Cyclic Voltammetry of the ECâ² Mechanism at Hemispherical Particles

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Cyclic Voltammetry of the EC0 Mechanism at Hemispherical Particles and Their Arrays: The Split Wave Kristopher R. Ward,† Nathan S. Lawrence,‡ R. Seth Hartshorne,‡ and Richard G. Compton*,† †

Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford, United Kingdom OX1 3QZ ‡ Schlumberger Cambridge Research Center, High Cross, Madingley Road, Cambridge, United Kingdom, CB3 0EL ABSTRACT: The EC0 (catalytic) mechanism, (1) A þ e h B; K2 (2) B þ X f A þ P, in which the reduction of ‘X’ to ‘P’ is mediated by the A/B redox couple, is studied at a regularly distributed array of hemispherical particles on a planar surface using simulated cyclic voltammetry. It is assumed that the supporting surface itself is not electroactive and therefore the electron transfer in process (1) occurs exclusively on the surface of the particles. Two-dimensional finite difference simulations are performed for a range of scan rates, particle surface coverages, and second-order rate constants, K2. Additionally, for the case of an isolated particle, the effect of the concentration of reactant species ‘X’ is also examined. Particular attention is paid to the ‘split-wave’ phenomenon, where two peaks are observed in the forward scan of a cyclic voltammogram, which tends to occur when the values of [A] and [X] are similar and the second-order rate constant, K2, is relatively high. The conditions under which two peaks are resolvable are elucidated and expressions are presented for the first peak current and potential for the isolated case.

1. INTRODUCTION 1.1. The EC0 Mechanism. The EC0 (catalytic) mechanism is defined as

Aþe h B

ð1.1Þ

K2

ð1.2Þ

BþXfA þP

as illustrated in Figure 1. The net result of the process is the transformation of reactant species ‘X’ to product species ‘P’, catalyzed by species ‘A’ (which is regenerated). It is important to distinguish this form of catalysis from the case where the particle catalyzes the electron transfer ‘A ( e h B’; rather in this case, the A/B redox couple catalyzes the transformation of X to P. An exemplar EC0 reaction is the oxidation of cysteine to cystine:1 K2

4 FeðCNÞ3 6 þ cysteine f FeðCNÞ6 þ cystine

ð1.4Þ

The same redox couple also has potential applications in gas sensing, for example of H2S.2 It is known that under certain conditions, a cyclic voltammogram of an EC0 system will show two distinct peaks in the forward sweep.1 This typically occurs when the initial concentrations of species ‘A’ and species ‘X’ are similar and when the second-order rate constant, K2, is above a certain value. Figures 2 and 3a) show the progression of cyclic voltammograms that result when cX = cA and K2 is increased, keeping all other system parameters constant. For low K2, the voltammetry is similar to that observed for a simple one-electron reduction (the E mechanism), as expected; r 2011 American Chemical Society

however, as the rate constant increases, so too does peak current response. The explanation is as follows: a faster rate constant means a more rapid regeneration of ‘A’ within the diffusion layer and hence greater current. As K2 is increased further, the waveform begins to split into two distinct peaks with greater K2, resulting in a more obvious separation. For a fast scan rate, this also leads to a decrease in peak height with increasing peak-topeak separation (Figure 2), although the peak currents tend toward limiting values as K2 increases and the peaks become more distinct. From Figure 3, it can be seen that even though the heights of both forward peaks decrease slightly with increasing K2, the integrated current in the forward sweep still increases. In the past, some attention has been given to the problem of simulating the EC0 mechanism, though often using the simplifying assumption that the chemical step is pseudo-first-order, and a few studies have examined the split-wave phenomenon. Simple one-dimensional systems (isolated microspheres or macroelectrodes) have been considered,3,4 and such simulations may easily be performed on commercially available software packages such as Digisim.5 Some characterization of the reaction at more complex electrode geometries such as the microdisk6 has also been reported. A series of papers by Saveant et al.712 examining the ‘homogeneous catalysis of electrochemical reactions’ explored systems similar to the EC0 mechanism described herein and Received: March 11, 2011 Revised: April 14, 2011 Published: May 17, 2011 11204

dx.doi.org/10.1021/jp2023204 | J. Phys. Chem. C 2011, 115, 11204–11215

The Journal of Physical Chemistry C noted the appearance of split polarographic waves resulting from a high second-order rate constant in experiment.10 Following this, Compton et al. studied the EC0 mechanism theoretically and experimentally at the channel electrode13,14 and at the rotatingdisk electrode.1517 Additionally, Dimarco et al.3 studied the second-order EC0 reaction on a planar surface in some detail, providing insight into the appearance of split-wave voltammetry. The purpose of the present paper is to examine the EC0 mechanism in detail both at isolated hemispherical particles and arrays of such particles, and to explore the split-wave phenomenon

