Theory and Experiment for the Collector-Generator Triple-Band Electrode Bruno Fosset and Christian Amatore* Ecole Normale Superieure, Laboratoire de Chimie, 24 Rue Lhomond, 75231 Paris, France Joan Bartelt and R. Mark Wightman* Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290
Thecurmntsobtahdat ablpkbandekctrockwlthacwtrai gemrator and flanking collectors have been cakulated by w a r .knubtkn. To determlm the current at steady state, a conformal map was used In which space is folded (10 that the geometry lo ohnllar to that of a thin-layer c d with opposing electrodeo. This geometry makes the flux h e 8 straighter and thus dmp#ller rapid, accurate calculatknr. I f flux through the a n a l gap between the coHectors in the conformal space is Ignored, a comlitlon that k tnn, at steady state, the current Is shown to approach that of a thin-layer cell with contlnuow, opposing electrodes. At infinite thnes, the collection efficiency will be one and the Inverse amplH1cation factor (defined as the current at a single band divided by the generator current) approaches zero. To evaluate the current for experhentally achievable t h e scales, a different conformal map was used for the slmulatlon. I n this case, space was folded on either side of the generator, resuttlng in a threeaidad geometry. Experhental data for the oxldatlan of ferrocene In acetonitrile are In good agreement with the simulated remits. Comparison of the remits reported here with pr’ work on the double- and rlngJe-bad gwmetrleo shows advantages In the present cam with respect to the t h e needed to achleve pseudo steady state.
INTRODUCTION In previous work, we have examined the faradaic current at the single band and the double band operated in the collector-generator configuration (I, 2). Both of these configurations have many of the advantages that other electrodes of very small dimensions exhibit (3). In addition, double-band electrodes used in the collector-generator mode provide information on the electrogenerated products, information not available at single-band microelectrodes operated under quasi-steady-state conditions (4,5). The usual mode of operation of collectol-generatordevices is to sweep the generator potential while maintaining the collector at a fixed potential. Voltammograms obtained in this way for a diffusion-limited process are a function of the electrode dimensions and the scan rate. The current at the generator electrode is enhanced by the feedback of electroactive material from the collector. This effect is most apparent at slow scan rates. Similarly, the collector current also increases with decreased scan rate because the diffusion layer of the generator more completely overlaps the region of the collector. Since the potential of the collector remains constant during the experiment, high gain can be used to amplify very small currents. This feature is especially useful when the electrogenerated product is unstable on the electrochemicaltime scan. As few as 106 molecules have been collected in such an experiment (6).
The focus of this paper is to present the theory and voltammetric data for triple-band electrodes, in which the outer two bands operate as collectors poised a t the same potential and the central band is the generator. We wish to examine the relative merits of the different types of microelectrode arrays because an extended future for these electrodes as electrochemical sensors seem certain, due to their wide applicability and high information content. The theory for interdigitated arrays of bands has been developed by using conformal maps (7),and these can be fabricated by microlithographic techniques (4, 7-10). However, the triple band seems particularly attractive, because it is simple to fabricate from metal foils and is capable of a high collection efficiency (11).
The theory shown here is based on the use of conformal maps, a procedure we have previously described (2). Two conformal maps are used; one that resembles a thin-layer cell is used to arrive at the steady-state currents. A different conformal map is then used for digital simulation of voltammetric data at experimentally achievable time scales. The simulated and experimental results allow a critical comparison to be made between the single-band and the double- and triple-band collector-generator arrays. THEORY Conformal Map for the Triple-Band Electrode. The generation of a conformal map for the triple band is shown in Figure 1. In x-y space, the electrode consists of a central generator electrode flanked by two collector electrodes. The generator is of width wg,and the collectors are both of equal width, w,. The gaps between the generator and collectors are of equal width, g. Use of the Schwarz-Christoffel transformation allows space to be transformed to a closed box, which is similar to a thin-layer cell. The transformation is accomplished with the following equation with a = w g / 2 ,b = a + g, and c = b + w,
z = KJ2l/((z2 0
(1)
where K is a scaling factor, 2 = r + j 0 is the coordinate in the transformed space, and z = x + j y is the coordinate in r-y space. The positions where space is folded are referred to as the poles of inversion and are at the edges of the generator and the inner edges of the collectors. The width of the generator in the transformed space is
wg* = K l a-al / ( ( z 2 - a2)(z2- b2))ll2dz
(2)
where the superscripted * indicates a dimension in the transformed space. The width of the gap is
and the width of the collector is
* To whom correspondence should be addressed. 0003-2700/9110363-1403$02.50/0
- a2)(22 - b 2 ) ) 1 / 2 dz
Q 1991 Amerlcan Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 14, JULY 15, 1991
B.
