Electrocatalysis at Microelectrodes: Geometrical Considerations

Jul 22, 2010 - starting with a disk (zero curvature) followed by a truncated ... number, ξo ) (k0)/(DOx)((A)/(4π))1/2 ) 2.5 × 10-2. For a ..... whe...
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J. Phys. Chem. C 2010, 114, 13650–13656

Electrocatalysis at Microelectrodes: Geometrical Considerations Huanfeng Zhu,† Yuriy V. Tolmachev,‡ and Daniel A. Scherson*,† Department of Chemistry, Case Western ReserVe UniVersity, CleVeland, Ohio 44106-7078, and Department of Chemistry, Kent State UniVersity, Kent, Ohio 44242 ReceiVed: March 5, 2010; ReVised Manuscript ReceiVed: May 29, 2010

The influence of the shape of individual electrocatalytically active, axisymmetric particles supported on an inactive substrate, on the current associated with heterogeneous redox reactions involving solution-phase species has been examined theoretically. In agreement with previous work, the diffusion-limited currents, ilim, for a disk and supported truncated spheres of the same area including a full sphere are very similar (within ca. 10%). Large enhancements in the predicted values of ilim were found, however, for spheroids of the prolate type, as the aspect ratio was increased. Analyses of arrays of electrocatalytically active disks embedded in a planar inactive support in the presence of a diffusion boundary layer of well-defined thickness, for example, a rotating disk electrode, revealed that values of ilim very close to those expected for fully catalytic surfaces could be achieved for disk coverages on the order of 1%. Similar conclusions were made for corresponding arrays of small disks embedded in microdisks or spherical inactive supports in quiescent media. This effect might explain observations made for iron porphyrins adsorbed on graphite surfaces in aqueous electrolytes, for which the experimentally determined onset for oxygen reduction is found to be at potentials well ahead of the onset of the voltammetric wave associated with the conversion of the inactive (ferric) to the active (ferrous) form of the macrocycle. Introduction Heterogeneous electron transfer reactions at electrode|electrolyte interfaces are often sensitive to the nature of the electrode material1,2 and, in the case of specific metals, to the crystallographic plane1-4 exposed to the electrolyte, which may not only affect reaction rates but also mechanistic pathways. Not surprisingly, the factors that control morphological aspects not only of metals,4,5 but also of metal oxides6,7 and other materials,8,9 are currently being thoroughly investigated. Of particular interest is the development of methods to grow welldefined particles10-13 of characteristic dimensions in the micrometer to nanometer scales in an effort to increase the surface to volume ratio and thus make better utilization of often scarce and expensive materials.14,15 From a general perspective, the overall rates of electrochemical processes are not only controlled by the intrinsic kinetics of electron transfer and other chemical steps, but also by the rates of mass transport.16 In this regard, forced convection techniques17-19 with well understood hydrodynamics characteristics, such as the rotating disk electrode (RDE)20-23 and its variant, the rotating ring disk electrode (RRDE),1-3,24 have been extensively employed in the study of reactions of both fundamental and technological interest. Analysis of data collected with this type of electrodes allows mass transport effects to be accounted for in a relatively straightforward fashion, making it possible to elucidate reaction mechanisms, and to determine values of rate constants and other parameters of kinetic relevance. Yet, another means of increasing mass transport is by reducing the size of the electrode. In fact, calculations show that for disk electrodes embedded in an insulating surface of a radius on the order of micrometer, the rates of mass transport * Corresponding author. E-mail: [email protected]. † Case Western Reserve University. ‡ Kent State University.

are equivalent to those achieved by conventional RDE at rotation rates on the order of several thousand rpm.25 Indeed, implementation of ultramicroelectrode-based tactics has made it possible to measure the rates of very fast electron transfer rates not accessible by other more conventional methodologies.26 This contribution is concerned with the effects of the shape of axisymmetric electrocatalytically active particles of characteristic dimensions in the micrometer to nanometer scales on the current-potential or polarization curves of redox reactions of the form Ox + ne f Red, where Ox and Red are solutionphase species, under strict steady-state conditions. Comparisons have been made for particles of different shape, but constant area, assuming first-order kinetics in the solution-phase reactant and, unless otherwise noted, neglecting the back reaction. Of particular interest was to determine whether changes in the geometric parameters of electrodes of a prescribed area can lead to gains in overall electrocatalytic performance. Some aspects of this problem, although from a different viewpoint, have been addressed by Myland and Oldham,27 who presented analytical solutions for diffusion-limited steady-state currents to a variety of axisymmetric microelectrodes resting on an insulating planar surface, including sphere-caps28 and a more general class of hemispheroids.27 More recently, Compton et al.29 employed strictly numerical methods to compare the linear voltammetric response of particles of various shapes. Also examined in our work are the effects of the coverage of electrocatalytic disks dispersed on either planar or spherical inactive supports on the observed polarization curves. Theoretical Aspects All simulations in this study were performed using judiciously selected dimensionless variables (see Table 1) within the COMSOL 64 bit platform on a laptop (Hewlett-Packard, Pavilion dv6). Because the primary goal of this work relates to the effects of the shape, the area of the electrocatalytically active

