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Electrochimica Acta 55 (2010) 5163–5172

Contents lists available at ScienceDirect

Electrochimica Acta

journa l homepage: www.e lsev ier .com/ locate /e lec tac ta

haracterization of slow charge transfer processes in differential pulseoltammetry at spherical electrodes and microelectrodes

ngela Molinaa,∗, Francisco Martínez-Ortiza, Eduardo Labordaa, Richard G. Comptonb,∗

Departamento de Química Física, Universidad de Murcia, Espinardo 30100, Murcia, SpainDepartment of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford, OX1 3QZ, United Kingdom

r t i c l e i n f o

rticle history:eceived 23 February 2010eceived in revised form 4 April 2010ccepted 7 April 2010

a b s t r a c t

The use of differential pulse voltammetry at spherical electrodes for the study of the kinetic of chargetransfer processes is analyzed. An analytical solution is presented, valid for any value of the electroderadius, the heterogeneous rate constant and the transfer coefficient.

Several reversibility criteria are established based on the variation of DPV peak with the duration of

vailable online 14 April 2010

eywords:ifferential pulse voltammetrylow charge transfer processeterogeneous rate constant

the potential pulses and the electrode radius. Moreover, general working curves for extraction of kineticparameters from DPV experiments are given.

The anomalous shape of DPV curves for quasireversible processes with small values of the trans-fer coefficient is reported. The effect of the presence of both electroactive species on DPV curves forquasireversible and irreversible systems is also studied.

ransfer coefficienticroelectrodes

. Introduction

Voltammetric differential pulse technique (DPV) is a very usefullectrochemical technique for characterization of electrode pro-esses, based on the application of successive double potentialulses (E1, E2 with �E = E2 − E1) and the sampling of the currentt the end of each potential step (I1, I2 with �I = I2 − I1) [1,2]. Thisechnique has been used at dropping mercury electrodes, beingalled in this case differential pulse polarography (DPP), and alsot solid electrodes where a long enough recovery time after eachouble potential pulse is imposed [3,4]. The �I–E response is aeak shaped curve characteristic of the electrochemical processnder study, and it can be considered as an approximation of therst derivative of the single pulse voltammetric I–E curve whenE << RT/nF [5]. Due to its great sensitivity, DPV has been exten-

ively applied to biological systems where very small concentrationf the electroactive species is available, the use of small sized elec-rodes being very appropriate for this purpose [6–9].

In a previous paper, we deduced an expression for DPV response

f reversible electrode processes at spherical electrodes of any size10]. Hitherto, explicit equations for non-reversible charge transfereactions in DPV have been derived, but only under planar diffusion11–14].

∗ Corresponding authors.E-mail addresses: amolina@um.es (Á. Molina), richard.compton@chem.ox.ac.uk

R.G. Compton).

013-4686/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2010.04.024

© 2010 Elsevier Ltd. All rights reserved.

The aim of this paper is to extend this study to spherical diffu-sion by deducing an explicit analytical expression applicable to DPVassuming that the duration of the first pulse is much longer thanthat of the second one (t2 ≤ t1/50), since under these conditionsthe solution can be easily analyzed to extract kinetic parame-ters. In order to show the validity of the analytical solution, wehave compared the analytical results with those obtained fromnumerical calculation, the difference being always smaller than0.5%.

The equation derived is valid for spherical electrodes of any sizeand it considers the presence of either or both electroactive speciesforming an O/R redox couple in bulk solution. From this, we deduce,as limiting cases, the expressions corresponding to reversible pro-cesses [10] as well as to planar electrodes (r0 → ∞) and sphericalultramicroelectrodes (r0 → 0). In this last case, the DPV station-ary response is obviously similar to that obtained in square wavevoltammetry (SWV) when the pulse height (�E) in DPV is equal tothe double of square wave potential, i.e. 2ESWV = �E.

The effects of the heterogeneous rate constant (ks = k0√

t2/D)and the transfer coefficient (˛) on the �I–E curves are analyzed,pointing out that for k0 values around 10−3 cm/s an anomaloussplitting in two peaks of DPV curve is observed for ˛ < 0.3, one peakbeing situated at more positive potentials than E0′

and the other

one at more negative potentials. This phenomenon is more appar-ent for positive �E values. This previously unobserved feature ofDPV is important to report since the peak splitting might easily bemistaken by the experimentalist as indicating the presence of extraredox active species.

