Resistance, Capacitance, and Electrode Kinetic Effects in Fourier

Anal. Chem. 2004, 76, 6214-6228

Resistance, Capacitance, and Electrode Kinetic Effects in Fourier-Transformed Large-Amplitude Sinusoidal Voltammetry: Emergence of Powerful and Intuitively Obvious Tools for Recognition of Patterns of Behavior Anna A. Sher,† Alan M. Bond,*,‡ David J. Gavaghan,*,† Kathryn Harriman,† Stephen W. Feldberg,§ Noel W. Duffy,‡ Si-Xuan Guo,‡ and Jie Zhang‡

Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, United Kingdom, School of Chemistry, Monash University, Clayton, Victoria 3800, Australia, and Energy Science and Technology Department, Brookhaven National Laboratory, Upton, New York 11973-500

Large-amplitude sinusoidal ac voltammetric techniques, when analyzed in the frequency domain using the Fourier transform-inverse Fourier transform sequence, produce the expected dc and fundamental harmonic ac responses in addition to very substantial second, third, and higher ac harmonics that arise from the presence of significant nonlinearity. A full numerical simulation of the process, Red a Ox + e-, incorporates terms for the uncompensated resistance (Ru), capacitance of the double layer (Cdl), and slow electron transfer kinetics (in particular, the reversible potential (E°), rate constant (k0), and charge transfer coefficient (r) from the Butler-Volmer model). Identification of intuitively obvious patterns of behavior (with characteristically different sensitivity regimes) in dc, fundamental, and higher harmonic terms enables simple protocols to be developed to estimate Ru, Cdl, E°, k0, and r. Thus, if large-amplitude sinusoidal cyclic voltammograms are obtained for two concentrations of the reduced species, data obtained from analysis of the recovered signals provide initial estimates of parameters as follows: (a) the dc cyclic component provides an estimate of E° (because the Ru and k0 effects are minimized); (b) the fundamental harmonic provides an estimate of Cdl (because it has a high capacitance-to-faradaic current ratio); and (c) the second harmonic provides an estimate of Ru, k0, and r (because the Cdl effect is minimized). Methods of refining the initial estimates are then implemented. As a check on the fidelity of the parameters (estimated on the basis of an essentially heuristic approach that solely utilizes the dc, fundamental, and second harmonic voltammograms), comparison of the predicted simulated and experimental third (or higher) harmonic voltammograms can be made to verify that * Corresponding authors. E-mail addresses: [email protected] (A.M.B.); [email protected] (D.J.G.). † Oxford University Computing Laboratory. ‡ Monash University. § Brookhaven National Laboratory.

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agreement between theory and experiment has been achieved at a predetermined level. The use of the heuristic pattern recognition approach to evaluate the oxidation of ferrocene at a platinum electrode (a reversible process) in the very high resistance solvent dichloromethane (0.1 M Bu4NPF6) and the reduction of [Fe(CN6)]3- at a glassy carbon electrode (a quasi-reversible process) in much lower resistance but higher capacitance conditions found in aqueous (0.5 M KCl) media is described and verifies the inherent advantages of employing large-amplitude sinusoidal techniques in quantitative studies of electrode processes. Quantitative studies of the mechanisms of electrode processes are frequently required in electrochemical studies (see refs 1-7 for example). At present, the tool most commonly employed for this task is probably dc cyclic voltammetry in which a triangular waveform is applied to an electrochemical cell. Typically, the scan rate is varied, then simulated, and experimental voltammograms are compared, in order to establish the mechanism. A vast theoretical literature is available to support this technique.1 Many considerably more sophisticated transient methods are available for electrode mechanism evaluation. For example, the (1) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; John Wiley and Sons: New York, 2001. (2) Bond, A. M. Modern Polarographic Methods in Analytical Chemistry; Marcel Dekker: New York, 1980; p 341. (3) Kissinger, P. T.; Heineman, W. R. Laboratory Techniques in Electroanalytical Chemistry; Marcel Dekker: New York, 1984. (4) Mansfeld, F.; Lorenz, W. J. In Techniques for Characterization of Electrodes and Electrochemical Processes; Varma, R., Selman, J. R., Eds.; John Wiley: New York, 1991; Chapter 12, p 581. (5) Sluyters-Rebach, M.; Sluyters, J. H. In Comprehensive Treatise of Electrochemistry; Yeager, E., Bockris, J. O’M., Conway, B. E., Sarangapani, S., Eds.; Plenum Press: New York, 1980; Vol. 9, p. 177. (6) Smith, D. E. Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1966; Vol. 1. (7) (a) Fisher, A. C. Electrode Dynamics; Oxford University Press: Oxford, 1996. (b) Park, S. M.; Yoo, J. S. Anal. Chem. 2003, 75, 455A. (c) Rubenstein, I., Ed. Physical Electrochemistry: Principles, Methods, and Applications; Marcel Dekker: New York, 1995. (d) Macdonald, J. R., Ed. Impedance Spectroscopy: Emphasizing Solid Materials and Systems; Wiley: New York, 1987. 10.1021/ac0495337 CCC: $27.50

© 2004 American Chemical Society Published on Web 09/25/2004

Figure 1. Sinusoidal ac voltammetric experiment. (a) Waveform employed in linear sweep ac voltammetry (sum of sine wave of any amplitude, (i), and dc linear sweep, (ii), produces the waveform used, (iii)). (b) Total current ac voltammogram produced by application of the cyclic version of the waveform.

