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Diagnostic Criteria for the Characterization of Electrode Reactions with Chemical Reactions Following Electron Transfer by Cyclic Square Wave Voltammetry
Electrochimica Acta 205 (2016) 20–28
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Electrochimica Acta journal homepage: www.elsevier.com/locate/electacta
Diagnostic Criteria for the Characterization of Electrode Reactions with Chemical Reactions Following Electron Transfer by Cyclic Square Wave Voltammetry John C. Helfrick Jr., Megan A. Mann, Lawrence A. Bottomley* School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, GA 30332-0400, USA
A R T I C L E I N F O
Article history: Received 13 November 2015 Received in revised form 30 March 2016 Accepted 2 April 2016 Available online 4 April 2016 Keywords: Square wave voltammetry theory chemically-coupled electrode reaction cyclic square wave voltammetry EC mechanism
A B S T R A C T
Theory for cyclic square wave voltammetry of electrode reactions with coupled chemical reactions following the electron transfer is presented. Theoretical voltammograms were calculated following systematic variation of empirical parameters to assess their impact on the shape of the voltammogram. 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. ã 2016 Published by Elsevier Ltd.
1. Introduction Electron transfer reactions that generate an unstable product are commonly referred to as an EC mechanism when the product of the chemical reaction following the electron transfer is electroinactive over the potential range examined. The chemical reaction may be reversible or irreversible; the electron transfer may be fast or kinetically-controlled. The EC mechanism can be identified with a variety of electrochemical techniques [1–8]. While theory to guide the experimentalist in identifying this electrode reaction using square wave voltammetry (SWV) has been reported [5,6,9–17], very few applications of this theory have been published [18–23]. Our current objective is to stimulate the use of SWV for determining electrode reaction mechanisms, especially by nonexperts in electrochemistry who make occasional use of voltammetry in characterizing new compounds. Interestingly, the technique most often used by these workers is cyclic voltammetry (CV). Most workers assign the electrode mechanism from observing shifts in peak potentials and changes in peak current magnitudes with increasing potential sweep rate as directed by the classic work by Nicholson and Shain [2]. The theory described in this paper was
* Corresponding author. E-mail address: [email protected] (L.A. Bottomley). http://dx.doi.org/10.1016/j.electacta.2016.04.006 0013-4686/ ã 2016 Published by Elsevier Ltd.
developed when analog instrumentation was commonplace. Now, digital instruments predominate. Cyclic voltammograms acquired with digital instruments are actually cyclic staircase voltammograms. The applicability of Nicholson and Shain theory depends upon when the current is sampled during the potential pulse [24–27]. The “correct” sampling point depends upon the electrode reaction mechanism [28–32]. This report focuses on the application of cyclic square wave voltammetry (CSWV) in characterizing an EC mechanism. CSWV is SWV in two directions; the potential is stepped through the region of the formal potential of the electroactive species under study and then back in an analogous fashion with CV. Readers unfamiliar with this waveform are directed to Table of Contents graphic. The immediate reverse potential sweep functions as a probe of the stability of the product generated on the forward potential sweep. The data display format is familiar to non-electrochemists who currently make extensive use of CV for compound characterization. Since the current is sampled at the end of each potential pulse, no correction factors are required to interpret shifts in peak potentials or changes in peak current magnitudes that occur following systematic adjustment of the waveform parameters. We assert that non-specialists in electrochemistry will appreciate the similarity in output of CSWV to CV and will use this technique to characterize electrode reactions if protocols for doing so are available. To this end, we have recently presented protocols for
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evaluating single and consecutive reversible electron transfer reactions [33], kinetically-controlled electron transfer reactions [34,35], and chemically-coupled reactions [36,37]. In these studies, signature trends resulting from systematic variation of the empirical parameters for CSWV, i.e. period, increment, switching potential, and amplitude, were identified and used to establish diagnostic criteria for identifying each mechanism. In this report, theory of CSWV for the EC mechanism is presented and the signature trends are identified. When properly applied, these trends enable the experimentalist to calculate both the equilibrium constant and rate of the following reaction. Experts in electrochemistry will appreciate the complementary nature of CSWV to SWV. In conventional SWV, the immediate reverse potential step functions as a probe of the stability of the product generated on the forward potential sweep. A chemical reaction following the electron transfer will diminish the current on the reverse potential pulse by a magnitude dependent upon the potential of the step, the kinetics of the following homogeneous reaction, and the duration of the potential step. Consequently, the net current will decrease. To assign the process as an EC mechanism requires careful examination of the individual currents on each potential step [5,11,14,17]. While SWV is fully capable of identifying an EC process, the CSWV protocol presented herein enables a more straightforward characterization of this mechanism and makes use of all empirical parameters rather than just frequency or period [5,9,11,15,17]. The key CSWV features that provide this capability involve comparison of the net peak magnitudes and potentials on the forward and reverse sweeps as well as systematic variation of the potential at which the forward sweep is reversed. The diagnostic criteria presented herein are novel and educe from the widely used trends commonly used to assign electrode reaction mechanisms in CV.
2. Theory The general reaction pathway for an electron transfer followed by a chemical reaction is: kf
Ox þ ne $Red ! Z
ð1Þ
kb
where Ox is the reactant, Red is the initial product of the electron transfer, Z is the electroinactive product of the following chemical reaction, kf is the rate constant for the conversion of Red to Z in s1, and kb is the rate constant for conversion of Z to Red in s1. All chemical reactions are treated herein as first order. The derivation of an equation that enables calculation of current at each applied potential for this electrode reaction starts from Fick’s laws of diffusion. Expressions for the concentrations of Ox and Red as a function of time and distance from the electrode are found using Laplace transformations following application of the boundary conditions. These expressions are related by the Nernst equation for a reversible electron transfer: RT COx ð0; tÞ ln ð2Þ Eapplied ¼ E0 þ nF CRed ð0; tÞ where n = number of electrons transferred, F = Faraday constant, A = area of the electrode, R = gas constant, T = temperature in Kelvin, E = applied potential, E = formal potential for the electron transfer reaction, DOx = diffusion coefficient of Ox (cm2/sec), DRed = diffusion coefficient of Red (cm2/sec), COx (0, t) = concentration of Ox at the electrode surface at any time t, and CRed (0, t) = concentration of Red at the electrode surface at any time t. Numerical approximation of the resultant integral equations were performed in the same manner put forth by Nicholson and
21
Olmstead [38]. The final equation used to compute theoretical voltammograms for the ErevC mechanism is ðktpÞ
1=2
e
1 Kþ1
þ 1e
Cm ¼
1=2 i¼m1 X 2kt
1 Kþ1
C i Sj
Lp
i¼1
þ 1e
2kt 1=2 Lp
i¼m1 X
K Kþ1
K R1 þ Kþ1
C i Rj
i¼1
ð3Þ
where L = number of subintervals on each potential, K = the equilibrium constant for the following chemical reaction and equal to kf/kb, Cm = dimensionless current for each time increment with the serial number m, t = period, k = the sum of the forward and reverse rate constants for the chemical reaction following the electron transfer, i.e. kf + kb, and nF Eapplied E0 e ¼ exp ð4Þ RT assuming DOx = DRed. To compute theoretical voltammograms for CSWV, we employed the cyclic waveform available with current commercial electrochemical instrumentation. The recursive calculation of current on each step for every step in the voltammogram was performed by systematic variation of period (t ), increment (dE), switching potential (El), and amplitude (ESW), over the following intervals: 1 ms t 5 s, 1 mV dE 25 mV, 100 El –1000 mV (relative to E ), 10 mV ESW 90 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 Data. Cyclic square wave voltammograms were calculated to examine the impact of the empirical parameters period, increment, switching potential, and amplitude on the characteristic features of the voltammogram for the singular case where the number of electrons transferred equals one. The predicted difference current, DC is determined by subtracting Creverse pulse from Cforward pulse and is plotted versus the average of the potentials (Estep) at which both currents were calculated. Throughout this work, the forward difference current, DCf will denote the difference currents acquired over the interval Einitial to the switching potential El, and the difference current, DCr, will denote difference currents acquired over the reverse potential sweep from El to the final potential, Efinal. To capture the effect of period as it relates to þ
current, DC is used throughout this work where . DCþ ¼ D C pffiffi t
ð5Þ
þ
The physical meaning of DC 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. Net peak currents on the forward and þ
þ
reverse sweeps are designated asDCp;f andDCp;r , respectively. þ
þ
Peak ratio is denoted asDCp;r =DCp;f . Similarly, Ep,f and Ep,r are used to represent peak potentials with peak separation DEp = Ep,r Ep,f. Peak widths (W1/2,f and W1/2,r) are measured at half peak currents. 3. Results and Discussion In our previous reports [33–37], we showed that mechanistic identification of an electrode reaction is made possible from an in-
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depth analysis of the shape of the voltammogram following changes in the empirical parameters of period, increment, switching potential, and amplitude. In the following sections, we present the results of our simulations in a similar fashion. This report concludes with a protocol for assigning a EC mechanism and a comparison of the diagnostic criteria between this and the reversible mechanism.
