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Inappropriate Use of the Quasi-Reversible Electrode Kinetic Model in Simulation-Experiment Comparisons of Voltammetric Processes That Approach the Reversible Limit Alexandr N. Simonov,‡ Graham P. Morris,† Elena A. Mashkina,‡ Blair Bethwaite,§ Kathryn Gillow,† Ruth E. Baker,† David J. Gavaghan,*,∥ and Alan M. Bond*,‡ ‡

School of Chemistry, Monash University, Clayton, Victoria 3800, Australia Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom § Monash eResearch Centre, Monash University, Clayton, Victoria 3800, Australia ∥ Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, United Kingdom †

S Supporting Information *

ABSTRACT: Many electrode processes that approach the “reversible” (infinitely fast) limit under voltammetric conditions have been inappropriately analyzed by comparison of experimental data and theory derived from the “quasireversible” model. Simulations based on “reversible” and “quasi-reversible” models have been fitted to an extensive series of a.c. voltammetric experiments undertaken at macrodisk glassy carbon (GC) electrodes for oxidation of ferrocene (Fc0/+) in CH3CN (0.10 M (n-Bu)4NPF6) and reduction of [Ru(NH3)6]3+ and [Fe(CN)6]3− in 1 M KCl aqueous electrolyte. The confidence with which parameters such as standard formal potential (E0), heterogeneous electron transfer rate constant at E0 (k0), charge transfer coefficient (α), uncompensated resistance (Ru), and double layer capacitance (CDL) can be reported using the “quasi-reversible” model has been assessed using bootstrapping and parameter sweep (contour plot) techniques. Underparameterization, such as that which occurs when modeling CDL with a potential independent value, results in a less than optimal level of experiment-theory agreement. Overparameterization may improve the agreement but easily results in generation of physically meaningful but incorrect values of the recovered parameters, as is the case with the very fast Fc0/+ and [Ru(NH3)6]3+/2+ processes. In summary, for fast electrode kinetics approaching the “reversible” limit, it is recommended that the “reversible” model be used for theoryexperiment comparisons with only E0, Ru, and CDL being quantified and a lower limit of k0 being reported; e.g., k0 ≥ 9 cm s−1 for the Fc0/+ process.

A

exercise are compared with experimental data in order to estimate system parameters.2,3 Commercial and in-house simulation packages are now readily available that accommodate the modeling of a vast array of electrochemical mechanisms.2,4−7 Nevertheless, detailed comparisons of experimental and simulated data that accommodate the electrode kinetics, uncompensated resistance, and background current have for too long remained in the domain of experienced electrochemists, who commonly decide on the basis of their own, but usually undefined, heuristic approaches when satisfactory agreement has been achieved. It is only in rare cases that statistical provision of the level of agreement between experimental and simulated voltammetric

s in spectroscopy and other branches of physical chemistry, a major goal in an electrochemical investigation is to quantify the experimental data obtained in terms of parameters that meaningfully define the system. In dynamic forms of electrochemistry, the potential−current−time or socalled voltammetric response typically contains contributions from Faradaic processes involving electron transfer, background current associated with capacitance and noise. Uncompensated resistance is also important and influences both the Faradaic and non-Faradaic contributions.1 In transient forms of voltammetry, almost invariably, except for the so-called “reversible” (infinitely fast) process that complies with the Nernst relationship, analytical mathematical solutions of the theory needed to describe the Faradaic current are not available. Consequently, simulations of the electrode kinetics that should ideally closely mimic the theory are normally employed and results derived from the modeling © 2014 American Chemical Society

Received: May 29, 2014 Accepted: July 20, 2014 Published: July 20, 2014 8408

dx.doi.org/10.1021/ac5019952 | Anal. Chem. 2014, 86, 8408−8417

Analytical Chemistry

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data is provided.8,9 This means that, unlike in X-ray crystallography where a rigorous report of theory and experiment are mandatory for publication of a crystal structure, a rather unsatisfactory situation prevails in reporting of dynamic electrochemical data. Indeed, it may be argued that the lack of compulsory use of theory-experiment comparisons has led to substantial laboratory to laboratory variation in reports of electrode kinetic parameters. A new issue that now complicates quantitative reporting of outcomes of voltammetric experiments is that the electrode materials employed are no longer predominantly the ideal liquid mercury that was in vogue when many theoretical models used to describe electrochemical experiments were developed and tested. Rather, voltammetric studies with highly heterogeneous electrode surfaces based on, for example, many forms of carbon are now dominant. This may introduce additional complexity into electrode kinetics and double layer background current models due to the presence of structural defects, impurities, or other imperfections.10,11 Expansion of electrochemistry into new environments found in room temperature ionic liquids also means that double layer models developed for use in molecular solvent (supporting electrolyte) media may no longer be relevant in modeling the background current in ionic liquids.12,13 In summary, theory-experiment comparisons in modern forms of dynamic electrochemistry can represent a daunting problem. d.c. cyclic voltammetry is undoubtedly the most widely used technique in dynamic electrochemistry. Typically, this method requires the use of a wide scan rate range. As a result, an extensive series of experiments are needed, during which time the treatment and history of, say, a carbon electrode can become important,14 making reliable data analysis reporting difficult. However, even though appropriate methodologies are available for the purpose, it is rare to find a detailed report of the state of the electrode for each experiment and statistical analysis of the level of agreement achieved between experimental and simulated d.c. cyclic voltammograms, as is needed. In the more sophisticated, but less widely used, technique of electrochemical impedance spectroscopy (EIS), where a small amplitude a.c. waveform containing many frequencies is usually superimposed onto a constant d.c. potential, quantitative comparison of theory based on equivalent circuits and experimental data is more prevalent but still restricted in the sense that, in most forms of the technique, data need to be collected at a wide range of d.c. potentials via a very extensive series of experiments.15 Additionally, the use of equivalent circuits rather than classical thermodynamic and electrochemical kinetic principles to solve the theory may represent a psychological barrier for chemists who are not closely familiar with this approach. Over the past few years, work from our laboratories has integrated the attributes of a.c. voltammetry and impedance spectroscopy that have historically been developed almost independently over many years.16 Thus, very large data sets are collected from experiments based on superimposition of large amplitude perturbations onto a d.c. ramp, with all data needed over a wide range of potentials and time scales becoming available from a single experiment. Typically, the total current− potential−time data are resolved by Fourier transform filtering into d.c. and a.c. components, taking advantage of the nonlinearity of the Faradaic current to generate higher order (usually 2nd to 10th) a.c. harmonic components that are very sensitive to both electrode kinetics and uncompensated

resistance but not capacitance current. Simulations of the theory in this approach have been developed in exactly the same way as in d.c. cyclic voltammetry.3 Thus, the simple process given in eq 1, E 0 , k0 , α

