1988
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978
(11) J. M. Markowitz and P. J. Elving, d . Am. Chem. Soc., 81, 3518 (1959). (12) D. P. Shoemaker and C.W. Garland, "Experiments in physical Chemistry". 2nd ed., McGraw-Hill, New York, 1967, pp 26-29, 34. (13) . . D. C. Grahame. Technical Reoott Number 1 to Office of Naval Research. March 9, 1950. (14) D. W. Marquardt, J . Soc. Ind. Appl. Math., 11, 431 (1963). (15) P. Delahay, "Double Layer and Electrode Kinetics". Wiley-Interscience, New York, 1965, pp 43-44.
(16) J. J. Lingane, J . Am. Chem. SOC.,7 5 , 788 (1953). (17) D. J. Macero and C. L. Rulfs. J . Electroanal. Chem., 7, 328 (1964).
RECEIVED for review July 12, 1978. Accepted September 14, lg7&The authors thank the Science Foundation) which helped support t h e work described.
Theoretical and Experimental Evaluation of Cyclic Staircase Voltammetry Lee-Hua L. Miaw, P. A. Boudreau, M. A. Pichler,' and S. P. Perone" Department of Chemistry, Purdue University, West Lafayette, Indiana 47907
The theory for Cyclic Staircase Voltammetry (CSCV) for the reversible case has been described and verified experimentally in this study. Voltammetric instrumentation was assembled and interfaced to a laboratory minicomputer which provided experimental control, data collection, ensemble averaging, and Fourier filtering functions. Experiments were conducted to test the effects of potential step size, A€, switching potential, E,,, and sampling parameter, a , on peak separation, A€p, ( E , , Epc),and peak current ratio, y, (ipa/ipc). Various data handling and data processing procedures were evaluated to provide the minimum of distortion in measuring peak separations and peak currents. Realistic error levels are reported. The results of the multiple experiments reported here confirm the predictions of CSCV theory.
-
In 1960, Barker and Gardner (I) reported improvements in polarographic performance obtainable by application of potential-step techniques in electrochemical analysis. Since then progress has been made in refining and extending pulse methods. Blutstein and Bond (2) described advantages of fast sweep differential pulse voltammetry over linear sweep and alternating current techniques applied to irreversible systems. Osteryoung and co-workers (3)compared the sensitivity and performance of several pulse voltammetric stripping methods a t t h e thin-film mercury electrode. Also, several new approaches for the complete elimination of charging current in voltammetric measurements have been reported (4-6). T h e appearance of t h e laboratory computer as a control device in electrochemical experiments has greatly facilitated the application of potential-step experiments in voltammetric investigations. Cyclic staircase voltammetry (CSCV) is a potential-step technique which is particularly important because of the widespread use of the corresponding analogue technique, cyclic voltammetry (CV). for mechanistic studies of electrode processes. Because of the distinct advantage of CSCV for discriminating against charging current, this technique should soon replace CV as the method of choice in mechanistic studies. Thus, a systematic examination of theoretical and experimental features of CSCV would seem t o be timely, and such a study is reported here. I Present address, Standard Oil Company, 4440 Warrensville Road, Warrensville Heights, Ohio 44128.
0003-2700/78/0350-1988$01 O O / O
T h e theory of staircase voltammetry (SCV) was first reported by Christie and Lingane (7) for reversible electrode reactions. Zipper and Perone (8) and Schroeder and coworkers (9) extended the theory by including t h e effect of varied current sampling time on SCV current-voltage waveforms. Schroeder's group (IO) also applied staircase voltammetry t o irreversible and quasi-reversible electrode processes. Ryan ( 1 1 ) generated theoretical CSC\' voltammograms using digital simulation and reported the use of CSCV in the study of kinetic mechanisms. Recently Reilley, et al. (12)described a deconvolution procedure employing the fast Fourier transform approach, where t h e influence of the data acquisition parameters can be removed from t h e SCV waveform allowing a direct comparison between data generated from SCV and classical linear sweep voltammetry. In the present work, we have presented a complete, systematic, theoretical, and experimental study of CSCV for reversible electrode processes (13). Included in the study are the effects of step height, sweep rate, switching potential, and sampling time. In addition, we have evaluated d a t a processing approaches in the analysis of CSCV current-voltage curves.
