Anal. Chem. 1998, 70, 1707-1720
Channel Microband Electrode Arrays for Mechanistic Electrochemistry. Two-Dimensional Voltammetry: Transport-Limited Currents John A. Alden, Mark A. Feldman, Emma Hill, Francisco Prieto, Munetaka Oyama, Barry A. Coles, and Richard G. Compton*
Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom Peter J. Dobson and Peter A. Leigh
Department of Engineering, Oxford University, South Parks Road, Oxford OX1 3PJ, United Kingdom
A channel electrode array, with electrodes ranging in size from the millimeter to the submicrometer scale, is used for the amperometric interrogation of mechanistically complex electrode processes. In this way, the transportlimited current, measured as a function of both electrode size and electrolyte flow rate (convection), is shown to provide a highly sensitive probe of mechanism and kinetics. The application of “two-dimensional voltammetry” to diverse electrode processes, including E, ECE, ECEE, EC′, and DISP2 reactions, is reported. Electrolytic reactions are rich in mechanistic diversity and complexity. Voltammetric methods for their interrogation have often employed variable mass transport to probe kinetic time scales and reaction mechanisms. One way in which this has been achieved is through variable convectionsfor example, by changing the solution flow rate at a channel1 or wall-jet2 electrode or the angular speed of a rotating disk electrode.3 A complementary approach is through the use of microelectrodes, where altering the dimensions of the electrodes changes the dynamic range of the voltammetry.4,5 The aim of the work reported in this paper is the use of both variable convection and electrode size in the same experiment. This will be shown to have an arguably unrivaled capacity for amperometric mechanistic discrimination through the evolution of a “two-dimensional voltammetry” experiment. This is accomplished by using a channel microband electrode array (Figures 1 and 2). At any particular flow rate, the rate of mass transport to each electrode differs, so each has its own characteristic kinetic “window”. The measurement of steady-state voltammograms, for a wide diversity of flow rates and different electrode sizes, provides a (1) Compton, R. G.; Dryfe, R. A. W. Prog. React. Kinet. 1995, 20, 245. (2) Brett, C. M. A.; Oliveira Brett, A. M. C. F. Compr. Chem. Kinet. 1986, 26, 355. (3) Albery, W. J.; Jones, C. C.; Mount, A. R. Compr. Chem. Kinet. 1989, 29, 129. (4) Wang, J. Microelectrodes; VCH: New York, 1990. (5) Montenegro, M. I. Res. Chem. Kinet. 1994, 2, 1. S0003-2700(97)01248-1 CCC: $15.00 Published on Web 03/28/1998
© 1998 American Chemical Society
pair of complementary mass transport components by which to study electrode processes.6 The experimental results can be most conveniently visualized as a surface of limiting current as a function these two parameters. Numerical simulations, surveyed in the following section, allow the prediction of the current as a function of both parameters. Comparison of the theoretical surface with the experimental surface is the essential basis of 2-D voltammetry, as is addressed below. To assist the wider availability of this and other information for the analysis of channel electrode and channel array electrodes, we have established a data analysis service on the World Wide Web7 (http://physchem.ox.ac.uk:8000/wwwda), in which working surfaces are stored using the results of the (computer intensive) simulations outlined in this work. Analyses for E, EC, EC2, ECE, EC2E, DISP1, DISP2, and EC′ mechanisms are available to all without the requirement of any knowledge of numerical modeling or expensive computer hardware. The results reported below make extensive use of this facility. THEORY To interpret the 2-D amperometric experiments reported below, it is necessary to predict the channel electrode response for a diversity of electrode reaction mechanisms in terms of various parameters. These describe the coupled homogeneous chemistry, the cell geometry, and, of course, the rate of mass transport as controlled by variation of both the electrode length and the solution flow rate. To simulate this response, the pertinent coupled convective-diffusion equations must be solvedsthis is accomplished through finite-difference simulations. This section summarizes the basis of these simulations and the results as applied to the mechanisms of interest below. Equation 1 gives the time dependence of the concentration of a species, A, at any point in solution in a channel flow cell, (6) Alden, J. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, 9741. (7) Alden, J. A.; Compton, R. G. Electroanalysis, in press.
