Modelling of cycling of lithium battery with microporous carbon electrode
Charge/discharge cycles of lithium cell with microporous carbon electrode under potentiodynamic control have been modelled. Predictions of the models with variable and constant diffusion coefficient neglecting the electric field inside the particle (CPM, DFM) are compared to the predictions of the...
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| Zitieren: | Modelling of cycling of lithium battery with microporous carbon electrode / D. Portnyagin // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 669-680. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1195792017-06-08T03:04:50Z Modelling of cycling of lithium battery with microporous carbon electrode Portnyagin, D. Charge/discharge cycles of lithium cell with microporous carbon electrode under potentiodynamic control have been modelled. Predictions of the models with variable and constant diffusion coefficient neglecting the electric field inside the particle (CPM, DFM) are compared to the predictions of the models with variable and constant diffusion coefficient in which electrostatic interaction inside the particles of carbon electrode (CPME, DFME) is taken into account. There is observed a considerable difference between both. Electrostatic interactions of lithium ions with each other and the charge distributed inside the particle promote intercalation during the discharge of the cell and deintercalation during the charge. The dependance of the effect of hysteresis during the cycling of the cell on the rate of change of the applied voltage is studied. The larger is the speed of change of the applied voltage the more effective is hysteresis. We have also obtained concentration profiles at different stages of charge/discharge process. Чисельно моделювалися розрядно/заряднi цикли лiтiєвої батареї iз вуглецевим мiкропористим електродом в режимi заданої напруги. Порiвнювалися передбачення двох видiв моделей: з постiйним та змiнним коефiцiєнтом дифузiї без врахування електростатичного поля, та з постiйним та змiнним коефiцiєнтом дифузiї з електростатичною взаємодiєю мiж iонами лiтiю та iз розподiленим зарядом всерединi частинок вуглецевого електрода. Спостерiгалася певна розбiжнiсть мiж ними. Електростатична взаємодiя сприяє iнтеркаляцiї пiд час розрядки батареї та деiнтеркаляцiї пiд час зарядки. Вивчалася залежнiсть вираженостi гiстерезису вiд швидкостi змiни прикладеної напруги. Гiстерезис проявляється тим бiльше, чим бiльша швидкiсть прикладеної напруги. Було також отримано профiлi концентрацiї на рiзних стадiях процесу розрядки/зарядки. 2008 Article Modelling of cycling of lithium battery with microporous carbon electrode / D. Portnyagin // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 669-680. — Бібліогр.: 7 назв. — англ. 1607-324X PACS: 82.47.Aa, 82.45.Gj, 82.45.Fk, 82.45.-h, 82.20.Wt DOI:10.5488/CMP.11.4.669 http://dspace.nbuv.gov.ua/handle/123456789/119579 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Charge/discharge cycles of lithium cell with microporous carbon electrode under potentiodynamic control have
been modelled. Predictions of the models with variable and constant diffusion coefficient neglecting the electric
field inside the particle (CPM, DFM) are compared to the predictions of the models with variable and constant
diffusion coefficient in which electrostatic interaction inside the particles of carbon electrode (CPME, DFME)
is taken into account. There is observed a considerable difference between both. Electrostatic interactions
of lithium ions with each other and the charge distributed inside the particle promote intercalation during the
discharge of the cell and deintercalation during the charge. The dependance of the effect of hysteresis during
the cycling of the cell on the rate of change of the applied voltage is studied. The larger is the speed of change
of the applied voltage the more effective is hysteresis. We have also obtained concentration profiles at different
stages of charge/discharge process. |
| format |
Article |
| author |
Portnyagin, D. |
| spellingShingle |
Portnyagin, D. Modelling of cycling of lithium battery with microporous carbon electrode Condensed Matter Physics |
| author_facet |
Portnyagin, D. |
| author_sort |
Portnyagin, D. |
| title |
Modelling of cycling of lithium battery with microporous carbon electrode |
| title_short |
Modelling of cycling of lithium battery with microporous carbon electrode |
| title_full |
Modelling of cycling of lithium battery with microporous carbon electrode |
| title_fullStr |
Modelling of cycling of lithium battery with microporous carbon electrode |