Figure 1. Schematic of the EC0 catalytic reaction mechanism.

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by characterizing the observed cyclic voltammetric behavior for a range of rate constants and particle surface coverages. 1.2. Arrays of Micro- and Nanoparticles. Throughout this study, it is assumed that the supporting surface itself is not electroactive for the range of potentials studied and therefore heterogeneous electron transfer occurs exclusively on the surface of the particles. The arrays are modeled using the well established diffusion domain approximation .18,19 Electrodes modified with particles with radii in the micrometer range can display a number of well documented20,21 experimentally useful properties that are not observed in the bulk material, including enhanced signal-to-noise ratio, suppression of charging current, and enhanced mass transport effects arising from the convergent nature of diffusion to microparticles.20,21 These benefits can be expected to be achieved to an even greater extent with particles in the nanometer range.22 A detailed review of the properties and uses of nanoparticles in relation to electroanalysis was given by Welch23 and has recently been updated by Campbell.24 In addition, Compton et al. have presented an overview25 of the design, fabrication, characterization, and applications of arrays of nanoelectrodes. Because of the high rate of mass transport to the individual particles, well-separated micro- and nanoparticles can display a high sensitivity,24 and electrodes modified with sparse distributions of nanoparticles are often used for this reason. Conversely, for arrays of micro- and nanoparticles that are somewhat denser, the current response approximates that observed for a macroelectrode of the same total geometric area26,27 despite only partial coverage by electroactive material. This allows expensive catalytic materials (gold, platinum, etc.) to be used more sparingly while still achieving the same results, thus lowering the electrode cost .23

Figure 2. Cyclic voltammograms for an isolated system with σ = 1000, cX = 1, and a variety of rate constants, k2. 11205

dx.doi.org/10.1021/jp2023204 |J. Phys. Chem. C 2011, 115, 11204–11215

The Journal of Physical Chemistry C

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Figure 3. (a) Cyclic voltammograms for an isolated system with σ = 1000, cX = 1, and a variety of rate constants, k2. (b) Integrated current response for the same.

Figure 4. (a) Regularly distributed hemispherical particles on a planar supporting surface, (b) coordinate system and diffusion domain for a spherical particle, (c) spatial mesh for a hemispherical particle, and (d) regular ΔΦ increments used to create a contracting R mesh across the electrode.

Recently, with regards to simulation of such systems, Davies et al. have examined the voltammetry of regular28 and random29 distributions of microdisc electrodes, Streeter has investigated the diffusion-limited current to isolated nanoparticles of a variety of geometries,30 and Belding has explored the problem of randomly distributed arrays of disk31 and spherical nanoparticles.26

This paper aims to extend the existing knowledge in this field through a detailed examination of a more complicated electrochemical reaction, namely the electrocatalytic reaction. 1.3. Diffusional Modes. Diffusion of material to the particle array may be divided into four limiting categories (Figure 5) which depend on the experimental time scale as well as the 11206



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Figure 5. The four categories of diffusion.

Table 1. Dimensionless Parameters parameter

Table 2. Boundary Conditions normalization

boundary

condition τ e 0, all R, Z R > 1, Z = 0

radial coordinate normal coordinate

R = r/re Z = z/re

time

τ = Dt/r2e

axial symmetry

R=0

∂c/∂R = 0

scan rate

σ = (F/RT)(r2e /D)ν

bulk (isolated)

R f ¥, Z f ¥

* = CN * /CA* cN = cN

potential

θ = (F/RT)(E EΘ f )

domain (array)

