Voltammetric Determination of the Geometrical Parameters of Inlaid

However, most of these investigations do not address the simultaneous determination .... In particular, it was ascertained that eq 2,40 which holds fo...
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Anal. Chem. 2004, 76, 6575-6581

Voltammetric Determination of the Geometrical Parameters of Inlaid Microdisks with Shields of Thickness Comparable to the Electrode Radius Ilenia Ciani and Salvatore Daniele*

Department of Physical Chemistry, University of Venice, Calle Larga, S. Marta, 2137, 30123 Venice Italy

The cyclic voltammetric behavior of disk microelectrodes surrounded by thin insulating shields (TSM) was investigated from both theoretical and experimental points of view. In particular, microdisks with shields of thickness (b - a), a few electrode radii (a) were considered. A finite difference simulation procedure with a nonuniform, expanding spatial grid, already available in the literature, was employed for predicting shape and height of the voltammograms. The parameters of this numerical simulation were optimized again, and the steady-state limiting currents found in this work for a range of TSMs compared well with previous publications. The steady-state limiting current at TSMs was enhanced with respect to microelectrodes surrounded by thick insulating sheaths (i.e., b . a), and steady-state conditions were achieved faster. Under these conditions, the difference in potential observed on the forward and backward waves, when the current is half of its maximum value (∆E1/2), was almost equal to zero. Nonsteady-state cyclic voltammograms were also simulated using the optimized parameters, and the effect of scan rate on ∆E1/2 was examined in detail. Based on this dependence, a voltammetric procedure for the simultaneous determination of microdisk radius and its insulator thickness was proposed. Experimental measurements were performed by using platinum wires, 1012.5-µm radius, and carbon fibers, 4-µm radius, coated with 4-6-µm-thick electrophoretic paint. The shield thickness produced around the wires was such that b/a < 3. The experimental results obtained were in general congruent with the theory and demonstrated the validity of the method proposed here for the simultaneous determination of radius and shield thickness of a TSM. Microelectrodes are attractive tools for undertaking a variety of electrochemical experiments1-3 and have found wide application * Corresponding author: (e-mail) [email protected]; (tel.) + 390412348630; (fax) + 390412348594. (1) Wightmann, R. M.; Wipf, D. O., In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1989; Vol. 15, pp 267-353. (2) Microelectrodes: Theory and Applications; Montenegro, M. I., Queiros, M. A., Daschbach, J. L., Eds.; NATO ASI Series, Kluwer Academic Publishers: Dordrecht, 1991. (3) Amatore, C. In Physical Electrochemistry; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995; p 131. 10.1021/ac049041u CCC: $27.50 Published on Web 10/09/2004

