Electrochemical Behavior of Two-Electron Redox Processes by

ARTICLE pubs.acs.org/JPCC

Electrochemical Behavior of Two-Electron Redox Processes by Differential Pulse Techniques at Microelectrodes A. Molina,*,† J. Gonzalez,† E. Laborda,†,‡ Q. Li,‡ C. Batchelor-McAuley,‡ and R. G. Compton*,‡ † ‡

Departamento de Química Física, Universidad de Murcia, 30100 Murcia, Spain Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom ABSTRACT: The application of differential pulse techniques (differential double pulse, differential multipulse, and differential normal pulse voltammetries) to the study of two-electron redox processes is carried out. The characterization of these requires the determination of the difference between the formal potentials of the electron transfer steps, which may reflect the interactions between the two different redox centers in the electroactive molecule. A new theory is developed for disk electrodes generating very simple analytical equations applicable to any electrode size and any double pulse technique. The influence of the technique parameters (pulse amplitude, pulse times) and the electrode geometry are examined. Procedures for the determination of the formal potentials are proposed from the values of the peak height and the half-peak width.

1. INTRODUCTION The potential applications of molecules which contain multiple redox centers are many and rising, for example, in energy conversion, molecular electronic devices, design of new materials, and chemiluminiscent analytical methods.18 These molecules include organometallic compounds, biological species (metalloproteins, enzymes, oligonucleotides, etc.), aromatic hydrocarbons, and so forth. In many examples, the electrochemical reactions are reversible such as alquil viologens,2 carotenoids,3,4 fullerenes,57 linear polymers, supramolecular species, and dendrimers.1,810 A major interest in this field is the study of the stability of the different oxidation states through the relative values of the formal potentials. In the case of molecules with several redox centers, this is indicative of the degree of interaction between them, and so it is very helpful for the characterization of the compound.1114 Differential pulse voltammetries are very attractive for the characterization of this kind of processes.1522 Besides the reduction of undesirable effects related to charging and background currents, the signal offers several advantages with respect to the use of cyclic voltammetry. Thus, under analogous experimental conditions, the peak obtained in differential techniques is better resolved and more symmetrical, making the quantitative determination of the experimental peak parameters more accurate and easier. Moreover, a peak-shaped response is obtained even when microelectrodes are used, which enable to minimize ohmic-drop and capacitative effects.17,23,24 On the other hand, the utilization of microelectrodes for the evaluation of multicenter redox molecules in cyclic voltammetry can be problematic since the peak-shaped response changes to sigmoidal voltammograms, compromising the analysis and the accuracy of the results. r 2011 American Chemical Society

Analytical explicit solutions for the multielectron redox chemistry of molecules with potential pulse techniques, such as abovementioned, have hitherto been generated only at electrodes with one-dimensional transport as is the case for planar and spherical electrodes.14,19,20,2527 Among the different types of microelectrodes, the disk geometry is the most commonly used in electrochemical experiments due to the easier fabrication and cleaning of the electrodes. However, the theoretical description is more complex and usually requires the use of numerical methods.2832 Accordingly to all of the above, in this paper we develop a new theory for the use of disk microelectrodes in combination with differential double pulse techniques for the study of twoelectron redox processes (EE). Easy to compute, analytical explicit equations are presented, permitting the simulation of the response at disk electrodes of any size in any double pulse technique. The results are particularized for the cases of differential double pulse voltammetry (DDPV) and differential double normal pulse voltammetry (DDNPV). An additional advantage when reversible processes are studied is that the response in DDPV is equivalent to that in differential multipulse voltammetry (DMPV),15 so the equilibration period between double pulses can be avoided (see Figure 1), reducing considerably the experimental time. From the expressions obtained, the use of these electrochemical methods for the characterization of two-electron reducible molecules is examined. The effect of the interaction degree of the redox centers (i.e., of the difference of the formal potentials), Received: October 18, 2011 Revised: November 17, 2011 Published: November 21, 2011 1070

dx.doi.org/10.1021/jp210009k | J. Phys. Chem. C 2012, 116, 1070–1079

The Journal of Physical Chemistry C

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diffusion coeffients are assumed as equal. For reversible electrode reactions and no solution phase reactions other than disproportionation, the following scheme (EE mechanism) applies: O1 þ e h O2

