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Anal. Chem. 1991, 63,2743-2750
and it can, in principle, be used to predict the error due to heterogeneity when an analysis is performed. In experimental practice, however, the use of the concept is limited. Only a very small fraction of the total number of inclusions has an intensity which is high enough to contribute significantly to the overall heterogeneity. The low abundance of the inclusions impedes the experimental determination of a sampling constant and gives rise to imprecise and inaccurate sampling constant values. The obtained value is dependent on the analytical conditions and underestimates the actual uncertainty due to heterogeneity. In practice, therefore, the sampling constant concept is limited to semiquantitative information. Apart from the difficulties mentioned above, it is impossible to transfer absolute values of sampling constants from SIMS to a volume-sampling technique. The values as measured in SIMS are transferable to other two-dimensional analysis techniques only to the extent that the intensity maps as measured in SIMS correspond to the actual concentration distributions. Matrix effects may influence the width and the shape of the intensity distributions to some extent. The distinction between two-dimensional and three-dimensional sampling techniques is related to the size of the inclusions.
If the size of the inclusions is appreciably smaller than the smallest dimension of the sampled volume, a technique will behave as a volume-sampling technique.
ACKNOWLEDGMENT We are indebted to S. Leigh (NIST) for the derivation of eq 6 and helpful discussions.
LITERATURE CITED Michieis, F. P. L.; Adams, F. C. V.; Brlght. D.; Simons. S. Anal. Chem., previous article In this issue. Scilla, G. J., Morrison, G. H. Anal. Chem. 1977, 49, 1529. Piessens, B. I n WaarschJnlild?&kskenhg en Statisti& ; Stop-Scientia: Gent-Leuven, 1969; p 100. Yakowitz. S. J. I n CMpUtetlonal Probabllky and SlmulaMon, AddlsonWesley Publishing Co.: Readlng. MA, 1977; p 63. Michlels. F. Ph.D. Thesis, University of Antwerp, Department of Chem istry, 1990.
Hendriks and Robey. Ann. Math. Stat. 1936, 7.
RECEIVED for review March 12, 1991. Revised manuscript received September 3, 1991. Accepted September 9, 1991. F.M. is a research associate of the National Science Foundation (NFWO/FNRS, Belgium). This research was sponsored by FKFO, Belgium, and DPWB, Belgium, in IUAP programs 11 and 12.
Pulse Voltammetry at Microcylinder Electrodes Mary M. Murphy, John J. O’Dea, and Janet Osteryoung* State University of New York at Buffalo, Department of Chemistry, 130 Acheson Hall, Buffalo, New York 14214
Mlcrocyllndrlcal electrodes are easler to construct and maintain than mlcrodlsk electrodes. I n the normal-pulse mode, ranges of time parameters and electrode slzes can be found wch that depletion of reactant Is unimportant and the response to the analysis pulse Is predicted by theory for planar condltlonr. Similarly, ranges of parameters are found for reverse-pulse voltammetry such that the potentlaldependent response can be treated as a sequence of Individual doubie-pulse responses. Cylindrical diffusion and convection act to replenkh reactant quickly near the electrode and thus permit overal experiment times In the range of seconds. For square-wave voltammetry the shape and position of the net current response are Independent of the extent of cyllndrkal diffusion.
