Journal of Electroanalytical Chemistry 660 (2011) 169–177
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A comparison of Marcus–Hush vs. Butler–Volmer electrode kinetics using potential pulse voltammetric techniques Eduardo Laborda a,b, Martin C. Henstridge a, Angela Molina b, Francisco Martínez-Ortiz b, Richard G. Compton a,⇑ a b
Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom 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 2011 Received in revised form 22 June 2011 Accepted 24 June 2011 Available online 12 July 2011 Keywords: Electrode kinetics Butler–Volmer model Marcus–Hush model Potential pulse techniques
a b s t r a c t Simulated voltammograms obtained by employing Butler–Volmer (BV) and Marcus–Hush (MH) models to describe the electrode kinetics are compared for commonly used potential pulse techniques: chronoamperometry, Normal Pulse Voltammetry, Differential Multi Pulse Voltammetry, Square Wave Voltammetry and Reverse Pulse Voltammetry. A comparison between both approaches is made as a function of the heterogeneous rate constant, the electrode size, the applied potential and the electrochemical method, establishing the conditions in which possible differences might be observed. The effect of these differences in the extraction of kinetic parameters, diffusion coefficients and electrode radii are examined, and criteria are given to detect possible deviations of the experimental system from Butler–Volmer kinetics from the behaviour of the chronoamperometric limiting current. The Butler–Volmer model predicts the appearance of an anodic peak in Reverse Pulse Voltammetry for irreversible processes and a peak split of differential pulse voltammograms for quasireversible processes with a value of the transfer coefficient very different from 0.5 (smaller than 0.3 for a reduction process). These striking phenomena are studied by using the Marcus–Hush approach, which also predicts the anodic peak for slow electrode reactions in Reverse Pulse Voltammetry but not the split of the curve in differential pulse techniques. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Different theories have been developed for the description of heterogeneous electron transfer processes between a reacting species in solution and a metal electrode. These can be divided into two main categories: the macroscopic, phenomelogical Butler–Volmer (BV) model [1,2], and the microscopic theories which include Marcus’ work [3,4] as well as other approaches based on quantum mechanics [5–8]. The Butler–Volmer model offers a simple and useful way to study and classify the kinetics of electrode reactions and so it has been preferably used in electrochemistry over many years. The microscopic approaches involve more complex fundamentals and formulations but they enable us to relate the electrode kinetics with the nature of the species involved, the solution composition 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 experimentally reveal the differences between the Butler–Volmer and Mar-
⇑ Corresponding author. Tel.: +44 (0) 1865 275413; fax: +44 (0) 1865 275410. E-mail address:
[email protected] (R.G. Compton). 1572-6657/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2011.06.027
cus–Hush kinetic models in the description and analysis of the electrochemical response [12–24]. It is theoretically predicted that significant discrepancies between both models exists in the case of slow charge transfer processes, which affects the values of kinetic parameters extracted. Suwatchara et al. have recently carried out the experimental assessment of both treatments with the slow reduction process of 2-nitropropane at high-speed channel microband electrodes finding, perhaps unexpectedly, a better agreement between the experimental results and those predicted by the Butler–Volmer approach [16]. Similar conclusions have been reached by Henstridge et al. [17] from the study of the one-electron reduction of 2-methyl-2-nitropropane and europium(III) at mercury microelectrodes. The aim of the present work is to deepen the comparison of Butler–Volmer and Marcus–Hush treatments by examining the differences in the response of the most popular potential pulse techniques: chronoamperometry, Reverse Pulse Voltammetry (RPV), Differential Multi Pulse Voltammetry (DMPV) and Square Wave Voltammetry (SWV) [25]. As well as in electroanalysis, these methods are used in the determination of diffusion coefficients, kinetic rate constants and reaction mechanisms and, as will be shown, the conclusions can differ depending on the kinetic model employed for the electron transfer reaction.