E01

O2 þ e h O3

E02

k1

2O2 a O1 þ O3 k2

0

0

K ¼

ð1Þ k1 k2

ð2Þ

0

where E0i (i 1, 2) are the formal potentials of each redox center and K the disproportionation constant (see also eq 7). In such conditions, the side reaction given by eq 2 does not affect the voltammetric response in any single or multipotential technique.25,3335 In this paper we discuss the electrochemical response of the above scheme to the application of two consecutive potential pulses E1 and E2 of length t1 and t2, respectively, to a disk electrode with radius rd. By following the procedure indicated in the Appendix, we find the following expressions corresponding to the currents of the first and second applied potential, I(1) and I(2), respectively: I ð1Þ ðt1 Þ ¼ FADcO1 gd ðt1 ÞΩ1

ð3Þ

I ð2Þ ¼ I ð1Þ ðt1 þ t2 Þ þ FADcO1 gd ðt2 ÞΩ2

ð4Þ

where: Ω1 ¼

J ð1Þ þ 2K ðJ ð1Þ Þ2 þ J ð1Þ þ K

ð5Þ

Ω2 ¼

J ð2Þ þ 2K J ð1Þ þ 2K 2 ðJ ð2Þ Þ þ J ð2Þ þ K ðJ ð1Þ Þ2 þ J ð1Þ þ K

ð6Þ

Figure 1. Potential-time program in (A) differential double pulse voltammetry (DDPV) and differential double normal pulse voltammetry (DDNPV) and (B) differential multipulse voltammetry (DMPV).

K ¼ exp

F 0 0 ðE0 E01 Þ RT 2

9 F > 00 > ðE1 E1 Þ > ¼ exp = RT F 0 > > ðE2 E01 Þ > ¼ exp ; RT

ð7Þ

the duration of the pulses, the electrode radius, and the pulse amplitude are analyzed with the aim of establishing optimal experimental conditions. The results are compared with those expected at spherical microelectrodes, showing that under steady state conditions the ratio between the differential currents for microdiscs and microspheres of the same radius is 4/π. The particular, interesting case where there is no interaction between the electroactive centers is analyzed. This corresponds to a difference of formal potentials of 35.6 mV,11,14,17 and the peak obtained has the same width of that for a one-electron transfer. In that case, the peak height is required to discriminate between both situations since the peak current for two-electron reducible molecules double that of one-electron reducible molecules. Finally, procedures for the determination of the formal potentials are proposed based on the values of the peak current and the half-peak width of the voltammograms.

2. THEORY We will consider the reduction of a solution soluble molecule with three possible oxidation states O1, O2, and O3, whose

J

ð1Þ

J ð2Þ

ð8Þ

Function gd(t) is obtained from the semiempirical Shoup and Szabo equation for disk electrodes derived by combining previous solutions for the chronoamperometric behavior of disk electrodes at long and short times with their results of digital simulations:36,37 gd ðtÞ ¼

4 rd rd 0:7854 þ 0:44315pffiffiffiffiffi þ 0:2146 exp 0:39115pffiffiffiffiffi πrd Dt Dt

ð9Þ 2.1. Application to Differential Double Pulse Techniques. Next, the analytical expressions derived above are applied to differential double pulse voltammetry (DDPV) and differential double normal pulse voltammetry (DDNPV). The corresponding potential-time program is shown in Figure 1, the difference between both techniques being the relative duration of the pulses.15,17 1071

dx.doi.org/10.1021/jp210009k |J. Phys. Chem. C 2012, 116, 1070–1079


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Thus, in DDNPV the durations of both pulses are similar: t1 ≈ t2. The current is sampled at the end of each pulse and the difference between them, ΔI = I2 I1, is plotted versus the average value of the applied potential (i.e., Eindex = E1 + E2/2), whose advantages have been indicated in ref 38. Thus, the expression for the DDNPV current is given by: ΔIDDNPV ¼ ½ gd ðt1 þ t2 Þ gd ðt1 ÞΩ1 þ gd ðt2 ÞΩ2 FADcO1

ð10Þ

In the case of the DDPV method, the duration of the second pulse is usually much shorter than that of the first one: t1 g 50t2. Under these conditions, it is the case that gd(t1 + t2) ≈ gd(t1) so that eq 10 simplifies to: ΔIDDPV ¼ gd ðt2 ÞΩ2 FADcO1

ð11Þ

where gd(t2) is dependent on both the electrode geometry and the second pulse duration, and Ω2 is only dependent on the applied double pulse waveform (E1 and E2). As was demonstrated in ref 15, in the case of reversible electrode reactions the signal in DDPV coincides with that in differential multipulse voltammetry (DMPV, see Figure 1B). This has been corroborated for the case of the EE mechanism by means of numerical simulation with a homemade program.39 It is found that the responses in double pulse and multipulse differential voltammetries differ less than 1% in the peak current when both electrochemical steps are reversible and t1/t2 g 50, whatever the electrode size. Consequently, the simple expression 11 holds when the DMPV potential-time program is applied, and the results discussed below for DDPV are also valid for DMPV.

