Characterization of the Electrocatalytic Response of Monolayer

Apr 30, 2012 - An approximate analytical, explicit, and easily manageable equation is given for the electrochemical response in square-wave voltammetr...
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Characterization of the Electrocatalytic Response of Monolayer-Modified Electrodes with Square-Wave Voltammetry J. Gonzalez,† A. Molina,*,† F. Martinez Ortiz,† and E. Laborda‡ †

Departamento de Química Física, Universidad de Murcia, 30100 Murcia, Spain. Regional Campus of Excellence Campus Mare Nostrum ‡ Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom ABSTRACT: An approximate analytical, explicit, and easily manageable equation is given for the electrochemical response in square-wave voltammetry at spherical electrodes and ultramicroelectrodes of the Nernstian redox conversion of a surface-attached couple that catalyzes the transformation of a substrate in solution. From this equation, simple expressions for limiting cases of interest correspoding to first-order catalysis and small electrodes will be deduced. Moreover, the influence of the electrode radius on the total current will be analyzed by showing that the response presents three different behaviors depending on the values of the rate constant of the chemical step and on the electrode radius. This has allowed us to construct a zone diagram for a complete characterization of the electrochemical response of these processes.

1. INTRODUCTION Electrocatalysis at modified electrodes is accomplished by an immobilized redox mediator, which is activated electrochemically by applying an electrical perturbation (potential or current) to the supporting electrode. As a result, the chemical or electrochemical conversion 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) takes place.1−4 Note that this approach contrasts with the “traditional” electrocatalytic processes where the electrocatalytic reactions involve materials as electrodes.5,6 The process considered here could be classified as “molecular electrocatalysis”, following the nomenclature employed by Saveant, given that the catalyst is a molecule immobilized in a monolayer.2 The main advantage of this kind of electrocatalised reactions lies in the great amount of synthetic procedures for modifying different electrode surfaces with electrochemically active molecules or biomolecules (e.g., proteins, enzymes).7−11 These procedures enable the design of new modified electrodes, which can be applied for allowing or enhancing the response of a great number of molecules of interest. Therefore, this approach has multiple applications in sensors and biosensors, preparative-scale transformations, fuel cells, environmental monitoring, and so on.1,2,5,9−18 The electrochemical characterization of these processes is complex because the response depends on the chemical kinetics of the reaction between the species in solution and the immobilized catalyst as well as on the mass transport of the former. The interplay of both processes is decisive for the characterization of the emerging current−potential curves. To simplify the experimental conditions for analyzing these processes, © 2012 American Chemical Society

stationary techniques such as rotating disk voltammetry have been broadly used to avoid the influence of mass transport. However, although the steady-state behavior is of interest, the understanding of these catalytic reactions requires having the theory for the evolution of the response from transient to stationary because this has dramatic effects on the characteristics of the electrochemical responses.1,3,7,19,20 Among the electrochemical techniques employed to study these electrocatalyzed reactions, cyclic voltammetry (CV) has been the most used. The first attempts to describe the CV response of these systems were carried out by Andrieux and Saveant21 and Aoki and coworkers22 in terms of nonexplicit integral equations. In a previous paper, we proposed an approximate analytical, explicit, and easily manageable equation (applicable to spherical electrodes and ultramicroelectrodes).19 The aim of this Article is to extend the application of this analytical solution to square-wave voltammetry to planar, spherical electrodes, and microelectrodes. This is a powerful electrochemical technique, which has the advantages of being a scanning voltammetric technique (such as CV) enabling quick analyses, low consumption of target species, and reduced fouling of the electrode surface. Moreover, SWV offers a better discrimination of the charging and background currents and better definition of the response than CV. Similar advantages are shown by differential pulse techniques (like differential double pulse voltammetry or additive differential double pulse voltammetry).23−25 Therefore, the SWV technique has been Received: April 3, 2012 Revised: April 30, 2012 Published: April 30, 2012 11206

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species are constant in agreement with eqs 1−3 (i.e., df O,m/ dt = −df R,m/dt = 0), in such a way that the current is given by

intensively examined for its application to analytical, kinetic and mechanistic studies.26−34 The SWV response is always peak-shaped, independently of its stationary or transient character. Therefore, from the analysis of the peak parameters for different values of the chemical rate constant and of the electrode size, it has been obtained that the response has three different limiting behaviors. Accordingly, a “zone diagram” has been constructed to gain deeper physical insight into these processes and carry out a complete characterization of the electrochemical responses. Moreover, simple expressions for the peak current under steady-state conditions are given to assist the determination of the homogeneous rate constants without recourse to further calculations. Also, the influence of the pulse amplitude, the frequency, and the electrode size are examined to establish the most appropriate conditions for the detection and study of the catalytic reaction coupled to the electrode reaction.

