Voltammetric modelling via extended semiintegrals

Journal of Electroanalytical Chemistry 445 (1998) 179–195

Voltammetric modelling via extended semiintegrals

Peter J. Mahon, Keith B. Oldham *

Chemistry Department, Trent Uni6ersity, Peterborough K9J 7B8, Canada

Received 19 August 1997; received in revised form 22 September 1997

Abstract

The modelling of many voltammetric experiments can be carried out expeditiously by making use of semiintegration or itsconverse, semidifferentiation. The virtue of this approach is that the modelling, be it algebraic, simulative or numerical, takesplace in one dimension only—that of time—rather than in the dual dimensions of space and time. However, the applicability ofpure semiintegration is limited to experiments in which transport is by planar semiinfinite diffusion, preceded by a state in whichno current flows, and without concurrent homogeneous reactions. In this article it is demonstrated that all these limitations maybe overcome by broadening the concept of semiintegration to include other convolutions that reduce to semiintegration in theshort-time limit. Appropriate convolutions are derived for spherical and cylindrical geometries, for thin-layer and Nernst-layerelectrodes, for faradaic processes complicated by homogeneous reactions of the EC, CE and ECE varieties, and for voltammetrypreceded by a steady state, but this list does not exhaust the possibilities. Although controlled-current experiments are mostreadily modelled by the extended semiintegral approach, a powerful procedure is described by which numerical one-dimensionalmodelling is applicable to controlled-potential voltammetry. Three worked examples are presented in detail: constant-currentchronopotentiometry at a wire electrode; a Nernst diffusion layer problem in which the current is shared by a faradaic path andby double-layer charging; and cyclic voltammetry complicated by a following chemical reaction. © 1998 Elsevier Science S.A. Allrights reserved.

Keywords: Voltammetric modelling; Semiintegrals; Constant-current chronopotentiometry; Nernst diffusion layer; Cyclic voltam-metry

1. Background

Semiintegration and its converse, semidifferentiation,have proved to be fruitful tools in predicting the out-come of voltammetric experiments, e.g. [1]. Let us firstreview the reasons for this.

Voltammetry concerns the interrelation of the threeprime electrochemical variables: cell current I, electrodepotential E, and time t. In the vast majority of voltam-metric experiments, one either imposes some simplepotential program and observes the resulting temporalvariation of current, or conversely a current program isimposed through the cell and the consequential evolu-tion of the electrode potential is monitored. The object

of the experiment may be, inter alia, to perform achemical analysis, to measure the kinetics of someprocess, to decipher a reaction mechanism, or purely tosatisfy intellectual curiosity. Irrespective of objective,there needs to be a comparison of the experimentaltemporal response function, I or E, with that predictedby modelling the voltammetry mathematically, if thevoltammogram is to be interpreted quantitatively.

In broad terms, there are three avenues by whichelectrochemical modelling may be accomplished. Themost useful, unfortunately limited to rather simple ex-periments, is the purely analytical route exemplified inthe classic work of Cottrell [2], Ilkovic' [3],Karaoglanoff [4] and others, in which an algebraicequation is derived that directly captures the soughtvoltammetric relationship. The most general modelling* Corresponding author. Fax: +1 705 7481625.

0022-0728/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.PII S 0 022 -0728 (97 )00535 -4

P.J. Mahon, K.B. Oldham / Journal of Electroanalytical Chemistry 445 (1998) 179–195180

Fig. 1. Classical approach to modelling controlled-potential voltammetry. The step labelled (1) in this and other figures is accomplished with theaid of the Nernst or Butler–Volmer equation.

route is simulation, in which the supposed behaviour ofthe electrochemical system, as it evolves in time, ismimicked in some other medium, nowadays digitally ina computer. A third route, which in some ways occu-pies a middle ground between the other two, relies onmathematical analysis of the voltammetric problem toproduces an algorithm for predicting experimental be-haviour and then implements that algorithm numeri-cally. The ground-breaking work of Nicholson andShain [5] is in this third category. For an excellentbibliography of this field see the introduction to thearticle by Pilo, Sanna and Seeber [6].

Irrespective of which of the three modelling routes isfollowed, a mathematical framework is needed onwhich to base the model. The usual framework formodelling voltammetry is illustrated in Fig. 1 for con-trolled-potential experiments and in Fig. 3 for con-trolled-current experiments. These diagrams are equallyappropriate whether the modelling is carried out ana-lytically, simulatively or numerically. Though they areelaborate, these flow-charts are nonetheless simplifica-tions in two respects. First, they relate to those rareexperiments in which only one electroactive species

experiences concentration variation, as is the case forthe

Cu(solid)−2e−�Cu2+(soln) (1)

reaction, rather than the more usual state of affairsrepresented by the generic equation

R(soln)+ne−�P(soln) (2)

in which both the electroreactant R and the product Pof its reduction or oxidation are at concentrations thatvary with time. Second, the procedure in Figs. 1 and 3ignores the usual need to resort to Laplace transforma-tion/inversion in solving Fick’s second law exactly.

In contrast, the flow-charts shown as Figs. 2 and 4for the semiintegral approach are eminently simple.Notice that this framework has completely bypassedthe need to consider the spatial coordinate, therebyeffectively reducing the dimensionality of the problemfrom two to one. It is easy to appreciate qualitativelywhy the distance coordinate can be dispensed with. Incolloquial terms, the electrode surface is ‘where theaction is’. Changes that occur elsewhere in the solutionare caused by, and respond quantitatively to, events at

P.J. Mahon, K.B. Oldham / Journal of Electroanalytical Chemistry 445 (1998) 179–195 181

the surface. The semiintegration and semidifferentiationoperators succinctly capture the interrelationship be-tween what occurs at the electrode surface and theconsequences elsewhere in solution.

Quantitatively, the key equations that forge theserelationships are

nAF(c sR−cb

R)DR=M (3)

and

nAF(cbP−c s

P)DP=M (4)

for the generic electrode reaction represented by Eq. (2)taking place at an electrode of area A. Because wepermit n to have either sign, reaction (Eq. (2)) may beeither an electroreduction or an electrooxidation. InEqs. (3) and (4), the c symbols superscripted b or s

denote concentration in the bulk solution and at theelectrode surface, respectively, while subscripts1

R and P

represent the reactant and product of the electrontransfer reaction2. D denotes the diffusivity (diffusion

coefficient) of the subscript species and F is Faraday’sconstant. One definition of the semiintegral M of thefaradaic current I is via the convolution integral

M=& t

0

I(t−t)� 1

pt

ndt=

& t

0

I(t)� 1

p(t−t)

ndt (5)

but there are others [7]. An alternative representation ofthis convolution integral utilizes an asterisk

M=I �1

pt(6)

This mathematical use of the asterisk must not, ofcourse, be confused with its programming use to signifymultiplication. Notice that bold type is being used toindicate quantities whose values change with time. Thefelicity of semiintegration arises because this simpleoperation is easily applied to the current and producesa quantity M that is linearly related to the concentra-tions of the electroactive species at the electrode sur-face.

