Anal. Chem. 2006, 78, 4383-4389
Characterization of ac Voltammetric Reaction-Diffusion Dynamics: From Patterns to Physical Parameters Costas A. Anastassiou,*,† Nicolas Ducros,‡,§ Kim H. Parker,‡ and Danny O’Hare‡
Institute of Biomedical Engineering and Department of Bioengineering, Imperial College London, Prince Consort Road, SW7 2AZ, London, U.K.
Despite the widespread use of electrochemical sensing techniques, the determination of the physical parameters from the current response of rapid voltammetric measurements has been difficult for two reasons: large capacitance contributions overwhelm the current response of transient measurements and the reaction dynamics are inherently nonlinear and nonstationary. In this work, we present a signal processing methodology that in combination with a large-amplitude/high-frequency voltage waveform method, ac voltammetry, is able to determine the underlying physical parameters in heterogeneous electrochemical reaction-diffusion processes. Through a large number of numerical calculations, we explore the effect of kinetic, thermodynamic, and mass transport parameters on two components of the current response, the even and the odd. We study the even component directly whereas for the odd component, which is considerably influenced by capacitance, we use the Hilbert transform, which is suitable for the analysis of nonstationary and nonlinear data sets, to minimize the capacitance contribution. The theoretical analysis is applied to measurements of well-characterized electrochemical reactions, Ru(NH3)62+/3+ and Fe(CN)64-/3-, using two different electrode materials, glassy carbon and platinum, and the physical parameters deduced are in excellent agreement with published results. Electrochemical methods of analysis offer excellent and tuneable spatial resolution, low cost, and extreme miniaturization and can therefore be used to considerable advantage in applications as diverse as neurochemistry, biosensors, scanning probe microscopy, and environmental analysis.1-5 Voltammetry, where the * Corresponding author. Tel.: +44 (0)207 589 5111 (XT) 55170. Fax: +44 (0)207 594 5177. E-mail:
[email protected]. † Institute of Biomedical Engineering. ‡ Department of Bioengineering. § Present address: Faculty of Engineering, Undergraduate Program, Ecole Nationale Superieure de Physique de Strasbourg, BP 10413, 67412 Illkirch Cedex, France. (1) Heien, M. L. A. V.; Khan, A. S.; Ariansen, J. L.; Cheer, J. F.; Phillips P. E. M.; Wassum, K. M.; Wightman, R. M. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 10023-10028. (2) Chi, Q.; Farver, O.; Ulstrup, J. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 16203-16208. (3) Chen, K.; Hirst, J.; Camba, R.; Bonagura, C. A.; Stout, C. D.; Burgess, B. K.; Armstrong F. A. Nature 2000, 405, 814-817. 10.1021/ac060122v CCC: $33.50 Published on Web 05/18/2006
© 2006 American Chemical Society
electrode voltage is perturbed in a predetermined manner, offers improved selectivity as well as improved discrimination against nonfaradaic contributions to the signal.6 Transient voltammetric methodologies, such as large-amplitude/high-frequency ac voltammetry, here simply referred to as ac voltammetry, have proven useful for gaining insight into different electrode mechanisms. Ac voltammetry, which uses the superposition of a dc signal, a ramp, with a large-amplitude/high-frequency ac signal as the voltage excitation, combines a number of attractive features such as the possibility to interrogate process dynamics on different time scales, to explore the kinetics and thermodynamics of different processes, or to selectively target specific process dynamics, such as parallel reactions.7 The challenge with such rapid voltammetric methodologies lies in the interpretation of the current response. Electrochemical signals are intrinsically nonlinear and nonstationary. Because this nonlinearity, usually attributed to electron transfer, is more pronounced for large-amplitude/high-frequency perturbations, ac voltammetry has generally been confined to small signal voltage excitations to linearize the current output. Under these conditions, techniques such as the fast Fourier transform (FFT) and the Laplace transform were used to extract electrochemical information, for instance, in energy conversion devices and corrosion.8,9 This is similar to electrochemical impedance spectroscopy where small-amplitude perturbation on a normally steady potential is used and the electrical impedance of the cell is obtained as a function of the perturbing frequency using Kramers-Kronig transform and FFT.6 For a number of applications though, it is important to measure transient phenomena occurring relatively fast. The measurement of such rapid events requires the application of rapid voltage waveforms that provide the necessary temporal resolution. However, the larger the voltage perturbation gradients, the larger the capacitance contribution becomes, which increases the difficulty in characterizing the underlying dynamics. (4) Zhang, B.; Zhang Y. H.; White, H. S. Anal. Chem. 2004, 76, 6229-6238. (5) Bard, A. J.; Mirkin, M. V.; Unwin, P. R.; Wipf, D. O. J. Phys. Chem. 1992, 96, 1861-1868. (6) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; Wiley: New York, 2001. (7) Bond, A. M.; Duffy, N. W.; Guo, S. X.; Zhang, J.; Elton, D. Anal. Chem. 2005, 77, 186A-195A. (8) Smyrl, W. J. Electrochem. Soc. 1985, 132, 1551-1555. (9) Yoo, J.-S.; Park, S.-M. Anal. Chem. 2000, 72, 2035-2041.