INTRODUCTION Microelectrodes have become a routinely used tool for electrochemical experimentation in recent years. Advantages of microelectrodes include the small size of the electrode assembly and improved performance with respect to electrodes of conventional size, especially in resistive media or for fast experiments. The most commonly used microelectrode is a circular conductor embedded in an insulating plane (a so-called disk electrode). Cylindrical microelectrodes, however, have special advantages with respect to ease of fabrication and theoretical properties which suggest they could be used to advantage more widely. In the case of a disk, the current density is in general nonuniform, with the highest current density at the edge (I). 0003-2700/9 1/0363-2743$02.50/0
Thus the uniformity of the circular geometry and quality of the seal between insulator and embedded conductor influence strongly the voltammetric response. In contrast, for cylindrical electrodes the configuration of the seal is unimportant to performance. Therefore fabrication of microcylindrical electrodes is technically undemanding. Microcylinders also have the practical advantage that area depends on length and radius, whereas diffusional properties depend only on radius (and time). Thus electrode area can be controlled independently over a fairly wide range of cylindrical radii. In harsh environments, for which there is no insulating material available, the electrode may be operated partly submerged without affecting the diffusional characteristics. Finally, the current density is uniform at a cylinder, which simplifies the interpretation of kinetic data (1). Voltammetry at cylindrical electrodes has been examined both theoretically and experimentally, primarily by Aoki and co-workers (2-8). Techniques which have been investigated include cyclic voltammetry (2-4),chronoamperometry (3,5, 6),chronopotentiometry (9, normal- and differential-pulse voltammetries (8), and staircase and square-wave voltammetries (9). Quasi-reversible (IO),irreversible (ll), and second-order catalytic (6) reactions have also been studied using cyclic voltammetry at microcylinder electrodes. Normal-pulse and square-wave voltammetry at a cylinder have been employed for chemical analysis (12),and normal- and reversepulse and staircase and square-wave voltammetry have been used for kinetic studies (13). In all cases interpretation of voltammograms has been complicated by the influence of cylindrical diffusion. The aim of the present study is to find procedures for using microcylinder electrodes which yield simplified results for a reversible system. The longer range 0 199 1 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 23, DECEMBER 1, 1991
n N o d pvlse
ReversePube
u
Figure 1. Pulse voltammetric waveforms. Parameters are indicated, and heavy dots show the points at which the current is sampled.
goal is to extend this approach to more complicated electrochemical systems and t o analytical applications. Diffusion at a cylinder is directly related to the dimensionless time, 8, defined as Dt,/r2, where D is the diffusion coefficient of diffusing material, T the radius of the cylinder, and t , a characteristic time, here the pulse width in pulse voltammetry. In the limiting case where diffusion is planar (0 < 0.01),current is given by the Cottrell equation. However, for typical values of a diffusion coefficient of 2.53 x cm2 s-l and a microcylinder of radius 12.5 pm, even pulse widths as short as 0.1 ms yield values of 0 which cause significant nonplanar diffusion. For t, = 0.1 ms, the current is 4% larger than the current a t a planar electrode of the same area. It is 11% larger when t , = 1 ms, 32% when t , = 10 ms, 89% when tp = 100 ms, and so on. Unlike the situation at a disk electrode, a steady state is not reached for large values of 0. Rather, the current decreases inversely with the logarithm of time. Theory for pulse voltammetry is greatly simplified in the normal- and reverse-pulse modes when the response to each analysis pulse is independent of the prior history of the experiment. This condition can be achieved even a t stationary electrodes if the waiting time, t,, a t the initial (zero current) potential is sufficiently long. The main objective of this paper is to establish these appropriate values of waiting times over a range of experimental conditions, together with the most suitable method of expressing the experimental results. For long waiting times and small radii, diffusion is enhanced and hence the waiting time is less than would be required under conditions of planar diffusion. The current in response to a single potential pulse to the diffusion-limitation region a t a microcylinder is ( 5 )
