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UNUSUAL ELECTROCHEMICAL IMMITTANCE SPECTRA
WITH NEGATIVE RESISTANCE AND THEIR VALIDATION
BY KRAMERS-KRONIG TRANSFORMATION
Andrzej Sadkowski - Institute of Physical Chemistry
of the Polish Academy of Sciences,
Kasprzaka 44/52, 01-224 Warsaw, PL
[email protected]
Local depassivation by creation of stable
centres of active dissolution on the passive electrode
is a crucial step for local processes including spatial
pattern formation and various forms of localized
corrosion, one of the most harmful forms of corrosion
damage.
The forewarning, much in advance of this to happen, is
essential for effective control.
Useful may be non-minimum-phase characteristics
of electrodes subject to local depassivation.
This feature is related to stability (metastability) of electrodes.
(to reference the title of this Symposium!)
Non-minimum phase electrodes show negative resistance
(„hidden” sometimes) in electrochemical impedance spectra (EIS).
This is demonstrated in the simplest case of 2nd order system:
( i  z1 )( i  z 2 )
Z ( i )  Z
( i  p1 )( i  p2 )
And modelled by electrical equivalent circuit:
R0
R1
C0
R2
C1
R1, R2 , C1 > 0 lub < 0. z1, z2 – zeros; p1, p2 - poles
In general case:
( s  z1 )( s  z 2 ).......( s  z n )
Z ( s )  Z
( s  p1 )( s  p2 ).......( s  pm )
zi - zeros,
pi - poles,
(m = n  n+1)
Z - hf limit
E
@
DA
‘ 
HL
In case of „bounded” or limited diffusion – approximation is possible:
@DH Lå 
H
HLL
fun0 x_ := Tanh
funup x_, k_ :=
t
x
k
1
1+ 4 t x
t
p2
i= 1
1+
x
1+ t x i p 2
4t x
2i + 1
p 2
F. Berthier et al. Electrochimica Acta 44 (1999) 2397; J. Electroanal. Chem. 502 (2001) 126.
Z(iω) – impedance = EAC / IAC
z1, z2, p1, p2 <0 - minimum-phase (mp) system, stable
unconditionally (under current or voltage control, with
any additional resistance added in series or in parallel).
-90o < phase angle of mp (ArcTg(Im(Z)/Re(Z)) < 90o
Non minimum-phase (nmp_ – when at least one of (zi, pi ) > 0
phase angle of nmp – any value !
mp – equivalent to „passive” electrical circuits with R, L, C > 0
Always stable!
nmp – includes negative resistance:Ri < 0 (sometimes „hidden”)
stability limited !
3
2.5
2
Z2 1.5
1
0.5
0
0.35
0.3
0.25
Y2 0.2
0.15
0.1
0.05
0
0.2
3
2
1
Z1
0
0
25
50
75
100
125
150
175
3
2
1 0 1
log10
2
0.2
1
log10 Z
Arg Z
4
0
3
4
0.4
Y1
0.6
0.8
0.6
0.5
0.4
0.3
0.2
0.1
0
3
2
1 0 1
log10
2
3
4
here Rdc = R1+R2 < 0 („explicite” negative resistance)
1
Fe armco, borate buffer, active-passive transition range.
„Explicit” negstive resistance represents negative slope of steady state
polarisation curve.
104
-7500
|Z|
103
102
101
10-3
-5000
10-2
10-1
100
101
102
103
104
105
Z''
Frequency (Hz)
-200
-2500
theta
-150
-100
-50
0
10-3
10-2
10-1
100
101
102
Frequency (Hz)
103
104
105
0
-2500
0
2500
Z'
5000
Nmp:
here: „hidden” negative resistance, seen only at certain frequencies.
This is the case most interesting for us !
Stability under control of the voltage i.e. source with
very small output resistance.
Instability under control of the current i.e. source with
very high output resistance.
Copper passivation in sulphates:
0.15 M CuSO4 + 5M H2SO4
RDE: 994 rpm, 99.4 rpm, T=298 K
0.075
-4
0.050
-3
Z''
0.025
-2
-1
0
0
0.5
-50
1.0
E (Volts)
0
1
0
-25
1
2
3
4
5
6
Z'
E= 20, 30, 40, 50, 60 mV 
0
Z''
I (Amps/cm2)
-5
 110, 140, 150, 170, 175 mV
25
50
75
-75
-50
-25
0
Z'
25
50
Negative resistance.
