Cyclic Biamperometry - Analytical Chemistry (ACS Publications)

Anal. Chem. 2010, 82, 1779–1785

Cyclic Biamperometry Mehdi Rahimi and Susan R. Mikkelsen* Department of Chemistry, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada Cyclic biamperometry has been investigated as a method for the quantitation of one form of a reversibly electroactive redox couple in the presence of the other form, using the ferri-ferrocyanide couple in aqueous KCl. A triangular voltage waveform applied across two equal-area, planar gold electrodes yields peak currents that depend on the square root of the applied voltage scan rate, when one form of the redox couple is present in excess. Independent measurement of electrode-to-solution potential during biamperometric scans allowed estimation of the fractional impedance at each electrode-solution interface, and these values allow calculation of the effective potential scan rates at each electrode. Results show that when one form is present in a 5-fold excess or greater, the potential scan rate for the limiting reaction is nearly identical to the applied voltage scan rate. Similar values were obtained from impedance calculations, but discrepancies between the predicted and experimental values are evident when the two forms are present at near equivalent concentrations. When one form of the redox couple is present in excess, cyclic biamperometric peak currents depend linearly on the concentration of the limiting form, and these currents can be amplified by using cells with one electrode much larger than the other. Because this method does not require a reference electrode, it can, in principle, be readily incorporated into new electrochemical array or lab-on-a-chip devices. Controlled-potential methods in electroanalytical chemistry commonly employ 3-electrode cells, with two conducting electrodes of variable potential (the working and auxiliary electrodes), and a third electrode, of constant potential (the reference electrode). This arrangement, used with a potentiostat, allows precise control of the potential difference between the working electrode surface and the electrolyte medium that physically separates the electrodes. Two-electrode cells, with one working and one reference electrode, are seen in single-use or low-current applications. Control of the potential difference across the working electrode-solution interface allows the properties of interfacial processes at this phase boundary to be studied in isolation from processes occurring at the counter or auxiliary electrode.1,2 Biamperometry, also called amperometry with two polarizable electrodes, was introduced more than 80 years ago as a method * To whom correspondence should be addressed. Fax: 519-746-0435. E-mail: [email protected]. (1) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd ed.; Wiley: New York, 2001. (2) Kissinger, P. T.; Heinemann, W. R. Laboratory Techniques in Electroanalytical Chemistry, 2nd ed.; Marcel Dekker: New York, 1996. 10.1021/ac902383w  2010 American Chemical Society Published on Web 02/03/2010

for the detecting titration end points.3 In this method, the voltage applied across two electrodes separated by an electrolyte is maintained at a constant, relatively small value (e100 mV) and current is measured as a function of the volume of titrant added to a stirred analyte solution. Typically, the electrolyte is an aqueous solution, and the electrodes are noble metals of approximately equal surface area. With this method, control of individual electrode-to-solution potentials is impossible, but the applied voltage is distributed between the electrodes such that current at one electrode is equal in magnitude but opposite in sign to that at the second electrode. The potential of the solution between the electrodes varies as the composition of the solution changes, i.e. as the [R]/[O] ratio for the detected redox couple changes due to the progress of the titration. In practice, this means that when only one redox couple is electrochemically reversible within the applied voltage window, the measured current results from this redox reaction proceeding in different directions at the two electrodes. For the relatively simple case in which the reduced form of a reversible couple is titrated with an electrochemically irreversible oxidant (for example, ferrocyanide titrated with permanganate), using two identical indicator electrodes and a small, fixed applied voltage (V), the relationship between dilution-corrected current (i) and the degree of completeness of the reaction (λ) has been described by eq 1, below.4,5 i ) (nF/2RT){(PP ′λ(1 - λ))/(P(1 - λ) + P ′λ)}C*V (1) In this equation, the terms n, F, R, and T have their usual meanings, C* is the initial concentration of the reduced form of the reversible couple, and λ varies from zero (before the titration has begun) to unity (at the equivalence point). The terms P and P′ are proportionality constants between current and concentration that depend on the mass transfer process for the reduced form (P) and the oxidized form (P ′). Usually, the values of P and P ′ are considered equal, although they are known to depend on diffusion coefficients and electrode areas as well as rate of stirring and electrode shape. For P ) P ′, eq 1 simplifies to eq 2: i ) (nF/2RT)(Pλ(1 - λ))C*V