R = Rdomain

∂c/∂R = 0

concentration of species N

cN = CN/CA*

electrode surface

R2 þ Z2 = 1

Nernst

electrode flux

j = i/(nFDCA* re)

second-order rate constant

k2 = K2CA* r2e /D

initial conditions insulating plane

cN = cN * = CN * /CA* ∂c/∂Z = 0

The total mass transport equation for each species is then, particle surface coverage.1,26,27 Category 1 describes diffusion that is planar to the microparticle surface and is observed at short times, e.g., with a fast scan rate in a voltammetry experiment. Category 2 occurs at longer times when the diffusion to the particle is convergent rather than linear. In both cases (1 and 2), adjacent particles are sufficiently far enough apart that they are diffusionally independent of the experimental time scale. In category 3, the diffusion layers of neighboring particles begin to overlap but are still small in size relative to the interparticle separation distance. Finally, for category 4, the diffusion fields strongly overlap such that diffusion to the entire surface is essentially planar. As either the time scale of the experiment increases or the particle separation decreases, the diffusional mode may transition from category 1 to category 4.

2. THEORETICAL MODEL 2.1. Mass Transport and Boundary Conditions. Hemispherical particles supported on a planar electrode surface, as shown schematically in Figure 4a), may be described using the (r,z,φ) cylindrical coordinate system, as shown in Figure 4b). While physically the particles are three-dimensional, the problem of diffusion to the particle is a two-dimensional one as the particles have angular (φ) isotropy: only a 2D ‘slice’ of the electrode needs to be considered in order to obtain a complete solution. The simulation model is normalized according to the set of dimensionless parameters shown in Table 1. In dimensionless units, Fick’s second law is,

Dcdiff D2 c 1 Dc D2 c þ ¼ 2þ DR R DR DZ2 Dτ

ð2.1Þ

DcA DcA;diff ¼ þ k2 cB cX Dτ Dτ DcX DcX;diff ¼ k2 cB cX Dτ Dτ

DcB DcB;diff ¼ k2 cB cX Dτ Dτ DcP DcP;diff ¼ þ k2 cB cX Dτ Dτ

ð2.2Þ ð2.3Þ

where k2 is the dimensionless second-order rate constant of the homogeneous reaction shown in eq 1.2 and as defined in Table 1. These equations are discretized to a two-dimensional mesh of spatial points using the alternating direction implicit (ADI) method. As the resulting set of equations are nonlinear, it is solved at each time step using the iterative NewtonRaphson method, subject to the boundary conditions described in Table 2. Additionally, for the electroactive surface of the particle, Nernstian kinetics are used as the boundary condition describing the process in eq 1.1. To simulate cyclic voltammetry, the potential, θ, is swept from an initial value, θi, to a more reducing potential followed by a reverse sweep, such that for the forward sweep, the value of θ at any time is given by, θ ¼ θi þ στ

ð2.4Þ

where σ is the dimensionless scan rate as defined in Table 1. It is assumed that the electrode kinetics are fast and the diffusion coefficients of all species are equal. 2.2. The Dimensionless Rate Constant. The dimensionless second-order rate constant, k2, is defined as:

k2 ¼

CA re2 K2 D

ð2.5Þ

where C*A is the initial bulk concentration of species ‘A’, re is the microparticle radius, and D is the diffusion coefficient which is equal for all species. For a typical diffusion coefficient of 106 cm2 s1 and concentration of 10 mM, the relationship 11207



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between k2 and K2 for a range of particle radii is as shown in Table 3. The smaller a particle is, the higher the rate constant needs to be in order to observe split wave behavior. The maximum attainable diffusion controlled rate constant is on the order of 1010 dm3 mol1 s1; for an electroactive particle with a radius of 100 nm, this corresponds to a dimensionless rate constant of k2 = 104. 2.3. Spatial Grid. A schematic of the discretized spatial grids used for hemispherical particles is shown in Figure 4c. In normalized space, the surface of a hemisphere is described by R 2 þ Z2 ¼ 1

where

Z>0

ð2.6Þ

Furthermore, for all points on the surface, Z ¼ R tan Φ

ð2.7Þ

where Φ is the angle in radians measured from the Z axis in the (R,Z) plane. Consequently, 1 R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ tan2 Φ

