Effect of Large-Amplitude Alternating Current Modulation on Apparent

Sep 8, 2010 - For completely reversible electrode processes the effect of the large-amplitude ac modulation is ... oxide formation/dissolution on Pt i...
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Anal. Chem. 2010, 82, 8137–8145

Effect of Large-Amplitude Alternating Current Modulation on Apparent Reversibility of Electrode Processes Andrzej S. Baranski* and Aliaksei Boika Department of Chemistry, University of Saskatchewan, 110 Science Place, Saskatoon, Saskatchewan, Canada, S7N 5C9 We examined the effect of a large-amplitude highfrequency alternating potential modulation on direct currents associated with irreversible, quasi-reversible, and reversible electron-transfer processes occurring at microelectrodes under voltammetric conditions. All irreversible processes appear to be accelerated by the superimposed ac modulation, and under certain conditions this may even lead to an electrochemical etching of noble metal electrodes. In the case of electrode processes which are reversible on the time scale of a dc polarization, but quasireversible on the time scale of the ac modulation, the distortion of voltammograms caused by the ac modulation can provide useful information about the kinetics of fast electron-transfer processes. For completely reversible electrode processes the effect of the large-amplitude ac modulation is essentially trivial; the distortion of voltammetric curves causes broadening of analytical signals without providing any useful information. Traditionally ac impedance and ac voltammetric measurements were carried out using low-amplitude ac perturbations (ideally with amplitudes much smaller than the RT/F term). The reason comes from convenience; when the modulation is very small the relationship between current and potential can be linearized and analytical relationships between various quantities can be derived without tremendous difficulty1 (although, in most cases, these derivations are still quite taxing). The wisdom of using low-amplitude perturbations in alternating current methods was challenged about 10 years ago by a number of authors.2-5 Indeed in analytical applications, sticking to low amplitudes does not make sense since larger amplitudes give better signal-to-noise ratios (and other benefits which will be discussed later), and mechanistic studies (for which exact mathematical description of the signals may be needed) are usually not of interest. Besides, these days, a quite accurate mathematical description of any electrode process can * To whom correspondence should be addressed. Phone: (306) 9664701. Fax: (306) 9664730. E-mail: [email protected]. (1) Sluyters-Rehbach, M.; Sluyters, J. H. In Comprehensive Treaties of Electrochemistry; Yeager, E. J., Bockris, J. O’M., Conway, B. E., Sarangapani, S., Eds.; Plenum Press: New York, 1984; Vol. 9, Chapter 4. (2) Singhal, P.; Kawagoe, K. T.; Christian, C. N.; Kuhr, W. G. Anal. Chem. 1997, 69, 1662–1668. (3) O’Connor, S. D.; Olsen, G. T.; Creager, S. E. J. Electroanal. Chem. 1999, 466, 197–202. (4) Engblom, S. O.; Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 2000, 480, 120–132. (5) Gavaghan, D. J.; Bond, A. M. J. Electroanal. Chem. 2000, 480, 133–149. 10.1021/ac1014222  2010 American Chemical Society Published on Web 09/08/2010

be obtained via numerical simulations, and in this case, handling nonlinear relationships is not a problem. This path was very successfully explored by Bond and co-workers,6,7 who created a method which they called “higher harmonic large-amplitude Fourier-transformed alternating current voltammetry”. In their method, the large amplitude of the modulation was primarily used to generate higher harmonics in the electrode response. This complex but very information-rich response can be processed to determine the electrode reaction mechanism,8,9 measure electrontransfer kinetics,10,11 or to obtain better analytical signals. In the last case, the authors claim that the fourth (or higher) harmonic response provides a few times better detection limit than either a dc or ac fundamental response, as well as better selectivity for multicomponent analysis.12,13 In our laboratory, a large-amplitude ac modulation was used to study faradaic rectification14 and heating of microelectrodes.15,16 We apply ac modulations at very high frequencies ranging from 0.1 MHz to 2 GHz; in this case, measurements of the ac response are very difficult and we can only measure changes in the dc response of the electrode and monitor associated phenomena (like heating and convection). In all experiments presented in this work, heating and convection were negligible, and we concentrated solely on the effect of high-frequency ac modulations on the dc current flowing through the electrode interface. It has been known for some time that ac modulation alters the dc current (i.e., current averaged within one ac cycle),17,18 but this work is broader in scope and more comprehensive than the previous ones. The phenomenon is related to faradaic rectification which was recently studied in (6) Sher, A. A.; Bond, A. M.; Gavaghan, D. J.; Harriman, K.; Feldberg, S. W.; Duffy, N. W.; Guo, S.-X.; Zhang, J. Anal. Chem. 2004, 76, 6214–6228. (7) Bond, A. M.; Duffy, N. D.; Guo, S.-X.; Zhang, J.; Elton, D. Anal. Chem. 2005, 77, 186A–195A. (8) Lertanantawong, B.; O’Mullane, A. P.; Surareungchai, W.; Somasundrum, M.; Declan, B. L.; Bond, A. M. Langmuir 2008, 24, 2856–2868. (9) Lee, C. Y.; Bond, A. M. Langmuir 2010, 26, 4614–4626. (10) Fleming, B. D.; Barlow, N. L.; Zhang, J.; Bond, A. M.; Armstrong, F. A. Anal. Chem. 2006, 78, 2948–2956. (11) Zhang, J.; Guo, S.-X.; Bond, A. M. Anal. Chem. 2007, 79, 2276–2288. (12) O’Mullane, A. P.; Zhang, J.; Brajter-Toth, A.; Bond, A. M. Anal. Chem. 2008, 80, 4614–4626. (13) Bond, A. M.; Duffy, N. W.; Elton, D. M.; Fleming, B. D. Anal. Chem. 2009, 81, 8801–8808. (14) Baranski, A. S.; Diakowski, P. M. J. Phys. Chem. B 2006, 110, 6776–6784. (15) Baranski, A. S. Anal. Chem. 2002, 74, 1294–1301. (16) Boika, A.; Baranski, A. S. Anal. Chem. 2008, 80, 7392–7400. (17) Baranski, A. S.; Szulborska, A. J. Electroanal. Chem. 1994, 373, 157–165. (18) Gavaghan, D. J.; Elton, D.; Oldham, K. B.; Bond, A. M. J. Electroanal. Chem. 2001, 512, 1–15.