| title_full_unstemmed |
Modelling of cycling of lithium battery with microporous carbon electrode |
| title_sort |
modelling of cycling of lithium battery with microporous carbon electrode |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2008 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119579 |
| citation_txt |
Modelling of cycling of lithium battery with microporous carbon electrode / D. Portnyagin // Condensed Matter Physics. — 2008. — Т. 11, № 4(56). — С. 669-680. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT portnyagind modellingofcyclingoflithiumbatterywithmicroporouscarbonelectrode |
| first_indexed |
2025-07-08T16:12:31Z |
| last_indexed |
2025-07-08T16:12:31Z |
| _version_ |
1837095878539608064 |
| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 4(56), pp. 669–680
Modelling of cycling of lithium battery with microporous
carbon electrode
D.Portnyagin
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received April 23, 2007, in final form September 17, 2008
Charge/discharge cycles of lithium cell with microporous carbon electrode under potentiodynamic control have
been modelled. Predictions of the models with variable and constant diffusion coefficient neglecting the electric
field inside the particle (CPM, DFM) are compared to the predictions of the models with variable and constant
diffusion coefficient in which electrostatic interaction inside the particles of carbon electrode (CPME, DFME)
is taken into account. There is observed a considerable difference between both. Electrostatic interactions
of lithium ions with each other and the charge distributed inside the particle promote intercalation during the
discharge of the cell and deintercalation during the charge. The dependance of the effect of hysteresis during
the cycling of the cell on the rate of change of the applied voltage is studied. The larger is the speed of change
of the applied voltage the more effective is hysteresis. We have also obtained concentration profiles at different
stages of charge/discharge process.
Key words: lithium battery, porous electrode, potentiometry
PACS: 82.47.Aa, 82.45.Gj, 82.45.Fk, 82.45.-h, 82.20.Wt
1. Introduction
The recent developments appearing in the market of mobile phones, laptop computers and
other portable devices and electric vehicles evoke the demand for high energy density portable
power sources. In portable batteries lithium often serves as a cathode material due to its low
electronegativity. Anodes are made of porous materials due to their large surface area associated
with high energy storage. Mathematical simulations of charge/discarge processes make it possible
to optimize a battery in order to obtain a higher performance. This can also be helpful in analysing
these processes to gain a deeper insight into the nature and courses of phenomena that occur during
the cycling of these devices. Recently, simulations of intercalation of lithium into the structure of
a porous electrode have been reported by several authors [1–3]. It is widely held that the main
driving force at the operation of the battery is diffusion and that the transport of ions across the
electrode is governed by Fick’s second law. The present paper presents an improvement to this
approach by taking into account electrostatic interaction between ions as well as the distribution
of charge in the bulk of porous electrode. Comparison of the predictions of the diffusive model and
the more realistic electrodynamic one suggests that there is a certain discrepancy between them.
2. Basic considerations. Cylindrical particles
We study the cycling of lithium battery under potentiodynamic control. In our research we
heavily rely upon the data from [2]. The battery consists of lithium foil, porous separator of
thickness Ls = 25 µm, porous carbon electrode of thickness L1 = 125 µm made of either cylindrical
or spherical particles of radius Rs = 3.5 µm, and current collector (see figure 1). The battery is
immersed in 1M solution of LiClO4 in propylene carbonate.
The applied potential changes linearly with time and is given by
Uapp = U0 + ωt,
c© D.Portnyagin 669
D.Portnyagin
where U0 is the initial applied potential, ω is the sweep rate, t is time. At sweep rates of 10, 5
and 1 mV/s the battery was discharged from its initial state to 0.075 V, then it was charged to
1.5 V, and finally it was again discharged to 0.075 V. These steps were repeated twice to reach
a periodic state. The periodic state is the state, at which the results are uniform and sustained
during consecutive cycles, when cycled under the same conditions.
Figure 1. Schematic diagram of the lithium cell. Arrows show the flux of lithium ions.
