A computer analysis of electrochemical impedance data

Corrosion Science, Vol. 23, No. 9, pp. 1007-1015, 1983 Printed in Great Britain.

0010-938X/83 $3.00 + 0.00 © 1983 Pergamon Press Ltd.

A COMPUTER ANALYSIS OF ELECTROCHEMICAL IMPEDANCE DATA M. W. KEND1G, E. M. MEYER, G, LINDBERG and F. MANSFELD Rockwell International Science Center, Thousand Oaks, CA 91360, U.S.A. Abstract--CIRFIT, a computer analysis of electrochemical impedance data, minimizes the sum of squares of the radial difference between observed and calculated impedance data presented in either the complex impedance or admittance plane. CIRFIT requires no assumptions of ideal capacitive frequency response for the double layer in order to extract the values of solution resistance and polarization resistance that are important to corrosion rate analysis. Examples are presented of the use of CIRFIT for the analysis of corroding iron in tapwater and for model pit electrodes that have two time constants in 0.5 M NaCI. INTRODUCTION A NEED exists for a general computerized analysis of electrochemical impedance data obtained for corroding metallic surfaces. As discussed elsewhere, 1~ electrochemical impedance measurements taken over a wide range of frequencies for many corroding surfaces show a circular arc in the complex impedance plane.* The high and low frequency intercepts of a system exhibiting a single arc define the respective solution resistance, R., and polarization resistance, R r A semicircular arc in the first quadrant, with its center on the real axis, results if the electrode surface can be modeled as a resistor, Rp, in parallel with a double layer capacitance, C a. However, experimental data for many systems give complex plane data that form circular arcs with centers lying below the real axis. l"a A program, CIRFIT, has been developed that fits complex plane data to a general circle characterized by a radius R o and center (Xo, Yo) in the complex plane. It finds the best fitting semicircle by minimizing the sum of squares of the radial distance between each observed point on the complex plane and the arc defined by R o, x o , and Yo- In addition to establishing the best-fit values of R o and (xo, Yo), the program calculates the high frequency intercept, Rs, the low frequency intercept, R s + Rp, and the residual sum of squares. The sum of squares characterizes the goodness of the fit and determines the errors in R mand Rp. Kleitz and Kennedy* have used a similar approach for cases involving more than one poorly resolved capacitive arc in the complex plane. Rather than simultaneously varying the center and radius to minimize the residual sum of squares, they preselect the center and then determine the radius as the average distance between the center and observed points. They do not explicitly present a criterion for goodness-of-fit. Others 5 have simultaneously fit the observed in-phase and out-of-phase impedance to nonlinear functions of frequency. The discussion presented here emphasizes the determination of the parameters R s and R v which have importance to corrosion analysis. Manuscript received 17 November 1982; in amended form 11 January 1983. *A typical presentation of complex electrochemical impedance takes the form of an Argand plot of the complex conjugate of the observed impedance. 1007

1008

M.W. KEI,~IOet at.

This communication will briefly describe the theoretical basis for CIRFIT and present some illustrative results obtained for corroding metallic electrodes exhibiting both one and two time constants. THEORETICAL DEVELOPMENT The expression for the complex impedance of a metallic surface exhibiting a polarization resistance, Rp, in parallel with a capacitance Cd, the combination being in series with a solution resistance, Rs, gives rise to a complex impedance: Z = R s -'l-

Rp

(1)

i +joiRpC d

which may be expressed as z

=

z'

-

jz"

(2)

where

z' = R, d- RJ(1 + (o2~l) and Z" = Rp0)l(1 + Ogr2), with ~

=

R p C d.

