darowicki - 1995 - the amplitude analysis of impedance spectra

8/6/2019 Darowicki - 1995 - The Amplitude Analysis of Impedance Spectra

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! PergamonElectrochtmrco ACIU. Vol 40, No 4. 439p. 445. 1995

Copyright ,c 1995 Elscwer Saence LtdPnnted in Great Britan. Allnghrs eserved

00134686195950+ 0.000013-46%(94)00303-3

THE AMPLITUDE ANALYSIS OF IMPEDANCE SPECTRA

KAZIMIERZ DAROWICKIDepartment of Anticorrosion Protection Technology, Faculty of Chemistry, Technical University of

Gdatisk, 80-952 Gdafisk, ul. Narutowicza 1 /12, Poland

(Received 5 July 1993; in revised form 4 July 1994)

Abstract-The amplitude analysis of a non-linear model of an electric system has been carried out. Thepossibility of linear and non-linear electric element differentiation has been stated on the basis of imped-ance measurements. The simultaneous frequency and amplitude analysis of impedance spectra allowedthe determination by extrapolation of electric element values corresponding to the zero value of theperturbation signal. The amplitude analysis of impedance spectra allows the determination of the effectiveamplitude value in an accurate way. The determination of the polarization resistance dependence on thefunction of the effective amplitude of the perturbation signal is the static characteristic of a tested system.

Key words: non-linear impedance, harmonic analysis

INTRODUCTION

Classic impedance investigations are carried out inthe conditions of stability, causality and linearity ofthe tested electrochemical system. Electrochemical

systems are non-linear systems. The linearity condi-tion is realized by applying low amplitude sine per-turbation signals with a sequentially changingfrequency. As a result, the electrochemical process inthe vicinity of the point at which impedance mea-surements are carried out, is described with linearequations. The carrying out of impedance measure-ments in linear conditions significantly simplifies themeasurement methodology and the frequencyanalysis of impedance spectra. It also allows the neg-ligence of the effect of the perturbation signal on thephysicochemical state of the tested system, and thusensures the stability condition. The use of highamplitude sine perturbation signals causes the neces-sity of the taking into account of furhter harmoniccomponents, apart from the basic component. Thiseffect is used in the Faradaic rectificationtechnique[ l-63 and the Fardaic distortion tech-nique. The determination of each harmonic com-ponent of the response signal of the tested systemallows its full characteristic.

The analysis of each harmonic component hasbeen carried out independently by Bertocci[ lo],Callow and coworkers[ll, 123, Deavy andMeszaros[ 131, and Gill et a/.[ 141.

The method of non-linear impedance measure-ments, proposed in his work, is different frommethods discussed in the literature and is based onthe analysis of the fundamental-harmonic com-ponent of the response signal of the tested system.The two-channel method of impedance measure-ments applied in ttransmittance analyzers allows thedetermination of the fundamental-harmonic com-ponent of the response signal. As a result, the appli-cation of the high amplitude sine perturbation signal

allows the determination of the fundamental-harmonic impedance, which is the function of theamplitude and frequency of the perturbation signal.The relationship between the perturbation signalamplitude and the fundamental-harmonic impedance

is another matter. The knowledge of this relationshipwould allow the determination by extrapolation ofimpedance parameter values, corresponding to thezero value of the sine voltage perturbation signalamplitude value.

The determination of the amplitude characteristicsof each element of the substitute electric schematicdiagram would allow the differentiation of linear andnon-linear electric elements. Additionally, the ampli-tude analysis of polarization resistance should enablethe determination of the static characteristic of thetested electrochemical system.

In this work I have attempted to describe the non-linear impedance, determined on the basis of the fun-damental harmonic analysis of the response signal ofthe investigated system. The suitability of the simul-taneous frequency amplitude analysis of impedancespectra has been checked for the complex character-ization of the tested system.