i = [ ~ F A D C * / ~ ] [ ( T+~0.422 ) - ~ /-~0.0675 log 8 + sgn {log (8) - 1.47)0.0058 [log (0) - 1.4712] (1) where sgn indicates the sign of the argument, n is the number of electrons transferred, F is the Faraday constant, A is the electrode area (=27rrl, where 1 is the length of the cylinder), and C* is the bulk concentration of reactant. In practice 1 is sufficiently large in comparison with T that the area of the exposed end is negligible. The dimension of current is contained in the factor nFADC*lr = 2mFDC*l, whereas the remainder of the right-hand side of eq 1 is dimensionless. Thus we can express the current as
i = 2rnFDC*lf(O)
(2)
where f(0) is the dimensionless current function. The response to a single pulse to an arbitrary potential is then (for a reversible reduction)
i = 2 m F D C * l f ( 0 ) / ( 1+ exp({))
(3)
where { = { ( E ) = nf(E and f = F / R T = 38.9 V-' a t 25 "C. Sujaritvanichpong et al. derived an equation for the current in normal-pulse voltammetry at a stationary cylinder taking into account the response to each potential step (eq 11 of ref 8):
ik = 2 m F D C * l { f ( O ) / [ l+ exp({,)]
k-1
+ jC[(f(Ci~/t, + =1
1)e) - f W / t , ) l / [ l + exp(L-j)lll ( 4 ) where T = t, + t,, t, is the waiting time at the initial potential (Ei)before each pulse, Ek = Ei - khE, where hE is the pulse increment, and fk = {(Ek). (This equation assumes that diffusion coefficients of oxidized and reduced forms are equal, Le., E"' = El$.) Figure 1shows the normal-pulse waveform and defines the parameters of potential and time. For sufficiently large values of the dimensionless time, 0, the sum in eq 4 has negligible value and the expression reduces to the limiting case of eq 3. The first objective of this work is to identify ranges of k r 0 / t , = kDT/r2, for which eq 3 may be applied. We consider also the current arising from the two reverse-pulse waveforms of Figure 1, as well as from square-wave and staircase voltammetry (Figure 1 ) . In addition, we wish to calculate the current at the end of each period of constant potential, not just the current at the end of each analysis pulse. Thus we adopt the general equation of Singleton et al. (9): b
i
Tj =
Et, m=l
t , is the period over which the potential has value E,, and Q, = (1 + exp(--Cj))-l for reductions. Equation 4 is a special case of eq 5. The potential-independent currents (limited by diffusion or homogeneous kinetics) of reverse-pulse voltammetry are free of artifacts arising from ohmic drop and usually can be
ANALYTICAL CHEMISTRY, VOL. 63,NO. 23, DECEMBER 1, 1991
measured much more accurately and precisely than peak currents obtained from cyclic voltammetry. Thus reversepulse voltammetry yields in a single experiment the qualitative information of cyclic voltammetry together with the quantitative information characteristic of double-potential step chonoamperometry. In order to apply simple interpretations to the resulting limiting currents at a stationary electrode, it is useful, as in the normal-pulse case, to employ conditions under which the reverse-pulse response is indistinguishable from that for the comparable double-pulse experiment. The second objective of this work is to define that range of conditions for a cylinder. The general approach is that applied by Sinru et al. (14) to the similar problem for a small-disk electrode. Finally, a third objective is to define under conventional conditions the range of parameters over which results are free of artifacts due to natural convection. The oxidation of ferrocene is used as a model reversible electrochemical system (15-18). Ferrocene is soluble in acetonitrile and in other nonaqueous solvents and undergoes a one-electron reversible oxidation a t about 0.1 V vs Ag/Ag+. Ferrocene has also been used as an internal standard and a reference material in nonaqueous and aqueous media (19,ZO). It has been suggested that ferrocene oxidation might be only quasi-reversible (21-24) and that, particularly a t high concentrations, there may be complicating side reactions (21). We did not encounter these problems in our experiments.