technique of electrochemical impedance spectroscopy (see refs 1 and 3-7 for example) most commonly employs a constant dc potential rather than a triangular waveform, and a periodic ac signal containing a wide range of frequencies is superimposed onto the dc signal. The amplitude of the periodic signal is maintained at a small value so that linear network analysis of equivalent circuit models can be used to quantify the phenomena associated with the interface formed in the electrochemical experiment. While this is a very powerful method, its use in mechanistic studies is limited by the inherent complexity of equivalent circuit analysis relative to the more general and intuitively obvious theory available for dc cyclic voltammetry.8 Pioneering work by Smith et al.2,6 demonstrated that smallamplitude ac voltammetric methods also provide a powerful method for analysis of electrode mechanisms. These studies usually were confined to the linear analysis region, as is the case with impedance spectroscopy. Recently, work from these and other laboratories9-13 has demonstrated that an analogue of dc linear sweep or cyclic voltammetry can be readily implemented both experimentally and theoretically (faradaic current only considered) by superimposing a periodic ac waveform of any amplitude onto a linear or triangular voltage (Figure 1a). Analysis of experimental data by Fourier transform methods enables the dc and ac components to be separated. Simulations employing the same well-known protocols applied in dc cyclic voltammetry8 combined with Fourier transform analysis then produce the dc and fundamental harmonic resolved responses, in addition to resolved higher ac harmonics that emerge from nonlinearities associated with the use of large-amplitude methods. Unlike the case with impedance spectroscopy, there is no need to limit the problem to the linear theoretical analysis regime and use of a (8) Rudolph, M.; Reddy, D. R.; Feldberg, S. W. Anal. Chem. 1994, 66, 589A. (9) Engblom, S. O.; Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 2000, 480, 120. (10) Gavaghan, D. J.; Bond, A. M. J. Electroanal. Chem. 2000, 480, 133. (11) Gavaghan, D. J.; Elton, D. M.; Bond, A. M. J. Electroanal. Chem. 2001, 513, 73. (12) Guo, S-X.; Zhang, J.; Elton, D. M.; Bond, A. M. Anal. Chem. 2004, 76, 166. (13) Rosvall, M.; Sharp, M. Electrochem. Commun. 2000, 2, 338.

swept potential instead of a series of constant dc potentials is trivial to implement from a theoretical perspective. Thus, the dc potential can be swept rapidly so that all required data are available in a very short time. Furthermore, the separation of the dc and ac harmonics gives rise to patterns of behavior that are intuitively obvious, unlike equivalent circuit plots that are employed to analyze data obtained in electrochemical impedance spectroscopy. To date, studies using Fourier-transformed large-amplitude ac voltammetry involving solution soluble processes of the kind considered in this paper have focused on reversible processes, and the influence of capacitance, resistance, and slow electron transfer have yet to be considered in any detail with respect to patterns of behavior that may readily enable all of these parameters to be quantified. In order to make experimental-theoretical comparisons fully realistic, we have now included charging current, uncompensated resistance, and electrode kinetics into the simulations and identified the patterns of behavior that emerge in an intuitively obvious manner for each of these terms. In a recent study involving surface-bound processes12 the influence of electrode kinetics and capacitance (but not resistance) was considered and will exhibit some relationships to the solution soluble mechanism which are of interest in the present case. In order to confirm the fidelity of large-amplitude sinusoidal voltammetric techniques analyzed by the Fourier transform method and detection of patterns of behavior, simulated ac voltammograms are compared against experimental data obtained for the

Fc h Fc+ + e-

(1)

(Fc ) ferrocene) process under very high resistance conditions in dichloromethane (0.1 M Bu4NPF6). The low resistance, quasireversible case considered is reduction of [Fe(CN)6]3- in aqueous media:

[Fe(CN)6]3- + e- h [Fe(CN)6]4-

(2)

Problems involving significant contributions from each of high Analytical Chemistry, Vol. 76, No. 21, November 1, 2004