A.
C
Table 11. Generator Current at a Triple-Band Electrode under Steady-State Diffusion Conditions" 0.5 1 2
I
U
"
*
~
I
I.,
(C) conformal space used for the timedependent simulations. In C are also represented the lines of equal concentration of the elecboactlve species duing a voltammogram at E = Eo lqRT/nF), for dimensionless scan rate p = ( ~ / P ) ( ~ F V / R %= ) ~0.01. '~
-
Table I. Comparison between the Steady-State Limiting Current at a Triple-Band Electrode Operated in the Collector-Generator Configuration, (i&, and Thin-Layer Approximation, iTLo triple band at steady state (im),/nFID@ C(*m)/CO
im/nFIDCOc
w,*/w,*
0.1 0.2 0.3 0.4 0.5 0.8 1 2 5
2.407 2.019 1.808 1.667 1.564 1.365 1.280 1.052 0.831
0.189 0.231 0.264 0.291 0.314 0.371 0.408 0.503 0.641
2.334 1.940 1.721 1.580 1.472 1.266 1.178 0.942 0.713
1.103 1.178 1.232 1.280
k
Flgure 1. Representation of a triple-band electrode (wp= w , = 9 ) in the real space (A) or in two different conformal spaces: (B) conforme1 space for steady-state conditions analopus to a thin-layer cell,
g/w,b
m
0.530 0.408 0.275 0
0.778 0.772 0.767 0.760 0.747 0.735 0.721
" Results are shown as a function of the ratio of the gap to generator width with equal widths of the bands (w, = W J . *Realspace relative dimensions. 'Thin-layer cell. In the conformal space, the triple band resembles a thinlayer cell with the generator on one side and the two collectors on the other side. The bulk of solution is located at the f m line, which is shown as a point in the two-dimensional representation in Figure 1. The distance between the outer edges of the collectors in the transformed space is w,* = wg* - 2w,* (5) Note that the value of w,* depends on the relative gap-towidth ratio in real space. Solution of the Flux at Steady State. At infinite time, steady-state behavior will be approached and, thus, Fick's second law approaches zero independent of which spatial coordinate system is considered. To approximate the solution for the triple band at steady state, consider the limiting flux when w,* = 0. This cell contains a generator on one surface and a continuous collector on the opposing surface. The collection efficiency of this thin-layer cell is unity a t steady state (i.e., the current at the collector and generator will be equal). The dimensionless current (Pm)is qn, = in,/nFIDcO = w,*/g* (6) where 1 is the length of the electrodes. The ratio wg*/g* is a ratio of elliptical integrals (eqs 2 and 3) and can be evaluated by Simpson's method. The results are given in Table I for different ratios of g/wr This solution will be approached by that of the triple band when the ratio of the gap to generator widths is small or the ratio of the collector to generator widths is large. To account for the steady-state limiting current with finite values of w,*, a finite difference solution for this geometry with d2C/dr2+ #C/dP = 0 was developed (Appendix I). The region comprised by w,* includes the f- line, which normally acts as a sink for products of the generator reaction and as
"Function of the ratio of the collection and generator widths with the width of the gap equal to the width of the generator (B = WJ. a source of the reactant. This line can be thought of as a small band, and the flux a t that band goes to zero at steady state (I). Therefore, the current at a tripleband collectowenerator assembly a t infinite time will be identical with a thin-layer cell of the dimensions in the transformed space, and the collection efficiency also will be unity. The steady-state dimensionless currents for this geometry (e,= P, = (im),/ nFDICa, where \kc is the s u m of the currents at the two collectors) are also given in Table I for various values of w,* with w, = wr A comparison of these values with those calculated with eq 6 shows that the approximation is suitable for geometries with small values of g/w,, since the results are within 3%. The differences increase as g/w, increases, with a 15% difference at g l w , = 5. Similar calculations were also made for the limiting, steady-state flux for the case where w, # wr The results for three values of w,/wg are given in Table 11, and it can be seen that, as the ratio increases, the flux approaches that given by eq 6, the last entry. The small variation of the steady-state current over a wide range of w,/wg shows that the edge of the collector closest to