MODELING OF ELECTROCHEMICAL IMPEDANCE
CHARACTERISTICS OF POROUS, COMPOSITE
INTERCALATION ELECTRODES
M.D. Levi and D. Aurbach
Department of Chemistry, Bar-Ilan University, 52900 Ramat-Gan, Israel
ABSTRACT. A new model considering two (or more) porous sublayers of different
thicknesses is proposed to explain the appearance of well-developed low-frequency
semicircles (LFSs) with very high capacitances at their maxima. Evidence is provided
that it is of key importance for relating the impedance characteristics of the porous
electrodes to that of single-particle electrodes. It is very helpful in distinguishing the
physical nature of the contributions to the total impedance of the porous electrodes.
1. Introduction.
Studies on the impedance of practical, composite electrodes composed of
intercalation particles with different particle size distribution (PSD) have recently
attracted meticulous attention. This is important from both theoretical and practical
point of view, especially, in detailed comparison with the impedance characteristics of
single-particle electrodes, which determination became recently available [1-8].
Meyers et al [9] proposed a model of the porous electrode, composed of spherical
particles with different kinds of PSD. The modeling is carried out in two steps:
i.
Firstly, the impedance of a single particle is derived
taking account slow, Li-ion migration through the passivation
layer (the so-called surface electrolyte interface, SEI) covering
the particles surfaces (the characteristic parameters: the
resistance due to the ionic migration, Rsl and a geometric
capacitance, Csl) and the limitation due to interfacial Li-ion
transfer (across the SEI/particle interface) with the
characteristic parameters as follows: the charge-transfer
resistance, Rct and the double layer capacitance, Cdl. This step
also includes a kind of the finite-space Warburg element, FSW
(specific of spherical particles), modeling the occupation of the
intercalation sites by Li-ions in the particle bulk.
ii.
Secondly, thus defined impedance of single particles
was incorporated into the expression for the impedance of
porous electrodes with four major parameters of the porous
structure, namely, the ratio of the true surface area of the
particles to their volume (a), the specific electronic
conductivities of the intercalation particles and the solution in
the pores space,  and , respectively, and the parameter 
characterizing the sharpness of the particle size distribution.
In a recent paper [10] we have proposed a generalization of the Meyers et al
model extending it to a situation frequently occurring with the practical composite
electrodes, when active electrode masses cover non-homogeneously the surface of the
current collectors. A simple case of two porous sublayers (each composed of spherical
2 - 46
intercalation particles with two different radii, Rs1 and Rs2), which thicknesses are
equal to L1 (fraction 1) and L2 (the corresponding fraction (1-1)). The implication of
this two-sublayers model was that at certain combination of the parameters of the
porous electrode structure, the third, low-frequency semicircle (LFS) with a very high
formal specific capacitance may appear in the related impedance spectra, in addition
to the high-frequency semicircle (HFS) presenting impedance of the surface layer and
the middle-frequency semicircle (MFS) due to interfacial kinetics limitation for the
Li-ion transfer.
The objective of the present work is to derive a system of equations describing
completely the properties of two-parallel porous sublayers model, providing a
reasonable explanation of the high capacitance values for the well-developed lowfrequency semicircles (LFSs). Electroanalytical behavior of porous, composite
graphite electrode with methyl silicate binder, prepared by sol-gel technique [11], is
probably the best experimental system for theoretical modeling of its impedance
characteristics since in this case, a decreased values of the both  and  are expected
(for typical organic electrolyte solutions containing Li salt).
2. Results and Discussion.
A sketch of the porous, composite electrode comprised of spherical particles of
different radii is shown in Fig. 1. For simplicity, only two sublayers, L1 and L2 thick
are considered. Although the active masses of practical composite electrodes present
particles with different size distribution (PSD), for the qualitative consideration we
assume that each sublayer is composed of two particles ("small" and "large" particles).
A consideration of three major contributions to the impedance of individual
intercalation particles mentioned in the introduction section lead to the following
expression for the overall particle’s impedance [9]:
Rct, i 
Z part, i 
Rpart, i
Ys, i

Rpart, i 
1  j C dl, i  Rct, i 

Ys, i 


Rsl, i
1  j Rsl, i C sl, i
(1)
with the finite-space diffusion resistive element, Rpart,i, of the form [9]:
Rpart,i 
and with
1

Ys, i

tanh

j  i
j   i  tanh


j  i
Rs,2 i
3Ds Cpart, i

i
3C part, i
.
(2)

where Ys,i-1 is a spherical analog of the characteristic function of frequency in the
expression for a linear finite-space Warburg element, i.e., coth ( j   i ) [10].
Here Cpart,i stands for the limiting low-frequency capacitance of a spherical particle,
Rs, i is the diameter of the spherical particle, Ds is the chemical diffusion coefficient, i
designates the related diffusion time constant, and  is the angular frequency of the
alternative current. Subscript i denotes the particles with two different radii, i.e. either
1 or 2. Rct and Rsl were previously defined (they relate to the diameters of the MFS
2 - 47
and HFS, respectively). The total admittance of an electrode comprising a mixture of
two types of particles in terms of different size, 1/Zmix, is regarded as an averaged sum
of the individual admittances 1/Z1 and 1/Z2 [9,10]:
1
Z mix

1
Z1

(1  1 )
,
Z2
(3)
where  1 is the fraction of the total capacity due to a contribution of the "small"
particles (for further consideration, it is assumed to be 0.5).
The distributed impedance of the porous electrode, Zporous, relates to the impedance
of the mixed particles electrode as is defined by Meyers et al. [9]:
Z porous


