Diagnostic Criteria for Identifying an ECE Mechanism with Cyclic Square Wave Voltammetry

Theory for cyclic square wave voltammetry of an irreversible ﬁrst-order chemical reaction coupling two electron transfers, i.e. an ECE mechanism, is presented. Theoretical voltammograms were calculated following systematic variation of empirical parameters to assess their impact on the shape of the voltammogram. Note that the results presented herein are applicable only to ECE processes where E 0 for the second electron transfer step is negative of that for the ﬁrst. Under this condition, disproportionation reactions do not occur. From the trends obtained, diagnostic criteria for this mechanism were deduced. When properly applied, these criteria will enable non-experts in voltammetry to assign the electrode reaction mechanism and accurately measure reaction kinetics over the range − 2 ≤ log k f ≤ 6. © The 2015. the terms Commons the any

Square wave voltammetry (SWV) has been shown to be particularly useful in identifying electrode reaction mechanisms especially those involving chemical reactions coupled to the electron transfer step. [1][2][3][4][5][6][7] Although a strong theoretical basis exists, 8 the use of SWV for identifying electrode reaction mechanisms has been limited to a small number of electrochemists. Our goal is to broaden the use of SWV for determining electrode reaction mechanisms especially by nonexperts in electrochemistry who make occasional use of voltammetry in characterizing new compounds. Cyclic square wave voltammetry (CSWV) is a modified form of SWV that steps through the region of the formal potential of the electroactive species under study and back to the initial potential as shown in Fig. 1a. The empirical parameters that comprise the CSWV waveform are period (τ), amplitude (E sw ), increment (δE), and the potential at which the direction of the sweep is reversed (E λ ).
Our current effort is focused on identifying the specific mechanisms CSWV provides a straightforward means for determining the mechanism and associated kinetic parameters. [9][10][11][12][13][14] In this work, we critically evaluate CSWV for the analysis of consecutive electron transfers coupled by a homogeneous irreversible chemical reaction, i.e. an ECE mechanism. The process used to evaluate CSWV is as follows. Theoretical voltammograms were computed by systematic variation of each empirical parameter. Figures of merit that describe the shape of the voltammogram as shown in Fig. 1b (peak potentials, peak currents, and peak widths) were compiled and correlated with the empirical parameter being varied. Trends characteristic of this mechanism were identified from these correlations and provide diagnostic criteria for assigning an electrode reaction as an ECE process. Application of these criteria will enable non-specialists to accurately assign the mechanism and, in many instances, quantify the rate constant of the chemical reaction.

H3102
Journal of The Electrochemical Society, 163 (4) H3101-H3109 (2016) where E 0 1 is the formal potential for reduction of Ox 1 and E 0 2 is the formal potential for the reduction of Ox 2 . For simplicity, we have arbitrarily set E 0 1 = 0 and reference all other potential values to it. We consider herein only the situation where E 0 2 ≤ E 0 1 and denote the difference in formal potentials as E 0 = E 0 1 -E 0 2 . This restriction enables discussion of the mechanism without considering the complication of homogeneous electron transfer (disproportionation [18][19][20][21]23,34,35 ) reaction, i.e. Ox 1 + Red 2 Red 1 + Ox 2 . Derivation of an equation for predicting current as a function of potential and time began by solving Fick's laws of diffusion using Laplace transformations following application of appropriate boundary conditions for each case. Expressions for the concentrations of Ox 1 , Red 1 , Ox 2 , and Red 2 are related by the Nernst equation for a reversible electron transfer for each redox pair: where n = number of electrons transferred, F = Faraday constant, A = area of the electrode, R = gas constant, T = temperature in Kelvin, E applied = applied potential, E 0 = formal potential for the designated electron transfer reaction, C Ox1 (x, t), C Ox2 (x, t), C Red1 (x, t), C Red2 (x, t) are concentrations of Ox 1 , Ox 2, Red 1, and Red 2, respectively at the electrode surface and any time t. Numerical approximation of the resultant integral equations were performed in the same manner put forth by Nicholson and Olmstead. 