A fresh approach to voltammetric modelling

Peter J. Mahon a, Jan C. Myland b, Keith B. Oldham b,�a Research School of Chemistry, Australian National University, Canberra ACT 0200, Australia

b Department of Chemistry, Trent University, Peterborough, ON, Canada K9J 7B8

Received 30 September 2002; accepted 9 October 2002

Abstract

A modular semianalytic procedure, alternative to simulation, is described for predicting the transient current response to any

applied potential signal under a wide variety of voltammetric conditions. Diverse geometries may be treated; the electrode reaction

may have any degree of reversibility; homogeneous chemical reactions may be involved. Conceptually, the method consists of

splitting the overall problem into three components, each of which is solved separately. Examples presented include cyclic

voltammetry and (Osteryoung) square-wave voltammetry.

# 2002 Elsevier Science B.V. All rights reserved.

Keywords: Convolution; Modelling; Voltammetry

1. Introduction

To interpret experimental voltammograms, one needs

to predict the voltammetric current accurately. Because

mathematical analysis is successful only in simple

instances [1], digital simulation has become the standard

workhorse by which predictions are made, this techni-

que having reached a high degree of sophistication in the

hands of skillful experts [2]. The less skilled are able to

make use of commercial software: either programs, such

as Digisim† [3] or Elsim [4,5], specially designed for

voltammetrists, or more general simulation packages,

such as Femlab† [6].

The same advances in computation that have facili-

tated simulations have also allowed the steady develop-

ment of semianalytical methods [7] of modelling

transient voltammetry. The seminal work of Nicholson

and Shain [8] who solved the cyclic voltammetry

problem via integral equations, may be considered an

early semianalytical model. Since then, the method has

slowly progressed under such names as ‘semiintegral

electroanalysis’ [9�/15], ‘convolution potential sweep

voltammetry’ [16�/20], ‘extended semiintegrals’ [21] and

‘convolutive modelling’ [22�/27], but has failed to find

wide acceptance, perhaps because of the unfamiliar

mathematics. This oversight may soon end, now that

commercial software [28], based upon convolution, is

available. The salient difference between simulative and

semianalytic methods is that, whereas simulations rely

on approximating the transport field by discrete ele-

ments which are handled arithmetically, transport is

solved exactly in the semianalytic approach, without

spatial discretization. Among the advantages of the

semianalytical technique are assured accuracy and

applicability to any voltage waveform, including those

not presently in the electrochemical repertoire. Ease of

programming makes it unnecessary to rely on ‘black-

box’ software. Among its disadvantages, in comparison

with digital simulation, is less kinetic flexibility and a

greater need for mathematical acumen.

This article seeks to promote semianalytic methodol-

ogy by exposing three significant advances: (a) demon-

strating how most transient voltammetric problems may

be dismembered into three simpler problems; (b) pre-

senting a simpler convolution algorithm than that used

formerly; (c) providing a library of the functions with

which current is convolved. These advances aid not only

the modelling of the voltammetry, but also the con-

ceptualization of the voltammetric process itself. We

shall show how three components contribute additively

� Corresponding author. Tel.: �/1-705-748-1336; fax: �/1-705-748-

1625

E-mail address: koldham@trentu.ca (K.B. Oldham).

Journal of Electroanalytical Chemistry 537 (2002) 1�/5

www.elsevier.com/locate/jelechem

0022-0728/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 0 2 2 - 0 7 2 8 ( 0 2 ) 0 1 2 6 3 - 9

to the electrochemical current, permitting a modular

approach to modelling.

We address controlled-potential voltammetry in

which an electroactive substrate, S, undergoes an n -electron quasireversible transfer, S(soln)�/ne�0/

P(soln), to give a product species, P, that is initially

absent.1 n can be positive or negative, according as the

reaction is anodic or cathodic. Transport of S and P is

usually by diffusion with diffusivities (or diffusion

coefficients) DS and DP. Either or both species may be

involved in homogeneous chemical reactions. The

electrode has a constant area A and no restriction isplaced on the way its potential E (t ) varies with time.

Assorted geometries will be treated.

