Journal of Electroanalytical Chemistry 690 (2013) 104–110

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Journal of Electroanalytical Chemistry

journal homepage: www.elsevier .com/locate / je lechem

Estimation of electrode kinetic and uncompensated resistance parametersand insights into their significance using Fourier transformed ac voltammetryand e-science software tools

Elena Mashkina a, Tom Peachey b, Chong-Yong Lee a, Alan M. Bond a,⇑, Gareth F. Kennedy a, Colin Enticott b,David Abramson b, Darrell Elton c

a School of Chemistry, Monash University, Clayton, Vic 3800, Australiab Faculty of Information Technology, Monash University, Clayton, Vic 3800, Australiac Department of Electronic Engineering, Latrobe University, Bundoora, Vic 3083, Australia

a r t i c l e i n f o

Article history:Received 8 August 2012Received in revised form 31 October 2012Accepted 2 November 2012Available online 24 November 2012

Keywords:Electrode kineticsSimulationsFourier transform techniquesE-science

1572-6657/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jelechem.2012.11.002

⇑ Corresponding author. Tel.: +61 3 9905 1338; faxE-mail address: alan.bond@monash.edu (A.M. Bon

a b s t r a c t

In transient forms of voltammetry, quantitative analysis of electrode kinetics and parameters such asuncompensated resistance (Ru) and double layer capacitance (Cdl) are usually undertaken by comparingexperimental and simulated data. Commonly, the skill of the experimentalist is heavily relied upon todecide when a good fit of simulated to experimental data has been achieved. As an alternative approach,it is now shown how data analysis can be based on implementation of e-science software tools. Previ-ously, a standard heuristic data analysis approach applied to the oxidation of ferrocene in acetonitrile(0.1 M Bu4NPF6) at a glassy carbon electrode using higher order harmonics available in Fourier trans-formed ac voltammetry implied that the heterogeneous charge transfer rate constant k0 is P0.25 cm s�1

with the charge transfer coefficient (a) lying in the range of 0.25–0.75. Application of e-science softwaretools to the same data set allows a more meaningful understanding of electrode kinetic data to be pro-vided and also offers greater insights into the sensitivity of the IRu (Ohmic drop) on these parameters.For example, computation of contour maps based on a sweep of two sets of parameters such as k0 andRu or a and k0 imply that a is 0.50 ± 0.05 and that k0 lays in the range 0.2–0.4 cm s�1 with Ru around130 Ohm. Quantitative evaluation of k0, a and Ru for the quasi-reversible ½FeðCNÞ6�

3� þ e � ½FeðCNÞ6�4�

process at a glassy carbon electrode in aqueous media is also facilitated by use of e-science software tools.In this case, when used in combination with large amplitude Fourier transformed ac voltammetry, it isfound for each harmonic that k0 for the electrode process lies close to 0.010 cm s�1, a is 0.50 ± 0.05and Ru is 610 Ohm.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

A detailed understanding of the mechanism of an electrode pro-cess is frequently required [1,2]. Unfortunately, quantitative anal-ysis of the kinetics and other parameters associated with anelectrode reaction is not trivial, particularly when analytical solu-tions to the theory are not available and simulations have to beemployed. At present, the experimental tool most commonly em-ployed for this task is dc cyclic voltammetry in which a triangularwaveform is applied to the electrochemical cell. Typically, the scanrate is varied over as wide a range as possible to provide a set ofvoltammograms. Subsequently, experimental and simulated dataare compared in order to establish the mechanism and quantitythe values of the relevant parameters.

ll rights reserved.

: +61 3 9905 4597.d).

Work from our laboratory [1,3–6] and others [7–10] has dem-onstrated that an analogue of dc linear sweep or cyclic voltamme-try can be advantageously implemented both experimentally andtheoretically by superimposing a sinusoidal or other periodic acwaveform onto the linear or triangular dc voltage. Analysis ofexperimental data, aided by Fourier or other transform methods[7–10] enables the dc and ac harmonic components to be sepa-rated. If access to the dc aperiodic term and say the first eightharmonics is available from a single experiment undertaken underlarge amplitude conditions [4–6], then this allows data to be con-veniently broken up into nine subsets that represent different time(or frequency) domains with different levels of sensitivity to theheterogeneous charge transfer rate constant (k0), charge transfercoefficient (a), uncompensated resistance (Ru), double layer capac-itance (Cdl), reversible potential (E0), diffusion coefficients (D) andthe other variables that contribute to the total voltammetric re-sponse. All these parameters need to be evaluated in quantitative

