Incorporating electrode kinetics into the convolutive modeling of reactions at planar, cylindrical and spherical electrodes

Electrochimica Acta 46 (2001) 953–965

Incorporating electrode kinetics into the convolutivemodeling of reactions at planar, cylindrical and spherical

electrodes

Peter J. Mahon, Keith B. Oldham *Department of Chemistry, Trent Uni6ersity, Peterborough, Ont., Canada K9J 7B8

Received 26 April 2000; received in revised form 14 September 2000

Abstract

Convolutive modeling provides a valuable alternative to digital simulation as a means of predicting the outcome ofa voltammetric experiment, for comparison with the laboratory version. Methods based on either of two conjugatefunctions are described, differing in the direction that the convolution takes. Other than the requirement of uniformaccessibility of the electrode surface, convolutive modeling places few restrictions on the range of experiments thatmay be modeled. The electrode reaction may have any degree of reversibility and may, or may not, be coupled to afirst-order chemical process. The diffusivities of the reactant and product may be equal or unequal. The current maybe the controlled electrical variable, or the current may be monitored in an experiment in which the potential iscontrolled. A range of experimental techniques is addressed in this article, including current-reversal chronopoten-tiometry and cyclic voltammetry without and with a following chemical reaction. Algorithms are reported for eachof the two convolution routes. Examples treated in detail include both planar and spherical diffusion fields. Theheterogeneous rate constant was varied in all instances, reversible, quasi-reversible, and near-irreversible cases beingconsidered. Differences between the predictions of the two routes was found to be insignificant, both of the theoreticalvoltammograms agreeing excellently with analytical formulas, where these are available for comparison. In the caseof quasi-reversible cyclic voltammetry, the prediction of the convolutive model was evaluated by global analysis: theinput parameters were recovered with only minor discrepancies. © 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Voltammetric modeling; Semi-integrals; Convolution integral; Cyclic voltammetry; Chronopotentiometry

www.elsevier.nl/locate/electacta

1. Introduction

We consider the following electrochemical reaction inwhich a solution soluble reactant, R, diffuses to anelectrode surface and is reduced or oxidized there toproduce a product, P, which then diffuses away fromthe electrode.

Rsoln−ne− X Psoln (1)

The number of electrons transferred between the elec-trode and the reactant is given by n, an integer of eithersign. Under conditions in which the rate of reaction iscontrolled by the kinetics of the electron transfer pro-cess and the surface concentrations are uniform, therelationship between the current and the surface con-centrations of the reactant and product is described bythe following form of the absolute rate equation.

I(t)=nFA [kfcRs (t)−kbcP

s (t)] (2)

* Corresponding author. Tel.: +1-705-7481336; fax: +1-705-7481625.

E-mail address: [email protected] (K.B. Oldham).

0013-4686/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0013 -4686 (00 )00680 -0

P.J. Mahon, K.B. Oldham / Electrochimica Acta 46 (2001) 953–965954

The current, I(t), is assigned a sign that is the same asn in accordance with the IUPAC convention, with cR

s (t)and cP

s (t) defined as the time-dependent concentrationsof the reactant and product, respectively, at the elec-trode surface. The symbol A refers to the electrode areaand F is Faraday’s constant.

To facilitate the modeling of the voltammetry it isnecessary to choose a theoretical framework whichdefines the potential-dependent electron-transfer rateparameters kf and kb, where the subscripts f and b,respectively refer to the forward and backward pro-cesses of reaction (1). We will use the Butler–Volmerformulation in which the rate parameters are given by

kf=k° exp!−gnF

RT[E°−E(t)]

"(3)

kb=k° exp!(1−g)nF

RT[E°−E(t)]

"(4)

where the sign of n determines again whether theforward or backward process is a reduction. If theforward process is a reduction, then g will be equal toa, the reductive transfer coefficient. Conversely, if theforward process is an oxidation, then g will be equal to1−a or b, the oxidative transfer coefficient. Thesetransfer coefficients may be weakly dependent upon theapplied potential [1,2] but they are treated here asconstants. In additional, E° is the standard or condi-tional potential, k° is the standard or conditional het-erogeneous rate constant, E(t) is the electrodepotential, R is the gas constant and T is thetemperature.

Generally, the surface concentrations of the reactantand the product during the course of an experiment willbe complicated time-dependent functions which com-bine elements reflecting the nature of the perturbingsignal and the geometry of the reaction space, as well asthe transport mode and electrochemical reactivity ofthe reactant and product. Therefore, an a priori knowl-edge of the surface concentrations necessary to modelthe current generated from reaction (1) via the directuse of Eq. (2) is seldom realized, especially when theelectron transfer is coupled to a homogeneous reaction.Only in some elementary instances, summarized byMacdonald [3], is it possible to linearize the dependenceof current, and its temporal convolute on potential.These linearization procedures are useful for analyzingexperimental linear scan voltammograms, but not forpredictive purposes.

However, procedures based upon extended semi-inte-grals have been developed recently [4,5] that enable awide range of voltammetric situations to be successfullymodeled under electrochemically reversible conditions.Using similar strategies, we will now further developthese procedures to include reactions where the kineticsof the electron transfer step significantly influence theoverall voltammetry.

There are generally three observable variables associ-ated with any electrochemical experiment, namely cur-rent, potential and time. It is common forelectrochemical methods to be categorized by the na-ture of the applied electrical signal and they are there-fore considered as either controlled-potential orcontrolled-current techniques. The second electricalvariable is then measured as a function of time or of theapplied signal. We will model examples from each ofthese two categories: controlled-potential in Section 4and controlled-current in Section 5.

