Journal of Electroanalytical Chemistry 655 (2011) 173–179

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Journal of Electroanalytical Chemistry

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Reaction layer thickness of a catalytic mechanism under transient and stationarychronopotentiometric conditions

Ángela Molina ⇑, Joaquín González, Carmen M. SotoDepartamento de Química Física, Universidad de Murcia, Espinardo, 30100 Murcia, Spain

a r t i c l e i n f o

Article history:Received 11 October 2010Received in revised form 13 January 2011Accepted 14 January 2011Available online 25 January 2011

Keywords:ChronopotentiometryReaction layersSteady stateExponential currentSpherical electrodes

1572-6657/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jelechem.2011.01.020

⇑ Corresponding author. Tel.: +34 868 887524; fax:E-mail address: amolina@um.es (Á. Molina).

a b s t r a c t

Expressions for the reaction layer thickness of a catalytic mechanism at planar and spherical electrodes,deduced for transient and stationary chronopotentiometric conditions for three types of programmedcurrents (constant current, power time current and exponential time current) are presented. From theseexpressions, the conditions for attaining a stationary behaviour, i.e., a time independent reaction layerthickness, are discussed, and the advantages derived form the use of exponential time currents in orderto accelerate the evolution of the response towards its stationary limit are analysed. Finally, expressionsfor the potential–time curves and their reciprocal derivatives are also given and the influence of the reac-tion layer thickness is studied.

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1. Introduction E–t response corresponding to a catalytic mechanism when differ-

The electrochemical study of the catalytic mechanism, bothwhen the redox mediator is soluble in the electrolytic solution orwhen it is immobilized at the electrode surface, is of great impor-tance for several reasons. First, it is the only mechanism in whichthe reaction product is simultaneously required for both thechemical and electrochemical reactions in order to regenerateand transform, respectively, the depolarizer [1–4]. Second, station-ary responses can be obtained for this mechanism even at largeelectrodes [4]. It also has a high number of practical applicationssince it appears in a lot of reactions in Biochemistry, Biology, Phar-macy and industrial and environmental processes [1–3,5–8].

In order to understand the electrochemical behaviour of thespecies involved in a pseudo-first order catalytic process and theevolution of the system from transient to stationary state it is ofgreat importance to study the reaction layer thickness, dr, whichcan be defined as the linear region disturbed by the chemical kinet-ics. Expressions for the reaction layer thickness have been obtainedunder steady state conditions dss

r in Voltammetry and Chronopo-tentiometry for planar and spherical electrodes [9,10]. Theseexpressions are time independent due to their stationary nature,such that, in most of the cases, dss

r is independent of the electro-chemical technique.

In recent papers we have studied the diffusion layer thicknessfor a simple charge transfer in Chronopotentiometry under tran-sient and stationary conditions [11]. We have also analyzed the

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+34 868 884148.

ent programmed currents such as a current step (I0), a power cur-rent–time function (I0tu), and an exponential current–timefunction (I0ewt) are applied to spherical and planar electrodes ofany size [4,10,12–14]. Chronopotentiometry with programmedcurrents presents several advantages such the possibility of select-ing a wide range of values of I0, u and w. Moreover, the capacitativeeffects observed in the E–t curves are practically negligible whenprogrammed currents are used [10,12,13].

The aim of this paper is to extend the concept of the reactionlayer thickness, dr, in a catalytic mechanism to chronopotentiomet-ric transient conditions, for which no expressions have been re-ported in the literature. Based on the results obtained here, weshall demonstrate that in the case of an exponential current–timefunction (I0ewt), it is possible to achieve a constant value of dss

r (i.e.steady state) faster than that corresponding to other programmedcurrents and even than that in the corresponding voltammetriccase, due to the dependence on w of dss

r . This means to that evolu-tion can be accelerated by simply increasing the value of w in theapplied current.

An analysis of the evolution from transient to steady statebehaviour of the potential–time (E–t) and derivative (dewt/dE) � Eresponses when the above programmed currents are applied atelectrodes of any size is also presented.

