Anal. Chem. 1994,66, 557-565
Square Wave Voltammetry of Reversible Systems at Ring Microelectrodes. 1 Theoretical Study
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Dennis E. Tallman Department of Chemistry, North Dakota State Universitv, Fargo, North Dakota 58 105 Ring microelectrodes can be fabricated so as to possess a high perimeter-to-area ratio. Such electrodes have been shown to exhibit high current density, a consequence of convergent diffusion, and should, therefore, display high signal-to-background and high signal-to-noise ratios when employed for analyticalvoltammetry. A particularly accuratecomputational method is employed to examine the square wave voltammetry of reversible systems at ring microelectrodes. The method involves a convolution (which for pulse forms of voltammetry reduces to a summation) of the transient current (in response toalargepotentialstep,computedinthis workusing theintegral equation method) and a function of the electrode potential. The influences of ring shape (thick to thin) and of square wave frequency, amplitude, base step height, and current sampling position on the resulting voltammogram are examined. Comparisonsof voltammograms obtainedfor pairs of ring and band and also for pairs of disk and band electrodes having equal area and equal perimeter are provided. Electrode geometry has negligible influence on peak position and peak width, but has a significant influence on peak height. Square wave voltammetry (SWV) of a reversible system at a conventional, arbitrarily shaped macroelectrode (Le., of millimeter dimension) yields a symmetrical, peak-shaped response centered on the half-wave potential.14 These attributes combined with a high discrimination against doublelayer charging current and the ability to achieve high effective sweep rates make square wave voltammetry an attractive electroanalytical technique capable of achieving low concentration detection limits. Microelectrodes (having at least one characteristic dimension on the order of micrometers) offer a number of important advantages compared to conventional macroelectrodes when employed for voltammetric measurement, including enhanced current density (a consequence of convergent diffusion), lower ohmic distortion in resistive media (a consequence of the very small current flow), and a shorter cell time constant (a consequence of the very low interfacial area and correspondingly low double-layer capacitance), making it possible to achieve very high scan rates5 Recently, square wave voltammetry at disk and spherical microelectrodes has been examined both experimentally and the~retically.”~In contrast to staircase or linear sweep voltammetry, where the (1) Ramaley, L.; Krause, M. S . Anal. Chem. 1969, 41, 1362. (2) Ramaley, L.; Tan, W. T. Can. J. Chem. 1981, 59, 3326. (3) ODea, J. J.; Osteryoung, J.; Osteryoung, R. A. Anal. Chem. 1981, 53, 695. (4) OD@, J. J.;Osteryoung, J.; Osteryoung, R. A. J.Phys. Chem. 1983,87,3911. ( 5 ) Wightman, R. M.; Wipf, D. 0. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker, Inc.: New York, 1989; Vol. 15, p 268. (6) O’Dea, J. J.; Wojciechowski,M.; Osteryoung, J.; Aoki, K. Anal. Chem. 1985,
57. 954.
0003-2700/94/0366-0557$04.50/0 0 1994 American Chemical Society
shape of the voltammogram is a function of D7/L2 (for diffusion coefficient D, electrode dimension L, and characteristic time of the measurement 7 ) , square wavevoltammetry yields voltammograms which are essentially invariant in shape over the entire practical range of these variable^.^ Our laboratory has been investigating the transient behavior of planar microelectrodes, with emphasis on band and ring These geometries are characterized by one microdimension (a width or thickness) and one macrodimension (a length or circumference). A ring (or band) microelectrode can be fabricated so as to possess an extremely high perimeter-to-area ratio (PAR) which leads to significantly higher current density than is obtainable at other electrode geometries, such as the sphere or the d i ~ k . l ~ This * ~ l higher current density should, in turn, lead to improved signal-tobackground and signal-to-noise ratios for many forms of transient ~oltammetry.~~,21 Thin (I1pm) ring electrodes of sizable diameter (110 pm) approach the steady state very ~ l o w l yand , ~the ~ ~square ~~ wave frequencies required to achieve true steady-state behavior are too low to be practical. At the other extreme, such thin ring electrodes require inordinately high square wave frequencies for linear diffusion approximations to apply.19 Consequently, a complete description of the square wave voltammetric behavior of ring microelectrodes must encompass the transient response under convergent diffusion conditions. This report describes the results of a computational study of square wave voltammetry for reversible electron transfer at ring microelectrodes of varying thickness. For comparisons, (7) Whelan, D. P.; ODea, J. J.; Osteryoung, J.; Aoki, K. J. Electroanal. Chem. Interfacial Electrochem. 