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A systematic approach to faradaic current, charging current and phase angle measurement by digital alternating current polarography
35
Chem, 222 (1987) 35-44 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
J. Electroanal.
A SYSTEMATIC APPROACH TO FARADAIC CURRENT, CHARGING CURRENT AND PHASE ANGLE MEASUREMENT BY DIGITAL ALTERNATING CURRENT POLAROGRAPHY *
A.M. BOND and I.D. HERITAGE ** Division of Chemical and Physical Sciences, Deakin University
Waurn Ponds 3217, Victoria (Awtralia)
(Received 11th November 1986)
ABSTRACT In microprocessor-based digital ac polarography, the simultaneous measurement of total alternating current polarograms at 10 o phase angle intervals allows the calculation of both the faradaic and charging current components of the experiment as well as their phase-angle relationship to the applied potential. The data evaluation procedure is based upon extrapolation of the background current at potentials removed from the faradaic process so as to enable the charging current to be calculated in the presence of the faradaic current. The faradaic current can then be calculated by subtraction of the charging current. Curve fitting of the separated faradaic and charging current functions to an equation of the kind Y = A sin (X + $I), where $J is the phase angle, enables interpolated phase angles for the faradaic response to be calculated with an accuracy of better than lo. Data are presented for the process [Fe(ox)s13- + e- + [F~(ox)~]~- (ox = oxalate) at a mercury electrode and provide a frequency-independent phase angle of (45 f 1)’ as expected theoretically for a reversible process. The two-electron reduction of copper(H) in 1 M NaN03 to produce a copper amalgam exhibits the theoretical frequency-dependent phase angle of less than 45O expected for a quasi-reversible process. The microprocessor-based digital ac method of phase-angle measurement is considered to be superior to conventional analog approaches, but not as accurate as the Fast Fourier Transform method developed by Smith and co-workers with more elaborate and expensive expensive laboratory computer-based instrumentation.
INTRODUCTION
Alternating current polarography, in which a small amplitude sinusoidal potential waveform is superimposed onto the direct current (dc) ramp and the alternating current measured as a function of applied dc potential, was developed over forty years ago as a logical extension of the classical technique of dc polarography [l]. The alternating current (ac) polarographic techniques found early use in novel
Dedicated to the memory of our friend and colleague, Donald E. Smith. ** Present address: Shell Company of Australia Ltd., Refinery Road, Corio 3214. Victoria, Australia. l
0022-0728/87/$03.50
0 1987 Elseviet Sequoia S.A.
36
applications of analytical voltammetry [l] via improved resolution and substantially different responses to surface phenomena (tensammetry). Additionally, extremely high sensitivity to both heterogeneous and homogeneous kinetic influences associated with electron transfer reactions [l] led to the extensive use of ac polarography in quantitative evaluation of electrode mechanisms. At the time Donald E. Smith commenced his investigations into ac polarography, a modest theory and understanding of the advantages of the technique were available. That is, an environment had been created in which a scientist of great vision and intellectual capacity could operate with enthusiasm and vigour. Pioneering work in implementing rigorous mathematical approaches to the theoretical description of the ac techniques, matched with the development of state-of-the-art instrumentation in both the analog and digital areas of electronics, has led to the term ac polarography and the work of Donald E. Smith becoming almost synonymous. The full extent of his contributions to ac polarography and related transient electrochemical techniques are conveyed in his own writings [2-51. In the laboratories of Deakin University, a microprocessor based technique called digital alternating current polarography [6] has been developed which is a “poor man’s” version of Fourier Transform Techniques developed by Smith [7]. Smith’s techniques require the use of elaborate and sophisticated laboratory computers. The digital ac technique developed in these laboratories uses inexpensive microprocessor-based technology. By replacing a small amplitude sine wave by a digital sine wave and using software to mimic the hardware, fundamental and second harmonic ac polarograms in phase-sensitive or total current versions can be generated from a single experiment as indeed can square wave and pulse polarograms [8].
Fig. 1. Phase-angle relationships in conventional ac polarography for a reversible process. E,, alternating potential, I, = faradaic current, Z, = charging current.