(2)

Equation 2 predicts a maximum current at λ ) 0.5, which is halfway to the equivalence point, with a value of ((nF/8RT)PC*V). (3) Foulk, C. W.; Bawden, A. T. J. Am. Chem. Soc. 1926, 48, 2045–2061. (4) Delahay, P. New Instrumental Methods in Electrochemistry; Interscience: New York, 1954; pp 254-264. (5) (a) Reilley, C. N.; Cooke, W. D.; Furman, N. H. Anal. Chem. 1951, 23, 1226–1229. (b) Reilley, C. N.; Cooke, W. D.; Furman, N. H. Anal. Chem. 1951, 23, 1223–1226.

Analytical Chemistry, Vol. 82, No. 5, March 1, 2010

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This is in reasonable agreement with results observed with equal area electrodes and when the diffusion coefficients of reduced and oxidized forms are similar. Equations 1 and 2 assume that the applied voltage V is equally distributed across the two electrode-solution interfaces, since the tangent (di/dE)i)0 is assumed constant (linear). The relationships between current and voltage shown in eqs 1 and 2 are derived from the assumption shown in eq 3: i ) (di/dE)i)0(V/2)

(3)

A more recent effort to model the shapes of fixed-voltage biamperometric titration curves has considered the effects of electrode area ratio, applied voltage and heterogeneous rate constant, beginning with either the Nernst (reversible) or the Butler-Volmer (irreversible) equation and considering linear diffusion (equal diffusion coefficients) and a stirred solution.6 These simulated titration curves are in qualitative agreement with experimental data for a variety of titration reactions and electrode area ratios and indicate that end point detection can be improved by using electrodes of different area. More complicated scenarios with multiple redox couples have also been studied, and relevant equations have been developed to describe constant-potential titration curves in the simple and more complicated cases; applications have been described for a wide variety of redox reactions.7-11 More recently, fixed-potential biamperometry has been applied to detectors for flow injection analysis12-19 as well as electrochemical arrays.20,21 In some of these reports,12-14 different redox reactions, of closely matched potentials, have been used for the anodic and cathodic half-reactions. In another,20 a conducting, variable-potential electrode was used as a counter electrode in a two-electrode cell and was called a “quasi-reference electrode”. Linear sweep biamperometry has been used to determine an appropriate fixed voltage for biamperometric measurements in flowing solutions and at interdigitated electrode pairs.16-18 The results of these studies show that the small diffusion layers that create renewal of reactant and removal of product from electrode surfaces (flowing and stirred solutions) and the effects of redox cycling at closely spaced interdigitated electrodes17,18 result in (6) Surmann, P.; Peter, B.; Stark, C. Fresenius J. Anal. Chem. 1996, 356, 173– 177. (7) Delahay, P. Anal. Chim. Acta 1950, 4, 635–640. (8) Duyckaerts, G. Anal. Chim. Acta 1953, 8, 57–64. (9) Gauguin, R.; Charlot, G. Anal. Chim. Acta 1953, 8, 65–87. (10) Kies, H. I. Anal. Chim. Acta 1958, 18, 14–28. (11) Tougas, T. P.; Jannetti, J. M.; Collier, W. G. Anal. Chem. 1985, 57, 1377– 1381. (12) Zhao, C.; Zhang, J.; Song, J. Anal. Biochem. 2001, 297, 170–176. (13) Zhao, C.; Song, J.-F.; Zhang, J.-C. Anal. Bioanal. Chem. 2002, 374, 498– 504. (14) Song, J.-F.; Chen, J.-Q. J. Pharm. Biomed. Anal. 2003, 33, 789–796. (15) Moreno Galvez, A.; Garcia Mateo, J. V.; Martinez Calatayud, J. Anal. Chim. Acta 1999, 396, 161–170. (16) Milardovic, S.; Ivekovic, D.; Rumenjak, V.; Grabaric, B. S. Electroanalysis 2005, 17, 1847–1853. (17) Milardovic, S.; Kerekovic, I.; Derrico, R.; Rumenjak, V. Talanta 2007, 71, 213–220. (18) Milardovic, S.; Kerekovic, I.; Nodilo, M. Talanta 2008, 77, 222–228. (19) Michalowski, J.; Trojanowicz, M. Anal. Chim. Acta 1993, 281, 299–304. (20) Tang, T.-C.; Deng, A.; Huang, H.-J. Anal. Chem. 2002, 74, 2617–2621. (21) Mann, T. S.; O’Hagan, L.; Ertl, P.; Sparkes, D. I.; Mikkelsen, S. R. Anal. Chem. 2008, 80, 2988–2992.