++ Rs – loss of stability under potential
control due to Z0
Complex Plane Graph of Demo Data
0.005
Z''
-5000
0.004
0.003
Y''
-2500
0.002
0
0.001
0
2500
-2500
0
2500
5000
Z'
-0.003
-0.002
-0.001
0
0.001
0.002
Y'
E=400 mV/Cu before and after
anodic (E=500 mV) polarisation.
104
103
|Z|
-0.001
-0.004
102
101
100
100
101
102
103
104
105
106
105
106
Frequency (Hz)
-200
theta
-100
0
100
200
100
101
102
103
104
Frequency (Hz)
„hidden” negative impedance = nmp system (black)
changes to stable unconditionally (mp – red) system
as a result of local depasivation.
The same plots:
2500
-1000
1500
-500
Z''
Z''
500
0
-500
-1500
500
1000
-1500
-2500
-1000
-500
0
0
500
1000
2000
3000
4000
5000
Z'
Z'
104
104
103
103
|Z|
|Z|
102
102
101
100
100
101
102
Frequency (Hz)
103
104
101
100
101
102
103
104
103
104
Frequency (Hz)
-100
-200
theta
-50
theta
-100
0
100
200
100
0
50
100
100
101
102
Frequency (Hz)
103
104
101
102
Frequency (Hz)
0.010
-30000
-20000
Y''
0.005
Z''
-10000
0
0
10000
-20000
-10000
0
Z'
10000
20000
-0.005
-0.010
-0.005
0
Y'
Impedance recorded almost exactly at the point of discontinuity.
Hopf bifurcation under current (galvanostatic) control.
0.005
All the more often reported are similar results:
Electrochimica Acta 47 (2001) 501–508
„On the origin of oscillations in the electrocatalytic
oxidation of HCOOH on a Pt electrode modified by Bi
deposition”.
Jaeyoung Lee *, Peter Strasser, Markus Eiswirth, Gerhard
Ertl
Oscillatory Peroxodisulfate Reduction on Pt and Au Electrodes under High Ionic Strength
Conditions, Caused by the Catalytic Effect of Adsorbed OH.
Shuji Nakanishi, Sho-ichiro Sakai, Michiru Hatou, Yoshiharu Mukouyama, and Yoshihiro
Nakato* J. Phys. Chem. B 2002, 106, 2287-2293
M. Bojinov – A model of the anodic oxidation of metals in concentrated solutions.
J. Electroanal. Chem. 405 (1996) 15
Electrochimica Acta 47 (2002) 2297_/2301
Electrochemical oscillations in the methanol oxidation on Pt
Jaeyoung Lee *, Christian Eickes, Markus Eiswirth, Gerhard Ertl
Mechanism of discontinuity (switching circles from left to right halfplanes, Hopf bifurcation under GC):
appearence of local conduction channels, active centers on passive
surface:
( s  2)( s  100)
Z ( s)  R0
( s  1)( s  50)
1
Y ( s) 
 Gp
Z ( s)
Gp – local conduction channel.
(local active center).
1 p1  p2
Gh  
R0 z1  z2
Gp = (0, 0.9, 1.0, 1.1, 1.2) * Gh
x=0
x = 0.9
x = 1.1
x = 1.4
Calculated Voltage-step (left) and current-step
(right column) responses for nmp electrode
with local depassivation represented by
parallel conductance Gp = 1/Rp = x * Gh
Some authors still deny validity of such data based on their
failing to comply with Kramers-Kronig transformation (KKT).
Imaginary part reconstructed from real part of the spectrum:
Z ' ' ( )  
2



0
Z '( x )  Z ' ( )
dx
2
2
x 
Real part reconstructed from imaginary part of the spectrum:
Z '( )  Z ' (  ) 
2



0
xZ '' ( x )   Z '' ( )
dx
x2   2
This rebuttal is evidently erroneous:
In case of nmp-type electrodes KKT fails for impedance data
but is successful for admittance representation of data.
Under voltage control it is admittance which is measured is
and it should be KK tested.
Failing of the KKT for data
transformed as impedance.
(lines– experimental data,
dots KKT data)
The same data KK transformed
in admittance representation and
back-calculated to impedance.
(lines– experimental data,
dots KKT data)
Good agreement !
To be quite honest: the agreement is good at peaks.
Much worse close to zero. Errors of integration!
KRAMERS-KRONIG TRANSFORMS AS VALIDATION OF
ELECTROCHEMICAL IMMITTANCE DATA NEAR DISCONTINUITY
A. Sadkowski1*, M. Dolata, J.-P. Diard
Journal of Electrochemical Society, in press (MS03.02.062)
Financial support by Research Grant No 7T08C 012 20 of the State
Committee for Scientific Research is gratefully acknowledged.