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resistance, slow electron transfer, and capacitance are considered on the basis of patterns of behavior found in simulated data. Importantly, we conclude our discussion by emphasizing the distinctly different patterns of behavior that emerge with respect to the influence of each of the capacitance, resistance, and electrode kinetic terms on dc and ac fundamental and higher harmonic terms. These patterns enhance the ability of an experimentalist to rapidly recognize the probable presence of a particular mechanism or nuance in a particular frequency (or time) domain and then quantitatively and systematically analyze the data obtained from Fourier-transformed large-amplitude ac techniques. EXPERIMENTAL SECTION Reagents. Ferrocene was of reagent grade purity and used as supplied by the manufacturer (Aldrich). The supporting electrolyte, Bu4NPF6, used for studies of ferrocene in organic solvents was obtained from GFS and purified by recrystallization as described in the literature.3 Dichloromethane and acetonitrile were HPLC grade and distilled and dried over basic alumina prior to use. For studies in aqueous media, K3[Fe(CN)6] was used as the source of [Fe(CN)6]3- and 0.5 M KCl was used as the supporting electrolyte. The deionized water was obtained from a MilliQMilliRho purification system (resistivity 18 MΩ cm). Instrumentation and Electrochemical Procedures. The Fourier transform (FT) ac voltammetric instrumentation was based on a conventional three-electrode potentiostat driven by a 19 bit delta sigma digital-to-analog converter. The electrode potentials were digitized by separate 18 bit delta sigma analogto-digital converters. The system was run synchronously at a sampling rate of 39063.5 samples s-1. All signal processing was performed using a desktop computer with a 550-MHz Pentium III processor running Windows 98, and the computer code was written in C++. Further, the sampling rate was selected to ensure that each of the data sets in the experiments was 215 bytes (32768) in size so as to avoid aliasing and to optimize the speed of the FT processing. A moving average of eight data points was used for data collection, meaning that the effective sampling rate was 4.88 × 103 samples s-1. The effect of this averaging was compensated for in the subsequent signal processing. Full details of the instrumentation and signal processing algorithms will be described elsewhere.14 Conventional dc cyclic voltammetric experiments and solution resistance measurements were undertaken with a Bioanalytical Systems (BAS model 100B) electrochemical workstation (Bioanalytical Systems, West Lafayette, IN). A conventional three-electrode cell was employed in all electrochemical measurements, with Pt or GC macrodisk working electrodes, Ag/AgCl (3 M NaCl) (BAS, West Lafayette, IN) (or Ag wire) as the reference (or quasi-reference) electrode, and a platinum wire as the auxiliary electrode. The working electrodes were polished with 0.3-µm alumina on a clean polishing cloth (Buehler, USA), rinsed with water, and dried with tissue paper. All experiments were carried out at (20 ( 1) °C, and solutions were purged with nitrogen for at least 10 min prior to commencement of a voltammetric experiment. (14) Manuscript in preparation.

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Parameters Employed in Experimental Studies. The working electrode areas were determined (A ) 0.0069 cm2 for Pt and A ) 0.0776 cm2 for GC) by linear sweep voltammetry using the Randles-Sevcik relationship1 and known diffusion coefficients (7.6 × 10-6 cm2 s-1 for [Fe(CN)6]3- in water (0.5 M KCl) and 2.3 × 10-5 cm2 s-1 for ferrocene in acetonitrile (see ref 1, p 813 and ref 15)). The definition of A and of all other symbols used in this paper is given in footnote 16. As a cross check on the fidelity of the new ac technique, uncompensated resistance values also were measured by the method available with the BAS workstation.17 Values of Ru obtained by this method were in the range of 3100-3400 Ω for Fc in DCM and e50 Ω for [Fe(CN)6]3- in water. For Fourier-transformed ac voltammograms, the scan rate was 397.36 mV s-1, the ac amplitude was 80 mV, and the frequency was 90.6 Hz. The experimental data, obtained with the FT form of instrumentation, yield current, time, and applied potential as the output information. The results are plotted either as current versus time or current versus dc applied potential. To obtain the dc applied potential, the expression Edc ) Estart + vt was employed. MODELING THE VOLTAMMETRIC EXPERIMENT In the experiments under consideration in this paper, a sinusoidal type waveform of any amplitude is superimposed onto the linear or triangular dc potential waveform which is swept at a finite scan rate, v. Figure 1a illustrates the waveform used for the linear sweep case, and Figure 1b shows the output when total current is plotted against time in the cyclic form of the experiment. We shall assume that the electrode reaction is a one electron oxidation process,18 where a species Red is oxidized to Ox by the loss of electrons to the electrode:

Red a Ox + e-

(3)

By IUPAC convention, the current for an oxidation process is a positive quantity. Further, we consider both completely reversible and quasireversible systems where Ox is initially absent under conditions, where the starting potential for the voltammetric experiment is significantly negative with respect to the reversible potential E°, and the initial scan direction is positive. We assume semi-infinite (15) Ikeuchi, H.; Kanakubo, M. Electrochemistry 2001, 69, 34. (16) Throughout this work, we shall use the following notational convention: all dimensional variables will be displayed in the usual italic mathematical typeface, as x, and the nondimensional variables will be displayed with a tilde sign, as ˜x. The symbols used in the text have the following definitions: A, electrode area; Cdl, capacitance of the double layer; Ru, uncompensated resistance; k0, heterogeneous charge transfer rate constant; E°, formal potential; R, charge transfer coefficient; f, frequency; ∆E, sine wave amplitude; v, scan rate; Estart, Eend, initial and final potential; D, diffusion coefficient; c, c∞, concentration and bulk concentration; P, period of sine wave; ω, angular frequency; Edc, Eac, dc and ac potential; Ic, Itot, capacitive and total current. (17) He, P.; Avery, J. P.; Faulkner, L. R. Anal. Chem. 1982, 54, 1313A. (18) Modeling of a reduction process or a cyclic voltammogram is readily achieved by choice of appropriate initial and boundary conditions (ref 1) and addressing issues related to the sign of the scan rate (ref 1). While we have not extensively exploited the cyclic form of voltammetry, significant additional information can be obtained by this version of the technique and indeed by repetitive cycling of the potential.