Eqs. (5) and (6) shows how I may be converted to Mby the operation of semiintegration. The converse oper-ation, semidifferentiation, is used to determine I fromM, and it is accomplished by carrying out the sameconvolution on M and then differentiating the resultwith respect to time

I=ddt�

M �1

pt

n(7)

Recognize that the convolutions in Eqs. (6) and (7) aredimensionally equivalent to integration with respect totime, so that both these equations confirm that the SIunit of M is A s1/2, while that of I is A. In other words,the � symbol is quite different from an arithmeticoperator in that the symbol itself has the dimensions oftime.

It will be convenient to consolidate Eqs. (3) and (4)into the single expression

−nAF �c si −cb

i �Di=M i=R or P (8)

The negative sign on the left-hand side of Eq. (8) arisesbecause the IUPAC convention assigns a sign to thecurrent (and thereby to M) opposite to that of n. Itcannot be too strongly emphasized that the validity ofEq. (8) is independent of the degree of reversibility ofthe electron transfer reaction and does not depend, inany way, on the nature of the electrical excitation(potential scan, current staircase, etc.) chosen for thevoltammetric experiment.

2. An example

To illustrate the ease with which the procedure ofFig. 4 may be implemented, consider the unfamiliarexperiment in which a L-shaped (or triangular) currentprogram

Fig. 2. Semiintegral approach to modelling controlled-potentialvoltammetry. In this and other figures, step �2 is accomplished withthe aid of Eq. (3) and step �3 by semidifferentiation.

1 ‘R’ is often used to designate the reduced member of a redoxcouple, but this is not necessarily so here. Likewise, when ‘O’ is usedto signify a solute species, it is not necessarily the oxidized member ofa redox couple.

2 As in most of the examples considered in this article, the bulkconcentration cb

P of the product species is often zero.


Fig. 3. Flow chart illustrating the classical approach to modelling controlled-current voltammetry.

I=Ir

tr− �t− tr�tr

(9)

is applied to an electrode at which reaction (Eq. (2))occurs. Such an experiment3 is the controlled-currentanalogue of cyclic voltammetry. Semiintegration of thecurrent [8] gives different formulas, namely

M=4Ir

t3/2

3trpt5 tr (10)

and

M=4Ir

t3/2−2(t− tr)3/2

3trpt] tr (11)

for the periods before and after reversal. Application ofEq. (8) then produces

�c si −cb

i �= −4Ir

t3/2

3nAFtrpDi

i=R or P (12)

before tr and a similar expression with t3/2 augmentedby −2(t− tr)3/2 after the reversal time. These resultsare independent of the kinetics of the R�P reaction.However, if the reaction is reversible, substitution ofthese expressions for the surface concentrations into theNernst equation then yields

E=E1/2−RTnF

ln!3p1/2nAFcb

RD1/2R tr

4Irt3/2 −1"

(13)

again with the same augmentation of the t3/2 term aftertr, as the final expression giving the temporal depen-dence of the potential. The same result can be obtained,though much more laboriously, via the scheme shownin Fig. 3.

3. Extensions

But how is it possible, as Figs. 2 and 4 appear tosuggest, to avoid compliance with Fick’s second law?Of course, it isn’t possible. The point that Fick’s laws,and other conditions, are fundamentally incorporatedinto Eqs. (3) and (4). How this incorporation has comeabout is explained in Appendix A.

From the foregoing, it may appear that the semiinte-gral approach is a veritable panacea for voltammetrists,providing easy solutions to their mathematical prob-lems. Unfortunately, that is not the case. The downsideis that there are a number of stringent conditions thatmust be satisfied to validate Eqs. (3) and (4). Theserequirements, which are inherent in the derivation inAppendix A, include:

(a) that the diffusion be planar, i.e. that the electrodesurface be a plane that is sufficiently large that theequiconcentration surfaces of the electroactive speciesin the vicinity of the electrode are themselves effec-tively planar at all experimental times.(b) that the diffusion be semiinfinite, i.e. that thediffusion zone in front of the electrode be unimpeded

3 Ir must not exceed nAFcbR27pDR/64tr as otherwise the reaction

(Eq. (2)) could not sustain the current.


and unstirred for distances large in comparison withDtmax, where D is the diffusivity of either electroac-tive species and tmax is the duration of the experi-ment.(c) that the electroactive species be homogeneouslyinert, i.e. that neither R nor P are involved in anychemical reaction in the solution phase.(d) that an equilibrium or a null state precede thevoltammetric experiment, so that the concentrationsof R and P are uniform initially.Whereas there are many instances in voltammetric

practice in which all these four conditions are met,there are many more in which they are not. This articleis devoted to explaining how these conditions maysometimes be circumvented. Penalties must be paidwhen any one of the four conditions is relaxed. Onepenalty is that an operation more complicated thansemiintegration is now required to generate a quantitythat is linearly related to the surface electroactive con-centrations via Eq. (8). This quantity will be denoted bya subscripted M symbol, even though it is not a semiin-tegral. A second penalty is that two (usually slightly)different quantities MR and MP, are needed in theformulas, namely


i �Di=Mi i=R or P (14)

that now express c sR and c s

P, rather than one quantitydoing double duty, as in Eq. (8). A third penalty is thatthe quantities analogous to the semiintegral are func-tions not only of time t, but also of other parameters. Afourth penalty is that, whereas there is a simple analogyof semiintegration, there is much greater difficulty inimplementing a semidifferentiation analogue.

In fact, the quantities MR and MP are the results ofconvolving the current I with some time-dependent andelectroactive-species-dependent function:

Mi=I�gi i=R or P (15)

The nature of the function gi, which has a dimensional-ity corresponding to the s−1/2 unit, depends on thecircumstances of the voltammetry, sometimes beingsimple and sometimes complicated. Occasionally anapproximate gi needs to be used. There are instances inwhich the mathematical operations in Eq. (15) can becarried out algebraically; in less tractable cases numeri-cal evaluation will be necessary. In the following sec-tions, cases will be discussed in which Eqs. (14) and (15)apply, and the identity of the gi function will be ex-posed in each case. These functions invariably reduce to1/pt in the t�0 limit. This legitimates our regardingthe quantities MR and MP as extensions of the semiinte-gral.

When the experiment in question is of the controlled-potential variety, the need is to determine the functionI from the known g function and an M which isaccessible via Eq. (14), i.e. analogous to the step la-belled �3 in Fig. 2. This is a deconvolution [9]. Thoughthis operation is accomplished formally via

I{I}=I{M}I{g}

(16)

where I{ } denotes Laplace or Fourier transformation,we are unfortunately unaware of any general nontrans-formational procedure, akin to Eq. (7), by which suchdeconvolutions may be undertaken for the g functionsof interest. Numerical deconvolution by Laplace inver-sion is notoriously difficult and rarely attempted.Voltammetric applications of numerical deconvolutionvia Fourier inversion have been proposed and imple-mented [10]. Later in this article, a numerical methodnot invoking (Eq. (16)) will be suggested by which Imay be found from M and g. This method is general inthe sense that it is not g-specific.

