Alternating Current Polarography in the Non-Coherent Wave Frequency Multiplex Mode Barry J. Huebertl and Donald E. Smith2 Department of Chemistry, Northwestern University, Evanston, 111. 60201
A number of experimental procedures which offer the possibility of rapid acquisition of the faradaic admittance-f requency response profile have been made feasible by the evolution of small, on-line, laboratory digital computers (minicomputers). As part of an experimental survey designed to assess the relative merits of such techniques, we have implemented and evaluated an approach based on: (a) the application of an excitation signal composed of a set of multiple, discrete, non-coherent frequency components; and (b) the simultaneous measurement of the cell response at each input frequency with the aid of appropriate tuned circuitry. The technique is referred to as “A.C. Polarography in the Non-Coherent Wave Frequency Multiplex Mode.” Experimental procedures invoked and results obtained are presented. The equivalence of faradaic admittance data obtained by use of the non-coherent wave frequency multiplex mode and conventional ac polarography is advanced as a demonstration of the new measurement scheme fidelity. A brief discussion of the major competing techniques and their relative merits also is given. As WITH MOST BRANCHES of scientific experimentation, the state of the art in electrochemical measurements has progressed considerably in recent years through the application of small laboratory digital computers (“minicomputers”) for on-line experiment control, data acquisition, and data analysis. One finds in the literature numerous reports (1-13) where on-line computerization of an electrochemical experiment has enabled dramatic gains in measurement speed, efficiency, and/or precision. The vitality of this area of endeavor is epitomized by the contents of a volume (11) devoted to a review of progress in “computerized electrochemistry.” It is evident from developments to date that most modern electrochemical techniques stand to benefit substantially from the utilization of on-line minicomputers as a key instrument Present address, Department of Chemistry, Colorado College, Colorado Springs, Colo. 80903. * To whom correspondence should be addressed. (1) G. Lauer, R. Abel, and F. C. Anson, ANAL.CHEM.,39, 765 (1967). (2) G. Lauer and R. A . Osteryoung, ibid., 40 (lo), 30A (1968). (3) S . P. Perone, J. E. Harrar, F. B. Stephens, and R. F. Anderson, ibid., p 899. (4) S. P. Perone, D. 0. Jones, and W. F. Gutknecht, ibid., 41, 1154 (1969). (5) F. B. Stephens, F. Jakob, L. P. Rigdon, and J. E. Harrar, ibid., 42, 764 (1970). (6) W. F. Gutknecht and S . P. Perone, ibid., p 906. (7) D. 0. Jones and S . P. Perone, ibid., p 1151. ( 8 ) L. B. Sybrandt and S. P. Perone, ibid., 43, 382 (1971). (9) H. E. Keller and R. A. Osteryoung, ibid., p 342. (10) J. Lawrence and D. M. Mohilner, J. Electrochem. Sac., 118,259 ( 1971 ). (11) “Computers in Chemistry and Instrumentation,” Vol. 2, J. S. Mattson, H. D. MacDonald, Jr., and H. B. Mark, Jr., Ed.
M. Dekker, Inc., New York, N. Y., in press.
(12) S. P. Perone, J. W. Frazer, and A. Kray, ANAL.CHEM., 43,1485 (1971). (13) H. Kojima and S . Fujiwara, Bull. Chem. Sac. Jup., 44, 2158 (1971).
component. Alternating current polarography and related techniques designed to characterize the faradaic admittance (14-16) appear to represent potentially one of the most fruitful areas for implementation of the minicomputer. The tedious repetitive nature of classical faradaic admittance measurements is well-recognized. This is true even in the extreme of highly automated analog instrumentation which enables direct recording of a faradaic admittance component us. dc potential at a particular frequency (17-19). To satisfy the need for information on two faradaic admittance components at each frequency (14,IS) and to obtain data at sufficientfrequencies to characterize the frequency response profile, such ‘‘ac polarograms” must be recorded under a variety of conditions. Consequently, even with modern automatic analog instrumentation the worker is faced with repetitive, redundant, experimental manipulations and data workup operations which consume notable time when admittance data are gathered by the usual approach of examining the cell response to a pure sinusoidal input. The implementation of an on-line computer aid is clearly a powerful and versatile approach to minimizing these demands of faradaic admittance measurements. Because of its precision in digital data acquisition and data analysis and the ease with which it can be programmed to control repetitive measurements (20), the on-line minicomputer is ideally applicable to faradaic admittance measurements. Using the same basic approach reported for interfacing a digital data acquisition system to an ac polarograph (21), workers in our laboratory encountered no difficulty in interfacing an operational amplifier ac polarograph to a Digital Equipment Corporation (DEC) PDP-8/S computer system (22). Programs were written which placed much of the burden of redundant experimental manipulations and data treatment on the computer. Faradaic admittance results obtained with the resulting computerized ac polarograph were generated with greater speed and precision than was normally observed with the same analog polarograph in absence of the on-line computer aid. This experience parallels that of other workers (1-13) who have implemented the on-line computer concept in various areas of chemical instrumentation, and it convinces us that computerization provides the path to reduc(14) D. E. Smith, in “Electroanalytical Chemistry,” A. J. Bard, Ed., M. Dekker, Inc., New York, N.Y., Vol. 1, 1966. (15) M. Sluyters-Rehbach and J. H . Sluyters, ibid., Vol. 4, 1970. (16) K. J. Vetter, “Electrochemical Kinetics,” Academic Press, New York, N.Y., 1967. (17) E. R. Brown, T. G. McCord, D. E. Smith, and D. D. DeFord, ANAL.CHEM.,38, 1119 (1966). (18) E. R. Brown, H. L. Hung, T. G. McCord, D. E. Smith, and G. L. Booman, ibid., 40, 1424 (1968). (19) R. deLevie and A. A. Husovsky, J. Electroanal. Chem., 20,181 (1969). (20) D. E. Smith, in “Computers in Chemistry and Instrumentation,’’ Vol. 2, J. S. Mattson, H. D. MacDonald, Jr., and H. B.
