A Method for Phase Angle Measurement in Second Harmonic Alternating Current Polarography Donald E. Glover and Donald E. Smith Department of Chemistry, Northwestern University, Euanston, Ill. 60201
RECENTWORK has established that second harmonic ac polarographic measurements may be utilized advantageously in the determination of rate parameters of fast electrode processes (1-3). Excellent agreement between experiment and theory has been observed for several examples of the socalled quasi-reversible case (1-3). Similar success has been realized with one system characterized by a fast follow-up chemical reaction (4). Rate parameters obtained in these studies agree favorably with those yielded by more familiar techniques, such as fundamental harmonic ac polarographic measurements (5, 6). The sensitivity of second harmonic ac polarographic currents to kinetic parameter variations led in one case to the suggestion that the charge transfer coefficient may be determined with three significant figure accuracy ( I ) . These and earlier investigations (see references 1-6 for a more complete bibliography) have been confined almost solely to measurement of the second harmonic current amplitude. Only brief reports of semi-quantitative attempts at second harmonic phase angle measurements may be found in the literature (7,s). In conventional (fundamental harmonic) ac polarography, it is well known that measurement of the phase angle not only provides a valuable complement to current amplitude data, but in most cases the phase angle is the preferred observable from the viewpoint of data analysis convenience (5). Theoretical studies have suggested that a similar situation exists for second harmonic measurements (9-11). Consequently, we have endeavored to develop an instrumental procedure for quantitative measurement of the phase angle between the second harmonic ac polarographic current and the applied potential. An approach based on phase-sensitive detection of the in-phase and quadrature components of the second harmonic current has proven applicable. Results are reported here. EXPERIMENTAL
The measurement principle employed is based on the fact that the phase angle cotangent may be calculated from in-
(1) J. E. B. Randles and D. R. Whitehouse, Trans. Faraday SOC., 64, 1376(1968). (2) T. G. McCord and D. E. Smith, ANAL.CHEM.,41, 131 (1969). (3) Ibid., 42, 126 (1970). (4) Ibid., p 2. (5) D. E. Smith in “Electroanalytical Chemistry,” A. J. Bard, Ed., Vol. 1 , Marcel Dekker, New York, N. Y . , 1966, Chap. 1. (6) B. Breyer and H. H. Bauer in “Chemical Analysis,” P. J. Elving and I. M. Kolthoff. Ed., Vol. 13, Interscience, New York, N. Y., 1963. (7) D. E. Smith, ANAL.CHEM.,35, 1811 (1963). (8) D. J. Kooijman and J. H. Sluyters, Rec. Traa. Chim., 83, 587 (1964). (9) T. G. McCord and D. E. Smith, ANAL.CHEM.,40,289 (1968). (10) T. G. McCord, H. L. Hung, and D. E. Smith, J. Electroaiial. Chem., 21,5 (1969). (11) T. G. McCord and D. E. Smith, ibid., 26,61 (1970).
phase and quadrature current measurements using the expression
where @ is the phase angle, L ( 2 w t ) is the component of the second harmonic current which is in-phase with the applied potential and ZQ(2wt) is the quadrature second harmonic component (5). The instrumental technique employed for phase-sensitive detection of second harmonic currents is basically the same as in previous reports (5, 7 , 12). The presently-employed equipment is distinguished from earlier versions by the provision for simultaneous readout of the inphase and quadrature second harmonic currents and by differences in individual circuit elements. Calibration of the phase relations of the current components detected, which is essential for absolute phase angle measurements, was performed by a method similar to that advanced by Britz and Bauer (12). However, the latter authors gave no data concerning the fidelity of their approach. A circuit schematic of the conditioning network is shown in Figure 1. The network invokes the usual procedure (5, 7 , 1 2 , 13) of tuned preamplification followed by phase-sensitive demodulation and low-pass filtering to obtain a dc response which is proportional to the magnitude of a second harmonic vectorial component. Second harmonic reference signals required by the phase-sensitive detector are generated from the sinusoidal oscillator (applied potential signal) by a nonlinear operation followed by tuned amplification of the second harmonic component (5, 7 , 12). In this study the nonlinear operator was a full-wave rectifier, a choice dictated by convenience (the circuits were available) rather than simplicity or cost considerations. Numerous other nonlinear operations would suffice-e.g., squaring, frequency doubling, etc. Phase shifters are provided in the reference signal paths to compensate for extraneous phase shift introduced by tuned amplifiers and other components. Absolute measurement of phase angles demands proper adjustment of the phase shifters so that the phase sensitive detectors readout the true quadrature and in-phase currents. The corresponding operation in fundamental harmonic