a
b
3.1. Effect of K and kf The shape of the voltammogram depends upon the magnitude of K as shown in Fig. 1a. When the equilibrium favors Red (log K < 1), the voltammogram resembles that obtained for a reversible process (e.g. peak separation is 0 mV, peak ratio is unity, etc.). The reader is referred to our previous publication for further details regarding reversible process characteristics [33]. When the equilibrium favors Z (log K 1), the shape of the voltammogram significantly departs from that of a reversible
c
þ
process. The peak parameters (DCp , Ep, and W1/2) for both the forward and reverse sweep depend upon K, kf, and kb. The shape of the voltammogram also depends upon the magnitude of kf as shown in Fig. 1b. At log kf 2, the peak currents are commensurate with a reversible process. At 2 < log þ
þ
kf 2, both DCp;f and DCp;r decrease; the resultant peak ratio drops from 1.0 to 0.2. At log kf > 2, the peak currents remain constant, and the peak ratio equals 0.2. In contrast, peak potentials remain constant at log kf 0. At larger values, both Ep,f and Ep,r shift positively. The peak separation is 19 mV for log kf > 2.
þ
þ
Fig. 2. The impact of K,kf, and kb on a)DCp;f , b)DCp;r , and c) peak ratio when ESW = 50 mV, t = 50 ms, El = 200 mV, and dE = 10 mV. Log K ranges from 3 (red) to 3 (purple) in steps of 1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
þ
Fig. 2a presents the dependence of DCp;f on both log (kf + kb) and K. This plot is similar to that presented as indicative of an EC þ
mechanism[5,11]. For all values of log K 1, DCp;f decreases, reaches a minimum, and then increases with increasing (kf + kb). The magnitude of the decrease and the span in (kf + kb) values where
DC
þ p;f remains
below
the
value
for
a
reversible
voltammogram depend upon K. At low values of (kf + kb), there is no appreciable chemical reaction following the electron transfer. þ
Thus, the process appears reversible. As kf increases, DCp;f þ
decreases until a minimum is reached. DCp;f increases when kb becomes sufficiently large for conversion of Z back to Red within þ
the time of the pulse. Fig. 2b presents the dependence of DCp;r versus log (kf + kb) for various values of K. In comparison to DC
a
þ p;f ,
þ
the impact of K, kf, and kb on DCp;r is more pronounced. The onset þ p;r
of the decrease in DC
magnitude occurs at lower values of þ
(kf + kb) and the magnitude of DCp;r at its minimum is much lower þ
þ
than DCp;f at its lowest value. The point at which DCp;r begins to increase in magnitude, however, occurs at the same the same þ
b
(kf + kb) value for DCp;f . Fig. 2c presents the dependence of peak ratio versus log (kf + kb) for various values of K. The trend in peak ratio with K, kf, and kb reflects the greater impact on the peak parameters for the reverse sweep relative to those on the forward sweep. Fig. S-1 presents the dependence of Ep,f, Ep,r, and DEp versus log (kf + kb) for various values of K. For log K 1, both Ep,f and Ep,r are equal to E and are thus independent of kf and kb. When log K > 1, both Ep,f and Ep,r shift positively by an amount that depends upon K, kf, and kb. Peak widths vary somewhat over the range in K, kf, and þ
kb investigated herein (see Fig. S-2). While the trends in DCp and Ep as a function of K and (kf + kb) suggest the utility of CSWV in identifying the presence of a chemical reaction following the electron transfer, they do not provide a means for determining the rate constants or equilibrium constant from analysis of a set of voltammograms. Fig. 1. The impact of a) K and b) kf + kb on the shape of the voltammogram when ESW = 50 mV, t = 50 ms, El = 200 mV, and dE = 10 mV. In panel a, log K ranges from 2 (red) to 2 (cyan) in steps of 1 when log (kf + kb) = 0. In panel b, log (kf + kb) ranges from 3 (red) to 6 (dark gray) in steps of 1 when log K = 6. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. Effect of Period (t ) A set of voltammograms in which period alone is varied is depicted in Fig. 3a. Ep,f and Ep,r are E at short periods but shift
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positively with increasing period; peak potential separation is þ
þ
initially zero increasing to 19 mV (Fig. 3b). DCp;f and DCp;r decrease as period increases. A characteristic of a diffusion controlled, reversible process is that peak currents are proportional to t1/2 [33–35]. As shown in Fig. 3c, a linear relationship was þ
þ
found for DCp;f but not forDCp;r. Curvilinearity is indicative of a following chemical reaction (vide infra). Fig. 3d presents the dependence of peak ratio on period. As period increases, the peak ratio decreases from 1.0 to 0.2. This trend is comparable to that observed in cyclic voltammetry since increasing period decreases the potential sweep rate and lengthens the time window for the conversion of Red to Z [2]. Fig. 4a presents the peak ratio as a function of both period and K when kf + kb = 1. At log K 2, the peak ratio is equal to unity and independent of period. This reflects little to no conversion of Red to Z within the time window determined by period. Over the range 2 < log K 1, the peak ratio decreases then increases with period; the span in peak ratio depends upon log K. A drop in peak ratio occurs because of increased conversion of Red to Z within the time window of the pulse. At longer periods, the peak ratio begins to rise with increasing conversion of Z back to Red and subsequently to Ox within the time window. Finally, at log K > 1, the peak ratio drops
a
23
from 1.0 to 0.2 indicative of little to no conversion of Z back to Red to Ox within the time window determined by period. þ
Fig. 4b presents the relationship between DCratio and log t as log (kf + kb) is varied from 3 to 6 and log K = 1 (the cyan trace in panel a). Vertical sectioning of this contour plot at any value of t þ
reveals that as log (kf + kb) increases, DCratio falls from 1 to a minimum before rising back to 1. Similarly, horizontal sectioning of þ
this contour plot reveals that as t increases DCratio may increase, decrease, or fall to a minimum before rising, depending upon on log (kf + kb). þ
Fig. S-3 demonstrates that the minimum in DCratio versus log t trace depends both on log K and log (kf + kb). Note: to facilitate the reader’s review of the evidence supporting this statement, two þ
versions of Fig. S-3 are provided; one that presents DCratio vs. log t traces for specific kf + kb values in an overlay format (Fig. S-3a) and the other in contour plot format (Fig. S-3b). At large K values, i.e., kf >> kb, peak ratio decreases with t in proportion to amount of Red converting to Z. For log kf > 2, the reaction proceeds so quickly that the peak ratio is 0.2 and independent of period. At intermediate K values (panels c-f in Fig. S-3a) although kf > kb, peak ratio drops and then rises with increasing t . The minimum value for peak ratio ranges from 0.2 (at log K = 4) to 0.4 (at log K = 1). A similar trend in peak ratio is observed at log K = 0 (panel g in Fig. S-3a). At this K, kf kb and the minimum value for peak ratio is 0.7. At log K 1 (panel h), the peak ratio is essentially unity and independent of period and log (kf + kb). Clearly, the dependence of peak ratio on t is an important indicator of an EC mechanism. This mechanism is indicated when the peak ratio 1 and varies with t . To experimentally determine K and kf + kb, a plot of peak ratio versus log period should be þ
constructed and examined. If a minimum in the DCratio - log t curve is observed, the value of K is directly obtained from the working curve presented in Fig. 4c. The value of log (kf + kb) is then determined by identifying the period at which the apex occurs in the appropriate panel in Fig. S-3. Note that extrapolation between minima presented in a given panel or between panels may be
b
þ
necessary. If a minimum in the DCratio - log t curve is not observed, both K and kf + kb can be estimated from overlaying the trace onto the best fitting trace in Fig. S-3a. In this situation, additional simulations at other log K and log (kf + kb) values than those reported in Fig. S-3 may be required for refinement of initial estimates. A second indicator of an EC mechanism is the relationship
c
þ
between DCp;f and t 1/2 (see Fig. S-4). At log K = 6 and low values of kf, the slope is identical to that found for a reversible mechanism. At high values of kf, the slope approximately half the value found for the reversible case. At intermediate values, the trace is þ
curvilinear. Similarly, the relationship between DCp;r follows þ p;f
the same trend as DC except that at high values of kf, the slope approaches zero reflecting that conversion of Red to Z is essentially complete. Again, at intermediate values, the trace is curvilinear but
d
þ
Fig. 3. The impact of period on the shape of the voltammogram when ESW = 50 mV, dE = 10 mV, El = 200 mV, log K = 3, and log (kf + kb) = 0. Panel a): Period ranges from þ
1 ms (red) to 5 s (brown). Panel b): Ep,f (red) and Ep,r (black). Panel c): DCp;f (red) þ
and DCp;r (black). Panel d): Peak ratio (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
with a larger bowing compared toDCp;f . A third indicator of an EC mechanism is the shift in peak potentials with period. This is shown in Figs. S-5 and S-6. The magnitude of shifts in Ep,f (Fig. S-5) and Ep,r (Fig. S-6) depend on K and kf + kb. At log K 0, the peak potentials remain essentially unchanged as period is varied. At 0 < log K < 4, the peak potential versus log period trace is curvilinear with slopes approaching 30 mV. At log K 4, the slope is 30 mV, consistent with the trend in peak potentials with scan rate found by Nicholson and Shain for cyclic voltammetry [2]. When varying period, several differences can be identified between the EC and reversible cases. Peak potentials shift
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positively from E with period for the EC case and remain at E for the reversible case. Current is curvilinearly related to t 1/2 for the EC case and linearly related to t 1/2 reversible case. Peak ratios vary with period for the EC case but are always unity for the reversible case. Readers are reminded that all trends for the EC case are dependent on K, kf, and kb. Thus, given these differences in the trends for the two mechanisms, variation of period can be used to distinguish between the reversible and EC mechanisms.
