Journal of Electroanalytical Chemistry 660 (2011) 169–177

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Journal of Electroanalytical Chemistry

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A comparison of Marcus–Hush vs. Butler–Volmer electrode kinetics usingpotential pulse voltammetric techniques

Eduardo Laborda a,b, Martin C. Henstridge a, Angela Molina b, Francisco Martínez-Ortiz b,Richard G. Compton a,⇑a Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdomb Departamento de Química Física, Universidad de Murcia, Espinardo 30100, Murcia, Spain

a r t i c l e i n f o

Article history:Received 25 May 2011Received in revised form 22 June 2011Accepted 24 June 2011Available online 12 July 2011

Keywords:Electrode kineticsButler–Volmer modelMarcus–Hush modelPotential pulse techniques

1572-6657/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jelechem.2011.06.027

⇑ Corresponding author. Tel.: +44 (0) 1865 275413;E-mail address: richard.compton@chem.ox.ac.uk (R

a b s t r a c t

Simulated voltammograms obtained by employing Butler–Volmer (BV) and Marcus–Hush (MH) modelsto describe the electrode kinetics are compared for commonly used potential pulse techniques: chrono-amperometry, Normal Pulse Voltammetry, Differential Multi Pulse Voltammetry, Square Wave Voltam-metry and Reverse Pulse Voltammetry. A comparison between both approaches is made as a functionof the heterogeneous rate constant, the electrode size, the applied potential and the electrochemicalmethod, establishing the conditions in which possible differences might be observed. The effect of thesedifferences in the extraction of kinetic parameters, diffusion coefficients and electrode radii are exam-ined, and criteria are given to detect possible deviations of the experimental system from Butler–Volmerkinetics from the behaviour of the chronoamperometric limiting current. The Butler–Volmer model pre-dicts the appearance of an anodic peak in Reverse Pulse Voltammetry for irreversible processes and apeak split of differential pulse voltammograms for quasireversible processes with a value of the transfercoefficient very different from 0.5 (smaller than 0.3 for a reduction process). These striking phenomenaare studied by using the Marcus–Hush approach, which also predicts the anodic peak for slow electrodereactions in Reverse Pulse Voltammetry but not the split of the curve in differential pulse techniques.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Different theories have been developed for the description ofheterogeneous electron transfer processes between a reacting spe-cies in solution and a metal electrode. These can be divided intotwo main categories: the macroscopic, phenomelogical Butler–Vol-mer (BV) model [1,2], and the microscopic theories which includeMarcus’ work [3,4] as well as other approaches based on quantummechanics [5–8]. The Butler–Volmer model offers a simple anduseful way to study and classify the kinetics of electrode reactionsand so it has been preferably used in electrochemistry over manyyears. The microscopic approaches involve more complex funda-mentals and formulations but they enable us to relate the electrodekinetics with the nature of the species involved, the solution com-position and the electrode material, and so to make predictions.Thus, the Marcus–Hush (MH) theory is being increasingly used[3,4,9–17].

Notable efforts have been made to theoretically and experimen-tally reveal the differences between the Butler–Volmer and Mar-

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fax: +44 (0) 1865 275410..G. Compton).

cus–Hush kinetic models in the description and analysis of theelectrochemical response [12–24]. It is theoretically predicted thatsignificant discrepancies between both models exists in the case ofslow charge transfer processes, which affects the values of kineticparameters extracted. Suwatchara et al. have recently carried outthe experimental assessment of both treatments with the slowreduction process of 2-nitropropane at high-speed channel micro-band electrodes finding, perhaps unexpectedly, a better agreementbetween the experimental results and those predicted by the But-ler–Volmer approach [16]. Similar conclusions have been reachedby Henstridge et al. [17] from the study of the one-electron reduc-tion of 2-methyl-2-nitropropane and europium(III) at mercurymicroelectrodes.

The aim of the present work is to deepen the comparison of But-ler–Volmer and Marcus–Hush treatments by examining the differ-ences in the response of the most popular potential pulsetechniques: chronoamperometry, Reverse Pulse Voltammetry(RPV), Differential Multi Pulse Voltammetry (DMPV) and SquareWave Voltammetry (SWV) [25]. As well as in electroanalysis, thesemethods are used in the determination of diffusion coefficients,kinetic rate constants and reaction mechanisms and, as will beshown, the conclusions can differ depending on the kinetic modelemployed for the electron transfer reaction.

170 E. Laborda et al. / Journal of Electroanalytical Chemistry 660 (2011) 169–177

The conditions in which important discrepancies between BVand MH formalisms can be observed are discussed as a functionof the heterogeneous rate constant, the electrode size and the elec-trochemical method. Moreover, criteria to detect deviations of theexperimental system from Butler–Volmer behaviour are discussedfrom the analysis of the behaviour of the chronoamperometric lim-iting current.