IMPEDANCE OF THE FUNDAMENTALHARMONIC

Transmittance analyzers are generally used forimmittance measurements of electrochemical pro-cesses. Two signals are generated in the transmit-tance analyzer[ 15, 161: the proper perturbationsignal, and the signal generated in parallel, shifted inphase by a n/2 angle, being the reference signal. Inthe multiplying block of the first channel theresponse signal of the investigated system is multi-plied by the proper perturbation signal. In the multi-plying block of the second channel the response

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440 K. DAROWICKI

signal of the investigated system is multiplied by thereference signal.

In the integrating blocks the signal in the firstchannel and the signal in the second channel areintegrated.

The multiplication and the integration of the

response signal by the perturbation signal and thereference signal eliminates its higher harmonics.Only the rspose signal fundamental harmonic of theinvestigated system is analyzed, which is divided intoa component in phase and a component shifted by an/2 angle in relation to the perturbation signal. As aresult, on the output of both analyzer channels,quantities are obtained, proportional to the real andimaginary parts of the investigated system immit-tance. To recapitulate, the two-channel analysis ofmeasurement signals allows the separation of thefundamental harmonic. The multiplying and averag-ing operations of the measurement signal in bothchannels are equivalent to filtration operations in anarrow-band filterC15, 163.

In classic electrochemical impedance spectrometrythe amplitude of the perturbation signal is so smallthat the response signal of the investigated electro-chemical system is proportional to the perturbationsignal. The immitance of the investigated system isthe proportionality coefficient, which is generally afunction of the perturbation signal frequency. Thelinearity condition is fulfilled if the measured immit-tance does not depend on the amplitude of the sineperturbation signal[15, 171.

The situation is different if the amplitude of thesine perturbation signal is so large that the investi-gated non-linear system cannot be estimated by alinear approximation. In this case the measuredimmittance is a function of the frequency and theamplitude of the sine perturbation signal.

Let us consider the immittance of the electrodeprocess, the stationary characteristic of which isdefined by the general equation:

i = i(E), (1)

where: i = current, E = potential.

The excitation of the analyzed non-linear systemby a signal described with a relation:

AE(wt) = E(t) - E, = AE, cos wt, (2)

where: AE(ot) = the voltage perturbation signal,E, = the bias potential, AE, = the amplitude of thesine voltage perturbation signal, w = the angular fre-quency, t = time, causes a direct current flow:

i, = i(E,). (3)

The alternating current of the electrode process maybe presented in the form of a Taylor power series:

For a sufficiently small amplitude of the sine voltageperturbation signal the alternating current is approx-imated by the linear part of equation (4). As a result,the immittance of the investigated procss depends onthe amplitude of the sine voltage perturbation signal.The application of higher amplitudes of the sine

voltage perturbation signal requires the taking intoaccount of higher terms than linear of the powerseries in the description of the alternating current (4).In this work we have limited the analysis to thirdorder terms. Equation (4) can be resolved in accord-ance with trigonometry principles into the currentcomponent independent of the frequency called theFaradaic rectification current[ 18-203, the fundamen-tal harmonic component and further harmoniccomponents[18,20] :

AI(t) z i,,, + Ai + Ai(2wt) + Ai(3wt) + . . (5)

The faradaic rectification current is equal to :

AE;.,

Equation (6) has been derived on the basis of equa-tion (4), with only third order terms taken intoaccount. The taking into account of further terms ofthe power series causes the necessity of using thenext quaternary term describing the faradaic rectifi-cation current:

AE;.

The range of sine voltage amplitudes of the pertur-bation signal, for which equation (6) is fulfilled, isstrictly connected with the potential region (E,- AE,, E, + AE,), for which equation (1) is fulfilled.