Table 1. Example Set of Reference Simulation Parameters Used to Theoretically Study the Influence of Different Parameters in ac Voltammetry

range of dc applied potential formal potential amplitude of applied potential period of applied potential frequency of applied potential angular frequency of applied potential scan rate bulk concentration of species Red rate constant area of electrode diffusion coefficient charge transfer coefficient uncompensated resistance double layer capacitance

linear diffusion with equal diffusion coefficients, D, for both Red (Dred) and Ox (Dox) species. Possible mass transport mechanisms are diffusion, convection, and migration. We concentrate only on the diffusion mechanism described by Fick’s second law of diffusion, since migration may be neglected, due to the presence of a high concentration of supporting electrolyte solution, as may convection, because we employ a stationary electrode and utilize short time-scale experiments that only have a total duration of 5 s or less. In Appendix A in the Supporting Information we give a detailed treatment of the derivation of the mathematical model used in all simulations in this paper. We use the standard approach of modeling a macroelectrode using the one-dimensional diffusion equation.19-22 Relationships between nondimensional and dimensional variables are summarized in Appendix A in the Supporting Information. Throughout this paper, numerical simulations are performed either for theoretical studies of the effect of varying different parameters or for comparison with experimental data; for dimensional and nondimensional parameter values for an example of a reference experiment see Table 1. Simulations of the electrode processes, Fourier transforms, and inverse Fourier transforms were undertaken using MATLAB.23 The software developed in this work will be made available on request to the authors. PATTERNS OF BEHAVIOR ASSOCIATED WITH THE INFLUENCE OF Cdl, Ru, and Electrode Kinetics Analysis of Total Current Voltammogram. Experimental and theoretical voltammograms of the kind shown in Figure 1b are obtained and then deconvoluted into dc and ac harmonics by a fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) sequence. (19) Gavaghan, D. J.; Elton, D. M.; Oldham, K. B.; Bond, A. M. J. Electroanal. Chem. 2001, 512, 1. (20) Gavaghan, D. J.; Myland, J. C.; Oldham, K. B. J. Electroanal. Chem. 2001, 516, 2. (21) Morton, K. W.; Mayers, D. F. Numerical Solution of Partial Differential Equations; Cambridge University Press: Cambridge, 1994. (22) Feldberg, S. W. J. Electroanal. Chem. 1981, 121, 1. (23) Matlab 6.5; The MathWorks Inc.: Natick, MA, 2002.

dimensional

nondimensional

Edc ) -0.2569 to 0.2569 V E° ) 0 V ∆E ) 25 mV P ) 0.007 s f ) 150 Hz ω ) 942.47 rad s-1 v ) 1 V s-1 c∞ ) 10-6 mol cm-3 ≡10-3 M k0 ) 6240.3 cm s-1 A ) 1 cm2 D ) 2 × 10-5 cm2 s-1 N/A Ru ) 25 Ω Cdl ) 10-4 F

E ˜ dc ) -10 to10 E ˜° ) 0 ∆E ˜ ) 0.97 P˜ ) 0.26 N/A ω ˜ ) 7.57π ˜v ) 1 u ˜∞ ) 1 k˜ 0 ) 3 × 105 N/A N/A R ) 0.5 R ˜ u ) 1.85 C ˜ dl ) 1.9 × 10-8

Similarly to Gavaghan et al.,10,11,19,20 the numerical study of the total current (dc + ac) voltammogram shown in Figure 1b is performed in the following manner: (1) Simulative Modeling. The parabolic partial differential equation with associated initial and boundary conditions is solved and the simulated current response is recorded as a function of potential (time). (2) Frequency Domain Analysis. The simulated or experimental signal is analyzed in the frequency domain by examining the power spectrum, filtering certain frequencies and inverting them back into the time domain, in order to study the recovered currents which represent dc and harmonics of the ac component of the original signal. (3) Data Analysis. The simulated and experimental results are compared. As the final stage of analysis, we achieve agreement between the simulated and experimental data at a level of accuracy which is defined by subtracting the numerical results from experimental. Patterns of Behavior in dc Voltammetry. Patterns of behavior are detected in conventional dc voltammetry but are far less powerful than those observed by examination of the Fouriertransformed ac voltammograms. Figure 2 contains examples of conventional dc cyclic voltammograms obtained theoretically as a function of capacitance, uncompensated resistance, and rate of electron transfer for a typical set of parameters that encompass ranges that are likely to be encountered experimentally. Simulations are presented at concentrations of 0.10 mM and 1.0 mM. Clearly, patterns of behavior are present that characterize the influence of each term. For example, the presence of capacitance leads to an offset in the dc current at long times and a decrease in the rise time (RuCdl time constant term) at the commencement of the experiment. Uncompensated resistance also affects the RuCdl time constant and distorts the capacitance and faradaic current terms in a characteristic manner.1 Slow electron transfer distorts the faradaic current term by broadening the voltammogram in a manner similar to the influence of Ru but does not affect the RuCdl time constant. Importantly, increasing the concentration from 0.10 mM to 1.0 mM leads to very characteristic changes. In particular, if ItotRu is negligible at both concentrations, simply a scaling of the faradaic Analytical Chemistry, Vol. 76, No. 21, November 1, 2004

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Figure 2. Simulated dc voltammograms obtained using a range of Ru and Cdl values with other input parameters from Table 1. The reversible processes are illustrated in blue Ru ) 0 Ω and Cdl ) 0 F, in black Ru ) 0 Ω and Cdl ) 10-4 F, in red Ru ) 25 Ω and Cdl ) 10-4 F. The quasi-reversible process is shown in green with k˜ 0 ) 0.01, Ru ) 25 Ω, and Cdl ) 10-4 F.