2. The basic idea

Consider the upper panel in Fig. 1 in which three

resistors are in series. It is easy to demonstrate that the

current I flowing through such a series combination

obeys the relationship

I

I1

�I

I2

�I

I3

�1 (1)

where I1 is the current that would flow if resistors R2

and R3 ceased to impede the current flow, as would be

the case if the second and third switches shown in the

diagram were closed. I2 and I3 are correspondingly

defined. There are strong similarities between the

‘seriesed’ resistors and the voltammetric circumstances

diagrammed in the lower panel of Fig. 1. This articlereports a quest for a tripartite formula, reminiscent of

Eq. (1), in which each left-hand term represents the

voltammetric behaviour if only one of the three im-

pediments*/the supply of the substrate to the electrode,

the transfer of electrons at the electrode, or the removal

of the product from the electrode*/were rate-determin-

ing. For reasons that will soon be apparent, it is

convenient to multiply each term in the analogue ofEq. (1) by a constant. Thus we seek a formulation with

the following pattern that applies to controlled-potential

transient voltammetry:

(supply term)�(transfer term)�(removal term)

�nFAcb (2)

Here F is Faraday’s constant, while cb is the bulk

concentration of the substrate2 S. Such a formula is

known to apply to steady-state voltammetry,3 but thetransient case is less obvious.

The three left-hand terms will be derived in turn,

starting with the second one. The final result will take

the form

I(t)+S(t)�I(t)

kf (t)�

I(t)P(t)

K(t)�nFAcb (3)

Notice that all terms in this equation have a dimen-

sionality corresponding to the C m�1 unit. The ‘+’symbol represents convolution, defined by I(t)+S (t )/�f

t

0I(t�t)S(t) dt: In Eq. (3), which we term the ‘master

equation’, kf(t) is the heterogeneous rate constant4 of the

forward reaction S�ne�?kf

kb

P and K (t) is the Nernstian

equilibrium constant4, defined by

K(t)�kf (t)

kb(t)�exp

�nF

RT[E(t)�E�]

�(4)

S (t) and P (t) are functions of time whose identities are

determined, respectively, by the supply and removal

scenarios. Some examples are given in the next section.To evaluate the transfer term, we seek to discover

what the faradaic current would be if the supply and

removal impediments did not exist. If, as usually holds,

the electron-transfer reactions are unimolecular, then

the equation

I(t)�nFA[kf (t)csS(t)�kb(t)cs

P(t)] (5)

applies and relates the current to the concentrations,

csS(t) and cs

P(t); of the substrate and product at the

electrode surface. Eq. (5) is general, but it simplifies in

the circumstance to which the ‘transfer term’ in Eq. (2)

relates. For then there is no impediment to the supply of

S to the electrode, which means that csS(t) will adopt the

bulk value cb of the concentration of species2 S. Like-

wise, there is no impediment to the removal of P fromthe electrode, and since P is absent from the bulk, this

means that csP(t) will be zero. Accordingly, I(t)�/

nFAkfcb in the transfer term. But, in the circumstance

1 with minor modifications, the treatment can be applied when

product is initially present.2 and/or of its precursor should there be a preceding chemical

reaction.

Fig. 1. Voltammetric constraints are analogous to resistors in series.

3 In the notation of this article it is (aI(t)=DS)�(I(t)=kf (t))�(aI(t)=DPK(t))�nFAcb for a hemispherical microelectrode of radius a

[29].4 Though kf and K depend directly on electrode potential, we treat

them as functions of time because the potential is itself time-dependent.

If Butler�/Volmer kinetics are adopted, note that kf�/ koKg where ko is

the standard heterogeneous rate constant and g is a transfer coefficient.

P.J. Mahon et al. / Journal of Electroanalytical Chemistry 537 (2002) 1�/52

when the supply and removal impediments are ‘switched

off’ the transport term must equal nFAcb on its own. It

follows that the transfer term is I (t )/kf(t), a remarkably

simple result.

The evaluation of the transfer term was simple

because the current in that case is not affected by the

‘history’ of the voltammetry; that is, the term reflects

only the present magnitude of I(t), not its earlier values.

This is not the case for the supply and removal terms,

which are more complicated in consequence.

The supply term must incorporate all the features that

influence the transport that leads to species S reaching

the electrode surface. These features include the diffu-

sivities of the species involved and parameters governing

homogeneous chemical reactions, if any. As well,

geometric parameters will often be present. The supply

term is determined by investigating how the current that

has flowed affects csS(t) for a particular supply scenario.

When csS(t) is set to zero, as is appropriate when neither

transfer nor removal pose an impediment, experience

[21] shows that the convolution I(t)+S (t )�/nFAcb

remains. Examples of S (t ) are included in Table 1.

Similar considerations apply to the supply and removal

terms. However, unlike the supply term, the removal

term reflects the electrode potential. Since, in construct-

ing the removal term, we are treating the rate of electron

transfer as imposing no impediment to current flow, the

electron-transfer reaction is at equilibrium, so that

csP(t)�K(t)cs

S(t): Moreover, because there is also no

impediment to the supply of the substrate S, csS(t) can be

replaced by cb and it follows that csP(t)�K(t)cb in

constructing the removal term. This term takes the

form [I(t)+P (t)]/K (t), where the P (t) functions are listed

in Table 1.

