The mirage effect: A sensitive probe for electrochemical cell calorimetry

119

J. Electroanal. Chem., 346 (1993) 119-133

Elsevier Sequoia S.A., Lausanne

JEC 02398

The mirage effect: a sensitive probe for electrochemical cell calorimetry J.M. Rosolen

l, M. Fracastoro-Decker

and F. Decker ’

Institute de Fisica, Unicamp, Campinas, SP (Brazil)

(Received 6 April 1992; in revised form 30 July 1992)

Abstract

Measurements of the mirage deflection in an electrochemical cell, where Cu electrodeposition and the ferro/ferricyanate reaction occurred, were performed in the dark under a potential step and in ac conditions with the aim of detecting thermal phenomena due to Joule and Peltier effects at the interface. A mathematical model based on the diffusion of thermal and ionic waves into the electrolyte allowed simulation of the mirage deflection in all cases. The experimental conditions under which the model is valid and the thermal effects can be separated from the ionic effects were discussed. Finally, the Peltier coefficient of the ferro/ferricyanate redox couple could also be measured using the ac technique.

INTRODUCTION

Refractive index changes in a quiescent electrolyte produced by an electrochemical reaction at a planar solid electrode have been detected both interferometritally [l] and by probe beam deflection (mirage effect) [2]. Although the probe beam deflection technique can be extremely sensitive to small temperature gradients in a liquid, and therefore is widely applied as a spectroscopic technique for weak absorption bands and thin-film materials [3-51, only few researchers have attempted to use the mirage effect as a calorimetric method to follow heat transfer

l

Present address: Dipartimento

0022-0728/93/$06.00

di Chimica, Universita “La Sapienza”, Rome, Italy.

0 1993 - Elsevier Sequoia S.A. All rights reserved

120

processes at an operating electrode. Wagner and Mandelis [61 reviewed the mirage effect, or photothermal deflection spectroscopy (PDS), as a photocalorimetric method, and proposed a model to quantify the PDS signal from an illuminated photoelectrochemical cell. More recently, we reported that the probe beam deflection (PBD) signal in a photoelectrochemical cell is due to both a temperature gradient and a concentration gradient in the cell electrolyte [7,8]. In the last few years, the mirage effect has frequently been applied to the study of ion transfer processes in an electrochemical cell in the dark [9-131. In the absence of illumination and of a light absorption process at the electrode, it is obvious that the thermal contribution to the PBD signal becomes very small. However, we believe that for many electrochemical reactions, this thermal contribution can readily be extracted by applying appropriate experimental conditions and carefully analysing the PBD signal. The aim of the present paper is to illustrate the applicability of the mirage effect as a calorimetric method to a simple electrochemical reaction occurring at a metal electrode in the dark. EXPERIMENTAL

The mirage set-up in this study uses radiation from an He-Ne laser (Hugues 3225 H-C, 0.5 mW) focused by a lens (doublet, f= 10 cm) in front of the electrode as the probe beam and a position sensor (UDT PIN SPOT/2D or LSD-SD) which measures the mirage deflection. The laser beam was set parallel to the electrode surface by autocollimating the beam reflected by the sample surface followed by 90” rotation. The electrochemical cell was a glass cuvette with parallel windows; the electrodes were plane Au (working electrode (WE)) and Pt (counter-electrode (CE)) foils (thickness, 120 pm). The reference electrode was a saturated calomel electrode (SCE) in the potential step experiments and a Pt wire for determining the Peltier coefficient in the ac experiments. The cell and the electrode can move vertically and horizontally relative to the probe beam in steps of 2.5 pm. The correlation between the mirage signal and the deflection angle can be determined if the detector-sample distance and the sensor output for a known displacement are known. The beginning and end of the potential pulses were controlled by a PC-XT microcomputer and the mirage deflection was digitalized onto a Tektronix 2430A oscilloscope. The modulation (ac) experiments were performed by applying to the cell a sinusoidal current of frequency varying from 0.7 to 30 Hz and with an amplitude such as to keep the reaction potential in an adequate range. The modulus of the mirage signal was measured as a function of the frequency at a fixed current amplitude by means of a lock-in amplifier (EG&G PAR 52101, whose reference was supplied by its calibrator, the probe beam being located either in front of the electrode or behind it. For the determination of the Peltier coefficient, in order to measure a purely thermal signal the beam was passed behind the electrode protected by an insulating film (Fig. 6, inset). In this case, the perturbation was a sinusoidal current (with fixed amplitude and frequency between 25 and 35 Hz) superposed on a dc current varying linearly with time and limited by

