Anal. Chem. 2000, 72, 2383-2390
Scanning Electrochemical Microscopy. Theory of the Feedback Mode for Hemispherical Ultramicroelectrodes: Steady-State and Transient Behavior Yoram Selzer and Daniel Mandler*
Department of Inorganic and Analytical Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
This contribution represents the first comprehensive attempt to treat complex geometry configurations of the scanning electrochemical microscope (SECM) using the alternating direction implicit finite difference method (ADIFDM). Specifically, ADIFDM is used to simulate the steady-state as well as the transient (chronoamperometric) behavior of a hemispherical ultramicroelectrode (UME) tip of the SECM. The feedback effect in this configuration is less pronounced as compared with a disk-shaped UME system. The differences between the two systems are discussed. Analytical approximations for the steady-state behavior and for characteristic features of the transient behavior are suggested. Finally, experimental feedback currents measured above a conductor and an insulator are in excellent agreement with the theory. Scanning Electrochemical microscopy (SECM) has matured and become a very powerful tool to investigate electrochemical and chemical systems.1-6 Although it has substantial advantages in high-resolution imaging and fast kinetics measurements, quantitative interpretation of results is not straightforward and relies on the ability to perform numerical simulations. The first theoretical treatment of the SECM response7 used a finite element method (FEM), but this technique has not been exploited further. Subsequent quantitative work has utilized the alternating direction implicit finite difference method (ADIFDM) that turned out to be extremely powerful for treating SECM problems6,8 as well as other electrochemical systems.9-11 Although by now several studies that utilize ultramicroelectrodes with other than disk(1) Bard, A. J.; Unwin, P. R.; Wipf, D. O.; Zhou, F. Am. Inst. Phys. Conf. Proc. 1992, 254, 235. (2) Bard, A. J.; Mirkin, M. V.; Fan, F.-R. F. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1993; vol. 18, p 243. (3) Arca, M.; Bard, A. J.; Horrocks, B. R.; Richards, T. C.; Treichel, D. A. Analyst (Cambridge, U.K.) 1994, 119, 719. (4) Macpherson, J. V.; Unwin, P. R. Chem. Ind. (London) 1995, 874. (5) Mirkin, M. V. Anal. Chem. 1996, 68, 177A. (6) Unwin, P. R. J. Chem. Soc., Faraday Trans. 1998, 94, 3183. (7) Kwak, J.; Bard, A. J. Anal. Chem. 1989, 61, 1221. (8) Unwin, P. R.; Bard, A. J. J. Phys. Chem. 1991, 95, 7814. (9) Heinze, J. J. Electroanal. Chem. 1981, 124, 73. (10) Heinze, J. Ber. Bunsen-Ges. Phys. Chem. 1986, 90, 1043. (11) Heinze, J. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, 1096. 10.1021/ac991061m CCC: $19.00 Published on Web 04/29/2000
© 2000 American Chemical Society
shaped geometry have already been published,12-17 most if not all theoretical treatments have considered only disk-shaped tip ultramicroelectrodes, mainly because of the simplicity by which a cylindrical coordinates system could then be used to characterize the tip-substrate domain. Recent contributions from Fisher et al.18,19 demonstrated the flexibility of the boundary element method (BEM) for simulating a broad range of SECM systems involving complex tip and substrate geometries. Although this method is highly suitable for steady-state problems, the authors failed to mention many contributions from Unwin’s group,4,6 which showed that a comprehensive treatment of the SECM response cannot be considered complete without numerical simulations of the transient. Significant data is hidden in the transient that cannot be extracted by steady-state experiments. The ADIFDM, on the other hand, is extremely suitable for transient experiments. This paper is a first step toward using the ADIFDM to simulate SECM systems with varying tip geometries. We believe that such treatment is appropriate at this stage of SECM development, i.e., before nano and sub-nanometer scale tips will become routine. Several important experimental steps in this direction