Determination of Kinetic and Thermodynamic Parameters of Surface

Anal. Chem. 2005, 77, 3357-3364

Determination of Kinetic and Thermodynamic Parameters of Surface Confined Species through ac Voltammetry and a Nonstationary Signal Processing Technique: The Hilbert Transform Costas A. Anastassiou,† Kim H. Parker,‡ and Danny O’Hare*,‡

Department of Bioengineering and Institute of Biomedical Engineering, Imperial College, London SW7 2AZ, U.K.

Data analysis of voltammetric responses has usually been done through application of the fast Fourier transform although it is widely accepted that electrochemical signals are intrinsically nonlinear and nonstationary. In this work, we present a time-series analysis based on the Hilbert transform (HT), a nonstationary signal processing technique, as an alternative tool that can overcome many of the difficulties associated with Fourier techniques. We use the HT to study the behavior of thin-film processes when the excitation perturbation is ac voltammetry. From the analysis of simulated data, we propose simple relations that enable species-specific kinetic and thermodynamic parameters to be estimated, without prior utilization of baseline subtraction even when double layer capacitance significantly influences the current response. We also propose a method to determine whether the characteristics of the applied voltage perturbation are adequate for the accurate estimation of these parameters. The methodology developed here will be applied to previously published experimental time series data (Guo, S. X.; Zhang, J.; Elton, D. M.; Bond, A. M. Anal. Chem. 2004, 76, 166-177.) obtained with ac voltammetry to show how a number of physical parameters can be directly extracted from the processed data. Various excitation waveforms of electrode potential, (e.g., linear, sawtooth, square wave, staircase, etc.,) have been studied in the past in analytical electrochemistry.2 Dc linear sweep, where the potential is swept linearly over the range of interest, and dc cyclic voltammetry (CV), where the scan rate is inverted at a predetermined point and the potential is scanned back cyclically, have emerged as the most frequently used input signals. Sinusoidal voltammetry (SV), which uses a sinusoid as excitation waveform, was recently shown to achieve excellent selectivity by optimizing the frequency and phase angle of the excitation to the analyte of interest and was used to study various biological molecules such as neurotransmitters and peptides.3 To apply these * Corresponding author. Fax: +44 (0) 207 594 5177. E-mail: [email protected]. † Institute of Biomedical Engineering. ‡ Department of Bioengineering. (1) Guo, S. X.; Zhang, J.; Elton, D. M.; Bond, A. M. Anal. Chem. 2004, 76, 166-177. (2) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; Wiley & Sons: New York, 2001. 10.1021/ac048137l CCC: $30.25 Published on Web 04/08/2005