current is observed, while if ItotRu is significant, the voltammetric wave shape broadens and the peak potential shifts. Detection of this latter pattern provides a clear distinction between the effects of electrode kinetics and ItotRu on voltammograms. Analysis of Sinusoidal ac Voltammograms and Emergence of Characteristic Patterns of Behavior. In a typical dc voltammetric evaluation of an electrode process the scan rate would be varied, and each of the Ru, Cdl, k0, E°, and R terms would be modified theoretically until “satisfactory” agreement is achieved with experiment. Reference to patterns of behavior noted in the previous section on Patterns of Behavior in dc Voltammetry would be used as a guide to achieving agreement. With the largeamplitude ac method, the range of patterns of behavior is expanded enormously: the dc term and above-mentioned patterns are retained (see the previous section on Patterns of Behavior in dc Voltammetry), but numerous additional easily identifiable patterns are now present in the power spectrum and the fundamental, second, third, and higher harmonics. Additionally, impedance and phase angle interrelationships become accessible from each experiment, which can be compared with simulated data. Each of these features exhibit characteristic Cdl, Ru, and k0 dependencies with different levels of sensitivity, so the prospect of obtaining a “unique” solution to the electrode mechanism is greatly enhanced. (1) Power Spectrum. Fourier analysis of total current ac voltammograms provides direct access to the power spectrum which enables an overview of the relative significance of the dc and ac terms to be rapidly assessed. Parts a-d of Figure 3 highlight the power spectrum dependencies on variation of each of the terms Ru, Cdl, and k0 under a designated set of conditions. The magnitude of the power spectrum in the region corresponding to the fundamental harmonic clearly increases as Cdl increases (Figure 3b), but decreases as Ru increases (Figure 3a) or k0 decreases (compare parts b and d of Figure 3). The power spectrum of the second (Figure 3) and higher harmonics (data not shown) have the very attractive feature of zero influence of capacitance when Ru ) 0 Ω but decrease as Ru increases or k0 decreases. As the ItotRu drop becomes more significant, so does 6218 Analytical Chemistry, Vol. 76, No. 21, November 1, 2004

the influence of capacitance on the second and higher harmonics. Patterns of behavior and the selected level of sensitivity therefore clearly emerge in the dependence of Ru, Cdl, and k0 in the power spectrum. With respect to the very low-frequency component of the power spectrum, or what is termed the dc component, addition of Ru and Cdl broadens the frequency range over which the dc signal is detected. If overlap of the dc and fundamental harmonic occurs in the power spectrum, then extraction of the necessary components may be attempted using the procedure described in Appendix B in the Supporting Information. (2) Recovered dc Component. The availability of very explicit “patterns” of behavior is clearly revealed after application of the inverse FT operation to recover the dc and ac harmonics components. Parts a-c of Figure 4 illustrate the influence of Ru, Cdl, and k0 on the dc component (similar to that found in a conventional dc voltammogram). Under some circumstances, noise (ringing24) is detected in addition to the signal of interest associated with the electrochemistry. The observed ringing phenomena that occurs near the boundaries of the potential (time) range used is an outcome of performing the inverse FFT operation. Appendix C in the Supporting Information gives details and information on how to minimize these ringing effects. (3) Recovered Fundamental Harmonic ac Component. Figure 5 illustrates the very high sensitivity of the recovered fundamental harmonic to changes in each of Ru, Cdl, k0, and c∞. Unlike the case with the power spectrum, obvious wave shape and/or position changes accompany peak height changes. Effectively, the fundamental harmonic frequency occurs at a different time scale to the dc component so that the relative influence of each of these terms is different from that found with the dc component. (4) Recovered Second Harmonic ac Component. In the case of the second harmonic, effectively an even shorter time scale (24) Boyce, W.; DiPrima, R. Elementary Differential Equations and Boundary Value Problems; John Wiley & Sons: New York, 1997.

Figure 3. Power spectrum obtained from a total current ac simulated voltammogram using the designated parameters together with the relevant parameters from Table 1.

is being probed than for a fundamental harmonic. Thus, since the effect of Cdl is small, the dependence on k0 is even greater than for the fundamental harmonic. The presence of Ru also may produce profound changes to second harmonic ac voltammograms. Figure 6 highlights the dependence of each of the major variables relevant to reversible and quasi-reversible processes on the recovered second harmonic and also the “patterns” of influence of each of Ru, k0, and c∞ that emerge. (5) Recovered Higher Harmonic Components and Other Data Evaluation Strategies Available. The third and higher harmonics are akin to the second harmonic in the sense that no charging current is present, but effectively these data are obtained at even shorter time scales so that simulated data are even more sensitive to the influence of k0, Ru, and c∞. The phase angles of the measured fundamental and higher harmonics data relative to the applied ac voltage also are changed in a characteristic manner that depends on the values of Ru and k0 as are the real and

imaginary components and hence impedance (admittance) all of which are accessible from FT analysis. SOLVING AN INVERSE PROBLEM VIA RECOGNITION OF PATTERNS OF BEHAVIOR Heuristic Approach. The results of the simulations above allow us to deduce intuitively obvious patterns of behavior with respect to the influence of E°, Cdl, Ru, k0, and R. These patterns suggest systematic strategies to solve an inverse problem which is based on the ability to utilize the readily measured nonlinear terms inherent in a large-amplitude system. In a practical experiment we would often wish to solve the inverse problem for reversible or quasi-reversible systems given information of the kind contained in Table 2. Note, we assume that we obtain experimental data for at least two different sets of initial concentration of species, e.g., 0.10 mM and 1.0 mM. Analytical Chemistry, Vol. 76, No. 21, November 1, 2004

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Figure 4. Recovered dc components obtained from ac voltammograms using the designated parameters together with the relevant parameters from Table 1.

