Radial Flow Microring Electrode: Investigation of Fast Heterogeneous

The application of the radial flow microring electrode (RFMRE) in the measurement of fast heterogeneous electron-transfer kinetics is described. In th...
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J. Phys. Chem. B 1998, 102, 9891-9897

9891

Radial Flow Microring Electrode: Investigation of Fast Heterogeneous Electron-Transfer Processes Julie V. Macpherson,* Claire E. Jones, and Patrick R. Unwin* Department of Chemistry, UniVersity of Warwick, CoVentry CV4 7AL, UK ReceiVed: June 29, 1998; In Final Form: September 21, 1998

The application of the radial flow microring electrode (RFMRE) in the measurement of fast heterogeneous electron-transfer kinetics is described. In this new hydrodynamic ultramicroelectrode, solution flows from a capillary, positioned close (5-40 µm) to a planar glass substrate, and then radially over a thin Pt ring electrode (0.1-0.5 µm thick) deposited around the outer edge of the capillary. The dimensions and geometry of electrodes are characterized with high precision using conducting atomic force microscopy. Because the thin electrode only sees the velocity profile in its immediate vicinity, the description of mass transport is effectively analogous to that to a rectangular channel electrode, but with the advantage that the mass transport rate can be changed by altering either the volume flow rate or cell height (nozzle/substrate separation). The high steady-state mass-transfer rates attainable with the RFMRE (coefficients in excess of 2 cm s-1) are used to investigate rapid heterogeneous electron-transfer kinetics. A treatment for quasi-reversible electron transfer at the RFMRE is developed from earlier theories for the tubular and channel electrodes. Theoretical results are presented in the form of kinetic indicator diagrams, which allow the standard rate constant, ko, and transfer coefficient to be derived by simple measurement of (i) the separation between the three-quarter and quarterwave potentials and (ii) the shift in half-wave potential from the formal potential. It is estimated that ko values up to 20 cm s-1 should be discernible through steady-state measurements at the RFMRE. The method is used to investigate the oxidation of (i) Fe(CN)64- in aqueous 0.2 mol dm-3 KCl and (ii) IrCl63- in aqueous 0.2 mol dm-3 KNO3. For the former system ko is 0.36 ( 0.06 cm s-1 (transfer coefficient, β ) 0.42 ( 0.02), while in the latter case ko is 1.9 ( 0.1 cm s-1 (β ) 0.56 ( 0.02).

Introduction The characterization of electrode reactions with increasingly faster heterogeneous and homogeneous kinetics continues to be a major challenge.1 Ultramicroelectrodes (UMEs), electrodes with at least one dimension in the micrometer or submicrometer range, have had a significant impact in this regard, by greatly extending the kinetic range of electrochemically accessible reactions, compared to electrodes of conventional size. In particular, as the steady-state diffusion-limited current density at a UME varies reciprocally with the characteristic electrode dimension,2 small electrodes enable fast kinetics to be measured under conditions where the current response is largely free from capacitative and resistive effects. UMEs have also allowed transient methods such as fast scan cyclic voltammetry (FSCV) and potential step chronoamperometry to be extended into the microsecond time domain,3 although capacitative and resistive effects may become significant at very short times,4 complicating the electrode response. Under steady-state conditions, the transport rate at a single UME in quiescent solution is fixed. A range of electrode sizes must therefore be employed to vary this parameter. As the characteristic dimension of the electrode approaches the nanometer scale, not only is there a diminution in the overall current signal but accurate geometric characterization becomes increasingly difficult, with the result that large uncertainties may be introduced in the determination of electron-transfer kinetics.5 In the past decade, several approaches have been adopted in an attempt to overcome the problems highlighted. By using the positive feedback mode of the scanning electrochemical