2. Theory

We consider the case of a one-electron reduction process takingplace on the surface of a (hemi)spherical electrode:

Oþ e� �E00 ;k0

R ð1Þwhere E00 is the formal potential of the redox couple O/R and k0 thestandard heterogeneous rate constant. When applying a constantpotential E where Reaction (1) occurs, the transport by diffusionof the electroactive species is described by Fick’s second law thatin the case of electrodes of spherical geometry is given by:

@cO@t ¼ DO

@2cO@r2 þ 2

r@cO@r

� �@cR@t ¼ DR

@2cR@r2 þ 2

r@cR@r

� �9>=>; ð2Þ

where cO and cR are the concentration profiles of the electroactivespecies, and DO and DR the diffusion coefficients. The boundary va-lue problem of the solutions of the differential equation system (2)is given by:

t ¼ 0; r P r0

t P 0; r !1

�cO ¼ c�O; cR ¼ c�R ð3Þ

t > 0, r = r0:

DO@cO

@r

� �r¼r0

¼ �DR@cR

@r

� �r¼r0

ð4Þ

DO@cO

@r

� �r¼r0

¼ kredcOðr ¼ r0Þ � koxcRðr ¼ r0Þ ð5Þ

where kred and kox are the reduction and oxidation rate constants,respectively, and c�O and c�R the bulk concentrations of the electroac-tive species.

Depending on the kinetic formalism assumed, the variation ofthe rate constants with the applied potential is described in differ-ent ways. According to the Butler–Volmer treatment this is givenby:

kBVred ¼ k0

BVe�ag

kBVox ¼ k0

BVebg

)ð6Þ

where g = F(E � E00)/RT and a and b are the transfer coefficients thatindicate the symmetry of the energy barrier, that is, if the transitionstate is reactant or product like [5,6]. According to Eq. (6), as the ap-plied potential is more negative the reduction rate constant (kBV

red)increases and the oxidation rate constant (kBV

ox ) decreases withoutlimit, and vice versa for positive potential values [14].

From the Marcus–Hush approach the following expressions arededuced for the rate constants [11,14]:

kMHred ¼ k0

MHe�g=2 Iðg;k�ÞIð0;k�Þ

kMHox ¼ k0

MHeg=2 Iðg;k�ÞIð0;k�Þ

9=; ð7Þ

where k� ¼ kF=RT , with k being the reorganization energy, andIðg; k�Þ is an integral of the form:

Iðg; k�Þ ¼Z 1

�1

exp �ðe�gÞ24k�

h i2coshðe=2Þ de ð8Þ

where e is an integral variable. The value of the reorganization en-ergy (k) corresponds to the energy necessary to adjust the configu-rations of the reactant and solvent to those of the product state. Itcan be separated into two contributions, the outer and inner reorga-nisation energies related to the reorganization of the solvent andthe electroactive species (bond lengths, angles,. . .), respectively [5].

For large values of the reorganization energy, the quotient ofintegrals Iðg; k�Þ=Ið0; k�Þ tends to unity such that the expressionsfor the rate constants in the MH model coincide with those inthe BV model for a = b = 0.5. Note that both approaches includethe Nernstian limit for large heterogeneous rate constant such thatfor fast electron transfer the results obtained from both formalismscoincide. Therefore, for reversible processes or large values of thereorganization energy, complete agreement is expected betweenthe Marcus–Hush model and the Butler–Volmer one witha = b = 0.5.

3. Results and discussion

3.1. Influence of the reorganization energy and the heterogeneous rateconstant

3.1.1. Single potential step chronoamperometry and Normal PulseVoltammetry (NPV)

First, the simplest case corresponding to the application of asingle potential step at a large overpotential for the reduction ofspecies O is considered. As is well known, taking into account theButler–Volmer model the value of the diffusion-controlled reduc-tion current at large overpotentials is given by the followingexpression for (hemi)spherical electrodes:

IBVlim ¼ FADOc�O

1ffiffiffiffiffiffiffiffiffiffiffiffipDOtp þ 1

r0

� �ð9Þ

Thus, the limiting current is exclusively controlled by the diffu-sion transport of species O towards the electrode surface, and it isindependent of the electrochemical reversibility of the process.

When the Marcus–Hush treatment is considered, the reductionrate constant is not predicted to increase continuously with the ap-plied potential but rather a maximum value exists. A simpleexpression for this value of the rate constant is given by [14]:

kMHmax ¼ k0

MH

ffiffiffiffiffiffiffiffiffiffiffi4pk�p

expðk�=4Þp� p3

4ðk�þ4:31Þð2:5 6 k� 6 80Þ ð10Þ

from which the following solution is derived for the limiting currentat (hemi)spherical electrodes, which gives rise to accurate results(error smaller than 1% with respect to Eq. (7)) in a wide range of val-ues of the reorganization energy, 2:5 6 k� 6 80:

IMHlim ¼ FADOc�O

1r0

KMHmax

1þ KMHmax

!

� 1þ KMHmax � exp

ffiffiffiffiffiffiffiffiDOtp

r01þ KMH

max

� �� �2

� erfcffiffiffiffiffiffiffiffiDOtp

r01þ KMH

max

� �� �" #

ð11Þ

where KMHmax ¼ kMH

max � r0=DO. Note that the value of the dimensionlessheterogeneous rate constant KMH

max increases with the standard rateconstant k0, the reorganization energy and the electrode radius,and it decreases with the diffusion coefficient.

