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a

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0h

Electrochimica Acta 74 (2012) 16– 22

Contents lists available at SciVerse ScienceDirect

Electrochimica Acta

j ourna l ho me pag e: www.elsev ier .com/ locate /e lec tac ta

lobal Analysis for the measurement of electrochemical parameters withylindrical electrodes

eter J. Mahon ∗,1, William R.C. Phillipswinburne University of Technology, Hawthorn, Victoria 3122, Australia

r t i c l e i n f o

rticle history:eceived 9 January 2012eceived in revised form 22 March 2012ccepted 22 March 2012vailable online 30 March 2012

a b s t r a c t

Global Analysis is applied to simulated cyclic voltammograms obtained at electrodes with cylindricalgeometry to calculate the characteristic transport, kinetic and thermodynamic parameters that describethe electron transfer process. The capacity to directly analyze for various electron transfer models is alsodemonstrated and the results indicate that the accuracy in obtaining the important transport, kinetic andthermodynamic parameters is excellent.

eywords:lobal Analysisylinder electrodeyclic voltammetrylectrochemical kineticsonvolution

© 2012 Elsevier Ltd. All rights reserved.

. Introduction

Global Analysis [1–6] relies on the existence of a unique surfacen (E(t), I(t), cs(t)) space that is independent of the experimentalechnique with the three time, t, dependent parameters being thelectrode potential, E(t), the current, I(t), and the surface concen-ration of the electroactive species, cs(t), respectively. Access to theurface concentration is obtained from a convolution involving theeasured current with a time dependent function derived for the

pecific electrode geometry that is used in the experiment. Theesulting convolved current2 is then combined with the measuredurrent and the applied potential so that the key transport, kineticnd thermodynamic parameters that describe the electron transfer

rocess can be estimated.

One of the acknowledged advantages of this technique beinghat the use of all data obtained from a single cyclic voltammo-ram can reveal the dependence of the electron transfer rate with

∗ Corresponding author at: Faculty of Life and Social Sciences, Swinburne Univer-ity of Technology, P.O. Box 218, Hawthorn, Victoria 3122, Australia.el.: +61 3 9214 4880; fax: +61 3 9819 0834.

E-mail address: pmahon@swin.edu.au (P.J. Mahon).1 ISE member.2 The convolved current has also been referred to as the semiintegral of the cur-

ent for planar electrodes [7] and as the extended semiintegral for other electrodeeometries [8] in circumstances where the surface concentrations can be exactlybtained by convolution of the current with the appropriate g(t).

013-4686/$ – see front matter © 2012 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.electacta.2012.03.117

potential [1–4]. Moreover, there has recently been much con-jectured on the interpretative framework when investigating thequantitative nature of this relationship [9–14]. It is noteworthythat all these investigations are based on comparisons of simulatedresponses derived from an assumption about the potential depen-dence of the electron transfer rate whereas Global Analysis willdirectly generate this information. While the comparative methodhas its merit, the Global Analysis method has been proven to bemore objective than trial and error based comparisons [2].

Global Analysis has been successfully demonstrated for planar[1] and spherical [3–5] electrode geometries where uniform sur-face concentrations arise due to the uniform accessibility of theelectroactive species to all regions of the electrode surface. It hasalso been applied to the disk [6] electrode geometry where the cir-cumstances that result in non-uniform surface concentrations ledto some inaccuracies. In this paper we will address the cylindricalelectrode, which also has the property that the access to the sur-face is uniform if the ends are appropriately capped to ensure thatpurely radial diffusion results.

The theory of cylindrical transport can be straightforwardlydescribed on the basis of a one-dimensional transport model how-ever analytical resolution is problematic and numerical solutionsneed to be employed [15–18]. Recent developments in this area [19]have enabled relatively accurate formulas to be obtained that can

be applied for the purpose of Global Analysis at cylindrical or wireelectrodes and demonstrating the application of these formulas isone of our objectives. Our approach is based on the developmentof formulas fitted to numerical data derived from evaluating well

ectrochimica Acta 74 (2012) 16– 22 17

ki

g

wctJoesicPnta

staa(ysm

2

sra

S

Tsac

tcbaa[

wakatba

wg

Table 1A list of important polynomial coefficients required for Eqs. (8), (9), (13) and (14).

