Fundamental-harmonic impedance of first-order electrode reactions

Elecrrochtmica

1Pergamoa

Acta, Vol. 39, No.

18, pp. 2157-21621994

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FUNDAMENTAL-HARMONIC IMPEDANCE OF FIRST-ORDER ELECTRODE REACTIONS KAZIMIERZDAROWICKI Department of Anticorrosion Protection Technology, Technical University of Gdabsk, 80-952Gdahsk,

ul. Narutowicza 1l/12, Poland (Received 12 January 1994; in revisedform 4 July 1994) Abstract-A theoretical description of the nonlinear impedance of a first-order electrode reaction has been presented. The dependencies of the charge transfer resistance and the Warburg impedance on the amplitude of the sinusoidal voltage perturbation have been determined. The amplitude analysis of the charge transfer resistance and the Warburg impedance enables determination by extrapolation of the charge transfer resistance and the Warburg impedance for the zero amplitude. Additionally, the amplitude analysis of the charge transfer resistance enables the determination of the global charge transfer coellkients of the cathodic and anodic processes. The amplitude analysis of the Warburg impedance allows determination of the diffusion coeffkients of the Ox and Red forms.

Key words: harmonic analysis, nonlinear impedance, first-order electrode reaction. INTRODUCTION Sinusoidal

current

or voltage

signals are commonly

used in investigations of electrode processes. The pioneer works of Breyer and Gutmann[l-31, Grahame[4], Ershler[S], Randles[6] and others created the basis of the elaboration of alternating current polarography. At present it is one of the more important electrochemical techniques. Basically, the ac polarographic experiment depends on the determination of the impedance of an electrolytic cell under polarographic conditions while applying a small-amplitude (< 10mV) alternating electrode potential. Theoretical research in UCpolarography or ac voltammetry is predominantly restricted to small amplitudes of the superimposed alternate voltage, as for higher amplitudes the rigorous mathematical treatment becomes extremely complex and the resulting formulae become highly unsolvable. The first contribution was from Matsuda[7], in which attention was paid even to growth of drops. Though his results were valid over a wide range of amplitudes, no explicit expression was presented with respect to the different harmonics present in the output signal. Smith[I] followed another way, and he obtained a solution to the problem in the form of a series. However, the solution corresponding to the fundamental frequency does not seem to be practical for amplitudes beyond 35/n mV, because of the weak convergence of the series. An identical problem was analysed by Mooring and Kies[9]. The transport of the reagents of a firstorder reaction was described by linear diffusion laws and the ac output signal took into account six harmonic components. In recent years there has been increased interest in the practical application of investigations of electrode processes using high amplitude perturbations.

Contributions of Bertocci[lO], Bertocci and Mullen[ 111, Devay and Meszaros[ 121, Gill et al.[ 141 and others[15-181 should be mentioned. The application of a high-amplitude sinusoidal perturbation in the case of impedance investigations leads to a situation where each electric element describing the electrode process depends on the amplitude of the sinusoidal perturbation. The determination of the character of this dependence makes estimation by extrapolation possible, of the electrode impedance corresponding to the zero value of the amplitude of the sinusoidal perturbation. Additionally, the simultaneous frequency and amplitude analysis of impedance spectra facilitates the unequivocal determination of the mechanism and kinetics of the investigated electrode process. The mentioned aspects of non-linear impedance measurements have been discussed by Darowicki[19], with respect to a model non-linear electric system. In this paper the description of non-linear impedance of a first-order electrode reaction has been presented. The description allows the determination of the basic extrapolation formulae: the dependencies of the charge transfer resistance and the Warburg coellicient from the amplitude of the sinusoidal voltage perturbation. The theoretical description of nonlinear impedance carried out should enable the determination, by extrapolation, of the charge transfer resistance and the Warburg impedance of the zero perturbation. Additionally, the derived extrapolation relations should enable the direct determination of the general charge transfer coefficients, the rate constants of the electrode reaction and the diffusion coefficients. The approach presented to the problem of non-linear impedance differs from that published in the literature[20]. The impedance is the fundamental-harmonic impedance, which it is possible to determine as the result of a two-channel measurement of the response signal[21].