According to Eq. (11), the value of the limiting current dependsnot only on the diffusion transport but also on the electrode kinet-ics so that it is a function of the reorganization energy and the het-erogeneous rate constant. For large values of KMH

max andffiffiffiffiffiffiffiffiDOtp

1þ KMHmax

� �=r0, the term KMH

max � expffiffiffiffiffiffiDOtp

r01þ KMH

max

� �� �2

�

(A)

1.0 1.5 2.0 2.5 3.0 3.5 4.0

Ι lim( t)

/ Ι lim

( t p)

1.0

1.5

2.0

2.5

3.0

3.5

4.0BV

MH (λ∗ = 40)MH, λ∗ = 20MH, λ∗ = 15MH, λ∗ = 10

(B)

0 2 4 6 8

c O /

c*

0.0

0.2

0.4

0.6

0.8

1.0

1t

Ot

x

D

Fig. 1. Single potential step chronoamperometry at large overpotentials. (A)Variation of the limiting current with time; (B) concentration profiles at the endof the pulse. Planar electrode, k0 ¼ 10�4 cm=s ¼ k0

BV ¼ k0MH

� �.

Table 1Study of the linearity of the curves Ilim(t)/Ilim (tp) vs. 1=

ffiffitp

with Marcus–Hush kinetics.Planar electrode, k0 ¼ 10�4 cm=s ð¼ k0

BV ¼ k0MHÞ, tp ¼ 1 s, t ¼ 0:01 s.

k� Marcus–Hush

Slope Intercept R2

>30 1 0 120 0.967 0.048 0.9996215 0.760 0.333 0.9828510 0.318 0.835 0.88022

Butler–Volmer (any k0): slope = 1. Intercept = 0. R2 = 1.

E. Laborda et al. / Journal of Electroanalytical Chemistry 660 (2011) 169–177 171

erfcffiffiffiffiffiffiDOtp

r01þ KMH

max

� �� �tends to r0ffiffiffiffiffiffiffiffi

pDOtp and the BV and MH expres-

sions then coincide. Thus, it can be concluded that these parame-ters set the discrepancy in the value of the limiting currentbetween the two kinetic models. Thus, greater differences are ex-pected for small k0 and/or k values, short t values and small elec-trode radius.

Under steady state conditions (r0 �ffiffiffiffiffiffiffiffiDOtp

), the expressions forthe limiting current simplify to:

IBVlim;ss ¼

FADOc�Or0

ð12Þ

IMHlim;ss ¼ FADOc�O

1r0

KMHmax

1þ KMHmax

!ð13Þ

The difference between the above expressions is the term

KMHmax

.1þ KMH

max that tends to unity for large KMHmax ¼ kMH

max � r0=DO

� �values such that both solutions coincide. Otherwise, the term

KMHmax

.1þ KMH

max is smaller than unity and the stationary current pre-

dicted by Marcus–Hush is less than by Butler–Volmer, the smallerthe electrode radius (i.e., the smaller the KMH

max value), the greaterthe difference between both solutions.

The steady state limiting current is usually employed in thedetermination of the radius of microelectrodes. From Eqs. (12)and (13) the ratio of the electrode radii determined with Butler–Volmer (rBV

0 ) and Marcus–Hush (rMH0 ) from an experimental current

value can be evaluated as a function of the dimensionless parame-ter KMH

max:

rBV0

rMH0

¼ KMHmax

1þ KMHmax

ð14Þ

The above expression reveals that for small KMHmax values (i.e.,

small radius, slow electrode reaction and/or small reorganizationenergy) the electrode radius estimated with the Butler–Volmerexpression for the stationary limiting current (Eq. (12)) is smallerthan the value obtained from the Marcus–Hush solution (Eq.(13)), such that for KMH

max < 20 the difference is greater than 5%.Note that this is not a significant practical problem since reversibleelectrode reactions (large KMH

max values) are usually employed for thecalibration of the electrode size.

In Fig. 1A the dimensionless limiting current Ilim(t)/Ilim(tp)(where tp is the total duration of the potential step) at a planarelectrode is plotted vs. 1=

ffiffitp

under the Butler–Volmer (solid line)and Marcus–Hush (dashed lines) treatments for an irreversibleprocess with k0 ¼ 10�4 cm=s ð¼ k0

BV ¼ k0MHÞ, where the differences

between both models are more apparent according to the abovediscussion.

Regarding the BV model (Eq. (9)), a unique curve is predictedindependently of the electrode kinetics with a Slope =

ffiffiffiffitp

pand an

Intercept = 0 (see Table 1). With respect to the MH model, for typ-ical values of the reorganization energy (k ¼ 0:5� 1 eV,k� � 20—40 [5]) the variation of the limiting current with timecompares well with that predicted by Butler–Volmer kinetics (Eq.(9)) (see Fig. 1A and Table 1). On the other hand, for small k values(k� < 20, see Table 1) and short times, differences between the BVand MH results are observed such that the current expected withthe MH model is smaller. In addition, a nonlinear dependence ofIlim(t)/Ilim(tp) with 1=

ffiffitp

is predicted, and any attempt at lineariza-tion would result in poor correlation coefficient and Slope <

ffiffiffiffitp

pand Intercept > 0.

In the case that kinetic control exists in the limiting current,according to Eq. (11) kMH

max can be determined by extrapolating thevalue of the current to t = 0:

IMHlim ðt ! 0Þ ¼ FAc�OkMH

max ð15Þ

The differences between BV and MH have also implications inthe concentration profiles of the electroactive species. Thus,whereas the BV model predicts a zero surface concentration ofthe oxidized species at the electrode surface, in the Marcus–Hushmodel the surface concentration of species O also depends on theelectrode kinetics such that for small values of the heterogeneousrate constant and reorganization energy this is not zero (seeFig. 1B); the smaller the reorganization energy, the greater the sur-face concentration of the reacting species.