2. Theory

Let us consider the application of a current–time perturbationI(t) such as a current step (I0), a power current–time function

174 Á. Molina et al. / Journal of Electroanalytical Chemistry 655 (2011) 173–179

(I0tu) or an exponential current–time function (I0ewt) to an elec-trode of spherical or planar geometry where a pseudo-first ordercatalytic process takes place in agreement with the followingScheme:

Aþ ne�¢ B ¢k1

k2

A ðIÞ

with k1 and k2 (s�1) being the homogeneous reaction rate constants.Under pure diffusion mass transport, the diffusive-kinetic dif-

ferential equation system for the species in Scheme (I) is givenby [15,16]:

@cAðr;tÞ@t ¼ D½@

2cAðr;tÞ@r2 þ 2

r@cAðr;tÞ@r � þ k1cBðr; tÞ � k2cAðr; tÞ

@cBðr;tÞ@t ¼ D½@

2cBðr;tÞ@r2 þ 2

r@cBðr;tÞ@r � � k1cBðr; tÞ þ k2cAðr; tÞ

9=; ð1Þ

with the following boundary value problem:

t ¼ 0; r P r0

t P 0; r !1

�; cA ¼ c�A; cB ¼ c�B ð2Þ

t > 0; r ¼ r0;@cAðr; tÞ@r

� �r0

¼ � @cBðr; tÞ@r

� �r0

ð3Þ

D@cAðr; tÞ@r

� �r0

¼ IðtÞnFA

ð4Þ

In the above equations we have assumed that DA = DB = D.To tackle the above problem, the following two variables, f and

/ss can be used [4]:

fðr; tÞ ¼ cAðr; tÞ þ cBðr; tÞ ð5Þ

/ssðr; tÞ ¼ cBðr; tÞ � KcAðr; tÞ ð6Þ

with K being the equilibrium constant K defined asK ¼ k2=k1 ¼ c�B=c�A.

Note that variable /ss(r, t) is an equilibrium perturbation func-tion, i.e., it is related with the deviation of concentrations fromthe situation of chemical equilibrium [17] in such a way that/ss(r ?1, t) = /ss(r, t = 0) = 0, since for t = 0 or r ?1 the concen-trations of species A and B take their bulk values and the equilib-rium holds. Moreover, in this mechanism f(r, t) is not anunknown function, since condition fðr; tÞ ¼ c�A þ c�B ¼ f� is fulfilledfor any values of r and t [4,18,19], and therefore, o f(r, t)/ o t = 0 issatisfied independently of whether the steady sate has beenreached or not.

Taking into account the variable /ss(r, t), the differential equa-tion system given by Eq. (1) becomes into:

@/ssðr; tÞ@t

¼ D@2/ssðr; tÞ

@r2 þ 2r@/ssðr; tÞ

@r

" #� j/ssðr; tÞ ð7Þ

being j = k1 + k2.Assuming that the chemical reaction is fast enough (i.e.,

j� 1 s�1), it is fulfilled that

@/ssðr; tÞ@t

¼ 0 ð8Þ

This assumption implies that the function /ss only depends onthe distance to or from the electrode, and under these conditions,the expression of /ss, whatever the perturbation, is given by [17]:

/ssðrÞ ¼ r0

r/ðr0Þe�

ffiffijD

pðr�r0Þ ð9Þ

From this expression we can immediately obtain that corre-sponding to the surface gradient of the equilibrium perturbationfunction /ss, which can be written:

@/ss

@r

� �r0

¼ /ssð1Þ � /ssðr0Þdss

r

ð10Þ

with dssr being the linear reaction layer thickness in spherical geom-

etry, i.e., the thickness of the region disturbed by the chemical reac-tion [9,12,20]. By introducing Eq. (9) into Eq. (10), and taking intoaccount that /ss(r ?1) = 0, we obtain:

dssr ¼

11r0þ

ffiffiffijD

p ð11Þ

This expression is independent of the nature of the electricalperturbation (i.e., of the particular form of the applied currentI(t) under the conditions of this paper), and therefore Eq. (11) coin-cides with that obtained in Voltammetry when the charge transferstep is reversible (see below) [17].