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sigmoid plots of current against applied voltage; these sigmoid plots are considered to be the normal shape of a linear-sweep biamperometric voltammogram. However, under quiet solution conditions, and in the absence of redox cycling, a response likened to a distorted voltammogram has also been described.22 We now report an investigation of the cyclic biamperometric method: the measurement of current as a function of voltage, with a triangular voltage waveform applied across two metal electrodes separated by an aqueous electrolyte. This work uses the wellstudied ferri-ferrocyanide couple, for which diffusion coefficients in aqueous electrolytes are known.24 Results with equal area electrodes show a linear dependence of cyclic biamperometric peak current on the concentration of the redox form present at lower concentrations; in addition, applied voltage scan rates are consistent with and nearly equal to independently measured potential scan rates under these conditions. Intermediate conditions have also been investigated and provide interesting insights. In principle, the absence of a reference electrode allows this method to be readily incorporated into lab-on-a-chip as well as electrochemical array instruments. EXPERIMENTAL SECTION Chemicals and Instrumentation. Potassium ferricyanide, potassium ferrocyanide, and potassium chloride were purchased from Sigma-Aldrich in the highest available quality and were used as received. All solutions were prepared in distilled, deionized water (Barnstead NanoPure). Buehler Micropolish II (1 µm) was purchased from Tech-Met Canada. Planar gold disk working (d ) 1.5 mm) and Ag/AgCl reference electrodes were obtained from Bioanalytical Systems Inc. (West Lafayette, IN). Temperature control for electrochemical experiments (25.0 ± 0.1 °C) was accomplished using a Haake D1 circulating bath and a water-jacketed electrochemical cell with an approximate total volume of 30 mL. Electrochemical measurements were made using a CHI650A potentiostat (CH Instruments). Typically, scans were initiated at the most positive applied potential (CV) or voltage (CB), and results are reported for steady-state voltammograms. Workingto-reference electrode potential measurements made during cyclic biamperometry experiments were made using a model NI-USB6251 data acquisition system with LabView software (National Instruments). Methods. Gold electrodes were prepared by polishing on a slurry of Buehler Micropolish II on a soft cloth supported by a flat glass plate. They were then rinsed with and sonicated (Branson 1200) in distilled, deionized water for 2 min and blotted dry with a KimWipe tissue. Electrode area measurements were carried out using chronoamperometry,23 by extrapolating it1/2 curves to t ) 0, with reduction of 4.00 mM ferricyanide (D ) 7.63 × 10-6 cm2/s)24 in 0.100 M KCl at 25 °C. Electrodes 1-5 had electroactive areas of 0.0240, 0.0239, 0.0237, 0.0252, and 0.0243 cm2, respectively. For biamperometric experiments, electrode 1 was always used (22) Reference 1, p 655. (23) Reference 1, p 163. (24) Adams, R. N. Electrochemistry at Solid Electrodes; Marcel Dekker: New York, 1969; p 219.

Table 1. Comparison of Experimental and Predicted Potential Scan Rates at W1 predicted υ, mV/s

measured υ, mV/s

[Fe(CN)64-], mM

[Fe(CN)63-], mM

(z1/z2)ox eq 8

(z1/z2)red eq 7

applied scan rate, mV/s

ox

red

ox

red

1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.55 0.60 0.70 0.80 0.90 1.00 2.50 5.00 10.00 25.00

0.40 0.60 0.80 1.00 1.20 1.40 1.60 25.00 10.00 5.00 2.50 1.00 0.90 0.80 0.70 0.60 0.55 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

0.073 0.215 0.520 1.169 2.630 6.364 18.703 2922.39 467.58 116.90 29.224 4.676 3.787 2.993 2.291 1.683 1.414 1.169 0.966 0.812 0.596 0.457 0.361 0.292 0.047 0.0117 0.0029 0.00047