EXPERIMENTAL Reagents. Chemicals used in this study were analytical reagent grade (Mallinckrodt Chemical Works, St. Louis, Mo.) and were used as received. Ferrioxalate solutions were prepared by dissolving ferric ammonium sulfate in 0.4 M potassium oxalate. Mercury was quadruply distilled and then scrubbed with 10% nitric acid by drawing air through the mercury layer with a vacuum aspirator. Water used was purified by passage through a mixed bed ion-exchange resin followed by distillation. Solutions were deaerated with high purity nitrogen. Traces of oxygen in the nitrogen were removed by passage through two gas washing bottles containing chromous chloride and zinc amalgam, one containing deionized water, and a fourth containing 0.4 M potassium oxalate. Cell and Electrodes. The cell was a 50-mL glass bottle with threaded Teflon top with holes drilled that snugly fit electrodes and nitrogen gas tubes. The working electrode was a Metrohm E-410 micrometer type hanging mercury drop electrode. Drop area was 1.80 f 0.05 mm2 throughout the study, and a new drop was used for each experiment. A Coleman 3-710 saturated calomel electrode was used as reference and the counter electrode was a platinum helix. The cell was mounted on a metal ring stand and placed inside a metal box which also housed the electronic circuitry. The temperature of the cell was laboratory ambient (23 f 2 "C). Blanks were run for each set of experimental conditions and were subtracted from voltammograms t o account for changes in background. C 1978 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978
1.0
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1989
1
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I
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]FLAG 2,
ADC
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i
1 CURRENT MEASUREMENT L
I
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Figure 1. Potentiostat and computer interface for CSCV instrument. A 3 is Analog Devices 8100. All other are Analog Devices 48K. Resistance values are in kR; capacitance values are in pF
Instrumentation. The three-electrode potentiostat and associated computer interface is shown in Figure 1. The electronics were assembled on two perforated hoards. The potentiostat is on one board and consists of operational amplifiers 1 through 4. The current measurement circuit is on a separate hoard. Cell current is measured by the voltage drop produced across the resistor in the potentiostat feedback loop. All amplifiers except 2 and 3 are provided with trim pots to null out any voltage offsets. .4mplifier 2 is the control amplifier and amplifier 3 is a current booster capable of supplbing the high currents produced by potential steps. Current compensation is provided a t the inverting input of the control amplifier to correct for voltage offsets from the control amplifier, booster, and electrochemical cell. Amplifier 1 is a 100-kHz low pass filter that discriminates against high frequency noise passed by the DAC. Amplifiers 5 , 6, and 7 compose an instrumentation amplifier of fixed gain 100. The overall gain of the current measurement circuit is controlled by varying the input resistance of inverting amplifier 8. The circuit has been stabilized by providing each amplifier with its own separate power and ground connections from the i l 5 - V power supply. In addition, points 1, 2, and 3 on the potentiostat are on a separate analog signal reference ground line to eliminate sources of instability and voltage offset in the potential control circuit. The noise of the system was measured in the following manner. A blank solution was placed in the cell and a cyclic staircase waveform of 2-mV step height at 500-Hz step frequency (f = 117) was placed on the cell. The sampling parameter, a , was set to 0.1. The results of four scans were then averaged. The standard deviation of the first 12 data points of the averaged scans was calculated. This value is considered the system noise and corresponds to approximately 12 nA of cell current in our studies. The staircase voltage waveform was software generated and applied to the potentiostat through an Analog Devices DAC14QM, 14-bit, &lo-V DAC. The DAC was calibrated, and the voltage output was found to be linear over its entire operating range. A Keithley model 160 digital voltmeter was used to monitor the voltage a t the reference electrode in the cell (output of amplifier 4) resulting from various digital codes applied to the DAC from the computer. Least squares analysis on the results yielded a correlation coefficient of 0.9999 and a slope factor relating the voltage produced in the cell to the digital code applied from the computer. This factor is passed to the control software and enables the user to accurately control the voltage on the cell.
Data acquisition and timing were provided by a general purpose interface (14) linking the computer to the experiment. The computer employed for data acquisition was a Hewlett-Packard 2116B with 16K core memory. The interface contained a track-and-hold amplifier (Analog Devices SHA-1A) with 40-ns aperture time and 5-ps settling time; an 11-bit, *lo-V ADC (Analog Devices-12QM);a 20-MHz crystal clock scaled down to available frequencies in the range 1 MHz to 0.1 Hz in steps of 1, 2, and 5; and various logic gates used to control timing. A Tektronix 601 storage oscilloscope was used to display the voltammograms, and a Hewlett-Packard 2895A high speed punch was used to store the digitized waveform on paper tape. The data were analyzed on a Hewlett-Packard 2100s digital computer with 32K core memory, floating-point firmware, 2.2 million word moving head disk, Centronix 306 line printer, Calcomp plotter, and Tektronix 4012 Graphics Terminal. This latter system was also used for all theoretical computations. Software for Experimental Studies. The data acquisition program was a Hewlett-Packard Assembly language subroutine which was appended to a modified version of Purdue Real Time Basic ( 1 5 ) . The staircast, waveform is software generated and applied to the cell through the DAC. The user enters staircase frequency (SCF),sampling time, initial potential, step size, number of steps in the scan, ana the number of scans that are to be ensembled. The switching potential, E,,, is determined by the step height and the num3er of steps in the forward scan. For example, if E , = 100 mV and if the desired switching potential is 500 mV, an experiment of 2-mV step size requires a total of 200 steps in the forward sweep. Data reduction software was written in FORTRAN I\'. The object of the programs is to accurately measure peak separations and calculate peak current ratios. This is accomplished in the following manner. First, the digitized waveforms from a sample and blank are read into the computer by a high speed reader. Both waveforms are then simultaneously displayed on a storage scope and: if the initial portions of each overlap, the analysis is continued. If not, the data are discarded. Figure 2A shows a typical CSCV voltammogram with blank scan superimposed. After blank subtraction the data are smoothed using a fast Fourier transformation (FFT) (26) with a translation-rotation modification ( 1 7 ) . A rectangular filter function is used to smooth the data. The width (5256 points) of this filter function must be set to reject noise frequencies without rejection of frequency components which characterize the signal. Comparison of peak