Im = kfR, m cCm(r0 , t ) FA ΓT

The term in the right-hand side of eq 5 depends on the chemical rate constant, the coverage of reduced species, and the surface concentration of species C (cmC(r0,t)). The mass transport of this last species for a spherical diffusion domain is defined by the following differential equation and boundary value problem (bvp)

2. THEORY Let us consider the following reaction scheme consisting of an electrocatalytic process between a species in solution (C) and a surface-inmobilized redox catalyst Oads + e− ⇌ R ads k

R ads + Csol → Oads

⎫ ⎪ ⎬ ⎪ + Dsol ⎭

fR, m =

ΓO, m ΓT

ΓR, m ΓT

(1)

=

1 1 + eη , m

(2)

⎫ ⎪ ⎪ ⎛ r−r ⎞ ⎪ 0 ⎟⎟ × erfc⎜⎜ ⎪ ⎪ ⎝ 2 DCt1 ⎠ ⎬ ⎪ cCm(r , t ) = cCm − 1(r , t ) + [cCm(r0 , tm) − cCm − 1(r0 , t )]⎪ ⎛ r−r ⎞ r ⎪ 0 ⎟⎟ m > 1 × 0 erfc⎜⎜ ⎪ r ⎝ 2 DCtm ⎠ ⎪ ⎭

dt

dt

r0 r

⎫ ⎪ ⎪ ⎬ m−1 ∏h = 1 [1 + ΛfR, h σh + 1, m] ⎪ DC ΛfR, m Im = FAcC* m > 1⎪ m ∏h = 1 [1 + ΛfR, h σh , m] τ ⎭

(3)

Im + kfR, m cCm(r0 , t ) FA ΓT

(8)

(9)

Taking into account eqs 5 and 9 and by following a procedure analogous to that discussed in ref 19, we obtain the following approximate expression for the current

with E being the formal potential of the O/R couple. The current obtained for the reaction scheme I is related both to the time variation of the surface coverages f O,m and f R,m and to the catalytic chemical reaction, with the following relationship being fulfilled:19,21,22 =−

⎛ ∂c m ⎞ DC⎜ C ⎟ = k ΓT fR, m cCm(r0 , t ) ⎝ ∂r ⎠r

cC1(r , t1) = cC* + [cC1(r0 , t1) − cC*]

0

dfR, m

(7)

I1 = FAcC*

F ηm = (Em − E 0) RT

=−

tm = 0, r ≥ r0 ⎫ cCm = cCm − 1 m > 1⎫ ⎪ ⎬ ⎬ ⎪ 1 tm > 0, r → ∞⎭ c = c * ⎭ C C

with 0 ≤ tm ≤ τ. tm is the time during which the mth potential step is applied and the total time elapsed between the application of the first and m potential steps is t = (m − 1)τ + tm. Equation 6 with the bvp (7)-(8) cannot be solved analytically. Here we will use an approximate solution that gives rise to very accurate results for small values of the electrode radius (see below). With this aim, we will assume that the concentration profile of species C when applying a potential Em can be written as a function of those obtained in previous potential pulses in the following approximate way19

where ΓO,m and ΓR,m are the surface excesses of species O and R respectively, for the mth step of the sequence, and ΓT = ΓO,m + ΓR,m, so f O,m + f R,m = 1, for any value of m. Moreover

dfO, m

(6)