Mark, Jr., Ed., M. Dekker, Inc., New York, N.Y., in press. (21) E. R. Brown, D. E. Smith, and D. D. DeFord, ANAL.CHEM., 38,1130 (1966). (22) H. L. Hung, B. J. Huebert, and D. E. Smith, Northwestern University, Evanston, Ill., unpublished work, 1968. ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972
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ing faradaic admittance measurements to a rapid, routine, convenient operation. The remarks in the preceding paragraph are addressed only to the rewards attending computerization of the conventional faradaic admittance measurement which employs the pure sinusoidal test signal. However, focusing attention solely on such relatively obvious on-line computer applications overlooks greater rewards to be gained by considering computerenabled procedures which invoke completely new dimensions in experimental procedure and philosophy. In the electrochemistry literature, one finds an excellent illustration of a powerful computer-enabled departure from the conventional experimental approach in the suggestion (23-26) that the faradaic admittance frequency response profile can be ascertained by Fourier Transformation of the electrochemical cell’s transient time-domain response to a step or impulse perturbation. The experimental approach is referred to as Transient Impedance (or Admittance) Analysis (25). Despite the magnitude of the mathematical manipulations attending Fourier Transformation, the operation can be performed readily with available laboratory computers in times which are short relative to, for example, a single drop life of a DME (27). Consequently, by the transient admittance analysis scheme, it is in principle possible to acquire the frequency response profile of the faradaic admittance at a particular dc potential in the life of a single mercury drop. This rather exciting and revolutionary potentiality is clearly not open to the conventional, single frequency faradaic admittance experiment, with or without computerization. Problems introduced by faradaic nonlinearity, which will be discussed below, may make the transient impedance analysis experiment fall somewhat short of the goal of obtaining a faradaic admittance frequency response with state-of-the-art precision in a few seconds, but it appears obvious that this particular departure from the conventional approach is highly advantageous, provided that a computer is available for the demanding mathematical analysis. -Transient admittance analysis is only one of several novel approaches to faradaic admittance measurement which are made feasible by an on-line digital computer and which should enable acquisition of faradaic admittance frequency response data with unprecedented rapidity. The common denominator shared by the various possibilities is the use of an input signal whose frequency domain spectrum (28, 29) consists of numerous components (the signal is either broadband or composed of an array of discrete frequency components), rather than the conventional single component spectrum. By suitable analysis of the cell response to such multiple-frequency inputs, the admittance is simultaneously acquired at enough frequencies to adequately reveal the frequency response profile. Although it has been mentioned only briefly in the electrochemical literature (26), the use of random noise (white, pink, etc.) as a test signal is probably the most frequently addressed alternative to transient analysis that one finds in the electronics (23) K. Doblhofer and A. A. Pilla, J. Electrochem. SOC.,submitted for publication. (24) H. P. Van Leeuwen, D. J. Kooijman, M. Sluyters-Rehbach, and J. H. Sluyters, J. Electroand. Chem., 23,475 (1969). (25) A. A. Pilla, J. Electrochem. SOC.,117,467 (1970). (26) R. L. Birke, ANAL.CHEM.,43, 1253 (1971). (27) Bulletin SP-360,Raytheon Computer Corp., Santa Ana, Calif., 1970. (28) R. Bracewell, “The Fourier Transform and Its Applications,” McGraw-Hill, New York, N.Y., 1965. (29) J. S. Bendat and A. G. Piersal, “Measurement and Analysis of Random Data,” John Wiley and Sons, New York, N.Y., 1966. 1180
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literature (29, 30). We will refer to the latter approach as “Noise Response Admittance Analysis.” In both transient and noise response analysis, a continuous broadband frequency spectrum characterizes the test Signal. Less attention has been devoted to the use of test signals which are composed of a finite set of discrete frequencies (the frequency spectrum is made up of a discontinuous finite ensemble of non-zero elements), even though such test signals may have certain advantages for the case of faradaic admittance measurements. In this category, one can envision two distinctly different types of test signal. The first and most conveniently accessible is a complex periodic signal, such as the common square, rectangular, triangular, or sawtooth waves or, preferably, pseudorandom noise (29, 30). Complex periodic signals are characterized by a frequency domain spectrum composed of discrete components whose frequencies are coherent with one another-Le., all frequency ratios are given by rational numbers (29). The second possibilityis a test signal comprised of a set of discrete frequency components whose frequencies are non-coherent-Le., the frequencies are not related by rational numbers. Such a signal waveform is often referred to as “almost periodic” (29) and usually results when one sums signals from a set of sinusoidal oscillators with arbitrarily selected frequencies. In general terms we refer to the latter two techniques as “Admittance Analysis in the Coherent Wave Frequency Multiplex Mode” and “Admittance Analysis in the Non-Coherent Wave Frequency Multiplex Mode,” respectively. The suitability and necessity of the term “Frequency Multiplex” is discussed elsewhere (20). In cases where the experiment is pursued in a manner analogous to ac polarography (measurement of the current response to a potential input as a function of dc potential) one may replace the more general term, “Admittance Analysis,” by “A. C. Polarography” in all of these titles. Each of the foregoing measurement concepts for rapid acquisition of faradaic admittance frequency response data possesses a variety of advantages and disadvantages relative to the others. The relevant factors are sufficiently numerous that selection of the particular technique offering the best combination of convenience, speed, accuracy, and bandpass in acquiring faradaic admittance data is not clear-cut on an a priori basis. Also, although convincing results have been reported in one instance (23, it remains to be established for most cases in question whether the concept is capable of yielding with actual electrochemical systems equivalent or improved data fidelity over that which characterizes conventional admittance measurement schemes. Consequently, we have undertaken a detailed experimental comparison of the measurement alternatives mentioned here in an effort to obtain some empirical evidence to supplement guidelines provided by information theory (31) regarding their relative and absolute merits. Because it is the only one of the four alternatives mentioned which retains the advantage of pure sinusoidal test signals with regard to “tuning out” effects of faradaic nonlinearity (14, 15) (see below for further discussion of this point) and because it was more readily implemented with the computer system originally available to this study, A. C. Polarography in the Non-Coherent Wave Frequency Multiplex Mode has been accorded the most detailed examination to date in our laboratory. The discussion to follow presents (30) Hewlett-Packard, Inc., “Determination of Transfer or Im-
pedance Functions Using Random Noise,” Technical Note No. 02-5951-1001,Hewlett-Packard, Inc., Santa Clara, Calif., 1970. (31) S. Goldman, “Information Theory,” Prentice-Hall, New York, N.Y., 1953.