a
3.3. Effect of Increment (dE) A set of voltammograms in which increment alone is varied is depicted in Fig. 5a for log K = 3 and log (kf + kb) = 0. Peak potentials shift toward the formal potential with increment (see Fig. 5b). The þ
þ
change in DCp;r with increment is greater than that for DCp;f though both values increase (see Fig. 5c). Peak ratio increases with increment from 0.25 to 0.7 as shown in Fig. 5d. Fig. 6 shows the effects of increment on peak ratio for selected values of K. For log K 2, the peak ratio is independent of increment. At greater values of log K, the peak ratio increases with increment. This trend is comparable to that observed in CV;
b
a
b
c c
d
Fig. 4. The impact of t on the peak ratio as a function of log K and log (kf + kb) when ESW = 50 mV, dE = 10 mV, and El = 200 mV. Panel a: illustrates the relationship þ
between DCratio and t as log K is varied from 3 (red) to 3 (purple), and log
(kf + kb) = 0. Panel b presents the relationship between DCratio and t as log (kf + kb) is varied from 3 to 6 and log K = 1 (the cyan trace in panel a). Panel
Fig. 5. The impact of dE on the shape of the voltammogram when ESW = 50 mV, t = 50 ms, El = 200 mV, log K = 3, and log (kf + kb) = 0. Panel a): dE ranges from 1 mV (red), 5 mV (orange), 10 mV (yellow), 15 mV (green), and 20 mV (cyan). Panel b): Ep,f
c relates the minimum in DCratio versus log K. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
ratio (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
þ
þ
þ
þ
(red) and Ep,r (black). Panel c): DCp;f (red) and DCp;r (black). Panel d): Peak
J.C. Helfrick et al. / Electrochimica Acta 205 (2016) 20–28
25
case but remain at E for the reversible case; the magnitude of the shift depends on K, kf, and kb. Peak currents increase with increment for the EC case but are unaffected by increment for the reversible. Peak ratios vary with increment for the EC case but are unity regardless of increment for the reversible case; the trend is dependent upon K, kf, and kb. Hence, the systematic variation of increment can be used to discern between the reversible and EC mechanisms. 3.4. Effect of Switching Potential (El) The dramatic effect of switching potential on the wave shape for an EC mechanism is shown in Figure 7a for log K = 3 and log þ
þ
(kf + kb) = 0. While DCp;f , Ep,f, and Ep,r are unaffected, DCp;r is þ
greatly affected by El (Fig. 7c). As El approaches E , DCp;r increases and consequently, the peak ratio increases (Fig. 7d). This occurs as less time is allowed for Red to convert to Z when El is closer to the þ
Fig. 6. The impact of increment on the peak ratio as log K is varied from 3 (red) to 3 (purple) in steps of 1 when ESW = 50 mV, t = 50 ms, El = 200 mV, and log (kf + kb) = 0. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
increasing increment raises the potential sweep rate and shortens the time window for the conversion of Red to Z. The relationship of peak ratio with increment is markedly dependent upon kf + kb and K. The complex dependency is shown in Fig. S-7. At log K 5 (panels a-c), the effect of kf + kb and increment on peak ratio is similar. When log (kf + kb) = 3, the trend for peak ratio vs. increment mirrors that for a reversible reaction. When 2 log (kf + kb) 0, peak ratio increases with increment. When log (kf + kb) > 0, peak ratio remains constant and independent of increment. For intermediate values of K (panels d & e), the effects of increment log (kf + kb) 0 remain the same as in panels a-c. For log (kf + kb) > 0, peak ratio decreases as increment is increased with a magnitude dependent upon the value of log (kf + kb). For 0 log K 1 (panels f & g), trends in peak ratio with increment and log (kf + kb) match those obtained for log K = 2 but over a smaller span in peak ratio. When log K 1 (shown for log K = 1 in panel h), the peak ratio is independent of increment and very slightly dependent on log (kf + kb). þ
Fig. S-8 presents the impact of increment on DCp;f (panel a) and
formal potential. The trends in DCp;r and peak ratio with El distinguishes the EC mechanism from others. The impact of K on the relationship between peak ratio and El is shown in Fig. 8 for log (kf + kb) = 0. At low K values where the
a
b
c
DCþp;r (panel b) as a function of kf + kb. Detailed inspection of this figure provides insight into the trends shown in Fig. S-7a. When log þ
þ
(kf + kb) > -2, both DCp;f and DCp;r increase in magnitude curvilinearly with increment; the slope depends upon log (kf + kb). þ
þ
But, for given value of log (kf + kb), DCp;f and DCp;r are not mirror images of each other. The difference reflects the rate of conversion of Red to Z. The impact of increment, K, and kf + kb on peak potentials is shown in Figs. S-9 and S-10. Regardless of the value of K, when the rate of the following reaction is slow (i.e. log (kf + kb) 1) Ep,f and Ep,r = E and are independent of increment. At faster rates of the reaction (i.e. log (kf + kb) > 1) and when the equilibrium lies strongly in favor of Z (i.e. log K 1, panels a-f), Ep values shift curvilinearly toward E with increment. The displacement of both peak potentials from E depends strongly on K and kf + kb. When the equilibrium begins to favor Red (i.e. log K < 1, panels g & h), the shift in Ep with increment is 10 mV over the range in increment from 1 to 25 mV and collapses on E when log K 1. In comparing the trends for the EC and reversible cases as a function of increment, several differences can be identified. Peak potentials shift negatively towards E with increment for the EC
d
Fig. 7. The impact of El on the shape of the voltammogram when ESW = 50 mV, dE = 10 mV, t = 50 ms, log K = 3, and log (kf + kb) = 0. Panel a): El: 1000 mV (red), 900 mV (orange), 800 mV (yellow), 700 mV (green), 600 mV (cyan), 500 mV (blue), 400 mV (purple), 300 mV (magenta), 200 mV (light gray), and 100 mV (dark gray). þ
þ
Panel b): Ep,f (red) and Ep,r (black). Panel c): DCp;f (red) and DCp;r (black). Panel d): Peak ratio (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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a
b
c
Fig. 8. The impact of El on peak ratio as log K is varied from 3 (red) to 3 (purple) in steps of 1 when ESW = 50 mV, dE = 10 mV, t = 50 ms, and log (kf + kb) = 0. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
equilibrium favors Red, peak ratio is unaffected by the switching potential and equal to unity just as in the reversible electron transfer mechanism. At higher K values, peak ratio decreases with increasing El. The magnitude of the decrease is proportional to K. The impact of K and kf + kb on the peak ratio vs. El is quite complex as shown in Fig. S-11. At low values of log (kf + kb) (i.e. 2), peak ratios are independent of El and equal to unity regardless of log K. The rate of the following reaction is too slow to impact the current within the time window of the period. At intermediate values (i.e. 2 < log (kf + kb) 2), peak ratios decrease curvilinearly with El; the span in peak ratio over the range in El depends upon log K. At high rates (log (kf + kb) > 2), the span in peak ratio is only 0.1 with increasing El. The magnitude of peak ratio at a given El increases with decreasing K. For example, at log (kf + kb) = 6 and El equal to 400 mV, the peak ratio is 0.5 for log K = 3 and 0.9 for log K = 2. The relationship between peak currents and El depends upon þ
both K (vide supra) and log (kf + kb) as shown in Fig. S-12. DCp;f remains constant while switching potential is varied though the þ
þ
value of DCp;f is dependent upon log (kf + kb). Similarly,DCp;r is unaffected by El when log (kf + kb) 2. At higher values of kf + kb,
DCþp;r increases dramatically as switching potential decreases. þ When log (kf + kb) 1, DCp;r approaches zero and remains
relatively constant as switching potential is varied. The effect of K and kf + kb on the relationship between peak potentials and El is shown in Figs. S-13 and S-14. Ep,f and Ep,r are unaffected by switching potential though their values are characteristic of K and kf + kb. There are contrasting trends for the effect of switching potential on the voltammograms of the reversible and EC cases. Ep,f and Ep,r are displaced from E but remain essentially constant with variation in switching potential for the EC mechanism whereas þ
Ep,f and Ep,r remain at E for the reversible case. DCp;f is unaffected þ
by switching potential for the EC case but DCp;r increases as þ
switching potential approaches E . The magnitude of both DCp;f þ