the generator is the major factor determining the amplitude of the steady-state current. Response away from Steady State. The steady-state approximation for the triple-band electrode given above was obtained by a u m i n g that a C / a r = 0 at f m . This is only true in the limit of infinite time, although the flux a t this point always will be less than when the collectors are not operating, i.e., when the generator acts as a single band with a flux \kb To calculate the triple-band case, we note that conservation of fluxes requires that the generator flux must equal the s u m of the other fluxes in space. Thus, as steady state is approached in the conformal space, P, = P,+ \k(*m,on) where \k(fm,on) specifies the dimensionless flux at f m when the collectors are operating. Because of conservation of flux, the flux at the infinity point for the single band as steady state is approached (\k(fm,off)) is equal to \kb, a value that approaches 0. Combination of these two relationships leads to We have used this equation in a previous treatment of the double band (2). Evaluation of P(f:..,on)/P(fm,off) allows examination of the relation between two experimentally measured quantities as steady state is approached the collection efficiency (PJP,) and the amplification factor (q,/\kd. By analogous arguments to those made previously (2),this ratio is equal to [ l - C(f-)/@I where C(fm) is the concentration in the conformal map of the triple band near the f m point, a concentration that differs from 0 because of the flux that occurs around that point. Values of C(fm)/C? at steady state were evaluated from the simulation of the closed box form of the conformal map described in Appendix I. The results of these calculationsare given in Table I for the case where w, = wr Over the range of g/wg considered, the value of 1 - C(*:co)/CO, and thus \k(fm,on)/P(fm,off), varies from 0.222 to 0.293. This is unlike the double band where this fador was equal to 0.5, irrespective of dimensions (2).
ANALYTICAL CHEMISTRY, VOL. 63,NO. 14, JULY 15, 1991
'
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I v
-0.50
i'
o.20 0.00 0.00
-1.00 - 1 50
0.20
0.40
0.60
0.80
1.00
-10
-5
0
5
10
F(E"-E)/RT Flguro 2. Simulated working curves for the triple-band electrode (we = w,) wlth the inverse amptlfication factor plotted as a fun& of the collectkn efficiency at E, = 4RTinF, for dmerent wkl!btogap rat& g l w , = 3 (01,2 ( O ) , 1.5 (O), 1 (A),and 0.5 (+). The limitlng behaviors at /c//e = 0 or 1 are indicated by the dashed lines (see text for the limit at /,//e = 1.
+
Thus,a plot of the inverse of the amplificationfactor versus the collection efficiency will lie between two weI1-defined limits. At low values of the collection efficiency when there is no feedback, the inverse amplification will be unity. In contrast, at complete steady state, 9Jqg= 1and 9 b / q g = 0. As this limit is approached, that is, near steady state, the slope of the curve will be approximately -4 (Le., between the ranges of 1/0.222 and 1/0.293 for the ratios of g / w , considered). The intersection of these two projections limits the range of valuea that can be obtained (Figure 2). The fact that the limiting slope is greater than with the double band (slope of -2) is simply a reflection of the fact that less flux escapes the triple band. Conformal Map for Digital Simulation. The closed box is not a convenient topography for digital simulations because of the complexity of expressing the space-dependent diffusion coefficient for this map. Therefore, we have employed the conformal map shown in Figure 1C which yields a three-sided, open box. The transformation is similar to the one used for the double band (21,but the poles of inversion are located between the generator and the collector, nearest to the generator. The distance between the two inversion poles is 2d, which is slightly larger than wg (see Appendix 11). The coordinate system is given by ~t= d(cosh r)(cos e) (7) y = d(sinh I')(sin e) (8) The spatially dependent diffusion parameter for this map is
D* = D/[2d(sinh2 r
+ sin2e)]
(9)
The diffusion equation
ac/at = D*[aZc/aez + a2c/arz]
(10)
in the conformal space was solved numerically by using the hopscotch algorithm (12).Although not shown, the addition of homogeneous kinetic schemes to the diffusion equations is straightforward as was shown for the double band (2).