3. RESULTS AND DISCUSSION Figure 2 shows the variation of the ΔI current with the dura0 0 0 tion of the second pulse, t2, for different ΔE0 (= E02 E01 ) values (and T = 298 K). Microelectrodes of disk (solid line, see eqs 10 and 11) and spherical (dashed line, see eqs 12 and 18 in ref 25) geometry are considered. As can be observed, as the duration of the second pulse increases, that is, as we turn from DDPV (or DMPV) to DDNPV, the peak current decreases, and the ΔI value at very negative values is not zero. Hence, the use of short pulse times (i.e., the use of DDPV/DMPV) is recommended since it offers greater sensitivity and symmetrical peaks. This facilitates the determination of the peak currents and potentials. Moreover, under DDPV conditions the analytical solution for the current simplifies to eq 11 for disk electrodes and eq 18 in ref 25 for spherical ones. An additional advantage of this technique when reversible processes are studied is that the response in DDPV is equivalent to that in differential multipulse voltammetry15 so the equilibration period between double pulses indicated in Figure 1 can be avoided, reducing considerably the experimental time. With respect to the position of the peaks, we can observe that this is independent of the duration of pulses (and therefore of the electrochemical technique) and of the electrode geometry. On the other hand, the dimensionless peak current does depend on the electrode shape, and the difference is more apparent for longer second pulses due to the different diffusion domains. The electrode size is another important variable to analyze since the use of microelectrodes is very relevant to experimental electrochemical studies enabling the reduction of capacitative and ohmic drop effects.17,23,24 Specifically, it is of great interest to

Figure 2. Influence of the duration of the second pulse (t2/t1 values indicated on the curves) on the differential pulse voltammograms for disk 0(solid line) and spherical (dashed line) electrodes. Three different ΔE0 values are considered: (A) 200 mV, (B) 0 mV, and (C) +200 mV. r0 = 10 μm, ΔE = 50 mV, t1 = 1 s, T = 298 K, and Id(t1) = FADc*/(πDt1)1/2.

check the behavior of the system when the size of the electrode is reduced. In Figure 3 the influence of the electrode radius on the DDPV/DMPV curves is shown for spherical and disk electrodes and different interaction degrees of the redox centers 0 (i.e., different ΔE0 values). As can be seen, independently of the electrode size, a peak-shaped response is obtained with the same peak potential and width (see the superimposed ΔI/ΔIpE curves in the inserted figures) since these responses are independent 1072



Figure 3. Influence of the electrode radius (r0 values indicated on the curves) on DDPV/DMPV curves for disk 0(solid line) and spherical (dashed line) electrodes. Three different ΔE0 values are considered: (A) 200 mV, (B) 0 mV, and (C) +200 mV. The curves normalized with respect to the peak current are also shown in the inserted graphs. t1/t2 = 50, ΔE = 50 mV, t1 = 1 s, T = 298 K, and Id(t1) = FADc*/(πDt1)1/2.

of the electrode geometry (see eq 11). This is a remarkable advantage in comparison with cyclic voltammetry where sigmoidal curves are obtained when small electrodes are employed, which makes data analysis more difficult and less precise. Moreover, given that the peak position and width are independent of the electrode size, the criteria given for the characterization of these molecules based on these DDPV/DMPV peak parameters (see Figure 8) are very general, and they can be applied to any electrode. It is important to highlight that, as expected, for ultramicroelectrodes (see curves with r0 = 5 μm in Figure 3 for which the steady state has been almost reached), the ratio between ΔI for disk and spheres (solid and dashed lines, respectively) tends to 4/π, since under these conditions we

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Figure 4. Influence of the pulse amplitude (ΔE values indicated on the curves) on the DDPV/DMPV curves for disk (solid line) and spherical 0 (dashed line) electrodes. Three different ΔE0 values are considered: (A) 200 mV, (B) 0 mV, and (C) +200 mV. r0 = 10 μm, t1/t2 = 50, t1 = 1 s, T = 298 K, and Id(t1) = FADc*/(πDt1)1/2.