have already been made.12-17 All of them utilized ultramicroelectrodes that were not necessarily disk-shaped, thus prohibiting any accurate quantitative studies. This work focuses on the application of hemispherical ultramicroelectrodes as tips for the SECM. A hemispherical shape was chosen initially because of several reasons: it is relatively simple to construct conventional-sized (micrometer-range) hemispherical ultramicroelectrodes,20 which are required to test our results experimentally. Second, a few theoretical treatments of this problem (a hemispherical electrode in close proximity to a (12) Lee, C.; Miller, C. J.; Bard, A. J. Anal. Chem. 1991, 63, 78. (13) Mirkin, M. V.; Fan, F.-R. F.; Bard, A. J. J. Electroanal. Chem. 1992, 328, 47. (14) Shao, Y.; Mirkin, M. V.; Fish, G.; Kokotov, S.; Palanker, D.; Lewis, A. Anal. Chem. 1997, 69, 1627. (15) Fan, F.-R. F.; Bard, A. J. Science (Washington, D.C.) 1995, 267, 871. (16) Fan, F.-R. F.; Kwak, J.; Bard, A. J. J. Am. Chem. Soc. 1996, 118, 9669. (17) Forouzan, F.; Bard, A. J. J. Phys. Chem. B 1997, 101, 10876. (18) Fulian, Q.; Fisher, A. C.; Denuault, G. J. Phys. Chem. B 1999, 103, 4387. (19) Fulian, Q.; Fisher, A. C.; Denuault, G. J. Phys. Chem. B 1999, 103, 4393. (20) Roland Alfred, L. C.; Oldham, K. B. J. Phys. Chem. 1996, 100, 2170, and references cited therein.
Analytical Chemistry, Vol. 72, No. 11, June 1, 2000 2383
conducting surface) have been reported which enable us to compare our results with previous work.13,21 Finally, considering other possibilities, this is a rather simple symmetry configuration, which can be used for demonstrating the necessary key steps involved in the implementation of the ADIFDM in complex SECM systems.
EXPERIMENTAL SECTION Reagents and Chemicals. All solutions were prepared from Milli-Q (Millipore Corp.) reagent water. The mercury plating solution was made of a mercury standard solution (Aldrich, 1000 mg/L in 0.1 M HNO3) and 0.5 M potassium nitrate (Fisons, A. R. grade). For the feedback experiments the solution consisted of 2 mM hexaamineruthenium(III) chloride (Ru(NH3)63+, Aldrich) and 0.1 M potassium nitrate. Instrumentation. The SECM apparatus has been described previously.22 A 25-µm diameter Pt ultramicroelectrode served as a working electrode in this study. A gold electrode (3-mm diameter) that was successively polished with 1-µm and then 0.05µm alumina slurry (Buehler) and finally rinsed with copious amounts of water served as the substrate in the feedback experiments. All the experiments utilized an Ag wire as a quasireference electrode. Procedure. Both mercury-plating and feedback experiments were made in deoxygenated solutions under a continuous stream of nitrogen. Mercury was deposited through the diffusioncontrolled reduction of Hg(I). The potential on the Pt electrode was held at -0.230 V for approximately 400 s, until the steadystate current was 1.57 times higher than the initial current, indicating a transformation from a disk-shaped into a hemispherical microelectrode. After mercury deposition, the cell was drained of the plating solution, rinsed three times with pure water, and refilled with the ruthenium solution. Feedback curves were measured by setting the potentials to -0.450 and 0.1 V at the ultramicroelectrode and the gold substrate, respectively, ensuring diffusion-controlled processes on both surfaces.
THEORY The geometrical configuration of a hemispherical ultramicroelectrode (UME) in close proximity to a surface is depicted schematically in Figure 1. The appropriate diffusion equation in hemispherical coordinates suitable for this configuration (written in polar coordinates: r, θ, φ) is:
∂2c ∂ ∂c 1 ∂c 1 ∂ 2∂c 1 1 ) 2 sin θ r + 2 + D ∂t r ∂r ∂r r sin2 φ ∂φ2 r2 sin θ ∂θ ∂θ
( )
(
)
(1)
Because of the spherical symmetry, there is no dependence of the concentration on φ, simplifying eq 1 to:
1 ∂c 1 ∂ 2∂c 1 ∂ ∂c r + 2 ) sin θ D ∂t r2 ∂r ∂r r sin θ ∂θ ∂θ
( )
(
)
(2)
(21) Davis, J. M.; Fan, F.-R. F.; Bard, A. J. J. Electroanal. Chem. 1987, 238, 9. (22) Selzer, Y.; Turyan, I.; Mandler, D. J. Phys. Chem. B 1999, 103, 1509.