3. Function library

Unless otherwise stated, the functions listed in theright-hand column of Table 1 apply equally to the

substrate S and the product P; replace D by DS or DP as

appropriate.

The procedure for creating entries to this table is

straightforward, so the library can be augmented as

necessary. We illustrate the procedure by recreating the

penultimate tabular entry.

Set up the appropriate version of Fick’s second law,then Laplace transform it:

DP

@2cP

@r2�

2DP

r

@cP

@r�kcP�

@cP

@tU

DP

d2cP

dr2�

2DP

r

dcP

dr�(s�k)cP�cb

P

(6)

Because P is absent from the bulk, the final term is zero.

The remaining ordinary differential equation can be

solved by standard methods to produce

cP�f (s)

rexp

��r

ffiffiffiffiffiffiffiffiffiffiffiffis � k

DP

s ��

g(s)

rexp

�r

ffiffiffiffiffiffiffiffiffiffiffiffis � k

DP

s �

�f (s)

rexp

��r

ffiffiffiffiffiffiffiffiffiffiffiffis � k

DP

s �(7)

Arbitrary functions are represented by f (s ) and g (s ), but

Table 1

Function library

Transport/geometry/chemical conditions S (t ) or P (t )

Semiinfinite planar diffusion [16,34] /1=ffiffiffiffiffiffiffiffipDt

p/

Semiinfinite diffusion to a cylinder of radius a [21]. The four reported terms in an

asymptotic series are usually adequate

/1=ffiffiffiffiffiffiffiffipDt

p�1=2a�(3=4a2)

ffiffiffiffiffiffiffiffiffiffiffiDt=p

p�3Dt=8a3 � � �/

Semiinfinite diffusion outside a (hemi)sphere of radius a [35] /1=ffiffiffiffiffiffiffiffipDt

p�(1=a)expfDt=a2gerfcf

ffiffiffiffiffiffiDt

p=ag/

Spherical diffusion within a (mercury) sphere of radius a [30]: the two formulae are

equivalent; use the first formula if 16Dt Ba2. In the second formula, yp is the pth positive

root of y� tan{y} [[36], p. 324]

/1=ffiffiffiffiffiffiffiffipDt

p�(1=a)exp(Dt=a2)�erfcf

ffiffiffiffiffiffiDt

p=ag/

/3=a�(2=a) a�p�1 expf�ypDt=a2g/

Advective diffusion of solution with constant velocity v to a porous planar electrode [37] /expf�v2t=4Dg=ffiffiffiffiffiffiffiffipDt

p�(v=2D)erfcf(v=2)

ffiffiffiffiffiffiffiffit=D

pg/

Planar diffusion in a ‘Nernst layer’ of width L ; homogeneous solution beyond

[30,21,38,36 (Secn 27:13)]

/(1=L)u2(0; Dt=L2)/

Planar diffusion in a thin-layer cell of width L ; impermeable barrier beyond [30,21,38,36

(Secn 27:13)]

/(1=L)u3(0; Dt=L2)/

Semiinfinite planar joint diffusion of substrate S and electro passive isomera S? with CE

reaction S??k

k?S 0 [21]

/[k�k? expf�(k�k?)tg]=(k�k?)ffiffiffiffiffiffiffiffiffiffiffipDSt

p/

Semiinfinite planar diffusion of product P with ECirr reaction 0 P0k [22�/24]: /expf�ktg=ffiffiffiffiffiffiffiffiffiffiffipDPt

p/

Semiinfinite planar diffusion of product P with EC reaction 0 P?k

k?P? to an isomera [21] /[k?�kexpf�(k�k?)tg]=(k�k?)

ffiffiffiffiffiffiffiffiffiffiffipDPt

p/

Semiinfinite diffusion of product P outside a (hemi)sphere of radius a , with with ECirr

reaction 0 P0k [30]

/expf�ktg=ffiffiffiffiffiffiffiffiffiffiffipDPt

p�(1=a)expfDPt=a2�ktgerfcf

ffiffiffiffiffiffiffiffiDPt

p=ag/

Any of the above at sufficiently small t /1=ffiffiffiffiffiffiffiffipDt

p/

a Isomers are assumed to share the same diffusivity D .