36 To compute theoretical voltammograms for CSWV, we employed the cyclic waveform and empirical parameters available with current commercial electrochemical instrumentation capable of CSWV as delineated in our previous reports. [9][10][11] The final equations used to compute theoretical voltammograms are: i2 S j ε 2 + S 1 [4] total = n 1 m1 + n 2 m2 [5] where L = number of subintervals on each potential, k f is the rate of the chemical reaction in s −1 , m = dimensionless current for each time increment with the serial number m, and The complete derivation of Eqs. 3 and 4 is available in Supplementary Material. The final expressions are in agreement with those previously presented by Miles and Compton. 3 The recursive calculation of current on each step for every step in the voltammogram was performed by systematic variation of period (τ), amplitude (E sw ), and increment (δE) over the following intervals: 1 ms ≤ τ ≤ 5 s, 10 mV ≤ E sw ≤ 90 mV, 1 mV ≤ δE ≤ 25 mV, and L = 20 over each period. Period limits were set in consideration of typical potentiostat rise times, commonly encountered solution resistances and electrode double layer capacitances as well as the time duration required per scan. Amplitude limits were set in accordance with the range typically used in SWV. Increment limits were set in consideration of the number of points to define the peak. Specific parameter levels for simulated data are denoted by open circles and listed in the captions of figures contained in this report and Supplementary Material. Voltammograms were calculated for the singular case where the number of electrons in both electron transfer steps equal one. The predicted difference current, , is computed from forward pulsereverse pulse . The corresponding potential for is the average of the corresponding forward and reverse potential pulses. The forward difference current, f denotes the difference currents acquired over the interval E initial to the switching potential E λ , and the difference current, r , denotes difference currents acquired over the reverse potential sweep from E λ to the final potential, E final . To capture the effect of period as it relates to current, + is used throughout this work where The physical meaning of + is the normalized faradaic current emanating from the electron transfer. The plotting convention used herein treats reduction currents as positive and oxidative currents as negative values. Figures of merit that describe the shape of the voltammogram are: + p,f and + p,r the net peak currents on the forward and reverse sweep, peak ratio + p,r / + p,f , E p,f and E p,r the peak potentials on the forward and reverse sweep, E p the peak separation defined as E p,r -E p,f , and W 1/2,f and W 1/2,r the peak widths measured at half peak maximum (see Fig. 1b).

Results and Discussion
Previously, we have shown that determination of an electrode reaction is made possible from an in-depth analysis of the shape of the voltammogram following changes in the empirical parameters of period, increment, switching potential and amplitude. [9][10][11][12][13][14] In the following sections, we present the results of our simulations in a similar fashion, first considering the impact of the chemical reaction rate on the shape of the voltammogram and then individually examining the impact of the each empirical parameter. We conclude with a set of instructions on how to identify an electrode reaction as an ECE mechanism and determining k f from empirical data. Where appropriate, we include comparison of the diagnostic trends for the ECE mechanism to other mechanisms.
The impact of chemical reactions on voltammetric features depends upon the extent of the reaction within the time window of the measurement. The overall effect is usually evaluated using a dimensionless kinetic parameter. In cyclic voltammetry (CV) this parameter is k f /v where v is the potential sweep rate. In CSWV, the effective sweep rate is the increment ÷ period; the dimensionless kinetic parameter is k f τ. We have chosen to separate k f from τ to enable consideration of the empirical parameters. In doing so, we present theoretical trends that directly match what experimentalists observe from systematic variation in period.