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Analytical Chemistry, Vol. 72, No. 11, June 1, 2000
Figure 1. A schematic presentation of a hemispherical UME in close proximity to a surface. The necessary variables to define the geometry of the problem are shown: the radius of the electrode, a, the distance to the surface, d, the radius of the insulating sheath rglass, and θ0, the angle corresponding to the point on the surface that is located at a distance of rglass.
After rearrangement we obtain:
(
1 ∂c ∂2c 2 ∂c 1 ∂2c cos θ ∂c + + ) + D ∂t ∂r2 r ∂r r2 ∂θ2 sin θ ∂θ
)
(3)
To get general results we normalize the above equation using
R)
r a
(4)
C)
c c*
(5)
T)
tD a2
(6)
where a, c*, D are the radius of the electrode, the bulk concentration of the redox mediator, and the diffusion coefficient of the redox mediator, respectively. The resulting normalized equation is:
(
)
∂C ∂2C 2 ∂C 1 ∂2C cos θ ∂C + ) 2+ + 2 ∂T ∂R R ∂R R ∂θ2 sin θ ∂θ
(7)
Assuming that the diffusion coefficients of both redox states of the mediator are equal allows mass conservation to be invoked:
CR + CO ) 1
(8)
Hence, it is sufficient to formulate and solve the equations for only one redox state. Accordingly, the subscripts R, O will be avoided for the rest of the treatment. The treatment presented will consider only positive-feedback conditions, i.e., a case where the mediator is regenerated at the surface at a diffusion-controlled rate. The changes necessary for dealing with negative-feedback conditions are straightforward.7
In a chronoamperometry experiment under feedback conditions, the electrode is located at a distance d above a surface. We assume that prior to the potential step only one state of the redox mediator is present and its concentration is uniform throughout the solution. The initial condition is therefore:
T ) 0, C ) 1
(9)
Subsequently, the electrode potential is stepped to a value in which the mediator undergoes a diffusion-controlled redox reaction. At the same time, the potential on the substrate is sufficient to drive the reverse reaction in a diffusion-controlled manner. The mediator is inert on the insulator (glass) around the electrode, and its concentration remains at bulk value, through all the experiment, beyond the tip-substrate domain. Because of the symmetry of the problem there is no flux in the θ direction for θ ) 0, 1 e R e 1 + L, where L is the normalized distance of the electrode from the substrate:
L)
d a
(10)
π (tip electrode) 2
π θ ) , 1 e R e RG (glass sheath) 2 1+L π (substrate) 0eθe ,R) 2 cosθ
[
F
)
e -1 b
2
1 + 2(1 + L)
)
e -1 + (1 + L)2 b
[
]
∂ C cosθ ∂C + ∂θ2 sinθ ∂θ
∂C )0 ∂θ
(12)
C)1
(13)
Accordingly, the concentration changes in zone I are calculated using eq 7 and those in zone II using eq 18. The grid and symbols that are used in the finite difference calculations are depicted in Figure 2. Points in the R,θ directions are denoted by k, j, respectively. The number of points in the R direction (NR, NR1 in zones I and II, respectively) was typically not less than 100. The number of points in the θ direction (NT) was 100. The separations between grid points in the angular and radial directions are denoted by:
C)1
∂C ) 0 (15) ∂θ
π 2 ∆θ ) NT
As in previous publications8 we use RG ) rglass/a ) 10. θ0 is defined as the angle for which:
∆F ) ∆R ) (16) ∆F1 )
It is difficult to predict the concentration profile as a function of R and θ; however, since it is likely to change rapidly as a function of L we decided to divide the tip-substrate domain into two zones (Figure 2). In the zone close to the conducting part of the UME, where rapid changes are expected, i.e., 1 e R e 1 + L, the R coordinate is not transformed. On the other hand, in the second zone, i.e., 1 + L < R e RG, a transformation (on R) was preformed using8
F ) ln(1 + b(R - (1 + L)))
(
F
2
(18)
(14)
1+L cos θ0 ) RG
]
∂C ∂2C 2b2e-F ∂C - b2e-2F ) b2e-2F 2 + F + ∂T ∂F ∂F e - 1 + b(1 + L)
(11)
C)0
θ g θ0, R g RG (beyond tip-substrate domain)
θ ) 0, 1 e R e 1 + L (axis of symmetry)
Substituting R in eq 7 by F from eq 17 gives:
(
The boundary conditions for our problem are therefore:
R)1, 0 e θ e
Figure 2. The coordinate system for the simulation. The tipsubstrate domain is divided into two zones: Zone I for I e R e 1 + L with uniform mesh differences and Zone II for 1 + L < R e RG with inflating mesh differences.