A XoooooooY B + e−

(1)

where species A and B are soluble in the solvent (electrolyte) of interest, can be modeled by a combination of heterogeneous electron transfer theory [requires reversible potential (E0), heterogeneous charge transfer rate constant (k0) and charge transfer coefficient (α) if the Butler−Volmer model is used, mass transport by diffusion (Fick’s Laws as appropriate to electrode geometry), Ohm’s law for uncompensated resistance (Ru), and double layer capacitance (CDL) theory for the background current], assuming parameters such as temperature, electrode area, diffusion coefficient, etc. are accurately known from independent measurements. Use of a potentialindependent CDL model for the background current is still problematic, being unrealistic in many situations encountered experimentally at solid electrodes.12 Despite all the advances in d.c. and a.c. voltammetric theory, the comparison of simulated and experimental results, with only a few exceptions, has remained heuristic even though numerous e-science based strategies for data analysis are now available for this purpose.8,9,16 In the a.c. method, Fourier transform or other methods of filtering have usually formed part of the data evaluation strategy. However, the experimentalist must choose how to filter the data and an understanding of the implications of doing so is needed. Philosophically, it would seem sensible to avoid any filtering and simply fit all parameters of interest to a very large total current−potential− time (frequency) data set. Ideally, via this modus operandi, for a given data set, any person in any laboratory analyzing the data would be able to estimate reproducibly the electrode kinetics and other parameters when using exactly the same method of data analysis. This is something that is unlikely to be accomplished by a heuristic approach that requires intervention of the experimentalist into the decision making process. However, filtering facilitates the introduction of sensitivity analysis, which can be advantageous, as shown in this study. In this paper, we examine the implications of applying different automated data analysis methods8,16 to the one electron oxidation of ferrocene (Fc) to its cation (Fc+) at a glassy carbon electrode in CH3CN containing 0.10 M (nBu)4NPF6 as the supporting electrolyte to give the Fc0/+ process (eq 2)

The oxidation of ferrocene has been studied for many years, and even though regarded as simple, the electrode kinetic parameters, particularly the reported k0 values, have ranged over many orders of magnitude even when studied under what appear to be identical conditions.17−22 In early studies, values of k0 in the 10−2 to 10−1 cm s−1 range were common, but now values are often in the 1 to 10 cm s−1 range. In this present study, the impact of the method of data analysis on the k0 value recovered is described. Additionally, results of global time domain fits of the a.c. data to models for the Fc0/+ process are compared to results after Fourier transformed filtering to ascertain the extent to which filtering provides additional 8409



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insights. Results from 10 consecutive experimental runs for each of two Fc concentrations with three sine wave frequencies are used to ascertain if the experimental data are better mimicked by the “reversible” or “quasi-reversible” models. Brief analyses of the a.c. voltammetric data obtained for the one electron reduction of [Ru(NH3)6]3+ and [Fe(CN)6]3− are also reported. The role of over- or underparameterization in the model used to mimic experiments and other issues that contribute to lack of laboratory-to-laboratory consistency in parameter reporting of processes near the “reversible” limit are considered as part of our ongoing objective to achieve quantitative reporting of data analysis in voltammetry more akin to that routinely used in X-ray crystallography and other methods of physical chemistry.

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Figure 1. Total current, aperiodic d.c. component, fundamental, 2nd, 4th, and 7th a.c. harmonic components of an a.c. cyclic voltammogram ( f = 72 Hz; ΔE = 0.080 V; v = 0.104 V s−1) for oxidation of 0.098 mM Fc in CH3CN (0.10 M (n-Bu)4NPF6) with a GC electrode at 297 K.

EXPERIMENTAL SECTION Electrochemical Instrumentation and Procedures. The details of a.c. voltammetric experiments for oxidation of Fc and reduction of [Ru(NH3)6]3+ and [Fe(CN)6]3− and use of EIS to independently measure values of Ru are provided in the Supporting Information. Theory. Simulations of a.c. voltammograms are based on the Butler−Volmer model for electron transfer and a semi-infinite planar diffusion model for mass-transport. Details of the theory and on the incorporation of Ru and CDL into the models are provided elsewhere.3,8,12 Simulations were undertaken using an in-house software written in C at the University of Oxford as in ref 8 or MECSim software7 at Monash University. The principles of the “Global Method of Parameter Recovery” used at Oxford and those of the Nimrod/O parameter optimization method used at Monash are detailed in ref 8. The Nimrod/G parameter sweep tool was used for experiment-simulation comparisons to define the quality of the fit as a function of k0 and Ru.23−25 To minimize the influence of the ringing artifact associated with inverse Fourier transformed data,24 the first and the last seconds of the voltammetric data set were ignored when experiment-theory fitting was undertaken with the aid of Nimrod based e-science tools.

initial and reversal d.c. potentials, d.c. scan rate (v), and frequency ( f) and amplitude (ΔE) of the a.c. component are assumed to be known at a level of accuracy where they do not contribute to significant uncertainties in estimation of parameters such as E0, k0, α, Ru, and CDL. Under conditions relevant to this study, values of these fixed parameters are as follows: T = 297 K, A = 0.70 cm2, C0 = 0.098 or 0.92 mM, DFc = DFc+ = 2.4·10−5 cm2 s−1,27 v = 0.10431 (for convenience referred to as 0.104) V s−1, ΔE = 0.080 V, f = 9.015, 72.05, and 219.05 (for convenience often referred to as 9, 72, or 219) Hz. The potential of the Pt quasi-reference electrode varied slightly from experiment to experiment, but the initial and reversal d.c. potentials in cyclic voltammograms were always approximately 0.25 V more negative and 0.25 V more positive than E0 for the Fc0/+ process, respectively. The use of a very low impedance Pt quasi-reference electrode is advantageous in high frequency a.c. voltammetry. If the Fc0/+ process is assumed to be “reversible”, the analytical problem is confined to the determination of E0, Ru, and CDL. If CDL is a constant, this means that only three unknown parameters need to be incorporated into the simulations. (This scenario is now designated as Model Rev3 where Rev stands for “reversible” and the numeral 3 for the number of unknown parameters. Later on, the notation Model QRx is introduced as shorthand for the “quasi-reversible” model with the number of unknown parameters being x.) However, even a casual inspection of the total current or the fundamental or second harmonic components of the a.c. voltammogram obtained with a glassy carbon (GC) electrode for oxidation of 0.098 mM Fc in CH3CN (0.10 M (nBu)4NPF6) (Figure 1) reveals that the experimental background current is potential dependent, so simulations based on CDL being constant will not mimic experimental data very well. Therefore, a CDL(E) model is needed.12 In this study, a third order polynomial function (eq 3) was used to simulate the background current,

