MODELING THE ELECTROCHEMICAL CHARACTERISTICS
OF GRAPHITE AND TRANSITION METAL OXIDE THIN FILM
ELECTRODES: A QUASI-METALLIC APPROACH
M.D.Levi and D.Aurbach
Department of Chemistry, Bar-Ilan University, 52900 Ramat-Gan, ISRAEL
ABSTRACT
This paper reviews our last findings related to application of a Frumkin-type
intercalation isotherm as a tool for a quantitative description of electrochemical
insertion of Li-ions into various Li-insertion anodes and cathodes. Four major
electroanalytical techniques, namely, slow-scan rate cyclic voltammetry (SSCV),
potentiostatic intermittent titration (PITT), galvanostatic intermittent titration (GITT)
and electrochemical impedance spectroscopy (EIS), are frequently used for the
determination of chemical diffusion coefficient of Li-ions, D. We have defined their
characteristic time-invariant functions (E), specific of each technique (I -1/2, I t1/2,
dE /dt1/2 and Aw, respectively) in such a way that the diffusion time constant  can be
expressed as a combination of the related function with the differential intercalation
capacitance, Cint. Such form of presentation allows (E) to be inter-related, thus
demonstrating the equivalence in application of three differential techniques -PITT,
GITT and EIS- for obtaining kinetic data. A common feature observed in the
experimental plots, D vs. E , for a variety of Li-insertion compounds, appears in the
form of deep minima corresponding to the peak values of the differential intercalation
capacitance. This was elucidated on the basis of an analysis which combines the
concept of Frumkin intercalation isotherm with a simple mechanistic diffusion model.
INTRODUCTION
Li-insertion cathodes and anodes, like lithiated transition metal oxides and graphites,
respectively, have recently attracted much attention as being the most promising
materials for high-energy Li-ion batteries. The first Li-ion battery appeared on the
market in 1991. The principle on which the Li-ion battery works can be described as
follows. The cathode material is usually one of the transition metal oxides of the
general formulae LixMO2 where M stands for Mn, Ni or Co, whereas the anode is
graphite or disordered hard carbon, LixC6. Here X=1 for the cathode materials and
graphite, and X>1 for the disordered carbon. During charging, Li+ is extracted from
the cathode and then transported through liquid or polymeric electrolyte solution and
finely inserted into the carbonaceous anode. During discharge, the process is reverted:
Li-ion is transferred from the anode to the cathode. It is not surprising that this kind of
battery is often called the “Rocking Chair Battery” or the “Swing Battery”. Numerous
attempts have been made to prepare materials with high specific capacity, good
rechargeability and fast Li-ion transfer kinetics.
Although theoretical and methodological basis for quantitative thermodynamic and
kinetic characterization of intercalation phenomena has been already reviewed (see an
excellent treatise on solid-state electrochemistry (B. G. Bruce, Ed. [1]) there still
remains the question related to the link between the lattice models, more precisely the
quantity of d/dX ( denotes the chemical potential of ions, whereas X is the
232
intercalation level), and the solid-state chemical diffusion coefficient, D [2]. In our
recent paper [3], we tried to take into account some kinetics effects, such as slow
interfacial ion transfer and its connection to intercalation isotherms. Practical Liinsertion electrodes, as evidenced by their electrochemical impedance spectra
measured in aprotic solvents, almost always reveal this interfacial kinetic contribution
in the medium frequency domain of their spectra [4-6]. Moreover, the picture of
intercalation processes becomes even more complicated if there is a preferential
orientation of powder particles in the composite electrode coatings, or in the case of a
broad particle size distribution [3].
The goal of this paper is to review Li insertion processes into thin-film electrodes,
both with respect to their kinetics and thermodynamics. We show that a first-order
phase transition during charge and discharge of Li-insertion compounds, complicated
by slow Li-ion transfer kinetics, can be described by a simple Frumkin-type isotherm,
taking into account the short-range interactions between the intercalation sites. The
proposed description of insertion processes in terms of intercalation isotherm easily
enables the comparison between the expressions for the solid-state diffusion time
constant obtained for different electroanalytical techniques, and for finding the
relationship between their characteristic time-invariant (but potential-dependent)
functions.
