Theory of Chronoamperometry at Cylindrical Microelectrodes and

J. Phys. Chem. C 2008, 112, 11637–11644

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Theory of Chronoamperometry at Cylindrical Microelectrodes and Their Arrays Edmund J. F. Dickinson, Ian Streeter, and Richard G. Compton* Department of Chemistry, Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford, United Kingdom OX1 3QZ ReceiVed: March 3, 2008; ReVised Manuscript ReceiVed: April 14, 2008

Potential step chronoamperometry at cylindrical electrodes is modeled using the finite difference method in a two-dimensional (2D) simulation space. Trends are discussed in terms of the cylinder height-to-radius ratio Ze for top-only, side-only, and fully conducting cylinders. The diffusion domain approximation is then used to simulate regular arrays of cylindrical microelectrodes supported on a conducting substrate, comparing the different shapes of chronoamperometric transients for varying Ze, surface coverage Θ, and top/side conductivity. R ) r/re ;

1. Introduction Cylindrical micro- and nanoelectrodes, especially when arrayed on conducting substrates, are important experimental systems. Wang and co-workers have produced electroactive carbon post arrays on glassy carbon substrates;1–3 “forests” of vertically aligned single-wall carbon nanotubes4 prepared on conducting substrates (Au or pyrolytic graphite) have been shown to be useful as nanoelectrode arrays5 and also have found use in biosensing.6 Cylindrical microelectrodes have also been constructed from conventional electrode materials such as Au and Pt wires, for such diverse applications as three-phase electrochemistry,7 fuel cell analysis,8 and glucose sensing.9 Past efforts at modeling Fickian diffusion to cylindrical electrodes have shied away from a comprehensive twodimensional (2D) treatment, however. The assumption that diffusion is negligible in the z direction, parallel to the cylindrical axis, reduces the system to a one-dimensional (1D) problem, which is easily solved: Amatore et al.10 and Weidner11 have discussed the properties of cylindrical microelectrodes using such a treatment. This approximation neglects convergent diffusion to the electrode top, however, which for shorter microand nanoelectrodes must be seen as important to both the observed current and the overall shape of the diffusion layer. This paper adapts techniques proven in previous work on 2D simulation of electrochemical processes12,13 to model chronoamperometry at the cylindrical electrode system, both for isolated cylinders and arrays of cylinders supported on a conducting substrate. The various combinations of cylinder top and cylinder side conductivity are also considered. Comparison is drawn throughout with the well-established behavior of the microdisc electrode, highlighting situations where an enhanced or otherwise interesting chronoamperometric response may be expected. 2. Theory for Simulating Cylindrical Electrodes 2.1. Cylindrical Electrode Geometry. A cylindrical electrode is naturally described by the cylindrical coordinate system, (r, z, φ), in which the cylinder is symmetric in φ. The problem is simplified by normalizing the linear coordinates r and z by the electrode radius re to dimensionless coordinates R and Z: * Corresponding author. Fax: +44 (0) 1865 275410. Tel.: +44 (0) 1865 275413. E-mail: [email protected].

Z ) z/re

(1)

The electrode radius Re is then equal to 1; the electrode height is given by a quantity Ze ) ze/re. The dimensionless area of the cylinder is the sum of contributions from the top and sides, i.e.,

Atop ) πRe2 ) π;

Aside ) 2πReZe ) 2πZe

(2)

and so, in comparison to a microdisc of the same radius re, the total dimensionless area is

Atot ) Adisc + 2πZe

(3)

Where only certain surfaces of the cylinder are electroactive, Aact is used to refer to this active dimensionless area. 2.2. General Simulation Procedure. The above dimensionless coordinate system may be extended to generate other dimensionless variables, presented in Table 1. Throughout, a simple E mechanism of the following form is assumed:

A a B + e-

(4)

This reaction has some formal potential relative to a standard hydrogen electrode, Ef. It is also assumed that the diffusion coefficients of the oxidized and reduced forms of the electroactive species are equal, i.e., DA ) DB in all cases, such that the normalized concentration of species B, b, is related to the normalized concentration of species A, a, by the equivalence b ) (1 - a) throughout the simulation space. In such a dimensionless system, Fick’s second law reduces to