13.574 4.619 1.909 0.848 0.377 0.156 0.053 0.00034 0.0021 0.0085 0.034 0.212 0.262 0.331 0.433 0.589 0.701 0.848 1.027 1.222 1.663 2.172 2.749 3.393 21.209 84.835 339.340 2120.88

25.0 25.0 25.0 25.0 25.0 25.0 25.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0 50.0

1.7 4.4 8.5 13.5 18.1 21.6 23.7 49.98 49.89 49.58 48.3 41.2 39.6 37.5 34.8 31.4 29.3 26.9 24.6 22.4 18.7 15.7 13.3 11.3 2.2 0.58 0.15 0.02

23.3 20.6 16.4 11.5 6.8 3.4 1.3 0.02 0.11 0.42 1.6 8.7 10.4 12.4 15.1 18.5 20.6 22.9 25.3 27.5 31.2 34.2 36.7 38.6 47.7 49.42 49.85 49.98

0.9 1.3 10.1 17.4 22.5 23.7 23.8 48.9 49.0 49.0 48.9 46.1 44.7 43.9 41.1 34.9 30.5 25.7 19.7 15.7 14.1 8.7 5.8 5.1 1.3 0.9 0.6 0.8

23.6 23.6 19.8 11.4 2.7 1.5 1.2 1.3 1.5 1.6 1.9 3.7 4.6 5.1 6.9 10.4 14.3 20.7 28.3 32.5 33.7 40.0 42.7 43.4 46.8 47.9 48.2 48.8

as W1 (see Figure 4) and was connected to the working electrode lead from the potentiostat, while W2 consisted of either electrode 2 alone or with sequential additional connections to electrodes 3-5, to increase the W2 area (edge effects were considered absent), and connection was to both the reference and auxiliary electrode leads of the potentiostat. For voltammetric experiments, electrode 1 was always used as the working electrode, in conjunction with a Ag/AgCl reference electrode and a large area Pt wire auxiliary electrode. For both CB and CV, steadystate data (obtained after the first two cycles) are shown. Potential profiles were monitored using two channels of the NI data acquisition system. One channel monitored the applied triangular voltage waveform, while the other channel measured the time-dependent potential difference between W1 and a Ag/ AgCl reference electrode. The data acquisition frequency on each channel was set to 100 Hz. The two time-dependent signals were converted to plots of measured potential vs applied voltage using Microsoft Excel. Slopes of these plots were determined by linear regression from -350 to -150 mV and from +150 to +350 mV for data shown in Figure 3 and Table 1. Potential profiles were monitored using two channels of the NI data acquisition system. One channel monitored the applied triangular voltage waveform, while the other channel measured the time-dependent potential difference between W1 and a Ag/ AgCl reference electrode. The data acquisition frequency on each channel was set to 100 Hz. The two time-dependent signals were converted to plots of measured potential vs applied voltage using Microsoft Excel. Slopes of these plots were determined by linear regression from -350 to -150 mV and from +150 to +350 mV for data shown in Figure 3 and Table 1.

RESULTS AND DISCUSSION A comparison of results obtained by cyclic biamperometry and cyclic voltammetry is shown in Figure 1. In the CV experiment (Figure 1(a)), the cathodic and anodic peak potentials are symmetrical about the formal potential of the redox couple (0.214 ± 0.001 V vs Ag/AgCl), while CB peaks (Figure 1(b)) are symmetrical about zero applied voltage (-0.003 ± 0.001 V). In both cases, a linear dependence of cathodic peak current on the square root of scan rate was obtained (insets to Figure 1(a),(b)). The slopes of these lines are 1.46 × 10-5 A/(V/s)1/2 (CV) and 2.62 - 10-5 (CB), with R2 values of 0.9999 and 0.9996, respectively. The Randles-Sevcik equation is commonly used to describe the dependence of cyclic voltammetric peak currents on scan rate for reversibly electroactive species and is shown in eq 4 below ip ) (2.69x105)n3/2AD1/2υ1/2C

