Simulation of Square-Wave Voltammetry: EC and ... - ACS Publications

J. Phys. Chem. B 2000, 104, 5331-5342

5331

Simulation of Square-Wave Voltammetry: EC and ECE Electrode Processes Adrian B. Miles and Richard G. Compton* Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, U.K. ReceiVed: February 22, 2000

A simulation of square-wave voltammetry (SWV), based on the backward implicit method, is developed to encompass EC and ECE mechanistic schemes. The processes considered are assumed to be electrochemically reversible, and the voltammetric propertiesspeak current, peak voltage, and half-height widthsare shown to give good agreement with approximate analytical solutions, where available. The simulations are computationally efficient and facilitate the construction of informative concentration profiles for all reacting species. Further, the inclusion of diffusional effects based on convective control of the diffusion layer thickness is shown to be possible, thereby permitting the quantitative use of SWV for mechanistic studies at hydrodynamic electrodes.

1. Introduction Square-wave voltammetry (SWV) is a highly attractive and especially sensitive method for the study of diverse electrochemical processes.1,2 The technique involves the application of the general potential waveform defined in Figure 1 and the measurement of the currents I(1) and I(2) in each cycle of the waveform at the points shown. The current difference

∆I ) I(1) - I(2) is highly discriminating against background currents, and the variation of ∆I with potential typically gives rise to peak-shaped voltammograms, which are highly sensitive to the electrode and to coupled homogeneous kinetics. In previous papers,3,4 an approach to the simulation of SWV that was based on the backward implicit method5,6 was introduced and applied to simple electron-transfer reactions. This was shown to give excellent agreement with independent theory, where available, and to be computationally efficient. The present paper reports the theoretical characterization of complex mechanistic electrode processes through the development of simulations for the EC and ECE mechanisms. The latter are defined as follows:

EC:

ECE:

RED - e- a OX OX f C

kf

C f OX

kb

RED - e- a OX

E1o′

OX f C

k

C - e- a D

E2o′

where kf, kb, and k are first-order rate constants and E1o′ and E2o′ are formal potentials of the RED/OX and C/D couples, respectively. Approximate analytical solutions exist only for the case of semi-infinite diffusion for both the EC7 and the ECE8 mechanisms (see Appendix). These solutions are based on the use of the step function method9 for solving integral equations that, for the EC case, were derived by Smith.10 We are unaware of any work that attempts to theoretically develop SWV for

Figure 1. Potential waveform for the SWV experiment. Ei is the initial voltage, ∆E the step voltage, ESW the square-wave voltage, and tp the duration of a pulse.

processes occurring at hydrodynamic electrodes, where convective dilution leads to finite diffusion effects. The major advantage of the work described in the following is the ability to allow for a diffusion layer thickness of an arbitrary size, thereby opening up the use of SWV for quantitative measurements at hydrodynamic electrodes. The finite difference simulation of SWV using the backward implicit method will now be developed for the EC and ECE mechanisms. The technique will be shown to give agreement with the analytical solutions for the limiting case of semi-infinite diffusion while also being computationally efficient. Further, this alternative method facilitates the production of concentration profiles so as to aid in the interpretation of the experiment. Most importantly, the methods show how simulations can allow the quantitative description of processes beyond the scope of existing analytical theory. 2. Theory Throughout this work, we consider the following electrode process to occur at the surface of a uniformly accessible

10.1021/jp0006882 CCC: $19.00 © 2000 American Chemical Society Published on Web 05/16/2000

5332 J. Phys. Chem. B, Vol. 104, No. 22, 2000

Miles and Compton the present work, a value of 20 was used in light of earlier optimization.3 It follows, from the definition of the distorted spatial function, that

∂ ∂ψ ∂ ) ∂y ∂y ∂ψ Figure 2. Diagram to show the back-to-back grid for the RED and OX species.

∂ψ a ) (1 - ψ2) ∂y δ and finally

()

electrode.