© 2004 American Chemical Society

in the study of fast heterogeneous and homogeneous reactions,1-4 for measurements in various microenvironments,5,6 and for highresolution electrochemical imaging.7,8 Although a variety of geometries exist, disk-shaped microelectrodes are mostly employed, because these are more easily and reproducibly fabricated.9,10 The most common fabrication procedure of inlaid microdisk electrodes involves encapsulating carbon fibers or metal wires in glass capillaries and then polishing the tips to expose the microdisk surfaces.9,10 Other methods have also been employed to produce an insulating sheath around the electrode material and include dip-coating metallic wires or carbon fibers with insulating polymers,9,11,12 deposition of silica thin films,13,14 and electrophoretic deposition of paint.15-20 With the latter construction methods, insulating shields with thickness (b - a) comparable to the electrode radius a are usually produced (Figure 1A). The current interest for such kinds of electrodes, other than for small-volume and microenvironment applications,5,6 is largely driven by the use of microelectrodes as tips in scanning electrochemical microscopy (4) Montenegro, M. I. In Research in Chemical Kinetics; Compton, R. C., Hancock G., Eds.; Elsevier: Amsterdam, 1999; p 1. (5) Kawagoe, K. T.; Zimmerman, J. B.; Wightman, R. M. J. Neurosci. Methods 1993, 48, 225-240. (6) Clark, R. A.; Zerby, S. E.; Ewing, A. G. In Electroanalytical Chemistry; Bard, A. J., Rubinstain, I., Eds.; Marcel Dekker: New York, 1998; Vol. 20, pp 227294. (7) Bard, A. J.; Fan, F. R. F.; Mirkin, M. V. In Electroanalytical Chemistry, Bard, A. J.; Rubistein, I., Eds.; Marcel Dekker: New York, 1994; Vol. 18, pp 243-373. (8) Scanning Electrochemical Microscopy; Bard, A. J., Mirkin, M. V., Eds.; Marcel Dekker: NewYork, 2001. (9) Ultramicroelectrodes; Fleischmann, M., Pons, S., Rolison, D. R., Schmidt, P. P., Eds.; Datatech Systems, Inc.: Morganton, NC, 1987. (10) Stulik, K.; Amatore, C.; Holub, K.; Marecek, V.; Kutner, W. Pure Appl. Chem. 2000, 72, 1483-1492. (11) Potje-Kmloth, K.; Janata, J.; Josowicz, M. Ber. Bunsen-Ges. J. Phys. Chem. 1989, 93, 1480-1485. (12) Strein, T. G.; Ewing, A. G. Anal. Chem. 1992, 64, 1368-1373. (13) Zhao, G.; Giolando, D. M.; Kirchoff, J. R. Anal. Chem. 1995, 67, 25922598. (14) Bozon, J. P.; Giolando, D. M.; Kirchoff, J. R. Electroanalysis 2001, 13, 911916. (15) Slevin, C. J.; Gray, N. J.; Macpherson, J. V.; Webb, M. A.; Unwin, P. R., Electrochem. Communn. 1999, 1, 282-288. (16) Zhang, X.; Ogorevc, B.; Rupnik, M.; Kreft, M.; Zorec, R. Anal. Chim Acta 1999, 378, 135-143. (17) Conyers, J. L. Jr.; White, H. S. Anal. Chem. 2000, 72, 4441-4446. (18) Schulte, A.; Chow, R. H. Anal. Chem. 1996, 68, 3054-3059. (19) Mao, B. W.; Ye, J. H.; Zhuo, X. D.; Mu, J. Q.; Fen, Z. D.; Tian, Z. W. Ultramicroscopy 1992, 42-44, 464-467. (20) Ciani, I.; Daniele, S. J. Electroanal. Chem. 2004, 564, 133-140.

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Figure 1. Schematic representation of (A) electrode dimensions and simulation domain and (B) diffusion at a TSMs. (C) SEM image of an electropainted 10-µm-radius platinum wire.

(SECM).7,8 In this technique, the parameter b/a, usually defined as RG, has strong influence on the Faradaic tip current, especially when the tip is moved in the vicinity of an inert substrate.21,22 One of the characteristic features of a thin-shielded microelectrode (TSM) is that, on the time scale of standard voltammetric measurements, the diffusion field undergoes a transition from linear to radial symmetry and is established radially behind the plane of the electrode and shield (Figure 1B). Under these conditions, the flux, and consequently the current, is enhanced to an extent that depends on the relative size of insulating shield and electrode radius (i.e., RG),20-26 and the equation for an inlaid microdisk electrode embedded on an infinite insulating plane27 (ISM)

id ) 4nFDca

(1)

is not longer valid for predicting the steady-state limiting current (id) at TSMs. In eq 1, n is the number of electrons, D and c are the diffusion coefficient and the bulk concentration of the electroactive species, respectively, and the other symbols have their usual meaning. Shoup and Szabo were the first to demonstrate, from a theoretical point of view, that at TSMs diffusion from behind the plane of the electrode enhances the flux to the inlaid disk.25 This has recently been confirmed by other researchers, who employed different digital simulation procedures to obtain either steady-state limiting currents22,26 or cyclic voltammograms23 for TSMs with a range of RG. From the simulation data, approximate analytical expressions for the steady-state limiting current (iss) as a function of RG have also been derived.13,23,26 Table 1 summarizes such equations (in a dimensionless form, i.e., Iss ) iss/id) and the limiting current values calculated for RG f 1, while Figure 2 displays either the graphical form of the latter equations over a wide RG range or currents at discrete RG values from ref 25. It is evident that the current values obtained from ref 23 differ to a relatively large extent with respect to those from refs 13, 25, and 26, for very thin-shielded microdisks (RG e 2). In all cases, however, (21) Kwak, J.; Bard, A. J. Anal. Chem. 1989, 61, 1221-1227. (22) Amphlett, J. L.; Denuault, G. J. Phys. Chem. B 1998, 102, 9946-9951. (23) Fang Y.; Leddy, J. Anal. Chem. 1995, 67, 1259-1270 (24) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1982, 140, 237-245 (25) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1984, 160, 27-31. (26) Zoski, C. G.; Mirkin, M. V. Anal. Chem. 2002, 74, 1986-1992. (27) Saito, Y. Rev. Polarogr. 1968, 15, 177.