(4)

where ip is the voltammetric peak current (A), n is the number of electrons transferred, A is the electroactive area of the working electrode (cm2), D is the diffusion coefficient of the reacting form (cm2/s), υ is the potential scan rate (V/s), and C is the concentration of the reacting species (mol/cm3).25 Equation 1 predicts a slope of 1.79 × 10-5 A/(V/s)1/2 for reversible ferricyanide reduction at 25 °C, while less reversible electron transfer would give a smaller value, depending on the rate constant (ko) and symmetry factor (R) for the heterogeneous reaction.25 The increase in ∆Ep with scan rate observed in the CV experiment (Figure 1(a), where values change from 77 mV at 10 mV/s to 92 mV at 150 mV/s) suggests quasireversibility, and the CV slope of 1.46 × 10-5 A/(V/s)1/2 is 18% less than the predicted value. (25) Reference 1, pp 231-239.


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Figure 2. Cyclic voltammetric (open circles) and cyclic biamperometric (closed circles) cathodic peak current (n ) 6) as a function of the mole fraction of ferricyanide, with a total concentration of ferricyanide plus ferrocyanide of 1.00 mM in 0.100 M KCl. The scan rate was 50 mV/s, and other conditions were the same as stated in Figure 1.

Figure 1. (a) Cyclic voltammograms of 1.00 mM potassium ferricyanide in 0.100 M KCl and (b) cyclic biamperograms of 1.00 mM potassium ferricyanide with 5.00 mM potassium ferrocyanide in 0.100 M KCl. Scan rates were 10, 25, 50, 75, 100, 125, and 150 mV/s, and measurements were made at 25 °C. The same gold electrode (0.0240 cm2) was used as the working electrode for CV and the W1 electrode for CB; the W2 area for CB was 0.0239 cm2. Insets show the resulting plots of cathodic peak current against the square root of the scan rate.

In the CB experiment (Figure 1(b)), the limiting reagent is ferricyanide, while ferrocyanide is present in 5-fold excess. Since the W1 and W2 electrodes are of similar electroactive area (0.0240 and 0.0239 cm2, respectively), and the diffusion coefficients of ferri- and ferrocyanide are similar (7.63 × 10-6 and 6.50 × 10-6 cm2/s, respectively, at 25 °C in 0.1 M KCl24), the limiting reaction is always ferricyanide reduction, which alternates between the W1 and W2 electrodes as the applied voltage alternates between negative and positive values. To clarify this point, consider the terms A, D, and C in eq 4, along with the stipulation that, with biamperometry of a single reversible redox couple, the same reaction occurs, in opposite directions, at W1 and W2. Because the currents at W1 and W2 are the same (but of opposite sign), the number of moles reduced (for example) at W1 must equal the number of moles oxidized at W2. For equal area electrodes and equal diffusion coefficients, if one form (reduced or oxidized) is present in excess, then the concentration of the other form will limit the magnitude of the current, so that this limiting form is completely consumed at one electrode, while the other form is incompletely consumed (in an equimolar quantity) at the other electrode; under these conditions, the limiting reaction is that involving the lower-concentration form of the redox couple, and under biamperometric cycling conditions this limiting reaction will alternate between the two electrodes. 1782


Application of this concept to the data shown in Figure 1 follows. Ferricyanide is produced at W1 while the applied voltage is positive, and the local increase in the quantity of ferricyanide at the W1 surface is equal to the quantity of ferrocyanide produced at the W2 electrode. Under these conditions, since the diffusion coefficients are similar, the sum of the W1 and W2 areas is expected to be the relevant value of A in eq 1. Thus the slope of the CB plot (2.62 × 10-5 A/(V/s)1/2) should be twice that of the CV plot (1.46 × 10-5 A/(V/s)1/2. The ratio of the slopes calculated from these experimental data is 1.8. The main reason that this lower value was obtained, we believe, is that the applied voltage scan rate in CB is distributed between two electrode-solution interfaces and cannot be the same as the working electrode-to-solution potential scan rate that is controlled in a CV experiment. The difference between these scan rates can be significant and has been addressed in further experiments described below. The concept of the limiting reaction involves the AD1/2C term in eq 4. When A values are equal for the two electrodes and D values are equal for the two forms of the redox couple, then the redox form present at lower concentration is readily identified as the reactant in the limiting reaction. However, when A and D values are not equal, the limiting reaction must be identified based on the AD1/2C product for each reaction at each electrode. For example, if concentrations and diffusion coefficients are equal for the two redox forms, then the reaction occurring at the smaller of the two electrodes will be the limiting reaction; in this case the limiting reaction will change from oxidation to reduction as the voltage applied to the smaller electrode alternates between positive and negative values. At fixed scan rate (50 mV/s), CB and CV data were collected for a series of solutions with a total ferricyanide plus ferrocyanide concentration of 1.00 mM. Since the total concentration is constant, CV peak currents are not expected to vary with the ferrito ferrocyanide ratio, and this was observed (Figure 2, open circles). CB data collected under identical conditions are also