  
2     cosh  

L
  
1 
,

  
 sinh 





(4)
with a complex parameter  of the form:
1
  2  a
  L
 
    Z mix
1
2
 ,

(5)
where  and  were earlier defined as the specific conductivities of the
electrolyte solution and solid particles, respectively, L is the thickness of the porous
electrode, a denotes the ratio of the total surface area to its volume, and Zmix is
obtained from Eq. 3.
The impedance of a parallel combination of two porous sublayers ( Zmix, L1 and
Zmix, L2 are the impedances of the two sublayers L1 and L2 thick considered), ZL1+L2, is
given by the formulae similar to Eq. 3 [10]:
1
Z L1 L2

 L1
Z mix, L1

(1   L1 )
.
Z mix, L2
(6)
In the following modeling, we consider the case when the thicker sublayer, L2
cover predominantly the surface of current collector, e.g. it was assumed  L2 = 0.85,
thus  L1 = 0.15
The impedance plot calculated according to Eqs. 1 – 6 (with the numerical
parameters indicated in the figure’s caption) is shown in Fig. 2. This plot reproduces
qualitatively all three semicircles (i.e. HFS, MFS and LFS) which are characteristic of
the experimental impedance spectra of graphite electrode [11]. The values of the
capacitances related to the HFS maxima may vary (depending on the capacitances of
the individual particles selected as the model parameter and the parameters of the
porous electrode structure  and ) between 0.1 and 10 Fcm-2, whereas the
capacitance related to the MFS varies between 0.01 and 1 mF cm-2. As the dominant
sublayer is thick, and the values of  and  are low, the capacitances at both
semicircles of the porous electrode impedance plot are lower compared to that related
to the individual particles. On the contrary, with thin electrodes, and for high values
of and  , the relevant impedance spectra exhibit a “congruent” behavior (as related
2 - 48
to the spectra of the individual particles) with a scaling factor reflecting the true surface
area of the porous electrode [].
3. Conclusion
The model proposed allows to reproduce semi-quantitatively all the three
semicircles (HFS, MFS and LFS) appearing in impedance spectra of many porous,
composite electrodes comprised of intercalation particles, and is of key importance for
relating the impedance characteristics of the porous electrodes with that of singleparticle electrodes.
REFERENCES
1. Nishizawa, M.; Hashitani, R.; Itoh, T.; Matsue, T.; Uchida, I. Electrochem. and
Solid-State Letters, 1998, 1, 10.
2. Umeda, M.; Dokko, K.; Fujita, Y.; Mohamedi, M.; Uchida, I.; Selman J.R.
Electrochim. Acta , 2001, 47, 885.
3. Sato, H.; Takahashi, D.; Nishina, T.; Uchida, I. J. Power Sources, 1997,68,
540.
4. Dokko, K.; Mohamedi, M.; Fujita, Y.; Itoh, T.; Nishizawa, M.; Umeda, M.;
Uchida, I. J. Electrochem. Soc., 2001,148, A422.
5. Dokko K.; Mohamedi, M.; Umeda, M.; Uchida, I. J. Electrochem. Soc.,
2001,150, A425.
6. Dokko, K.; Nishizawa, M.; Horikoshi, S.; Itoh, T.; Mohamedi, M.; Uchida, I.
Electrochem. and Solid-State Letters, 2000, 3, 125.
7. Uchida, I.; Mohamedi, M.; Dokko, K.; Nishizawa, M.; Itoh, T.; Umeda, M. J.
Power Sources, 2001, 97-98, 518.
8. Shu D.; Chung, Kyung Yoon; Cho, Won Il; Kim, Kwang-Bum. J. Power
Sources, 2003,114, 253.
9. Meyers, J.P.; Doyle, M.; Darling, R.M.; Newman, J. J. Electrochem. Soc,
2000, 147, 2930.
10. Levi, M.D.; Aurbach, D.. J.Phys. Chem (in print).
11. Aurbach,D; Levi, M.D.; Lev,
J. Appl. Electrochem., 1998, 28, 1051.
2 - 49
O.;
Gun,
J.;
Rabinovich,
L.
1600
L1
2
-Z '' / cm
Solution
Current collector
1200
L2
800
HFS
f*=2.9 kHz,
Csl=0.3 Fcm-2
MFS
f*=24 Hz,
Cdl=52 Fcm-2
400
0
0
R1
LFS,
f*=2 mHz,
CLFS=160mFcm-2
400
R2
Fig. 1. The model of a simple,
non-homogeneous coating of the
current collector with two porous
sublayers (thicknesses L1 and L 2)
comprised of intercalation particles
of two different radii (R1 and R2);
800
1200
1600
Z ' / cm
2
Fig. 2. Nyquist plot calculated according to the
model using Eqs. 1-6. The following parameters were
involved in the calculation. “Large” particles: Rs,1 = 4
m, (Rpart = 336 cm2, Cdl = 100 Fcm-2, Rct = 88.1
cm2, Csl = 1 Fcm-2, Rsl = 80 cm2). “Small”
particles: Rs,1 = 2 m, (Rpart = 84 cm2 , all other
parameters as that for “large” particles). The porous
structure parameters: a = 5x103 cm-1,  = 5.5x10-5 1
cm-1,  = 10-5 -1cm-1, L1 = 0.04 cm, L2 = 0.0003
cm, L1 = 0.15 and L2 = 0.85.
2 - 50