J. Phys. Chem. C 2007, 111, 15053-15058
15053
Linear Sweep Voltammetry at Randomly Distributed Arrays of Microband Electrodes Ian Streeter and Richard G. Compton* Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford, United Kingdom OX1 3QZ ReceiVed: June 18, 2007; In Final Form: August 2, 2007
Randomly distributed microband arrays are characterized in terms of their voltammetric response to a reversible electron transfer in a linear sweep experiment. The conditions required for diffusional independence of the bands are quantified. It is shown that bands must be packed less densely at a random array compared with a regular array to achieve an equivalent level of current sensitivity.
1. Introduction Ultramicroelectrodes are routinely used in analytical electrochemistry because of their many advantageous properties.1 Nonlinear diffusion and edge effects lead to an increase in current density and enhanced sensitivity. Their small size reduces the problems associated with capacitance and ohmic distortions, allowing faster scan rates to be reached and therefore the study of ultrafast kinetic processes.2 An array of ultramicroelectrodes wired in parallel can offer most of the advantages of a single microelectrode but with the added benefit of a higher total current output.3-6 The packing of the electrodes within the array determines the diffusional behavior of the electroactive species. Space efficiency is maximized with a dense packing of the electrodes. However, to take advantage of the microelectrodes’ attractive properties, they must retain sufficient separation to be diffusionally independent. An array may be described as randomly or regularly distributed, referring to the spacing of electrodes within the array. Typically, the measured current is reduced if the distribution of microelectrodes is random as compared with when they are regularly packed. This phenomenon has been described analytically7 and by numerical solution of diffusion at microdisc arrays.5 This work considers randomly distributed arrays of microband electrodes. Numerical simulations are performed for a fast reversible electron transfer in a potential sweep experiment. Regularly distributed arrays of microbands have been previously used in experiments8 and have been characterized by analytical methods9,10 and by numerical simulations.11 The randomly distributed array of microbands is of interest because of its relative ease of fabrication. 2. Theory 2.1. Simulating Random Arrays. Figure 1a shows a schematic diagram of the electrode arrays considered in this paper. The constituent microband electrodes are uniform in terms of their dimensions, with their length being many times greater than their width. They all lie in the x,y plane, with their longer edges parallel to the y coordinate. The separation distance of bands along the x coordinate is described by a one-dimensional probability distribution function. * Corresponding author. Fax: +44 (0) 1865 275410. Phone: +44 (0) 1865 275413. E-mail:
[email protected].
Figure 1. (a) Schematic diagram of a section of a randomly distributed microband array. The theoretical array extends to infinity in both the x and the y directions. (b) Cross section through the x,z plane.
In this work, we consider two different distributions. Distribution 1 is described by eq 1, where s is the center-to-center separation of two neighboring microbands, 〈s〉 is the mean separation, and f1(s) ds gives the probability of finding two neighboring points with separation between s and s + ds.
( )
f1(s) ) exp -
s 〈s〉
(1)
Distribution 1 is appropriate when the microbands are positioned randomly and independently along the x coordinate. However, the function f1(s) is nonzero at small s, where the center-tocenter separation is less than the width of a microband. This leads to overlap of some bands, decreasing the array’s total electrode surface area. Distribution 2 is described by eqs 2 and 3, where w is the width of the microband.
s g w:
(
f2(s) ) exp s < w:
s-w 〈s〉 - w
f2(s) ) 0
)
(2) (3)
Distribution 2 is appropriate for randomly positioned bands when it is physically impossible for overlap to occur. In previous work on regular microband arrays,11 the array dimensions were described by the parameters xe and xgap, which represented half the microband width and half the band gap width, respectively (Figure 1b). In this work, we describe a
10.1021/jp0747205 CCC: $37.00 © 2007 American Chemical Society Published on Web 09/26/2007
15054 J. Phys. Chem. C, Vol. 111, No. 41, 2007
Streeter and Compton
Figure 2. Schematic diagram of diffusion categories 1-4 at a microband array.