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Table 1. Steady-State Limiting Equations for Microdisk Electrode with Thin Shields equation (ref) (RG)-1

Iss for RG f 1 (RG)-2

Iss ) 1.000 + 0.234 + 0.255 (23) Issa ) 1.000 + 0.379 (RG)-2.342 (13) Issa ) 1.000 + 0.1380 (RG - 0.6723)-0.8686 (26) a

a

1.489 1.379 1.364

Iss ) iss/id.

the current at a TSM is at least 36% greater than that predicted by eq 1, when the insulating sheath thickness tends toward zero. In some cases, the equations displayed in Table 1, along with experimental steady-state limiting currents, have been employed for the electrochemical characterization of TSMs,13,16,20,23 and in particular for determining b if a was known, or vice versa. However, for these electrodes, often, neither b nor a is known with accuracy and their simultaneous determination by electrochemical measurements is not straightforward. SECM has been used to evaluate tip shapes and the RG parameter.7,8,26,28-36 The method is based on the measurement of tip current as a function of tip-substrate separation (approach curve21) over either a conducting or an insulating substrate. Experimental approach curves are then compared to a family of theoretical working curves computed for different electrode aspect ratios and RG values. Several microelectrode geometries, including inlaid disks,8,29 have been characterized in this way. However, most of these investigations do not address the simultaneous determination of electrode radius and its insulator thickness specifically. For the simultaneous determination of the b and a of TSMs, a cyclic voltammetric method was proposed by Fang and Leddy.23 It makes use of both steady-state and non-steady-state currents, (28) Shao, Y.; Mirkin, M. V.; Fish, G.; Kokotov, S.; Palanker, D.; Lewis, A. Anal. Chem. 1997, 69, 1627-1634. (29) Selzer, Y,; Mandler, D. Anal. Chem. 2000, 72, 2383-2390. (30) Mauzzeroll, J.; Hueske, E. A.; Bard, A. J. Anal. Chem. 2003, 75, 38803889. (31) Demaille, C,; Brust, M. B.; Tsionsky, M.; Bard, A. J. Anal. Chem. 1997, 69, 2323-2328. (32) Lee, Y.; Amemiya, S.; Bard, A. J. Anal. Chem. 2001, 73, 2261-2267. (33) Liljeroth, P.; Johans, C.; Slevin, C. J.; Quinn, B. M.; Kontturi, K. Anal. Chem. 2002, 74, 1972-1978. (34) Mirkin, M. V.; Fan, F.-R. F.; Bard, A. J. J. Electroanal. Chem. 1992, 328, 47-62. (35) Machpherson, J. V.; Unwin, P. R. Anal. Chem. 2000, 72, 276-285. (36) Zoski, C. G.; B. Liu.; Bard A. J. Anal. Chem. 2004, 76, 3646-3654.

Figure 2. Theoretical steady-state dimensionless currents against the parameter RG: (- - -) from refs 23, (‚ ‚ ‚) 13, (s) 26, and (*) 25 and (O) this work.