Each of the next eight sections will address a distinctvoltammetric circumstance to which gR and gP func-tions may be assigned. Some of these applications havebeen exposed in an earlier publication [11], where stillothers will be found; most are exposed here for the firsttime, as far as the authors are aware. Each applicationis to some voltammetric circumstance that meets onlythree of the conditions (a)− (d) enumerated above. Forexample, the first three instances violate condition (a).

Fig. 4. Semiintegral approach to modelling controlled-currentvoltammetry. The step labelled �4 is accomplished by semiintegra-tion.


There is no inherent restriction requiring three of thefour conditions to be satisfied; indeed only two weremet in some of the cases considered earlier [11].

4. Convex spherical diffusion

When voltammetry is conducted at the surface of aspherical (or hemispherical) electrode of radius a, withboth electroactive species being dissolved in the solu-tion that surrounds the electrode, then the quantity thatreplaces the semiintegral is defined by

Mi=I�gi with gi=Di

aexp

!Dita2

"erfc

!Dita

"i=R or P (17)

The derivation of this result is in the literature [12] in asomewhat different formulation.

5. Concave spherical diffusion

When metal ions undergo the amalgamation reaction

Mn+(aq)+ne−�M(amal) (18)

at a spherical mercury electrode, the metal atoms dif-fuse into the mercury with a diffusivity DP, whereas themetal ions diffuse in the solution. The ions dissolved insolution obey the equation discussed in the previoussection, but because the diffusion field is of limitedextent inside the mercury electrode, the amalgamatedatoms obey a more complicated relationship. The ap-propriate function for them is

gP=DP

a�

3+2 %�

j=1

exp!−y2

j DPta2

"n(19)

in this case, where yj is the jth positive root of theequation y= tan{y} [13] and a is the radius of themercury sphere. This expression, which was derivedpreviously [12], converges rapidly4 for large values ofDPt/a2, but for small values of this parameter Eq. (19)is computationally inferior to the equivalent expression.

gP=DP

aexp

!DPta2

"erfc

!−DPta

"+ insignificant terms (20)

which gives excellent precision for t50.062a2/DP. Thissolution differs from Eq. (17) only by a sign.

6. Cylindrical diffusion

When reaction (Eq. (2)) occurs at the surface of awire electrode or some other electrode structure ofconvex cylindrical symmetry, the relationship betweenthe surface concentration excursions and the current isonce again subsumed by Eqs. (14) and (15). In Ap-pendix B, it is demonstrated that the appropriate quan-tities that mimic the semiintegral involve convolutionwith a power series in the square-root of time, namely

gi=1

pt−

Di

2a+

3Di

4a2' t

p−

3t8a3 D3

i +…

i=P or R (21)

where a is now the radius of the cylinder. This asymp-totic series converges rapidly if the maximum time ofinterest is sufficiently smaller than the term a2/D, as isusually the case in electrochemical applications.

There is an analogous treatment of a concave cylin-drical electrode that could be applied to a tubularelectrode, the inside walls of which support reaction(Eq. (2)). However, this will not be pursued here.

7. Finite diffusion zone (finite diffusion with a blockedboundary)

As described above under (b), the simple semiintegra-tion formula requires that the x-dimension in the direc-tion fronting the electrode be occupied by solution to asufficient distance that a reservoir of undepleted reac-tant exists throughout the experiment. Here we addressa situation in which this requirement is violated by theplacement of an impermeable wall at a distance d fromthe electrode. Many ‘thin layer’ cells have this configu-ration.

In Appendix C, it is demonstrated that the function

gi=Di

du3�

0;Ditd2

�i=R or P (22)

is appropriate for a reaction taking place at the junc-tion of an electrode with a thin layer of solution. Theu3(0; Dit/d2) function is expressible as either of twoseries

u3�

0;Ditd2

�=

d

pDit

�1+2 %

�

j=1

exp!− j2d2

Dit"n

=1+2 %�

j=1

exp!− j2p2Dit

d2

"(23)

which converge so rapidly that only one summand isever needed if the first representation is used for Dit/d2

less that 1/p and the second when t]d2/pDi. The errorin excluding the j=2, 3, 4,... terms then never exceeds0.0007%.

4 Four summands, involving the values y12=20.191, y2

2=59.68,y3

2=118.9 and y42=198, are sufficient to give MP to a precision of

one part in 3×105 for t]0.062a2/DP.


8. Nernst diffusion layer (finite diffusion with an openboundary)

In applications to membrane-covered electrodes, forexample, diffusion occurs in a restricted zone of lengthd, beyond which exists a region of unchanging ‘bulk’concentration. Such a model, which again violates con-dition (b), is often adopted to provide an approximatedescription of transport mediated by diffusion and con-vection.

In Appendix D, it is demonstrated that the appropri-ate function for a Nernst diffusion layer takes the form

gi=Di

du2�

0;Ditd2

�i=R or P (24)

which differs from Eq. (22) only in the replacement ofthe theta-3 function by theta-2. The latter is anothercomputationally felicitous function, namely

u2�

0;Ditd2

�=

d

pDit

�1+2 %

�

j=1

(− )j exp!− j2d2

Dit"n

=2 %�

j=0

exp!− (j+1/2)2p2Dit

d2

"(25)

Again, if the first representation is used when Dit/d2 isless than 1/p and the second when t]d2/pDi, thenterms beyond j=1 may be ignored.

9. EC reaction

Condition (c) precludes the product P of electrodereaction (Eq. (2)) being involved in any homogeneousreaction in solution. But the semiintegration conceptcan be extended to include the possibility that P isinvolved in an isomerization reaction

P(soln)k?k %

O(soln) (26)

to produce an electropassive isomer O. k and k % are thefirst-order rate constants for the homogeneous inter-conversion reactions. Of course, the transport of thereactant R is unaffected by the occurrence of thisreaction and therefore cR

s obeys the simple Eq. (8).However, the homogeneous reaction modifies the trans-port of P away from the electrode. If it is assumed thatP and O share the same diffusivity D, then it is demon-strated in Appendix E that

−nAF(c sP−cb

P) D=MP=I � gP (27)

where

gP=k %+k exp{− (k+k %)t}

(k+k %) pt(28)

The case where cPb =0=k % was first exposed by

Woodard et al. [14]. An application was described byBlagg et al. [15].