Table I. Response Frequency Components Obtained from Terms of First-, Second-, and Third-Order in AC Polarographic Theory for Case of Four Input Frequency Components, fi, fz, fa,f4
Frequencies generated by Frequencies generated by second-order terms first-order terms (Proportional to AEz, (proportional to AE, relative component Component relative component magnitude 50.10 classification magnitude = 1.OO) for AE 5 10 mV) Fundamental harmonic A, f2,fa, f 4 None components
results of a feasibility demonstration and evaluation of the latter technique. The observations in question were sufficiently satisfactory that the instrumentation developed for this purpose has subsequently received routine use in several electrochemical kinetic investigations (32, 33). THEORETICAL
Our interest in the non-coherent wave frequency multiplex mode as an approach to faradaic admittance measurements is based on the following key concept: If an input composed of multiple discrete frequencies is applied to an electrolytic cell, with appropriate instrumentation and experimental conditions one can measure simultaneously the fundamental harmonic responses corresponding to each input frequency and the response at each frequency will be identical to what one would observe in a conventional experiment involving a single input component of the frequency in question. In other words, interactions between the various applied frequency components are either negligible or they cannot influence the fundamental response measurement so that conventional faradaic admittance theory is applicable. This concept is supported by a sound theoretical foundation whose starting point may be found in the theory of intermodulation polarography (34, 3 3 , a technique based on the use of two applied frequencies. Although algebraically cumbersome, it is a relatively simple matter to extend this theory to experiments involving a significantly larger number of discrete input frequencies (36). Because of the routine nature of this theoretical problem and because the main thrust of the present report is to validate the above-stated concept on an empirical basis, we will not dwell on quantitative theoretical details at this point. Rather, our remarks will be confined to a brief survey of key qualitative (32) B. J. Huebert and D. E. Smith, J. Electroanal. Chem., 31, 333
(1971). (33) W. E. Geiger and D. E. Smith, Northwestern University, Evanston, Ill., unpublished work, 1970. (34)W.H. Reinmuth, ANAL.CHEM., 36, 211R (1964). (35)J. Paynter, Ph.D. Thesis, Columbia University, New York, N.Y. 1964. (36)B. J. Huebert, Ph.D. Thesis, Northwestern University, Evanston, Ill., 1971.
Frequencies generated by third-order terms (proportional to AE3,relative component magnitude 5 0.010 for AE 5 10 mV)
A, A, fa,f 4
and semiquantitative conclusions one draws upon extending existing ac and intermodulation polarographic theory to situations involving more than two applied frequency components. As has been shown for the case of two applied frequencies (35), in the first-order (linear approximation-ac response signal amplitude proportional to input signal amplitude, A@ the mathematical treatment yields only ac components whose frequencies correspond to each applied frequency (fundamental harmonics) and whose magnitudes and phase angles are given by the same expressions one obtains for the conventional single frequency measurement. This rather simple, but important result manifests the well-known response additivity principle which is rigorously applicable to any circuit component with a linear transfer function and which, of course, always results when one invokes a linear approximation with a non-linear system. The effects of the well-known non-linearity of the faradaic admittance become evident if one extends the mathematical treatment to the second-order (34, 35) (response amplitude proportional to AE2). At this level of rigor, one predicts the existence of a variety of minor (but significant) additional components whose frequencies correspond either to twice the input frequencies (second harmonics) or to the sum and difference of input frequency pairs (modulation components). With AE values which are typical for kinetic applications of faradaic admittance measurements (usually, A E I 10 mV), the second-order components are usually characterized by magnitudes which are less than about 10% of the first-order terms (14, 15). If one extends the theory to the third-order (response terms proportional to AEa),a rich array of new frequency components is predicted, some with frequencies which are three times the input frequencies (third harmonics) and some “modulation components” with frequencies given by various linear combinations of the input frequencies (37). An additional contribution to the fundamental harmonic is also noted. These third-order components normally (for AE I 10 mV) represent a fraction of a per cent of the first-order terms and for most purposes can be considered negligible. Table I provides an illustration of (37)D. E. Smith, Northwestern University, Evanston, Ill., unpublished work, 1971. ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972
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A. C. SIGNAL CONDITIONING NETWORK NO. 1
A. C. SIGNAL CONDITIONING NETWORK NO. 2
A. C. SIGNAL CONDITIONING NE W O R K NO. 3
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NETWORK NO. 4
D. C. SIGNAL CONDITIONING NETWORK
11 MULTIPLEXER
IW ANALOG-TO-DIGITAL
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ASR-33 TELETYPE
CONVERTER
Figure 1. Instrument schematic