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The conditions in which important discrepancies between BV and MH formalisms can be observed are discussed as a function of the heterogeneous rate constant, the electrode size and the electrochemical method. Moreover, criteria to detect deviations of the experimental system from Butler–Volmer behaviour are discussed from the analysis of the behaviour of the chronoamperometric limiting current. 2. Theory We consider the case of a one-electron reduction process taking place on the surface of a (hemi)spherical electrode: E00 ;k0
O þ e R
ð1Þ
where E00 is the formal potential of the redox couple O/R and k0 the standard heterogeneous rate constant. When applying a constant potential E where Reaction (1) occurs, the transport by diffusion of the electroactive species is described by Fick’s second law that in the case of electrodes of spherical geometry is given by: @cO @t @cR @t
2 9 = ¼ DO @@rc2O þ 2r @c@rO > 2 ; ¼ DR @@rc2R þ 2r @c@rR >
ð2Þ
where cO and cR are the concentration profiles of the electroactive species, and DO and DR the diffusion coefficients. The boundary value problem of the solutions of the differential equation system (2) is given by:
t ¼ 0; r P r 0 t P 0; r ! 1
cO ¼ cO ;
cR ¼ cR
ð3Þ
t > 0, r = r0:
@cO @cR DO ¼ DR @r r¼r0 @r r¼r0
ð4Þ
@cO ¼ kred cO ðr ¼ r 0 Þ kox cR ðr ¼ r 0 Þ DO @r r¼r0
ð5Þ
where kred and kox are the reduction and oxidation rate constants, respectively, and cO and cR the bulk concentrations of the electroactive species. Depending on the kinetic formalism assumed, the variation of the rate constants with the applied potential is described in different ways. According to the Butler–Volmer treatment this is given by: BV
0
BV
0
kred ¼ kBV eag
) ð6Þ
kox ¼ kBV ebg
where g = F(E E00 )/RT and a and b are the transfer coefficients that indicate the symmetry of the energy barrier, that is, if the transition state is reactant or product like [5,6]. According to Eq. (6), as the apBV plied potential is more negative the reduction rate constant (kred ) BV increases and the oxidation rate constant (kox ) decreases without limit, and vice versa for positive potential values [14]. From the Marcus–Hush approach the following expressions are deduced for the rate constants [11,14]:
9
0 g;k Þ = kMH eg=2 IðIð0;k Þ
MH kred
¼
MH kox
0 g;k Þ ; ¼ kMH eg=2 IðIð0;k Þ
ð7Þ
where k ¼ kF=RT, with k being the reorganization energy, and Iðg; k Þ is an integral of the form:
Iðg; k Þ ¼
Z
1
1
exp
h
ðegÞ2 4k
i
2coshðe=2Þ
de
ð8Þ
where e is an integral variable. The value of the reorganization energy (k) corresponds to the energy necessary to adjust the configurations of the reactant and solvent to those of the product state. It can be separated into two contributions, the outer and inner reorganisation energies related to the reorganization of the solvent and the electroactive species (bond lengths, angles,. . .), respectively [5]. For large values of the reorganization energy, the quotient of integrals Iðg; k Þ=Ið0; k Þ tends to unity such that the expressions for the rate constants in the MH model coincide with those in the BV model for a = b = 0.5. Note that both approaches include the Nernstian limit for large heterogeneous rate constant such that for fast electron transfer the results obtained from both formalisms coincide. Therefore, for reversible processes or large values of the reorganization energy, complete agreement is expected between the Marcus–Hush model and the Butler–Volmer one with a = b = 0.5. 3. Results and discussion 3.1. Influence of the reorganization energy and the heterogeneous rate constant 3.1.1. Single potential step chronoamperometry and Normal Pulse Voltammetry (NPV) First, the simplest case corresponding to the application of a single potential step at a large overpotential for the reduction of species O is considered. As is well known, taking into account the Butler–Volmer model the value of the diffusion-controlled reduction current at large overpotentials is given by the following expression for (hemi)spherical electrodes:
1 1 IBV lim ¼ FADO c O pffiffiffiffiffiffiffiffiffiffiffiffi þ pDO t r0
ð9Þ
Thus, the limiting current is exclusively controlled by the diffusion transport of species O towards the electrode surface, and it is independent of the electrochemical reversibility of the process. When the Marcus–Hush treatment is considered, the reduction rate constant is not predicted to increase continuously with the applied potential but rather a maximum value exists. A simple expression for this value of the rate constant is given by [14]: MH kmax
¼
0 kMH
pffiffiffiffiffiffiffiffiffiffiffi 4pk expðk =4Þ ð2:5 6 k 6 80Þ p3 p 4ðk þ4:31Þ
ð10Þ
from which the following solution is derived for the limiting current at (hemi)spherical electrodes, which gives rise to accurate results (error smaller than 1% with respect to Eq. (7)) in a wide range of values of the reorganization energy, 2:5 6 k 6 80: ! K MH 1 max IMH ¼ FAD c O O lim r0 1 þ K MH max " # pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi 2 DO t DO t MH MH 1 þ K max exp 1 þ K MH erfc 1 þ K max max r0 r0
ð11Þ MH
where K MH max ¼ kmax r 0 =DO . Note that the value of the dimensionless heterogeneous rate constant K MH max increases with the standard rate constant 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 depends not only on the diffusion transport but also on the electrode kinetics so that it is a function of the reorganization energy and the heterogeneous rate constant. For pffiffiffiffiffiffiffiffi the term DO t 1 þ K MH max =r 0 ,
values of K MH and max pffiffiffiffiffiffi 2 DO t K MH 1 þ K MH max exp max r0
large
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erfc
pffiffiffiffiffiffi DO t r0
1 þ K MH max
r0 tends to pffiffiffiffiffiffiffiffi and the BV and MH expres-
pDO t
sions then coincide. Thus, it can be concluded that these parameters set the discrepancy in the value of the limiting current between the two kinetic models. Thus, greater differences are expected for small k0 and/or k values, short t values and small electrode radius. pffiffiffiffiffiffiffiffi Under steady state conditions (r0 DO t), the expressions for the limiting current simplify to:
IBV lim;ss ¼
FADO cO r0
IMH lim;ss
FADO cO
¼
ð12Þ 1 K MH max r 0 1 þ K MH max
! ð13Þ
The difference between the above expressions is the term . MH MH 1 þ K MH ¼ kmax r 0 =DO max that tends to unity for large K max
K MH max
The differences between BV and MH have also implications in the concentration profiles of the electroactive species. Thus, whereas the BV model predicts a zero surface concentration of the oxidized species at the electrode surface, in the Marcus–Hush model the surface concentration of species O also depends on the electrode kinetics such that for small values of the heterogeneous rate constant and reorganization energy this is not zero (see Fig. 1B); the smaller the reorganization energy, the greater the surface concentration of the reacting species. Next we will consider the application of a sequence of independent pulses at different potential values, that is, the case of Normal Pulse Voltammetry (Fig. 2A). Fig. 3A shows the voltammograms 0 0 0 0 obtained for k ¼ 104 cm=s (where kMH ¼ kBV ¼ k ) and different values of the reorganization energy. In the inset, the difference be-
(A)
values such that both solutions coincide. Otherwise, the term . MH K MH max 1 þ K max is smaller than unity and the stationary current pre-
BV MH ( λ∗ = 40)
3.5
Ι lim(t) / Ιlim(t p )
dicted by Marcus–Hush is less than by Butler–Volmer, the smaller the electrode radius (i.e., the smaller the K MH max value), the greater the difference between both solutions. The steady state limiting current is usually employed in the determination of the radius of microelectrodes. From Eqs. (12) and (13) the ratio of the electrode radii determined with Butler– MH Volmer (r BV 0 ) and Marcus–Hush (r 0 ) from an experimental current value can be evaluated as a function of the dimensionless parameter K MH max :
r BV K MH 0 max ¼ MH r0 1 þ K MH max
4.0
MH, λ∗ = 20 MH, λ∗ = 15
3.0
MH, λ∗ = 10
2.5 2.0 1.5 1.0 1.0
ð14Þ
MH