which are combined with the length δ ) [RTD/nFv]1/2 (v is the scan rate, T is the temperature in kelvin, R is the gas constant), and the parameters a and b. The proposed procedure has been applied to characterize inlaid microdisks with RG > 3.5. Considering a cyclic voltammogram obtained with a microdisk, the approach to steady state can also be characterized on the basis of the difference in potential observed on the forward and backward waves, when the current is half of its maximum value (∆E1/2).23,37,38 In fact, at steady state ∆E1/2 ) 0; that is, the forward and backward current-potential profiles exactly retrace one another.23,36 Otherwise ∆E1/2 > 0, and for a given δ, it depends on RG, or vice versa. The latter dependence could therefore be exploited for the simultaneous determination of the b and a geometrical parameters. The present paper focuses on TSMs with RG < 3, where the current enhancement is significant, may be useful for analytical purposes,20 and has three main objectives: first, to optimize the Fang and Leddy simulation procedure in order to obtain steadystate limiting currents that better agree with other theoretical approaches; second, to propose a new voltammetric method, based on ∆E1/2, for the simultaneous determination of radius and shield thickness; third, to provide experimental current data and cyclic voltammograms to test the theory. EXPERIMENTAL SECTION Chemicals. All chemicals were of analytical-reagent grade. Hexammineruthenium(III) chloride (from J. Matthey), ferrocene (Fc), potassium chloride, and acetonitrile (from Aldrich) were used as received. All aqueous solutions were prepared with deionized water purified via a Milli-Q unit (Millipore, Bedford, MA). Cathodic electrophoresis paint Stollaquid D 1330 was supplied by Herberts. Before use, the paint was diluted 1:6 with Milli-Q water and shaken for at least 24 h. All voltammetric measurements were carried out in solutions that had been deaerated with pure nitrogen (99,99%) (from SIAD). Apparatus. Cyclic voltammetry and other electrochemical measurements were performed using an M 283 potentiostat/ (37) Zoski, C. G.; Bond, A. M.; Colyer, C. L.; Myland, J. C.; Oldham, K. B. J. Electroanal. Chem., 1989, 263, 1-21. (38) Zoski, C. G. J. Electroanal. Chem., 1990, 296, 317-333.

galvanostat (EG & G PAR, Princeton, NJ) and the M 270 electrochemical software (EG&G PAR). Unless stated otherwise, the experiments were performed at room temperature in a onecompartment electrochemical cell placed in a Faraday cage using a two-electrode cell configuration. Unless otherwise stated, the measurements were carried out at room temperature (22 ( 1 °C). Electrodes. Working microelelectrodes were fabricated starting from 10- and 12.5-µm-radius platinum wires and 4-µm-radius carbon fibers. Those electrodes that had to be coated with paint were prepared as described elsewhere.16,20 In particular, 2-3-cm Pt wires or carbon fibers were mounted at the end of a copper wire (0.2-mm diameter) and fixed by a silver conducting paint. These mounted electrodes were inserted in glass capillaries (0.5mm inner diameter) and ∼1 cm of wire (or fiber) was left protruding. The cylindrical length of the electrode was insulated by deposition of the cathodic electrophoretic paint in a twoelectrode cell containing the Stollaquid/water mixture. The cell was thermostated at 29 ( 1 °C, and the film deposition was triggered by applying a potential step starting from 0 to -8 V for 5 min. The coated electrodes were allowed to cure in an oven at 180 °C for 20 min. Full coverage of the wires (or fibers) was assessed by performing cyclic voltammograms in a Ru(NH3)6Cl3 aqueous solution and from the lack of any current signal over the potential range -1.5 to 1.5 V. To expose the microdisk, the end part of the paint-insulated electrode was cut off with a microsurgical scalpel blade. A scanning electron microscope (SEM) image of a TSM is shown in Figure 1C. Platinum microelectrodes with larger insulating shields RG ≈ 100 were prepared by sealing the wires following standard procedures.1 Unless otherwise stated, the reference electrode was an Ag/ AgCl saturated with KCl. RESULTS AND DISCUSSION Optimization of the Simulation Parameters. Theoretical cyclic voltammograms for a range of TSMs were obtained by using the simulation procedure described in ref 23. It relies on the explicit finite difference (EFD) method with a fixed time grid and exponentially expanding spatial grid (Supporting Information). Since this simulation procedure provides, as mentioned in introduction, steady-state limiting currents, which differ to some extent from those reported in other works,13,22,24-26 relevant simulation parameters were optimized again (Supporting Information). The main differences found between our simulation data and those reported previously23 lied in the number of the grid elements (Na) that were necessary for an accurate prediction of steady-state limiting currents. In ref 23, Na ) 6 was used throughout regardless of the RG values. We verified (Supporting Information) that simulated steady-state limiting currents converged to a constant value (within 1%) only for rather larger Na values, (i.e., g 25, Tables S2 and S3). Moreover, the currents obtained in this work agree within 3.5 and 1.5% with those from refs 13 and 26, respectively, whereas discrepancies up to ∼9% arise with respect to those from ref 23 (see Table S3). A series of simulated steady-state limiting currents, obtained in this work for TSMs, with RG over the range 1.05-18, is included for comparison in Figure 2 (circles). Analysis of Simulated Cyclic Votammograms and Effect of Scan Rate. The optimized simulation parameters were used Analytical Chemistry, Vol. 76, No. 22, November 15, 2004