1990
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978 ,STEP
‘“i E
n
A
“I 440
- E , MV
d=l-iT/ri
Figure 3. Applied potential as function of time for cyclic SCV. E , = initial potential, E,, = switching potential, step M = switching step, A € = step size, T = step width, T‘ = sampling time, a = sampling
parameter
48
-E,
MV
440
Figure 2. (A) CSCV voltammogram and blank superimposed. Conditions: a = 0.1; SCF = 500 Hz; €, = -0.48V vs. SCE; Step size = 2.06 mV; ensembles = 4. (B) CSCV voltammogram after blank subtraction and Fourier filtering
separations and visual inspection of the inverse transformed voltammograms for a series of filter functions gave a measure of the effects of the F F T on noise discrimination and waveform distortion. The results indicated that, for the electrochemical data generated in this investigation, the peak separation was constant and no waveform distortion was evident for filter function widths as low as 20. To ensure against distortion of the voltammograms, it was decided to set the width of the filter function to 50 for all of the data analyzed, except for the experiments with potential step size of 10 mV, where the width was set to 60. The latter was done since larger step sizes produce voltammograms of higher frequency content. Figure 2B is the backgroundcorrected, Fourier-filtered voltammogram pictured in Figure 2A. The tick marks show where the base-line current was measured and the location of the cathodic peak, Epcrand anodic peak, E,,, by analysis of the first derivative of the data as described below. Peak positions are located by taking the first derivative of the data and locating the points at which the derivative equals zero. The first derivative was taken by finding the difference between neighboring data points. In addition, the method of Savitzky and Golay (18, 19) was applied by calculating 5, 9, 15, and 25-point quadratic first derivatives of the data. The Savitzky treatment (28) evaluates the derivative of the least squares best fit and thus results in a “smoothed” first derivative. For the electrochemical data generated in this study, whenever greater than a 5-point smooth was used, the Savitzky treatment yielded calculated values of peak separation that were greater than those calculated from ordinary first derivatives, Le., those obtained by taking the difference between consecutive data points. Thus, it was concluded that only the ordinary or the 5-point quadratic derivative could be used in calculation of the first derivative to minimize bias in the computed results. In routine data analysis, an ordinary derivative was followed by a 5-point quadratic derivative, and the peak separations calculated from each were compared. In cases where the peak separation resulting from the 5point derivative was smaller than that calculated from the ordinary derivative, the smaller peak separation was retained, since it was concluded that the smoothing
contributed less error to the calculation of peak separation than the residual noise on the data with an ordinary derivative. The currents at the peak potentials were calculated in the following manner. The background-corrected, Fourier-filtered data were used to measure the base-line current. The magnitude of the base line is defined as the average of the data obtained during the first 30 mV of the forward scan. Next, the data in the vicinity of the peaks were fit to a quadratic equation (20). From this fit, the total anodic peak current, i,, and cathodic peak current, i,,, were calculated, and then corrected for the base-line current. The ratio of the anodic peak current to the cathodic peak current was then calculated.
THEORY Figure 3 illustrates the applied potential in cyclic staircase voltammetry as a function of time. Starting from t h e initial potential, E,, t h e potential is first varied step-wise in one direction until t h e switching potential, E,,, is reached, a n d then is swept back in t h e same manner. For each step, t h e and t h e step width, T. One step size is designated as S, current datum may be sampled for each step to form a cyclic staircase voltammogram. T h e time interval between the beginning of the step and the moment t h e datum is taken is designated as the sampling time, T’. Theoretically, the shape of a cyclic voltammogram is affected by t h e ratio of T’ to T , whereas t h e absolute magnitude of t h e voltammogram is directly related to t h e respective values of T’ and 7. T h e parameter determined by the ratio of T’ to T is called t h e sampling parameter, a , and is defined as a = 1-
(T’/T)
(1)
n ranges from 0 to 1 depending on t h e time a t which t h e d a t u m is sampled. Besides a , t h e step size, AE, and t h e switching potential, E,,, play a n important role in the construction of a cyclic voltammogram. For staircase voltammetry at a planar stationary electrode, Zipper and Perone (8) showed that the current, i, sampled at some time during the J t h step, could be represented by t h e expression:
which describes the current function for step J as dependent n is the on the sampling parameter, a , and the step size, S. number of electrons involved in the redox reaction, F t h e faradaic constant, A the surface area of the electrode, and Do the diffusion coefficient of the electroactive species. Equation 3 defines t h e current function $ for all t h e steps preceding
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978
the switching step M , as well as for the switching step M itself; Le., 1 5 J 5 M .
where t J = exp(-(nF/RT)(E, + J A E E l J); J = 1, 2 , 3, ... M , a n d AE = EJ+l - EJ Equation 4 describes t h e current function 4 for all steps beyond t h e switching step M , where J > M. Here, tJ = exp(-(nF/RT)(E, MAE-IJ MlAE E l J ) . -