IMH lim ðt ! 0Þ ¼ FAc O kmax
ð15Þ
2.0
2.5
3.0
3.5
4.0
1 t
(B)
1.0
0.8
cO / c*
The above expression reveals that for small K MH max values (i.e., small radius, slow electrode reaction and/or small reorganization energy) the electrode radius estimated with the Butler–Volmer expression for the stationary limiting current (Eq. (12)) is smaller than the value obtained from the Marcus–Hush solution (Eq. (13)), such that for K MH max < 20 the difference is greater than 5%. Note that this is not a significant practical problem since reversible electrode reactions (large K MH max values) are usually employed for the calibration 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 planar pffiffi electrode is plotted vs. 1= t under the Butler–Volmer (solid line) and Marcus–Hush (dashed lines) treatments for an irreversible 0 0 0 process with k ¼ 104 cm=s ð¼ kBV ¼ kMH Þ, where the differences between both models are more apparent according to the above discussion. Regarding the BV model (Eq. (9)), a unique curve is predicted pffiffiffiffi independently of the electrode kinetics with a Slope = t p and an Intercept = 0 (see Table 1). With respect to the MH model, for typical values of the reorganization energy (k ¼ 0:5 1 eV, k 20—40 [5]) the variation of the limiting current with time compares 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 BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of pffiffi Ilim(t)/Ilim(tp) with 1= t is predicted, and any attempt at linearizapffiffiffiffi tion would result in poor correlation coefficient and Slope < t p and Intercept > 0. In the case that kinetic control exists in the limiting current, MH according to Eq. (11) kmax can be determined by extrapolating the value of the current to t = 0:
1.5
0.6
0.4
0.2
0.0 0
2
4
6
8
x DO t Fig. 1. Single potential step chronoamperometry at large overpotentials. (A) Variation of the limiting current with time; (B) profiles at the end concentration 0 0 0 of the pulse. Planar electrode, k ¼ 104 cm=s ¼ kBV ¼ kMH .
Table 1 pffiffi Study of the linearity of the curves Ilim(t)/Ilim (tp) vs. 1= t with Marcus–Hush kinetics. 0 0 0 4 Planar electrode, k ¼ 10 cm=s ð¼ kBV ¼ kMH Þ, t p ¼ 1 s, t ¼ 0:01 s. k
>30 20 15 10
Marcus–Hush Slope
Intercept
R2
1 0.967 0.760 0.318
0 0.048 0.333 0.835
1 0.99962 0.98285 0.88022
Butler–Volmer (any k0): slope = 1. Intercept = 0. R2 = 1.
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185 mV for the conditions considered in the figure) such that in circumstances where the difference in the limiting current is not significant, it may be at these potentials. So, the peak current in the differential pulse techniques is expected to be more sensitive to differences between the two kinetic models (see below).
tween the results obtained with the MH model and the BV model for a = b = 0.5 are shown. When differences are observed, the current predicted by Marcus–Hush kinetics is smaller than for Butler–Volmer kinetics. Note that the most significant discrepancy is found at intermediate potential values (between 165 and
(B) Reverse Pulse Voltammetry (RPV)
(+)
( -)
(A) Normal Pulse Voltammetry (NPV) Current sample
Current sample
Ι2
Potential
Potential
Recovery of initial t2 equilibrium conditions
Eindex = E2 t1
Recovery of initial equilibrium conditions
tp Eindex E1
Time
(C) Differential Multi Pulse Voltammetry (DMPV)
(D) Square Wave Voltammetry (SWV) scan
( -)
( -)
Time
scan
Current sample
Ιp
Ιf
Current sample
t1 ΔE
1 2 tp
ESWV Eindex_
Ι1
_E
f =
Es
Potential
Potential
tp
tp
index
Ιb
Es
Time
Time
Fig. 2. Potential waveform in the pulse techniques: NPV, DMPV, SWV and RPV.
60
1.0 50 40
ε (%)
Ι(t) / Ιd(tp)
0.8 0.6
30 20
0.4
10
BV ΜΗ, λ∗ = 40 ΜΗ, λ∗ = 20 ΜΗ, λ∗ = 10
0.2 0.0 -15
-10
0 -20
-15
-10
-5
0
η
-5
η
0
5
Fig. 3. Comparison of the response with BV (a = b = 0.5) and MH in Normal The inset shows the relative difference (e ð%Þ) between both results in Pulse Voltammetry. pffiffiffiffiffiffiffiffiffiffiffi 0 0 0 function of the applied potential. Planar electrode, tp = 1 s, k ¼ 104 cm=s ¼ kBV ¼ kMH . Id ðt p Þ ¼ FADcO = pDt p .