measurements is readily accomplished by a simple calibration step involving the use of a precision “dummy cell” such as a precision resistor or capacitor which provides a cell current reference signal of precisely known phase angle relative to the applied potential. For example, if a resistive dummy cell is employed, the phase shifters are adjusted so that one phase-sensitive detector output is maximized (in-phase current detector), while the second produces a null signal (quadrature current detector). The difficulty with this procedure for the case of second harmonics is that the usual precision passive components used in dummy cells are linear and do not generate a second harmonic component which could be used as a phase reference. Non-linear passive components, such as diodes, generate higher harmonics (12) H. H. Bauer and D. Britz, Chem. Inst., 2, 361 (1970). (13) R. D. Moore and 0. C. Chaykowsky, Princeton Applied Research Technical Bulletin 109, Princeton Applied Research Corp., Princeton, N. J., 1963. ANALYTICAL CHEMISTRY, VOL. 43, NO. 6, MAY 1971
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ANALYTICAL CHEMISTRY, VOL. 43, NO. 6, MAY 1971
with poorly-defined phase responses-e.g., the phase response depends on signal amplitude for signal levels of interest. Consequently, we adopted a phase calibration procedure whereby a precision active circuit element was used to generate a second harmonic reference signal. A precision full-wave rectifier (5) was used for this purpose. With this device the phase angle between a fundamental harmonic input and the second harmonic component of the output is -90” (i.e., an input signal, sin w t yields a second harmonic output of cos 2wt) (14). Second harmonic phase calibration amounted to using a resistive dummy cell to generate a fundamental harmonic signal at the current output of the potentiostat (zero phase shift) which was passed through the full-wave rectifier to generate the second harmonic phase reference. The phase shifters then were adjusted as stated above for fundamental harmonic measurements. Of course, in this case the 90” phase shift between the second harmonic reference signal and the applied alternating potential implies that the phase-sensitive detector whose output is maximized in the calibration step is the quadrature current detector, whereas the one whose output is nulled is the in-phase current detector. The use of a full-wave rectifier in the calibration operation is explicitly shown in Figure 1 as part of the current signal path when the mode control is in the calibrate position. Our test of the viability of this procedure for phase angle calibration was based on whether or not second harmonic phase angle measurements yield electrochemical kinetic parameters for well-characterized model systems which are consistent with those obtained by other measurement techniques. Other aspects of utilizing the signal conditioning network of Figure 1 are standard (5, 7, 15, 16), including the method of current amplitude response calibration (16). Most measurements discussed employed identical gain factors for the inphase and quadrature signal conditioning paths. This enabled convenient calculation of phase angles from Equation 1 without the necessity of introducing the scaling factors which relate recorder chart deflection to absolute current magnitude. The potentiostat employed in this work is identical to a previously described device (17, 18), except that Zeltex Model 145L operational amplifiers were used in place of the Analog Devices Model 210 units. The potentiostat features positive feedback elimination of iR drop effects. This fact, combined with the negligible second harmonic charging current contribution, allows one to assume that the instrument readout represents a pure faradaic response, unless significant adsorption effects are operative (19,20). Because the latter problem was not encountered with the systems studied here, all results reported represent direct comparison of instrument readout with faradaic current theory. All other aspects of the electrochemical instrument, the peripheral supporting equipment and various experimental procedures are the same as reported elsewhere (2, 15, 18). Measurements were performed at 24.6 + 0.1 “C. The polarographic cell consisted of a dropping mercury working electrode, a platinum wire auxiliary electrode and a Ag/AgCl reference electrode. Measurements involved the redox systems, Cd2+/Cd(Hg) in 1 M Na2S04 and Fe(C204)33-/Fe(14) R . D. Stuart, “Fourier Analysis,” John Wiley and Sons, New York, N. Y., 1961. (15) E. R. Brown, T. G. McCord, D. E. Smith, and D. D. DeFord, ANAL.CHEM.,38, 1119 (1966). (16) D. E. Smith and W. H. Reinmuth, ibid.,33,482 (1961). (17) E. R. Brown, D. E. Smith, and G. L. Booman, ibid.,40, 1411 (1968). (18) E. R. Brown, H. L. Hung, T. G . McCord, D. E. Smith, and G . L. Booman, ibid.,p 1424. (19) M. Sluyters-Rehbach and J. H. Sluyters, in “Electroanalytical Chemistry,” A. J. Bard, Ed., Vol. 4, Marcel Dekker, New York, N. Y., 1970. (20) K. Holub, G. Tessari, and P. Delahay, J. Phys. Chem., 71,2612 (1967).