EXPERIMENTAL SECTION Chemicals. Ferrocene (FeCp,) (Aesar, Johnson-Matthey, Seabrook, NH)was purified by sublimation before use. Solutions of FeCp, were prepared in acetonitrile (UV grade, Burdick & Jackson, Muskegan, MI) with 0.2 M tetrabutylammonium perchlorate (TBAP). TBAP (Aldrich, Milwaukee, WI)was recrystallized twice from a pentaneethyl acetate mixture. Solutions were deoxygenated with acetonitrile-saturated nitrogen. The diffusion coefficient for ferrocene in 0.2 M TBAP/CH&N is 2.2 x 10-6 cm2 s-l (13). Electrodes. The triple-band electrodes were prepared by placing one or two layers of mylar film (2-pm thickness, Polaron
Flgue 3. Simulated cyclic voltammograms for a triple-band electrode (we = w,; g = 0.5 we), for p = ( ~ / ~ ) ( ~ F v / R T=D 0.005. )~/*
Instrumenta, Hatfield, PA) as insulators between three sheets of Pt foil (11). The Pt foil was purchased from Aesar in sheets 4 f 2 pm thick. The electrodes were constructed by applying a small amount of epoxy (Epon 828 with 14% m-phenylenediamine, Miller-Stephenson Co., Danbury, CT) first to the bottom glass slide and then between all layers as the electrode assembly proceeds. Pressure was applied during the curing process (150 "C) with a large spring clip to keep the spacing between the Pt foils as small aa possible. Electrical connection was made to the back of each foil with silver epoxy (Epo-Tek H20E, Epoxy Technology, Billerica, MA), and then TORSeal (Varian Associates, Lexington, MA) was applied as a barrier to solutions. The edge of the assembly was ground smooth with a slurry of 1000-grit carborundum powder on a glass plate. Electrodes were polished with a slurry of l-pm cerium oxide and fine polished with 0.25-pm diamond paste on a napless cloth (all, Buehler). The surface of the electrode was initially cleaned by sonication in both methanol and water. The surface was cleaned between experiments by wiping with a tissue soaked in acetonitrile. Electrode lengths were estimaied microscopically with a Bausch & Lomb Stereo Zoom microscope. Widths were estimated from photographs obtained with an inverted stage light microscope (Zeias,Axiovert 35) calibrated with 2-pm divisions. The electrode widths and interelectrode gap widths varied between 4 and 7 pm. It was apparent from these photographs that the bands were misaligned lengthwise by a maximum of 5 % in some cases. To correct for this, the values of ib (single band current) and ig (generator current) were reduced by a proportional amount (2). Instrumentation. Potential control for the triple-band electrodes was maintained with a bipotentiostat of conventional design. Potential waveforms were generated with a PAR 175 universal programmer (Princeton Applied Research, Princeton, NJ), and voltammograms were recorded on a x-yy'recorder or digital oscilloscope. A conventional three-electrode cell was employed with a large Pt counter electrode and either an SSCE reference electrode or Ag wire quasi-reference electrode. Simulations. The programs were written in Pascal (Professional Pascal from Metaware, Inc. (Santa Cruz))and performed on a Compaq 386/20e (clock speed 20 MHz) equipped with a Weitek 1167 numerical coprocessor and on mainframe Gould computers of the Centre de Calcul Recherche of the University Pierre et Marie Curie (Paris, France). Programs are available on request from Christian Amatore.
RESULTS AND DISCUSSION Evaluation of the Simulation. Figure 3 shows simulated voltammograms for the generator and collector. At this scan rate, the triple band gives a nearly steady-state response a t the generator, whereas the collector current slightly increases with time. The accuracy of the simulation was tested by increasing the number of time and space elements over those given in Appendix II, and less than 0.5% change in the results was found. A second test of the program was made in which the collectors were considered as insulators. In this case, the result obtained was identical with that published previously for the single band (1). This demonstrates that moving the pole of inversion slightly away from the edge of the generator
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 14, JULY 15, 1991
Table 111. Approach of Triple-Band Electrode to Steady State with Various Values of g/w,. Calculations for w, = wa I
/
dimensionless generator
u, 0
0.0
current
.
I
-0.5 -2
-1
0
2
1
log{w(nFv/RTD)"2)
collection
g/w,
steady state
simulated"
efficiency
0.5 1 2 3
1.472 1.178 0.942 0.832
1.487 1.203 0.976 0.870
0.830 0.785 0.720 0.672
Calculated for p = 0.01.