When the diffusion field is planar (i.e., r0 ?1), the linear reac-tion layer thickness is given by [9,10,12,20–23]:

dplan;ssr ¼

ffiffiffiffiDj

rð12Þ

Note that the above expression of the linear reaction layerthickness (Eq. (11)) coincides with that obtained for the linear dif-fusion layer thickness, i.e., dss

r ¼ dssd , calculated from the expression

of the concentration profile of oxidized species A. This is easily de-duced if we take into account that from Eqs. (1)–(6), the stationaryconcentration profile of species A in this mechanism is given by:

cAðr; tÞ ¼f�

ð1þ KÞ �r0

r1

1r0þ

ffiffiffijD

p IðtÞnFAD

e�ffiffijD

pðr�r0Þ ð13Þ

From this expression we can immediately obtain that corre-sponding to the surface gradient of cA, which can be written:

@cA

@r

� �r¼r0

¼ cAð1Þ � cAðr0Þdss

d

ð14Þ

with dssd being the linear diffusion layer thickness in spherical geom-

etry for this reaction Scheme [9,11]. By considering that cAð1Þ ¼ c�A,it is obtained that dss

r ¼ dssd , independently of the geometry of the

electrode.This fact is a characteristic of the catalytic mechanism. More-

over, for very small radii (i.e. r0 ? 0), the linear reaction layerthickness (or the diffusion layer thickness) takes the followingvalue:

dmicro;ssr ¼ dmicro;ss

d ¼ r0 ð15Þ

3. Rigorous treatment of the problem

The variables f and /ss, given by Eqs. (5) and (6), are the mostadequate for analyzing the stationary behaviour of the catalyticmechanism given in Scheme (I) (see Eqs. (8), (9), and (13)). f canbe used again to obtain the transient response of this mechanism,but we should introduce the new variable /(r, t),

/ðr; tÞ ¼ ðcBðr; tÞ � KcAðr; tÞÞejt ð16Þ

With these two variables, the problem has been solved in Refs.[13,19] (constant current), [12] (power time current) and [10](exponential time current). As in the stationary case discussedabove, we can deduce from Eq. (10) an expression for the linearreaction layer thickness dr, although in this case dr has a timedependence which is a function of the particular programmed cur-rent used. The expressions of dr for the three current perturbationsconsidered in this paper are summarized in Table 1 for planar andspherical geometries. We have also included in Table 1 thevoltammetric expressions for dr corresponding to the application

Table 1Expressions for the reaction layer thickness dr and dss

r at planar and spherical electrodes obtained under chronopotentiometric conditions for three current–time perturbations (I0,I0tu and I0ewt) [10,12,13], and under voltammetric conditions (application of a current step when the charge transfer process is reversible) [15,16]. The functions Tvolt(v), S(n, v),Tu(n, v), Tplan

u ðvÞ and G(n, h) are defined in Appendix B.

Linear reactionlayer dr, dss

r

Geometry Constant potential Voltammetry I0 I0tu I0ewt

Plane Transient ffiffiffiffiDj

rTvoltðvÞ ðT:1Þ

ffiffiffiffiDj

rerf ð ffiffiffivp Þ ðT:2Þ

ffiffiffiffiDj

r2e�jtTplan

u ðvÞ ðT:3Þffiffiffiffiffiffiffiffiffiffiffiffiffi

Dwþ j

rerf ð

ffiffiffihpÞ ðT:4Þ

Stationary

v� 1h� 1

ffiffiffiffiDj

rðT:5Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiD

wþ j

rðT:6Þ

Sphere Transient 11

Tvolt ðvÞ

ffiffiffijD

pþ 1

r0

ðT:7Þ 2ffiffiffiffiffiffiDtp

Sðn;vÞ ðT:8Þ 2ffiffiffiffiffiffiDtp

Tuðn;vÞ ðT:9Þ 2ffiffiffiffiffiffiDtp

Gðn; hÞ ðT:10Þ

Stationary

v� 1h� 1

11r0þ

ffiffiffijD

p ðT:11Þ 11r0þ

ffiffiffiffiffiffiffiffiwþj

D

p ðT:12Þ

Á. Molina et al. / Journal of Electroanalytical Chemistry 655 (2011) 173–179 175

of a potential step when the charge transfer process is reversible[9,15,16].