Unimolecular isomerization reactions described byEq. (23) are rare. More common are bimolecular reac-tions such as

P(soln)+X(soln)k¦?k %

O(soln) (29)

between the electroproduct and some species, such asthe solvent or a component of the supporting elec-trolyte, present at an effectively constant concentrationcX

b . The treatment above may be modified to cater tosuch a mechanism simply be replacing k by k¦cX

b , wherek¦ is the second-order rate constant of the forwardmember of scheme (Eq. (29)). A semiintegral approachto the case of catalytic regeneration, in which thechemical step recreates the reactant irreversibly

P(soln)+X(soln)k¦� R(soln)+another product (30)

has been treated in the literature [11].

10. CE reaction

Here we address the case in which a species S presentin the bulk solution is itself electropassive, but is con-vertible via the homogeneous isomerization

S(soln)k?k %

R(soln) (31)

into the electroactive R. Extension to instances inwhich the conversion is bimolecular can be handled asin the preceding paragraph.

Recognize that S and R will be present in the bulksolution at concentrations k %cb/(k+k %) and kcb/(k+k %)respectively, where cb is the total analyte concentration,a more significant experimental quantity than the bulkconcentration of either isomer singly. The treatment ofCE reactions parallels that of EC reactions and it willsuffice to quote the result, namely that

−nAF�

c sR−

kcb

k+k %� D=MR=I � gR (32)

where

gR=k+k % exp{− (k+k %)t}

(k+k %) pt(33)

when S and R are assumed to share the same diffusivityD. Of course cP

s is given by Eq. (8) because the trans-port of the electroproduct P is uninfluenced by thehomogeneous reaction.


11. ECE reaction

It requires no major adaptation of the original semi-integration concept to treat an electrode reaction thatoccurs in two steps. Thus for the EE reaction

R(soln) �n1e−

Q(soln) �n2e−

P(soln) (34)

one has straightforwardly

AF(c sR−cb

R) DR=M1

n1

=I1

n1

� g (35)

−AFc sQ DQ=

M1

n1

−M2

n2

=�I1

n1

−I2

n2

n� g (36)

and

−AF(c sP−cb

P) DP=M2

n2

=I2

n2

� g (37)

where the M ’s are true semiintegrals

Mi=Ii�g=Ii �1

pti=1 or 2 (38)

and the total current is I1+I2. In Eq. (36), it has beenassumed that the intermediate Q is absent from thebulk, as will commonly be the case.

The ECE reaction that is addressed in this section is

R1(soln) �n1e−

P1(soln) ?k R2(soln) �n2e−1

P2(soln)k %(39)

in which the solutes P1 and R2 share the same diffusiv-ity D and are absent from the bulk solution. Thederivation of the results

−AFc sP1D=

MP1

n1

(40)

where

MP1=I1 �k %+k exp {− (k+k %)t}

(k+k %) pt

−I2 � n1k %1−exp {− (k+k %)t}

n2(k+k %) pt(41)

and

−AFcR2s D= −

MR2

n2

(42)

where

MR2= −I1�n2k1−exp{− (k+k %)t}

n1(k+k %) pt

+I2 �k+k % exp{− (k+k %)t}

(k+k %) pt(43)

though quite elaborate, will not be reported because noprinciples beyond those exposed in Appendix E areinvolved. Expressions for cR1

s and cP2s are strictly

analogous to Eqs. (35) and (37), respectively, because

the homogeneous reactions do not influence the trans-port of these species.

12. Preexisting steady state

Previous sections have exemplified violations of con-ditions (a), (b) and (c). Now it is the turn of condition(d) to be violated. Voltammetric experiments generallycommence from an equilibrium or null state, in whichno current flows. Here, however, the experiment ispreceded by a steady state characterized by a constantcurrent I0 and by linear concentration gradients of thereactant

cR=c s0R −

I0xnAFDR

(44)

and product

cP=c s0P +

I0xnAFDP

(45)

in the vicinity of the electrode. The symbol c is0 denotes

the preexisting steady-state surface concentration ofspecies i. Because I0 and n have opposite signs, theconcentration gradients of R and P are seen to berespectively positive and negative.

The derivation in Appendix F shows that a voltam-metric experiment that starts from such a preexistingsteady state obeys the equation

−nAF �c si −c s0

i �Di=M−2I0 ' tp= (I−I0) �

1

pti=R or P (46)

Notice that, in this unusual case, the g function is thesimple 1/pt of conventional semiintegration, but it isthe I term, with which g is convolved, that is subtrac-tively modified.

The bulk concentrations of R and P do not appear inEq. (46). That is because the preexisting linear concen-tration gradients were treated unrealistically as extend-ing indefinitely into the solution. This lack of realismmeans that the equations of this section can be expectedto apply only for rather brief experimental times.

13. Summary

The original semiintegration formula


i � Di=M=I � g=I �1

pt=

d−1/2Idt−1/2

i=R or P (47)

is valid only if all of the four conditions set out earlierare met. However, we have provided ample examples todemonstrate that some of these conditions may often be


relaxed, while retaining the essence of the semiintegra-tion formulation. An M function still applies, though itis now given by a different convolution and it maydiffer between the reactant and product. In the examplein the section immediately preceding this summary, itwas the I component of the convolution that wasmodified, but more usually it is the g component thatchanges:


i �Di=Mi=I�gi i=R or P (48)

The gi function requires customization to the particularcircumstances, though it invariably reduces to 1/pt atshort times.

14. Applications

By their very nature, the semiintegral approach andits extensions cannot be used to solve problems that areotherwise insoluble. Nevertheless, these approaches areuseful in curtailing the effort required and in providinginsight into relationships involved.

Two scenarios can be envisaged for the application ofthe extended semiintegral approach: when the convolu-tion integral I*gi can be evaluated analytically, andwhen this is not feasible. Because I occurs in theintegrand of the convolution, the first scenario is alwaysmore likely in a controlled-current experiment, than incontrolled-potential voltammetry. Numerical evalua-tion of the convolution will be necessary in all but thesimplest applications to controlled-potential voltamme-try.

Three modelling examples will now be presented. Thefirst is an interesting controlled-current problem, whichwill be treated analytically. The second is a moredifficult controlled-current example, in which the con-volution will be carried out numerically. The third is aproblem from cyclic voltammetry, which will be tacklednumerically.

15. Analytical example

From well-established chronopotentiometric princi-ples [16], it is known that when the constant anodiccurrent I is passed through a planar electrode of area Asupporting the one-electron oxidation R(soln)−e−�P(soln), then a transition is encountered at the timeinstant

tplane=pA2F2(cb

R)2DR

4I2 (49)

How is this result changed if the electrode is in theshape of a wire of radius a and length A/2pa?