During the discharge of the battery, lithium is dissolved into lithium ions from the negative elec-
trode, migrates through the separator and finally intercalates into the carbon electrode. During the
charge the reverse process takes place (see figure 2, 3). We neglect the expansion and contraction of
the electrodes. There exist two approaches for modelling the lithium insertion into the particle, both
of which lead to the solution of the diffusion equation in a particle. In the first approach, the driving
force is the gradient of concentration while the diffusion coefficient remains constant. However, it
has been reported that there is a strong dependence of the diffusion coefficient on concentration
due to the ‘lithium ion’-‘lithium ion’ interactions inside the particle, which cannot be ignored to
obtain good agreement with experimental data. In the second approach, Verbrugge and Koch [4]
considered the gradient of the chemical potential of the inserted lithium ions as the driving force.
Figure 2. Lithium intercalates into the carbon
fiber.
Figure 3. Lithium intercalates into spherical
particle.
670
Modelling of cycling
In the present section we consider cylindrical particles with the ratio of length to radius suffi-
ciently large, for which the concentration of lithium inside the particle is a function only of radial
distance, governed by the equation
∂y
∂τ
=
1
R
∂
∂R
(
Rf(y)
∂y
∂R
)
, (1)
y = y0 at τ = 0, ∀R, (2)
∂y
∂R
= 0 at R = 0, ∀τ, (3)
∂y
∂R
= −
j+
n
Ds
Rs
Cs,maxf
at R = 1, ∀τ, (4)
where τ = tDs/R2
s , y = Cs/Cs,max, R = r/Rs; are dimensionless variables. Ds is the diffusion
coefficient in the solid phase, assumed to be constant, Rs is the radius of the particle, Cs is
the concentration of lithium ions inside the particle, Cs,max is the maximum concentration of
lithium ions inside the particle, f is the activity factor dependant on the intercalation fraction and
calculated by Verbrugge and Koch [4], j+
n is the flux of lithium ions at the surface of the particle.
The initial value of y is equal to 0.01. The flux of lithium ions at the surface of the particle is
equal to the electrochemical reaction rate per unit of surface area of the particle as given by a
Butler-Volmer reaction rate expression
j+
n = K
(
C(1 − y
∣
∣
R=1
)
)β−1 (
y
∣
∣
R=1
)β
{
exp
[
(1 − β)F
<T
(η − U)
]
− exp
[
−βF
<T
(η − U)
]}
,
where C is the concentration of the electrolyte, K is the reaction rate constant, β is the symmetry
factor taken to be 0.5, reflecting a strongly solvated ion, F is the Faraday constant, < is universal
gas constant, T is temperature, η is the potential between solid phase and electrolyte, and U
represents the open-circuit cell potential with respect to a metallic lithium electrode which is
evaluated at the surface of the particle where the electrochemical reaction takes place and which
is given by [4]
U = Us +
<T
F
ln
(
1 − y
∣
∣
R=1
y
∣
∣
R=1
)
−
7
∑
s=2
Ωs
F
s(y
∣
∣
R=1
)s−1 for 0 < y|R=1 < 0.985, (5)
where Us is the standard cell potential with respect to a metallic lithium electrode, and Ωs are the
self-interaction energies. This expression is obtained by least square polynomial regression of the
experimental curve open-circuit potential U vs. concentration on the surface of the particle y
∣
∣
R=1
(see [4]). Ωs describes the deviation from the ideal behavior due to s−nary (i. e., binary, ternary,
. . . ) interactions between ions, in the expression for free energy. We restrict ourselves to s = 7, in
all the other respects formulae (5), (7) are exact. We take the activity coefficient
f = 1 (6)
for a purely diffusive model, and
f =
(
1 +
d ln γ+
d ln y
)
= 1 +
7
∑
s=2
Ωs
<T
s(s − 1)(ys−1 − ys) (7)
for a chemical potential model [4] (at low lithium concentrations, f increases with the lithium ion
concentration increasing due to repulsive effects, takes on its maximum at y = 0.2, and decreases
with the lithium ion concentration increasing due to a low ion mobility at higher concentrations).
Our amendment to the aforementioned models consists in adding the current coursed by electric
field to the righthand side of equation (1).