Next we will consider the application of a sequence of indepen-dent pulses at different potential values, that is, the case of NormalPulse Voltammetry (Fig. 2A). Fig. 3A shows the voltammogramsobtained for k0 ¼ 10�4 cm=s (where k0

MH ¼ k0BV ¼ k0) and different

values of the reorganization energy. In the inset, the difference be-

172 E. Laborda et al. / Journal of Electroanalytical Chemistry 660 (2011) 169–177

tween the results obtained with the MH model and the BV modelfor a = b = 0.5 are shown. When differences are observed, the cur-rent predicted by Marcus–Hush kinetics is smaller than for But-ler–Volmer kinetics. Note that the most significant discrepancy isfound at intermediate potential values (between �165 and

scan

(C) Differential Multi Pulse Voltammetry (DMPV)

Time

Pote

ntia

l

t1

tp

Es

ΔE Eindex

(-)

Ι1

ΙpCurrent sample

_

(A) Normal Pulse Voltammetry (NPV)

Time

Pote

ntia

l( -)

Current sample

Recovery of initial equilibrium conditions

tp

Eindex

Fig. 2. Potential waveform in the pulse te

η-15 -10 -5 0

Ι(t)

/ Ιd(

t p)

0.0

0.2

0.4

0.6

0.8

1.0

BVΜΗ, λ∗ = 40ΜΗ, λ∗ = 20ΜΗ, λ∗ = 10 -

ε(%

)

0

10

20

30

40

50

60

Fig. 3. Comparison of the response with BV (a = b = 0.5) and MH in Normal Pulse Voltfunction of the applied potential. Planar electrode, tp = 1 s, k0 ¼ 10�4 cm=s ¼ k0

BV ¼ k0MH

�

�185 mV for the conditions considered in the figure) such that incircumstances where the difference in the limiting current is notsignificant, it may be at these potentials. So, the peak current inthe differential pulse techniques is expected to be more sensitiveto differences between the two kinetic models (see below).

(B) Reverse Pulse Voltammetry (RPV)

TimePo

tent

ial

Recovery of initial equilibrium conditions

t1

t2

scan(D) Square Wave Voltammetry (SWV)

Time

Pote

ntia

l

tp

E1

Eindex = E2

Es

ESWV

Eindex

(+)

( -)

p

1

2f

t=Ιf

Ιb

Ι2

_

Current sample

Current sample

chniques: NPV, DMPV, SWV and RPV.

5

η20 -15 -10 -5 0

ammetry. The inset shows the relative difference (e ð%Þ) between both results in�. IdðtpÞ ¼ FADc�O=

ffiffiffiffiffiffiffiffiffiffiffipDtp

p.

E. Laborda et al. / Journal of Electroanalytical Chemistry 660 (2011) 169–177 173

3.1.2. Potential step techniquesIn this section we will consider the difference between the BV

and MH models in the most common potential pulse techniques:Reverse Pulse Voltammetry (RPV), Differential Multi Pulse Voltam-metry (DMPV) and Square Wave Voltammetry (SWV) [5,6]. The po-tential-time program applied in each method is shown in Fig. 2.

For the theoretical study of the response in RPV the analyticalsolution available for spherical electrodes are employed [26],which was initially deduced assuming the Butler–Volmer modelbut it can be modified to the Marcus–Hush model by substitution

(A) k0 = 0.01 cm/s

ηindex

-8 -6 -4 -2 0 2 4 6 8

Ι 2 / Ι d

(t 2)

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Butler-Volmer Marcus-Hush

λ* > 10

15

λ* > 30080

20

10

5

(B) k0 = 0.001 cm/s

ηindex

-15 -10 -5 0 5 10 15

Ι 2 / Ι d

(t 2)

-0.8

-0.6

-0.4

-0.2

0.0

0.2

λ* > 10020

10

5

(C) k0 = 0.0001 cm/s

ηindex

-20 -10 0 10 20

Ι 2 / Ι d

(t 2)

-0.8

-0.6

-0.4

-0.2

0.0

0.2

λ* >100040

20

10

15

Fig. 4. Comparison of the response with BV (a = b = 0.5) and MH in Reverse PulseVoltammetry for different values of k0 ð¼ k0

BV ¼ k0MHÞ and k� (indicated on the

graphs). Planar electrode, t1 ¼ 1 s; t2=t1 ¼ 0:05. Idðt2Þ ¼ FADc�O=ffiffiffiffiffiffiffiffiffiffiffipDt2p

.

of the expressions for the oxidation and reduction rate constantsby the expressions given in Eq. (7). The responses in DMPV andSWV were obtained by means of finite difference numerical meth-ods with a homemade program employing EXTRAP4 algorithm fortime integration and an exponentially expanding grid with highexpansion factors and four-point formulae for the spatial discreti-zation according to the results presented in Refs. [27–29]. In allcases the value of the integral of the MH model given by Eq. (8)has been evaluated numerically using the trapezium rule.