L NR
(19) (zone I)
ln(1 + b(RG - (1 + L))) NR1
(zone II)
(20)
(21)
To ensure that the derivative at R ) 1 + L (the radial point at which the functions are changed) will be continuous, the value of b was calculated so that the first ∆R in zone II will equal the ∆R (∆F) in zone I. The value of b was calculated by solving numerically the following equation:
NR1 ln(1 + b∆R) - ln(1 + b(RG - (1 + L))) ) 0
(22)
(17)
where b is a constant that is calculated, vide infra. To ensure high accuracy of calculation, the θ coordinate was not transformed throughout the entire space.
Because the plane of the substrate breaks the hemispherical symmetry of the system, the number of radial points (NRADP(θ)) in the second zone for θ < θ0 is less than NR1. It is also evident that the separation between k ) NRADP(θ) and k ) Analytical Chemistry, Vol. 72, No. 11, June 1, 2000
2385
NRADP(θ) - 1, abbreviated as ∆Ff1, is different and should satisfy ∆Ff1 < ∆F1 (Figure 2). NRADP(θ) and ∆Ff1 are calculated in the following manner. For every angle θ the distance to the substrate is:
R)
1+L cos θ
(23)
// / // // // Cj,k - Cj,k Cj,k + 1 - 2Cj,k + Cj,k - 1 ) + ∆T/2 (∆R)2 //
//
Cj,k + 1 - Cj,k - 1 1 2 + 1 + k∆R 2∆R (1 + k∆R)2
[
/ Cj/+ 1,k - 2Cj,k + Cj/- 1,k
(∆θ)2
+
]
/ / cos(j∆θ) Cj + 1,k - Cj - 1,k (31) 2∆θ sin(j∆θ)
R is transformed to F and then:
F NRADP(θ) ) int +1 ∆F1
( ) F ∆F 1 ) F - int( ∆F1 ∆F1)
(24) (25)
f
The intersecting substrate plane causes the number of angular points for each distance R in zone II to be lower than NT. For each distance we can define a minimal angle θmin. Points that are located at the same distance but at angles that are lower than θmin do not have to be considered because they are located inside the substrate (Figure 2). θmin has a corresponding index IANGP(R) and an initial mesh difference ∆θi, that are defined as follows:
1+L cos θmin ) R
(26)
θmin ) cos-1 θmin
(27)
( )
θmin +1 ∆θ
(28)
∆θi ) IANGP(R)∆θ - θmin
(29)
IANGP(R) ) int
// are the new concentrations that are evaluated using where Cj,k / the old concentrations Cj,k . For Zone II, eq 18 transforms into the following:
/ C j,NR+ k - Cj,NR + k ) ∆T/2 Cj,NR + k + 1 - 2Cj,NR + k + Cj,NR + k - 1 b2e-2k∆F1 + (∆F1)2
[
e
[
[
/
/
]
cos(j∆θ) Cj + 1,k- Cj - 1,k + (30) 2∆θ sin(j∆θ)
Cj,k denotes the normalized concentration at point j,k; whereas, C/j,k identifies those concentrations which are to be evaluated at the new time, T + ∆T/2, from the former concentrations at T. For the second time step, eq 7 gives the following 2386
/ / - 2Cj,NR + k + Cj - 1,NR + k
(∆θ)2
+
]
/ / cos(j∆θ) Cj + 1,NR + k - Cj - 1,NR + k (32) 2∆θ sin(j∆θ)
// / Cj,NR + k - Cj,NR + k ) ∆T/2 // // // Cj,NR + k + 1 - 2Cj,NR + k + Cj,Nr + k - 1
(∆F1)2
[
k∆F1
e
]
b2e-2k∆F1
[
1 R2
+
2b2e-k∆F1 - 1 + b(1 + L)
// // Cj,NR + k + 1 - C j, NR + k- 1 + 2∆F1
