1506
Anal. Chem. 1980, 52, 1506-1511
Double Layer Capacitance Measurements with Digital Synchronous Detection at a Dropping Mercury Electrode Paul F. Seelig and Robert de Levie" Department of Chemistry, Georgetown University, Washington, D.C. 20057
A method is described to perform digital synchronous detection, using a Hadamard transform algorithm. The method has in common with Fourier transform ac polarography the simultaneous application of a number of frequencies, and the convenience of a compact algorithm which requires little space beyond that for the original data array. A drawback of the method is its restriction of the measurement frequencies to powers of two times a base frequency, a restriction which is of little consequence in the present application. The major advantage of the method is that it Is significantly faster than Fourier transform analysis and is therefore more readily applicable in "real-time" ac polarography.
errors, whereas floating-point routines, at least in our minicomputer (Digital Equipment PDP-11/20), require a lengthy computation. Most likely, use of a double wordlength fixed-point algorithm could be used to overcome this difficulty. Here, however, we report on an alternative approach, based on the recent work of Hayakawa et al. ( 1 5 ) , which is considerably faster than the Fourier transform method while retaining the advantages of frequency multiplexing without sacrificing precision. Digital Synchronous Detection. In a lock-in amplifier, a sinusoidal input signal A sin (wet + I$) of amplitude A , angular frequency wo and phase angle I$ is multipled by a square wave of unit amplitude and angular frequency w. Since a square wave can be represented by its Fourier series, we can write the product as
Measurements of the electrode admittance Y or of its inbelong to the standard reperverse, the impedance 2 = Y-', toire of electrochemical techniques, in both homogeneous and interfacial electrochemistry. In the former, it is the method of choice to determine electrolyte conductance; in the latter, it is used to measure rate parameters of electrode reactions as well as double layer capacitance. We will focus here on the latter application, especially in conjunction with a dropping mercury electrode, which is the more demanding one because the measured quantity varies with time. The first significant admittance measurements at dropping mercury electrodes were made using an ac bridge with additional timing circuitry to facilitate bridge balancing at a fixed drop age (1). The first attempts to automation (2, 3) merely recorded the absolute magnitude of the alternating current. Modern automatic ac-polarographic instrumentation started with the work of Jessop, who introduced phase-sensitive detection ( 4 ) and automatic compensation for series (solution) resistance by positive feedback ( 5 ) . Unfortunately, the commercial instrument incorporating these innovations, the Cambridge "Univector", used heavy damping, so that only an average resistance compemation could be achieved. The same limitation applied to a subsequent design (6) using a commercial lock-in amplifier. Alternative approaches were introduced by Smith, who recorded magnitude and phase angle (7)rather than in-phase and quadrature components, and by Hayes and Reilley, who used an analog multiplier as phaseselective detector and elegantly avoided the need for (and noise introduced by) positive feedback in their resistance compensation method (8). Most commercial instruments presently available, however, use the lock-in technique in combination with positive feedback and sampling at the end of drop life, along the lines of designs developed in this laboratory (9, IO). The advantages of frequency multiplexing were realized by Smith et al. (11,12) and further exploited by combination with the Fourier transform method (13, 14). With a hardwired fast Fourier transformation accessory, or with an array processor, data analysis and reduction can then be performed while the next mercury droplet is forming. Even without such additional hardware, sufficiently fast software routines are available. We have found, however, that the common single-precision algorithms often introduce uncomfortably large truncation
{ A sin ( q t + 4)i Esin (an + l)ot = n=02n+ l 1 " 1 -AXcos {(oo- (2n + l ) w ] t + 2 n=02n+ 1
0003-2700/80/0352- 1506$0 1 .OO/O
-
1
-
1
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+1
2 n=02n
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+ (2n + l ) w ] t +
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If the multiplication process is followed by.a low-pass filter which rejects all frequencies but zero (dc), then only when wo = (2n + 1 ) w will we obtain a nonzero output, of magnitude ( A cos @)/2(2n+ 1). (Since filters with zero bandwidth would require an infinitely large averaging time, practical filters strike a compromise between bandwidth and averaging time, especially in ac polarography, where the measured cell admittance varies with time. In that case, the condition relaxes to wo = (2n + l)o,with an uncertainty of the order of the reciprocal of the filter time constant used.) Although lock-in amplification is a well-established analog method, its principle of operation is of a binary nature: I t involves multiplication of successive samples of the signal waveform by +1 and -1; Le., it alternately adds and substracts such samples. As addition and subtraction are among the most efficient of computer operations, such a procedure should be quite amenable to digital implementation. If one wants to use various frequencies simultaneously, in order to benefit from Fellgett's multiplex advantage, one must take into account the fact that Equation 1 exhibits complete frequency selectivity except at odd harmonics. The latter must therefore by excluded, which restricts the excitation frequencies to 2" times the base frequency w, where n = 0, 1, 2, 3 , . . . , N . Although this limits the number of measurement frequencies to about 31,f3per decade, this is usually adequate in ac polarography since the electrode admittance exhibits little fine structure as a function of frequency. One may not even want to use all frequencies but delete one or more from the excitation signals in order to measure (or test for) harmonic distortion. In order to extract information a t a given frequency and phase by digital synchronous detection from 2N input data, a total of 2 x 2" additions and subtractions are required, i.e., 2" at two orthogonal phase angles. For N different fre-
C
1980 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 52, NO. 9, AUGUST 1980
1507
detection of the in-phase and quadrature components a t N - 1 frequencies would require 2(N - 1)2Noperations, whereas the fast Hadamard transform requires N X 2“. Thus, the specific advantage of using the fast Hadamard transform instead of straightforward digital detection is a factor of 2(N - l ) / N , which approaches 2 as N increases.