microscope (SECM)6 operating as a variable gap thin-layer cell, Bard and co-workers have shown that it is possible to achieve variable and high mass-transfer rates under steady-state conditions at a single UME. This technique has been used successfully to investigate the kinetics of heterogeneous electron transfer7 and coupled chemical reactions.8 Variable mass transfer can also be attained through the employment of UMEs in convective systems, provided that sufficiently high convective rates can be generated. This concept is well illustrated by the microjet electrode (MJE),9 in which a high velocity jet of solution is fired through a fine nozzle positioned directly over a disk UME in a wall tube arrangement. With the MJE, it was shown that the steady-state mass-transfer rate to a 25 µm diameter electrode could be enhanced by over 2 orders of magnitude.9a,b Alternatively, Compton and co-workers have developed a high-pressure flow system in which solution is flowed at high volume flow rates (approaching 10 cm3 s-1) over a microband electrode in a channel geometry,10 with resulting mass transport coefficients up to ca. 1 cm s-1. We have recently introduced the radial flow microring electrode (RFMRE),11 as a new, easy-to-use hydrodynamic UME, which has variable and high mass transport rates. In the RFMRE arrangement (Figure 1), solution flows from a capillary (typical internal diameter, dn, of 100 µm), which is positioned very close to a planar substrate using a micropositioner. The ring electrode, of typical thickness 0.1-0.5 µm, is formed around the outer edge of the capillary, by deposition of a metal film followed by an epoxy resin insulating sheath. As fluid leaves the capillary, it impinges on the substrate below

10.1021/jp9827936 CCC: $15.00 © 1998 American Chemical Society Published on Web 11/10/1998

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Figure 1. Schematic cross section of the RFMRE arrangement employed for heterogeneous electron-transfer kinetic measurements.

and is forced into the nozzle/substrate gap (typical separation of 5-40 µm; in this study 4.6-20.6 µm) whereupon it flows radially past the ring electrode. It is well-documented that flow in a confined wall-jet-type configuration can be conveniently described in terms of three regions.12 In the impingement zone (A in Figure 1) the flow changes from axial to largely radial. In region C, the flow is purely radial, characterized by a parabolic velocity profile, given by eq 1, in which the center-line velocity (at z ) h) is r-dependent:

Vr )

z-h2 3 Vf 18 πrh h

[ ( )]

(1)

where Vr is the velocity in the radial direction, Vf is the volume flow rate, r is the radial coordinate, z is the axial coordinate, and h is the half-height of the nozzle/substrate gap. The extent of the transition zone (B in Figure 1) between these two regions (effectively an entry length for the establishment of the parabolic velocity profile, given by eq 1) has been shown to be dependent on Vf, h, dn, and the kinematic viscosity of the solution, υ.12 Numerical calculations, for flow conditions that approach those employed in the studies herein (with h , dn), indicate that the transition zone is short, with an effective parabolic profile established at r e 0.7dn.12 Alternatively, the establishment of a fully developed parabolic flow profile can be considered to occur in the region where the nozzle wall (at z ) 2h) begins to restrict Prandtl layer development.13 The Prandtl layer thickness, δo, for the impinging jet is given by13

()

δo ) 6.11

υ Vf

3/4

dn1/2r5/4

(2)

The entrance length can be considered as the r distance for which δo e 2h. For the range of cell heights and the volume flow rate employed in this study, the corresponding entry length is within r e 0.8dn. Given the experimental conditions outlined later, the ring electrode of the RFMRE is generally located in region C where radial flow is fully developed and parabolic. Moreover, given the small thickness of the ring electrodes deployed in the RFMRE, the electrode effectively “sees” a constant local velocity profile (at a specific r value), and mass transport can be considered analogous to a conventional rectangular channel electrode.14 This is consistent with previous experimental observations11 and the present studies, where we have shown that convective diffusion to the RFMRE can be described in terms of the Levich equation for the channel electrode14 with negligible contributions from diffusional edge effects.11