[

]

RED(aq) - e a OX(aq)

a 2 ∂2 ∂ ∂2 ) (1 - ψ2) (1 - ψ2) 2 - 2ψ 2 δ ∂ψ ∂y ∂ψ

We also assume fast electrode kinetics and, therefore, electrochemically reversible redox couples. The Nernst equation is applied at the electrode surface

Labeling the y coordinate and temporal positions with j and t, respectively, the spatially transformed finite difference form of the RED mass transport equation is given by

-

[RED(aq)]y)0 [OX(aq)]y)0

) exp(-θ)

(1)

(

t [RED]j+1

F (E - E°′) where θ ) RT

( )

and where E is the square-wave scan potential, E°′ is the formal potential of the redox couple, and y is the coordinate normal to the electrode surface. The simulation of both the EC and ECE mechanisms is achieved by the formulation of mass transport equations describing the species present. The mass transport equations are cast into finite difference form and solved using the backward implicit method. At each temporal node, a two-step process is required. The first step involves simulation of the OX and RED concentrations, while the second step involves simulations of C concentrations. For simplicity, we use a backto-back grid,11,12 as shown in Figure 2, where OX species are assigned negative y coordinates and RED species occupy positive y coordinates. At the electrode, y ) 0, we simulate the concentration of RED, while we consider the OX concentration implicitly. The C species are also assigned negative y values so as to coincide with the OX species with which they kinetically interconvert. The Thomas algorithm is used to solve the tridiagonal matrices in both steps of the simulation cycle. The systems we consider are initially composed entirely of RED species. The first reaction step in both mechanisms is the electrochemical equilibrium between RED and OX species. The RED species has no part in any other processes and, hence, is always described by the same mass transport equation, as given by Fick’s second law.

∂2[RED] ∂[RED] ) DRED ∂t ∂y2

(2)

The spatial transformation function used by Brookes et al.3 is implemented so as to focus on the area closest to the electrode, where the concentrations of all species change most rapidly.

ψ ) tanh

(ayδ )

(3)

In eq 3, a is an adjustable parameter, and δ is the diffusion layer thickness, which can be taken large enough to correspond to semi-infinite diffusion. The parameter a can be increased so as to focus more closely on the area near the electrode, and for

()

[

2 [RED]tj - [RED]t-1 j a ) DRED (1 - ψ2j ) (1 - ψ2j ) × ∆t δ

-

2[RED]tj 2 (∆ψ)

)

t + [RED]j-1

(

-

)]

t t [RED]j+1 - [RED]j-1 2ψj 2∆ψ

(4)

where ∆ψ ) tanh(a)/NJ and NJ is the number of spatial nodes. By defining

λRED )

DRED∆t(1 - ψ2j )

(δa)

2

(∆ψ)2

(5)

eq 4 can be simplified to t ) -λRED(1 - ψ2j + ψj∆ψ)[RED]j-1 + [RED]t-1 j

(1 + 2λRED(1 - ψ2j ))[RED]tj t (6) λRED(1 - ψ2j - ψj∆ψ)[RED]j+1

Similar equations exist for the OX and C species. However, the mechanism studied alters the formulation, and the two mechanisms must be considered separately. We return to this issue subsequently. The progression of the concentration profile for the RED species is restricted by two boundary conditions. First, we define a finite diffusion layer thickness, δ, at which essentially semiinfinite diffusion, or not for hydrodynamic electrodes, operates. At this boundary, the concentrations of the species can be assigned their bulk values.

[RED]y)δ ) [RED]j)NJ ) [RED]BULK [All other species]y)-δ ) [All other species]j)-NJ ) 0 Normalizing all concentrations with respect to the bulk concentration of RED, eq 6 at this boundary reduces to t-1 2 t ) -λRED(1 - ψNJ-1 + ψNJ-1∆ψ)[RED]NJ-2 + [RED]NJ-1 2 t ))[RED]NJ-1 (1 + 2λRED(1 - ψNJ-1 2 - ψNJ-1∆ψ) (7) λRED(1 - ψNJ-1

In addition, the flux of RED to the electrode must be conserved

Simulation of Square-Wave Voltammetry

J. Phys. Chem. B, Vol. 104, No. 22, 2000 5333

CHART 1. Matrix Equation Used to Simulate the Concentrations of the RED and OX Species in the First Step at Each Temporal Node for the EC Mechanism

CHART 2. Matrix Equation Used to Simulate the Concentration of the C Species in the Second Step at Each Temporal Node for the EC Mechanism

with that of OX.