the foregoing remarks for the special case of four discrete input frequencies. Information of the type illustrated in Table I amounts to a simple, but important set of theoretical guidelines for determining instrument requirements for the experiment under consideration. To obtain accurate measurements of the fundamental harmonic response at one input frequency, one must be able to adequately “tune out” the manifestations of faradaic non-linearity at the second harmonic and modulation frequencies as well as the more dominant fundamental harmonic responses at the other applied frequencies. Because the second-order components are usually not negligible, one concludes that it is, in general, inadequate to design tuned circuitry which is simply capable of resolving the major fundamental harmonic responses. One also must reckon with the harmonic and modulation components, paying particular attention to their location in the frequency domain. For example, if fi = 200 Hz and fi = 403 Hz, relatively crude tuned circuitry is capable of resolving responses at these frequencies. However, the second-order terms at the frequencies, 2f, and fi - fi correspond to 400 Hz and 203 Hz, respectively. Each of the latter frequencies is located only 3 Hz from a fundamental harmonic frequency of interest and, 1182
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even though relatively small, these components will perturb the measurement unless, for example, a sufficiently narrow band detection system is employed. Because the response times of bandpass detection systems increase as the bandwidth decreases, one will encounter problems with a time-dependent faradaic admittance such as with a DME. If a true instantaneous faradaic admittance measurement is desired, rather than some weighted time average, one must place a lower limit on detector bandwidth. Meeting the latter requirement may cause difficulties in adequately rejecting a second-order component which is located only a few Hz from a fundamental harmonic of interest. Fortunately, one need not rely solely on achieving a sufficiently narrow bandpass detector, as there are two other experimental parameters which one can beneficially adjust to meet experimental requirements. First, one can judiciously choose the applied frequencies such that the separation between fundamental harmonic and second-order response components is maximized. This can be quite effective, but the gains realizable through this approach obviously are decreased as the number of applied frequencies is increased. Second, one may reduce the relative importance of the second-order components by reducing AE, a process which carried far enough, will allow one to ignore all
effects of faradaic non-linearity (AE < 1.5 mV is usually adequate). However, by reducing AE one increases the possibility of measurement error due to extraneous noise in the system, so that implementation of this idea also has limitations. From considerations such as the foregoing, we conclude that optimum operating conditions will amount to a compromise in which one relies on the combined use of narrow-band detection, small amplitude input signals and appropriate input frequency selection without pushing any of these procedures near its limit. EXPERIMENTAL
Instrumentation. The primary objective of this investigation was to effect a feasibility study of the non-coherent wave frequency multiplexing concept. Consequently, to avoid overcommitting resources in a possibly disappointing endeavor, it was decided that an instrument should be developed with the minimum capability which could be considered consistent with definitively evaluating the experimental concept in question. On this basis an instrument with the ability to simultaneously readout two ac response components at each of four input frequencies was considered a reasonable compromise between the ultimate application of the non-coherent wave frequency multiplex mode and caution in resource utilization. By comparison, whenever the faradaic admittance-frequency profile contains some structure (38), at least four frequency points per decade is a realistic requirement and, since a frequency range of at least three decades (e.g., 10 Hz-10 KHz) characterizes many careful kinetic investigations, a capability of simultaneously measuring the response at twelve frequencies (at least) might be considered a reasonable ultimate goal. In the meantime, the instrument described here can be used to acquire data at more than four frequencies by redundant experimental runs, each involving a different set of four frequencies. Incidentally, one should note that in not-infrequent cases where the faradaic admittance exhibits a simple frequency dependence [e.g., a linear cot+w*’* profile (14)], the limited four frequency capability we have developed may be adequate to characterize an electrode process in a single experimental run (39). Figure 1 provides a schematic and signal flow diagram of the instrument system utilized in this investigation. The analog portion of the instrument features a dual potentiostat system to implement subtractive compensation of double-layer charging current (17, 18). The potentiostats are equipped with positive feedback iR compensation which enables elimination of ohmic potential drop effects over a reasonable, but finite frequency range (18,40). Thus, for a well-defined set of conditions, the signal at the output of the subtractor circuit may be taken as a pure faradaic response (17, 18), a situation which applied to all work described here. The potentiostats are identical to those described previously (40), except that Zeltex 145L amplifiers have been employed as control and current amplifiers. Front-panel switches allow one to conveniently connect each potentiostat to either a chemical cell or a dummy cell. The dc potential applied to the potentiostat was derived from a conventional initial voltage source (17) and a computer-controlled incremental dc sweep source (digital-to-analog converter plus a signal conditioning amplifier) whose precise operation mode is described below. The four sinusoidal signals to be applied to the cell are selected from twelve discrete frequency Wein-bridge oscillators (41), (38) H. L. Hung, J. R. Delmastro and D. E. Smith, J . Electround. Chem., 7, l(1964). (39) M. E. Peover and J. S. Powell, ibid., 20, 427 (1969). (40) E. R. Brown, D. E. Smith, and G . L. Booman, ANAL.CHEM., 40, 1411 (1968). (41) Burr-Brown Research Corp. “Handbook of Operational Amplifier Applications,” Burr-Brown Research Corp., Tucson, Ariz., 1963, p 66.