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Attempts to develop signal processing methods that overcome the effect of capacitance and relate specific patterns of the current response to the underlying physical parameters have mainly used baseline subtraction or the FFT. The fundamental assumption behind baseline subtraction is that capacitance interference does not change substantially during the course of the experiment. In rapid voltammetric measurements, where capacitance contributions most commonly overwhelm the charge-transfer response, slight changes in the capacitance characteristics have a detrimental impact on the analysis and often result in statistical artifacts.10 More recently, ac voltammetry was combined with FFT to discriminate capacitance contributions, mainly present in the fundamental harmonic of the ac spectrum, from other processes. Engblom et al.11 and Gavaghan and Bond12 applied the FFT to the current output signal of heterogeneous electrochemical reactions in diffusive media and revealed that while electrochemical reaction kinetics manifest themselves in higher ac harmonics the effect of capacitance and resistance differs between fundamental and harmonic frequencies. Additionally, it was shown that in order to recover the higher harmonics of the ac signal accurately, the power of the ac component of the frequency has to be sufficiently high. Zhang et al.13 showed that combining ac voltammetry with the FFT can enhance separation between the reversible reaction of Ru(NH3)62+/3+ from an overlapping irreversible oxygen reaction process as well as improve kinetic resolution on electrode materials such as boron-doped diamond. A more detailed theoretical study was published by Sher and co-workers where the effects of resistance, capacitance, and electrode kinetic effects were analyzed with the FFT to propose a heuristic approach to evaluate the unknown parameters of the system.14 The effect of different physical parameter values on the higher harmonics was investigated in a series of simulations, and quantification of the resulting patterns led to the suggestion of a self-correcting fitting algorithm to be used to deduce all of the parameters involved. The disadvantage in using an FFT-based analysis on voltammetric data is that Fourier techniques assume periodicity, whereas transient voltammetric data sets are nonstationary and nonlinear. A nonlinear process does not obey the principle of superposition, nor does it have the property of frequency preservation.15 A signal processing technique that has been often used for the analysis of nonstationary and nonlinear processes is the Hilbert transform (HT).16 The HT has been typically used to calculate instantaneous attributes of time domain signals, for instance, to measure the synchronization characteristics in voltammetric time series of electrochemical oscillators or changes in pulmonary blood pressure in response to step changes of oxygen concentration in (10) Heien, M. L. A. V.; Johnson, M. A.; Wightman R. M. Anal. Chem. 2004, 76, 5697-5704. (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. (13) Zhang, J.; Guo, S.-X.; Bond, A. M.; Marken, F. Anal. Chem. 2004, 76, 36193629. (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) Director, S. W.; Rohrer R. A. Introduction to system theory; McGraw-Hill: New York, 1972. (16) Gabor, D. Proc. IEE 1946, 93, 429-457.