= applied
37
Consequently, ‘r~taircase” or step function equations can be used to provide a theoretical description of digital ac polarography [9]. That is, the instrumentation and theory of ac polarography have been simplified to the point where the relationships with dc and other transient methods are very obvious. Bioanalytical Systems have incorporated the digital ac technique in their commercially available BAS 100 ~~~~h~~ Analyzer event so that wider use of the method is now possible. The concepts of digital ac technique have been expanded on by Faulkner and co-workers [lo,11 ] as part of their stimulating articles on “cybernetic” voltammetry. In the present paier which is dedicated to the memory of Donald E. Smith, a simple and systematic strategy for measurement of the phase angle of an electrode process by digital ac polarography is described. The use of phase-angle measurement has formed an important strategy in many of Smith’s papers on ac polarography [2,3]. Traditionally, this datum is derived from measurement of the in-phase and quadrature components of phase-selective ac polarograms, with phase-selective detectors or lock-in amplifiers forming au important aspect of the analogue instrumentation. Figure 1 shows diagrammatically the well known relationships in conventional alternating current pol~o~aphy. ~a~erna~~y, the relationshipscan be expressed as
IT = ((h$
cotcp= 4,/r,
+ uQ)*y2
(1)
(2)
where tp, the phase angle, is the parameter of interest in many kinetic inv~ti~tions and I,, I,, and 1o are the total, in-phase and quadrature current components, respectively. In the technique of digital alternating current polarography, data over a wide range of dc potentials are collected at an equivalent of every 10 o (assuming that a 36 step digital sine wave has been applied [ti]). In conventional analog instrumentation, the in-phase and quadrature component measurementsmust be obtained from two separate experiments. However, digital ac polarographic measurements are more information rich and consequently this datum can be derived from a single experiment. Thus, numerous additional strategies for phase-angle measurement exist, which greatly improves the precision and accuracy of measurement. in this paper, it is now shown how the versatility of digital ac ~l~~ap~c rne~~ern~~ can be extended to the calculation of the phase angle. This contribution therefore supplements the capability for simultaneousgeneration of square wave, alternating current, direct current and pulse polarograms demonstrated previously. The relationship to both analog and FFT instrumeutation pioneered by Donald Smith will become apparent during the presentation of the data. It is therefore with great pleasure that this paper is dedicated to the memory of our friend and colleague, Donald E. Smith. Without his pioneering work and personal interest, work on the technique of digital ac polarography would not have commenced in our laboratories.
38 EXPERIMENTAL
All chemicals were of Analytical Reagent grade and distilled/deionised water was used throughout. Solutions were degassed with high purity nitrogen for 10 min prior to performing the polarographic experiments. Polarograms were recorded in a temperature-controlled laboratory at (23 f 1)’ C. Data were transferred to a “Sphere” personal computer for evaluation via an RS232 interface. This “Sphere” microcomputer system is based upon a Motorola 6809 microprocessor as described previously [12]. The microprocessor based equipment for digital ac function generation and data collection has been described elsewhere [6]. This system was interfaced to an AMEL (Milan) Model 551 Potentiostat and used in the conventional three-electrode mode with positive feedback circuitry. The three electrodes used were a static mercury drop working electrode (SMDE), Model 303 from EG & G Princeton Applied Research Corp. (PARC), Princeton, NJ, interfaced to the AMEL Potentiostat via a PARC Model 306 interface accessory, together with a platinum auxiliary electrode and a Ag/AgCl (3 M KCl) reference electrode. RESULTS
AND DISCUSSION
The technique of digital ac polarography as previously described [6] consists of superimposing a digital sine wave of 36 steps onto the dc staircase ramp via the summing amplifier associated with the potentiostat. The polarograms may be presented in a total or phase-selective format but both methods contain phase-angle information. In this work, current measurements were made at the end of each of the 36 steps and the total current format used. Each step can therefore be considered to be equivalent to a measurement at a given phase angle at 10” intervals. Current measurements were averaged over 16 cycles to eliminate the unwanted dc terms and to provide an adequate signal-to-noise ratio. Figure 2 shows the digital ac polarograms for the one-electron reduction of ferric oxalate in oxalate media [ Fe(III)ox,]
3-+ e- + [ Fe(III)ox,]
4-
(ox = oxalate)
(3)
in the total alternating current format. The data are presented as 36 individual ac polarograms with each consecutive polarogram corresponding to measurements with a 10 o phase angle shift from the previous polarogram. It has been shown [5,13] that the heterogeneous charge transfer rate constant for this electrode process is > 1 cm s-l and therefore this can be used as a model which is close to a reversible process under the low-frequency ac conditions of experiments reported in this paper. Perusal of Fig. 2 indicates that the total data set generated for data evaluation is very extensive and substantially greater than available in analog experiments. Indeed, the data set is closely related in magnitude and information content to that available in the FFT measurement mode described by Smith [7]. Consequently, extensive and accurate data treatment procedures can be developed for extraction of the required phase angle information.
39
Fig. 2. Digital (total current) ac polarogram from 0 to 360 o for reduction of 2.25 X 10e4 M Fe@) in 0.3 M K&O4 +O.l M H&O4 at 23OC. The digital sine wave frequency is 66.6 Hz with a peak-to-peak amplitude of 6 mV. The dc potential scan is over the range 0.0 to 0.5 V vs. Ag/AgCl with 5 mV steps.