ð2.8Þ

A spatial grid of successive R points across the hemispherical particle may thus be generated by specifying an angle increment, ΔΦ, and then determining R at each increment, with the same spacing also used in the Z direction, as shown in Figure 4d). The Table 3. Rate Constants re

k2/K2

10 nm 100 nm

108 mol dm3 s 106 mol dm3 s

1 μm

104 mol dm3 s

10 μm

102 mol dm3 s

100 μm

1 mol dm3 s

grid then expands from the particle surface (R = 1; Z = 1) to the bulk solution or edge of the diffusion domain as appropriate according to: ΔRi ¼ γΔRi1

ð2.9Þ

where γ and ΔR0 can be chosen for an appropriate level of accuracy. In the case of a diffusion domain, the singularity at the edge of the domain requires that there be a greater mesh density in this region. Therefore, such a mesh will be required to expand both from R = Relectrode and from R = Rdomain into the center of the diffusion domain. 2.4. Arrays of Particles. The Diffusion Domain Approximation. For the case of an isolated particle, the particle is assumed to ‘sit’ on a plane which is infinite in extent. A bulk solution condition, c = c*, is implemented at a distance of 6(τmax)1/2 from the particle surface, where τmax is the total dimensionless time that the scan takes. This value is sufficient to exceed the diffusion horizon. For an array, the insulating surface is split into a regular array (either cubic or hexagonal) of cells, each containing one particle as shown in Figure 6a and 6c. The walls of each cell are equidistant from their nearest neighboring particles, and so at all times during a voltammetric experiment, zero net reactant flux will pass through them. Consequently, each cubic or hexagonal cell is diffusionally independent. This means that in order to simulate an array, one need only simulate a single cell (containing a single particle); the current response may then be scaled by the total number of such particles to give an accurate account of the voltammetry of the whole surface. Unfortunately, this is a three-dimensional simulation problem and is therefore very computationally demanding, such that performing a series of accurate simulations would currently be feasible only on a supercomputer.

Figure 6. The diffusion domain approximation for cubic and hexagonal arrays of particles. 11208



Figure 7. Evolution of surface concentrations of species ‘A’ and ‘X’ with overlaid linear sweep voltammetry for an isolated system with σ = 1000; k2 = 107; cX = 1. Note that LSV is scaled in the vertical axis for clarity.

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Figure 9. Steady-state voltammograms for a system with σ = 0.001, k2 = 107, and a variety of concentrations, cX.

the results to be in good agreement. The diffusion domain approximation has also been used to simulate a variety of other cylindrically symmetric electroactive particles including microcones,33 microcylinders,34 and spheres on a surface .35,26 It should be noted that the diffusion domain approximation is not appropriate for particles on the edge of a finite array.36 Therefore, to ensure accurate results, the number of particles in the array should greatly exceed the number of particles on the perimeter. The approximation is also not appropriate in the case where the particle size varies significantly, because the symmetry which the approximation relies upon is lost .19,37 Therefore, it is assumed that all of the particles are of uniform size. The density of the particles may be defined in terms of the surface coverage, Θ: Θ¼ Figure 8. Cyclic voltammograms for a system with σ = 1000, k2 = 107, and a variety of concentrations, cX.

To avoid this problem, a useful simplification may be employed: the diffusion domain approximation. Under this approximation, which was first introduced in papers by Reller18 and Amatore,19 the cubic or hexagonal cells are substituted with circular cells of exactly the same area, as shown in Figure 6b and 6d. Hence, the problem is reduced to a two-dimensional one, as the simulation space for each identical particle is a radially symmetric cylindrical unit cell (a diffusion domain). This differs from the isolated case only in terms of the outer boundary condition: in the isolated case, the space is semi-infinite, whereas in the array the space extends only as far as Rdomain and a zero-flux boundary condition is employed. From Figure 6, it is clear that the hexagonal packing provides a better fit to the diffusion domain approximation. Work by Brookes et al.32 examined simulations of both geometric models (cubic and hexagonal) using partially blocked electrodes, the inverse of the situation described in this paper, and found both models to be in reasonable agreement with experiment, though the hexagonal model provided a slightly better fit, as expected. Additionally, Davies et al.28 compared simulations of cubic arrays of microelectrodes with experiment and found