Cj/+ 1,NR + k
/ / - 2Cj,NR + k + Cj - 1,NR + k
(∆θ)2
+
]
/ / cos(j∆θ) Cj + 1,NR + k - Cj - 1,NR + k (33) 2∆θ sin(j∆θ)
where
Cj,k + 1 - Cj,k - 1 1 2 + 1 + k∆R 2∆R (1 + k∆R)2 (∆θ)2
2b2e-k∆F1 - 1 + b(1 + L)
Cj,NR + k + 1 - Cj,NR + k - 1 + 2∆F1
Cj/+ 1,NR + k
1 R2
The basic concept behind the ADIFDM is to divide the appropriate finite difference diffusion equation over the time step ∆T into two successive equations, each covering a half time step of ∆T/2. The derivatives in eqs 7, 18 are replaced by finite difference analogues, formulated in the first equation in terms of known concentration values in the R direction and unknowns in the θ direction and vice versa for the second equation. The equation for the first half time step in zone I is as follows:
/ Cj/+ 1,k - 2Cj,k + Cj/- 1,k
]
b2e-2k∆F1
b2e-2k∆F1
/ Cj,k - Cj,k Cj,k + 1 - 2Cj,k + Cj,k - 1 ) + ∆T/2 (∆R)2
k∆F1
Analytical Chemistry, Vol. 72, No. 11, June 1, 2000
1 ) k∆F1 R2 e -1 b
(
)
2
1 + 2(1 + L)
(
)
ek∆F1 - 1 + (1 + L)2 b
Notice that the index k is defined between 1,NR in zone I and between 1,NR1 in zone II. Equations 32 and 33 need to be slightly modified in the case where k ) NRADP(θ) - 1 or j ) IANGP(R) because at these points the mesh differences are not equal on both sides of the grid point. The following equations describe the case where the
mesh varies both in the radial and angular directions. Obviously, there are points for which the necessary changes should be made in only one direction. For these cases the appropriate direction should be changed while treating the other direction according to eqs 32, 33. / Cj,NR +k - Cj,NR + k ) ∆T/2 Cj,NR + k + 1 - Cj,NR + k Cj,NR + k - Cj,NR + k - 1 ∆Ff1 ∆F1 2 -2k∆F1 be + ∆Ff1 + ∆F1 2
[
2b2e-k∆F1 ek∆F1 - 1 + b(1 + L) Cj,NR + k + 1 - Cj,NR + k - 1 + b2e-2k∆F1 ∆Ff1 + ∆F1
[
Figure 3. A comparison between the feedback curves for a disk UME (0) and a hemispherical UME (-).
]
Cj/+ 1,NR + k
1 R2
/ / / Cj,NR - Cj,NR +k + k - C j- 1,NR + k ∆θ ∆θi + ∆θi + ∆θ 2
]
/ / cos(j∆θ) Cj + 1,NR + k - Cj - 1,NR + k (34) ∆θ + ∆θi sin(j∆θ) // Cj,NR +k
/ Cj,NR +k
∆T/2
iT ) nFD
∫
π/2
2πa2
0
(∂c∂r)
r)0
sin θdθ
(36)
In dimensionless form this becomes
iT iT,∞
) 2π
∫
π/2
0
sin θ
(∂C ∂R)
R)0
dθ
(37)
)
// // // // Cj,NR Cj,NR + k + 1- Cj,NR + k + k - Cj,NR + k - 1 ∆Ff1 ∆F1 b2e-2k∆F1 + ∆Ff1 + ∆F1 2
[
2b2e-k∆F1 ek∆F1 - 1 + b(1 + L)
[
thorough explanation is available as Supporting Information. Current at the end of the second half time step was calculated according to the following equation:
// // Cj,NR + k + 1 - Cj,NR + k - 1 + ∆Ff1 + ∆F1
/ / / Cj,NR Cj/+ 1,NR + k - Cj,NR + + k - Cj - 1,NR + k ∆θ ∆θi 1 + 2 ∆θ + ∆θ R i 2
/
/
iT,∞ ) 2πnFDc/a
(38)
The above formulation was coded in Fortran in double precision, and the program was executed on the mainframe facility of the university.