+
-
-
Table I. Cathodic Peak Current Function f i q Various Step Sizes. CY > 0.95 -nAE, mV
0.99
0.999
2.05 3.08 4.10 5.13 6.16 10.27 12.83 15.40 17.97 20.53 23.10
0.37445 0.5035 0.6252 0.7425 0.8560 1.290 1.552 1.806 2.10 2.29 2.54
0.7966 1.1398 1.4807 1.814 2.142 3.442 4.24 5.02 5.77 6.55 7.30
THEORETICAL RESULTS For all calculations, the initial potential was kept constant a t 192/n mV anodic of the half-wave potential. T h i s value is anodic enough to fulfill t h e assumption t h a t t h e reduced species is not present a t t h e beginning of the potential scan. T h e potential is first swept cathodically, and then swept back anodically. T h e effects of each of the three variables, Le., 2, CY,and E,,, were explored by changing the variable of interest while keeping the other two constant. T h e ranges of the three variables used in the study are as follows: a ranges from 0.001 to a value approaching 1, (0.999 999 99). (This corresponds to values of the sampling ratio from 0.999 to and covers the range from the end of the step to the beginning.) Different values falling between -2.05/n mV to -23.10/n mV were used for 1E;and the switching potential, E,,, was varied from (Es,
CY=
a=
CY=
2.163 3.190 4.208 5.222 6.235 10.24 12.75 15.20 17.60 20.00 22.30
6.486 9.674 12.85 16.03 19.19 31.81 :39.52 47.40 54.90 62.60 70.00
0.99
0.999
0 , 9 9 9 9 0,99999
2.05 3.08 4.10 5.13 6.16 10.27 12.83 15.40 17.97 20.53 23.10
12.59 11.66 11.23 10.96 10.85 11.54 12.18 12.70 14.11 15.13 15.96
5.46 3.86 5.21 5.47 5.78 7.31 8.17 9.23 10.07 11.15 12.94
2.49 2.78 3.10 3.59 3.92 5.77 6.89 8.08 9.39 10.77 12.00
cy=
a=
200.01 299.92 399.78 499.52 599.2 997 1246 1488 1732 1968 2200
- E,,,), mV, for Various
-n AE, mV
CY=
for
0.9999 0.99999 0.99999999
Table 11. Cathodic Peak +(E,, Step Sizes. cy > 0.95 a=
a=
1.49 1.88 2.41 2.88 3.38 5.26 6.57 7.69 8.79 10.26 11.63
cy=
0.999 999 99 1.00 1.54 2.05 2.56 3.08 5.26 6.41 7.69 8.49 10.00 11.35
El J = -120/n mV to (E,* = -300/n mV. The shapes of the theoretical current-voltage curves are characterized in terms of the peak current ratio 7 ,( ~ ~ ~ /and i ~ t~ h e) peak , (Epa- EPJ. (ips and i, are measured relative separation 2,, to zero current.) Because of t h e low d a t a density for large 1E values, a graphical curve-fitting technique using a drafting template was employed in the determination of peak locations and peak heights in the cyclic staircase voltammograms, assuming that peak tops are symmetrical. In most cases absolute magnitudes are not given here, b u t these can be calculated by referring to Ref. 8 where absolute values for current functions and (EP- El,?)are given. For those values of (Y not considered in Ref. 8 we have included tables of absolute values here (Table I and Table 11). Results are summarized in &he following sections. Effect of S w i t c h i n g Potential. In this study, sets of data corresponding to different CY values and varying switching potentials were generated. T h e potential step size was kept a t -2.05/n m V for all cases. Figures 4 and 5 illustrate t h e effects of E,, on the peak current ratio and t h e peak separation. Effect on Peak Current Ratio y. I t is obvious t h a t t h e influence of t h e switching potential upon t h e quantity y decreases as the sampling parameter CY is increased toward unity (Figure 4). Moreover, as 01 approaches 1.00, the quantity y stays a t a value very close to unity for the entire range of the switching potentials used. Therefore, if the cyclic staircase voltammogram is constructed by taking each data point very close to the beginning of each step, peak current ratios will be nearly independent of t h e switching potential over t h e range studied here. Effect on Peak Separation LEp. Table I11 shows t h e limiting peak separations, AEPf,for various cy values. T h e limiting peak separation is t h e peak separation with a very cathodic switching potential (E,,- El/z= -350/n mV). As the switching potential becomes less negative, the anodic peak is shifted positive, resulting in a n increased peak separation (Figure 5 ) . This effect is more obvious in t h e cases with -
Computer programs were written to calculate the current function for various values of 01 and 1E.These programs were written in FORTRAN IV and executed on a Hewlett-Packard 2100s computer. Because of the 16-bit word size, doubleprecision arguments were used for cases where a exceeded 0.999 ( u p to a limit of T ' / T = lo-'). These programs are available upon request. Although t h e parameter, a , is used in t h e numerical computations, we will often refer to the sampling ratio, T ' / T , for convenience in further discussions. Also, in all computations R T / F is evaluated for 25 "C.
CY=
pc
1991
1992
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978
1 1OO.OC
15O.OC -I* E S h
,
Figure 5. Effect of switching potential on peak separation n l E = -2 05 m V , = peak separation, = limiting peak separation
! 1 3 0 . 0 0 IEC.00 230.00 2 8 0 . 0 0 -NIESk-E1/21 MV
080.00
Z O C - 0 0 250.00 3 0 0 . 0 0 350.00 1 / 2 1 fli
330.00
(See Table 111 for values of A€,, for various a values ) d -
Figure 4. Effect of switching potential on peak current ratio. n l € = -2.05 m V
Table 111. Limiting Peak Separations for Various Values. n A E = -2.05 mV nAE,*, mV
01
69.42 68.27 62.88 54.11 48.24 25.49 11.02 5.00 3.00 2.33 2.18
0.001 0.1 0.5
0.8 0.9 0.99 0.999 0.999 0.999 0.999 0.999
cy
9 99 999 999 9
0 8 0.7
0 .G
smaller N values. As N is increased, the predicted change in peak separation is decreased. Ultimately, no change is predicted when T ’ / T < ( N > 0.99999). Because of t h e dependence described here, a fixed value of t h e switching potential, (Esw- El,*)equal to -200/n m y , was chosen for the remaining studies of t h e effect of t h e potential step size, LE, and the effect of t h e sampling parameter, a. This value is not cathodic enough to completely eliminate the effect of E,,, but it is at least fixed and realistic for cyclic voltammetry experiments. Effect of P o t e n t i a l S t e p Size AE. To explore t h e influence of different potential step sizes, theoretical cyclic SCV curves were calculated for values of 1E between -2.05/n and -23.10/n mV. T h e switching potential was kept constant as described above. T h e predicted peak separations a t t h a t switching potential deviate less than 1.0 mV from the limiting peak separations.
0 1 0.001
I
si00
?Ji03
lb.00 -*iAE.
1;m
1 eb.00 25.00
FV
Figure 6. Effect of potential step size on peak current ratio n(€,, - E,,2) = -200 mV (Refer to Table IV for tabulated values.)
