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

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  • V

    o

    ersitain

    Received in revised form 22 June 2011Accepted 24 June 2011Available online 12 July 2011

    Keywords:Electrode kinetics

    amperometry, Normal Pulse Voltammetry, Differential Multi Pulse Voltammetry, Square Wave Voltam-

    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 MarcusHush (MH) theory is being increasingly used[3,4,917].

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

    microelectrodes.The aim of the present work is to deepen the comparison of But-

    lerVolmer and MarcusHush 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 coefcients,kinetic rate constants and reaction mechanisms and, as will beshown, the conclusions can differ depending on the kinetic modelemployed for the electron transfer reaction.

    Corresponding author. Tel.: +44 (0) 1865 275413; fax: +44 (0) 1865 275410.

    Journal of Electroanalytical Chemistry 660 (2011) 169177

    Contents lists availab

    n

    lseE-mail address: richard.compton@chem.ox.ac.uk (R.G. Compton).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 ButlerVol-mer (BV) model [1,2], and the microscopic theories which includeMarcus work [3,4] as well as other approaches based on quantummechanics [58]. The ButlerVolmer 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-

    electrochemical response [1224]. It is theoretically predicted thatsignicant 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 nding, perhaps unexpectedly, a better agreementbetween the experimental results and those predicted by the But-lerVolmer 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 mercuryButlerVolmer modelMarcusHush modelPotential pulse techniques

    1. Introduction1572-6657/$ - see front matter 2011 Elsevier B.V. Adoi:10.1016/j.jelechem.2011.06.027metry 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 coefcients and electrode radii are exam-ined, and criteria are given to detect possible deviations of the experimental system from ButlerVolmerkinetics from the behaviour of the chronoamperometric limiting current. The ButlerVolmer 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 transfercoefcient very different from 0.5 (smaller than 0.3 for a reduction process). These striking phenomenaare studied by using the MarcusHush 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.

    cusHush kinetic models in the description and analysis of theArticle history:Received 25 May 2011

    Simulated voltammograms obtained by employing ButlerVolmer (BV) and MarcusHush (MH) modelsto describe the electrode kinetics are compared for commonly used potential pulse techniques: chrono-A comparison of MarcusHush vs. Butlerpotential pulse voltammetric techniques

    Eduardo Laborda a,b, Martin C. Henstridge a, Angela MRichard G. Compton a,aDepartment of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford UnivbDepartamento de Qumica Fsica, Universidad de Murcia, Espinardo 30100, Murcia, Sp

    a r t i c l e i n f o a b s t r a c t

    Journal of Electroa

    journal homepage: www.ell rights reserved.olmer electrode kinetics using

    lina b, Francisco Martnez-Ortiz b,

    y, South Parks Road, Oxford OX1 3QZ, United Kingdom

    le at ScienceDirect

    alytical Chemistry

    vier .com/locate / je lechem

  • at (hemi)spherical electrodes, which gives rise to accurate results

    alytThe 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 ButlerVolmer 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:

    O e E00 ;k0

    R 1where 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 Ficks second law thatin the case of electrodes of spherical geometry is given by:

    @cO@t DO @

    2cO@r2 2r @cO@r

    @cR@t DR @

    2cR@r2 2r @cR@r

    9>=>; 2

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

    t 0; r P r0t P 0; r !1

    cO cO; cR cR 3

    t > 0, r = r0:

    DO@cO@r

    rr0

    DR @cR@r

    rr0

    4

    DO@cO@r

    rr0

    kredcOr r0 koxcRr r0 5

    where kred and kox are the reduction and oxidation rate constants,respectively, and cO and c

    R the bulk concentrations of the electroac-

    tive species.Depending on the kinetic formalism assumed, the variation of

    the rate constants with the applied potential is described in differ-ent ways. According to the ButlerVolmer treatment this is givenby:

    kBVred k0BVeagkBVox k0BVebg

    )6

    where g = F(E E00)/RT and a and b are the transfer coefcients 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 (kBVred)increases and the oxidation rate constant (kBVox ) decreases withoutlimit, and vice versa for positive potential values [14].

    From the MarcusHush approach the following expressions arededuced for the rate constants [11,14]:

    kMHred k0MHeg=2 Ig;k

    I0;k

    kMHox k0MHeg=2 Ig;k

    I0;k

    9=; 7

    where k kF=RT , with k being the reorganization energy, andIg; k is an integral of the form:

    2h i

    170 E. Laborda et al. / Journal of ElectroanIg; k Z 11

    exp eg4k

    2coshe=2 de 8(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 FADOcO1r0

    KMHmax1 KMHmax

    !