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Figure 3. Simulated cyclic voltammograms for a TSM with RG ) 1.45 at different scan rates.

to compute cyclic voltammograms at different scan rates (or length δ), to include both steady-state and non-steady-state behaviors, for a series of TSMs with RG over the range 1.1-3.1. Figure 3 shows typical cyclic voltammograms obtained at different scan rates for a TSM with RG ) 1.45. As expected,23 the waves from the sigmoidal shape, with the forward and backward curves retracing one another, become peak shaped, the maximum current increases, and ∆E1/2 is sensibly greater than zero, as v increases (or δ decreases). This follows for the larger contribution of linear diffusion to the electrode surface, relative to radial flux. This makes the achievement of the steady state less rapidly. From several sets of simulated cyclic voltammograms of the type shown in Figure 3, the quantitative effect of b, a, and δ on ∆E1/2 was evaluated. Figure 4A displays the plots obtained from combinations of b/a and (b - a)/δ, parametrized by ∆E1/2. These dependences can be regarded as working curves, which can be employed for the simultaneous determination of a and b for TSMs characterized by RG over the above interval (see below). The shape of the steady-state voltammograms was also examined in terms of the Tomesˇ criterion,39 (E1/4 - E3/4), which is related to the reversibility of the electrode process. In particular, it was ascertained that eq 2,40 which holds for a reversible

(E1/4 - E3/4) ) RT/nF ln 9

(2)

electrode process, taking place at a microdisk embedded in an infinite insulating sheath (symbols have their usual meanings), is also valid for a TSM. In fact, simulated cyclic voltammograms (setting n ) 1 and T ) 298 K) provided (E1/4 - E3/4) ) 56.33 mV, irrespective of the RG value, which is very close to 56.38 mV calculated by eq 2. Therefore, the Tomesˇ criterion can also be used to establish the reversibility of an electrode process at TSMs. Simultaneous Determination of a and b by Voltammetry Based on ∆E1/2. For the simultaneous determination of a and b, the working curves displayed in Figure 4A can be employed. (39) Tomes, J. Collect. Czech. Chem. Commun. 1927, 9, 150. (40) Bond, A. M., Oldham, K. B.; Zoski, C. G. Anal. Chim. Acta 1989, 216, 177-230.

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Figure 4. (A) Several series of (O) simulated data in the plane b/a vs (b - a)/δ for different values of ∆E1/2: from left to right, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, and 18 mV. The solid lines are the polynomial fit reported in eq 3. (B) Example of working curves for the simultaneous determination of b and a of a TSM; M ) log(b/a) and N ) log(δ2/δ1). Inset: N vs log(b/a) plot.

To facilitate their use, third-order polynomials, of general eq 3, were derived from data fitting:

b b-a b-a 2 b-a + A2 + A3 ) A0 + A1 a δ δ δ

(

)

(

)

(

3

)

(3)

where the coefficients A0,1,2,3 depend on the ∆E1/2. Table 2 summarizes A0,1,2,3 values (relative error within 0.5%) for a range of ∆E1/2. For the simultaneous determination of the unknowns b and a, two specific equations of the type 3 are needed. These can be obtained by recording experimental cyclic voltammograms at two different scan rates (or δ), so that two corresponding ∆E1/2 values and two sets of A0,1,2,3 coefficients (from Table 2) are established. The so-obtained equations solved simultaneously provide b and a. A graphical procedure for an estimate of b and a could also be exploited. To this purpose, data shown in Figure 4A were replotted in a logarithm form, as shown in Figure 4B, which for simplicity

Table 2. Coefficients To Evaluate a and b According to Cubic Eq 3 (R h 2 ) 0.99992) ∆E1/2 (mV)

A0

A1

A2

A3

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1.012 60 1.010 85 1.010 32 1.010 18 1.010 20 1.010 27 1.010 37 1.010 47 1.010 58 1.010 68 1.010 78 1.010 88 1.010 98 1.001 10 1.011 16 1.011 25