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our laboratory, but again, this work is very different from the published one. In the published work, we used sinusoidal ac perturbations with square-wave modulated amplitude and the data was processed in a very complex way to obtain information about electron-transfer kinetics of very fast redox reactions. In this work, the amplitude of the ac modulation was constant during the measurement and also our primary interest was in irreversible electrode processes. The present method is experimentally very simple and does not require any specialized equipment (except for an inexpensive and easily available ac generator with a frequency output up to 10 MHz); at the same time the results are spectacular and often surprising. EXPERIMENTAL SECTION Reagents. All solutions were prepared using Millipore water and ACS grade chemicals, which were used without any further purification. Most electrochemical measurements were done without the removal of dissolved oxygen; only in experiments involving the reduction of Eu3+ on Hg/Au electrode dissolved oxygen was removed by bubbling pure Ar through the solution for 10 min prior to the measurements. Electrochemical Cell. In most experiments a standard threeelectrode electrochemical cell was used. The auxiliary and pseudoreference electrodes were made of a platinum wire ∼0.3 mm in diameter (Alfa Aesar). In some cases a standard Ag|AgCl|KCl(sat.) reference electrode was employed. The disk working electrodes (with a diameter of 10 µm or larger) were prepared by sealing platinum or gold microwires (Goodfellow Metals Ltd.) into glass tubing (World Precision Instruments). A lead was made by connecting a copper wire with a microwire with the aid of a small piece of Pb-Sn solder. Then the electrode tip was cut, and the electrode was polished with 3 and 0.3 µm finishing films on a micropipet beveller (WPI, model 48000) to achieve a mirror-like surface. In some experiments electrochemically sharpened electrodes were used. Preparation of these electrodes was described in our previous publication.16 Electronic Circuit. The block diagram of the electronic setup used in this work is shown in Figure 1. The ac modulation was produced by a custom-made generator which was based on a direct digital synthesis (DDS) integrated circuit (Analog Devices AD9910). This generator was capable of synthesizing a sinusoidal waveform with frequency ranging from 0 to 400 MHz (with 0.23 Hz resolution) and amplitude ranging from 0 to 3 Vrms (with 0.18 mV resolution). However, in experiments presented in this paper, only amplitudes lower than 0.5 Vrms and frequencies ranging from 0.1 to 10 MHz were used. The ac generator was coupled with the electrochemical cell via a 1:1 telecommunication transformer with the primary inductance of 100 µH (Pulse T3001NL). Another important element of a circuit is a 0.1 µF capacitor C1 (a capacitor with low inductance and resistance >1010 Ω should be used) between the working electrode and the potentiostat ground. This capacitor provides a pathway for ac currents to ground. A filter shown in Figure 1 is optional, but it reduces the noise level when working with microelectrodes. The filter is made of a network of 50 Ω resistors and 0.1 µF capacitors. This filter increases the uncompensated cell resistance by 150 Ω, which is completely negligible when working with microelectrodes, but it may be unacceptable when working with standard size electrodes. 8138

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Figure 1. Interface between a potentiostat, an ac generator, and an electrochemical cell (W. is the working, Aux. is the auxiliary, and Ref. is the reference electrode). C1 is a 0.1 µF capacitor, R1 is a 50 Ω resistor, and T1 is a 1:1 telecommunication transformer.