2. The extended semi-integral

Through the Butler–Volmer formulas, Eq. (2) pro-vides the link among the current, the potential and thetwo surface concentrations. An obvious strategy is toseek a substitution that enables the individual surfaceconcentrations to be described as functions of either thecurrent alone or the potential alone. The extendedsemi-integral has this capability; it is linearly related tothe surface concentration and is also related to thecurrent via a convolution integral. However, it is im-portant to note that the approach described in thispaper is only possible when the concentrations of thereactant and product are uniform at the electrode sur-face and is therefore restricted to appropriate planar,spherical and cylindrical electrodes [5].

The relationships between the extended semi-inte-grals, Mi(t), and the surface concentrations of thereactant and the product are [4,5]

MR(t)=nFADR[cRb −cR

s (t)] (5)

MP(t)=nFADP[cPs (t)−cP

b] (6)

The subscripts R and P refer to the reactant and theproduct, D is the diffusivity of the subscripted speciesand the superscript b refers to the bulk concentration ofeach species. It is possible to recast Eq. (2) in terms ofthe extended semi-integral.

I(t)=kf[MR−MR(t)]

DR

−kb[MP+MP(t)]

DP

(7)

where the positive constants MR and MP are defined as

Mi=nFADic ib i=R, P (8)

Commonly, the product is initially absent from thesolution and MP will then be equal to zero.

Alternatively, each M(t) can be described by thefollowing convolutions [6]

Mi(t)=I(t) � gi(t) i=R, P (9)

where the asterisk represents convolution with respectto time. The g(t) function is independent of the re-versibility of reaction (1) and the nature of the excita-

P.J. Mahon, K.B. Oldham / Electrochimica Acta 46 (2001) 953–965 955

tion signal. However, the nature of each g(t) function isdependent upon the presence or absence of chemicalreactions associated with each species and the ge-ometries of the electrode and the diffusion space. Theform of the g(t) function is therefore application spe-cific and many g(t) functions have been developed tocover the most common voltammetric situations [5,6].In the absence of homogeneous chemical complicationsand when the diffusivities of R and P are equal, MR(t)and MP(t) will be identical and considerable simplifica-tion then results. The simplest example of g(t) is for thecase of planar diffusion over an electrode where thereactant and the product are both stable.

g(t)=1

pt(10)

In this case convolution reduces to straightforwardsemi-integration.

Conversely, the current can be expressed in the formof a convolution that contains the extended semi-inte-gral. Separate convolutions exist for the reactant andthe product

I(t)=dMi(t)

dt� hi(t) i=R, P (11)

The h(t) function is the conjugate function [5] of thecorresponding g(t) and therefore it is also dependentupon the presence of homogeneous chemical processesand the nature of the diffusion space. For the casecorresponding to the example given in Eq. (10), aunique result is observed where g(t) and h(t) areidentical.

h(t)=1

pt(12)

We have demonstrated [5] that the convolution inte-grals given in Eq. (11) are more useful in modelingapplications if they are applied in the following equiva-lent way.

I(t)=Mi(t) *dhi(t)

dti=R, P (13)

3. The convolution algorithm

An efficient convolution algorithm has been previ-ously developed which enables the convolute of twofunctions to be calculated [4–6]. That algorithm re-quires that values of the two functions be known atevenly spaced intervals and the double integral of oneof the functions must be calculable for all times. Forthe general case in which the two arbitrary time-depen-dent functions y(t) and z(t) are known at time instantsD, 2D, 3D, …, jD, …, ND, where D is a brief timeinterval and j is an index such that yj=y( jD), theconvolute can be calculated at each point t=JD from

[y(t) * z(t)]t=JD

=1D

�yJZ1+ %

J−1

j=1

yJ− j(Zj−1−2Zj+Zj+1)n

J=1, 2, …,N (14)

Here N is the total number of points and Zj is thedouble integral of the z(t) function with respect to time,evaluated at t= jD

Zj=& jD

0

& t

0

z(t)dt dt (15)

This algorithm is valid only if y0 is zero, as it is in allthe applications discussed here.

Two separate applications of the algorithm can bedeveloped. The first converts the current via g(t) intothe extended semi-integral and is based on Eq. (9)

(MJ)i=1D

�IJ(G1)i+ %

J−1

j=1

IJ− j(Gj−1−2Gj+Gj+1)i

nJ=1, 2, …, N i=R, P (16)

where MJ is the value of M(t) at t=JD and Gj=0

jD 0t g(t)dt dt. The second algorithm enables the con-

version of the extended semi-integral into current viah(t) and is based on Eq. (13)

IJ=1D

�(MJ)i(H1)i+ %

J−1

j=1

(MJ− j)i(Hj−1−2Hj+Hj+1)i

nJ=1,2,…,N i=R, P (17)

where Hj=0jD h(t)dt is the single integral of h(t).

4. Modeling a controlled-potential experiment

Perhaps the most commonly applied form of a con-trolled-potential experiment is cyclic voltammetrywhere the current is recorded as a function of a linearlyvaried potential. However, the theoretical description ofthis experiment is very difficult and only a few limitingcases have been evaluated analytically [7,8] to give I(t)as an explicit function of E(t). It is usual to model thisexperiment using one of the various methods of finitedifferences to obtain a digitally simulated [9,10] form ofthe cyclic voltammogram; this is then compared toexperimental data. The accuracy of a digital simulationtechnique is usually ascertained from a comparison ofI(E) with theory, using the known limiting cases thatare available. Alternatively, for the case of quasi-re-versible electron transfer at electrodes which are planaror spherical, a procedure known as global analysis[2,11–14] exists which uses every potential–currentdata pair from a cyclic voltammogram to calculate thekinetic, thermodynamic and transport parameters thatdescribe the process. This global method is based on aconvolution procedure in which M(t) is calculated from


known values of I(t) using the convolution algorithm(Eq. (16)) and the appropriate g(t) function.