10
B1
0.0
0.2
1
8 0 0
00
-3
-2
-1
0
n
I 1
B
log(p)
0.4
Flgure 4. (A) Varlatbns of the dlmensbnlessgenerator peak current, as a function of the dlmnsbnless scan rate ( ~ ( n F v l R 7 D ) "and ~ gap-to wldth ratio. (B) Collection efficiency as a function of p = ( g 1 2 ) ( n F ~ l R T D )and ' ~ ~ gap-to-wldth ratio. I n A and 6, theoretical values are indicated by open symbols for w, = w, and g l w , = 0.5 (+), 1 (A),1.5 (O), 2 (01, and 3 (01, and experlmental values are Indicated by filled symbols (V)single band: (A)triple band; w, = w, = 6.9 pm, g = 6.4 pm, 1 = 0.42 cm) for a 1.1 mM FeCp, solution with 0.2 M TBAP in acetonMle at 20 'C. Theoretical and experhnental collection efficiencies are given at E, = Eo 4RTlnF.
*,,,
+
does not alter the results. A simulation with equivalent accuracy using a real space coordinate system and a linear grid would need 550 points on the half of the generator used for calculations. The conformal map simulation has a spacing of 2AI'/w, = 0.0383 at the outer edge of the generator. This would require that 2Ax/w, = 0.0018 in real space. The number of points required is the inverse of the latter number. Approach to Steady State. The simulator was used to evaluate how rapidly steady state is approached. An example set of calculations is shown in Figure 4A and is compared to the scan rate behavior of a single band. At high scan rates, little interaction occurs between the collector and the generator, and the generator current becomes identical with that of the single band. As the scan rate is lowered, the generator current approaches steady-state behavior, and the transition to this behavior depends on the gap size. With small gaps, the diffusion layer of the collector and generator interact more readily, leading to a higher scan rate where the transition occurs. The steady-state value reached is inversely proportional to the gap-to-width ratio in the conformal space and can be approximated by eq 6. The degree to which true steady-state behavior is attained is shown by the results in Table 111. The simulator was used to calculate the currents at a scan rate corresponding to 1mV for w, = wg = 4 pm and D = 2 X lod cm2s-l). The results were compared to those calculated for true steady-state conditions (Appendix I). The values of the generator currents are within 1% of the steady-state value for small values of g/w,, but for larger values, greater deviation is seen. These results demonstrate that steady state is achieved more rapidly with small values of g/w,. Note also that the collection efficiency, which is unity at steady state, approaches this value more slowly for large values of g/w,. Effect of Scan Rate. As shown in the Theory section, the relation between the inverse amplification factor and the collection efficiency should fall within the limits given by the
-1.21 0.6
0.4
0.2
0.0
I
E (V vs. SSCE)
Flgure 5. Voltammetry of 1.1 mM FeCp, with 0.2 M TBAP In acetonltrlle at single-band, doublaband, and triple-band electrodes. In the triple-band case the central band operates as the generator with the two outer bands as collectors. The central band was used as a single band and as the g e m t o r for the double-band case. Electrode length, 0.41 cm;electrode wldth, 6.9 pm; gap width, 6.4 pm. Scan rates are (A) 0.01 V s-' and (B) 1 V s-'.
solid lines in Figure 2, irrespectiveof scan rate. The simulator was used to calculate a series of points for different values of p and glw,, all with w, = wr As seen, the values all lie within the predicted limits. In contrast with the double band, the points do not lie on a single curve, but rather form a family of curves that depend upon the value of glw,. For the smallest value of this parameter, the limiting behavior is most closely approached. Shown in Figure 4B is the collection efficiency as a function of dimensionless scan rate for various values of g/ wr Again, the curves do not superimpose, a result in direct contrast for that previously shown at the double band (2). This is because collection becomes much more efficient at the triple-band electrode as g l w , becomes small, and the collector-generator assembly acts more like a thin-layer cell. For this reason, the general expression reported elsewhere (4) should only be used for electrode assemblies of specific dimensions used in those simulations. Oxidation of Ferrocene at a Triple Band. The voltammetry of ferrocene in acetonitrile obtained at a triple-band electrode (10 mV s-l) is shown in Figure 5A. Included in the figure are the single-band, generator, and collector voltammograms. The outer collectors were maintained at 0.0 V, while the potential of the central generator was swept. The collection efficiency for this triple band, with the generator scanned at 10 mV s-l, is 0.75. In contrast, the collection efficiency with this electrode operated in the two possible double-band configurations using the central band as the generator was both 0.50 at 10 mV s-l. Voltammograms obtained with the same electrode when the generator is scanned 100 times faster are shown in Figure
::pq :p=, ANALYTICAL CHEMISTRY, VOL. 63,
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A
0 00
-0 50 - 1 00
-10
-5
0
5
10
-5
0
5
10
0 50
O W -0 50 -10
(E"-E)F/RT
Figwe 6. Comparisonof the sknuhted voltammograms at a single (S), double (Oe),or triple (TB)band electrode, for p = 0.1 (A) or at steady state (B). Simulation conditions: w g = w, = w b = g; the generator currents are labeled 0, and the collector cunents are labeled C in A. I n B, the cunent is also shown for an IDA wlth electrodes of identical dknensbns. Only generator cunents are shown because the collection efflclencies are unlty.