E. Mashkina et al. / Journal of Electroanalytical Chemistry 690 (2013) 104–110 105

electrode kinetic studies. Notably, the advantage of this Fouriertransformed (FT) ac voltammetric approach is that each ac har-monic data subset, derived from a single experiment, allows de-tailed analyses to be undertaken on each component, eachhaving a different level of sensitivity to the parameters requiringquantification. Additionally, the Fourier transform method facili-tates filtering of unwanted mains frequency and other noisecomponents.

Nowadays, procedures for the voltammetric determination ofparameters needed to quantify an electrode process commencewith a computer controlled physical experiment in order to obtaina data set. Then, the relevant computational model, believed to mi-mic the experiment, is executed repeatedly with variation inparameters, and results quantified by minimizing the differencesbetween the physical and simulated systems in the, so called, heu-ristic approach [3,5]. Simulated and experimental data compari-sons are used in this way to predict electrode kinetic parameterssuch as E0, k0, a (e.g. using Butler–Volmer model with mass trans-port described by linear diffusion) along with (Ru,) and double layercapacitance (Cdl). Other parameters such as the concentration (C),D, midpoint potential (Em) or E0 and effective electrode area (A)are usually assumed to be known from the literature or determinedfrom independent experiments. For example in experiments de-scribed in this paper, the electrode area was determined from lin-ear sweep voltammetry on a well characterized system and use ofthe Randles–Sevcik equation [6].

In the heuristic approach, as commonly employed in transientforms of voltammetry, the experimenter adjusts parameters in asimulation until a good fit is believed to have been achieved be-tween experimental and model. In ideal, but very rare cases whereanalytical solutions to the theory are available, well known statis-tical methods can be used to compare experiment and theory. Asituation of this kind is described in Ref. [11], although, the analyt-ical solution does not accommodate the IRu drop issue, so it needsto be assumed that perfect IRu compensation has been achieved byan instrumental method.

Recently [6], it was shown that when experimental data setscan be mimicked by simulation, that a more formal procedureusing readily available optimization algorithms can expediteparameter fitting relative to the situation prevailing when thepurely heuristic approach is employed. This study introduced theuse of the Nimrod tool set [12,13] into electrochemical data anal-ysis. The e-science approach can be accessed via http://messag-elab.monash.edu.au. The present paper now extends this workand describes in detail how the Nimrod e-science tool kit may beused to achieve quantitative electrode kinetic analysis in a conve-nient manner. The protocol described uses experimental and sim-ulated data derived from technique of large amplitude FT acvoltammetry in combination with data analysis using the Nimrodtool kit. The approach is applied to the oxidation of ferrocene(Fc) in acetonitrile (a very fast and close to reversible system, Eq.(1)) and the reduction of ferricyanide ½FeðCNÞ6�

3� (a slower processthat is regarded as being quasi-reversible, Eq. (2)).

Fc � Fcþ þ e� ð1Þ

½FeðCNÞ6�3� þ e� � ½FeðCNÞ6�

4� ð2Þ

This paper expands on the analysis of data provided in Ref. [6]where a detailed description of the use of the potential Nimrode-science tool kit was the major point of emphasis.

2. Nimrod and the computational model

The term ‘e-science’ has recently gained currency for scientificresearch that involves substantial digital technology in planning,

data acquisition or analysis. A common example is the use of acomputational model that incorporates the underlying theory,simulating the experiment, and typically requires many inputparameters. The term ‘parametric modeling’, when used in this pa-per, describes the exploration of how the model outputs vary withthe various inputs; Nimrod has been developed to assist suchexploration. Nimrod works with a variety of applications. It in-vokes the code supplied, using standard command line functions,and thus is generally applicable with user supplied software pack-ages that provide simulations of experimental data as long as theydo not have a graphical user interface. Thus, DigiSim [14,15], acommercially available computer code (distributed by Bioanalyti-cal Systems), and other commercially software packages availablefor simulations of voltammetric theory that have a graphical userinterface are not likely to be directly suitable for use with Nimrod.