deduce from eqs 9 and 11: ss ΔIdisc 4 ¼ Ω2 πrd FADcO1

ð12Þ

and from eq 18 in ref 25: ss ΔIsphere

FADcO1 1073

¼

1 Ω2 rs

ð13Þ



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Figure 5. Variation of the DDPV/DMPV curve at a disk microelectrode with0 the difference between the formal potentials of the redox centers, 0 0 ΔE0 (= E02 E01 ) (values indicated on the curves). The case corresponding to a two-electron E process is also plotted (gray dotted line). r0 = 10 μm, ΔE = 50 mV, t1/t2 = 50, t1 = 1 s, T = 298 K, and Id(t1) = FADc*/(πDt1)1/2.

in such a way that for rd = rs we find: ss ΔIdisc 4 ¼ ss π ΔIsphere

ð14Þ

The influence of the potential pulse amplitude, ΔE = E2 E1, is considered in Figure 4. Again, the peak potentials are not affected by the ΔE value. On the other hand, the peak current and the peak width increase with ΔE, along with the interference of the charging current. These last two effects are detrimental in electrochemical studies, and so a compromise value for ΔE must be selected to have satisfactory sensitivity and well-resolved peaks (typically of 50 mV). Next, we will consider different degrees of interaction between the two redox centers of the molecule. As above-mentioned, this 0is related to0 the relative value of the formal potentials 0 ΔE0 (= E02 E01 ). According to the definition of K given in eq 7, when the formal potential 0of the first step is much larger than that of the second one, ΔE0 e 200 mV, the intermediate oxidation state, O2, is stable. This corresponds with the existence of strong repulsive intramolecular interactions between centers. As can be seen in Figure 5, under these conditions two well-separate peaks are obtained, centered on the formal potential of each process and with the same features of the signal corresponding to a simple 0 one-electron reaction. As the ΔE0 value increases, the peaks are closer since the intermediate species is less stable. Thus, a transition from two peaks to a single0 peak is found. As will be seen below, independently of the ΔE0 value and the form of the signal, this is symmetrical with respect to the average value of the 0 0 formal potentials, (E02 + E01 )/2 (see below). This value corresponds to the minimum of the valley between peaks for very 0 0 negative values of ΔE0 and to the peak potential for ΔE0 > 70 mV. This behavior is independent of the electrode size and geometry and of the pulse durations. Eventually, when the formal potential of the second step is much larger than0 that of the first one (attractive intramolecular interactions), ΔE0 g 200 mV, the system behaves in an identical way to a simple charge transfer process of two electrons (gray dotted curve).

Figure 6. Comparison of the differential pulse voltammograms corre0 sponding to two-electron reducible molecules with ΔE0 = 35.6 mV (black line) and one-electron E mechanism (gray line) at disk (solid line), spherical (dashed line), and planar (dotted line) electrodes. Two values of the duration of the second pulse are considered: (A) t1/t2 = 50 and (B) t1/t2 = 10. r0 = 25 μm, ΔE = 50 mV, t1 = 1 s, T = 298 K, and Id(t1) = FADc*/(πDt1)1/2.

In the transition from two peaks to a single peak (that is from repulsive to attractive intramolecular interactions between cen0 ters), the case of ΔE0 =35.6 mV is particularly interesting since it corresponds to noninteracting centers.11,14 This situation is analyzed for different pulse times and electrode shapes (including disk, spherical and planar electrodes) in Figure 6. The 0shape of the peak of a two-electron reducible molecule with ΔE0 = 35.6 mV is the same as that corresponding to singlecenter molecules such that they have equal peak widths (see below), but the current is double, independently of the electrode geometry, the technique parameters (duration of pulses, pulse amplitude), and the voltammetric method. This fact emphasizes the importance of having at our disposal analytical explicit equations for the DDPV/DMPV response. Figures 7 and 8 analyze the evolution of the DDPV/DMPV 7) and dimensionless peak height hpeak = ΔIpeak/I0 d(t1) (Figure 0 0 half-peak width W1/2 (Figure 8) with ΔE0 (= E02 E01 ). Thus, Figure 7 shows the DDPV/DMPV peak height of a two-electron reducible molecule calculated from eq 11 for three values of the disk radius and three potential pulse amplitudes 1074



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Figure 8. 0 Evolution of the DDPV/DMPV half-peak width W1/2 with 0 0 ΔE0 = E02 E01 , corresponding to a two-electron reducible molecule at a disk (solid line) and spherical (dashed line) electrodes. ΔE values are: (a) 50 mV, (b) 25 mV, and (c) 10 0mV. t1 = 1 s, t1/t2 = 50, and T = 298 K. Dashed line marks the value ΔE0 = 35.6 mV.