Fig. 4 shows the RPV curves obtained under the BV (solid line,a = b = 0.5) and MH (dashed lines) formalisms for different valuesof the standard rate constant (k0 ¼ k0

MH ¼ k0BV) and the dimension-

less reorganization energy (k� ¼ kF=RT). For fast electrode reactionsor large reorganization energies the results coincide in the wholeregion of potentials whereas when the heterogeneous rate con-stant and/or the reorganization energy of the system decrease,the value of the current in MH is smaller in absolute value. Sincethe second pulse is usually shorter than the first one (t1/t2 = 20),the differences in the anodic limiting current are more apparent.

The values of the cathodic and anodic limiting currents in RPVare commonly employed for the determination of the diffusioncoefficients of the electroactive species and so it is interesting toanalyze the possible effect of the kinetic model on them. Accordingto the BV model the value of the limiting currents is independent ofthe electrode kinetics and, in the case of a planar electrode, also ofthe diffusion coefficient of the product species [28]. Thus, in lineardiffusion the following simple linear relationship applies for thenormalized anodic limiting current (Id,RPV):

IBVd;RPV

IBVlimðt2Þ

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

t2

t1 þ t2

r� 1 ð16Þ

Assuming the MH formalism, it is found that for k� > 25 the re-sults coincides satisfactorily with those predicted by Eq. (16) (seeFig. 5). For smaller k� values, the anodic limiting current is smallerin absolute value than that predicted by Butler–Volmer, it isdependent on the values of both diffusion coefficients (the smallerthe diffusion coefficient of species R, the greater the absolute valueof the anodic limiting current) and it deviates from the linearbehaviour described by Eq. (16), as can be observed in Fig. 5.

Next, an analogous study is performed for DMPV and SWV tech-niques. According to that discussed in Section 3.1.1, given the re-gion of potentials where the peak appears and that the pulse

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-1.0

-0.8

-0.6

-0.4

-0.2

0.0

BV MH (λ∗ > 30)MH (λ∗ = 20)MH (λ∗ = 15)MH (λ∗ = 10)

d,RPV

BVlim 2

I

I (t )

2

1 2

t

t t+

Fig. 5. Study of the linearity of the curves Id;RPV=IBVlimðt2Þ vs.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2=t1 þ t2

ppredicted

by Butler–Volmer (solid line) and Marcus–Hush (dashed lines) for different valuesof the reorganization energy. Planar electrode, t1 ¼ 1 s; k0 ¼ 10�4 cm=s¼ k0

BV ¼ k0MH

� �.

174 E. Laborda et al. / Journal of Electroanalytical Chemistry 660 (2011) 169–177

length is usually very short in these techniques [5,6] the responseis expected to be more dependent on the kinetic model used thanthe value of the limiting currents. Thus, whereas a good agreementin the value of the limiting current has been always found fork� > 20, in DMPV and SWV the peaks corresponding to slow elec-trode processes (k0 < 10�3 cm/s) are different in BV and MH in thiscommon range of values of the reorganization energy (k� > 20), the

(A) k0 = 0.01 cm/s

ηindex

-4 0 2 4 6 8

ΔΙ /

Ι d(t p

)

0.0

0.1

0.2

0.3Butler-VolmerMarcus-Hush

λ* >101

(B) k0 = 0.001 cm/s

ηindex

-12 -10 -8 -6 -4 -2 0 2 4 6 8

ΔΙ /

Ι d(t p

)

0.00

0.02

0.04

0.06

0.08

0.10λ* >200

802010

5

(C) k0 = 0.0001 cm/s

ηindex

-16 -12 -8 -4 0 4

ΔΙ /

Ι d(t p

)

0.00

0.01

0.02

0.03

0.04

0.05λ* >1000

80

40

20

10

-8 -6 -2

Fig. 6. Comparison of the response with BV (a = b = 0.5) and MH in DifferentialMulti Pulse Voltammetry for different values of k0 ¼ k0

BV ¼ k0MH

� �and k (indicated

on the graphs). Planar electrode, DE ¼ �50 mV, ES ¼ �5 mV, t1 = 1 s, tp/t1 = 0.02.IdðtpÞ ¼ FADc�O=

ffiffiffiffiffiffiffiffiffiffiffipDtp

p.

peak current being smaller in the former as can be observed inFigs. 6 and 7. Consequently, the values of the kinetic parameters

k0MH; k

n oor k0

BV;an o� �

of sluggish processes extracted in DMPVand SWV may differ depending on the model assumed. In thissense, an analysis of the consistency of the results obtained withdifferent techniques and different time scales can be useful to re-

(A) k0 = 0.01 cm/s

ηindex

-8 -6 -4 -2 0 2 4 6 8ΔΙ

/ Ι d

(t p)

0.0

0.2

0.4

0.6

Butler-VolmerMarcus-Hush

λ* > 40101

(B) k0 = 0.001 cm/s

ηindex

-12 -8 0-4 4 8

ΔΙ /

Ι d(t p

)

0.00

0.05

0.10

0.15

0.20

0.25λ* > 300

80

20

10

5

(C) k0 = 0.0001 cm/s

ηindex

-20 -15 -10 -5 0 5

ΔΙ /

Ι d(t p

)

0.00

0.05

0.10

0.15

0.20

λ* >100080

40

20

10

Fig. 7. Comparison of the response with BV (a = b = 0.5) and MH in Square WaveVoltammetry for different values of k0 ¼ k0

BV ¼ k0MH

� �and k� (indicated on the

graphs). Planar electrode, f ¼ 25 Hz, ESWV ¼ 50 mV, ES ¼ �5 mV.IdðtpÞ ¼ FADc�O=

ffiffiffiffiffiffiffiffiffiffiffipDtp

p.