Nimrod/O allows the user to run an arbitrary computationalmodel as the core of a non-linear optimization process. In the pres-ent voltammetric context, optimization algorithms are used toautomate the parameter fitting by minimizing some measure ofdifference between the simulated results and experimental data.Typically, these algorithms require a small batch of jobs for eachiteration and each iteration depends on the results of the previousone, so advantages provided by computational parallelism are notas great as in other applications. However, Nimrod/O can executeparallel jobs via Nimrod/G, using multiple cores on a local machine,or by submitting jobs to a cluster queue. It can also perform multi-ple optimizations in parallel when that is required. Nimrod/G per-forms ‘sweep’ experiments where each parameter of interest isassigned a set of values; Nimrod/G (another part of the Nimrodtoolkit) [16] generates jobs for each combination of these values,controls the execution, and aggregates the results into a responsesurface. In cases where the computational time is lengthy, thetasks required can be automatically distributed to multiple proces-sors (either within a computer cluster or on the Internet), and theresults can be computed in parallel. This makes it possible to per-form very large and extensive computational tasks. Even complexparametric experiments can be defined and run with virtually noprogrammer effort.

The present work used simulations of FT ac voltammetry de-rived from MECSim (Monash Electrochemistry Simulator), an in-house simulation package written in Fortran code, to model the re-sponse of the electrochemical cell to the applied potential. Matlabcode was used to isolate individual harmonic components for boththe modeled response and the experimental data. Since these twodatasets may be out of phase, an envelope function was created foreach (a simple linear spline) and the area between the two enve-lopes computed as the measure of the difference. This result, the‘metric’, is written to a file which Nimrod accesses. Analogous out-comes could be achieved with any other combination of a voltam-metric model and a system for computing a metric for thecomparison of simulated and experimental data.

The commonly used procedure for the determination of elec-trode kinetic parameters using simulations is empirical and timeconsuming. The relevant computational model is executed repeat-edly with systematic variation in parameters. Results are evaluatedrather arbitrarily by observing the differences between the physi-cal and simulated systems in the, so called, ‘heuristic approach’.The experimentalists use their experience to determine when agood quality of simulation-experiment agreement has beenachieved. In essence, with large amplitude FT cyclic ac voltamme-try, the aperiodic dc component provides Em and hence a good esti-mate of E0, while the fundamental harmonic provides the basicinformation on Cdl. In the second and higher harmonics, effectivelyshorter time scales are being probed than for the fundamental har-monic, so that the dependence of the Faradaic current on k0, Ru anda is far greater that for the fundamental harmonic. Furthermore,

Table 1Outcome of the simplex optimization of k0, Ru and a for the first to eighth harmonics for the oxidation of 0.99 mM and 0.49 mM Fc at a glassy carbon electrode in acetonitrile(0.1 M Bu4NPF6). Optimization range for optimal parameters were restricted to 0 < k0 < 0.5 cm s�1, 0 < Ru < 200 Ohm, 0 < a < 1. The term metric is defined as an area differencebetween experimental and simulated outputs and represents the quality of the fit.

Fc [0.99 mM] Fc [4.99 mM]

Harmonic Optimal parameters Metric (amp) Harmonic Optimal parameters Metric (amp)

k0 (cm s�1) Ru (Ohm) a k0 (cm s�1) Ru (Ohm) a

1 0.34 131.9 1.0 1.40e�08 1 0.31 161.1 0.99 6.10e�072 0.33 144.6 1.0 2.12e�09 2 0.28 151.4 0.96 3.92e�083 0.42 145.7 0.84 4.08e�10 3 0.41 150.9 0.81 6.23e�094 0.43 145.5 0.78 9.91e�11 4 0.11 137.6 0.49 2.94e�105 0.50 145.9 0.53 1.57e�11 5 0.13 142.2 0.49 1.09e�106 0.50 149.0 0.52 5.00e�12 6 0.16 147.3 0.49 6.96e�117 0.50 141.7 0.48 2.25e�12 7 0.23 156.6 0.48 6.54e�118 0.50 139.4 0.50 1.92e�12 8 0.48 156.9 0.41 7.08e�11

Table 2Outcome of five simplex optimizations with the third harmonic for k0 and Ru.Optimization range for optimal parameters were restricted to 0 < k0 < 0.5 cm s�1,0 < Ru < 200 Ohm when a is fixed to 0.5 for oxidation of 0.99 mM Fc at a glassy carbonelectrode in acetonitrile (0.1 M Bu4NPF6).