10.1021/jp102011t  2010 American Chemical Society Published on Web 07/22/2010

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TABLE 1: Governing Differential Equations, Boundary Conditions, and Dimensionless Variables for Mass Transport to a Truncated Sphere Supported on an Unreactive Planar Substrate dimensioned variables

dimensionless variables

cOx cbulk

I.1

r z ,Z ) rs rs

I.2

cOx

COx )

r,z

R)

∇2cOx

( )

Governing Differential Equation

∂2cOx 1 ∂ ∂cOx ≡ r + ≡0 r ∂r ∂r ∂z2

∇2COx )

(

)

∂2COx ∂2COx 1 ∂COx + ≡0 +R R ∂R ∂R2 ∂Z2

I.3

Boundary Conditions

∂cOx(r, z) DOx ∂r ∂cOx(r, z) DOx ∂z

|

r)0

|

z)0

Boundary I (Axial)

) 0, 0 < z < zmax

∂COx(R, Z) ∂R

) 0, 0 < Z < Zmax

I.4

) 0, 0 < R < Rmax

I.5

COx ) 1, Z ) Zmax, 0 < R < Rmax

I.6

Boundary II (Insulator)

) 0, 0 < r < rmax

cOx ) cbulk, z ) zmax , 0 < r < rmax cOx ) cbulk,r ) rmax , 0 < z < zmax

|

∂COx(R, Z) ∂Z

R)0

|

Z)0

Boundary III (Infinite)

Boundary IV (Infinite)

COx ) 1, R ) Rmax, 0 < Z < Zmax

I.7

Boundary V (Electrode Surface)

j ) -DOx∇cOx · nˆ ) -k0ce-Rnfη

J ) -∇COx · nˆ ) -ξC k0 DOx COx(R, Z) ) 0

where ξ ) cOx(r, z) ) 0

electrodes, denoted as A, was kept fixed. To this end, dimensionless distances, when appropriate, were defined using rs, that is, the radius of a (full) electrocatalytic sphere, as a normalizing factor (see eq I.2 in Table 1). The relative sizes of the domain as well as of the mesh were changed until the results obtained did not vary by more than a fraction of a percent. An assessment of the accuracy of the calculations was made by comparing the results with analytical ones, when available, yielding differences on the order of a percent. This work has been organized into four sections: Section A presents polarization curves for geometrical shapes associated with a sphere emerging from a flat, electronically conducting, albeit nonelectrocatalytic surface, starting with a disk (zero curvature) followed by a truncated sphere through a hemisphere and ending with a full sphere sitting on the flat surface. Data are also shown for a full sphere as a function of the distance from a planar surface to assess the effects of shadowing on the steady-state currents. Section B focuses on particles of other shapes, including spheroids as a function of their aspect ratios. Section C presents results for arrays of active disks dispersed on a larger inactive disk or spherical surfaces to examine the effects of the coverage on the polarization curves with emphasis on the limiting currents. These concepts are extended in section D to account for redox active molecular electrocatalysts supported on inactive substrates. A. Polarization Curves for Truncated Spheres: From a Disk to a Full Sphere. The governing differential equations and boundary conditions for mass transport to a truncated or full sphere supported on an inactive infinite planar substrate assuming irreversible first-order kinetics in the solution-phase reactant, Ox, are listed in Table 1 in both dimensioned (left column) and dimensionless variables (right column). The

 4πA e

I.8

-Rnε

,ε )