Effect on Peak Current Ratio y. Figure 6 provides plots of y vs. n l E for various values of cy. First of all, it is important to note t h a t when N is set to 0.7, t h e potential step size has no effect on the peak current ratio; the value of y stays a t 0.691 throughout the entire 1E range. It is curious also t h a t this value of y is close to t h a t predicted for linear sweep cyclic voltammetry (0.689) with the same switching potential (21). Note also t h a t a t N = 0.7, the magnitude of peak currents is proportional to (1E)’ (8). With regard t o the dependence on N , opposite trends are observed for N values greater or less than 0.7. For CI < 0.7, -( decreases as 1E is increased; for cr > 0.7, y increases as S
Table IV. Peak Current Ratio y for Various Step Sizes. n(E,, - E , , > )= -200 mV -nAE,
cy=
cy=
a=
01=
cy=
a=
a=
cy=
mV
0.001
0.1
0.6
0.7
0.8
0.9
0.99
0.999
0,999 9
0.999 99
0.999 999 99
2.05 3.08 4.10 5.13 6.16 10.27 12.83 15.40 17.97 20.53 23.10
0.662 0.655 0.648 0.642 0.637 0.621 0.61 2
0.665 0.658 0.652 0.646 0.641 0.626 0.619 0.614 0.609 0.605 0.602
0.686 0.683 0.681 0.678 0.676 0.671 0.670 0.668 0.667 0.666
0.691 0.691 0.691 0.691 0.691 0.691 0.691 0.691 0.691 0.691 0.691
0.700 0.701 0.702 0.703 0.705 0.710 0.713 0.716 0.720 0.7 23 0.726
0.721 0.726 0.730 0.734 0.737 0.750 0.757 0.762
0.815 0.828 0.840 0.849 0.856 0.877 0.885 0.892 0.898 0.901 0.903
0.920 0.926 0.932 0.936 0.940 0.952 0.956
0.965 0.972 0.976 0.978 0.981 0.985 0.986 0.987 0.988 0.988 0.989
0.987 0.990 0.992 0.995 0.996 0.997 0.997 0.998 0.999 1.000 1.000
1.000 1.000 1.000 1.000 1.000 1.000 1,000 1.000 1.000 1.000 1.000
0.606 0.601 0.597 0.595
0.666
cy=
0.767
0.772 0.776
0.960
0.962 0.963 0.965
cy=
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER Table
-nAE, mV
2.05 3.08 4.10 5.13 6.16 10.27 12.83 15.40 17.97 20.53 23.10
V.
Peak Separations,
nAE,, for
a=
oi=
a=
0.001
0.1
0.6
0.7
0.8
0.9
0.99
0.999
70.1 72.7 75.0 76.9 78.6 84.5 87.6 90.5 93.1 95.5 97.6
69.1 71.3 73.7 75.5 77.0 82.3 85.2 87.8 90.1 92.3 94.1
61.4 62.2 63.1 63.9 64.7 67.7 69.3 70.9 72.3 73.7 75.1
59.1 59.5
55.1 54.8 54.6 54.7 54.9 55.9 56.7 57.6 58.6 59.8 61.3
48.2 47.1 46.3 45.9 45.7 45.8 46.5 47.4 48.6 50.0 51.4
25.6 23.7 22.6 22.2 22.1 23.2 24.3 25.7 27.1 28.3 30.1
12.1 10.6 10.5
5.0
11.1 11.9
7.4 8.4 12.0 14.0 16.1 18.2 20.1 22.2
CY=
60.0
60.5 61.0
62.9 64.1 65.3 66.5 67.7 68.9
1993
n(E,, - E , , ? )= -200 mV
V a r i o u s S t e p Sizes.
Q =
CY=
1978
CY=
CY=
CY=
0 , 9 9 9 9 0,99999 5.6 6.6
14.4 16.0 17.8 19.6 21.6 23.5
3.0 3.8 4.8 5.6 6.8 10.6 12.9 15.2 17.5 19.7 21.8
a=
0,99999999 2.0 3.1 4.0 4.8 5.8 9.6 12.0 14.4 16.8 19.2 21.6
on AE is observed. An exception to this trend occurs when
a -
a approaches unity, as a linear relationship is once again
observed. I t is interesting to note that, for a = 0.7, n l E , approaches a limiting value of 58.2 mV as AE approaches zero. This is comparable to a nAE, of 58.3 mV predicted for cyclic voltammetry with a triangular waveform using (EsR- E l , J = -200/n mV (22). T h e effects of AE on LE, are summarized in detail in Table V. One should be aware of the fact that increasing the potential step size LE while keeping t h e step-width T constant is comparable to adjusting the sweep rate in linear sweep cyclic voltammetry. Also, varying T while keeping A E constant changes the effective "sweep rate". For a and T constant, the effect of changes in 1E is described above. But, when a and 1E are held constant and T is varied, there is no effect on 1E, or 7. On t h e other hand. t h e magnitude of the cyclic voltammograms does vary with T as described by Equation 2. Varying AE also affects the absolute magnitude of t h e waveforms, but not with analytic dependence, except for a = 0.7, as described earlier. Figures 8 and 9 demonstrate the effect of 1E on cyclic voltammograms for two different values of a. Effect of Sampling Parameter a. T h e sampling parameter, a , is a measure of the relative sampling position. An n value close to unity means that data points are taken shortly after t h e beginning of t h e step, whereas an n value close to 0 indicates the d a t a points are sampled near t h e end of each step. T h e effects of the sampling parameter on the resulting current-voltage curves, in terms of the observed peak current ratio 7 and peak separation AEp,are illustrated in Figures 10 and 11. and in Tables IV and V.