DRED

|

∂[RED] ∂y

y)0

) -DOX

|

∂[OX] ∂y

remaining two species, we must incorporate terms to describe the chemical reaction steps.

(8)

y)0

This condition requires that, at the electrode, the finite difference mass transport equation is always given by t - κ[RED]t1 0 ) [RED]t0(κ + eθ) - [OX]-1

(9)

where κ is DRED/DOX. Case 1: The EC Mechanism. First, we consider the EC mechanism, as defined above. The mass transport equation for the RED species has already been described and is composed exclusively of a term derived from Fick’s second law. For the

∂[OX] ∂2[OX] ) DOX - kf[OX] + kb[C] ∂t ∂y2

(10)

∂[C] ∂2[C] + kf[OX] - kb[C] ) DC ∂t ∂y2

(11)

The system we consider is initially composed entirely of RED species. OX is formed at the electrode and may, in turn, react to form C. The finite difference OX and C equations are analogous to those derived for the RED species above. However, terms for the forward and backward chemical steps are,

5334 J. Phys. Chem. B, Vol. 104, No. 22, 2000 obviously, included.

()

[

Miles and Compton For t > 0; y ) δ (j ) NJ)

2 [OX]tj - [OX]t-1 j a ) DOX (1 - ψ2j ) (1 - ψ2j ) × ∆t δ

(

)

t t [OX]j+1 - 2[OX]tj + [OX]j-1

(∆ψ)2 t t [OX]j+1 - [OX]j-1 2ψj 2∆ψ

(

[C]tj

(

∆t

[C]t-1 j

)]

()

2

) DC

[

(∆ψ)2

[OX]-NJ ) [C]-NJ ) 0

- kf[OX]tj + kb[C]tj (12)

) ( - 2ψj

For t > 0; y ) δ (j ) -NJ)

-

a (1 - ψ2j ) (1 - ψ2j ) × δ

t t [C]j+1 - 2[C]tj + [C]j-1

[RED]NJ ) [RED]BULK

)]

t t [C]j+1 - [C]j-1 2∆ψ

+

kf[OX]tj - kb[C]tj (13)

By defining λOX and λC similarly to λRED above, we have t [OX]t-1 + kb∆t[C]tj ) -λOX(1 - ψ2j + ψj∆ψ)[OX]j-1 + j

(1 + 2λOX(1 - ψ2j ) + kf∆t)[OX]tj t (14) λOX(1 - ψ2j - ψj∆ψ)[OX]j+1

Knowledge of the concentrations of all three species allows simulation, in the first step, of the concentrations of OX and RED at the next point in time. The second step involves simulation of the C concentrations given the concentrations of OX from the current point in time and of C from the previous temporal node. The finite difference equations are organized into two matrices, as shown in Charts 1 and 2. The first matrix is solved for the RED and OX species, while the second yields concentrations of the C species. The SWV current difference can be calculated from the currents, I(1) and I(2), at the end of each half cycle. The flux of RED on the electrode enables the calculation of the current at a given point in time.

( )(

IRED/OX{t} ) nFADRED

and

(1 + 2λC(1 - ψ2j ) + kb∆t)[C]tj t (15) λC(1 - ψ2j - ψj∆ψ)[C]j+1

In our two-step backward implicit method, we in fact use the concentration of C from the previous time point to solve for the concentration of OX. That is, we assume that

[C]tj ≈ [C]t-1 j This approximation is valid assuming either that the concentration of C does not change rapidly with time or that the temporal step we employ is sufficiently small. As discussed above, the concentrations of the OX and C species at semi-infinite diffusion are zero. Further, the C species is electrochemically inert and cannot react at the electrode. There must, therefore, be no flux of C on the electrode.