The transition time corresponds to the surface con-centration of species R having become zero and, ac-cording to Eqs. (48) and (21), this corresponds to

AFcbR D=I�

� 1

pt−

DR

2a+

3DR

4a2

' tp−

3t8a3 D3

R

+ ...n

(50)

Convolution with a constant devolves into ordinaryintegration, so one easily finds

AFcbR D=I

�2' t

p−

t DR

2a+

DR

2a2

't3

p−

3t2

16a3 D3R

+ ...n

(51)

The transition time twire is the value of t that satisfiesthis equation. The right-hand side of this equation is apower series in t. Reverting this series [13 Secn 11:14]leads to

twire= t=a2

DR

�l+

p

4l2+

p−28

l3

−p(18−5p)

64l4 ...

n2

(52)

where l=pAFcbRDR/2aI. For a wire that is not too

thin, this formula may be approximated by

twire=tplane�

1+pDRtplane

2an

(53)

These expressions show how the cylindricity of the wireelectrode augments the chronopotentiometric transitiontime. For DR=10−9 m2 s−1 and a=5×10−4 m, a 10s planar transition time is predicted by the secondformula to increase by 18%, while using Eq. (52) in itsentirety refines this estimate to 19.6%.

16. Numerical implementation

An efficient convolution algorithm was publishedearlier [11] and has found application [12]. It requiresevenly-spaced current data Il, where l=0,1,2,....L, cor-responding to the time instants 0,D,2D,3D,...,lD,...,LD.Here D is a very brief time interval, typically one-hun-dredth or less of the total duration of the experiment.The algorithm, namely


i � Di=Mi=I�gi

=1D

[ILG1+ %L−1

l=1

IL−l(Gl−1−2Gl+Gl+1)i ]

i=R or P (54)

delivers the surface concentration c si of species i, either

the reactant or product, at the time instant t=LD. TheG function is the double integral of the gi function withrespect to time


Gi=& t

0

& t

0

gidtdt (55)

and (Gl)i denotes the value of this function at t=lD.Note that G has a dimensionality corresponding to theunit (seconds)3/2. Eq. (54) forms the basis of our convo-lution algorithm.

It was noted earlier that all g functions reduce to1/pt at short enough times. The instant t=D iscertainly an example of such a short time and thereforeEq. (54) may be rewritten as


i � Di

=4IL

3'D

p+

1D

%L−1

l=1

IL−l(Gl−1−2Gl+Gl+1)i

i=R or P (56)

on replacing G1 in the isolated term by 4D3/2/3p,which is the double integral of 1/pt evaluated at t=D.Eq. (56) may be reformulated as

I=IL= −34

nAF �c si −cb

i � 'pDi

D

−34' p

D3 %L−1

l=1

IL−l(Gl−1−2Gl+Gl+1)i

i=R or P (57)

to provide an explicit expression for the current at timet in terms of prior currents and of the surface concentra-tion of either the reactant R or the product P.

17. Numerical chronopotentiometric example

Here we predict the potential evolution when reaction(Eq. (2)) is reversible and occurs at an electrode that issubjected to an anodic current-step. However, becauseof double-layer charging with a time constant of Y, thefaradaic current is not constant but has a magnitudegiven by

I=I��

1−exp!

−tY"n

(58)

at times subsequent to t=0. In this example, the reac-tant R reaches the electrode through a Nernst diffusionlayer of width d and the electroproduct P leaves by thesame route. For simplicity only, it will be assumed thatboth diffusivities equal D and that P is absent from thebulk solution. These simplifications validate the rela-tionship c s

R+c sP=cb

R and cause Nernst’s law to become

E=E1/2+RTnF

ln!c s

R

c sP

"=E1/2+

RTnF

ln!cb

R

c sP

−1"

(59)

We seek to determine the electrode potential E at timet= LD, for L=1,2,3,.... For this purpose, c s

P in Eq.(59) is replaced by use of the equation

c sP=

−1

nAFD D

�ILG1

+ %L−1

l=1

IL−l(Gl−1−2Gl+Gl+1)n

(60)

which has its origin in Eq. (54). It remains to identifythe Gl functions.

For algebraic felicity, we select the algorithmic smalltime interval as

D=d2

ND(61)

where N is a convenient large integer. From Eqs. (24)and (25), it then follows that

gR=gP=2

plD�1

2+ %

�

j=1

(− )j exp!− j2N2

l

"n(62)

at the time t=lD, when we select the first seriesrepresentation of the theta-two function. At this instantthe current is

Il=I��

1−exp!−lD

Y"n

(63)

In Appendix G, the single and double integrals of thetheta-two function are reported. The double integralwhen t=lD is

Gl=8(lD)3/2

3p

�12+ %

�

j=1

(− )j�1+j2Nl

nexp

!− j2N2

l

"− (− )jj

�32+

j2Nl

n'pNl

erfc!

j'N

l

"n(64)

All the requirements of the algorithmic convolutionare now in place. Numerical values calculated from Eqs.(63) and (64) are substituted into the convolution for-mula (Eq. (60)), whereby the surface concentration of Pis calculated. The potential then follows via Eq. (59).Fig. 5 displays an example of a chronopotentiogramcalculated in this way, and compares it with chronopo-tentiograms of simpler experiments.

18. Numerical cyclo-voltammetric example

When the forward branch is positive-going, the po-tential waveform in cyclic voltammetry obeys the equa-tion

E=Er−n �t− tr�=Er−n �L−N �D (65)

where Er and tr are the reversal potential and reversaltime respectively and n is the unsigned sweep-rate. Ldenotes the serial number of the time datum of interest.Consecutive data are separated by the time interval D,which equals tr divided by a convenient large integer N.No oxidized species is present initially, and the initialpotential Er−ntr is chosen to be sufficiently negativethat the electroreactant R then undergoes oxidation at


Fig. 5. Reversible chronopotentiograms modelled by the procedure described in Section 17. Curves A and C are constant-faradaic-currentchronopotentiograms with diffusion occurring semiinfinitely (curve A) or in a Nernst layer of width d (curve C). Curves B and D represent themodifications to curves A and C, respectively, caused by a finite rise-time Y of the faradaic current. The following parameters were adopted:electrode area=A=7.844×10−5 m2; total current I�=3.000×10−4 A; electron number=n= −1; bulk reactant concentration=cb

R=1.000mol m−3; half wave potential=E1/2=0.000 V, diffusivities DR=Dp=1.000×10−9 m2 s−1; Nernst diffusion layer thickness=d=� (curves Aand B) or 2.236×10−6 m (curves C and D); rise time constant=Y=0 (curves A and C) or 2.000×10−3 s (curves B or D).

a negligible rate. We seek to predict the current engen-dered by such a waveform when the reaction scheme is

R(soln)−e−

?+e−

P(soln)k?k %

O(soln) (66)

consisting of a reversible one-electron oxidation fol-lowed by a bidirectional chemical isomerization.