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RESULTS AND DISCUSSION Oxidation of Ferrocene: Modeling Assuming “Reversible” Charge Transfer. Studies on the electrode kinetics of the Fc0/+ process (eq 2) in molecular solvents such as CH3CN containing excess supporting electrolyte (e.g., 0.10 M (nBu)4NPF6), assuming a “quasi-reversible” model, produce k0 values that range from 0.1 to 10 cm s−1.17,18,20−22 However, given the substantial magnitude of uncompensated resistance inevitably encountered in organic media along with instrumental and technique limitations, it is plausible that the Fc0/+ process should be regarded as “reversible” in all reported studies.26 On this basis, the a.c. voltammetric data obtained for oxidation of Fc in CH3CN (0.10 M (n-Bu)4NPF6) (exemplified in Figure 1 for 0.098 mM Fc) was initially simulated assuming “reversible” charge transfer. This model was implemented using simulations derived from the “quasi-reversible” model with fixed values of k0 = 1.0 × 103 cm s−1 and α = 0.50. Under these conditions, the modeling complies with the Nernst equation and is independent of the electrode kinetics. In carefully executed experiments, parameters such as temperature (T), electrode surface area (A), initial bulk Fc concentration (C0), and Fc and Fc+ diffusion coefficients (DFc and DFc+), as well as instrumentally set parameters such as

C DL(E) = C DL0 + C DL2(E′(t ) − ECdl)2 + C DL3(E′(t ) − ECdl)3

(3)

where E′(t) is the potential which incorporates IRu drop and ECdl, CDL0, CDL2, and CDL3 are constants for a given frequency to which no physical significance should be attached. Apart from being potential dependent, CDL also is a function of frequency. This feature is well recognized in EIS studies, where constant phase elements have been introduced15 to mimic the frequency dependence. In essence, the double layer 8410



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When using the “reversible” models Rev3 or Rev6, in principle, all relevant unknown parameters can be recovered from the comparisons with experimental data on the basis of the “best fit” aided by methods such as those available in the Nimrod/O tool kit or by the “Global Method” (GM) as employed previously.8 Nimrod/O uses parallel processing resources to optimize the parameters sought using a Simplex algorithm8 for minimization of a single objective. In the initial optimization studies detailed below, this is arbitrarily calculated as the equally weighted sum of individual objective functions (eq 4) for the fundamental, second, third, fourth, and fifth harmonic components, which make the dominant contribution to the total current of the a.c. voltammogram. Therefore, systematic noise from mains frequency, random noise, and contributions from d.c. and 6th and higher order harmonics have been removed. Analysis of the contribution from random and systematic noise present in the a.c. voltammetric data is provided in the Supporting Information (Figure S1 and related text). This approach allows a close to direct comparison to be made with the GM approach where no filtering of any kind is introduced, but as will be shown later, optimization can be applied to individual harmonics resolved by Fourier transformation in order to facilitate the introduction of sensitivity analysis with respect to selected unknown parameters. In the case of the Nimrod based analysis, the objective function ΨNim is defined as

behaves like an imperfect capacitor.15 In this FTAC voltammetric study, modeling of the frequency dependence is accommodated by independent evaluation of the constants appropriate to the polynomial described in eq 3 whenever the frequency is changed. Indeed, and as expected, the use of a unique combination of parameters in eq 3 is needed to model the background current at each frequency. With CDL(E) defined as in eq 3, simulations can now well mimic the experimental background current for both the fundamental and second a.c. harmonic components (Figure 2), but the number of unknown parameters needed in simulations is now increased to six (E0, Ru, CDL0, CDL2, CDL3, ECdl) for even the simplest “reversible” case (model Rev6).

H

ΨNim =

1 ·∑ H h=1

N

∑i = 1 (fhExp (xi) − f hSim (xi))2 N

∑i = 1 fhExp (xi)2

(4)

where h indicates the number of the harmonic component, H is the number of harmonic components evaluated, f Exp (x) and h f Sim are the experimental and simulated functions in the h corresponding harmonic, respectively, and N is the number of data points. The GM recovers the unknown parameters with the aid of a Quasi-Newton method,28 which minimizes the difference (ΨGM) between the experimental ( f Exp) and simulated ( f Sim) data expressed as a sum of squares of the errors:

Figure 2. Comparison of data simulated using models Rev3 (red) and Rev6 (black) (Table 1) and experimental fundamental, 2nd and 4th a.c. harmonics derived from a.c. cyclic voltammogram ( f = 72 Hz; ΔE = 0.08 V; v = 0.104 V s−1) for oxidation of 0.098 mM Fc in CH3CN (0.10 M (n-Bu)4NPF6) using a GC electrode (experiment No. 5 in Table 1) (blue). Note that the experimental and simulated 4th a.c. harmonic components are indistinguishable at the resolution of this figure.

Table 1. Parameters and Objective Functions (ΨNim) Recovered by Nimrod/O Using the “Reversible” Modelsa with Constant CDL (Rev3) and Potential Dependent CDLb (Rev6) to Fit Experimental a.c. Voltammograms ( f = 72 Hz, ΔE = 0.080 V) for Oxidation of 0.098 mM Fc CDL/μF cm−2 ΨNim