Algebraic rearrangement, combining z" and z', results in equation

(3): (3) Equation 3 can be recognized as that of a circle with center at xo -----Rs q- ½ Rp and Yo = 0, and intercepts at R~ and R, q- Rp on the real axis. Experimentally, however, the center may not be on the real axis. Thus, a routine designed to fit the experimental data that eliminates the constraint that Yo = 0, while retaining the definitions of Rs and R, q- Rp as real axis intercepts, requires that equation (1) be modified to contain the phenomenoiogical coefficients 13and lr:

z = R, +

Rp 1 + (jm)a

(4)

The occurrence of a depressed seimcircle requires replacing the frequency dependent t e r m j t o R p C a in equation (1) by the frequency dependent term (j¢0x)a in equation (4). Equation (4) is a unique description for the frequency response since the electrochemical response is assumed to be linear. Others have presented an analogous equation for the frequency dependence of a transfer function giving a circular arc and have rationalized the uniqueness of equation (4) for describing its frequency

A computer analysisof electrochemicalimpedancedata

1009

dependence, e The physical processes leading to this behavior could result from electrode geometry as discussed in general by Scheide: and for specific cases where ~ = 0.5 by DeLevie) In addition, electrode kinetics characterized by a finite number of closely spaced time constants can provide impedance plane curves that resemble depressed semicircles. 9 Equation (4) satisfies Kramers-Kronig criteria7 demanded of electrochemical impedance,t° A detailed theoretical discussion of the entire number of physical phenomena that can lead to depressed semicircular complex impedance curves is beyond the scope of this paper. In fact, the approach described here fits experimental data to semicircles with centers below the real axis to give the solution resistance, Rs, and the polarization resistance, Rp, without requiring a physical model that explains the depressed semicircle. The values Rp and R s have considerable importance to electrochemical corrosion rate measurements. By allowing the center of the curves in the complex plane to fall below the real axis, experimental data exhibiting one time constant can be fit to the three parameters, P'v, R= and ~ by minimizing the sum of squares of the radial distances between the observed points (z", z') and the best fit circle. Referring to Fig. 1, the sum of residual squares, D, over N data points can be calculated as: N

z

(x~ + y 2 )

(5)

xt = z~ -

xo -

(6)

Yi

Yo -- Ro sin 0i

D =

i=1

where x t and y~ can be found as:

"~i

--

Ro c o s 0 i

(7)

with cos

O,

-

z'~ -

xo

(8)

R' sinO i -

z~--yo

(9)

R r

where R ' = a/[(z~ -

Xo) ~ +

(z;-yo)']

Minimization of D with respect to Xo, Yo and R o requires that 8D ~Xo

8D ~Yo

c~D --0. ~Ro

(I0)

M . W . I¢.m~[G et al.

I010

~.112-11g73

2"

_

Z'|'i

;t°

(xo, Yo) FI¢~. 1.

Schematic for a complex p l a n e i m p e d a n c e plot.

Hence, the set of equations

~D --NXo÷Rt ~X o

~D --NYo+Ro ~Yo ~D

c~Ro

N 7. i- 1

cos0 i -

N y i.,,1

sin0 i -

N E

-- N R o ~ x o

t-I

N

Y~ z;----0

(11)

l- I N

Y

z~==0

(12)

l.-I N

cos Oi -I- Yo

N

Z t-I

sin 0 i

N

z;cosO t Iml

Z z;sinO t--O

(13)

Isl

are solved iteratively using a Newton-Raphson approach, x1.12 The iteration is continued to a preset value for a change in the sum of residual squares. The resulting minimum sum of squares is in units of (fl) -~ and represents the experimental error when normalized as:

As a criterion for the goodness-of-fit, 8 R / R o can be used. From geometric considerations, the uncertainty in R s and Rp can be determined from tiR as: ~R ~R 8R s ---= (14b)

oos( l -

cos

\

~Rp --- 28R,

\Xoll

(14c)