1986, 202, 23. ( 8 ) Aoki, K.; Tokuda, K.; Matsuda, H.; Osteryoung, J. J. Electroawl. Chem. Interfacial Electrochem. 1986, 207, 25. (9) Komorsky-Lovric, S.; Lovric. M.; Bond, A. M. Electroanalysis 1993, 5, 29. (10) Coen, S.; Cope, D. K.; Tallman, D. E. J. Electroanal. Chem. Interfacial Electrochem. 1986, 215, 29. (1 1) Szabo, A.; Cope, D. K.; Tallman, D. E.; Kovach, P. M.; Wightman, R. M. J. Electroanal. Chem. Interfacial Electrochem. 1987, 217, 417. (12) Cope, D. K.; Tallman, D. E. J. Electroanal. Chem. Interfacial Electrochem. 1987, 235, 97. (13) Cope, D. K.; Scott, C. H.; Kalapathy, U.; Tallman, D. E. J. Electroanal. Chem. Interfacial Electrochem. 1990, 280, 27. (14) Cope, D. K.; Scott, C. H.; Tallman, D. E.J. Electroanal. Chem. Interfacial Electrochem. 1990, 285.49. (15 ) Kalapathy, U.; Tallman, D. E.; Cope, D. K. J. Electroanal. Chem. Interfacial Electrochem. 1990, 285, 71. (16) Cope, D. K.; Tallman, D. E . J. Electroanal. Chem. Interfacial Electrochem. 1990, 285, 79. (17) Cope, D. K.; Tallman, D. E. J. Electroanal. Chem. Interfacial Electrochem. 1990, 285, 85. (18) Cope, D. K.; Tallman, D. E. J. Electroawl. Chem. Interfacial Electrochem. 1991, 303, 1. (19) Kalapathy, U.; Tallman, D. E.; Hagen, S. J. Electroanal. Chem. Interfacial Electrochem. 1992, 325, 65. (20) Kalapathy, U.; Tallman, D. E. Anal. Chem. 1992, 64, 2693. (21) Sleszynski, N.; Osteryoung, J.; Carter, M. Anal. Chem. 1984, 56, 130. (22) Zoski, C. G. J. Electroanal. Chem. Interfacial Electrochem. 1990,296,317.
Analytical Chemistry, Vot. 66,No. 4, February 15, 1994 557
results are also presented for disk and band electrodes, geometries which may be considered limiting forms of the ring geometry.lg The influence of square wave frequency (spanning seven decades), amplitude, base step height, and current sampling position is also assessed. Of particular interest is the dependence of the peak current on ring shape (thickness) and on square wave frequency. In a subsequent paper we will presents results of an experimental study, including the extent to which experimental square wave voltammograms conform to theoretical predictions at ring microelectrodes.23
COMPUTATIONAL METHODS The model assumes a reversible electron transfer of the type
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0 + ne- R where the mass transfer of 0 and R within the cell occurs only by semiinfinite (convergent) diffusion. The cell contains only the species 0 initially (although extension to the case where both 0 and R are initially present is straightforwardI8), and no ohmic loss occurs in the cell. The electrode may be of any arbitrary geometry, and the potential may be any arbitrary function of time (although ring electrodes and square wave waveforms are the focus of this report). One further restriction is necessary to obtain a tractable solution, namely, that the diffusion coefficients of the oxidized species and the reduced species are equal (Do = DR = D ) . The current is then given by the convolution integra18J8 i R ( t ) = d S l i d ( t- u)O(u) du dt 0 where iR(t) is the time- (and potential-) dependent reversible current in response to a potential-time function d ( t ) , and id(t) is the transient diffusion-limited current at the same electrode in response to a potential step (from an initial potential at which no current flows to a final potential sufficiently beyond Eo' that the current reaches its maximum diffusion-limited value). Note that the convolution integral itself provides the time-dependent charge, the derivative of which with respect to time yields the current. All variables in eq 1 are dimensionless and have the same definitions in terms of physical variables that we employed in previous ~ o r k : ' ~ , ~ ~ i ( t ) = Z(T)/nFDC*L
(2a)
The solution of eq 1 requires two time-dependent quantities, id(t) and e(t). For convergent diffusion, id(t) values were
computed using the integral equation method (IEM), the accuracy and versatility of which has been discussed previAt the lowest squarewavefrequenciesexamined in this work, id(t) values were required out to very long times t. Such values were obtained from Szabo's long time expression for current at ring microelectrode^,^^ using the approach we described previously for linear sweep voltammetry.20 For rectilinear (or one-dimensional) diffusion, the values of id(t) were computed using the dimensionless form of the Cottrell equation id(t) = ( * / t ) ' / 2