(16)

for j ) 0, 1, 2, ..., NJ

[OX]j ) [C]j ) 0 for j ) -1, -2, ..., -NJ For t > 0; y ) 0 (j ) 0)

[C]-1 ) [C]0 [RED]0 ) [OX]0 exp(-θ) ∂[RED] ∂[OX] DRED | ) DRED | ∂y 0 ∂y 0

( )(

IRED/OX(i) ) nFADRED

)

t t a [RED]1 - [RED]0 δ ∆ψ

(18)

where n ) 1 and i ) 1 or 2, corresponding to the value of t. For the presentation and discussion of results, it is helpful to define the dimensionless current

IDL{t} ) I{t}

xπτ nFA[RED]BULKxDRED

(19)

and correspondingly

(i) ) IRED/OX DL

The relevant boundary conditions for the EC mechanism are summarized as follows. For t ) 0

[RED]j ) [RED]BULK

(17)

I(1) and I(2) can, therefore, be calculated using this formulation at the relevant temporal nodes

t + kf∆t[OX]tj ) -λC(1 - ψ2j + ψj∆ψ)[C]j-1 + [C]t-1 j

∂[C] | )0 DC ∂y y)0

)


xπτDRED [RED]BULK

( )(

)


(20)

Further, the square-wave voltammetry experiment reports the current difference. This is given by

) IRED/OX (1) - IRED/OX (2) ∆IRED/OX DL DL DL

(21)

Finally, the current for the EC mechanism is solely dependent on the RED/OX couple, and we can, therefore, define the total current difference as

∆IDL ) ∆IRED/OX DL

(22)

Case 2: The ECE Mechanism. We now consider the ECE mechanism, as described earlier. Again, the mass transport equation for the RED species is as derived above. The remaining



CHART 3. Matrix Equation Used to Simulate the Concentrations of the RED and OX Species in the First Step at Each Temporal Node for the ECE Mechanism

three species have the following mass transport equations:

∂[OX] ∂2[OX] - k[OX] ) DOX ∂t ∂y2

(23)

∂[C] ∂2[C] ) DC + k[OX] ∂t ∂y2

(24)

∂2[D] ∂[D] ) DD ∂t ∂y2

(25)

and, in finite difference form

()

(

-

2[OX]tj 2

[

) )]

t + [OX]j-1

()

[

- k[OX]tj (26)

2 [C]tj - [C]t-1 j a ) DC (1 - ψ2j ) (1 - ψ2j ) × ∆t δ

(

t [C]j+1

()

-

2[C]tj + 2

)

t [C]j-1

-

(∆ψ) t t [C]j+1 - [C]j-1 2ψj 2∆ψ

(

[

)]

2 [D]tj - [D]t-1 j a ) DD (1 - ψ2j ) (1 - ψ2j ) × ∆t δ

(

t [D]j+1

-

2[D]tj + 2

(∆ψ)

) (

t [D]j-1

(1 + 2λOX(1 - ψ2j ) + k∆t)[OX]tj t (29) λOX(1 - ψ2j - ψj∆ψ)[OX]j+1

and

t (30) (1 + 2λC(1 - ψ2j ))[C]tj - λC(1 - ψ2j - ψj∆ψ)[C]j+1

-

(∆ψ) t t [OX]j+1 - [OX]j-1 2ψj 2∆ψ

(

t [OX]t-1 ) -λOX(1 - ψ2j + ψj∆ψ)[OX]j-1 + j

t + k∆t[OX]tj ) -λC(1 - ψ2j + ψj∆ψ)[C]j-1 + [C]t-1 j

2 [OX]tj - [OX]t-1 j a (1 - ψ2j ) (1 - ψ2j ) × ) DOX ∆t δ t [OX]j+1

of λOX and λC, the first two finite difference equations simplify to

+ k[OX]tj (27)

In the ECE mechanism, the C species is electrochemically active, and consequently, unlike the EC case, the flux of C on the electrode cannot be set equal to zero. For the purpose of this work, we assume that E2o′ , E1o′. Therefore, C is oxidized far more easily than RED, and the concentration of C at the electrode is assumed to be equal to zero. Further, as we assume all C species at the electrode are converted to D, we have no requirement to simulate the concentrations of the D species. We can therefore discard the mass transport equation for D. The ECE simulation is carried out in a manner identical to that of the EC simulation. In the first step of each cycle, we employ a back-to-back grid for the RED and OX species. The second step is, again, purely concerned with the simulation of the concentration of C. The ECE boundary conditions can be summarized as follows: For t ) 0

)]

t t [D]j+1 - [D]j-1 - 2ψj 2∆ψ

(28)

Unlike the EC case, the mass transport equation for OX is independent of the concentration of C. With suitable definitions

[RED]j ) [RED]BULK [OX]j ) [C]j ) 0 For t > 0; y ) 0 (j ) 0)

for j ) 0, 1, 2, ..., NJ for j ) -1, -2, ..., -NJ


Miles and Compton

CHART 4. Matrix Equation Used to Simulate the Concentration of the C Species in the Second Step at Each Temporal Node for the ECE Mechanism

Figure 3. Plot of ∆Ip as a function of log(2kECtp) for the EC mechanism and various values of K ) (A) 0.01, (B) 0.1, (C) 1, (D) 10, (E) 100, and (F) 1000.