Electrochemical measurements were carried out with a customdesigned data acquisition system based on a microcontroller (Microchip PIC18F2550), which was linked with a computer via a USB interface. A three-electrode potentiostat was based on a classical design,19 where the working electrode is at virtual ground potential and the dc potential is applied to the solution via auxiliary and reference electrodes. The dc potential of the cell was controlled by a 16-bit digital-to-analog converter (Linear LTC1655). Scanning of the dc potential range was performed in a staircase fashion. The potential steps in the scanning waveform were always less than 1 mV. The current was sampled with a 16-bit analogto-digital converter (Texas-Instruments TLC4545) after passing through an antialiasing fourth order low-pass filter, made of two operational amplifiers, four capacitors, and four digital potentiometers. The software for the microcontroller was written in assembly language (with an assembler provided by Microchip), and the PC software (for data acquisition and processing) was written using Microsoft Visual C++, version 6.0 and Visual Basic, version 6.0. The experimental data were further processed in Microsoft Excel. Convolution and deconvolution procedures were performed using fast Fourier transformation. Although measurements described in this paper were performed with custom-made devices, we believe that they can be easily repeated with any instruments commonly used in electrochemical laboratories.

(19) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; Wiley: New York, 2001; p 640.

RESULTS AND DISCUSSION In the case of a faradaic process kf

f Ox + ner

R

kb

the electrical current passing through the electrode solution interface can be described as i(t) S ) kf(E)CSR(t) - kb(E)COx nFA

(1)

where R and Ox represent the reduced and the oxidized form of S a depolarizer, CRS and COx are the surface concentrations of R and Ox, kf and kb are the forward and the reverse rate constants of the electron-transfer process, n is the number of exchanged electrons, F is the Faraday constant, and A is the surface area of the electrode. The same general equation is valid for species dissolved in solution and for species confined to the electrode surface. Usually the electrode potential E is a function of time, and certainly this is the case when the electrode potential is modulated by an alternating (ac) potential. The ac modulation can alter kf and kb, as well as the surface concentrations of the reactant and the product. However, the situation is much simpler when the electrode process is very slow; then we can neglect changes in the surface concentrations occurring during an ac cycle. This will be the first case considered in our paper. Another simple case (which will be discussed next) involves very fast (fully reversible) processes. Here, kinetics is unimportant (because the system is at equilibrium), but the current is affected by changes in the surface concentrations. Finally, the last part of this section deals with the most complicated case when changes in the rate constants and surface concentrations during an ac cycle are significant. Irreversible Electron-Transfer Processes. Let us consider first the case when the electron-transfer process is so slow that the electrode reaction appears irreversible (i.e., cannot reach equilibrium) during a relatively slow change of dc potential. Of course, in this case, the process also will not be able to reach equilibrium during a very short ac cycle when a high-frequency modulation of the electrode potential is applied. We do not need to make any additional assumptions about the nature of the electrode process, because in all cases the effect is the same and quite surprising: all irreversible processes appear faster (more reversible) when a large-amplitude ac modulation is applied. Figure 2 shows the effect of a superimposed ac potential on electrode processes occurring at gold microelectrode (25 µm in diameter) in 0.3 M H2SO4 at the scan rate of 20 V/s. On these curves the anodic peak is due to the formation of a gold oxide layer, the cathodic peak is due to gold oxide reduction, and the change of the current at the most positive and the most negative potentials is caused by O2 and H2 formation, respectively. When electrodes are small and currents not too large O2 and H2 dissolve in solution and diffuse away from the electrode without forming gas bubbles. The gold oxide formation is a complex surface process, involving a sequence of electron- and proton-transfer steps and a turnover of AuOH

Figure 2. Cyclic voltammograms obtained with a 25 µm Au disk microelectrode in 0.3 M H2SO4, without ac modulation (black) and with ac modulation amplitudes of 100 mVrms (blue) and 200 mVrms (orange). The frequency of the superimposed ac modulation was 0.1 MHz, and the scan rate was 20 V/s.

Figure 3. Cyclic voltammograms obtained with a 25 µm hemispherical Hg microelectrode (Hg electrodeposited on a Au disk) in acetic buffer solution (0.5 M NaAc, 0.1 M HAc) containing 5 mM Eu3+, without ac modulation (black) and with ac modulation amplitudes of 100 mVrms (blue) and 250 mVrms (orange). The frequency of the superimposed ac modulation was 0.1 MHz, and the scan rate was 10 V/s.

surface dipoles.20 Figure 2 clearly shows that all listed processes appear to be accelerated when the ac modulation is applied; the gold oxide formation/reduction process looks more reversible with sharper peaks and small difference between the anodic and cathodic peak potentials. At the same time formation of O2 and H2 occurs at less positive and less negative potentials, respectively. It should be emphasized that these dramatic changes are caused by quite moderate ac amplitudes (in this case, 100 and 200 mVrms). The frequency of the modulation is unimportant (100 kHz were used) as long as the applied signal is not substantially affected by the ohmic drop. A very similar effect was observed during the reduction of 5 mM Eu3+ on Hg/Au microelectrode in an acetic buffer solution (0.5 M CH3COONa, 0.1 M CH3COOH). This electrode process involves a simple outer-sphere electron transfer with the standard rate constant of about 7 × 10-5 cm/s.21 In this case, the peak due to Eu3+ reduction appears at less negative potentials when amplitude of the ac modulation increases, and also, when the process becomes more reversible, a small peak due to the reverse process (Eu2+ f Eu3+ + e-) becomes visible (Figure 3). (20) Conway, B. E.; Barnett, B.; Angerstein-Kozlowska, H. J. Chem. Phys. 1990, 93, 8361–8373. (21) Weaver, M. J. J. Phys. Chem. 1980, 84, 568–578.