∂a ∂2a 1 ∂a ∂2a + ) + ∂τ ∂Z2 R ∂R ∂R2

(5)

This is independent of all real parameters and so is easily solved by discretization in (R, Z) to a 2D grid of points over the simulation space. A series of coupled simultaneous equations is generated using an alternating direction implicit (ADI) finite difference method.14 These equations are solved using the Thomas (tridiagonal matrix) algorithm,15 implemented in C++, at successive steps in dimensionless time, τ, as defined in Table 1. The limits of the grid for an isolated electrode are chosen to exceed the electrode surface by 6
τmax, at which distance the diffusion layer is not apparent, so the grid limits at R ) Rmax

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Dickinson et al. To establish a steady-state current, the current as τ f ∞ was explored, as the ADI method does not permit direct solution of the steady-state equation (∂a/∂τ) ) 0 across the simulation space. The simulation parameters used were converged to e0.2% accuracy for the resulting steady-state current jss. Chronoamperometric transients were also simulated directly, using very short timesteps to ensure a correct result. Convergence to the Cottrellian slope, (∂log j/∂log τ) ) -1/2, was used to assess accuracy. The following boundary conditions then apply, for an isolated cylindrical electrode:

TABLE 1: Dimensionless Variables Used in Numerical Simulation, Where cA and cB are the Concentrations of A and B, Respectively parameter

definition

R Z τ a b j

r/re z/re DAt/re2 cA/cA,bulk cB/cA,bulk i/2πFDAcA,bulkre

and Z ) Zmax may be considered equivalent to bulk solution. For arrays, a diffusion domain approach is used, as described below. An expanding grid of points in (R, Z) is used, in order to maximize the density of points close to the singularities16 where the cylinder meets a symmetry axis or an insulating boundary and also close to the singularity where the top of the cylinder meets the side. The grid used is similar to that used in past work,17 expanding in both directions from R ) Re and Z ) Ze. Formally:

Ri+1 - Ri ) hi ; hi ) γRhi-1 ;

Zi+1 - Zi ) ki

(6)

ki ) γZki-1

(7)

In all simulations conducted, h0 ) k0 and γR ) γZ. An example of the grid is shown in Figure 1; values of γR ) 1.125 and h0 ) 10-5 were typical. An expanding time grid is also used, where the timesteps expand as
τ, in line with diffusion layer thickness:

τ < τs

τi ) τi-1 + γT√τs ;

τ g τs

τi ) τi-1 + γT√τi-1

(8)

This also effectively reduces simulation time without harming accuracy, provided the parameters γT and τs are properly converged. 2.3. Simulation of Chronoamperometry. A typical chronoamperometric experiment involves a rapid potential step to a highly oxidizing potential, E . EQf . A potential step is modeled simply by setting the concentration of A, the oxidized species A at the electrode surface, ael, to be zero. This generates a singularity at τ ) 0 where the flux is analytically infinite; in all cases, timesteps were sufficiently dense to render any oscillations resulting from the τ ) 0 singularity negligible in the region of interest.

Zmax ) Ze + 6√τ;

Rmax ) 1 + 6√τ

Z ) Zmax

a)1

R ) Rmax

a)1

R < 1, Z < Ze

a)0

R g 1, Z g Ze, τ ) 0

a)1 ∂a )0 ∂R ∂a )0 ∂Z

R ) 0, Z g Ze Z ) 0, R g 1

(9)

(10)

The electrode top (Z ) Ze, R < 1) and electrode side (R ) 1, Z < Ze) have varying boundary conditions, depending on whether they are considered insulating or conducting surfaces. In the insulating case, these surfaces have zero-flux boundary conditions of the form (∂a/∂R)el ) 0 or (∂a/∂Z)el ) 0. In the conducting case, the boundary condition ael ) 0 is used. 2.4. Calculation of Flux. Dimensionless flux for a microdisc is defined as

j)

∂a ∫01 ( ∂Z )elR dR ) 2πFDAicA,bulkre

(11)