EXPERIMENTAL SECTION Chemicals. Acetonitrile was HPLC grade and was used as received. Propylene carbonate was purified by passing argon through it for 24 h. The supporting electrolyte, tetrabutylammonium hexafluorophosphate (TBAHFP) was recrystallized at least twice from ethanol/water. Ferrocene was purified by sublimation. Sulfuric acid was fumed to remove impurities. Deionized water was obtained using a four-cartridge Millipore Milli-Qsystem. A BASF copper catalyst was used to remove traces of oxygen from the argon gas. A drying agent (Drierite)was used to remove water from the argon. Solutions were deoxygenated by sparging with dry argon (Linde-Union Carbide, prepurified grade) saturated with acetonitrile. Electrodes. Working electrodes were made from Pt or Pt-Ir (80/20) wire of 25" diameter (9). The Pt-Ir alloy in this application is electrochemically equivalent to Pt but is easier to work with because it is stiffer. These electrodes are more fragile than embedded electrodes, but are incomparably simpler to make well. Although the simplest method of cleaning a microcylinder is to pass it through a flame for several seconds, that method did not yield quantitatively reproducible results. A number of other chemical cleaning methods were tried. An effective and reliable method was to place the electrode in KOH/EtOH cleaning solution for 10-15 min and then to activate it by cycling the potential in 1M H2S04(aq).The success of the procedure was monitored by observing the characteristic anodic and cathodic peaks. Silver wire was used as a pseudoreference electrode in propylene carbonate. The counter electrode was made of platinum foil (ca. 2 cm2). With regard to the reversible potential for oxidation of ferrocene, we note the value of Kolthoff and Thomas for the system PtlCp2Fe,Cp2Fe+,0.1 M TEAPIIO.1 M TEAP, 0.01 M AgNO,/Ag of 0.0672 V in acetonitrile and the value for the same Ag/Ag+ reference electrode of E = 0.274 V vs NHE (as) (25). We measured potentiometrically the value in acetonitrile of 0.310 V for Ag/O.Ol M AgN03, 0.4M TBAHFP vs SCE (aq). Kolthoff and Thomas report 0.2588V for the similar system with 0.1 M TEAP (25). By normal-pulse voltammetry we obtain El = +0.060 V in 0.4M TBAHFP vs Ag/O.Ol M AgN03, 0.4 M TbAHFP (AN) and +0.364 V vs SCE (aq). The corresponding value in 0.1 M TBAHFP/AN was 0.380V. The 95% confidence intervals on these values of E!/2are all less than 5 mV. In the work reported below in acetonitrile the reference electrode was Ag/O.Ol M AgNO,, 0.4 M TBAHFP (AN) and the analyte solution was 0.1 M TBAHFP. The reference electrode compartment was separated from the analyte solution by a plug of unglazed Vycor, and the
2745
reference electrode was assembled afresh for each set of experiments. Our focus was on the shape and amplitude of the volammetric response and on the precision of measurement of its position, rather than on the absolute value of the potential for ferrocene. We did not calibrate the reference electrode for every set of experiments. Therefore the values of El$ for ferrocene reported below, in contrast with those reported in this paragraph, may include systematic error due to uncertainty in the value of the reference potential. Instruments. Some experiments were carried out using an EG&G PARC 273 potentiostat interfaced to a DEC PDP-8/e minicomputer. Faster response times were possible using a homemade potentiostat which permitted pulse widths of less than 1 ms or pulse/waiting time combinations that could not be achieved reliably using commercially available instrumentation. An IBM potentiostat and x-y recorder (Omnigraphic 2000) were used for activating the cylinder in sulfuric acid. Other Experimental Details. All experiments were performed using a small SMDE cell (EG&G PARC). Experiments were performed on a vibrationless table (TMC 61-462) using a Haake FE2 circulator thermostated at 25 O C . The cell was sealed with a 1/4 in. thick Buna N rubber cell cover with three holes for the electrodes and an additional hole for degassing. A blanket of argon was kept over the solution for all experiments. Molecular sieves were added to the gas washing bottle to keep the acetonitrile contained therein free from water. Chronoamperometry was performed at a platinum-disk electrode of diameter 1.6 mm (Bioanalytical Systems) to obtain a reliable value for the diffusion coefficient. A value of 2.53 X loa cm2s-l was calculated for ferrocene oxidatior in acetonitrile from a plot of i vs t-lI2. This value agrees well with those of other reports (16,18) and was used for all other calculations involving ferrocene in acetonitrile. Computations. Normal- and reverse-pulse and staircase and square-wave voltammograms were coded according to eq 5 in Fortran, and computation of the voltammograms was implemented on a MASSCOMP 5500 Series or DEC PDP8/e computer. For analysis of voltammograms the limiting equations for t , >> t,, t , >> t , were used. Analysis was carried out by means of a nonlinear least-squares procedure implemented by the COOL algorithm (26).