5 mica A

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2

O

smes

ı

w

ı

ws

d

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at which no current flows [4]. Lovric has shown for reversible pro-cesses [3], and we have proved by numerical simulations, that awaiting period ≤5·t1 is long enough to achieve this condition atspherical electrodes; the smaller the electrode radius, the shorterthe waiting period required. So, the expression for DPV response isimmediately particularized from Eq. (12):

164 Á. Molina et al. / Electrochi

The characterization of electrode processes, that is, the extrac-ion of kinetic parameters, is a major issue in electrochemistry1,2,13–16]. In this paper several criteria of reversibility are given,or example, based on the asymmetry of the DPV peaks for �E < 0nd �E > 0 at different electrode radii, which depends on k0 value.nother criterion could be based on the great influence of theariation of the double pulse duration on the peak current foruasireversible systems, whereas for totally irreversible ones thehift of the position of DPV peak towards more negative potentials ishe most remarkable effect. It is also worth highlighting that whilsthe initial presence of both electroactive species does not affect thehape of DPV curve for reversible processes, this is considerablyistorted for non-reversible systems in such a way that for totally

rreversible processes two net separated peaks are obtained, oneorresponding to the reduction of species O and the other one tohe oxidation of species R. The greater the distortion is, the smallerhe k0 value. Finally, from the mathematical solutions presented,orking curves are given for extraction of kinetic parameters from

xperimental data.

. Theory

Let us consider the following charge transfer reaction:

+ e−kf

�kb

R (1)

First, the applied potential is set at a value E1 at a stationarypherical electrode during the interval 0 ≤ �1 ≤ t1. The diffusionass transport of the electroactive species towards or from the

lectrode surface is described by the following differential equationystem:

ˆc(1)O (r, t) = ıc(1)

R (r, t) = 0 (2)

ith:

ˆ = ∂

∂t− D

(∂2

∂r2+ 2

r

∂

∂r

)(3)

here we have assumed that both electroactive species have theame value of the diffusion coefficient (DO = DR = D).

The boundary conditions to be fulfilled by the solutions of theifferential equations are given by:

�1 ≥ 0, r → ∞�1 = 0, r ≥ r0

}c(1)

O = c∗O, c(1)

R = c∗R (4)

1 > 0, r = r0 :

(∂c(1)

O (r, t)

∂r

)r=r0

= −(

∂c(1)R (r, t)

∂r

)r=r0

(5)

(∂c(1)

O

∂r

)r=r0

= k0K−˛1 c(1)

O (r0, t) − k0K1−˛1 c(1)

R (r0, t) (6)

here the different parameters are defined in Appendix A and Table.

By means of the mathematical procedure described in Appendix, the expression for the current is derived:

1(t) = Id(∞) · (�1 − 1)(1 − �K1)�1(1 + K1)

[1 + (�1 − 1) · F

(�1

2

)](7)

here all the variables and functions are given Appendix A. Thisxpression coincides with that obtained by Delmastro and Smith17] when DO = DR = D.

cta 55 (2010) 5163–5172

At the time t ≥ t1, the applied potential is stepped from E1 to E2and, analogously to the above, during this second period (t = t1 +�2; 0 ≤ �2 ≤ t2) the mass transport of species O and R is describedby:

ıc(2)O (r, t) = ıc(2)

R (r, t) = 0 (8)

with the following boundary value problem:

�2 ≥ 0, r → ∞�2 = 0, r ≥ r0

}c(2)

O = c(1)O , c(2)

R = c(1)R (9)

�2 > 0, r = r0 :

(∂c(2)

O (r, t)

∂r

)r=r0

= −(

∂c(2)R (r, t)

∂r

)r=r0

(10)

D

(∂c(2)

O

∂r

)r=r0

= k0K−˛2 c(2)

O (r0, t) − k0K1−˛2 c(2)

R (r0, t) (11)

Eq. (11) assumes that the transition state does not change withpotential and so the same value of ˛ applies to the forward andreverse processes.

From the mathematical procedure given in Appendix B, theexpression for the current of the second potential pulse is obtained:

I2 = I1(t1 + �2) + Id(∞)(�2 − 1) · f (�1/2)

�2(1 + K2)

[1 + (�2 − 1) · F

(�2

2

)](12)

with all the variables and functions given in Appendix A.The DPV technique consists of the application of successive dou-

ble potential pulses, so that the initial equilibrium conditions arere-established after each one and the duration of the second pulseis much shorter than that of the first one (t2 << t1). The current isregistered twice, at the end of the first pulse (I1(t1)) and at the endof the second one (I2(t1 + t2)), and DPV curve is a plot of the differ-ence between the two current samples (�I = I2(t1 + t2) − I1(t1)) vsa potential x-axis [1,2]. In a previous paper [10], we discussed theconvenience of choosing the arithmetic average of both potentialvalues (E1,2 = (E1 + E2)/2, see Fig. 1) as the potential-axis.

As mentioned above, in DPV technique initial equilibrium con-ditions are re-established before the application of each doublepotential pulse, for example, by renewal of the electrode (mer-cury drop electrode), by open circuiting the working electrode fora waiting period [10] or by setting the applied potential at a value

�I

Id(∞)= (�2 − 1) · f (�′

1/2)�2 (1 + K2)

[1 + (�2 − 1) · F

(�2

2

)](13)

where:

�′1 = 2

√D · t1

r0+ 2

√t1

Dk0K−˛

1 (1 + K1) (14)

Á. Molina et al. / Electrochimica Acta 55 (2010) 5163–5172 5165

F ble puc s.