j a∗j

b∗j

1 −0.49189591 −0.083742030842 5.9364142 −0.015695228203 −11.996254 −0.0077923713074 13.793311 −0.0037163816205 −4.9158711 −0.0025132433446 −6.6424451 −0.0015476542407 4.9968820 −0.0011804745968 5.0561800 −0.00082090563249 −6.9447112 −0.0006676990269

r 2� 4�2 ⎞

P.J. Mahon, W.R.C. Phillips / El

nown but difficult integral equations [20,21]. The relevant integraln this circumstance being

(t) = 4�2

∫ ∞

0

e−tu2

J21 (u) + Y2

1 (u)

du

u(1)

hich represents the diffusion of heat from an infinitely longircular cylinder [22] but has also found application in chronopo-entiometry [23,24]. In this equation, u is the integration variable,1(u) and Y1(u) are first order Bessel functions of the first and sec-nd kind, respectively. The numerical results were obtained bymploying generalized Laguerre–Gauss quadrature specific to theemi-infinite interval [0, ∞) [25], which is most appropriate for thisntegral compared to previous evaluations [17,26] and the resultsoncur precisely with values recently published by Britz et al. [17].hillips and Mahon [19] further provide accurate (within 0.1% of theumerical results) approximate solutions that have the advantagehat they can be straightforwardly integrated, which is required forpplication in the convolution algorithm [27] as described later.

In this paper Global Analysis will be applied to a wide array ofimulated cyclic voltammograms obtained using a cylindrical elec-rode for parameters that enable a range of diffusion conditionsnd kinetic parameters to be fully characterized. The simulationsre obtained using the commercial DigiElch simulation packagehttp://www.elchsoft.com) [28–32]. The application of Global Anal-sis will therefore provide a valuable test of the accuracy of thisimulator and we will also examine the accuracy of the kineticodels that are available within the software.

. General theory

If we consider the following electrochemical reaction in which aolution soluble substrate, S, diffuses to an electrode surface and iseduced or oxidized to produce a product, P, which can then diffuseway from the electrode.

− ne− � P (2)

he number of electrons transferred between electrode and sub-trate is given by n, an integer of either sign that has the same signs the current so that the direction of the reaction and sign of theurrent are in accord with IUPAC convention [33].

There has been much recent activity [9,11–13,34,35] regardinghe quantitative nature of the reaction rate for process (2) in electro-hemical systems with the focus on applying the model describedy Chidsey [36] based on earlier descriptions of Marcus [37,38]nd Hush [39,40]. This has been contrasted with the conventionalpproach based on the phenomenological Butler–Volmer model41,42]. Their common origin begins with

I(t)nFA

= rate = kf csS(t) − kbcs

P(t) (3)

here F is Faraday’s constant, A is the electrode area, csS(t) and cs

P(t)re the surface concentrations of the substrate and the product withf and kb being potential dependent rate constants for the forwardnd backward processes. The exact form of kf and kb is where thewo electron transfer models differ and this will be detailed laterut at equilibrium, the current will be zero and the Nernst equation

pplies so that

kf

kb= exp

{nF(E(t) − Eo′

)RT

}(4)

here Eo′is the formal (or conditional) potential, R is the universal

as constant and T is the absolute temperature.

10 2.2120080 −0.0004987783596

2.1. Convolutions to obtain surface concentrations at acylindrical electrode

As described earlier, the surface concentration of the substratecan be obtained from the current when it is convolved with anappropriate function, g(t), that is specific for a particular electrodegeometry

csS(t) = cb

S − I(t) ∗ g(t)

nFA√

D(5)

where D is the diffusivity (or diffusion coefficient) of S, which wewill assume to be the same for the substrate and the product andcb

S is the bulk concentration of the substrate. The asterisk symbolrepresents the convolution operation and has units of time; it isequivalent to the following integral

I(t) ∗ g(t) =∫ t

0

I(�)g(t − �)d� (6)

An analogous equation can be written for the product under thecondition that the product is initially absent from solution, whichis very common in practice

csP(t) = I(t) ∗ g(t)

nFA√

D(7)