The current data at each potential in ac polarography represents the sum of the faradaic and charging currents, modified by their respective phase-angle relationships. It is clear from both Figs. 1 and 2 that two equivalent but unique data sets 180” apart exist in both the conventional and digital ac technique in which the measured response is solely the charging current as a function of potential. Similarly, two unique data sets, 180° apart, represent the measurement of purely Faradaic current in the absence of charging current. In principle therefore, the charging current data in digital ac polarography can be measured directly in the absence of the faradaic current at either of two particular phase angles. Subsequent calculation and correction of charging current via standard trigonometric functions for all phase-angle responses could be applied, enabling the pure faradaic response to be obtained as a function of phase angle. Inspection of data contained in Fig. 2 also leads to the conclusion that the charging and faradaic current differ in phase angle by about 45 O. That is, the phase angle for the faradaic process for reduction of ferric oxalate under conditions of digital ac polarography can be identified as being 45 o for a reversible process as is the case with conventional ac polarography [14,15]. In conventional methods of measurement of the ac response, for which extensive theory is available, the phase angle at which the charging current attains its maximum value is defined to be the quadrature or 90 o phassangle component of the experiment (see Fig. 1). This point can be used as an internal reference in digital ac polarography for calculating the phase angle of the faradaic current. Use of this reference point avoids errors from instrumental artefacts which may introduce a phase shift and provides a conversion of data in the total current digital ac format to that conventionally obtained as in Fig. 1. Unfortunately, a resolution of 10 O, as obtained in Fig. 2, is totally inadequate for simple and accurate measurement of the phase angle and accurate calculation of the charging current. The faradaic current
40
and phase-angle measurement require the introduction of curve fitting and mathematical interpolation of data. The procedure for evaluating both the faradaic and charging currents plus their phase relationships with high precision requires that all of the polarographic data depicted in Fig. 2 is transferred from the function generator/data-acquisition system to the data evaluation system described in the Experimental Section. The data evaluation program, written in BASIC, interpolates the faradaic peak current, It, and charging current, I,, in the following way. Initially the background current where faradaic current is present is approximated from data selected before and after the ac polarographic peak. A quadratic least-squares regression analysis is then performed to determine the “best fit” of the background or charging current over the entire potential range. 1, may then be calculated from the background approximation. This background current is subtracted to generate a purely faradaic current from which the faradaic peak position and peak current is calculated using mathematical procedures described in ref. 16. A particular data set from Fig. 2 is shown in Fig. 3 in which the symbols and extrapolation procedures are defined. This procedure is performed upon all 36 ac polarograms to form two sets of data: the faradaic and charging current components at each phase angle. With the previously described two sets of data collected as in Fig. 4, the data evaluation program can then be used to calculate the function each current
.
I
0.0
01
a2 -E/V
.
a3
1
0.4
0.5
Fig. 3. Expanded view of an ac polarogram at 45 o relative to the applied potential. Data obtained from the (total current) ac polarogram in Fig. 2. Zr = faradaic current, Z, = charging current, both calculated at the peak potential. Fig. 4. Least-squares analysis of the function Y = A sin (X + 9) for both the faradaic current, I,, and the charging current, Z,, from the total current digital ac polarograms in Fig. 2.
41
component (I, and I,) displays. This is achieved by apply&least-squares and fitting the data to the trigonometric function
analysis
Y=A sin(X++) (4) thereby enabling the phase angle, (p, for both I, and 1, to be calculated with respect to the applied alternating potential as required in conventional ac polarography (Fig. 1). The result of this procedure is shown in Fig. 4 for the reduction of the ferric oxalate system at a frequency of 66.6 Hz. Calculations based on eqn. (4) from the data in Fig. 4 show that the phase angles are (45.0 f 0.5)’ and (90 f 0.5)” for I, and 1, respectively, relative to the applied signal for the [Fe(ox),] 4-/3- reduction process. This result is exactly as expected [2,14,15] for a reversible process as measured under conditions of conventional ac polarography. Figure 5 shows the dependence for the ferric oxalate system of I, on w’/* at 45 o and 1, on o at 90’ where w is the angular frequency. In both cases a linear dependence is observed as expected theoretically [2,14,15]. Figure 6 shows the results of applying the digital ac experimental procedure to the reversible [F~(ox)~]~-/‘- system over a range of frequencies. This result demonstrates that within the limit of experimental error, the expected phase angle of 45” for a reversible faradaic process is obtained over the frequency range available with the present instrumentation. The very small deviation of the plot from a constant at higher frequencies could imply a minor contribution from heterogeneous charge transfer kinetics. Phase angle measurement errors are calculated at the three standard deviation levels and are included in Fig. 6. Errors in faradaic phase-angle
Fig. 5. Faradaic and charging cm-rent responses at 45O and !90”, respectively, calculated from the total current digital ac polarograms for reduction of [Fe(ox),]‘- as functions of o’fl and o, where o is the angular frequency. Other experimental parameters are as specified in Fig. 2
42
50
+
40
_,
k
I
100
300
200
+d
400
500
a-’
Fig. 6. Phase-angle mcapuranen t (9) as a function of angular frequency (0) for the [F~(ox),]~-‘~system as determined by digital ac polarography. Unspecified parameters and conditions as for Fig. 2.