© 2005 American Chemical Society

attractive features of SV to systems of unknown composition, ac voltammetry emerges as a good alternative by superimposing a “slower” ramp potential of CV with a “faster” sinusoidal perturbation. Ac voltammetry is an elegant and accurate method of electroanalysis when studying electrode processes, and in many cases, it has been shown to be superior to the widely used dc voltammetric methods,4 allowing the investigation of phenomena occurring on different time scales. For instance, rapid processes such as heterogeneous kinetics are detected by the “fast” ac component while slower processes, such as diffusion, are detected by the “slow” dc component. The main reason that ac voltammetry has not become as popular as other electrochemical techniques is probably due to the lack of an appropriate theoretical framework to deduce kinetic and thermodynamic parameters of the electrochemical reaction. Hitherto, the method of choice to process voltammetric signals has been the fast Fourier transform (FFT).5 Because large perturbation amplitudes enhance the nonlinear faradaic response,6,7 ac voltammetry was confined for a long time to applying rather small excitation amplitudes. It received a similar treatment as ac impedance, a method that uses a small-amplitude excitation perturbation very similar to ac voltammetry, where the data have often been analyzed with FFT and the Laplace transform to study the state of the electrodes in energy conversion devices as well as in corrosion.8-10 Engblom et al.11 and Gavaghan and Bond12 were the first to present a detailed study on the effect of large voltage amplitudes by calculating the voltammetric responses analytically and numerically. In their work, linear diffusion was the governing process and electrochemical reaction appeared as a boundary condition. The authors used FFT for the analysis of the output signal and observed that for high frequencies the kinetic effects manifest themselves in the higher harmonics of (3) Brazill, S. A.; Bender, S. E.; Hebert, N. E.; Cullison, J. K.; Kristensen, E. W.; Kuhr, W. G. J. Electroanal. Chem. 2002, 531, 119-132. (4) Bond, A. M. Modern Polarographic Methods in Analytical Chemistry; Marcel Dekker: New York., 1980. (5) McCord, T. G.; Smith, D. E. Anal. Chem. 1968, 289. (6) Smith, D. E. Electroanal. Chem. 1966, 1, 1. (7) Anta, J. A.; Marcelli, G.; Meunier, M.; Quirke, N. J. Appl. Phys. 2002, 92, 1002-1008. (8) Yoo, J.-S.; Park, S.-M. Anal. Chem. 2000, 72, 2035-2041. (9) Jurczakowski, R.; Lasia, A. Anal. Chem. 2004, 76, 5033-5038. (10) Smyrl, W. J. Electrochem. Soc. 1985, 132, 1551-1555. (11) Engblom, S. O.; Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 2000, 480, 120-132. (12) Gavaghan, D. J.; Bond, A. M. J. Electroanal. Chem. 2000, 480, 133-149.

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Table 1. Nomenclature Latin symbols A a C E/0 / E,* E/ac,E/dc, Ein F f f* I, i, if j k0 kf, kb O R Rg T t tsw v

electrode surface (m2) instantaneous amplitude (.) capacitance (F m-2) formal oxidation potential (V) applied, harmonic, ramp and initial potential (V) Faraday constant (C mol-1) instantaneous frequency (.) driving frequency (Hz) overall (A), dimensionless overall and dimensionless faradaic current response imaginary number standard electron-transfer constant (s-1) Forlward and back(rd electron-transfer rate constants (s-1) oxidized species form reduced species form universal gas constant (C V mol-1 K-1) temperature (K) time (s) ramp switching time (s) scan rate (V s-1)

the ac spectrum. Honeychurch and Bond13 used FFT for the investigation of voltammetric responses of species that were permanently adsorbed on the electrode surface. They applied their methodology to azurin molecules adsorbed on a paraffinimpregnated graphite electrode1 and came to the conclusion that even if the analysis provided valuable insights into various process parameters it could not adequately quantify the nonlinear behavior related to kinetic or thermodynamic dispersion or other phenomena that influence the thin-film process. A methodology based on the FFT for developing simple protocols to estimate various parameters of an electrochemical process was recently presented by Sher et al.14 A heuristic approach for data analysis was utilized to deduce mechanistic information that can be associated with reversible or quasi-reversible electrode processes. The authors investigated the impact of various parameters on the fundamental and higher harmonics and proposed a strategy of extracting similar information from experimental data through a selfcorrecting algorithm. The most common method of estimating electrochemical kinetic constants as a function of the potential is by using the Butler-Volmer kinetic equation, which is highly nonlinear. Using tools developed for periodic, stationary, and linear data sets, such as the FFT, becomes problematic when they are applied to data obtained from such a highly nonlinear process. In general, a nonlinear process does not obey the principle of superposition, nor does it have the property of frequency preservation.15 A nonstationary signal processing technique that has emerged as an alternative tool for the analysis of nonlinear phenomena is the Hilbert transform.16 Kiss and co-workers17 applied it to voltammetric time series of a population of electrochemical oscillators in order to measure emerging coherence. Arundell et al.18 were the first to use this technique for purely (13) Honeychurch, M. J.; Bond, A. M. J. Electroanal. Chem. 2002, 529, 3-11. (14) 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. (15) Lynn, P. A. An introduction to the analysis and processing of signals, 3rd ed.; Macmillan: New York, 1989. (16) Bendat, J. S.; Piersol, A. G. Random Data, 3rd ed.; Wiley & Sons: New York, 2000. (17) Kiss, I. Z.; Zhai, Y. M.; Hudson, J. L. Science 2002, 296, 1676-1678.