Starting parameters Optimal parameters Metric (amp)

k0 (cm s�1) Ru (Ohm) k0 (cm s�1) Ru (Ohm)

0.46 140.9 0.29 137.7 1.33e�090.45 164.9 0.46 144.6 1.35e�090.05 163.5 0.29 136. 6 1.33e�090.29 163.2 0.33 139.3 1.33e�090.23 156.4 0.32 139.1 1.33e�09

Table 3Outcome of simplex optimization for the third harmonic for k0 and Ru using differentk0 optimization ranges when a is fixed at 0.5. The example given is for oxidation of0.99 mM Fc at a glassy carbon electrode in acetonitrile (0.1 M Bu4NPF6).

k0 range Optimal parameters Metric (amp)

k0 (cm s�1) Ru (Ohm)

0 6 k0 6 0.1 cm s�1 0.10 94.7 1.79e�090 6 k0 6 0.3 cm s�1 0.27 135.4 1.33e�090 6 k0 6 0.5 cm s�1 0.48 145.8 1.35e�090 6 k0 6 0.8 cm s�1 0.37 142.2 1.33e�090 6 k0 6 1 cm s�1 0.46 146.2 1.35e�090 6 k0 6 2 cm s�1 1.55 155.9 1.42e�090 6 k0 6 5 cm s�1 0.79 152.2 1.38e�090 6 k0 6 10 cm s�1 2.24 157.7 1.44e�09

Fig. 1. Work flow diagram for optimization procedure based on use of Nimrod/O.

106 E. Mashkina et al. / Journal of Electroanalytical Chemistry 690 (2013) 104–110

the higher harmonics have minimal contribution from Cdl. Ideally,each parameter needed should be derived from data that are sen-sitive to the parameter of interest. Nimrod/O allows this outcometo be achieved [6] since parameter fitting is expedited by the useof standard optimization algorithms with distributed executionbeing undertaken on computer grid resources. Ultimately, all infor-mation is collected and displayed for examination by the user.

Fig. 1 represents the work flow diagram that summarizes optimi-zation procedures using Nimrod/O. In the Nimrod/G protocol, in-stead of finding the best optimization result, Nimrod/G collectsthe set of metrics for floating parameters restricted by the searchspace.

Typically, the execution time for a single run, using experimen-tal data together with MECSim and the Matlab metric computa-tions, was approximately 2–20 min depending on the number ofexperimental data points, parameters to be optimized and thenumber of users on the cluster. A single cluster of 36 nodes withmultiple cores was sufficient for the data analysis outcomes de-scribed in this paper.

3. Experimental

Experimental details related to the study of the oxidation of fer-rocene in acetonitrile (0.1 M Bu4NPF6) are as described in Ref. [17].The data for ferrocene (0.99 mM and 4.99 mM concentrations)were obtained with home built Fourier transform voltammetricinstrumentation [4] at 293 K using a sine wave perturbation witha frequency of 9.02 Hz and an amplitude of 80 mV. 16,384 datapoints were collected when cycling the dc potential from �0.3 Vto 1.0 V and back to �0.3 V vs a silver wire quasi-reference elec-trode at a scan rate 0.9686 V s�1. Initial uncompensated resistance(Ru) determinations were undertaken with a Bioanalytical System(BAS model 100B) electrochemical work station [18]. Ru valuesestimated from the analysis of the RuC time constant response[18] were found to lie in the range of 120–156 Ohm.

The ½FeðCNÞ6�3�=4� experimental data were derived from

0.90 mM potassium ferricyanide (K3[Fe(CN)6]) in aqueous 0.5 MKCl at 293 K using the same frequency, amplitude and number of16,384 data points as for the ferrocene study, but the potentialwas ramped from 0.6 V to �0.1 V and back to 0.6 V vs Ag/AgCl(3 M KCl) using a scan rate of 0.05215 V s�1 as described in [6].In this case, estimates of Ru, from analysis of the RuC time constantresponse, showed a low value of less than 20 Ohm.