In this paper we demonstrate how the RFMRE can be used to characterize fast electron-transfer kinetics under steady-state conditions. This is illustrated through studies of the Fe(CN)64-/ Fe(CN)63- and IrCl63-/IrCl62- redox couples in KCl and KNO3 supporting electrolytes, respectively, at platinum electrodes. The former system has been studied extensively in quiescent solution with UMEs, under both steady-state and transient conditions.15 In most cases the standard heterogeneous electron-transfer rate constant, ko, was found to vary between 0.05 and 0.3 cm s-1 but was reported to depend strongly on electrode pretreatment and the ionic strength of the solution. This system represents an ideal first test for the RFMRE, as the documented ko values lie well within the range of achievable mass-transfer coefficients. In contrast, the reported ko values for the IrCl63-/IrCl62couple are higher. One investigation,16 employing both FSCV and ultrasound to increase mass transport rates to an UME, determined ko to be greater than 1 cm s-1. Given the impressive and well-defined mass transport rates demonstrated in initial studies with the RFMRE, we aim to demonstrate here that fast electron-transfer processes with standard rate constants in excess of 1 cm s-1 can be readily characterized, under steady-state conditions, with high precision. Theory In this section the steady-state current-voltage characteristics of a quasi-reversible redox couple (eq 3) in the RFMRE configuration are considered. kb

A y\ z B + nek

(3)

f

Both A and B are soluble; A is initially the only species present in solution, and n ()1 for the systems herein) is the number of electrons transferred per redox event. Equation 3 is written as an oxidation, for consistency with the reported experiments. kf and kb are the potential-dependent rate constants:

kf ) ko exp[-R(nF/RT)(E - E°′)]

(4)

kb ) ko exp[β(nF/RT)(E - E°′)]

(5)

where R and β are transfer coefficients for the reduction and oxidation steps (R + β ) 1), Eo′ is the formal potential for the redox couple, and F, R, and T have their usual meaning. This problem has been treated for the hydrodynamic tubular electrode17 and was recently modified for the analogous channel flow geometry.10a Since mass transport to the RFMRE can also be considered to be equivalent to that for the channel electrode,11 the earlier treatment for the tubular electrode can also be adopted here. However, as outlined below, we choose a different method of analysis to earlier work with channel and tubular electrodes, which implicitly assumed R ) β ) 0.5 in the determination of kinetic parameters. By drawing an analogy with the channel electrode, the current, i, which flows at the RFMRE, normalized by the current that would flow at the same electrode if charge transfer was reversible, irev, is given by17

i/irev ) 1 - 2u + 2u2 ln(1 + u-1)

(6)

where10a,17

u)

0.6783DB2/3(3Vf/4h2wxe)1/3 ko[exp{Rθ} + (DB/DA)2/3 exp{-βθ}]

(7)

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DB and DA are the diffusion coefficients of species B and A, respectively, w is the circumference of the thin ring electrode, and xe is the ring thickness. The parameter θ, related to the applied electrode potential, E, is defined by

θ)

F (E°′ - E) RT

(8)

Finally, irev is given by17

irev )

ilim 1 + (DA/DB)2/3 exp{θ}

(9)

ilim is the convective mass transport-limited current flowing at the ring electrode, in the RFMRE geometry, in the absence of diffusional edge effects. ilim can be written by simple modification of the Levich equation18 for the channel electrode, giving

ilim ) 0.925nFc*A

(

)

DAxew h

2/3

Vf1/3

(10)

where c*A denotes the bulk concentration of A. A simple, but powerful, method for the kinetic analysis of quasi-reversible steady-state voltammograms is to measure quartile potential shifts,19,20 such as ∆E ) (E3/4 - E1/4) and (E1/2 - E°′) where Em/4 represents the potential at milim/4 (where m ) 1, 2, or 3). This approach has been widely applied to UME voltammetry in quiescent solution,2b,15b,c,20 where it has been shown that, by simply measuring these two parameters, ko (normalized with respect to the mean mass-transfer coefficient) and β can be determined independently, provided the ratio DA/DB is known. Application of this method to the RFMRE involved theoretically generating current-voltage curves (for a particular DA/ DB ratio) by solving eqs 6-10 numerically as a function of β and λ, where

λ ) ko/kn

(11)

In eq 11, kn is the numerator of eq 7, which is related to the mean mass-transfer coefficient (kt) for the RFMRE:

( )

kt ) 0.925

DA2Vf 1/3 h2wxe

) 1.50kn

(12)