0

(I)

e ηm 1 + e ηm

=

m ⎛ ∂ 2c m ∂cCm 2 ∂cC ⎞ ⎟ = DC⎜ 2C + ∂tm r ∂r ⎠ ⎝ ∂r

tm > 0, r = r0}

where Oads and Rads are the oxidized and reduced forms of the adsorbed catalyst, Csol and Dsol represent the species in solution, and k is the second-order rate constant of the chemical step. We will assume that the redox catalyst is adsorbed at an electrode of spherical/hemispherical geometry and that the surface electrode transfer takes place reversibly, so the expressions for the coverages of O and R when a sequence of potential steps E1, ..., Em, ..., Ej of duration τ is applied are given by19 fO, m =

(5)

DC 1 ΛfR,1 τ 1 + ΛfR,1 σ1,1

(10)

with fR,m given by eq 2 and Λ = k ΓT

(4)

σh , m =

cmC(r0,t)

with denoting the surface concentration of species C corresponding to the application of Em to the spherical electrode of radius r0 and A denoting the electrode area. Note that when discrete constant potentials are applied and the surface electrode process is fast, the surface coverages of both

ξ= 11207

τ DC

(11)

1 ξ+

1 π (m − h + 1)

(12)

DCτ r0

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The dimensionless parameter Λ accounts for what we could denote as “chemical demand” on the response due to the second step of the reaction scheme I, that is, on the depletion of species C at the electrode surface due to its reaction with the adsorbed species R. The parameter ξ is related to the “sphericity” of the electrode, that is, to the radial component of the mass transport of species C. The relationship between both parameters (which informs about the rate of depletion of species C at the monolayer/ solution interface and the speed at which this species arrives at the interface) characterizes the shape of the overall response. 2.1. Stationary Limits. 2.1.1. First-Order Limit. In the limit case Λ → 0 (i.e., very low demand from the chemical process at the monolayer/solution interface), the current given by eq 10 would become (I m)Λ→ 0 ≅ FAcC*

DC ΛfR, m = FAcC*k ΓTfR, m τ

Taking into account the expression for the step current given by eq 10, the square-wave current can be written as ISW = FAcC*

− fR,2p

p = 1, 2, ...np/2

Note that the above equations can be expressed as a function of the frequency (f) by simply considering that: f = 1/(2τ).27,31−34 The response obtained in SWV is peak-shaped, and because of the complexity of eq 20 the peak coordinates cannot be analytically deduced with the exception of simpler situations like stationary conditions (see below). 2.2.1. Stationary Limits. First-Order Behavior. By considering the simplified expression for the current given by eq 14, the square-wave response is given by the following expression

(14)

(ISW )Λ→ 0 ≅ FAcC*

DC Λ{fR,2p − 1 − fR,2p } τ

= FAcC*k ΓT{fR,2p − 1 − fR,2p }

(21)

Under these conditions, the peak potential would coincide with the formal potential of the immobilized redox couple (E0) and the peak current is given by35

(15)

DC ⎛η ⎞ Λ tanh⎜ SW ⎟ ⎝ 2 ⎠ τ η ⎛ ⎞ = ±FAcC*k ΓT tanh⎜ SW ⎟ ⎝ 2 ⎠

(ISW, p)Λ→ 0 = ±FAcC*

with Θ=

2p − 1 ∏h = 1 [1 + ΛfR, h σh + 1,2p] ⎫ ⎪ ⎬ 2p ∏h = 1 [1 + ΛfR, h σh ,2p] ⎪ ⎭

(20)

which corresponds to a first-order catalytic reaction in agreement with ref 35. Note that the expression for the current given in eq 14 is independent of the electrode geometry and size. 2.1.2. Small Electrodes. By introducing the limiting condition ξ ≫ 1 (i.e., small values of the radius) in eq 10 (i.e., σh,m → (1/ξ), see eqs 12 and 13), the following stationary limit is found kfR, m ΛfR, m D = FA ΓTcC* Im = FAcC* C 1 + ΘfR, m τ 1 + ΘfR, m

2p − 2 ⎧ ∏h = 1 [1 + ΛfR, h σh + 1,2p − 1] DC ⎪ − Λ⎨fR,2p − 1 2p − 1 τ ⎪ ∏h = 1 [1 + ΛfR, h σh ,2p − 1] ⎩

kΓ r Λ = T0 ξ DC

(16)