-NAE. t l V
Figure 7. Effect of potential step size on peak separation. n ( E S , E , , 2 ) = -200 m V (Refer to Table V for tabulated values.)
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, 61
, -69
-:DO
Ib2
6;
-c9
-&IO
lb2
6;
-A9
-io0
N[E-EI/ZI. nV
Figure 8. Effect of potential step size on shape of cyclic staircase voltammogram. n(Esw- E,,,) = -200 m V . cy = 0.99. (A) nAE = -2.05 m V , y = 0.815, nAE, = 25.6 m V ; (B) n l E = - 1 2 83 mV, 7 = 0.885, n A E , = 24.3 m V ; (C) n A E = -20.53 m V , y = 0.901, nAE, = 28.3 mV
1994
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978
XI0 XI0 0 .
*
I
R
9
$1 w
6
1
2
-
1
0
0
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61
-A9
HIE-EI/ZI.
Ih2
-400
6;
-cg
-100
nv
Figure 9. Effect of potential step size on shape of cyclic staircase voltammogram. n(Esw- E,,*) = -200 mV; a = 0.1. (A) n l E = -2.05 mV, y = 0.665, n l E , = 69.1 mV; (B) n l E = -12.83 m V , 3 = 0 619, n l E , = 85.2 mV; (C) n l E = -20.53 mV, y = 0.605, n l E , = 92.3 mV 0 D
.o
1 - 0 G I I-di 0 1
0.00
,
-1.50
I
-3.00
I
-1.50
,
-6.00
1
-7.50
,
-9.00
LOG1 1-4)
Figure 10. Effect of sampling parameter N on peak current ratio. /?(Esw- E,,,) = -200 m V , n l E = -2.05 mV (Refer to Table I V for tabulated values.) T h e approach to unity of the peak current ratio and the diminishment of t h e peak separation as the sampling parameter approaches unity indicates the formation of rather symmetrical cyclic voltammograms. This can be observed in Figure 12 which shows t h e trend in the wave shape with N.
EXPERIMENTAL RESULTS CSCV data were obtained here with the ferrioxalate system for a representative range of experimental parameters to verify the theoretical predictions. The starting potential of each scan was 192 mV anodic of t h e half wave potential of the ferrioxalate system (-0.24 V vs. SCE). In the experiments which tested the effect of step size the switching potential was held fixed at a potential 200 mV cathodic of E,,'*. T h u s the total voltage sweep was 392 mV and the step size chosen dictated t h e total number of steps to be taken during t h e scan. T o discriminate against low frequency noise t h a t would not be rejected by Fourier filtering in later data analysis, the results of four scans were ensemble averaged. T h e cell time constant was 21 ~s and was measured by calculating the negative reciprocal of the slope of a plot of the
Figure 11. Effect of sampling parameter ct on peak separation. n(E,,E,,,) = -200 mV, n l E = -2.05 mV (Refer to Table V for tabulated values.) logarithm of the blank current vs. time (23). Current sampling time after a step was limited to times greater than 25 cell time constants to allow time for the distortive effects of induced charging current ( 2 4 ) to subside. Obviously this placed limitations on staircase frequency and on CY values that were experimentally accessible with our system. For example, for rn = 0.99 and current samples to be taken a t least 500 k s after a step requires that the length of each step be a t least 50 ms. This corresponds to a staircase frequency of 20 Hz; for a 390-mV cyclic sweep with a step size of 2 mV, the experiment requires 18 s to complete. For a n experiment of this duration a t a mercury drop electrode, the theoretical assumption of diffusion to a planar electrode may be invalid. Another problem arises from experiments with large step size, in that the data density is low. For example, for a step size of 10 mV, only 76 points describe the cyclic staircase voltammogram. Despite these limitations, for t h e sake of completeness, experiments were conducted under t h e conditions of N = 0.99 and also with low information density. T h e deviations from theory observed for these particular experiments were not totally unexpected. T h e d a t a reported below are a n average of t h e results of experiments conducted over a period of one month. T h e
1995
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1978
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9
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-is
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Figure 12. Effect of sampling parameter N on shape of cyclic staircase voltammogram. n(E,,- E,,J = -200 m V , n l E = -2.05 m V . (A) cy = 0.9, 7 = 0 721, n l E , = 48.2 m V ; (B) a = 0.999, 1 = 0.920, n l E , = 12 1 m V ; (C) CI = 0 , 9 9 9 9 9 , y = 0.987, n l E , = 3.0 mV
Table VI. Effect of Step Size o n Peak Separations and Peak Current Ratios' step size, mV
no. of points in scan
theor.
2.06 3.09 5.14 6.17 10.26
386 254 158 126 76
69.1 71.3 75.5 77.0 82.3
' a = 0.1; Ew
=
AE,, mV
A%,
theor.
0.1
73.7 63.1 60.0 54.6 46.3 22.6
0.6a 0.7a 0. 8a O.gb 0.99'
69.7 71.8 76.0 76.5 85.2
I
i
i
i i
theor.
0.6 0.8 0.5 1.3 0.6
measd
0.665 0.658 0.646 0.641 0.626
0.659 A 0.652 i0.651 i 0.642 z 0.634 z
-0.440 V vs. SCE; staircase frequency = 500 Hz; Ei = -0.48 V vs. SCE.