An equivalent definition is suitable for the top of a cylindrical electrode; for the sides, where dimensional area A ) 2πzere, the following applies:

i ) FDcA,bulk

dA ∫A ( ∂a ∂r )el

(12)

z ∂a ∂a z ∂a ∂A ∂R ∂z dA ) ∫0 ( ) dz ) ∫0 ( ) dZ ∫A ( ∂a ) ∂r el ∂r el ∂r ∂r el ∂z ∂r ∂z e

e

(13) Substituting

i ) 2πFDAcA,bulkre

∂a ∫0z ( ∂R )el dZ e

(14)

Therefore, provided the respective surfaces are conducting

jtotal ) jtop + jside jtop )

∂a ∫01 ( ∂Z )elR dR ∂a ∫0Z ( ∂R )el dZ

(15)

i 2πFDAcA,bulkre

(16)

jside )

e

where in both cases

j)

Figure 1. Expanding grid of points used in this work (some lines excluded for clarity).

3. Isolated Cylindrical Electrodes 3.1. Mass-Transport-Limited Currents, jss. Steady-state currents were considered in the limit τ f ∞, as explained in

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Figure 4. jssAact-1 vs log10 Ze for a side-only conducting cylinder. Figure 2. Diagram presenting the various conductivity cases for the isolated cylinders studied in section 3.

Figure 3. jssAtot-1 vs log10 Ze for fully conducting and side-only conducting cylinders.

section 2.3, for a fully conducting cylinder, a top-only conducting cylinder, and a side-only conducting cylinder. These conductivity combinations are presented at Figure 2. In all cases, the cylinder was assumed to be supported on an insulating surface. Steady-state or mass-transport-limited currents, jss, were calculated for a range of Ze values over 4 orders of magnitude, from squat, disklike cylinders with Ze ≈ 0.1 to very tall, macroscopic cylinders with Ze ≈ 2000. The significant limits for all cases are Ze ) 0, a disk, and Ze ) ∞, which is an infinite 1D plane, described analytically by the Cottrell equation. For the fully conducting case, jss ) 2/π for a disk,18 and jssAtot-1 ) 0 for a Cottrellian plane. The plot of jssAtot-1 versus Ze at Figure 3 demonstrates that both limits are approached in the range of Ze examined. There is no maximum in this plot: for no value of Ze > 0 does the mass-transport-limited current density at a cylindrical electrode exceed that at a disk. It is clear, then, that the increased contact area with solution available to a cylinder is not efficiently converted to current density; rather, the planarity of diffusion to the cylinder side, compared to its top, causes suboptimal current density at steady state. The diffusion layer shape at steady state is shown at Figure 6. It is important to note here that the time τ required to achieve

this steady state increases with Ze, as upward diffusion to the area of accelerated, nonplanar diffusion at the cylinder top takes longer for a taller cylinder. Indeed, due to the
τ proportionality of mean diffusion distance, a τ ∝ Ze2 dependence seems likely. Consequently, the practical limits placed on simulation time may cause some fractional overestimation of jss for the highest Ze values shown. For the side-only conducting cylinder, the limit as Ze f ∞ should tend toward equivalence with the fully conducting case, as the effect of the top becomes ever more negligible with increasing Ze. At the Ze f 0 limit, high current density is observed; jss is similar to that for a microdisc, suggesting a hemispherical diffusion layer but with much smaller electroactive surface area. A plot of jssAtot-1 versus Ze is shown at Figure 3, normalized to the full top and side area; a plot normalized to side area alone is at Figure 4. The expected coincidence of jss for fully and side-only conducting cylinders is demonstrated to occur at Ze ≈ 20, whereafter a common hemispherical diffusion layer above the cylinder top occurs, despite the lack of direct oxidation at this surface. This is shown at Figure 7; although this is a much taller cylinder and a consequently larger diffusion layer than Figure 6, the similarity in shape is clear. The microdisc jss ) 2/π limit as Ze f 0 applies for a toponly conducting cylinder, but in this case the Ze f ∞ limit is not clearly defined. A plot of jss versus Ze is at Figure 5, which shows a clear upper limit for the steady-state current as Ze f ∞ of jss,max ) 0.917. At some height Ze the effect of the insulating base becomes negligible, as it is so distant from the conducting top of the cylinder. This height is found by simulation to be Ze ≈ 4-95% of jss,max, Ze ≈ 20-99% of jss,max, and Ze ≈ 200-99.9% of jss,max. The mass-transport-limited diffusion layer here is roughly spherical around the cylinder top (Figure 8), and so replenishment of material by diffusion in the R direction from bulk solution is sufficient to limit the diffusion layer such that it does not reach the base. Consequently, once past this Ze limit, the exact height of the cylinder ceases to affect its steady-state current. At lower Ze, the interaction of the insulating base with the near-spherically expanding diffusion layer limits the amount of solution available to the conducting surface, and hence jss does not achieve the upper limit. 3.2. Extent of the Cottrellian Region for Isolated Cylinders. Plots of log j versus log τ for the fully, top-only, and side-only conducting cases are shown at Figure 9. These show