A computer analysis of electrochemical impedance data

I011

A P P L I C A T I O N TO E X P E R I M E N T A L D A T A

Two systems serve as examples for how CIRFIT may be used. In the first case, the a.c. impedance of an iron surface exposed to tapwater was measured using a transfer function analyzer and potentiostat as described previously.X,2 In order to test the feasibility of applying CIRFIT to a system exhibiting two time constants, the a.c. impedance for a "model pit" was also obtained. Figure 2 shows the complex plane plots for the a.c. impedance of iron in tapwater at several exposures times,x8 The complex plane data points between the high frequency limit and the point of maximum z"were selected in order to eliminatesystematic error in the low frequeny data due to drift in the corrosion rate with time. CIRFIT applied to this data set produced the best fit values of R,, Rp and ~ listed in Table 1. The uncertainties in Rp and R s have been determined from the sum of radial residual squares according to equations (14a)-(14c). Since the experimental high frequency points follow quite closely the form of a circle, ~R/R o values are all very small ( ~ 0.002) indicating good agreement of the calculated circle to these observed points. The values for ~ are less than 1.0 since the semicircles have their centers below the real axis of the complex plane. The values for Rp decrease with time indicating an increase of corrosion rates. The decrease of R, is considered to be due to the generation of dissolved corrosion products. The example of iron in tapwater results in impedance data giving one arc in the complex plane, x In certain cases, localized corrosion processes occurring on metallic surfaces give rise to an electrochemical impedance that exhibits two time constants, TAm~I. Rp, P~(in ohm) AND~ DETERMINEDnYCIRFITx~oR IRONIN TAPWATER

Time

R,

Rp

0.5 4.5 7.5

2841 -~ 2 1645 -4- 4 1287 ± 2

~

325 ± 1 297 ± 2 294 ± 1

0.847 ± 0.0004 0.851 -t- 0.001 0.858 ± 0.001

TAP WATER 2000

I

(b)

|

8R/Re 0.0006 0.0018 0.0017 SC81-13497

I

A = 1.2 cm 2 ( ~ :0.5 h ( ~ : 4.5 h (~) : 7.5 h

.

t0.31 HZ

1000

~ m ~ ~ . , . ~ . ~

~(I~

3000

Z" (OHMS) F I o . 2.

Complex plane plot for iron in tapwater. I'

40OO

1012

M.W.

K E N D I G et al.

as discussed earlier for AI alloys, z Such is the case for the MPE placed vertically in 0.5 M NaCI. The MPE was made from a 0.005 cm thick electroless Ni coating on a (2.5 × 2.5) cm 2 steel flag shaped specimen containing three 0.013 cm dia holes drilled to a depth of 0.025 cm through the Ni plate, thereby exposing an active iron surface." The side of the MPE not containing the holes was masked. The a.c. impedance of the MPE in 0.5 M NaCI exposed to air was measured for 24 h during exposure to the electrolyte. Two time constants occur which correspond to two arcs in the complex plane. The Bode plot of the impedance modulus IZI and phase angle G provides a clearer illustration of the existence of two time constants which give rise to two maxima for the phase angle, as shown in Fig. 3. A general model for the two time constant behavior appears as the inset in Fig. 3, where R, is the solution resistance, Ro is the ohmic resistance in the pit, Rp is the polarization resistance in the pit, Co is the capacitance for the passive Ni coating and Z k = Rp (.jt0z) -a is an element representing charging of the surface within the pit. Rp

Rs = 0.5;; Cd

128 ~,F

R£t= 15'.' Rp = 1135.~. ~ Cd

i

i

t

I Z"-'k /"

oo 3

105

:

=

,i

= 0.79

t

I

MPE - 3 PITS 4.5 hr$ ~ --CALCULATED • OBSERVED

75

60

A ud uJ

er 2

45

]•

coo

(3 UJ K3

30

15

-1

-"

I

1

t

7

]

0

1

2

3

4

"I 5

LOG

F I G . 3.

Bode plot for MPE in 0.5 M NaCI at 4.5 h.

The numerical values of the elements in the model of Fig. 3 have been determined using CIRFIT. The procedure takes the following steps. First, the complex plane impedance data are mapped on to the admittance plane to give a curve containing two arcs as shown in Fig. 4. This inverse transformation of impedance plane data maps the high frequency data on to a second, larger semicircle in the admittance plane (Fig. 4).

A computer analysis of electrochemical impedance data

1013

SC82-18774 I

I

1.5

MPE - 3 PITS 4.5 HRS

1.0 E

0.5

0

J 0.5

I

I

1.0

1.5

.