The g(t) function for the cylinder has been derived [19]; it is abipartite expression that approximates equation (1) and has thefollowing form at short times

g(t) =√

D

r

⎛⎝ 1√

��− 1

2+ 3

4

√�

�+ �̄� + 4

�2

10∑j=1

b∗j sin

(j��

2

)⎞⎠(8)

where r is the radius of the cylindrical electrode, � is the dimen-sionless time parameter equal to Dt/r2 with �̄ = − 0.16321 and thecoefficients b∗

j are listed in Table 1. The long-time expression is

g(t) =√

D(

1 − ln 4� − �

−4 + �2 + 4(ln 4� − �) − 6(ln 4� − �)2

32�3+ 4

�2

10∑j=1

a∗j �−(j+2)⎠ (9)

with � being Euler’s constant equal to 0.5772156649. . ., the a∗j coef-

ficients are also listed in Table 1 and the transition from the shortto long time formula occurs at � = 2.000, where they coincide.

1 ectroc

2

i

M

sI

wmf

ct

M

w∫Gv

G

a

G

wa7cbn

i

avtfwti

8 P.J. Mahon, W.R.C. Phillips / El

.2. Global Analysis

A simplifying notation is possible by applying the following def-nitions

(t) = I(t) ∗ g(t) = nFAD1/2∣∣cs

i (t) − cbi

∣∣ , i = S or P (10)

o that the convolved current is identified by M(t) (see footnote 2).t is now possible to combine Eqs. (3)–(5), (7) and (10) into

I(t)√

D

kf+ M(t)

[1 + exp

(nF(Eo′ − E(t))

RT

)]= M (11)

here M is a constant equal to nFAcbS

√D. Eq. (11) contains the two

easured quantities I(t) and E(t), as well as M(t) which is derivedrom the current via the convolution in Eq. (10).

The convolved current is readily calculated using an efficientonvolution algorithm applicable to data with regularly spacedime intervals [27]

(t) = [I(t) ∗ g(t)]t=J� = 1�

⎛⎝IJG1 +

J−1∑j=1

IJ−j(Gj−1 − 2Gj + Gj+1)

⎞⎠(12)

here � is a small increment of the total time, G(t) =t

0

∫ �

0g(s) dsd� and the abbreviations IJ and GJ refer to I(J�) and

(J�). The double integral of g(t) at short times for use in the con-olution algorithm is obtained from

(t) = r3

D3/2

(4�3/2

3√

�− �2

4+ �5/2

5√

�+ �̄�3

6

+ 4�2

10∑j=1

4b∗j

(j�)2

[j��

2− sin

(j��

2

)]⎞⎠ (13)

nd for the long time series

(t) = r3

D3/2

(0 + 4�(ln 4� − � − 1) + 2(1 − �) ln 4� + (ln 4�)2

8

+1 + (7 − 6�) ln 4� + 3(ln 4�)2

32�+ 4

�2

10∑j=1

a∗j �−j

j(j + 1)

⎞⎠ (14)

ith transition occurring when � = 2.739. The coefficients 0nd 1 are integration constants that are equal to −�/10 and–(3 + �2)/2–7� + 3�2, respectively. The application of G(t) in theonvolution algorithm requires an estimate of D and r but they cane combined in the following way so that only a single parametereeds to be provided

=√

D

r(15)

n this way � = 2t and the factor r3/D3/2 in Eqs. (13) and (14) is 1/3.Global Analysis requires that two sets of I(t) and M(t) data are

vailable for a single value of potential and cyclic voltammetry pro-ides the simplest potential waveform where two values of I(t) (andherefore M(t)) can be obtained at the same potential during theorward and reverse scan of potential. When E(t) during the for-ard and reverse scans has a common value, designated as E, then

wo versions of Eq. (11) can be written where the forward arrow

ndicates values obtained during the forward scan at a particular E

�I√D

kf+ �M

[1 + exp

nF(Eo′ − E)RT

]= M (16)

himica Acta 74 (2012) 16– 22

and a backward arrow refers to values obtained during the reversescan

�I√

D

kf+ �M

[1 + exp

nF(Eo′ − E)RT

]= M (17)

At this given potential, the simultaneous equations (16) and (17)can be rearranged to eliminate kf.