whose frequencies range from 9.5 Hz to 8892 Hz. The four selected oscillator outputs are added in a conventional operational-amplifier summing circuit (the “adder”) which also attenuates each signal’s amplitude to 10 mV peak-to-peak (AE = 5 mV). The faradaic current signal from the subtractor is coupled to the inputs of four ac signal conditioning networks, each tuned to one of the four applied frequencies. Each of these networks provides two dc output signals, one proportional to the total faradaic current amplitude at the frequency of interest (generated via full-wave rectification), and one proportional to the corresponding in-phase current (obtained by phasesensitive demodulation). The detailed circuit configuration used in the ac signal conditioning networks is shown in Figure 2. A preferable alternative to readout of total and in-phase signals would be to use two phase-sensitive demodulators to obtain in-phase and quadrature signal components. In this manner, the excellent frequency selectivity of the phase-sensitive demodulator (also known as a lock-in amplifier) would be operative in both signal paths, putting less demands on the tuned preamplifier. Our choice of total signal measurement was based on economic considerations which favored the fullwave rectifier at the time this work was undertaken. However, this consideration is now obsolete as an operational amplifier phase-sensitive demodulator now can be constructed at a cost which only slightly exceeds that of the full-wave rectifier. For this reason we would employ phase-sensitive detection of in-phase and quadrature ac signals in any new construction of analog ac signal conditioning networks for the purpose in question. In addition to readout of the eight ac signals, a dc signal proportional to the dc polarographic current was generated by preamplification and low-pass filtering using a fourth-order Butterworth response (see dc signal conditioning network, Figure 1). This was done not only to acquire in the same experiment the system dc response, whose utility is above question, but also to provide a simple, independent “check” on solution stability, capillary behavior, the presence of dc maxima, etc. As indicated in Figure 1, the nine cell current signal channels are applied to nine sample-and-hold (S/H) amplifiers to enable simultaneous measurement of each signal at a point in the mercury drop life which is precisely controlled by an external timing circuit (17, 20) which also controls the mercury drop life by activating electromechanical “drop-knockers.’’ The S/H amplifier outputs are applied to the inputs of a computercontrolled multiplexer which sequentially switches these outputs to the analog-to-digital (A/D) converter input for A/D conversion and storage in the PDP-8/S computer core memory. The timing circuit control pulse is monitored by the computer via a multiplexer channel, enabling precise synchronization of the computer digital data acquisition process with drop growth and analog data sampling. Digital data readout is effected with the aid of the ASR-33 Teletype. Provision also was made for obtaining analog point plots of digital data arrays by interfacing an X-Y recorder to the computer (not shown in Figure 1). The reader is referred to previous accounts for full details of: (a) analog circuits, their implementation and individual performance characteristics (14, 17-20, 40); (b) the PDP-8/S computer and peripherals (42, 43); and (c) a description of the modular approach used to package the rather extensive analog circuitry (36). These features are quite standard and, indeed, the analog portion of the instrument differs from conventional single-frequency counterparts primarily in the application of a circuit redundancy principle. In effect, the instrument depicted in Figure 1 achieves more rapid acquisi(42) S. C. Creason, R. J. Loyd, and D. E. Smith, ANAL.CHEM., 44, 1159 (1972). (43) Digital Equipment Corporation, “Small Computer Handbook,” Digital Equipment Corp., Maynard, Mass., 1966.
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NETWORK SCHEMATIC
A
CELL CURRENT SIGNAL INPUT (FROM SUBTRACTOR)
@
TOTAL CURRENT OUTPUT SkNAL
I
I I
PHASE SHIFTER
I LOW-PASS
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II
I I
I
I I
REFERENCE SlGNPL INPUT (FROM PPPLIED SINUSOIOPL POTENTIPL SOURCE)
IN PUPSE CURRENT OUTPUT SIGNAL ~
I
B
+
‘ I
SQUARING CIRCUIT
Complementary outputs Referenc Input
Figure 2. AC signal conditioning network A . Complete network schematic Amplifiers 2, 5,6, and 8 = Burr-Brown 3057 Amplifiers 3 and 4 = Burr-Brown 1506 Amplifier 7 = Burr-Brown 3013 Amplifier 1 = Burr-Brown3077 B. Squaring circuit detail Transistors = Texas Instruments 2N1305 Capacitors = 15 pf
tion of frequency response data by replacing redundancy in experimental manipulation (the conventional means) by redundancy in electronic circuitry. Although the redundancy concept is simple and was considered some time ago (21), its implementation to any appreciable degree was a rather remote and unappealing prospect until recently when two significant developments made the idea more accessible. First, the evolution of the minicomputer provided a sufficiently precise, 1184
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trouble-free, and efficient means of monitoring and storing data from a large number of analog signal channels (e.g., imagine the problems attending the acquisition, utilization, and maintenance of nine X-Y recorders in place of the computer in Figure 1). Second, dramatic improvements in the performance-cost factor associated with operational amplifiers, together with the attendant miniaturization of these units, has made the construction and simultaneous operation of large