2

c

-

-

-

1 2All the calculations have been performed in quadruple preci-

sion with homemade programs written in FORTRAN 90 (availableas Supporting information).

ig. 1. Differential pulse voltammetry: (a) Potential-time function; after each douurrent samples; (d) signals for normal ((�E < 0)) and reverse (�E > 0) DPV mode

.1. Particular cases

From Eq. (13), expressions under some interesting conditionsan be derived:

Planar electrodes (r0 → ∞). Expression (13) becomes:

�Iplane

Id(t2)=

√t2

D· k0 · K−˛

2 · F(�p2/2)

×{

1 − �K2 − (1 − �K1)(

1 + K2

1 + K1

)

×[

1 +(

1 + K1

1 + K2

(K2

K1

)˛

− 1

)· F(�p

1/2)

]}(15)

Ultramicroelectrodes (UME). When r0 <<√

Dt2, Eq. (13) tendsto:

�IUME

Id(∞)= kmicro

s

[K−˛

2 (1 − �K2)�2

− K−˛1 (1 − �K1)

�1

](16)

which, in the case of reversible systems (k0 → ∞), is equivalentto that obtained in Square Wave Voltammetry under steady-state conditions [18]. Note that the use of this equation is quiterestricted since the reach of steady state conditions in DPVrequires the use of very small electrodes due to that the durationof the second potential pulse (t2) is very short.Reversible process. By making k0 → ∞ we obtain:( )

�Irev

Id(∞)= (1 + �)

(K1 − K2)(1 + K1)(1 + K2)

1 + r0√Dt2

(17)

which coincides with that deduced in Ref. [10] for DO = DR.

lse, the initial equilibrium conditions are restored; (b) technique parameters; (c)

3. Results and discussion

As discussed in Appendix B, for the resolution of the problem wehave supposed that the mathematical concentration profiles of thefirst pulse for the total time t1 + t2 is not disturbed by the applicationof the second one (Eq. (B35)). This assumption is fully valid for anyelectrode radius in DPV technique, where the duration of the secondpulse is much shorter than that of the first one (t1 >> t2). We haveconfirmed the validity of the analytical Eqs. (7) and (12) in DPVconditions by comparison with numerical results [19], since non-significant deviations are obtained (less than 0.5%) around the peakpotential when t /t > 50.

Fig. 2. Influence of the heterogeneous rate constant (k0) on DPV curves (Eq. (13)).k0 values are marked on the curves. r0 = 30 �m, ˛ = 0.5, �E = +20 mV, t1 = 1s,t1/t2 = 100, D = 10−5 cm2/s, � = 0.

5166 Á. Molina et al. / Electrochimica Acta 55 (2010) 5163–5172

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per

oosptop

˛f

tipsraons

as

Fig. 4. Influence of the initial presence of the electrode reaction product (speciesR) on (a) DPV curves (Eq. (13)) and (b) normal pulse voltammetry curves (Eq. (7))

ig. 3. Anomalous behaviour of DPV curves for small values of the transfer coeffi-ient (˛) (Eq. (13)). �E = +50 mV. (a) Different ˛ values are considered (marked onhe curves); r0 = 30 �m; (b) different electrode sizes are considered (marked on theurves); ˛ = 0.2. Other conditions as in Fig. 2.

In Fig. 2 the effect of the reversibility of the electron transferrocess on DPV peaks is studied at conventional spherical micro-lectrodes (2 = 0.2, which corresponds to an electrode radius0 = 30 �m for t2 = 10 ms and D = 10−5 cm2/s).

As previously depicted for planar electrodes [1,14], the decreasef the heterogeneous rate constant gives rise to the diminutionf the peak current, the increase of the peak half width and thehift of the peak potential towards more negative values. For com-letely irreversible systems (very small k0 values), it is observedhat the current peak and the peak half width become independentf the rate constant, whereas the position of the peak (i.e., the peakotential Epeak) is sensitive to k0.

For the above study the usual value of the transfer coefficient= 0.5 has been considered. With small ˛ values, DPV peaks are

ound to show a special shape under certain conditions.As can be seen in Fig. 3a, for ˛ < 0.3 �I − E curves corresponding

o quasireversible processes with k0∼10−3 cm/s presents a strik-ng splitting of the peak, a sharper peak appearing at more anodicotentials. This phenomenon is promoted by small transfer con-tants and it is more obvious for positive pulse heights (�E > 0,everse mode) and for planar electrodes (see Fig. 3b), where thenodic peak is even greater than the cathodic one. The descriptionf this phenomenon is of great interest since this could lead to erro-

eous interpretation of experimental data for real quasireversibleystems with low ˛ values [14,20].

It is worth highlighting that this anomalous behaviour is char-cteristic of quasireversible processes with k0∼10−3 cm/s, and theplitting is not observed for greater (k0 ≥ 10−2 cm/s) or smaller

for reversible (k0 = 0.3 cm/s, ), quasireversible (k0 = 1.5 · 10−3 cm/s, – – –) andirreversible (k0 = 3 · 10−5 cm/s, · · ·) systems. � = 1, �E = −50 mV. Other conditionsas in Fig. 2.