shown in Figure 2 (filled circles). These results show a maximum peak current for equimolar concentrations of oxidized and reduced forms and a linear dependence of peak current on mole fraction at low and high mole fractions. With equal area W1 and W2 electrodes, the limiting CB reaction switches from ferricyanide reduction on the left side of Figure 2 to ferrocyanide oxidation on the right side. These results are consistent with those reported for fixed-potential biamperometric titrations with one reversible redox couple.4-10 The maximum CB peak currents (at mole fraction 0.5) are about 25% smaller than the CV peak currents obtained under identical conditions (Figure 2). As with the difference between the Randles-Sevcik slopes for CB and CV (Figure 1), this is attributed to a scan rate effect that occurs in CB but not in CV. This was the topic of a further experiment to understand how the applied voltage is distributed between the two electrodesolution interfaces. To determine the electrode-to-solution scan rates that occur during cyclic biamperometric scans (where electrode-to-electrode scan rates are applied), potential measurements were made between W1 and a Ag/AgCl reference electrode during the course of CB experiments performed at a fixed voltage scan rate, with equal area W1 and W2 electrodes (0.0240 and 0.0239 cm2), for solutions containing different ferri- to ferrocyanide concentration ratios. Typical results for a series of solutions with a constant total concentration of 2.00 mM are shown in Figure 3. Remarkably, each curve contains two distinct linear regions that correspond to either oxidation (at positive applied voltage) or reduction (at negative applied voltage) at W1, although curves c and e of Figure 3 show slight nonlinearity between -0.3 and -0.4 V. A more subtle but important observation involves the curvature of these plots near zero applied voltage; earlier theoretical treatments of biamperometry,4,5 and the resulting derivation of equations to describe biamperometric titration curves, have considered the term (di/dE)i=0 (eq 3) to be constant and linear, with an equal distribution of potential differences across the two electrode-solution interfaces. These data (Figure 3) show that this is generally not the case, since the slopes are different for positive and negative applied voltages, and curvature is observed near zero volts applied. The y-intercepts of each curve in Figure 3 represent the potential of the solution as defined by the Nernst equation for the bulk concentration ratio under investigation; using the Eo′ value of +0.214 ± 0.001 V vs Ag/AgCl obtained from CV data (Figure 1(a)), these intercepts are in good agreement (±3 mV) with predicted values. Furthermore, in each linear region of each curve (positive or negative voltage) the slopes may be taken as the fractions of the 25 mV/s applied voltage scan rate that occur across the W1-solution interface. In principle, the values of these slopes can vary from zero to unity; in these extreme situations, the applied voltage (and thus the applied scan rate) would occur entirely across the solution-to-W2 (zero slope) or the W1-tosolution (unit slope) interface. The applied voltage scan rate (across two electrodes separated by the solution) is thus distributed between the two working electrode-solution interfaces, and the potential scan rate at W1 clearly changes between the oxidative and reductive regions of each applied voltage scan.

Figure 3. Plots of measured W1 potential (vs Ag/AgCl) against applied voltage, for (a) a total [ferricyanide]+[ferrocyanide] concentration of 2.00 mM, with an applied scan rate of 25 mV/s. Concentration ratios ([ferricyanide]:[ferrocyanide] are 4:1, 2.33:1, 1.5:1, 1:1, 1:1.5, 1:2.33, and 1:4, for curves a to g, respectively (see the first seven entries in Table 1). Solutions were prepared in 0.100 M KCl; measurements were made at 25 °C with gold electrodes of 0.0240 cm2 (W1) and 0.0239 cm2 (W2) areas.