η RT ,f ) f F

boundaries of the domain involved in the simulations are defined in Scheme 1. A convenient way of representing the results of our calculations is by defining a normalized dimensionless flux, HS,lim ), where IDim is the dimensionless integrated INor ) (IDim)/(IDim flux for a specific shape, IDim ) AA J · nˆ dA, where J is the surface flux of Ox and nˆ is a unit vector normal to the HS,lim is the corresponding infinitesimal area element dA, and IDim analytical dimensionless integrated flux for a hemisphere under HS,lim ) 2π2. strict diffusion control, that is, IDim Shown in Figure 1 are dimensionless polarization curves in the form of INor versus -ε, where ε ) (ηF)/(RT) ) (1)/(nR) ln ξo - (1)/(nR) ln ξ is a dimensionless overpotential (see eq I.8 in Table 1), obtained for a hemisphere (curve a), a full sphere (b), two shallow truncated spheres Rt ) 5 (c) and Rt ) 10 (d), where Rt is the actual radius of the truncated sphere divided by rs, and an embedded disk electrode (e), all of a common area A, for a rather arbitrary value of the (dimensionless) Sherwood number, ξo ) (k0)/(DOx)((A)/(4π))1/2 ) 2.5 × 10-2. For a spherical electrode of radius rs ) 5 × 10-5 cm, assuming a diffusion coefficient DOx ) 1.93 × 10-5 cm2/s, the ξo selected would correspond to a relatively slow redox process, that is, k0 ) 10-2 cm/s. This representation makes it possible to compare the magnitudes of INor for particles of various shapes upon polarization at the same potential by simply drawing a vertical line at a specific ε and determining the points of intersection with each of the curves. Cursory inspection of these plots, where the solid curves are best fits to the calculated data (in scattered symbols), indicates that INor is virtually independent of the electrode shape in the region controlled by kinetics, that is, small overpotentials. As the overpotential increases, the diffusional contribution becomes more significant, imparting the curves their

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SCHEME 1: Schematic Diagram of the Domain Used in the Simulations Showing the Various Boundaries Involved (See Table 1 for Boundary Conditions), Where the Dimensions Have Been Altered for Clarity

Figure 2. Plot of Ilim Nor vs ζ for the geometries shown in the inset, where the horizontal dashed line represents the limiting current for an lim for the sphere unsupported sphere, where the arrow represents the INor sitting on the surface (see arrow in the inset).

sigmoidal character, to reach, for very negative ε, a diffusionlimited plateau (see below). Based on these results, the hemisphere (curve a) and the disk (scattered points, curve e) yield the largest and smallest values of INor, respectively, with the supported sphere (curve b) somewhere in between.

Figure 1. Plots of INor vs -ε, for various particle shapes of the same area a hemisphere (curve a), sphere (b), two shallow truncated spheres, that is, Rt ) 5 (c) and Rt ) 10 (d) (see text for details), and a disk (e) assuming k0 ) 10-2 cm/s, D ) 1.93 × 10-5 cm2/s, rs ) 5 × 10-5 cm, R ) 0.5, f ) 38.94 V-1, and n ) 1, where the solid curves represent best fits to the calculated points. Also shown in this figure is the simulated (COMSOL) polarization curve (half filled symbols in curve f) for a much faster redox reaction, that is, k0 ) 1 cm/s (see dashed line, curve f). The dashed curves for the disks were calculated on the basis of expressions proposed by Oldham et al.30 (see text for details).