8.843 12 6.378 17 4.953 85 4.031 35 3.387 00 2.912 17 2.548 02 2.259 98 2.026 47 1.833 33 1.670 91 1.532 40 1.412 84 1.308 56 1.216 80 1.353 90

-5.877 11 -1.676 48 -0.175 51 0.434 52 0.696 04 0.805 35 0.843 30 0.846 38 0.832 23 0.809 78 0.783 72 0.756 55 0.729 60 0.703 57 0.678 80 0.655 43

18.174 96 7.460 69 3.461 29 1.726 38 0.887 18 0.448 03 0.204 76 0.064 47 -0.018 61 -0.068 47 -0.098 36 -0.115 93 -0.125 74 -0.130 57 -0.132 20 -0.131 76

refers to two working curves. For a given value of log(b/a), a separation distance (N) between the two working curves applies (see inset in Figure 4B). Two sets of experimental δ1, δ2, and the corresponding (∆E1/2)1 and (∆E1/2)2 are again necessary. The specific ∆E1/2 values allow us to identify the two working curves, while δ1 and δ2 make it possible to obtain the specific distance N ) log(δ2/δ1) (see Figure 4B). This segment, slid down along the two working curves until it fits their separation distance, sets the ordinate M ) log(b/a) and the abscissa Q2 ) log((b - a)/δ2) (or Q1 ) log((b - a)/δ1). Thereafter, a simple algebraic treatment of both the identified M ) log(b/a) and Q2 ) log((b - a)/δ2) (orQ1 ) log((b - a)/δ1)) quantities provides the b and a values simultaneously. It must be noticed that this simpler and fast method is, in general, less accurate than that based on eq 3, especially when N changes little as a function of log(b/a). Experimental Cyclic Voltammograms. A series of platinum (Pt-TSM) and carbon (C-TSM) disk microelectrodes with thin insulating shields (RG was over the range 1.40-2.60), prepared by the electrophoretic deposition of paint, were investigated in both aqueous and nonaqueous solutions containing Ru(NH3)63+ and Fc, respectively, as electroactive species. Both reactants undergo a one-electron reversible process and are therefore well suited to test the above theory. Figure 5A shows typical cyclic voltammograms recorded at 1 mV/s in 1.3 mM Ru(NH3)63+ at a Pt-TSM ( RG ) 1.48, b ) 18.5 µm, a ) 12. 5 µm) and a C-TSM ( RG ) 2.55, b ) 10.2 µm, a ) 4.0 µm). The diffusion coefficient of Ru(NH3)63+ is 6.3 × 10-6 cm2 s-1,41 so that, under the experimental conditions employed, δ/a ) 10.1 and δ/b ) 7.0 for Pt-TSM and δ/a ) 32.0 and δ/b ) 12.4 for C-TSM. These conditions should provide a flux from behind the plane of the electrode, and steady-state conditions should be reached faster. In fact, as is evident in Figure 5A, the predicted sigmoidal shape with the forward and backward waves almost superimposed is observed at both electrodes. Moreover, the current is larger than that predicted from eq 1, by about 19 and 10% for Pt-TSM and C-TSM, respectively. These current enhancements are congruent with theoretical expectation (see Figure 2). (41) Daniele, S.; Ugo, P.; Mazzocchin, G. A.; Rudello, D. In Electrochemistry in Colloids and Dispersions; Mackay, R. A., Texter, J., Eds.; VCH: New York, 1992; pp 55-69.

Figure 5. Experimental cyclic voltammograms recorded in 1.1 mM Ru(NH3)63+ + 0.1 M KCl solution; scan rate 1 mV s-1. (A) (1) Pt-TSM with RG ) 1.48 (b ) 18.5 µm, a ) 12. 5 µm), (2) C-TSM RG ) 2.55 (b ) 10.2 µm, a ) 4.0 µm); (B) Pt-TSM (a ) 12.5 µm and RG ) 1.48) and at a ISM 12.5-µm radius. Inset: magnification of the indicated zone.

