When the impedance is measured on a battery, an inductive impedance is often observed in a high frequency range. This inductance is frequently related to the cell geometry and electrical leads. However, certain authors claimed that this inductance is due to the concentration distribution of reacting species through the pores of battery electrodes. Their argument is based on a paper in which a fundamental error was committed. Hence, the impedance is re-calculated on the basis of the same principle. The model shows that though the diffusion process plays an outstanding role, the overall reaction rate is never completely limited by this process. The faradaic impedance due to the concentration distribution is capacitive. Therefore, the inductive impedance observed on battery systems cannot be, by any means, attributed to the concentration distribution inside the pores. Little frequency distribution is found and the impedance is close to a semi-circle. Therefore depressed impedance diagrams in porous electrodes without forced convection cannot be ascribed to either a Warburg nor a Warburg-de Levie behaviour.Nomenclature
A
D¦
C¦ (mole cm s
–1)
-
B
j+K¦
C¦ (mole cm s
–1)
-
b
Tafel coefficient (V
–1)
-
C(x)
Concentration of
S in a pore at depth
x (mole cm
–3)
-
C
0
Concentration of
S in the solution bulk (mole cm
–3)
-
C C(x)
change under a voltage perturbation (mole cm
–3)
- ¦
C¦
Amplitude of
C (mole cm
–3)
-
D
Diffusion coefficient (cm
2 s
–1)
-
E
Electrode potential (V)
-
E
Small perturbation in
E namely a sine-wave signal (V)
- ¦
E¦
Amplitude of
E(V)
-
F
Faraday constant (96500 A s mol
–1)
-
F(x)
Space separate variable for
C
-
f
Frequency in Hz (s
–1)
-
g(x)
KC(x)¦
E¦(mole cm s
–1)
-
I
Apparent current density (A cm
–2)
-
I
st
Steady-state current per unit surface of pore aperture (A cm
–2)
- j
Imaginary unit [(–1)
1/2]
-
K
Pseudo-homogeneous rate constant (s
–1)
-
K
Potential derivative of
K, d
K/d
E (s
–1 V
–1)
-
K
*
Heterogeneous reaction rate constant (cm s
–1)
-
L
Pore depth (cm)
-
n
Reaction order
- P
Reaction product
-
p
Parameter for
F(x), see Equation 13
-
q
Parameter for
F(x), see Equation 13
-
R
e
Electrolyte resistance (ohm cm)
-
R
p
Polarization resistance per unit surface of pore aperture (ohm cm
2)
-
R
t
Charge transfer resistance per unit surface of pore aperture (ohm cm
2)
- S
Reacting species
-
S
a
Total surface of pore apertures (cm
2)
-
S
0
Geometrical surface area
-
S
p
Developed surface area of porous electrode per unit volume (cm
2 cm
–3)
-
s
Concentration gradient (mole cm
–3 cm
–1)
-
t
Time
-
U
Ohmic drop
-
x
Distance from pore aperture (cm)
-
Z
Faradaic impedance per unit surface of pore aperture (ohm cm
2)
-
Z
x
Local impedance per unit pore length (ohm cm
3)
-
z
Charge transfer number
-
Porosity
-
Thickness of Nernst diffusion layer
-
Penetration depth of reacting species (cm)
-
Penetration depth of a.c. signal determined by the potential distribution (cm)
-
Electrolyte (solution) resistivity (ohm cm)
-
0
Flow of S at the pore aperture (mole cm
2 s
–1)
-
Angular freqeuncy of a.c. signal, 2
f(s
–1)
-
Integration constant
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