From this table it can be observed that for v( = jt)� 1, i.e., forfast kinetics, the expression of dr shown in Table 1 correspondingto constant current, power time current and also that obtained un-der voltammetric conditions coincide with Eqs. (11) and (12) forspherical and planar electrodes, respectively, whereas in the caseof exponential current (w + j) appears instead of j (see Refs.[4,10,16,21]). This is in agreement with the discussion inSection 2.

4. Results and discussion

Fig. 1 shows the temporal evolution of the reaction layer thick-ness of a catalytic mechanism at planar electrodes, dplan

r , calculatedfrom Eqs. (T2)–(T4) of Table 1, for the programmed currents I0ewt

(a), I0 (b) and I0tu (c), and different values of the parameter j.From these curves it can be seen that dplan

r always increases withtime until it reaches its stationary limit given by Eqs. (T5) and (T6)in Table 1 (dotted lines). The time required to attain this constantvalue depends on the j value and on the type of programmed cur-rent used. Thus, higher values of j give rise to higher values of thedimensionless parameters v = (jt) and h = ((w + j)t) which appearin the expressions of dplan

r (see Eqs. (B.2)–(B.5) of Appendix) and,therefore, to a faster evolution towards the stationary limit valueof the reaction layer thickness. Note also that the increase of j alsocauses a decrease of the thickness of the stationary reaction layer(see Eqs. (T.5) and (T.6)) in such a way it enables that this limitingvalue could be reached earlier. Concerning the type of time vari-able current used, it can be observed for a fixed j value that theexponential current gives rise to a faster increase of dplan

r followedby the constant current and the power time current. This behav-iour is related with the dependence of dplan

r on the h parameter, in-stead of v in Eq. (T.4).

Fig. 2 shows the temporal evolution of the reaction layer thick-ness of a catalytic mechanism with j ¼ 1 s�1 at spherical elec-trodes, dr, calculated for an exponential time current I0ewt (Eq.(T10) of Table 1), three values of w (0, 1 and 10 s�1), and differentelectrode radii, including the limiting case of planar electrodes(r0 ?1).

As in the previous case, dr increases up to its stationary limit gi-ven by Eq. (T12), with this increase being faster as r0 decreases.Note that for very low values of the electrode radius, the reactionlayer thickness coincides with r0 (i.e., ðr0ÞP

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD=ðwþ jÞ

p), in

agreement with Eq. (T.12)).Moreover, the increase of the exponent w in the current

applied (and, therefore, of (w + j)) causes a faster evolution of

the behaviour of the system towards the stationary response. Thus,for example for r0 = 0.01 cm and w = 1 s�1, values of t > 0.8 s areneeded to guarantee the validity of Eq. (T.12) with an error smallerthan 1%, whereas for w = 10 s�1 this limit is t > 0:25 s. An increaseof w causes a decrease of the thickness of the stationary reactionlayer (see Eq. (T. 2)) in such a way it makes easier to reach thisvalue, as in the case discussed in Fig. 1. Note that when theelectrode radius decreases the reaction layer thickness becomescomparable to r0 in such a way that the diffusive transport isdominant, being the reaction layer identical to the diffusion layerunder these conditions (see Eq. (15)).

In Fig. 3 we show the comparison between the temporalevolution of the ratio ðdplan

r =dssr Þ obtained for the three programmed

currents considered (I0ewt (dotted line), I0 (dashed line) and I0tu

(dash-dotted line)), and also that corresponding to the voltammet-ric case, calculated by using Eq. (T.7) (solid lines). All the abovechronopotentiometric curves are independent of the currentamplitude I0, and of the applied potential E in the case of the vol-tammetric one. Note that in the case of a constant current or apower time current, the chronopotentiometric reaction layersevolve towards the constant steady state value more slowly thanthe voltammetric one. For the exponential current it possible toaccelerate this evolution, compared to the voltammetric one, bysimply increasing the value of w (compare curves with w = 1 and10 s�1). This is due to the decrease of dr as w increases (compare(T.11) and (T.12)).

