Concerning the use of the Kramers-Kronig transforms for the validation of impedance data

Corrosion Science, Voi. 28, No. 9, pp. 9 3 3 - 9 3 8 , 1988 Printed in Great Britain

0010-938X/88 $3.00 + 0.00 Pergamon Press plc.

SHORT COMMUNICATION CONCERNING

THE USE OF THE KRAMERS-KRONIG

TRANSFORMS

F O R THE VALIDATION OF IMPEDANCE DATA

H O N G S H I N AND FLORIAN M A N S F E L D

University of Southern California, Materials Science Department University Park, Los Angeles, CA

90089-0241

Abstract: The a p p l i c a t i o n o f t h e K r a m e r s - K r o n i g t r a n s f o r m s t o a number typical corrosion systems has shown that severe proble m s e xis t with the ir for validation of experimental impedance data. Even f o r t h e o r e t i c a l data agreement with the transforms is observed if the impedance data have r e a c h e d t h e i r dc l i m i t w i t h i n t h e f r e q u e n c y r a n g e u s e d f o r t h e t r a n s f o r m .

of use no not

INTRODUCTION The i n c r e a s i n g to

the

study

application

of a variety

means o f v a l i d a t i o n the

EIS

models. will

data

of such data.

have

a

produce semicircles

preted fulness

out

form

that

or other

cannot

experimental

curves

(3,4) have given examples

important be

data

Such c u r v e s al

(K-K) t r a n s f o r m s

of electrochemical

necessary

with

simple

can be o b t a i n e d

"which

impedance

plot,

cannot,

data.

b u t w h i c h do

then,

(2) h a v e p o i n t e d

as a tool

(EIS)

to define

in cases where

explained

in a complex plane

Van N e i r h a e g h e e t

of the Kramsrs-Kronig

interpretation

especially

and

the Kramsrs-Kronig relationsl

as impedance = (1).

impedance spectroscopy

p h e n o m e n a makes i t

This is

complicated

Cahan h a s p o i n t e d

not satisfy

of electrochemical

of corrosion

be i n t e r -

out the use-

for the evaluation

Recently,

and

Macdonald et

al

for the use of the K-K relations as diagnostic cri-

teria for determining the validlty of EIS data. The application of the K-K relations to EIS data for a number of corrosion systems obtained in this laboratory has shown that serious problems exist with this approach of validation of impedance data. sion resistant systems,

In many studies of very corro-

such as inhibited solutions,

polymer coated metals and

anodized surfaces,

the impedance data do not show a d c

limit within the lowest

frequencies

which

result

range

demonstrates

can

the excellent

be

studied.

corrosion

While

resistance

this

by

itself

of the system studied,

been found that the K-K relations suggest that the data are not valid. apparently

due

to

the

fact

that

the

impedance

Manuscript ~.eived for publication 22 June 1988. 933

data

were

not

finite

often it has

This is in

the

934

HONG SHIH and FLORIAN MANSFELD

investigated frequency range. involve

a Warburg

impedance

the K-K relations results,

It has also been observed that for systems which (Randles-circuit)

again suggest

it is proposed

that

or a transmission

that the data are not valid.

the use of

the K-K relations

line element

Based on these

for validation

of

experimental EIS data be considered only with severe reservations until further analyses

of

this problem have been made.

In the following

some examples

for

the erroneous transforms will be given. RESULTS AND DISCUSSION The first example involves the simple case of a resistance Rp in parallel with a capacitance Cd and in series with a resistance R s.

If the lowest fre-

quency used for the recording of EIS data for this system is 0.01 Hz, then the spectrum will quency

end

in the capacitive

is about 0.0016 Hz

transforms

from

the

(5).

region,

since

the lower break-point

fre-

The real part Z' of the impedance obtained by

imaginary

part

Z"

shows

large

discrepancies

from

the

"measured" value of Z' at the lowest frequencies which could be interpreted as being

due

to errors

in these

(theoretical)

data

(Fig.

1).

The same analysis

with the lowest frequency at 10 -5 Hz results in perfect agreement between both curves.

When ~

was

lowered from 106 ohm to 104 ohm, very good agreement was

observed for a low frequency limit of the K-K transform at 0.01 Hz. The second e x ~ p l e impedance. quencies

deals with the Randles-circuit which involves a Warburg

As shown in Fig.

which

Nyquist-plots transforms.

2 a discrepancy is found again at the lowest fre-

is especially for

The

the

curve

pronounced

theoretical calculated

data with

for Z" and

(Fig.

those

2b).

Fig.

calculated

the K-K relations

3 shows

using

the

the K-K

does not agree with

that for a Randles-circuit. The

third

passivated pitting

example

in CeC13

is observed

and after

uses then

experimental exposed

this time.

data

for

to 0.5 N NaCl

AI

6061

for

7 days

The experimental

which

had

(6,7).

been Some

data can be fitted to

the pitting model described recently which involves a transmission line element for the reactions

in the pits

(6,7).