2157

K. DAROWICKI

2758 MATHEMATICAL DESCRIPTION DISCUSSION

AND

The first-order electrode reaction is schematically described by the equation: VC

Ox + ne - -Red, VA

(1)

where Vc is the reduction rate, V’ is the oxidation rate, n is the number of exchanged electrons, Ox is the oxidized form, and Red is the reduced form. It is assumed that the electrode reaction reagents are not adsorbed on the surface of the electrode. Substance Ox reaches the electrode by diffusion. The formed substance Red is soluble in the electrolyte solution or the material electrode. Transport of this substance proceeds in accordance with diffusion laws. Additionally, the condition of electroneutrality is fulfilled. For such a defined process the general equations describing the reduction and oxidation rates are valid : v, = KG, cd,

v* = UC

(2)

d,

where c,, is the concentration of substance Ox, cR is the concentration of substance Red. By expanding equation (2) in a Taylor series, one obtains a relation describing the changes of reduction and oxidation reactions caused by the voltage perturbation. The expansion in the Taylor series has been limited to third-order terms:

where AVdt) are changes in reduction rate, AV,(t) are changes in oxidation rate, AE(t) is voltage perturbation, Ac,,(O,t) are changes of substance Ox concentration on the electrode surface, Ac,(O, t) are changes of substance Red concentration on the electrode surface, and t is time. The electrode reaction is of the first order, thus the partial derivatives (aY+“V/dE”W) for u > 1 are equal to zero. Substance Ox reaches the electrode surface by diffusion. The transport of substance Red also occurs by diffusion. Thus, concentration changes of substances Ox and Red are described by the second Fick equation:

aAco(x, t) _ at

AW) + (z)EAco(O,

0

ax2 ’

d2Ac,(x, t) aA&, 0 _ R ax2 ’ at D

(5)

where Do, DR are diffusion coefficients of Ox and Red forms, x is the spatial coordinate. For the sinusoidal voltage perturbation: AE(t) = AE, cos ot,

(6)

where AE, is the amplitude of the sinusoidal voltage perturbation. The solution of the second Fick equation for substance Ox takes the form: Aco(x, t) = i

exp (

m=l

A&#) = @,

a2Ac,(x, t)

D

Js)

t) x [Ao.cos(mwr

azv, AE2(t) +i (> dEZ,,

-Ex)

I

+

B,,

sin(mwt -&x)1,

(7)

where Ao, , B,, are integration constants. The solution of the Fick equation for the Red substance takes the form: AcR(x, t) = i

Ill=, (3) AV’(t) = (z)C,

AW) + (2)B

aw, +2 (> aE2_ I

Ad4

exp ( &)

X pRmcos(m+x)

t) + B,sin(mwr

-/zx)],

@)

AE’(t)

(4)

where AR,,,, B,,,, are mtegratlon constants. The expansion of the reduction and oxidation reaction into a Taylor series has been limited to third-order terms. As a result, the solution of the Fick equation for Ox and Red substances has been limited to the third harmonic. Equations analogous to equations (7) and (8) have been used by Carslaw and Jaeger[22] to describe the transport of heat. Mooring and Kies[9] used the Fick equation solutions in the form of a Fourier series to describe high-amplitude UCvoltammetry.

2159

Fundamental-harmonic impedance The changes of Ox and Red substances electrode surface are given by equations:

on the

Ac,,(O,t) = i Ac,(O, mot) m=l A,, cos mot + B,, sin mot),

(9) -

AC,@,t) = m&A~a(O, mot) =

m&t.cos mot + B,,

sin mot).

(10)

Equations (9) and (10) are Fourier series and are the first boundary condition for concentration waves of substances Ox and Red. The second boundary condition for substances Ox and Red is given by relations :

J

+(A,,

- B,,) sin ot,

where AV(AE,, wt) = AVdAE,, wt) - AV*(AE, , @-the fundamental-harmonic of the resulting electrode reaction rate. Equation (14) allows the facile determination of A,, and B,, integration constants, and simultaneous determination of the fundamental-harmonic of the resulting electrode reaction rate. The presented method of A,, and B,, integration constant determination is not accurate because higher than thirdorder terms have been ignored. This simplification is justified because higher than third-order terms complicate the form of obtained relations. The effect of these terms on the amplitude of the fundamentalharmonic of the resulting electrode reaction rate is insignificant. The carrying out of impedance measurements in non-linear conditions causes the first harmonic of the electrode reaction current, given by relation (15), to be a function of the amplitude and the frequency of the sinusoidal voltage perturbation:

AC&L t)x=0(11) 1 bdx, 0x=0(12) 1

A&(t) - AV, = D, ax [

AV,(t) - Al” = D, ax [

By connecting equations (11) and (12) one obtains the relations between the integration constants:

@ham+ @horn= 0, ~&ln + &Born = 0.