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Greek symbols R γ θ κ ξ σ τ, τavg ψ ∆τ

electron-transfer coefficient symmetry factor in a fraction of the electrode covered by oxidized species O dimensionless standard electron-transfer constant dimensionless excitation potential offset in a dimensionless and dimensionless average peak separation time offset in f dimensionless peak separation time

φ ω Ω Γ ∆E*

instantaneous phase (rad) harmonic perturbation frequency (s-1] dimensionless harmonic perturbation frequency Electrlode surface concentration [mol m-2] harmonic perturbation amplitude [V]

superscripts ∧ characteristic property

electroanalytical purposes, showing that the instantaneous amplitude and frequency resulting from the HT can be related to electrochemical phenomena occurring on the electrode surface. Other electrode processes such as passivation could also be detected. In this work, we introduce the HT as a tool for the characterization of various electrochemical parameters. The simulated reversible and quasi-reversible electrode surface processes were analyzed for three excitation waveforms: cyclic, sinusoidal, and ac voltammetry. We show that the kinetic and thermodynamic parameters of the process can be estimated from simple parameters related to the HT of the ac voltammetry current. THEORY Kinetic and Thermodynamic Equations. (For symbols used throughout, see Table 1.) We assume a monolayer of electrochemical active species permanently adsorbed on the electrode surface. The adsorbed molecules do not interact with each other but can undergo a one-electron reaction:

O+eaR

(1)

Electrode surface heterogeneity is neglected. For reversible reaction kinetics (Nernst equation), the current response I (A) including a constant double layer capacitance is given by19

I)

exp{-F(E* - E/0)/RgT} dE* dE* F2AΓ + CA RgT [1 + exp{-F(E* - E/)/R T}]2 dt dt 0 g (2)

( )

( )

where C (F m-2) is the double layer capacitance, E* (V) the applied potential, A (m2) the area of the electrode surface, E/0 (V) the formal reversible potential, and Γ (mol m-2) the electrode surface concentration. Equation 2 is derived from the Nernst (18) Arundell, M.; Patel, B. A.; Yeoman, M. S.; Parker, K. H.; O’Hare, D. Electrochem. Commun. 2004, 6, 366-372. (19) Newman, J. S. Electrochemical Systems, 2nd ed.; Prentice Hall: Upple Saddle River, NJ, 1991.

Figure 1. Typical excitation waveforms for CV and SV shown on upper left and right, respectively. The superposition of these two excitation waveforms results in the perturbation used for ac voltammetry, shown in the lower figure.

equation and depends only on E*. In the case of a quasi-reversible surface reaction

I ) FAΓ[kb(1 - θ) - kfθ] + CA

(dE* dt )

(3)

frequency of the superimposed sinusoidal waveform, and ∆Ε*(V) is its amplitude. Equations 7 and 8 are used when applying CV, eq 9 for SV, and eq 6 for ac voltammetry. To transform these equations into dimensionless quantities, we define three characteristic properties:

where θ is the fraction of the electrode covered by oxidized species O and kb and kf (s-1) are the backward and forward electron-transfer rate constants, respectively. We assume ButlerVolmer kinetics apply

kf ) k0 exp kb ) k0 exp

(

(

-RF (E* - E/0) Rg T

)

(4)

)