Figure 7. Evolution of the DDPV/DMPV dimensionless peak height 0 0 0 hpeak = ΔIpeak/Id(t1) with ΔE0 (= E02 E01 ), corresponding to a twoelectron reducible molecule at disk (solid line) and spherical (dashed line) electrodes. The electrode radius values (in μm) are: (A) 1, (B) 10, and (C) 100. ΔE values (in mV) are marked on the graphs. t1 = 1 s, t1/t2 = 50, and T = 298 K.

ΔE = E2 E1. We have also included in this figure the values of hpeak corresponding to spherical electrodes calculated from eq 18 of ref 25 (dashed lines). From this figure it0 can be seen that the peak height always corresponding to oneincreases with ΔE0 from the value 0 electron charge transfers for ΔE0 < 120 mV, until it reaches 0 that corresponding to two-electron processes when ΔE0 > 200 mV. Note that, as previously indicated, the peak height increases with the potential amplitude for both geometries and the differences between disk and sphere are less obvious as the radius increases. Thus, for electrode radii larger than 50 μm the same dimensionless peak current, ΔIpeak/Id(t1), is obtained at both electrodes.

Figure 8 shows the variation of the DDPV/DMPV half-peak width W1/2 (calculated for T = 298 K) for three potential pulse amplitudes at spherical and disk electrodes. Note that the curves for planar electrodes, discs, and spheres are superimposed since the value of the peak width is independent of the electrode size0 and shape, being determined by the pulse amplitude and ΔE0 . Therefore, the results presented in this figure are very general and enable the characterization of two-electron reducible molecule whatever the electrode employed. As can be observed, W1/2 takes a constant value close to 90 mV 0 when two separate monoelectronic peaks0 are obtained for ΔE0 < 0 150 mV, showing a sharp jump at ΔE around 140 mV. This value corresponds to the case at which the height 0of the central valley coincides with the half-peak height. For ΔE0 > 150 mV, two unresolved peaks or a single peak are obtained, and W1/2 0 decreases with ΔE0 until it reaches a value close to 45 mV, corresponding to a two-electron transfer reaction. As mentioned 0 above, a special case is that of ΔE0 = 35.6 mV (K = 1/4), for which W1/2 is the same as that corresponding to a simple oneelectron charge transfer (see dashed line in the figure), whereas the response has double the intensity. This particular case corresponds to the absence of interactions between the two redox centers, and it applies for any electrochemical technique and for any electrode geometry (see also Figure 6 and refs 11, 14, and 25). The dependence of the peak height and half peak width on 0 ΔE0 can be usefully treated analytically. By equating to zero the derivative of eq 11 with respect to Eindex(E1 + E2)/2,38 a eighth degree polynomial is obtained, for which the following three real positive roots for Eindex are obtained: 0

EI ¼ E01 þ

RT pffiffiffiffi lnð K Þ F

any value of K ðaÞ

9 > > ! > pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi > 2 > 2K f2 ðKÞ A f1 ðKÞ þ 2A K RT > ln þ ðbÞ > = K < 0:06 F 2A 0 ðΔE0 EIII ¼ E1 > ! > pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi > > 2K þ f2 ðKÞ A f1 ðKÞ þ 2A2 K RT > ln ðcÞ > þ ; F 2A 0

EII ¼ E01

ð15Þ 1075



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with K given in eq 7 and, f1 ðKÞ ¼

ð4K 1ÞðK A2 þ 2A2 K þ A4 KÞ A2

potentials of the voltammograms for the characterization of twoelectron reducible molecules. ð16Þ

f2 ðKÞ ¼ A2 K þ 8K 2 þ 16A2 K 2 þ 8A4 K 2 10A2 K pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi A4 K 4AK f1 ðKÞ 4A3 K f1 ðKÞ A ¼ exp

F ΔE 2RT

ð17Þ

0

ðA1Þ

ð18Þ 0

0

ðE01 þ E02 Þ 2

i ¼ 1, 2, 3 p ¼ 1, 2

Ô cðpÞ ¼ 0; δ i Oi

The first root (EI) is valid for any ΔE0 value and corresponds to the following applied potential: EI ¼

’ APPENDIX When a double potential pulse is applied to a disk electrode, the consequent mass transport by diffusion for the reaction scheme (1) is described by Fick's second law:

ð19Þ

The physical meaning of EI depends on the K value considered. 0 For K > 0.06 (ΔE0 > 70 mV), it corresponds to the peak potential of the single response obtained (see Figure 5). 0 For ΔE0 < 70 mV (K < 0.06, separated peaks), EI corresponds to the potential of the valley between the two peaks (see Figure 5). 0 The roots EII and EIII, only operative for ΔE0 < 70 mV, correspond to the peak0 potentials of separate signals (i.e., to 0 0 E01 and E02 ) when ΔE0 < 200 mV. In these conditions, the peak heights tend to that corresponding to 0 a single electron 0 E mechanism as ΔE0 decreases (see ΔE0 < 120 mV in Figure 7).