�I − �I�M�I − �M�I

= exp(nFEo′/RT)

Mexp(−nFE

RT

)+ 1

M(18)

When written in this way it is clear that regression analysis of theleft hand term versus exp(− nFE/RT) for a set of potentials will pro-vide a linear equation of best fit that has an intercept equal to 1/Mwith a slope exp(nFEo′

/RT)/M. The value of Eo′is readily obtained

from the slope and the intercept. The calculated value of M can beused along with the known values of the cylinder length, h, cb andr to refine the input parameter with the convolution procedurerepeated until the recovered value of is unchanged from the valueused as input.

= M

n2�hr2Fcb(19)

The reciprocal of the left hand term in Eq. (18) is the valuethat the convolved current (i.e. proportional to the concentrationexcursion) would be in the absence of kinetic influence.

M∗ =�M�I − �M�I

�I − �I(20)

An alternative rearrangement of Eqs. (16) and (17) provides

kf =√

D( �M�I − �M�I)

M( �M − �M)(21)

and if a value of kf is calculated at each value of potential tra-versed during the cyclic voltammogram then it is possible that thepotential dependence of the forward rate of electron transfer can bedirectly determined. Analogous to Eq. (20), Eq. (21) can be writtenin a form that predicts the current that would result in the absenceof transport impediments

I∗ =�M�I − �M�I�M − �M

(22)

2.3. Kinetic models

It is now necessary to consider the explicit dependence of kf withpotential and the potential dependent rate constant according tothe Butler–Volmer formulation is described as

kf = ko exp

{�BVnF(E − Eo′

)RT

}(23)

where ko is the reference value of the heterogeneous rate constantat Eo′

. If the forward process is a reduction (i.e. n = −1), then �BV willbe equal to ˛, the reductive transfer coefficient. Conversely, if theforward process is an oxidation (i.e. n = 1), then �BV will be equal to1 − ̨ or ˇ, the oxidative transfer coefficient. This equation can berewritten as

ln(kf ) = ln(ko) + �BVnF(E − Eo′)

RT(24)

so that linear regression analysis of ln(kf) versus (E − Eo′) will pro-

vide an equation with an intercept of ln(ko) and a slope of �BVnF/RT.

Adaption of the more advanced Marcus–Hush theory to the

Butler–Volmer model predicts that �BV is also potential dependentand this has been confirmed experimentally [43–46]. The straight-forward modification of the Butler–Volmer equation to incorporate

ectroc

te

�

wap

l

Rvb

d

k

w�rp

k)√

�∗

∗)

ws

S

a

S

Ttt

e

Twi

(bean(p

3

b

�

wd

P.J. Mahon, W.R.C. Phillips / El

his subtle feature [2] is accomplished by substituting the followingquation into Eq. (24)

BV = �oBV + ı(E − Eo′

) (25)

here �oBV is the reference value of the transfer coefficient at Eo′

nd ı is a constant equal to d�BV/dE. An analogous treatment thatroduced Eq. (24) results in the following quadratic equation

n (kf ) = ln (ko) + �oBVnF(E − Eo′

)RT

+ ınF(E − Eo′)2

RT(26)

egression analysis based on finding the best parabolic fit of ln(kf)ersus (E − Eo′

) enables estimates of ln(ko), �oBVnF/RT and ınF/RT to

e obtained.The Chidsey formulation of Marcus–Hush for the potential

ependent rate is described in the following way [9,36]

f = ko

∫ ∞−∞ exp

{−(

2x−�∗+nF(E−Eo′)/RT

2√

�∗

)2}

dx1+exp{2x}

∫ ∞−∞ exp

{−(

2x−�∗2√

�∗

)2}

dx1+exp{2x}

(27)

here x is the integration variable and �* is equal to �/RT with being the molar reorganization energy. This formulation hasecently been examined by Oldham and Myland [35] and they pro-osed the following simplifying approximation

f = ko

∑∞j=0(−)j

[eerfc

{(j + 1

2 )√

�∗ − nF(E−Eo′)/RT

2√

�∗

}+ eerfc

{(j + 1

2

2 exp

{[nF(E−Eo′ )/RT]2

4�∗ − nF(E−Eo′ )/RT2

}S (�

here the integrals have been replaced by summations and theummation, S(�*), is approximated as follows for �* ≤ �

(�∗) = 12

[1 − �∗

5 + 3�∗2

65 − 141�∗3

9139 + 3�∗4

189 − �∗5

1670

](29)

nd for �* ≥ �

(�∗) = 12

√��∗

[1 − �2

5�∗ + 3�4

65�∗2 − 141�6

9139�∗3 + �8

189�∗4 − �10

1670�∗5

](30)

he symbol eerfc in Eq. (28) is the product of an exponential func-ion and a complimentary error function [47], which correspondso

erfc(y) = exp(y2)erfc(y) (31)

he approximate function S(�*) was found to generate valuesithin 2% of the corresponding summation results and the remain-

ng components of Eq. (28) can be considered as exact [35].The potential dependent kf values that are calculated using Eq.