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Figure 8. Steady-state concentration profile for a top-only conducting cylinder, Ze ) 10. Figure 5. jss vs log10 Ze for a top-only conducting cylinder, showing the Ze f 0 and Ze f ∞ limits.

Figure 9. Log10 j vs log10 τ for various isolated cylinders, showing the extent of the Cottrellian region, Ze ) 1. Figure 6. Steady-state concentration profile for a fully conducting cylinder, Ze ) 5.

Figure 10. (∂log10 j/∂log10 τ) vs log10 τ for various isolated cylinders, Ze ) 1. Figure 7. Steady-state concentration profile for a side-only conducting cylinder, Ze ) 100.

a straight-line Cottrellian region with gradient ≈ -1/2, corresponding to an entirely planar diffusion layer, at τ < 2 × 10-4. A further plot of (∂log j/∂log τ) versus log τ is shown at Figure 10; the lower limit of 0.5 here again corresponds to the Cottrellian case. It is clear that in the case where either top-

only or side-only conduction is active, nonplanar diffusion is observed earlier; this effect is presumed to arise from the cylinder top edge at R ) 1, Z ) Ze, where edge diffusion around this “corner” is magnified by the greater availability of electroactive material to the edge, should either top or side not conduct.

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Figure 11. Diagram presenting the various conductivity cases for the arrays of cylinders studied in section 4.

4. Regular Arrays of Cylindrical Electrodes on a Conducting Substrate 4.1. The Diffusion Domain Approximation. The three cases treated for the isolated electrode (full, top-only, and side-only conductivity) were also simulated for a regular array of electrodes supported on a conducting substrate. Additionally, the case of an entirely insulating cylinder was considered. These four cases are presented at Figure 11. A diffusion domain approach19 was used to simulate the array: the regular array is divided into equivalent unit cells, each with an electrode at its center and a dimensionless area Add ) dsep2, where dsep is the separation of electrodes in the array. The cell is then approximated by a cylindrical volume with the same basal area, i.e., Add ) dsep2 ) πR02, where R0 is the radius of the cylindrical simulation space in dimensionless units. This volume is then φ-symmetric and so reduces to a 2D simulation, with

response equivalent to a plane of the same area. Case 2 describes a situation where j exceeds the Cottrellian value due to nonplanar diffusion effects but where the diffusion layers for neighboring electrodes in the array remain separate. In case 3, these diffusion layers begin to overlap but retain nonplanar character. Case 4 describes the τ f ∞ situation where diffusion layer overlap causes an overall planar diffusion layer above the surface, returning to a Cottrellian situation; in a chronoamperometric potential step experiment, this implies a mass-transport-limited current of jss ) 0 for any array, irrespective of electrode size or geometry. Chronoamperometry is consequently a useful experiment to simulate, as a single chronoamperometric transient may be sufficient to observe all of cases 1-4, as τ increases. Conductivity, geometry, and array spacing determine the current and τ dependence of all of cases 1-3 and so directly affect the shape of the transient. 4.3. Analytical Approach to Short and Long τ Limits. As they are dependent solely on absolute or effective electroactive area, the Cottrellian predictions for observed current to an array in the case 1 and 4 limits may be approached analytically. The case 4 limit has an effective dimensionless electroactive area equivalent to the size of the diffusion domain, i.e., Adim ) Add ) πR02, and so is independent of which surfaces, if any, of the cylinder are conducting; as explained below, however, this does not mean that all systems attain case 4 in the same time or even approach it from a common state of higher or lower flux. The case 1 limit has Adim ) Aact where Aact is the sum of the areas of all electroactive surfaces, including the conducting base, which has area