∂y
∂t
=
Ds
R2
s
1
R
∂
∂R
(
Rf(y)
∂y
∂R
)
−
1
FCmax,s
div(σ(y)E), (8)
671
D.Portnyagin
where E is the electric field, σ is the ionic conductivity given by Einstein relation
σ = yCmax,sNaDse
2/kT,
k is Boltzman constant, e is elementary charge, Na is Avogadro number, and here div(A(R)) =
1
R
∂
∂R
(RA(R)).
In the first approximation we assume that the current of positive ions through the surface
of the particle is entirely due to the uniform distribution within the particle of negative charge,
which carbon, being more electronegative, draws from lithium, as well as due to the distribution of
the charge caused by imposed external electric field. However, X-ray photoelectron spectroscopy
(XPS) proved [5] that after insertion, lithium retains only a fraction of the positive charge +δ, while
carbon takes a negative charge −δ. Therefore, we add the term associated with the nonuniform
distribution of lithium ions to the distribution of charge in the bulk of the particle. This results in
div(E) =
2
Rsσeff,1
j+
n − δ
FCmax,s
ε0
(yavr − y), (9)
where σeff,1 is the effective conductivity of electrolyte in the carbon electrode, ε0 is the dielectric
constant, yavr =
∫
ydV/V = 2
1
∫
0
yRdR (or = 3
1
∫
0
yR2dR for spherical particles) is the average
concentration of ions in the particle, δ is the delocalization factor which equals 1 when we have
naked lithium ions and negative charge, drawn from lithium, uniformly spread over carbon sites,
and equals 0 when negative charge is maximally localized on lithium ions. Within the framework of
our model the first term in (9) is the density of a uniformly distributed charge that yields the current
j+
n on the surface of a particle. Herein we add the term associated with nonuniform distribution
of lithium ions inside the particle, multiplied by the factor that characterizes the amount to which
the electron cloud is drawn from lithium by a more electronegative carbon. The mean value of the
second term is zero.
We shall refer to the insertion of lithium ions as a process given by the solution of (1)–(4) with
f given by (6) the (DFM) model, and with f given by (7) – the (CPM) model. We shall call the
diffusion process described by (8), (2)–(4) with f given by (6) the (DFME) model, and with f
given by (7) – the (CPME) model.
The ionic current across the carbon electrode i2 is equal to the external current through the
battery itotal at the contact with separator, and is zero at the current collector. Between these two
values the current is assumed to be distributed according to the equation:
∂i
∂x
= aFj+
n ,
where a is the interfacial area of particles per unit volume of porous electrode, calculated by
a = 0.03 · 2(1− ε1)/Rs,
for the case of cylindrical particles, or by
a = 0.02 · 3(1 − ε1)/Rs
for spherical particles; ε1 is the porosity of carbon electrode. It appears quite obvious that after we
have pressed and baked the carbon material, only a fraction of the particle’s surface will be exposed
to electrolyte, so we have introduced a suitable factor in the formula for the interfacial area. The
equation for the concentration of the electrolyte in the solution phase of the carbon electrode is as
follows:
εk
∂C
∂t
= ∇ (εkDeff,k∇C) + a(1 − t0+)j+
n ,
where k = 1, s (1 corresponds to electrode, s to separator), t0+ is transfer number, Deff,k = ε0.5
k D,
D is the diffusion coefficient of electrolyte, Cinitial = 1000 mol/m3. We impose on C the following
672
Modelling of cycling
boundary conditions: (i) that the flux of ions at lithium electrode is equal to the total current
through the cell
εsDeff,s∇C
∣
∣
x=Ls+L1
= itotal/F,
(ii) that the flux of mass is continuous at the separator-electrode interface
εsDeff,s∇C
∣
∣
x=L1+0
= ε1Deff,1∇C
∣
∣
x=L1−0
,
and (iii) that it is equal to zero at the current collector
ε1Deff,1∇C
∣
∣
x=0
= 0.
Table 1. Standard cell potential, interaction energies, model parameters for the carbon-lithium
cell and physical constants.