Quasi-reversible cyclic voltammograms will be mod-eled here by the techniques of the present paper. Wewill consider the general methods for modeling a quasi-reversible reaction and describe how some commonconditions enable considerable simplification. It will beseen that there are two routes that enable the current tobe calculated. The accuracy of these modeling proce-dures will be verified by the application of globalanalysis under spherical diffusion conditions. An addi-tional example will be modeled in which the quasi-re-versible electrochemical step is followed by anirreversible homogeneous chemical reaction; this is of-ten designated as an EqCi mechanism.

4.1. Method based upon g(t)

In the more direct method, the functions MR(t) andMP(t) in Eq. (7) are replaced by the appropriate convo-lution algorithm given by the general equation (Eq.(16)). After some rearrangement, the following equa-tion is obtained which enables the current at t=JD tobe calculated from previously calculated values of thecurrent in the interval D5 t5 (J−1)D

IJ=

kfDP

�MRD− %

J−1

j=1

IJ− j(G. j)R

n−kbDR

�MPD+ %

J−1

j=1

IJ− j(G. j)P

nDDRDP+kfDP(G1)R+kbDR(G1)P

(18)

Here we use the abbreviation

(G. j)i= (Gj−1−2Gj+Gj+1)i i=R, P (19)

For the rather common situation in which the condi-tions MP=0 and DR=DP=D can be assumed, so that(Gj)R= (Gj)P=Gj, Eq. (18) simplifies considerably togive

IJ=

kfMRD− [kf+kb] %J−1

j=1

IJ− j(Gj−1−2Gj+Gj+1)

DD+ [kf+kb]G1(20)

Although it is not explicit in Eq. (18) or Eq. (20), thedependence of the current upon the applied potential is

a consequence of the potential dependence of kf and kb

via Eqs. (3) and (4).

4.2. Method based upon h(t)

In the second and less direct route to modeling thecurrent, the calculation of MR(t) or MP(t) is necessaryand then application of the convolution algorithmgiven by Eq. (17) with the appropriate single integral ofthe h(t) function is used to calculate I(t). Once againwe begin with Eq. (7) and substitute the convolutionalgorithm for the reactant given in Eq. (17) for thecurrent function I(t). Next we write an equation inwhich the convolution algorithm for the product, asdescribed by Eq. (17), is substituted for I(t) in aseparate application of Eq. (7). We find that two simul-taneous equations

1D

�(MJ)R(H1)R+ %

J−1

j=1

(MJ− j)R(H. j)R

n=

kf[MR− (MJ)R]

DR

−kb[MP+ (MJ)P]

DP

(21)

and

1D

�(MJ)P(H1)P+ %

J−1

j=1

(MJ− j)P(H. j)P

n=

kf[MR− (MJ)R]

DR

−kb[MP+ (MJ)P]

DP

(22)

result. Here

(H. j)i= (Hj−1−2Hj+Hj+1)i i=R, P (23)

The solution of simultaneous Eqs. (21) and (22) for(MJ)Rand (MJ)P, which are the discretized versions ofMR(t) and MP(t) at t=JD, yields

(MJ)R=kfDP(H1)PDMR−kbDDR

�(H1)PMP+ %

J−1

j=1

(MJ− j)R(H. j)R− %J−1

j=1

(MJ− j)P(H. j)P

n− (H1)PDRDP %

J−1

j=1

(MJ− j)R(H. j)R

(H1)R(H1)PDRDP+kfDP(H1)PD+kbDR(H1)RD(24)

and

(MJ)P=kfDDP

�(H1)RMR+ %

J−1

j=1

(MJ− j)R(H. j)R− %J−1

j=1

(MJ− j)P(H. j)P

n−kbDR(H1)RDMP− (H1)RDRDP %

J−1

j=1

(MJ− j)P(H. j)P

(H1)R(H1)PDRDP+kfDP(H1)PD+kbDR(H1)RD(25)


In the rather common situation where MP=0 andDR=DP=D can be assumed, so that (Hj)R= (Hj)P=Hj, Eqs. (24) and (25) simplify massively to give thesingular result

MJ=

kfDMR−D %J−1

j=1

MJ− j(Hj−1−2Hj+Hj+1)

H1D+D [kf+kb](26)

4.3. Modeling cyclic 6oltammetry at sphericalelectrodes

It is now possible to turn attention to the modelingof voltammetry at a spherical electrode, the product Pdissolving in the solution. The simplifying conditionsMP=0 and DR=DP=D apply because we will assumethat the product is initially absent from the solutionand has the same diffusivity as the reactant. Furthersimplification occurs because we are modeling a processwithout homogeneous chemical reactions. In this exam-ple, the following conditions apply gR(t)=gP(t)=g(t)and hR(t)=hP(t)=h(t).

4.3.1. Algorithm based on g(t)The appropriate g(t) function is [5,6]

g(t)=1

pt−

D

rexp

!Dt

r2

"erfc

!Dt

r"

(27)

where r is the radius of the spherical electrode anderfc{} is the error function complement [15]. The dou-ble integral of g(t), required for use in Eq. (20), is

G(t)=r3

D3/2

�1−

2r'Dt

p+

Dt

r2 −exp!Dt

r2

"erfc

!Dt

r"n(28)

and the algorithm in Appendix A.2 is sufficient tocalculate the required voltammogram.