Figure 5. Plot of -∆Ep as a function of log(2kECtp) for the EC mechanism and various values of K ) (A) 0.01, (B) 0.1, (C) 1, (D) 10, (E) 100, and (F) 1000.

[OX]-NJ ) [C]-NJ ) 0 The finite difference equations and boundary conditions can, again, be organized into two matrices (Charts 3 and 4) that are solved to yield the variations of concentrations of the RED, OX, and C species with time. Finally, the current difference observed for the ECE mechanism has two components. The first component, arising from the RED/OX couple, is calculated as for the EC mechanism. The second component arises from the electrochemical activity of C and has an analogous formulation.

Figure 4. Plot of W1/2 as a function of log(2kECtp) for the EC mechanism and various values of K ) (A) 0.01, (B) 0.1, (C) 1, (D) 10, (E) 100, and (F) 1000.

[C]0 ) 0

I

C/D

( )(

For t > 0; y ) δ (j ) NJ)

[RED]NJ ) [RED]BULK For t > 0; y ) δ (j ) -NJ)

(31)

Again, n ) 1 and i ) 1 or 2 corresponding to the value of t. However, as the concentration of C is always zero at the electrode, this equation can be simplified to the following:

( )( )

[RED]0 ) [OX]0 exp(-θ) ∂[RED] ∂[OX] DRED |0 ) -DOX | ∂t ∂y 0

)

t t a [C]-1 - [C]0 (i) ) nFADC δ ∆ψ

IC/D(i) ) nFADC

t a [C]-1 δ ∆ψ

(32)

The dimensionless current for the C/D couple is given by

IC/D DL (i) )

xπτDC [RED]BULK

( )( ) t a [C]-1 δ ∆ψ

(33)



Figure 6. (i) OX and (ii) C species concentration profiles for the EC mechanism during the (a) 45th, (b) 50th, and (c) 55th square-wave cycles for K ) 0.01 and kEC ) 5 × 10+4 s-1.

and the current difference can be calculated as C/D C/D ∆IC/D DL ) IDL (1) - IDL (2)

k in the following ranges:

(34)

Finally, the total current for the ECE mechanism is given by the sum of the current contributions from the RED/OX and C/D couples.

∆IDL ) ∆IRED/OX + ∆IC/D DL DL

(35)

Simulations and approximate analytical solutions were programmed in Fortran 77, and the computation was performed on a Silicon Graphics Origin 2000 server. Excel 2000 and IDL 5 were used to analyze the results. The simulations were converged in order to find the number of spatial nodes (NJ) and time nodes per pulse (NT) required to give suitable accuracy. Each variable was converged in turn, holding the other constant at 1000, to give agreement to within 1% of the limiting values (determined by running heavily “over-converged” jobs) of peak height, width, and position. We consider values of kEC, K, and

5 × 10-2 < kEC/s-1 < 5 × 10+4 (10-3 < 2×kEC×tp < 10+3) 10-2 < K < 10+3 5 × 10-2 < k/s-1 < 5 × 10+4

(10-3 < 2×k×tp < 10+3)

For the EC mechanism, NT ) 100 and NJ ) 100 were found to be suitable for all values of K when kEC is small. For large kEC, the value of NT required varies from 100 for large K to greater than 2000 for small K, while the required value of NJ varies from 200 for small K to 1000 for large K. A value of NT ) 100 is suitable for the ECE mechanism throughout the entire range of k considered, while NJ varies from 100 for small k to 1000 for large k. 3. Results and Discussion The values of NJ and NT required to give suitable agreement with the results of “over-converged” jobs have been reported