The accuracy of these simulations was assessed by comparing the values obtained for the polarization curve for the microdisk (see scattered points, curve e) with those determined on the basis of an expression proposed by Oldham30 et al. (following normalization) for irreversible processes under steady-state lim HS,lim conditions, (Idisk)/(IHS,lim Dim ) ) (IDim)/(IDim ){1 + (π)/(κ)((2κ + 3π)/ 2 -1 (4κ + 3π ))} (see dashed line, curve e), where κ ) (πkfrd)/ (8DOx) ) (π)/(4)ξ, and rd is the diameter of the disk, for the same set of parameters. As may be inferred from the results, the agreement between the two methods is very good. Virtually overlapping curves were also obtained for a close to reversible reaction, that is, k0 ) 1 cm/s, which require for the Butler-Volmer term associated with the back reaction (not shown in Table 1) to be explicitly accounted for, using the corresponding applicable HS,lim lim ) ) (IDim )/ equation proposed by Oldham et al., (γIdisk)/(IDim HS,lim ){1 + (π)/(κγ)((2κγ + 3π)/(4κγ + 3π2))}-1 (see dashed (IDim line, curve f), where γ ) 1 + exp(-nε), assuming a common diffusion coefficient for both redox species. Shown in Figure 2 is a plot of diffusion-limited currents, Ilim Nor, versus the distance between the top of the sphere and the flat surface normalized by the radius of the sphere, rs, denoted as ζ lim for (see inset in this figure), where the arrow represents the INor the sphere sitting on the surface (see arrow in the inset). Note that in the case of a truncated sphere, ζ represents its height. As pointed out by Myland and Oldham,27 the steady-state diffusion-limited currents for this type of electrodes (except those to be discussed in the next section) are rather insensitive lim for to their shape. Also included in Figure 2 are values of INor the full sphere at various distances from the flat substrate. Not surprisingly, as ζ increases beyond 2 (see vertical arrows), the underside of the sphere becomes more accessible, and, consequently, the integrated flux approaches the value for a sphere in the open solution (see horizontal dashed line in Figure 2). In lim for the full sphere far away from the inert substrate is fact, INor more than 40% larger than that of the same sphere sitting on the plane, indicating that the blocking effects are quite substantial. It may thus be concluded that the most efficient geometry, that is, the highest diffusion-controlled current, would be that of a sphere supported on an infinitely thin (sets of) wire(s), which would provide both mechanical integrity and electrical conductivity. It should be noted that some of the geometries analyzed by Compton et al. using strictly numerical techniques have been examined by analytical methods earlier

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Figure 3. INor vs -ε for a sphere (large solid circles, curve A) and various oblate and prolate spheroids of difference aspect ratios, β ) 20 (a), 10 (b), 5 (c), 2 (d), 0.5 (e), 0.2 (f), and 0.1 (g) (see text for details), calculated using the same parameters as in the caption of Figure 1.

by other workers.28,31,32 In particular, the diffusion-limited current for a sphere sitting on a flat, inactive surface has been shown to be given by Ilim ) 4π(ln 2)nFDCrs, where the coefficient 4π(ln 2) ) 8.71 is precisely equal to that found by Compton by numerical techniques. B. Polarization Curves for Spheroids. Shown in Figure 3 are plots of INor versus -ε for a sphere (large solid circles, curve A) and various oblate and prolate spheroids supported on a flat surface of difference aspect ratio, β defined as the ratio of the polar, a, over the equatorial, b, lengths of the spheroid, that is, a/b, for prolates (β > 1) and oblates (β < 1) of exactly the same area calculated using the same parameters given in the caption of Figure 1. The data above and below curve A correspond to prolates and oblates, respectively. It follows from these results that an increase in β for prolates gives rise to enhancements in the current, whereas exactly the opposite is found for oblates due to the blocking effect, which in this case is far more pronounced than for the sphere above. It is interesting to note that for β ) 20, Ilim Nor is more than twice the corresponding value for the hemisphere and larger than that found for a fully unsupported sphere of the same area (see Figure 2). Obviously, larger values can be obtained as the aspect ratio is increased. An interesting means of assessing the blocking effect of the substrate on the limiting currents for the various spheroids is provided by defining a new parameter, Ξ ) 1 - χ, where χ is lim for a specific β in contact and infinitely far the ratio of INor away from the surface, shown in curves a and b, Figure 4 (left ordinate), respectively. The calculated values in curve b were found to be in excellent agreement with those determined from the analytical expressions derived by Bruckenstein and Janiszewska,33 which are twice those for hemispheroids supported on an insulated plane reported by Myland and Oldham.27 As expected (see plot of Ξ vs β in this figure, see curve c, right ordinate), this effect is most pronounced for small β (oblates) and decreases as β increases. For example, for β ) 0.1, Ξ ) 0.45, which means that about one-half the surface of the oblate is diffusionally blocked by the presence of the underlying support. lim initially increases as a As indicated in curve b, Figure 4, INor function of β, reaching a virtual plateau in the region 0.5 < β < 2, as was also noted by Compton et al. for hemispheroids.29 As lim is clearly evident from our results, however, INor begins to lim increase for β > 2 to reach β ) 20, values of INor ca. 35% larger.

lim Figure 4. Plots of INor vs β for different spheroids in contact with a large planar substrate (curve a) and infinitely far from the surface (b) (left ordinate). Curve c (see right ordinate) provides values of the blocking effect, Ξ, as a function of the aspect ratio β.