data can be seen to be of the order of lo (3 standard deviations) in data presented in Fig. 6, despite the fact that the resolution of the original experiment is only 10 “. Reduction of Cu(I1) in 1 A4 NaNOS occurs according to the two-electron process Cu(I1) + 2 e- %u(Hg)
(5)
The rate of electron transfer has been reported as being in the range of 3-5 x 10m2 CIIlS -I [17,18] and this can therefore be regarded as a quasi-reversibleelectrochemical process under conditions of ac polarography. Examination of the reduction of 2.0 x 10e4 M copper by digital ac polarography and the calculation procedures described above produce the data presented in Table 1 and Fig. 7. Unlike the reversible [F~(ox)~]~-‘~- process, the phase angle is now a function of w and also considerably less than the 45’ found for a reversible process. Data are consistent with the quasi-reversiblemodel with a k, value of approximately 5 X lo-* cm s-i
TABLE 1 Faradaic and charging current and phawangle data at the peak potential for reduction of 2.0 X 10m4 M copper in 1 M NaNO, as measured by digital ac polarography o/rad s-l 210 350 480
0.211 0.253 0.319
0.187 0.311 0.462
9 /degrees
CQt+
28.0 23.8 22.0
1.88 2.27 2.48
43
Fig. 7. Plot of cot Q, at the peak potential versus o ‘I2 for the quasi-reversible two-electron reduction of 2.0 x 10v4 M copper in 1 M NaNOS at 23OC. Amplitude of the digital sine wave is 6 mV peak-to-peak.
(o = 0.5 assumed). That is, a plot of cot $3 versus &* unity where cot(+)
= 1+ G/h
is linear with an intercept of
(6)
and X=
( ks/D1'2)(e-a~
+ eS’)
(7)
with other symbols being defined in refs. 2 and 19. This result confirms the validity of the phase-angle method of measurement described in this paper. Whilst cot(+) was calculated at the peak potential in the above example, calculations can be made at any potential to obtain [cot(+], which is the parameter which should be used in a rigorous calculation [2]. The digital ac method for phaseangle measurement described in this paper can be seen to be inherently more accurate than conventional analog procedures undertaken with a lock-in amplifier or analog phase-sensitive detector. However, the frequency range available is restricted by the speed of the microprocessor to a much greater degree than with analog instrumentation. Whilst the digital ac method has some of the features of the FFT method for phase-angle measurement described by Smith [7] in that measurement of the charging current and correction by subtraction is possible, the accuracy, precision and frequency range of the FFT method are still unrivalled. The digital ac method has the advantage of requiring far less complex and expensive instrumentation than the FFT method. The challenge for the future surely must be to translate the high-performance but expensive laboratory computer technology utilized by Smith for FIT electrochemistry [20] into a low-cost microprocessor-based format without lowering the unparalleled performance which have characterized the data produced from Smith’s laboratory at Northwestern University.
44
ACKNOWLEDGEMENT
The authors gratefully acknowledge the Australian Research Grants Scheme for financial assistance in support of this project. REFERENCES 1 B. Breyer and H.H. Bauer in P.J. Elving and I.M. Kolthoff (Eds.), Alternating Current Polarography a_ndTensammetry, Chemical AnaIysis Series, Vol. 13, Interscience, New York, 1963. 2 D.E. Smith in A.J. Bard (Ed.), EIectroanaIyticaIChemistry, Vol. 1, Marcel Dekker, New York, 1966, Ch. 1. 3 D.E. Smith, C.RC. Crit. Rev. Anal Chem., 2 (1971) 247. 4 D.E. Smith in J.S. Mattson, H.D. MacDonald, Jr. and H.B. Mark, Jr. (Eds.), Computers in Chemistry and Instrumentation, Vol. 2, Marcel Dekker, New York, 1972, p. 369. 5 D.E. Smith in S.K. Rangarajan (Ed.), Applications of Minicomputers to Measurement of Faradaic Admittance. Topics in Pure and Applied Electrochemistry, SAEST, Karaikudi, India, 1975, pp. 43-67. 6 J.E. Anderson and A.M. Bond, Anal Chem., 53 (1981) 1394. 7 D.E. Smith, Anal. Chem., 48 (1976) 221A. 8 J.E. Anderson and A.M. Bond, Anal Chem., 55 (1983) 1934. 9 J.E. Anderson and A.M. Bond, Anal Chem., 54 (1982) 1575. 10 P. He, J.P. Avery and L.R Fatdkner, Anal Chem., 54 (1982) 1313A. 11 P. He and L.R. Faulkner, J. Chem. Inf. Comput. Sci., 25 (1985) 275. 12 A.M. Bond, H.B. GreenhiB, I.D. Heritage and J.B. Reust, Anal. Chim. Acta, 165 (1984) 209. 13 J. Heyrovsky and J. K&a, Principles of Polarography, Academic Press, New York/London, 1%6. 14 A.M. Bond, Modern Polarographic Methods in Analytical Chemistry, Marcel Dekker, New York 1980, Ch. 7, pp. 288-389. 15 A.M. Bond, Anal Chem., 44 (1972) 315. 16 A.M. Bond and I.D. Heritage, Anal. Chem., 57 (1985) 174. 17 J.E.B. Randles and K.W. Somerton, Trans. Faraday Sot., 48 (1952) 951. 18 A.M. Bond and D.R. Canterford, Anal. Chem., 44 (1972) 721. 19 R.J. Schwa& A.M. Bond and D.E. Smith, Anal. Chem., 49 (1977) 1805. 20 R.J. Schwa& A.M. Bond, R.J. Loyd, J.G. Larsen and D.E. Smith, Anal. Chem., 49 (1977) 1797.