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breathing gas.17,18 Recently, we showed that the HT in combination with ac voltammetry can significantly minimize capacitance influences and provide novel kinetic and thermodynamic information in surface-confined voltammetric processes.19 In this work, we show a method based on simple algebraic manipulations and the HT that can accurately determine the physical parameters involved in reaction-diffusion processes from ac voltammetry data sets. MATERIALS AND METHODS All chemicals used in the voltammetric measurements (Ru(NH3)6Cl2, K4Fe(CN6), KCl) were purchased from Sigma-Aldrich and used as received. In all experiments, analyte bulk concentration was 1 mM. The 1 M KCl was the supporting electrolyte in a standard three-electrode cell. We used two inlaid disk electrodes, one glassy carbon (GC) where A ) 4.7 × 10-6 m2 and one Pt where A ) 3.1 × 10-6 m2, both purchased from CH Instruments (Austin, TX). The pretreatment of the working electrodes included the following: polishing with aqueous 1- and 0.3-µm alumina on polishing pads (Buehler) with sonication between each grade and a final electropolish of 500 2-V cycles in 1 M sulfuric acid. The reference electrode, a Ag/AgCl wire in a 3 M KCl solution, and the Pt counter electrodes were manufactured in-house. Prior to experiments with Ru(NH3)6Cl2, the solution was bubbled for 20 min with nitrogen to remove dissolved oxygen from solution. The ramped harmonic waveforms were digitally generated using Labview 7.0 software (National Instruments, Austin, TX) and converted to an analog signal through a NI PCI 6036E card interfaced with a Bivolt PK30 potentiostat (Device Technology Ltd.), which was also used to record the current output signal (16-bit, 200-kHz bandwidth). RESULTS AND DISCUSSION Theoretical Modeling of the Process Dynamics. To simulate the spatiotemporal dynamics of ac voltammetry, we assume the one-electron electrochemical reaction occurring on the electrode surface: kf
Ox + e {\ } Red k b
(1)
where Ox and Red are the oxidized and reduced species and kf and kb (m s-1) are the forward and backward nonlinear kinetic rate expressions quantitatively described by the Butler-Volmer theory.6 (See Table 1 for nomenclature used in this section.) The typical dimensionless 1-D formulation describing the dynamics of heterogeneous electrochemical reaction processes occurring in diffusive media is
∂u/∂τ ) ∂2u/∂x2
(2)
(17) Kiss, I. Z.; Zhai, Y. M.; Hudson, J. L. Science 2002, 296, 1676-1678. (18) Huang, W.; Shen, Z.; Huang, N.; Fung, Y. C. Proc. Natl. Acad. Sci. U.S.A. 1998, 96, 1834-1839. (19) Anastassiou, C. A.; Parker, K. H.; O’Hare, D. Anal. Chem. 2005, 77, 33573364.
Table 1. Nomenclature electrochemical dynamics electrode surface area
A
a w, a w V-1
double layer capacitance (C concentration of reduced species, initial (bulk) concentration (mol m-3)
D E0
diffusion coefficient (m2 s-1) formal oxidation potential (V)
ifar, icap, i
dimensionless faradaic, capacitance, and overall current current (A) forward and backward reaction rate parameter (m s-1)
k0
kinetic rate constant (m s-1)
t u
time (s) dimensionless concentration of the reduced species spatial coordinate (m) charge-transfer coefficient (-) capacitance contribution (C m mol-1 V-1) dimensionless time superscripts
x R λ τ ∧ *
characteristic property dimensional properties
-, +
left and right envelope feature physical constants Faraday constant (96485.3 C mol-1) gas constant (8.31 C V mol-1 K-1) temperature (298.15 K)
F Rg T
+
dimensionless instantaneous amplitude of w, position of the envelope maximums
aw
m-2)
Cdl cRed, cRed,0
I kf, kb
pattern recognition -,
(m2)
aw,cap, aw,diff
offset of the dimensionless instantaneous amplitude of w due to capacitance and diffusion influences
g, g-, g+
dimensionless even component of i, magnitude of the minimum and maximum of g
j w
imaginary number odd component of i
z γg, γw ∆ξg, ∆ξw
analytic signal of w envelope extrema ratio of g and w dimensionless voltage difference of extrema in g and w
ξg, ξw
dimensionless voltage in the g and w envelopes applied voltage waveform initial applied voltage (V) ac harmonic frequency (Hz)
Ein* f*
with the boundary conditions
v ∆E*, ∆ξ
scan rate of the dc ramp (V s-1) ac harmonic amplitude (V) and dimensionless amplitude
ξ, ξin, ξn
dimensionless applied, initial dc ramp and normalized applied voltage
τsw
dimensionless switch time of the dc ramp
less properties:
|