The potential program conspires with Nernst’s law toenforce a particular ratio, namely

c sP

c sR

=exp! F

RT(Er−E1/2−n �L−N �D)

"=r (67)

of the surface concentrations of the electroactive spe-cies. As shown, we will use r to represent this ratio atthe time t of interest. Each of the individual surfaceconcentrations is related by a distinct convolution to

the same current I : by straightforward semiintegrationfor the electroreactant

AFc sR D=AFcb

R D−I�gR=AFcbR D−I�

1

pt(68)

but by the extended semiintegration formula

AFcPs D=I�gP=I�

k %+k exp{− (k+k %)t}

(k+k %) pt(69)

in the case of the product, as evidenced by Eq. (28).Notice that, because the latter equation assumes acommon diffusivity D for P and its isomer O, we haveadopted this diffusivity for R also. The double integralsof the two g functions, at the time instant t=lD, areeasily found to be

(Gl)R=43'l3D3

p(70)


and

(Gl)P

=4k %3k

'l3D3

p+

kk5/2

�'klDp

exp{−klD}

+ (klD−1/2) erfc{klD}n

(71)

where k=k+k %. Recall Eq. (57) and that it appliesto each of the electroactive species. Hence, simplybe setting n= −1 and cb

P=0, and after some re-organization, the following two expressions arefound to hold for the current: in the present prob-lem

I=IL=34' p

D3

�AF(cb

R−c sR)D D

− %L−1

l=1

IL−l(Gl−1−2Gl+Gl+1)Rn

(72)

and

I=IL=34' p

D3

�AFc s

PD D

− %L−1

l=1

IL−l(Gl−1−2Gl+Gl+1)Pn

(73)

By invoking the definition c sP/c s

R=r, it is possible toeliminate both surface concentration terms from Eqs.(72) and (73), to derive

I=IL=34

(1+r)' p

D3

�rAFcb

RD D

− %L−1

l=1

IL−l{r(Gl−1−2Gl+Gl+1)R

+ (Gl−1−2Gl+Gl+1)P}n

(74)

which is the final result.A summary algorithm to calculate the voltam-

mogram involves the following steps:(1) Input experimental values of: the Nernstian

voltage unit, RT/F ; the sweep-rate, n ; the half-wavepotential, E1/2; the reversal potential, Er; the compositeconstant, AFcb

R D ; and the rate constants k and k %.Also select and input D, the time interval between datapoints, and N, the number of such intervals per voltam-metric branch. Note that the algorithm does not em-ploy the initial potential (which is also the finalpotential), but this equals Er−nND and should besufficiently negative to engender a negligible initial cur-rent.

(2) Dedicate storage for seven files, each of size2N+2, though the zeroth and/or last locations in thesefiles are not always used.

(3) For l=0, 1, 2, 3,....2N, 2N+1 calculate (Gl)R

and (Gl)P from Eqs. (70) and (71) and store the numer-ical values of these quantities, which have s3/2 units.

(4) For l=1, 2, 3,....2N calculate and store (Gl−1−2Gl+Gl+1)i for i=R and P.

(5) For L=0, 1, 2, 3,..., 2N calculate r via Eq. (67)and store these values. Note that, here and elsewhere,no special action need to taken at the reversal point.

(6) Set L=0, calculate E via Eq. (65) and I via Eq.(74). The summation in the latter equation is emptywhen L is zero and the sum is interpreted as zero. Storethese initial coordinates of the cyclic voltammogram inthe zeroth locations of the sixth and seventh files.

(7) Progressively increment L, recalculating and stor-ing E and I each time. Note that each calculation of Imakes use of the corresponding r value (i.e. that calcu-lated from the present L) and all prior I values.

(8) Quit after the E and I values have been recordedfor L=2N. These are the coordinates of the final pointon the cyclic voltammogram.

Fig. 6 was calculated using this algorithm and thenumerical values listed in the figure’s legend. Fig. 6 andTable 1 compare our result with that generated by theDigisim® software [17] at 20 mV intervals. Evidentlythe two approaches agree excellently.

Acknowledgements

The assistance of Jan Myland is acknowledged withgratitude, as is the financial support of the NaturalSciences and Engineering Research Council of Canada.We are grateful for the Digisim data provided by AlanBond and Graeme Snook.

Appendix A

In these appendices c(x,t) or c(r,t) will be used torepresent the concentration of an electroactive speciesat a position identified by the spatial coordinate x or rat time t. The x-coordinate applies when the electrodeis effectively planar and denotes distance measuredfrom the electrode surface into the solution. The rcoordinate applies to spherical and cylindrical ge-ometries and measures distance from the centre of thesphere or the axis on the cylinder.

Fick’s second law of planar diffusion5

5 The phrase ‘linear diffusion’ is often used in this context todistinguish from diffusion in other geometries, such as those withspherical symmetry. The adjective ‘linear’, however, is inappropriatein that the diffusion path is equally linear in spherical diffusion as inplanar diffusion.


Fig. 6. Cyclic voltammograms for the EC reaction described in the text, with the following parameters: electrode area=A=2.000×10−6 m2;electron number=n= −1; bulk reactant concentration=cb

R=1.000 mol m−3; half-wave potential=E1/2=0.000 V; scan rate=n=1.000 V s−1;reversal potential=Er=0.250 V; reversal time= tr=0.500 s; diffusivities DR=DP=DO=1.000×10−9 m2 s−1; forward rate constant=k=3.000 s−1; backward rate constant=k %=1.000 s−1. Though no distinction is apparent, there are two curves overlaid on this diagram, onemodelled by the procedure described in Section 18, and one generated by the Digisim® software. The lower panel contains a plot of the differencein current between the data calculated via Eq. (74) and the Digisim® software (i.e. Iresidual=Imodel−IDigisim).

D(2c(x2 (x,t)=

(c(t

(x,t) (A1)

Laplace transforms to

Dd2c̄dx2 (x,s)=sc̄(x,s)−c(�,0) (A2)

after the initial condition

c(x,0)=constant=bulk concentration=c(�,0) (A3)

is incorporated. Here c̄(x,s) denotes the Laplace trans-form of c(x,t), s being the ‘dummy’ variable of trans-formation. The most general solution of the ordinarydifferential Eq. (A2) is

c̄(x,s)=U(s) exp!

−x' s

D"

+V(s) exp!

x' s

D"

+c(�,0)

s(A4)

where unknown functions of s are denoted by U(s) andV(s). The second of these must be identically zero,however, if the remote boundary condition

c(�,t)=finite constant=c(�,0) (A5)

is to be satisfied. On differentiating Eq. (A4), withV(s)=0, the result

dc̄dx

(x,s)= −' s

DU(s) exp

!−x

' sD"

= −' s

D�

c̄(x,s)−c(�,0)

sn

(A6)

is then obtained, where the second step arises fromsubstituting for U(s) from Eq. (A4), recalling that V(s)is zero. Next, Eq. (A6) is specialized to the electrodesurface and rearranged to

c̄(0,s)−c(�,0)/s

D=

−1

s

dc̄dx

(0,s) (A7)


The convolution theorem of Laplace inversion allowsEq. (A7) to be inverted to

c(0,t)−c(�,0)

D= −

& t

0

(c(x

(0,t−t)dt

pt

= −d1/2

dt1/2

(c(x

(0,t) (A8)

Here d−1/2/dt−1/2 denotes the semiintegration operatorwith respect to time.