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but the time-average rate constant, kapp f (Edc), is larger than kf(Edc): kfapp(Edc) )

kf(Edc - ∆E) + kf(Edc + ∆E) ) 2 kf(Edc)cosh

Figure 4. Explanation of the apparent acceleration effect. If kf depends exponentially on potential (blue line) then, in the case of square-wave modulation of potential, the time-average kf (shown as kfappsred line) is always larger than kf.

In the Supporting Information we included three additional figures showing the effect of a superimposed ac potential on electrode processes occurring at a Pt microelectrode in H2SO4 (Supporting Information Figure S-1), redox processes of arsenic, As(0)/As(III), adsorbed on gold (Supporting Information Figure S-2), and reduction of dissolved oxygen at a gold microelectrode (Supporting Information Figure S-3). In all cases the effect is qualitatively the same, except for currents associated with the adsorption/desorption of hydrogen on Pt, but this last process is quite fast, and it is affected by the ac modulation in a different way. Changes shown in Figure 2 are very similar to those caused by an increase of temperature, but the heating of the electrode has to be ruled out. The power dissipated at the electrode during the experiment with the largest amplitude of ac modulation presented in Figure 2 was less than 50 µW. On the basis of our studies of hot microelectrodes,16 this power would be sufficient to raise the electrode temperature by only 0.04 K. The absence of heating was also proven experimentally: we added 5 mM of Fe(II) to the solution and measured changes in the limiting current of Fe(II) oxidation during the experiments with ac modulations. A change of the electrode temperature would cause a change of viscosity of the solution surrounding the microelectrode and consequently a change in the limiting current. We were capable of detecting changes in the limiting current smaller than 0.2% which would correspond to a change of temperature by less than 0.1 K, but no change in the limiting current (other than random noise) was observed. Still the phenomenon shown in Figures 2 and 3 can be easily explained on different grounds. The basic idea is presented in Figure 4. For simplicity we will consider first a square-wave modulation. Figure 4 shows an exponential dependence of the kf on potential given by the Butler-Volmer equation:

kf(E) ) ks exp

(E - E°)] [ RnF RT

(2)

where E° is the standard potential, ks is the standard rate constant, and R is the electron-transfer coefficient. Let us assume that before applying the ac modulation the electrode potential is Edc and the forward rate constant is kf(Edc). After applying the square-wave modulation of amplitude ±∆E the time-average electrode potential will remain the same (Edc), 8140

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∆E) ( RnF RT

(3)

We call cosh[(RnF/RT)∆E] the apparent acceleration factor. This factor is equal to 1 if R∆E ) 0 and greater than 1 in all other cases. In the case of sinusoidal modulation the situation is only little bit more complicated. It can be easily shown that, in this case, the apparent acceleration factor is given by ξ

1 ∆E) ) ( RnF RT 2π ∫

2π

0

exp

∆E sin(φ)] dφ [ RnF RT

(4)

The ξ(x) function is similar to the cosh(x) function, and we established that it can be approximated as

ξ(x) )

1 N

N-1

∑ cosh[x sin(π 2k4N+ 1 )]

(5)

k)0

For 0 e x e 10 and N ) 4 this approximation gives errors smaller than 0.003% (ξ(10) ) 2815.72). We call this kind of acceleration “apparent” because the superimposed ac modulation does not actually affect the standard rate constant of the electron-transfer process and the relationship between the instantaneous electrode potential and the instantaneous rate constant remains the same. However, it should be noted that it is possible to avoid the use of the convoluted phrase “apparent acceleration” by characterizing the processes in terms of the average (or dc) rate constant, which indeed increases in the presence of ac modulation. In practical terms the acceleration is real; for example, a gold electrode polarized to +1 V in H2SO4 (see Figure 2) will produce much more oxygen when an ac modulation of 200 mVrms is superimposed on the dc potential. At this stage our theoretical model involves only simple electron-transfer processes. However, we expect even more interesting relationships in cases when the electrode reactions go through a sequence of steps and involve intermediate products. It is expected that in such cases the superimposed ac modulation will alter the concentration of intermediates and that, under certain circumstances, may lead even to changes in the electrode reaction mechanism. Probably both the electron-transfer acceleration and alteration of the electrode reaction mechanism is responsible for electrochemical etching of platinum and other noble metals, which is typically performed in CaCl2 aqueous solution using a lowfrequency alternating voltage of a few volts in amplitude.22 The electrochemical etching of noble metal electrodes by a largeamplitude ac modulation occurs even at very high frequencies. Ghanem et al.23 observed damage of Pt microelectrodes after experiments which involved heating of microelectrodes with focused microwave radiation. We used atomic force microscopy (22) Zhang, B.; Zhang, Y.; White, H. S. Anal. Chem. 2004, 76, 6229–6238. (23) Ghanem, M. A.; Thompson, M.; Compton, R. G.; Coles, B. A.; Harvey, S.; Parker, K. H.; O’Hare, D.; Marken, F. J. Phys. Chem. B 2006, 110, 17589– 17594.