The fundamental harmonic component of the elec-trode process current is given by equation (7)[21]:

Ai z[( >

$ AE,ES

+;(-&$AE;+...]coswt. (7)

The two-channel analysis of the response signal ofthe investigated system allows the separation of thefundamental alternate current harmonic. Theremaining harmonic components are filtered away.As a result, the polarization resistance of the investi-gated electrode process is determined by therelationship :

Equation (8) is the static characteristic of theinvestigated system. The knowledge of this charac-teristic and the frequency characteristics allow thedetermination of the transmission properties of agiven process. In reference to electrochemical pro-cesses, the knowledge of frequency and static charac-teristics enables the unequivocal determination of


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Amplitude analysis of impedance spectra 441

the mechanism and kinetics. The impedance mea-surements of the investigated electrode process as afunction of the amplitude of the sine voltage pertur-bation signal enable the determination of the valuesof the first and third current derivatives in relation tothe potential, determined for the potential E, . Addi-tionally, the determination of the second currentderivative in relation to the potential and the valuesof the current for potential E, are possible on thebasis of the direct current as the function of theamplitude of the sine voltage of the perturbationsignal.

Let us consider the non-linar model of an electricsystem presented in Fig. 1. This system is character-ized by two time constants. It contains one non-linear element D, and three linear elements C,, C,and R,. From the electrochemical point of view theelectric system can be treated as a non-linear, simpli-fied analogue of a substitute electric diagram of atwo-step electrode reaction with adsorption of theintermediate product. As a result, the C, capacitorrepresents the capacitance of the electric doublelayer. The R, resistor and the C, capacitor representthe adsorption process of the intermediate product.The R, resistance represents the charge transferprocess. The connection of a diode in the model elec-tric system in the conductivity direction causes adirect current flow, which is an exponential functionof the voltage:

i, = i exp(bE,) (9)

where b = the voltage constant of the diode andi = the current constant of the diode.

The faradaic rectification current is equal to [7] :

(10)

The resistance of the diode is given by the relation-ship :

1- - i i , b+cAEi+....ROW,)

(11)

Taking into account the remaining electric elementsand introducing the imaginary module, the imped-ance of the analyzed non-linear electric system iseasily determined :

ZCjw, AR,,) = R, +1

1joC, + -

jwc,

R,(AE,)+ 1 +jwC, R,

(12)where: j2 = - I and AE,, = the generated ampli-tude of the sine voltage perturbation signal.

, , C,=2OpF

Fig. 1. The investigated non-linear electric system. C,,C,--capacitors, R,, R,-resistors, D,4iode.

The effective amplitude of the perturbation signalrequires additional comment. When determining theR, resistances, it has been assumed that the gener-ated and effective amplitude of the sine voltage per-turbation signal are qua]. In reality, due to theprsence of the R, resistance describing the electrolyte

resistance, the effective and generated amplitudesshould be differentiated. To do this, the sine voltageperturbation signal diagram, presented in Fig. 2,should be analyzed. The amplitude of the generatedsine voltage perturbation signal in the transmittanceanalyzer system is equal to AE,,. The amplitude ofthe effective sine voltage perturbation signal is deter-mined by the relationship:

AE,(wt) = Zow AEo,) - R, AEZow, AE,,)

= AR;(w) + jAEb/(w) (13)

where: AEb(w) the real part of the effective amplitudeof the sine voltage perturbation signal, AER(w) theimaginary part of the amplitude of the sine voltageperturbation signal and R, the electrolyte resistance.

The amplitude of the generated sine perturbationsignal AE,, is a real quantity. The effective ampli-tude of the sine perturbation signal AE,(wt) is acomplex quantity. The magnitude of the effectiveamplitude of the sine voltage perturbation signaldepends on the amplitude of the generated sinevoltage signal and on the relation between theimpedance of the investigated system Z(jw, AE,,)and the R, electrolyte resistance. For limiting lowfrequencies equation (13) takes the form :

R2lim AE,(wt) = AE, = ___

&+R2AEo, (14)

0-0

where: AE, = the effective amplitude of the sinevoltage perturbation signal determined for limitinglow frequencies.

The polarization resistance of the investigatedsystem is defined by the relationship:

R,(AE,) = lim Z(jw, AE,,) - lim Z(jw, AE,,)w-0 w 'L

(15)

The polarization resistance of the non-linear elec-tric system is equal to the resistance of the diode forthe considered model.