Analytical Chemistry, Vol. 70, No. 9, May 1, 1998 1707
Figure 1. The 13-electrode array. The dotted lines delineate the position of the cell. The electrode lengths (xe values) double from 0.5 µm to 2 mm.
Figure 2. Schematic diagram of the assembled channel electrode array, showing the coordinate system used to characterize the flow cell. Note that some authors, particularly in diffusion-only work, use “width” to refer to the band’s shortest dimension. Here we use the hydrodynamic convention: “length” is the distance, xe, in the direction of flow, and “width” is in the z-coordinate.
∂2[A] ∂2[A] ∂[A] ) DA + + DA - vx ∂t ∂x ∂x2 ∂y2
∂[A]
steady state asymptotically.13 For sufficiently large electrodes and fast flow rates, if
n
∑(homogeneous step)
i
(1)
DAh/2voxe2 , 1
(3)
i)1
where x is the coordinate in the direction of flow and y is that perpendicular to the electrode surface (Figure 2). DA is the diffusion coefficient of species A, and vx is the flow velocity in the x direction, which is a parabolic function of y:
vx ) vo{1 - (h - y) /h } 2
2
(2)
where vo is the velocity at the center of the channel and 2h is the channel depth. Equation 1 has been solved computationally using the hopscotch and alternating direction implicit (ADI) methods,8 the strongly implicit procedure (SIP),9 the multigrid method,10 and preconditioned Krylov subspace (PKS) methods.11,12 Of these the SIP, PKS, and multigrid methods allow the steady-state response to be simulated directly, which is vastly more efficient than using relaxation methods such as ADI or hopscotch to approach the (8) Heinze, J. J. Electroanal. Chem. 1981, 124, 73. (9) Compton, R. G.; Dryfe, R. A. W.; Wellington, R. G.; Hirst, J. J. Electroanal. Chem. 1995, 383, 13. (10) Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 415, 1. (11) Alden, J. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, 8941. (12) Alden, J. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, 9606.
1708 Analytical Chemistry, Vol. 70, No. 9, May 1, 1998
(where xe is the electrode length), it can be shown that diffusion in the x-coordinate is negligible in comparison with convective transport,14 so that, under steady-state conditions, eq 1 becomes
∂2[A] ∂[A] + 0 ) DA - vx ∂x ∂y2
n
∑(homogeneous step)
i
(4)
i)1
where the last term includes the possibility of species A reacting homogeneously. n denotes the number of distinct homogeneous reactions in which it participates. Condition 3 shows that diffusion in the x-coordinate may be neglected for macroelectrodes but not for microelectrodes.16 In such cases, the favored numerical technique is the backward implicit (BI) method,15 which is highly efficient, allowing the steady-state response to be simulated without iteration. Mechanisms already addressed in the literature for macroelectrodes include simple electron transfers for electrochemically reversible, quasi-reversible, and irreversible cases;16 EC, ECE, (13) Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 402, 1. (14) Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 404, 27. (15) Compton, R. G.; Pilkington, M. B. G.; Stearn, G. M. J. Chem. Soc., Faraday Trans. 1988, 84, 2155. (16) Compton, R. G.; Wellington, R. G. Electroanalysis 1992, 4, 695.