P.J. Mahon et al. / Journal of Electroanalytical Chemistry 537 (2002) 1�/5 3

the latter was set to zero as otherwise the concentration

of P would increase without limit as r approaches

infinity. The current is related to the gradient of P’s

concentration at the electrode surface by I (t )�/

�/nFADP(@cP/@r)r�a and the transform of this relation-

ship, on combination with Eq. (7), yields

I(s)��nFADP

�dcP

dr

�r�a

�nFA

�DP

a2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(s � k)DP

pa

f (s)exp

��a

ffiffiffiffiffiffiffiffiffiffiffiffis � k

DP

s �(8)

This formula enables f(s) to be expressed in terms of

I(s); and this expression may be inserted back into the

r�/a version of Eq. (7). After rearrangement, the result,

and its invert are

I(s)ffiffiffiffiffiffiDP

p[

ffiffiffiffiffiffiffiffiffiffiffiffis � k

p�

ffiffiffiffiffiffiDP

p=a]

�nFA(cP)r�a

UI(t)+P(t)�nFAcsP�nFAcbK(t)

(9)

where P (t) is the penultimate term in the table.Though the function library covers many scenarios of

voltammetric interest, it is not exhaustive. The semi-

analytic method requires the functions S (t) and P(t) to

be known. Currently, the semianalytic approach can

address only uniformly accessible geometries, except

when the reaction is reversible or the concentration

polarization is extreme [24].

The serial configuration of the three ‘impediments’lies at the heart of the semianalytic method, so direct

interaction between species S and P might compromise

the master equation. Nevertheless, the semianalytic

method has been applied [30] to the catalytic regenera-

tion of S from P. In the absence of excess supporting

electrolyte, S and P may influence each other through

the electric field in solution; however the semianalytic

method becomes feasible if either S or P is uncharged.The ECE mechanism involves two transfer steps and

three transport steps; it can be handled by an analogue

of the master equation in which there are five left-hand

terms.

4. Implementation

The master equation simplifies under conditions

when: (a) the electrode reaction is totally irreversible,

the removal term disappears; (b) the electrode reaction is

reversible, the transfer term disappears; (c) polarization

is modest, i.e. at rather negative potentials for anoxidation, the supply term is negligible; (d) the polariza-

tion is intense, in what is often called the ‘plateau region’

of the voltammogram, both the transfer and removal

terms may be ignored; (e) the reaction is slow, especiallyif the solution is stirred, it may be possible to ignore the

supply and removal terms.

The following algorithm [31] may be used to imple-

ment the master equation

IJ �nFAcb=D�

PJ�1

j�1 Ij[SJ�j � PJ�j=KJ ]

(0:824=ffiffiffiffiD

p)(1=

ffiffiffiffiffiffiDS

p� (1=KJ=

ffiffiffiffiffiffiDP

p)) � 1=(kf )JD

(10)

Here, D is a small time interval, J is the large integer t /

D , I(jD ) is abbreviated Ij and similar subscripts have

similar significance. In accuracy, this algorithm may beinferior to one previously reported [32], but has ad-

vantages of simplicity and speed. To calculate a 1000

point voltammogram takes less than 1 s on a typical

personal computer.

To test algorithm Eq. (10), we compared its output

with that of DigiSim†, for a quasireversible cyclic

voltammetry experiment, at a spherical electrode, with

a following chemical reaction. Fig. 2 shows that theagreement is excellent5, a tribute to the accuracy of

DigiSim†.

5. Summary

Wide ranges of geometric, chemical, electrochemical

and transport conditions can be solved by the semiana-

lytic procedure. It can be applied equally to any type of

Fig. 2. Cyclic voltammogram for a quasireversible oxidation,

S(soln)�/e�0/P(soln), with ko�/10�4 m s�1and a�/0.5, followed

by a homogeneous decomposition, P(soln)0/Q(soln), with k�/0.1 s�1,

at a spherical electrode of radius 10�3 m. Here cb�/0.001 mol m�3;

DS�/DP�/10�9 m2 s�1; starting potential�/10RT /F ; sweep rate�/

4RT /F s�1; reversal potential�/�/10RT /F ; final potential�/10RT /F ;

1000 points with D�/0.01 s. The solid line is the output from algorithm

Eq. (10); the dots are the DigiSim results.

5 The worst discrepancy is less than 0.05% of the peak height.

P.J. Mahon et al. / Journal of Electroanalytical Chemistry 537 (2002) 1�/54

voltage program. Fig. 3, for example, shows an applica-

tion to Osteryoung square-wave voltammetry [33].

Acknowledgements

The Natural Sciences and Engineering Research

Council of Canada provided generous support.

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Fig. 3. A normalized Osteryoung squarewave voltammogram, for a

reversible reduction with no following reaction, at a planar electrode.

DS�/DP�/10�9 m2 s�1; initial potential�/�/255 mV; step potential�/

10 mV; squarewave amplitude�/50 mV; pulse time�/0.05 s; 1000

points/pulse; D�/0.00005 s. The peak height of the Dc curve is 0.903,

which differs from that tabulated by Osteryoung and O’Dea [33] by

2.6%.

P.J. Mahon et al. / Journal of Electroanalytical Chemistry 537 (2002) 1�/5 5