121

the reaction potential. Other electronic equipment included an x-y recorder (HP 7046B), a potentiostat (EG&G PAR 273) and a signal generator (EG&G PAR 193). MODEL

The mirage deflection angle t) of a probe beam passing close to an electrode of width h into an electrolyte of refractive index IZ can be understood as a distortion of the wavefronts constituting the beam, whose wavelength increases as the phase velocity u = c/n increases. Accordingly, the refractive index of an electrolyte usually increases with decreasing temperature and increasing concentration. When appropriate conditions are met, the deflection angle $ can be determined using the optic ray approximation. Since in an electrochemical reaction there is a thermal contribution (due to the Peltier and Joule effects at the interface) and an ionic contribution due to the concentration variations in the electrolyte, + is proportional to the sum of the temperature and concentration gradients of the electrolyte refractive index: h *=,

an 3T

an ac

i aTax+acax

1

Calculation of the above expression requires a knowledge of the temperature and concentration profiles in the electrolyte. These can be obtained by solving the thermal diffusion equation for the heat evolved (or absorbed) at the interface and the ionic transport equation in which migration and convection effects are neglected. In quiescent electrolytes (i.e. at short times in potential steps or at “high” frequencies in ac conditions) no convection occurs and migration of the electroactive ions can be minimized provided that a large quantity of support electrolyte is present in the solution. However, this support electrolyte should not contribute to the mirage deflection, and this is only possible if its an/at value is small, as in the case of NH,OH for example 1141. Concentration profiles for redox reactions can be calculated from the profiles of the reacting ions and of the soluble products, taking into account the electroneutrality condition. In the following, we shall always consider the mirage angle to be positive when the beam is deflected away from the electrode, as a result of a decrease in the electrolyte refractive index, and negative as an approximation of the beam occurs in the opposite case. Potential step measurements

We present potential step measurements of the mirage deflection when three different processes take place in the electrochemical cell: (a) Cu electrodeposition; (b) oxidation of the ferro/ferricyanate system; (cl reduction of the ferro/ferricyanate system. In all these cases the mirage deflection was measured as a function of time in front of and behind the electrode in the dark.

122

In the case when a negative potential step is applied to the cell the concentration profiles of the ionic species are [15]

(2a)

cg( X, t) = co*+

CR(X, t)=c;-

(2b)

where the symbols have their usual meanings. The above expressions are the solutions of the ionic diffusion equation (Fick’s second law) with boundary conditions as in ref. 15. The temperature profile within the electrolyte as the thermal power Z’(t) is released (or absorbed) at the electrochemical interface is

Gg( g-[ &)]

T(x,t)=T,T

(3)

If the only thermal effect present in the cell is the Peltier effect, then P(t) = IIZ, where II is the Peltier coefficient of the electrochemical reaction and Z is the current which obeys the Cottrell equation (0 + 0). In the above expression T, is the electrolyte bulk temperature, p and cp are the electrolyte density and specific heat capacity respectively, Dth is the electrolyte thermal diffusion coefficient and A is the electrode area. Equation (3) was obtained from the thermal diffusion equation with the boundary conditions T(x, 0) = T,

DthaT = ax x=0

limT(x,

X’rn T-

P(t)

t) = T,

(4)

APC,

Therefore the mirage deflection in potential step conditions when only one species is deposited at the electrode, as in the case of Cu2+ reduction, is

(5) where &z/ClT and &r/&z are the temperature and concentration gradients of the refractive index and Dion is the ionic diffusion coefficient. n, is the number of electrons in the reaction and F is the Faraday constant. For the ferro/ferricyanate redox system under the same negative potential step conditions we have