4.3.2. Algorithm based on h(t)For an electrode with spherical geometry, the appro-

priate h(t) function is [5]

h(t)=1

pt+

D

r(29)

and the single integral of this function required for usein Eq. (26) is

H(t)=2' t

p+

tD

r(30)

Notice that this method does not require the errorfunction complement. To calculate a cyclic voltam-mogram by this method see the algorithm in AppendixA.3.

4.3.3. Modeling resultsThe extended semi-integrals calculated as functions

of the potential for three different heterogeneous rateconstants are displayed in Fig. 1. As expected for areversible reaction, the extended semi-integral is identi-cal during the forward and backward sweeps in (a). The

Fig. 1. A plot of M(t) calculated with three different values ofthe heterogeneous rate constant (a) L=104 (reversible); (b)L=1.00000 and (c) L=0.100000. Additional parameters in-clude g=0.50000, D=1.000×10−9 m2 s−1, s=0.1, cR

b =1.000 mM, E°=0.000 V, Er=0.250 V, y=1 V s−1, n=1,T=298.15 K, N=501 and D=2.000×10−3 s.

Fig. 2. Cyclic voltammograms corresponding to the conditionsgiven in Fig. 1. The solid circles were calculated based on theg(t) method and the open circles are based on the h(t)method.


influence of decreasing the heterogeneous rate constantis demonstrated for the series (a) through (c). Cyclicvoltammograms modeled using each approach are dis-played in Fig. 2 and no perceptible difference betweenthe two numerical methods can be seen.

4.4. Global analysis

As mentioned earlier, it is possible to verify theaccuracy of the modeling procedure via the applicationof global analysis to recover the input parameters. Theconcepts and application of this analysis have beendescribed previously [2,11–14]. This method involvestwo separate linear regressions in which four quantities,the currents, and extended semi-integrals at a commonpotential encountered during the forward and back-ward branches of a cyclic voltammogram are analyzedin a manner which allows E°, k°, g and D to becalculated.

When Eqs. (3) and (4) are substituted into Eq. (7), anequation results which more explicitly demonstrates therelationship between the current, potential and the ex-tended semi-integrals of the reactant and product whenthe electrode kinetics are important

I(t)DRDP

k°−DPu(t)−g[MR−MR(t)]

+DRu(t)(1−g)[Mp+MP(t)]=0 (31)

where u(t) is the potential-dependent term

u(t)=exp!nF

RT[E(t)−E°]

"(32)

In the rather common situation where MP=0, DR=DP=D, gR(t)=gP(t)=g(t) and MR(t)=MP(t)=M(t), then Eq. (31) simplifies to

I(t)D u(t)g

k°+M(t)[1+u(t)]−MR=0 (33)

If Ib and I� denote the currents encountered at acommon potential, E, during the forward and back-ward sweeps and Mb and M� represent the correspondingextended semi-integrals then Eq. (33) can be written as:

Ib D

kf

+Mb �1+o exp!nFE

RT"n

=MR (34)

I� D

kf

+M� �1+o exp!nFE

RT"n

=MR (35)

where o=exp{−nFE°/RT}. It is possible to combinethese two equations to eliminate kf and this gives

Ib −I�M� Ib −Mb I�

=1

MR

+o exp{nFE/RT}

MR

(36)

If the left-hand quantity is evaluated at each potential,then a regression analysis can be performed based onEq. (36) when the left-hand quantity is plotted againstexp{nFE/RT}. From this MR can be obtained as the

reciprocal of the intercept and E° can then be calcu-lated from the slope and the intercept as follows:

E°=RT

nFln!intercept

slope"

(37)

The diffusivity can be obtained from MR because D=[MR/nFAcR

b ]2 and all the right-hand quantities areknown. Additionally, Eqs. (34) and (35) can be com-bined to give the second regression equation.

ln!D [M� Ib −Mb I� ]

MR[M� −Mb ]"

= ln{k°}−gnF [E−E°]

RT(38)

A plot of the left-hand term of Eq. (38) versus −nF [E−E°]/RT permits the calculation of the transfercoefficient from the slope and the heterogeneous rateconstant from the intercept.

There are two important dimensionless parametersthat are often used when describing the electrochem-istry that we are considering. One is the dimensionlessheterogeneous rate constant, L, defined as [16]

L=k°' RT

nFDy(39)

where y is the sweep rate. Quasi-reversible behavior isobserved for a range of about 10−35L515. Theother dimensionless parameter, known as the sphericityfactor, s, describes the degree of convergence in the fluxlines of the diffusive species normal to the electrodesurface.

s=1r'RTD

nFy(40)

An essentially steady-state current will be observed [17]for s\40 and the global analysis procedure will fail inthis range because Ib:I� and Mb :M� . Conversely, avalue of sB0.012 leads to a less than 2% increase inthe forward peak current compared with the planarcase. Under these conditions, the flux lines will beapproximately parallel and the planar diffusion limitresults. We are most interested in values of s betweenthese limits.