21) were subjected to weighted complex non-linear optimizationased on the Marquardt algorithm [48,49] to estimate the param-ters ko and �* using Eq. (28). The weighting for this data was alsopplied for the regression analysis of Eqs. (24) and (26) and an alter-ative weighting function was applied to the data obtained from Eq.18). This is in accordance with the weighting scheme that has beenreviously described [4,6].

. Simulations

The dimensionless curvature of the cylindrical electrode, �, cane parameterized√

RTD

= |n| F�r2

(32)

here � is the scan rate (V s−1). In addition, it is usual to define theimensionless rate parameter, , for linear sweep voltammetry to

himica Acta 74 (2012) 16– 22 19

+ nF(E−Eo′)/RT

2√

�∗

}](28)

describe the degree of reversibility of the electron transfer process[50]

= ko

√RT

|n| F�D(33)

with quasireversibility observed for 10−3 ≤ ≤ 15. This normaliza-tion can also be applied to the potential dependent rate parameterkf to obtain the dimensionless potential dependent rate parameterf.

Simulations of cyclic voltammograms based on cylindricalelectrodes were obtained using the electrochemical simulationsoftware DigiElch Ver 4.0 (http://www.elchsoft.com) [28–32]. Thedefault model parameters were used and the electrochemicalparameters were selected so that a range of geometric curvatures,rate parameters and reorganization energies could be examined.Due to slow electron transfer dilating the cyclic voltammograms,the initial and switching potentials were varied so that the poten-tial range was extended at least 150 mV beyond the potentials thatcorrespond to the peak currents. The Global Analysis procedure wasapplied to these simulated cyclic voltammograms.

4. Results and discussion

The results for the application of Global Analysis where thecyclic voltammograms were generated based on Butler–Volmer

kinetics and this framework was also used as the kinetic model inthe analysis are given in Table 2. Generally the outcomes are excel-lent with the analysis being able to recover the input parametersand this implies that the g(t) functions developed for cylindricalelectrodes are sufficiently accurate for this purpose. One of the lim-itations of Global Analysis when applied to cyclic voltammogramsbased on Butler–Volmer kinetics for near reversible reactions is thatthe differences between the convolved currents during the forwardand reverse branches are very small due to the lack of kinetic hys-teresis [5]. This results in large inaccuracies when using Eq. (21)due to the difference term in the denominator. The results for the = 10 case can also be improved if the default model parametersin DigiElch are altered to increase the number of grid points ordecrease the relative truncation error but this is for expert users toinvestigate.

Table 3 contains the corresponding analysis where the cyclicvoltammograms were generated using the Marcus–Hush kineticformalism, as programmed in the DigiElch simulator, with thismodel, as described by Eq. (28), being used to recover the kineticparameters. Again the outcomes are generally very good andimprove with an increasing degree of quasi-reversibility indicatingthat the approximate Eq. (28) is sufficiently accurate for use in theoptimization algorithm and that the Marcus–Hush implementationin DigiElch is also reliable.

It is also apparent that Global Analysis is very sensitive forfast electron transfer reactions (i.e. large ) with small reor-ganization energies and the reason for this is contained inFigs. 1 and 2. These figures contain simulated cyclic voltammo-grams produced by DigiElch for the two different electron transfermodels based on the listed parameters. In addition, the convolvedcurrents resulting from the application of the convolution algo-rithm to the corresponding cyclic voltammograms are shown. The

potential dependent rate for each voltammogram is also displayed.Fig. 1 shows that for small reorganization energies the convolvedcurrents are not coincident at extreme potentials and the differ-ences between the forward and reverse scans are significant at all

20 P.J. Mahon, W.R.C. Phillips / Electrochimica Acta 74 (2012) 16– 22

Table 2Results from applying Global Analysis to a range of simulated cyclic voltammogramscalculated based on the Butler–Volmer model.