123

When a positive potential step is applied, the mirage deflection will be given by an expression similar to eqn. (6), with 0 and R interchanged. Equations (5) and (6) show that thermal contributions to the mirage signal are proportional to the thermal power dissipated (or absorbed) at the interface. These thermal effects are expected to be noticeable at very short times, since thermal waves are two orders of magnitude faster than ionic waves. However, ionic effects dominate the overall behavior of the deflection at longer times. It is important to emphasize that, when the potential step applied to the cell is inverted, the mirage deflection due to ionic effects will be inverted (processes B and C> as well as the current. However, depending on its origin, the deflection due to thermal effects will react differently to the inversion of the potential step. If it is due to the Joule effect, it will remain the same since the thermal power is proportional to the square of the current. Conversely, thermal phenomena due to the Peltier effect are expected to give rise to a mirage signal which is inverted on inversion of the potential step, since in this case the power dissipated is linearly dependent on the current (see also eqn. (9)). AC experiments

We know from our previous work on photoelectrochemical cells [7,8] that the logarithm of the amplitude of the mirage deflection plotted as a function of the square root of the light modulation frequency exhibits a typical two-slope behavior. This is due to the mixed ionic and thermal contributions at low frequencies, and to the dominant thermal phenomena in the high frequency range. Even in the case of a mirage experiment in the dark, with an alternating current applied to the cell, the signal observed is a vector signal resulting from the phasor sum of deflections due to thermal and ionic contributions. If the product of the reaction is not soluble, as in Cu reduction, the vector sum of the ionic and thermal contributions to the ac mirage signal is given by the following equation [8]: In Saln

IJIl =0.5ln

[

A2 exp -2 1 [

+2~~A*,,i-[~EEl+{~]~)

xcos([&El

If there is no ionic contribution, 1nSaln

I#] =ln A,=

- \iGE#)j

the above equation simplifies to

(7a)

124

where A,=--

h dn

I,

n ac neAFDion

and A,=--

h an

PO

n 3T pAc,D,,,

(8)

In the above expressions I,, and w are the amplitude and frequency of the sinusoidal current applied to the cell, and P,, is the amplitude of the resulting ac thermal power at the electrochemical interface. In the case of a redox reaction, in which we consider two different electrolytic species diffusing in the cell (Fe(CN) 4- 13-), the ac mirage deflection results from the sum of three phasors. The generalized form of this equation is given in the Appendix. The Peltier coefficient of the ferro/ferricyanate redox reaction can also be determined by means of an ac calorimetric experiment, as previously shown using an acoustic technique [16]. The procedure is as follows. The mirage deflection is proportional to the thermal power developed at the interface: P=I-Il+Rz*

(9)

where I = Zdc+ Z,, sincot) is the sum of an ac and a dc current applied to the cell, where 1,, is the amplitude of the sinusoidal current and w is its angular frequency. II is the Peltier coefficient of the reaction occurring at the electrode, and R is the sum of the charge transfer and mass transfer pseudoresistance of the cell. This equation is valid only for “small” values of the overpotential, so that the latter is proportional to the current. The mirage signal measured at a frequency f is therefore s, = CYPf= a( IIZ,, + 2&Z,,)

(10)

where a is a correlation coefficient. The plot of S, vs. Zdc is therefore expected to be a straight line, with slope 2aRZ,, and intercept with the vertical axis equal to &I,,. The Peltier coefficient (in volts) is therefore II = S&Z,, = O)/(aZ,,>. Since R is known from the current vs. potential curve, it is also possible to calculate the correlation coefficient cr from the slope of the experimental line. RESULTS

Figure 1 shows the mirage deflection as a function of the time when a potential pulse is applied to a cell where Cu electrodeposition occurs. The experimental curve does not show any modification at short times, as we would expect from the presence of thermal effects. Data were fitted according to eqn. (3, neglecting the thermal contribution and using the electrode-beam distance and the Cu*+ diffi-

125 1.6 14 I.2 I.o cd*+zs-c” n IOna

E 0.8

CUSO.