Table 1 contains the results of applying global analy-sis to cyclic voltammograms modeled using the ap-proaches presented in Appendices A.2 and A.3. Threedifferent values of L have been chosen which lie in thequasi-reversible range and these have been coupled withthree values of s that traverse the range from thenear-steady state to almost planar diffusion regimes.The results indicate that both methods are sufficientlyaccurate to enable comparisons with experimental data.However, the method based upon g(t) appears to besuperior to the less direct method which uses h(t) andthe extra computation is most likely responsible for theslight errors that have been detected. It is interesting tonote that the evaluation of the special function erfc{}that appears in Eqs. (27) and (28) as part of g(t) and itsdouble integral does not contribute any significant error


Table 1Parameters recovered by global analysis from cyclic voltammograms a modelled using the convolution algorithms based on the g(t)and h(t) functions

Sphericity s=0.1s=1 s=0.01

h(t) g(t) h(t)g(t) g(t)Method h(t)

L=10.00000 10.00000 9.97941 10.00008 9.91344 10.00048 9.893280.50004 0.50000 0.50004 0.49999 0.50004g 0.50000

−0.00001 0.00000 −0.000030.00000 0.00000E° (V) −0.000031.0000109D (m2 s−1) 1.0004 1.0000 1.0009 1.0000 1.0009

0.99978 1.00000L=1.00000 0.999011.00000 1.00000 0.998790.50005 0.50000 0.500220.50000 0.50000g 0.50025

0.00000E° (V) −0.00001 0.00000 −0.00002 0.00000 −0.000021.0002 1.0000 1.0004 1.0000109D (m2 s−1) 1.00051.0000

0.098511 0.100000L=0.100000 0.0999650.100000 0.100000 0.099959g 0.50000 0.49999 0.50000 0.50004 0.50000 0.50005

0.00000 0.00000 0.00000 0.00000E° (V) 0.000000.000001.0001 1.0000 1.0001 1.00001.0000 1.0001109D (m2 s−1)

a Cyclic voltammograms calculated with input parameter values of g=0.50000, E°=0.00000 V and D=1.0000×10−9 m2 s−1.

to the analysis. This indicates that the algorithm [15]which was used to calculate erfc{} is sufficiently accu-rate for this purpose.

4.5. Modeling cyclic 6oltammograms for an EqCi reactionat planar electrodes

In this section we will consider the following reactionmechanism:

Rsoln−ne− X Psoln

Psoln �k

Osoln (41)

where k is the first-order chemical rate constant withunits of s−1. The extended semi-integrals for theproduct and the reactant will differ in this case becausethe chemical process decreases the concentration of theproduct at the electrode surface.

4.5.1. Method using the g(t) functionThe appropriate g(t) functions are [5,6]

gR(t)=1

pt(42)

gP(t)=exp{−kt}

pt(43)

The double integrals of these g(t) functions, requiredfor use in Eq. (18), are

GR(t)=4t3/2

3p(44)

GP(t)=exp{−kt}

k' t

p+� t

k−

12k3/2

nerf{kt}

(45)

where erf{} is the error function [15]. The algorithm inAppendix A.4 is sufficient to calculate the requiredvoltammogram.

4.5.2. Method using the h(t) functionThe appropriate h(t) functions are [5]

hR(t)=1

pt(46)

hP(t)=exp{−kt}

pt+k erf{kt} (47)

and the single integrals of these functions required byEqs. (24) and (25) are

HR(t)=2' t

p(48)

HP(t)=' t

pexp{−kt}+

� 1

2k+k t

nerf{kt}

(49)

Appendix A.5 should be consulted in order to calculatea voltammogram by this method.

4.5.3. Modeling resultsThe importance of the EqCi reaction mechanism,

because of the interesting interplay between the homo-geneous and heterogeneous kinetics, has resulted insome excellent attempts to characterize the generalbehavior of this process [18,19]. The seven-segmentkinetic-zone diagram [18] clearly delineates the condi-tions under which the influence of the two kineticfactors is most significant. In the context of voltamme-try, two important dimensionless kinetic parameters arenecessary when describing this mechanism. For theheterogeneous electron transfer process, L from Eq.


(39) is used and the homogeneous kinetic process isdescribed by another dimensionless parameter

l=RTk

nFy(50)

which contains the chemical rate constant and thepotential sweep rate.

Fig. 3 contains plots of MR(t) and MP(t) calculatedas a function of potential based on the h(t) method andFig. 4 shows the corresponding cyclic voltammogramscalculated using the convolutive modeling methodsbased on either g(t) or h(t). The three separate panelsin each figure typify three of the important scenariospossible within the EqCi reaction mechanism. The situa-tion where l�0 and log L= −0.3 is given in (a); thesevalues are consistent with a pure quasi-reversible (QR)process where MR(t) and MP(t) are virtually identical.The combined effect of the two kinetic processes isshown in (b); the value of the dimensionless parameterscorresponds to the KG region of the kinetic zonediagram of Nadjo and Saveant [19]. The third panel (c)corresponds to the KO region of the kinetic zonediagram where L�� and log l= −0.6, the electro-chemical step is reversible and the overall process isunder mixed chemical-diffusion control.

5. Modeling a controlled-current experiment

In a controlled-current experiment the function I(t)will, of course, be known. The extended semi-integralsMR(t) and MP(t) can then be calculated from theknown I(t) using convolution algorithms based on Eq.(9) with the double integrals of the appropriate g(t)functions. It is therefore possible to numerically modelthe chronopotentiogram by solving Eq. (31) at eachtime increment to find u(t) and then using Eq. (32) tocalculate E(t). However, except for the simple caseswhere g=0, 1/2 and 1, it will be necessary to use aniterative method to solve for u(t) in Eq. (31). It isnoteworthy that many computer software packagessuch as MAPLE V® and MATHEMATICA®, designed toperform mathematical manipulations, can readily solveEq. (31) or Eq. (33) to calculate u(t) with minimalprogramming effort by the user.