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3.1.2. Potential step techniques In 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 Voltammetry (DMPV) and Square Wave Voltammetry (SWV) [5,6]. The potential-time program applied in each method is shown in Fig. 2. For the theoretical study of the response in RPV the analytical solution available for spherical electrodes are employed [26], which was initially deduced assuming the Butler–Volmer model but it can be modified to the Marcus–Hush model by substitution
(A) k0 = 0.01 cm/s
0.2
0.0
Ι2 / Ιd(t2)
Butler-Volmer Marcus-Hush -0.2
-0.4
-0.6
1 5 λ* > 10
-0.8 -8
-6
-4
-2
0
2
4
6
8
ηindex
(B) k
0.2
0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffi t2 1 ¼ t þ t2 1 IBV ðt Þ lim 2 IBV d;RPV
= 0.001 cm/s
λ* > 300 20
Ι2 / Ιd(t2)
10
-0.2
5
-0.4 5
-0.6 λ* > 100
-0.8 -15
-10
-5
0
20
5
10
10
15
0.0
ηindex
-0.2
(C) k0 = 0.0001 cm/s
0.2
-0.4
0.0
Ι2 / Ιd(t2)
ð16Þ
Assuming the MH formalism, it is found that for k > 25 the results coincides satisfactorily with those predicted by Eq. (16) (see Fig. 5). For smaller k values, the anodic limiting current is smaller in absolute value than that predicted by Butler–Volmer, it is dependent on the values of both diffusion coefficients (the smaller the diffusion coefficient of species R, the greater the absolute value of the anodic limiting current) and it deviates from the linear behaviour described by Eq. (16), as can be observed in Fig. 5. Next, an analogous study is performed for DMPV and SWV techniques. According to that discussed in Section 3.1.1, given the region of potentials where the peak appears and that the pulse
80
0.0
of the expressions for the oxidation and reduction rate constants by the expressions given in Eq. (7). The responses in DMPV and SWV were obtained by means of finite difference numerical methods with a homemade program employing EXTRAP4 algorithm for time integration and an exponentially expanding grid with high expansion factors and four-point formulae for the spatial discretization according to the results presented in Refs. [27–29]. In all cases 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 values 0 0 0 of the standard rate constant (k ¼ kMH ¼ kBV ) and the dimensionless reorganization energy (k ¼ kF=RT). For fast electrode reactions or large reorganization energies the results coincide in the whole region of potentials whereas when the heterogeneous rate constant and/or the reorganization energy of the system decrease, the value of the current in MH is smaller in absolute value. Since the 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 RPV are commonly employed for the determination of the diffusion coefficients of the electroactive species and so it is interesting to analyze the possible effect of the kinetic model on them. According to the BV model the value of the limiting currents is independent of the electrode kinetics and, in the case of a planar electrode, also of the diffusion coefficient of the product species [28]. Thus, in linear diffusion the following simple linear relationship applies for the normalized anodic limiting current (Id,RPV):
Id,RPV BV (t 2 ) Ilim
-0.2
10
-0.4
-0.6 BV MH (λ∗ > 30) MH (λ∗ = 20) MH (λ∗ = 15) MH (λ∗ = 10)
-0.8 15
-0.6 40
-1.0 0.0
20
λ* >1000
-0.8 -20
-10
0
ηindex
10
20
Fig. 4. Comparison of the response with BV (a = b = 0.5) and MH in Reverse Pulse 0 0 0 Voltammetry for different values of k ð¼ kBV ¼ kMH Þ and k (indicated on the pffiffiffiffiffiffiffiffiffiffiffi graphs). Planar electrode, t 1 ¼ 1 s; t2 =t1 ¼ 0:05. Id ðt2 Þ ¼ FADcO = pDt 2 .
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t2 t1 + t 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 5. Study of the linearity of the curves Id;RPV =IBV t2 =t 1 þ t2 predicted lim ðt2 Þ vs. by Butler–Volmer (solid line) and Marcus–Hush (dashed lines) for different values 0 of the reorganization energy. Planar electrode, t1 ¼ 1 s; k ¼ 104 cm=s 0 0 ¼ kBV ¼ kMH .