Rev6b

Ru/Ω #

EIS

Rev3

Rev6

Rev3

ECdl

CDL0

CDL2

CDL3

E /V

1 2 3 4 5 6 7 8 9 10

66 61 56 54 51 50 37 64 49 56

60 54 46 48 45 47 24 55 37 43

63 57 48 50 47 49 25 58 40 46

19.4 18.6 19.6 16.1 16.9 19.1 18.7 17.3 19.1 20.3

0.14 0.25 0.26 0.36 0.32 0.35 0.36 0.31 0.36 0.33

13.9 14.0 14.4 12.8 13.1 15.4 15.2 13.7 15.5 15.9

5.5 6.6 7.3 7.2 6.4 7.1 6.8 5.5 7.6 7.0

1.2 1.4 1.4 1.9 2.5 2.2 3.0 2.7 2.3 2.5

0.584 0.629 0.630 0.617 0.627 0.629 0.629 0.637 0.641 0.644

c

0

d

Rev3

Rev6

0.129 0.124 0.131 0.122 0.124 0.128 0.122 0.124 0.128 0.131

0.052 0.041 0.043 0.037 0.039 0.048 0.037 0.037 0.037 0.037

Invariant simulation parameters: f = 72.05 Hz; ΔE = 0.080 V; v = 0.10431 V s−1; A = 0.070 cm2; T = 297 K; DFc = DFc+ = 2.4·10−5 cm2 s−1. Defined by eq 3. cDerived from EIS data. dE0 values recovered with Rev3 and Rev6 models are identical. Since E0 was measured versus a quasireference electrode, the value is unique for each experiment. a b

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N

ΨGM =

∑ (f Exp (xi) − f Sim (xi))2

Table 3. Mean Values and Standard Deviations of the Differences in Rua Derived from EIS Data and Ru Recovered by “Best” Fits of Experimental a.c. Voltammograms ( f = 9 or 72 Hz) for Oxidation of 0.098 or 0.92 mM Fc (10 Independent Experiments at Each C0) with Simulations Based on Designated Modelsb Using Nimrod/O (Nim) and Global Method (GM)

(5)

i=1

Table 1 specifies the parameters derived using Nimrod/O when simulated data based on the Rev3 and Rev6 models are fitted to 10 independent experimental a.c. voltammograms, each obtained for oxidation of 0.098 mM Fc at 72 Hz with the same Fc solution and GC electrode whose surface is freshly polished between each experiment. Experiment-theory agreement achieved with the Rev3 and Rev6 models was highly reproducible (Tables 1 and 2), except for E0 where variation in the potential of the Pt quasi-reference electrode is evident (although this does not affect k0, α, Ru, or CDL).

ΔRu/Ω 9 Hz

9 Hz C0/mM

model

0.098

a

0.92

Rev3 Rev6a QR5/QR8b QR4/QR7c Rev3a QR5b QR4c

108·ΨGM 1.5 ± n.a.d 1.5 ± 1.5 ± 1.9 ± 1.9 ± 2.2 ±

0.3 0.3 0.3 0.4 0.4 0.5

ΨNim

± ± ± ± ± ± ±

Nim

0.098

Rev3 Rev6 QR5/QR8c Rev3 QR5

−19 ± 16 n.a.d 4.5 ± 56 11 ± 4 15 ± 4

n.a. n.a. n.a. 6.3 ± 1.7 10 ± 3

Nim 8.5 6.1 18 4.4 4.4

± ± ± ± ±

3.4 3.5 7.2 1.2 1.1

frequency independent RC model assumed). For calculation of Ru, the a.c. voltammetric method uses the total current which is the sum of Faradaic and capacitive terms, and sensitivity to the IRu drop depends on experimental conditions. Indeed, with use of the low frequency (9 Hz) and low concentration (0.098 mM) combination, current values are relatively insensitive to IRu drop so that variation of the Ru parameter recovered using the GM and Rev3 model is substantial (Tables 3 and S1, Supporting Information). Besides, imperfections introduced in the use of a constant CDL model still contribute to error in the recovered Ru values in this scenario. At higher Fc concentration (0.92 mM), the Faradaic current dominates the total current (Figure S2, Supporting Information) unlike experiments undertaken with the lower C0 value of 0.098 mM (Figures 1 and 2). Having a much higher Faradaic to background a.c. current ratio means that imperfections introduced when using a potential independent CDL model are now relatively unimportant in recovery of Ru with Rev3 parameterization at both lower (9 Hz) and higher f (72 Hz) (Tables 3 and S2 and S3, Supporting Information). Under these circumstances, the use of the more accurate Rev6 model produces similar Ru values to the use of Rev3 and only minimally decreases the objective function ΨNim. Application of the GM to the low frequency a.c. voltammetric data for oxidation of 0.92 mM Fc leads to recovery of E0 values that differ by less than 1 mV with those recovered using Nimrod/O (Table S3, Supporting Information). However, GM analysis produces CDL and Ru values that are lower than those recovered by the Nimrod/O method in most cases (Tables 3 and S3, Supporting Information). The Ru values recovered from the a.c. voltammetric data for oxidation of 0.92 mM at 72 Hz using Nimrod/O are only slightly below those calculated from EIS data (Table 3). In summary, under low (9 Hz) and moderate (72 Hz) a.c. frequency conditions, oxidation of Fc at 0.098 and 0.92 mM can be well modeled as an ideal “reversible” process. Oxidation of Ferrocene: Modeling Based on the Assumption of “Quasi-Reversible” Charge Transfer. Typically, the voltammetric oxidation of Fc has been modeled assuming “quasi-reversible” behavior to generate k0 and α values rather than as a “reversible” process which is independent of these parameters. However, since the electron transfer kinetics closely approach the “reversible” limit even

ΨNim 0.126 0.041 0.039 0.045 0.053 0.054 0.070

GM

a Calculated as ΔRu = (REIS − RRecovered ). bSimulation models are u u specified in Table 2 and the text. cWith C0 = 0.098 mM, QR5 model refers to the use of GM and QR8 to Nimrod/O. dNot analyzed.

72 Hz

n.a. n.a. n.a. n.a. 0.015 ± 0.003 0.016 ± 0.003 0.034 ± 0.007

model

0.92

Table 2. Mean Values and Standard Deviations of Nimrod/ O (ΨNim) and Global Method (ΨGM) Objective Functions Derived from the Comparison of Experimental a.c. Voltammograms (f = 9 or 72 Hz) for Oxidation of 0.098 or 0.92 mM Fc (10 Independent Experiments for Each C0) with Simulations Based on Designated Models

72 Hz

C0/mM

0.003 0.005 0.005 0.004 0.006 0.002 0.010

a Optimized parameters (O.p.) are Ru, E0, and CDL (Rev3) or ECdl, CDL0, CDL2, and CDL3 (Rev6). bO.p. are E0, k0app, αapp, Ru, and CDL (QR5) or ECdl, CDL0, CDL2, and CDL3 (QR8); with C0 = 0.098 mM, QR5 model was used by GM and QR8 was used by Nimrod/O. cO.p. are E0, k0app, αapp, and CDL (QR4) or ECdl, CDL0, CDL2, and CDL3 (QR7); in this case, Ru values were set equal to those derived from EIS data; with C0 = 0.098 mM, QR4 model was used by GM and QR7 was used by Nimrod/O. dNot analyzed.