to monitor changes in the Au and Pt electrodes morphology after applying for 1 min ac modulation (0.3 Vrms, 0.1 MHz) in 0.2 M HCl at Edc ) 1.6 V versus SCE. After this treatment both electrodes showed clear evidence of electrochemical etching (see Supporting Information Figures S-4 and S-5). It should be noted, however, that such positive dc potentials are rarely used in normal electrochemical experiments (under similar conditions but with Edc ) 1.3 V the electrochemical etching was not detected). However, the change of the electrode reaction mechanism probably does not occur when the ac amplitude is relatively small, like in the case presented in Figure 2. In this case, all observed changes are continuous (increase steadily with an increase of the amplitude) and all these changes are quantitative, not qualitative (new features do not appear on CV curves). It should be noted that the superimposed ac modulation affects the apparent rates of both forward and reverse electron-transfer processes. If R * 0.5, then the forward and the reverse electrode processes will have different apparent acceleration factors, and this means the apparent standard potential will also be affected. Both the forward and the reverse rates will become equal at potential E0app:

Figure 5. Effect of superimposed ac modulation on steady-state voltammograms for reversible redox processes. Solid lines represent the results of simulations carried out for a 3.6 µm hemispherical electrode, ks ) 100 cm/s, R ) 0.5, scan rate ) 0.001 V/s, and frequency of modulation 1000 Hz. Amplitudes of ac modulation (peak to mean in RT/nF units): 0 (black), 2.5 (blue), 5 (green), 7.5 (orange), and 10 (red). Markers represent results obtained using eq 13 with N ) 10 and ∆E the same as for underlying lines.

These predictions were verified by running numerical simulations, and the results are given in the Supporting Information. Figure S-6 in the Supporting Information shows two simulated cyclic voltammograms for a redox process with E° ) 0 mV, ks ) 0.001 cm/s, R ) 0.65, and the scan rate 20 V/s, with (red) and without (blue) superimposed ac modulation at frequency 100 kHz and amplitude 250 mV (peak to mean). These two curves were subsequently analyzed by a convolution method to determine kinetic parameters.24 The results are presented in Figure S-7 in the Supporting Information. In the absence of ac modulation the analysis gave E° ) 0.9 mV, ks ) 1.07 × 10-3 cm/ s, and R ) 0.65 (which confirms the fidelity of the simulation and the data processing procedure). In the presence of ac modulation the same data analysis gave E° ) -66.0 mV, ks ) 1.74 × 10-2 cm/s, and R(E°) ) 0.68. This is in very good agreement with eq 6, which predicts an apparent shift in the standard potential by -66.4 mV, and eq 7, which predicts an apparent increase of ks by a factor 16.9. However, in the presence of ac modulation, the dependence of ln(kf) versus E was not completely linear. Most likely the acceleration of the electrode process was so large that the assumption of constant surface concentration during the ac modulation cycle was not completely satisfied.

Fully Reversible Electron-Transfer Processes. Here we would like to briefly discuss a case when the electron-transfer process is completely reversible during dc and ac polarization. The effect of superimposed ac modulation on reversible electrode reactions under cyclic voltammetric conditions was previously described by Oldham and co-workers,18,25 and some experimental results were shown in our previous paper.15 In brief, an increasing amplitude of ac modulation causes some broadening and, in extreme cases, splitting of voltammetric peaks. In the case of microelectrodes, the easiest to perform are steady-state voltammetric measurements. In the presence of ac modulation the term steady state describes a situation when the average current, as well as average concentrations of reactants and products, become independent of time, but, of course, the instantaneous current and the concentration vary periodically following oscillations of the electrode potential. A distortion of steady-state voltammograms caused by a sinusoidal ac modulation is illustrated in Figure 5. Interestingly, the effect appears to be the opposite to one observed for irreversible processesswith an increasing amplitude of the modulation the slope of steady-state curves decreases, so they resemble curves recorded for a quasi-reversible redox process, or curves recorded at a much higher temperature. Actually, this distortion caused some errors in the determination of the temperature of hot microelectrodes in our previous paper,15 and tracking these errors led directly to the present study. Some curves in Figure 5 were obtained by standard numerical simulations, which involve the defining of the boundary conditions at the electrode surface based on the Nernst equation and then solving the mass transport equation through iterations. We used 32 time increments per each ac cycle, and this led to a very long simulation (often exceeding 1 h on a very fast desktop computer) when the frequency was high and the scan rate was low. However, an equivalent mathematical description can also be obtained much faster in a different way, which actually provides more insight into the nature of the phenomenon. Let us consider a simple redox process R f Ox + ne-. For simplicity we will assume that both forms R and Ox are soluble and have the same diffusion coefficient. In addition, at the

(24) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; Wiley: New York, 2001; p 247.

(25) Oldham, K. B.; Gavaghan, D. J.; Bond, A. M. J. Phys. Chem. B 2002, 106, 152–157.