Again,

that the data are not valid as shown in Fig. parameters

were

used

to produce

theoretical

the K-K relations would suggest 4 and 5. data down

Only when the fitting to 10 -7 Hz

(!)

it was

possible to validate these data with the K-K relatlons. CONCLUSIONS A transfer function can only be called an impedance when the conditions of causallty,

linearity#

and stability

are

fulfilled

and the

impedance

data

are

Short C o m m u n i c a t i o n

o

•

2

~

2

935

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~r

m--

'

<

o

° ~ 4--

°~

T

3

T

I

~

°

t~

(~1..IO 1

°

t~l

o

o (t~q0)

~ued 6ewz.Z-

._

al

~Jed Ieau-z

N

"

.u

m

C

~o

J

0 0

0 0

.4

,,~

_j

I 0

I

0 ~'

00wlwqolSetu~Z-

O

I11

00$/(tut.10

ul;

Z)~eauz

~

936

HONG SHIn and FLORIAN MANSFELD 0 0

w~

,A 0 0

~

l

¢

¢

0 0

0 0

0 0

o

~

o

0 0

0 0

o

0 0

o

o

0 0

~

o

t

0

(mClO ul Z) ~emIZel

..;=

A

A

e

d

•I-

~e

~o

0 ~

0 0 0

(.,q0)

0 0 0

~Jecl

0 0 0

0 0 0

o 0 0

0

0 o o

(gqO)

OemIz-

o o o

!

o 0 o

~Jed

leeu-Z

o

E

w ..Jol

0

0

i

I

0

0

i

i

o

o

(mqo uT Z)

eie,.1:Z-

&

Short Communication finite.

The

present

criterion

in the use of the K-K transforms

impedance data. which K-K

is due

shows

that

a problem

with

the

last

for the validation of experimental

the

results

problems

indicated

in real systems, were subjected to the

that

these

data were

the K-K transforms agree with the experimental

in those cases where these data approach a d c used for the K-K transform. apparently

exists

When theoretical data calculated for a lower frequency limit, to experimental

analysis,

ently,

discussion

937

cannot

detect

not

valid.

Appar-

or theoretical data only

limit within the frequency range

These results and the fact that the K-K analysis

errors

due

to

non-linearity

justify

the

suggestion

that at the present state of knowledge the K-K relations should not be used for validation

of

impedance

experimental

data.

Further

work

is

in progress

evaluate solutions to these problems and to define alternative approaches.

to The

question of non-linearity will be addressed next in this evaluation. The authors still agree with the necessity for criteria for validation of experimental which data

impedance

are being

data

reported

especially

for

many

of

in the rapidly expanding

do not seem to agree with

the

complicated

literature.

systems

Many of these

any of the models which describe

the physical

realities of such systems. Acknowledgment The

authors

acknowledge

pointed out some deficiencies

the

comments

by

in the original

W.J.

Lorenz

and

M.

Ebert

authors for the use of the K-K transforms and suggested some improvements. initial

approach

Macdonald.

was

based

on

the

computer

who

computer program written by the

program

kindly

supplied

The

by D.D.

This work has been performed under Contract No. DAAL 03-86-K-0156.

with the U.S. Army Research Office. REFERENCES 1.

B.D. Cahan and C.-T. Chen, J. Electrochem.

2.

R.L. van Meirhaeghe, Acta 21, 39 (1976).

3.

D.D. Macdonald and M. Urquidi, J. Electrochem.

4.

M. Urquidi-Macdonald,

5.

F. Mansfeld, Corrosion 37, 301

6.

F. Mansfeld,

7.

F. Mansfeld, S. Lin, S. Kim and H. Shih, Corrosion/88, Mo., paper No. 380, submitted to Corrosion.

E.C.

Dutoit,

Soc. 129, 474

F. Cardon

and W.P.

(1982).

Gomes,

Electrochim.

Soc. 132, 2316

S. Real, and D.D. Macdonald,

(1985).

ibid. 133, 2018

(1986).

(1981).

S. Lin, S. Kim and H. Shih, Corr. Sci. 27, 997

(1987).

NACE,

St.

Louis,

938

HONG SHIH and FLORIANMANSFELD

Figure Captions

1.

Theoretical and K-K transform data Cd = 10-4F).

(Rs = 10 ohm, Rp = 106 ohm,

2.

Theoretical and K-K transform data for Randles-circuit.

3.

Nyquist-plots for data in Fig. 2.

4.

Experimental and K-K transform data for A1 6061, passivated in 1000 ppm CeCl 3, after exposure in 0.5 N NaC1 for 7 days.

5.

Nyquist-plots for data in Fig. 4.