RESULTS AND DISCUSSION
1. The Frumkin-Type Intercalation Isotherm
The approach used for the description of Li-intercalation into inorganic hosts will be
illustrated here, using the insertion of Li into CoO2 as an example. A clear
voltammetric peak (and hence, the corresponding plateau on the charge and discharge
curves) is well documented [3,5] and relates to X ranging from 1 to 0.75 (referring to
LixCoO2). The number of Li-ions per unit intercalation site is assumed to be one
corresponding to the following stoichiometry:
4 LiCoO2 = 4 L 0.75CoO2 + e- + Li+
(1)
Definition of the stoichiometry of the intercalation reaction is an important step in the
kinetic description. This is because the intercalation site plays the role of a molecule,
if one formally compares reaction (1) with the conventional localized redox-species
reactions [7]. The stoichiometry following Eq. 1 relates exactly to a two-phase coexistence region (3.6 – 4.0 V vs. Li/Li+), as is evident from the in-situ XRD
characterizations [8]. Thus, advancement of the above insertion/deinsertion reaction
can be realized in terms of occupation of intercalation sites with Li+. Concerning the
graphite anode, a single intercalation site includes 6 carbon atoms, and thus, the
occupation of sites is defined by X in LixC6.
A schematic view of a thin inorganic matrix, capable of incorporating of Li+ cations
(anions are completely excluded) which thus exhibits permselective behavior is
shown in Fig.1. One of the important principles of thermodynamic equilibrium
applied to such an intercalation electrode demands that the total charge due to cations
inserted into the matrix bulk from the solution be compensated (locally) by an equal
amount of electronic species transferred from the substrate metal (see Fig. 1).
The presence of two kinds of species during intercalation results in a rather
complicated picture of potential distribution in the host bulk and across the both
233
current collector / intercalation electrode and intercalation electrode / solution
interfaces, as compared with the distribution in the classical case of metal/solution
interface. Two kinds of mobile species in the insertion compounds evidently create
two possible kinetic limitations. These limitations are due to the transfer of the
electronic species across the current collector / intercalation electrode interface and
ionic species across the intercalation electrode / solution interface. Moreover, two
additional mass-transport steps may appear due to the movement of both species from
the electrode host’s boundaries to its interior.
Significantly, the equilibrium potential distribution may change during Li insertion
processes, as intercalation proceeds. Fig. 2 contains a schematic view of a typical
case of a dielectric film possessing different partition constants for the electronic
equilibrium across the Me/host interface [9].
A dielectric film possessing a very low concentration of mobile electronic species
reveals a potential distributed within the whole film. As the electronic partition
constant becomes higher, the potential in the host starts to drop in close proximity to
the electrode host/solution interface. This activity reflects behavior typical of
inorganic semiconductors in contact with electrolyte solutions. Further advancement
of the intercalation process with a variation of electrode potential results in equal
potential drops across both interfaces (symmetrical “electron-counterion’ case [9]).
This result appears as a consequence of the local electroneutrality in the host bulk,
assuming a 1:1 ratio between the elementary charges of the electronic and ionic
species [9,10], and an equilibrium for both the electronic species across the current
collector / electrode host interface and the ionic species across the electrode
host/solution interface. Finally, when the electronic partition constant is high, typical
of a metal, the distribution of interfacial potential becomes similar to the classical
distribution, with a single potential drop in the part of solution close to its boundary
with the host material. We designate this case as a quasi-metallic approximation to
intercalation electrodes. Mathematical expressions describing equilibrium and kinetic
characteristics of these electrodes are then reduced to the simplest form (see below).