Fig. 2. Fundamental to eighth harmonics contour maps derived from a sweep of the Ru (ordinate) and k0 (abscissa) values with a fixed at 0.5 for oxidation of 4.99 mM and0.99 mM Fc at a glassy carbon electrode in acetonitrile (0.1 M Bu4NPF6). (For interpretation to colours in this figure, the reader is referred to the web version of this paper.)

E. Mashkina et al. / Journal of Electroanalytical Chemistry 690 (2013) 104–110 107

4. Results and discussion

4.1. Optimization procedure

Optimization analysis was initially applied to the entire exper-imental data set for oxidation of 0.99 mM and 4.99 mM Fc at aglassy carbon macrodisk electrode in acetonitrile (0.1 M Bu4NPF6).In the previous study [17] the data sets were truncated to avoidproblems introduced by ringing observed near the initial and finalpotential boundaries as the result of approximating the periodicextension of a function with a distinct boundary value by trigono-metric polynomials in the FT data analysis.

The value of k0 reported for oxidation of Fc varies widely[7,19,20], but the process is known to be fast [17]. Since close prox-imity to reversibility is expected, the data are anticipated to be rel-atively insensitive to k0 and a. Input parameters needed in thesimulation such as temperature, frequency and amplitude of theapplied ac signal, dc scan rate, voltage range, bulk Fc solution con-centration, electrode area, diffusion coefficient, E0 and fourth orderpolynomial coefficients describing the double layer capacitance

background current were taken from Ref. [17] and included inthe one electron transfer model. Butler–Volmer electrode kinetics,with mass transport governed by linear diffusion, was assumed.Thus k0, a and Ru are the parameters to be derived from analysisof data, although in principle any of the parameters could be trea-ted as unknowns.

The data analysis method employed independent optimizationsfor each of the first to eighth harmonic components. The startingpoints for searches were selected randomly in the domains 0 6 k0 -6 0.5 cm s�1, 0 6 Ru 6 200 Ohm and 0 6 a 6 1. Optimizationparameters resulting from the Nimrod/O tool set are summarizedin Table 1 and illustrate the significance of undertaking optimiza-tion searches with multiple parameters and use of several harmon-ics. Although the resistance values derived from analysis of eachharmonic are moderately consistent (up to 5% parameter varia-tion), the k0 and a electrode kinetic terms vary dramatically (upto 60% parameter variation) from the first to fourth harmonics.One reasonable interpretation is that k0 and a are not rigorouslydefined by the search of these harmonics (too close to the revers-ible limit). However, it may be noted that the a value consistently

Fig. 3. Fundamental to eighth harmonic contour map derived from the sweep of a (ordinate) and k0 (abscissa) values for oxidation of 4.99 mM and 0.99 mM Fc at a glassycarbon electrode in acetonitrile (0.1 M Bu4NPF6) when Ru is fixed at 130 Ohm and 150 Ohm for 0.99 mM and 4.99 mM Fc respectively. (For interpretation to colours in thisfigure, the reader is referred to the web version of this paper.)

108 E. Mashkina et al. / Journal of Electroanalytical Chemistry 690 (2013) 104–110

finds its optimum value around 0.5 from analysis of the kineticallymore sensitive fifth to eight harmonics. Intriguingly, k0 varies in anapparently random fashion in the case of 4.99 mM Fc or else hitsthe upper limit of 0.5 cm s�1 chosen for the optimization rangein the case of 0.99 mM Fc. This implies that kinetically more sensi-tive higher order harmonics provide realistic information on a, butthe k0 value still remains relatively undefined. In multiple searchesof Ru and k0 parameters for only one harmonic, with a set at 0.50,the k0 value still varies considerably while Ru stays within therange of 140–160 Ohm (Table 2) as expected on the basis of inde-pendently measured values of 120–156 Ohm (see Experimentalsection). However, it should be noted that reported k0 values varywith the optimization range (Table 3), which implies that thisparameter is not well defined, for the given experimentalconditions.