Parameter Value
U0 0.91489 V
Us 0.8170 V
Ω2/F 0.9926 V
Ω3/F 0.8981 V
Ω4/F −5.630 V
Ω5/F 8.585 V
Ω6/F −5.784 V
Ω7/F 1.468 V
Cs,max 18, 000 mol/m3
β 0.5
K 3.28 × 10−6 mol1/2/m1/2s
KLi 4.1 × 10−6 mol1/2/m1/2s
Cinitial 1000 mol/m3
T 298 K
yinitial 0.01
D 2.6 × 10−10 m2/s
Ds 1.0 × 10−14 m2/s
t0+ 0.2
σeff,k 0.0001 + ε1.5
k C0.855(0.00179 exp(−0.08(0.00083C − 0.6616)2−
0.0010733C + 0.855))
σel 100 S/m
Rs 3.5 × 10−6 m
Ls 25 × 10−6 m
L1 125 × 10−6 m
k 1.381 × 10−23 J/K
Na 6.022 × 1023 mol−1
< 8.314 J/(mol· K)
F 96, 487 C/mol
ε0 8.854 × 10−12 C2/(N· m2)
e 1.9 × 10−19 C
δ 2 × 10−10
ε1 0.35
εs 0.55
The potential in the solution phase is (see [6])
∇φ1 = −
i2
σeff,k
+
<T (1− t0+)
FC
∇C,
673
D.Portnyagin
where σeff,k is the effective conductivity of electrolyte given in table 1. The potential in the solid
phase of the electrode is
∇φ2 = −
(itotal − i2)
σeff
,
where σeff = ε1.5
1 σel is the effective conductivity of electrode. The local surface overpotential is
given by
η = φ1 − φ2 .
The applied potential of the cell is related to η by
Uapp = η
∣
∣
x=Ls+L1
+ (φ1 − φ2)kin +
x=L1
∫
x=0
[itotal − i2(x)]
σeff
,
where (φ1 − φ2)kin is given by kinetic expression (see [6])
itotal = FKLiC
0.5(exp((0.5F/(<T ))(φ1 − φ2)) − exp(−(0.5F/(<T ))(φ1 − φ2)))
with KLi being the reaction rate constant at the lithium electrode.
All the parameters of the cell are evaluated at T = 298 K for the reasons explained in [7]. The
values of the standard cell potential, the self-interaction energies, and the kinetic parameters are
given in table 1.
The corresponding set of equations from table 2 has been solved numerically.
Table 2. System of model equations and boundary conditions.
Region Value Equation or boundary condition
x = Ls + L1 y y = 0
C εsDeff,s∇C = itotal/F
i2 i2 = itotal
Ls + L1 > x > L1 y y = 0
C εs
∂C
∂t
= ∇ (εsDeff,s∇C)
i2 i2 = itotal
η ∇η =
i2
σeff,s
−
<T (1 − t0+)
FC
∇C
x = L1 C εsDeff,s∇C
∣
∣
L1+0
= ε1Deff,1∇C
∣
∣
L1−0
L1 > x > 0 y For cylindrical particles (1) with (2)–(4), or (8) with (2)–(4);
For spherical particles (13) with (2)–(4), or (14) with (2)–(4).
C ε1
∂C
∂t
= ∇ (ε1Deff,1∇C) + a(1 − t0+)j+
n
i2 ∇i2 = aFj+
n
η ∇η =
−itotal
σeff
+ i2
(
1
σeff
+
1
σeff,1
)
−
<T (1 − t0+)
FC
∇C
x = 0 C ε1Deff,1∇C = 0
i2 i2 = 0
674
Modelling of cycling
Let us clear up how the speed of changing of j+
n during the cycling depends on the speed of
the change of the applied voltage Uapp. We have, approximately,
d
dt
j+
n = KCβ−1
[
(β − 1)
(
1 − y
∣
∣
R=1
)
− β
(
y
∣
∣
R=1
)] (
(1 − y
∣
∣
R=1
)
)β−2 (
y
∣
∣
R=1
)β−1
×
{
exp
[
(1 − β)F
<T
(Uapp − U)
]
− exp
[
−βF
<T
(Uapp − U)
]}
dy
dt
+
+ KCβ−1 F
<T
(
(1 − y
∣
∣
R=1
)
)β−1 (
y
∣
∣
R=1
)β
{
(1 − β) exp
[
(1 − β)F
<T
(Uapp − U)
]
+
+ β exp
[
−βF
<T
(Uapp − U)
]}[
Uapp
dt
− U ′
dy
dt
]
.