2 πRelectrode 2 πRdomain

ð2.10Þ

The (dimensionless) diffusion domain radius is therefore: pffiffiffiffiffiffiffiffiffi Rdomain ¼ 1=Θ ð2.11Þ Although random (as opposed to regular) distributions of particles have been studied in the past for a simple one-electron transfer,26,31 the relative complexity of the EC0 mechanism, along with the range of variables (K2, σ, Θ) under investigation here, renders such a study computationally unfeasible in this case. Regardless, it is not expected that the differences between the two distributions will lead to any significant deviation in the general results. 2.5. Flux Calculations. The current in amps, i, is related to the dimensionless flux, j, of species ‘A’ through the electrode (as in Table 1), which, by Fick’s first law, is equal to the concentration gradient at the particle surface. As the simulation grid is rectangular, the flux at each point on the surface is found by summing its components in the R and Z directions. The dimensionless flux density, J, is defined as, J ¼ 11209

DcA DcA cos Φ þ sin Φ DR DZ

ð2.12Þ



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Figure 10. Cyclic voltammograms of isolated systems with k2 = 5 104, cX = 1, and a variety of scan rates, σ.

where Φ is as defined in section 2.3. The total flux, j, is found by integration of J over the electrode surface: Z π=2 ð2.13Þ Jcos ΦdΦ j ¼ 2πN π=2

where N is the number of particles. 2.6. Computation. Simulations were performed with a Dell Precision T5500 with two quad core hyperthreaded Intel Xeon E5520 processors (16 logical cores, 2.23 GHz). Simulations were written in Cþþ using Microsoft Visual Cþþ 2008 with OpenMP for multithreading. The alternating direction implicit (ADI) method which is used for two-dimensional simulations is particularly amenable to parallelization: at each time step, multiple calculations may be carried out simultaneously (provided the machine has multiple processors), which can decrease runtime considerably. However, because of the coupling of the mass transport equations, the convergence demands38 for simulation of the EC0 mechanism can be quite high. A relatively dense spatial grid (compared to, for example, simulation of a one-electron transfer at a microdisc) must be used in order to ensure accurate results, with a higher rate constant, k2 necessitating a denser grid. For lower rate constants (1103), ADI simulations took 10 min to 1 h; however, for values of k2 = 106, runtime could be in excess of 6 h for a single simulation, even on a machine with 16 logical cores.

3. RESULTS AND DISCUSSION 3.1. Isolated Particle. For the case of an isolated hemispherical particle on a surface, the system is angularly isotropic and so simulations may be reduced to one spatial coordinate: the radial coordinate r. One-dimensional (1D) simulations compute relatively rapidly, with runtimes on the order of seconds rather than hours which two-dimensional simulations require. Approximately 5000 1D simulations were performed by varying the dimensionless scan rate, σ, in the range 103 to 104, the secondorder rate constant, k2, in the range 11010, and the concentration of species X, cX, in the range 110. 3.1.1. The Effect of k2: The Split Wave. As described in the Introduction, a high second-order rate constant can lead to voltammetry with a split wave. Figure 7 shows the evolution of the concentrations of species ‘A’ and ‘X’ at the particle surface overlaid with the forward sweep of the cyclic voltammogram. It can be seen that the first peak corresponds to the transition from kinetic to diffusion control of species ‘X’ whereas the second corresponds to the same transition for species ‘A’. In the same manner as the case of a planar surface,17 it may be seen that the shape of the voltammogram is determined by the superposition of these two transitions: if they occur on a similar time scale, then only one peak is observed whereas when the ‘X’ transition is made more rapid (by increasing k2), two distinct peaks may be observed. As an intermediate case, the ‘X’ peak may appear as a 11210



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Figure 11. Variation of first peak current (a and b) and potential (c and d) with cX and k2 for an isolated system with σ = 1000.