Once reaction products, stoichiometry and the type of the potential distribution across
the interfaces are defined, one can proceed with formal kinetic analysis of
intercalation reactions. The combination of the Frumkin-type isotherm with the
Butler-Volmer equation for slow charge/discharge transfer (here we suggest Li+
transfer across the electrode host/solution interface) results in an equation which is
valid for description of a quasi-equilibrium intercalation/deintercalation reaction
[5,11,12]:
Idim=(ko/f){(1-X) exp[-(1-)gX) exp [(1-)f(E-Eo)]-X exp(gX) exp[-f(E-Eo)] (2)
Here Idim is the dimensionless current; k = (ko/f) is the dimensionless rate constant,
with ko and  representing the standard heterogeneous rate constant (cm/s) and the
potential scan rate (V/s), respectively,  is the thickness of the host matrix (cm); g is
the dimensionless interaction parameter, and f is defined as f = F/RT (39.8 V-1 at
room temperature). The charge-transfer coefficient  in Eq. (2) is taken symmetrically
for the anodic and cathodic reactions, which is a reasonable initial approximation for
many electrochemical reactions:  = 0.5. X in Eq. 2 denotes the Li content in Li1-X
CoO2. The rate of anodic (deintercalation) reaction is proportional to X in LiXCoO2
(or (1-X) in Li1-X CoO2).
234
At equilibrium, the net current passed through the intercalation electrode should be
zero. By equalizing both terms of Eq. 2, one immediately obtains the Frumkin
isotherm:
X/(1-X) = exp [f (E - Eo)] exp (-gX)
(3)
Frumkin isotherm (Eq. 3), in contrast to Langmuir one, includes the interaction term
exp (-gX). This makes Frumkin isotherm steeper than the Langmuir one for attractive
interactions (g < 0), or, on the contrary, flatter for repulsive interactions between the
intercalation sites (g >0). There appears a critical value of gcrit = -4 such that as g <
gcrit attractive interactions become extremely intensive, resulting in first-order phase
transition. Thus gcrit is appropriate for distinguishing between monotonous and nonmonotonous character of intercalation reactions.
The extent to which the intercalation reaction may deviate from equilibrium is is
expressed by the dimensionless constant K = ko/fThe larger is K, the less is the
deviation of the charge and discharge process from equilibrium conditions. For a
given intercalation reaction (characterized by ko), the system is closer to equilibrium
at lower scan rates, and for thinner coatings.
We have recently successfully applied kinetic equation (2) for the simulation of cyclic
voltammetric response for Li intercalation reactions [3,5,12]. One important result is
worth to be reviewed here. Fig. 3 shows a best fit of a SSCV curve of LixCoO2
electrode with Eq. 2, in the range of potentials from 3.5 to 4.4V. The scan rate was 10
V/s, the fitted parameters were g = - 4.2 and ko = 8.0 10-7 cm/s. In this case, as is
seen from the above figure, a satisfactory agreement between the experimental and
theoretical curves is observed only for the anodic peak (see Fig. 3). The height of the
theoretical cathodic peak is somewhat larger than that of the experimental peak,
whereas the peak-potential separation agrees with the theoretical one. Simulation of a
(capacitive-like) plateau on the SSCV curve located at higher anodic potentials is
beyond the scope of Eq. 2.
Fig.4a presents as an example a family of theoretical SSCV curves simulated
according to Eq.2 with g = - 4.2 and different effective scan rates (K between 0.4 and
400). Experimental SSCV curves for the same range of  are shown in Fig.4 b. Both
families of curves are in broad agreement with each other in the range of  between
10 and 50 V/s. We noted two limiting ranges of  covering  > 50 V/s and possibly
 <10V/s in which a considerable difference between both sets of curves had been
observed. At higher scan rates, experimental SSCV curves revealed a deviation
towards a diffusion-controlled behavior. The deviation of experimental curves from
the theoretical ones in the limit of low scan rates may be of principal importance: at g
< - 4 , as discussed above, Li intercalation proceeds via first-order phase transition,
thus the differential capacity peak starts to increase enormously approaching a deltafunction behavior (see Fig. 4a). Deviation of the experimental curves from this
limiting behavior can be explained in terms of large Ohmic potential drops developed
as the current increases steeply (flattens the actual voltammetric peak).