Curvature factor Parameter = 10 = 1 = 0.1 = 0.01

� = 0.005 Eo′ (V) 0.00001 0.00001 0.00000 0.00000109 D (m2 s−1) 0.9997 0.9997 0.9998 0.9999

Linear 9.98 0.998 0.0999 0.00998�BV 0.499 0.500 0.500 0.500

Quadratic 9.97 0.998 0.0999 0.00999�o

BV 0.499 0.500 0.500 0.500ı (V−1) −0.033 −0.002 0.000 0.000

� = 0.05 Eo′ (V) 0.00000 0.00001 0.00000 0.00000109 D (m2 s−1) 0.9991 0.9991 0.9992 0.9992

Linear 10.03 0.999 0.0999 0.00998�BV 0.499 0.500 0.500 0.500

Quadratic 10.00 0.999 0.0999 0.01000�o

BV 0.499 0.500 0.500 0.500ı (V−1) −0.075 −0.006 −0.001 −0.001

� = 0.5 Eo′ (V) 0.00001 0.00000 0.00000 0.00000109 D (m2 s−1) 0.9975 0.9954 0.9954 0.9955

Linear 10.32 1.005 0.1000 0.00997�BV 0.497 0.500 0.500 0.501

Quadratic 10.23 1.002 0.1000 0.01002�o

BV 0.498 0.500 0.500 0.499ı (V−1) −0.219 −0.018 −0.004 −0.004

� = 5 Eo′ (V) 0.00000 0.00001 0.00002 0.00003109 D (m2 s−1) 0.9971 0.9974 0.9977 0.9973

Linear 9.92 0.999 0.1000 0.01001�BV 0.501 0.500 0.500 0.500

Quadratic 9.95 0.999 0.0998 0.00995�o

BV 0.501 0.500 0.500 0.501ı (V−1) 0.083 0.006 0.004 0.003

� = 50 Eo′ (V) 0.00001 0.00001 0.00002 0.00002109 D (m2 s−1) 0.9994 0.9996 0.9996 0.9996

Linear 9.96 0.998 0.0999 0.01000�BV 0.500 0.500 0.500 0.500

Quadratic 9.97 0.998 0.0998 0.00995�o

BV 0.500 0.500 0.500 0.501ı (V−1) 0.019 0.004 0.002 0.002

Linear and Quadratic refer to the use of Eqs. (24) and (26) as the basis for the respec-ta

pcado[

TREM

T

Pote ntial /V

0.40.20.0-0.2-0.4

ln( Λf)

-15

-10

-5

0

5

Curr

ent

/ μA

-20

-10

0

10

20

30

40 λ* = 1

λ* = 10

λ* = 33

BV

Co

nv

olv

ed C

urr

ent

/ μA

s-1

/2

0

5

10

15

20

Fig. 1. Simulated cyclic voltammograms, convolved currents and dimensionlessrate parameters for a range of kinetic conditions where the substrate concentration

ive regression analyses to obtain these parameters. Simulation input parametersre: Eo′ = 0.00000 V, D = 1.0000 × 10−9 m2 s−1, �o

BV = 0.500 and ı = 0.000 V−1.

otentials. The implication for this observation is that the appli-ation of Global Analysis to systems that are close to reversiblend have small reorganization energies could directly reveal kineticetails that have been difficult to observe using other meth-

ds where the electron transfer rate must be assumed a priori11,12,14].

able 3esults from the optimized fitting of Eq. (28) for the kinetic data calculated usingq. (21) for a range of simulated cyclic voltammograms calculated based on thearcus–Hush model.

Parameter = 10 = 1 = 0.1 = 0.01

� = 0.05 9.99 0.997 0.0997 0.00997�* = 1 �* 1.039 1.003 1.007 1.006�* = 10 10.03 0.999 0.0998 0.00998

�* 11.18 9.98 10.01 10.01�* = 33 10.06 1.006 0.1005 0.01005

�* 47.43 33.56 33.04 33.01�* = 80 10.01 1.002 0.1002 0.01002

�* 237.2 83.19 80.18 80.02� = 5 9.93 0.997 0.0997 0.00997�* = 1 �* 0.970 0.998 1.006 1.007�* = 10 9.91 0.997 0.0998 0.00998

�* 8.97 9.85 10.00 10.00�* = 33 10.02 1.005 0.1005 0.01005

�* 24.93 32.17 32.80 32.95�* = 80 10.00 1.002 0.1002 0.01002

�* 45.69 75.40 78.65 79.43

hermodynamic and transport parameters are given in Tables 4 and 5.

is 1 mM, � = 0.05 and = 0.1.