Chem, 222 (1987) 35-44 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
J. Electroanal.
A SYSTEMATIC APPROACH TO FARADAIC CURRENT, CHARGING CURRENT AND PHASE ANGLE MEASUREMENT BY DIGITAL ALTERNATING CURRENT POLAROGRAPHY *
A.M. BOND and I.D. HERITAGE ** Division of Chemical and Physical Sciences, Deakin University
Waurn Ponds 3217, Victoria (Awtralia)
(Received 11th November 1986)
ABSTRACT In microprocessor-based digital ac polarography, the simultaneous measurement of total alternating current polarograms at 10 o phase angle intervals allows the calculation of both the faradaic and charging current components of the experiment as well as their phase-angle relationship to the applied potential. The data evaluation procedure is based upon extrapolation of the background current at potentials removed from the faradaic process so as to enable the charging current to be calculated in the presence of the faradaic current. The faradaic current can then be calculated by subtraction of the charging current. Curve fitting of the separated faradaic and charging current functions to an equation of the kind Y = A sin (X + $I), where $J is the phase angle, enables interpolated phase angles for the faradaic response to be calculated with an accuracy of better than lo. Data are presented for the process [Fe(ox)s13- + e- + [F~(ox)~]~- (ox = oxalate) at a mercury electrode and provide a frequency-independent phase angle of (45 f 1)’ as expected theoretically for a reversible process. The two-electron reduction of copper(H) in 1 M NaN03 to produce a copper amalgam exhibits the theoretical frequency-dependent phase angle of less than 45O expected for a quasi-reversible process. The microprocessor-based digital ac method of phase-angle measurement is considered to be superior to conventional analog approaches, but not as accurate as the Fast Fourier Transform method developed by Smith and co-workers with more elaborate and expensive expensive laboratory computer-based instrumentation.
INTRODUCTION
Alternating current polarography, in which a small amplitude sinusoidal potential waveform is superimposed onto the direct current (dc) ramp and the alternating current measured as a function of applied dc potential, was developed over forty years ago as a logical extension of the classical technique of dc polarography [l]. The alternating current (ac) polarographic techniques found early use in novel
Dedicated to the memory of our friend and colleague, Donald E. Smith. ** Present address: Shell Company of Australia Ltd., Refinery Road, Corio 3214. Victoria, Australia. l
0022-0728/87/$03.50
0 1987 Elseviet Sequoia S.A.
36
applications of analytical voltammetry [l] via improved resolution and substantially different responses to surface phenomena (tensammetry). Additionally, extremely high sensitivity to both heterogeneous and homogeneous kinetic influences associated with electron transfer reactions [l] led to the extensive use of ac polarography in quantitative evaluation of electrode mechanisms. At the time Donald E. Smith commenced his investigations into ac polarography, a modest theory and understanding of the advantages of the technique were available. That is, an environment had been created in which a scientist of great vision and intellectual capacity could operate with enthusiasm and vigour. Pioneering work in implementing rigorous mathematical approaches to the theoretical description of the ac techniques, matched with the development of state-of-the-art instrumentation in both the analog and digital areas of electronics, has led to the term ac polarography and the work of Donald E. Smith becoming almost synonymous. The full extent of his contributions to ac polarography and related transient electrochemical techniques are conveyed in his own writings [2-51. In the laboratories of Deakin University, a microprocessor based technique called digital alternating current polarography [6] has been developed which is a “poor man’s” version of Fourier Transform Techniques developed by Smith [7]. Smith’s techniques require the use of elaborate and sophisticated laboratory computers. The digital ac technique developed in these laboratories uses inexpensive microprocessor-based technology. By replacing a small amplitude sine wave by a digital sine wave and using software to mimic the hardware, fundamental and second harmonic ac polarograms in phase-sensitive or total current versions can be generated from a single experiment as indeed can square wave and pulse polarograms [8].
Fig. 1. Phase-angle relationships in conventional ac polarography for a reversible process. E,, alternating potential, I, = faradaic current, Z, = charging current.