5B. The collection efficiency of the electrode is decreased to 0.56 at 1V s-l. Single-band and generator dimensionless peak currents are compared to the simulated values in Figure 4A for data obtained at scan rates from 0.01 to 1 V s-l. Figure 4B compares the collection efficiency obtained in these experiments to the simulation. The agreement with the simulation is seen to be within experimental error.
Comparison between Single-, Double-, and TripleBand and Interdigitated Array Electrodes. In previous work, we have evaluated the current at a single band and the double band operated in the collector-generator mode (I, 2). The triple-band electrode is advantageous over the double band in that higher collection efficiencies are achieved than with a double band with identical gap size. Another advantage of the triple band is that it approaches steady state more rapidly than the other two types of electrodes. This is illustrated in Figure 6A, where simulated voltammograms are compared at identical scan rates. The current at the single band is clearly not steady state; the triple band has a greater degree of steady-state character than the double band. For the double and triple band, steady state is achieved because of feedback from the collector. The greater feedback that exists at the triple band results in its more rapid approach to steady state. This is directly apparent in the amplification factor, which is found to have greater values at the triple band for fixed values of the collection efficiency (compare Figure 2 with Figure 5 of ref 2). However, in many instances, the double band is the electrode of choice because of the increased experimental difficulty in fabrication of triple bands compared to double bands. Furthermore, although the triple band is a more symmetric structure with respect to the generator, the evaluation of the theory is more complicated and does not lead to concise relationships as were found for the double band (2). Another point of interest is a comparison of these results with the electrochemical response of interdigitated arrays (IDAs). Since the IDAs have been considered under steadystate conditions (7),this discussion is limited to conditions where the collection efficiencies are unity for the double-band and triple-band electrodes and IDAs. This is illustrated in Figure 6B, where simulated steady-state voltammograms (limit for p 0) are represented for a single band (I),a double band (2),a triple band (this work), and one element of IDAs (one cathode + one anode) (7). For a single band, the steady-state
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NO. 14, JULY 15, 1991
1407
current is zero. The steady-state voltammogram for an IDA was determined by using eq 28 of ref 7 (note that there is a typographical error in eq 21 of the same reference: H is the difference and not the sum of the two potential dependent terms). The larger generator current observed at the triple band in comparison to that observed at one element of an IDA may seem surprising. However, this arises because in each element of an IDA, one generator is associated with one collector (as in the double band) rather than with two collectors (as in the triple band). Moreover, lateral diffusion to the generator is shielded by the other IDA elements. Therefore, an IDA element can be viewed as more similar to a double band than a triple band. However, since there are many generator-collector pairs in an IDA, the total current will be larger for an IDA than for a triple band with elements of identical dimensions.