Substituting for
dy
dt
its expression from the diffusion equation in which we neglect electro-
static interaction, and substituting 1 for y and its spacial derivatives, since they are dimensionless
magnitudes, we get the following estimate for the order of magnitude:
d
dt
j+
n ' j+
n
(
1 +
7
∑
s=2
Ωs
<T
)
Ds
R2
s
± j+
n
βF
<T
dUapp
dt
− β
(
1 +
7
∑
s=2
Ωs
<T
)2
Ds
R2
s
, (10)
where we take “+” when j+
n is positive, and “−” when it is negative. Hence, we can conclude that
the steeper is the relation graph of j+
n vs. the applied voltage, and the stronger is the following
inequality
∣
∣
∣
∣
dUapp
dt
∣
∣
∣
∣
�
<T
F
(
1 +
7
∑
s=2
Ωs
<T
)2
Ds
R2
s
(11)
the more effective is hysteresis.
Figure 4. Comparison of current density
(A/m2) vs. the applied voltage (V) at scan
rates 10 mV/s, 5 mV/s and 1 mV/s for cyli-
ndrical particles for chemical potential model
with electric field.
Figure 5. Comparison of current density
(A/m2) vs. the applied voltage (V) at scan
rates 10 mV/s, 5 mV/s and 1 mV/s for spheri-
cal particles for chemical potential model with
electric field.
And vice versa, the stronger is the reverse inequality
∣
∣
∣
∣
dUapp
dt
∣
∣
∣
∣
�
<T
F
(
1 +
7
∑
s=2
Ωs
<T
)2
Ds
R2
s
(12)
675
D.Portnyagin
the less effective is hysteresis, (in this case
d
dt
j+
n does not become large, because the whole righthand
side in (10) is multiplied by the derivative of y with respect to radial distance, which is small, since
the concentration profile is sloping). The above mentioned is verified by figures 4 and 5.
Figure 6. Current density (A/m2) vs. the ap-
plied voltage (V) at scan rate 10 mV/s for cyli-
ndrical particles for purely diffusive model with
(DFME) and without electric field (DFM).
Figure 7. Current density (A/m2) vs. the
applied voltage (V) at scan rate 10 mV/s
for cylindrical particles for chemical potential
model with (CPME) and without electric field
(CPM).
Figure 8. Current density (A/m2) vs. the
applied voltage (V) at scan rate 5 mV/s
for cylindrical particles for chemical potential
model with (CPME) and without electric field
(CPM).
Figure 9. Current density (A/m2) vs. the
applied voltage (V) at scan rate 1 mV/s
for cylindrical particles for chemical potential
model with (CPME) and without electric field
(CPM).
Though we do not consider this in the present paper, but numerical simulations suggest that
hysteresis also depends on the size of particles of which the electrode is made. In order to reduce the
irreversible losses and to make hysteresis less significant, the size of particles has to be diminished.
One can see from figure 6 that in the case of a constant diffusion coefficient, the graph of
electrodynamic model lies below the graph of a purely diffusive model at the interval corresponding
to the discharge of a cell, and is above the latter at large values of the applied voltage. This is
due to the fact that the presence of a negative charge distributed inside the particle enhances the
insertion of lithium ions, while the positive charge enhances, correspondingly, their outlet. At the
same time, the graph of electrodynamic model is below the graph of a purely diffusive model in
the vicinity of zero because the concentration profile is more sloping in electrodynamic case during
676
Modelling of cycling
the switching of the regime. In the case of a variable diffusion coefficient, the graph of the model
with electric field, as figures 7–9 show, is generally closer to the graph of chemical potential model
than in the previous case, because the ‘lithium ion’-‘lithium ion’ interactions governed by the
activity coefficient promote a more uniform distribution of lithium ions inside a particle and make
the term proportional to ∇y in div σE in the diffusion equation not that significant. Comparison
of figures 7–9 shows that the relative difference between electrodynamic and non-electrodynamic
models becomes more significant as the sweep rate of the applied voltage decreases. It seems
that the electric field has its stable contribution to j+
n which becomes more pronounced with the
suppression of hysteresis.