= applied
37
Consequently, ‘r~taircase” or step function equations can be used to provide a theoretical description of digital ac polarography [9]. That is, the instrumentation and theory of ac polarography have been simplified to the point where the relationships with dc and other transient methods are very obvious. Bioanalytical Systems have incorporated the digital ac technique in their commercially available BAS 100 ~~~~h~~ Analyzer event so that wider use of the method is now possible. The concepts of digital ac technique have been expanded on by Faulkner and co-workers [lo,11 ] as part of their stimulating articles on “cybernetic” voltammetry. In the present paier which is dedicated to the memory of Donald E. Smith, a simple and systematic strategy for measurement of the phase angle of an electrode process by digital ac polarography is described. The use of phase-angle measurement has formed an important strategy in many of Smith’s papers on ac polarography [2,3]. Traditionally, this datum is derived from measurement of the in-phase and quadrature components of phase-selective ac polarograms, with phase-selective detectors or lock-in amplifiers forming au important aspect of the analogue instrumentation. Figure 1 shows diagrammatically the well known relationships in conventional alternating current pol~o~aphy. ~a~erna~~y, the relationshipscan be expressed as
IT = ((h$
cotcp= 4,/r,
+ uQ)*y2
(1)
(2)
where tp, the phase angle, is the parameter of interest in many kinetic inv~ti~tions and I,, I,, and 1o are the total, in-phase and quadrature current components, respectively. In the technique of digital alternating current polarography, data over a wide range of dc potentials are collected at an equivalent of every 10 o (assuming that a 36 step digital sine wave has been applied [ti]). In conventional analog instrumentation, the in-phase and quadrature component measurementsmust be obtained from two separate experiments. However, digital ac polarographic measurements are more information rich and consequently this datum can be derived from a single experiment. Thus, numerous additional strategies for phase-angle measurement exist, which greatly improves the precision and accuracy of measurement. in this paper, it is now shown how the versatility of digital ac ~l~~ap~c rne~~ern~~ can be extended to the calculation of the phase angle. This contribution therefore supplements the capability for simultaneousgeneration of square wave, alternating current, direct current and pulse polarograms demonstrated previously. The relationship to both analog and FFT instrumeutation pioneered by Donald Smith will become apparent during the presentation of the data. It is therefore with great pleasure that this paper is dedicated to the memory of our friend and colleague, Donald E. Smith. Without his pioneering work and personal interest, work on the technique of digital ac polarography would not have commenced in our laboratories.
38 EXPERIMENTAL
All chemicals were of Analytical Reagent grade and distilled/deionised water was used throughout. Solutions were degassed with high purity nitrogen for 10 min prior to performing the polarographic experiments. Polarograms were recorded in a temperature-controlled laboratory at (23 f 1)’ C. Data were transferred to a “Sphere” personal computer for evaluation via an RS232 interface. This “Sphere” microcomputer system is based upon a Motorola 6809 microprocessor as described previously [12]. The microprocessor based equipment for digital ac function generation and data collection has been described elsewhere [6]. This system was interfaced to an AMEL (Milan) Model 551 Potentiostat and used in the conventional three-electrode mode with positive feedback circuitry. The three electrodes used were a static mercury drop working electrode (SMDE), Model 303 from EG & G Princeton Applied Research Corp. (PARC), Princeton, NJ, interfaced to the AMEL Potentiostat via a PARC Model 306 interface accessory, together with a platinum auxiliary electrode and a Ag/AgCl (3 M KCl) reference electrode. RESULTS
AND DISCUSSION
The technique of digital ac polarography as previously described [6] consists of superimposing a digital sine wave of 36 steps onto the dc staircase ramp via the summing amplifier associated with the potentiostat. The polarograms may be presented in a total or phase-selective format but both methods contain phase-angle information. In this work, current measurements were made at the end of each of the 36 steps and the total current format used. Each step can therefore be considered to be equivalent to a measurement at a given phase angle at 10” intervals. Current measurements were averaged over 16 cycles to eliminate the unwanted dc terms and to provide an adequate signal-to-noise ratio. Figure 2 shows the digital ac polarograms for the one-electron reduction of ferric oxalate in oxalate media [ Fe(III)ox,]
3-+ e- + [ Fe(III)ox,]
4-
(ox = oxalate)
(3)
in the total alternating current format. The data are presented as 36 individual ac polarograms with each consecutive polarogram corresponding to measurements with a 10 o phase angle shift from the previous polarogram. It has been shown [5,13] that the heterogeneous charge transfer rate constant for this electrode process is > 1 cm s-l and therefore this can be used as a model which is close to a reversible process under the low-frequency ac conditions of experiments reported in this paper. Perusal of Fig. 2 indicates that the total data set generated for data evaluation is very extensive and substantially greater than available in analog experiments. Indeed, the data set is closely related in magnitude and information content to that available in the FFT measurement mode described by Smith [7]. Consequently, extensive and accurate data treatment procedures can be developed for extraction of the required phase angle information.
39
Fig. 2. Digital (total current) ac polarogram from 0 to 360 o for reduction of 2.25 X 10e4 M Fe@) in 0.3 M K&O4 +O.l M H&O4 at 23OC. The digital sine wave frequency is 66.6 Hz with a peak-to-peak amplitude of 6 mV. The dc potential scan is over the range 0.0 to 0.5 V vs. Ag/AgCl with 5 mV steps.