Moreover, it is also apparent from a comparison of Figs. 1 and 2that the increase in geometric curvature corresponds to an increasethe kinetic sensitivity and this is most obvious when comparingthe kinetically limited current for �* = 1 in Fig. 2 with the corre-sponding voltammogram in Fig. 1. This is similar to other electrodeshapes, such as the sphere and disk where the maximum sensitivityoccurs under steady-state conditions [9,51] where the forward andreverse scans of a cyclic voltammogram are coincident. However,the cylinder does not reach a true steady state [52] and this is anadvantage when applying Global Analysis, which relies on differ-ences in current between the forward and reverse scans of a cyclicvoltammogram.

It is also of interest to see how the more physically relevantMarcus–Hush behavior may have manifested itself in voltam-mograms that are then interpreted based on the traditionalButler–Volmer kinetic model [9]. Tables 4 and 5 contain data fromthe application of Global Analysis based on Butler–Volmer to cyclicvoltammograms generated using the Marcus–Hush kinetic model.In general, the thermodynamic and transport parameters obtainedby regression analysis of Eq. (18) are accurately recovered indicat-ing the robust nature of this component of Global Analysis. The datawas obtained with two different curvature factors to see the extentof this parameter for cylindrical electrodes. Apart from the obviousinappropriateness of the linear analysis, the most noticeable effectis that the value of ı decreases with a decrease in reversibility forthe largest curvature (i.e. � = 5) but seems to have a maximum in theintermediate range of values examined for � = 0.05. Furthermore,for larger reorganization energies the value of ı is independent

of the rate parameter and approaches a limiting value as the ratedecreases with the apparent transfer coefficient measured to be 0.5.

P.J. Mahon, W.R.C. Phillips / Electrochimica Acta 74 (2012) 16– 22 21

Table 4Results from applying Global Analysis to a range of simulated cyclic voltammogramscalculated based on the application of the Butler–Volmer model to data created usingMarcus–Hush kinetics for � = 0.05.

� = 0.05 Parameter = 10 = 1 = 0.1 = 0.01

�* = 1 Eo′ (V) 0.00000 0.00001 0.00000 0.00001109 D (m2 s−1) 0.9991 0.9991 0.9991 0.9990

Linear 8.29 0.891 0.1388 0.01803�BV 0.474 0.331 0.124 0.063

Quadratic 9.74 0.959 0.1043 0.01253�o

BV 0.504 0.486 0.339 0.225ı (V−1) 2.030 1.845 0.911 0.475

�* = 10 Eo′ (V) 0.00000 0.00001 0.00000 0.00000109 D (m2 s−1) 0.9991 0.9991 0.9992 0.9992

Linear 9.73 0.962 0.1342 0.03704�BV 0.500 0.476 0.329 0.181

Quadratic 10.00 0.999 0.1005 0.01155�o

BV 0.499 0.500 0.493 0.448ı (V−1) 0.642 0.697 0.651 0.533

�* = 33 Eo′ (V) 0.00000 0.00001 0.00000 0.00000109 D (m2 s−1) 0.9991 0.9991 0.9992 0.9992

Linear 9.93 0.986 0.1078 0.01689�BV 0.499 0.494 0.447 0.377

Quadratic 10.00 1.000 0.0999 0.01003�o

BV 0.499 0.500 0.500 0.499ı (V−1) 0.188 0.255 0.258 0.256

�* = 80 Eo′ (V) 0.00000 0.00001 0.00002 0.00000109 D (m2 s−1) 0.9991 0.9991 0.9992 0.9992

Linear 9.99 0.993 0.1029 0.01222�BV 0.499 0.497 0.478 0.450

Quadratic 10.00 0.999 0.0999 0.01000�o

BV 0.499 0.500 0.500 0.500ı (V−1) 0.041 0.110 0.114 0.114

Linear and Quadratic refer to the use of Eqs. (24) and (26) as the basis for the respec-tive regression analyses to obtain these parameters. Simulation input parametersare: Eo′ = 0.00000 V and D = 1.0000 × 10−9 m2 s−1.