TABLE 1: Boundary Conditions for Eq 6 time
boundary
condition
description
t)0 t>0 t>0 t>0 t>0
all x, z x)0 x ) xe + xgap z ) 6xDt z ) 0, x > xe
[A] ) [A]bulk ∂[A]/∂x ) 0 ∂[A]/∂x ) 0 [A] ) [A]bulk ∂[A]/∂z ) 0
initial conditions zero flux through domain boundary zero flux through domain boundary bulk solution zero flux through insulating boundary
TABLE 2: Dimensionless Parameters Used for Numerical Simulation parameter
expression
concentration of species A lateral coordinate normal coordinate time mean band gap width scan rate potential electrode flux
a ) [A]/[A]bulk X ) x/xe Z ) z/xe τ ) Dt/x2e X h gap ) xjgap/xe σ ) F/RT νx2e /D θ ) F/RT (E - E Qf ) j ) -i/2NFDl[A]bulk
randomly distributed array using the parameters xe and xjgap, where xjgap is the mean value of xgap across the whole array. The number of microbands in the array, N, and their macroscopic length in the y direction, l, are considered to be sufficiently large that the contribution to current at the edge of the array may be ignored. 2.2. Diffusion at Microband Arrays. As current is driven across a microband array, a “diffusional zone” is created at each microband as the electroactive species is depleted. As the time of the experiment progresses, these diffusional zones extend further into solution, often to the extent that they overlap with those of neighboring microbands. In the cylindrical diffusion limit, the diffusion layer thickness increases with the logarithm of time.12 The diffusional behavior of the electroactive species depends upon the extent of the diffusion zone overlap, which in turn depends upon an interplay of the relative sizes of the diffusion layer, the microband width, and the band separations. For the potential sweep experiments considered in this work, the diffusion layer thickness is dependent on the scan rate used. The length of the microbands has no effect on the nature of the diffusion, since it is orders of magnitude greater than a diffusion layer thickness. Regular microband arrays can exhibit five different categories of diffusion, each readily distinguishable by a measure of the voltammetric current.11 Since the arrays considered in this work have a random distribution, it is possible that different regions of the same array will exhibit different diffusional behavior. However, a randomly distributed array may still be categorized according to its overall diffusional behavior. A description of the diffusion categories is given here, including how the voltammetric current is used to identify such overall behavior
Figure 3. Simulated linear sweep voltammetry of an electrochemically reversible electron transfer at a randomly distributed microband array. Distribution 1 is used; σ ) 0.1; X h gap varies from 0.01 to 1000.
TABLE 3: Categorization of Diffusion Regimes at a Microband Array category 1 2 3 4 5
condition 1/2 -1
jp e 0.4463σ and jp g jiso p jp g jiso p jp < 0.4463σ1/2(1 + Xgap) and jp < jiso p jp g 0.4463σ1/2(1 + Xgap) jp g 0.4463σ1/2(1 + Xgap) and jp e 0.4463σ1/2-1
at a random array. Diffusion categories 1-4 are illustrated in Figure 2. Category 1: The total current measured at the array is that expected for a macroelectrode with a geometric area equal to the sum of its constituent microbands. This behavior occurs when the diffusion layer thickness is much smaller than the electrode width such that mass transport to a band is dominated by planar diffusion. The average separation of bands must be large to minimize the electrode surface area lost to band overlap by the random distribution. Category 2: The total current measured at the array is that expected for a series of diffusionally independent microband electrodes. The constituent bands must have an average spacing large enough that there is no significant overlapping of either the bands themselves or their associated diffusional zones. This is typically the preferred behavior of a microband array since it allows for rapid diffusion to the band edge, which leads to the heightened sensitivity typically associated with a single microelectrode. Category 3: The individual microband electrodes are no longer diffusionally independent. There is significant overlapping of neighboring diffusion zones, reducing the flux density along the band edges. This is the behavior that an analytical electrochemist would typically like to avoid.
Randomly Distributed Microband Arrays
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Figure 4. Zone diagrams showing the category of diffusional behavior exhibited in a simulated linear sweep for varying σ and X h gap. Categorization is described in Table 3 with the value of shown in the figure. Solid lines are the boundaries for the randomly distributed array. Dashed line shows the boundary between categories 2 and 3 for a regularly distributed array.