(k0 ≤ 10−4 cm/s) rate constants, where the usual single peak isobtained regardless of ˛ value. In the same way, for ˛ ≥ 0.5 thetypical shape of DPV curves is found for any k0.

In Fig. 4 we compare the variation of DPV (Fig. 4a) and nor-mal pulse voltammetry (NPV, Fig. 4b) curves with k0 when bothelectroactive species are initially present in the solution (� =c∗

R/c∗O /= 0).

From this figure we can observe that the initial presence of thereaction product has no influence on the shape of DPV curves forreversible processes, though the signal is proportional to the totalbulk concentration c∗

O + c∗R according to Eq. (17).

However, the shape of DPV curve for non-reversible systemsis considerably affected by the initial presence of the reducedspecies, the greater the distortion, the smaller the k0 value. For com-pletely irreversible systems there exist two well-separated peaks:the cathodic one associated to the reduction process of species Oand the anodic one to the oxidation of species R, as can be deducedby comparing DPV and NPV results (Fig. 4b). As expected, eachpeak only depends on the corresponding reactive species, since thebackward process does not take place at such extreme potentials.

Note that due to the subtractive character of DPV technique,both peaks exhibit positive values of DPV current since the sign of

�I depends on the sign of the pulse height (�E) and not on theprocess taking place (reduction or oxidation). So, the initial pres-ence of both electroactive species may be also mistaken for an EEmechanism in DPV.

Á. Molina et al. / Electrochimica Acta 55 (2010) 5163–5172 5167

Fig. 5. Influence of the double pulse duration on normalized DPV curves (Eq. (15))for reversible (k0 = 4 cm/s, corresponding to anthracene/anthracene radical anioncouple on Pt electrode [2]), quasireversible (k0 = 7 · 10−3 cm/s, corresponding toFe(III)/Fe(II) couple on Pt electrode [2]) and irreversible (k0 = 10−5 cm/s, corre-so�

3t

rtrpa

ton1bOtrtn

mtddtvewgvawp

sssw(o

EUME = Eirrev,UME = E0′ + RTln(kmicro) (20)

ponding to Cr(III)/Cr(II) couple on Hg electrode [23]) systems. t2 values are markedn the curves, while keeping the ratio t1/t2 = 100. Planar electrode (r0 → ∞),E = +50 mV. Other conditions as in Fig. 2.

.1. Characterization and determination of kinetic parameters ofhe charge transfer process

From the equations obtained for DPV, it is deduced that theeversibility of an electrode process can be characterized by varyinghe time scale of the DPV experiment or by varying the electrodeadius. As shown in the following figures, when the charge transferrocess is not reversible, the peak current and/or the peak potentialre affected by these variables.

Fig. 5 demonstrates that, when the ratio t1/t2 remains constant,he double pulse duration has not influence either on �Ipeak/Id(t2)r the peak potential for reversible electrode processes at pla-ar electrodes. For quasireversible processes (10−2 < k0(cm/s) <0−3) the peak height is greatly affected by the duration of the dou-le potential step (t1 + t2), while the peak potential slightly varies.n the other hand, for totally irreversible processes the peak poten-

ial notably varies with the time scale of the experiment. So, theeversibility degree of the charge transfer process is easily charac-erizable from the time behaviour of the height and position of theormalized DPV peak.

The dependence of �Ipeak with 1/√

t2, keeping constant experi-ental variables, provides a very simple and reliable criterion to

est reversibility in DPV. As shown in Fig. 6 and Eq. (17), for aiffusion-controlled reversible process �Ipeak is linearly depen-ent on 1/

√t2 for any electrode radius, while this dependence

urns no-linear and always lower for quasi-reversible or totally irre-ersible charge transfer reaction. This criterion is very easy to applyxperimentally, since from measurements at different t2-values,e can obtain different differential pulse voltammograms in a sin-

le experiment. The behaviour of the peak current with t2 is alsoaluable to discard adsorption, because in presence of adsorptioncharacteristic linear dependence of ln

(∣∣�Ipeak

∣∣) with t2 is found,hich does not hold when both electroactive species are in solutionhase.

For quantitative characterization of the system, in Fig. 7a wetudy the variation of the peak position with respect to the timecale at planar electrodes. It can be seen that for totally irreversible

ystems the position of the DPV peak has a linear relationshipith respect to the logarithm of the dimensionless rate constant

ks = k0√

t2/D). Thus, a simple expression for the peak potential isbtained, which permits us to readily determine kinetic parame-

Fig. 6. Variation of the peak current �Ipeak with 1/√

t2 for reversible ((Eq. (17)),k0 = 1 cm/s), quasireversible ((Eq. (13)), k0 = 3 · 10−3 cm/s) and irreversible ((Eq.(13)), k0 < 10−4 cm/s) systems. t1 = 1s, r0 = 30 �m, �E = +50 mV, D = 10−5 cm2/s,� = 0.

ters:

Eplanepeak = A

(˛,

t1

t2, �E

)+ Eirrev,plane

1/2 (18)

where

Eirrev,plane1/2 = E0′ + RT

˛Fln(2.31ks) (19)

and A = 49.7 mV for the typical values ˛ = 0.5, �E = +50 mV andt1/t2 = 100.