In order to clarify the relationship between the reaction layerthickness and the concentration profiles of the different speciesof the catalytic mechanism, Figs. 4 and 5 show the profiles ofcA(x, t), cB(x, t) (solid lines, Fig. 4) and pseudospecies /(x, t) (solidlines, Fig. 5), obtained for the application of an exponential currentto planar electrodes for two values of w and a fixed value of time(Figs. 4a and 5a), and also for w = 1 s�1 and different values of time(Figs. 4b and 5b). We have also plotted the tangent straight time tothe profiles at the surface (dotted lines) and also the stationaryprofiles of / calculated from Eq. (9) (symbols in Fig. 5a and b).

Thus, we can see in Figs. 4 and 5 that the reaction layer thick-ness, dplan

r , is in agreement with Eqs. (10) and (14), the value of xat which the abscissa of the tangent straight time reaches a valueequal to that of the equilibrium (i.e., equal to c�i for ci(x, t), andequal to zero for /). These values have been marked in the figurewith large black dots).

All the curves in Figs. 4a and 5a have been obtained for timevalues at which the steady state has been already achieved suchthat the values of dplan

r in these figures coincide with theirstationary value dplan;ss

r given by Eq. (T.12) (solid vertical lines). Notealso that the reaction layer thickness decreases with w in line with

Fig. 2. Temporal evolution of the reaction layer thickness dr (in cm) at sphericalelectrodes of various radii, including the planar electrode case (r0 ?1) obtained inchronopotentiometry for an exponential current–time perturbation, I0ewt, withw = 0 (a), 1 (b) and 10 s�1 (c). These curves have been calculated from Eq. (T.10) inTable 1. The value of r0 are shown in the figures. Dotted lines mark the limitingstationary values of dss

r (Eq. (T12).

Fig. 1. Temporal evolution of the reaction layer thickness dr (in cm) at planarelectrodes (r0 ?1) obtained in chronopotentiometry for an exponential current–time perturbation (I0ewt, w = 1 s�1) (a), a current step perturbation (I0) (b), and apower current–time perturbation (I0tu, u = 1) (c), for different value of the catalyticconstant j. These curves have been calculated from Eqs. (T2)–(T4) in Table 1. Thevalues of j are shown in the figures. Dotted lines mark the stationary values for dss

r

(Eqs. (T.5) and (T.6)). n = 1. T = 298 K.

176 Á. Molina et al. / Journal of Electroanalytical Chemistry 655 (2011) 173–179

Eq. (T.12) in such a way that the higher w is, the more easily thesteady state is reached (see Figs. 2 and 3).

Concerning the temporal dependence of dplanr shown in Figs. 4b

and 5b, it is clear that for low values of time, a transient behaviouris observed (see curves with t = 0.1 in Figs. 4b and 5b), and there-fore the value of dplan

r does not coincide with the stationary value(dplan;ss

r ; solid vertical line). However, for time values t P 1 s, thesteady state behaviour is reached, and although the surface valuesand concentration profiles of cA, cB and /(x, t) change with time,the diffusion or reaction layer thickness is constant, i.e. it coincideswith dplan;ss

r . It is worthwhile mentioning that, although the steadystate behaviour for dr has been reached for t P 1 s, the profilesare not in steady state (i.e., they do not coincide with their

stationary limit plotted with symbols), and greater values of timeare required for this coincidence to be obtained (see Fig. 5b) [4].

Finally, we will analyze the behaviour of the potential–timecurve and its derivatives in order to get a deeper insight aboutthe steady state behaviour. When the charge transfer reaction ofScheme I is reversible, the stationary E–t response of the systemat spherical electrodes is given by [10]:

EðtÞ ¼ E0 þ RTnF

lnf�

1þK � drIðtÞnFA

Kf�

1þK þ drIðtÞnFA

!ð17Þ

Fig. 3. Temporal evolution of the ratio dplanr =dss

r at planar electrode (r0 ?1)obtained under chronopotentiometric conditions for an exponential current–timeperturbation (I0ewt) (solid lines, with w shown in the figure), a current stepperturbation (I0) (dotted-dashed lines), and a power current–time perturbation(I0tu) with u = 1 (dashed lines) (Eqs. (T2)–(T4)), and also under voltammetricconditions (Eq. (T.1) (dotted lines). j ¼ 5 s�1.