Effect of k f .-At the onset, an ECE mechanism can be readily distinguished from case where the analyte undergoes two sequential electron transfers, i.e. the EE mechanism. Two separate reversible processes are observed in the voltammogram for the EE mechanism so long at the E 0 ≥ 150 mV. The peak potentials for each process remain at E 0 , peak currents scale linearly with τ −1/2 , and peak ratios for both processes are unity. 9 In cyclic voltammetry and CSWV, the number and magnitude of peaks observed for the ECE mechanism is a function of the rate of the chemical reaction interposed between the two electron transfers, the difference in formal potentials, and the effective potential sweep rate. 17 Fig. 2 presents the impact of log k f and E 0 on the voltammogram. For very slow rates of reaction, the voltammogram is comprised of a single peak on each sweep direction whose shape and magnitude resembles that found for a reversible electron transfer reaction. At intermediate rates (e.g. −2 < log k f < 4), the number of peaks observed on each sweep depends on E 0 2 relative to E 0 1 . Two peaks are observed on each sweep direction when E 0 2 = −300 mV. At E 0 2 = −100 mV, on the forward sweep the first peak broadens and splits into two peaks while only one peak is observed on the reverse sweep with increasing k f . When E 0 2 = E 0 1 , only one peak is observed on each sweep. At fast rates (e.g. log k f ≥ 4), two peaks are observed on the forward sweep and only one peak on the reverse sweep when E 0 2 ≤ E 0 1 . Comparison of the voltammograms when E 0 2 < E 0 1 reveals that the peak potentials for the first electron transfer step shift positively as log k f is increased from −3 to 6 whereas the peak potentials for the second remain constant at E 0 2 as shown in Fig. 2 and more quantitatively in Fig. S-1 of Supplementary Material. The peak currents for the first process (both + p,f1 and + p,r1 ) diminish in magnitude before holding constant with increasing log k f . In contrast, the peak currents for the second process (both + p,f2 and + p,r2 ) increase in magnitude before holding constant with increasing log k f (see Fig. S-1). Similarly, comparison of the voltammograms when E 0 2 = E 0 1 (bottom panels in Fig. 2 and Fig. S-1) reveals that at low to moderate values of log k f , one peak is observed on each sweep. At log k f > 3, a second peak emerges at potentials positive of E 0 1 . The peak current for this process remains constant with increasing log k f . A qualitative tool for identifying an ECE process with cyclic voltammetry involves performing multiple scans over the potential range. The same holds true for CSWV. For a reversible mechanism, there is no difference in the shape of the voltammogram on the first and second cycle. However, in the presence of coupled chemical reactions, the difference in peak currents between the first and second scans is a function of log k f and E 0 . 2 second relative to the first scan at log k f ≥ −1; at E 0 2 = 0, measurable differences are observed at log k f ≥ 3.

Effect of period (τ).-
The effect of period on the shape of the voltammogram is a complex function of the separation between E 0 2 and E 0 1 and log k f . Figure 3 presents the impact of varying period on the shape of the voltammogram as a function of E 0 2 for log k f = 3. On both the forward and reverse sweeps, the peak potential for the first process shifts positively with increasing period while the second process (at E 0 2 ) is invariant for E 0 2 = −300 and −100 mV. In contrast, when E 0 2 = 0, the peak potentials on both the forward and reverse sweeps are invariant with period at this particular value of log k f . Figure 4 presents a detailed description of the relationships between peak potentials and currents on period and log k f when E 0 2 = −300 mV. Both E p,f1 and E p,r1 shift positively by 30 mV per decade with increasing period (for log k f ≥ 0) whereas E p,f2 and E p,r2 remain essentially constant at E 0 2 . Peak currents are proportional to the τ −1/2 over a wide range in log k f . The + p,f1 vs. τ −1/2 trace is linear at log k f ≤ 1, curvilinear over the range 2 ≤ log k f ≤ 3, and again linear at log k f ≥ 4. The slopes of the linear traces decrease with increasing log k f . The ratio of + p,r1 / + p,f1 ranges from 0.1 to 1 depending upon log k f and period. This is shown in the upper right panel of Fig. 4 and is comparable to the trend for an EC mechanism. 13 Similarly, the ratio of + p,r2 / + p,f2 ranges between 1 and 2 depending upon log k f and period. At log k f ≤ 0, this ratio increases to a value of two and then decreases to a value of one. At 0 < log k f ≤ 2, the peak ratio decreases to a value of one over the range in period investigated herein. For log k f > 2, the ratio is invariant at unity. The trend shown in is comparable to the trend for a CE mechanism where peak ratios > 1 reflect the amount of conversion of Red 1 to Ox 2 and subsequent reduction to Red 2 over the period of the potential pulse. 12 Finally, the ratio of  . Plots of peak current versus τ −1/2 and peak potentials and peak ratios versus log period for E 0 2 = −300 mV as a function of log k f ranging from −3 (red) to 6 (dark gray) using the color scheme described in the caption of Fig. 2. Parameter values are: amplitude = 50 mV, increment = 10 mV, and switching potential is 200 mV negative of E 0 2 . Open circles denote the specific parameter levels for simulated data.