This solution is a limiting case of eq 10, and it can also be deduced by assuming ∂cmC/∂tm = 0 in eq 6. According to eqs 2 and 15, for small spherical electrodes the current is only dependent on the applied potential. Note that eq 14 obtained for the first-order behavior is a particular case of eq 15 for Θ ≪ 1 (i.e., Λ ≪ ξ). 2.2. Cyclic Square Wave Voltammetry (CSWV). In this technique, a square wave potential waveform is applied, with the potential sequence being described in the following way:23,31 ⎫ ⎡ ⎛ p + 1⎞ ⎤ ⎟ − 1⎥|ΔE | ⎪ Ep = E initial + ⎢Int ⎜ s ⎣ ⎝ 2 ⎠ ⎦ ⎪ r + (− 1) p + 1|ESW |; p = 1, 2, ..., np /2 ⎬ ⎪ Ep = Enp − p + 1; p = (np /2) + 1, ..., np ⎪ ⎭

2.2.2. Stationary Limits. Small Electrodes. Considering the simplified expression for the current given by eq 15, eq 20 becomes ISW = FAcC*

ISW = FAcC* (17)

−

(23)

⎛ DC ⎜ 1 Λ⎜ τ ⎝ 1 + Θ + exp(η2p − 1)

⎞ 1 ⎟ 1 + Θ + exp(η2p − 1 ± 2ηSW ) ⎟⎠

(24)

with ηj = ηSW

(18)

The SWV current is plotted versus the index potential, defined as the arithmetic average value of the potentials applied in each pair of pulses (2p−1, 2p) E index, p = Ep − ( −1) p ESW ; p = 1, 2, ...np/2

⎛ fR,2p − 1 fR,2p ⎞ DC ⎜ ⎟ Λ⎜ − τ ⎝ 1 + ΘfR,2p − 1 1 + ΘfR,2p ⎟⎠

with Θ given in eq 16. By taking into account that E2p−1 = E2p ∓ 2ESW,23,31 with the upper sign being applied in the first (cathodic) sweep and the lower corresponding to the second (anodic) one (see eq 19), we can rewrite eq 23 as

where |ΔEs| is the potential step of the staircase, |ESW| is the squarewave amplitude, and Int(x) is the integer part of the argument x. The current is sampled at the end of each potential pulse, and the net current is the difference between the current corresponding to a pulse with odd index (forward, If) and the signal of the following pulse with even index (reverse, Ir), such that the ISW−E square-wave voltammetric curve is defined as ISW = I2p − 1 − I2p = If − Ir ; p = 1, 2, ...np /2

(22)

⎫ F (Ej − E 0)⎪ ⎪ RT ⎬ F ⎪ = ESW ⎪ ⎭ RT

(25)

In this case, it is possible to obtain the following simple expressions for the peak parameters ss Epeak = E0 +

(19) 11208

RT ln(1 + Θ) F

(26)

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The Journal of Physical Chemistry C DC ⎛η ⎞ Λ tanh⎜ SW ⎟ ⎝ 2 ⎠ τ 1+Θ ⎛η ⎞ 1 tanh⎜ SW ⎟ = ±FAcC*k ΓT ⎝ 2 ⎠ 1+Θ

Article

ss Ipeak = ±FAcC*

(27)

We will discuss the applicability of eqs 21 and 24 in Section 3.