Table VII. Effect of Sampling Parameter, a , o n Peak Separation and Peak Current Ratiod a
Y
measd
mV measd 73.6 i 63.4 t 60.4 t 54.2 + 46.0 = 20.7 i
1.1
1.5 0.5 0.5 0.6 0.6
Y
theor.
measd
0.652 0.681 0.691 0.702 0.730 0.840
0.656 i 0.003 0.678 z 0.003 0.688 i 0.005 0.705 s_ 0.003 0.730 r 0.006 0.846 i 0.010
a Staircase frequency (SCF) = 200 Hz. SCF = 100 Hz. ' SCF = 20 Hz. Step size = 4.12 mV; E , = -0.048 V VS. SCE; E,, = -0.440 V VS. SCE.
results are considered to be reliable because of the low noise of t h e system a n d t h e method of d a t a analysis. They also reflect t h e day-to-day precision, which includes varying environmental conditions, different electrodes and cell geometry, as well as amplifier drift. Each point represents a n average of five to ten voltammograms (obtained as described in the Experimental section) collected over two to four days. E f f e c t of S t e p Size. T h e effect of potential step size on peak separation, M,, and peak current ratio, 7 , is listed in Table VI. T h e effect of t h e current sampling parameter, ct, measured a t constant step size a n d constant switching potential is contained in Table VII. As can be seen from the data, there is excellent agreement between theory and experiment, except for t h e cases of cy = 0.99 or potential step size of 10 mV, as discussed previously. E f f e c t of Switching P o t e n t i a l . T h e effect of switching potential on peak separation and current ratio is described in Table VIII. T h e effect of current sampling parameter, a . for a step size of 2.05 mV, a t constant switching potential. is
0.003 0.004 0.003 0.003 0.002
~-
Table VIII. Effect of Switching Potential on Peak Separation and Peak Current Ratio' -(Ew El,,), mV
AE,, mV theor. measd
-
Y
-
theor.
measd
0.561 t 0.010 3 70.8 71.0 i 1.1 0.565 3 69.7 69.8 i 1.7 0.617 0.617 i 0.006 4 68.5 69.2 i 1 . 2 0.706 0.704 i 0.002 5 68.3 68.8 i 0.8 0.730 0.726 i 0.003 ' 01 = 0.1; SCF = 500 Hz; Ei = .0.048 V vs. SCE; step size = 2.06 mV. 113 i 154 t 267 i 318 i
Table IX. Effect of Sampling Parameter, a , o n Peak Separation and Peak Current Ratio"
a
O.la
0.5' 0.8b 0.9' 0.9gd
-(Ew A E , , mV El,,), mV theor. measd
206 206 206 206 206
i i
t i i
5 4 3 3 4
69.0 63.4 54.4 48.5 25.6
' SCF = 500 Hz.
SCF = 20 Hz. vs. SCE.
68.8 t 62.9 = 54.2 i 47.7 i 25.6 i
Y
theor.
0.7 0.667 2.0 0.682 1.9 0.704 1.0 0.722 2.7 0.813
measd 0.662: 0.681 i 0.700 i 0.719 t 0.825 i
0.003 0.002 0.004 0.008
0.012
SCF = 200 Hz. SCF = 100 Hz. Step size = 2.06 m V ; Ei = -0.048 V
described in Table IX. T h e confidence intervals for t h e switching potentials reported in Tables VI11 and I X were computed in the following manner. T h e results of 65 experiments were used to tabulate results presented in Tables VI11 and IX. The average of the cathodic and anodic peak positions for all 65 experiments was -0.239 V vs. SCE with a standard deviation of h0.0056 V. T h e
1996
ANALYTICAL CHEMISTRY, VOL. 50, NO. 14, DECEMBER 1 9 7 8
literature value for the halfwave potential of the ferrioxalate system is 4 . 2 4 V (25). The variance of the calculated average above is t h e sum of the variances of two experimentally measured points.
What we want to define here is t h e ability to reproduce any desired switching potential. If t h e assumption is made that the variance of the switching potential is equal to that of either peak potential, we can substitute in Equation 5 and calculate it from t h e relationship:
Thus, t h e standard deviation of the switching potential, for all 65 measurements was h0.004 V. T h e 95% confidence interval of t h e mean, X f ( t n / t / n ) , for each value of E,, was t h e n determined by using uESwfor all 65 experiments. T h e t value used corresponds to the total number of experiments, n, conducted in the particular subset (of the 65 experiments) which produced t h a t d a t a point. T h e widths of the confidence intervals for E,, reported in Tables VI11 and IX suggest that potential control within the cell (on a day-to-day basis) is not as precise as it is often assumed to be. Despite t h e fact that within-day precision might be excellent, ( a = 0.001 V), it is unrealistic to report only that value. The effects of day-to-day precision are clearly shown in our data, and this underscores t h e need of running replicate analyses over several days t o minimize bias in the results.