Figure 12. Variation of first peak current with σ. The dotted line indicates the apparent high scan rate limit.

shoulder to the ‘A’ peak; it may be described qualitatively, but there is no quantifiable peak current that can be measured. This behavior is in good agreement to that which is observed in the case of a planar electrode .3 3.1.2. The Effect of cX. It can be seen in Figure 8 that for a given rate constant, increasing the concentration of ‘X’ leads to a substantial increase in the height of both peaks but also that the first increases far more than the second, totally obscuring it at

Figure 13. Cyclic voltammograms of systems with σ = 1, k2 = 1000, and a variety of surface coverages, Θ. Peak heights decrease with increasing surface coverage, and split wave behavior becomes resolvable.

higher concentrations of ‘X’ for the reasons justified in section 3.1.1. Such a large current response is possible even though the potential is significantly below E0 (θ = 0) because the continual depletion of ‘B’ and production of ‘A’ in the chemical step drives the equilibrium at the electrode surface more in favor of ‘B’ by Le Chatelier’s principle (assuming the A/B redox couple is electrochemically reversible). This feedback means that as soon as the 11211



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Figure 14. Concentration profiles of species ‘A’ taken at the end (θ = 10) of the forward potential sweep of a CV for systems with k2 = 10: (a and b) σ = 1000, Θ = 0.1; (c) σ = 100, Θ = 0.5; (d) σ = 1, Θ = 0.7. Panels a and b show the same profile for a diffusionally isolated (case 2) particle in one (radial) and two (R,Z) dimensions, respectively. Panels c and d show case 3 and case 4 behavior, respectively. Hashed areas represent the electrode.

Figure 15. Forward sweep peak-to-peak separation for systems with a variety of surface coverages, rate constants, and (a) σ = 0.1, (b) σ = 1, (c) σ = 10, and (d) σ = 100. 11212



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Figure 17. Cyclic voltammograms for systems with σ = 10, k2 = 106, and a range of surface coverages, Θ.

Figure 16. Variation of the ratio of forward sweep peak heights, jpeak,A/ jpeak,X, with scan rate and diffusion domain radius Rdomain (= (1/Θ)1/2). for systems with k2 = 105. Similar trends are observed for other values of k2 assuming peak resolution is possible.

potential reaches a level where the electrochemical reduction of ‘A’ to ‘B’ is kinetically viable, species ‘X’ will be rapidly consumed in the vicinity of the electrode and a sharp current peak will be observed. 3.1.3. The Effect of Scan Rate and Steady-State Voltammetry. For a simple E process at a hemispherical particle, a very low scan rate (σ e 103) typically gives steady-state cyclic voltammetry because diffusion to the particle is convergent. Similar behavior is also observed under the EC0 mechanism; however, instead of the characteristic sigmoid normally associated with steady-state voltammetry, a ‘double-sigmoid’ is observed (for high k2) with the first plateau being associated with the depletion of species ‘X’ and the second with that of species ‘A’ as per section 3.1.1. Figure 9 depicts a series of steady-state voltammograms for increasing concentrations of ‘X’. As the limiting current is due to the diffusion of both ‘A’ and ‘X’, the value measured is equivalent to that in an E system with one electroactive species with initial concentration c = cA þ cX (assuming equal diffusion coefficients). Note that in the dimensionless unit system used in this investigation, this limiting current is simply equal to the sum of the concentrations of ‘A’ and ‘X’ as demonstrated in Figure 9. From Figure 10, it is clear that for a split wave system, as the scan rate is increased, the peak to peak separation (of the forward scan peaks) tends to decrease, as is observed in a planar system.3 This is because (a) the first peak moves toward a more negative potential with decreasing scan rate because the scan takes a longer time and ‘X’ is therefore depleted at a lower potential, and (b) the second peak moves to a more positive potential with decreasing scan rate as expected for a hemispherical electrode.39 An interesting consequence of this peak movement is that for a certain range of rate constants, k2, the voltammogram will transition from a single peak to a split wave as the scan rate is decreased, as shown for the example of k2 = 5 104 in Figure 10. Of course as the system moves to very low scan rates (σ e 0.001), the cyclic voltammograms tend toward a more sigmoidal shape, and thus no peaks are observed at all. For the range of

isolated particle systems studied, it was not generally possible to resolve the two peaks at any value of k2 < 3 104 for any scan rate. For high scan rates (σ g 100), resolution was generally only possible for k2 > 106. Consequently, for a typical system as described in section 2.2, split wave voltammetry is likely to be observed only for isolated particles with radii in excess of 10 μm. 3.1.4. Expressions for Peak Potentials and Currents. Figure 11 displays the variation of the first (or ‘only’ for cases where the peaks cannot be resolved) peak current and potential for variety of values of cX and k2 at a fixed scan rate (σ = 1000). For cases where k2 < 106, the two peaks are generally not resolvable so the peak characteristics (Figure 11 a 11c) are determined by their superposition and are thus difficult to predict. However, as can be seen in Figure 11 b and 11d, when k2 > 106 (i.e., when peak resolution is possible), relatively simple relationships exist between k2, cX, and peak current and potential. From careful analysis of the simulated data, the relationship for θ is deduced to be of the form: θpeak ¼ ðR1 log k2 þ R2 Þlog cX R3 log k2 þ R4