4. CONCLUSIONS Explicit analytical expressions, which are very easy to apply, have been presented for the study of two-electron reducible molecules by means of double and multipulse differential voltammetries with disk microelectrodes. They permit the determination of the interaction between the redox centers as shown through the difference between the formal potentials of the twoelectron transfers. The influence of the technique parameters (i.e., pulse times and pulse amplitude) and the electrode geometry on the differential voltammograms has been analyzed. It has been shown that the peak potential and the half-peak width are independent of the electrode size and shape. So, the criteria given for the determination of the interaction degree from the peak width can be applied to any electrode. Differential voltammetries have the advantage versus cyclic voltammetry that they present peak-shaped responses even when microelectrodes are used. Moreover, when a steady state ΔIE curve is obtained, the ratio between ΔI for microdiscs and microspheres of the same radius tends to 4/π. The peak height is also informative about the interaction between centers. In the particular case of noninteracting centers 0 0 (E02 E01 = 35.6 mV) this is essential since the shape of the signal is exactly the same of that of one-electron processes, but the current is double than that of single center molecules, which enables the experimentalist to discriminate between these two situations. Besides working curves based on the peak height and half peak width, analytical expressions are reported for the peak and valley

where c(p) Oi are the concentration profiles of species Oi in the first or second potential pulse (p = 1 or 2), and δÔi is the diffusion operator which depends on the electrode geometry. For a disk electrode it takes the following expression:17,24 ! 2 2 ∂ ∂ 1 ∂ ∂ ^ O ¼ DO þ 2 δ þ ðA2Þ i i ∂t r ∂r ∂r 2 ∂z with DOi being the diffusion coefficient of species Oi. If we assume that DO1 = DO2 = DO3 = D, only species O1 is initially present, and the electron transfer reactions are reversible, the boundary value problem for the concentration profiles for the first potential pulse, E1, is given by: ) t1 ¼ 0, r g 0, z g 0 ð1Þ ð1Þ ð1Þ c ¼ cO1 ; cO2 ¼ cO3 ¼ 0 ðA3Þ t1 ¼ 0, r g 0, z f 0 O1 0

1 ð1Þ ∂cOi A t1 > 0, r > r d , z ¼ 0 : @ ∂z

¼0 z¼0

t1 > 0, 0 e r e rd , z ¼ 0 : 0 1 0 1 0 1 ð1Þ ð1Þ ð1Þ ∂cO3 ∂cO1 ∂cO2 @ A A A þ @ þ @ ∂z ∂z ∂z z¼0

z¼0

ð1Þ

ð1Þ

ð1Þ

ð1Þ

cO1 ¼ J ð1Þ cO2

ðA4Þ

9 =

¼0

ðA5Þ

z¼0

ðA6Þ

cO2 ¼ J ð1Þ cO3 =K ; where: 0

J ð1Þ

FðE1 E01 Þ ¼ exp RT

!

0

FðE1 E02 Þ J ð1Þ =K ¼ exp RT

!

9 > > > > > = > > > > > ;

ðA7Þ

K is the disproportionation constant (see eq 7), and cO1* is the bulk concentration of the two-electron reducible molecule. This problem can be expressed in a simpler way by defining the following variable: ð1Þ

ð1Þ

ð1Þ

ζð1Þ ¼ cO1 þ cO2 þ cO3 1076

ðA8Þ



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such that the differential equation system and the boundary problem turn into: ^ ð1Þ ¼ 0 δζ t1 ¼ 0, r g 0, z g 0 t1 > 0, r g 0, z f ∞