E. Laborda et al. / Journal of Electroanalytical Chemistry 660 (2011) 169–177 175

veal the most suitable approach for the description of the kineticsof the experimental system.

If we consider the case a – 0:5 in the Butler–Volmer kinetics,the discrepancy with Marcus–Hush is more apparent given thateven for very large values of the reorganization energy there isnot a direct correspondence between both models and the re-sponses do not converge like in Figs. 4–7. In these cases, it is nec-essary to adjust the values of the kinetic parameters and the formalpotential to obtain a satisfactory agreement between the responsesin both approaches (see Fig. 8 and Table 2). Thus, for a < 0:5 asmaller value of the heterogeneous rate constant is required inMH for a good fit, and a greater k0 value for a > 0:5.

Finally, the effect of the electrode size on the difference be-tween the responses in the different techniques here consideredis shown in Fig. 9 for a slow electrode process withk0 ¼ 10�4 cm=s ð¼ k0

BV ¼ k0MHÞ. As was deduced above, the smaller

the electrode radius, the greater the difference between the kineticformalisms. In general, we can infer that any change promoting theirreversibility of the electrode reaction, that is, shorter pulse times,smaller heterogeneous rate constant and/or smaller electrode ra-dius, increases the difference between the two kinetic models.

(A) DMPV

ηindex

ΔΙ /

Ι d(t p)

0.00

0.02

0.04

0.06

0.08BVMH

0 -4BV 10 cm / sk =

α = 0.6

α = 0.4

(B) SWV

ηindex

-15 -10 -5 0

-15 -10 -5 0

ΔΙ /

Ι d(t p)

0.00

0.05

0.10

0.15

0.20

0.25

0.30BVMH

0 -4BV 10 cm / sk =

α = 0.6

α = 0.4

Fig. 8. Comparison of the response with BV (a + b = 1) and MH in (A) DifferentialMulti Pulse Voltammetry and (B) Square Wave Voltammetry for a = 0.4 and a = 0.6.The values of the kinetic parameters and formal potential corresponding to eachcase are indicated in Table 2. Other conditions as in Figs. 6 and 7.IdðtpÞ ¼ FADc�O=

ffiffiffiffiffiffiffiffiffiffiffipDtp

p.

For the conditions considered in Figs. 4–7 and a hemisphericalmicroelectrode with r0 ¼ 5 lm it is found that the agreement inthe value of the limiting currents (Ilim, Id,RPV) in both models is verysatisfactory for k� P 40 (k P 1 eV). Regarding the DMPV and SWVtechnique, for quasireversible and irreversible processes(k0

< 10�2 cm=s) the peak current depend on the kinetic treatmentassumed and so the applicability of the kinetic model must be crit-ically examined in each case.

3.1.3. Striking phenomena predicted by the Butler–Volmer modelIn this section some anomalous features of DMPV, SWV and RPV

curves predicted with the BV treatment will be considered byemploying the MH one.

First, the peak split in differential pulse voltammetries (DDPV,DMPV, SWV, ADPV) for quasireversible processes is considered, aphenomenon predicted by several authors in the literature [30–34] assuming the Butler–Volmer treatment. For a reduction pro-cess the split takes place for small values of the transfer coefficient(a < 0:3), it is more apparent at planar electrodes and it consists oftwo peaks: a sharp peak situated at more positive potentials thanE00 and a broad peak at negative overpotentials. In Fig. 10 DMPVand SWV curves were also obtained with the Marcus–Hush modelfor a wide range of values of the reorganization energy under the

Table 2Values of the heterogeneous rate constant, transfer coefficient and formal potentialcorresponding to the DMPV and SWV curves shown in Fig. 8.

Fig. 8A. DMPV: t1 = 1 s, t1/t2 = 50, DE = �50 mV, Es = �5 mVButler–Volmer k0

BV ¼ 10�4 cm=s k0BV ¼ 10�4 cm=s

a ¼ 0:6 a ¼ 0:4

E00BV ¼ 0 mV E00

BV ¼ 0 mVMarcus–Hush k0

MH ¼ 8:2� 10�4 cm=s k0MH ¼ 0:19� 10�4 cm=s

k� ¼ 7 k� ¼ 45

E00MH ¼ �105 mV E00

MH ¼ þ65:5 mV

Fig. 8B. SWV: f = 25 Hz, ESWV = 50 mV, Es = �5 mVButler–Volmer k0

BV ¼ 10�4 cm=s k0BV ¼ 10�4 cm=s

a ¼ 0:6 a ¼ 0:4

E00BV ¼ 0 mV E00

BV ¼ 0 mVMarcus–Hush k0

MH ¼ 10:7� 10�4 cm=s k0MH ¼ 0:7� 10�4 cm=s

k� ¼ 120 k� ¼ 40

E00MH ¼ �92 mV E00

MH ¼ �7:7 mV

0.1 1 10 100

Ι max

(BV)

/ Ι m

ax(M

H)

2

4

6

8

10

DMPV: ΔΙpeak

SWV: ΔΙpeak

RPV: Ιd,RPV

NPV: Ιlim

00

p/2

rRDt

=

Fig. 9. Effect of the electrode size on the differences between the results obtainedfor the maximum currents in the different electrochemical techniques for a slowelectrode process with k0 ¼ 10�4 cm=s ¼ k0

BV ¼ k0MH

� �from BV (a = b = 0.5) and MH

(k� ¼ 20) models. Other conditions as in Figs. 4–7.