4.2. Sweep procedure

To shed more light on the significance of the results describedabove, a was fixed at 0.50 and a parameter sweep was performedwith Nimrod/G tool kit using 20 by 15 evaluations equally spaced

through the search space in the range 0 6 k0 6 1 cm s�1 and0 6 Ru 6 200 Ohm. The results presented in the form of a contourmap for the objective metric were k0 and Ru are used as floatingparameters. The contour maps for the first to eighth harmonics(Fig. 2) show the elevation of the two variables (k0 and Ru) bymeans of contour lines. Contour lines represent a curve alongwhich the metric has a constant value. The region giving the bestoptimum established by the tolerance value of 10�3 for all experi-ments is indicated in white.

Examination of the contour plot in Fig. 2 suggests that Ru isaround 140 Ohm in experiments for 0.99 mM Fc, and 150 Ohmfor 4.99 mM Fc, with a lower limit of k0 of about 0.25 cm s�1. ThisRu outcome is consistent with the independent measurements va-lue [18] of 120–156 Ohm reported in the Experimental section. Therate constant of k0 > 0.25 cm s�1 is larger than some values pro-posed in the literature [19,20] when determined by dc cyclic vol-tammetry at a microdisk electrode. However, this k0 value isconsistent with results obtained with a heuristic approach and atmicroelectrodes under steady state conditions which minimizethe impact of the IRu drop [17]. Notably, the best optimizationsare obtained for the higher Fc concentration. The amplitudes [21]

Fig. 4. Fundamental to eighth harmonic contour maps derived from a sweep of the Ru (ordinate) and k0 (abscissa) values with a fixed at 0.5 for reduction of 0.99 mM½FeðCNÞ6�

3� at a glassy carbon electrode in aqueous 0.5 M KCl electrolyte. (For interpretation to colours in this figure, the reader is referred to the web version of this paper.)

Fig. 5. Fundamental to eighth harmonic contour maps derived from the sweep of a (ordinate) and k0 (abscissa) values for reduction of 0.99 mM ½FeðCNÞ6�3� at a glassy carbon

electrode in aqueous 0.5 M KCl electrolyte when Ru is fixed at 10 Ohm. (For interpretation to colours in this figure, the reader is referred to the web version of this paper.)

E. Mashkina et al. / Journal of Electroanalytical Chemistry 690 (2013) 104–110 109

of the poor signal-to-noise ratio of ca. 7:5 for the eighth harmonic,with 0.99 mM Fc, translates into an apparent shift in k0 to a largerlower limit.

The major conclusion drawn from the earlier study [17] is thatthe k0 value plausibly lies in the range 0.25–0.50 cm s�1 with a inthe range of 0.25–0.75. Further attention is now focused on therange of a as a function of k0. In order to achieve this goal, a sweepprocedure was performed with the value of Ru fixed at 130 Ohm forthe 0.99 mM Fc case and 150 Ohm for the 4.99 mM Fc case andother parameters as in Ref. [17]. Contour plots provided in Fig. 3now suggest that for the higher harmonics, a is 0.50 ± 0.05, withk0 being in the range of 0.20–0.40 cm s�1, regardless of concentra-tion. This is plausibly the best description of the system yet pro-vided, but reveals the strong interplay of k0, a and Ru and

highlights the difficulty in quantifying the electrode kinetics in fastprocesses that are very close to the reversible limit under theexperimental conditions employed [22].

The second example, for which experimental data are describedin [6] and now chosen for more detailed analysis, aided bye-science, is for the slower quasi-reversible one-electron reductionof ½FeðCN6Þ�3� to ½FeðCN6Þ�4� in aqueous 0.5 M KCl electrolyte. Theresults of optimization (Nimrod/O), as described in Ref. [6], consis-tently provided estimates of k0 around 0.01 cm s�1 and a of0.50 ± 0.05 whereas estimates of Ru showed no pattern at all. Con-tour maps of Ru vs k0 for the search space range 0 Ohm < Ru < 100 -Ohm (Nimrod/G) show that the Ru value is not defined by analysisof up to the fourth harmonic, while the value of k0 is consistentlyfound to be around 0.010 cm s�1. In fact, both k0 and a are very

110 E. Mashkina et al. / Journal of Electroanalytical Chemistry 690 (2013) 104–110

insensitive to the Ru value in the contour plots shown in Fig. 4. Incontrast, in the ferrocene case, the much larger IRu value that isapplicable in acetonitrile meaning that Ru is clearly defined evenfrom the first harmonic data analysis. With fifth, sixth and seventhharmonics, analysis of the optimization procedure for reduction of½FeðCNÞ6�

3� yields a localized estimate of Ru 6 10 Ohm (Fig. 4)which is consistent with the value of <20 Ohm obtained by inde-pendent measurement [18] of this parameter (see Experimentalsection). Due to the low Faradaic current magnitude approachingthe background noise, the optimization sweep result is less certainwith respect to the limit of Ru in the eighth harmonic. Contourmaps derived from the sweep of a and k0 values with Ru fixed at10 Ohm, all suggest that a and k0 are close to 0.50 and 0.010 cm s�1

respectively, regardless of the harmonic examined (Fig. 5). Thesevalues are therefore regarded as reliable for the ½FeðCN6Þ�3�=4� pro-cess with respect to the conditions used, noting that electrode pre-treatment and history can contribute to the electrode kinetics forthis inner sphere process [23].

The fidelity of the insights gained in k0 and a values is con-firmed by data in Figs. 4 and 5. Thus, for a quasi-reversible process,in which the electrode kinetics are well removed from the revers-ible regime, and where iRu does not play a key role, then k0 and avalues are readily quantified by the optimization tools.

The common tendencies in the contour map evaluation methodare as follows. For the first harmonic, the determination of param-eters tends to be dominated by the substantial contribution frombackground current derived from the double layer capacitance,so this harmonic component is not highly informative for quantify-ing the electrode kinetic parameters. For the higher harmonics,where the background current is negligible, the region for goodoptimization outcome for the parameters of interest shrink to amore compact region as uncertainty of the variable being soughtis decreased. Optimal results are likely to be obtained for aboutthe fifth, sixth and seventh harmonics where the Faradaic to back-ground current (noise) ratio is excellent. However, if the Faradaiccurrent is of the same order of magnitude as instrumental noise,then optimization derived from the contour maps for higher orderharmonic are likely to be less informative and may even have nophysical meaning. In the present study, the eighth harmonic ap-pears to represent a situation where usefulness in electrode kineticdata evaluation starts to decline.

5. Conclusions and outlook

FT ac voltammetric studies on the very fast and essentiallyreversible ferrocene oxidation process and quasi-reversible reduc-tion of ferricyanide demonstrate the value of e-science optimiza-tion tools in establishing the significance of estimated values ofk0, a and Ru and their interplay in studies of electrode kinetics. Rou-tine incorporation of e-science infrastructure into electrode kinet-ics evaluations therefore represents a significant opportunity toadvance knowledge in this area. Introduction of e-science toolsinto the evaluation of more complex electrochemical mechanismswhere chemical steps are coupled to electron transfer and for reac-tion schemes involving surface confined reactions also should pro-vide new insights. While this work highlights the use of FT acvoltammetry, Buttler–Volmer electrode kinetics and linear diffu-sion, it is also possible to implement the optimization routine withdc cyclic voltammetry, the spherical or cylindrical stationary elec-trodes, the rotating disk electrode or Marcus–Hush electron trans-fer theory, all of which can be simulated via use of MECSim.

In the present study, advantages of access to higher order acharmonics, all obtained from a single experiment have been dem-onstrated. Since e-science methods and optimization strategies aregeneric, they should be applicable to all forms of voltammetry with

considerable benefit. However, the large amplitude FT ac methodhas an inherent advantage in that each harmonic has a differentlevel of sensitivity to k0, a and Ru which enhances the ability toconfirm the fidelity of a deduced parameter.

This work focus on e-science optimization tools only but thereare other approaches [11,24,25] which accommodate simulationand statistical analysis. For example, in Ref. [24], the authors pro-pose a search for the optimal parameters with the Simplex algo-rithm as well as provide statistical error analysis by minimizingthe sum of squared deviations. Alternatively, the estimation of con-fidence intervals has been performed empirically, based on infor-mation furnished by the Simplex calculation [25]. Commerciallyavailable DigiSim simulation software also provides least-squarev2 tests to qualify the quality of the fit, but does not report anyconfidence limits for the parameters [26]. However, these methodsdo not introduce the IRu drop issue, and it is common for electrodekinetic data to be affected by systematic errors originating fromincomplete IRu drop compensation. In our previous study [17] weestimated the quality of the fit between experimental and simu-lated data by the examination the residual sum of squares (RSS)and by the mean percentage error (MPE). This work did not providestatistical analysis related to errors for the parameters of interestbut IRu drop was considered.