5.1. Method based upon g(t)

It is instructive to consider, as an example, the exper-iment of current-reversal chronopotentiometry underplanar diffusion conditions where the theoretical de-scriptions for many electrode kinetic regimes are wellestablished [20,21]. Although the perturbation signalfor this technique involves the application of an appar-ently uncomplicated sequence of constant currents withopposite sign, analytical solutions for this problemappear to exist only for the fully reversible and irre-versible limit [20,21].

In this example, if all the simplifying conditionsincorporated into Eq. (33) are valid, then the appropri-ate g(t) function is:

Fig. 3. A plot of MR(t) for the reactant (solid circles) andMP(t) for the product (open circles) involved in an EqCi

reaction calculated with three sets of values for the dimension-less chemical rate parameter and dimensionless heterogeneousrate constant (a) l=10−4 and L=0.50; (b) l=0.25 andL=0.50; (c) l=0.25 and L=104 (reversible). Additionalparameters include g=0.500, DR=DP=1.000×10−9 m2

s−1, A=2.000×10−6 m2, cRb =1.000 mM, E°=0.000 V,

Er=0.200 V, y=1.000 V s−1, n=1, T=298.15 K, N=801and D=1.000×10−3 s.

Fig. 4. Cyclic voltammograms corresponding to the conditionsgiven in Fig. 3. The solid circles were calculated using the g(t)method and the open circles are based on the h(t) method.


Fig. 5. (a) A plot of M(t) calculated based on g(t) (solidcircles) and based on h(t) (open circles). Parameters includeD=1.000×10−9 m2 s−1, A=2.000×10−6 m2, cR

b =1.000mM, E°=0.000 V, I=50 mA, n=1, T=298.15 K, N=1333and D=1.170×10−5 s. (b) A plot of the percentage error forboth methods with the symbols as for (a).

and the single integral of h(t) required by the convolu-tion algorithm (Eq. (53)) is

H(t)=2' t

p(55)

The algorithm described in Appendix B.2 should besufficient to calculate the chronopotentiogram.

5.3. Modeling results

M(t) is independent of the electrode kinetics becauseit responds solely to the applied current. Fig. 5 containsa plot of M(t) resulting from the application of thecurrent-reversal program. There are actually three over-laid plots in this diagram: they are M(t) calculated byEq. (16) based on g(t), M(t) calculated by Eq. (53)based on the single integral of h(t) and the theoreticalcurve [20,21], M(t)th, described by

M(t)th=2It

pfor t5 tr (56)

M(t)th=2I [t−2t− tr]

pfor t] tr (57)

It is apparent that the two methods based on theconvolution algorithm reproduce the theoretical curvewith imperceptible differences. The lower portion ofthis figure shows the percentage error calculated as thedifference between the modeled data and the theoreticalcurve, which is then scaled to the maximum value, MR,that M(t) can reach. An error of about 1% occurs forthe first point after each current step; this decreasesrapidly to an error of 0.1% or less. The errors for bothconvolution methods are plotted in section (b) of Fig. 5and it is interesting to note that they completely over-lap. It can therefore be concluded that either the convo-lution algorithm can be chosen with confidence andthat the accuracy of the calculation for M(t) will beindependent of this choice.

Chronopotentiograms are plotted in Fig. 6 for vari-ous values of k°, curve A being the reversible wave. Thelower section of this diagram shows the difference inpotential between curve A and the theoreticalchronopotentiogram. The theoretical relationship is:

E(t)th=E°+RT

nFln!'tr

t−1

"for t5 tr (58)

E(t)th=E°+RT

nFln! tr

t−2t− tr

−1"

for t] tr

(59)

The error is generally on the order of 0.1 mV; howeverthis increases to as much as 10 mV when dE(t)/dt isvery large and E(t) approaches infinity at t=0, tr and4tr/3. These are regions in which the experimental dataare already badly corrupted by interference from dou-ble layer charging and larger errors can be tolerated.

g(t)=1

pt(51)

and the double integral of g(t) required by the convolu-tion algorithm (Eq. (16)) is

G(t)=4t3/2

3p(52)

The algorithm in Appendix B.1 is sufficient to calculatethe chronopotentiogram for this case.

5.2. Method based on h(t)

An alternative method based on the convolutiondescribed by Eq. (13) can also be used to calculateM(t). It is possible to arrange the algorithm (Eq. (17))in the following way

(MJ)i=1

(H1)i

�IJD− %

J−1

j=1

(MJ− j)i(Hj−1−2Hj+Hj+1)i

nJ=1,2, …, N i=R, P (53)

where the extended semi-integral at each new time JD iscalculated from I(JD) and the appropriately weightedvalues of M(t) calculated for the previous instantst=D to (J−1)D.

For the chronopotentiometric example where planardiffusion is appropriate

h(t)=1

pt(54)


Curves B and C demonstrate the expected behavior[20,21] for a kinetically controlled reaction. We regardthe model-theory concordances, demonstrated in Figs.5 and 6, as amply validating the proposed convolutivemodeling of controlled-current electrochemical experi-ments.

6. Discussion

One of the advantages of the convolutive modelingprocedure is that there is always more than one routethat can be followed when attempting to model aparticular electrochemical problem. However, factorssuch as speed and accuracy are an ever-present consid-eration when developing a numerical model and someimportant choices within the convolutive modeling pro-cedure may strongly influence these factors.

For a controlled-current experiment there appears tobe little difference in speed and accuracy between themodeling methods based on g(t) and h(t). We havedemonstrated that modeling a controlled-potential ex-periment to calculate the current is intrinsically moreaccurate if the convolution algorithm based on g(t) isused. In the spherical electrode example, when G(t) iscompared with the relatively simple equation for H(t),the time penalty associated with employing the morecomplex expression associated with the g(t) convolu-

tion algorithm could become an important factor whenchoosing between the two methods. However, in thesecond example involving a coupled chemical reaction,the G(t) and H(t) functions were of equal complexityand the more direct method based on the g(t) would befavored in this case.

