Electrocatalysis at Modified Microelectrodes: A Theoretical Approach

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Electrocatalysis at Modified Microelectrodes: A Theoretical Approach to Cyclic Voltammetry ´ ngela Molina,*,† Joaquıń Gonza´lez,† Eduardo Laborda,† Francisco Martıńez-Ortiz,† and A Lesław K. Bieniasz‡,§ Departamento de Quı´mica Fı´sica, UniVersidad de Murcia, Espinardo 30100, Murcia, Spain, Department of Complex Systems and Chemical Processing of Information, Institute of Physical Chemistry of the Polish Academy of Sciences, ul. Niezapominajek 8, 30-239 Cracow, Poland, and Faculty of Physics, Mathematics, and Applied Computer Science, Cracow UniVersity of Technology, ul. Warszawska 24, 31-155 Cracow, Poland ReceiVed: May 27, 2010; ReVised Manuscript ReceiVed: July 12, 2010

An analytical and easily manageable equation is given for the cyclic voltammetric current of catalytic reactions taking place at modified electrodes and microelectrodes under conditions of spherical diffusion. This approximate expression gives accurate results for spherical electrodes of small size and ultramicroelectrodes and provides accurate values of the peak potentials, even for planar electrodes. Interesting limiting cases for the rate constant of the catalytic step and for ultramicroelectrodes are obtained, and their physical interpretation is discussed by considering the behavior of the surface concentration of the substrate in solution in each case. Easy methods for determining the rate constant from the use of working curves are proposed. In the case of ultramicroelectrodes, the catalytic rate constant can be directly measured from the current plateau or the half wave potential of the voltammogram. 1. Introduction A great number of surface synthetic procedures have been elaborated in recent years for developing modified electrodes by means of the immobilization of monomolecular or multimolecular films of electrochemically active molecules or biomolecules (e.g., proteins and enzymes) at different electrode surfaces.1-5 Electrocatalysis at modified electrodes is accomplished by an immobilized redox substance, which is activated electrochemically by applying an electrical perturbation (potential or current) to the supporting electrode. As a result, the oxidation or reduction of other species located in the solution adjacent to the electrode surface (which does not occur or occurs very slowly in the absence of the immobilized catalyst) can take place.1,6-8 From the use of these modified electrodes, a large number of applications have emerged for sensors and biosensors, preparative-scale transformations, fuel cells, environmental monitoring, arrays of modified microelectrodes, modified nanoparticles, etc.1,2,5,9-14 These electrocatalyzed reactions have been studied by using different electrochemical methods.1,7,15-17 Among these, cyclic voltammetry (CV) has been the most used, and theoretical methods based on the numerical solution of integral equations have been developed to explain the behavior of the voltammetric curves. A first attempt to describe the CV response of these systems in terms of nonexplicit integral equations was carried out by Andrieux and Saveant18 and later extended by Aoki and co-workers.19 In both cases, the solutions were restricted to linear diffusion of the catalyzed species in solution. Because of the complexity of the response, stationary techniques, such as rotating disk voltammetry, have been extensively used to avoid the influence of mass transport,7,20,21 and more recently, ap* To whom correspondence should be addressed. E-mail: [email protected]. † Universidad de Murcia. ‡ Institute of Physical Chemistry of the Polish Academy of Sciences. § Cracow University of Technology.

proximate solutions for different nonlinear reaction-diffusion processes taking place at polymer-modified electrodes under steady state conditions have been reported.22 However, although the steady behavior shown by the rotating disk electrode is of interest, it is important to have solutions that could explain the evolution of the CV curve from a transient to a stationary behavior. This change, which could be reached by simply varying the scan rate, implies the transformation of a peak-shaped response into a sigmoidal one, and it is of great interest in understanding the mechanism of these reactions, as is clearly indicated in refs 23-30. The aim of this paper is to present an analytical, explicit, and easily manageable equation for the CV response corresponding to the Nernstian redox conversion of a surfaceimmobilized catalyst involved in the direct catalysis of a solution substrate. This equation is applicable to spherical electrodes and ultramicroelectrodes (UMEs) for which, as far as we know, no solution has been reported. From this analytical equation, simple expressions for limiting cases of interest will be deduced and physically interpreted. Moreover, the influence of the electrode radius on the total current will be analyzed by showing that for high values of the chemical rate constant, the cyclic voltammogram evolves from two peaks to a single response as the electrode radius decreases. It will also be shown that the smaller the electrode radius is, the smaller the influence of the sweep rate on the catalytic CV curves. The variation of the surface concentration of the substrate with the electrode radius and the catalytic rate constant will also be analyzed to show its influence on the CV response. Finally, a comparison of the CV catalytic curves with those corresponding to the direct reduction of the substrate at a naked electrode will be carried out. The variation of the peak potential and the dimensionless maximum current versus log (F) (the F parameter includes the scan rate and the electrode radius) are proposed as working curves for the extraction of the catalytic rate constant from the

10.1021/jp104860y  2010 American Chemical Society Published on Web 08/06/2010

Electrocatalysis at Modified Microelectrodes

J. Phys. Chem. C, Vol. 114, No. 34, 2010 14543

experimental CV data. Likewise, from the expressions corresponding to UMEs, we propose simple procedures for obtaining the rate constant from the measurement of the current plateau or the half wave potential of the voltammogram, depending on the relationship between the rate constant and the electrode radius.