RESULTS AND DISCUSSION Normal-Pulse Voltammetry. Normal-pulse voltammetric experiments were carried out according to Figure 1 for pulse widths ranging from 0.2 ms to 10 s. A series of voltammograms obtained for pulse widths ranging from 1 to 500 ms is presented in Figure 2. The plateau current values are proportional to t;1/2 (see inset); the only effect of nonplanar diffusion in this range of 0 is the non-zero y intercept. Thus the linearity of the traditional plot of i, vs t;1/2 serves as a test for diffusion control in the case of cylindrical diffusion as well as in the case of planar diffusion. From the slope of the time dependence (inset of Figure 2) a value of 2.55 (f0.04)X 10" cm2 s-l is calculated for the diffusion coefficient, assuming that the slope is given by nFAC*(D/7r)1/2 Therefore, even when the absolute contribution of cylindrical diffusion to the current is large, the time dependence is Cottrellian and yields accurate values of diffusion coefficient. The second important attribute of the normal-pulse response is the shape and position on the potential scale. Table I displays the results of analysis of the shapes of the voltammograms of Figure 2. The Ellz values vary by several millivolts, but the narrow confidence interval obtained from the nonlinear least-squares analysis (25) indicates that there is little change in shape of the voltammograms even when nonplanar contributions to current are substantial. The traditional semilogarithmic plot is a much less sensitive test for shape. The values of Table I are nearly equal to the predicted reversible value of 59 mV (decade)-' for a oneelectron transfer. The biggest problem encountered with these experiments over a wide time range was that current was not reproducible
2746
ANALYTICAL CHEMISTRY, VOL. 63,NO. 23, DECEMBER 1, 1991
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Table I. Influence of Cylindrical Diffusion on the Shape and Position of Normal-Pulse Voltammogramsa
t,/ms 1
2 5 10 20 50 100 200 500
graphical analysisb El12/V slope/mV (decade)-'
nonlinear curve fittingC E1,dV
0.0982 0.0963 0.0958 0.0969 0.0966 0.0955 0.0964 0.0978 0.1005
0.0980 0.0968 0.0965 0.0969 0.0966 0.0960 0.0971 0.0975 0.1002
64.8 62.9 60.5 61.2 58.8 57.4 57.8 58.5 64.1
(0.0012)d (0.0006) (0.0005) (0.0004) (0.0008) (0.0007) (0.0012) (0.0010) (0.0014)
"Data of Figure 2. Range 0.016 I0 5 8. bStraight-linesegments are fit to the current before the wave, on the plateau, and on the central portion of wave. The limiting current is taken to be the difference between currents at the two points of intersection, and El,* is taken to be the value of potential at which the current is half the limiting value. The slope is bE/6 log [il - i)i-l]. 'Results of nonlinear least-squares fit to a model for reversible oxidation under planar diffusion without depletion (i.e., single potential step chronoamperometry)employing the COOL algorithm as described in ref 23. The only parameter is Ell> Confidence interval (&) at 95% confidence. Typical correlation coefficient 0.999 92; typical sianal/noise ratio 170. a t longer pulse widths. The absolute value of the limiting current was sometimes larger than that predicted by theory (eq 5 ) by 25% or more (in one case the value was 54% larger). Values of diffusion coefficient obtained a t a cylinder using chronoamperometry were too large also, as much as twice as large as expected, even at short times. However, for t, 5 20 ms, limiting currents are reproducible and agree reasonably well with theoretical predictions, as shown in Table 11. At short times, the plateau current is only slightly larger than, and hence can be used as an estimate for, the Cottrell current. This situation contrasts with that for chronoamperometry, in which results free of artifacts due to natural convection are obtained up to about 8 s (for typical experimental conditions) (3,5,9). The normal pulse currents of Table 11, though close to the theoretical values, are consistently too large, even a t pulse widths as short as 0.2 ms, and the relative error is
Table 11. Comparison of Theory and Experiment for Normal-Pulse VoltammetryO i -. d d
t,/ms
t,/ms
experiment
theory
re1 % error
0.2 0.4 0.6 0.8 1.0 2.0 5.0 10.0
100 100-200 100-200 200 100-200 100-200 500 1000
96.1 (2.80)b 68.3 i m o j 56.84 (1.23) 50.30 (-)' 44.89 (1.44) 33.03 (1.13) 22.22 (0.55) 17.09 (0.537)
89.1 64.2 53.4 46.7 42.1 30.9 20.9 15.8
7.86 6.39 6.44 7.71 6.63 6.89 6.32 8.16
Experimental conditions: ferrocene oxidation; Pt cylinder, r = cm; C = 0.594 mM; 7' = 24 O C . bValuesin parentheses are confidence intervals (&) at 95% confidence. cOnly two values. a
12.5 pm, 1 = 0.938