Miles and Compton

Figure 7. (i) OX and (ii) C species concentration profiles for the EC mechanism during the (a) 45th, (b) 50th, and (c) 55th square-wave cycles for K ) 0.01 and kEC ) 5 × 10-2 s-1.

above. These results are next compared with those from approximate analytical solutions.8,9 In particular, we consider the values of the dimensionless peak current, ∆IDL,p, the peak width at half-height, W1/2, and the peak position, ∆Ep. ∆Ep is defined as

∆Ep ) Ep - Eo′ where Ep is the potential at which the maximum current is observed in the square-wave voltammogram and Eo′ is the formal potential of the RED/OX couple for both the EC and the ECE mechanisms. In addition, the generation of concentration profiles provides further insight into the experiment. Concentration profiles are generated for the OX and C species in both the EC and the ECE mechanisms, and those illustrated display the variation of concentration over a complete squarewave cycle, including both forward and backward pulses. In

general, the following parameters are utilized, in addition to the values of kEC, K, and k previously discussed: ESW ) 50 mV, ∆E ) 10 mV, tp ) 0.01 s (corresponding to a frequency of 50 Hz), Eo′ ) 0 V, Ei ) -0.5 V, and Ef ) 0.5 V, where Ei and Ef are the initial and final voltages, respectively. Further, the diffusion layer thickness, δ, is defined as 200 µm, except when the introduction of a finite diffusion layer is considered. The results are reported for each mechanism in turn. Case 1: The EC Mechanism. Voltammograms for the range of kEC and K values described above were simulated using NT ) 1000 and NJ ) 1000. For most experiments, this combination of NT and NJ corresponds to full-convergence and often overconvergence for at least one of the two values. Corresponding approximate analytical solutions for semi-infinite diffusion were calculated using 100 subintervals per half square-wave period (l ) 100). Values of ∆IDL,p and W1/2 calculated with l ) 100



Figure 11. Plot of W1/2 as a function of log(2ktp) for the ECE mechanism.

Figure 8. Plot showing the dependence of ∆Ip on diffusion layer thickness, δ, and log(2kECtp) for the EC mechanism when K ) 100.

Figure 12. Plot of -∆Ep as a function of log(2ktp) for the ECE mechanism.

Figure 9. Plot showing the dependence of -∆Ep on diffusion layer thickness, δ, and log(2kECtp) for the EC mechanism when K ) 100.

Figure 10. Plot of ∆Ip as a function of log(2ktp) for the ECE mechanism.

typically show agreement with results using l ) 500 to within -0.4 ( 0.3% and ( 0.03%, respectively. ∆Ep tends to deviate more significantly but is generally within (0.1 mV. The variation of the simulation values of ∆IDL,p, W1/2, and ∆Ep with kEC and K is shown in Figures 3-5. The values of ∆IDL,p typically agree with the approximate analytical solution8 within the range +0.31 to +1.05%, while the agreement for W1/2 is generally between -0.03 and +0.10%. ∆Ep is usually within (0.01 mV for small kEC (∼5 × 10-2 s-1), while its deviation is no more than (1.6 mV for large kEC (∼5 × 10+4 s-1). In general, the agreement between the simulation and approximate analytical solution is more than adequate for all experimental purposes.

Representative concentrations profiles are given in Figures 6 and 7 and illustrate the dependence on kEC and K. Figure 6 displays concentration profiles for the OX and C species when K ) 0.01 and kEC ) 5 × 10+4 s-1. The concentration profiles for the two species have almost identical forms, showing that large rate constants for the forward and backward C reaction steps (large kEC) facilitate rapid equilibriation between the OX and C species. The absolute values of the concentrations are, however, different and are in good agreement with the value of K. The concentration profiles for K ) 0.01 and kEC ) 5 × 10-2 s-1 are presented in Figure 7. The concentration profiles for the OX species are very similar to those already discussed. The C concentration profiles have very different forms. The much smaller concentration of C and the much slower response to the gradual increase in concentration of OX are illustrative of the small value of kEC. Finally, we consider the introduction of a finite diffusion layer and its effect on the voltammetric properties. The dependence of ∆IDL,p and ∆Ep on diffusion layer thickness, δ, and kEC (for K ) 100) are illustrated in Figures 8 and 9, respectively. The edges of these surfaces corresponding to δ ) 200 µm replicate the data for ∆IDL,p and ∆Ep shown in Figures 3 and 5, respectively. Decreasing the diffusion layer thickness has little effect on ∆IDL,p until δ is less than 50 µm, at which point ∆IDL,p is seen to increase rapidly. The semi-infinite diffusion values of ∆Ep are exhibited when δ is greater than 100 µm. Below 100 µm, ∆Ep decreases rapidly with decreasing δ. Case 2: The ECE Mechanism. The values NT ) 100 and NJ ) 1000 were used to simulate voltammograms, over the considered range of k, for the ECE mechanism. The voltammetric properties are displayed in Figures 10-12. The corresponding approximate analytical solution9 results are generated using the corrected analytical equations given in the Appendix. We consider all C at the electrode surface to be converted to D, and hence, the value of 2 is set equal to zero. Again, l )