Category 4: The total current measured is that expected for a macroelectrode with a geometric area equal to the entire microband array. This behavior is observed when neighboring diffusional zones overlap to the extent that the overall concentration profile is planar. Within this limit, it is possible for neighboring microbands to overlap each other significantly, reducing electrode surface area, while having very little effect on the overall diffusion profile. Category 5: This is the unusual case where each microband experiences a planar diffusional flux, and the overall array also behaves as a macroelectrode. This is only possible at an array with a very high surface coverage of electrode. Category 5 diffusion provides a link between categories 1 and 4. 2.3. Diffusion Domain Approach. In our previous work on microband arrays, the simulation problem was simplified by identifying the presence of a diffusionally independent unit cell.11 A y,z plane of symmetry was present at the center of each microband and at the midpoint between two neighboring microbands, across which the net diffusional flux must be equal to zero. Diffusion only needed to be simulated at a single diffusion domain between these planes of symmetry. A randomly distributed microband array does not contain any planes of symmetry. However, the array can be partitioned in an analogous manner into smaller units, the boundaries being the y,z planes at the center of each microband and at the midpoint between any two neighboring microbands. The approximation is made of zero net flux across these planes, so
that each resulting diffusion domain may be considered independently. In contrast with the regular array, these diffusion domains will all be of different sizes with a distribution described by eq 1 or eq 2. The first step in calculating the current response of a randomly distributed microband array is to simulate the mass transport at a range of different sized diffusion domains. The current response of each domain is then weighted according to the probability distribution function (eq 1 or 2). The sum of these weighted current responses gives the voltammetric response for the whole array. This approach of partitioning a randomly distributed array into diffusion domains of varying size has been previously used in simulations of microdisc arrays5,13 and partially blocked electrodes.14-16 The (z ) 0) boundary of a diffusion domain typically consists of two regions, one corresponding to the microband electrode and one to the surrounding insulating surface. It was noted in section 2.1 that partial overlapping of microbands is possible in distribution 1. The diffusion domains that arise from this situation are unusual in that the (z ) 0) boundary consists entirely of the electrode region. The contributing current of these domains is found by simulating the one-dimensional linear diffusion response.17 2.4. Electron Transfer. Equation 4 shows the electron transfer considered in these numerical simulations. Only species A is present in bulk solution.
A + e- h B
(4)
15056 J. Phys. Chem. C, Vol. 111, No. 41, 2007
Streeter and Compton
The reduction is fast and reversible, such that the electrode surface concentrations of species A and B are described at all times by the Nernst equation:
[A]elec [B]elec
) exp
(RTF (E - E )) Q f
(5)
where E is the electrode potential and E Qf is the formal electrode potential. A potential sweep experiment is considered whereby the potential is varied at a uniform scan rate, ν. The diffusion of species A is described by Fick’s second law:
(
)
∂2[A] ∂2[A] ∂[A] + )D ∂t ∂x2 ∂z2
(6)
The diffusion coefficient, D, is assumed to be equal for both species, such that their concentrations at any point in solution may be described by eq 7, and the concentration profile of species A may be simulated independently from species B.18
[A] + [B] ) [A]bulk
(7)
The concentration profile of species A is found for a series of time steps by solving eq 6 subject to eq 5 and the additional boundary conditions in Table 1. The current, i, is found at any time using eq 8, which is an integral over the electrode surface area:
i ) -F
∫ ∫elec
∂[A] dx dy ∂z
(8)
2.5. Normalized Model and Numerical Simulation. The model described in section 2.4 is normalized using the dimensionless parameters in Table 2. Using this normalized approach, we describe a microband array by the parameter X h gap, and a potential sweep experiment is described by the normalized scan rate, σ. The voltammetric output is described by a normalized diffusional flux density through the electrode, j, and its variation with normalized potential, θ. The value of j applies to a single microband in the array; it is not normalized per unit area. The alternating direction implicit finite diffference method19 is used to simulate the voltammetric response of the diffusion domains. An expanding space grid is used as discussed previously,11,20 and the same procedures were followed in this paper. The simulation program is tested for convergence to ensure that the spatial and temporal grids lead to less than 1% variation from the asymptotic peak current. 3. Simulation Results Potential sweep experiments were simulated for a reversible electron transfer at a randomly distributed array of microband electrodes. The scan rate and mean band separation width were varied across the range -5 e log10 σ e 2 and -2 e log10 X h gap e 3. The equivalent set of data for a regularly distributed array has been previously recorded.11 The range of scan rates used allows the response of the arrays to be found for varying diffusion layer thickness. Figure 3 shows a selection of the fluxpotential plots for distribution 1 using a scan rate of σ ) 0.1. There is an alteration in appearance of the voltammetry as the mean band separation, X h gap, is varied. This resembles the simulated voltammetry reported for a regular array of microbands,11 where the variation was attributed to a procession through different categories of diffusion. The simulated peak flux can be used to categorize the overall diffusional behavior at a microband array. Table 3 summarizes
reg Figure 5. Ratio jran simulated for varying σ and X h gap. (a) p /jp Distribution 1. (b) Distribution 2.