(

1 ∂cC1 ∂2cC1 2 ∂cC + ) DC ∂t r ∂r ∂r2

t ) 0, r g r0 t > 0, r f ∞

2. Theory Let us consider the following reaction scheme consisting of a surface electrode process and a catalytic reaction:

Oads + e- h Rads k

Rads + Csol 98 Oads + Dsol

(I)

where Oads and Rads are the oxidized and reduced forms of the surface immobilized catalyst, Csol and Dsol represent the species in solution, and k is the second order rate constant. We will assume that the surface electrode transfer takes place reversibly, so the expressions for the coverages of O and R when a potential step E1 is applied during a time 0 e t e τ are given by:

fO,1

ΓO,1 eη1 ) ) ΓT 1 + eη1

(1)

ΓR,1 1 ) ΓT 1 + eη1

(2)

fR,1 )

where ΓO,1 and ΓR,1 are the surface excesses of species O and R, respectively, and ΓT ) ΓO,1 + ΓR,1, so fO,1 + fR,1 ) 1. Moreover

η1 )

F (E - E0) RT 1

(3)

with E0 being the formal potential of the O/R couple. The time variation of both surface coverages (fO,1 and fR,1) is null due to the independence on time of the applied potential (i.e., dfO,1/dt ) -dfR,1/dt ) 0); in such a way, it is fulfilled that

I1 0)+ kfR,1cC1(r0, t) FAΓT

(4)

with cC1(r0, t) denoting the surface concentration of species C corresponding to the application of E1 to a spherical electrode of radius r0 and A denoting the electrode surface. By rearrangement of eq 4 the current is given by:

I1 ) kfR,1cC1(r0, t) FAΓT

t > 0, r ) r0}

∂cC1 ∂r

(6)

cC1 ) cC*

( )

DC

)

r0

(7)

) kΓT fR,1cC1(r0, t)

(8)

We will assume that the concentration profile of species C when applying a potential E1 is given by the following expression (see the Appendix):29

cC1(r, t) ) cC* + [cC1(r0) - cC*]

( )

r0 r - r0 erfc r 2√DCt

(9)

By introducing eq 9 into condition 8, the surface concentration of species C is easily obtained

cC1(r0, t) ) cC*

1 kΓT fR,1 1 1 1+ + DC r0 √πDCt

(

)

-1

(10)

2.1. Linear Sweep Voltammetry (LSV) and CV. We will consider below the application of a series of successive potential steps E1, E2, ..., Ep of amplitude ∆E and duration τ such that

Ej ) Einitial - j · ∆E;j ) 1, 2, ..., (p/2) Ej ) Efinal + j · ∆E;j ) (p/2) + 1, ..., p

(11)

with the time elapsed between the beginning of the m-th and the end of the j-th potential step given by:

tm,j ) (j - m + 1)τ, m ) 1, 2, ..., j

(12)

Moreover, the total time elapsed between the application of the first and j-th potential step is t ) t1,j ) jτ. In the case of LSV and CV, the pulse amplitude (∆E) that appears in eq 11 tends to zero for a fixed scan rate (i.e., ∆E < 0.01 mV in the practice see refs 29 and 31), so the applied potential waveform given by eq 11 behaves as a continuous function of time in the way:32

E(t) ) Einitial - V · t for t e λ E(t) ) Efinal + V · t for t > λ

(13)

(5)

The right-hand side of eq 5 depends on the surface concentration of species C (cC1(r0, t)). The mass transport of this species when supposing spherical diffusion is defined by the following boundary value problem:

where λ is the time at which the scan is reversed and V ) dE/dt. By applying the superposition principle for any potential pulse of the scan Ej,29 the concentration profile of species C, cCj(r, t), can be written as a function of those obtained in previous potential pulses in the following approximate way:

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cCj(r, t) ) cCj-1(r, t) + [cCj(r0, t) - cCj-1(r0, t)] × r0 r - r0 for j > 1 (14) erfc r 2 D t

( ) √

and

ΓT dfR,j eηj ) cC*√aDC dηj cC*√aDC (1 + eηj)2 ΓT

Ψjsurf ) -

C j

Using the procedure detailed in the Appendix, an expression for the surface concentration of species C for the j-th pulse is deduced

j-1

cat

j-1

cC (r0, Ej) ) cC*

Ψj

∏ [1 + ΛfR,mσm+1,j]

j

m)1 j

(15)