For each voltammetric wave formulated, values of ∆E and ∆E1/2 were extracted, and the results are presented in the form of Figure 2, i.e., as “kinetic indicator plots” of the type advocated by Oldham et al.,20 in which a series of contour lines link points of equal λ and β. A computer program for carrying out this procedure (written in Fortran 77) is available from the authors upon request. In experimental practice, the quartile shifts from a recorded voltammetric curve supply the x and y coordinates of a point on the plot. This in turn denotes, unequivocally, λ and β, with the former parameter yielding ko (eq 11). Figure 2a demonstrates that for low λ values, where mass transport effects strongly dominate, kinetic limitations are very evident in the wave shape. For example, considering the case where λ ) 0.1 and β ) 0.5, E1/2 is shifted 141.9 mV from E°′ (at 298 K) while ∆E ) 116.6 mV. The Tomesˇ criterion for a reversible system21 at 298 K dictates that ∆E ) 56.5 mV. As the λ value increases, ∆E1/2 tends toward zero and ∆E approaches 56.5 mV. A similar effect is observed by increasing the β value. Even when the rate of electron transfer becomes comparable to the rate of mass transfer, kinetic measurements,

Figure 2. Kinetic indicator plots simulated for DA ) DB illustrating the relationship between ∆E and ∆E1/2 at constant λ and β for (a) λ values between 0.1 and 1.0 and β values between 0.3 and 0.7 and (b) λ values between 1.0 and 10.0 (in 1.0 steps) and β values between 0.3 and 0.7.

elucidated from quartile potential shifts, can still be easily made; for example, when λ ) 1.0 and β ) 0.5, ∆E1/2 ) 38.5 mV and ∆E ) 93.6 mV. Figure 2b can be used to estimate the maximum value of ko that should be measurable with the RFMRE technique. Two main experimental factors come into play: the maximum rate of mass transport that can be obtained in the RFMRE arrangement and the accuracy with which potential shift measurements can be made. In the absence of potential drift, a reasonable estimate for the latter would be (1 mV. Bond et al.22 quoted measured differences of 3 ( 2 mV from the reversible halfwave potential for the oxidation of ferrocene in acetonitrile at 0.3 and 0.5 µm radii Pt disk electrodes and used these data to estimate a lower limit of 6 cm s-1 for ko. In the RFMRE arrangement kn values up to ca. 2 cm s-1 can be readily achieved.11 Figure 2b demonstrates that when λ ) 10.0 and β ) 0.5, ∆E1/2 ) 4.7 mV and ∆E ) 61.3 mV; potential shift values which should be experimentally discernible from the reversible case, thus making possible the determination of ko reliably up to ca. 20 cm s-1. Experimental Section Chemicals. Solutions were prepared from potassium ferrocyanide trihydrate (Fisons, AR) at a concentration of 5 × 10-4 mol dm-3 in 0.2 mol dm-3 potassium chloride (Fisons, AR) or

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Figure 3. (a) Conductivity AFM image of the ring capillary electrode surface and (b) the corresponding height image. Both images were recorded simultaneously with a platinum-coated AFM tip.

potassium iridate hexachloride (Aldrich) at a concentration of 5 × 10-4 mol dm-3 in 0.2 mol dm-3 potassium nitrate (Fisons, AR). Potassium nitrate and potassium chloride both served as supporting electrolytes. All solutions were prepared using Milli-Q (Millipore Corp.) reagent water. Electrodes. The procedure for the fabrication, preparation, and polishing of thin platinum ring capillary electrodes has been described in detail.11 General inspection of ring electrodes, polished to a finish of 0.05 µm, was accomplished with an Olympus BH2 light microscope equipped with a PM-10AK photomicrographic system (magnifications ×100-×1000). For accurate simultaneous determination of the width and topography of the ring electrodes, a Nanoscope E atomic force microscope (AFM: Digital Instruments, Santa Barbara, CA) was used with conducting tips.23,24 The AFM was equipped with a scan head allowing a maximum scan range of 100 µm by 100 µm. AFM probes (Nanoprobe, Park Scientific, Sunnyvale, CA) consisted of silicon nitride cantilevers (length 200 µm, manufacturer’s nominal spring constant 0.06 N m-1) with integrated pyramidal tips, which had a height of 2.86 µm and a base width of 4 µm. To make the AFM tip conducting, the entire cantilever and support was sputter-coated first with a chromium anchor (30 Å) and then with platinum (600 Å). Although measurements were made in air, a facile method for making electrical connection to the probe was to secure it in position in a standard Digital Instruments fluid cell. Electrical contact was then made to the cantilever via the metal tip holder, external to the cell. To measure the tip-sample resistance, a potential difference