the conditions that must be satisfied for a categorization. In each case, is a parameter that alters how stringent the conditions are: the lower its value, the less stringent the conditions. The condition for category 2 compares the simulated values of jp with jiso p , its value for an isolated band, found by the limit of X h gap tending to infinity. Category 4 behavior is satisfied when the peak current is that expected for planar diffusion to a macroelectrode with geometric area equal to the microband array. The condition for category 4 in Table 3 compares jp with a normalized Randles-Sˇ evcˇ´ık expression for the peak flux.21 The category 1 condition compares the peak flux with that predicted by Randles-Sˇevcˇ´ık for planar diffusion to an electrode the size of a single microband. Diffusion is labeled as category 5 when the planar diffusion result is satisfied at the overall array and also at each individual microband. Category 3 behavior is simply that which cannot be described by the other four categories. Figure 4 presents zone diagrams for distributions 1 and 2, showing the behavior that was found from the simulated voltammetry using the conditions in Table 3 and values of ) 0.9 and ) 0.99. The equivalent zone diagrams for a regularly distributed array have been previously published.11 Category 2 diffusion is observed for large values of X h gap where the bands are so sparsely distributed that overlap of neighboring diffusional zones is negligible. Category 1 diffusion occurs at the fast scan rate end of the category 2 region, where each microband acts as an independent macroelectrode. Category 4 diffusion is observed at slow scan rates for which the diffusion layer thickness is large, especially at arrays with a high density of
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Figure 6. Simulated voltammetry of an electrochemically reversible electron transfer at a microband array. Curves are shown for the regular distribution and for both the random distributions for selected values of σ and X h gap. For b and c, all three curves overlay; for d, the curves for the two random distributions overlay.
microbands, where there is extreme overlapping of neighboring diffusional zones. Distribution 2 shows a transition into category 5 behavior as X h gap decreases and the fractional surface coverage of electrode approaches unity. Distribution 1 has no such transition because there is significant overlapping of the electrodes and therefore larger gaps are present between some microbands. An experimental electrochemist will typically want to work in region 2, where the presence of edge effects leads to heightened sensitivity; hence, it is important to know the position of the boundary between region 2 and region 3. The dashed lines in Figure 4 show the position of this boundary previously reported for a regular array of microbands.11 Changing an array’s distribution from regular to random places a greater restriction on the range of scan rates that may be used while still working in the type 2 region. For arrays with the same density of microbands, a random distribution has more significant overlapping of diffusion zones than a regular distribution. The overall diffusional behavior at a microband array is affected by the distribution of bands within the array. The magnitude of this effect can be described by comparing the simulated peak fluxes for the different distributions. Figure 5 reg shows the value of (jran p /jp ), which is the ratio of the peak fluxes recorded at the randomly and regularly distributed arrays, respectively, over the full range of X h gap and σ. It can be seen that for both distribution 1 and distribution 2 the region associated with category 3 behavior shows the largest relative difference in jp between the random and the regular arrays. Furthermore, for all values of X h gap and σ, the peak flux recorded at a random array of microbands does not exceed that for a regular array. This is in accordance with studies of regular and
random microdisc arrays.5,7 Figure 6 presents a comparison of voltammetry at regular and random arrays of microbands, for a selection of X h gap and σ values. 4. Conclusions Diffusional behavior at a microband array depends on the electrode width, the relative size of the diffusion layer, and the nature of the electrode distribution. In a linear sweep experiment, different diffusional behaviors may be observed by varying the dimensionless scan rate, σ, whose value also depends on the electrode width and the diffusion coefficient of the electroactive species. In electrochemical experiments, it is usually desirable to use microband arrays that will operate with category 2 diffusion. The conditions required for this behavior have been quantified for randomly distributed microband arrays. The conditions are more restrictive than for a regularly distributed array: bands must be packed less densely at a random array compared with those at a regular array to achieve an equivalent level of current sensitivity. The arrays considered in this paper have been for completely random distributions of the electrode when overlapping is either allowed or prohibited. The exact distribution of bands in a real array will depend on the method of fabrication, and distributions other than the two considered here are possible. However, it is a general result that for a fixed number of microbands over a fixed array area, the regular distribution gives the maximum possible current output. Random arrays give a reduced current because of more significant overlapping of diffusion zones in regions of high local electrode density.