The saw-like oscillations in figures 6, 7, 8, 14–16 are due to a computational error and should
vanish as we decrease the step of integration. But this surpasses our computational facilities.
Figure 10. Profile of dimensionless concentra-
tion vs. dimensionless radial distance, i. e., the
distance from the axis of symmetry of the par-
ticle, for cylindrical particles located at the dis-
tance L1/2 from current collector, for chemical
potential model with electric field at the ap-
plied voltage Uapp=0.91489 V.
Figure 11. Profile of dimensionless concen-
tration vs. dimensionless radial distance for
cylindrical particles located at distance L1/2
from current collector, for chemical potential
model with electric field at the applied voltage
Uapp=0.31489 V.
Figure 12. Profile of dimensionless concen-
tration vs. dimensionless radial distance for
cylindrical particles located at distance L1/2
from current collector, for chemical potential
model with electric field at the applied voltage
Uapp=0.51489 V.
Figure 13. Profile of dimensionless concen-
tration vs. dimensionless radial distance for
cylindrical particles located at distance L1/2
from current collector, for chemical potential
model with electric field at the applied voltage
Uapp=0.81489 V.
677
D.Portnyagin
Figures 10–13 show the profiles of dimensional concentration y vs. dimensional radial distance R
at Stage 1, Stage 2, Stage 3, Stage 4 of the cycle, correspondingly. Figure 10 corresponds to the
discharge of the cell and the lithium ions going inside the particle (intercalation). At this stage,
ions are concentrated near the surface of the particle and their concentration decreases towards the
center. Figure 11 corresponds to the switching of the regime, when we have stopped to discharge
the cell and have begun to charge it. At this stage, lithium ion concentration still increases with
radial distance, i. e., the distance from the axis of symmetry of the particle, near the center of
the particle, takes on its maximum, and decreases with radial distance near the surface, where the
process of the lithium ions outlet (deintercalation) has begun. Figure 12 corresponds to the charge,
where the concentration is larger at the center of the particle and decreases with radial distance.
Figure 13 corresponds to the switching of the regime. At this stage, we have ceased to charge the
battery and have begun to discharge it again. The lithium ion concentration still decreases with
radial distance near the center, takes on its minimum, and increases towards the surface, where
intercalation (insertion) of lithium ions takes place.
3. Spherical particles
For spherical particles, equations (1) and (8) are replaced by
∂y
∂τ
=
1
R2
∂
∂R
(
R2f(y)
∂y
∂R
)
, (13)
∂y
∂t
=
Ds
R2
s
1
R2
∂
∂R
(
R2f(y)
∂y
∂R
)
−
1
FCmax,s
div(σ(y)E), (14)
div(E) =
3
Rsσeff,1
j+
n − δ
FCmax,s
ε0
(yavr − y),
where div(A(R)) =
1
R2
∂
∂R
(
R2A(R)
)
.
Figure 14. Current density (A/m2) vs. the
applied voltage (V) at scan rate 10 mV/s
for spherical particles for purely diffusive
model with (DFME) and without electric field
(DFM).
Figure 15. Current density (A/m2) vs. the
applied voltage (V) at scan rate 10 mV/s
for spherical particles for chemical potential
model with (CPME) and without electric field
(CPM).
The corresponding set of equations from table 2 has been solved numerically.
In the same way as for cylindrical particles, figure 14 shows that in the case of a constant
diffusion coefficient, the electrodynamic model admits lager discharge currents during the discharge,
and at higher voltages during the charge. In the case of a variable diffusion coefficient, the model
with electric field admits, in general, lager amplitudes of current in hysteresis, as figures 15–17
678
Modelling of cycling
Figure 16. Current density (A/m2) vs. the
applied voltage (V) at scan rate 5 mV/s
for spherical particles for chemical potential
model with (CPME) and without electric field
(CPM).