∏ [1 + ΛfR,mσm,j]

m)1

in which we have used potential instead of time since they are variables linearly linked through eqs 11-13 (for example, in LSV, Ej ) Einitial - νt1,j ) Einitial - j∆E). Moreover

kΓT

Λ)

)

∏ (1 + ΛfR,mσm+1,j)

m)1 ΛfR,j j

(25)

∏ (1 + ΛfR,mσm,j)

m)1

Equation 24 corresponds to the contribution of the surface electrode process, and eq 25 corresponds to the catalytic conversion of the dissolved substrate C. 2.2. Particular Cases. From eq 25 for the catalytic current, simplified expressions for some particular cases can be immediately derived. 2.2.1. Limiting Values for G. 2.2.1.1. Planar Electrodes (r0 f ∞, F f 0).

(16)

√aDC

j-1

1 1 + eηm

fR,m )

(24)

Ψjcat,plane ) ΛfR,j

(17)

p ) ∏ (1 + ΛfR,mσm+1,j

m)1 j

(26)

∏ (1 +

p ΛfR,mσm,j

)

m)1

1

σm,j )

F)

(18)

1

F+

√πδm,j

p σm,j ) √πδm,j

( )

(19)

FV RT

(20)

DC

1/2

ar02

a)

F - Einitial | m e p/2 |E RT j-m+1 F ) |-Ej-m+1 - Einitial + 2Efinal | RT

δm,j ) δm,j

where

m > p/2

(21) with ηm ) F(Em - E0)/RT and V ()∆E/τ) being the scan rate. Thus, according to eqs 5 and 15 and taking into account that under LSV or CV conditions the variation with time of the surface coverages of species O and R is not null due to the continuous dependence of the potential perturbation E(t) with time (see eq 13), the approximate expression for the current contains two contributions:

2.2.1.2. UME (σm,j f 1/F).

Ψjcat,UME )

Ψj ) Ψj

+ Ψj

cat

(22)

ΛfR,j Λ 1 + fR,j F

(28)

This solution is a limiting case of eq 25, and it can also be deduced by assuming ∂cC/∂t ) 0 in eq 6. According to eq 28, for an UME, the current is only dependent on the applied potential, and the cyclic voltammetric curve Ψjcat,UME - ηj has a sigmoidal shape with a current plateau (Ej f -∞, fR,j f 1) and a half-wave potential given by:

Ψjcat,UME(plateau) )

E1/2 ) E0 + surf

(27)

Λ 1+

Λ F

RT Λ ln 1 + F F

(

)

(29)

(30)

Equation 28 can be simplified in the two following cases. 2.2.1.3. Λ , F. In this situation, the effects of the applied potential and of the catalytic reaction on the current are separated

with

Ψj )

Ij FAcC*√aDC

(23)

Ψjcat,UME(Λ , F) ) ΛfR,j

(31)



in such a way that the response is identical to that obtained for a pseudofirst order catalytic reaction (i.e., cCj(r0, t) = cC*). Therefore, the catalytic rate constant can be directly obtained from the current plateau (Ψjcat,UME(plateau)Λ,F ) Λ). Under these conditions, the half wave potential coincides with E0. 2.2.1.4. Λ . F. In this case, the current plateau of the sigmoidal Ψjcat,UME - ηj curve is not sensitive to the catalysis since, in line with eq 29, Ψjcat,UME(plateau)Λ.F ) F. However, the half wave potential can be used to obtain the catalytic rate constant because E1/2 ) E0 + (RT/F) ln (Λ/F). 2.2.2. Limiting Values for Λ. 2.2.2.1. Slow Catalytic Kinetics. From eq 15, we deduce that for small values of Λ, the surface concentration of species C is scarcely affected by the catalysis for any value of the electrode radius (i.e., in the expression of cCj(r0, t) given by eq 15, we can neglect the terms ΛfR,mσm,j) in such a way that

cCj(r0, t) f cC*

(32)

and the current becomes identical to that corresponding to a pseudofirst order catalytic mechanism (see refs 33 and 34 for comparison),

Ijcat ) kcC*fR,j FAΓT

(33)

2.2.2.2. Fast Catalytic Kinetics and fR,j f 1. In this situation (cathodic limit), the rate-determining step is the mass transport. Therefore, by introducing the condition ΛfR,mσm,j . 1 in eq 15, we obtain for any electrode radius:

Ijcat 1 1 ) + FAcC*DC r0 √πDCt1,j

(34)

where t1,j is the time elapsed from the beginning of the scan. In this case, the catalytic limiting current coincides with that obtained for a simple charge transfer process (see eq 22 in ref 35). Note that eq 34 turns into eq 29 for very small electrodes (r0 , (πDCt1,j)1/2, Λ . F). 3. Results and Discussion To check the validity of the analytical equations presented here, the comparison with numerical calculations (carried out in the way discussed in ref 36) has been performed (Figure 9). As will be seen below, a very good agreement between analytical and numerical results for spherical electrodes of small size has been found. In the case of larger electrodes, these equations also provide accurate results for the value of the peak potential. In the following, we will denote the current as ψ or I instead of ψj or Ij since we will consider the response of LSV or CV as a continuous function of the potential. Figure 1 shows the effects of the electrode size (r0) and the catalytic rate constant (Λ) on the dimensionless catalytic component of the voltammogram (Ψcat - E curves, eq 25). For a given electrode radius, the increase of Λ (i.e., the increase of the rate constant k) gives rise to the enhancement of Ψcat and to the shift of the voltammogram toward more anodic potentials. For very fast catalytic reactions (Λ g 100), the peak current becomes independent of Λ in an analogous way to that

Figure 1. Catalytic CV Ψcat - (E - E0) curves for spherical microelectrodes calculated from eq 25 for different values of Λ (shown in the curves). These curves have been calculated for ∆E ) 10-2 mV, a scan rate V ) 0.1 V s-1, and three electrode radii r0 (in µm): (a) 50, (b) 20, and (c) 10.

previously reported for planar electrodes.18,19 On the other hand, the peak potential is always sensitive to the kinetics. The shape of the voltammograms is affected both by the electrode radius and by the Λ. Thus, a transition from a peakshaped response to a sigmoidal wave is observed as Λ and/or the electrode radius decrease, although this behavior has different causes in each case (see below). Figure 2 shows the influence of the sweep rate on the catalytic component of the current (Icat/FAcC*) at spherical microelectrodes of two radii (r0 ) 20 and 5 µm) for a fast catalytic reaction (Λ/F ) 100).

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Figure 2. Influence of the scan rate V on the catalytic component of the current (Icat/FAcC*) calculated from eq 25 for spherical microelectrodes of two radii (r0 ) 20 and 5 µm, panels a and b, respectively). The values of V in V s-1 are (from top to bottom) 1.0, 0.5, 0.1, and 0.01. Λ/F ) 100. ΓT ) 10-10 mol cm-2. Other conditions are as in Figure 1.

As can be seen, for conventional microelectrodes (r0 ) 20 µm, Figure 2a), the catalytic current increases with the sweep rate (V), although the effect of V is less apparent than that reported at planar electrode, at which the peak current Ipeakcat increases proportionally to V1/2 for fast catalytic reactions (see ref 17). When considering a smaller microelectrode (r0 ) 5 µm, Figure 2b), the influence of the scan rate on Icat notably diminishes, with the current becoming independent of the scan rate for UMEs, in agreement with eq 28. To discuss the behavior of the process (I) at spherical UMEs, in Figure 3, we have plotted the catalytic component of the voltammograms obtained at a spherical UME with r0 ) 1 µm for two values of the Λ/F ratio (100 and 0.1) and different scan rates. When Λ/F , 1 (Figure 3a), a unique sigmoidal response independent of the scan rate is obtained according to eq 31 (i.e., Icat/(FAcC*) = kΓTfR,j), the half wave potential coinciding with the formal potential of the surface redox couple E0 (see also the inner Figure 3a). Under these conditions, the value of the rate constant can be immediately determined from the plateau current (see eq 33). On the other hand, when Λ/F . 1 (see Figure 3b), sigmoidal responses are also obtained although they are shifted toward more anodic potentials as Λ increases (in this figure, E1/2 - E0 ) 118 mV). To obtain the rate constant from the measurement of E1/2, it

Molina et al.

Figure 3. Catalytic CV Icat/FAcC* - (E - E0) curves obtained at a spherical UME with r0 ) 1 µm for two values of the Λ/F ratio, 100 (a) and 0.1 (b), and for different scan rates (from top to bottom): 1.0, 0.5, 0.1, and 0.01 V s-1. Inner figures: d(Icat/FAcC*)/dE - (E - E0) curves calculated from the stationary voltammograms of panels a and b. Other conditions are as in Figure 2.

is convenient to differentiate the stationary Ψ - E curve and obtain E1/2 from the peak potential in line with eq 28 (see the inner Figure 3b). In this case, the current plateau does not depend on the catalytic rate constant, and it tends to the limiting value DC/r0 for scan rates slower than 0.1 V/s. Figure 4 shows the catalytic component of the CV response of mechanism (I) when Λ . F, as compared with that corresponding to a simple Nernstian charge transfer involving soluble species (Csol + e- h Dsol). It is worth highlighting that the peak current of the catalytic contribution is greater than the peak current of the voltammogram corresponding to the direct reduction of species C at a naked electrode. This behavior is similar to that described in ref 37 for planar electrode, although the difference between both responses decreases with the electrode radius. Thus, at UMEs, the current plateau of a process following mechanism (I) with Λ . F coincides with that corresponding to a simple charge transfer reaction (see eq 15 in ref 31). In Figure 5, the LSV voltammograms of the total current I - E calculated from eqs 22-25 are shown for different electrode sizes and different Λ values. For fast kinetics (Λ ) 103), two signals can be distinguished in the voltammograms, one situated around the formal potential of the O/R couple and another at more positive values; the former is related to the surface redox process and the latter with the catalytic process. By comparing the curves for Λ ) 103 in