The substantially higher values of ΨNim obtained with the Rev3 model relative to the Rev6 parameterization (Tables 1 and 2) are consistent with the inadequacy of using a constant CDL value for modeling the experimental background current (Figure 2). This is an example of underparameterization in a simulation model resulting in less than optimal agreement between experiment and theory. Nevertheless, the E0 values recovered with the Rev3 and Rev6 models by Nimrod/O are identical at the three significant figure level (Table 1) notwithstanding the imperfections in the potential independent CDL model. Moreover, both the Rev3 and Rev6 models provide Ru values close to those determined on the basis of the EIS measurements (RC model) for all 10 experiments processed (Tables 1 and 3), but the Rev6 values are always closer to the values measured by EIS, reflecting the use of a better model of CDL. In essence, Ru is the key parameter in the “reversible” model as it is the only one that provides variation in the magnitude of the simulated Faradaic a.c. current, when other parameters are assumed to be exactly known. Systematic differences of Ru recovered by Nimrod/O with respect to the EIS values by ΔRu ≈ 6 Ω (Table 3, model Rev6) probably emphasize experimental uncertainties in A, C0, and DFc, which were assumed to be exactly known, and in Ru derived from the EIS method (simple 8412



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Table 4. Parameters and Objective Functions (Ψ) Recovered by Nimrod/O (Nim) and the Global Method (GM) Using the “Quasi-Reversible” Model (QR5)a to Fit Experimental a.c. Voltammograms for Oxidation of 0.92 mM Fc in CH3CN (0.10 M (n-Bu)4NPF6) at 9 Hz CDL/μF cm−2

Ru/Ω #

EIS

1 2 3 4 5 6 7 8 9 10

61 66 55 63 63 54 71 53 58 85

b

k0app/cm s−1

E0/Vc

αapp c

Ψ

Nim

GM

Nim

GM

Nim

GM

Nim

GM

Nim

GM

Nim

GMd

50 60 46 55 53 43 65 38 47 76

38 55 41 53 49 36 61 37 45 67

14.8 18.0 22.1 22.9 23.5 23.8 21.5 23.9 21.9 24.1

15.2 18.9 20.9 21.6 21.4 21.0 20.3 20.7 20.4 24.6

0.527 0.558 0.567 0.559 0.541 0.531 0.600 0.584 0.587 0.589

0.527 0.558 0.567 0.559 0.542 0.531 0.601 0.585 0.586 0.589

1.3 3.0 2.9 3.0 2.3 3.0 1.9 1.0 1.4 2.5

0.45 1.8 2.8 5.7 14 162 3.6 8.3 3.3 1.6

0.53 0.50 0.50 0.48 0.48 0.48 0.48 0.51 0.51 0.48

0.51 0.54 0.54 0.00 0.65 0.98 0.72 0.57 0.80 0.49

0.013 0.014 0.015 0.015 0.018 0.016 0.014 0.023 0.014 0.017

1.1 1.7 2.1 2.2 1.9 2.2 2.0 2.0 1.6 2.5

a Optimized parameters: E0, k0app, αapp, Ru, and CDL. bValues derived from EIS data. cSimulations are insensitive to variation in αapp. The close proximity to 0.50 is an artifact (see the Supporting Information, page S18). dΨGM data are multiplied by 108.

under kinetically sensitive a.c. voltammetric conditions,22,26 recovered kinetic parameters can have considerable uncertainties and hence are often designated below as apparent k0app and αapp values. To ascertain the implications of this kinetic rather than thermodynamic form of analysis near the reversible limit, the a.c. voltammograms obtained for oxidation of Fc were reprocessed but now with k0app and αapp being regarded as unknowns. That is, the optimized recovered parameters were E0, k0app, αapp, Ru, and CDL (model QR5) or ECdl, CDL0, CDL2, and CDL3 (model QR8). At low frequency ( f = 9 Hz), E0 values recovered by Nimrod/O and GM using all versions of the “reversible” and “quasi-reversible” models are essentially indistinguishable (Tables 4 and S3, Supporting Information). Differences in CDL are also insignificant (Tables 4 and S3, Supporting Information). However, Ru values recovered from the lowfrequency data with 0.92 mM Fc using either Nimrod/O or GM with the QR5 model are systematically lower than those recovered using either “reversible” parameterization or EIS (Tables 3 and 4). GM and Nimrod methods of data analysis based on the “quasi-reversible” QR5 model give k0app values that are mostly greater than 1 cm s−1 (Tables 4 and 5). The problem is that k0app values are highly variable from experiment

to experiment, with an outcome from any given experimental data set that historically and inadvertently may have been reported as a “correct solution”. Another feature to emerge from data in Table 4 is that implausible values of αapp are recovered by the GM for some experiments. The close proximity of αapp to 0.50 found with Nimrod/O is a data treatment artifact as explained in the Supporting Information. On the basis of the objective function, the level of the experiment-theory agreement achieved for low-frequency data with the QR5 model is essentially the same as that found with the “reversible” model (Table 2), notwithstanding the introduction of the two additional adjustable kinetic parameters which would be expected to lead to superior agreement. The pitfalls and variability in outcomes of electrode kinetic data analysis based on use of the “quasi-reversible” model near the “reversible” limit can be identified by performing experiments over a range of concentrations and frequencies, as well as by imposing constraints on the simulation model. In the latter context, if the Ru value derived from the relevant EIS data is used as a known rather than unknown parameter, in what are referred to as the QR4 (optimized parameters: E0, k0app, αapp, and CDL) or QR7 models (optimized parameters: E0, k0app, αapp, ECdl, CDL0, CDL2, and CDL3), the k0app values recovered from oxidation of 0.92 mM Fc using the GM and Nimrod/O protocols with the QR4 model are now significantly higher (Tables 5 and S4, Supporting Information). At the lower 0.098 mM concentration, and even when the contribution from IRu is relatively unimportant, the use of the “quasi-reversible” model with the GM can result in large variations of both k0app (Tables 5, 6, and S5, Supporting Information) and ΔRu (Tables 3 and 6) derived from the analysis of the 10 experimental low-frequency data sets. In a few cases, realistic parameter values (e.g., experiments 1 or 3 in Table 6) similar to those found with higher concentration data at the same f are achieved, while in other cases the parameters recovered are manifestly wrong (e.g., experiments 2 or 10 in Table 6). Processing higher frequency (72 Hz) and higher concentration (0.92 mM) data sets using Nimrod/O and the QR5 model (Table S6, Supporting Information) produced Ru values similar to those obtained with “reversible” models (Table 3) along with k0app values that are much higher (Table 5) and Ψ values similar to those obtained by use of Rev3 (Table 2). That is, the “solution” to the “quasi-reversible” model has effectively