∂u ) ifar ) kfu - kb(1 - u) ) ∂x x)0 exp{(1 - R)ξn}u - exp{-Rξn}(1 - u) (3) u(x,0) ) 1
and
u(∞,τ) ) 1
(4)
with u being the concentration of the reduced species, τ the time, x the space coordinate, ifar the faradaic current response, R the charge-transfer coefficient, and ξn the voltage. These dimensionless variables are based on the process characteristic quantities:
ˆt dyn )
D k02
E ˆ )
RgT F
ˆI ) FACRed,0k0
(5)
u)
τ)
t ˆt dyn
ξn )
x)
E* - E0 E ˆ
x*k0 D
)ξ-
(6) E0 E ˆ
i)
I ˆI
where E* (V) is the applied voltage, E0 (V) the formal oxidation potential, I (A) the current response, x* (m) the spatial coordinate, t (s) the time, and cRed (mol m-3) the concentration of reduced species. The overall current response i is defined as i ) ifar + icap, where icap is the capacitance current response:
icap ) where ˆtdyn (s) is the characteristic time of the dynamics of the diffusion-reaction process, Eˆ (V) the characteristic voltage, and ˆI (A) the characteristic current. In these expressions, D (m2 s-1) is the diffusion coefficient, k0 (m s-1) the kinetic reaction constant, Rg (8.31 C V mol-1 K-1) the universal gas constant, T (298.15 K) the absolute temperature, F (96 485.31 C mol-1) the Faraday constant, A (m2) the electrode surface area, and cRed,0 (mol m-3) the initial concentration of the reduced species. The process characteristic quantities in eq 5 are used to define the dimension-
cRed cRed,0
( )
E ˆ ∂ξ λ Fk0ˆt ∂τ
with
λ)
Cdl CRed,0
(7)
with Cdl (C V-1 m-2) being the double layer capacitance and λ (C m mol-1 V-1) the capacitance contribution parameter. Although strictly not true, the independent calculation of the faradaic and capacitance current often provides a good approximation of the overall current response particularly for outer-sphere electron transfer and has been widely used to study the capacitance interference in voltammetry.6,7,11-13,19,20 Analytical Chemistry, Vol. 78, No. 13, July 1, 2006
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follows: the ac harmonic amplitude ∆E* ) 0.2 V, the ac harmonic frequency f* ) 100 Hz, and the dc scan rate v ) 1 V s-1. We propose a method to deduce E0, ˆtdyn ) D/k02 and R from the ac voltammetry response i. To do so, we conducted a numerical sensitivity analysis. In our model simulations, eqs 2-4 are solved numerically with input parameter ξ, according to eqs 8-11, for a set of parameters E0, ˆtdyn, R, and λ to calculate ifar and icap and give the overall current response i. All reaction-diffusion simulations were conducted using a commercial software package (Femlab 3.1) on a 2.8-GHz Intel Pentium IV personal computer. SYMMETRY Since ξ is a temporally even function, i.e., symmetrical about τsw, it is convenient to define the two components, temporally “even” and “odd”, of the output current. The even component g(τ)and the odd component w(τ)
Figure 1. (A) Dimensionless applied voltage ξ in ac voltammetry vs τ. The current response of an ac voltammetry experiment for (B) a 1 mM solution of Ru(NH3)62+/3+ on a GC electrode and (C) a 1 mM solution of Fe(CN)64-/3- on a Pt electrode. Note that capacitance overwhelms the faradaic current in both experiments. The applied voltage parameters were ∆E* ) 0.2 V, f* ) 100 Hz, v ) 1 V s-1, and tsw ) 0.8 s for both experiments and for (B) E/in ) -0.6 V and (C) E/in ) -0.2 V.
In ac voltammetry, the externally controlled electrode dimensionless voltage ξ is prescribed by
ξ(τ) ) ξdc(τ) + ξac(τ)
(8)
1 g(τ) ) [i(τ) + i(2τsw - τ)], 2
0 e τ e τsw
(12)
1 w(τ) ) [i(τ) - i(2τsw - τ)], 2
0 e τ e τsw
(13)
It follows that i(τ) ) g(τ) + w(τ). One advantage of this decomposition is that the capacitance contributions, as described through eq 7, are absent from g(τ) and present only in w(τ). Hilbert Transform. In this work, we use the HT of a realvalued time series as defined by Gabor.16 The HT of w(τ), w j (τ), is the time series that makes the complex time series z(τ) ) w(τ) + jw j (τ) analytic (i.e., it satisfies the Cauchy-Riemann conditions), where j is the imaginary number. The HT of w(τ) can be released through a convolution integral as follows:
w j (τ) ) HT[w(τ)] )
∫
∞
-∞
w(u) du π(τ - u)
(14)
where
ξdc(τ) ) ξin + τ ) (Ein* + vt)/E ˆ,
0 e τ e τsw
(9)
ˆ, ) ξin(τ) + 2τsw - τ ) (E/in + 2vtsw - vt)/E τsw < τ e 2τsw (10) ˆ ) cos(2πf*t) ξac(τ) ) ∆ξ cos(Ωτ) ) (∆E/E
(11)
Like the Fourier transform, the Hilbert transform is a linear operator and a number of its properties are described in ref 21. The instantaneous attributes of z(τ) of the “odd component” w(τ), the instantaneous amplitude aw(τ), and the instantaneous phase φw(τ), are then calculated from the polar form:
z(τ) ) aw(τ)ejφw(τ)
(15)
In eqs 8-11, the dimensionless variables are as follows: ξin the initial voltage, τsw the dc switching time when the ramp direction is inverted, ∆ξ the amplitude of the superimposed harmonic waveform, and Ω its harmonic angular frequency. Moreover, E/in (V) is the initial dc voltage, v (V s-1) the scan rate of the applied voltage dc ramp, ∆E* (V) the harmonic oscillation amplitude, and f* (s-1) the oscillatory, driving, frequency. In Figure 1 we show two ac voltammetry experiments. The dimensionless applied voltage ξ is shown as a function of τ in (A) as well as the current response I from two electrochemical systems: (B) Ru(NH3)62+/3+ on a GC electrode and (C) Fe(CN)64-/3- on a platinum electrode. In both experiments, the applied voltage parameters were as
While the Fourier transform of w(τ) is a complex-valued frequency domain signal, w j (τ) and the analytic signal z(τ) are defined in the time domain τ. It is due to this property that the HT is suitable for studying nonstationary processes and has often been used for the analysis of nonlinear phenomena.17,18,21 Moreover, it was shown that the HT can be used to minimize the influence of capacitance on ac voltammetry data.19 Pattern Recognition. The results of the transformations described by eqs 12-15 are shown in Figure 2. We use the characteristics of the envelopes plotted over the dimensionless applied voltage ξ(τ), (ξg-, g-), (ξg+, g+) from the even component g(τ) shown in Figure 2A and (ξw-, aw-), (ξw+, aw+) from the odd component w(τ) shown in Figure 2B, to deduce the underlying
(20) Lai Miaw, L.-H.; Perone, S. P. Anal. Chem. 1979, 51, 1645-1650.
(21) Bendat, J. S.; Piersol, A. G. Random Data, 2nd ed.; Wiley: New York, 2000.
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Figure 2. Characteristic envelopes resulting when eqs 12-15 are applied on a simulated ac voltammogram. (A) In g(τ) vs ξ(τ), capacitance contributions are absent due to the definition of icap. The characteristic envelope properties used in the analysis are (ξg-,g-) and (ξg+,g+). (B) From aw(τ) vs ξ(τ) we use (ξw-,aw-) and (ξw+,aw+) for our analysis. The offset aw,cap is related to double layer capacitance contributions while aw,diff relates to the effect of diffusion and strongly depends on the physical parameters of the process.
physical parameters E0, ˆtdyn, and R. To do so, we define the voltage difference ∆ξg and ∆ξw as well as the relative ratios γg and γw:
∆ξg ) ξg+ - ξgγg )
|g-| - |g+| |g-| + |g+|
and
∆ξw ) |ξw- - ξw+|
and γw )
aw- - aw+ aw- + aw+ -2aw,diff
(16) (17)
In Figure 2B, the offset aw,cap is directly attributed to double layer capacitance contributions, as described by eq 7. aw,diff is a measure of the spread of the capacitance contribution (aw,diff - aw,cap) and is related to the mobility of the species. More specifically, for small ˆtdyn, i.e., slow diffusion, the forward and backward waves of the current response with respect to τsw are symmetric and result in a small (aw,diff - aw,cap). When ˆtdyn is large, i.e., fast diffusion, the forward and backward waves are less symmetric and aw,diff increases. We determined the formal oxidation potential E0 (V) from the midway potential in both signal components:
E0 ξg- + ξg+ ) E ˆ 2
and
E0 ξw- + ξw+ ) E ˆ 2
(18)
This proved to be very robust, and negligible differences were found between the estimates of E0 from g or w. From eq 7, large capacitance contributions due to large Cdl have the same impact as low cRed,0. The value of parameter λ that indicates the capacitance contribution can be deduced from eqs 7, 15 and aw,cap:
λ)
Fk0ˆt Ω∆ξE ˆ
aw,cap
Figure 3. Results of the theoretical analysis shown where points indicate statistical means and bars the range mainly due to variations of 0.3 e R e 0.7. (A) ∆ξg vs ˆtdyn for (i) ∆E* ) 0.2 V and (ii) ∆E* ) 0.1 V. The different colors indicate (D/m2 s-1): 10-10 (gray), 5 × 10-10 (cyan), 10-9 (purple), 5 × 10-9 (green), and 10-8 (black). Capacitance does not affect component g(τ). (B) ∆ξw vs ˆtdyn for different levels of capacitance contribution (λ/C m mol-1 V-1) ) 1 (cyan), 10 (gray), and 100 (black) when ∆E* ) 0.2 V. Each point represents up to three variations of (D/m2 s-1) ) [10-10, 10-8] and (k0/m s-1) ) [10-5, 10-3] giving a specific ˆtdyn. We observe how for slow electron-transfer kinetics both ∆ξg and ∆ξw increase.