The basic concept is that we develop a protocol that employs recognition of patterns of behavior (based on our knowledge of the electrochemical system), conduct a set of numerical experiments (simulative modeling with Fourier analysis) for each variable and/or a few variables together, and optimize the set of all possible outcomes that reproduce the experimental results. The found optimal set of variables is the solution to the inverse problem. The method presented below is essentially heuristic in concept, but statistical error minimizing approaches and mathematically based pattern recognition concepts will be developed in future studies. Below we illustrate one possible approach to solving the inverse problem for a reversible or quasi-reversible process, which implies E°, Cdl, and Ru or E°, Cdl, Ru, k0, and R need to be determined, respectively. Experimentally, the oxidation of Fc in dichloromethane (DCM) is used as an example of a reversible process with a large ItotRu drop, while the reduction of [Fe(CN)6]36220


in water represents an example of a quasi-reversible case with negligible ItotRu drop. We emphasize that the present work illustrates our ability to heuristically solve an inverse problem. In particular, we (1) identify the patterns of behavior which clearly emerged in the dependence of Ru, Cdl, and k0 in the power spectrum and recovered dc and ac harmonic currents, (2) select a level of sensitivity and follow specified steps iteratively to make initial estimates of values for the unknown parameters of the system, and finally, (3) we perform numerical simulations, comparing them against experimental results until the level of desired accuracy is reached (accuracy is checked each time by subtracting the simulated results from the experimental ones). The inverse problem is the following: given the known parameters in Table 2 and two experimental data sets for two different concentrations of ∼10-4 M and ∼10-3 M, find E°, Cdl, Ru, k0, and R (or just E°, Cdl, and Ru for the reversible case). One

Figure 5. Recovered fundamental harmonics obtained from ac voltammograms using the designated parameters together with the relevant parameters from Table 1.

Table 2. The Inverse Problem given

find

current response, Itot frequency of applied signal, f amplitude of applied signal, ∆E scan rate, v voltage range, Estart and Eend initial concentration of species, cs electrode area, A diffusion coefficients of species, Ds

formal potential, Eo double layer capacitance, Cdl uncompensated resistance, Ru rate constant (i.e., reversibility), k0 transfer coefficient, R

protocol for the solution of this problem is described below. From the mathematical point of view, the patterns of behavior identified imply that we will have to solve a system of up to five unknowns. Figure 7 illustrates certain steps used in our data evaluation strategy.

Step 1. We start by examining the power spectrum in order to confirm that there is no overlap between any dc or ac terms. Overlap is most likely to contribute to inaccuracies in the recovered dc and ac components. Appendix B in the Supporting Information describes how to deal with overlap detected in the power spectrum. In the case of experimental data obtained for 0.12 or 1.0 mM Fc in DCM and 1.0 mM [Fe(CN)6]3- in water, no overlap is observed (e.g., Figures 8b and 9b represent the DCM case and Figure 10b represents the water case). Step 2. Next, we make an estimate of the value of E°. Figure 7a illustrates the strategy using E° ) (Emax + Emin)/2, where Emax and Emin are the potentials at which the peak values of oxidation and reduction currents occur. Simulations have shown that the use of the cyclic form of data minimizes errors from the Ru and k0 terms. To estimate the value of E°, we use the recovered dc cyclic voltammogram of the higher concentration data set, since Analytical Chemistry, Vol. 76, No. 21, November 1, 2004

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Figure 6. Recovered second harmonics obtained from ac voltammograms using the designated parameters together with the relevant parameters from Table 1.

this is a less noisy data set (i.e., less influenced by the charging current) than that obtained from the lower concentration data set (e.g., see 0.12 mM Fc in DCM, Figure 8c), although the lower concentration data set is sometimes highly suited for this task if the value of Cdl is small. In the case of 1.0 mM Fc in DCM and 1.0 mM [Fe(CN)6]3- in water (Figures 9c and 10c, respectively) we find that initial estimates are E° ) 0.259 V versus Ag and E° ) 0.278 V versus Ag/AgCl, respectively. Step 3. We make an initial (nonzero) estimate of the double layer capacitance value using the recovered fundamental harmonic. Cdl can be estimated directly from the simulation as follows: we run a simulation using the estimated value C sim dl and obtain a corresponding value for the charging current, I sim c employing the recovered fundamental harmonic. Since the charging currents ratios (see Figure 7b) of the experimental charging sim current value I exp and the numerically obtained one, I exp c c /I c , 6222


sim should be equal to the capacitance value ratio C exp dl /C dl , we can determine the value of the experimental capacitance C exp dl ) exp sim C sim I /I . Alternatively, we may use the formula C dl ) dl c c |Ic|/(∆Eω) where |Ic| is the amplitude of the capacitance current prior to the onset of the faradaic current. In practice, the values found from both concentration data sets are averaged and this is taken to be the Cdl value. In the cases considered Cdl ) 6.5 × 10-8 F for Fc in DCM at a Pt electrode and Cdl ) 24 × 10-7 F for [Fe(CN)6]3- in water at a glassy carbon electrode over the potential range of interest. The values of Cdl/A are 9.4 ( 0.1 µF cm-2 in DCM and 31.3 ( 0.1 µF cm-2 in water. A significantly larger capacitance current per unit area in aqueous medium studies explains why the faradaic-to-charging current ratio in water is much poorer than that for the dichloromethane system (compare Figures 9d and 10d).