So, by varying the potential pulse duration (keeping the ratiot1/t2 constant) the value of the transfer coefficient (˛) can be

directly determined from the slope of the plot Eplanepeak vs ln

(√t2/D

).

Once ˛ value is known, the formal potential (E0′) or the hetero-

geneous rate constant (k0) is accessible from the intercept of thestraight line.

In a similar way, at ultramicroelectrodes a linear relationshipis obtained between the peak potential and the logarithm of theelectrode radius, as observed in Fig. 7b. A simple expression is alsodeduced for this case:

peak 1/2 ˛F s

which is totally coincident with the half-wave potential forirreversible processes due to the time-independent response.According to this equation, the kinetic parameters could be

5168 Á. Molina et al. / Electrochimica Acta 55 (2010) 5163–5172

Fa˛+

om

itf(wm

aE

ptfaic

rpiaaf

Fig. 8. Influence of electrode sphericity on DPV curves for different k0 values: (a)0.3 cm/s, (b) 3 · 10−3 cm/s and (c) 3 · 10−5 cm/s. Three electrode sizes are consid-

ig. 7. (a) Epeak − E0′vs Ln(2.31 · ks) curves for planar electrodes (Eq. (15), r0 → ∞)

nd (b) Epeak − E0′vs ln(kmicro

s ) for ultramicroelectrodes (Eq. (16), r0 < 0.1 �m). Threevalues are considered: ˛ = 0.4 (–·· –·· –··), ˛ = 0.5 ( ) and ˛ = 0.7 (– – –). �E =50 mV. Other conditions as in Fig. 2.

btained from the variation of the peak potential for different ultra-icroelectrode radii.Besides the time scale of the experiment, the electrode radius

s other variable we can work with in order to elucidate the sys-em reversibility. In Fig. 8 we study its influence on DPV curvesor the three general situations: reversible (Fig. 8a), quasireversibleFig. 8b) and irreversible (Fig. 8c) electrode processes. In all cases,e have considered the application of negative (�E < 0, normalode) and positive (�E > 0, reverse mode) pulse heights.For a reversible system (Fig. 8a), the peak potential is not

ffected by the electrode sphericity (2), and it takes the valuerev1,2peak = E0′

[10]. Moreover, for a given electrode radius, DPVeaks corresponding to the normal and the reverse modes havehe same peak potential and the same peak current so that they areully symmetrical. This complete symmetry of the peaks, as wells the independence of the peak potential with r0, are character-stic features of DPV curves of reversible behaviour, and so theyonstitute simple diagnostic criteria of reversibility.

In Fig. 8b and c the quasireversible and irreversible cases areespectively analyzed. Contrarily to that observed for reversiblerocesses, the peak potential depends on the electrode size, tak-

ng more negative values as 2 increases, that is, as r0 diminishesnd/or the second pulse duration is longer. Both experimental vari-bles can be readily modified in such a way that the shift of the peakor different 2 values gives information about the reversibility of

ered: planar electrode (Eq. (15), 2 → 0, ), microelectrode (Eq. (13), 2 = 0.4,

– – –) and ultramicroelectrode (Eq. (13), 2 = 1, · · ·· · · ·· · · ··).∣∣�E

∣∣ = 50 mV. Other

conditions as in Fig. 2.

the electrode reaction. Moreover, the values of the peak current innormal and reverse mode for a given 2 value are different.

For determination of kinetic parameter with DPV technique, thevariation of the peak potential and the normalized peak current�Ipeak/Id(t2) with the determining dimensionless parameters areproposed as working surfaces in Figs. 9a and 10a, respectively. Theplots vs log(2 = 2

√Dt2/r0) are also plotted in Figs. 9b and 10b for

different kmicro = k0r /D values. Note that given a kmicro value, the

s 0 svariation of the parameter 2 is equivalent to the variation of thedouble pulse duration; so, they are general working curves fromwhich the kinetic parameters of the electrode process can be eas-ily obtained, once D value is known, by measurement of the peak

Á. Molina et al. / Electrochimica Acta 55 (2010) 5163–5172 5169

Fw˛

pp

ptistbk

c�vit

gtfap

s

ig. 9. (a) Epeak − E0′ – log(2) – log(kmicros ) working surface; (b) Epeak − E0′

vs log(2)orking curves for several kmicro

s values (marked on curves). Eq. (13), �E = +50 mV,= 0.5, t1/t2 = 100.

otential and the peak current for different values of the doubleulse duration, keeping constant the ratio t1/t2.

Regarding the peak potential (Fig. 9), the duration of the doubleulse has no influence on the peak position for reversible processes,aking the E0′

value which corresponds to the “plateau” observedn Fig. 9a. On the other hand, for quasireversible and irreversibleystems the peak potential shifts towards less negative values whenhe double pulse duration increases. So, the analysis of the timeehaviour of the peak potential is very appropriate to determineinetic parameters for irreversible and quasireversible systems.