numbers of analog circuits, such as those shown in Figure 2, a prospect which is realistic from the viewpoints of cost, maintenance, and space requirements. For example, an adequate version of the circuit shown in Figure 2 can be constructed from less than $300 in components. As will be pointed out below, this second development becomes unimportant if a minicomputer which is faster and more modern than the PDP-8/S is available, as such a device enables one to implement digital signal conditioning procedures. Programmed Experiment Sequence. The program used to implement the non-coherent wave frequency multiplexing experiment was written using the PAL-I11 assembly language (43). The program listing and the source and binary tapes (punched paper tapes) are available from the authors on request. A brief description of the program’s functions is as follows: (a) It asks the operator for information pertinent to the experiment, such as sample, temperature, solvent, whether or not averaging is to be employed, the frequencies to be used, the dc potential range to be employed, etc. (b) It informs the operator of unacceptable input, whether it is an overloaded A/D channel or unacceptable operator input. Diagnostic messages are printed out which tell the operator how to correct the problem. (c) It instructs the operator in the conduct of the experiment and reminds him of easily-forgotten operations. (d) It calibrates via measurements with precision dummy cells the nine A/D input channels, relating each input voltage to a single current component in the chemical cell. (e) It acquires and stores digital cell current data. (f) It applies an incremental dc ramp potential to the cells. This potential is held constant over the life of a mercury drop, then incremented to take data at the next potential. This potential may be removed from any number of drops before data are to be taken to eliminate the depletion effect (44, 45), or it may be held constant for several drops. (8) It can average the currents from any number of drops at a particular dc potential before moving on to the next dc potential. (h) It calculates from calibration factors the dc current, inphase ac, and total ac current components for each frequency, as well as the four cotangents. All of these are printed out in real-time for each dc potential, keeping the operator informed about the status of the experiment as it proceeds. (i) It stores the data so that they may be printed out later in blocks according to frequency. This or the real-time output could be punched on paper tape to enable further computer analysis at some future time, (j) The data acquisition and print-out segments of the operation are totally automatic. A more detailed description of how the foregoing functions are implemented is available elsewhere (36). A number of aspects of the program described above, such as real-time data print-out, error and overload messages, reminders to the operator, etc., might be considered extra features which are not necessary for implementation of the frequency multiplexing concept. While, strictly speaking, this observation is valid, one must recognize that the instrumentation employed (Figure l) contains an analog circuit array whose various required adjustments place demands on the operator’s memory which are somewhat unprecedented. Consequently, we have found that these “extras” are responsible for reducing to a low level the number of runs which are invalidated by some form of improper instrument adjustment. Electrode Processes. The redox systems used to test the non-coherent wave frequency multiplexing concept were : Cr(CN)sa-/Cr(CN)B4- in 1M KCN (the “chromium cyanide” (44) J. Heyrovsky and J. Kuta, “Principles of Polarography,” Academic Press, New York, N.Y., 1966, pp 96, 99-103. (45) J. F. Coetzee, J. M. Simon, and R. J. Bertozzi, ANAL.CHEM., 41,766 (1969).
Table 11. Sampled Full-Wave Rectifier Outputs with Various Applied Frequencies Full-Wave Rectifier Outputs Frequen- Frequencies cies Measured Frequencies present present Frequencies frequency = M = M 2 present present channel (M) = M alone lowest lowest = All four 9.57Hz 0.532 0.530 0.530 0.530 44.9 Hz 0.854 0.854 0.855 0.854 105.1 Hz 1.418 1.417 1.429 1.428 439.0 Hz 2.817 2.820 2.826 2.829 22.3 Hz 1.001 1.002 1.002 1.002 84.5 Hz 1.om 0.998 0.997 0.997 230.5 Hz 1.001 1.002 1.002 1.ooo 845.0 1,008 1.008 1.008 1.008 1060.0 Hz 0.816 0.817 0.816 0.817 2237.0 1.285 1.284 1,284 1.284 4598.0 1.635 1.641 1.651 1.654 8930.0 Hz 0.807 0.811 0.819 0.823 Outputs given in volts; measured with Vidar Model 500 digital voltmeter.
+
+
system), CdZ+/Cd(Hg) in 1MNa2S04(the “cadmium” system, and Fe(C204)33-/Fe(C204)34in 0.5MK G 0 4 (the “iron oxalate” system). Methods of compound and solution preparation, purification of nitrogen for cell degassing and solvent (H2O)purification have been described previously (17, 18). Measurements were performed at 25.0 =k 0.1 “C. The polarographic cell contained a DME working electrode (Sargent S-29417 capillary), a platinum wire auxiliary electrode, and a saturated calomel reference electrode. Peripheral supporting equipment employed in these measurements, such as constant temperature baths, etc., also have been described elsewhere (17,18). RESULTS AND DISCUSSION
Dummy Cell Tests. To establish on a purely electrical basis, the quantitative accuracy of the instrument outlined in Figure 1, many tests were run using precision dummy cells in place of the chemical cell. Because of the nature of the experimental procedure described above, the most critical aspects of the analog circuitry performance are linearity, dc offset levels, and frequency selectivity. Particular attention was paid to these characteristics. High linearity and negligible dc offsets are requirements because a single point calibration routine is used. The demands on these response characteristics could be relaxed by a more sophisticated calibration routine, which the computer can readily implement, but this was found to be unnecessary for our immediate purposes. The response curve for each signal channel was examined using precision (=t 0.1 %) resistive dummy cells whose values were varied so that the S/H amplifier output signals varied from -9 volts to -0.09 volt (approximately). Over this range the assumption of a linear response with zero intercepts was found to be accurate to within +0.4 for the ac signal conditioning networks. A typical experiment designed to evaluate the ability of the analog instrumentation to resolve the four different frequency components was one in which the total cell current output (using a resistive dummy cell) at a particular frequency was measured, first with only that frequency applied to the cell, then in the presence of one, two, and three additional applied frequencies. This process was repeated for each of the twelve frequencies. The results, shown in Table 11, are divided into three frequency groups. These same frequency groups were ANALYTICAL CHEMISTRY, VOL. 44, NO. 7, JUNE 1972