Fig. 2. The schematic presentation of the distribution ofthe amplitude of the sine voltage perturbation signal.A&o-the amplitude of the generated signal, AR,-theeffective sine voltage perturbation signal. Z(jo)-the imped-

ance of the investigated process.


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442 K. DAROWICKI

A relatively high value of the electrolyte resistancein relation to the polarization resistance affects, in abasic way, the value of the effective amplitude of theperturbation signal. To obtain a correct static char-acteristic in the form R =f(AE,), it is essential totake into account the effect of the electrolyte resist-

ance on the value of the effective amplitude of theperturbation signal.

EXPERIMENTAL

The analysis has been carried out of the equivalentcircuit, presented in Fig. 1. The impedance measure-ments have been carried out in the lOmHz-1OkHzfrequency range, using a measurement assemblymade up of the ATLAS 8511 transmittance analyzerand the ATLAS 8531 potentiostat. The value of theamplitude of the sine voltage perturbation signal waschanged in the 14-86mV range. The amplitude ofthe sine voltage perturbation signal was controlledwith an oscilloscope. The polarization investigationsof model electric systems were carried out using apotential scan rate of 1 mV s- .

RESULTS AND DISCUSSION

The polarization curve of the analyzed non-linearelectric system is presented in Fig. 3. Curve A corre-sponds to the polarization relationship without

elimination of the ohmic drop iR,. Curve B is thepolarization characteristic of the analyzed systemafter ohmic drop iR, elimination. In accordance withexpectations the real relationship of the logarithm ofthe current from the potential is a straight line. Thedirection coefficient of the determined relation ini =f(E) is equal to the diode voltage coefficient.

In points A (E, = 0,400), B (E, = 0,500), C (E, =0,600) and D (E, = 0,700) of the polarization depen-

-4.01

-6.0 -f

400 600 600

E/mVFig. 3. The polarization characteristic of the investigatednon-linear electric system. (A) with no IR, ohmic drop cor-

rection. (B) with iR, ohmic drop correction.

00~~0 14 mVa=*** 20 mV ~~nn42 mVmm ... 56 mVa*.&& 70 mv

lOOO- l &AA04 mV

$ 8 .o 0

19 : : ;Aa b.p*o

500: Nh I an

: 8 . a0 l

_ .. :. .O

.i :&$

0 - ~ ,I.III~I,,,I,,,,,,I,,,,,,,,,~0 500 1000 1500 2000

Re(Z)/OhmFig. 4. The impedance spectra of the investigated non-linear electric system detrmied for E, = 0.4OOV and each

amplitude of the sine voltage perturbation signal.

dence the non-linear impedance investigations havebeen carried out. In Figs 4-7, the impedance spectraof the investigated electric system have been present-ed, determined for various amplitudes of the sinevoltage perturbation signal and different E, poten-tials. The impedance spectra are characterized bytwo time constants and take the form of two capac-

Re(Z)/Ohm

Fig. 5. The impedance spectra of the investigated non-linear electric system determined for E, = 0.500 V and eachamplitude of the sine voltage perturbation signal.

Re(Z)/OhmFig. 6. The impedance spectra of the investigated non-linear electric system determined for E, = 0.600 V and each



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20.00 21 .oo 22.00 23.00 24.00 25.00

Re(Z)/Ohm

Fig. 7. The impedance spectra of the investigated non-linear electric system determined for E, = 0.7OOV and each


itance semicircles. A decrease of the polarizationresistance is observed, together with an increase ofthe amplitude of the sine voltage perturbation signal.

The impedance spectra determined for each ampli-tude of the sine voltage perturbation signal has beensubjected to frequency analysis. The frequencyanalysis of the impedance spectra has been carriedout on the basis of the EQUIVCRT.PASalgorithm[22], proposed by Boukamp. In accord-ance with expectations, the C,, C, and R, values,determined on the basis of frequency analysis, do notdepend on the amplitude of the sine voltage pertur-

bation signal because they are linear elements. Astrong dependence from the amplitude of the sinevoltage perturbation signal is exhibited by the R,

D)0.240

E

Yi

cz 0.220 y.