With respect to the normalized peak current, in Fig. 10 wean see that for both reversible and totally irreversible processesIpeak/Id(t2) is not sensitive to k0 value. Contrarily, for quasire-

ersible processes the duration of the double pulse has a greatnfluence on �Ipeak/Id(t2) (see also Fig. 5) so that we can determinehe kinetic parameters from Fig. 10b.

Note that, in all cases, the diminution of the electrode radiusives rise to the diminution of the “apparent” reversibility ofhe system under study, due to the enhancement of the dif-

usion transport [1,2]. So, the range of applicability of thebove-proposed characterization methods reaches faster electroderocesses (greater k0 values) when smaller electrodes are used.

For the working curves, the general case ˛ = 0.5 has been con-idered. Nevertheless, the proposed methodology is general, so

Fig. 10. (a) �Ipeak/Id(t2) – log(2) – log(kmicros ) working surface; (b) �Ipeak/Id(t2) vs

log(2) working curves for several kmicros values (marked on curves). Eq. (13), �E =

+50 mV, ˛ = 0.5, t1/t2 = 100.

that from the simultaneous analysis of the variation of the peakpotential and the peak current with the double pulse duration, thecomplete characterization of a charge transfer process is possible,permitting us to obtain the values of both the rate constant and thetransfer coefficient.

4. Conclusions

Analytical expressions for the response of slow charge transferreactions at spherical electrodes in differential pulse voltammetryare obtained, valid whatever the electrode size and the reversibilitydegree of the electron transfer process.

From this solution, the influence of the electrode kinetics, theelectrode sphericity and the initial presence of the reaction prod-uct are studied. Thus, the anomalous shape of DPV curves forquasireversible processes with k0∼10−3 cm/s and small values ofthe transfer coefficient is reported, which can lead to misinterpre-tation of experimental data. When both electroactive species areinitially present two peaks are obtained when the electrode process

is quasireversible or irreversible, corresponding to the reductionand oxidation processes.

The value of DPV for elucidation of the electrode processreversibility is also demonstrated. Several diagnosis criteria areproposed, based on the variation of the DPV curve with the duration

5 mica A

oudo

A

pNFt

A

F

�

I

I

�

K

�

�

�

�

�

�

f

T

170 Á. Molina et al. / Electrochi

f the double pulse, the electrode radius and the reaction prod-ct concentration. Moreover, general working curves are given foretermination of kinetic parameters from the position and heightf DPV peak.

cknowledgements

A.M., F. M-O and E.L. greatly appreciate the financial supportrovided by the Dirección General de Investigación (MEC) (Projectumbers CTQ2006-12522/BQU and CTQ2009-13023) and by theundación SENECA (Project Number 08813/PI/08). Also, E. L. thankshe Ministerio de Ciencia e Innovacion for the grant received.

ppendix A. Variables and functions

(x) = exp (x)2 · erfc(x) (A.1)

= c∗R

c∗O

(A.2)

d(∞) = FAc∗OD

r0(A.3)

d(t2) = FAc∗O

√D√

t2(A.4)

j = F

RT(Ej − E0′

) j ≡ 1, 2 (A.5)

j = exp(�j) j ≡ 1, 2 (A.6)

1 = 1 + kmicros K−˛

1 (1 + K1) (A.7)

2 = 1 + kmicros K−˛

2 (1 + K2) (A.8)

1 = 1 + �p1 (A.9)

1 = 2√

D · (t1 + t2)r0

(A.10)

p1 = 2

√t1 + t2

Dk0K−˛

1 (1 + K1) = (�1)1=0 (A.11)

2 = 2 + �p2 (A.12)

2 = 2√

D · t2

r0(A.13)√

p2 = 2

t2

Dk0K−˛

2 (1 + K2) = (�2)2=0 (A.14)

(x) = 1 − �K2 − (1 − �K1)(

K2

K1

)˛[

�2

�1+(

1 − �2

�1

)· F(x)

](A.15)

able 1. Definitions

r0 radius of the electrodetj, j ≡ 1, 2 duration of the j-th potential pulse�j, j ≡ 1, 2 time during which the j-th potential pulse is appliedD diffusion coefficient of the electroactive speciesc∗

i, i ≡ O, R bulk concentration of species i

c(j)i

,i ≡ O, Ri ≡ 1, 2

concentration profile of species i for the j-th potentialpulse

kf , kb heterogeneous rate constants of reduction, oxidationprocesses

k0 standard heterogeneous rate constant

ks

(= k0

√t2D

)kmicro

s

(= k0 · r0

D

)˛ transfer coefficientEj, j ≡ 1, 2 potential applied during the j-th pulse�E pulse height

cta 55 (2010) 5163–5172

Appendix B.