(1 - R)F (E* - E/0) Rg T

(5)

where R is the electron-transfer coefficient and k0 (s-1) is the standard electron-transfer constant. In all equations, Rg is the universal gas constant (8.31 C V mol-1 K-1), T the absolute temperature (298.15 K), and F the Faraday constant (96 485.3 C mol-1). The total applied potential E* in ac voltammetry consists of two components, a ramp and a high-frequency sinusoid, and is shown in Figure 1. The applied potential E* is defined as follows:

E*(t) )

E/dc(t)

+

E/ac(t)

(6)

E/dc )

E/in

+ 2vtsw - vt

tsw < t e 2tsw

E/ac(t) ) ∆E* sin (ωt)

(7) (8)

ˆt ) E ˆ /v

(11)

ˆI ) FAΓ/tˆ

(12)

τ ) t/tˆ

(13)

Ω ) ωtˆ

(14)

κ ) k0ˆt

(15)

ξ ) E* - E/0/E ˆ

(16)

i ) I/Iˆ

(17)

With these definitions, the nondimensional faradaic current if for a reversible electrochemical reaction is

where

0 e t e tsw

(10)

ˆt (s) is the characteristic time, Eˆ (V) the characteristic potential, and ˆI (A) the characteristic current. We now define the nondimensional parameters:

if ) E/dc(t) ) E/in + vt

E ˆ ) RT/F

exp{-ξ}

dξ [1 + exp{-ξ}]2 dτ

( )

(18)

while for quasi-reversible kinetics

if )

dθ ) κ exp{(1 - R)ξ}(1 - θ) - κ exp{-Rξ}θ (19) dτ

(9)

where v (V s-1) is the dc-scan rate, tsw (s) the dc switching time when the ramp direction is inverted, ω (rad s-1) is the angular

eq 19 is a linear first-order ODE for θ with respect to τ with nonconstant coefficients; ξ, which is the nondimensional applied voltage as defined by eq 16, is an independent parameter, Analytical Chemistry, Vol. 77, No. 10, May 15, 2005

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sometimes referred to as the control input. In the numerical calculations accounting for the overall quasi-reversible current response, eq 19 was solved by MATLAB differential equation subroutines20 and the result was added to the capacitance response according to eq 2 to give the overall current i. Hilbert Transform (HT) The HT of a real function of time x(τ), as introduced by Gabor,21,22 is defined as the real function y(τ) for which the complex function z(τ) ) x(τ) + jy(τ) is analytic with j being the imaginary number. Then y(τ) ) H[x(τ)]. Our analysis will be based upon the complex, analytic function

z(τ) ) x(τ) + jH[x(τ)]

(20)

which can be written in the polar form

z(τ) ) a(τ)ejφ(τ)

(21)

where a(τ) (.) is referred to as the instantaneous amplitude and φ(τ) (rad) the instantaneous phase of z(τ). From eq 21 we derive the frequency f(τ) (.) of z(τ) through differentiation:

f(τ) )

1 dφ(τ) 2π dτ

(22)

An advantage of the analytic signal approach is that a(τ) and f(τ) can be easily calculated from experimental time series.18 Furthermore, the Signal Processing Toolbox in MATLAB20 contains the HT algorithm, while f(τ) is conveniently calculated using a Savitzky-Golay differentiation filter.23 We will show that information about the electrochemical surface processes can be easily obtained from a(τ) and f(τ). All simulations were performed on a 2.8-GHz Pentium IV personal computer. RESULTS AND DISCUSSION To introduce the use of HT for analyzing voltammetric time series, we will initially apply CV and SV and then ac voltammetry. The initial case will involve reversible kinetics (Figure 2) and then we will look at the quasi-reversible case (Figure 3). Throughout this work, the CV parameters are E/in ) -0.4 V and v ) 0.5 V s-1 while the SV parameters are ∆E* ) 0.2 V and f* ) ω/2π ) 1 Hz. When applying ac voltammetry E/in, v, and ∆E* are kept constant and the driving frequency of the superimposed sinusoidal excitation will be taken to be f* ) 100 Hz. In all cases, E/0 ) 0 V, Γ ) 3 × 10-7 mol m-2, and the electrode radius r ) 3 mm. In Figure 3, for the analysis of quasi-reversible cases k0 ) 70 s-1 and R ) 0.5. In all cases where double layer capacitance was included in the dynamics of the system, C ) 10 µF cm-2. Since it is our intention to compare the HT to the FFT, we chose physical properties and electrode dimensions similar to the ones used by Guo et al.,1 who based the analysis of their simulations and experimental measurements on FFT. (20) Matlab 6.5; The MathWorks Inc.: Natick, MA, 2002. (21) Gabor, D. Proc. IEE 1946, 93, 429-457. (22) Boashash, B. Proc. IEEE 1992, 80, 4, 520-537. (23) Vetterling, W. T.; Teukolsky, S. A.; Press, W. H.; Flannery, B. P. Numerical Recipes, 2nd ed.; Cambridge University Press: Cambridge, 1992.