0 Eapp

RT ln )E + nF 0

ξ

[ (1 -RTR)nF ∆E] RnF ∆E) ξ( RT

(6)

and the apparent standard rate constant will be ∆E)] { [ (1 -RTR)nF ∆E]} [ξ( RnF RT

ksapp ) ks ξ

R

1-R

(7)

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starting point the concentration of R (CR) is uniform and equal to Cb and the concentration of Ox (Cox) is zero. We also assume a semi-infinite diffusion and complete reversibility of the process. In this case, at any moment, at the surface of the electrode the equilibrium is reached and the concentration of R at the surface of the electrode (CRs) can be derived from the Nernst equation:

CsR(t) )

]

E(t) ) Edc + ∆E sin(ωt)

(9)

where Edc is the dc potential, ∆E is the peak-to-mean amplitude of the modulation, and ω is the angular frequency (frequency multiplied by 2π) of the modulation. Note that normally Edc changes slowly with time, but at truly steady-state conditions that change can be entirely neglected. Now, we may be interested in calculating the average value of CRs within an ac cycle: Cb 2π

∫

2π

0

dφ nF nF∆E E + sin(φ) 1 + exp RT dc RT

[

] (10)

We do not know an analytical solution to the integral appearing in eq 10, but a numerical integration in this case is straightforward, and it can be performed even in a spreadsheet program. Consequently, the steady-state current (averaged within each ac cycle) can be described as

[

¯i (Edc) ) ilim 1 -

¯ sR(Edc) C Cb

nF 2k - N + 1 ∆E)] E + [ RT ( N f ∑ nF 2k - N + 1 ∆E)] 1 + exp[ (E + RT N exp

N-1

dc

k

k)0

dc

(8)

where E is the electrode potential and other symbols have their normal meaning. Now, in the case of superimposed ac modulation

¯ sR(Edc) ) C

¯i (Edc) ) ilim

(13)

Cb nF E(t) 1 + exp RT

[

The weights represent the fraction of time needed to change the electrode potential by 2∆E/N during the sinusoidal ac polarization. The complete algorithm in this case is

]

(11)

In Figure 5 we compared distortions of reversible steady-state voltammograms predicted by standard numerical simulations (lines) and the moving window averaging according to eq 13 with N ) 10 (markers); in all cases the agreement was very good. It should be noted that describing the shape of steady-state voltammograms is in some sense fundamental, because we can easily generate cyclic voltammograms for planar diffusion by semidifferentiation (deconvolution of 1/t) of steady-state curves, and similarly, curves for surface-confined redox species can be generated by ordinary differentiation of steady-state curves. However, for fully reversible processes, all of this is trivial. The whole effect arises from a mathematical transformation (moving window averaging). This obviously does not provide any additional information about the studied system (and does not reduce random noise during experiments) but, rather, obscures the existing information by broadening the signal. Quasi-Reversible Process. Now we will focus on processes which reach equilibrium during a slow dc polarization but cannot reach equilibrium during an ac cycle. This case is not trivial, but qualitative predictions are not difficult. The outcome depends on the degree to which ac modulation can disturb the equilibrium at the electrode, and this is determined by the kinetics of the electron-transfer process and the frequency and the amplitude of the ac modulation. In this case we did not work out any mathematical tricks which could simplify a quantitative description; consequently, we have to resort to standard numerical simulations and experiments. Simulations presented in Figure 6 reveal that the current distortion induced on steady-state voltammograms by a high-frequency ac modulation depends on the kinetics of the electron-transfer process: the standard rate constant, the electron-transfer coefficient (R), as well as the

where ilim is the limiting current for a given system. Equation 11 is valid for any system which can reach a steady state (e.g., a disk or a hemispherical microelectrode) and fulfills the starting assumptions. It should also be noted that the mathematical operation described by eq 10 is actually equivalent to a moving window weighted averaging of the original (undistorted) steady-state curve with a moving window containing N data points (equally spaced within the potential range ±∆E) with weights, fk, assigned according to the following equation:

fk )

1 2k 2k + 2 arcsin 1 - arcsin 1 π N N

[

(

)

(

)]

(12)

where k ) 0, 1, ..., N - 1 is the data point number within the moving window. 8142

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Figure 6. Simulated steady-state voltammograms for a hemispherical electrode 3.6 µm in diameter and for the same ac modulation (196.6 mVrms at 10 MHz). Kinetic parameters: R ) 0.6, dR/dE ) 0, and ks (in cm/s) equal to 15 (black), 1.5 (blue), and 0.15 (orange); ks ) 1.5 cm/s, R ) 0.5, dR/dE ) 0 (red); ks ) 1.5 cm/s, R(E°) ) 0.6, and dR/dE ) 0.02nF/RT (green).