]
b2e-2k∆F1
where iT,∞ denotes the steady-state current at a simple hemispherical electrode:
]
cos(j∆θ) Cj + 1,NR + k - Cj - 1,NR + k (35) ∆θ + ∆θi sin(j∆θ) Equations 30,32,34 are implicit in the θ direction, and eqs 31,33,35 are implicit in the R direction, making the overall calculation stable for any time step value. This is a significant advantage because it allows varying the value of ∆T during the simulation. When the changes in concentrations are rapid, in short times, calculations are preformed with small ∆T values. As the changes in concentrations become more gradual, ∆T can be increased without affecting the precision of calculation. Equations 30-35 should be transformed into matrices that are solved according to the appropriate boundary conditions. A
RESULTS AND DISCUSSION Steady-State Behavior. Figure 3 shows simulated positive and negative feedback curves, of hemispherical and disk ultramicroelectrodes with the same radius a. Obviously the feedback effect is much less pronounced in the case of a hemispherical UME. While under positive feedback conditions the mass transfer to a disk UME is enhanced more than 8 times at a normalized distance of L ) 0.1, it is enhanced only 3 times at a hemispherical UME. On the other hand, under negative feedback conditions the current at a disk UME is essentially zero for any practical purposes at L ) 0.1, whereas it is approximately 0.4 at a hemispherical UME. Some may argue that it is possible to approach a surface to a distance less than 0.1 when a hemispherical UME is used.24 Nevertheless we have limited our discussion at this point to L ) 0.1 because of the rather thick insulator (RG ) 10) that was used in all simulations that is likely to prevent a closer approach because of practical experimental considerations. The simulated positive (23) Selzer, Y.; Mandler, D. Electrochem. Commun. 1999, 1, 569. (24) Demaille, C.; Brust, M.; Tsionsky, M.; Bard, A. J. Anal. Chem. 1997, 69, 2323.
Analytical Chemistry, Vol. 72, No. 11, June 1, 2000
2387
Figure 4. Calculated current densities on a disk UME as a function of R, the radial distance from the center of the UME, for an UME that is located at a normalized distance of: (a) L ) 10, (b) L ) 0.1 above a conducting substrate, (c) L ) 0.1 above a nonconducting substrate.
Figure 5. Calculated current densities on a hemispherical UME as a function of θ, for an UME that is located at a normalized distance of: (a) L ) 10, (b) L ) 0.1 above a conducting substrate, (c) L ) 0.1 above a nonconducting substrate.
and negative feedback curves can be accurately described (with an error of less than 1%) by eqs 39-40, respectively
I ) 0.873 + ln(1 + L-1) 0.20986 exp[-(L - 0.1)/0.55032] (39) I ) 0.39603 + 0.42412L - 0.09406L2
(40)
for 0.1 e L e 2. The results in Figure 3 are understandable by considering the effect of the distance L on the profile of current densities at the electrodes. Figure 4a shows the current densities at a disk UME that is located at a normalized distance of L ) 10, as a function of the distance R from the center of the electrode. As it has already been demonstrated previously,25 the current density is nonuniform and may attain an infinite value at the edge of the electrode. As the electrode is pushed toward a conducting surface to a normalized distance of L ) 0.1 (Figure 4b), the current density increases. The enhancement is similar for all values of R, i.e., curve 4a is basically shifted upward as the distance between the disk UME and the conducting substrate decreases. Similar behavior can be seen as the electrode is pushed toward a nonconducting surface to the same normalized distance: curve 4a as a whole is shifted to lower values. For all practical purposes, the current density in this case is zero all over the electrode except for a short tale of higher density at the edge. While the effect of the substrate is quite uniform for a disk UME, the situation is entirely different in the case of a hemispherical UME. Figure 5a shows the current density on such an electrode as a function of θ when it is located far above a substrate (L ) 10). As long as the substrate is sufficiently far not to distort the symmetry of the system, a uniform flux is expected on the basis of the well-established theory of hemispherical electrodes.26 Figure 5 parts b-c clearly demonstrate that flux uniformity is perturbed when the electrode is brought in close proximity to conducting and insulating surfaces, respectively. The effect is (25) Amatore, C. In Physical Electrochemistry; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995; p 131. (26) Wipf, D. O.; Whiteman, M. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1989; vol. 15, p 267.