3. RESULTS AND DISCUSSION The validity of the analytical solution given in eq 20 has been checked by comparison with numerical calculations (as discussed in appendix B of ref 36). As will be seen below, very good agreement between analytical and numerical results for spherical electrodes has been found. In the case of planar electrodes, the analytical equations lead to significant errors in the peak parameters of the SWV response and therefore only the numerical results will be analyzed. A more detailed analysis of this comparison is presented in the Appendix. First, the main features of the SWV response of reaction scheme I at planar electrodes will be discussed. After discussing the overall behavior in planar diffusion, the case of spherical electrodes will be considered. To understand the influence of the kinetic parameter Λ (= kΓT (τ/DC)1/2) on the voltammetric behavior of the catalytic process given in scheme I, Figure 1 shows the SWV ψ−Eindex curves (with ψ = I/FAc*C (D/τ)1/2) obtained for different values of Λ at planar electrodes. Note that in SWV the net response is bell-shaped under both transient and stationary conditions. This is beneficial for analytical purposes, but it is not informative about the nature of the response. So, for further understanding, we have included in this Figure the forward (ψf) and reverse (ψr) components of the square-wave current. Therefore, from Figure 1a−c, it can be seen that both forward and reverse components of the current increase with Λ, the increase being much more apparent in the forward current. Actually, the net peak current almost coincides with ψf with an error 1. The reverse component ψr is also peakshaped for high values of Λ and becomes quasi-sigmoidal for Λ < 1. The net response is situated at potentials close to E0 (i.e., the formal potential of the immobilized catalyst) for the lowest Λ value (Figure 1a), and it is shifted toward more positive potential as the chemical demand (i.e., the chemical rate constant) increases. (See Figure 1b,c.) This shift of the response toward more anodic potential values with Λ confirms the catalytic nature of the process. The two effects discussed above (i.e., the increase of the peak current and the shift of the peak potential toward more anodic values) are related to the behavior of the surface concentration of species C, cC(x = 0,t), which is a function of Λ, applied potential, and time. Therefore, for a better comprehension of the above responses, in Figure 2 we have plotted the values of cC(x = 0,t) for different applied potentials at planar electrodes with the Λ values used in Figure 1 (shown on the curves). We have also included in this Figure the evolution of the coverage of species R, f R,m, with the applied potential, given by eq 2 (white circles). For low values of Λ (i.e., curves with Λ = 0.1), the surface concentration of species C remains close to its bulk value until potentials around 100 mV, f R,m increasing at more negative potentials. The peak of the square-wave voltammogram

Figure 1. ψ−(Eindex−E0) (solid lines), ψf−(Eindex−E0) (dashed lines), and ψr−(Eindex−E0) curves (dashed dotted lines) at a planar electrode for different values of Λ (indicated on the graphs), calculated by using the procedure discussed in ref 36. ESW = 50 mV, Es = −5 mV, τ = 10 ms, T = 298.15 K, and D = 10−5 cm2 s−1. Dotted line marks the formal potential of the immobilized catalyst.

corresponds to the maximum difference between the forward and reverse currents, and for Λ = 0.1, it is located at potentials slightly greater than E0. (See Figure 1a.) In the limit Λ → 0, the chemical demand from the chemical process at the monolayer/ solution interface is very low, and the condition cC(x = 0,t) ≅ cC* holds for the entire range of potentials. Therefore, the current is given by eq 21, which corresponds to a first-order catalytic reaction in agreement with ref 35. As the chemical reaction is more demanding, the surface concentration of species C begins to decrease at more positive potentials (i.e., lower values of f R,m). Also, the decrease in cC(x = 0,t) is steeper (see curves for Λ = 10, for which cC(x = 0,t) falls to zero for potentials below 20 mV). In this case, the differences between the forward and reverse values of the concentration, and therefore the current, are greater (compare Figure 1a,c). In the cyclic mode of SWV, two sweeps can be analyzed in an analogous way to CV.3,23,27 In Figure 3, we have plotted the 11209

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Figure 2. Variation of the surface concentration of species C (cmC(x = 0,E)/c*C ) (lines) and of the surface coverage of species R (white dots) with the forward (dashed lines) and reverse potentials (dashed-dotted lines) for planar electrodes, for three values of Λ (shown in the curves), calculated by using the procedure discussed in ref 36. Other conditions as in Figure 1.

cyclic SWV curves (a) of the catalytic process given in reaction I for different values of Λ at planar electrodes, together with the evolution of the peak currents (b) and peak potentials (c) of the first (1) and second (2) sweeps toward cathodic and anodic potentials, respectively, as a function of log Λ. From these curves, we can see that the increase in Λ gives rise to the increase in both responses up to a value of Λ ≥ 5, from which no influence is observed on the peak currents. (See also Figure 3b.) For low Λ values, identical peak heights proportional to Λ are obtained in both sweeps. This feature is typical of first-order catalytic behavior (see figure embedded in Figure 3b).35 As Λ increases, a strong asymmetry appears between the responses of the first and second sweeps, with the former being higher than the latter. For log Λ > 1, the ratio between both peaks takes the constant value 7.42. We have obtained a numerical expression for the dependence of the both peak currents with log Λ, which covers the whole range of values shown in Figure 3b ⎫ −(log Λ+ 0.5891)/0.4037 ⎪ ⎪ 1+e ⎬ −0.0301 ⎪ ψp2 ≅ −(log Λ+ 1.4655)/0.3715 ⎪ ⎭ 1+e ψp1 ≅