B

0.6

0.0

50

I 5.0

IO.0

200

TIME/s

Fig. 1. Experimental deflection (full points) after a negative potential step to - 0.4 V/SCE for Cu2+ reduction and the best fit (full curve): WE, Au foil; h = 2 mm; fitting parameters, see text.

sion coefficient as free parameters (all adjusted curves were obtained from a non-linear least-squares procedure which uses the Marquardt algorithm [ 171). From the fitting procedure we obtained x =i 110 pm and D,, = 8.5 x 10W6cm’/s. Figures 2 and 3 show the experimental data for the reduction and the oxidation of the ferro/ferricyanate system under a potential step. In both cases a small peak

x=

120 ILm

12mM

K,Fe(CN)e

0.5M

NH.OH

0 E 2

Fs(CNjd+Ie--Fe(CNj;

Fig. 2. Experimental deflection (full points) after a negative potential step to -0.6 V/SCE for Fe(CN)i- reduction and the best fit (full curve): WE, Au foil; h = 2 mm; fitting parameters, see text.

126

r 2.5 2.0 1.5

0.0

5.0

10.0 TIME

I 5.0

20.0

/ s

Fig. 3. Experimental deflection (full points) after a positive potential step to +0.6 V/SCE for Fe(CN)i- oxidation and the best fit (full curve): WE, Au foil; h = 2 mm; fitting parameters, see text.

appears at the beginning of the experiment, always opposite the overall beam deflection at longer times. The experimental points in Fig. 2 were fitted according to eqn. (61, with no thermal contribution, leaving x, Do and D, as free parameters. The data in Fig. 3 were fitted according to the equation for the positive potential step with the same free parameters. The average values obtained from the fitting of the two sets of experimental data were x = 120 pm, Do = 8.5 x 10e6 cm’/s and D, = 7.6 X lO-‘j cm2/s. The results of the mirage measurements for ac Cu deposition and for the ferro/ferricyanate oxidation and reduction are shown in Figs. 4 and 5 respectively. In the case of the Cu reaction (Fig. 4) the mirage signal was measured in front of the electrode, whereas in the experiment with the ferro/ferricyanate redox species (Fig. 5) the signal was measured both in front of and behind it. The signal measured in front of the electrode results from the contribution of ionic and thermal phenomena, and indeed the figures show the typical two-slope behavior expected from eqn. (7a). For detection behind the electrode (Fig. 5, upper curve) only one straight line is observed, as expected from eqn. (7b), with the same slope as the lower curve at high frequencies. The experimental data in Fig. 4 were fitted according to eqn. (7a). The values of the fitting parameters were x = 170 pm and Dion = 2 X 10e5 cm2/s. The results in the lower curve of Fig. 5 were fitted using the equation presented in the Appendix. In this case of the lower curve the fitting parameter values were x = 160 pm, Do = 5 x 10m5 cm*/s and D, = 4 x lo-’ cm2/s. The upper curve was fitted by eqn. (7b), leaving the electrode-beam distance as the only adjustable parameter, with a value of x = 170 pm. The data in

127

2.0 0.1 M

I.0 -

0.6M Ilc=

0.0 -

x.

cuso* H,SO. 7.4 mA/cm2

170ym

> -1.0 E L -2.0 t - -3.0 < -4.0 -5.0 I I I I 2.0 3.0 4.0 5.0 I/i-/ Hr”’ Fig. 4. Experimental deflection amplitude (full points) for the reaction Cuzf +2e- it Cu at fixed current amplitude Iac and best fit (full curve): WE, Au foil; h = 2 mm; fitting parameters, see text. -6.0 ’ 0.0

I .o

0

- 1.0

I 1.0

F

X=

175~m.I,*75mA/cm2

>

E -2.0 \ *= -3.0 G

-4.0

0.15 MK.,Fe

( CN)e

K,Fa(CN),

, 2M

, 0.15

M

KCL

I

-5.0

-6.0

t ’ 0.0

I

I

1.0

2.0

I

I

I

I

4.0 6.0 3.0 5.0 fi / Hz”’ Fig. 5. Experimental deflection amplitude (full points) for the redox reaction Fe(CN)i- +e- z+ Fenat fixed current amplitude Iac and best fit (full curve): It = 3 mm; probe beam behind (upper curve) and in front of (lower curve), the Au working electrode.