Similar conclusions were reached much earlier by Myland and Oldham for supported hemispheroids.27 In other words, it becomes possible to increase the limiting current without changing the area, an observation with potential relevance to practical applications. Also of interest from this latter viewpoint is to consider the effect on the limiting current upon changing the shape of the electrodes, keeping the volume as opposed to the area constant. Preliminary calculations based on the analytical solutions reported by Bruckenstein and Janiszewska33 have shown that the limiting current increases with β for hemispheroids of the prolate type, a behavior analogous to the constant area results described above. C. Electrocatalytic Disks Embedded in Planar and Spherical Noncatalytic Supports. The primary objective of this section is to seek correlations between the magnitude of the diffusion-limited currents and the coverage of catalytic inlaid disks of a fixed area dispersed onto the surface of a flat inactive support under conditions in which a diffusion boundary layer of well-defined thickness is formed by forced convection (large electrodes) or spontaneously (microelectrodes). Also to be examined is the behavior of disk arrays on spherical inactive supports, ignoring in the latter case effects due to the (local) microcurvature of the substrate. Some aspects of these problems

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Figure 5. Plot of If vs θ for σ ) 1 × 10-4 (curve a), 2 × 10-4 (b), and 4 × 10-4 (c) calculated on the basis of eq 3 (solid lines). The solid circle in curve a corresponds to θ ) 0.002, for which the value of If is ca. 0.95. Corresponding If vs log θ plots in curves a′, b′, and c′, upper abscissa in this figure, to highlight the low coverage behavior.

have been addressed in the context of other systems particularly in the biological area.34 Planar Supports. According to Morf,35 the diffusion-limited current for an ensemble of catalytic disks of radius ro embedded into a planar support of area Asubs, iarray, is given by:

[

iarray ) nFDC

π(1 - Ael1/2/d)2ro δ + 4NAel Asubs

]

-1

(1)

In this equation, Ael is the area of a catalytic disk and N is the number of disks, d ) (Asubs)/(N) is the distance between the centers of neighboring disks, δ is the thickness of the diffusion boundary layer, assumed to be uniform along the entire surface of the substrate, and other symbols have their customary meaning. This expression can be rearranged to read:

1 iarray

)

π(1 - θ1/2)2ro δ 1 + nFDCAsubs 4nFDCAsubs θ

(2) where θ ) (Ael)/(d2) ) (NAel)/(Asubs) is the net fractional coverage of disks on the surface. A comparison between the current due to the array and that of the fully covered surface iA ) nFDAsubsC/δ can be conveniently made by defining a dimensionless quantity

If )

iarray ) iA

expanded view to emphasize the differences found for very small values of θ (see also corresponding If vs log θ plots in curves a′, b′, and c′, upper abscissa in this figure). In the case of a rotating disk electrode, for which δ is independent of the radius of the disk, this value is on the order of tens of micrometers. For small N, Iarray ) N · 4nFDCro, that is, the sum of the contributions due to each of the tiny disks. An interesting implementation of this strategy with relevance to electroanalysis was reported by Fletcher and co-workers.36 Specifically, these authors constructed and tested random arrays of microelectrodes (RAM) using a few thousand carbon fibers a few micrometers in diameter, embedded in epoxy resin to yield composite microdisks, where the area of the exposed carbon microdisks was only a fraction of the total area. Assuming the catalytic disks to be of only a few nanometers in radius, σ would be on the order of 10-4. Based on the results displayed in curve a, Figure 5 (see solid circle), for ro ) 3 nm, the required coverage of Pt to achieve ca. 95% of the diffusion-limited current is on the order of about 0.2%, and thus an increase in the amount of Pt, and thus in the cost, would not result in significant improvements in performance. It should be borne in mind, however, that these effects may not be as marked as in a real situation, because the reactions under typical operation conditions are not purely controlled by diffusion. As the radius of the inert disk substrate, to be denoted as Fo, is decreased, steady state is achieved without any form of convection, and the thickness of the diffusion boundary layer is given by δ ) (π)/(4)Fo.37 Under these conditions, If can be shown to be given by:

If )

1 σ′(1 - θ1/2)2 1+ θ

(4)

where σ′ ) (ro)/(Fo). Shown in Figure 6 (curves a and b, solid lines, left ordinate) are plots of If versus θ for nanodisks embedded in a microdisk for different values of σ′ (see also corresponding If vs log θ plots in curves a′ and b′, upper abscissa in this figure). As may be expected, the current rises steeply as σ′ decreases, that is, for ro , Fo. Spherical Supports. A rather similar situation is also found for arrays of small disks uniformly distributed on the surface of the inactive spherical support. As shown by Zwanzig,38 the total steady-state flux toward such an arrangement normalized by that of a fully covered sphere is given by:

KE ) Is ) KSM

1 θ ) ) 1 π 1 1+π θ + µ(4 - θ) - µ µN 4 4 1 (5) π 4 1+ µ -1 4 θ

(

)

(

1 1 ) 1/2 2 π(1 - θ ) ro σ(1 - θ1/2)2 1+ 1+ θ 4θδ

(3) where σ ) (πro)/(4δ) is a dimensionless parameter that provides a relative measure of the radius of the disks and the thickness of the diffusion boundary layer. Shown in Figure 5 is a plot of If versus θ for σ ) 1 × 10-4 (curve a), 2 × 10-4 (b), and 4 × 10-4 (c) (solid lines) calculated on the basis of eq 3, in an

)

In this expression, µ ) ro/Ro, where Ro is the radius of the spherical support. A plot of Is versus θ for µ ) 1 × 10-4 (curve c) and 2 × 10-4 (d) calculated on the basis of eq 5 is shown in Figure 6 (dashed lines, right ordinate). As indicated, rather small coverages are needed to achieve fluxes close to those associated with a fully covered sphere (see also corresponding If vs log θ plots in curves c′ and d′, upper abscissa in this figure). For example, for µ ) 2 × 10-4, a coverage θ ) 0.02 would yield

Electrocatalysis at Microelectrodes a normalized total flux Is ) 0.97. Based on the results in Figure 6, as µ decreases, that is, the radius of the particles decreases, keeping the substrate area constant, the corresponding θ required to achieve the same performance would decrease. An interesting aspect emerging from this discussion relates to the effect of the shape of the inactive substrate on the overall performance of supported catalysts. As pointed out earlier, the diffusion-limited current toward a fully catalytic hemisphere is about 10% higher than that toward a microdisk of the same area (see curves a and e in Figure 1). Consider now an inert hemisphere and an embedded and a microdisk both of the same area, Asubs, decorated by the same number of small catalytic disks, N, of precisely identical radii, ro, so that the total catalytic surface becomes NAel. It thus follows, based on simple geometrical considerations, that F0 ) 2Rs, µ ) 2σ′, where Rs is the radius of the hemisphere and all other symbols have the f same meaning as before. Shown in Figure 7 are plots of INor versus θ () (NAel)/(Asubs)) for catalytic nanodisks embedded in an inert microdisk for σ′ ) 2 × 10-4 (left ordinate, curve a) s versus θ for the catalytic nanodisks embedded in an and of INor inert hemispherical substrate, that is, µ ) 22 × 10-4 (right f s and INor are normalized by the ordinate, curve b), where INor diffusion-limited current for a hemisphere of the area (see also f vs log θ plots in curves a′ and b′, upper corresponding INor abscissa in this figure). For exceedingly small coverages, these two curves should overlap, as the nanodisks would be virtually isolated and the current would only depend on their number. As θ increases, the current associated with the array supported on the microdisk substrate increases at a much faster rate than that on the hemispherical counterpart. However, for θ of ca. 0.6% (the crossing point in the diagram), the spherical substrate achieves a larger current and thus much better performance. It seems conceivable that the very high activities for dioxygen reduction in acidic media reported by Adzic et al.39 for very small coverages of Pt nanoparticles supported on niobium oxide may be due in part to this type of geometrical effect. D. Extensions to Molecular Electrocatalysts Adsorbed on Inactive Substrates. Some of the concepts discussed in the previous section can be applied to systems involving molecular

Figure 6. Plots of If vs θ for nanodisks embedded in a microdisk (solid lines, left ordinate) for σ′ ) 1 × 10-4 (curve a, thick red line) and 2 × 10-4 (curve b, thin blue line). Corresponding plots for nanodisks, embedded in a spherical surface, Is vs θ, are shown in dashed dotted lines (right ordinate) for different values of µ ) 1 × 10-4 (curve c, thick red dash line) and 2 × 10-4 (curve d, thin blue dash line). Corresponding If vs log θ plots based on these data are given in curves a′, b′, c′, and d′, upper abscissa in this figure, to highlight the low coverage behavior.