The reproducibility of the voltammograms recorded with the same electrode was excellent (the relative standard deviation was within 0.5% from at least five replicates). The shape of the cyclic voltammograms recorded at 1 mV/s in both aqueous and nonaqueous solutions containing Ru(NH3)63+ and Fc, respectively, with 10 different Pt-TSMs and C-TSMs (the nominal radius of each electrode was the same), prepared using the same paint mixture, was examined in detail. The wave analysis in terms of ∆E1/2 provided average values of 0.7 ( 0.2 mV, for Pt-TSMs and 0.4 ( 0.2 mV for C-TSMs, in the Ru (NH3)63+ solution, and of 0.5 ( 0.2 mV, for Pt-TSMs and 0.3 ( 0.1 mV for C-TSMs, in the Fc solution. These values are very close to ∆E1/2 ) 0, theoretical expectation for true steady-state conditions. The lower ∆E1/2 values obtained at C-TSMs and in Fc solutions (diffusion coefficient of Fc, 2.5 × 10-5 cm2 s-1 42) are also congruent with the faster achievement of a steady state, the corresponding δ/a being greater than those of the Pt-TSMs or Ru(NH3)63+ solution. The Tomesˇ parameter evaluated from all cyclic voltammograms recorded at 1 mV s-1 provided, on average, a value equal to 55.5 ( 0.5 mV, irrespective of the TSM or (42) Adams, R. N. Electrochemistry at Solid Electrodes; M. Dekker: New York, 1969; p 193.

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Figure 6. Cyclic voltammograms recorded at a Pt-TSM (RG ) 1.42) in 1.35 mM Fc in acetonitrile at different scan rates. Inset: magnification of the indicated zone. Table 3. Experimental and Theoretical Geometrical Parameters and Steady-State Currents TSM

a (µm)a

b (µm)a

RGa

iss (nA)b

iss (nA)c

C (mM)d

∆ (%)

1e 2e 3e 4e 5e 6e 7f 8f 9f

13.5((0.3) 12.0((0.2) 12.9((0.2) 13.1((0.2) 12.9((0.2) 10.5((0.2) 3.5((0.1) 4.0((0.1) 3.5((0.1)

19.6((0.3) 17.5((0.2) 18.3((0.2) 17.3((0.2) 18.7((0.2) 15.5((0.2) 10.0((0.1) 10.0((0.1) 9.5((0.1)

1.45 1.46 1.42 1.32 1.45 1.47 2.86 2.50 2.50

2.70((0.02) 2.10((0.02) 2.21((0.02) 2.58((0.02) 2.51((0.02) 2.41((0.02) 1.31((0.01) 1.42((0.01) 1.38((0.01)

2.66 2.04 2.20 2.48 2.55 2.32 1.27 1.47 1.29

0.70 0.60 0.60 0.65 0.70 0.78 1.20 1.20 1.20

1.5 2.9 3.9 4.0 1.5 3.9 3.1 3.4 4.6

a Obtained by voltammetry with the proposed method. b Experimental current. c Simulated current. d Concentration of the electroactive species. Measurements performed as noted in footnotes e and f. e Measurements performed in aqueous solutions containing Ru(NH3)6Cl3 at the indicated concentration by using Pt-TSMs. ∆% ) (issb - issc)/issb × 100. f Measurements performed in aqueous solutions containing Ru(NH3)6Cl3 at the indicated concentration by using C-TSMs. ∆% ) (issb - issc)/issb × 100.

electroactive species employed. This is in excellent agreement with the theoretical value of 55.8 mV at 22 °C, calculated by eq 2. Figure 5B contrasts the cyclic voltammograms obtained at 1 mV s-1 with a Pt-TSM (RG ≈ 1.48) and a Pt-ISM (RG ≈ 100), fabricated starting from a wire, with nominal radius equal to 12.5 µm. It can be seen that although at both electrodes sigmoidal waves are obtained, the current response is higher and ∆E1/2 is approaching zero at the Pt-TSM. Instead, at the Pt-ISM, ∆E1/2 is higher than 5 mV, despite the very low scan rate employed. This latter finding agrees with earlier theoretical calculations,37,38 which indicated that, with ISM, having radii similar to those employed here, scan rates much lower than 1 mV s-1 would be necessary to attain true steady-state conditions (i.e., ∆E1/2 ) 0). Figure 6 shows the effect of scan rate on the experimental cyclic voltammograms obtained at a Pt-TSM in 1.35 mM Fc in acetonitrile. As expected, both current and ∆E1/2 became larger, as the scan rate increases. Typical ∆E1/2 values found upon changing the scan rate were 0.5, 5, 9, 14, and 22 mV at, respectively, 1, 20, 60, 120, and 160 mV s-1. This ∆E1/2 variation is sufficiently large to ensure sensitivity in the measurements. Similar results were obtained in a Ru(NH3)6Cl3 aqueous solution (voltammograms not shown). Experimental Evaluation of the Method for the Simultaneous Determination of b and a Based on ∆E1/2. The geometrical parameters of several Pt-TSMs and C-TSMs were 6580 Analytical Chemistry, Vol. 76, No. 22, November 15, 2004

