A Comparison of the Time Scales Accessible and Kinetic

Anal. Chem. 1999, 71, 806-810

A Comparison of the Time Scales Accessible and Kinetic Discrimination Obtainable Using Steady-State Voltammetry at Common Hydrodynamic and Microelectrode Geometries J. A. Alden, S. Hakoura, and R. G. Compton*

Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, U.K.

The kinetic discrimination of steady-state voltammetry at microdisk, microhemispherical, rotating disk, channel (microband), wall-jet, and wall-tube electrodes was compared by measuring the broadness of numerically simulated ECE and EC2E working curves and superimposition of numerically simulated EC and EC2 working curves by equating the average diffusion thicknesses. This together with the range of time scales calculated from the span of typical experimental parameters (electrode size, flow rate, etc.) allows the range of rate constants measurable with each geometry to be compared. It has been argued1 that electrode geometries that are nonuniformly accessible should have a greater inherent kinetic discrimination than uniformly accessible electrodes, since in the former the kinetic time scale changes across the electrode surface, which should lead to a “stretching” of the working curvesa plot of the experimental observable (limiting current, shift in half-wave potential, etc.) vs dimensionless rate constant. Comparison of working curves for a wall-jet and a rotating disk electrode for an ECE reaction2 seems to support this argument (this mechanism, together with others discussed in this paper, is outlined in Table 1). However, Unwin and Compton3 showed that the inherent kinetic discrimination of rotating disk and channel electrodes was virtually identical for first-order processes, though they postulated that this might change for a second-order process. A number of recently computed working curves and surfaces4-8 allow a quantitative comparison of the kinetic discrimination of common electrode geometries for both first- and second-order homogeneous processes. RESULTS Table 2 shows the approximate range of time scales and rate constants (for an ECE and EC2E reaction) that may be measured (1) Compton, R. G.; Unwin, P. R. J. Electroanal. Chem. 1986, 205, 1. (2) Compton, R. G.; Fisher, A. C.; Tyley, G. P. J. Appl. Electrochem. 1991, 21, 295. (3) Unwin, P. R.; Compton, R. G. J. Electroanal. Chem. 1988, 245, 287. (4) Alden, J. A. D.Phil. Thesis, Oxford University, 1998. Available on-line at http://physchem.ox.ac.uk: 8000/john/Thesis. (5) Alden, J. A.; Compton, R. G. Electroanalysis 1998, 10, 207. (6) Alden, J. A.; Compton, R. G. J. Phys. Chem. B. 1997, 101, 9751. (7) Alden, J. A.; Compton, R. G. J. Phys. Chem. B. 1997, 101, 9606. (8) Alden, J. A.; Hakoura, S.; Compton, R. G. Anal. Chem., in press.

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Table 1. Common First- and Second-Order Homogeneous Processes Encountered in Electrochemical Reaction Mechanisms process EC EC2

stepsa

rate law of homogeneous steps

A+e fB k B 98 C A+e fB

∂[B]/dt ) -k[B]; ∂[C]/dt ) k[B]

2B 98 C A+e fB k B 98 C C+e fD A+e fB

∂[B]/dt ) -2k[B]2; ∂[C]/ dt ) k[B]2

k

ECE

EC2E

2B

∂[B]/dt ) -k[B]; ∂[C]/dt ) k[B]

∂[B]/dt ) -k2[B]2; ∂[C]/ dt ) k[B]2

k

98 C C+e fD

a The working curves used here were computed for a Nernstian (fully reversible) electron transfer. In the case of ECE and EC2E reactions, the working curve is for the transport-limited current.