Figure 4. Catalytic CV curve Ψcat - (E - E0) (solid line, eq 25) and CV curve ΨE - (E - E0) corresponding to a simple Nernstian charge transfer involving soluble species (dashed lined, eqs 13-15 of ref 31) obtained for a spherical microelectrode of radius 40 µm. The scan rate is V ) 0.01 V s-1 in both cases. In the catalytic response Λ/F ) 2 × 103.

Figures 5a-c, it can be seen that the shape of the voltammogram varies with the electrode size so that for larger microelectrodes (r0 ) 50 µm) two well-defined peaks are observed, whereas for smaller ones (r0 ) 10 µm) the signal corresponding to the catalysis gradually turns into a “shoulder”. This behavior reflects the transition of the catalytic contribution from a peak to a sigmoidal wave when the electrode radius decreases, as described in Figure 1. For slow kinetics (Λ ) 1, 0.2) a single signal is observed, which comprises a peak, related to the surface redox process, and a current plateau, associated with the catalytic process. Note that the transport of species C gives rise to a shift of the peak potential toward more cathodic potentials in relation to that expected for a pseudo first-order surface catalytic mechanism (i.e., E0).33,34 This behavior is more apparent for intermediate kinetics (Λ ) 1) and small electrodes. To understand the above results, it is interesting to consider the variation of the surface concentration of species C (cCj (r0, E)) with the electrode radius and the catalytic rate constant. In Figure 6, the curves cCj (r0, E) - E calculated from eq 15 are plotted for two values of Λ (10 and 1) and three electrode radii (50, 20, and 10 µm). It can be seen that when the rate constant k (and therefore Λ) decreases, the surface concentration of species C increases, so for slow kinetics, it is closer to the bulk value cCj (r0, E) ≈ cC*, as has been pointed out in section 2.1.2. Regarding the electrode size, it is found that the decrease of the electrode radius also leads to the increase of the surface concentration of species C. It is worth highlighting that although both slow kinetics and small electrodes give rise to the increase of the surface concentration of species C, their effects on the catalytic current are the opposite. Thus, when k decreases, the extent of the catalytic reaction is smaller, as is the demand of species C at the electrode surface (see section 2.1.2). Therefore, the catalytic contribution to the current (Ψcat) decreases. On the other hand, for a given rate constant k, the diffusion transport is more efficient when the electrode radius decreases, so cCj (r0, E) increases and more species C is available at the electrode surface. As a consequence, Ψcat increases when the electrode radius decreases.

Figure 5. LSV I/FAΓT - (E - E0) curves calculated from eqs 22-25 for spherical microelectrodes and different Λ values (shown in the curves). The values of the electrode radius (in µm) are (a) 50, (b) 20, and (c) 10. The dotted line marks the formal potential of the O/R redox couple. V ) 0.1 V s-1, and cC* ) 0.1 mM.

In accordance with all of the above results, for very small electrodes and/or very slow catalytic processes (i.e., Λ , F), the surface concentration of species C tends to be that of the bulk solution (cCj (r0, E) f cC*). Under these conditions, it is deduced from eq 5 that the catalytic current would be given by Ijcat ) FAΓTkfR,jcC* (see eq 31). This expression coincides with that corresponding to a pseudofirst order catalytic mechanism,33,34 which is independent of the geometry and size of the electrode, and it can be considered as a limiting case of the reaction scheme (I).

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Figure 6. Variation of the surface concentration of species C [cCj(r0, E)/cC*] with the potential for spherical microelectrodes, calculated from eq 15 for three values of the electrode radius (shown in the curves). The ratio Λ/F is (a) 10 and (b) 1. Other conditions are as in Figure 1.

Figure 7. LSV I/FAΓT - (E - E0) curves calculated from eqs 22-25 for spherical microelectrodes and different Λ values (shown in the curves). The values of the electrode radius are (in µm) 10 (dashed lines) and 50 (dotted lines). The I/FAΓT - (E - E0) curves corresponding to a pseudofirst order catalytic mechanism, calculated from eq 2 of ref 34 (solid lines), have been included for comparison. Other conditions are as in Figure 5.

In Figure 7, this situation is analyzed by plotting the voltammograms corresponding to the total current {[I/(FAΓT)] - E curves, see eqs 22-25} for two small values of the Λ parameter (0.1 and 0.05) and two microelectrodes. The [I/(FAΓT)] - E curves

Molina et al.