Figure 3. (a) ψ−(Eindex−E0) (solid lines) at a planar electrode for different values of Λ (indicated on the graphs) calculated by using the procedure discussed in ref 36. Peak currents (b) and potentials (c) of the first (black circles) and second (white triangles) sweeps as a function of log Λ. ESW = ± 50 mV, Es = ∓5 mV, with the upper sign corresponding to the first sweep (cathodic) and the lower to the second (anodic), respectively. Inner figure in panel b: peak currents of the first sweep as a function of Λ. Other conditions as in Figure 1.

0.2191

(28)

from which the obtaining of Λ is immediate. In the case of the anodic peaks, the errors are higher because of the small values of ψp2 (the relative error is higher than 5% for log Λ ≤ −1) such that the use of the expression for the peak heights of the first sweep is more adequate. Concerning the peak potentials, both values coincide with E0 for log Λ < −2 (first-order behavior35), and both responses shift toward more positive potentials as Λ increases. Moreover, the peak potentials of both responses depend linearly on log Λ for Λ > 1 with a slope that is approximately equal to 59 mV, in agreement with the following expressions (T = 298 K) Ep1 − E 0 ≅ 51.6 + 59.4 log(Λ) ⎫ ⎪ ⎬Λ > 1 Ep2 − E 0 ≅ 26.4 + 58.4 log(Λ)⎪ ⎭

On the basis of these results, three different behaviors can be distinguished in the SWV response at planar electrodes: (a) First-order region (Λ < 0.01). In this region, the chemical rate constant k is low and therefore the surface concentration is approximately equal to the bulk value. The peak current is proportional to Λ (ψp1 = Λ tanh(ηSW/2), see eq 22 and Figure embedded in Figure 3b), and the peak potential coincides with the formal potential of the surface-attached redox catalyst, E0. (b) Mixed region (0.01 < Λ < 10). For intermediate values of Λ, the surface concentration of species C decreases more rapidly and at more positive potentials as Λ increases. Under these conditions, the current increases, and the SWV peak potentials of both sweeps shift toward more anodic values when Λ increases.

(29)

Note that the difference between peak potentials of the two sweeps becomes constant (ΔEp ≅ 25 mV) for log Λ > 1. 11210

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(c) Mass transport control region (Λ > 10). This zone corresponds to high values of the chemical rate constant (i.e., very fast conversion of species C by the immobilized redox catalyst). The chemical conversion of Csol through the assistance of the attached mediator R is very fast so that the diffusive mass transport is the limiting step. In this case, the peak currents become independent of Λ and the peak potentials show a logarithmic dependence on the chemical rate constant (i.e., Λ). The behavior of the peak potential is similar to the variation of the peak potential of a totally irreversible charge-transfer reaction with the logarithm of the heterogeneous rate constant.3 Note that all of the above considerations about Figure 2 and eqs 28 and 29 have been deduced for a square-wave potential |ESW| = 50 mV and a staircase potential |ΔES| = 5 mV, which are the optimal values of these parameters for SWV in line with ref 27. (See also Figures 7 and 8.) When spherical electrodes are considered, the mass transport is enhanced as compared with planar electrodes due to the occurrence of radial transport, this effect being more notorious as the electrode size decreases.3,37 Therefore, the values of the parameter Λ that define the transition between the three regions of the SWV response above-mentioned are logically affected. This fact can be clearly seen in Figures 4 and 5, where the ψ−(Eindex−E0) curves obtained for different Λ values at spherical electrodes are shown in Figure 4, and the evolution of the SWV peak currents and potentials with log(Λ) for planar and spherical electrodes with three values of the radius are shown in Figure 5. In Figure 4, it can be seen that although the increase in Λ has the same effects on the SWV curves at spherical and planar electrodes, the relative magnitude of these effects changes with the electrode radius. Therefore, the shift of the peak potential of the first sweep with respect to E0 for Λ = 103 decreases from 224 to 144 mV when the radius is diminished from 100 to 1 μm. The asymmetry between both SWV responses also decreases with the ratio between peak currents changing from a value |ψp1/ψp2| ≅ 4 for a radius of 100 μm to |ψp1/ψp2| ≅ 1 for r0 = 1 μm. To conclude, the reduction of the electrode size leads to symmetrical waves and less shift of the current responses. The maximum current of the SWV curves for the first sweep, ψp1,max (corresponding to very large Λ values, mass transport control region) is only a function of the parameter ξ. We have obtained the following expressions from numerical fitting ⎫ ψp1,max ≅ 0.2177 + 0.7963(ξ) for log(ξ)⎪ ⎪ ≤2 ⎪ ⎬ 0.6732 ψp1,max ≅ −9.6125 + 3.6815(ξ) for 2 ⎪ ⎪ ⎪ < log(ξ) ≤ 3 ⎭