of 1 V was applied between the platinum-coated tip and the thin platinum ring electrode, across a current-limiting resistor (1 MΩ). Simultaneous images of the substrate topography and conductivity were then recorded, with a signal access module (Digital Instruments) facilitating the ready acquisition of both sets of data. RFMRE Apparatus. The RFMRE cell comprised a fully detachable Teflon base, cylindrical glass body, and Teflon lid, with a total volume of 25 cm3. The cell base contained a small recess which could securely accommodate a glass disk. The glass body contained an outlet pipe to prevent solution overflow in the cell and an optical window (15 mm diameter) so that video microscopy could be used to monitor the position of the capillary nozzle relative to the glass substrate. The video microscope comprised a zoom microscope with a CCD camera attachment that offered maximum on-screen resolution of 2.2 µm per pixel. The flow of electrolyte at a rate of 1.67 × 10-2 cm3 s-1 through the RFMRE nozzle was achieved with a Gilson (Villiers-Le-Bel, France) model 305 HPLC pump equipped with a 25 W Ti pump head and model 806 manometric module. Instrumentation. The position of the RFMRE capillary normal to the glass substrate (z axis) was controlled with 0.1 µm spatial resolution by mounting the nozzle on a piezoelectric translator, incorporating a strain gauge sensor (translator model P843 and controller E501, Physik Instrumente, Waldbronn, Germany). The positioner was attached to a Newport Corp. (Fountain Valley, CA) model 461 x, y, z stage. The stages were,

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in turn, located on a vibrationally isolated Newport CSD series breadboard, and the cell was shielded using a home-built Faraday cage. Current-potential characteristics were recorded with an EcoChemie (Utrecht, Holland) Autolab Electrochemical Workstation, incorporating a preamplifier (model ECD) for low-current measurements. All voltammetric measurements were made in a three-electrode arrangement with a saturated calomel electrode (SCE: Radiometer, Copenhagen) as a reference and a platinum gauze counter electrode. The thin platinum ring served as the working electrode in RFMRE experiments. For the determination of Eo′, a 50 µm diameter Pt UME, fabricated according to standard procedures,25 was employed as the working electrode. Procedure. Linear sweep voltammograms were recorded for the oxidation of Fe(CN)64- and Ir(Cl)63- at the RFMRE, at 298 K, as a function of the nozzle/substrate separation, for a fixed volume flow rate of 1.67 × 10-2 cm3 s-1. The nozzle/substrate separation was established by contacting the end of the capillary with the glass disk and then retracting the nozzle from the substrate a set distance. Results and Discussion Characterization of the Platinum Ring Capillary Electrode with Conducting AFM. In our previous study,11 we employed steady-state voltammetry in quiescent solution and scanning electron microscopy to determine the ring electrode thickness, with the latter technique also providing some information on the geometric quality of the electrode. Ideally the ring should sit coplanar with the insulator, in order for the electrode to see the expected laminar, pseudo-Poiseuille flow profile when employed in a RFMRE arrangement. A Vf1/3 or h-2/3 dependence of the mass-transport-controlled current (eq 10) is an indirect route for this assessment. If the electrode is significantly raised or recessed with respect to the insulating plane, deviations from this predicted behavior may be observed. Such effects have been reported for a variety of hydrodynamic electrodes26,27 where the electrode is significantly raised or recessed due to the polishing procedures or manufacturing processes adopted. Conducting AFM is a powerful method both for determining directly the ring thickness and assessing the geometry of the electrode with greater precision than attainable from SEM. Figure 3 shows (a) a conductivity image of the surface of a ring capillary electrode and (b) the corresponding height image, recorded simultaneously. The conductivity image provides an absolute value for the ring thickness of 540 ( 30 nm for this particular electrode. In parallel with the conductivity data, the height image demonstrates unequivocally that the electrode is not appreciably raised or recessed. There is a small degree of roughness over the insulator and electrode (ca. (50 nm), but this is the level expected with the polishing procedure adopted.11 The mass-transport-limited current response of this electrode was linear with respect to h-2/3 (for h over the range 5.0-15.0 µm), in agreement with eq 10, demonstrating that small irregularities in the surface do not significantly affect the flow hydrodynamics. This result is not unexpected given the low Reynolds numbers, Re,28 inherent in the RFMRE geometry11 defined as