235
2. Simultaneous Application of SSCV, PITT, GITT and EIS for Studying LiIon Solid-State Diffusion Kinetics
As mentioned above (see Introduction section), thin composite electrodes are not
well-defined systems compared to conventional metal and semiconductor electrodes,
thus in order to obtain reliable results one has to utilize for their characterization a
variety of electroanalytical techniques. We recently demonstrated the advantage of
simultaneous application of SSCV, PITT, EIS and GITT for characterization of
lithiated graphite and several transition metal oxides with subsequent modeling of
their electroanalytical responses [3,4,5,11,12]. Here we present a short review of the
results obtained.
All four electroanalytical techniques under consideration are related to the same
finite-space solid-state diffusion appearing in the host material after application of the
input signal specific of the technique used: linear potential scan in SSCV, small
potential steps in PITT, current pulse in GITT and small ac voltage in EIS. The
fundamentals of one-dimensional finite-space diffusion problem and the routes for
quantitative treatments of the output responses were developed by K Aoki et al. [13]
(SSCV), W.Weppner and R.A.Huggins [14,15] (PITT and GITT), and C.Ho et al [16]
(EIS). A primary diffusion parameter obtained by these techniques is the
characteristic diffusion time  which is defined for one-dimensional case as
 = l2/D
(4)
where l is the characteristic diffusion length, D denotes the chemical diffusion
coefficient connected through thermodynamics to the Frumkin intercalation isotherm.
Note that correct calculations of D through Eq. 4 depend on the adopted values of l
which for the powdered composite electrode is expected to correlate with the average
particle size rather than with the electrode’s thickness. This should be taken into
account when comparing D for the same materials obtained by different authors.
Table 1 summarizes expressions for the characteristic time-invariant (but potentialdependent) functions (E) and the differential intercalation capacity Cint that were
derived for four basic electroanalytical techniques. In this Table, GITT 1 corresponds
to a limiting case of short current pulses, whereas GITT 2 refers to small current
pulses of longer duration (for further details see below).
The first line in Table 1 presents (E) characteristic of each technique. For SSCV,
(E) is defined for the entire range of intercalation electrode potentials. Note that
(E) in Table 1 corresponds to a short-time domain of the responses except for
SSCV, which is a large-amplitude technique. The second line specifies the expression
for the differential intercalation capacity Cint. In a recent paper [3] we presented a
rigorous proof that  for each specific technique used is expressed through
combinations (within the accuracy of a constant) of the terms listed in the two lines of
Table 1, (e.g. (E) and Cint(E)).
(E) is a specific form of presentation of experimental data for each technique. For
example, for SSCV this function is defined as a hight of the voltammetric peak
normalized with respect to square-root of the scan rate . For PITT, (E) is the so236
called Cottrell slope I t1/2, i.e. the product of gradually decaying current and squareroot of time elapsing after application of a small potential step to intercalation
electrode. In the case of GITT 1, a current pulse applied to the system during a period
of time  t perturbes the system from equilibrium; t and s are the changes in the
electrode potential during the pulse (dynamic characteristic) and after its switchingoff (equilibrium or steady-state characteristic), respectively.The slope of the plot E
vs. t1/2 characterizes time dependence of the potential measured after application of a
small amplitude constant current (GITT 2). Finally, Aw is known in the theory of
electrochemical impedance as Warburg slope equal to A w = Re/-1/2 = Im/-1/2
(Re and Im are the differences in the real and imaginary components of the
impedance, respectively, corresponding to a finite variation in the angular frequency
of the alternative current, ).
Table 1. Characteristic time-invariant function (E) and differential intercalation
capacity, Cint for four basic electroanalytical techniques used for determination of
solid-state diffusion time constant.