and DCp;r depend on K, kf, and kb. Peak currents are unaffected by switching potential for the reversible case. Subsequently, peak ratio increases as the switching potential approaches E for the EC case while peak ratio is invariant of switching potential and equal
d
Fig. 9. The impact of ESW on the shape of the voltammogram when dE = 10 mV, t = 50 ms, El = 200 mV, log K = 3, and log (kf + kb) = 0. Panel a): ESW ranges from 10 mV (red), 20 mV (orange), 30 mV (yellow), 40 mV (green), 50 mV (cyan), 60 mV (blue), 70 mV (purple), 80 mV (magenta), and 90 mV (light gray). Panel b): Ep,f (red) þ
þ
and Ep,r (black). Panel c): DCp;f (red) and DCp;r (black). Panel (d): Peak ratio (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
to unity for the reversible case. Thus, the effect of switching potential on the peak properties can be used to identify EC vs. reversible mechanisms. 3.5. Effect of Amplitude (ESW) The effect of amplitude on the voltammogram is illustrated in Fig. 9a and explored in detail in Fig. 9b-d for log K = 3 and log (kf + kb) = 0. Peak potentials shift slightly with amplitude (Fig. 9b). þ
þ
Both DCp;f and DCp;r increase with amplitude though the effect þ p;f
on DC is more drastic (Fig. 9c). Amplitude has a negligible effect on peak ratio as shown in Fig. 9d where the peak ratio increases by only 0.06 when the amplitude is increased from 10 to 90 mV. The value of peak ratio is governed by log K and log (kf + kb) as shown in Fig. S-15. Fig. S-16 provides insight into the peak ratio-amplitude þ
þ
trends shown in Fig. S-15a. Both DCp;f and DCp;r increase curvilinearly with amplitude however the effect of amplitude on
DCþp;f is greater than that on DCþp;r . Figs. S-17 and S-18 present the impact of amplitude on peak potentials as a function of log (kf + kb) and log K. When log (kf + kb) 1 (red, orange and yellow traces in both figures), both
J.C. Helfrick et al. / Electrochimica Acta 205 (2016) 20–28
27
Table 1 Diagnostic Plots and Protocol for Assessing an EC Electrode Reaction by CSWV. Empirical variables Waveform Period, Ï„ parameters Plot Trend Peak currents
DIp vs.
Peak DEp separation vs. log t
Peak ratio 1 and varies with log t ; magnitude depends on log K and log (kf + kb) (see Figs. 3 d, 4, and S-3) Ep shifts positively with log t ; (see Figs. 3 b, S-5, and S-6) magnitude of Ep depends upon log K and log (kf + kb) Complex relationship, may increase or decrease with log t (compare Figs. S-5 and S-6)
Peak widths
W1/2 is generally independent of log t
Peak ratio vs. log t Peak Ep vs. potentials log t
W1/2 vs. log t
Switching potential, Eλ
Amplitude, ESW
Plot
Plot
Trend
Plot
Trend
DIp,f is independent of El; DIp,r increases with El
DIp
DIp increases with ESW (see Figs. 9 c and S-16)
Trend
DIp increases nonlinearly DIp DIp increases with dE; DIp with t 1/2 (see Figs. 3 c and vs. dE magnitude depends on log K vs.
t1/2 S-4) Peak ratio
Increment, δE
vs. and log (kf + kb); (see Fig. 5c El approaching E (see Figs. 7 c ESW and S-8) and S-12) Peak Peak ratio 1 and varies with Peak Peak ratio 1 and increases Peak ratio ratio dE depending upon K and ratio as El approaches E (see vs. vs. dE kf + kb (see Figs. Fig. 5d, 6, and vs. Figs. 7 d, 8, and S-11) ESW El S-7) Ep vs. Ep shifts negatively with dE; Ep vs. Ep is independent of El; Ep vs. magnitude of Ep depends El magnitude of Ep depends ESW dE upon log K and log (kf + kb) upon log K and log (kf + kb) (see Fig. 5b, S-9, and S-10) (see Figs. 7 b, S-13, and S-14) DEp DEp decreases with dE; DEp DEp is independent of El D Ep vs. dE magnitude of Ep depends vs. vs. El ESW upon log K and log (kf + kb) (compare Figs. S-9 and S-10) W1/2 W1/2 varies by 10 mV with d W1/2 W1/2 is independent of El W1/2 vs. dE E vs. vs. El ESW
peak potentials are essentially at E , regardless of log K and amplitude. When log (kf + kb) 0 and log K 3, both Ep,f and Ep,r shift positively with amplitude; peak potentials values and magnitude of change depends on log (kf + kb) (see panels a-d in both figures). As K becomes smaller (panels e-g in both figures), larger log (kf + kb) values are invariant with amplitude while intermediate vales of log (kf + kb) shift positively with amplitude. When log K 1, peak potentials become invariant of amplitude, regardless of kf + kb. Several contrasting trends can be identified when comparing the EC and reversible mechanisms. For the EC mechanism, peak potentials shift positively from E with amplitude. For the reversible mechanism, peak potentials are independent of amplitude. Peak currents increase with amplitude for both cases, but the increase is dependent upon K, kf, and kb for the EC mechanism. Peak ratios are either less than unity or increase towards unity for the EC mechanism but are always unity for the reversible case. Thus, the variation of amplitude can be used to distinguish between the reversible and EC mechanisms.
Peak ratio 1 and increases slightly with ESW (see Figs. 9 d and S-15) Ep shifts positively with ESW; magnitude of Ep depends upon log K and log (kf + kb) (see Figs. 9 b, S-17, and S-18) DEp increases with ESW for select values of log (kf + kb) but otherwise is invariant with ESW W1/2 increases with ESW
summarized in Table 1. The diagnostic criteria presented in this Table are novel. It is based on analyzing voltammetric features rather than individual currents thereby enabling rapid identification of EC mechanisms using CSWV. Close examination of the trends presented in supplementary data reveals the following limits. When the following chemical reaction is irreversible, i.e. log K 3, measureable log kf values are limited to the range 2 < log kf < 2. When the following chemical reaction is reversible, i.e. 0 log K < 3, measureable log (kf + kb) values span a much broader range 2 < log (kf + kb) < 6. We are currently investigating kineticallycontrolled electron transfers coupled to chemical reactions and will report diagnostic criteria for these EquasiC and CEquasi mechanisms elsewhere. Acknowledgments Acknowledgment is made to CMMI1130739 for financial support.
NSF
grant
number
Appendix A. Supplementary data 4. Conclusions CSWV is SWV in two directions without delay or loss of control on the working electrode potential paralleling the complementary nature of CV to linear sweep voltammetry. In CV, the experimenter identifies the electrode reaction mechanism by systematic variation in potential sweep rate and evaluating commensurate shifts in peak currents and potentials. In SWV, the experimenter varies ESW and t (or frequency) and tracks shifts in peak current and potential. In CSWV, the experimenter make use of four empirical parameters (ESW, dE, t , and El) and identifies the mechanism from characteristic shifts in six peak features þ
þ
þ
(DCp;f , DCp;r , DCratio , Ep,f, Ep,r, and DEp). Thus, the cyclic waveforms (both CV and CSWV) provide additional empirical parameters and peak features that afford greater discrimination between mechanisms compared to linear sweep voltammetry or conventional SWV. In this report, we have shown that the peak currents, potentials, and shapes of cyclic square wave voltammograms for the EC mechanism are a complex function K, kf + kb, increment, period, switching potential, and amplitude. The effects of systematic variation in these empirical parameters on the voltammograms are
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Contents lists available at ScienceDirect
Electrochimica Acta journal homepage: www.elsevier.com/locate/electacta
Diagnostic Criteria for the Characterization of Electrode Reactions with Chemical Reactions Following Electron Transfer by Cyclic Square Wave Voltammetry John C. Helfrick Jr., Megan A. Mann, Lawrence A. Bottomley* School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, GA 30332-0400, USA
A R T I C L E I N F O
Article history: Received 13 November 2015 Received in revised form 30 March 2016 Accepted 2 April 2016 Available online 4 April 2016 Keywords: Square wave voltammetry theory chemically-coupled electrode reaction cyclic square wave voltammetry EC mechanism
A B S T R A C T
Theory for cyclic square wave voltammetry of electrode reactions with coupled chemical reactions following the electron transfer is presented. Theoretical voltammograms were calculated following systematic variation of empirical parameters to assess their impact on the shape of the voltammogram. 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. ã 2016 Published by Elsevier Ltd.