128

I M NapSO,

0.0

“““I

30

20

‘1 1.0

‘1

0.0 -1.0 -20 h/mA

““‘I -30

-4.0 -50

Fig. 6. Experimental deflection amplitude (full curves) for the redox reaction as in Fig. 5 for four current amplitudes lac at fiied frequency f = w /27r: probe beam behind the electrode; Au WE area, 0.12 cm’.

Figs. 4 and 5 were fitted assuming that the value of the electrolyte thermal diffusion coefficient was Dtb = 2 x 10e3 cm2/s. Figure 6 shows the experimental data for a modulation experiment performed by applying an ac current superposed on a dc current, varying linearly with time, to the cell where the ferro/ferricyanate reaction occurs. The ac mirage signal was measured behind the electrode in order to avoid the contribution of ionic effects. The mirage signal measured as a fixed frequency using a lock-in amplifier was registered as a function of the dc current. The different curves correspond to different values of the ac amplitude. As predicted by eqn. (10) these curves are straight lines in the current range where only one reaction occurs. We obtained the value of the correlation coefficient (cy = - 1.2 mV/mW) and of the Peltier coefficient of the electrochemical reaction (Il = - 0.3 V) from the slope and intercept of these lines with the vertical axis. DISCUSSION

We have noted above that thermal effects should be important, in potential step experiments at very short times. Indeed, the small pre-peak in the 4 vs. t curves, observed at the beginning of the experiment on the ferro/ferricyanate system (Figs. 2 and 31, may be attributable to a thermal phenomenon. In this case, the experiments would be in qualitative agreement with our’expectation of a Peltier

129

effect at the interface. In fact, the Peltier coefficient of the reaction is negative (positive) for the reduction (oxidation) process, and hence this thermal contribution to the mirage deflection is always opposite to that of the ionic effect. However, it is quite surprising that the dc experiment on the Cu system (Fig. 1) does not show a similar pre-peak at short times, since in this case the Peltier coefficient is approximately equal to 130 J/K mol [181, only half that for the ferro/ferricyanate reaction. Furthermore, potential step experiments performed with the probe beam behind the electrode did not show any deflection whatsoever. Therefore the fact that the pre-peak appears only in the ferro/ferricyanate experiment and with the probe beam passing in front of the electrode cannot be explained as being due to a thermal effect, but to the presence of two species with slightly different coefficients of diffusion into the electrolyte. Numerical simulations of the deflection due to a potential step according to eqn. (6) without the thermal contribution and using the literature values of the diffusion coefficients (D, = 8.9 X 10e6 cm2/s and D, = 7.4 X lO-‘j cm2/s) resulted in theoretical curves identical to those observed in our experiments. More recently, Barber0 et al. [19] have calculated potentiostatic transients for the beam deflection during Fe(CN)zoxidation, and have obtained similar results. Furthermore, our simulations of the potential step experiment using only one value of the ionic diffusion coefficient and including the thermal contribution were not able to reproduce the pre-peak observed in the experiments. Therefore we conclude that the pre-peaks recorded in the dc potential step measurements with the ferro/ferricyanate redox couple arise from the difference between the diffusion coefficients of the two species and not from a thermal effect. The values of the fitting parameters were in very good agreement with those in the literature, and the agreement between the experimental and calculated curves was excellent. Since the refractive index gradient an/&z of the support electrolyte is small (0.77 cm3/mol) composed with that of the electroactive electrolyte (about 50 cm3/mol for K,Fe(CN), and 70 cm3/mol for K,Fe(CN),), this fitting confirms that the influence of the NH,OH electrolyte on the deflection angle can be neglected. In the case of Cu electrodeposition (Fig. 11, the fitting procedure yields a value of the diffusion coefficient (D,” = 8.5 X low6 cm2/s) slightly larger than that reported in the literature. We believe that this is due to neglect of the influence of the supporting electrolyte in our model, because its refractive index gradient is not negligible in this case. In order to stress the importance of the mirage technique as a calorimetric tool we discuss ac measurements, since we know from our previous work [8] that thermal effects, when present, are clearly observed in this kind of experiment. In fact, at high frequencies only thermal waves contribute to the signal because of their much larger diffusion coefficient. The upper curve in Fig. 5 indicates that a thermal phenomenon occurs behind the electrode and is detected by the ac technique, since no reaction is possible on this side because of the presence of the insulating paint. We observe that the