Table 5. Mean Values and Standard Deviations of k0app Recovered by Nimrod/O and GM Assisted Optimization of the Simulation Parameters Using the “Quasi-Reversible” Modela to Provide the “Best” Fit of the Experimental a.c. Voltammetric Data for Oxidation of 0.098 or 0.92 mM Fc (10 Independent Experiments for Each C0) at 9 and 72 Hz k0app/cm s−1 9 Hz C0/mM

model

0.098

QR5/QR8b QR4/QR7b QR5 QR4

0.92

0.89 1.4 4.6 34

72 Hz

GM

Nim

± ± ± ±

n.a.d n.a. 2.2 ± 0.8 13 ± 2

0.66c 1.1 4.3c 33

Nim 20 35 51 84

± ± ± ±

5 4 10 10

a

Simulation models are specified in Table 2. bAt C0 = 0.098 mM, QR5 and QR4 models were used with GM, while QR8 and QR7 were used with Nimrod/O. ck0app values of 24 and 162 (Tables 6 and 4) were not used in calculation of the mean values and the standard deviations. d Not analyzed. 8413



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Table 6. Parameters and Objective Functions ΨGM Recovered by GM Using the “Quasi-Reversible” Model (QR5)a to Fit Experimental a.c. Voltammograms for Oxidation of 0.098 mM Fc in CH3CN (0.10 M (n-Bu)4NPF6) at 9 Hz Ru/Ω

a

#

EISb

GM

CDL/μF cm−2

E0/V

k0app/cm s−1

αapp

108·ΨGM

1 2 3 4 5 6 7 8 9 10

66 61 56 54 51 50 37 64 49 56

71 0 67 0 2 84 33 40 18 184

24.3 22.4 24.5 19.1 19.9 23.8 23.2 21.4 23.2 24.6

0.587 0.632 0.635 0.621 0.629 0.630 0.630 0.638 0.644 0.647

2.18 0.46 1.47 0.32 0.39 24 1.48 0.75 0.66 0.32

0.92 0.92 0.92 0.92 0.93 0.96 0.94 0.93 0.94 0.95

1.8 1.5 2.0 1.1 1.2 1.2 1.4 1.4 1.4 1.6

Optimized parameters: E0, k0app, αapp, Ru, and CDL. bValues derived from EIS data.

converged to that for the “reversible” case, but the question remains as to whether the recovered k0app values have any real significance. Other features to emerge are that with the QR4 and QR7 models, which are based on experimentally determined Ru values, the quality of the fit, in terms of the objective function, becomes worse as compared to QR5 and QR8 (Table 2). Analogously with the use of the “reversible” models, Ru values recovered at f = 72 Hz from the low-concentration data using Nimrod/O with the QR8 model are lower than those recovered at higher Fc concentration (model QR5) and found by EIS (Table 3). Accordingly, k0app values recovered with “quasireversible” models from the experiments undertaken at low C0 and 72 Hz are lower than those found with 0.92 mM Fc at the same frequency (Table 5). In summary, the use of the “quasi-reversible” model near the “reversible” limit gives rise to substantial concentration, frequency, and data analysis method variation in the recovered values of k0app, αapp and Ru, and in this sense is unsatisfactory relative to use of the far more robust “reversible” model. Moreover, when experimental data are analyzed with the QR models, even a very small uncertainty in the current magnitude arising from uncertainties in C0, DFc, or A values (experimental uncertainty of ca. 2%) has a major impact on the analysis of the electrode kinetics and Ru. Significantly, from the heuristic data analysis point of view as used in most studies, it should be emphasized that the differences between the simulated curves generated from Rev3(6), QR5(8), and QR4(7) and experimental ones are indistinguishable at the resolution available in Figure S3, Supporting Information. Improper use of the “quasireversible” model explains a great deal of the large variation in electrode kinetic parameters reported for oxidation of Fc. Insights into the Origin of the Data Analysis Ambiguities When Oxidation of Ferrocene Is Modeled as a “Quasi-Reversible” Process. Confidence Intervals for the Recovered Parameters Based on the Bootstrapping Algorithm. To examine the statistical significance of E0, k0app, αapp, Ru, and CDL values derived from use of the “quasireversible” model, the bootstrapping technique (BST)29 was used to provide confidence intervals for the parameters recovered with GM optimization. The BST resamples the residuals from a parameter optimization exercise to generate surrogate data sets by redistributing the existing experimental noise randomly throughout the data. In practice, for a given experimental data set (IM), the best-fit simulated data (IS) is generated, which then gives, for each discrete measurement

S point j, IM j = Ij + rj, where rj represents the residuals. New data sets are further generated as IG = ISj + rk, where k is selected randomly with replacement from the range of possible values of j. This allows the construction of as many data sets as required. By performing a parameter optimization on each data set and provided the operations are repeated a sufficient number of times, reliable bounds on the parameter confidence intervals are obtained. The usefulness of the BST was first tested using data published for the reduction of [Fe(CN)6]3− at a GC electrode in aqueous 1 M KCl electrolyte.8 For the [Fe(CN)6]3−/4− process, the fit of the QR4 model to the experimental data with either Nimrod/O or GM produces k0 and α data fully concordant with a “quasi-reversible” process. The acceptability of the solution for the electrode kinetics of a process well removed from the “reversible” limit is confirmed by very narrow confidence intervals for each of the recovered parameters E0 = 0.2131 ± 0.0001 V vs Ag|AgCl|1 M KCl, k0 = 0.00955 ± 0.00003 cm s−1, α = 0.521 ± 0.002, and CDL = 24.0 ± 0.01 μF cm−2 derived from 1000 iterations of the BST (Figure S4, Supporting Information). In contrast, in the case of a.c. voltammetric data for oxidation of 0.92 or 0.098 mM Fc at 9 Hz (Table 4, experiment 7; Table 6, experiment 1), k0app and αapp confidence intervals derived from 1000 iterations of the BST can be very broad (Figures 3 and S5, Supporting Information). For example, αapp can appear to have any value between 0.0 and 1.0. While an acceptable outcome with respect to Ru (60 ± 1 Ω) is obtained at higher concentration (Figure S5, Supporting Information), highly variable and unrealistically high Ru values are derived from the lower concentration data derived from use of the BST (Figure 3). The poor distributions obtained with some parameters emphasize that no well-defined unique set of electrode kinetic parameters is obtained on varying the concentration and frequency conditions, and only CDL and E0 values are accurately estimable from the low frequency experimental data. In essence, these Fc oxidation data should not be fitted with the “quasi-reversible” model. Localized Minima Problem in k0app and Ru Combinations Revealed by Parameter Sweep (Contour Plot) Analysis. A method of data analysis that clearly reveals the likelihood of having nonunique [k0, Ru] combinations in the analysis of the voltammetric data with the “quasi-reversible” model is available in the Nimrod/G sweep tool.9,23−25 Now, objective functions displayed in the form of contour plots are compared on the basis of selected combinations of k0 and Ru, having fixed the values of all other parameters (QR2 model). In this procedure,