dependence of R on potential. All these simulations were carried out for the same ac modulation: 196.6 mVrms at 10 MHz for a hemispherical electrode 3.6 µm in diameter. The simulations were done for the scan rate of 10 V/s, which was too high to get steady-state curves directly, but steady-state voltammograms can be easily obtained from the surface concentration array generated by the simulation program (if the process is fast enough, the surface concentration is independent of the scan rate). In this case, the simulation of a single curve took about 10 min and involved 3.2 × 107 iterations (clearly, generation of a steady-state voltammogram for 10 MHz at say 0.01 V/s scan rate would be an impossible task for our desktop computer). Three curves shown in Figure 6 (black, blue, and orange) were simulated for the same R ) 0.6 and dR/dE ) 0, but for different ks values (15, 1.5, and 0.15 cm/s, respectively); these curves cross in (approximately) a single point, and the main difference between them is in the slope (di/dE) at the crossing point, which increases with a decrease of ks. The other group of three curves (red, blue, and green) were obtained for ks ) 1.5 cm/s, but different values of R. In the first two cases dR/dE ) 0, and in the last one dR/dE ) 0.02nF/RT and R(E°) ) 0.6. When R is changed from 0.5 to 0.6 (red and blue) the main change is a horizontal shift of the middle part of the curve toward more negative potentials; the dependence of R with potential causes more complex changes (green curve), which will not be discussed in detail in this paper. It should be stressed that, without the high-frequency ac modulation, simulated steady-state voltammograms for all cases shown in Figure 6 are identical with exception of a curve for ks ) 0.15 cm/s (the electrode process in this last case shows very slight irreversibility at a 3.6 µm electrode under steady-state conditions, but all other processes are completely reversible). In Figure 7, parts a and b, we presented families of steadystate voltammograms for the oxidation of 20 mM Fe(CN)64- in 2 M KCl and 40 mM KCN (KCN was added to prevent formation of Prussian blue deposits on the electrode surface26). The curves were recorded at a platinum disk microelectrode (3.6 µm in diameter) at the scan rate of 10 mV/s. Curves in Figure 7, parts a and b, were obtained using ac modulation of 100 kHz and 1 MHz, respectively; the amplitudes of ac modulation are listed in the figure’s legend. The distortion caused by the ac modulation is similar to one observed for fully reversible processes, but the magnitude of distortion is smaller and it decreases with an increase of frequency, which clearly indicates the kinetic nature of this phenomenon. Also, the crossing point for a family of curves does not occur at 0.5ilim, as in the case of reversible processes, but at higher currents, which indicates that the electron-transfer coefficient is somewhat larger than 0.5. In the Supporting Information we also included experimental curves obtained with 10 MHz ac modulation (Supporting Information Figure S-8) and simulated curves for ks ) 1.5, R ) 0.6 and other simulation parameters similar to experimental conditions (Supporting Information Figures S-9 to S-11). One of many possible data analysis methods is presented in Figure 8. We determined a normalized slope for each curve at the half of the limiting current and plotted it versus the square of the normalized ac amplitude ((nF/RT)∆Erms). (26) Huang, W.; McCreery, R. J. Electroanal. Chem. 1992, 326, 1.

Figure 7. (a) Steady-state voltammograms for the oxidation of 20 mM Fe(CN)64- in 2 M KCl and 40 mM KCN at a Pt disk microelectrode (3.6 µm in diameter). The scan rate was 0.01 V/s, and the ac modulation was 0.1 MHz with amplitudes (in Vrms) of 0 (black), 0.053, 0.075, 0.092, 0.107, 0.12, 0.13, 0.14, and 0.15 (red). (b) Steadystate voltammograms for the oxidation of 20 mM Fe(CN)64- in 2 M KCl and 40 mM KCN at a Pt disk microelectrode (3.6 µm in diameter). The scan rate was 0.01 V/s, and the ac modulation was 1 MHz with amplitudes (in Vrms) of 0 (black), 0.053, 0.075, 0.092, 0.107, 0.12, 0.13, 0.14, and 0.15 (red).

Figure 8. Normalized slope of steady-state voltammograms (see eq 14) plotted vs the square of the normalized ac amplitude. Lines obtained from numerical simulations for R ) 0.6, dR/dE ) 0, and various values of the kinetic parameter χ (as shown in the figure). Markers are based on experimental data obtained for frequencies of ac modulation of 0.1 (O), 1 (4), and 10(]) MHz and presented in Figure 7, parts a and b, and Supporting Information Figure S-8.

The normalized slope at the half of the limiting current is defined as S1/2 )

dE nF i 4RT lim di

( )

i)(1/2)ilim

(14)

where ilim is the limiting steady-state current and other parameters have their usual meaning. Analytical Chemistry, Vol. 82, No. 19, October 1, 2010