130

slopes of the upper and lower curves at high frequencies are very similar. Since the two values of the electrode-beam distance obtained from the fitting differed only slightly, the results suggest that the same thermal wave phenomenon is occurring in the electrolyte in the vicinity of both sides of the electrode. Although the agreement between the calculated and experimental curves was very good, and the adjusted value of the electrode-beam distance was reasonable, the fitted values of the ionic diffusion coefficients for both species in solution were much larger than expected. This can be attributed to several factors. First, the mirage experiments should ideally be performed at sufficiently low frequencies (f < 4D/x*) to satisfy the ray optic limit [20], which is an assumption implicit in our equations. Although the above condition is fulfilled for the thermal waves, for the ionic waves all ac experiments should be performed at frequencies below 0.3 Hz (with x = 0.01 cm and D = Dion z 7 x 10T6 cm2/s), but this was not possible with the analog lock-in technique. Therefore the deduction of Dion is not rigorous. Moreover, in these ac experiments the electrolyte concentration is much higher than in the potential step experiments to allow higher ac stationary currents, so that the condition of an infinitely dilute solution is not satisfied and the refractive index gradient of the support electrolyte may affect the results. In our ac experiments on the Cu system (Fig. 4) the two-slope character of the experimental curve again confirms that an ionic effect as well as a thermal phenomenon is present in this reaction. Again, the fitted value of Q, is much larger than the literature data, and this can be attributed to the same reasons as in the above case. Therefore we conclude that thermal diffusion occurring at an electrochemical interface can be detected and calculated using the ac mirage technique and distinguished from ionic effects under appropriate experimental conditions. The ionic diffusion phenomenon can be detected and compared with the model much more accurately in a transient experiment than under ac conditions, in agreement with what we have already suggested in previous work [8]. The sensitivity of the mirage technique as an ac calorimetric technique which can yield quantitative information is shown by the results presented in Fig. 6. In a previous paper we demonstrated that it is possible to determine the Peltier coefficient of an electrochemical reaction using an electroacoustic technique [16] under very similar current modulation conditions. In this paper we have shown that this can also be done in an optical experiment using a mirage set-up, which is more convenient for most electrochemical experiments. The results obtained from both types of measurement are in reasonable agreement. The value obtained in the present work agrees better with that reported by Kuz’minskii and Gorodyskii [21]; the difference between this and the previous result could well be due to the different electrolytes used in the two experiments. Kuz’minskii and Gorodyskii report different Peltier heats depending on the electrolyte. We believe that the mirage refractometry has the advantage over acoustic calorimetry of being a sensitive probe for thermal waves and concentration waves simultaneously, and of being both a front-wall and a back-wall technique, allowing the choice between a thick or a thin electrode as required.

131 CONCLUSIONS

We performed dc and ac measurements of mirage deflection on two typical electrochemical systems: the Cu*+/Cu couple and the ferro/ferricyanate redox system. Potential step experiments are in good agreement with mathematical simulation and numerical fitting, provided that diffusion conditions are satisfied by the addition of an excess of indifferent support electrolyte with negligible effect on the optical measurement (i.e. with small &r/&z). No thermal phenomena have been detected in this kind of experiment. It is important to stress that the diffusion of both the reactant and the product species has been taken into account in order to describe precisely the feature of the experimental curves. Further work is in progress, and will be published elsewhere, on the important effect of the support electrolytes and on how to take this into account during mirage experiments. Thermal phenomena in dark electrochemical reactions can only be clearly observed using the ac technique, since under appropriate experimental conditions they can be distinguished from ionic contributions. The detection of a mirage signal behind the electrode (where no reaction could take place because of the presence of insulating paint) left no doubt that a thermal phenomenon, arising from both the Peltier and the Joule effects, occurs at the electrochemical interface. Mirage calorimetry is more convenient than other calorimetric techniques because it is contactless, is not limited by any cell constant or coefficient and has the advantages of a high frequency modulation technique. It can be extended to semiconductor and metal electrodes, both under illumination and in the dark, and is a quantitative technique for reversible electrochemical reactions. ACKNOWLEDGEMENTS