ð3.1Þ

where the coefficients, R1, R2, R3, and R4 depend on the scan rate. For a scan rate of σ = 1000, this relationship is found to be: θpeak ¼ ð0:02 log k2 þ 1:01Þlog cX 1:18 log k2 þ 4:20 ð3.2Þ For all simulated cases (for σ = 1000; k2 > 10 ; 1 e cX e 10), this expression was accurate to within 1%. Where the peaks are sufficiently separated (k2 > 5 106), the peak current is approximately independent of the value of k2 and is simply, 6

jpeak ¼ βcX

ð3.3Þ

where β depends on the scan rate as shown in Figure 12. In the limit of fast scan rate, pffiffiffi jpeak ¼ 0:62cX σ ð3.4Þ in the same manner as the well-known RandlesSevcik equation .40,41 3.2. Particle Arrays. When particles have sufficiently small interparticle separation, they are no longer diffusionally independent of the experimental time scale. Consequently, the 11213



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Figure 18. Cyclic voltammograms for systems with k2 = 104, Θ = 0.6, and a variety of scan rates, σ. Similar behavior is seen for higher surface coverages.

simulation space is no longer spherically symmetric and a simple one-dimensional simulation is no longer appropriate. Using the two-dimensional ‘alternating direction implicit’ (ADI), a range of simulations were performed, varying the scan rate, σ, from 0.1 to 1000, the surface coverage, Θ, from 0.1 to 0.9, and the secondorder rate constant, k2, from 1 to 106. 3.2.1. General Observations. In a manner consistent with previous studies on particle arrays,31 as the size of the diffusion domain decreases, the peak current of all peaks decreases substantially as adjacent diffusion fields overlap and material in the diffusion domain is depleted as evidenced in Figure 13. Figure 14 shows concentration profiles of species ‘A’ for several systems. It can be seen that decreasing the scan rate and increasing the surface coverage leads to a change in diffusional behavior, as described in section 1.3, from Case 2, in which diffusion is convergent to each particle, to Case 4, in which diffusion is linear to the whole surface. 3.2.2. Surface Coverage. As Θ is increased, the amount of time for which radial diffusion dominates is reduced; as a CV scan progresses, the mode of diffusion transitions from case 2 to case 3, and then possibly to case 4. This transition will occur sooner for a smaller diffusion domain, and so the amounts of ‘A’ and ‘X’ available to the electrode through radial diffusion are reduced. Consequently, both peaks move to a more negative potential since material is depleted more rapidly. In section 3.1.3, it was seen that for an isolated particle, decreasing the scan rate while keeping all other parameters fixed could lead to a transition from single peak to split peak behavior. Figure 13, which shows cyclic voltammetry for a system with a low scan rate (σ = 1) and a range of surface coverages, demonstrates that increasing Θ can also lead to a similar transition. In this example, this splitting is attributable to two separate factors. The first is the tendency for Δθ (the peak-to-peak separation of the