Miles and Compton

Figure 13. (i) OX and (ii) C species concentration profiles for the ECE mechanism during the (a) 45th, (b) 50th, and (c) 55th square-wave cycles for k ) 5 × 10-2 s-1.

100 is employed in the approximate analytical solution, and agreement with the results using l ) 500 are similar to those observed for the EC mechanism. The simulation and approximate analytical solution results typically agree within +0.6 ( 0.2% and (0.05% for ∆IDL,p and W1/2, respectively. The greatest discrepancies are again seen for ∆Ep, for which the deviation can be as much as -0.12 or +0.79 mV for large k. A comparison of the results for the ECE mechanism and the EC case, using K ) 1000 (i.e., kf ≈ k), reveals that the values of ∆Ep are almost identical. W1/2 also shows good agreement for small k (and kEC), while greater deviation is seen for large k. The W1/2 values for large rate constants are, however, found to agree more closely when a very large value of K is employed in the EC simulation. The dimensionless peak current, ∆IDL,p, is also very similar for the two mechanisms in the limit of small k. For large k, ∆IDL,p is consistently greater for the ECE mechanism than for the EC case, irrespective of the value of K. This increase in ∆IDL,p is clearly the current contribution from the second electrochemical step (C/D) and is consequently

observed to have a greater significance for large k, corresponding to more rapid formation of C, which can, in turn, be oxidized to D. The OX and C concentration profiles, for k ) 5 × 10-2 s-1, are given in Figure 13. The profiles for the OX species are very similar to those illustrated for the EC mechanism (Figures 6 and 7). The C concentration tends to zero both at large distances from the electrode and at the electrode surface. This behavior near the electrode surface is very different from that of the EC case and is, obviously, characteristic of the oxidation of C to D at the electrode surface. Further, the maximum concentration of C is seen to be greater in Figure 13 than it is in Figure 7. This suggests that the depletion of C by oxidation at the electrode is overwhelmed by the more extensive conversion of OX to C in the ECE mechanism than in the EC case (K ) 0.01) illustrated. The dependences of ∆IDL,p and ∆Ep on diffusion layer thickness, δ, and on k are shown in Figures 14 and 15, respectively. The variation of ∆IDL,p and ∆Ep with δ is very


J. Phys. Chem. B, Vol. 104, No. 22, 2000 5341 Introducing8 kEC ) kf + kb and K ) kf/kb, the finite sum approximation for the dimensionless current corresponding to the EC mechanism is given by

IRED/OX {t} ≈ bm ) DL πxkECτ

-

Kxπ m-1

∑

1 + K i)1

biRj′ -

xπ + KR1′ 1+K

x ( 2kECτ

1

l

1+K

x ( 2kECτ

1

l

1+K

+

+

)∑

1

m-1

i)1

biSj′

)

1

(A2)

where Figure 14. Plot showing the dependence of ∆Ip on diffusion layer thickness, δ, and log(2ktp) for the ECE mechanism.