period as shown in Fig. 4. The slope of the linear portions of the traces depicted in this panel of the figure are consistently 0.778. Note that peak ratios presented in Fig. 4 are specific to the increment, amplitude and switching potential values given in the caption. At switching potentials further negative of E 0 2 , each trace shifts to the left. Figure 5 presents the relationships between peak potentials and currents on period and log k f when E 0 2 = −100 mV. E p,f1 shifts positively by 30 mV per decade increase in period (for log k f ≥ 0). E p,f2 emerges when log k f ≥ 0, and remains essentially constant at E 0 2 . This trend is not observed for E p,r . At log k f ≥ 3, the peak is located at E 0 2 over all periods. At log k f < 3, the peak moves from E 0 1 to E 0 2 as period increases. Peak currents are linearly or curvilinearly related to τ −1/2 depending upon log k f . The log Period Figure 5. Plots of peak current versus τ −1/2 and peak potentials and peak ratios versus log period for E 0 2 = −100 mV as a function of log k f ranging from −3 (red) to 6 (dark gray) using the color scheme described in the caption of Fig. 2. Parameter values are: amplitude = 50 mV, increment = 10 mV, and switching potential is 200 mV negative of E 0 2 . Open circles denote the specific parameter levels for simulated data.
) unless CC License in place (see abstract  the ratio begins to increase to ∼1. 15. At log k f > 0, the shift in E p,r to −100 mV occurs at shorter periods and the forward peak separates into two. When this occurs, the ratio + p,r / + p,f1 increases from 1.15 to 1.8 whereas the ratio + p,r / + p,f2 decreases from 1.15 to unity with increasing period. Thus, the transition in the shape of the voltammogram from one to two peaks on the forward sweep occurs for all log k f values but at different periods; the higher the log k f value, the shorter the period at which the transition occurs. Figure 6 provides insight into this transition by presenting the individual and difference currents (i.e. and ) as a function of period for log k f = 1. until log k f ≥ 3 when both E p,f1 and E p,f2 are resolved. At this point, E p,f1 shifts positively by 30 mV per decade with increasing period. E p,f2 remains at E 0 2 . E p,r remains essentially constant at 0 mV over the entire range in log k f . + p,f1 vs. τ −1/2 is linear for all log k f values. However, the slopes of the traces marginally vary with log k f . The ratio of + p,r / + p,f1 ranges from 0.9 to 1.0 depending upon log k f and period until E p,f1 and E p,f2 are resolved. At this point, the ratio jumps to a value between 1.7 and 1.8 and the ratio of + p,r / + p,f2 is unity. At −3 ≤ log k f ≤ −1, the peak width is invariant with period and equivalent to that for a reversible process. At all other values of log k f , the W 1/2,f increases with period with a magnitude dependent on k f until two processes are resolved along the potential axis (see Fig.  S-4).