Figure 7. Some aspects of the data evaluation strategy.

Step 4. In order to identify the reversibility or quasireversibility of our system, we examine25 the recovered second harmonic26 of the lower concentration data set, where the ItotRu influence should be minimal,27 and compare it against the theoretically reversible case (i.e., the simulation is obtained using k˜ 0 ) 106). If there is a good agreement, then the electrochemical system under study is concluded to be reversible. Fc in DCM is found to be a reversible system (thus step 6 can be skipped in this case), while [Fe(CN)6]3- in water is found to be a quasireversible system. Step 5. We check for the presence of ItotRu by considering the ratio between the peak currents of the recovered second (25) We run the simulations using values of Cdl estimated in step 3, Ru ) 0 Ω, k0 ) 6240 cm s-1, R ) 0.5 unless otherwise specified. (26) In principle, the recovered dc or fundamental harmonic terms could be used in an analogous way to identify the degree of reversibility. However, the presence of the significant charging current makes use of the recovered second harmonic highly advantageous. (27) Simulations under the experimentally used conditions employed in this study show that the ItotRu influence is minimal for the 0.1 mM concentration data sets.

harmonic signal of both concentrations data sets. If the ItotRu drop is negligible then the ratio of the current peak values is the same as the concentration ratio (8.3 to 1 for Fc and 10 to 1 for [Fe(CN)6]3-). Additionally, if ItotRu drop is present, the peak locations for the recovered dc and all ac harmonics differ in the lower and higher concentration data sets (higher Ru value yields larger shift). In the dichloromethane case the ratio of the recovered second harmonic peak currents is 2.7 to 1 and is well removed from 8.3 to 1. We therefore conclude there is a significant ItotRu drop associated with oxidation of Fc in DCM. In contrast, the ratio of 9.8 to 1 suggests that there is negligible ItotRu drop for reduction of [Fe(CN)6]3- in water. These conclusions are confirmed by noting that peak positions in DCM are a function of Fc concentration (see Figures 8 and 9) while those in water are independent of [Fe(CN)6]3- concentration. Step 6. We estimate k0 using the lower concentration data set28 by examining the peak current, the shape, and the crossover (28) For the initial estimate of k0 we work with the lower concentration set since this region is sensitive to k0 but not in this case to ItotRu.


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Figure 8. The ac voltammogram and FT-recovered responses for oxidation of 0.12 mM Fc in DCM. The experimental data (red) and numerical results (blue) were obtained using the relevant parameters in Table 3.

potential positions of the recovered second harmonic current (see Figure 7c). The estimate for k0 is obtained by choosing any 6224


nonzero value of k0 and iterating until acceptable agreement is reached.

Figure 9. The ac voltammogram and FT-recovered responses for oxidation of 1.0 mM Fc in DCM. The experimental data (red) and numerical results (blue) were obtained using the relevant parameters in Table 3.

Step 7. We make an initial estimate for Rusany small nonzero value.29 We run the simulation for the higher concentration data

set30 and compare the peak current, the shape, and the crossover potential position to the experimental data employing the recovAnalytical Chemistry, Vol. 76, No. 21, November 1, 2004

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Figure 10. The ac voltammogram and FT-recovered responses for reduction of 1.0 mM [Fe(CN)6]3- in water. The experimental data (red) and numerical results (blue) were obtained using the relevant parameters in Table 3.

ered second harmonic current. The simulation is rerun for different Ru values until acceptable agreement is reached. 6226 Analytical Chemistry, Vol. 76, No. 21, November 1, 2004

Step 8. If our system is quasi-reversible, then we need to evaluate R. R ) 0.50 is used as an initial estimate. To refine the

Table 3. Summary of Parameters Derived from Studies on the Oxidation of Fc and the Reduction of [Fe(CN)6]3- a

E° Cdl Cdl/A Ru k0 R

Fc in DCM

[Fe(CN)6]3- in water

0.259 V 0.065 µF 9.4 µF cm-2 3400 Ω reversible reversible

0.278 V 2.4 µF 31.3 µF cm-2 e50 Ω 0.10 cm s-1 0.50

a The values of other parameters used in the code are v ) 397 mV s-1, f ) 90.6 Hz, ∆E ) 80 mV, D(Fc) ) 2.3 × 10-5 cm2 s-1, D([Fe(CN)6]3-) ) 7.6 × 10-6 cm2 s-1, A(Fc) ) 0.0069 cm2, and A([Fe(CN)6]3-) ) 0.0776 cm2.