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Reversible Surface Reaction. When applying the three excitation potentials shown in Figure 1, namely CV, SV, and ac voltammetry, to surface-confined species undergoing a reversible reaction step according to eq 1, we obtain the voltammograms shown in the left column of Figure 2. The a and f, as calculated with the HT for these voltammograms are shown to the right. In the first row of Figure 2 we applied CV, in the second SV, while in the third and fourth ac voltammetry. In the first three rows, only the faradaic component is shown, i.e., the solution and analysis of eq 18. In the fourth row, double layer capacitance is included in the dynamics of the process. In the first two rows, the time-series analysis of the faradaic response to CV and SV immediately reveals the moment when ξ ) 0, and therefore E/0, by the maximums of both a and f. The symmetry of the peaks in shape and in absolute value implies the reversibility of the reaction step. One notices also that there are specific periods when i ) 0 but the instantaneous amplitude a * 0, for example, when τ ) 25-40 for CV. This is because we do not analyze the current response signal x(τ) directly but its corresponding analytic signal z(τ), as defined by eq 21. The nonzero a at these periods is in agreement with the HT theory.16 In the third and fourth rows, we observe how when applying ac voltammetry the cyclic crossing of the formal oxidation potential in the presence of reversible kinetics results in patterns that can be described by their envelopes. The fact that the current output signal for the faradaic case, third row, always preserves a local zero mean for each oscillation results in a zero offset. We see that ξ ) 0 is now indicated by the position of the peaks of the envelopes. When capacitance is neglected for the reversible ac voltammogram, f is dominated by numerical noise during the periods when the output signal is very small. Double layer capacitance has a significant effect on the calculated current, as seen in the fourth row. In this case, the local mean is again zero but the linear capacitance component, as defined in eq 2, leads to an offset, “baseline” σ in a, with σ ) 22.50. The effect of capacitance on f is more drastic and is seen when comparing the results shown in the third row with the well-defined envelopes in the fourth row, which have similar behavior with a. This phenomenon is attributed to the presence of a fundamental linear oscillatory component in the current response resulting in the instantaneous frequency baseline ψ ) 5.15. From the above we come to the conclusion that σ and ψ are properties characterizing the capacitance effect. Based on eq 2 if

∆E*ω . v

(23)

we can deduce C directly from a and σ:

C ) σIˆˆt /∆EΩE ˆ

(24)

Additionally, the driving frequency f* can be extracted from the data through ψ by

f* ) ψ/tˆ

(25)

The utilization of the HT for the analysis of surface-confined reversible electrochemical reactions when the excitation technique is ac voltammetry allows not only the clear definition of E/0 but

Figure 2. Calculated voltammograms of the reversible case for the three different excitations (first row, CV; second row, SV; third and fourth rows, ac voltammetry). In the left column, the voltammetric response to each perturbation is shown while the second and third columns show the instantaneous amplitude a and instantaneous frequency f after applying the HT to the current response. In the first three rows, only faradaic events are considered whereas in the fourth row the effect of capacitance is included resulting in the capacitance-characteristic offsets σ and ψ.