Future developments are likely to include the implementationof statistical analysis based, for example, on Bayesian probabilitiesinto parameter evaluations. In future, it is planned to exploremore complex mechanisms, examine global search optimizationmethods other than Simplex algorithm that are available in theNimrod tool kit and implement metric normalization, so one candirectly compare the optimization evaluation outcome for eachharmonic.

References

[1] K. Oldham, J. Myland, A. Bond, Electrochemical Science and Technology:Fundamentals and Applications, John Wiley & Sons, 2011.

[2] A.J. Bard, L.R. Faulkner, Electrochemical Methods: Fundamentals andApplications, Wiley, 2000.

[3] A.A. Sher, A.M. Bond, D.J. Gavaghan, K. Gillow, N.W. Duffy, S.X. Guo, J. Zhang,Electroanalysis 17 (2005) 1450–1462.

[4] A.M. Bond, N.W. Duffy, S.X. Guo, J. Zhang, D. Elton, Anal. Chem. 77 (2005)186a–195a.

[5] A.A. Sher, A.M. Bond, D.J. Gavaghan, K. Harriman, S.W. Feldberg, M.W. Duffy,S.X. Guo, J. Zhang, Anal. Chem. 76 (2004) 6214–6228.

[6] T. Peachey, E. Mashkina, C.Y. Lee, C. Enticott, D. Abramson, A.M. Bond, D. Elton,D.J. Gavaghan, G.P. Stevenson, G.F. Kennedy, Philos. Trans. R. Soc. A 369 (2011)3336–3352.

[7] C.A. Anastassiou, K.H. Parker, D. O’Hare, Anal. Chem. 77 (2005) 3357–3364.[8] C.A. Anastassiou, K.H. Parker, D. O’Hare, J. Phys. Chem. A 111 (2007) 13053–

13060.[9] J.E. Garland, C.M. Pettit, D. Roy, Electrochim. Acta 49 (2004) 2623–2635.

[10] L. Wang, X.J. Huang, Electroanalysis 19 (2007) 1421–1428.[11] M.C. Henstridge, R.G. Compton, J. Electroanal. Chem. 681 (2012) 109–112.[12] R. Buyya, D. Abramson, J. Giddy, H. Stockinger, Concurr. Comp.–Pract. Exp. 14

(2002) 1507–1542.[13] W. Sudholt, K.K. Baldridge, D. Abramson, C. Enticott, S. Garic, C. Kondric, D.

Nguyen, Future Gener. Comp. Sys. 21 (2005) 27–35.[14] M. Rudolph, J. Electroanal. Chem. 503 (2001) 15–27.[15] J.B. Ketter, S.P. Forry, R.M. Wightman, S.W. Feldberg, Electrochem. Solid State

Lett. 7 (2004) E18–E22.[16] D. Abramson, B. Bethwaite, C. Enticott, S. Garic, T. Peachey, IEEE Trans. Parall.

Distr. 22 (2011) 960–973.[17] E. Mashkina, A.M. Bond, Anal. Chem. 83 (2011) 1791–1799.[18] P.X. He, L.R. Faulkner, Anal. Chem. 58 (1986) 517–523.[19] N.G. Tsierkezos, U. Ritter, J. Appl. Electrochem. 40 (2010) 409–417.[20] L. Xiao, E.J.F. Dickinson, G.G. Wildgoose, R.G. Compton, Electroanalysis 22

(2010) 269–276.[21] M.A. Choma, M.V. Sarunic, C.H. Yang, J.A. Izatt, Opt. Express 11 (2003) 2183–

2189.[22] T. Kuwana, D.E. Bublitz, G. Hoh, J. Am. Chem. Soc. 82 (1960) 5811–5817.[23] R.L. McCreery, M.T. McDermott, Anal. Chem. 84 (2012) 2602–2605.[24] L.K. Bieniasz, B. Speiser, J. Electroanal. Chem. 458 (1998) 209–229.[25] J.J. Odea, J. Osteryoung, R.A. Osteryoung, J. Phys. Chem.-Us 87 (1983) 3911–

3918.[26] E.G. Jager, M. Rudolph, J. Electroanal. Chem. 434 (1997) 1–18.