Figure 4. Equivalent circuit for a biamperometry cell. W1 and W2 represent the two working electrodes with double-layer capacitances Cdl,1 and Cdl,2 and electrochemical impedances z1 and z2; solution resistance is shown as RΩ.

The biamperometric electrochemical cell can be represented by the equivalent circuit shown in Figure 4.26 The two working electrode-solution interfaces are separated by a solution resistance, and each interface is characterized by a double-layer capacitance in parallel with an electrochemical impedance. The electrochemical impedance can be further broken down into a charge-transfer resistance in series with a Warburg impedance, which is a masstransfer based impedance. At the low-frequency limit, for smallamplitude bidirectional (sinusoidal) perturbations from a constant applied potential, electrochemical impedance has been described by eq 5, below 1/2 z ) RΩ + Rct + Cdl((RT/n2F2A)((1/DO CO*) +

(1/DR1/2CR*)))2 (5) where z is the impedance, RΩ is the solution resistance, Rct is the charge-transfer resistance, Cdl is the double-layer capacitance, and R, T, n, F, A, D, and C* have their usual meanings.27 In an attempt to understand the results shown in Figure 3, we have considered each portion (positive or negative applied voltage) of the slow, linear-scan CB waveform to be a unidirectional perturbation. This allows the simplification of eq 5 to the following form z ) RΩ + Rct + Cdl((RT/n2F2A)(1/D1/2C*))2

(6)

(26) (a) Pettit, C. M.; Goonetilleke, P. C.; Sulyma, C. M.; Roy, D. Anal. Chem. 2006, 78, 3723–3729. (b) Chang, B.-Y.; Park, S.-M. Anal. Chem. 2006, 78, 1052–1060. (27) Reference 1, pp 380-387.


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Figure 5. Calculated (as z1/z2/(1+z1/z2) from eq 7 or 8) (lines, dotted for reduction and solid for oxidation at W1) and measured fractional impedance at W1 (as the slopes of the plots shown in Figure 3) (symbols, shown as open circles for reduction and closed circles for oxidation) as a function of the mole fraction of ferricyanide in 0.100 M KCl. Data are shown in Table 1.

where the values of D and C* are those of the electrochemical reactant (e.g., those of the oxidized species for a reduction). Further simplification can occur if the values of RΩ and Rct are negligible with respect to the third (Warburg) term; this simplification should be valid at high electrolyte concentration and for rapid electron-transfer kinetics. With these assumptions, the ratio of impedances z1/z2 (Figure 4) can be approximated by eq 7 (for reduction at W1) and by eq 8 (for oxidation at W1) z1 /z2 ) (A2DRCR*2)/(A1DOCO*2)

(7)

z1 /z2 ) (A2DOCO*2)/(A1DRCR*2)

(8)

where the Cdl/A values (eq 6) for the two electrodes have been taken to be the same. The impedance ratio z1/z2 allows approximation of electrode-to-solution potential differences for a given applied voltage as well as potential scan rates at each electrode-solution interface, for a given applied voltage scan rate. The general linearity that we have observed experimentally between measured potential and applied voltage (Figure 3) demonstrates that the relative impedance values (z1/z2) are constant with applied voltage but are different for the oxidative and reductive portions of the biamperograms. Table 1 shows a comparison of potential scan rates calculated using either eq 7 or eq 8 with the corresponding experimental values determined for the reductive or oxidative segments of the solutions studied (Figure 3 and additional data obtained at an applied scan rate of 50 mV/s). Potential scan rates were calculated as fractional impedance (or slope from Figure 3) multiplied by the applied voltage scan rate. There is general agreement between predictions and experimental results for the most extreme concentration ratios studied, where one form of the redox couple contributes minimally to the total impedance of the cell. Under these conditions, the measured potential scan rate is nearly equal to the applied voltage scan rate when the limiting reaction occurs at W1. Figure 5 shows a comparison of calculated fractional impedance values ((z1/z2)/(1+z1/z2)) with the experimental values determined from the slopes of the oxidative and reductive 1784


