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A Simulation-based Approach to Determining Electron Transfer Rates using Square-Wave Voltammetry Philippe Dauphin-Ducharme, Netzahualcoyotl Arroyo-Curras, Martin Kurnik, Gabriel Ortega, Hui Li, and Kevin W. Plaxco Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b00359 • Publication Date (Web): 09 Apr 2017 Downloaded from http://pubs.acs.org on April 11, 2017
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A Simulation-based Approach to Determining
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Electron Transfer Rates using Square-Wave
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Voltammetry
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Philippe Dauphin-Ducharme1,2,‡, Netzahualcóyotl Arroyo-Currás1,2,‡, Martin Kurnik1,2, Gabriel
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Ortega1,2,3, Hui Li1,2 and Kevin W. Plaxco1,2*
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Barbara, CA 93106, USA.
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USA.
Department of Chemistry and Biochemistry, University of California Santa Barbara, Santa
Center for Bioengineering, University of California Santa Barbara, Santa Barbara, CA 93106,
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‡These authors contributed equally.
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CIC bioGUNE, Bizkaia Technology Park, Building 801 A, 48170 Derio, Spain
Corresponding author.
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ABSTRACT
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The efficiency with which square-wave voltammetry (SWV) differentiates faradaic and charging
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currents renders it a particularly sensitive electroanalytical approach, as evidenced by its ability
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to measure nanomolar or even picomolar concentrations of electroactive analytes. Due to the
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relative complexity of the potential sweep it employs, however, the extraction of detailed kinetic
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and mechanistic information from square-wave data remains challenging. In response we
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demonstrate here a numerical approach by which square-wave data can be used to determine
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electron transfer rates. Specifically, we have developed a numerical approach in which we model
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the height and shape of voltammograms collected over a range of square-wave frequencies and
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amplitudes to simulated voltammograms as functions of the heterogeneous rate constant and the
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electron transfer coefficient. As validation of the approach we have employed it to determine
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electron transfer kinetics in both freely diffusing and diffusion-less surface-tethered species,
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obtaining electron transfer kinetics in all cases in good agreement with values derived using non-
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square-wave methods.
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KEYWORDS. Square-wave voltammetry, numerical simulations, electron transfer rate.
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MAIN TEXT. Square-wave voltammetry (SWV), an electroanalytical technique that involves
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the application of repeating square-shaped potential pulses superimposed on a staircase potential
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sweep, enables the effective discrimination of faradaic processes from charging currents.1 It
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achieves this by imposing two opposite square-wave pulses of the same height at each staircase
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potential step and measuring the resultant current at the end of each pulse, ifwd and ibwd. Taking
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the difference between these currents returns a net voltammogram with improved signal-to-noise
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ratios relative to those of, for example, cyclic voltammetry.1-2 This, in turn, renders SWV one of
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the most sensitive electroanalytical techniques, enabling the detection of metal ions and organic
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molecules at nanomolar and picomolar concentrations, respectively.1
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In this work we build on prior approaches aimed at determining electron transfer kinetics using
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SWV by modulating the height (amplitude) or width (frequency) of the square-wave pulses.1, 3
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This earlier work includes that of Lovric4-5 who, for example, developed a mathematical method
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termed “quasi-reversible maximum” that extracts electron transfer kinetics from the maximum in
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a plot of SWV peak current versus square-wave frequency. Mirceski and co-workers,6-9 in
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contrast, have developed a mathematical method that extracts electron transfer kinetics from the
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relationship between SWV peak current and square-wave amplitude, with the maximum on this
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plot again correlating with the electron transfer rate. These methods have allowed the
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determination of the electron transfer rates of a range of freely diffusing8 and surface anchored6
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redox couples, including linear and duplex DNA strands modified with a redox reporter10-11.
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Historically, however, these approaches have been limited to the measurement of relatively slow
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transfer rates of redox couples undergoing quasi-reversible or irreversible electron transfer
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reactions.3, 8, 12 This leaves uncharacterized many important and fully reversible redox couples,
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which transfer electrons more rapidly. Here we have expanded upon this prior literature and
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developed a method of extracting electron transfer rates from SWV data that overcomes this
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limitation, enabling access to much more rapid electron transfer kinetics than is possible with the
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prior approaches.