1. Introduction Electron transfer reactions that generate an unstable product are commonly referred to as an EC mechanism when the product of the chemical reaction following the electron transfer is electroinactive over the potential range examined. The chemical reaction may be reversible or irreversible; the electron transfer may be fast or kinetically-controlled. The EC mechanism can be identified with a variety of electrochemical techniques [1–8]. While theory to guide the experimentalist in identifying this electrode reaction using square wave voltammetry (SWV) has been reported [5,6,9–17], very few applications of this theory have been published [18–23]. Our current objective is to stimulate the use of SWV for determining electrode reaction mechanisms, especially by nonexperts in electrochemistry who make occasional use of voltammetry in characterizing new compounds. Interestingly, the technique most often used by these workers is cyclic voltammetry (CV). Most workers assign the electrode mechanism from observing shifts in peak potentials and changes in peak current magnitudes with increasing potential sweep rate as directed by the classic work by Nicholson and Shain [2]. The theory described in this paper was
* Corresponding author. E-mail address: [email protected] (L.A. Bottomley). http://dx.doi.org/10.1016/j.electacta.2016.04.006 0013-4686/ ã 2016 Published by Elsevier Ltd.
developed when analog instrumentation was commonplace. Now, digital instruments predominate. Cyclic voltammograms acquired with digital instruments are actually cyclic staircase voltammograms. The applicability of Nicholson and Shain theory depends upon when the current is sampled during the potential pulse [24–27]. The “correct” sampling point depends upon the electrode reaction mechanism [28–32]. This report focuses on the application of cyclic square wave voltammetry (CSWV) in characterizing an EC mechanism. CSWV is SWV in two directions; the potential is stepped through the region of the formal potential of the electroactive species under study and then back in an analogous fashion with CV. Readers unfamiliar with this waveform are directed to Table of Contents graphic. The immediate reverse potential sweep functions as a probe of the stability of the product generated on the forward potential sweep. The data display format is familiar to non-electrochemists who currently make extensive use of CV for compound characterization. Since the current is sampled at the end of each potential pulse, no correction factors are required to interpret shifts in peak potentials or changes in peak current magnitudes that occur following systematic adjustment of the waveform parameters. We assert that non-specialists in electrochemistry will appreciate the similarity in output of CSWV to CV and will use this technique to characterize electrode reactions if protocols for doing so are available. To this end, we have recently presented protocols for
J.C. Helfrick et al. / Electrochimica Acta 205 (2016) 20–28
evaluating single and consecutive reversible electron transfer reactions [33], kinetically-controlled electron transfer reactions [34,35], and chemically-coupled reactions [36,37]. In these studies, signature trends resulting from systematic variation of the empirical parameters for CSWV, i.e. period, increment, switching potential, and amplitude, were identified and used to establish diagnostic criteria for identifying each mechanism. In this report, theory of CSWV for the EC mechanism is presented and the signature trends are identified. When properly applied, these trends enable the experimentalist to calculate both the equilibrium constant and rate of the following reaction. Experts in electrochemistry will appreciate the complementary nature of CSWV to SWV. In conventional SWV, the immediate reverse potential step functions as a probe of the stability of the product generated on the forward potential sweep. A chemical reaction following the electron transfer will diminish the current on the reverse potential pulse by a magnitude dependent upon the potential of the step, the kinetics of the following homogeneous reaction, and the duration of the potential step. Consequently, the net current will decrease. To assign the process as an EC mechanism requires careful examination of the individual currents on each potential step [5,11,14,17]. While SWV is fully capable of identifying an EC process, the CSWV protocol presented herein enables a more straightforward characterization of this mechanism and makes use of all empirical parameters rather than just frequency or period [5,9,11,15,17]. The key CSWV features that provide this capability involve comparison of the net peak magnitudes and potentials on the forward and reverse sweeps as well as systematic variation of the potential at which the forward sweep is reversed. The diagnostic criteria presented herein are novel and educe from the widely used trends commonly used to assign electrode reaction mechanisms in CV.
2. Theory The general reaction pathway for an electron transfer followed by a chemical reaction is: kf
Ox þ ne $Red ! Z
ð1Þ
kb
where Ox is the reactant, Red is the initial product of the electron transfer, Z is the electroinactive product of the following chemical reaction, kf is the rate constant for the conversion of Red to Z in s1, and kb is the rate constant for conversion of Z to Red in s1. All chemical reactions are treated herein as first order. The derivation of an equation that enables calculation of current at each applied potential for this electrode reaction starts from Fick’s laws of diffusion. Expressions for the concentrations of Ox and Red as a function of time and distance from the electrode are found using Laplace transformations following application of the boundary conditions. These expressions are related by the Nernst equation for a reversible electron transfer: RT COx ð0; tÞ ln ð2Þ Eapplied ¼ E0 þ nF CRed ð0; tÞ where n = number of electrons transferred, F = Faraday constant, A = area of the electrode, R = gas constant, T = temperature in Kelvin, E = applied potential, E = formal potential for the electron transfer reaction, DOx = diffusion coefficient of Ox (cm2/sec), DRed = diffusion coefficient of Red (cm2/sec), COx (0, t) = concentration of Ox at the electrode surface at any time t, and CRed (0, t) = concentration of Red at the electrode surface at any time t. Numerical approximation of the resultant integral equations were performed in the same manner put forth by Nicholson and
21
Olmstead [38]. The final equation used to compute theoretical voltammograms for the ErevC mechanism is ðktpÞ
1=2
e
1 Kþ1
þ 1e
Cm ¼
1=2 i¼m1 X 2kt
1 Kþ1
C i Sj
Lp
i¼1
þ 1e
2kt 1=2 Lp
i¼m1 X
K Kþ1
K R1 þ Kþ1
C i Rj
i¼1
ð3Þ
where L = number of subintervals on each potential, K = the equilibrium constant for the following chemical reaction and equal to kf/kb, Cm = dimensionless current for each time increment with the serial number m, t = period, k = the sum of the forward and reverse rate constants for the chemical reaction following the electron transfer, i.e. kf + kb, and nF Eapplied E0 e ¼ exp ð4Þ RT assuming DOx = DRed. To compute theoretical voltammograms for CSWV, we employed the cyclic waveform available with current commercial electrochemical instrumentation. The recursive calculation of current on each step for every step in the voltammogram was performed by systematic variation of period (t ), increment (dE), switching potential (El), and amplitude (ESW), over the following intervals: 1 ms t 5 s, 1 mV dE 25 mV, 100 El –1000 mV (relative to E ), 10 mV ESW 90 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 Data. Cyclic square wave voltammograms were calculated to examine the impact of the empirical parameters period, increment, switching potential, and amplitude on the characteristic features of the voltammogram for the singular case where the number of electrons transferred equals one. The predicted difference current, DC is determined by subtracting Creverse pulse from Cforward pulse and is plotted versus the average of the potentials (Estep) at which both currents were calculated. Throughout this work, the forward difference current, DCf will denote the difference currents acquired over the interval Einitial to the switching potential El, and the difference current, DCr, will denote difference currents acquired over the reverse potential sweep from El to the final potential, Efinal. To capture the effect of period as it relates to þ
current, DC is used throughout this work where . DCþ ¼ D C pffiffi t
ð5Þ
þ
The physical meaning of DC 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. Net peak currents on the forward and þ
þ
reverse sweeps are designated asDCp;f andDCp;r , respectively. þ
þ
Peak ratio is denoted asDCp;r =DCp;f . Similarly, Ep,f and Ep,r are used to represent peak potentials with peak separation DEp = Ep,r Ep,f. Peak widths (W1/2,f and W1/2,r) are measured at half peak currents. 3. Results and Discussion In our previous reports [33–37], we showed that mechanistic identification of an electrode reaction is made possible from an in-
22
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depth analysis of the shape of the voltammogram following changes in the empirical parameters of period, increment, switching potential, and amplitude. In the following sections, we present the results of our simulations in a similar fashion. This report concludes with a protocol for assigning a EC mechanism and a comparison of the diagnostic criteria between this and the reversible mechanism.