Applied to reaction (Eq. (2)), Faraday’s law relatesthe electric current to the fluxes of the electroactivespecies at the electrode and thence, via Fick’s first law,to the concentration gradients of R and P at theelectrode surface. The relationships are

I(t)nAF

= −DR

(cR

(x(0,t)=DP

(cP

(x(0,t) (A9)

This result may now be semiintegrated and combinedwith Eq. (A8) which, it will be recalled, applies to eitherelectroactive species. Accordingly

1nAF

d1/2

dt1/2 I(t)=DR[cR(0,t)−cR(�,0)]

= −DP[cP(0,t)−cP(�,0)] (A10)

which expression is identical, apart from notation, withEq. (3) of the main text.

Appendix B

For diffusion to an effectively infinitely long cylinder,Fick’s second law takes the form

(2c(r2 (r,t)+

1r(c(r

(r,t)=1D(c(t

(r,t) (B1)

Following a route parallel to that taken in Appendix Aleads to the equation

c̄(r,s)=U(s)I0!

r' s

D"

+V(s)K0!

r' s

D"

+c(�,0)

s(B2)

as a replacement for Eq. (A4). Here I0{ }and K0{ } arethe zeroth-order instances of the hyperbolic Bessel andBasset functions (see [13], ch. 49 and 51). BecauseI0{r s/D} approaches infinity as r��, whereasK0{r s/D} vanishes in that limit, U(s) must be zeroto meet the requirement that c̄(�,s) equal c(�,0)/s.Accordingly, on differentiation, we find

dc̄dr

(r,s)= −V(s)' s

DK1

!r' s

D"

= −' s

DK1

!r' s

D"c̄(r,s)−c(�,0)

sK0

!r' s

D"

(B3)

where K1{ } denotes the Basset function of unity order(see [13], ch. 49 and 51). This leads to

c̄(a,s)−c(�,0)/s

D=

−1

s

K0!

a' s

D"

K1!

a' s

D" dc̄

dr(a,s) (B4)

on rearrangement and specialization to the electrodesurface r=a. Substantively, this equation differs fromEq. (A7) only by the presence of the ratio of Bassetfunctions and this ratio is expressible as the asymptoticseries

K0!

a' s

D"

K1!

a' s

D"=1−

D/s2a

+3D

8a2s−

3D3/2

8a3s3/2 + ... (B5)

Under the conditions generally encountered in voltam-metry, only the first two or three terms in this series aresignificant. After substituting Eq. (B5), inversion of Eq.(B4) leads to

c(a,t)− (�,0)D

= −& t

0

� 1

pt−

D2a

+3D4a2

't

p− ...

ndcdr

(a,t−t) dt

(B6)

Table 1Comparison of the predictions of the extended semiintegral approachwith those of Digisim® for the cyclic voltammogram illustrated inFig. 6

E/mV I/nA

ModelDigisimModel Digisim

4−240 4 −1557 −15588−220 8 −1681 −1681

17 17−200 −1834 −183436 −2026 −2026−180 3678 −2274 −2274−160 78

−2601−2601169−140 169−120 365365 −3037 −3038−100 −3619−3619783782

−43631648 −43631648−80−60 33463345 −5200 −5200

6324 6324−40 −5832 −5831−20 10562 10560 −5627 −5626

−4008−4010147570 1476016965 1696320 −1323 −1321

40 16713 16712 1197 11981513260 15132 3193 319413333 1333380 4327 432711788 11788100 4980 498010575 10575120 5378 5378

9640 5655140 56569641588358848911 8912160

60988330 60988330180200 7854 7855 6318 6318

7456220 7456 6580 65807116 68186818240 7116


via the convolution theorem. Incorporation of the ex-pression

I(t)nAF

= −DR

(cR

(r(a,t)=DP

(cP

(r(a,t) (B7)

analogous to Eq. (A9), now leads to

cR(a,t)−cR(�,0)

DR

=& t

0

� 1

pt−

DR

2a+

3DR

4a2

't

p− ...

nI(t−t)nAF

dt (B8)

for species R, and a corresponding expression for P.These results are equivalent to Eqs. (14), (15) and (21)of the main text.

Appendix C

Here we solve the same problem as in Appendix Aexcept that the remote boundary condition Eq. (A5) isreplaced by

(c(x

(d,t)=0 (C1)

which asserts that there be no flux of electroactivesolute across the plane at x=d. Eq. (A4) still holds,but we cannot now set V(s) to zero. Instead, differenti-ate Eq. (A4):

dc̄dx

(x,s)=' s

D�

−U(s) exp!

−x' s

D"

+V(s) exp!

x' s

D"n

(C2)

When x is specialized to d and relation (Eq. (C1))invoked, it becomes evident that

V(s)=U(s) exp!

−2d' s

D"

(C3)

With the aid of this result, V(s) may be eliminatedbetween Eqs. (A4) and (C2). Then, following specializa-tion to x=0, we combine these two equations into

c̄(0,s)−c(�,0)/s

D=

−1

scoth

!d' s

D" dc̄

dx(0,s) (C4)

by eliminating U(s) and invoking the definition of thehyperbolic cotangent in terms of exponentials. Now,the inverse Laplace transform of (−1/s)coth {d s/D} is in terms of the theta-3 function of zero parameter(see [13], sect. 27:13), so that the invert of Eq. (C4) isthe convolution integral

c(0,t) − c(�,0)

D=

−Dd

& t

0

u3�

0;Dt

d2

� dcdx

(0,t−t)dt

(C5)

From here, the route to Eq. (22) of the main textfollows the path blazed in Appendix B.

Appendix D

Here we solve the same problem as in Appendix Aexcept that the remote boundary condition (Eq. (A5)) isreplaced by

c(d,t)=c(�,0) (D1)

which asserts that at the edge of the Nernst diffusionlayer (and beyond) the concentration is unchanging.Eq. (A4) still holds, and on specializing it to x=d wediscover that the left-hand side cancels with the thirdright-hand term. Thereby

V(s)= −U(s) exp!

−2d' s

D"

(D2)

This result differs only by a sign from that in AppendixC and, proceeding as in that appendix, we find

c̄(0,s)−c(�,0)/s

D=

−1

stanh

!d' s

D" dc̄

dx(0,s) (D3)

and thence

c(0,t)−c(�,0)

D=

−Dd

& t

0

u2�

0;Dt

d2

� dcdx

(0,t−t)dt

(D4)

The only difference from Eq. (C5) is that the theta-2function replaces theta-3. Eq. (24) of the main textmakes use of this result.