(3)
with i d ( t ) and r defined as in eq 2. The numerical accuracy of id( t ) values for convergent diffusion computed from the IEM were within 0.1%. Of course, those computed for rectilinear diffusion from eq 3 were exact. Values for e(t) were computed from eq 2c, with E ( T ) representing the square wave waveform. The waveform employed in this work is that described by Osteryoung et al.3 and is characterized by an amplitude E,,, a frequency F,, or a period T = l/Fsw,and a base staircase step height AE (see Figure 1 of ref 3). One cycle of the square wave is superimposed on each step of the staircase, and current is sampled on the forward and on the reverse pulse of each cycle. The forward, the reverse, and the net (forward minus reverse) currents may be displayed. In this work, only symmetrical square waves are considered, with the forward pulse corresponding to a negative going pulse, the reverse pulse to a positive going pulse, since the potential sweep is in the negative direction for a reduction. The forward current is sampled at a fraction afof the way along the forward pulse, and the reverse current is sampled at a fraction arof the way along the reverse pulse. In this work, only the case for which af= ar = a is considered, with 0 I a I 1. The potential-time function E( Tj for the square wave waveform starting from an initial potential EI may be written as
E ( T ) = E , - n A E - E,, for (n - 1 ) 50 000. For a specified area, the perimeter of the disk electrode is invariant, whereas that of the band electrode (as for the ring) can be made arbitrarily large. Thus, band electrodes share the same analytical advantages discussed in the previous sections for ring electrodes. The difference in behavior of ring and band electrodes having identical area and active perimeter is both subtle and complex. In this case, the outer perimeter of the ring exhibits greater accessibility than does an equal length of the band perimeter, whereas the inner perimeter of the ring exhibits lower accessibility (Figure 7). Comparisons of transient currents at ring and band electrodes (of equal area and active perimeter) show that at short times the current at the band is greater than that at the ring (though not much greater) but eventually drops below thecurrent at the ring.” The crossover point moves to longer time as y increases. This phenomenon was attributed to transannular shielding” and reflects the relative accessibilities described above. The individual forward and reverse currents of the square wave voltammograms exhibit analogous behavior. The differences between ring and band currents are greatest at small y and at low frequency, the forward and reverse currents at the ring being larger than those at the band (e.g., Figure 8 displays the largest relative difference observed in this work, representing the smallest y and fswinvestigated). As either y or fsw increases, the gap between ring and band currents decreases (for example, compare Figure 8 with Figure 9A,B). For each y (other than 0.5, the disk), there is a crossover frequency above which the individual currents at the band are slightly greater than those at the ring. Figure 9 illustrates this behavior for a ring of y Analytical Chemistty, Vol. 66,No. 4, February 15. 1994
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the net band current is larger than the net ring current even though the individual forward and reverse currents are larger for the ring. In this case (as also for a y = 5 ring) the crossover frequency is lower than 0.05, the lowest frequency investigated. For y = 2 and y = 1 rings, the crossover frequency lies in the range 0.05 0.05). A ring of y = 40 exhibits a current that is virtually identical to that observed at the corresponding band electrode at all frequencies investigated (Table 1).
'- 7
7--1
I
8.00
4.00
8.00
4.m
'
I
'
0.00
I 400
0.00
400
(E - E")nF/RT
'
I -800
12
b LL
r
(E - E")nF/RT
-800
Figure Q. Square wave voltammograms for a ring of y = 10 and a band electrode having equal area and equal active perimeter. Conditions: fa, = 0.05 (A), 0.5 (B); 6e, caw, a,and N a s in Figure 1. Net current (solid lines), forward current (upper dashed lines of each set), and reverse current (lower dashed lines) are displayed. For this pair of electrodes, the width of the band (We) Is equal to the width of the ring (W, = R2 R,),and the length of the band (4) is equal to the average circumference of the ring (r(R1 -t &)), where R,and R2are the inner and outer radii, respectively, of the ring electrode.