a
b
3.1. Effect of K and kf The shape of the voltammogram depends upon the magnitude of K as shown in Fig. 1a. When the equilibrium favors Red (log K < 1), the voltammogram resembles that obtained for a reversible process (e.g. peak separation is 0 mV, peak ratio is unity, etc.). The reader is referred to our previous publication for further details regarding reversible process characteristics [33]. When the equilibrium favors Z (log K 1), the shape of the voltammogram significantly departs from that of a reversible
c
þ
process. The peak parameters (DCp , Ep, and W1/2) for both the forward and reverse sweep depend upon K, kf, and kb. The shape of the voltammogram also depends upon the magnitude of kf as shown in Fig. 1b. At log kf 2, the peak currents are commensurate with a reversible process. At 2 < log þ
þ
kf 2, both DCp;f and DCp;r decrease; the resultant peak ratio drops from 1.0 to 0.2. At log kf > 2, the peak currents remain constant, and the peak ratio equals 0.2. In contrast, peak potentials remain constant at log kf 0. At larger values, both Ep,f and Ep,r shift positively. The peak separation is 19 mV for log kf > 2.
þ
þ
Fig. 2. The impact of K,kf, and kb on a)DCp;f , b)DCp;r , and c) peak ratio when ESW = 50 mV, t = 50 ms, El = 200 mV, and dE = 10 mV. Log K ranges from 3 (red) to 3 (purple) in steps of 1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
þ
Fig. 2a presents the dependence of DCp;f on both log (kf + kb) and K. This plot is similar to that presented as indicative of an EC þ
mechanism[5,11]. For all values of log K 1, DCp;f decreases, reaches a minimum, and then increases with increasing (kf + kb). The magnitude of the decrease and the span in (kf + kb) values where
DC
þ p;f remains
below
the
value
for
a
reversible
voltammogram depend upon K. At low values of (kf + kb), there is no appreciable chemical reaction following the electron transfer. þ
Thus, the process appears reversible. As kf increases, DCp;f þ
decreases until a minimum is reached. DCp;f increases when kb becomes sufficiently large for conversion of Z back to Red within þ
the time of the pulse. Fig. 2b presents the dependence of DCp;r versus log (kf + kb) for various values of K. In comparison to DC
a
þ p;f ,
þ
the impact of K, kf, and kb on DCp;r is more pronounced. The onset þ p;r
of the decrease in DC
magnitude occurs at lower values of þ
(kf + kb) and the magnitude of DCp;r at its minimum is much lower þ
þ
than DCp;f at its lowest value. The point at which DCp;r begins to increase in magnitude, however, occurs at the same the same þ
b
(kf + kb) value for DCp;f . Fig. 2c presents the dependence of peak ratio versus log (kf + kb) for various values of K. The trend in peak ratio with K, kf, and kb reflects the greater impact on the peak parameters for the reverse sweep relative to those on the forward sweep. Fig. S-1 presents the dependence of Ep,f, Ep,r, and DEp versus log (kf + kb) for various values of K. For log K 1, both Ep,f and Ep,r are equal to E and are thus independent of kf and kb. When log K > 1, both Ep,f and Ep,r shift positively by an amount that depends upon K, kf, and kb. Peak widths vary somewhat over the range in K, kf, and þ
kb investigated herein (see Fig. S-2). While the trends in DCp and Ep as a function of K and (kf + kb) suggest the utility of CSWV in identifying the presence of a chemical reaction following the electron transfer, they do not provide a means for determining the rate constants or equilibrium constant from analysis of a set of voltammograms. Fig. 1. The impact of a) K and b) kf + kb on the shape of the voltammogram when ESW = 50 mV, t = 50 ms, El = 200 mV, and dE = 10 mV. In panel a, log K ranges from 2 (red) to 2 (cyan) in steps of 1 when log (kf + kb) = 0. In panel b, log (kf + kb) ranges from 3 (red) to 6 (dark gray) in steps of 1 when log K = 6. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. Effect of Period (t ) A set of voltammograms in which period alone is varied is depicted in Fig. 3a. Ep,f and Ep,r are E at short periods but shift
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positively with increasing period; peak potential separation is þ
þ
initially zero increasing to 19 mV (Fig. 3b). DCp;f and DCp;r decrease as period increases. A characteristic of a diffusion controlled, reversible process is that peak currents are proportional to t1/2 [33–35]. As shown in Fig. 3c, a linear relationship was þ
þ
found for DCp;f but not forDCp;r. Curvilinearity is indicative of a following chemical reaction (vide infra). Fig. 3d presents the dependence of peak ratio on period. As period increases, the peak ratio decreases from 1.0 to 0.2. This trend is comparable to that observed in cyclic voltammetry since increasing period decreases the potential sweep rate and lengthens the time window for the conversion of Red to Z [2]. Fig. 4a presents the peak ratio as a function of both period and K when kf + kb = 1. At log K 2, the peak ratio is equal to unity and independent of period. This reflects little to no conversion of Red to Z within the time window determined by period. Over the range 2 < log K 1, the peak ratio decreases then increases with period; the span in peak ratio depends upon log K. A drop in peak ratio occurs because of increased conversion of Red to Z within the time window of the pulse. At longer periods, the peak ratio begins to rise with increasing conversion of Z back to Red and subsequently to Ox within the time window. Finally, at log K > 1, the peak ratio drops
a
23
from 1.0 to 0.2 indicative of little to no conversion of Z back to Red to Ox within the time window determined by period. þ
Fig. 4b presents the relationship between DCratio and log t as log (kf + kb) is varied from 3 to 6 and log K = 1 (the cyan trace in panel a). Vertical sectioning of this contour plot at any value of t þ
reveals that as log (kf + kb) increases, DCratio falls from 1 to a minimum before rising back to 1. Similarly, horizontal sectioning of þ
this contour plot reveals that as t increases DCratio may increase, decrease, or fall to a minimum before rising, depending upon on log (kf + kb). þ
Fig. S-3 demonstrates that the minimum in DCratio versus log t trace depends both on log K and log (kf + kb). Note: to facilitate the reader’s review of the evidence supporting this statement, two þ
versions of Fig. S-3 are provided; one that presents DCratio vs. log t traces for specific kf + kb values in an overlay format (Fig. S-3a) and the other in contour plot format (Fig. S-3b). At large K values, i.e., kf >> kb, peak ratio decreases with t in proportion to amount of Red converting to Z. For log kf > 2, the reaction proceeds so quickly that the peak ratio is 0.2 and independent of period. At intermediate K values (panels c-f in Fig. S-3a) although kf > kb, peak ratio drops and then rises with increasing t . The minimum value for peak ratio ranges from 0.2 (at log K = 4) to 0.4 (at log K = 1). A similar trend in peak ratio is observed at log K = 0 (panel g in Fig. S-3a). At this K, kf kb and the minimum value for peak ratio is 0.7. At log K 1 (panel h), the peak ratio is essentially unity and independent of period and log (kf + kb). Clearly, the dependence of peak ratio on t is an important indicator of an EC mechanism. This mechanism is indicated when the peak ratio 1 and varies with t . To experimentally determine K and kf + kb, a plot of peak ratio versus log period should be þ
constructed and examined. If a minimum in the DCratio - log t curve is observed, the value of K is directly obtained from the working curve presented in Fig. 4c. The value of log (kf + kb) is then determined by identifying the period at which the apex occurs in the appropriate panel in Fig. S-3. Note that extrapolation between minima presented in a given panel or between panels may be
b
þ
necessary. If a minimum in the DCratio - log t curve is not observed, both K and kf + kb can be estimated from overlaying the trace onto the best fitting trace in Fig. S-3a. In this situation, additional simulations at other log K and log (kf + kb) values than those reported in Fig. S-3 may be required for refinement of initial estimates. A second indicator of an EC mechanism is the relationship
c
þ
between DCp;f and t 1/2 (see Fig. S-4). At log K = 6 and low values of kf, the slope is identical to that found for a reversible mechanism. At high values of kf, the slope approximately half the value found for the reversible case. At intermediate values, the trace is þ
curvilinear. Similarly, the relationship between DCp;r follows þ p;f
the same trend as DC except that at high values of kf, the slope approaches zero reflecting that conversion of Red to Z is essentially complete. Again, at intermediate values, the trace is curvilinear but
d
þ
Fig. 3. The impact of period on the shape of the voltammogram when ESW = 50 mV, dE = 10 mV, El = 200 mV, log K = 3, and log (kf + kb) = 0. Panel a): Period ranges from þ
1 ms (red) to 5 s (brown). Panel b): Ep,f (red) and Ep,r (black). Panel c): DCp;f (red) þ
and DCp;r (black). Panel d): Peak ratio (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
with a larger bowing compared toDCp;f . A third indicator of an EC mechanism is the shift in peak potentials with period. This is shown in Figs. S-5 and S-6. The magnitude of shifts in Ep,f (Fig. S-5) and Ep,r (Fig. S-6) depend on K and kf + kb. At log K 0, the peak potentials remain essentially unchanged as period is varied. At 0 < log K < 4, the peak potential versus log period trace is curvilinear with slopes approaching 30 mV. At log K 4, the slope is 30 mV, consistent with the trend in peak potentials with scan rate found by Nicholson and Shain for cyclic voltammetry [2]. When varying period, several differences can be identified between the EC and reversible cases. Peak potentials shift
24
J.C. Helfrick et al. / Electrochimica Acta 205 (2016) 20–28
positively from E with period for the EC case and remain at E for the reversible case. Current is curvilinearly related to t 1/2 for the EC case and linearly related to t 1/2 reversible case. Peak ratios vary with period for the EC case but are always unity for the reversible case. Readers are reminded that all trends for the EC case are dependent on K, kf, and kb. Thus, given these differences in the trends for the two mechanisms, variation of period can be used to distinguish between the reversible and EC mechanisms.