Technique
Time-invariant
SSCV
PITT
GITT 1
GITT 2
EIS
function,
Ip -1/2
I t1/2
t/s
dE/dt1/2
Aw
I(E)/ 
QmX
It/s
ItmX(E)/E
-1/Z
(E)
Cint=Qm dX/dE
The second line in Table 1 lists expressions for the differential intercalation capacity
Cint=Qm dX/dE with Qm standing for the maximum intercalation charge. The form of
dX/dE can be easily specified for each involved technique [3].
The above approach was rigorously checked by us for a variety of lithiated
intercalation compounds including LixC6 [10], LixCoO2 [5], LixNiO2 [4],
LixCo0.2Ni0.8O2 [17], LixMn2O4 [4] and proved completely its validity for calculation
of diffusion time constant  and chemical diffusion coefficient D. As an example,
Fig.7a and b shows Cint vs. E and D vs. E curves obtained for thin LixCo0.2Ni0.8O2
electrode. Fig. 7a demonstrates generally a reasonable agreement between Cint vs. E
relationships obtained from SSCV, PITT and GITT 2 measurements. In addition, it is
clear that differential (incremental) techniques such as PITT and GITT provide
obtaining of more resolved curves compared to SSCV (which is a long-amplitude
voltage scanning technique).
A pronounced minimum appears on the D vs. E curves (see Fig 5b) corresponding to
the maximum in Cint of thin LixCo0.2Ni0.8O2 electrode. Similar observations have been
also well-documented for LixC6 [10], LixCoO2 [5], LixNiO2 [4,17], LixMn2O4 [4].
Moreover, the peaks in both Cint vs. E and D vs. E curves were very narrow for those
electrodes, in which Li-ion intercalation occurred in the form of first-order phase
transition.
The above correspondence can be easily substatiated with the use of thermodynamics
and simple model of ion diffusion according to which chemical diffusion coefficient
is defined by the following equation [2]:
237
D = (a2k*)(1-X)X(Li+ /X)(kT)
(5)
Where Mo = a2 k* denotes ionic mobility of pure phase (X = 1) in terms of the product
of the hopping rate constant k* and the nearest neighbor separation a .
The product L = X (Li+ /X)(kT)-1 is usually called the enhancement factor [14,15],
which reflects the influence of interactions between intercalation sites on the chemical
diffusion coefficient. Of course, this quantity should be specified for each particular
form of intercaltion isotherm. It was found [3] that for the Frumkin-type isotherm L =
(1-X)-1[1 + g (1-X) X], hence, the chemical diffusion potential normalized by Mo
takes the form:
Dnorm/(a2k*) = 1 + g (1-X) X
(6)
General conclusion related to prediction of the shape of D vs. X (or E) curves can be
summarized as follows: Eq. 6 describes the shape of log D vs X curves as a function
of the interaction parameter g. As g > gcrit D vs. X curve has a single maximum at X =
0.5. At g = 0 (Langmuir isotherm) D is independent of X. At 0 > g > gcrit D vs X plot
possesses a minimum at X = 0.5 with D having always positive values. At the critical
value gcrit = - 4, both L and D reach zero values. As g < gcrit (non-monotonous
charging due to first-order phase transition), L and D imply formally negative values
(physically unreasonable) along the unstable branch of the Frumkin isotherm.
Experimental plots shown in Fig.5b for LixCo0.2Ni0.8O2 electrode as well as for a
variety of lithiated cathode and anode materials [3], reflect this kind of behavior.
CONCLUSION
In this paper, we presented an integrated view on Li-ion intercalation process into
various anode and cathode materials for Li-ion batteries. The intercalation reaction
was shown to be dependent, to very much extent, on the short-range interactions
amongst intercalation sites. Generally, thermodynamics and kinetics of intercalation
reactions can be described in terms of a Frumkin-type intercalation isotherm
combined with the Butler-Volmer equation for slow charge-transfer kinetics. Two
kinds of species, electronic and ionic, participate in thermodynamic equilibrium
resulting, generally, in a complicated potential profile across the cell, including its
both interfaces (i.e. current collector / electrode and electrode / electrolyte solution ).