also of double layer capacitance C. A significant advantage when using this method is that one avoids baseline subtraction, a process that can prove problematic for all cases where faradaic events cannot be clearly distinguished from capacitance response. In the next section, we will concentrate on quasi-reversible kinetics and how compound-specific parameters can be directly quantified through the HT analysis. Quasi-Reversible Surface Reaction. In Figure 3, we show the analysis of a quasi-reversible process reacting to ac voltammetry perturbation, without double layer capacitance in the upper row and with capacitance in the lower row. The asymmetric envelopes observed in the left column are typical when kinetic effects affect the electrochemical process. This asymmetry has an impact on the HT analysis. While a single peak in the envelope indicated where ξ ) 0 for the reversible case, for quasi-reversible cases, two intersecting envelopes occur. From a number of simulations it was observed that in order to estimate E/0 the

midway potential, as used in CV, can be adopted:

E/0 ) E*(τav)

with

τav ) (τ1 + τ2)/2

(26)

where τ1 and τ2 are the dimensionless time coordinates of the first and second peaks of the envelope of a. In f, similar behavior with the reversible capacitance-free case is observed; even close to ξ ) 0 peak separation is also visible (as for a) and is attributed to the kinetic limitations of the process. Away from ξ ) 0, numerical noise dominates. With increasing k0, the spikes become larger. As can be seen in the lower row, capacitance C and driving frequency f* can be determined directly from the HT using eqs 24 and 25 just as in the case of reversible kinetics. In the Butler-Volmer model, there are two macroscopic phenomenological parameters, k0 (or κ), a measure of the kinetic facility of the redox couple and, R, the charge-transfer coefficient, Analytical Chemistry, Vol. 77, No. 10, May 15, 2005

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Figure 3. Output signal calculated for a quasi-reversible surface reaction as well as a and f resulting from ac voltammetry, the upper for purely faradaic response, the lower including capacitance. The compound-specific parameters used for both simulations are k0 ) 70 s-1 and R ) 0.50. Kinetic limitations result in asymmetric voltammograms and intersecting envelopes. Based on the characteristic shapes of the envelopes, we introduce factors to deduce kinetic and thermodynamic information about the reaction. As can be seen in the last row, C and f* can be evaluated directly from the time domain analysis using eqs 24 and 25. We observe that numerical noise dominates f in the absence of capacitance.

which is a measure of the symmetry of the energy barrier in the reaction coordinate. Since the electron-transfer coefficient R is defined as a symmetry measure, we chose to estimate it through the factor γ representing the symmetry in a

γ)

γ 1 - γ2 γ1 + γ2

(27)

where γ1 and γ2 are the values of the two peaks of the a-envelopes measured relative to the capacitance baseline. In Figure 4, the top figure illustrates the exact definition of the attributes of γ. In the middle of Figure 4, we show the relation between γ and R for different values of k0 (k0 ) 10-104 s-1). The dots represent simulated data and the lines the linear regressions for each data set. It can be seen that there is a linear relation between relative peak height difference and R for different values of k0. The more R deviates from 0.5 the larger the difference in the peaks becomes. For fast kinetics (large k0) the effect of R on eq 19 is negligible compared to k0 and the difference in envelope height remains small. For small k0, the effect of R is no longer negligible and the difference in heights grows. To determine k0, we introduce a second factor, ∆τ:

∆τ ) τ2 - τ1

(28)

which relates peak separation to k0, a method first proposed by Laviron24 for CV. We conducted a large number of simulations, and we found that ∆τ was nearly independent of R over the range of R for which an envelope of the current signal was clearly defined, 0.5 < R < 0.7. This is seen in Figure 4c, where the dots represent the average ∆τ determined for R ) 0.5, 0.6, 0.65, and 0.7 and the error bars the range of the four values. Analogous (24) Laviron, E. J. Electroanal. Chem. 1979, 101, 19-28.

