Anal. Chem. 1982, 5 4 , 1575-1578
Welfare: U.S. Goverlnment Printing Offlce, Washlngton, DC, 1978, Publlcatlon No. 76-21!;. (2) Marcali, K. Anal. Chem. 1957, 2 9 , 552. (3) Grim, K. E.; Linch, A. I_. Am. Ind. Hyg. Assoc. J . 1964, 25, 285. (4) Keller, J.; Dunlap, K. I..; Sandrldge, R. L. Anal. Chem. 1974, 46, 1845.
(5) Dunlap, K. L.; Sandridge, R. L.; Keller, J. Anal. chem. 1978, 48,497. (6) Levine, S. P.; Hoggatt, J.; Chladek. E.; Jungclaus, G.; Gerlock, J. L.
Anal. Chem. 1979, 51, 1106. (7) Bagon, G. A.; Purnell, C:. J. J . Chrornatogr. 1980, 790, 175. (8) Graham, J. D. J. Chrornatogr. Scl. 1980, 78, 384.
1575
(9) Ebell, G. F.; Fleming, D. E.; Genovese, J. H.; Taylor, G. A. Ann. Oc-
cup. Hyg. 1980, 2 3 , 185.
RECEIVED for review March 2, 1982. Accepted M~~ 3, 1982. The opinions or assertions contained herein are the private views of the authors and are not to be construed as reflecting the views of the Department of the Army of the Department of Defense.
Theory of Digital Alternating Current Polarographic Techniques J. E. Anderson and 14. M. Bond* Division of Chemical and Physical Sciences, Deakin Universiv, Waurn Ponds, Vlctoria 32 17, Australia
I n the technique of dllgitai ac polarography, a digital step function sine wave is applied to the cell rather than an analog sine wave. By modification of existing theory for step functional changes in potential, a theory for reversible systems may be obtained. The theory enables the effect of the number of steps in the digital sine wave to be examined and results are compared with conventional ac theory. The theory lndlcates the number a l steps in the digital sine wave Is Important In determining the magnitude of the alternating current. Phase angle relationships are affected by the number of steps as well as the point in the time at which current measurements are made during the potentlai steps. Experlmental data for both frindamentai and second harmonic responses confirm the thieoretlcal predictions.
The technique of digild ac polarography has recently been developed (I) as a means of simplifying ac polarographic techniques from an instrumental point of view. The technique has some of the characteristics of digital fast Fourier transform (FFT)ac polarography but is more closely related to the analog techniques. The basis of digital 8ic polarography is the use of microprocessor-based instrumentation to simulate the electronic components found in conventional ac instrumentation. The simulation begins with the use of a digital sine wave consisting of a fixed number of potential steps (e.g., 36) and the current is measured a t a fixed point in time for each potential step. Although in digital ac polarography the data are clearly measured in the time domain, it was found that the collection of data could be considered to be taken once every 10" in the frequency domain (once per step; 36 steps). Once the data are collected, software can be used to manipulate the data and simulate the high pass filter input and phase-sensitive detector commonly used in ac polarography. The subsequent digital ac polarograms which can be generated (either total ac current, phase-sensitive fundamental, or second harmonic) show the approximate phase relationships between the applied potential and the faradaic and capacitance components predicted from conventional sinusoidal theory (1). Furthermore, the ac peak shapes were as theoretically predicted (for reversible systems) and the peak currents showed a linear increase as a function of the square root of t h e angular frequency ( I ) . Reported here is the modification of general voltammetric theory for any stepwise potential wave form (2)to comply with the conditions existing in the digital ac technique. The theory 0003-2700/82/0354-1575$01.25/0
yields satisfactory results when compared with the digital ac experiments (for a reversible system) and can be used to examine the effect of the number of potential steps used to define the digital sine wave. The digital ac theory and experiment are compared with conventional ac theory which can take into account sphericity, amalgam formation, and slow electron transfer.