Figure 4. ψ−(Eindex−E0) at spherical electrodes for different values of Λ and three values of the electrode radius (indicated on the graphs) calculated from eq 20. Other conditions as in Figure 3.

and 2.) When the electrode size decreases and the mass transport is much more efficient, the transition from the firstorder zone logically starts at higher chemical demand (Λ) values (compare curves for planar electrode and r0 = 1 μm). An analogous behavior is observed for the Λ value required to attain the mass transport control region. Note that the decrease in the electrode radius has a stronger effect on the peak of the second sweep (both on the peak potential and the peak current). From the above data, a “zone diagram” is plotted in Figure 6 for the three different regions previously defined in function of parameters Λ and ξ. The dashed lines correspond to those values of Λ for which the peak current is equal to 0.95 ψp1|max (black dots) and to 0.05 ψp1|max (white squares), with ψp1|max being the maximum SWV peak current obtained at the mass transport control region for a given ξ value. (See eq 30.) These lines separate the first-order (ψp1 ≤ 0.05ψp1|max), mixed (0.95 ψp1|max < ψp1 < 0.05ψp1|max), and mass transport

0.9727

(30)

An additional verification of the effect of the sphericity of the diffusion field can be seen in Figure 5. The normalized peak height of the SWV response increases with log Λ as in the planar case (black circles), although this increase takes place at higher Λ values as the radius r0 is decreased. The evolution of the SWV response from the first-order region to the mass transport control region is related to the increase in the chemical demand and the inability of the diffusion mass transport to provide enough amount of species C for the reaction at the monolayer/solution interface. (See Figures 1 11211

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Figure 5. Normalized peak currents (a) and peak potentials (b) of the first (1) and second (2) SWV sweeps obtained for planar and spherical electrodes as a function of log Λ, calculated by using the procedure discussed in ref 36 and eq 20. The values of the electrode radius (in micrometers) are: circles, planar electrode; squares, 100; triangles, 10; diamonds, 1. Other conditions as in Figure 3.

Figure 7. Influence of the staircase potential on the SWV responses of planar and spherical electrodes (the radius is shown in the curves), calculated by using the procedure discussed in ref 36 and eq 20. The values of |ΔEs| (in mV) appear on the curves. Other conditions as in Figure 1.

As we point out in the Appendix, the approximate solution proposed in eq 20 leads to errors in the peak current smaller than 5% for ξ ≥ 0.1 (dotted region in Figure 6). Moreover, we have checked that the limiting stationary solution given in eq 24 coincides with that of eq 20, with errors below 5% in the gray region of Figure 6. Therefore, for values of Λ and ξ inside this zone, the peak parameters can be obtained from eqs 26 and 27. Note that from eq 27 the expressions for the peak current in the three regions above-mentioned are given by: - Stationary first-order region (Θ ≪ 1):

Figure 6. Zone diagram of the transient and stationary response of the catalytic process given in reacion 1 in function of the dimensionless parameters ξ = (DCτ)1/2/r0 and Λ = kΓT(τ/DC)1/2. The three zones correspond to those values of Λ for which the peak current is equal to the 0.95 ψp1|max (black dots) and to the 0.05ψp1|max (white squares), with ψp1|max being the maximum SWV peak current obtained at the mass transport control region for a given ξ value. Other conditions as in Figure 1.