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behavior is observed for 0.3 < R < 0.5. It has to be mentioned that the larger the difference between ac and dc excitation, as defined by eq 23, the more precise the definition of the envelopes becomes and the range in the graphs (such as the ones shown in Figure 4) further decreases. The fact that k0 ) 200 s-1 represents a limit of our excitation waveform is manifested in Figure 5, where the relation between γ and k0-1 is shown. For k0 < 200 s-1, we observe a clear separation between the different cases and we can estimate R from the asymptotes (toward larger k0-1) on the γ -axis through:

R ) (γ + 1)/2

(29)

For k0 > 200 s-1, it becomes difficult to distinguish between different cases and estimations for R and k0, have to be done cautiously. Nevertheless, one has to keep in mind that the results on Figures 4 and 5 are based on the characteristic properties of the excitation waveform (∆E* ) 0.2 V, f* ) 100 Hz). To analyze an adsorbed species with k0 > 200 s-1, faster f* can be applied leading to graphs similar to Figures 4 and 5 that discriminate cases of R for higher k0. The analysis on Figures 4 and 5 is based on parameters of the ac voltammetry waveform that are easily applied with standard instrumentation. In our analysis, we have not used the frequency information, except for the offset as an indicator of f*. However, we anticipate that this information will be valuable for the interpretation of more complex processes, for example, when diffusion is important. In this work, it is not our intention to analyze all possible variations of ac voltammetry but much more to prove that the HT can provide a powerful tool to analyze time series resulting from ac voltammetry experiments. To illustrate the usefulness of the HT method, we applied it to the experimental time series obtained with ac voltammetry by Guo et al.1 for a thin film of azurin, an electron-transporting protein belonging to the family of “blue” copper proteins (their Figure

Figure 5. γ vs k0-1 plot for (b) R ) 0.50, (2) R ) 0.60, (9) R ) 0.65, and (1) R ) 0.70 to determine if the applied excitation perturbation is adequate to estimate the kinetic parameters of the thin-film process under investigation. For the given waveform, k0 ) 200 s-1 represents a limit of accuracy and is indicated by the dashed line. The region to the right of the dashed line represents the region where estimations are accurate, as seen from the linear asymptotic behavior of the curves. In this case, eq 29 can be used safely. To the left of the dashed line, predictions should be made cautiously. To be able to make safe predictions for k0 > 200 s-1, we have to increase the excitation frequency f*.

Figure 4. (a) Parameters used for the estimation of R and k0 for a quasi-reversible redox couple defined. (In the simulation shown, k0 ) 70 s-1 and E/0 ) 0 V). The parameters γ1, γ2, τ1, and τ2 are defined via the peak positions of the envelope of a. Potential waveform parameters: E/in ) -0.4 V, v ) 0.5 V s-1, ∆E* ) 0.2 V, and f* ) 100 Hz. The analysis is based on the forward sweep, but due to symmetry, similar conclusions can be derived from the backward sweep. (b) The linear relation between peak symmetry factor γ, as defined by eq 27, and R is shown for (1) k0 ) 10 s-1, (2) k0 ) 102 s-1, (3) k0 ) 103 s-1, and (4) k0 ) 104 s-1. (c) ∆τ, as defined by eq 28, is shown as a function of k0 for R ) 0.50, R ) 0.60, R ) 0.65, and R ) 0.70. Dots represent the mean and the error bars the range of the simulated data points.