Thus, key indicators of an ECE mechanism are the differential dependences of peak potentials, currents, and current ratios with period. The shift in peak potential with period is diagnostic for the presence of a chemical reaction following the first reduction process. 17 A plot of peak current versus τ −1/2 confirms that the electrode reaction is diffu-sional rather than surface-confined 11,37,38 and identifies the presence of a chemical reaction following the first reduction process. Peak ratios deviate from unity and are indicative of both a following and preceding chemical reaction. 17 It is interesting to note that the trends for the first process mirror those for an EC mechanism; 13 however, the trends for the second process are unlike the trends for a CE mechanism. 12 Taken collectively, systematic variation of period provides a diagnostic trend characteristic of the ECE mechanism from which the magnitude of k f can be determined.

Effect of increment (δE).-
The effect of increment on the shape of the voltammogram depends on the separation between E 0 2 and E 0 1 and log k f . Figure 7 presents the impact of varying increment on the shape of the voltammogram as a function of E 0 2 for log k f = 3 and 1. On the forward scan, E p,f1 shifts negatively and + p,f1 increases with increment while E p,f2 and + p,f2 are invariant at E 0 2 = −300 and−100 mV and log k f = 3 and 1. When E 0 2 = 0 mV at log k f = 3, E p,f and E p,r are essentially unchanged, + p increases slightly, and the apparent peak width increases with increment. At log k f = 1, E p,f = E p,r and the peak currents are independent of increment. The effective scan rate is increment divided by period. Thus, increasing increment versus increasing period shifts E p,1 in opposite directions (see Figs. 3 and 7). Figure 8 presents the dependence of peak potentials, currents, and current ratios on increment as a function of log k f when E 0 2 = −300 mV. E p,f1 and E p,r2 shifts negatively ∼28 mV per decade in increment whereas E p,f2 and E p,r2 remain constant at E 0 2 . At log k f ≤ −2, + p,f1 and + p,r1 are independent of increment. At higher k f values, + p,f1 and + p,r1 increase curvilinearly with increment whereas + p,f2 and + p,r2 decrease with increment at low values of k f but become independent of increment at log k f ≥ 2. The ratio of + p,r1 / + p,f1 ranges from 0.1 to 1 whereas the ratio of + p,r2 / + p,f2 ranges between 1 and 2.2 depending upon log k f and increment. Finally, the ratio + p,f2 / + p,f1 decreases with increment for log k f ≥ −1. This trend is comparable to that observed in cyclic voltammetry 17 where increasing the potential sweep rate shortens the time window for the conversion of Red 1 to Ox 2 . Figure S-5 presents the dependence of peak potentials, currents, and current ratios on increment as a function of log k f when E 0 2 = −100 mV. E p,f1 shifts negatively ∼28 mV per decade in increment whereas E p,f2 , when it emerges, remains constant at E 0 2 . At log k f < 1, E p,r moves from E 0 2 to E 0 1 as increment increases whereas at log k f ≥ 1, E p,r is located at E 0 2 over all increments. At log k f ≤ −2, Figure S-6 presents the dependence of peak potentials, currents, and current ratios on increment as a function of log k f when E 0 2 = 0 mV. E p,f1 remains constant (within the value of the increment) until it becomes resolved from E p,f2 . Then, E p,f1 shifts negatively by ∼28 mV per decade in increment whereas E p,f2 remains constant at E 0 2 . E p,r remains constant at E 0 1 (within the value of the increment) over all increments.  The differential dependences of peak potentials and current ratios with increment are also indicators of an ECE mechanism. When the two processes are sufficient separated along the potential axis, E p,f1 and E p,r1 shift negatively towards E 0 with increment similar to an EC mechanism. 17 In contrast to the CE mechanism, E p,f2 and E p,r2 are independent of increment. 12 Peak ratios for the first process are ≤1 (similar to an EC mechanism) and for the second process ≥1 (similar to a CE process). 12,13 Peak potential dependence on increment can provide a measure of k f if E 0 1 is known. When the two processes are not resolved along the potential axis, trends in peak potentials and peak current ratios distinguish this mechanism from a reversible, quasireversible, EC, and CE mechanism (vide supra).