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The methods of Lovric and Mirceski rely on the fact that SWV peak height is defined by the
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interplay between the forward and backward electron transfer rates and the frequency and
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amplitude of the square-wave pulse.1 Specifically, Lovric’s approach5 extracts rates using the
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relationship between square-wave frequency and the height of the resulting voltammetric peaks.
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This works because changing the square-wave frequency optimizes the time scale of the
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measurement to the time scale of electron transfer and, in turn, influences the peak height.
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Mirceski’s approach,8 in contrast, extracts kinetic information from the relationship between
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square-wave amplitude and the height of the resulting voltammetric peaks. This works because
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changing the square-wave amplitude alters the driving force underlying electron transfer,
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changing the rates of the forward and backward electron transfer reactions and thus altering peak
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splitting and, in turn, peak height.
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The success of the methods of Lovric and Mirceski demonstrates that information regarding
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electron transfer kinetics can be extracted by observing peak height as a function of either
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square-wave frequency or square-wave amplitude. Building on this we describe here an approach
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that captures both effects simultaneously, potentially providing more information regarding rates
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than methods that extract rates by varying only one or the other. Specifically, we present a
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numerical approach that, by simultaneously modeling both the heights and shapes of
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voltammetric peaks, is able to measure electron transfer kinetics for reactions that had proven too
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rapid to measure using prior SWV methods.
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Figure 1. Description of the numerical model. Our approach starts with one-dimensional numerical models;
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shown are models for (A) a freely diffusing redox couple and (B) a diffusion-less, surface-bound redox couple. (C)
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We then simulated the response of these to SWV using a square waveform of frequency, f, and amplitude, ESW. This
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is superimposed on a staircase that ramps the mean potential (the mean potential per pulse pair, E) by a potential
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step size per cycle, Estep. In response to this varying potential the system produces an oscillating faradaic current
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(grey trace in D). We deconvolute this into “forward” and “backward” voltammograms (red and black curves in D)
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by extracting the current at specific times (ifwd and ibwd). The difference between these two voltammograms then
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returns a “net” voltammogram (blue curve in D).
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Our approach to extracting electron transfer kinetics uses numerical simulations to model the
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height and shape of experimental SWV scans collected over a range of frequencies and
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amplitudes. To do so we first numerically model the experimental set-up as a simple, one-
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dimensional electrochemical cell. As examples we explore here two such cases: 1) a freely
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diffusing redox couple, and 2) a redox couple that is confined to the electrode surface via a short,
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flexible linker. We define two boundaries in each model, one representing the surface of the
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working electrode and the other representing either the bulk of the solution (Figure 1A) or the
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cell wall (Figure 1B). The reaction considered in all cases is reversible and defined as: + ⇌
Eq. 1
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where n represents the number of electrons transferred. The electrode boundary condition for
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both oxidized, O, and reduced, R, species is set in all cases to “flux” as defined by Fick’s second
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law of diffusion:
= ∇
= ∇
Eq. 2
Eq. 3
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where DO and DR are experimentally determined diffusion coefficients (for ferrocenemethanol,
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for example, which we employ below, both DO and DR are 7.8 x 10-6 cm s-1)13 and CO and CR
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represent the concentration of the oxidized and reduced species, respectively, and are both
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functions of time, t, and distance from the electrode surface, x. Per Eq. 1, the initial conditions
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are CO = CO,initial* and CR = 0 and the right boundary conditions are set as follows:
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Case 1: diffusive redox couple ∗ lim→ (, ) = , ,
Eq. 4
lim→ " (, ) = 0 108
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Case 2: surface-confined redox couple lim→ (, ) = , lim→ " (, ) = "
Eq. 5
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Eq. 4 describes the case of the bulk boundary that is positioned far from the electrode surface so
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that CO = CO, initial* (bulk concentration of oxidized species) and where CR = 0 remains constant at
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all t and is unaffected by the electrode reaction. Eq. 5, in contrast, describes the case of a thin-
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layer cell with an inert wall, which mimics the physics of an electroactive monolayer on an
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electrode surface. It does so by limiting the length of the electrochemical cell to x