EXPERIMENTAL SECTION Instrumentation. The microprocessor-based instrumentation used to perform the digital ac experiments was essentially the same as described previously ( I ) . Further details of this system, based on a Motorola 6800 D2 microprocessor "kit", are also available elsewhere (3,4). The only modification made from ref 1was the substitution of a 1-MHz clock for the 614.4-kHz clock normally available with the 6800 processor. This clock was changed to increase the upper ac frequency previously (I) attainable. In the present work, data from the digital ac experiments were converted to decimal code and sent to a DEC 20/50 computer system via a RS 232 interface on the microprocessor. This procedure facilitated experiment-theory correlations. All computer programs associated with the theoretical work were written in FORTRAN and used with the DEC-20 computer. Computer programs are available on request. Data from experiment and theory were plotted on a Tektronix 4662 digital plotter. A static mercury drop electrode (SMDE), Model 303, from EG&G Princeton Applied Research Corp., Princeton, NJ, was used as the working electrode in the digital ac experiments. A platinum auxiliary electrode and a Ag/AgCl (saturated KC1) reference electrode completed the three-electrode potentiostat system. Reagents. Analytical reagent grade chemicals were used throughout as was distilled water. Prior to the experiments all solutions were degassed for at least 5 min with high-puritynitrogen saturated with water. Polarograms were recorded at ambient temperatures of 21 A 1"C. Electrode areas were obtained from the weighing of 400 mercury drops collected during the actual experiments.
RESULTS AND DISCUSSION The general equation for voltammetry with step-functional potential changes given by Rifkin and Evans (2) was modified so as to apply to digital ac polarography a t the constant electrode area SMDE. This equation applies to planar stationary electrodes, and it was assumed that sphericity does not significantly affect the ac current for reversible systems. This was confirmed by using a digital simulation program for the SMDE which showed that the spherical correction terms are usually small. The theory for digital ac polarography for a reversible process based on the above assumption is given by 1-5. 0 1982 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982
i = nFADo1l2C0/&(
r
$A $ ; . / f i ) 1 4 5.0
where
8, = exp{(nF/RT)(E, - E o ) ] t9j-l = exp[(nF/RT)(Ej-l
- EO)}
= EA sin (0’ - 1)(360/m)l
(3) (4)
(5)
All symbols are as given by Rifkin and Evans (2) where m is now the number of steps in the digital sine wave. Equations 1-4 are identical with those available in the literature (2). E A of eq 5 is the half peak to peak amplitude of the sine wave. Note also that eq 5 is written in terms of degrees rather than radians. The value of the initial potential, Eo, was held constant for all “experiments” (drops). In this respect, the theory actually defines a potential step experiment with a digital sine wave superimposed, which is appropriate for the SMDE. The dc component is eliminated from the calculation since it is not included in the summation term of eq 1. Instead of measuring the total current at any step in the digital sine wave the data can be treated theoretically in an equivalent fashion to the operation of an analog lock-in-amplifier to produce phase-selective detection of the various harmonics. For example, if the currents are multiplied by a square wave of the same frequency with an amplitude of f1.0 (with the desired phase relationship with respect to the applied sine wave), then the resulting points can be averaged to obtain the phase-selective fundamental harmonic current. Both total current and phase-selective approaches are examined theoretically and experimentally in the present work. Theoretical results enable predictions to be made as to the optimum way of performing experiments. Calculations indicate that five cycles of the digital sine wave need to be “applied” before the absolute value of the minimum and maximum of the calculated current become constant to within 0.5%. A transient response occurs upon the initial application of the digital sine wave irrespective of when the sine wave is applied in the ramp. Consequently, data from the fifth and sixth cycle were used as the calculated alternating current at a given applied potential (ramp step) to eliminatethe transient response associated with the initial application of the digital sine