Table 4. Results from SEM Measurements and from Applying the Proposed Method to Experimental Measurements Obtained in Either Ru(NH3)6Cl3 or Ferrocene Solutions electrode

parameter

SEM

Ru(NH3)6Cl3

Fc

1

a (µm) b (µm) RG a (µm) b (µm) RG

12.5 18.5 1.48 12.4 18.1 1.46

12.9 ((0.3) 18.7((0.3) 1.45 12.7((0.2) 18.1((0.2) 1.43

13.0((0.2) 18.2((0.2) 1.40 12.8((0.2) 18.2((0.2) 1.42

2

determined using either Ru(NH3)63+ in water or Fc in acetonitrile. For each electrode investigated, a series of cyclic voltammograms over the scan rate range 5-200 mV s-1 was recorded, and a series of ∆E1/2 was obtained accordingly. Two or three scan rate pairs that fit two or three ∆E1/2 values reported in Table 2 were chosen, and the appropriated A0,1,2,3 coefficients were defined. This procedure provided several pairs of particular polynomial equations (eq 3), whose solutions provided simultaneously a and b. Table 3 summarizes the results obtained for the geometrical parameters, experimental currents evaluated at 1 mV s-1, and theoretical values simulated by setting the a and b pairs thus calculated for each TSM. As is evident from Table 3, a general agreement within 4.6% exists between theoretical and experimental

steady-state limiting currents. Comparing b and a determined by voltammetry and those obtained by SEM also assessed the accuracy of the method, and data obtained are shown in Table 4. Also, in this case a satisfactory agreement (within 5%) was found. CONCLUSIONS In this paper, we have studied from a theoretical and experimental point of view the cyclic voltammetric behavior of a series of disk microelectrodes with shields of thickness comparable to the electrode radius. Our attention has been focused on TSMs with RG < 3, where the current is enhanced to a significant extent with respect to microdisk electrodes embedded in an infinite insulating sheath. From a theoretical point of view, it has been verified that the EFD simulation method, first employed in ref 23, allows accounting with accuracy the flux to the microdisk provided that a sufficiently high number of grid elements across the electrode surface is considered. Here, we have verified that Na needed, strongly depended on the RG, and for TSMs with RG e 2, Na g 25 was required. When proper conditions are fulfilled, the diffusion-limited steady-state currents obtained by EFD simulations agree within 3.4% with those reported in earlier papers,13,22,24-26 which were based on different theoretical models. Experimental measurements performed with a range of TSMs of platinum and carbon fibers were congruent with the theoretical expectation. A new voltammetric method, for the simultaneous determination of the geometrical parameters a and b has also been proposed. This is based on the difference in potential observed on the forward and backward waves, when the current is half of its maximum value. This procedure may present an advantage with respect to the method proposed in ref 23, which is based on

measurements of steady-state and un-steady-state limiting currents. In fact, if the sweep rate during the potential scan is not quite slow enough, a departure from the true steady-state current occurs.37,38 Therefore, the degree to which steady-state conditions are attained is often in experimental doubt. This may lead to uncertainties in the determination of the experimental steady-state limiting current. Instead, the procedure proposed here, based on ∆E1/2, only requires two non-steady-state voltammograms, therefore overcoming the problem related to the uncertainty in assessing the experimental steady-state conditions. Finally, it must be noted the larger duffusional flux that applies to TSMs with respect to ISMs and the faster achievement of a steady state may be beneficial for better discriminating between the diffusion current and that due to surface process, the latter being relatively important in transient voltammetry.20 ACKNOWLEDGMENT Financial support by MURST, Rome (Piano “servizi al cittadino ed al territorio” cluster C 22, progetto 20) is gratefully acknowledged. SUPPORTING INFORMATION AVAILABLE Simulation model, optimization of the simulation parameters and Appendix. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review June 30, 2004. Accepted August 27, 2004. AC049041U

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