(13)

The expansion into the Taylor series of the reduction and oxidation rates may easily be changed into the Fourier series, taking into account that the perturbation is a sinusoidal voltage signal given by equation (6) and the Ox and Red form concentration waves on the electrode surface are given by relations (9) and (10). The reduction and oxidation rate changes can be resolved into each harmonic component. The taking into account of equations (1 l)(13) allows the determination of integration constants Ao,, AR,,,, Bo,, BRm. The integration constants determined in this way are described by very complicated expressions. As a consequence, the relations are of little practical use. Much simpler expressions describing the integration constants are obtained by omitting higher than third-order terms. In this case the fundamental-harmonic of the resulting electrode change rate is equal to:

Ai(AE,, wt) = nF AV(AE,, of),

(15)

where F is the Faraday constant. In the measurement system the fundamentalharmonic of the electrode reaction current is resolved into the in-phase component and a component shifted by a n/2 angle in relation to the sinusoidal voltage perturbation. The in-phase component, in other words the real part of the fundamental-harmonic, is determined by the cosine function. The component shifted by a x/2 angle is the imaginary part of the fundamental-harmonic and is determined by the sine function. To shorten the notation the following designations are introduced:

The real component Ai’(AE,, w) and the imaginary component Ai”(AEo, o) of the first harmonic of the electrode reaction current are described by the rela-

2760

K. DNU3WICKI

tions given below : Ai’(A&, w) = + p - q) AE,

nFr,_&&/~

-2~PvmG+~o~+wNP+~~l

(16)

Ai”(AE,, w) = nFr,,h&i&

- q) AE,

(17)

-2[P@z+D,o+@-qMP+q)l’

The knowledge of the real and imaginary components of the first harmonic of the electrode reaction and the amplitude of the sinusoidal voltage perturbation allows the determination of the electrode reaction impedance :

where Rcr(0) is the charge transfer resistance determined for the limiting low amplitude of the sinusoidal perturbation. The intercept value of relation (21) is equal to the charge transfer resistance for the zero perturbation. The angle of inclination of relation (21) allows the determination of the global charge transfer coellicients for the cathodic and anodic reactions. The diffusion resistance of Ox and Red substances are characterized by the Warburg impedance. This impedance describes the reagents transport through infinite-length diffusion layer. The real part of the Warburg impedance is equal to the imaginary part. The quantities are given by the equation: Zw(AE,, co)= Z;(AE,

Ai’(AE,, co)AE, Z’(AE, 94 = CAI”tAE ,,, co)]’ + [Ai”(AE,, o)]’ = Rc,JAE,) + Zw(AE,, o),

(18)

Ai”(AE,, co)AE,

= Z&(AE, , w),

(23)

=-F’

Z”(A& $4 = CAI’fAEo, co)]’ + [Ai”(AE,, a)]’ (1%

where Z’(AE,, o), Z”(AE,, o) are the real and imaginary components, respectively, of the electrode reaction impedance, Zw(AE,, CD), Zw(AE,, w) are the real and imaginary components of the Warburg impedance, and R,(AE,) is the charge transfer resistance. The extrapolation of the real component of impedance for frequencies approaching infinity allows the determination of the charge transfer resistance:

where W(AE,) is the Warburg coefficient. In reality, the thickness of the diffusion layer is finite. In effect, the Warburg impedance described by equation (23) does not characterize the impedance spectrum for low range frequency. The minus sign, as in the Warburg impedance, results from the adopted sign convention, in accordance with which the cathodic current is positive. For small amplitudes of the sinusoidal voltage perturbation the expression describing the Warburg impedance can be reduced to the form: Z&(0, w) = Z&(0,0)