⎛η ⎞ ss = Λ tanh⎜ SW ⎟ ψp1 ⎝ 2 ⎠ - Stationary mixed region:

(ψp1 ≥ 0.95ψp1|max) regions. From this plot, it can be seen that the Λ values defining these regions increase with ξ.

ss = ψp1

11212

⎛η ⎞ Λ tanh⎜ SW ⎟ ⎝ 2 ⎠ 1+Θ

(31)

(32)

dx.doi.org/10.1021/jp303171f | J. Phys. Chem. C 2012, 116, 11206−11215

The Journal of Physical Chemistry C

Article

Figure 8. Influence of the square-wave potential on the SWV responses of planar (a,b) and spherical electrodes (c,d) (r0 = 1 μm) calculated by using the procedure discussed in ref 36 and eq 20 for two values of Λ: 1 (a,c) and 10 (b,d). The values of ESW (in mV) appear on the curves. Other conditions as in Figure 1.

(34)

have chosen the value ΔEs = 5 mV in the different Figures plotted here because although a value of ΔEs = 10 mV would give rise to higher currents in the case of planar electrodes it would also produce a loss of definition on the peak parameters. Concerning the effect of ESW, higher values of this parameter also involve higher currents together with a distortion of the SWV curve (and the appearance of a shoulder), particularly at planar electrodes (Figure 8a,b). At spherical electrodes the distortion is smaller and both peaks turn into broad plateaus for ESW > 100 mV (although for the response of the first (cathodic) sweep this plateau is not well-defined, especially for high Λ values).

This equation clearly shows a logaritmic dependence between the peak potential and Λ, as it has been pointed out above. (See Figure 5.) To apply eqs 26 and 27 and 31−34 (i.e., to work under conditions corresponding to the gray region of Figure 6), it is convenient to decrease the SWV frequency (i.e., to increase the pulse length) achieving higher ξ values. Note also that, as in the planar case discussed in Figure 3, all of these considerations have been obtained for a square wave potential |ESW|=50 mV and a staicase potential |ΔES| = 5 mV, which are the optimal values of these parameters for SWV in line with ref 27. (See also Figures 7 and 8.) Finally, the influence of the staircase potential (ΔEs) and the square-wave amplitude (ESW) of the SWV perturbation have been analyzed, and the results are shown in Figures 7 and 8, respectively. From Figure 7a, it can be seen that for planar electrodes the increase in ΔEs leads to higher currents and to a shift of the SWV response toward more negative potentials. For spherical electrodes (Figure 7b,c), this influence is less apparent, such that for the smallest radius (r0 = 1 μm) no effects in the voltammograms are observed. On the basis of these results, we

4. CONCLUSIONS The electrochemical response of the catalytic mechanism given in scheme I results from the combination of the kinetics of the catalytic chemical reaction at the monolayer/solution interface and of the mass transport. Stationary responses can be achieved in the following situations: - First- or pseudo-first-order response. In this case, the chemical demand is low, and it gives rise to a practically constant value of the surface concentration of species C. - Small electrodes, for which the mass transport is enhanced. In both cases it is possible to characterize the current− potential curves obtained in a very simple way. Square-wave voltammograms for this reaction scheme are always peak-shaped, independently of the transient or stationary nature of the current. This fact, which is not observed in the case of CV, enables the complete characterization of the process from the values of the peak parameters. From the peak potentials and currents, three different regions for the SWV curves can be established, which are defined by the relationship between Λ (the chemical demand) and ξ (the electrode

- Stationary mass transport control region (Θ ≫ 1): ⎛η ⎞ ss ψp1 = ξ tanh⎜ SW ⎟ ⎝ 2 ⎠

(33)

In line with these results and eq 33, in the stationary mass transport region the kinetic information of the peak current is lost. The chemical influence is, however, very important in the peak potential when the condition Θ ≫ 1 holds. In this case, from eq 26, one obtains (Ep1)ξ