(A) DMPV

ηindex

-20 -10 0 10

ΔΙ /

Ι d(t p)

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

BV, α = 0.3, β = 0.7MH, λ∗ = 40MH, λ∗ = 10MH, λ∗ = 4

BV, α = 0.2, β = 0.8MH, λ∗ = 40MH, λ∗ = 10MH, λ∗ = 4

(B) SWV

ηindex

-30 -20 -10 0 10

ΔΙ /

Ι d(t p)

0.00

0.05

0.10

0.15

0.20

0.25

Fig. 10. Peak split in (A) DMPV (k0 ¼ 10�3 cm=s ¼ k0BV ¼ k0

MH) and (B) SWV(k0 ¼ 10�4 cm=s ¼ k0

BV ¼ k0MH) curves predicted by Butler–Volmer (solid line) for

quasireversible processes with small values of the a transfer coefficient togetherwith the results predicted by Marcus–Hush (dashed lines) for different values of thedimensionless reorganization energy (indicated on the graphs). Planar electrode,ESWV = 50 mV, DE ¼ �50 mV, ES ¼ �5 mV, t1 ¼ 1 s, t2 ¼ tp ¼ 0:02 s.Idðtp=2Þ ¼ FADc�O=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipDtp=2

p.

ηindex

-30 -20 -10 0 10 20 30

Ι 2/Ιd(

t 2)

-0.2

0.0

0.2

0.4

0.6

0.8

BVMH, λ∗ = 40MH, λ∗ = 20MH, λ∗ = 10 ηindex

25

ΙRPV

/Ι d(t 2)

-0.3

-0.2

-0.1

0.0

0.1

0 5 10 15 20

Fig. 11. Anodic peak in RPV curves for irreversible electrode processes predicted byButler–Volmer (solid line, a ¼ b ¼ 0:5) together with the results obtained withMarcus–Hush (dashed lines) for different values of the dimensionless reorganiza-tion energy (indicated on the graphs). Planar electrode, k0 ¼ 10�4 cm=s¼ k0

BV ¼ k0MH

� �, t1 ¼ 1 s; t2=t1 ¼ 2: Idðt2Þ ¼ FADc�O=

ffiffiffiffiffiffiffiffiffiffiffipDt2p

.

176 E. Laborda et al. / Journal of Electroanalytical Chemistry 660 (2011) 169–177

conditions where the split is predicted by BV. As can be observed,the MH model predicts an asymmetrical peak for intermediate k�

values but not the peak split. An analogous study has been per-formed for smaller and larger k0 values than those included inFig. 10 and no evidences of the peak split with Marcus–Hush havebeen found. This can be understood by taking into account that thesplit in the BV model is predicted for such small a values that haveno direct correspondence in the Marcus–Hush formalism. Oster-young et al. [31] reported an experimental evidence of this splitin SWV for the reduction of Zn(II) to Zn(0) at mercury electrodesthat would support a better parametrisation of this system by But-ler–Volmer kinetics rather than Marcus–Hush kinetics.

The second anomalous feature of the voltammograms predictedby Butler–Volmer is the appearance of a peak in the anodic branchfor sluggish electrode processes when the duration of the secondpulse is similar to that of the first one. As can be seen in Fig. 11,the presence of the peak is predicted by both models when thereorganization energy is large (k� > 20 for k0 ¼ 10�4cm=s). This in-volves a crossing of the chronoamperograms [26] such that afterthe crossing time higher currents are recorded at smalleroverpotentials, which is related with the more rapid depletion ofthe species electrogenerated at the electrode surface at largeoverpotentials.

4. Conclusions

The comparison between the results obtained with Butler–Vol-mer and Marcus–Hush approaches have been carried out for themain potential pulse techniques. Whereas the former is easier tohandle and compute, the latter enables us to make predictions ofthe electrode kinetics from the molecular nature of the electroac-tive species, electrode and solution. The difference between thetwo models can be significant for small values of the heteroge-neous rate constant and/or the reorganization energy, especiallywhen small electrodes and/or short pulse times are used.

The discrepancy is also dependent on the applied potential.Thus, under diffusion limiting conditions the results obtained withboth approaches are in good agreement for k > 0:6 eV at planarelectrodes and k P 1 eV at hemispherical microelectrodes withr0 > 5 lm. At intermediate overpotentials the differences aregreater such that the peak current of slow electron transfers in dif-ferential pulse techniques is more sensitive to the kinetic modelemployed. Consequently, the kinetic parameters extracted maydiffer, necessitating the study of the consistency of the results un-der different experimental conditions to check the suitability of theapproach employed.