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Figure 4. Contour plots for the objective function Ψ calculated using eq 4 for (a, b) 1st to 5th, (c) 2nd to 8th, and (d) 2nd to 7th a.c. harmonic components for experimental data obtained with (a−c) 0.92 mM Fc (experiment No. 3 in Tables 4 and S7, Supporting Information) and (d) 0.098 mM Fc (experiment No. 1 in Table 1) in CH3CN (0.1 M (n-Bu)4NPF6) at f = 9 (a), 72 (b, c), or 219 Hz (d) and data simulated using model QR2 with Ru varied over the range of 0−100 Ω (10 Ω steps) and k0 varied over the range of (a) 0−50 cm s−1 (0.5 cm s−1 steps), (b, c) 0−100 cm s−1 (2 cm s−1 steps), or (d) 0− 150 cm s−1 (2 cm s−1 steps). E0 and CDL were fixed at values recovered from use of Nimrod/O optimization, and α was fixed at 0.50, while other parameters are as defined in relevant tables and text.

about 3 cm s−1. Flatness of the Ψ(k0, Ru) surface between these localized minima (difference in ΨNim does not exceed 0.005 as shown in Figure 4a) and absence of the upper limit for the second Ψmin region are now seen to be the origin of unacceptable variability of k0app recovered by Nimrod/O and GM. Ψ1−5(k0, Ru) contour plots derived from 0.92 mM Fc at the kinetically more sensitive higher frequency of 72 Hz exhibit a major area of Ψmin with acceptable Ru values, but with k0app now ≥9 cm s−1 (Figure 4b). Inclusion of the objective functions for the kinetically more sensitive higher order harmonics (6th to 8th) to the aggregate Ψ value and removal of the least kinetically sensitive first harmonic (Ψ2−8) produces a new isolated Ψmin region on the [k0, Ru] surface with a reliable Ru value and k0app in the range of about 3−11 cm s−1 (Figure 4c), which accommodates k0 values reported by Sun and Mirkin on the basis of analysis of steady state data obtained with nanoelectrodes.22 Ψ2−7(k0, Ru) surface derived from the 0.098 mM Fc data set at an even higher frequency of 219 Hz, which represents the highest level of kinetic sensitivity examined in this study (Figure 4d), is qualitatively similar to the Ψ1−5(k0, Ru) contour plot obtained for the 0.92 mM and 9 Hz data. However, the lower and upper k0app limits (k0min and k0max, respectively) for the first Ψmin region located at lower Ru and k0min for the second Ψmin region with undefined k0max are now shifted to higher values. Examination of Ψ(k0, Ru) contour plots at 9, 72, and 219 Hz (Figure 4) now reveal that, as expected, k0min is a function of the kinetic and Ru sensitivity of the technique which are defined by f and C0 (IRu drop). That is, under the conditions of Figure 4, parameter sweep (contour plot) analysis with Ψ combinations given above suggests that any k0app value from k0min of about 3 to 9 up to about 150 cm s−1 can provide the “best fit” of the experimental data with simulations based on the “quasireversible” model. In the present transient a.c. voltammmetric context, the k0 value based on modeling as a “quasi-reversible”

Figure 3. Distribution of (a) E0, (b) k0app, (c) αapp, (d) CDL0, and (e) Ru values optimized using GM to fit an experimental a.c. voltammogram for oxidation of 0.098 mM Fc at 9 Hz (experiment No. 1 in Table 6) with simulations based on the “quasi-reversible” model (QR8) derived from 1000 iterations of the bootstrapping technique. Red lines provide the 95% confidence interval. The potential dependent model (eq 3) was used to simulate CDL.

unknown parameters are assigned fixed values determined by the parameter search space and the sweep step selected (e.g., k0 range from 0 to 5 cm s−1 with 0.1 cm s−1 steps in Figure 4a). Unlike the parameter optimizations based on a Simplex algorithm used above, where any value of k0 within the defined search space is available as the optimized reported value, only a series of discrete parameter values are considered (e.g., 0.0, 0.1, 0.2, 0.3, 0.4, ... cm s−1). Consequently, Ψ values obtained by sweep analysis can differ from those reported from application of the Simplex optimization. In the oxidation of 0.92 mM Fc, the 9 Hz low frequency Ψ(k0, Ru) contour map constructed from the first five a.c. harmonics (Ψ1−5) reveals two regions of minimum objective function (Ψmin) in the [k0, Ru] surface. The first localized Ψ1−5 minimum in the contour plot is associated with Ru values of ca. 40 Ω and k0app values of about 1−2 cm s−1, as repeatedly found using Simplex optimization with the QR5 model (Table 4). The second Ψmin region is associated with Ru, values that are closer to those measured by EIS or derived from optimization using the “reversible” model (Table S3, experiment 3, Supporting Information), and with any k0app values above 8415