Effect of amplitude (E SW ).-
The effect of amplitude on the shape of the voltammogram also depends on the separation between E 0 2 and E 0 1 and log k f . Figure 9 presents the impact of varying amplitude on the shape of the voltammogram as a function of E 0 2 for log k f = 3. On both the forward and reverse sweeps, the peak potentials for the first process shift positively with amplitude while the peak potentials for second process (at E 0 2 ) are invariant for all three E 0 2 values investigated herein. Increment (mV) Figure 8. Plots of peak currents, potentials, and ratios versus increment for E 0 2 = −300 mV as a function of log k f ranging from −3 (red) to 6 (dark gray) using the color scheme described in the caption of Fig. 2. Parameter values are: period = 50 ms, amplitude = 50 mV, and switching potential is −500 mV. Open circles denote the specific parameter levels for simulated data. The dependence of peak potentials, currents, and current ratios on amplitude is also indicative of log k f . When E 0 2 = −300 mV, E p,f1 and E p,r1 shift positively whereas E p,f2 and E p,r2 remain constant with increasing amplitude (see Fig. 10). The peak current magnitudes increase curvilinearly with amplitude; the degree of curvature depends upon log k f . In contrast, peak ratios are almost always independent of amplitude but the value changes with log k f . When E 0 2 = −100 mV, E p,f1 shifts positively when log k f ≥ 2 whereas E p,f2 remains essentially constant with increasing amplitude (see Fig. S-7) Note that at log k f ≤ −2 E p,r remains at E 0 1 and at log k f ≥ 2 E p,r remains at E 0 2 . In between these limits in log k f , E p,r shifts with amplitude (see Fig. S-7). The peak current magnitudes increase curvilinearly with amplitude; the degree of curvature depends somewhat upon log k f . In contrast to when E 0 2 = −300 mV, the peak ratio + p,r / + p,f1 depends upon both amplitude and log k f . At log k f ≤ −2, + p,r / + p,f1 equals unity at all amplitudes. At −2 < log k f ≤ 0, + p,r / + p,f1 is less than unity and increases towards unity with amplitude. At log k f = 1, two forward peaks are resolved along the potential axis at amplitudes ≤ 50 mV. At log k f ≥ 2, increases from 1.4 to 2.2 with amplitude. When E p,f2 is resolved from E p,f1 , the ratio + p,r / + p,f2 is measureable and equal to unity except with log k f = 1 (see Fig. S-7).
Figure S-8 presents the dependence of peak potentials, currents, and current ratios on amplitude as a function of log k f when E 0 2 = 0 mV. E p,f1 remains essentially constant with increasing amplitude up to log k f ≤ 3. When log k f = 4, E p,f2 becomes resolved from E p,f1 at low amplitudes. At log k f > 4, E p,f2 is resolved from E p,f1 at all amplitudes and both shift positively with increasing amplitude but with differing slopes. In contrast, E p,r remains essentially constant at E 0 1 regardless of k f and amplitude. The peak current magnitudes increase curvilinearly with amplitude; the degree of curvature depends somewhat upon log k f and whether the two processes are resolved. The dependence of peak ratios on amplitude mirrors that presented above for E 0 2 = −100 mV (see Fig. S-8). Thus, the shift in peak potentials and the magnitude of peak ratios as a function of amplitude can be used to identify the value of k f . Amplitude (mV) Figure 10. Plots of peak currents, potentials, and ratios versus amplitude for E 0 2 = −300 mV as a function of log k f ranging from −3 (red) to 6 (dark gray) using the color scheme described in the caption of Fig. 2. Parameter values are: period = 50 ms, increment = 10 mV, and switching potential is −500 mV. Open circles denote the specific parameter levels for simulated data.