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length is usually very short in these techniques [5,6] the response is expected to be more dependent on the kinetic model used than the value of the limiting currents. Thus, whereas a good agreement in the value of the limiting current has been always found for k > 20, in DMPV and SWV the peaks corresponding to slow electrode processes (k0 < 103 cm/s) are different in BV and MH in this common range of values of the reorganization energy (k > 20), the
peak current being smaller in the former as can be observed in Figs. the values of the kinetic parameters o 7.nConsequently, n 6 and o 0 0 kMH ; k or kBV ; a of sluggish processes extracted in DMPV and SWV may differ depending on the model assumed. In this sense, an analysis of the consistency of the results obtained with different techniques and different time scales can be useful to re-
(A) k0 = 0.01 cm/s
(A) k0 = 0.01 cm/s
λ* > 40
λ* >10
0.6
ΔΙ / Ιd(tp)
ΔΙ / Ιd(tp)
0.3
0.2
0.4
0.2
0.1
0.0
0.0 -8
-6
-4
-2
0
2
4
6
-8
8
-6
-4
-2
0
λ* >200
ΔΙ / Ιd(tp)
ΔΙ / Ιd(tp)
20
0.04
5
0.10 0.05
0.00 -12
0.00 -6
-4
-2
ηindex
0
2
4
6
10
0.15
0.02
-8
-12
8
-8
-4
80
0.20
ΔΙ / Ιd(tp)
20
10
0.02
8
λ* >1000 80 40
40
0.03
4
(C) k0 = 0.0001 cm/s
λ* >1000
0.04
0
ηindex
(C) k0 = 0.0001 cm/s 0.05
8
80
0.20
0.06
-10
6
λ* > 300
0.25
80 20 10 5
0.08
4
(B) k0 = 0.001 cm/s
(B) k0 = 0.001 cm/s
0.10
2
ηindex
ηindex
ΔΙ / Ιd(tp)
Butler-Volmer Marcus-Hush
10 1
Butler-Volmer Marcus-Hush
1
0.15
20
0.10 10
0.05
0.01
0.00 -16
-12
-8
-4
ηindex
0
4
Fig. 6. Comparison of the response with BV (a = b = 0.5) and MH in Differential 0 0 0 Multi Pulse Voltammetry for different values of k ¼ kBV ¼ kMH and k (indicated on the graphs). Planar electrode, DE ¼ 50 mV, ES ¼ 5 mV, t1 = 1 s, tp/t1 = 0.02. pffiffiffiffiffiffiffiffiffiffiffi Id ðtp Þ ¼ FADcO = pDt p .
0.00 -20
-15
-10
-5
ηindex
0
5
Fig. 7. Comparison of the response with BV (a = b = 0.5) and MH in Square Wave 0 0 0 Voltammetry for different values of k ¼ kBV ¼ kMH and k (indicated on the graphs). Planar electrode, f ¼ 25 Hz, ESWV ¼ 50 mV, ES ¼ 5 mV. pffiffiffiffiffiffiffiffiffiffiffi Id ðtp Þ ¼ FADcO = pDt p .
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veal the most suitable approach for the description of the kinetics of 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 that even for very large values of the reorganization energy there is not a direct correspondence between both models and the responses do not converge like in Figs. 4–7. In these cases, it is necessary to adjust the values of the kinetic parameters and the formal potential to obtain a satisfactory agreement between the responses in both approaches (see Fig. 8 and Table 2). Thus, for a < 0:5 a smaller value of the heterogeneous rate constant is required in MH for a good fit, and a greater k0 value for a > 0:5. Finally, the effect of the electrode size on the difference between the responses in the different techniques here considered is shown in Fig. 9 for a slow electrode process with 0 0 0 k ¼ 104 cm=s ð¼ kBV ¼ kMH Þ. As was deduced above, the smaller the electrode radius, the greater the difference between the kinetic formalisms. In general, we can infer that any change promoting the irreversibility of the electrode reaction, that is, shorter pulse times, smaller heterogeneous rate constant and/or smaller electrode radius, increases the difference between the two kinetic models.