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In the case of an undistorted steady-state voltammogram for a reversible process S1/2 ) 1. Lines in Figure 8 represent results of numerical simulations for R ) 0.6, dR/dE ) 0, and various kinetic parameters (χ ) (ks)/ ((2πfacD)1/2), where D is the diffusion coefficient and fac is the frequency of ac modulation). The markers represent experimental results for frequencies 0.1, 1, and 10 MHz (based on curves shown in Figure 7, parts a and b, and Supporting Information Figure S-8). The match between the experimental results and simulations is pretty good and indicates that the standard rate constant for Fe(CN)63-/Fe(CN)64- redox couple is about 1.5 cm/s. This value is about 3 times larger than previously reported.14,26 This difference cannot be explained by the apparent acceleration effect presented in the first part of the paper, because this effect is accounted for in the numerical simulations. Also a change in the reaction mechanism cannot be taken into account since this is a simple redox process, and besides the match between experimental results and simulations is equally good for small and large amplitudes (alternation of the mechanism cannot be reasonably postulated for small amplitudes). Nevertheless our results may involve small systematic errors; therefore, they should be treated as preliminary. Experiments were performed with a disk microelectrode (which exhibits nonuniform distribution of currents), whereas simulations were carried out for a hemispherical electrode. In a very near future we will check whether this difference could lead to errors. In addition, we do not know whether the match between experimental and theoretical curves is unique (perhaps another good match could be obtained for somewhat different ks, R, and dR/ dE * 0). The method obviously requires more extensive testing before it can be used with confidence, but at present, it seems to be very promising for studying very fast electrode kinetics. The ability to assess fast kinetics is due, of course, to the high frequency of the ac modulation. However, since we monitor only the dc component of the electrode current, high-bandwidth instruments (data acquisition systems and potentiostats) are not required; in addition, from an experimental stand point the method is very simple. There is no obvious upper frequency limit in such measurements; we observed effects similar to ones described in this paper even at 2 GHz frequency, but still higher frequencies could be used (if ever needed). Of course, such measurements (and even more their interpretation) are quite challenging. Detailed discussion of all these challenges is premature and exceeds the scope of the paper, but we would like to briefly mention some of them. The alternating potential applied to the electrode (∆E) is partitioned between a potential drop across the double layer (∆Edl) and a potential drop across the solution resistance (∆ERs). Only ∆Edl is responsible for effects shown in Figures 6, 7, and 8; on the other hand, large ∆ERs causes heating of the electrode. When electrodes are very small, concentration of the supporting electrolyte high, and the frequency relatively low, ∆Edl ≈ ∆E and ∆ERs is negligiblesthis was the case in experiments presented in Figure 7, parts a and b, but in other cases ∆Edl has to be determined. For measurements described in this paper we estimated the ∆Edl/∆E ratios based on the solution resistance (which is easy to measure) and the double layer capacitance (assumed to be 25 µF/cm2). These ratios were 8144

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quite high: 0.999, 0.995, and 0.71 for 0.1, 1, and 10 MHz, respectively; hence, in this case, we did not have to know the double layer capacitance with high accuracy. The problem is much more serious at higher frequencies: it is quite likely that the double layer capacitance changes with frequency, but we do not know how (since there is no method to measure it directly above 20 MHz), and yet accurate values of the double layer capacitance are needed to make reliable estimates of ∆Edl/∆E ratios. Fortunately, the apparent acceleration of irreversible processes (described in the first part of this section) could be used to gauge the double layer polarization (and the double layer capacitance) even at GHz frequencies. The heating of the electrodes affects the electrode kinetics and the dc currents (by altering both the kinetics and the mass transport). In typical faradaic rectification experiments the amplitude of a sinusoidal ac perturbation is modulated by a lowfrequency square wave; consequently, if heating occurs, changes in the electrode temperature follow this modulation pattern. Under these conditions, even a small (fraction of K) change in the electrode temperature, which is periodic and synchronous with the amplitude modulation, makes the interpretation of results virtually impossible. However, in the method presented in this paper, there is no amplitude modulation; if heating occurs, the electrode temperature will be constant during the measurement (the proof of this statement is in our studies of hot microelectrodes). In this case, even large changes in the electrode temperature can be tolerated (if accurately measured and accounted for in the interpretation of results). Besides, various strategies can be developed to minimize heating of the electrode. CONCLUSIONS It has been shown that the effect of large-amplitude ac modulation on reversible redox processes is essentially trivial. The distortion of voltammetric curves causes broadening of the analytical signal without providing any useful information about the studied system. Still, any researcher using large-amplitude ac modulation has to be aware of these effects, because they may affect the interpretation of experimental results. The situation is very different in the case of completely irreversible processes, which after superimposing ac modulation on the electrode potential appear faster, often producing sharper peaks (which may have some advantage in electrochemical analysis). Furthermore, these effects probably play a key role in electrochemical etching of noble metals under ac conditions. We also speculated that, in the case of complex multistep electrode processes, a largeamplitude ac modulation may substantially alter the concentration of intermediates and that, under certain circumstances, may lead even to changes in the electrode reaction mechanism. According to our current understanding of the phenomenon, the effect of large-amplitude ac modulation on single-step fully irreversible processes should be independent of frequency. This opens a completely new possibility of determining the magnitude of double layer polarization at very high frequencies, which may lead to studies of the double layer capacitance and double layer relaxation phenomena at gigahertz frequencies. Note the magnitude of the distortion observed on reversible curves cannot be used for the same purpose because no electrode process can be fully reversible at gigahertz frequencies.

We also pointed out that in the case of fast electrode processes (reversible within the time scale of dc polarization and fast in comparison with the steady-state mass transport) the distortion caused by a high-frequency and large-amplitude ac modulation can provide information about the kinetics of the electron-transfer process. When frequencies