Re ) Uh/υ ) Vf/4πaυ

(13)

where U h is the mean solution velocity at the location of the ring electrode. For this case, υ ) 0.01 cm2 s-1, a ) 71 µm,

Figure 4. (a) Kinetic indicator plot for the oxidation of Fe(CN)64- at a RFMRE characterized by dn ) 100 µm, a ) 79 µm, xe ) 0.25 µm, Vf ) 1.67 × 10-2 cm3 s-1, and h ) 2.8 (f), 3.3 (0), 3.8 (b), 4.3 (4), and 4.8 (1) µm. The β and λ values range from 0.40 to 0.44 in steps of 0.01 and from 0.26 to 0.40 in steps of 0.02, respectively. Experimental current-voltage curves (s), corresponding to theoretical behavior (- - -) based on ko and β values determined from the kinetic indicator plot (a) and the predicted reversible response (‚‚‚) for the RFMRE recorded at h separations of (b) 3.3 and (c) 4.3 µm. The insets show Tafel analyses of the data.

and Vf ) 1.67 × 10-2 cm3 s-1; hence, over the range of h values quoted above Re ) 19. RFMRE Voltammetry. Fe(CN)64-/Fe(CN)63- Couple. Figure 4a is a kinetic indicator plot, marked with the ∆E and ∆E1/2 coordinates derived from steady-state current-voltage curves for the oxidation of Fe(CN)64- at a RFMRE. The RFMRE system was characterized by an internal nozzle diameter, dn ) 100 µm, radial distance from the axis of the nozzle to the ring, a ) 79 µm, xe ) 0.25 µm, and h ) 2.8 (f), 3.3 (0), 3.8 (b), 4.3 (4), and 4.8 µm (1). A volume flow rate Vf ) 1.67 ×

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TABLE 1: Kinetic Analysis of Voltammetric Data for Fe(CN)64- Oxidation at a RFMRE Characterized by dn ) 100 µm, a ) 79 µm, xe ) 0.25 µm, and Vf ) 1.67 × 10-2 cm3 s-1 h/µm

λ

kt/cm s-1

ko/cm s-1

β

2.8 3.3 3.8 4.3 4.8

0.310 0.290 0.280 0.360 0.400

2.04 1.82 1.61 1.50 1.32

0.42 0.35 0.30 0.36 0.35

0.42 0.43 0.43 0.41 0.40

10-2 cm3 s-1 was employed, together with a potential scan rate of 10 mV s-1. The ilim-h-2/3 behavior for these voltammograms closely obeyed the predicted theoretical response (eq 10), with a correlation coefficient of 0.997. The locations of the y coordinate of the experimental points in Figure 4a required a knowledge of E°′. For a reversible redox couple, where the solution species is initially in its reduced form, eq 14 applies:29

E1/2 ) E°′ - (RT/nF) ln(DA/DB)

(14)