Abase ) π(R02 - Re2) ) π(R02 - 1)

(20)

The implications of these varying electroactive areas may be considered by deriving a dimensionless Cottrell equation. In its dimensional form, the equation states that for an infinite plane21

i(t) ) nFAcA,bulk




DA πt

(21)

Then, substituting with the dimensionless variables in Table 1

j(τ) )

n A 1 2π r 2 √πτ e

Rmax ) R0 )

dsep

√π

(17)

A/re2 corresponds to Adim. The Cottrell equation may then be written, for n ) 1

As the array is regular, and so all cells are assumed to be equivalent, a zero-flux boundary condition applies here:

∂a )0 ∂R

R ) Rmax

(18)

replacing the bulk boundary condition at R ) Rmax used for an isolated electrode. The diffusion domain radius R0 is easily related to total electrode surface coverage Θ:

Θ)

πRe2 πR02

)

1 R02

j(τ) )

χ 2√πτ

(22)

where

χ)

Adim π

(23)

Then

χ4 ) R02 (19)

4.2. General Theory of Electrode Arrays. The theory of the current response of microdisc electrode arrays has been wellcharacterized by past theoretical work by Davies et al.,20 and the four cases presented in that work have been shown to be equally applicable to a variety of geometries.12,13 Case 1 describes the τ f 0 situation where diffusion layers are approximately planar, and any geometry gives a Cottrellian

χ1,Ins ) R02 - 1 χ1,Top ) R02 χ1,Side ) R02 - 1 + 2Ze χ1,Full ) R02 + 2Ze

(24)

which in combination with eq 22 analytically describe the expected τ f 0 and τ f ∞ limits for the transients in all cases.

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Figure 12. Log jAdd-1 vs log τ for an array of insulating cylinders with varying R0 at (a) Ze ) 1 and (b) Ze ) 10.

Figure 13. Log jAdd-1 vs log τ for an array of top-only conducting cylinders with varying R0 at (a) Ze ) 1 and (b) Ze ) 10.

4.4. Simulated Results for Chronoamperometric Transients. The case of an array of insulating cylinders on a conducting substrate, being in effect a partially blocked planar electrode, is easy to understand. Plots of log jAdd-1 versus log τ for varying R0 and Ze ) 1 and 10 are shown at Figure 12. The analytical currents described above are in close agreement with simulated results. A clear transition from case 1 to 4 is noted for Ze ) 1, with the current increasing as soon as the diffusion layer exceeds the cylinder top at Z ) Ze; this occurs at longer τ for a taller cylinder and so is not observed for Ze ) 10 within this τ range. The effect of the cylinders is also not perceptible at surface coverage Θ < 0.04, i.e., R0 > 5. Plots for a top-only conducting cylinder are shown at Figure 13. In this case, the predicted case 1 and case 4 currents are equivalent, as the cylinder top entirely compensates for the blockage at the electrode surface. As diffusion to the cylinder top is nonplanar, however, a case 2 regime is quickly reached, with the transition occurring between τ ) 10-3 and τ ) 10-2. The interaction between cases 2-4 is strongly dependent on Ze and R0. For R0 > 10, the enhanced current from the cylinder top is negligible compared to the Cottrellian substrate current at τ < 10, and so longer time scales are required to observe the case 1 to case 2 transition. At low Ze, the case 4 limit is more rapidly attained, as the diffusion layer extending upward from the substrate exceeds the cylinder top, and hence any geometric effects resulting from the cylinder are swamped. This distinction is clear from the two values of Ze in Figure 13. Greater Ze is then required at greater R0 in order to observe any deviation