wave. In the experimentalprocedure described previously (I),data from the f i i t two cycles of the digital sine wave were discarded to eliminate any transient response and the results from the following 16 cycles averaged. In light of the theoretical results, this estimate of the required settling time may have been marginally optimistic. However, the influence of any transient response would have been minimized in the experimentaldata presented because of the averaging of the subsequent 16 cycles. The theory predicts that the time of current measurement in each step affects the amplitude of the resulting ac polarogram and the phase angle (step number in the digital sine) a t which the maximum signal is obtained. Figure 1shows a plot of alternating current as a function of phase angle for a number of measurement times on each step of the sine wave (20%, 40%, 60%, 80%, and 95% of the way through each step). Note that the step at which the applied alternating potential is 0 V (in the ascending direction) is denoted as 0’. Experimental results indicate that the theoretically predicted enhancement in faradaic current gained by taking measurementa very early in the step could not be completely realized because of the RC time constant of the electrochemical cell and measurement capabilities of the iiistrumentation (5). For
-5.0
0
I
I
I
32
64
96
I
126
I \
160
1
DEGREES
Flgure 1. Effect of sampling time during potential steps on ac amplitude and phase relationshlp. Measurement times were (a) 20, (b) 40, (c) 60, (d) 80, and (e) 95% of the way through each step. A 100 Hz digital sine wave with amplitude of 6.25 mV peak to peak was used. Theoretical parameters used in calculations were 5 X M electroactive species with n = 2, D = 0.78 X IO“ cm2/s,and A = 0.0158 om2. T = 21 O C .
example, if a 100 Hz digital sine wave is used and the measurement time is 20% of the way through each step, the current needs to be measured 56 ps after each step. On this time scale, accurate measurements would require a cell time constant of approximately 2 ps (5). The cell time constant is certainly larger than this since the rise time of the potentiostat itself is approximately 40 ps (to 99%). Measurement accuracy therefore increases at longer step times. In addition, Figure 1 indicates that the phase angle of maximum current is closer to 4 5 O (as expected for a reversible system based on sinusoidal theory) for current measurements made late in the potential step. Although the measurement of current as late as possible in the potential step reduces the amplitude of the Faradaic current, it provides a more favorable Faradaic to charging current ratio. The measurement of the current late in the potential step also ensures that distortions due to the time constant of the electrochemical cell are minimized. As a result of this theoretical investigation all experimental measurements reported in this paper were made 140 ps before the end of each step, rather than midway through the step as in earlier work (1). This time (140 ps) is a limitation of the instrument used and is the time required for the microprocessor to obtain and store the current data in memory. Figure 2 shows a comparison of the theoretical and experimental digital ac polarograms for the reduction of 1.0 x M ferric oxalate in 1.0 M Na2C204/0.05M H2C204. This is a reversible one-electron reduction [Fe(C20,)3]3- e- + [Fe(C204)314(6)
+
The capacitance current from the experimental data was minimized by substraction of the currents obtained from a blank containing only the supporting electrolyte. Figure 2a is the total current polarogram obtained at 90° with respect to the applied digital sine wave and Figure 2b is a phase selective fundamental harmonic polarogram ( O O ) . Agreement between theory and experiment is satisfactory. It is clear that the lock-in-amplifier approach applied to the experimental data is effective in decreasing the noise as would be expected. Since the software lock-in-amplifier appears to adequately simulate ita analog counterpart in other respects, the increase in signal to noise obtained should be similar to that obtained by a conventional analog lock-in-amplifier. Figure 3 shows a comparison of the theoretical and experimental ac peak heights as a function of phase angle. There appears to be a slight positive offset of the experimental total current curve (Figure 3a) which may be due to incomplete
ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982
*
1577
2.5
0.022.0
K K I
0.5 0
J
.