peaks of the forward sweep) to increase with increasing surface coverage. As previously stated, increasing Θ results in a shift of both peaks to more negative potential; however, as shown in Figure 15, for a low scan rate, Δθ increases with increasing Θ because the ‘X’ peak tends to shift more than the ‘A’ peak. It should be noted that at higher scan rates this may not necessarily be the case. The second factor is the change in relative heights of the two peaks. In the example of Figure 13, it is obvious that as the surface coverage is increased, the height of the second peak, jpeak,A, decreases relative to that of the first, jpeak,X. From Figure 16, it can be seen that for a given scan rate and rate constant, the ratio of the peak heights, jpeak,A/jpeak,X, tends to increase with increasing diffusion domain radius (decreasing surface coverage), and tends toward some limiting value as Rdomain f ¥ (i.e., as the particle becomes isolated). If jpeak,A is sufficiently large, it will obscure jpeak,X and no split-wave behavior will be observed. A similar variation in jpeak,A/jpeak,X with σ can be seen in Figure 17. As a consequence of these two factors, split-wave behavior in an array of particles may be seen to occur for substantially lower rate constants, k2, than are possible in the isolated particle case. Therefore, following the discussion in section 2.2, arrays of particles may display split-wave behavior with particle radii in the 10100 nm range, and such behavior will be easily accessible in the micrometer range. 3.2.3. Scan Rate. For arrays of particles, a change in scan rate has an effect similar to that seen in the isolated case. As shown in Figure 18, peak separation becomes more apparent as the scan rate is decreased, and there may be a transition from a single peak to a split wave. The main difference is that as the scan rate is decreased, there is no tendency toward steady-state behavior (i.e., sigmoidal cyclic voltammograms) for the case of an array. This difference can be seen by comparing Figure 18 with 11214


The Journal of Physical Chemistry C Figure 10. As the scan rate is decreased, the scan takes more time and thus case 4 diffusional behavior is seen. As discussed in section 3.2.2, this means that radial diffusion to the particles can only continue for a finite time; when the diffusion fields overlap, the diffusion becomes planar and no steady-state voltammetry can be observed.

4. CONCLUSIONS The effect of a number of parameters, namely scan rate, σ, rate constant, k2, surface coverage, Θ, and concentration of species ‘X’, cX, on the cyclic voltammetry of the EC0 mechanism on particle-modified electrodes has been investigated. The occurrence of resolvable split wave behavior is dependent upon a number of competing factors. First, a high rate constant, k2, ensures that the presence of species ‘X’ makes a substantial contribution to the voltammogram (i.e., a large current response), as well as ensuring that the peaks are well separated. Related to this is the particle size: for an isolated particle, it will be possible to observe split waves in voltammetry of particles with a radius on the order of 10 μm, but not significantly lower, whereas for a particle array, a split wave may be observable for particles in the 10100 nm range. A low scan rate causes the peak separation, Δθ, to increase, making resolution easier, though for the isolated case, a very low scan rate leads to steady-state behavior where no peaks are observed. Additionally, the concentration of ‘X’ must be similar to that of ‘A’; too low and little current response is seen, too high and jpeak,X will completely obscure jpeak,A. Finally, a high surface coverage generally leads to better peak resolution as jpeak,A/jpeak,X becomes closer to unity and Δθ increases. It is hoped that the insights offered by this work will aid in the analysis and deconvolution of experimentally obtained cyclic voltammetry of the EC0 mechanism. ’ AUTHOR INFORMATION Corresponding Author

*Tel: þ44 (0) 1865 275413. Fax: þ44 (0) 1865 275410. E-mail: [email protected].

’ ACKNOWLEDGMENT K.R.W. thanks Schlumberger Research Cambridge (SCR) for funding. ’ REFERENCES (1) Compton, R. G.; Banks, C. E. Understanding Voltammetry, 2nd ed.; ICP: London, 2010. (2) Jeroschewski, P.; Haase, K.; Trommer, A.; Gruendler, P. Electroanalysis 1994, 6, 769–72. (3) Dimarco, D. M.; Forshey, P. A.; Kuwana, T. ACS Symp. Ser. 1982, 192, 71–97. (4) Britz, D. Int. J. Electrochem. Sci. 2006, 1, 1–11. (5) Feldberg, S. W.; Campbell, J. F. Anal. Chem. 2009, 81, 8797–8800. (6) Harriman, K.; Gavaghan, D. J.; Suli, E. J. Electroanal. Chem. 2004, 569, 35–46. (7) Andrieux, C. P.; Dumas-Bouchiat, J. M.; Saveant, J. M. J. Electroanal. Chem. 1978, 87, 39–53. (8) Andrieux, C. P.; Dumas-Bouchiat, J. M.; Saveant, J. M. J. Electroanal. Chem. 1978, 87, 55–65. (9) Andrieux, C. P.; Dumas-Bouchiat, J. M.; Saveant, J. M. J. Electroanal. Chem. 1978, 88, 43–8.

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Cyclic Voltammetry of the ECâ² Mechanism at Hemispherical Particles

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