OBSERVATIONS T h e results of the experimental work reported here confirm t h e predictions of CSCV theory. All of the predicted trends have been followed. T h e deviations a t n equal 0.99 and for large s t e p size are attributed t o t h e response limitations of our experimental system and the effects of low d a t a density on d a t a processing algorithms, respectively. I n contrast to linear sweep cyclic voltammetry, cyclic SCV is a much more complex technique. T h e dramatic effects of A E and LY must be recognized before attempting to use t h e method for fundamental studies of electrode processes. On t h e other hand, it is clear t h a t some of these effects are desirable-for example, the production of symmetrical forward/reverse peaks when cy approaches unity. Another subtle advantage is that waveforms corresponding to various a values can be obtained during a single experiment. T h a t is, by saving multiple d a t a samplings from each step in a cyclic SCV experiment, a series of voltammograms can be reconstructed a n d displayed subsequently which represent variations of c( over t h e entire range of 0 to 1 under t h e conditions of that single experiment. Thus, it is possible to observe with a single cyclic sweep the effects of the dynamic p q e s s e s which affect t h e experiment over a broad time domain. This feature will be particularly valuable for studies of electrode processes involving coupled chemical reactions or activation-controlled electron transfer. Theoretical descriptions of these waveforms will be presented at a later time. I t should be pointed out t h a t t h e experimental implementation of t h e variations in sampling parameter, 0, described here theoretically, might not be a trivial problem. Associated with each step in t h e sweep is a charging pulse which must decay significantly before meaningful faradaic currents are measured. Traditionally, it has been assumed t h a t a delay time of about 4 cell time constants was required before the double-layer potential is charged to the proper step value and the corresponding charging current has approached
zero so that faradaic currents can be measured. Recent work in our laboratory (13,24)has shown that the time delay period required is more like 20 to 30 cell time constants before undistorted faradaic currents can be measured. This is because of a combination of effects: the time delay required to reach t h e desired double layer potential, t h e flow of potential-step charging current, and the flow of induced charging current. These observations suggest that for a typical cell time constant of 50 I J S , for example, t h e shortest feasible r' would be -1 ms. For a maximum T of about 50 ms (to allow completion of a 200-step experiment in 10 s), the maximum value of cy attainable would be 0.98. Thus, this poses a serious practical limitation on sampling parameter variations in experimental studies. We are currently exploring instrumental methods to circumvent this problem. One of t h e more intriguing observations made here is the unique effect of using a sampling parameter of N = 0.7. Under these conditions, cyclic staircase voltammetry exhibits behavior very nearly identical to linear sweep cyclic voltammetry. Specifically, the peak current ratio stays constant for N = 0.7 with changing LE ("effective" sweep rate), and assumes a value of 0.691 which is very close to t h e predicted value of 0.689 in conventional cyclic voltammetry. Although the peak separation does vary with 1E it has a linear dependence only a t n = 0.7, and the limiting value of 58.2/n mV a t AE = 0 mV is comparable to the predicted value of 58.3/n mV for conventional cyclic voltammetry. These observations suggest some underlying fundamental correlation between potential-step and linear sweep voltammetry for a = 0.7. This point was also observed by Reilley and co-workers (12) in describing a Fourier transform convolution procedure for simulating staircase voltammograms independent of t h e sampling parameter.
LITERATURE CITED (1) G. C. Barker and A W. Gardner, fresenius Z . Anal. Chem., 173, 79 (1960). (2) H. Blutstein and A. M. Bond, Anal. Chem., 48, 248 (1976). (3) J. A. Turner, U. Eisner, and R. A. Osteryoung, Anal. Chim. Acta, 90, 25 (1977). (4) N. Klein and Ch. Yarnitzky, J . flectroanal. Chem. InterfacialEkctrochem., 61. 1 (1975). (5) J. H. Christie, L L. Jackson, and R. A. Osteryoung, Anal. Chem., 48, 242 (1976). (6) W. P. van Bennekom and J. B. Schute, Anal. Chim. Acta, 89, 71 (1977). (7) J. H. Christie and P. J. Lingane, J . Nectroanal. Chem., 10, 176 (1965). (8) J. J. Zipper and S. P. Perone, Anal. Chem., 45, 452 (1973). (9) D. R. Ferrier and R. R. Schroeder, J . Electroanal. Chem. Interfacial Electrochem., 45, 343 (1973). (10) D. R. Ferrier. D. H. Chklester, and R. R. Schroeder, J. Electroanal. Chem. Interfacial flectrochem., 45, 361 (1973). (11) M. D. Ryan, J . flectroanal. Chem., 79, 105 (1977). (12) H.L. Suprenant, T. H. Ridgway, and C. N. Reilley, J . Electroanal. Chem., 75, 125 (1977). (13) L. H. L. Miaw, Ph.D. Thesis, Purdue University, 1978. (14) E. D. Schmidlin, in "Digital Logic and Laboratory Computer Experiments", by C. L. Wilkins, S . P. Perone, C. E. Klopfenstein, R. C. Williams, and D. E. Jones, Plenum Press, New York. 1975, Appendix F. (15) J. F. Eagleston and S . P. Perone, J . Chem. Educ., 48, 317 (1971). (16) SUBROL~NEFORT obtained from Computing Center, Purdue University, West Lafayette, Ind. 47907. (17) J, W. Hayes, D. E. Glover, D. E. Smith, and M. W. Overtin, Anal. Chem., 45, 277 (1973). (18) A. Savitzky and M. J. E. Golay, Anal. Chem., 36, 1627 (1964). (19) J. Steiner, J. Termonia, and J. Deltour, Anal. Chem., 44, 1906 (1972). (20) P. R. Bevington, "Data Reduction and Error Analysis for the Physical Sciences", McGraw-Hill Book Co., New York, 1969. (21) R. S . Nicholson. Anal. Chem., 38, 1406 (1966). (22) R. S . Nicholson. I . Shain, Anal. Chem., 36, 706 (1964). (23) E. P. Parry and R. A. Osteryoung, Anal. Chem., 37, 1634 (1965). (24) S . S. Fratoni, Jr., and S . P. Perone, Anal. Chem., 48, 287 (1976). (25) D. T. Sawyer and J. L. Roberts, Jr., "Experimental Electrochemistry for Chemists", John Wiley and Sons, New York, 1974, p 364.
RECEIVED for review October 11. 1977. Resubmitted August 14,1978. Accepted August 2 2 , 1978. T h e financial support of t h e Office of Naval Research is greatefully acknowledged.