Figure 8. F(Ep - E0)/RT - log(F) (a) and Ψmaxcat - log(F) (b) working curves for spherical microelectrodes calculated from eq 25 for different values of the log(Λ/F) (values shown in the curves). We have considered that the voltammogram shows a current plateau when the difference between the peak current and the current at very negative potentials is lower or equal to the 5%.

corresponding to a pseudofirst order catalytic mechanism33 are also plotted for comparison (solid lines). The LSV curves in Figure 7 show the typical shape described for slow kinetics in Figure 5 (see curves for Λ ) 0.2), with a peak located around the formal potential of the O/R redox couple and a current plateau smaller than that corresponding to a pseudofirst order catalytic reaction (kcC*, solid lines), tending to this value as the electrode radius and the rate constant decrease. As was discussed for Figure 1, the peak potential and the maximum current of the voltammogram are sensitive to the catalytic rate constant so that they can be useful for kinetic studies. With this aim, Figure 8 shows working curves for the determination of the catalytic rate constant (k) from the variation of the peak potential [F(Ep - E0)/RT] and the maximum current (Ψmaxcat, i.e., peak current or plateau current) with log(F). Note that for a given value of the electrode radius, the F parameter varies with the scan rate (eq 19). The F(Ep - E0)/RT - log(F) and Ψmaxcat - log(F) curves have been plotted for different values of the ratio Λ/F ()kΓTr0/ DC). To obtain the kinetic information from these working curves, voltammograms at different sweep rates have to be recorded, and the variation of the peak parameters has to be fitted with the theoretical curves.


J. Phys. Chem. C, Vol. 114, No. 34, 2010 14549 numerical results are always smaller than 7 mV in all of the range of F values corresponding to the peak region of interest (see Figure 8a). For slower kinetics (log(Λ/F) < 3), F values (i.e., electrode size and scan rate) in the range -3 < log(F) < -1 must be selected to obtain deviations smaller than 4 mV in the peak potential (see Figure 9a). Concerning the maximum current (see Figure 9b), for any value of the catalytic rate constant, the relative difference between analytical and numerical results is smaller than 5% when F > 0.6, which corresponds to an electrode radius smaller than 30 µm for V ) 0.1 V s-1. The accuracy of the results improves for slow kinetics, so for log(Λ/F) < 1 deviations smaller than 5% are obtained for any electrode size and scan rate, including planar electrodes. 4. Conclusions

Figure 9. Differences found between the analytical and the numerical results for the peak potential (a) and the maximum current (b) calculated by using eq 25 and the procedure discussed in ref 36 for different values of the log(Λ/F) (shown in the curves).

In Figure 8a, two regions can be differentiated. In the peak region [log(F) < 0.5, ηpeak > -5], a well-defined peak that varies linearly with log(F) for F < -0.3 is obtained. Therefore, the rate constant can be determined from the potential peak obtained at different F by means of these working curves. These conditions can be experimentally achieved by using high scan rates. In the plateau region [log(F) > 0.5, ηpeak < -5], a sigmoidal response is obtained, so the peak potential is not well-defined, and it is not appropriate for studying the system. With regard to the maximum current (see Figure 8b), it increases with F. For log(F) > -0.1, a great increase in the maximum current is observed, and the curves obtained for log(Λ/F) g 2 are coincident. This region corresponds to a current plateau, which shows a linear dependence with F according to eq 29. This relationship is confirmed by the inner Figure 8b. The validity of the analytical solution deduced in this paper (eq 25) has been tested by comparison with results obtained from numerical calculations carried out with a homemade program.36 In Figure 9, the differences found between the analytical and the numerical results for the peak potential (Figure 9a) and the maximum current (Figure 9b) are plotted. Regarding the peak potential, it is observed that the analytical expression here deduced gives very accurate results. Thus, for fast kinetics (log(Λ/F) > 4), the deviations with respect to the