The current data at each potential in ac polarography represents the sum of the faradaic and charging currents, modified by their respective phase-angle relationships. It is clear from both Figs. 1 and 2 that two equivalent but unique data sets 180” apart exist in both the conventional and digital ac technique in which the measured response is solely the charging current as a function of potential. Similarly, two unique data sets, 180° apart, represent the measurement of purely Faradaic current in the absence of charging current. In principle therefore, the charging current data in digital ac polarography can be measured directly in the absence of the faradaic current at either of two particular phase angles. Subsequent calculation and correction of charging current via standard trigonometric functions for all phase-angle responses could be applied, enabling the pure faradaic response to be obtained as a function of phase angle. Inspection of data contained in Fig. 2 also leads to the conclusion that the charging and faradaic current differ in phase angle by about 45 O. That is, the phase angle for the faradaic process for reduction of ferric oxalate under conditions of digital ac polarography can be identified as being 45 o for a reversible process as is the case with conventional ac polarography [14,15]. In conventional methods of measurement of the ac response, for which extensive theory is available, the phase angle at which the charging current attains its maximum value is defined to be the quadrature or 90 o phassangle component of the experiment (see Fig. 1). This point can be used as an internal reference in digital ac polarography for calculating the phase angle of the faradaic current. Use of this reference point avoids errors from instrumental artefacts which may introduce a phase shift and provides a conversion of data in the total current digital ac format to that conventionally obtained as in Fig. 1. Unfortunately, a resolution of 10 O, as obtained in Fig. 2, is totally inadequate for simple and accurate measurement of the phase angle and accurate calculation of the charging current. The faradaic current
40
and phase-angle measurement require the introduction of curve fitting and mathematical interpolation of data. The procedure for evaluating both the faradaic and charging currents plus their phase relationships with high precision requires that all of the polarographic data depicted in Fig. 2 is transferred from the function generator/data-acquisition system to the data evaluation system described in the Experimental Section. The data evaluation program, written in BASIC, interpolates the faradaic peak current, It, and charging current, I,, in the following way. Initially the background current where faradaic current is present is approximated from data selected before and after the ac polarographic peak. A quadratic least-squares regression analysis is then performed to determine the “best fit” of the background or charging current over the entire potential range. 1, may then be calculated from the background approximation. This background current is subtracted to generate a purely faradaic current from which the faradaic peak position and peak current is calculated using mathematical procedures described in ref. 16. A particular data set from Fig. 2 is shown in Fig. 3 in which the symbols and extrapolation procedures are defined. This procedure is performed upon all 36 ac polarograms to form two sets of data: the faradaic and charging current components at each phase angle. With the previously described two sets of data collected as in Fig. 4, the data evaluation program can then be used to calculate the function each current
.
I
0.0
01
a2 -E/V
.
a3
1
0.4
0.5
Fig. 3. Expanded view of an ac polarogram at 45 o relative to the applied potential. Data obtained from the (total current) ac polarogram in Fig. 2. Zr = faradaic current, Z, = charging current, both calculated at the peak potential. Fig. 4. Least-squares analysis of the function Y = A sin (X + 9) for both the faradaic current, I,, and the charging current, Z,, from the total current digital ac polarograms in Fig. 2.
41
component (I, and I,) displays. This is achieved by apply&least-squares and fitting the data to the trigonometric function
analysis
Y=A sin(X++) (4) thereby enabling the phase angle, (p, for both I, and 1, to be calculated with respect to the applied alternating potential as required in conventional ac polarography (Fig. 1). The result of this procedure is shown in Fig. 4 for the reduction of the ferric oxalate system at a frequency of 66.6 Hz. Calculations based on eqn. (4) from the data in Fig. 4 show that the phase angles are (45.0 f 0.5)’ and (90 f 0.5)” for I, and 1, respectively, relative to the applied signal for the [Fe(ox),] 4-/3- reduction process. This result is exactly as expected [2,14,15] for a reversible process as measured under conditions of conventional ac polarography. Figure 5 shows the dependence for the ferric oxalate system of I, on w’/* at 45 o and 1, on o at 90’ where w is the angular frequency. In both cases a linear dependence is observed as expected theoretically [2,14,15]. Figure 6 shows the results of applying the digital ac experimental procedure to the reversible [F~(ox)~]~-/‘- system over a range of frequencies. This result demonstrates that within the limit of experimental error, the expected phase angle of 45” for a reversible faradaic process is obtained over the frequency range available with the present instrumentation. The very small deviation of the plot from a constant at higher frequencies could imply a minor contribution from heterogeneous charge transfer kinetics. Phase angle measurement errors are calculated at the three standard deviation levels and are included in Fig. 6. Errors in faradaic phase-angle
Fig. 5. Faradaic and charging cm-rent responses at 45O and !90”, respectively, calculated from the total current digital ac polarograms for reduction of [Fe(ox),]‘- as functions of o’fl and o, where o is the angular frequency. Other experimental parameters are as specified in Fig. 2
42
50
+
40
_,
k
I
100
300
200
+d
400
500
a-’
Fig. 6. Phase-angle mcapuranen t (9) as a function of angular frequency (0) for the [F~(ox),]~-‘~system as determined by digital ac polarography. Unspecified parameters and conditions as for Fig. 2.