independent of pulse width. The shortest total experimental time in Table I1 is 6 s. For t, = 20 ms and t, = 200 ms, the duration of the experiment is 13.2 s. I t appears, then, that the factor which determines the onset of convective enhancement of the current is the duration of the experiment, not the pulse width. This is physically reasonable in light of optical characterization of the diffusion field (27,281. The main effect of convection is to shift the location of the diffusion field with respect to the position of the electrode (28). For the results presented thus far, t, > 10 t,, and thus equilibrium is reestablished prior to each pulse (14). However, enhanced mass transport for 0 > 1 should permit one to employ smaller values of t,, and hence complete experiments more quickly (8). For small values of t,/t,, both experimental and computed (eq 5) normal-pulse voltammograms display a maximum rather than a plateau value of current. Plateau currents result when the waiting time between pulses is at least twice the pulse width. The minimum waiting time between pulses needed to obtain a plateau value within 10% of the single-pulse limiting value (eq 2) was determined experimentally for pulse widths of 1, 10, and 100 ms, where the waiting time between pulses was the only variable. Plots comparing experiment and theory (eq 4) are presented in Figure 3. The value of, i is the maximum current for a given
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 23, DECEMBER 1, 1991
iDc/(
i 0 C - i
2747
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Figure 4. Reverse-pulse plateau current ratios. f, = 0.5 s and t , = 5, 10, 20, 50, and 100 ms (current ratio decreases with decreasing
0.44
2
8
4
a
J IO
L/t,
Flgure 3. Normalized maximum currents for various waiting times: experiment (O), theory (eq 4) (-). i, is the value of i,, for t , = lot,. Conditions: 1 mM ferrocene in 0.1 M TBAHFP in acetonitrile; 25-pmdiameter Pt or Pt-Ir. Each point is an average value from two-four experiments on different days. t,lms = 1 (upper), 10 (middle), 100 (bottom).
fp). Each point is the average value from seven-twelve experiments on different days. Error bars indicate the standard devlatlon. The d i d line indicates perfect agreement (zero intercept and unit slope). Inset: RPV voltammograms for t , = 0.5 s and t , = 20, 50, and 100 ms (largest to smallest total current). Solution: 0.15 mM ferrocene in 0.1 M TBAHFP in propylene carbonate. Electrode: Pt-Ir, r = 12.5 pm, and I = 0.242 cm.
0.5I 0
t,/t, combination, as indicated in the figure. When t, = t, = 1 ms, ,,i is a peak current value. For all other values of the ratio t,/t,, i,, is a plateau current value. The value of iL, is the value of the plateau current for t,/t, = 10. The results of Figure 3 demonstrate two points. First, experimental data agree reasonably well with the predictions of eq 4. Second, the ratio t,/t, needs to be only 2-3 (for this range of 0) for the value of the plateau current to be within 10% of the value expected for the single-pulse experiment. These results are similar to those obtained at a microdisk electrode under similar conditions but somewhat different from prior results for NPV at microcylinders (B), where it was found that t,/t, needed to be 5 or larger to obtain the proper plateau current. However, in that case both the waiting time and the step height were varied to maintain a constant effective scan rate for a given t, value. This is not the way, in practice, this experiment would be carried out, and the resulting restriction on t,/t, is misleadingly high. Reverse-Pulse Voltammetry. Two kinds of reverse-pulse experiment were performed (Figure 1). In the first, the initial potential is held at a value that is well out on the diffusionlimited plateau for the forward (NPV) reaction for a period of time t, (the generation time). The potential is then stepped back in a regular manner, as shown in Figure 1,toward a value that is located at the base of the NPV wave (Eifor the NPV experiment). Implementation of this waveform is staightforward experimentally, but voltammograms obtained using this technique are difficult to analyze, since current at the end of each pulse depends on the entire prior course of the reaction. Reverse-pulse voltammograms were calculated using the theory of eq 5. Two plateau currents are seen (Figure 4, inset). The current that flows due to the initial oxidation of ferrocene at the generation potential (which has the same sign as the plateau current in the normal-pulse experiment) is called im. The limiting current due to reduction of the ferrocenium ion when the potential is stepped back sufficiently far from the generation potential is called im A ratio of these two plateau current values provides the same kind of information as a peak current ratio does for cyclic voltammetry. For comparison of theory and experiment the ratio iDc/(iDc - i l ~ p is ) used to minimize measurement errors associated with