ðA9Þ

) ζð1Þ ¼ cO1

∂ζð1Þ t1 > 0, r g 0, z ¼ 0 : ∂z

ðA10Þ

O3

! ¼0

ðA11Þ

z¼0

The solution for the problem given by eqs A9A11 is: ð1Þ

ð1Þ

ð1Þ

ζð1Þ ¼ cO1 þ cO2 þ cO3 ¼ cO1

i ¼ 1, 2, 3

t1 ¼ 0, r g 0, z g 0 t1 > 0, r g 0, z f ∞

) ð1Þ

0 t1 > 0, r > rd , z ¼ 0 :

ð1Þ

ð1Þ

cO1 ¼ cO1 ; cO2 ¼ cO3 ¼ 0

t1 > 0, 0 e r e rd , z ¼ 0 :

d

O3

1 gp ðtÞ ¼ pffiffiffiffiffiffiffiffi πDt

ðA17Þ

1 1 gs ðtÞ ¼ pffiffiffiffiffiffiffiffi þ rs πDt

ðA18Þ

where rs is the radius of the spherical electrode. For this scheme reaction, the following relationships hold (1) between the currents of the first and second steps, I(1) 1 and I2 , respectively, corresponding to the application of E1, and the surface gradients of the two-electron reducible molecule in its different oxidation states: 9 0 1 ð1Þ > > ∂cO1 > ð1Þ > A > I1 ¼ FAD@ > > ∂z = 0z ¼ 0 1 ðA19Þ ð1Þ > > ∂cO2 > ð1Þ ð1Þ > A > I2 ¼ I1 þ FAD@ > > ∂z ; z¼0

1

ð1Þ ∂c @ Oi A

∂z

1

with c(1) Oi (0 e r e rd, z = 0) given by eq A14. On the other hand, the expression for the function g depends on time and on the shape and size of the electrode, and it is given by eq 9 for disk electrodes, and by the following expressions for planar and spherical ones, respectively:

ðA12Þ

for any r, z, and t values. This is valid independently of the reversibility of the electrode reactions and of the electrode geometry. By considering eqs A6, A9, and A12, we conclude that the surface concentrations of the electroactive species are timeindependent and they can be easily determined, independently of the electrode geometry (see eq A14). Under these conditions, the initial problem of three (1) (1) variables c (1) O 1 , c O 2 , and c O 3 becomes into three independent problems of one variable with constant initial and surface conditions: ^ cð1Þ δ Oi ¼ 0;

where the potential function is independent of the electrode geometry and is given by: 9 ð1Þ ð1Þ fO1 ðE1 Þ ¼ cO1 cO1 ð0 e r e r d , z ¼ 0Þ > > = ð1Þ ð1Þ fO2 ðE1 Þ ¼ cO2 ð0 e r e r d , z ¼ 0Þ ðA16Þ > > ð1Þ ð1Þ ; f ðE Þ ¼ c ð0 e r e r , z ¼ 0Þ

¼0

ðA13Þ

z¼0

8 > ðJ ð1Þ Þ2 ð1Þ > > > cO1 ¼ cO1 > > K þ J ð1Þ þ ðJ ð1Þ Þ2 > > < J ð1Þ ð1Þ cO2 ¼ cO1 ð1Þ > K þ J þ ðJ ð1Þ Þ2 > > > > K > ð1Þ > > : cO3 ¼ cO1 K þ J ð1Þ þ ðJ ð1Þ Þ2

ðA14Þ The above differential equation problem is similar to that corresponding to the application of a potential pulse under limiting current conditions, only differing in the constant value of the surface concentrations. Therefore, the solution for the surface flux of any electroactive species can be written as a product of a function dependent on the applied potential, f(1) Oi (E1), and another function that depends on time and the electrode geometry,40,41 that is: 0 1 ð1Þ ∂c ð1Þ @ Oi A ¼ fOi ðE1 Þ gd ðtÞ; i ¼ 1, 2, 3 ðA15Þ ∂z

with A being the disk electrode area. By taking into account eqs A15A16, we can rewrite eq A19 in the way, 9 ð1Þ ð1Þ I1 ¼ FADgd ðt1 ÞðcO1 cO1 ð0 e r e r d , z ¼ 0ÞÞ > > > = ð1Þ ð1Þ I2 ¼ FADgd ðt1 ÞððcO1 cO1 ð0 e r e r d , z ¼ 0ÞÞ > > > ð1Þ ; cO2 ð0 e r e r d , z ¼ 0ÞÞ ðA20Þ The total current is the sum of those corresponding to the first and second steps, ð1Þ