20

G'. mho

FIG. 4. Admittance plane plot for data from Fig. 3. The high frequency intercept at the real axis G' in the admittance plane (Fig. 4) equals 1]R s. By applying CIRFIT only to the points on the high frequency arc, an accurate determination of R, is possible. R~ is then subtracted from the impedance using a computer program to give the corrected admittance which may be expressed in the high frequency limit as: (Z

--

Rs) -1

(15)

=jo3C d + R~

This corrected complex admittance (Fig. 5), produces a line perpendicular to the real axis in the high frequency limit. From the imaginary component of the admittance, G"(¢o), on the high frequency line perpendicular to the real axis, the capacitance can be determined as: (16)

C d = G"(co)/to

Knowing Cd and R s, a further deconvolution of the impedance data can be made leaving the low frequency impedance, ZLF, attributed solely to the pit: ZLF = { ( Z - Rs) -1 --jcoCe} -1 = Rn +

Rp

1 -at- (jco'r) B

(17)

CIRFIT can fit complex plane values for the impedance ZLF described by equation 17, since the fight hand side of the equation takes the form of equation 4. The pit impedance exhibits non-ideal behavior, giving a ~ equal to 0.79, which approximate~ that for a first order branched R C network that would theoretically describe the

1014

M . W . I~NDIO et al. SC82-18775 0.6

....

I

" I

MPE - 3 P I T S 4.5 HRS

0.4

493 Hz

0.2

247

.[o

J

Hz

.I 0.2

J 0.4

I 0.6

0+8

G', rnho

FIo. 5. Admittance plane data from Fig. 3 corrected for solution resistance R,. response for an occluded surface. ~ Significantly, the ohmic resistance of the pit is about 15 f2, while the polarization resistance is 1135 f2. With realistic constants for relating/?p to the corrosion current xs, corrosion currents in the order o f 10-20 ~A within the pit are calculated. Such currents would give an ohmic drop o f less than 1 mV across the ohmic resistance in the pit, Rn. The results for the M P E illustrate the use of C I R F I T in conjunction with a deconvolution procedure for extracting meaningful data from systems exhibiting more than one well-resolved time constant. SUMMARY The computer program, C I R F I T , can extract solution resistance and polarization resistance data from electrochemical impedance data that need not show ideal capacitive behavior for the double layer. This approach provides estimates for the goodness-of-fit as 8R/Ro and can be applied to systems exhibiting two time constants where observable separation of time constants exists. REFERENCES 1. F. MANSI~LD,M. I¢,~IO and S. Ts~, Corrosion 38, 570 (1982). 2. F. MANSl~LD,Corrosion 37, 301 (1981). 3. W. J. LORENZand F. MANS~LD, Cortes. Sci. 21, 647 (1981). 4. M. KLEXTZand J. H. K ~ Y , Proceedings of Fast Ion Transport in Solids (eds. P. VASmSHTA, J. N. Mv~qVYand O. K. SH~NOY),pp. 185-188. Elsevier-North Holland, New York (1979). 5. J. R. MACDONALDand J. A. G~n.~, J. electrochem. Soc. 124, 1023 (1977). 6. K. S. CoLn and R. COLE,J. chem. Phys. 9, 341 (1941). 7. W. SCH~rDER,J. phys. Chem. 79, 127 (1975). 8. R. DELEv~, Electrochemical response of porous and rough surfaces, in Advances in EJectrochemistry and Electrochemical Engineering (ed. P. DV.LAHAY),Vol. 6, p. 329 (1967). 9. H. SCHWExCK~T,W. J. LORENZand H. FRIF.DBURO,J. electrochem. Soc. 127, 1693 (1980). 10. R. L. VANM£IRHAEOH~,E. C. DUTOIT,F. C~RDONand W. P. Gob~s, Electrochim. Acta 20, 995 (1975). 11. P. HENR[CI,Elements of Numeric Analysis, Wiley, New York (1964).

A computer analysis of electrochemical impedance data

1015

12. D. H. K~LSt~, F. B. M A ~ , M. K~NDIO and C. L~UNG, Study Prosram for Encapsulation Materials Interface for Low-Cost Solar Array, Annual Report, SC5106.104AR (February 1981). 13. F. MJocS~J.D, M. I¢~,~DIGand S. TSAI, Corros. Sci. 22, 455 (1982). 14. M. ~ I o and F. M A ~ L D , AC Electrochemical Impedance of a Model Pit 162rid Meeting, The Electrochemical Society, Detroit, Oct. 1982, Abstract No. 64. 15. F. MAWS~LV,in Advances in Corrosion Science and Technology, Vol, 6, p. 163 Plenum Press (1976).