC charg-
ing time and increase of the CV charging time as a function of cycle
number. The model results indicated that a gradual decrease in total
charging time occurs with cycling. This phenomenon was also ob-
served experimentally.
15
As a result of the side reaction, the film
resistance continued to increase with cycling, which reduced the
constant current charging time due to continuous increase of the
voltage drop at the interface. The SOC of the cathode material de-
creased for each cycle 共Eq. 19, 20兲, which also contributes the cell
voltage to reach the cutoff value earlier resulting in a decrease in
total charging time with cycling.
Figure 3 presents the simulated charge curves that show the de-
crease in the capacity with cycling. This includes both constant cur-
Table I. Electrode parameters for intercalation model.
Symbol Units
Anode
共graphite兲
Cathode
(LiCoO
2
)
L
i
␮m88 80
␴ S/m 100 100
␧
1
0.49 0.59
␧
2
0.485 0.385
brug 4 4
␦␮m2 2
c
1
max
mol/m
3
30555 51555
␪
0
0.03 0.95
D
1
m
2
/s
3.9 ⫻ 10
⫺14
1.0 ⫻ 10
⫺14
k A/m
2
/
(mol/m
3
)
3/2
4.854 ⫻ 10
⫺6
2.252 ⫻ 10
⫺6
␣
a
0.5 0.5
␣
c
0.5 0.5
c
2
0
mol/m
3
1000
D
2
m
2
/s
7.5 ⫻ 10
⫺10
t
⫹
0.363
R
SEI
⍀ m
2
0.01 0
Table II. Parameters for the solvent reduction side reaction.
Symbol Units Value
Uref
s
V 0.4
M
P
mol/kg
7.3 ⫻ 10
4
␳
P
kg/m
3
2.1 ⫻ 10
3
i
os
A/m
2
1.5 ⫻ 10
⫺6
␬
P
S/m 1
Journal of The Electrochemical Society, 151 共2兲 A196-A203 共2004兲A198
Downloaded 01 Aug 2011 to 129.252.86.83. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp

rent and constant voltage parts of the charge cycle and the capacity
was calculated using Eq. 16. During the CC part, the current was
constant and the capacity was obtained by the product of current and
charge time. During the CV part, the current decayed during charge
as shown in Fig. 2b. Hence, the capacity was calculated using Eq.
16. The SOC was corrected at the end of each cycle by using Eq. 20,
which accounts for the capacity loss due to the side reaction.
Simulation of discharge characteristics.—Figure 4 shows the
simulated discharge curves after 1, 50, and 100 cycles. Due to the
loss of the active material as a result of side reaction, the SOC of the
electrode material decreased while the capacity loss increased with
the cycle number. Because the capacity loss due to the side reactions
was assumed to occur only during charging the cell, the capacity
fade model was programmed to simulate only the charging perfor-
mance for every cycle. Thus, to simulate the discharge performance
of the cell for any cycle number, it is necessary to run the model for
the required number of charge cycles, which updates the capacity
fade parameters, based on the extent of side reaction and number of
cycles.
While the charge simulation was in progress, the parameter val-
ues contributing to the cell capacity loss and the cell voltage drop
could be collected at the end of every cycle. Thus, to estimate the
discharge performance after any cycle number, the Li-ion intercala-
tion model could be run only once with the updated parameters.
Apart from the capacity loss with continued cycling simulations, the
voltage plateau of simulated discharge curves continued to decrease
which is attributed to the continuous increase in the film resistance
during charging as a result of the side reaction.
The variation of film resistance over the particle surface of nega-
tive electrode has been shown in Fig. 5. the solid line of Fig. 5 共top
x axis and right y axis兲 presents the increase in the film resistance
during CC-CV charging estimated for cycle number 40 by using Eq.
12, 13. The dotted line of Fig. 5 共bottom x axis and left y axis兲
presents the variation of film resistance with cycling which in-
creased almost linearly with increase in cycle number. The film re-
sistance after any charge cycle was calculated using Eq. 14. Thus as
shown in Fig. 5, due to the side Reaction 1, the film resistance
continuously increased with cycling thereby causing an increased
drop in the voltage plateau in the simulated discharge curves.
Capacity fade with cycling.—The variations of cell capacity
(Q
p
) with number of cycles, capacity lost per cycle (Q
s
), and the
SOC lost per cycle (␪
l
) are shown in Fig. 6. Since in the model, the
capacity loss was assumed to occur only during charging, the de-
Figure 2. 共a兲 Variation of cell voltage during CC-CV charging for various
cycles. 共b兲 Variation of current during CC-CV charging for various cycles.
Figure 3. Charge curves of Li-ion cells for various cycles.
Figure 4. Discharge characteristics of Li-ion cells for various cycles.
Figure 5. Variation of film resistance during charging for 共solid line兲 cycle
40 and 共dotted lines兲 variation of film resistance with cycle number.
Journal of The Electrochemical Society, 151 共2兲 A196-A203 共2004兲 A199
Downloaded 01 Aug 2011 to 129.252.86.83. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Development of first principles capacity fade model for li-ion cells" ?