Electrocatalyzed reactions have been studied by using CV at spherical electrodes and microelectrodes. Approximate analytical explicit equations have been obtained for the redox conversion of an immobilized species involved in the catalysis of a soluble solution substrate in a spherical diffusion field. The influence of the scan rate and of the electrode size has also been analyzed, indicating the transformation of the peakshaped response into a sigmoidal wave as the scan rate and/or the electrode radius decrease. From these equations, interesting limit cases have been obtained corresponding to the case of UMEs and slow catalytic kinetics. The study of the surface concentration of the substrate with the electrode radius and the catalytic rate constant gives us an insight about the sharpness of the concentration profile of this species, which is related to the transition from second order kinetics to a pseudofirst order one. Working curves have been proposed based on the variation of the dimensionless maximum current and peak potential with F parameter, which depends on the electrode radius and on the scan rate. For UMEs, easy methods have been presented for the determination of the catalytic rate constant from the current plateau or the half wave potential of the stationary voltammogram, depending on whether the Λ/F ratio is greater or smaller than the unity. Acknowledgment. We greatly appreciate the financial support provided by the Direccioń General de Investigacioń Cientı´fica y Tećnica (Project Number CTQ2009-13023) and the Fundacioń SENECA (Project Number 08813/PI/08). E.L. thanks the Ministerio de Educacioń y Ciencia for the grant received. L.K.B. thanks the Fundacioń Seneca. Appendix The current corresponding to the process I when a linear sweep of the potential is applied as in LSV or CV can be considered as a limit of the current obtained for a sequence of potential steps E1, E2, ..., Ej, ..., Ep, of pulse amplitude ∆E and duration τ when the difference between two consecutive steps tends to zero (∆E f 0) for a fixed scan rate V ()∆E/τ).29,31 In this case, we will assume that the concentration profile for the application of a constant potential E1 is given by the approximate expression of eq 9 and also that the superposition principle is applicable in this case. Under these conditions, the boundary value problem given by eqs 6-8 for the j-th potential step applied during a time 0 e tj e τ can be written in the following general way:

14550


(

j ∂cCj ∂cCj ∂2cCj 2 ∂cC ) DC + ) ∂t ∂tj r ∂r ∂r2

tj ) 0, r g r0 tj > 0, r f ∞ tj > 0, r ) r0:

Molina et al.

)

(A1)

cCj(r, t) ) cCj-1(r, t1,j-1)

(A2)

( )

(A3)

DC

∂cCj ∂r

r0

) kΓT fR,jcCj(r0, t) j

We will suppose that the solution of eq A1, cC (r, t), can be written as the following linear combination of the solutions of the previous potential steps, cC1(r, t), cC2(r, t), ..., cCj-1(r, t) j

cCj(r, t) ) cCj-1(r, t) + c˜Cj(r, t) ) cC1(r, t) +

∑ c˜Ci(r, t) i)2

(A4) with c˜Cj(r, t) being new unknown functions to be determined at each potential step and cC1(r, t) given by eq 9. By taking into account the expression of cCj(r, t) given by eq A4 and that cC1(r, t), cC2(r, t), ..., cCj-1(r, t) fulfill differential eq A1, we can rewrite eqs A1 and A2 only in terms of c˜Cj(r, t) functions

(

)

(A5)

c˜Cj(r, t) ) 0

(A6)

j ∂c˜Cj ∂2c˜Cj 2 ∂c˜C ) DC + ∂tj r ∂r ∂r2

tj ) 0, r g r0 tj > 0, r f ∞

The solution of eqs A5 and A6 takes the following general expression:

( )

r0 r - r0 ) erfc r 2√DCtj r r - r0 [cCj(r0, t) - cCj-1(r0, t)] r0 erfc 2 D t

c˜Cj(r, t) ) c˜Cj(r0, t)

( ) √

j > 1 (A7)

C j

By introducing eq A7 into eq A4, we obtain the concentration profile of species C corresponding to Ej, cCj(r, t), given in eq 14, which can be rewritten as: j

cCj(r, t) ) cC* +

∑ [cCm(r0, tm,j) - cCm-1(r0, tm-1,j)] ×

m)1

(

r - r0 r0 erfc r 2√DCtm,j

)

(A8)

with cCm(r0, tj), m ) 1, 2, ..., j, being the expression of the surface concentration of species C for the m-th potential step and cC0(r, t) ) cC*. tm,j is given by eq 12. Note that the total time elapsed between the application of E1 and Ej is t ) t1,j-1 + tj ) (j - 1)τ + tj. By differentiating eq A8 with respect to the distance, the surface flux of species C for Ej can be written as:

( ) ∂cCj ∂r

j

)

r0

∑

a [c m(r , t ) - cCm-1(r0, tm-1,j)]σm,j DC m)1 C 0 m,j (A9)

with σm,j given by eq 18. By substituting eq A9 into eq A3, it is possible to deduce the expression of cCj by proceeding in a recursive manner. Thus, it is immediately obtained that

cCm(r0, tm,j) )

1 + ΛfR,m-1σm,j m-1 c (r0, tm-1,j) 1 + ΛfR,mσm,j C m ) 2, 3, ..., j (A10)