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Figure 4 and iron systems (c) 3.00 X 10”M Fe(C@&- in 0.50M K&Os
25 “C
Applied: Run 1. Sum of 22.3 Hz, 84.5 Hz, 230.5 Hz, and 845. Hz. Sine waves of 10 mV peak-t*peak amplitude Run 2. Sum of 9.57 Hz, 44.9 Hz, 105.1 Hz, and 439 H z Sine waves of 10 mV peak-to-peak amplitude Measured: Peak phase angle cotangent far faradaic alternating current; average offour replicate measurements e.0-Run 1,Run 2, cadmium system m,O-Run 1,Run 2, chromium cyanide system r,A-Run 1, Run 2, iron oxalate system ~= theoretical response cadmium system __- = theoretical response chromium cyanide system _ _ _ - - _ = theoretical response iron oxalate system
TIME Figure 3. Scope display photos showing potentiostat signals and unfiltered phase-sensitive demodulator output
clheoretical responses calculated using k. and u values given in Table Ill, except for iron oxalate system u value which was taken from Reference (49)(a = 0.86)1
Potentiostat applied potential signal (top) and wlI current signal (bottom) with four applied frequency components. nents, the unfiltered phase-sensitive demodulator shows the Vertical sensitivity = 50 mV/cm relatively “pure” single frequency response illustrated in B. Phase-sensitive demodulator output with 0” (top) and %I9 Figure 38. (bottom) phase angles between reference signal and current Tests of the ability of the combined digital-analog instrusignal at frequency ofinterest ment system t o correctly assess absolute currents and phase angles were implemented using RC dummy cells, a process generally employed when studying chemical systems (see which was performed regularly as a means of detecting degbelow). In each case, each frequency was applied alone, then radation in circuit performance. With four simultaneously with the lowest other frequency in its group, then with the two applied frequencies, such tests indicate that measurement accuracy is at least as good as with typical single frequency lowest and, finally, all four frequencies were applied. As shown in Table 11, in only two cases (the two highest freanalog instrumentation. For example, with purely resistive quencies) is there evidence of interaction between frequencies dummy cells, where greatest faith can be placed on component at a level exceeding 1%, and generally it is much less than this tolerance resistors were employed), an accuracy (jz0.l figure. The crosstalk between the two highest frequency average apparent error of 0.6Zwas observed for both total and in-phase currents in a typical set of 20 measurements. channels resulted from the close physical proximity of the With RC dummy cells employing capacitance values in the tuned amplifiers and the small frequency difference between range 0.50 to 15.0 mF, average errors of 1.5% and 2.7% were them (less than a factor of two). One should recognize that observed for total and in-phase currents, respectively, i n a this total-current experiment represents the worst case. The similar set of measurements. A significant contribution t o the additional frequency selectivity provided by the phase-sensilarger apparent errors in the latter case probably arises from tive demodulators in the in-phase signal channels eliminated all evidence of crosstalk in experiments of the type just described. inaccuracy in the capacitance values which, although caliA more pictorial (but qualitative) demonstration of the tuned brated to + 1 originally, had a several-year period in which amplifier-demodulator combination’s ability to resolve a parto “drift.” ticular component from a four frequency set is illustrated in Measurements with Model Electrochemical Systems. BeFigure 3. Typical potentiostat input (applied potential) and cause no simple RC dummy cell represents an adequate simuoutput (cell current) signals are shown in Figure 3.4. When lation of the faradaic admittance, ultimate conclusions rethe network is tuned to one of the lower frequency compogarding measurement system fidelity must be based on results A.
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Figure 5. Typical cot@
-
DC KITENTIAL, VULTS
Edc results with chromium cyanide system
System: 2.00 X 10-aM Cr(CN)e-Sin 1.00M KCN, 25 “C Applied: sum of 22.3 Hz, 84.5 Hz, 230.5 Hz, and 845 Hz. Sine waves of 10.0 mV peak-to-peak amplitude; incremental dc scan as described in text Data shown: Results at 845 Hz using average of four replicate measurements at each dc potential 0 = data = theory for a = 0.59 and k, = 0.28 cm sec-l (solid curve drawn through these points) 0 = points where data and theory match perfectly
observed with well-characterized “model” electrochemical systems. The above-mentioned “cadmium,” “chromium cyanide,” and “iron oxalate” systems served this purpose in the present investigation. Some typical faradaic admittance phase angle data are shown in Figures 4 and 5 . Results of heterogeneous charge transfer kinetic parameter measurements are given in Table 111. The cot @ - ~ 1 ’ 2data observed with the model systems (Figure 4) agree closely with expectations based on results of single frequency measurements (17, 18, 46-49) with respect to both absolute magnitude and dispersion. If anything, data dispersion may be less than normal. This is particularly evident in the cot@ - E d 0 data shown in Figure 5, if one notes that the ordinate has been expanded considerably relative to normal presentations of such data (14). The average deviation between experimental points and the theoretical cot @ Edc profile in Figure 5 is 0,008 (absolute cot@ units). Finally, the k , and CY values obtained by this application of the noncoherent wave frequency multiplex concept (Table 111) are consistent with most literature reports (17,18,46-49). Significantly, all essential data were recorded and reduced to the three sets of rate parameters given in Table I11 in a time corresponding to a “typical” working day (8-9 hours). From results such as these, we conclude that ac polarography in the non-coherent wave frequency multiplex mode yields faradaic admittance data which are as valid as those obtained by more conventional procedures, at least for the (46) J. E. B. Randles and K. W. Somerton, Trans. Faraday SOC.,48, 937 (1952). (47) T. G. McCord and D. E. Smith, ANAL.CHEM.,41,131 (1969). (48) D. E. Glover and D. E. Smith, ibid., 43, 775 (1971). (49) R. deleeuwe, M. Sluyters-Rehbach, and J. H. Sluyters, Electrochim. Acta, 14, 1183 (1969).