Figure 6. (a) Cathodic peak current measured at 25 °C at W1 as a function of the W2 electrode area, for [ferricyanide]/[ferrocyanide] ratios of 1:1 (a), 3:1 (b), and 7:1 (c) (total concentration 2.00 mM) in 0.100 M KCl at an applied scan rate of 25 mV/s and (b) cyclic biamperograms observed for a [ferricyanide]/[ferrocyanide] ratio of 7:1 with a W1 area of 0.0240 cm2 and W2 areas of 0.0239, 0.0476, 0.0728, and 0.0971 cm2 for curves a to d, respectively.

portions of the curves shown in Figure 3, as a function of the fractional composition of the solution. Although eqs 7 and 8 correctly predict the increase (or decrease) of W1 potential scan rate with increasing ferricyanide, and also predict the shapes of these curves, the experimental curves show sharper ascents (or descents) when the limiting reaction switches from one working electrode to the other. This discrepancy is not surprising, since eq 5 strictly applies to impedance studies that use small-amplitude, sine wave perturbations;27 furthermore, the assumptions inherent in eqs 7 and 8 (RΩ ) 0, Rct ) 0, and equal Cdl/A values at W1 and W2) are problematic. However, the experimental results show that only a small excess of one form of the redox couple over the other provides a sufficiently high impedance ratio to allow an essentially constant and controllable W1 potential scan rate, and they also demonstrate that the W2 impedance is insignificant under these conditions. Figure 6(a) shows the dependence of cathodic CB peak current, measured at a fixed voltage scan rate (25 mV/s), as a function of the electroactive surface area of the W2 electrode, for different ratios of ferri- to ferrocyanide. The W1 electrode (0.0240 cm2) was fixed, while additional gold electrodes of approximately the same area (see the Experimental Section) were connected to the W2 potentiostat inputs and were geometrically coplanar in the solution. At the most extreme concentration ratio (7:1), a linear dependence of ip,c on W2 area was observed, and these results imply that further signal amplification should be possible for extreme concentration

ratios by increasing the W2/W1 area ratio beyond our maximum value of about 4. By way of comparison, it should be noted that increasing the size of the auxiliary electrode in CV has no effect on current magnitudes, and only the working electrode area is important (eq 4). The curvature observed for the 1:3 and 1:1 solutions indicates that more extreme concentration ratios are needed for linear signal enhancement through increased W2 area; in these cases, calculated impedance ratios (the z1/z2 ratio in eq 7) suggest that W1 potential scan rates are not constant but increase as the W2 area increases. Measured W1 potential scan rates for the reductive sections of the applied voltage scans for the 1:1 solution were 16.1, 24.1, 24.5, and 24.7 mV/s for the smallest to largest W2 areas, respectively. The cyclic biamperograms obtained at the most extreme concentration ratio studied, 7:1, are shown in Figure 6(b), where the effect of increased W2 area on the appearance and symmetry of the biamperograms is evident. Under these conditions, oxidation of ferrocyanide at W2 is the limiting process despite its larger area (i.e., z2 > z1). These biamperograms also exhibit a slight shift toward negative applied voltage values, with the apparent average peak potentials ((Ep,c + Ep,a)/2) shifting from -5 to -57 mV as the W2 area is increased from smallest to largest. This is likely an artifact that results from the increased charging current that must exist when the limiting reaction occurs at the larger W2 electrode.

Taken together, these results have significant implications for the application of cyclic biamperometry to analytical problems. First, cyclic biamperometric peak currents depend on the concentration of the limiting form of the redox couple, rather than the total concentration of oxidized and reduced forms (as CV peak currents do); thus the concentration of one form is readily measurable in the presence of an excess of the other. Second, because the limiting reactant is generated, from the form present in excess, during the reverse scan, it should be possible, with an extreme enough concentration ratio, to amplify CB peak current magnitudes through the use of a large-area W2 electrode. Third, CB requires no reference electrode, so that the manufacture of instrumentation and electrodes for electrochemical arrays or micro/nano devices incorporating this method should be considerably simpler than the manufacture of devices requiring a reference potential. ACKNOWLEDGMENT Financial support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. Helpful comments from reviewers of this work were also appreciated. Received for review October 20, 2009. Accepted January 19, 2010. AC902383W


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Cyclic Biamperometry - Analytical Chemistry (ACS Publications)

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