Fig. 4. Concentration profiles ci(x, t)/f⁄ with i = A and B (solid lines) and linearconcentration profiles ciðx; tÞ ¼ ððc�i � ci;surf Þ=diÞxþ ci;surf (dotted lines) of the oxi-dised and reduced species obtained for a catalytic mechanism with j ¼ 1 s�1, K = 1and D = 10�5 cm2/s for an exponential current–time perturbation with t = 0.5 s,I0 ¼ 0:5 lA and two different values of w (s�1) (a), and w = 1 s�1, I0 ¼ 10 lA and twodifferent values of time (b) at planar electrodes. These profiles have been calculatedfrom Eqs. (A.1) and (A.2) of Appendix A. The values of the exponent w (in s�1) andtime (in s) are provided in the figures.

Fig. 5. Profile of /(x, t)/f⁄ (solid lines), /ss(x, t)/f⁄ (triangles) and linear concentra-tion profiles (dotted lines) for a catalytic mechanism calculated for an exponentialcurrent–time perturbation with t = 0.5 s, I0 ¼ 0:5 lA, two different values of w (s�1)(a) and w = 1 s�1, I0 ¼ 10 lA and four different values of t: 0.1 s (circles), 0.5 s(triangles), 1 s (squares) and 1.5 s (rhombus) (b) at planar electrode. The stationaryprofiles in (b) have been calculated from Eqs. (16), (A.1), and (A.2). The values of theexponent w (in s�1) and time (in s) are provided in the figures. Other conditions asin Fig. 3.

Á. Molina et al. / Journal of Electroanalytical Chemistry 655 (2011) 173–179 177

In Fig. 6a we have plotted the transient (solid lines) and station-ary (circles) E–t responses for a reversible catalytic process withK = 0 obtained at different electrode radii for w = 1 s�1 in the ap-plied current I0ewt, and Fig. 6b shows the reciprocal derivativecurves (dewt/dE) vs. (E � E0) corresponding to the E–t responses inFig. 6a.

From Fig. 6a it can be noticed that, in agreement with results inFig. 2, by increasing the value of r0, the E–t curve reaches a station-ary behaviour even at short times (i.e. a stationary response is ob-tained, since both transient and stationary curves are perfectlysuperimposable over the whole range of time) [10]. However, thedifferences at short times between transient and stationary re-sponses are higher as the r0 value increases.

In spite of this, the differences at short times do not affect thepeak of the derivative curves, as can be seen in Fig. 6b. Moreover,the peak potential is independent of the electrode radius.

5. Conclusions

– We have deduced expressions for the reaction layer thickness ofa catalytic mechanism under transient and stationary condi-tions when three types of programmed currents (constant cur-rent, power time current and exponential time current) areapplied to planar and spherical electrodes.

Fig. 6. Chronopotentiometric transient (solid lines) and stationary (symbols)(E � E0) � t (a) and (dewt/dE) � (E � E0) curves (b) for a catalytic mechanism witha reversible charge transfer step at a spherical electrode and K = 0. These curvescorrespond to exponential current–time perturbation I0ewt with w = 1 s�1 anddifferent values of electrode radius, including the limit case of planar electrodes.They have been calculated from Eq. (12) in [10] (transient) and Eq. (17) (stationary)assuming n = 1, D = 10�5 cm2/s, I0 ¼ 100 lA and T = 298 K. The values of electroderadius (in lm) are provided in the figures.

178 Á. Molina et al. / Journal of Electroanalytical Chemistry 655 (2011) 173–179

– Under steady state conditions, the expression of the reactionlayer thickness does not depend on the electrochemical tech-nique (except in the case of an exponential current). However,under transient conditions, dr is strongly dependent on the typeof programmed current used.