Table 111. Heterogenous Charge Transfer Rate Parameters Obtained by AC Polarography in the Non-Coherent Wave Frequency Multiplex Mode
System k , (cm sec-lp a” 1. Cd*/Cd(Hg) in 1.00M 0.15 i 0.01 0.30 =k 0.03 Na2S04at 25 “C 2. Cr(cN)~~-/cr(CN)e~in 0.27 i 0.02 0.59 =t0.03 1.OOM KCN at 25 “C 3. Fe(C,04)33-/Fe(C,04)a4-in 1.2 =k 0.03 > O . 5b 0.500M K2C204at 25 “C Rate parameters calculated from peak cot9 measurements (14); gives average value from measurements at eight frequencies; uncertainties = average deviations. cot peak too shallow for accurate estimate of peak potential, 0
case of four simultaneously applied frequencies. To ascertain in a preliminary way whether similarly satisfactory results could be realized with a substantially larger number of applied frequencies, all of the twelve available discrete frequencies were simultaneously applied to a cell containing the chromicyanide system. Regardless of which set of four frequencies was monitored by the available signal conditioning circuits, results essentially equivalent to those acquired with four applied frequencies were obtained. A definite increase in data dispersion was observed, but this was readily suppressed by increasing the ensemble size in the averaging routine. Thus, we feel that the non-coherent wave frequency multiplexing concept can be expanded considerably beyond the four frequency “test case” invoked in the present work. A More Appealing Approach to Non-Coherent Wave Frequency Multiplexing. The dependence on extensive analog
equipment which characterizes the investigation reported here should not be considered an essential requirement for impleANALYTICAL CHEMISTRY, VOL. 44,
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mentation of ac polarography in the non-coherent wave frequency multiplex mode. In our situation, the relative slowness of the PDP-8/S computer system (particularly with regard to A/D conversion) did necessitate the type of approach adopted where the major signal conditioning load is assigned to analog circuitry. The latter statement also applies to most situations where an ac polarograph is interfaced to a more capable computer in a low-priority, time-sharing status. However, in a situation where a minicomputer of substantially greater speed than the PDP-8/S (Le., most of the newer systems) is available for dedicated on-line work, a more desirable approach becomes possible whereby most of the signal conditioning procedures are assigned to the digital computer. What is required is an A/D conversion-data storage capability which enables digital data acquisition with sufficient rapidity that the detailed current-time waveform (Figure 3 4 is defined with good resolution by an array of sequential digital data points. This data array can then be analyzed by digital mathematical operations in the computer’s central processor (Le., an appropriate form of Fourier analysis) to obtain the desired individual ac signal characteristics. Such implementation of digital signal conditioning possesses substantial advantages over the analog procedures discussed here and should make the technique under consideration more conveniently applicable. With digital signal conditioning, one avoids the cost and time associated with purchasing and constructing analog signal conditioning networks. Also eliminated is the inconvenience of having to frequently check, trim, and tune the numerous analog circuits because of component drift, particularly in the tuned circuitry-Le., a digital program does not “drift.” Certain types of “tuning” are required with digital signal conditioning operations (50), but this type of procedure can be automated. Because of the considerations given in the previous paragraph, we view the primary role of the work described here as firmly establishing that ac polarography in the frequency multiplex mode will yield faradaic admittance data with accuracy at least equivalent to classical procedures and with much greater speed, As far as the means to achieving this end is concerned (the instrumentation of Figures 1 and 2), it is considered a somewhat primitive temporary expedient, rather than the “last word.” Relationship of Non-Coherent Wave Frequency Multiplexing to Alternative Techniques for Rapid Acquisition of Faradaic (50) Hewlett-Packard, Inc. “Fourier Analyzer Training Manual,”
Application Note 140-0, Hewlett-Packard, Inc., Santa Clara, Calif.
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Admittance Data. In addition to non-coherent wave frequency multiplexing, we identified in the introduction three alternative, and probably more widely recognized, possible approaches to rapid acquisition of faradaic admittance data: (1) admittance analysis in the coherent wave frequency multiplex mode, (2) transient admittance analysis, and (3) noise response admittance analysis. Among the many obvious differences, the major and possibly crucial distinction between non-coherent wave frequency multiplexing and the alternatives arises because of faradaic non-linearity . Namely, noncoherent wave frequency multiplexing is the only member of this group of four techniques which provides the opportunity to “tune out” many effects of faradaic non-linearity so that one can operate with applied signal amplitudes at which faradaic non-linearity manifestations are significant, without undesirable consequences. The alternatives (Techniques 1-3) are all characterized by a test signal whose frequency domain spectrum contains numerous components, including many with frequencies related by integers. Under such conditions, harmonics and modulation components generated by faradaic non-linearity will have frequencies identical to the fundamental harmonic responses of interest so that the former cannot be eliminated by tuned circuitry (e.g.,consider the implications of Table I for the case where 45 = 3f;? = 2f3 = f4). Consequently, with Techniques 1-3, one is forced to operate with substantially smaller amplitudes than necessary with noncoherent wave frequency multiplexing to ensure that faradaic non-linearity effects are negligible. This reduction in allowable test signal amplitude means that it will be necessary to devote a correspondingly larger time to ensemble averaging to reduce to a particular level effects of extraneous noise (noise not contained on input signal). Although these considerations imply a longer experiment time for Techniques 1-3, it is open to question whether the cost will be serious. Correlation techniques (26-31,50) provide a means in addition to ensemble averaging for reducing extraneous noise effects so that the added time investment may be sufficiently small that the advantages of Techniques 1-3 (better frequency resolution, simpler signal generators, etc.) may make one of them the method of choice. We are now making an effort to obtain experimental data relevant to this question and hope to publish on this matter subsequently.
RECEIVED for review November 10,1971. Accepted February 16, 1972. Presented in part at the 21st Meeting of CITCE, Prague, Czechoslovakia, October 1970. Work supported by National Science Foundation Grant GP-16281. B. J. H. was an NIH Graduate Fellow, 1968-70.