Table 5Results from applying Global Analysis to a range of simulated cyclic voltammogramscalculated based on the application of the Butler–Volmer model to data created usingMarcus–Hush kinetics for � = 5.

� = 5 Parameter = 10 = 1 = 0.1 = 0.01

�* = 1 Eo′ (V) 0.00001 0.00001 0.00001 0.00002109 D (m2 s−1) 0.9971 0.9971 0.9977 0.9978

Linear 7.84 0.888 0.1310 0.01497�BV 0.438 0.301 0.120 0.084

Quadratic 9.56 0.949 0.1017 0.01109�o

BV 0.507 0.479 0.321 0.256ı (V−1) 2.009 1.759 0.748 0.503

�* = 10 Eo′ (V) 0.00001 0.00001 0.00002 0.00002109 D (m2 s−1) 0.9971 0.9972 0.9976 0.9977

Linear 9.53 0.966 0.1775 0.04394�BV 0.500 0.451 0.270 0.159

Quadratic 9.94 0.998 0.1029 0.01256�o

BV 0.500 0.500 0.480 0.425ı (V−1) 0.756 0.701 0.604 0.481

�* = 33 Eo′ (V) 0.00001 0.00001 0.00002 0.00003109 D (m2 s−1) 0.9971 0.9972 0.9976 0.9975

Linear 9.79 0.985 0.1161 0.02084�BV 0.501 0.486 0.429 0.355

Quadratic 9.95 0.999 0.0998 0.01004�o

BV 0.500 0.500 0.500 0.498ı (V−1) 0.333 0.266 0.261 0.255

�* = 80 Eo′ (V) 0.00001 0.00001 0.00002 0.00003109 D (m2 s−1) 0.9971 0.9972 0.9977 0.9976

Linear 9.87 0.993 0.1059 0.01309�BV 0.501 0.494 0.471 0.442

Quadratic 9.95 0.999 0.0998 0.00995�o

BV 0.500 0.500 0.500 0.501ı (V−1) 0.194 0.121 0.118 0.117

Linear and Quadratic refer to the use of Eqs. (24) and (26) as the basis for the respec-tive regression analyses to obtain these parameters. Simulation input parametersare: Eo′ = 0.00000 V and D = 1.0000 × 10−9 m2 s−1.

Potential /V

0.40.20.0-0.2-0.4

ln( Λf)

-15

-10

-5

0

5

Curr

ent

/ μA

0.0

0.5

1.0

1.5

2.0 λ* = 1

λ* = 10

λ* = 33

BV

Convolv

ed C

urr

ent

/ μA

s-1

/2

0.00

0.05

0.10

0.15

0.20

Fig. 2. Simulated cyclic voltammograms, convolved currents and dimensionless

rate parameters for a range of kinetic conditions where the substrate concentrationis 1 mM, � = 5 and = 0.1.

The relationship between ı and �* can be derived based on def-initions given by Weaver and Anson [53] and it follows that

ı = F

4RT�∗ (34)

For the conditions used for the simulation of the voltammograms,this equation becomes ı = 9.73/�* which means for �* = 80 thatı should be 0.121and this is similar to the limiting values inTables 4 and 5. However, the discrepancy becomes much greaterfor smaller values of �* and this is clearly due to the non-parabolicshape for the potential dependent ln (f) curve as displayed inFigs. 1 and 2.

5. Conclusions

The application of Global Analysis to voltammograms obtainedat cylindrical electrodes has been demonstrated over a range of geo-metric and kinetic conditions that are experimentally accessible. Itwas shown to be effective in recovering the parameters used to sim-ulate cyclic voltammograms and has been able to accurately extractkinetic parameters directly from voltammograms simulated basedon the Marcus–Hush electron transfer model. In doing so, the accu-racy of the cylindrical diffusion formulae that are described hasbeen established and the reliability of DigiElch as a simulator has

also been confirmed. It was also demonstrated that the combinationof small reorganization energies and cylindrical electrodes shouldprove to be particularly beneficial when applying Global Analysis.

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eferences

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