APPENDIX I Resolution of the Laplace Equation in the Closed Box: Steady-State Approximation for the Triple Band. The Laplace equation (vzC= 0) was solved for the closed box with the boundary conditions of C = 0 at the generator and C = CO at the collector with zero flux at the other surfaces, including the fa, line. Because of the symmetry of the sptem, the calculation was limited to positive values of r (corresponding to positive values of x ) . The zero flux condition was imposed and concentrations calculated with a parabolic approximation of the second derivative. The region where the concentrationprofiles are most sharply curved is at the outer edges of the collector. Thus, a space grid was chosen that included 85 points across the region 0 < r < 1.07wg*/2,with 15 more points across the remainder of the collector. A total of 58 points were calculated across the 8 axis with 30 of these located adjacent to the collector. The initial concentration profiles were set so that they were linear between 0 = 0 (generator surface) and 0 = g* (collector surface). From these initial profiles, iterations were performed until li, - icl/(ig + ie) < 10-4. APPENDIX I1 Diffusion Equations in the Conformal Space Used for Simulations of the Triple-Band Current. To solve for the current at the triple-band electrode, Fick's second law
L
must be solved. The following boundary conditions define the reversible, diffusion-controlled case where the diffusion coefficients of the reactant (A) and product (B) are equal and where wg = w, = w: att=O y 2 0,
o;
r-m,
0 < e < ~ / 2 ; U-i,b-o
generator
r = 0;
cos-' (w/2d)
< e < a / 2 ; b = uet
collector cosh-' ((w/2
+ g ) / d ) < r < cosh-'
((3w/2 O=O;
+ g)/d); a=0, b = l
insulator
r = 0, o < e < COS-1 ( ~ 1 2 4 ;(aa/ar),,, = 0; o < r < cosh-1 ( ( ~ / 2+ g ) / d ) , e = 0; (aa/ae),,, = 0, r > cash-1 ((3w/2 + g ) / d ) , e = 0; (eqae),=, symmetry condition
r > 0; e = */2;
( a a / d ~ ) , , , ~=~
o
=o
The grid employed in the conformal space needs to have a high definition at the poles of inversion to account for curvature in the concentration profiles in these regions. In this work, the 0 axis (from 0 = 0 (the pole) to ~ / 2 was ) divided into 10 equal elements, and the element nearest the pole was subdivided into 10 equally spaced elements, giving a total of 19 points in the 0 axis. The edge of the generator is located in the middle of the well-defined region because of the selection of the pole of inversion (d) in the gap. Along the r axis, a double scale is also used with 40 equally spaced points across the collector (only one collector needs to be considered, again because of symmetry). Forty grid elements defined the remainder of the axis until the outer bound which was given empirically by 7 - p In p based on previous results at the single band. The initial dimensionless potential employed was -10 and the final potential +lo, and 10000 iterations were employed. Registry No. Pt, 7440-06-4; FeCp,, 102-54-5. LITERATURE CITED Deakin, M. R.; Wlghtman, R. M.; Amatwe, C. A. J . Electroanel. Chem. 1988, 215, 49-61. Fosset, E.; Amatore, C. A.; Bertelt, J. E.;Michael, A. C.; Wightman, R. M. Anal. Chem. 1991, 63, 306-314. Wightman, R. M.; Wipf, D. 0. Voltammetry at Mlcroelectrodss. In Electroanalyticel Chembtry; Bard, A. J.; Ed.; Marcel Dekker: New York, 1988; Vol. 15, Chapter 3, pp 267-353. Bard, A. J.; Clayston, J. A.; Kittlesen, G. P.; Varco Shea, T.; Wrighton, M. S. Anal. Chem. 1988, 58, 2321-2331. Varco Shea, T.; Bard, A. J . Ana/. Chem. 1987, 59, 2101-2111. Licht, S.; Cammarata, V.; Wrighton, M. S. Sclence 1989, 243, 1 176- 1170. Add, K.; Morlte, M.; Niwa, 0.:Tabei, H. J . Ektroanal. Chem. 1988. 256, 269-282. Chldsey, C. E.; Feldman, E. J.; Lundgren, C.; Murray, R. W. Anal. W m . 1988, 58, 601-607. Sanderson, D. G.; Anderson, L. 6. Anal. Chem. 1985, 57. 2388-2393. Fosdick, L. E.; Anderson, J. L.; Baginski, T. A.; Jaeger, R. C. Ana/. Chem. 1088, 58, 2750-2756. Barteit, J. E.; Deakin, M. R.; Amatore, C. A,; Wightman, R. M. Anal. Chem. 1988, 60, 2167-2169. Shoup, D.; Szebo, A. J . Electfoanal. Chem. 1984, 160, 1-17. b u r . J. E.; Wightman, R. M. J . Electraa~l.Chem., in press.
RECEIVED for review November 9,1990. Accepted March 22, 1991. This research was supported by the National Science Foundation (CHE-8996213) and CNRS (U.A. 1110, "Activation Moleculaire"). Financial support from NATO (International Grants Division) is also acknowledged.