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(C204)34-in 0.5M KzC204. These systems are characterized by simple quasi-reversible or reversible (diffusion-controlled) behavior, depending on experimental conditions (2, 5, 16, 18, 21). RESULTS AND DISCUSSION
Figure 2 depicts a typical recording of the in-phase and quadrature second harmonic current components obtained with the aid of the circuit in Figure 1, sample-and-hold amplifiers and an X-Y, Y ‘ recorder. The polarogram showing the largest signal excursions is the quadrature component. Results of second harmonic phase angle us. dc potential measurements [phase angle polarograms (9)1 with CdZ+/ Cd(Hg) in 1 M Na2S04 are compiled in Figure 3. Solid circles represent experimental points. The solid curves represent theoretical phase angle polarograms for a quasireversible system ( 9 ) with k , = 0.15 cm sec-’ and a = 0.30. These rate parameter values match those previously obtained in our laboratory for this system using other observables (2, 18,22). They provide approximately the best fit of theory and experiment for the phase angle data. The overall good agreement between theory and experiment provides convincing evidence for the fidelity of the second harmonic phase angle measurement procedure and the underlying theory. These data also extend the previous demonstration of excellent self-consistency between rate parameters derived from various ac polarographic observables (2, 3, 18, 22). In terms of the phase angle polarograms, theoretical predictions for rate parameters outside the range k , = 0.15 + 0.02 cm sec-l and a = 0.30 + 0.03 yield unquestionably poorer agreement with experiment. Thus, the uncertainty in the calculation of k , and cy from the data is definitely no greater than h0.02 and +0.03, respectively. Second harmonic complex plane polarograms, which manifest both the amplitude and phase response, have been (21) R. deleeuwe, M. Sluyters-Rehbach, and J. H. Sluyters, Electrochim. Acta, 14, 1183 (1969). (22) B. J. Huebert and D. E. Smith, unpublished work, Northwestern Univ., Evanston, Ill., 1970. ANALYTICAL CHEMISTRY, VOL. 43, NO. 6, MAY 1971
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Figure 3. Second harmonic ac polarographic phase angle-dc potential results for Cd2+/Cd(Hg) system System: Same as Figure 2 Applied: Same as Figure 2 except frequencies are listed below Measured: second harmonic phase angle at end of mechanically controlled drop life. Readout aided by sample-and-hold circuitry - - - Theoretical second harmonic phase angle polarograms for CY = 0.30, k , = 0.15 cm. sec-l, D o = 6.00 X cm2 sec-I, D R = 1.60 X 10-5cm2sec-', t = 5.0 sec., A = 0.035 cm2, T = 24.6 "C,A E = 5.00 mV, n = 2, C,* = 3.19 X lO+M, and appropriate w value 0 = experimental phase angles A. Applied frequency = 18 Hz B. Applied frequency = 25 Hz C. Applied frequency = 68 Hz 778
ANALYTICAL CHEMISTRY, VOL. 43, NO. 6, MAY 1971
Applied frequency = 1% Hz Applied frequency = 395 H z F. Applied frequency = 1000 Hz D. E.
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Figure 4. Second harmonic complex plane polarogram for Cd2+/Cd(Hg) system System and Applied: Same as Figure 2, except frequency = 190Hz Measured : second harmonic in-phase current magnitude us. second harmonic quadrature current magnitude a t end of mechanically-controlled drop life using sample-and-hold readout mode ~- - Theoretical complex plane polarogram for parameters listed in Figure 3, except frequency = 190 Hz - _ _ - Theoretical complex plane polarogram for diffusion-controlled (nernstian) system 0 = experimental results
suggested as a useful data presentation format (9-11). The instrument used in this work allows automatic recording of complex plane polarograms simply by driving the recorder X and Y inputs with the phase-sensitive detector outputs (Figure 1). Alternatively, the complex plane polarograms may be constructed manually from the in-phase and quadrature second harmonic polarograms. Figure 4 shows a typical theory-experiment comparison for a complex plane polarogram of the cadmium system. For reference, Figure 4 also shows the theoretical complex plane polarogram of a strictly diffusion-controlled system which is a straight line of -45 O slope (9). It is apparent from Figure 4 that the complex plane data and theory not only agree reasonably well, but deviate markedly from the limiting diffusion-controlled response. This occurs even though the reaction in question is moderately fast while the frequency employed (190 Hz)
Second harmonic complex plane polaro-
gram for Fe(C204)3a-/Fe(Cz04)34system System: 2.% X 10-3MFe3f in 0.50MK2C204 Applied: same as Figure 2, except frequency = 17.5 Hz Measured: same as Figure 4 - = Theoretical complex plane polarogram for diffusion-controlled (nernstian) system 0 = experimental points a t dc potentials between the two second harmonic current peaks A = experimental points a t dc potentials outside the two current peaks-Le., on the “wings” of the polarogram
is low. This profound influence of charge transfer kinetics under the relatively “mild” experimental conditions depicted in Figure 4 is another example of the previously-mentioned extreme sensitivity of second harmonic measurements, particularly the phase response, to the kinetic status of the electrode process. Even the Fe(C204)33-/Fe(C204)34system, whose k , value exceeds that of the cadmium system by an order of magnitude (ZI), did not yield a strictly diffusioncontrolled response, except at the very lowest frequencies. Under the latter conditions, one did obtain the expected linear complex plane polarogram as shown in Figure 5. We frel that the foregoing results establish that one can obtain valid second harmonic phase angle data utilizing the relatively standard procedure of phase-selective current detection in combination with an unusual, but effective, technique for phase calibration. RECEIVED for review December 30, 1970. Accepted January 28, 1971. Work supported by National Science Foundation Grant GP-16281
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