DISP1, and DISP2 reactions;15 dimerizations;17,18 and complexation reactions.19 Channel microband electrodes have been simulated for reversible, irreversible, and quasi-reversible electron transfer.25 Also, ECE and EC2E homogeneous processes have been simulated.6 Simulation results are most usefully stored in the form of working curves and surfaces.6 The basis for these can be appreciated if we introduce the Le´veˆque approximation,20 which is valid near y ) 0:
(
(h - y)2
(
(h - y) (h - y) 1+ h h
vx ) vo 1 -
) vo 1 ≈
2voy h
h2
)
)(
∂2[C] ∂2[C] ∂[C] ∂[C] ) D C 2 + D C 2 - vx + k(ii)[B] ) 0 ∂t ∂x ∂y ∂x
where k(ii) is the rate constant corresponding to reaction ii. The transport-controlled current for the ECE process is characterized by the effective number of electrons transferred, Neff, which lies between 1 and 2.21-23 Dimensional analysis shows that, under the Le´veˆque approximation, Neff is a sole function of two parameters if D ) DA ) DB ) DC. These are the shear Peclet number,
Ps ) 3xe2Vf/2h2dD
)
for y ≈ 0
{(2h) /DA} . (xe/vo)
(10)
where Vf ()4vohd/3) is the volume flow rate, and a dimensionless rate constant,
KECE ) {k(ii)xe2/D}{Ps}-2/3
(5)
This linearizes the parabolic velocity profile and is valid when the time taken to diffuse across the depth (2h) of the cell is long compared to the time taken to be transported along the length of the electrode, so that 2
(9)
(11)
Alternatively, if the Le´veˆque approximation is inapplicable (although as in the work reported below it is easy to design experimental cells for which it is valid),1 then Neff depends on KECE and the parameters
p1 ) xe/h
(12)
(6) which is the aspect ratio, and the Peclet number,
The merits of this approximation can be seen when other than a simple electrode transfer process is examined. For example, consider an ECE reaction, -
E step
A(e fB
(i)
C step
BfC
(ii)
E step
C ( e- f products
(iii)
where C is more readily reduced/oxidized than A. The steadystate coupled mass transport and kinetic equations are
∂[A] ∂2[A] ∂2[A] ∂[A] ) DA 2 + DA 2 - vx )0 ∂t ∂x ∂y ∂x ∂[B] ∂2[B] ∂2[B] ∂[B] ) DB 2 + D B 2 - vx - k(ii)[B] ) 0 ∂t ∂x ∂y ∂x
(7)
(8)
(17) Compton, R. G.; Pilkington, B. G. J. Chem. Soc., Faraday Trans. 1 1989, 85, 2255. (18) Compton, R. G.; Eklund, J. C.; Nei, L.; Bond, A. M.; Colton, R.; Mah, Y. A. J. Electroanal. Chem. 1995, 385, 249. (19) Cooper, J. A.; Alden, J. A.; Oyama, M.; Compton, R. G.; Okazaki, S. J. Electroanal. Chem., in press. (20) Le´veˆque, M. A. Ann. Mines. Mem. Ser. 1928, 12/13, 201. (21) Amatore, C.; Save´ant, J. M. J. Electroanal. Chem. 1977, 85, 27. (22) Amatore, C.; Gareil, M.; Save´ant, J. M. J. Electroanal. Chem. 1983, 147, 1. (23) Leslie, W. M.; Alden, J. A.; Compton, R. G.; Silk, T. J. Phys. Chem. 1996, 100, 14130. (24) Britz, D. Digital Simulation in Electrochemistry, 2nd ed.; Springer-Verlag: Berlin, 1988. (25) Bidwell, M. J.; Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 404, 27.
p2 ) Vf/dD
(13)
Ps ) (3/2)p2(p1)2
(14)
Note that
It is therefore evident that simulation results can be stored as working curves or surfaces relating observables such as Neff to a small number of independent (mechanism- and geometry-dependent) parameters.19 References 1 and 9 give typical examples. The simulations results, reduced to working surfaces, provide a readily accessible means by which experimentalists can interpret data without direct recourse to computation. Interpolation allows, for each experimental data point characterized by a flow rate and an electrode width, the appropriate theoretical parameter to be generated. For example, for a simple electron transfer, the limiting current may be interpolated from a surface of p1 and p2. For an ECE process (at a macroelectrode under Le´veˆque conditions), a working surface of Ps and KECE permit the interpolation of theoretical values of Neff. A measure of the fit of the experimental and theoretical surfaces is obtained through evaluation of the mean scaled absolute deviation (MSAD). In the case of transport-limited current data, this is defined as follows
MSAD )
() 1
∑| N N
Ilim(expt) - Ilim(theory) | Ilim(theory)
(15)
where N is the number of experimental points that lie within the boundaries of the simulated working surface. By definition, the Analytical Chemistry, Vol. 70, No. 9, May 1, 1998
1709
best fit of the experimental and theoretical values corresponds to the case of minimum MSAD. Here, BI simulations were used to generate working curves and surfaces for macroelectrodes, and multigrid simulations were used to extend the ECE and EC2E working surfaces given in ref 18. These were of high enough resolution that the (bi)linear interpolation error was negligible (
