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

ansad@ichf.edu.pl

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 characteristicsof 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).

Z i Zi z i z

i p i p( )

( )( )

( )( )

1 2

1 2

This is demonstrated in the simplest case of 2nd order system:

R0 R1 R2

C0 C1

And modelled by electrical equivalent circuit:

R1, R2 , C1 > 0 lub < 0. z1, z2 – zeros; p1, p2 - poles

Z s Zs z s z s z

s p s p s pn

m

( )( )( ) . . . . . . . ( )

( )( ) . . . . . . . ( )

1 2

1 2

(m = n n+1)

In general case:

zi - zeros, pi - poles, Z - hf limit

fun0@x_D:= TanhAt xE‘t x

In case of „bounded” or limited diffusion – approximation is possible:

funup@x_, k_D:=1H1+4 t xp2L å

i=1

k1+t xHi pL2

1 +4 t xHH2 i +1L pL2F. 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, stableunconditionally (under current or voltage control, withany additional resistance added in series or in parallel).

Non minimum-phase (nmp_ – when at least one of (zi, pi ) > 0

phase angle of nmp – any value !

-90o < phase angle of mp (ArcTg(Im(Z)/Re(Z)) < 90o

mp – equivalent to „passive” electrical circuits with R, L, C > 0

Always stable!

3 2 1 0 1 2 3 4log10

20

15

10

5

0

grA

Z

3 2 1 0 1 2 3 4log10

00.10.20.30.40.50.6

gol01Z

0 1 2 3 4Z1

0.250.5

0.751

1.251.5

1.752

Z2

0 0.2 0.4 0.6 0.8 1Y1

0.050.1

0.150.2

0.250.3

0.350.4

Y2

nmp – includes negative resistance:Ri < 0 (sometimes „hidden”)

stability limited !

3 2 1 0 1 2 3 4log10

175150125100755025

0

grA

Z

3 2 1 0 1 2 3 4log10

00.10.20.30.40.50.6

gol01Z

4 3 2 1 0 1Z1

00.5

11.5

22.5

3

Z2

0.2 0 0.2 0.4 0.6 0.8 1Y1

00.050.1

0.150.2

0.250.3

0.35

Y2

here Rdc = R1+R2 < 0 („explicite” negative resistance)

10-3 10-2 10-1 100 101 102 103 104 105101

102

103

104

Frequency (Hz)

|Z|

10-3 10-2 10-1 100 101 102 103 104 105

-200

-150

-100

-50

0

Frequency (Hz)

the

ta

-2500 0 2500 5000

-7500

-5000

-2500

0

Z'

Z''

Fe armco, borate buffer, active-passive transition range.„Explicit” negstive resistance represents negative slope of steady statepolarisation curve.

1.5 1 0.5 0 0.5 1 1.5Z1

1.5

1

0.5

0

0.5

1

Z2

3 2 1 0 1 2 3 4log10

0

0.05

0.1

0.15

0.2

0.25

gol01Z

3 2 1 0 1 2 3 4log10

150 100 50

050

100150

grA

Z

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.

0 0.5 1.00

0.025

0.050

0.075

E (Volts)

I (A

mps

/cm

2 )

Copper passivation in sulphates:

0 1 2 3 4 5 6

-5

-4

-3

-2

-1

0

1

Z'

Z''

-75 -50 -25 0 25 50

-50

-25

0

25

50

75

Z'

Z''

Negative resistance.++ Rs – loss of stability under potentialcontrol due to Z0

0.15 M CuSO4 + 5M H2SO4

RDE: 994 rpm, 99.4 rpm, T=298 K

E= 20, 30, 40, 50, 60 mV

110, 140, 150, 170, 175 mV

-2500 0 2500 5000

-5000

-2500

0

2500

Z'

Z''

Complex Plane Graph of Demo Data

-0.004 -0.003 -0.002 -0.001 0 0.001 0.002-0.001

0

0.001

0.002

0.003

0.004

0.005

Y'

Y''

100 101 102 103 104 105 106100

101

102

103

104

Frequency (Hz)

|Z|

100 101 102 103 104 105 106

-200

-100

0

100

200

Frequency (Hz)

thet

a

E=400 mV/Cu before and afteranodic (E=500 mV) polarisation.

„hidden” negative impedance = nmp system (black)changes to stable unconditionally (mp – red) systemas a result of local depasivation.

-1500 -1000 -500 0 500

-1000

-500

0

500

1000

Z'

Z''

0 1000 2000 3000 4000 5000-2500

-1500

-500

500

1500

2500

Z'

Z''

100 101 102 103 104100

101

102

103

104

Frequency (Hz)

|Z|

100 101 102 103 104

-200

-100

0

100

200

Frequency (Hz)

thet

a

100 101 102 103 104101

102

103

104

Frequency (Hz)|Z

|

100 101 102 103 104

-100

-50

0

50

100

Frequency (Hz)

thet

a

The same plots:

-20000 -10000 0 10000 20000

-30000

-20000

-10000

0

10000

Z'

Z''

-0.010 -0.005 0 0.005-0.005

0

0.005

0.010

Y'

Y''

Impedance recorded almost exactly at the point of discontinuity.Hopf bifurcation under current (galvanostatic) control.

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 PtJaeyoung Lee *, Christian Eickes, Markus Eiswirth, Gerhard Ertl

Gp = (0, 0.9, 1.0, 1.1, 1.2) * Gh

Z s Rs s

s s( )

( )( )

( )( )

0

2 1 0 0

1 5 0

Y sZ s

G p( )( )

1

Gp – local conduction channel.(local active center).

GR

p p

z zh

1

0

1 2

1 2

Mechanism of discontinuity (switching circles from left to right half-planes, Hopf bifurcation under GC):appearence of local conduction channels, active centers on passive surface:

Calculated Voltage-step (left) and current-step(right column) responses for nmp electrodewith local depassivation represented byparallel conductance Gp = 1/Rp = x * Gh

x = 0

x = 0.9

x = 1.1

x = 1.4

Some authors still deny validity of such data based on theirfailing to comply with Kramers-Kronig transformation (KKT).

ZZ x Z

xd x' ' ( )

' ( ) ' ( )

2

2 20

Z ZxZ x Z

xdx' ( ) ' ( )

' ' ( ) ' ' ( )

2

2 20

Imaginary part reconstructed from real part of the spectrum:

Real part reconstructed from imaginary part of the spectrum:

This rebuttal is evidently erroneous:In case of nmp-type electrodes KKT fails for impedance databut is successful for admittance representation of data.

Under voltage control it is admittance which is measured isand it should be KK tested.

The same data KK transformedin admittance representation andback-calculated to impedance.(lines– experimental data,dots KKT data)

Good agreement !

Failing of the KKT for datatransformed as impedance.(lines– experimental data,dots KKT data)

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.