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process is therefore concluded to be ≥9 cm s−1, which is equivalent to stating that the process is “reversible” under the conditions employed. The virtue of Fourier filtering the a.c. voltammetric data for quantification of very fast electrode kinetics is now obvious; it allows data from the higher order harmonics, which have maximum kinetic sensitivity, to be analyzed in isolation and hence is akin to the use of very fast scan rates in d.c. studies but with the major attribute of having no capacitance background current. Modeling Reduction of [Ru(NH3)6]3+ and [Fe(CN)6]3−. 0 kapp values reported for the [Ru(NH3)6]3+/2+ process have also progressively increased with time, with recent nanoelectrode data from Sun and Mirkin22 suggesting that the value is in the range of 14−19 cm s−1, which is too fast to be measured accurately under the transient conditions used in this study, i.e., very close to or at the “reversible” limit. Even though quantification of the [Ru(NH3)6]3+/2+ electrode kinetics in aqueous electrolyte media is not hampered by significant IRu drop, processing a.c. voltammetric data with the “quasireversible” model still leads to the same ambiguities found with analysis of the Fc0/+ process. Thus, with f = 9 Hz, the use of Nimrod/O and “reversible” parameterization gives ΨNim = 0.025 whereas the “quasi-reversible” (QR8) model produces an unlikely value of k0app = 0.40 cm s−1 with ΨNim = 0.017. With f = 219 Hz, Nimrod/O produces Ru values consistent with EIS data and similar ΨNim values with the use of either the Rev6 or QR8 models (Table S7, Supporting Information). However, analysis of seven runs of the Simplex optimization on this high frequency data set produced highly variable k0app = 48 ± 45 cm s−1 and αapp = 0.59 ± 0.39 parameter values. The variability produced under these low IRu drop conditions is clearly unacceptable. That is, again the “reversible” model should be used to simulate the a.c. voltammetry of the [Ru(NH3)]3+/2+ process with the conclusion being reached that k0 ≥ 10 cm s−1. In contrast to the Fc0/+ and [Ru(NH3)6]3+/2+ processes, reduction of [Fe(CN)6]3− represents an example of a “quasireversible” process well removed from the “reversible” limit.8 Under conditions of small IRu drop with 1 M KCl aqueous electrolyte, reliable values of k0, α, and Ru parameters are readily recovered by the automated methods of data analysis (Table S8, Supporting Information) with ΨNim slightly decreasing as the number of the optimized parameters increases. In this case, the use of the now underparameterized “reversible” model for the [Fe(CN)6]3−/4− process gives absurd recovered Ru values of 754 Ω and the quality of fit is extremely poor and unacceptable (Table S8, Supporting Information).

and S8, Supporting Information) is to be expected, solely because it requires optimization of two additional parameters. However, overparameterization pitfalls can be revealed by data analysis which includes the variation of concentration of the electroactive species and kinetic sensitivity (frequency in a.c. voltammetry) along with bootstrapping to provide a statistical analysis of the parameters recovered from experiment-theory comparisons and parameter sweep (contour plot) analysis to probe the sensitivity of the analytical technique to the parameters being sought. Quantitative experiment-theory comparisons applied to the individual higher order harmonic components of a.c. voltammetric data for the Fc0/+ process with a conventional macroelectrode show that the “reversible” model is superior to the “quasi-reversible” one with the implication being that k0 ≥ 9 cm s−1. This report of this lower limit is in agreement with studies by Sun and Mirkin22 undertaken with nanoelectrodes. In the past few years, significant advances in the fabrication of the nanoscale electrodes and, most importantly, accurate determination of their geometries have been achieved.30 Apart from providing very high electrode kinetic sensitivity, studies at the nanoscale using the technique of scanning electrochemical microscopy have also proven to be of the utmost importance in the rationalization of kinetic dispersion that can occur with carbon based and other forms of heterogeneous macroscale electrodes.11,14 Arguably, technological advances will ultimately make nanoscale electrodes the most promising platform for studies of very fast electrochemical kinetics. While our present study uses data derived from macroelectrodes and the kinetically sensitive and experimentally efficient technique of a.c. voltammetry, analogous data analysis issues and the possibility of misinterpretation of data will almost certainly apply to the widely employed d.c. methods, other voltammetric techniques, and electrodes of all dimensions, whenever the reversible limit is approached.

■

ASSOCIATED CONTENT

S Supporting Information *

Experimental section. Tables reporting the parameters recovered by the Global Method or Nimrod/O from analysis of a.c. voltammograms for oxidation of Fc or reduction of [Ru(NH3)6]3+ and [Fe(CN)6]3−. Figure S1 showing a histogram of random noise recovered from an a.c. voltammogram for oxidation of 0.92 M Fc. Figures S2 and S3 comparing a.c. voltammograms for oxidation of 0.92 mM Fc and simulations based on different models. Figures S4 and S5 showing the results of the bootstrapping method of statistical analysis of parameters recovered from a.c. voltammograms for reduction of [Fe(CN)6]3− and oxidation of 0.92 mM Fc. Text providing an explanation of the charge transfer coefficient recovery artifact. This material is available free of charge via the Internet at http://pubs.acs.org.

■

CONCLUSIONS The risk of recovering incorrect, but physically meaningful, values of k0, α, and Ru from experimental and simulated voltammetric data comparisons can be substantial when an electrode process is close to the “reversible” limit, as demonstrated by applying a range of data analysis methods to a.c. voltammetric data for oxidation of ferrocene and reduction of [Ru(NH3)6]3+. Underparameterization in a simulation model is usually identifiable, as it leads to a poor fit of experimental with theoretical data. Overparameterization can be more difficult to detect. This situation can arise if a “quasi-reversible” simulation is used to model what is in reality a “reversible” process. Apparently improved agreement between experimental and simulated data found in some cases when using the “quasireversible” rather than “reversible” simulation model (Tables 2

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest. 8416



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Article

(27) Janisch, J.; Ruff, A.; Speiser, B.; Wolff, C.; Zigelli, J.; Benthin, S.; Feldmann, V.; Mayer, H. A. J. Solid State Electrochem. 2011, 15, 2083− 2094 and references therein. (28) Gill, P. E.; Murray, W. SIAM J. Numer. Anal. 1978, 15, 977−992. (29) Efron, B.; Tibshirani, R. J. An Introduction to the Bootstrap; Chapman and Hall: New York, 1993. (30) (a) Nogala, W.; Velmurugan, J.; Mirkin, M. V. Anal. Chem. 2012, 84, 5192−5197. (b) Nioradze, N.; Chen, R.; Kim, J.; Shen, M.; Santhosh, P.; Amemiya, S. Anal. Chem. 2013, 85, 6198−6202. (c) Güell, A. G.; Meadows, K. E.; Dudin, P. V.; Ebejer, N.; Macpherson, J. V.; Unwin, P. R. Nano Lett. 2014, 14, 220−224. (d) Chen, S.; Liu, Y. Phys. Chem. Chem. Phys. 2014, 16, 635−652 and references therein.

ACKNOWLEDGMENTS This publication is based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). Financial support from the Australian Research Council is also gratefully acknowledged.

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REFERENCES

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