– We have demonstrated that, in the case of an exponential cur-rent–time function (I0ewt), it is possible to achieve a constantvalue of the reaction layer thickness (i.e. a steady state) fasterthan when other programmed currents or a constant potentialis applied by simply increasing the value of w in the appliedcurrent.

– We have presented expressions for the chronopotentiometricpotential–time (E–t) and reciprocal derivative (dewt/dE) � Eresponses when different programmed currents are applied atelectrodes of any size, and the influence of dr on these curveshas been analyzed.

Appendix A

The expressions for the concentration profiles of the electroac-tive species involved in a catalytic process represented by Scheme(I) when a programmed current of the form I0ewt is applied to aspherical electrode of any size can be obtained by solving the equa-tion system given by Eq. (1),

cAðr; tÞn�

¼ 11þK

� r0

r2I0

nFAffiffiffiffiDp

f�t1=2e�v

X1j¼0

X1i¼0

ð�1Þini

Pi

l¼0p2jþ1þl

0BB@

1CCAhj

j!ðA:1Þ

cBðr;tÞn�

¼ K1þK

þ r0

r2I0

nFAffiffiffiffiDp

f�t1=2e�v

X1j¼0

X1i¼0

ð�1Þini

Pi

l¼0p2jþ1þl

0BB@

1CCAhj

j!ðA:2Þ

with

n ¼ 2ffiffiffiffiffiffiDtp

r0ðA:3Þ

h ¼ ðwþ jÞt ðA:4Þ

p2jþ1þl ¼2C 1þ ð2jþ1þlÞ

2

� C ð2jþ1þlÞ

2

� ðA:5Þ

where C(x) is the Euler Gamma function, and w(s) are Koutecky’sfunctions [24].

The corresponding concentrations profiles for the current stepare obtained by making w = 0 in Eqs. (A.1) and (A.2).

In the case of planar electrodes, the concentration profiles canbe obtained by making r0 ?1 in Eqs. (A.1) and (A.2). Finally, theprofiles corresponding to a simple charge transfer process can bededuced by making j = 0 in the above equations.

In the case of the power time current, the concentration profilesof species A and B in Scheme (I), are given by Eqs. (A20) and (A21)in Ref [12].

Appendix B

The functions Tvolt(v), S(n, v), Tu(n, v), Tplanu ðvÞ and G(n, h), which

appear in Table 1, have been obtained previously and are given by:

– Constant potential Voltammetry [14,15]

TvoltðvÞ ¼ 1e�vffiffiffiffiffipvp þ erf ð ffiffiffivp Þ ðB:1Þ

– Current step I0 [13,19]

Sðn;vÞ ¼ expð�vÞn4v� n2 expðn=2Þ2erfcðn=2Þ�expðvÞ 1�2

ffiffiffivpn

erf ð ffiffiffivp Þ� � �ðB:2Þ

– Power time current I0tu [12]

Tuðn;vÞ ¼ e�vX1j¼0

X1i¼0

ð�1Þini

Qil¼0

p2uþ2jþlþ1

0BBB@

1CCCAvj

j!ðB:3Þ

where the exponent u can take any real value u P �1/2 for the sur-face concentration to have physical meaning.

For a planar electrode (r0 ?1), function Tplanu ðvÞ in Table 1 has

the following form:

Tplanu ðvÞ ¼ e�v

X1j¼0

1p2uþ2jþ1

vjþ1=2

j!ðB:4Þ

– Exponential time current I0ewt [10]

Gðn;hÞ ¼ expð�hÞn4h� n2 expðn=2Þ2erfcðn=2Þ�expðhÞ 1�2

ffiffiffihp

nerf ð

ffiffiffihpÞ

!( )

ðB:5Þwhere h is a dimensionless kinetic variable,

h ¼ ðwþ jÞt ðB:6Þwith

j ¼ k1 þ k2 ðB:7Þ

Á. Molina et al. / Journal of Electroanalytical Chemistry 655 (2011) 173–179 179

Note that, in the h expression, the coupled pseudo-first orderrate constant (w + j) appears instead of the usual j value [4,10,16].

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