We would like to thank CNPq and FAPESP for financial support, and Professor B. Scrosati for providing laboratory facilities during part of this work. REFERENCES 1 R.H. Muller in P. Delahay and C.W. Tobias (Eds.1, Advances in Electrochemistry and Electrochemical Engineering, Vol. 9, Wiley, New York, 1973. 2 B.S.H. Royce, D. Voss and A. Bocarsly, J. Phys. (Paris), Colloq. 6,44 (1983) 325. 3 A.C. Boccara, D. Fournier and J. Badoz, Appl. Phys. L&t., 36 (1980) 130. 4 J.P. Roger, D. Fournier, A.C. Boccara, R. Noufi and D. Cahen, Thin Solid Films, 128 (1985) 11. 5 B.S.H. Royce, F.S. Sinencio, R. Goldstein, R. Muratore, R. Williams and W.M. Yim, J. Electrochem. Sot., 129 (1982) 2393. 6 R. Wagner and A. Mandelis in A. Mandelis (Ed.), Photoacoustic and Thermal Wave Phenomena in Semiconductors, North-Holland, New York, 1987, p. 323. 7 F. Decker and M. Fracastoro-Decker, J. Electroanal. Chem., 243 (1988) 187. 8 M. Fracastoro-Decker and F. Decker, J. Electroanal. Chem., 266 (1989) 215. 9 M.A. Tamor and M. Zanini, J. Electrochem. Sot., 133 (1986) 1399. 10 R.E. Russo, F.R. McLarnon, J.D. Spear and J.E. Cairns, J. Electrochem. Sot., 134 (1987) 2783.

132 11 F. Decker, R. Neuenschwander, C.L. Cesar and A.F.S. Penna, J. Electroanal. Chem., 228 (1987) 481. 12 V. Plichon and S. Be&es, J. Electroanal. Chem., 284 (1990) 141. 13 C. Barbero, MC. Miras, 0. Haas and R. Kdtz, J. Electrochem. Sot., 138 (1991) 669. 14 R.C. Weast (Ed.), CRC Handbook for Chemistry and Physics (70th edn.), CRC Press, Cleveland, OH, 1989. 15 A.J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980. 16 F. Decker, M. Fracastoro-Decker, N. Cella and H. Vargas, Electrochim. Acta, 35 (1990) 25. 17 D.W. Marquardt, J. Sot. Ind. Appl. Mat, 11 (1963) 431. 18 R.M. Garrels and C.L. Christ, Solutions, Minerals and Equilibria, Harper & Row, New York, 1965, p. 403. 19 C. Barbero, M.C. Miras and R. Kdtz, Electrochim. Acta, 37 (1992) 429. 20 A. Mandelis and B.S.H. Royce, Appl. Opt., 23 (1984) 2892. 21 Y.V. Kuz’minskii and A.V. Gorodyskii, J. Electroanal. Chem., 252 (1988) 21. APPENDIX

In the case of a redox reaction occurring in the cell under ac conditions, we considered the mirage deflection to be due to thermal phenomena and to an ionic effect arising from the contribution of both electroactive species in solution. Taking into account that these two effects have opposite contributions to the mirage effect, the complete expression of the deflection amplitude can be expressed as follows: In If&l =0.5ln

+4

(

A2 exp -2 l

4

[

/i,i.]+.:+J$y)~]

-2&3x]

-2Al~~e~(-[~Ta+~EJ]~)

x_i

[/la-

+++Q(

xcos(

&#)

-[/la+

[GJ-

-2A2A+-(-[/Ta+/G#

x-([im-im]~)

&g)]j

dra]j

133

In the case when there is no ionic contribution, identical with that given in eqn. (7b).

the mirage deflection

becomes