15058 J. Phys. Chem. C, Vol. 111, No. 41, 2007 Acknowledgment. I.S. thanks the EPSRC for financial support via a DTA studentship. References and Notes (1) Forster, R. J. Chem. Soc. ReV. 1994, 23, 289-297. (2) Andrieux, C. P.; Hapiot, P.; Save´ant, J. M. Chem. ReV. 1990, 90, 723-738. (3) Feeney, R.; Kounaves, S. P. Electroanalysis 2000, 12, 677-684. (4) Reller, H.; Kirowa-Eisna, E.; Gileadi, E. J. Electroanal. Chem. 1982, 138, 65-77. (5) Davies, T. J.; Compton, R. G. J. Electroanal. Chem. 2005, 585, 63-82. (6) Chevallier, F. G.; Compton, R. G. Electroanalysis 2006, 18, 23692374. (7) Scharifker, B. R. J. Electroanal. Chem. 1988, 240, 61-76. (8) Le Drogoff, B.; El Khakani, M. A.; Silva, P. R. M.; Chaker, M.; Vijh, A. K. Electroanalysis 2001, 13, 1491-1496. (9) Amatore, C.; Save´ant, D.; Tessier, D. J. Electroanal. Chem. 1983, 147, 39-51.
Streeter and Compton (10) Huangxian, J.; Hongyuan, C.; Hong, G. J. Electroanal. Chem. 1992, 341, 35-46. (11) Streeter, I.; Fietkau, N.; del Campo, J.; Mas. R.; Mun˜oz, F. X.; Compton, R. G. J. Phys. Chem. C 2007, 111, 12058-12066. (12) Aoki, K.; Tokuda, K.; Matsuda, H. J. Electroanal. Chem. 1987, 225, 19-32. (13) Ordeig, O.; Banks, C. E.; Davies, T. J.; del Campo, J.; Mun˜oz, F. X.; Compton, R. G. J. Electroanal. Chem. 2006, 592, 126-130. (14) Davies, T. J.; Banks, C. E.; Compton, R. G. J. Solid State Electrochem. 2005, 9, 797-808. (15) Chevallier, F. G.; Davies, T. J.; Klymenko, O. V.; Jiang, L.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2005, 580, 265-274. (16) Davies, T. J.; Brookes, B. A.; Fisher, A. C.; Yunus, K.; Wilkins, S. J.; Greene, P. R.; Wadhawan, J. D.; Compton, R. G. J. Phys. Chem. B 2003, 107, 6431-6444. (17) Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 36, 706-723. (18) Streeter, I.; Compton, R. G. Phys. Chem. Chem. Phys. 2007, 9, 862-870. (19) Heinze, J. J. Electronanal. Chem. 1981, 124, 73-86. (20) Gavaghan, D. J. J. Electroanal. Chem. 1998, 456, 1-12. (21) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications; John Wiley and Sons: New York, 2001.