Figs. 3 and 4 demonstrate clearly the capability ofthe convolutive modeling procedure to numericallymodel complicated electrochemical processes. Consider-ation of this reaction mechanism also demonstrated oneof the great advantages of this modeling technique, thatis, the time required to model each 6oltammogram isindependent of the 6alue of the chemical rate constant.Large chemical rate constants are a severe problem forfinite difference simulations because the reaction layercan become considerably smaller than the diffusionlayer [9,10,22–25]. The difficulty in overcoming thisproblem has been mainly responsible for the number ofdifferent finite-difference simulation methods in use. Toconfront this difficulty, it is usual to increase the num-ber of space elements and/or decrease the size of thetime increment so that the reaction layer is more fullycharacterized; depending on the simulation method, asubstantial increase in the computation time canthereby result.

7. Summary

The powerful convolutive modeling procedure hasbeen extended to treat situations in which the kineticsof the electrochemical reaction are important and al-gorithms for three specific examples have been de-scribed. General strategies were presented which enablemany controlled-potential and controlled-current prob-lems to be modeled at planar, spherical and cylindricalelectrodes.

The modeling of cyclic voltammetry for electrodeswith spherical symmetry was demonstrated and vali-dated via the application of global analysis. In general,excellent accuracy was observed for a wide range ofvalues for the dimensionless kinetic and sphericity fac-tors. It was also demonstrated that the interestinghomogeneous chemical EqCi mechanism could also bemodeled using the convolutive procedures developed inthis paper.

Comparison of the modeled results for current-rever-sal chronopotentiometry with the analytical solutionfor M(t) indicates that there is no significant differencebetween using the convolution algorithms based oneither g(t) or h(t). The iterative procedure to calculateE(t) demonstrated that sufficient accuracy could beobtained when compared to the analytical result at thereversible limit.

Fig. 6. (a) Current-reversal chronopotentiograms calculatedwith three different values of the heterogeneous rate constant(A) k°=104 m s−1 (reversible); (B) k°=5.00×10−4 m s−1

and (C) k°=1.00×10−4 m s−1 with g=0.500 in all cases.(b) A plot of the difference in potential between curve A andthe theoretical response obtained at the reversible limit.


Acknowledgements

The financial support of the Natural Sciences andEngineering Research Council of Canada is greatlyappreciated.

Appendix A

A.1. The potential program for cyclic 6oltammetry

The potential waveform for cyclic voltammetry obeysthe following equation

E(t)=Er−y � t− tr � (A1)

where at time tr, the reversal potential, Er, is reached. Ifthe forward branch is divided into, L time incrementsof duration, D, where L is some large integer, then thediscrete version of Eq. (A1) is

EJ=Er−y � J−L−1 � D (A2)

where J is an index such that EJ=E(JD). The initialpotential E1 will be equal to Er−yLD and should bechosen to be sufficiently negative that negligible currentis generated initially. The total number of points, N, isequal to 2L+1 and the final potential will be encoun-tered when J=N.

A.2. The algorithm for cyclic 6oltammetry based ong(t)

1. Input values of: the nernstian voltage unit, RT/F ;the sweep-rate, y ; the conditional potential, E°; thereversal potential, Er; the electrode radius, r ; thecomposite constant, FAD cR

b ; and the number, n,of electrons transferred between the electrode andthe reactant. Also, input the time increment D andthe number, L, of increments per branch of thecyclic voltammogram. Additionally, the kineticparameters, the heterogeneous rate constant, k°;and the charge transfer coefficient, g ; are required

2. Create storage, each of size N=2L+1, for thefollowing data arrays: E, G, I.

3. For J=1,2, …, N calculate EJ from Eq. (A2) witht=JD.

4. For J=1,2, …, N calculate GJ from Eq. (28).5. For J=1,2, …, N calculate IJ from Eq. (20) using

stored values of GJ and the values of kf from Eq. (3)and kb from Eq. (4) based on values for EJ, recalcu-lating the summation in Eq. (20) for each J.

A.3. The algorithm for cyclic 6oltammetry based onh(t)

To calculate the voltammogram as described in Sec-tion 4.3.2, the algorithm in Appendix A.2 is modified sothat storage for M and H replaces the storage for G instep 2. Steps 4 and 5 are replaced by the following threesteps.4. For J=1,2, …, N calculate HJ from Eq. (30).5. For J=1,2, …, N calculate MJ from Eq. (26) using

stored values of HJ and the values of kf from Eq. (3)and kb from Eq. (4) based on values for EJ, recalcu-lating the summation for each J.

6. For J=1,2, …, N calculate IJ from Eq. (17) usingstored values of HJ and recalculating the summationfor each J.

Although steps 5 and 6 of this algorithm describe theseparate calculations of MJ and IJ, they can be com-bined into a single step so that the summation termcommon to both Eqs. (17) and (26) is calculated onlyonce. MJ is calculated based upon this summation andEq. (26) and IJ is finally calculated using this MJ value,the same summation term and Eq. (17).

A.4. The algorithm for cyclic 6oltammetry of the EqCi

mechanism based on g(t)

1. Input values of: the nernstian voltage unit, RT/F ;the sweep-rate, y ; the diffusivities, DR and DP; theconditional potential, E°; the reversal potential, Er

and the number, n, of electrons transferred betweenthe electrode and the reactant. Also, input the timeincrement D and the number, L, of increments perbranch of the cyclic voltammogram. In addition, thekinetic parameters, the heterogeneous rate constant,k°; the charge transfer coefficient, g ; and k, thechemical rate constant, are required.