R value, we again employ the recovered second harmonic current using the lower concentration data set.31 A symmetrical recovered second harmonic current implies R ) 0.50. If a nonsymmetrical second harmonic is obtained, the value of R is made > 0.50 or < 0.50 (as appropriate). Step 9. After obtaining estimates of k0, Ru, and R from steps 6-8, we find the combination that gives the best agreement between simulated and experimental data for both concentration data sets. To refine k0, Ru, and R values we again simultaneously examine the shape and crossover potential position of the recovered second harmonic current. From the mathematical point of view, we are solving a system with three unknowns. Step 10. At this stage, we refine the estimates of E°, Ru, k0, and R. To do this we simultaneously compare the simulated and experimental peak currents, wave shapes, and crossover potential positions employing the recovered second harmonic current for both concentration data sets. Additionally, we compare the peak heights and positions of the recovered dc component current for higher concentration data sets. Step 11. We verify the values found using the recovered third harmonic data set. Figures 8f, 9f, and 10f show the agreement achieved between simulated and experimental data for designated low- and high-concentration data sets. The third harmonic is exceptionally sensitive to Ru and k0. This is highlighted by the recovered third harmonic current data for 1.0 mM Fc in DCM (see Figure 9f), where highly unusual shapes detected experimentally are confirmed by simulation and are attributable to the very high Ru value of 3400 Ω. Step 12. We repeat steps 2 and 6-11 until a desired level of accuracy is reached. In the present paper, a heuristic approach is employed to all aspects of the problem. In future studies statistical methods of error analysis will be introduced to quantify the level of agreement between theory and experiment. Table 3 summarizes the final estimates for the Fc and [Fe(CN)6]3- based on this protocol. The estimated values of Ru, (29) Note that Ru ) 0 is possible only for an ideal electrochemical system. When ItotRu is negligible, we can only conclude that Ru is smaller than a certain value. The precise value cannot be found because the system is insensitive to very low uncompensated resistance values. This situation prevails in the [Fe(CN)6]3- case in water. (30) The higher concentration data set is more sensitive to the influence of ItotRu. Thus, we use the higher concentration data set since otherwise we may encounter a nonsensitive ItotRu region. (31) As before, minimal influence of ItotRu on the simulation at the lower concentration data set is advantageous for the estimation of R.

Cdl, E°, k0, and R give an excellent agreement with the experimental data (Figures 8-10). We emphasize that the above protocol represents only one heuristic approach, commonly based on intuitively obvious forms of recognition of patterns of behavior. The power of the approach is revealed by perusal of Figures 8-10. For example, we specifically point out that the experimental and simulated third harmonic responses for oxidation of ferrocene in DCM vary from the expected three lobes for the 0.12 mM to four at the 1.0 mM ferrocene concentration level. This is a pattern of behavior that is easily recognized and fully accounted for by a change from a low ItotRu drop (0.12 mM) to a high ItotRu (1.0 mM) situation. In the studies of the Fc in DCM and the [Fe(CN)6]3- in water cases, a significant improvement in theory versus experimental correlations would be achieved by removing the assumption that Cdl is independent of potential. We have calculated the value of Cdl from experimental data obtained in the potential region near the initial potential. However, the poorer agreement achieved in the switching potential region probably reflects the inadequacy of taking Cdl to be independent of potential. A detailed discussion of the theoretical dependence of Cdl on potential is available elsewhere.1 CONCLUSIONS Large-amplitude ac methods with Fourier analysis to recover dc and higher harmonics signal in the frequency domain provides a highly efficient data analysis technique. In particular, we have shown that employment of intuitively obvious patterns of behavior enables the rapid analysis of mechanistic nuances associated with reversible or quasi-reversible electrode processes in the presence of Ru and Cdl. This is one of the primary reasons for adopting the heuristic approach to data analysis employed in this paper. By establishing rules for visual recognition of the pattern behaviors we have been able to demonstrate how our results can be interpreted intuitively and straightforwardly by an electrochemist. Systematic analysis of different sets of experimental data (e.g., in our case, two sets at different concentrations of initially present electroactive species) allows us to identify behavioral patterns and, consequently, develop the strategies for evaluation of unknown variables of Ru, Cdl, E°, k0, and R for a quasi-reversible process. The ability to employ pattern recognition and sensitivity analysis in an intuitively obvious manner using large-amplitude ac methods clearly enhances the fidelity of the data evaluation process. The fact that the theoretical modeling and the experiments are undertaken in an almost identical manner implies that all mechanisms which are amenable to study by the conventional dc voltammetric method and for which almost universal simulation packages are available8,32 can be adapted with significant advantage to the large-amplitude ac technique. Furthermore, automated computational approaches to achieving a statistically optimal evaluation are accessible and will form the basis of further studies in our laboratories. ACKNOWLEDGMENT The research described in this paper was generously supported by a Grant from the Australian Research Council. A. M. Bond gratefully acknowledges colleagues at the University of Oxford (32) Bieniasz, L. Comput. Chem. 1997, 21, 1.


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for their hospitality during his study leave period that enabled this collaborative research project to be undertaken. The authors are also pleased to acknowledge the financial support of the EPSRC (Grant No. GR/R17041 for K.H.). SUPPORTING INFORMATION AVAILABLE Appendices A-C, together with Figures S1-S3, provide more details concerning the mathematical modeling, overlap in the

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power spectra, and ringing phenomena. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review March 24, 2004. Accepted July 4, 2004. AC0495337

Resistance, Capacitance, and Electrode Kinetic Effects in Fourier

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