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= 10, where at f,, = 0.05 the ring currents are larger (maximum difference ca. 13%), but at f,, = 0.5 the band currents are slightly larger (ca. 2%). As fSw is increased to 50 000, the small gap between these individual currents again closes and they become superimposed. The difference in the net current at a ring and at a band of equal area and active perimeter varies with y andf,, in a qualitatively similar fashion. At low frequency the net ring current is higher than the net band current, dropping below the net band current above some crossover frequency (which decreases with increasing y) and then merging with the net band current at very high frequency. However, the crossover frequency is lower than that for the individual forward and reverse currents for a given y. For the example in Figure 9A, 564
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CONCLUSIONS The computational approach used in this work is both efficient and accurate, permitting an extensive examination of the theoretical response of reversible systems to square wave voltammetry at ring microelectrodes. Rings and bands of widely varying shape yield net current voltammograms of essentially invariant shape, symmetrical peaks with constant width at half-height centered on the formal potential. Substantial enhancements in current density are obtainable at thin ring and thin band microelectrodes relative to that obtainable at a macroelectrode or even a disk microelectrode. These enhancements, which persist to rather high frequencies, should result in comparable improvements in signal-tobackground ratio. A square wave amplitude of SO/n mV provides nearly optimum response in terms of sensitivity and resolution. Variation of the staircase step height has little effect on net current voltammograms obtained at constant frequency. Increasing the square wave frequency, even while maintaining a constant sweep rate by appropriately decreasing staircase step size, results in voltammograms exhibiting higher net currents. The differences in the square wave voltammograms obtained at ring and band electrodes having identical area and active perimeter are interesting but are too subtle to be of practical consequence, especially for thin ring/band electrodes (y > 5). Only at thick rings approaching the disk geometry are substantial differences observed and even then only at the lowest frequencies. The charging current which flows in the square wave voltammetry experiment is an important consideration in optimizing the applied waveform. If the electrochemical cell is modeled as a series RC circuit, consisting of the uncompensated solution resistance R, and the interfacial capacitance c d (assumed independent of potential), then the superposition theorem of electronics states that the total charging current is the sum of two contributions, a contribution from the base staircase component and a contribution from the square wave component of the applied waveform, each contribution consisting of the appropriate difference current. We have verified this by computing these individual contributions as well as the total charging current over a range of frequencies and RC time constants and for various combinations of staircase step size and square wave amplitude. As expected, the magnitude of each component of the charging current depends on the relationship between the RC time constant and the waveform period, 7 . However, the percentage of each of the two components depends only on the relative magnitudes of the staircase step height, de, and the square wave amplitude,
esw. For example, the percentage of the total charging current arising from the base staircase ranges from 0.5% to 20% as the ratio 6e:eSwis varied from 1 5 0 to 1:1. Clearly, the square wave component of the waveform is responsible for most of the charging current but the base staircase does make a contribution. In view of the above, there may be situations where decreasing the base staircase step size (maintaining frequency constant) may be of analytical advantage, resulting in somewhat reduced charging current and in higher voltammogram definition, with little or no decrease in the net faradaic current (e.g., see Figure SA). The major disadvantage of such a decrease in step size would be an increase in sweep time, although this would be of less concern at higher frequencies where the experimental time scale is short. The increase in the number of data points which must be acquired (and perhaps stored) may also be considered a disadvantage. However, the peak current and/or peak position are usually the analytically significant quantities extracted from the net current voltammogram of a reversible system, and the larger
number of points defining the voltammogram, particularly in the vicinity of the peak, improves the accuracy with which these quantities can be determined. Furthermore, powerful personal computers with mass storage are now found in virtually every analytical laboratory and make the acquisition, manipulation, and storage of large amounts of data routine. Experiments are in progress to test the extent to which the analytical advantages of ring microelectrodes predicted from this theoretical study are realized in practice, and results will be reported in due course.
ACKNOWLEDGMENT The support for this research provided by the National Science Foundation through Grant CHE-9 108921is gratefully acknowledged. Received for review September 16, 1993. Accepted December 3, 1993.' Abstract published in Aduance ACS Absrracrs, January 15, 1994.
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