Figure 17. Current density (A/m2) vs. the
applied voltage (V) at scan rate 1 mV/s
for spherical particles for chemical potential
model with (CPME) and without electric field
(CPM).
show. In this case, the graphs of electrodynamic model lie closer to non-electrodynamic ones than
in the previous. The relative difference between electrodynamic and non-electrodynamic models is
more significant as the sweep rate of the applied voltage becomes smaller.
Comparing cyclic voltammograms for spherical (figures 15, 16, 17) and for cylindrical particles
(figures 7–9) one may see that the effect of electrostatic field is more significant in the case of
cylindrical particles because in that case the charge that produces this field occupies a larger part
of a space.
4. Conclusions
We have made a simulation of the cycling of lithium cell with microporous carbon electrode
under potentiodynamic control. We have compared the predictions of the models in which electric
field is not considered (CPM, DFM) and the ones in which electrostatic interaction of lithium ions
between each other and with the distribution of charge in the bulk of carbon electrode is taken
into account (CPME, DFME). We have observed that there is a considerable difference between
the results predicted by both models. The form of the particles does not have a significant effect on
the predictions of both models. In the case of a constant diffusion coefficient, the electrodynamic
model allows for lager discharge currents. In the case of a variable diffusion coefficient, the model
with electric field allows for a lager sweep of current in general. The models without electric field
predict steeper profiles of concentration of lithium ions inside the particle. The results indicate
that the electrostatic interactions do matter, and the kinetic parameters obtained with the purely
diffusive (DFM) or chemical potential model (CPM) may not represent the real kinetics of the
system.
679
D.Portnyagin
References
1. Kovalyuk Z.D., Polyakov I.O., Dugaev V.K., Litvinov V.I., Russian Journal of Electrochemistry, 1997,
33, No. 1, 21–25.
2. Botte G.G., White R.E., J. Electrochem. Soc., 2001, 148, A54–A66.
3. Lee S.-I., Kim Y.-S., Chun H.-S., Electrochim. Acta, 2002, 47, 1055–1067.
4. Verbrugge M.W., Koch B.J., J. Electrochem. Soc., 1996, 143, 600.
5. Kanno R., Kawamoto Y., Takeda Y., Ohashi S., Imanishi N., Yamamoto O., Electrochem. Soc., 1992,
139, 3397.
6. Doyle M., Ph.D. Thesis. University of California, Berkeley, CA, 1995.
7. Johnson B.A., Botte G.G., White R.E., J. Electrochem. Soc., 1999, 146, 914.
Моделювання циклiчної роботи лiтiєвої батареї з
мiкропористим вуглецевим електродом
Д.В.Портнягiн
Iнститут фiзики конденсованих систем НАН України,
79011 Львiв, вул. Свєнцiцького, 1
Отримано 23 квiтня 2007 р., в остаточному виглядi – 17 вересня 2008 р.
Чисельно моделювалися розрядно/заряднi цикли лiтiєвої батареї iз вуглецевим мiкропористим еле-
ктродом в режимi заданої напруги. Порiвнювалися передбачення двох видiв моделей: з постiйним
та змiнним коефiцiєнтом дифузiї без врахування електростатичного поля, та з постiйним та змiнним
коефiцiєнтом дифузiї з електростатичною взаємодiєю мiж iонами лiтiю та iз розподiленим зарядом
всерединi частинок вуглецевого електрода. Спостерiгалася певна розбiжнiсть мiж ними. Електро-
статична взаємодiя сприяє iнтеркаляцiї пiд час розрядки батареї та деiнтеркаляцiї пiд час зарядки.
Вивчалася залежнiсть вираженостi гiстерезису вiд швидкостi змiни прикладеної напруги. Гiстерезис
проявляється тим бiльше, чим бiльша швидкiсть прикладеної напруги. Було також отримано профiлi
концентрацiї на рiзних стадiях процесу розрядки/зарядки.
Ключовi слова: лiтiєва батарея, пористий електрод, потенцiометрiя
PACS: 82.47.Aa, 82.45.Gj, 82.45.Fk, 82.45.-h, 82.20.Wt
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