from Cottrellian behavior and, hence, any indication of toponly conductivity at the cylinder. Plots for the fully conducting case are shown at Figure 14. Unlike the former two cases, the fully conducting case displays greater case 1 current than case 4 current, as the cylinder sides increase the electroactive surface area. At the base of the cylinder, however, the diffusion layer resulting from electroactivity at the substrate interacts with that from the cylinder side at all τ > 0 after a potential step. True Cottrellian behavior matching eq 22 is difficult to observe, as a result; the loss of current here is compensated for, however, by the edge effect at the cylinder top, resulting in a case 1 to case 2 transition with enhanced current density. This transition is seen clearly for a cylinder with Ze ) 10 but is not observed at Ze ) 1 where the competing case 1 to case 4 transition dominates: a dramatic fall in current to case 4 behavior is noted as soon as diffusion layers from the cylinder sides of neighboring electrodes overlap, causing a predominantly planar diffusion layer. For lower Ze, this arises more quickly, as less time is required for the diffusion layer to exceed the cylinder top. Exhaustion of electroactive solution from early nonplanar diffusion appears to render this case 3 current less than the case 4 limit, with diffusion from bulk being required to attain this precisely as τ f ∞. A comparison with exemplar side conducting cylinder arrays for R0 ) 2 are at Figure 15: only at such high surface coverage Θ does the cylinder top generate sufficient current relative to the base to make the distinction between these cases clearly observable. Otherwise, and also where Ze is large, cylinder top

Chronoamperometry at Cylindrical Microelectrodes

J. Phys. Chem. C, Vol. 112, No. 31, 2008 11643 conductivity suggest that chronoamperometry could act as a diagnostic experiment: the relative top and side conductivities of arrayed cylindrical structures on a conducting substrate might be easily assessed. The above analysis poses certain problems with such an approach, however. From roughly the 10 µm scale upward, times less than τ ) 10 are experimentally accessible. Then, Θ < 0.01 cases are very difficult to analyze, as the substrate effectively swamps all contributions from the cylinder. For large values of Ze, side-only and full conductivity are also impossible to extricate at short τ. In the ultramicro- and nanoscales, however, τ < 10 corresponds to milli- and microsecond ranges that are not feasibly observable in a potential step experiment: much longer τ must then be considered, so short Ze cases will rapidly achieve case 4 and cannot be studied. For very large Ze, toponly, side-only, or full and insulating cases could all potentially be separated; larger Ze and longer τ also lengthen simulation time, however, limiting a thorough test of these limits. 5. Conclusions

Figure 14. Log jAdd-1 vs log τ for an array of fully conducting cylinders with varying R0 at (a) Ze ) 1 and (b) Ze ) 10.

The rigorous theory of chronoamperometry at cylindrical electrodes has been presented and described thoroughly. For isolated electrodes, steady-state currents and the shape of the mass-transport-limited diffusion layer were presented; for electrode arrays, variations in the shape of the chronoamperometric transient were considered. In all cases, trends with varying cylinder height Ze, and, for arrays, surface coverage Θ, have been presented and rationalized. It has been shown that a fully conducting cylinder never exceeds the steady-state current density of a microdisc electrode, but that this current density may be enhanced by supporting the microdisc on an insulating cylinder, increasing the availability of electroactive material to the region of nonplanar diffusion. The utility of chronoamperometry as a diagnostic tool for discovering the conducting surfaces of a cylinder has also been considered: certain constraints in Ze and Θ apply. Although the limitations of studying transients at high τ must be noted, it has again been demonstrated that modern computing techniques permit complex 2D simulations, such that 1D approximations, as in past work on cylinders, may be in general superseded. In conjunction with past work on conical electrodes,13 an increased understanding of the properties of threedimensional microelectrodes has been achieved. The development of these models through either incorporation of coupled homogeneous kinetics or consideration of other possible enhanced geometries is expected in future research. Acknowledgment. I.S. thanks the EPSRC for a studentship. References and Notes

Figure 15. Comparison of log jAdd-1 vs log τ for arrays of side-only and fully conducting cylinders with R0 ) 2.

currents are negligible compared to side and base currents, and edge effects at the cylinder top rapidly compensate for the lack of direct conduction at this surface, rendering side-only conducting cylinder arrays indistinguishable from the fully conducting case. 4.5. Implications for Chronoamperometry as an Experimental Diagnostic for Cylinder Properties. The significant differences in shape of a log j versus log τ plot following a potential step for the different possible combinations of cylinder

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