using steady-state voltammetry at various electrode geometriess definitions of dimensionless rate constants are given in Table 3. In these mechanisms, the range of kinetic discrimination may be quantified by the broadness of working curvessplots of the effective number of electrons transferred (Neff) vs dimensionless rate constant. The range of rate constants was calculated from the values of the dimensionless rate constant, which give values of 1.1 and 1.9 for Neff from a working curve for each geometry. These are regarded as thresholds for kinetic discrimination,9 outside which the experimental error in Neff gives rise to unacceptably large errors in the rate constant. The calculations are based on the following typical experimental parameters: (1) A representative diffusion coefficient of 1 × 10-5 cm2 s-1 in nonaqueous solvents was used, together with a concentration of 1 × 10-6 mol cm-3. At room temperature, kinematic viscosities for common solvents (such as water, acetonitrile, and DMF) are in the range 1 × 10-3-1 × 10-2 cm2 s-1. (2) Commercially available10 microdisk electrodes of radii 0.670 µm may be used for steady-state measurements without (9) Amatore, C. A.; Save´ant, J. M. J. Electroanal. Chem. 1978, 86, 227. (10) Microglass instruments, Greensborough, Victoria, Australia. (e-mail: chemk@ lure.latrobe.edu.au). 10.1021/ac9809128 CCC: $18.00

© 1999 American Chemical Society Published on Web 01/05/1999

Table 2. Comparison of the Kinetic Time Scales Accessible with Steady-State Voltammetry Using Common Electrode Geometries

electrode geometry hemispherical microdisk rotating disk wall-jet wall-tube channel conventional microband fast flow a

range of time scales (tc) accessible

range of log dimensionless rate const (K) ECE EC2E

range of rate const (k) that can be measured ECE EC2E

400 µs-5 s 200 µs-3 s 10-50 s 50 µs-100 s 10 µs-5 ms

3.82 3.93 2.47 3.03 2.47a

4.31 4.47 3.71 3.93 3.71a

2 × 10-3-2 × 105 6 × 10-3-7 × 105 9 × 10-2-1 × 103 9 × 10-5-5 × 105 4 × 102-3 × 107a

6 × 103-2 × 1012 1 × 104-6 × 1012 6 × 104-2 × 1010 4 × 101-8 × 1012 3 × 108-4 × 1014 a

0.1-10 s 5 ms-1 s 30 µs-10 ms

2.45 2.75 2.45

3.43 3.47 3.43

1 × 10-2-5 × 102 2 × 10-1-1 × 106 20-4 × 106

1 × 104-4 × 109 3 × 105-4 × 1011 1 × 107-3 × 1013

Assuming the analogy with the RDE is valid.

Table 3. Definitions of Dimensionless Rate Constant electrode geometry spherical electrode microdisk electrode rotating disk electrodea wall-jet electrodeb wall-tube electrodec channel electroded

Peclet no.

Pe ) Sc‚Re3/2 Pe ) CWJE/re3/4D Pe ) CMJEre3/D Pe ) (3/2)(vf/Dd)(xe/h)2

dimensionless rate const K ) kre2/D K ) kre2/D K ) (kre2/D)Pe-2/3 K ) (kre2/D)[9Pe/8]-2/3 K ) (kre2/D)Pe-2/3 K ) (kxe2/D)Pe-2/3

a For the rotating disk electrode, the Reynolds number is given by16 Re ) re2f/ν, where f is the rotation frequency (in Hz); the Schmidt number, Sc ) ν/D; re is the electrode radius; ν is the kinematic viscosity; D is the diffusion coefficient. b For the wall-jet electrode, the hydrodynamic constant CWJE is given by17 CWJE ) [125M3/(216ν5)]1/4. The constant M is given by M ) kc4vf3/(2π3rjet2); kc is an experimentally determined constant; rjet is the nozzle radius. c For the wall-tube electrode, the hydrodynamic constant CMJE is given by18 CMJE ) 0.54 [(rjet/zjet)-0.054 (vf/rjet3)1/2]3 ν-1/2; zjet is the electrode-jet separation.d xe is the length of the channel electrode in the direction of flow.