10). Azurin experiments were conducted on a paraffin-impregnated graphite electrode (PGE) in a buffer solution (pH 8.0) containing 0.02 M (HOCH2)3CNH2 and 0.1 M NaCl. It has been shown that dynamics of the process can realistically be modeled by the Butler-Volmer kinetics.25 The main difficulty when studying the electrochemical behavior of azurin is the strong influence of capacitance which overwhelms the faradaic response signal. This problem is particularly apparent for a PGE, which has a rough surface with numerous complex chemical groups.26 The results of our analysis are presented in Figure 6, where the left column shows on the top the experimentally measured output signal and at the bottom the theoretical voltammogram resulting from the simulation, which used parameters estimated through the HT analysis. The authors measured Γ ) 3 × 10-11 mol cm-2 using (25) Jeuken, L. J. C.; Armstrong, F. A. J. Phys. Chem. B 2001, 105, 5271-5282. (26) Jeuken, L. J. C.; Armstrong, F. A. J. Phys. Chem. B 2002, 106, 2304-2313.

Figure 6. Left column: nondimensional experimental voltammogram of the azurin thin film obtained from Guo et al.1 as well as the theoretically calculated voltammogram based on the kinetic and thermodynamic parameters deduced from the HT analysis. Right column: a (top) and f (bottom) calculated from the experimental current output using the HT. These were then used to estimate R, k0, and E0 as described in the text.

CV ,and the experimental data were nondimensionalised via eqs 10-17 using the waveform characteristic properties reported (∆E* ) 80 mV, f* ) 9.54 Hz, and v ) 50 mV s-1). In the right column, we show a and f after applying the HT on the experimental data. From the offset σ, we estimate (through eq 24) C ) 88 µF cm-2. The typical ranges of capacitance for carbon electrodes lies within 10-70 µF cm-2 27 implying a high surface roughness factor, which is not unusual for porous electrodes.28 From eq 26, we estimate E/0 ) 280 mV versus a standard hydrogen electrode (SHE) which is in good agreement with Jeukes and Armstrong;25 they reported E/0 ) 252 mV versus SHE for pH 9.0. Using the dimensionless quantities γ and ∆τ as defined by eqs 28 and 29 (27) Kinoshita, K. Carbon: Electrochemical-Physicochemical Properties; Wiley & Sons: New York, 1988. (28) McCreery, R. L.; Rice, K. K. In Laboratory Techniques in Electroanalytical Chemistry; Kissinger, P. T., Heineman, W. R., Eds.; Marcel Dekker: New York, 1996; pp 293-332.

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and an analysis identical to the one shown in Figures 4 and 5, we estimate R ) 0.55 and k0 ) 30 s-1 ((10 s-1). Despite the fact that these waveform parameters are within the region where estimations are difficult (the critical value of k0, as presented for Figure 5, is 30 s-1), the voltammogram resulting from the simulation, which used the above-reported parameters, is in good agreement with the experimental data. CONCLUSIONS Ac voltammetry in combination with the HT, a nonstationary signal processing technique, can provide remarkable insight into surface-confined electrochemical processes. An advantage of the application of this time domain analysis, which results in the instantaneous amplitude and frequency of the output signal, is that it overcomes the difficulties of baseline subtraction. Moreover, kinetic and thermodynamic parameters can easily be estimated from the HT analysis. The formal oxidation potential and capacitance can be deduced from the data and for quasi-reversible kinetics obeying the Butler-Volmer kinetics, simple relations can

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Analytical Chemistry, Vol. 77, No. 10, May 15, 2005

be applied to estimate R and k0. Whether the waveform is suitable to quantify the kinetic parameters of the compound under investigation can also be obtained with the analysis presented in Figure 5. This, together with the fact that the driving frequency f* can be monitored through f at every moment, suggests that the HT can be used to analyze time series of excitation waveforms where the parameters of the perturbation are a function of time, as for example in chirps. ACKNOWLEDGMENT We thank S. Guo, J. Zhang, and A.M. Bond for providing the experimental data. C.A.A. thanks the Institute of Biomedical Engineering at Imperial College and the EPSRC Life Science Interface grant GR/R89127/01 for funding.

Received for review December 16, 2004. Accepted March 17, 2005. AC048137L