2388 Analytical Chemistry, Vol. 72, No. 11, June 1, 2000
Figure 6. A comparison between steady-state currents, calculated in this work, and the analytical approximations of ref 13.
pronounced at the points at which the hemisphere is closest to the substrate, i.e., at small values of θ, and it gradually decreases as θ increases. The current density at θ ) 0 behaves quite similarly to that of a disk UME at R ) 0. Nevertheless, the contribution of points at small θ values to the overall current is rather poor because of the small differential area at these θ values. On the other hand, points close to the insulating sheath around the electrode (θ = 90°) almost do not feel the presence of a substrate and, at the same time, their corresponding differential area is quite large. The steady-state behavior of a hemispherical UME in a SECM configuration has previously been addressed using analytical approximations.13 Comparisons between these approximations and our calculations are depicted in Figure 6. We find that while the error in the case of a positive feedback is 7%, the error in the case of a negative feedback is as high as 66% at short distances. The analytical approximation overestimates the effect of an insulating surface on the current. The analytical approximations are based on dividing the electrode into infinitesimal segments and addressing each one of them separately using the feedback theory for disk UMEs. It is possible to divide the flux vector to the UME into two orthogonal contributions: one normal to and one parallel to the substrate. The rather good agreement between the simulated positive feedback curve and the analytical approximation implies that, in this case, the normal contribution predominates. In the case of a negative feedback there is no preferred contribution.
Figure 7. Experimental (0) and theoretical (-) feedback currents of a hemispherical ultramicroelectrode over a conductor and an insulator.
Figure 9. Figure 7 is replotted as a function of the inverse of the square root of the normalized time to emphasize the behavior at short times.
only when a steady concentration profile after depletion exists all the way to the edge of the tip-substrate domain where bulk conditions prevail. At short times, before the expanding diffusion layer is perturbed by the substrate, semi-infinite radial diffusion applies. Under these conditions, the normalized transient current should follow:26
iT 1 +1 ) iT,∞ xπT
Figure 8. Normalized chronoamperometric responses of a hemispherical UME located at log(L) ) -1.0, -0.8, -0.6, -0.4, -0.2, 0.0. Curves denoted by (c) are for a conducting substrate, and curves denoted by (i) are for a nonconducting substrate.
For extremely short L values it has already been suggested that the positive feedback can be regarded as a one-dimensional system21 (neglecting any parallel contribution). The appropriate analytical solution is then: I ) ln(1 + L-1). The relative error between this equation and our simulations for L values of 0.1 and 0.001 are 21 and 7%, respectively, suggesting that the parallel contribution is indeed less important and diminishes as L becomes smaller. The calculations were supported by experimental measurements. A Hg hemispherical ultramicroelectrode was prepared by electrodeposition of mercury onto a Pt disk electrode. Figure 7 shows the steady-state feedback currents measured over a conductor, under diffusion-controlled conditions, and above an insulator. For comparison, the theoretical feedback currents are also depicted. It is evident that the experimental results are in excellent agreement with our calculations. Figure 8 shows simulated transients in a dimensionless form for a conducting (c) and insulating (i) substrate for different L values, from the top curve downward, log(L) ) -1.0, -0.8, -0.6,...,0 for a conductor, and from the bottom curve upward, log(L) ) -1.0, -0.8, -0.6,...,0 for an insulator. At long times, the transients converge with the steady-state currents described above. While the current at the UME reaches steady state rapidly above a conducting substrate, it exhibits a quasi steady state above an insulator. A true steady state is attained
(41)
The simulated transient currents are replotted in Figure 9 as a function of the inverse square root of the dimensionless time, highlighting the behavior at short times. At sufficiently short times all the chronoamperometric curves behave identically and according to eq 41. At longer time, the curves rapidly deviate toward a constant iT > iT,∞ above a conducting substrate and iT < iT,∞ above an insulating substrate. As it has already been suggested,27 a critical time, tc, corresponding to a transition from a pure radial diffusion to a thinlayer cell regime, can be defined. Obviously, tc is related to the time it takes for the redox mediator to diffuse to and from the UME in the presence of a substrate. For of a disk UME, tc for an insulating substrate is shorter than that for a conducting surface. This can be explained semiquantitatively by considering the time-of-flight of the redox mediator between the substrate and the UME in both cases. Although the symmetry of a hemispherical UME system is much more complex, the same reasoning should also be applicable here. Because L is of the order of the electrode radius or less, we expect the normal contribution of the flux to change first as a function of time. In the case of an insulating substrate it will occur after a time in the order of d2/D, which is the time-of-flight of the redox mediator from the UME to the substrate. In the case of a conducting substrate, tc should be twice that for an insulating substrate, i.e., the time it takes the mediator to diffuse to the substrate and back to the UME (assuming DO ) DR). Experimentally measured tc and d can be used to evaluate D even without knowing the exact value of the concentration. (27) Bard, A. J.; Denuault, G.; Friesner, R. A.; Dornblaser, B. C.; Tuckerman, L. S. Anal. Chem. 1991, 63, 1282.