a
3.3. Effect of Increment (dE) A set of voltammograms in which increment alone is varied is depicted in Fig. 5a for log K = 3 and log (kf + kb) = 0. Peak potentials shift toward the formal potential with increment (see Fig. 5b). The þ
þ
change in DCp;r with increment is greater than that for DCp;f though both values increase (see Fig. 5c). Peak ratio increases with increment from 0.25 to 0.7 as shown in Fig. 5d. Fig. 6 shows the effects of increment on peak ratio for selected values of K. For log K 2, the peak ratio is independent of increment. At greater values of log K, the peak ratio increases with increment. This trend is comparable to that observed in CV;
b
a
b
c c
d
Fig. 4. The impact of t on the peak ratio as a function of log K and log (kf + kb) when ESW = 50 mV, dE = 10 mV, and El = 200 mV. Panel a: illustrates the relationship þ
between DCratio and t as log K is varied from 3 (red) to 3 (purple), and log
(kf + kb) = 0. Panel b presents the relationship between DCratio and t as log (kf + kb) is varied from 3 to 6 and log K = 1 (the cyan trace in panel a). Panel
Fig. 5. The impact of dE on the shape of the voltammogram when ESW = 50 mV, t = 50 ms, El = 200 mV, log K = 3, and log (kf + kb) = 0. Panel a): dE ranges from 1 mV (red), 5 mV (orange), 10 mV (yellow), 15 mV (green), and 20 mV (cyan). Panel b): Ep,f
c relates the minimum in DCratio versus log K. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
ratio (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
þ
þ
þ
þ
(red) and Ep,r (black). Panel c): DCp;f (red) and DCp;r (black). Panel d): Peak
J.C. Helfrick et al. / Electrochimica Acta 205 (2016) 20–28
25
case but remain at E for the reversible case; the magnitude of the shift depends on K, kf, and kb. Peak currents increase with increment for the EC case but are unaffected by increment for the reversible. Peak ratios vary with increment for the EC case but are unity regardless of increment for the reversible case; the trend is dependent upon K, kf, and kb. Hence, the systematic variation of increment can be used to discern between the reversible and EC mechanisms. 3.4. Effect of Switching Potential (El) The dramatic effect of switching potential on the wave shape for an EC mechanism is shown in Figure 7a for log K = 3 and log þ
þ
(kf + kb) = 0. While DCp;f , Ep,f, and Ep,r are unaffected, DCp;r is þ
greatly affected by El (Fig. 7c). As El approaches E , DCp;r increases and consequently, the peak ratio increases (Fig. 7d). This occurs as less time is allowed for Red to convert to Z when El is closer to the þ
Fig. 6. The impact of increment on the peak ratio as log K is varied from 3 (red) to 3 (purple) in steps of 1 when ESW = 50 mV, t = 50 ms, El = 200 mV, and log (kf + kb) = 0. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
increasing increment raises the potential sweep rate and shortens the time window for the conversion of Red to Z. The relationship of peak ratio with increment is markedly dependent upon kf + kb and K. The complex dependency is shown in Fig. S-7. At log K 5 (panels a-c), the effect of kf + kb and increment on peak ratio is similar. When log (kf + kb) = 3, the trend for peak ratio vs. increment mirrors that for a reversible reaction. When 2 log (kf + kb) 0, peak ratio increases with increment. When log (kf + kb) > 0, peak ratio remains constant and independent of increment. For intermediate values of K (panels d & e), the effects of increment log (kf + kb) 0 remain the same as in panels a-c. For log (kf + kb) > 0, peak ratio decreases as increment is increased with a magnitude dependent upon the value of log (kf + kb). For 0 log K 1 (panels f & g), trends in peak ratio with increment and log (kf + kb) match those obtained for log K = 2 but over a smaller span in peak ratio. When log K 1 (shown for log K = 1 in panel h), the peak ratio is independent of increment and very slightly dependent on log (kf + kb). þ
Fig. S-8 presents the impact of increment on DCp;f (panel a) and
formal potential. The trends in DCp;r and peak ratio with El distinguishes the EC mechanism from others. The impact of K on the relationship between peak ratio and El is shown in Fig. 8 for log (kf + kb) = 0. At low K values where the
a
b
c
DCþp;r (panel b) as a function of kf + kb. Detailed inspection of this figure provides insight into the trends shown in Fig. S-7a. When log þ
þ
(kf + kb) > -2, both DCp;f and DCp;r increase in magnitude curvilinearly with increment; the slope depends upon log (kf + kb). þ
þ
But, for given value of log (kf + kb), DCp;f and DCp;r are not mirror images of each other. The difference reflects the rate of conversion of Red to Z. The impact of increment, K, and kf + kb on peak potentials is shown in Figs. S-9 and S-10. Regardless of the value of K, when the rate of the following reaction is slow (i.e. log (kf + kb) 1) Ep,f and Ep,r = E and are independent of increment. At faster rates of the reaction (i.e. log (kf + kb) > 1) and when the equilibrium lies strongly in favor of Z (i.e. log K 1, panels a-f), Ep values shift curvilinearly toward E with increment. The displacement of both peak potentials from E depends strongly on K and kf + kb. When the equilibrium begins to favor Red (i.e. log K < 1, panels g & h), the shift in Ep with increment is 10 mV over the range in increment from 1 to 25 mV and collapses on E when log K 1. In comparing the trends for the EC and reversible cases as a function of increment, several differences can be identified. Peak potentials shift negatively towards E with increment for the EC
d
Fig. 7. The impact of El on the shape of the voltammogram when ESW = 50 mV, dE = 10 mV, t = 50 ms, log K = 3, and log (kf + kb) = 0. Panel a): El: 1000 mV (red), 900 mV (orange), 800 mV (yellow), 700 mV (green), 600 mV (cyan), 500 mV (blue), 400 mV (purple), 300 mV (magenta), 200 mV (light gray), and 100 mV (dark gray). þ
þ
Panel b): Ep,f (red) and Ep,r (black). Panel c): DCp;f (red) and DCp;r (black). Panel d): Peak ratio (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
26
J.C. Helfrick et al. / Electrochimica Acta 205 (2016) 20–28
a
b
c
Fig. 8. The impact of El on peak ratio as log K is varied from 3 (red) to 3 (purple) in steps of 1 when ESW = 50 mV, dE = 10 mV, t = 50 ms, and log (kf + kb) = 0. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
equilibrium favors Red, peak ratio is unaffected by the switching potential and equal to unity just as in the reversible electron transfer mechanism. At higher K values, peak ratio decreases with increasing El. The magnitude of the decrease is proportional to K. The impact of K and kf + kb on the peak ratio vs. El is quite complex as shown in Fig. S-11. At low values of log (kf + kb) (i.e. 2), peak ratios are independent of El and equal to unity regardless of log K. The rate of the following reaction is too slow to impact the current within the time window of the period. At intermediate values (i.e. 2 < log (kf + kb) 2), peak ratios decrease curvilinearly with El; the span in peak ratio over the range in El depends upon log K. At high rates (log (kf + kb) > 2), the span in peak ratio is only 0.1 with increasing El. The magnitude of peak ratio at a given El increases with decreasing K. For example, at log (kf + kb) = 6 and El equal to 400 mV, the peak ratio is 0.5 for log K = 3 and 0.9 for log K = 2. The relationship between peak currents and El depends upon þ
both K (vide supra) and log (kf + kb) as shown in Fig. S-12. DCp;f remains constant while switching potential is varied though the þ
þ
value of DCp;f is dependent upon log (kf + kb). Similarly,DCp;r is unaffected by El when log (kf + kb) 2. At higher values of kf + kb,
DCþp;r increases dramatically as switching potential decreases. þ When log (kf + kb) 1, DCp;r approaches zero and remains