0.0
5
u
L L - ~ L . - L L - -
L ----
-0.44
-0.36
-0.28
VOLT
-0.20
VS
-0.12
-0.44
0.04
-0.36
-0.28
0.20
0.12
0.04
VOLT vs Ag/AgCl
Ag/AgCI
Figure 4. Comparison of experimental (0) and theoretical (-) phase selective second harmonic polarograms at 20’. All conditions and experimental parameters are the same as in Figure 2.
b
0.0
-0.44
-0.36
-0.28 -0.20 VOLT vs Ag/AgCI
-012
-0.04
Figure 2. Comparison of experimental (average of 16 cycles) and theoretical digital ac polerograms for reduction of 1 X M ferric oxalate in 1.O M Na2C20,,b0.05 M H2C20, (+, experimental; -, theoretical). Curves in a are1 the total current ac polarograms measured at 90’ with respect to the applied potential; curves In b are the Inphase phase-selective fundamental harmonic polarograms. Digbl sine wave of 84.1 Hz, 0.006125 V peak to peak. Parameters used in theoretical calculations: M = 1, A = 0.0158 om2, D = 0.494 X cm2/s, E o = 0.240 V, potential scanned 0.0 to -0.45 V (0.005 V steps), T = 21 O C . I
..*.
0
60
120
180
240
300
DEGREES
Figure 5. Effect of the number of potential steps In digital ac polarography, plot of ac response vs. phase angle (DEG). Number of steps in the sine wave are 180, 120, 90, 60, 36, 18, and 10 in order of decreaslng current amplitude. Theoretical parameters used in calcuiatlons: n = 2, T = 21 OC, 5 X M electroactive species, D = 0.78 X cm2/s, A = 0.0158 cm2. Digital sine wave was 100 Hz with an amplltude of 0.00625 V peak to peak.
1
oretical and experimental data. Figure 4 shows a comparison of the experimental and theoretical second harmonic polarograms for the ferric oxalate system. Another system studied was the reduction of cadmium
20.1-
Cd(J.1)
0
60
120
I
I
I
180
240
300
I
-1
DEGREES
Figure 3. Comparison of experimental and theoretical digltal ac response at various phase angles. Curves for a are the total current ac response: curves for b are the phase selective fundamental harmonic ac response. Symbols refer to experlmental data. All conditions are the same as In Figure 2.
removal of the dc coniponent or an artifact of the transient response due to the iniiiial application of the digital sine wave. However, the lock-in-amplifier approach used to obtain the fundamental harmonic (Figure 3b) eliminates this offset. The digital ac theory also predicts the existence of a second harmonic component in the total current. By use of the software lock-in-ampllifiertechnique with a reference signal of twice the frequency of the applied sine wave, the second harmonic polarograms may be generated from both the the-
+ 2e- + Cd(Hg)
(7)
which is essentially reversible. However, this system involved amalgam formation and the experimental alternating current peak amplitudes were approximately 13.4%larger (electrode area = 0.0158 cm2,amplitude = 0.003125 V, frequency = 84.1 Hz)than the peak amplitude predicted by the theory presented in this paper. This is in close agreement to the increase in amplitude of 13.9% predicted theoretically when using conventional ac theory with and without amalgam formation (6). Theoretical sinusoidal alternating currents which allow for spherical diffusion and amalgam formation were calculated by extending a recently developed digital simulation program for dc experiments which can be applied to the SMDE (7). This computer program uses Feldberg’s improved approach to simulating the hydrodynamics occurring at a growing drop electrode (8)and calculates the surface concentrations. From the surface concentrations derived from the dc theory, the alternating current was calculated by using theory which has been described previously (6). In view of the generally excellent agreement between theory and experiment, it is believed that the theory can be adequately used to examine the effect of increasing the number
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ANALYTICAL CHEMISTRY, VOL. 54, NO. 9, AUGUST 1982
although the peak current is approximately 23.8% lower. Even with 180 steps, the digital alternating current is 9.9% lower than that predicted by sinusoidal theory.