In order to solve the problem we introduce the following vari-able change,

u(j)i

(r, t) = c(j)i

(r, t) · r

c∗O · r0

i ≡ O, Rj ≡ 1, 2

(B1)

Considering the new variable, u(j)i

(r, t), the differential equationsystem and the corresponding boundary value problem for the firstpotential pulse (t = �1; 0 ≤ �1 ≤ t1) become into,

∂u(1)O (r, t)

∂t= D

∂2u(1)O (r, t)

∂r2

∂u(1)R (r, t)

∂t= D

∂2u(1)R (r, t)

∂r2

⎫⎪⎪⎬⎪⎪⎭ (B2)

t ≥ 0, r → ∞t = 0, r ≥ r0

}u(1)

O = r

r0; u(1)

R = �r

r0(B3)

t > 0r = r0

}(∂u(1)

O

∂r

)r=r0

− u(1)O (r0)

r0= −

⎡⎣(

∂u(1)R

∂r

)r=r0

− u(1)R (r0)

r0

⎤⎦(B4)

D

⎡⎣(

∂u(1)O

∂r

)r=r0

− u(1)O (r0)

r0

⎤⎦ = k0K−˛

1 u(1)O (r0) − k0K1−˛

1 u(1)R (r0)

(B5)

By applying Koutecky’s dimensionless parameter method[21,22], we suppose that solutions are functional series of the

dimensionless variable (

= 2√

D·tr0

),

u(1)O (r, t) = u(1)

O (s, ) =∑

k

�(1)k

(s)k (B6)

u(1)R (r, t) = u(1)

R (s, ) =∑

k

(1)k

(s)k (B7)

Taking into account the dimensionless variables s(

= r−r02√

Dt

)and ,

the differential equation system and the boundary value problemturn into,

∂2u(1)O

∂s2+ 2s

∂u(1)O

∂s− 2

∂u(1)O

∂= 0

∂2u(1)R

∂s2+ 2s

∂u(1)R

∂s− 2

∂u(1)R

∂= 0

⎫⎪⎪⎬⎪⎪⎭

s → ∞

(B8)

u(1)O (∞) = 1 + s; u(1)

R (∞) = �(1 + s)s = 0

(B9)

(∂u(1)

O

∂s

)− u(1)

O (0) = −[(

∂u(1)R

∂s

)− u(1)

R (0)

](B10)

s=0 s=0(∂u(1)

O

∂s

)s=0

− u(1)O (0) = r0k0K−˛

1D

[u(1)

O (0) − K1u(1)R (0)

] (B11)

mica A

d

t

wt�t

�

�

�

�

wp

�

�

�

wtp

�(b

u

bc

Á. Molina et al. / Electrochi

By introducing the expressions (B6) and (B7) into Eq. (B8), theifferential equation system becomes,

�(1)′′k

(s) + 2s�(1)′k

(s) − 2k�k(s) = 0

(1)′′k

(s) + 2s (1)′k

(s) − 2k k(s) = 0

}(B12)

he solutions of which have the following form,

�(1)k

(s) = �(1)k

(∞)

lim Lks→∞

Lk + a(1)k

· �k(s)

�(1)k

(s) = �(1)k

(∞)

lim Lks→∞

Lk + b(1)k

· �k(s)

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(B13)

here a(1)k

and b(1)k

are constants that will be determined fromhe boundary value problem, Lj are s1-powers numeric series, and

k(s1) are Koutecky’s functions which have the following proper-ies,

k(0) = 1 (B14)

k(∞) = 0 (B15)

′k(x) = −pk�k−1(x) (B16)

0(x) = 1 − erf (x) (B17)

Taking into account the form of the solutions (Eq. (B13)) alongith the properties of Koutecky’s functions, the boundary valueroblem is given by,

s → ∞�(1)

0 (∞) = 1 (1)0 (∞) = �

�(1)1 (∞) = s; (1)

1 (∞) = �s

�(1)k

(∞) = 0 (1)k

(∞) = 0 k > 1s = 0

(B18)

(1)′k

(0) − �(1)k−1(0) = − (1)′

k(0) + (1)

k−1(0) (B19)

(1)′k

(0) − �(1)k−1(0) = r0k0K−˛

1D

[�(1)

k−1(0) − K1 (1)k−1(0)

](B20)

By applying Eqs. (B18)–(B20), we obtain the expressions for(1)k

(s) and (1)k

(s), and by introducing them in Eqs. (B6) and (B7)

e find the solutions of the problem (u(1)O (r, t) and u(1)

R (r, t)). Fromhese, the I–E expression for the application of the first potentialulse at spherical electrodes (Eq. (7)) is obtained.