I ð1Þ ðt1 Þ ¼ I1

ð1Þ

þ I2

ð1Þ

¼ FADgd ðt1 Þf2ðcO1 cO1 ð0 e r e r d , z ¼ 0ÞÞ ð1Þ

cO2 ð0 e r e r d , z ¼ 0Þg

ðA21Þ

By introducing the expressions of the surface concentrations for species O1 and O2 given in eq A14, the expression for the current of the first pulse (eq 3) is finally obtained. Next, we will consider the problem corresponding to the application of the second potential pulse, E2: Ô cð2Þ ¼ 0; i ¼ 1, 2, 3 δ i Oi

z¼0

1077

ðA22Þ



ARTICLE

) t2 ¼ 0, r g 0, z g 0 ð2Þ ð1Þ c ¼ cOi ; i ¼ 1, 2, 3 t2 > 0, r g 0, z f ∞ Oi 1 ð2Þ ∂cOi A t2 > 0, r > r d , z ¼ 0 : @ ∂z

0 ðA23Þ

t2 > 0, r > r d , z ¼ 0 :

0

¼0

ð2Þ

ð2Þ

ð2Þ

ð2Þ

¼0

ðA25Þ

ðA26Þ

with J ð2Þ

!

0

FðE2 E02 Þ J =K ¼ exp RT ð2Þ

!

9 > > > > > = > > > > > ;

ðA27Þ

ð2Þ

ð2Þ

ζð2Þ ¼ cO1 þ cO2 þ cO3 ¼ cO1

ðA28Þ

So, by combining eqs A26 and A28, the following timeindependent expressions are obtained for the surface concentrations of the electroactive species during the second pulse: 9 > ðJ ð2Þ Þ2 ð2Þ > > cO1 ð0 e r e r d , z ¼ 0Þ ¼ cO1 2> ð2Þ ð2Þ > K þ J þ ðJ Þ > > > = ð2Þ J ð2Þ cO2 ð0 e r e r d , z ¼ 0Þ ¼ cO1 > K þ J ð2Þ þ ðJ ð2Þ Þ2 > > > > K > ð2Þ > cO3 ð0 e r e r d , z ¼ 0Þ ¼ cO1 2> ð2Þ ð2Þ K þ J þ ðJ Þ ; ðA29Þ Given that the diffusion operator is linear, the solutions corresponding to the second potential step can be written as a linear combination of solution functions: ð2Þ

ð1Þ

ð2Þ

ð1Þ

ðA33Þ

ð2Þ

cOi ¼ cOi þ ~cOi ; i 1, 2, 3

ðA30Þ

where cOi(1) are the solutions corresponding to the first pulse and ~cOi(2) the new unknown partial solutions of three independent problems with constant initial and surface conditions as occurs for the first pulse (eq A14): ) t2 ¼ 0, r g 0, z g 0 ð2Þ ~c ¼ 0; i ¼ 1, 2, 3 t2 > 0, r g 0, z f ∞ Oi

’ AUTHOR INFORMATION *Tel.: +34 868 88 7524. Fax: +34 868 88 4148. E-mail: [email protected]. Tel.: +44 (0) 1865 275 413. Fax: +44 (0) 1865 275 410. E-mail: [email protected].

’ ACKNOWLEDGMENT A.M. and J.G. greatly appreciate the financial support provided by the Direccion General de Investigacion Científica y Tecnica (Project Number CTQ2009-13023) and the Fundacion SENECA (Project Number 08813/PI/08). E.L. thanks the Fundacion SENECA for the grant received. Q.L. thanks Schlumberger Cambridge Research Limited for funding. ’ REFERENCES

From analogous considerations as those made for the first pulse, it is fulfilled that, ð2Þ

z¼0

Corresponding Author

z¼0

cO2 ¼ J ð2Þ cO3 =K ;

0

ðA32Þ

By following a procedure analogous to that described for the first potential pulse, the expression for current corresponding to the second potential pulse given in eq 4 is obtained.

9 =

FðE2 E01 Þ ¼ exp RT

¼0

t2 > 0, 0 e r e rd , z ¼ 0 : ~cOi ¼ cOi cOi ; i 1, 2, 3

ðA24Þ

z¼0

z¼0

cO1 ¼ J ð2Þ cO2

∂z

ð2Þ

t2 > 0, 0 e r e rd , z ¼ 0 : 0 1 0 1 0 1 ð2Þ ð2Þ ð2Þ ∂cO ∂c ∂c @ O1 A þ @ O2 A þ @ 3A ∂z ∂z ∂z z¼0

1

ð2Þ ∂~c @ Oi A

ðA31Þ

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Electrochemical Behavior of Two-Electron Redox Processes by

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