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Figure 10. Definition of the critical time, tc, shown for the case when L ) 0.1. The projection from point A to the time axis defines the tc in the case of a conducting surface, and the projection from point B defines tc in the case of a nonconducting surface. For explanations see text.
Figure 12. A comparison between tc values for (A) a disk UME and (B) a hemispherical UME, located at a normalized distance L ) 0.1. Point A is calculated on a tangent line to the short time behavior of the disk UME. Point B is calculated on a line according to eq 41.
is significantly shorter than the corresponding value for a hemispherical UME (point B).
Figure 11. Working curves for tc determination. The linear fits are suitable for normalized distances in the range of 0.1 e L e 1.0. (A) nonconducting substrate, (B) conducting substrate.
Although true values of tc are not accessible experimentally, it is still possible to devise a procedure to extract arbitrary tc values that can be used to calculate D. Figure 10 demonstrates, using simulated curves, how these tc values are defined. The transient curves are drawn along with the transient according to eq 41 (curve a). For a conducting substrate (curve b), the normalized tc is defined as the time when the current according to eq 41 equals the steady-state current (point A). For a nonconducting surface we define tc, as a matter of convenience, as the time when eq 41 equals one (point B). Figure 11 shows two working curves of the square root of Tc (normalized tc) as a function of L, which can be used to extract the diffusion coefficient from chronoamperometric curves using a procedure that has been suggested previously.27 Although two excellent linear curves are obtained (Figure 11), the physical meaning of their slopes as well as of their intercepts is not clear. Nevertheless, it is worth noticing that the values of critical time at L ) 0 are not zero due to the shape of the UME. This is in contrast with the behavior of a disk UME. Finally, Figure 12 shows two transients calculated for a disk UME (curve a) and a hemispherical UME (curve b) located at L ) 0.1 above a conductor. Clearly, the critical time, tc, for a disk UME (point A) (28) Nann, T.; Heinze, J. Electrochem. Comm. 1999, 1, 289.
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CONCLUSIONS The steady-state and the transient currents at a hemispherical UME operating in the feedback mode of the SECM have been simulated using the ADIFDM. As expected, we find that the presence of a surface (either conducting or nonconducting) in close proximity to such an UME alters its current to a much lesser extent than in the case of a disk UME. Nevertheless, the effect is quite substantial, and hemispherical UMEs can be used as SECM tips. The ADIFDM method is shown once more to be highly suitable to simulate SECM problems, and it can easily be adapted to handle complex geometry configurations. At the same time, other methods, such as the adaptive finite element (AFE) approach, have recently been introduced,28 which might better deal with boundary singularities. Extensive work in our group is currently being undertaken to elaborate the above method to calculate the steady-state and transient characteristics of subhemispherical and conical microelectrodes both in general and under SECM feedback conditions. ACKNOWLEDGMENT This research was supported by the Foundation for Basic Research administrated by the Israel Academy of Sciences and Humanities (597/97-1) and the Ministry of Science (8931). SUPPORTING INFORMATION AVAILABLE Three pages of description of how to construct the necessary matrices of the ADIFDM and the algorithm to solve them is available as Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.
Received for review September 13, 1999. Accepted February 17, 2000. AC991061M