Figure 4. Determination of the charge-transfer coefficient R from the ratios γg and γw. (A) γg vs R for (ˆtdyn/s): (1) 10-3, (2) 10-2, (3) 10-1, and (4) 1. Range bars due to 0.3 e R e 0.7 are very narrow. Dashed lines indicate linear regressions. (D) γw vs R for (ˆtdyn/s): (1) 10-3, (2) 10-2, (3) 10-1, (4) 1, and (5) 10. Each point in (A) and (B) represents up to three combinations of (D/m2 s-1) ) [10-10, 10-8] and (k0/m s-1) ) [10-5, 10-3] that give the same ˆtdyn. This effect was small compared to that variations due to R and λ.
(19)
For the sensitivity analyses shown in Figures 3 and 4, we used a wide parameter range, i.e., 10-4 < (tˆdyn/s) < 102, to generalize
our conclusions. To determine ˆtdyn ) D/k02, we plot ∆ξg, defined by eq 16, versus ˆtdyn, as seen in Figure 3A. The colors indicate the different values for D, with 10-10 e D/(m2 s-1) e 10-8. Variations due to different values of the electron-transfer coefficient Analytical Chemistry, Vol. 78, No. 13, July 1, 2006
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R (0.3 e R e 0.7) are indicated through the statistical mean and the bars that represent the range. We show the results for (i) ∆E* ) 0.2 V and (ii) ∆E* ) 0.1 V. First, we note that the larger ˆtdyn, i.e., the slower the electron-transfer kinetics, the larger ∆ξg becomes. This is equivalent to the well-known peak distance broadening observed for sluggish kinetics, for instance, in cyclic voltammetry.6 Second, for fast reaction kinetics and slow diffusion, i.e., ˆtdyn < 5 × 10-3 s, and vice versa, i.e., ˆtdyn > 2, the differences between the two perturbation waveforms are undistinguishable. It is only in the intermediate region that a larger perturbation amplitude results in larger ∆ξg and thus provides a better resolution of ˆtdyn. This conclusion was also supported by a number of simulations where the effect of different voltage perturbation parameters was studied. In fact, for v ) 1 V s-1, we found that the value of the driving frequency f* of the harmonic ac perturbation does not significantly affect the outcome of this analysis within the range 70-250 Hz. For f* > 250 Hz, large capacitance contributions complicate the analysis. Third, the range bars for ˆtdyn > 1 s increase significantly compared to ˆtdyn < 1 s for both ∆E*. Ac voltage amplitude ∆E* ) 0.2 V offered better resolution, especially in the analysis of R, so we adopt it for the remaining sections. In Figure 3B, we show ∆ξw, as defined by eq 16, versus ˆtdyn when ∆E* ) 0.2 V. Since capacitance contributions are present in the current output component w(τ), the analysis is shown for three different levels of capacitance, λ ) 1, 10, and 100 C m mol-1 V-1. Range bars are due to 0.3 e R e 0.7 and remain relatively uniform for all λ. Each point represents up to three different values of D and k0 that give the same ratio ˆtdyn, with (D/m2 s-1) ) [10-10, 10-8] and (k0/m s-1) ) [10-5, 10-3]. In general, the differences due to these variations proved subtle and they are also included in the range. In total, Figure 3B shows the results of ∼400 simulations. As for ∆ξg versus ˆtdyn, sluggish kinetics cause ∆ξw to increase. For λ ) 1 C m mol-1 V-1, two regions emerge: the first for 10-3 e (tˆdyn/s) e 1 and the second for 1 e (tˆdyn/s) e102. The slope in these two regions differs slightly. This effect is smeared out by capacitance for larger λ. For λ < 1 C m mol-1 V-1, the trends are identical to λ ) 1 C m mol-1 V-1 since capacitance contributions are small compared to the faradaic signal. Thus, based on the analysis shown in Figure 3, both components of the current output signal, g(τ) and w(τ), are used to determine the underlying physical parameter ˆtdyn ) D/k02 and comparison between them indicates the quality of the prediction. In Figure 4, the analysis for the determination of the electrontransfer coefficient R is shown. As in Figure 3, we varied D and k0 and found that their specific values are, in general, of no importance for the outcome of the analysis, only the value of the ratio ˆtdyn ) D/k02. In Figure 4A, we show the ratio γg between the extrema in g(τ), as defined by eq 17, versus R for 10-3 e (tˆdyn/s) e 1. The linear relationship between γg and R is observed throughout the range of R, and the dashed lines indicate linear regressions. For R < 0.3 and R > 0.7, the reaction is considered irreversible.6 The larger the ˆtdyn the larger the slope; i.e., the faster the kinetics of the electrochemical reaction the more pronounced the effect of R on the symmetry of g(τ). We found that for ˆtdyn > 3 the relation between R and γg remains linear but the crossing point of the regression does not coincide with the one observed at R ) 0.6. The points shown in γg versus R include range bars indicating variations due to different D and k0, which are very narrow. 4388