For the conditions considered in Figs. 4–7 and a hemispherical microelectrode with r 0 ¼ 5 lm it is found that the agreement in the value of the limiting currents (Ilim, Id,RPV) in both models is very satisfactory for k P 40 (k P 1 eV). Regarding the DMPV and SWV technique, for quasireversible and irreversible processes 0 (k < 102 cm=s) the peak current depend on the kinetic treatment assumed and so the applicability of the kinetic model must be critically examined in each case. 3.1.3. Striking phenomena predicted by the Butler–Volmer model In this section some anomalous features of DMPV, SWV and RPV curves predicted with the BV treatment will be considered by employing the MH one. First, the peak split in differential pulse voltammetries (DDPV, DMPV, SWV, ADPV) for quasireversible processes is considered, a phenomenon predicted by several authors in the literature [30– 34] assuming the Butler–Volmer treatment. For a reduction process the split takes place for small values of the transfer coefficient (a < 0:3), it is more apparent at planar electrodes and it consists of two peaks: a sharp peak situated at more positive potentials than E00 and a broad peak at negative overpotentials. In Fig. 10 DMPV and SWV curves were also obtained with the Marcus–Hush model for a wide range of values of the reorganization energy under the
(A) DMPV 0 kBV = 10-4 cm / s
0.08 BV MH
Table 2 Values of the heterogeneous rate constant, transfer coefficient and formal potential corresponding to the DMPV and SWV curves shown in Fig. 8.
α = 0.6
Fig. 8A. DMPV: t1 = 1 s, t1/t2 = 50, DE = 50 mV, Es = 5 mV 0 0 Butler–Volmer kBV ¼ 104 cm=s kBV ¼ 104 cm=s a ¼ 0:6 a ¼ 0:4
ΔΙ / Ιd(tp)
0.06
E00 BV ¼ 0 mV 0 kMH
Marcus–Hush
0.04
E00 BV ¼ 0 mV 4
¼ 8:2 10 k ¼7
α = 0.4
0
kMH ¼ 0:19 104 cm=s k ¼ 45
cm=s
E00 MH ¼ 105 mV
E00 MH ¼ þ65:5 mV
Fig. 8B. SWV: f = 25 Hz, ESWV = 50 mV, Es = 5 mV 0 Butler–Volmer k ¼ 104 cm=s
0.02
BV
Marcus–Hush
0.00 -15
-10
-5
η index
a ¼ 0:4
E00 BV ¼ 0 mV
E00 BV ¼ 0 mV
0
0
0
kBV ¼ 104 cm=s
a ¼ 0:6
0
kMH ¼ 10:7 104 cm=s k ¼ 120
kMH ¼ 0:7 104 cm=s k ¼ 40
E00 MH ¼ 92 mV
E00 MH ¼ 7:7 mV
(B) SWV 0 kBV = 10-4 cm / s
0.30 BV MH
10
α = 0.6
0.25
Ιmax(BV) / Ιmax(MH)
8
ΔΙ / Ιd(tp)
0.20 0.15
α = 0.4
0.10 0.05
DMPV: ΔΙpeak SWV: ΔΙpeak
6
RPV: Ιd,RPV NPV: Ιlim
4
2
0.00 -15
-10
-5
0
η index Fig. 8. Comparison of the response with BV (a + b = 1) and MH in (A) Differential Multi 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 each case are indicated in Table 2. Other conditions as in Figs. 6 and 7. pffiffiffiffiffiffiffiffiffiffiffi Id ðtp Þ ¼ FADcO = pDt p .
0.1
1
10
R0 =
100
r0 Dt p/2
Fig. 9. Effect of the electrode size on the differences between the results obtained for the maximum currents in the different electrochemical techniques for a slow 0 0 0 electrode process with k ¼ 104 cm=s ¼ kBV ¼ kMH from BV (a = b = 0.5) and MH (k ¼ 20) models. Other conditions as in Figs. 4–7.
E. Laborda et al. / Journal of Electroanalytical Chemistry 660 (2011) 169–177
(A) DMPV BV, α = 0.3, β = 0.7 MH, λ∗ = 40 MH, λ∗ = 10 MH, λ∗ = 4
-0.06
0.0
0.6
Ι2/Ιd(t2)
ΔΙ / Ιd(tp)
-0.05
0.1
0.8
-0.04 -0.03
ΙRPV/Ιd(t2)
176
0.4
BV MH, λ∗ = 40 MH, λ∗ = 20 MH, λ∗ = 10
0.2
-0.1
-0.2
-0.3
0
5
0.0
10
15
ηindex
20
25
-0.02 -0.2 -0.01 -30
-20
-20
-10
ηindex
0
10
0
10
20
30
Fig. 11. Anodic peak in RPV curves for irreversible electrode processes predicted by Butler–Volmer (solid line, a ¼ b ¼ 0:5) together with the results obtained with Marcus–Hush (dashed lines) for different values of the dimensionless reorganiza0 tion energy (indicated on the graphs). Planar electrode, k ¼ 104 cm=s pffiffiffiffiffiffiffiffiffiffiffi 0 0 ¼ kBV ¼ kMH , t1 ¼ 1 s; t 2 =t1 ¼ 2: Id ðt 2 Þ ¼ FADcO = pDt 2 .