RJAE,)

= lim Z’(AE,, w) = - -&. uo-m

(20)

The negative value of the charge transfer resistance results from the adopted sign convention, in accordance with which the cathodic current is positive. Equation (20) can be presented in another form:

The inverse of the charge transfer resistance is a parabolic function of the amplitude of the sinusoidal voltage perturbation. As the result, the amplitude analysis of impedance spectra enables the determination, by extrapolation in accordance with formula (21), of the charge transfer resistance for the zero value of the sinusoidal perturbation. The extrapolated value of the charge transfer resistance is given by the well-known relation[$ 6, 21,23-25](22): -&,.(O) = -

lim Rc,-(AEo) A,%+0

= nF[(,..‘-

(z)cd’

(22)

(24) where Z&(0, w), Z&(0, w) are the real and imaginary parts of the Warburg impedance determined for limiting low amplitude of the sinusoidal perturbation. Equation (24) is the limiting form of equation (23) and is a well-known relation describing the linear Warburg impedance[21,23-25-j. BY transforming relation (23) the following expres-

AE;.

(25)

Relation (21) enables the determination of the general charge transfer coefficients and the reduction and oxidation reaction rate constants. The determination of the intercept value and the the dependence inclination degree with simultaneous W@EoY%Wo) =W:;f

Fundamental-harmonic

Fia. 1. Equivalent electrical circuit of fundamentalharmonic-impedance of first-order electrode reaction. C,, is the double layer capacity, Rer(AE,) is the charge transfer resistance, Zw(AE,, jo) is the Warburg impedance.

of l/R&A&) =f(AEg) allows the direct determination of the diffusion coefficient of the Ox form or the diffusion coefficient of the Red form. The high-amplitude sinusoidal voltage perturbation causes a faradaic current flow, which is the sum of all harmonics. At the same time a nonfaradaic current flows, connected with the charging and discharging of the capacitance of the electric double layer. If the capacitance of the electric double layer does not depend on the amplitude of the sinusoidal voltage perturbation, then the Randles substitute schematic diagram presented in Fig. 1 corresponds to the first-order reaction. The substitute electric schematic diagram has been determined for model conditions, assuming the independent courses of the faradaic and non-faradaic processes. In reality there is a coupling between the faradaic and non-faradaic processes[26-281. Impedance measurements in a wide frequency range in the function of the amplitude of the sinusoidal voltage perturbation create the possibility of investigating the mutual interference of both processes. knowledge

SUMMARY Classical impedance measurements are carried out in linear conditions. The linearity condition is fulfilled if the measured impedance does not depend on the amplitude of the sinusoidal perturbation. The frequency analysis of electrode impedance in linear conditions allows the determination of the charge transfer resistance, independent from the amplitude of the sinusoidal perturbation, as well as the Warburg impedance and the capacitance of the electric double layer. Impedance measurements in non-linear conditions allow the simultaneous frequency and amplitude analysis of impedance spectra. The amplitude analysis of the charge transfer resistance allows the determination by extrapolation of the charge transfer resistance value for the zero perturbation. Additionally, the determination of the charge transfer resistance dependence from the amplitude of the sinusoidal voltage perturbation allows the determination of the cathodic or anodic charge transfer coefficient. The determination of the R&A&) and W(AE,)/R&AE,) dependencies from the amplitude of the sinusoidal voltage perturbation allows the establishment of the diffusion coefficient values of the Ox form or the Red forms.

2761

impedance

To recapitulate, the simultaneous amplitude and frequency analysis allows the direct and unequivocal determination of the general charge transfer coefficients, the Ox and Red form diffusion coefficients and the cathodic and anodic process rate constants. The parameters of the first-order electrode reactions, determined by extrapolation, correspond to the zero value of the perturbation. First-order electrode reaction is non-linear sinusoidal perturbation process. High-amplitude causes the formation of zero harmonic of faradaic current. It is called rectification current. Simultaneously, the effect of amplitude on measured immittance is observed. These two effects are the consequence of non-linearity of the first-order electrode reaction. Acknowledgement-This BW 9390411047.

work has been financed by grant

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