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Figure 5. Reverse-pulse plateau current ratios. The straight line indicates perfect agreement. t , = 5 s, t , = 5, 10, 20, 50, 100, 500 ms. Each point is the average value from five-seven separate voitammograms. Other conditions are as Figure 4. uncertainty in the zero of current. The influence of natural convection encountered above in normal-pulse voltammetry was clearly manifest in the reverse-pulse experiments as anomalously large current ratios for RPV experiments. However, the current ratio is less sensitive to this anomaly than is the absolute current. A number of different methods for ameliorating this problem were attempted. The distal end of the cylinder was fixed to a glass support or a baffle was placed around the cylinder, and experiments were carried out on a vibration-free table. None of these factors influenced the results systematically to an extent that made the extra trouble worthwhile. The stiffer Pt-Ir electrodes behaved as the more flexible Pt electrodes. In propylene carbonate, the diffusion coefficient for ferrocene is about 1 order of magnitude lower (3 X lo4 cm2 s-l) (17)than it is in acetonitrile (2.5 X cm2 s-l). Thus the effects of natural convection should be less pronounced in propylene carbonate. The traditional reversepulse experiment in which the generation time is much longer than the pulse width was performed in propylene carbonate for generation times of 0.5 and 5 s, respectively, for various t, values (Figures 4 and 5). For a generation time of 0.5 s (experiment duration 30-36 s), there is reasonable agreement between theory and experiment, as shown in Figure 4. The line of Figure 4 is that
2748
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ANALYTICAL CHEMISTRY, VOL. 63, NO. 23, DECEMBER 1, 1991
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average value from two-four separate voltammograms. Other conditions are as Figure 4. expected for perfect agreement of experiment and theory. The experimental points yield a linear relationship with slope 0.87, intercept 0.021, and correlation coefficient 0.991. For these conditions, the value of im,and thus the ratio im/(im - i,Rp), is so small that significant error can be introduced from noise in the voltammograms, as is apparent from the voltammograms in the inset of Figure 4. For longer generation times (experimental duration 300-330 s), the experimental current ratio is too large, and the effect is more pronounced at longer pulse widths, as shown in Figure 5. Only for t , < 20 ms is there quantitative agreement with theory. Data for pulse widths ranging from 1 ms to 1 s and various generation times are compiled in Figure 6 for both acetonitrile and propylene carbonate. The results for acetonitrile and propylene carbonate are identical (within experimental error) in this format; that is, the adherance to theory depends on Dt,. Thus the advantage of the more viscous solvent is that experimental results agree with theoretical predictions at longer times. Propylene carbonate provides good values for the current ratio for pulse widths 510 ms (experiment duration 566 9). Another series of experiments was carried out using reverse-pulse voltammetry with boundary conditions renewed between pulses, RPW in the nomenclature of Sinru et al. (14). Note, in the waveform of Figure 1, a waiting time between each set of generation and analysis pulses, during which initial boundary conditions are renewed. Experimentally, this technique is slightly more difficult to implement than RPV, but data obtained using RPW are easier to analyze, since the response to each set of analysis pulses is independent of previous history if t, is sufficiently long (see above discussion for NPV). Microcylinders have a particular advantage in this application since convection as well as cylindrical diffusion can play a role in renewing boundary conditions quickly. Figure 7 represents the high quality of data that can be obtained using this waveform. Pulse width is kept constant a t 10 ms, and t, is varied from 10 to 100 ms in the example shown here. Noise is insignificant, current ratios are fairly large and easy to measure, and correspondence with theory is excellent. The current ratio is a simple function of t J t , and 8 easily derived from eq 5. Cyclic Staircase and Square-Wave Voltammetry. A variety of other voltammetric techniques were also performed to test microcylinder electrode response. Some ohmic drop was seen when long cylinders (ca. 1 cm) and high concentrations of ferrocene (1-2 mM) were used for faster experimenta (>lo00 Hz).Results were better a t shorter cylinders (