Except as provided under U. S. copyright law, this work may not be reproduced, resold, distributed, or modified without the express permission of The Electrochemical Society ( ECS ). The archival version of this work was published in the Journal of the Electrochemical Society. 

Q2. What other reactions could be included in the capacity fade model?

Other reactions such as electrolyte oxidation and phase transformation etc., that are specific to electrode materials could also be included in the capacity fade model for better predictions. 

Q3. What is the reason why the voltage plateau of simulated discharge curves decreased?

Apart from the capacity loss with continued cycling simulations, the voltage plateau of simulated discharge curves continued to decrease which is attributed to the continuous increase in the film resistance during charging as a result of the side reaction. 

Q4. Why was the capacity fade model programmed to simulate only the charging performance for every cycle?

Because the capacity loss due to the side reactions was assumed to occur only during charging the cell, the capacityfade model was programmed to simulate only the charging performance for every cycle. 

Q5. How to estimate the capacity fade parameters?

In order to match the simulated charge and discharge performance with the experimental cycling data, it would be critical to estimate the capacity fade parameters by using a nonlinear parameter estimation method. 

Q6. What is the simplest reaction scheme for modeling capacity loss?

as one of the organic solvent for the electrolyte, the simplest reaction scheme that can be considered for modeling capacity loss is the reduction of EC. 

Q7. What is the model of the side reaction of lithium ion batteries?

The model was based upon diffusion of the organic solvent present in the battery electrolyte followed by reduction near the negative electrode surface thereby forming unwanted products which form as a passive film ~SEI!. 

Q8. What is the percentage capacity fade value after 10 cycles?

The percentage capacity fade values after 10 cycles were estimated to be 7.2, 4.4, and 3.8%, respectively, for EOCV 4.2, 4.0, and 3.9 V. 

Q9. What is the effect of a lower cutoff voltage on the capacity fade of Li-i?

This suggests that for applications where 100% of the cell capacity may not be needed, cycling the cells to lower cutoff potentials results in increased cycle life and smaller capacity loss. 

Q10. What is the governing equation for the solid phase conductivity?

The solution phase conductivity as a function of concentration c2 ~in mol/dm 3! is20keff 5 k«2 4.0 5 S 4.1253 3 1024 1 5.007c2 2 4.7212 3 103c2211.5094 3 106c23 2 1.6018 3 108c24 D «24.0 @B-6#The model equation that describes the solid phase lithium concentration is given byDownloaded 01 Aug 2011 to 129.252.86.83. 

Q11. What is the effect of the side reaction on the cell?

This indicates that the active material loss due to the side reaction is more pronounced during initial phases of cycling and becomes progressively lower with cycling. 

Q12. Why is the lithium carbonate a poor conductor?

The reason behind this is that, if the authors consider the entire product formed as lithium carbonate, it would result in overestimation of film resistance with cycling because it is a very poor conductor. 

Q13. What is the general initial condition for SOC of cathode for any cycle number?

For simulating the capacity in the next charge cycle, the SOC of the positive electrode has to be updated and hence the general initial condition for SOC of cathode for any cycle number ~N! is given byup ouN 5 up ouN21 2 u luN21 @20# 

Q14. What is the effect of ios on the capacity fade?

It is clear from the plot that increasing the value of ios by even one order of magnitude dramatically increased the capacity loss with charging.