(B) SWV
0.25
BV, α = 0.2, β = 0.8 MH, λ∗ = 40 MH, λ∗ = 10 MH, λ∗ = 4
0.20
4. Conclusions
0.15
ΔΙ / Ιd(tp)
-10
ηindex
0.00
0.10
0.05
0.00 -30
-20
-10
0
ηindex 0
0
10
0
Fig. 10. Peak split in (A) DMPV (k ¼ 103 cm=s ¼ kBV ¼ kMH ) and (B) SWV 0 0 0 (k ¼ 104 cm=s ¼ kBV ¼ kMH ) curves predicted by Butler–Volmer (solid line) for quasireversible processes with small values of the a transfer coefficient together with the results predicted by Marcus–Hush (dashed lines) for different values of the dimensionless reorganization energy (indicated on the graphs). Planar electrode, ESWV = 50 mV, DE ¼ 50 mV, ES ¼ 5 mV, t1 ¼ 1 s, t 2 ¼ t p ¼ 0:02 s. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Id ðtp=2 Þ ¼ FADcO = pDt p=2 .
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 performed for smaller and larger k0 values than those included in Fig. 10 and no evidences of the peak split with Marcus–Hush have been found. This can be understood by taking into account that the split in the BV model is predicted for such small a values that have no direct correspondence in the Marcus–Hush formalism. Osteryoung et al. [31] reported an experimental evidence of this split in SWV for the reduction of Zn(II) to Zn(0) at mercury electrodes that would support a better parametrisation of this system by Butler–Volmer kinetics rather than Marcus–Hush kinetics. The second anomalous feature of the voltammograms predicted by Butler–Volmer is the appearance of a peak in the anodic branch for sluggish electrode processes when the duration of the second pulse 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 the 0 reorganization energy is large (k > 20 for k ¼ 104 cm=s). This involves a crossing of the chronoamperograms [26] such that after the crossing time higher currents are recorded at smaller overpotentials, which is related with the more rapid depletion of the species electrogenerated at the electrode surface at large overpotentials.
The comparison between the results obtained with Butler–Volmer and Marcus–Hush approaches have been carried out for the main potential pulse techniques. Whereas the former is easier to handle and compute, the latter enables us to make predictions of the electrode kinetics from the molecular nature of the electroactive species, electrode and solution. The difference between the two models can be significant for small values of the heterogeneous rate constant and/or the reorganization energy, especially when 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 with both approaches are in good agreement for k > 0:6 eV at planar electrodes and k P 1 eV at hemispherical microelectrodes with r0 > 5 lm. At intermediate overpotentials the differences are greater such that the peak current of slow electron transfers in differential pulse techniques is more sensitive to the kinetic model employed. Consequently, the kinetic parameters extracted may differ, necessitating the study of the consistency of the results under different experimental conditions to check the suitability of the approach employed. Finally, two striking phenomena predicted by the Butler–Volmer model have been studied with the Marcus–Hush approach: the anodic peak in reverse pulse voltammograms of slow charge transfer processes and the peak split of the differential pulse voltammograms of quasireversible processes with small values of the transfer coefficient. With both models the appearance of the anodic peak is predicted for sluggish processes. However, no evidence of the peak split has been found in Marcus–Hush, although this is experimentally reported in the literature. Acknowledgements A.M. and F.M.-O. greatly appreciate the financial support provided by the Dirección General de Investigación Científica y Técnica (Project Number CTQ2009-13023), and the Fundación SENECA (Project Number 11989). E.L. also thanks the Fundación SENECA for the grant received. M.C.H. would like to thank EPSRC for 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 [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 and Applications, 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.
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