    1 KMHmax expDOt

    p

    r01 KMHmax 2

    erfcDOt

    p

    r01 KMHmax " #

    11

    where KMHmax kMHmax r0=DO. Note that the value of the dimensionlessheterogeneous rate constant KMHmax increases with the standard rateconstant k0, the reorganization energy and the electrode radius,and it decreases with the diffusion coefcient.

    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-where e is an integral variable. The value of the reorganization en-ergy (k) corresponds to the energy necessary to adjust the congu-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 Ig; k=I0; k 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 MarcusHush model and the ButlerVolmer one witha = b = 0.5.

    3. Results and discussion

    3.1. Inuence 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 theButlerVolmer model the value of the diffusion-controlled reduc-tion current at large overpotentials is given by the followingexpression for (hemi)spherical electrodes:

    IBVlim FADOcO1pDOt

    p 1r0

    9

    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 MarcusHush 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 k0MH4pk

    pexpk=4

    p p34k4:312:5 6 k 6 80 10

    fromwhich the following solution is derived for the limiting current

    ical Chemistry 660 (2011) 169177erogeneous rate constant. For large values of KMHmax andDOt

    p1 KMHmax

    =r0, the term KMHmax exp

    DOt

    pr0

    1 KMHmax 2

  • The differences between BV and MH have also implications inthe concentration proles of the electroactive species. Thus,whereas the BV model predicts a zero surface concentration ofthe oxidized species at the electrode surface, in the MarcusHushmodel 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 104 cm=s (where k0MH k0BV k0) and differentvalues of the reorganization energy. In the inset, the difference be-

    (A)

    1.0 1.5 2.0 2.5 3.0 3.5 4.0

    lim

    ( t) /

    lim( t p

    )

    1.0

    1.5

    2.0

    2.5

    3.0

    3.5

    4.0BV

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

    (B)

    0 2 4 6 8

    c O /

    c*

    0.0

    0.2

    0.4

    0.6

    0.8

    1.0

    1t

    x

    alytical Chemistry 660 (2011) 169177 171erfcDOt

    pr0

    1 KMHmax

    tends to r0pDOt

    p 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 DOt

    p), the expressions for

    the limiting current simplify to:

    IBVlim;ss FADOcO

    r012

    IMHlim;ss FADOcO1r0

    KMHmax1 KMHmax

    !13

    The difference between the above expressions is the term

    KMHmax.1 KMHmax that tends to unity for large KMHmax kMHmax r0=DO

    values such that both solutions coincide. Otherwise, the term

    KMHmax.1 KMHmax is smaller than unity and the stationary current pre-

    dicted by MarcusHush is less than by ButlerVolmer, the smallerthe electrode radius (i.e., the smaller the KMHmax 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 ButlerVolmer (rBV0 ) and MarcusHush (r

    MH0 ) from an experimental current

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

    rBV0rMH0

    KMHmax

    1 KMHmax14

    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 ButlerVolmerexpression for the stationary limiting current (Eq. (12)) is smallerthan the value obtained from the MarcusHush solution (Eq.(13)), such that for KMHmax < 20 the difference is greater than 5%.Note that this is not a signicant practical problem since reversibleelectrode reactions (large KMHmax 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=

    t

    punder the ButlerVolmer (solid line)

    and MarcusHush (dashed lines) treatments for an irreversibleprocess with k0 104 cm=s k0BV k0MH, where the differencesbetween 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 =

    tp

    pand an

    Intercept = 0 (see Table 1). With respect to the MH model, for typ-ical values of the reorganization energy (k 0:5 1 eV,k 2040 [5]) the variation of the limiting current with timecompares well with that predicted by ButlerVolmer 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=

    t

    pis predicted, and any attempt at lineariza-

    tion would result in poor correlation coefcient and Slope 0.

    In the case that kinetic control exists in the limiting current,according to Eq. (11) kMHmax can be determined by extrapolating the

    E. Laborda et al. / Journal of Electroanvalue of the current to t = 0:

    IMHlim t ! 0 FAcOkMHmax 15OtD

    Fig. 1. Single potential step chronoamperometry at large overpotentials. (A)Variation of the limiting current with time; (B) concentration proles at the endof the pulse. Planar electrode, k0 104 cm=s k0BV k0MH

    .

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

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