problems associated with natural convection. Dimensionless rate constants were interpolated from the working curve simulated in ref 7. (3) Hemispherical electrodes may be realized experimentally using hanging mercury drops for macroelectrodes and mercurycoated microdisk electrodes for microhemispheres. The lower radius limit is thus governed by the microdisk radius (0.6 µm representing the smallest radius currently available commercially); the upper limit has been chosen as 70 µm, above which natural convection is likely to become significant. (4) Macpherson and Unwin11,12,18 have recently developed a microjet electrode (MJE). In their work, the jet is positioned off to one side of the disk electrode, shown in Figure 1a, rather than coaxially corresponding to a wall-tube electrode (WTE), shown (11) Macpherson, J. V.; Beaston, M. A.; Unwin, P. R. J. Chem. Soc., Faraday. Trans. 1995, 91, 899. (12) Martin, R. D.; Unwin, P. R. J. Electroanal. Chem. 1995, 397, 325. (13) Chin, D. T.; Tsang, C. H. J. Electrochem. Soc. 1978, 125, 1461. (14) Rees, N. V.; Alden, J. A.; Dryfe, R. A. W.; Compton, R. G.; Coles, B. A. J. Phys. Chem. 1995, 99, 14813. (15) Compton, R. G.; Fisher, A. C.; Wellington, R. G.; Dobson, P. J.; Leigh, P. A. J. Phys. Chem. 1993, 97, 10410. (16) Pleskov, Yu. V.; Filinovski, V. Yu. The rotating disc electrode; Plenum: New York, 1976; p 17. (17) Compton, R. G.; Greaves, C. R.; Waller, A. M. J. Appl. Electrochem. 1990, 20, 575. (18) Macpherson, J. V.;.Marcar, S.; Unwin, P. R. Anal. Chem. 1994, 66, 2175.

Figure 1. (a) Microjet geometry used by Unwin et al.11,12,18 in order to achieve the maximum mass transport rate (typical dimensions are given). (b) Uniformly accessible wall-tube geometry.

in Figure 1b. In this work, we assume that the jet and electrode have been aligned coaxially so that the electrode is uniformly accessible. Unwin et al. reported flow rates in the range 2 × 10-35 × 10-2 cm3 s-1 through a 30-60-µm-radius nozzle at distances varying from tens to hundreds of micrometers from the microdisk electrode (30-720 µm were used in these calculations, since Chin and Tsang13 found their empirical equation for a WTE to be valid in the region 0.4 < zjet/rjet < 12). Dimensionless rate constants were interpolated from the rotating disk working curve presented in ref 5 and converted by equating diffusion layer thicknesses. (5) For the rotating disk electrode (RDE), the operating range of rotation speed is between approximately 1 and 50 Hz and a typical radius is 0.25 cm. Dimensionless rate constants were interpolated from the working curve presented in ref 5. Analytical Chemistry, Vol. 71, No. 4, February 15, 1999

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Figure 2. Steady-state voltammetric time scales of some common electrode geometries. The gray zones show the effect of using a band electrode of 1-µm length (xe) in the channel flow cell.

(6) In the case of the wall-jet electrode (WJE), experimental flow rates are in the range 1 × 10-3-1 cm3 s-1 with a typical jet diameter of 0.3 mm impinging on an electrode of radius 0.1-1 cm. Dimensionless rate constants were interpolated from the working curves in ref 8. (7) For the channel electrode (ChE), the following typical parameters were used: d ) 0.6 cm; 2h ) 0.06 cm; w ) 0.4 cm; xe ) 0.1-0.4 cm; vf ) 1 × 10-3-0.3 cm3 s-1. The smallest microband that has been fabricated reliably either by a sandwiching method14 or lithography15 has a length of approximately 3-5 µm (xe ) 5 µm was used). The effect of using a 1-µm band has also been calculated. Dimensions for the fast-flow cell are as follows: d ) 0.2 cm; 2h ) 0.01 cm; w ) 0.15 cm. This can accommodate electrodes of 1-100 µm and flow rates of 1 × 10-2-2.5 cm3 s-1. Dimensionless rate constants were interpolated from the surfaces given in ref 6. The results are summarized graphically in Figures 2-4. The overall rate constant “window” (Figure 4) of each geometry is the product of the range of kinetic visibility at a particular geometry (Figure 3) and the range of time scales that can be accessed (Figure 2). It is clear from Figure 3 that the hydrodynamic electrodes have a narrower kinetic “window” (i.e., less inherent kinetic discrimination) than diffusion-only systems, but convection allows faster time scales to be accessed so the effect is more than offset. This effect can be seen in the working surfaces for both the ChE6 and the WJE8 in the broadening of the working curve with decreasing Peclet number. The conventional ChE, using a macroelectrode, and fast-flow channel both operate in the limit of negligible axial diffusion. The increased kinetic discrimination of the channel microband electrode arises from the use a of small microband at slow flow rates, thus incurring a significant amount of axial diffusion. By equating the average diffusion layer thickness, an approximate “equivalence” relationship19,20 may be deduced between electrode geometries, summarized in Table 4. Such an approach has, for example, allowed reaction layer theory derived for spherical21,22 and rotating disk electrodes23-26 to be applied to other (19) Oldham, K. B.; Zoski, C. G. J. Electroanal. Chem. 1988, 256, 11. (20) Amatore, C. A.; Fosset, B. Anal. Chem. 1996, 68, 4377. (21) Zhuang, Q.; Chen, H. J. Electroanal. Chem. 1993, 346, 29. (22) Zhuang, Q.; Sun, D. J. Electroanal. Chem. 1997, 440, 103.