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f Figure 7. Plots of INor vs θ for catalytic nanodisks embedded in an s inert microdisk for σ ′ ) 2 × 10-4 (curve a, left ordinate) and of INor vs θ for catalytic nanodisks embedded in an inert hemispherical substrate, that is, µ ) 22 × 10-4 (curve b, right ordinate). f Corresponding INor vs log θ plots in curves a′ and b′, upper abscissa in this figure, to highlight the low coverage behavior.

Figure 8. Plots of Θ vs E - Eo′ (left ordinate) and 0.1 × (dΘ)/(d(E - Eo′)) vs E - Eo′ (right ordinate, inverted bell shaped curve) for σ ) (πro)/(4δ) ) 2 × 10-4 (see text for details). Also shown in this figure is a plot of the current due to the reduction of the solution-phase species, -If, vs E - Eo.

redox active electrocatalysts adsorbed on otherwise inert supports, where the activity is associated with only one of the redox forms of the catalyst. One of the most intriguing experimental observations for systems of this type involves the reduction of dioxygen mediated by the reduced form of certain macrocycles of the porphyrin type bearing iron centers. As thoroughly studied by Anson and co-workers,40 a few of these materials adsorbed on ordinary pyrolytic graphite (OPG) at coverages on the order of a single monolayer display an onset for the reduction of O2 in aqueous acidic electrolytes at potentials far more positive than those associated with the reduction of the macrocycle itself, to yield formally an active iron center in the ferrous state. One possible explanation for this rather unique effect may be found in the rates of mass transport to the active sites. The analysis to follow is based on a number of simplifying assumptions, which serve to illustrate the nature of the phenomena involved. Specifically, (i) The reaction between dioxygen and the reduced form of the macrocycle is sufficiently fast to render overall rates to be purely diffusion controlled. (ii) The coverage of the reduced form of the adsorbed species, Θ, follows a potential-dependent Langmuir type isotherm:

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Θ)

1 F 1 + exp (E - E°′) RT

[

]

Zhu et al.

(6)

where E°′ is the formal potential of the adsorbed redox couple. Shown in Figure 8 are plots of If and Θ as a function of E o E ′, assuming ro ) 3 nm and a rotation rate of ca. 1000 rpm, that is, δ of ca. 12 µm, yielding σ ) (πro)/(4δ) ) 2 × 10-4. The bell-shaped curve was obtained by taking the derivative of Θ with respect to E - Eo′, which is proportional to the voltammetric curve. As clearly indicated, and, in qualitative agreement with the experimental observations, the onset potential for the reduction of dioxygen can indeed be far more positive, that is, ca. 200 mV in this case, than that associated with the reduction of the macrocycle. It is interesting to note that many of the macrocycles investigated by Anson et al.40 and other researchers exhibit non Langmuir type isotherms, yielding voltammetric peaks that, in general, are broader than those associated with such ideal behavior. Under such conditions, the onset for the reduction of O2 would be displaced toward more positive potentials as compared to Eo′. Summary The most important conclusions emerging from this study may be summarized as follows: (i) The diffusion-limiting current, ilim, for spheroids of the prolate type of a common area appears to reach a plateau for aspect ratios, β () a/b, where a and b are the polar and equatorial lengths of the spheroid) in the range 0.5 < β < 2, but undergo further increases for larger values of β. This finding suggests that simple changes in the shape of the electrocatalytic surface can lead to gains in overall performance, that is, larger currents, while keeping the area constant. (ii) Electrocatalytic particles of small dimensions supported on otherwise inactive substrates under conditions in which a diffusion boundary layer of fixed thickness is formed yield for relatively small coverages values of ilim approaching those expected for full coverage. This affords a solid theoretical foundation consistent with very recent results published in the literature, as well as affords for the first time a plausible explanation for a phenomenon described two to three decades ago by Anson and co-workers, which had remained unresolved up to now. Acknowledgment. This work is supported by the NSF (grant no. 0616800). References and Notes (1) Markovic, N. M.; Gasteiger, H. A.; Ross, P. N. J. Phys. Chem. 1995, 99, 3411–3415. (2) Markovic, N. M.; Adzic, R. R.; Vesovic, V. B. J. Electroanal. Chem. 1984, 165, 121–133. (3) Adzic, R. R.; Markovic, N. M.; Vesovic, V. B. J. Electroanal. Chem. 1984, 165, 105–120.

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