0.200

0.00 2.00 4.00

lObE-/V 103AEoa/V2

diode resistance, which is equal to the value of thepolarization resistance. The dependence of theinverse of the perturbation resistance from thesquare of the amplitude of the sine voltage pertur-bation signal has been presented in Fig. 8. In accord-ance with the derived equation (11) the obtained

relations take the form of straight lines. The direc-tion coefficients of these straight lines and the inter-cept values enable the determination of the diodevoltage coetlicient. The values of the diode voltagecoefficients and the values of the polarization resist-ance have been gathered in Table 1.

The impedance spectra corresponding to the zerovalue of the sine voltage perturbation signal can beobtained by determining, by extrapolation, thedependence of the real and imaginary part of imped-ance from the amplitude of the sine voltage pertur-bation signal for each frequency. In reference toelectrochemical systems, the spectra correspond tothe free course of electrode processes. The extrapo-lated impedance spectra of the model electric systemare presented in Fig. 9. The extrapolated impedancecorresponds to the spectral power density, which is

Table 1. The values of the diode voltage coefficients deter-mined by the method of non-linear impedance spectros-

copy and the method of potentiostatic polarization

&IV 0.4 0.5 0.6 0.7

b/5- NIS 19.80 19.82 20.12 20.31Pot. P. 19.80

NIS-Non-linear impedance spectroscopy.Pot. P.-Potentiostatic polarization.

4.60

3.60 10.00 2.00 4.00

10 3A E /I

Fig. 8. The dependence of the inverse of the polarization resistance and the square of the amplitudedetermined for each E, value. (A) E, = 0.4OOV, (B) E, = 0.500 V, (C) E, = 0.600 V, (D) E, = 0.700 V.


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444 K. DAROWICKI

AI

~~~ Re(Z)/Ohm $_0 Re(Z)/Ohm

D)

lim Re(Z)/OhmAr-0

lim Re(Z)/OhmAE-0

Fig. 9. The extrapolated impedance spectra determined for each E, value. (A) E, = 0.4OOV, (B) E, =OSOOV, (C) E, = 0.6OOV, (D) E, = 0.7OOV.

determined bytechnique[23].

the electrochemical noise

SUMMARY

The carried out non-linear impedance investiga-tions of the model electric system simulating theelectrode process confirms the correctness of thetheoretical analysis. They also confirm the possibilityof using transmittance analyzers in non-linearimpedance investigations. The immittance of theanalyzed electric system depends on the frequencyand the amplitude of the sine voltage perturbationsignal. Non-linear impedance investigations enablenot only the frequency but also the amplitudeanalysis of immittance spectra. The amplitudeanalysis of immittance spectra enables the differen-tiation of linear and non-linear quantities describingthe investigated electrode process. Immittance mea-surements allow the facile elimination of the electro-lye resistance and the determination of the effectiveamplitude of the sine voltage perturbation signal.

The dependence of the inverse of the polarization

resistance on the square of the amplitude of the sinevoltage perturbation signal is the static characteristicof the investigated electrode process. The character-istic is determined in the potential range from E,- AE, to E, + AE, and is the differential form of

the polarization characteristic i =j(E). It is possibleto determine the values of each electric element cor-responding to the zero value of the sine voltage per-turbation signal, based on the amplitudecharacteristic. The values determined in this way

correspond to the free course of the electrodeprocess.

The presented considerations and measurementsconcerned the non-linear model of an electric system,simulating an electrode process. The stability condi-tion of the investigated electric systems was thus ful-filled. The effect of the magnitude of the amplitude ofthe perturbation signal on the physicochemical stateof the system, and thus its stability, is a differentmatter. This matter will be discussed in the followingarticles on the amplitude analysis of impedancespectra of specific electrochemical systems.

Acknowledyements-This work was financed by theBW93904/047 grant.

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5.6.7.

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9.

10.

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