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In Figure 4B, we show the dependence between γw and R for different ˆtdyn and λ. When compared to γg versus R, we observe that for all ˆtdyn the linear relation between γw and R is preserved but the range is larger. This increase is mainly attributed to λ. For ˆtdyn < 10-2 s, the slope is small mainly due to the large capacitance contributions, which makes the determination of R impossible. To gain a better resolution in this region voltage waveforms of shorter characteristic time, i.e., larger v, ∆E* and f*, have to be applied. To determine the parameters E0, ˆtdyn ) D/k02, and R of an electrochemical reaction-diffusion process using ac voltammetry, we proposed the following algorithm: decompose the current response into the components g(τ) and w(τ) according to eqs 12 and 13. From the representation g(τ) versus ξ(τ), we define ∆ξg, eq 16, and the ratio γg, eq 17. We then calculate the instantaneous amplitude aw(τ) of the analytic signal of w(τ) to suppress the influence of capacitance and from the representation aw(τ) versus ξ(τ) we define the voltage ∆ξw, eq 16, and the ratio γw, eq 17. Using those four characteristic values, ∆ξg, γg, ∆ξw, and γw, we can deduce ˆtdyn and R from Figures 3 and 4. We tested the proposed methodology with ac voltammetry experiments using two electrochemical couples, Ru(NH3)62+/3+ and Fe(CN)64-/3-, on two electrode materials, GC and Pt with 1 M KCl as supporting electrolyte, data sets shown in Figure 1B and C. Both species are well studied and show outer-sphere electron transfer and similar values of the diffusion coefficient in the reduced and oxidized state.22-25 The analysis, according to eqs 12-17, is shown in the left column of Figure 5 for Ru(NH3)62+/3+ on a GC electrode, and on the right column of Figure 5 for Fe(CN)64-/3- on a Pt electrode. We show g versus ξ for both experiments in the first row, Figure 5A and B. Inserted we illustrate the cyclic voltammogram I versus ξdc when v ) 1 V s-1 (cyan). We observe that the representation g versus ξ has obvious similarities to the cyclic voltammogram regarding shape characteristics such as peak separation and symmetry. aw versus ξ is shown for both systems in the lower row, Figure 5C and D. Capacitance significantly influences the current response, both the overall current response, shown in Figure 1B and C, and the individual components g and aw, and does not limit itself to one frequency as so often assumed. We observe how the HT minimizes the effect of capacitance in aw versus ξ. The values of the physical parameters determined from our analysis introduced in Figures 3 and 4 are presented in Table 2 and are in close agreement with published values for both electrochemical couples. For Ru(NH3)62+/3+ on GC, reported values of (tˆdyn/s) ) 6.3 × 10-2 - 6.3 have been determined with detailed cyclic voltammetry studies using ultramicroelectrodes.22,23 From our analysis, ˆtdyn ) 10-1 s and R = 0.50 using γg and γw (λ = 1 C m mol-1 V-1). For Fe(CN)64-/3- on the Pt electrode, we observe small ∆ξg and ∆ξw, which is an indication of very fast kinetics. The determined ˆtdyn and R are in excellent agreement with reported ultramicroelectrode studies.24,25 It has to be noted that while the diffusion coefficient D in aqueous solutions is typically within 1 order of magnitude, 10-10 < (D/m2 s-1) < 10-9, the kinetic constant k0 varies within much wider ranges. From eq 19, we calculate Cdl ) (22) Deakin, M. R.; Stutts, K. J.; Wightman, R. M. J. Electroanal. Chem. Interfacial Electrochem. 1985, 182, 113-122. (23) McCreery, R. L. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1991; p 221. (24) Beriet, C.; Pletcher, D. J. Electroanal. Chem. 1993, 361, 93-101. (25) Beriet, C.; Pletcher, D. J. Electroanal. Chem. 1994, 375, 213-218.