) exp

[

]

F(E - Eo′) RT

Rj′ ) erfxjkECτ/2l - erfx(j - 1)kECτ/2l Sj′ ) xj - xj - 1 m)

Figure 15. Plot showing the dependence of -∆Ep on diffusion layer thickness, δ, and log(2ktp) for the ECE mechanism.

and l is the number of subintervals per half-period. This equation differs from that published8 by a factor of xπ in the first term of the numerator. This correction is supported by the results of the simulations reported in the text. For the ECE mechanism, defined as above, the current contribution from the first couple (RED/OX) is that given by the EC case in the limiting example of infinite K. This can be expressed as

{t} ≈ bm ) IRED/OX DL

similar to the behavior observed for the EC case. Both voltammetric properties, again, reach their semi-infinite diffusion layer limits when δ is much less than 200 µm. Interestingly, the greater values of ∆IDL,p at large k (or kEC) are more pronounced with small δ for the ECE mechanism.

πxkτ - 1xπ

5. Appendix: Approximate Analytical Solutions We define the dimensionless current

IDL{t} ) I{t}

xπτ FA[RED]BULKxDRED

x [( ) ] x

m-1

∑ i)1

1xπ erf

4. Conclusion The simulation of square-wave voltammetry for complex mechanistic schemes has been shown to give accurate results that display excellent agreement with existing approximate analytical theory, where available. Immediate realization of concentration profiles for the chemical species involved is also possible and provides further insight into the experiment. In addition, the ability to introduce a finite diffusion layer thickness enables the quantitative investigation of hydrodynamic electrodes.

2lt τ

2kτm-1

biRj′ -

kτ

1/2

biSj′ ∑ i)1

l

(A3)

2kτ

+

2l

l

where

Rj′ ) erfxjkτ/2l - erfx(j - 1)kτ/2l

[

]

F(E - E1o′) 1 ) exp RT

Discrepancies from the previously published equation9 are, again, noted. The current contribution from the second couple (C/D) is given by9

IC/D DL {t} ≈ hm ) (A1)

where I(t) is the current, τ is the square-wave period, F is the Faraday constant, A is the area of the electrode surface, [RED]BULK is the concentration of RED in the bulk solution, and DRED is the diffusion coefficient of the RED species.

(bm(S1 - xπR1′) + S1

m-1

m-1

biSj′ - xπ ∑ biRj′) ∑ i)1 i)1

S1(1 + 2)

m-1

-

hiSj′ ∑ i)1 (A4)

where


S1 ) 2 ) exp

[

(2kτl )

1/2

]

F(E - E2o′) RT

The second term in this equation does not appear in the original publication.9 Results obtained for the ECE mechanism support the above amendments to the equations describing the RED/ OX and C/D current contributions. References and Notes (1) Ç akir, S.; Biç er, E.; Ç akir, O. Electrochem. Commun. 1999, 1, 257. (2) Xiao, Z.; Lavery, M. J.; Bond, A. M.; Wedd, A. G. Electrochem. Commun. 1999, 1, 309. (3) Brookes, B. A.; Ball, J. C.; Compton, R. G. J. Phys. Chem. B 1999, 103, 5289.

Miles and Compton (4) Brookes, B. A.; Compton, R. G. J. Phys. Chem. B 1999, 103, 9020. (5) Anderson, J. L.; Moldoveanu, S. J. Electroanal. Chem. 1984, 179, 109. (6) Laasonen, P. Acta Math. 1949, 81, 30917. (7) Compton, R. G.; Pilkinton, M. B. G.; Stearn, G. M. J. Chem. Soc., Faraday Trans. 1 1988, 84, 2155. (8) O’Dea, J. J.; Osteryoung, J.; Osteryoung, R. A. Anal. Chem. 1981, 53, 695. (9) O’Dea, J. J.; Wikiel, K.; Osteryoung, J. J. Phys. Chem. 1990, 94, 3628. (10) Nicholson, R. S.; Olmstead, M. L. In Electrochemistry: Calculations, Simulation and Instrumentation; Mattson, J. S., Mark, H. B., MacDonald, H. C., Eds.; Marcel Dekker: New York, 1972; Vol. 2, Chapter 5. (11) Smith, D. E. Anal. Chem. 1963, 35, 602. (12) Dryfe, R. A. W. DPhil Thesis, Oxford University, Oxford, U.K., 1995; p 82. (13) Bidwell, M. J.; Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 417, 119. (14) Alden, J. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, 8941.

Simulation of Square-Wave Voltammetry: EC and ... - ACS Publications

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