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Figure 3. Broadness of the steady-state working curves (of Neff vs logarithm of the dimensionless rate constant) for ECE and EC2E processes, providing a measure of the inherent kinetic discrimination of some common hydrodynamic and microelectrode geometries.

electrode geometries.3 However, in the present context, the approximation provides a useful protocol by which to compare working curves of the shift in dimensionless half-wave potential, which tend to a linear limit corresponding to the response predicted by simple reaction layer theory. The values in the “equivalence” relationship depend on the constants included in the definition of the dimensionless rate constant, detailed in Table 3. Note that the dangers of using an equivalence approach to translate working curves between electrode geometries for quantitative kinetic analysis27,28 have been emphasized:7,29 the quality of the approximation is mechanism dependent;7,20,30 the approximation may not hold over the full working curve, especially at lower rate constants;7 although the approximation may hold well at steady-state, it may break down for time-dependent (23) Galus, Z.; Adams, R. N. J. Electroanal. Chem. 1962, 4, 248. (24) Albery, W. J.; Bruckenstein, S. J. Chem. Soc., Faraday Trans. 1 1966, 62, 1948, 2584. (25) Andrieux, C. P.; Nadjo, L.; Save´ant, J. M. J. Electroanal. Chem. 1973, 42, 223. (26) Compton, R. G.; Harland R. G.; Unwin P. R.; Waller A. M. J. Chem. Soc., Faraday Trans. 1 1987, 83, 1261. (27) Fleischmann, M.; Lasserre, F.; Robinson, J.; Swan, D. J. Electroanal. Chem. 1984, 177, 97. (28) Fleischmann, M.; Lasserre, F.; Robinson, J. J. Electroanal. Chem. 1984, 177, 115. (29) Alden, J. A.; Hutchinson, F.; Compton, R. G. J. Phys. Chem. B 1997, 101, 949. (30) Phillips, C. G. J. Electroanal. Chem. 1990, 296, 255. (31) Denuault, G.; Mirkin, M. V.; Bard, A. J. J. Electroanal. Chem. 1991, 308, 27.

Figure 4. Range of ECE and EC2E rate constants that may be measured at common hydrodynamic and microelectrodes. Table 4. Equivalence between Dimensionless Rate Constants at Various Electrode Geometries geometry relationship

difference in K

value

disk/sphere RDE/sphere ChE/sphere WJE/sphere WTE/sphere ChE/RDE wall-jet/RDE wall-tube/RDE

log KDisc - log Ksphere log KRDE - log Ksphere log KChE - log Ksphere log KWJE - log Ksphere log KWTE - log Ksphere log KChE - log KRDE log KWJE - log KRDE log KWTE - log KRDE

-3 log(4/π) -0.384 0.185 -0.310 0.237 0.569 0.074 0.621

experiments.20,29,31 The authors are unaware of any applications of equivalence relationships to quasi-reversible electrode processes. The equivalence transformations are applied in Figure 5 to compare the EC and EC2 working curves which were computed in refs 4-8. Note that the small difference between the WJE and RDE is about the same for both EC and EC2 processes (correcting earlier work,2 where a marked difference was observed for the EC2 process). One might expect second-order kinetics to pronounce the nonuniform concentration distribution at the electrode surface. This effect does not appear to be “felt” significantly in the shift in the half-wave potential. Comparing the hydrodynamic electrodes, the ECE data in Figure 4 support the findings of Compton et al.2 that the WJE offers slightly more kinetic discrimination than the ChE. Similarly Unwin and Compton3 found virtually identical working curves for the ChE and RDE, as are evident in Figure 4. Figure 5 shows, for EC and EC2 mechanisms, that the responses of the hydrodynamic