Electroanalytical techniques frequently used for the characterizations of intercalation
electrodes can be usefully divided into two following groups: large-amplitude
technique such as SSCV and differential techniques (PITT, GITT and EIS). For each
differential technique we have found his characteristic time-invariant (but potentialdependent) function (E) such that solid-state diffusion time constant  (and hence D)
can be presented as a unique combination of the related function (E) with the
differential intercalation capacity, Cint. In this way, the functions (E) can be interrelated, thus demonstrating equivalence in application of these techniques, provided
that the corresponding input signals are small enough.
We have presented clear evidence that both the experimental plots of Cint and log D vs
E can be modeled utilizing the Frumkin isotherm and taking into account Li-ion
transfer limitations.
Theoretical analysis of chemical diffusion coefficient based on the concept of
Frumkin intercalation isotherm and simple mechanistic ion diffusion model allowed
us to elucidate appearance of narrow minima on D vs. E curves and maxima on Cint vs.
238
E curves as a consequence of short-range attractive interactions between the
intercalation sites. Experimental evidence for this phenomenon has been well
documented for various lithiated anodes and cathodes: LixC6, LixCoO2, LixNiO2,
LixCo0.2Ni0.8O2 and LixMn2O4.
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and (b) Chapter 8 by W. Weppner “Electrode performance”, p.199-228.
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intercalation” in: Modern Aspects in Electrochemistry; Plenum Press,
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239
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240
6
6
0
x
1
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x
p
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i
m
e
n
t
a
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o
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e
t
i
c
a
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V
6
4
0
x
1
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6
2
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6
2
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6
4
0
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0
3
.
4 3
.
6 3
.
8 4
.
0 4
.
2 4
.
4 4
.
6
E
/
V
Fig. 3 Experimental SSCV peak for a thin LiCoO2 electrode ( = 10V/s) fitted
with Eq. 2 with K =ko/f = 40 and g=-4.2 (refer to ref. [5]).
6
0
5
0
i
n
f
i
n
i
t
e
l
y
l
a
r
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e
K
4
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1
6
8
1
.
6
4
0
.
8
Idim
3
0
1
0
0
1
0
0
.
20
.
10
.
0 0
.
1 0
.
2 0
.
3 0
.
4
(
E
E
)
/
V
o
Fig. 4 (a) A family of I vs. (E-Eo) curves calculated according to Eq. (2),
using different values of the dimensionless constant rate K (as indicated
in the figure) and g=-4.2
241
4
1
1
0

V
s
2
0
3
5
0
1
0
0
2
0
0
5
0
0
Cint /F
2
1
0
1
2
3
3
.
6
3
.
8
4
.
0
4
.
2
4
.
4
E
/
V
Fig. 4 (b) Experimental plots of differential intercalation capacity, Cint = I/
vs. electrode potential measured at different scan rates, as indicated (refer to
ref. [5]).
3
L
i
C
o
N
i
O
a
x
0
.
2
0
.
8
2
Cint /F
2
G
I
T
T
P
I
T
T
3
.
6
2
S
S
C
V
3
.
7
0
3
.
5
7
4
.
0
0
1
0
3
.
43
.
63
.
84
.
04
.
2
+
E
/
V
(
v
s
.
L
i
I
L
i
)
Fig. 5(a) Comparison between incremental capacity curves, Cint vs. E obtained for
LixCo0.2Ni0.8O2 electrode by SSCV, PITT, and GITT.
242
1
1
4
.
0
V
b
log D/cm 2s-1
L
i
C
o
N
i
O
x
0
.
2
0
.
8
2
1
2
3
.
6
8
V
G
I
T
T
P
I
T
T
3
.
5
7
V
1
3
3
.
4 3
.
6 3
.
8 4
.
0
+
E
/
V
(
v
s
.
L
i
I
L
i
)
Fig. 5(b) Comparison between the plots of log D vs. E obtained for LixCo0.2Ni0.8O2
electrode by PITT and GITT
243