CONCLUSIONS
4
I
30
I
60 NUMBER
I
I
120
90 OF
d-i 150
STEPS
Figure 6. Plot of digital ac polarogram peak current vs. number of steps in digital sine wave. Same theoretical parameters as in Figure 5, solid line at 5.9 pA is the peak current predicted from sinusoidal ac theory.
of steps in the digital sine wave. This task is more difficult to undertake experimentally because of excessive memory requirements and assembler language software would have to be altered significantly to retain high speed (frequency) capabilities. Figure 5 shows the theoretical digital ac peak heights as a function of p h e angle for several digital sine waves of various numbers of steps (10, 18, 36, 60, 90, 120, 180). In these calculations, the current was measured 90% of the way through each step. It is apparent that as the number of steps increases the maximum peak heights increase and the phase angle more closely approaches the expected values of 4 5 O (sinusoidal theory) for a reversible system for total current ac polarograms (-45' for the phase selective fundamental harmonics). Presumably, as the number of steps increase, the closer the response should be to that obtained in conventional sinusoidal ac polarography. As a test of this hypothesis the theoretical alternating current was calculated for a reversible electrode process using both sinusoidal and digital wave forms. Figure 6 shows a plot of the maximum ac current of the total current ac polarograms as a function of the number of steps in the digital sine wave. The solid line at 5.9 pA is the expected ac peak height from sinusoidal ac theory. As may be seen, the digital ac current initially increases rapidly as the number of steps is increased. However, it is apparent that a very large number of steps would be required before the peak currents would be identical within the limit of experimental error (2%) because of the asymptotic nature of the curve. The plot shown in Figure 6 is very similar to that reported previously in a study digital simulation of ac voltammetry (9). As suggested in both these studies, the number of steps in the sine wave is far more important to the current amplitude than to the wave shape. The wave shape obtained in digital ac polarography is consistent with sinusoidal theory for 36 steps (I)
The theory presented appears to be adequate for examining different aspects of the technique of digital ac polarography. Calculations suggest that although the 36 potential steps initially used to define the digital sine wave is completely adequate for most purposes (I) it is an insufficient number to simulate a pure sine wave with respect to theoretically predicted ac amplitudes to better than the 20% level. Because of the large number of steps required to obtain current amplitudes to within 2% of those obtained in conventional ac polarography, the instrumentation which would be required to achieve this could easily approach the sophistication required in FFT techniques. In this case there may be little point in pursuing the digital ac technique in favor of the FFT approach. In this sense it is probably far simpler to use a theory based on a step function as used in this work if predictions of current amplitudes are required. Fortunately, the theory for the step function is relatively simple. However, even in the use of ac polarography as a tool for studying electrode kinetics the use of absolute current magnitudes is quite rare. Rather it is the phase angle, peak position, and curve shape information which are most useful. In this regard, inspection of the data presented here suggests that the number of steps required to accurately (with interpolation) extract phase angle information may be as low as 72 (one point every 5O), a task which is certainly within the range of simple microprocessor-based instrumentation.
ACKNOWLEDGMENT The authors thank Howard B. Greenhill for assistance in developing the hardware and software associated with the transfer of experimental data to the DEC 20 computer.
LITERATURE CITED Anderson, J. E.; Bond, A. M. Anal. Chem. 1981, 53. 1394-1398. Rifkin, S.C.; Evans, D. H. Anal. Chem. 1976, 48, 1616-1618. Anderson, J. E.; Bond, A. M. Anal. Chem. 1981. 5 3 , 504-508. Anderson, J. E.; Bagchi, R. N.; Bond, A. M.; Greenhill, H. B.; Henderson, R. L.; Walter, F. L. Am. Lab. (Falrfleld, Conn.) 1981, 13 (12), 21-32. Miaw, L. H. L.; Perone, S. P. Anal. Chem. 1979, 51, 1645-1650. Bond, A. M.; O'Halloran, R. J.; Ruzic, I.; Smlth, D. E. Anal. Chem. 1978, 48, 872-883. Anderson, J. E.; Bond, A. M.; Jones, R. D.; O'Halloran, R. J. J . flectroanal. Chem. 1981, 130, 113-122. Feldberg, S . W. J . Necfroanal. Chem. 1980, 109, 69-82. Bond, A. M.; O'Halloran, R. J.; Ruzic, I.; Smith, D. E. J . flectroanal. Chem. 1978, 90, 381-388.
RECEIVED for review November 12, 1981. Accepted April 5, 1982. The financial assistance of the Australian Research Grants Committee in support of this work is gratefully acknowledged.