2. Create storage, each of size N=2L+1, for thefollowing data arrays: E, (G)R, (G)P, I.

3. For J=1,2, …, N calculate EJ from Eq. (A2) witht=JD.

4. For J=1,2, …, N calculate (GJ)R from Eq. (44).5. For J=1,2, …, N calculate (GJ)P from Eq. (45).6. For J=1,2, …, N calculate IJ from Eq. (18) using

stored values of (GJ)R, (GJ)P and the values of kf

from Eq. (3) and kb from Eq. (4) based on valuesfor EJ, recalculating the summations for each J.

A.5. The algorithm for cyclic 6oltammetry of the EqCi

mechanism based on h(t)

To calculate a cyclic voltammogram by this method,in which MR(t) and/or MP(t) are calculated en route toI(t), the algorithm in Appendix A.4 is modified so thatstorage for MR, MP, (H)R and (H)P replaces the storagefor (G)R and (G)P in step 2. Steps 4, 5 and 6 arereplaced by the following four steps.1. For J=1,2, …, N calculate (HJ)R from Eq. (48).


5. For J=1,2, …, N calculate (HJ)P from Eq. (49).6. For J=1,2, …, N calculate (MJ)R from Eq. (24)

using stored values of (HJ)R and (HJ)P with the valuesof kf from Eq. (3) and kb from Eq. (4) based on valuesfor EJ, recalculating the summations for each J.

7. For J=1,2, …, N calculate IJ from Eq. (17) usingstored values of (HJ)R and recalculating the summa-tion for each J.

In step 6, it is also possible to calculate (MJ)P usingEq. (25) as a route to calculating IJ with (HJ)P requiredin step 7. The short-cut method where MJ and IJ arecalculated in one step can alternatively be performed inplace of step 6 and 7 of this algorithm.

Appendix B

B.1. Iterati6e equation sol6ing

The iterative solution of Eq. (33) at each time instantcan be achieved in many ways. We have chosen to usea straightforward method [15] that initially requires twoestimates, u0 and u1, of u(t) at each time instant. Animproved estimate u2 is then calculated and the processis repeated using u1 and u2 to generate a further improvedestimate u3, and so on, until uj+1 satisfies some pre-scribed solution criterion. Each new estimate uj+1 iscalculated from the previous estimates as follows:

uj+1=uj−1 f(uj)−uj f(uj−1)

f(uj)− f(uj−1)(B1)

where f(u) is the function which requires minimization,in this case

f(u)=I(t)D ug

k°+M(t)[1+u ]−MR (B2)

The rate of convergence is strongly dependent upon theinitial estimates. When the solution of Eq. (33) is requiredas a function of increasing time it is most efficient to usethe value of u that provided the final solution for theprevious time instant as the basis for selecting u0 and u1

for the present time instant. This is incorporated into step13 of the algorithm below.

B.2. Method based on g(t)

The following algorithm can be used to calculate thecurrent-reversal chronopotentiogram. We first calculateM(t) using g(t) in steps 4 and 5.

1. Input values of: the nernstian voltage unit, RT/F ;the current magnitude, I ; the conditional potential,E°; the normalized heterogeneous rate parameter,k°/D; the charge transfer coefficient, g ; the rever-sal time, tr; and the composite constant MR equalto nFAD cR

b . Also input the number of incre-ments, L, into which tr will be divided; tr/L will be

equal to D. The total number of points, N, will beequal to 4L/3 [20,21]. The initial estimates for u0

and u1 must also be input, we chose u0=1 andu1=1.01.

2. Create storage, each of size N, for the followingdata arrays: t, G, M, u.

3. For J=1,2, …, N calculate tJ as JD.4. For J=1,2, …, N calculate GJ from Eq. (52) with

t=JD.5. For J=1,2, …, N calculate MJ from Eq. (16) using

the following criteria for IJ :

If tj5 tr then Ij=I

If tj\ tr then Ij= −ISpecific for current-

re6ersal chronopotentiometry6. For J=1,2, …, N

(A For…Next loop to step 14 to sol6e Eq. (33))Set F=1099: Set x=u0

(and find uJ based on Eq. (B1)).Go to 10

7. If F"1099 Go to 8Set r=u1

Go to 98. Replace r by (Rf−rF)/( f−F)9. Set R=x : Set x=r : Set F= f

10. Calculate f= f(x) from Eq. (B2) with u=x11. If f"0, Go to 7 (Has the 6alue of f(u) con-

6erged to be equal to zero?)12. Set u(JD)=x13. Set u0=x : Set u1=1.01x14. Next J (Increment J and loop back to step 6).15. Stop

The convergence test for the iterative algorithm tofind the best estimate of u(JD) at particular time in-stant JD is based on whether evaluation of Eq. (33)after substitution of u(JD) produces zero as the answer.Computationally, it is possible that due to rounding,this criterion may not be satisfied in some cases and itmay be necessary to set some minimum level thatapproximates zero at the step labeled 11 in the al-gorithm. The number of significant digits carried in thecomputation will determine this limit and in turn, thiswill influence the error associated with the estimate ofu(JD).

B.3. Method based on h(t)

To implement this alternative method requires modifi-cation of the algorithm in B.2 to provide storage for thearray H instead of G in step 2 and the calculation of HJ

using Eq. (55) in place of step 4.

References

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ÌÂ

Å


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Documents

Incorporating electrode kinetics into the convolutive modeling of reactions at planar, cylindrical and spherical electrodes