data can be seen to be of the order of lo (3 standard deviations) in data presented in Fig. 6, despite the fact that the resolution of the original experiment is only 10 “. Reduction of Cu(I1) in 1 A4 NaNOS occurs according to the two-electron process Cu(I1) + 2 e- %u(Hg)
(5)
The rate of electron transfer has been reported as being in the range of 3-5 x 10m2 CIIlS -I [17,18] and this can therefore be regarded as a quasi-reversibleelectrochemical process under conditions of ac polarography. Examination of the reduction of 2.0 x 10e4 M copper by digital ac polarography and the calculation procedures described above produce the data presented in Table 1 and Fig. 7. Unlike the reversible [F~(ox)~]~-‘~- process, the phase angle is now a function of w and also considerably less than the 45’ found for a reversible process. Data are consistent with the quasi-reversiblemodel with a k, value of approximately 5 X lo-* cm s-i
TABLE 1 Faradaic and charging current and phawangle data at the peak potential for reduction of 2.0 X 10m4 M copper in 1 M NaNO, as measured by digital ac polarography o/rad s-l 210 350 480
0.211 0.253 0.319
0.187 0.311 0.462
9 /degrees
CQt+
28.0 23.8 22.0
1.88 2.27 2.48
43
Fig. 7. Plot of cot Q, at the peak potential versus o ‘I2 for the quasi-reversible two-electron reduction of 2.0 x 10v4 M copper in 1 M NaNOS at 23OC. Amplitude of the digital sine wave is 6 mV peak-to-peak.
(o = 0.5 assumed). That is, a plot of cot $3 versus &* unity where cot(+)
= 1+ G/h
is linear with an intercept of
(6)
and X=
( ks/D1'2)(e-a~
+ eS’)
(7)
with other symbols being defined in refs. 2 and 19. This result confirms the validity of the phase-angle method of measurement described in this paper. Whilst cot(+) was calculated at the peak potential in the above example, calculations can be made at any potential to obtain [cot(+], which is the parameter which should be used in a rigorous calculation [2]. The digital ac method for phaseangle measurement described in this paper can be seen to be inherently more accurate than conventional analog procedures undertaken with a lock-in amplifier or analog phase-sensitive detector. However, the frequency range available is restricted by the speed of the microprocessor to a much greater degree than with analog instrumentation. Whilst the digital ac method has some of the features of the FFT method for phase-angle measurement described by Smith [7] in that measurement of the charging current and correction by subtraction is possible, the accuracy, precision and frequency range of the FFT method are still unrivalled. The digital ac method has the advantage of requiring far less complex and expensive instrumentation than the FFT method. The challenge for the future surely must be to translate the high-performance but expensive laboratory computer technology utilized by Smith for FIT electrochemistry [20] into a low-cost microprocessor-based format without lowering the unparalleled performance which have characterized the data produced from Smith’s laboratory at Northwestern University.
44
ACKNOWLEDGEMENT
The authors gratefully acknowledge the Australian Research Grants Scheme for financial assistance in support of this project. REFERENCES 1 B. Breyer and H.H. Bauer in P.J. Elving and I.M. Kolthoff (Eds.), Alternating Current Polarography a_ndTensammetry, Chemical AnaIysis Series, Vol. 13, Interscience, New York, 1963. 2 D.E. Smith in A.J. Bard (Ed.), EIectroanaIyticaIChemistry, Vol. 1, Marcel Dekker, New York, 1966, Ch. 1. 3 D.E. Smith, C.RC. Crit. Rev. Anal Chem., 2 (1971) 247. 4 D.E. Smith in J.S. Mattson, H.D. MacDonald, Jr. and H.B. Mark, Jr. (Eds.), Computers in Chemistry and Instrumentation, Vol. 2, Marcel Dekker, New York, 1972, p. 369. 5 D.E. Smith in S.K. Rangarajan (Ed.), Applications of Minicomputers to Measurement of Faradaic Admittance. Topics in Pure and Applied Electrochemistry, SAEST, Karaikudi, India, 1975, pp. 43-67. 6 J.E. Anderson and A.M. Bond, Anal Chem., 53 (1981) 1394. 7 D.E. Smith, Anal. Chem., 48 (1976) 221A. 8 J.E. Anderson and A.M. Bond, Anal Chem., 55 (1983) 1934. 9 J.E. Anderson and A.M. Bond, Anal Chem., 54 (1982) 1575. 10 P. He, J.P. Avery and L.R Fatdkner, Anal Chem., 54 (1982) 1313A. 11 P. He and L.R. Faulkner, J. Chem. Inf. Comput. Sci., 25 (1985) 275. 12 A.M. Bond, H.B. GreenhiB, I.D. Heritage and J.B. Reust, Anal. Chim. Acta, 165 (1984) 209. 13 J. Heyrovsky and J. K&a, Principles of Polarography, Academic Press, New York/London, 1%6. 14 A.M. Bond, Modern Polarographic Methods in Analytical Chemistry, Marcel Dekker, New York 1980, Ch. 7, pp. 288-389. 15 A.M. Bond, Anal Chem., 44 (1972) 315. 16 A.M. Bond and I.D. Heritage, Anal. Chem., 57 (1985) 174. 17 J.E.B. Randles and K.W. Somerton, Trans. Faraday Sot., 48 (1952) 951. 18 A.M. Bond and D.R. Canterford, Anal. Chem., 44 (1972) 721. 19 R.J. Schwa& A.M. Bond and D.E. Smith, Anal. Chem., 49 (1977) 1805. 20 R.J. Schwa& A.M. Bond, R.J. Loyd, J.G. Larsen and D.E. Smith, Anal. Chem., 49 (1977) 1797.