Finally, two striking phenomena predicted by the Butler–Vol-mer model have been studied with the Marcus–Hush approach:the anodic peak in reverse pulse voltammograms of slow chargetransfer processes and the peak split of the differential pulse vol-tammograms of quasireversible processes with small values ofthe transfer coefficient. With both models the appearance of theanodic peak is predicted for sluggish processes. However, no evi-dence of the peak split has been found in Marcus–Hush, althoughthis is experimentally reported in the literature.

Acknowledgements

A.M. and F.M.-O. greatly appreciate the financial support pro-vided by the Dirección General de Investigación Científica y Téc-nica (Project Number CTQ2009-13023), and the FundaciónSENECA (Project Number 11989). E.L. also thanks the FundaciónSENECA for the grant received. M.C.H. would like to thank EPSRCfor funding (EP/H002413/1).

References

[1] J.A.V. Butler, Trans. Faraday Soc. 19 (1924) 729.[2] T. Erdey-Gruz, M. Volmer, Z. Phys. Chem. 150A (1930) 203.

E. Laborda et al. / Journal of Electroanalytical Chemistry 660 (2011) 169–177 177

[3] R.A. Marcus, J. Chem Phys. 24 (1956) 4966.[4] R.A. Marcus, Annu. Rev. Phys. Chem. 15 (1964) 155.[5] A.J. Bard, L.R. Faulkner, Electrochemical Methods, second ed., Fundamentals

and Applications, Wiley, New York, 2001.[6] R.G. Compton, C.E. Banks, Understanding Voltammetry, World Scientific,

Singapore, 2007.[7] S. Fletcher, J. Solid State Electrochem. 11 (2007) 965.[8] S. Fletcher, J. Solid State Electrochem. 14 (2010) 705.[9] N.S. Hush, J. Chem. Phys. 28 (1958) 962.

[10] N.S. Hush, J. Electroanal. Chem. 470 (1999) 170.[11] C.E.D. Chidsey, Science 215 (1991) 919.[12] C.J. Miller, Homogeneous electron transfer at metallic electrodes, in: I.

Rubinstein (Ed.), Physical Electrochemistry, Principles, Methods andApplications, Marcel Dekker, New York, 1995, p. 27.

[13] L. Tender, M.T. Carter, R.W. Murray, Anal. Chem. 66 (1994) 3173.[14] S.W. Feldberg, Anal. Chem. 82 (2010) 5176.[15] K.B. Oldham, J.C. Myland, J. Electroanal. Chem. 655 (2011) 65.[16] D. Suwatchara, M.C. Henstridge, N.V. Rees, R.G. Compton, J. Phys. Chem. C,

2011, doi:10.1021/jp203426.[17] M.C. Henstridge, Y. Wang, J.G. Limon-Petersen, R.G. Compton, J. Phys. Chem. C,

submitted for publication.[18] S. Fletcher, T.S. Varley, Phys. Chem. Chem. Phys. 13 (2011) 5359.[19] J.-M. Savéant, D. Tessier, Faraday Discuss. Chem. Soc. 74 (1982) 57.[20] S. Amemiya, Z. Ding, J. Zhou, A.J. Bard, J. Electroanal. Chem. 483 (2000) 7.

[21] Z. Ding, B.M. Quinn, A.J. Bard, J. Phys. Chem. B 105 (2001) 6367.[22] P. Sun, F. Li, Y. Chen, M. Zhang, Z. Zhang, Z. Gao, Y. Shao, J. Am. Chem. Soc. 125

(2003) 9600.[23] M.J. Weaver, F.C. Anson, J. Phys. Chem. 80 (1976) 1861.[24] A.D. Clegg, N.V. Rees, O.V. Klymenko, B.A. Coles, R.G. Compton, J. Am. Chem.

Soc. 126 (2004) 6185.[25] A. Molina, E. Laborda, F. Martínez-Ortiz, D.F. Bradley, D.J. Schiffrin, R.G.

Compton, J. Electroanal. Chem. 659 (2011) 12.[26] A. Molina, F. Martínez-Ortiz, E. Laborda, R.G. Compton, J. Electroanal. Chem.

648 (2010) 67.[27] F. Martínez-Ortiz, N. Zoroa, A. Molina, C. Serna, E. Laborda, Electrochim. Acta

54 (2009) 1042.[28] E. Laborda, E.I. Rogers, F. Martínez-Ortiz, J.G. Limon-Petersen, N.V. Rees, A.

Molina, R.G. Compton, J. Electroanal. Chem. 634 (2009) 1.[29] F. Martínez-Ortiz, A. Molina, E. Laborda, Electrochim. Acta 56 (2011) 5707.[30] V. Mirceski, S. Komorsky-Lovric, M. Lovric, Square-wave voltammetry. Theory

and application, in: F. Scholz (Ed.), Monographs in Electrochemistry, Springer-Verlag, Berlin, 2007.

[31] W.S. Go, J.J. O’Dea, J. Osteryoung, J. Electroanal. Chem. 255 (1988) 21.[32] A. Molina, F. Martínez-Ortiz, E. Laborda, R.G. Compton, Electrochim. Acta 55

(2010) 5163.[33] H. Matsuda, Bull. Chem. Soc. Jpn. 53 (1980) 3439.[34] E. Laborda, E.I. Rogers, F. Martínez-Ortiz, A. Molina, R.G. Compton,

Electroanalysis 22 (2010) 2784.