Regarding the application of the second potential step (t = t1 +2; 0 ≤ �2 ≤ t2), due to that the diffusion operator ı (given by Eq.3)) is linear, the solutions of the differential equation system can

e written as,

(2)i

(r, t) = u(1)i

(r, t) + u(2)i

(r, �2) i ≡ O, R (B21)

eing u(2)i

(r, �2) the new unknown partial solutions with null initialonditions,

∂u(2)O (r, �2)

∂�2= D

∂2u(2)O (r, �2)

∂r2

∂u(2)R (r, �2)

∂�2= D

∂2u(2)R (r, �2)

∂r2

⎫⎪⎪⎬⎪⎪⎭ (B22)

cta 55 (2010) 5163–5172 5171

�2 ≥ 0, r → ∞�2 = 0, r ≥ r0

}

u(2)O = 0; u(2)

R = 0 (B23)

�2 > 0r = r0

}

(∂u(2)

O

∂r

)r=r0

− u(2)O (r0)

r0= −

⎡⎣(

∂u(2)R

∂r

)r=r0

− u(2)R (r0)

r0

⎤⎦ (B24)

D

[(∂u(2)

O

∂r

)r=r0

−u(2)

O (r0)

r0

]− k0K−˛

2 u(2)O (r0) + k0K1−˛

2 u(2)R (r0)

= −

{D

[(∂u(1)

O

∂r

)r=r0

−u(1)

O (r0)

r0

]− k0K−˛

2 u(1)O (r0) + k0K1−˛

2 u(1)R (r0)

}(B25)

Again we solve this problem by means of Koutecky’s dimen-sionless parameter method. Thus, we introduce the dimensionless

variables s2

(= r−r0

2√

D�2

)and 2

(= 2

√D�2

r0

), so that the solutions are

given by,

u(2)O (r, �2) = u(2)

O (s2, 2) =∑

k

�(2)k

(s2)k2 (B26)

u(2)R (r, �2) = u(2)

R (s2, 2) =∑

k

(2)k

(s2)k2 (B27)

and the differential equation system and the boundary value prob-lem turn into,

∂2u(2)O

∂s22

+ 2s2∂u(2)

O

∂s2− 22

∂u(2)O

∂2= 0

∂2u(2)R

∂s22

+ 2s2∂u(2)

R∂s2

− 22∂u(2)

R

∂2= 0

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(B28)

s2 → ∞u(2)

O (∞) = 0; u(2)R (∞) = 0

s2 = 0(B29)

(∂u(2)

O

∂s2

)s2=0

− 2u(2)O (0) = −

⎡⎣(

∂u(2)R

∂s2

)s2=0

− 2u(2)R (0)

⎤⎦ (B30)

(∂u(2)

O

∂s2

)s2=0

− 2u(2)O (0) − r0k0K−˛

2

D

[u(2)

O (0) − K2u(2)R (0)

]2

= −

{(∂u(1)

O

∂s1

)s1=0

11

− u(1)O (0) − r0k0K−˛

2

D

[u(1)

O (0) − K2u(1)R (0)

]}2 (B31)

The solutions of the system (B28) have the following form,

�(2)k

(s2) = �(2)k

(∞)Lk + a(2)

k· �k(s2)

⎫⎪⎪⎪⎪

lim Lks2→∞

�(2)k

(s2) = �(2)k

(∞)

lim Lks2→∞

Lk + b(2)k

· �k(s2)

⎪⎬⎪⎪⎪⎪⎪⎭

(B32)

5 mica A

w

�

dhsn

t

(tp

A

t

[

[[[[[

[

[[[

[20] L.M. Abrantes, J. González, A. Molina, F. Saavedra, Electrochim. Acta 45 (1999)457.

[21] J. Koutecky, Czech. J. Phys. 2 (1953) 50.

172 Á. Molina et al. / Electrochi

hich must fulfil the following limit and surface conditions:

s2 → ∞�(2)

k(∞) = 0; (2)

k(∞) = 0 ∀k

s2 = 0(B33)

(2)′k

(0) − �(2)k−1(0) = − (2)′

k(0) + (2)

k−1(0) (B34)

• �(2)′k

(0) − �(2)k−1

(0) − r0k0K−˛2

D

[�(2)

k−1(0) − K2 (2)

k−1(0)

]= 0 k /= 1

• �(2)′1 (0) − �(2)

0 (0) − r0k0K−˛2

D

[�(2)

0 (0) − K2 (2)0 (0)

]= −

{(∂u(1)

O

∂s1

)s1=0

11

− u(1)O (0) − r0k0K−˛

2

D

[u(1)

O (0) − K2u(1)R (0)

]}k = 1

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(B35)

As can be seen, the surface condition (B35) introduces depen-ence with the first potential pulse. Given that in DPV technique weave that t1 >> t2, we assume that the mathematical form of theolutions corresponding to the first pulse (u(1)

O (r, t) and u(1)R (r, t)) is

ot disturbed by the application of the second one.By applying the conditions given by Eqs. (B33)–(B35), we deduce

he expressions for �(2)k

(s2) and (2)k

(s2), and taking into account Eqs.

B21), (B26) and (B27), the solutions u(2)i

(r, t) are obtained. Fromhese, the equation for the current corresponding to the secondotential pulse (Eq. (12)) is derived.

ppendix C. Supplementary data

Supplementary data associated with this article can be found, inhe online version, at doi:10.1016/j.electacta.2010.04.024.

[[

cta 55 (2010) 5163–5172

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