By recovering the expressions of the previous cCm-1(r0, tm-1,j) functions, we finally obtain eq 15. All of the expressions given throughout this paper give accurate results for spherical electrodes and microelectrodes. For large spherical electrodes (including planar electrodes), not too high values of Λ should be considered to determine the peak current (see Figure 9). References and Notes (1) Bard, A. J., Stratmann, M., Fujihira, M., Rusling, J. F., Rubinstein, I., Eds. Encyclopedia of Electrochemistry; Willey-VCH: Weinheim, 2007; Vol. 10. (2) Bartlett, P. N. Bioelectrochemistry, Fundamentals, Expermiental Techniques and Applications; Wiley: Chichester, 2008. (3) Christopher Love, J.; Estroff, L. A.; Kriebel, J. K.; Nuzzo, R. G. Chem. ReV. 2005, 105, 1103. (4) Downard, A. J. Electroanalysis 2000, 12, 1085. (5) Alkire, R. C., Kolb, D. M., Lipkowski, J., Ross, P., Eds. AdVances in Electrochemical Science and Engineering Vol. 11: Chemically Modified Electrodes; Wiley-VCH: Weinheim, 2009. (6) Saveánt, J. M. Chem. ReV. 2008, 108, 2348. (7) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamental and Applications; Wiley: New York, 2001. (8) Murray, R. W. Philos. Trans. R. Soc. London 1981, A 302, 253. (9) Alleman, K. S.; Weber, K.; Creager, S. E. J. Phys. Chem. 1999, 100, 17050. (10) Wang, J. Electroanalysis 2005, 17, 7. (11) Ulgut, B.; Abrunã, H. D. Chem. ReV. 2008, 108, 2721. (12) Wang, J. Analytical Electrochemistry, 3rd ed.; Wiley-VCH: Weinheim, 2006. (13) Campbell, F. W.; Compton, R. G. Anal. Bioanal. Chem. 2010, 296, 241. (14) Willner, I.; Yan, Y.-M.; Willner, B.; Tel-Vered, R. Fuel Cells 2009, 9, 7. (15) Flinkea, H. O. In Electroanalytical Chemistry; Bard, A. J., Rubinstein, I., Eds.; Marcel Dekker: New York, 1996; Vol. 19. (16) Laviron, E. In Electroanalytical Chemistry; Bard, A. J., Rubinstein, I., Eds.; Marcel Dekker: New York, 1989; Vol. 12. (17) Lyons, M. E. G. Sensors 2002, 2, 339. (18) Andrieux, C. P.; Saveant, J. M. J. Electroanal. Chem. 1978, 93, 163. (19) Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1986, 199, 69. (20) Kulesza, P. J.; Marassi, R.; Karnicka, K.; Wlodarczyk, R.; Miecznikowski, K.; Skunik, M.; Kowalewska, B.; Chojak, M.; Baranowska, B.; Kolary-Zurowska, A.; Ginalska, G. ReV. AdV. Mater. Sci. 2007, 15, 225. (21) Kutner, W.; Wang, J.; L’Her, M.; Buck, R. P. Pure Appl. Chem. 1998, 70, 1301. (22) Sethamarai, R.; Rajendran, L. Electrochim. Acta 2010, 55, 3223. (23) Molina, A.; Martıńez-Ortiz, F.; Laborda, E.; Puy, J. Phys. Chem. Chem. Phys. 2010, 12, 5396. (24) Molina, A.; Martinez-Ortiz, F.; Laborda, E.; Morales, I. J. Electroanal. Chem. 2009, 633, 7. (25) Molina, A.; Martinez-Ortiz, F.; Laborda, E. Electrochem. Commun. 2009, 11, 562. (26) Amatore, C.; Arbault, S.; Maisonhaute, E.; Szunerits, S.; Thouin, L. In Trends in Molecular Electrochemistry; Pombeiro, A. J. L., Amatore, C., Eds.; Fontis MediasMarcel Dekker: Lausanne, 2004. (27) Molina, A.; Serna, C.; Martinez-Ortiz, F.; Laborda, E. Electrochem. Commun. 2008, 10, 376. (28) Molina, A.; Morales, I. J. Electroanal. Chem. 2005, 583, 193.

Electrocatalysis at Modified Microelectrodes (29) Molina, A.; Serna, C.; Camacho, L. J. Electroanal. Chem. 1995, 394, 1. (30) Amatore, C. In Physical Electrochemistry: Principles, Methods and Applications; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995; Chapter 4. (31) Molina, A.; Moreno, M. M. Collect. Czech. Chem. Commun. 2005, 70, 133. (32) Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 36, 706. (33) Gonzalez, J.; Soto, C. M.; Molina, A. Electrochim. Acta 2009, 26, 6154.

J. Phys. Chem. C, Vol. 114, No. 34, 2010 14551 (34) Molina, A.; Soto, C. M.; Gonzalez, J. Anal. Chem. 2009, 81, 6830. (35) Molina, A.; Gonzalez, J.; Martinez-Ortiz, F.; Compton, R. G. J. Phys. Chem. C 2010, 114, 4093. (36) Molina, A.; Serna, C.; Martinez-Ortiz, F.; Laborda, E. J. Electroanal. Chem. 2008, 617, 14. (37) Xie, Y.; Anson, F. C. J. Electroanal. Chem. 1995, 384, 145.

JP104860Y

Electrocatalysis at Modified Microelectrodes: A Theoretical Approach

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