Since ko for the Fe(CN)64-/Fe(CN)63- redox couple is moderately fast (vide supra), the steady-state current-voltage characteristics at a large diameter UME in quiescent solution should show reversible behavior, as transport of species to the UME will be relatively slow. A linear sweep voltammogram recorded at a potential sweep rate of 2 mV s-1 for the oxidation of Fe(CN)64- at a 50 µm diameter UME in quiescent solution gave a Tomesˇ separation of 57 mV, indicative of reversible behavior. After taking into account the unequal diffusion coefficients of Fe(CN)64- and Fe(CN)63-, E°′Fe(CN)64-/Fe(CN)63- was determined as 0.200 ( 0.001 V vs SCE, using eq 14. The diffusion coefficients of Fe(CN)64- and Fe(CN)63- were determined as (6.7 ( 0.1) × 10-6 and (7.6 ( 0.1) × 10-6 cm2 s-1, respectively, from the steady-state diffusion-limited current recorded at a 50 µm diameter electrode in a solution containing 5 × 10-4 mol dm-3 of either Fe(CN)64- or Fe(CN)63-, together with 0.2 mol dm-3 KCl. The λ values determined for the oxidation of Fe(CN)64-, shown in Figure 4a, demonstrate admirably that the mass transport rates of the RFMRE are sufficiently high to compete with, and indeed significantly exceed, the standard rate constant for the oxidation of Fe(CN)64-. This allows kinetic measurements to be made with very high precision. Table 1 provides a compilation of the kinetic data, obtained from the potential shifts derived in Figure 4a. The value of ko falls within a narrow range with a mean value 0.36 ( 0.06 cm s-1, while β is in the range 0.40-0.43. The standard rate constant is at the upper end of the range measured previously,15 validating measurements made with the RFMRE. To indicate the extent to which the kinetic data, derived from the above analysis, predicts the entire waveshape, Figure 4b,c shows the full experimental (s) current-voltage curves recorded at h separations of (b) 3.3 and (c) 4.3 µm, along with the corresponding theoretical behavior (- - -) based on the ko and β values derived in Table 1. Excellent agreement between theory and experiment is evident over a very wide range of potentials. The high level of correlation is further demonstrated by the Tafel analyses of the data, shown as inserts. Also included in Figure 4b,c are the current-voltage characteristics predicted for a reversible system (‚‚‚). Comparison of these latter theoretical voltammograms with those measured shows clearly the extent to which the RFMRE is able to push the voltammetry of the Fe(CN)64-/Fe(CN)63- couple away from a position of reversibility.

Figure 5. (a) Analysis of ilim-h-2/3 data in terms of eq 10 for a RFMRE system characterized by dn ) 95 µm, a ) 70 µm, xe ) 0.23 µm, and Vf ) 1.67 × 10-2 cm3 s-1. (b) Experimental current-voltage curves (s), the corresponding theoretical behavior (- - -) based on ko and β values determined from a kinetic indicator plot and the predicted reversible response (‚‚‚) for the RFMRE, recorded at h separations of (i) 2.3, (ii) 3.3, and (iii) 3.8 µm.

TABLE 2: Kinetic Analysis of Voltammetric Data for IrCl63- Oxidation at a RFMRE Characterized by dn ) 95 µm, a ) 70 µm, xe ) 0.23 µm, and Vf ) 1.67 × 10-2 cm3 s-1 h/µm

E1/2 Eo′/mV

E3/4 E1/4/mV

λ

kt/cm s-1

ko/cm s-1

β

2.8 3.3 3.8

27.4 25.4 26.4

84.0 78.8 81.9

1.39 1.55 1.54

2.14 1.94 1.76

2.0 2.0 1.8

0.57 0.56 0.54

IrCl63-/IrCl62- Couple. Figure 5a shows ilim-h-2/3 data for a series of steady-state voltammograms (potential sweep rate of 10 mV s-1) recorded at a RFMRE, at a fixed volume flow rate, Vf ) 1.67 × 10-2 cm3 s-1. These results were obtained with a range of nozzle/glass substrate separations between 2.3 µm (highest limiting current) and 10.3 µm (lowest limiting current). The system was characterized by dn ) 95 µm, DIrCl63) 7.5 × 10-6 cm2 s-1, a ) 70 µm, and xe ) 0.23 µm. The results closely obey the predicted theoretical response (eq 10), demonstrating that mass transport in this particular system is well-defined and characterized. Table 2 provides a summary of the information derived from a kinetic indicator plot appropriate to this system. Kinetic data were only extracted from voltammograms recorded at the three closest nozzle/substrate distances, representing the highest mass transport rates in the RFMRE arrangement, sufficient to compete with the fast kinetics associated with this couple. E°′ was determined from voltammetry at a disk UME, using the method above, for which there is a direct relation between the formal and half-wave potential due to the approximately equal diffusion coefficients of the oxidized and reduced forms of the couple,29