Figure 5. In the first row we show g vs ξ for the ac voltammetry experiment using (A) 1 mM Ru(NH3)62+/3+ on GC (current response in Figure 1B) and (B) 1 mM Fe(CN)64-/3- on Pt (current response in Figure 1C). In the second row we show aw vs ξ for (C) 1 mM Ru(NH3)62+/3+ on GC and (D) 1 mM Fe(CN)64-/3- on Pt. The cyclic voltammetry response I vs ξdc of both systems is also shown in g vs ξ for ν ) 1 V s-1 (cyan).
Table 2. Experimental Results 2+/3+on
Ru(NH3)6
D/k02/s R E0/mV vs Ag/AgCl
GC
g
awt
refs 22 and 23
10-1 0.55 -200
10-1 0.50 -231
6.3 × 10-2 - 6.3 0.50 -231
Fe(CN)64-/3- on Pt g 2/s
D/k0 R E0/mV vs Ag/AgCl
10-4 0.67 231
aw
refs 24 and 25
10-3
10-4-10-3 0.67 267
2 231
0.6 C V-1 m-2 for Ru(NH3)62+/3+ on GC, typical values for GC 0.1 < (Cdl/C V-1 m-2) < 0.7,26 and Cdl ) 0.8 C V-1 m-2 for Fe(CN)64-/3- on Pt, typical values for Pt 0.1 < (Cdl/C V-1 m-2) < 0.5.27,28 The values from eq 19 underestimate the effect of capacitance, especially for Ru(NH3)62+/3+ on GC, since they do not account for the capacitance influence in g(τ). For both electrochemical couples, the formal oxidation potential E0 is determined with millivolt accuracy despite large capacitance contributions. The fact that with this method we use two components, g and aw, that result in narrow ranges of ˆtdyn, R, and E0 that are in excellent agreement with reported values enhances the level of confidence in our prediction. CONCLUSIONS In this work, we use tools valid for nonstationary signal processing to extract physical information from electrochemical (26) Kinoshita, K. Carbon: Electrochemical-Physicochemical Properties; Wiley: New York, 1988. (27) Franks, W.; Schenker, I.; Schmutz, P.; Hierlemann, A. IEEE Trans. Biomed. Eng. 2005, 52, 1295-1302. (28) Los, P.; Zabinska, G.; Kisza, A.; Christie, L.; Mount, A.; Bruce, P. G. Phys. Chem. Chem. Phys. 2000, 2, 5449-5454.
reaction-diffusion processes using large-amplitude/high-frequency ac voltammetry. We show that the proposed method adequately minimizes the significant capacitance contributions related to such rapid voltammetric techniques, and moreover, it is capable of determining the underlying physical parameters with high precision. The analysis is based on two components of the current response: the even, which is used directly and shows remarkable similarities to cyclic voltammograms, and the instantaneous amplitude of the odd, where capacitance contributions are suppressed. We conducted a large number of numerical simulations and derived the behavior of the two signal components with respect to order of magnitude variations of the physical parameters. The characteristic patterns that emerged from this sensitivity analysis allows for the determination of these parameters. The fact that separate predictions from two components of the same data set are utilized enhances the confidence of this method. Macroscopic experiments with two model electrochemical species and two electrode materials acquired in less than 2 s showed good agreement with published values obtained using much lengthier and experimentally tedious methods suggesting possible applications in biological, environmental, and chemical sensing. ACKNOWLEDGMENT C.A.A. thanks Severin Harvey and Hong Zhao for experimental assistance and Martin Arundell for the Labview program and Bhavik A. Patel for initial measurements. Financial support from the Institute of Biomedical Engineering and the EPSRC is greatly appreciated. Note Added after ASAP Publication. There was an error in the caption of Figure 5 in the version published ASAP May 18, 2006; the corrected version was published ASAP June 13, 2006. Received for review January 17, 2006. Accepted March 30, 2006. AC060122V Analytical Chemistry, Vol. 78, No. 13, July 1, 2006
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