relatively constant as switching potential is varied. The effect of K and kf + kb on the relationship between peak potentials and El is shown in Figs. S-13 and S-14. Ep,f and Ep,r are unaffected by switching potential though their values are characteristic of K and kf + kb. There are contrasting trends for the effect of switching potential on the voltammograms of the reversible and EC cases. Ep,f and Ep,r are displaced from E but remain essentially constant with variation in switching potential for the EC mechanism whereas þ
Ep,f and Ep,r remain at E for the reversible case. DCp;f is unaffected þ
by switching potential for the EC case but DCp;r increases as þ
switching potential approaches E . The magnitude of both DCp;f þ
and DCp;r depend on K, kf, and kb. Peak currents are unaffected by switching potential for the reversible case. Subsequently, peak ratio increases as the switching potential approaches E for the EC case while peak ratio is invariant of switching potential and equal
d
Fig. 9. The impact of ESW on the shape of the voltammogram when dE = 10 mV, t = 50 ms, El = 200 mV, log K = 3, and log (kf + kb) = 0. Panel a): ESW ranges from 10 mV (red), 20 mV (orange), 30 mV (yellow), 40 mV (green), 50 mV (cyan), 60 mV (blue), 70 mV (purple), 80 mV (magenta), and 90 mV (light gray). Panel b): Ep,f (red) þ
þ
and Ep,r (black). Panel c): DCp;f (red) and DCp;r (black). Panel (d): Peak ratio (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
to unity for the reversible case. Thus, the effect of switching potential on the peak properties can be used to identify EC vs. reversible mechanisms. 3.5. Effect of Amplitude (ESW) The effect of amplitude on the voltammogram is illustrated in Fig. 9a and explored in detail in Fig. 9b-d for log K = 3 and log (kf + kb) = 0. Peak potentials shift slightly with amplitude (Fig. 9b). þ
þ
Both DCp;f and DCp;r increase with amplitude though the effect þ p;f
on DC is more drastic (Fig. 9c). Amplitude has a negligible effect on peak ratio as shown in Fig. 9d where the peak ratio increases by only 0.06 when the amplitude is increased from 10 to 90 mV. The value of peak ratio is governed by log K and log (kf + kb) as shown in Fig. S-15. Fig. S-16 provides insight into the peak ratio-amplitude þ
þ
trends shown in Fig. S-15a. Both DCp;f and DCp;r increase curvilinearly with amplitude however the effect of amplitude on
DCþp;f is greater than that on DCþp;r . Figs. S-17 and S-18 present the impact of amplitude on peak potentials as a function of log (kf + kb) and log K. When log (kf + kb) 1 (red, orange and yellow traces in both figures), both
J.C. Helfrick et al. / Electrochimica Acta 205 (2016) 20–28
27
Table 1 Diagnostic Plots and Protocol for Assessing an EC Electrode Reaction by CSWV. Empirical variables Waveform Period, Ï„ parameters Plot Trend Peak currents
DIp vs.
Peak DEp separation vs. log t
Peak ratio 1 and varies with log t ; magnitude depends on log K and log (kf + kb) (see Figs. 3 d, 4, and S-3) Ep shifts positively with log t ; (see Figs. 3 b, S-5, and S-6) magnitude of Ep depends upon log K and log (kf + kb) Complex relationship, may increase or decrease with log t (compare Figs. S-5 and S-6)
Peak widths
W1/2 is generally independent of log t
Peak ratio vs. log t Peak Ep vs. potentials log t
W1/2 vs. log t
Switching potential, Eλ
Amplitude, ESW
Plot
Plot
Trend
Plot
Trend
DIp,f is independent of El; DIp,r increases with El
DIp
DIp increases with ESW (see Figs. 9 c and S-16)
Trend
DIp increases nonlinearly DIp DIp increases with dE; DIp with t 1/2 (see Figs. 3 c and vs. dE magnitude depends on log K vs.
t1/2 S-4) Peak ratio
Increment, δE
vs. and log (kf + kb); (see Fig. 5c El approaching E (see Figs. 7 c ESW and S-8) and S-12) Peak Peak ratio 1 and varies with Peak Peak ratio 1 and increases Peak ratio ratio dE depending upon K and ratio as El approaches E (see vs. vs. dE kf + kb (see Figs. Fig. 5d, 6, and vs. Figs. 7 d, 8, and S-11) ESW El S-7) Ep vs. Ep shifts negatively with dE; Ep vs. Ep is independent of El; Ep vs. magnitude of Ep depends El magnitude of Ep depends ESW dE upon log K and log (kf + kb) upon log K and log (kf + kb) (see Fig. 5b, S-9, and S-10) (see Figs. 7 b, S-13, and S-14) DEp DEp decreases with dE; DEp DEp is independent of El D Ep vs. dE magnitude of Ep depends vs. vs. El ESW upon log K and log (kf + kb) (compare Figs. S-9 and S-10) W1/2 W1/2 varies by 10 mV with d W1/2 W1/2 is independent of El W1/2 vs. dE E vs. vs. El ESW
peak potentials are essentially at E , regardless of log K and amplitude. When log (kf + kb) 0 and log K 3, both Ep,f and Ep,r shift positively with amplitude; peak potentials values and magnitude of change depends on log (kf + kb) (see panels a-d in both figures). As K becomes smaller (panels e-g in both figures), larger log (kf + kb) values are invariant with amplitude while intermediate vales of log (kf + kb) shift positively with amplitude. When log K 1, peak potentials become invariant of amplitude, regardless of kf + kb. Several contrasting trends can be identified when comparing the EC and reversible mechanisms. For the EC mechanism, peak potentials shift positively from E with amplitude. For the reversible mechanism, peak potentials are independent of amplitude. Peak currents increase with amplitude for both cases, but the increase is dependent upon K, kf, and kb for the EC mechanism. Peak ratios are either less than unity or increase towards unity for the EC mechanism but are always unity for the reversible case. Thus, the variation of amplitude can be used to distinguish between the reversible and EC mechanisms.
Peak ratio 1 and increases slightly with ESW (see Figs. 9 d and S-15) Ep shifts positively with ESW; magnitude of Ep depends upon log K and log (kf + kb) (see Figs. 9 b, S-17, and S-18) DEp increases with ESW for select values of log (kf + kb) but otherwise is invariant with ESW W1/2 increases with ESW
summarized in Table 1. The diagnostic criteria presented in this Table are novel. It is based on analyzing voltammetric features rather than individual currents thereby enabling rapid identification of EC mechanisms using CSWV. Close examination of the trends presented in supplementary data reveals the following limits. When the following chemical reaction is irreversible, i.e. log K 3, measureable log kf values are limited to the range 2 < log kf < 2. When the following chemical reaction is reversible, i.e. 0 log K < 3, measureable log (kf + kb) values span a much broader range 2 < log (kf + kb) < 6. We are currently investigating kineticallycontrolled electron transfers coupled to chemical reactions and will report diagnostic criteria for these EquasiC and CEquasi mechanisms elsewhere. Acknowledgments Acknowledgment is made to CMMI1130739 for financial support.
NSF
grant
number
Appendix A. Supplementary data 4. Conclusions CSWV is SWV in two directions without delay or loss of control on the working electrode potential paralleling the complementary nature of CV to linear sweep voltammetry. In CV, the experimenter identifies the electrode reaction mechanism by systematic variation in potential sweep rate and evaluating commensurate shifts in peak currents and potentials. In SWV, the experimenter varies ESW and t (or frequency) and tracks shifts in peak current and potential. In CSWV, the experimenter make use of four empirical parameters (ESW, dE, t , and El) and identifies the mechanism from characteristic shifts in six peak features þ
þ
þ
(DCp;f , DCp;r , DCratio , Ep,f, Ep,r, and DEp). Thus, the cyclic waveforms (both CV and CSWV) provide additional empirical parameters and peak features that afford greater discrimination between mechanisms compared to linear sweep voltammetry or conventional SWV. In this report, we have shown that the peak currents, potentials, and shapes of cyclic square wave voltammograms for the EC mechanism are a complex function K, kf + kb, increment, period, switching potential, and amplitude. The effects of systematic variation in these empirical parameters on the voltammograms are
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