Radial Flow Microring Electrode E°′ ) E1/2 ) 0.693 V vs SCE. It is evident from Table 2 that the electron-transfer kinetics for this particular system are rapid, with ko ) 1.9 ( 0.1 cm s-1 and β in the range 0.54-0.57. Figure 5b gives the full experimental (s) voltammograms recorded at h ) (a) 2.8, (b) 3.3, and (c) 3.8 µm, which correlate well with the corresponding theoretical characteristics (- - -) based on the ko and β values listed in Table 2. Figure 5b also shows the reversible current-voltage behavior for all three mass transport rates, which demonstrate that, as a result of the high mass transport rates generated in the RFMRE arrangement, fast electron-transfer kinetics can be measured with high precision. For this system, the half-wave potential has shifted by up to 30 mV from E°′, which represents a readily measurable quantity. Conclusions The results presented in this paper demonstrate that the RFMRE is a powerful device for characterizing fast heterogeneous electron-transfer kinetics under steady-state voltammetric conditions. The method has been used to determine the oxidation kinetics at a platinum electrode of (i) Fe(CN)64- in aqueous 0.2 mol dm-3 KCl, ko ) 0.36 ( 0.06 cm s-1 (transfer coefficient β ) 0.42 ( 0.02), and (ii) IrCl63- in aqueous 0.2 mol dm-3 KNO3, ko ) 1.9 ( 0.1 cm s-1 (β ) 0.56 ( 0.02). Given that high mass-transfer coefficients in excess of 2 cm s-1 can readily be attained in the RFMRE geometry, it is estimated that ko values up to 20 cm s-1 should be measurable. In addition to the investigation of heterogeneous kinetics, the RFMRE should find considerable application in the measurements of fast homogeneous kinetics coupled to electron-transfer processes. Acknowledgment. We appreciate support from the EPSRC (GR/K97011 and GR/L71377) and Unilever Research Port Sunlight Laboratory (Dental Division). We also thank Drs. Robert Somekh and Zoe Barber (Department of Materials Science and Metallurgy, Cambridge University) for platinum coating the AFM tips used in this study. References and Notes (1) Andrieux, C. P.; Hapiot, P.; Save´ant, J. M. Chem. ReV. 1990, 90, 723 and references therein. (2) (a) Saito, Y. ReV. Polarogr. 1968, 15, 177. (b) Penner, R. M.; Heben, M. J.; Longin, T. L.; Lewis, N. S. Science 1990, 250, 1118. (c) Shao, Y.; Mirkin, M. V.; Fish, G.; Kokotov, S.; Palanker, A.; Lewis, A. Anal. Chem. 1997, 69, 1627. (3) (a) Howell, J. O.; Wightman, R. M. Anal. Chem. 1985, 56, 524. (b) Howell, J. O.; Kuhr, W. G.; Ensman, R. E.; Wightman, R. M. J. Electroanal. Chem. 1986, 209, 77. (c) Andrieux, C. P.; Garreau, D.; Hapiot, P.; Pinson, J.; Save´ant, J. M. J. Electroanal. Chem. 1988, 243, 321. (d) Wipf, D. O.; Kristensen, E. W.; Deakin, M. R.; Wightman, R. M. Anal. Chem. 1988, 60, 306. (e) Wipf, D. O.; Wightman, R. M. J. Phys. Chem. 1989, 93, 4286. (4) See for example: Wightman, R. M.; Wipf, D. O. Acc. Chem. Res. 1990, 23, 64. (5) (a) Baranski, A. S. J. Electroanal. Chem. 1991, 307, 287. (b) Oldham, K. B. Anal. Chem. 1992, 64, 646. (c) Smith, C. P.; White, H. S. Anal. Chem. 1993, 65, 3343.

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