Appendix E

Fick’s second law, for solutes P and O, becomesaugmented by kinetic terms, namely

D(2cP

(x2 (x,t)−kcP(x,t)+k %cO(x,t)=(cP

(t(x,t) (E1)

and

D(2cO

(x2 (x,t)+kcP(x,t)−k %cO(x,t)=(cO

(t(x,t) (E2)

when P and O homogeneously interconvert accordingto Eq. (26). It is useful to define cS as the sum of thetotal concentration of the two isomers

cS(x,t)=cP(x,t)+cO(x,t) (E3)

and cD as their weighted difference:

cD(x,t)=kcP(x,t)−k %cO(x,t)

k+k %(E4)

With the aid of these definitions, Eqs. (E1) and (E2)may be combined into


D(2cS

(x2 (x,t)=(cS

(t(x,t) (E5)

and

D(2cD

(x2 (x,t)=(cDS

(t(x,t)+ (k+k %)cD(x,t) (E6)

Eq. (E5) resembles Eq. (A1) and leads, as in AppendixA, to

cS(�,0)−cS(0,t)

D=& t

0

(cS

(x(0,t−t)

dt

pt

=& t

0

(cP

(x(0,t−t)

dt

pt(E7)

The final step in Eq. (E7) follows from recognition that,because of the electropassivity of species O, (#cO/(x)(0,t) is zero. A different treatment is needed for Eq.(E6). Its Laplace transform is

Dd2c̄D

dx2 (x,s)= (s+k+k %)c̄D(x,s)−cD(�,0) (E8)

Now, it is evident from definition Eq. (E4) that cD

represents a departure from equilibrium and we canassume that equilibrium prevails in the bulk of thesolution. The final term in Eq. (E8) is therefore zero. Ageneral solution of the thereby truncated equation is

dc̄D

dx(x,s)=U(s) exp

!−x

's+k+k %D

"+V(s) exp

!x's+k+k %

D"

(E9)

We can argue, as in Appendix A, that V(s) must bezero and then eliminate U(s) between Eq. (E9) and itsx-derivative, so producing

−cD(0, t)

D=& t

0

exp{− (k+k %)t}(cD

(x(0, t−t)

dt

pt

=k

k+k %& t

0

exp{− (k+k %)t}(cP

(x(0, t−t)

dt

pt(E10)

after Laplace inversion and specialization to x=0. Asin Eq. (E7), it is the electropassivity of O that justifiesthe second equality.

Because of the equilibrium that exists in the bulksolution, k %cS(�,0)/(k+k %)=cP(�,0). Use of this iden-tity allows a suitably weighted addition of Eqs. (E7)and (E10) to generate an expression for the concentra-tion excursion of the electroproduct species

cP(�,0)−cP(0,t)

D

=1

k+k %& t

0

[k %+k exp{− (k+k %)t}](cP

(x(0,t−t)

dt

pt

(E11)

As usual, Eq. (A9) provides a link between the currentand the gradient of P’s concentration at the electrodesurface. Thereby, Eq. (27) of the main text is estab-lished.

Appendix F

Eq. (A2), the Laplace transform of Fick’s secondlaw, takes the form

DR

d2c̄R

dx2 (x, s)=sc̄R(x, s)−c s0R +

I0xnAFDR

(F1)

for species R when the initial condition of the experi-ment is the preexisting steady state described by Eq.(44). The general solution of this ordinary differentialequation is

c̄R(x,s)=U(s) exp!

−x' s

DR

"+

c s0R

s−

I0xnAFDRs

(F2)

after V(s) has, for the usual reason, been set to zero.Following the standard procedure in these appendices,the U(s) term is next eliminated between Eq. (F2) andits x-derivative. This gives

c̄(0,s)−c s0R /s

D= −

1

s

dc̄R

dx(0,s)+

I0

nAFDRs3/2 (F3)

after specialization to x=0 and rearrangement.Laplace inversion and incorporation of Eq. (A9) nowlead to the R-version of Eq. (46) of the main text.

Appendix G

The single and double integrals of the theta-twofunction that occurs in the theory of Nernst-layervoltammetry are& z

0

u2(0; z) dz

=4'z

p

�12+ %

�

j=1

(− )j exp!− j2

z"

− (− )jj'p

zerfc

! j

z

"n(G1)

& z

0

& z

0

u2(0; z) dz dz

=8z3/2

3p

�12+ %

�

j=1

(− )j�1+j2

zn

exp!− j2

z"

− (− )jj�3

2+

j2

zn'p

zerfc

! j

z

"n(G2)


This formulation is based solely on the first of the twoseries in Eq. (25). Simpler single and double integralsare available from the second formulation, but theseare of less utility when a lower limit of zero is pertinent.

It is the theta-3 function that occurs in thin-layervoltammetry. Its single and double integrals are identi-cal with the right-hand sides of Eqs. (G1) and (G2)except that the (− )j multipliers are absent. Please notethat the double integrals of the theta-2 and theta-3functions were previously reported [11] incorrectly.

References

[1] J.C. Myland, K.B. Oldham, J. Electroanal. Chem. 153 (1983) 43.[2] F.G. Cottrell, Z. Physik. Chem. 42 (1902) 385.[3] D. Ilkovic' , Collect. Czech. Chem. Commun. 6 (1934) 498.[4] Z. Karaoglanoff, Z. Elecktrochem. 12 (1906) 5.[5] R.S. Nicholson, I. Shain, Anal. Chem. 36 (1964) 706.

[6] M.I. Pilo, G. Sanna, R. Seeber, J. Electroanal. Chem. 323 (1992)103.

[7] K.B. Oldham, J. Spanier, The Fractional Calculus: Theory andApplication of Differentiation and Integration to Arbitrary Or-der, Ch. 3, Academic Press, New York, 1974.

[8] K.B. Oldham, J. Electroanal. Chem. 430 (1997) 1.[9] P.A. Jansson, Deconvolution, Academic Press, New York, 1984.

[10] H.L. Surprenant, T.H. Ridgway, C.H. Reilley, J. Electroanal.Chem. 75 (1977) 125.

[11] K.B. Oldham, Anal. Chem. 58 (1986) 2296.[12] J.C. Myland, K.B. Oldham, C.G. Zoski, J. Electroanal. Chem.

193 (1985) 3.[13] J. Spanier, K.B. Oldham, An Atlas of Functions, sect. 34: 7,

Hemisphere and Springer-Verlag, Berlin, 1987.[14] F.E. Woodard, R.D. Goodin, P. Kinlen, Anal. Chem. 56 (1984)

1920.[15] B. Blagg, S.W. Carr, C.R. Dobson, J.B. Gill, D.C. Goodall, B.L.

Shaw, N. Taylor, T. Boddington, J. Chem. Soc. Dalton Trans. 1(1985) 1213.

[16] K.B. Oldham, J.C. Myland, Fundamentals of ElectrochemicalScience, sect. 11:3, Academic Press, San Diego, 1994.

[17] M. Rudolph, D.P. Reddy, S.W. Feldberg, (Default values wereused), Anal. Chem. 66 (1994) 589A.

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Voltammetric modelling via extended semiintegrals