Figure 5. Working curves of dimensionless half-wave potential shift as a function of the dimensionless rate constant, superimposed by equivalent diffusion layer thickness for (a) EC mechanism (reaction layer equation:21 ∆Θ1/2 ) ln(1 + K1/2)) and (b) EC2 mechanism (reaction layer equation22 adapted for the rate law shown in Table 1: ∆Θ1/2 ) ln(K/21/2)/3).

electrodes are again virtually identical (although it could be argued that there is a marginal trend in increasing broadness in the order RDE < ChE < WJE). CONCLUSIONS The figures presented should allow at-a-glance selection of an appropriate electrode geometry to study a given rate constant using steady-state voltammetry. The microdisk, wall-jet, and channel electrodes may be used to span a broad range of rate constantss∼8 orders of magnitude. The latter geometries obviously offer more convenience/reproducibility in that the apparatus does not need to be disassembled and the working electrode changed to modulate the rate of mass transport. In the case of the channel, an array of bands of increasing thickness32-34 has been used to span the full range of rate constants within a single piece of apparatus. The fast-flow channel and microjet electrodes are candidates for reaching submicrosecond time scales using steady-state voltammetry. In the case of the fast-flow channel microband system, the electrode length may be reduced. However, assuming the present cell design and flow velocity are not optimized further, the length of the electrode in the flow direction would need to be (32) Alden, J. A.; Feldman, M. A.; Hill, E.; Prieto, F.; Oyama, M.; Coles, B. A.; Compton, R. G. Anal. Chem. 1998, 70, 1707. (33) Prieto, F.; Oyama, M.; Coles, B. A.; Alden, J. A.; Compton, R. G.; Okazaki, S. Electroanalysis 1998, 10, 685. (34) Cooper, J. A.; Compton, R. G. Electroanalysis 1998, 10, 141.

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reduced by 2 orders of magnitude to ∼14 nm to reach submicrosecond time scales in a steady-state experiment. Alternatively one could either increase the flow velocity via volume flow rate (again 2 orders of magnitude would be necessary) or reduce the cell height (reducing this parameter alone would require a factor of 10 resulting in a cell height of ∼10 µm). Either way, a substantial increase in the driving pressure would be required. Also, unless the cell height is reduced, the Reynolds number would increase, which may result in turbulent flow over the electrode. In the case of the MJE, the only practical option is to increase the jet velocity (since for uniformly accessible conditions, the time scale is independent of electrode radius) by either increasing the volume flow rate or reducing the nozzle radius. To reach submicrosecond time scales, the volume flow rate would need to be increased by a factor of ∼14, yet the same could be achieved by reducing the jet radius and separation a factor of ∼2.5, which seems very attractive. However, as with the channel, higher pressures would be required to overcome the increased friction in the nozzle. A practical drawback to reduction of the cell height or jet diameter is the susceptibility to blockages, which has already

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presented difficulties in preliminary experiments on a 60-µmdiameter jet (similar to Macpherson’s18). A major advantage of the channel geometry is the wellcharacterized laminar flow. For the MJE, one must either perform a numerical simulation of the hydrodynamics or rely on Chin and Tsang’s empirical description of the flow.13 The channel geometry also removes practical difficulties associated with jet alignment precision and electrode erosion. ACKNOWLEDGMENT J.A.A. thanks the EPSRC for a Quota Award and Keble College for a scholarship. We thank Chris Brett, Robert Dryfe, and Jon Cooper for correspondence concerning practical electrode dimensions and operating conditions. We also thank Julie Macpherson for her correspondence regarding microjet electrodes and Joanne Aixill for bringing practical difficulties of flow through small nozzles to our attention. Received for review August 13, 1998. Accepted November 11, 1998. AC9809128