EXPERIMENTAL
- EX/ST>/VG
L
The same PD-11/20 computer is used as in our earlier Fourier-transform work (16),and a similar substitution method is used to calculate the components of the cell impedance. A programmable clock based on a 20-MHz crystal oscillator controls the precise timing of input and output signals (on alternate clock pulses); synchronization is facilitated by the implementation of an additional computer instruction, “pause”, which halts the computer until the next clock pulse (or other trigger pulse) is received; restarting the program requires less than 50 ns. The corresponding computer modifications are shown in Figures 1 and 2. The use of a precision clock and of the additional computer instruction are convenient though by no means necessary for the implementation of the method. The analog circuitry used is shown in Figure 3; it is similar to that used earlier (16) and includes a low-pass filter to avoid aliasing, a potentiostat, a current amplifier, and a postamplifier. A standard, thermostated polarographic cell is used with facilities for deaeration. A customary, blunt polarographic capillary is used as the dropping mercury electrode, concentrically surrounded by a cylindrical Pt gauze auxiliary electrode. The external reference electrode is connected through a solution bridge. A drop hammer under computer control can dislodge the mercury droplets by causing a slight, momentary horizontal displacement of the capillary tip. As excitation signal we use a “song” of 2N data points, generated in advance of the experiment and stored in memory. The song consists of the algebraic sum of N - 1 sine waves, at a number of cycles of 2% where n = 0, 1,2, 3, . . . , N - 2 , and of selectable amplitude and phase. During measurement, the song is sequentially read into a 12-bit digital-tu-analog converter. The song can be repeated any number of times, and the resulting response added coherently to the preceding one; the song can also be applied to the cell before measurements are made, as a “presong”, in order to allow the transient response upon application of the excitation wave form to dissipate before measurements are taken. The frequencies at which the song is to be applied, and the number of repetitions of song and presong, can be specified from the
YES
SEVEYEG L rVES
..***
-433ED
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Figure 1. Modifications of t h e central processing unit of a PDP 11/20 to incorporate a PAUSE instruction, using the otherwise unused computer instruction 007 quencies, the corresponding number is 2N X 2v. This is twice the number of operations as that required in fast Fourier or Hadamard transform algorithms. In fact, the Hadamard transform is eminently suitable because the Walsh functions include all of the required square waves; see the Appendix. Therefore it is convenient t o implement digital synchronous detection by using a fast Hadamard transform algorithm. I t uses the original data array to store all intermediate and final data, so t h a t its memory requirements are minimal. T h e advantage of using Hadamard over Fourier transformation derives from the higher calculational speed, since no multiplications by sine functions are needed. Even if calculation of the sine functions can be circumvented by the use of a look-up table, one cannot avoid the corresponding multiplications in the fast Fourier transform algorithm. As a consequence of the Nyquist limitation, information on only N - 1 frequencies can be extracted from 2“ data points, viz., a t n = 0, 1, 2, . . . , N - 2 . Digital synchronous ENABLE
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2 7
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Figure 2. Additional circuitry, mounted on a separate board, to implement the PAUSE instruction. Signal labels and specific implementation may vary, but similar modifications of other members of the PDP-11 family may well b e possible since the instruction code 007 appears not to have been used there either
1508
ANALYTICAL CHEMISTRY, VOL. 52, NO. 9, AUGUST 1980
Table 11. Dummy Cell Measurements Testing the Response Linearitya resistance,
capacitance, nF
resistance,
kii
k rl
capacitance, nF
30.02
100.3
30.00
200.5
30.00 30.00 29.99
401.0
29.99 29.98 29.98 29.98 29.98
601.3 701.1 800.1 899.3 1000.2
300.6 500.9
Dummy cell used:
a
1.0 uF: standard used:
Figure 3. The analog circuit used. The "song" signal passes a low-pass fitter (Krohn-Hite3202)and is then added to a dc signal and, if wanted, to a positive feedback signal to compensate for time-independent iR drop such as that of the capillary. Amplifiers 1, 2, and 3 comprise the
potentiostat; the cell current is converted into a voltage by amplifier 4, further amplified by postamplifier 5 in order to utilize the full dynamic range of the analog-to-digital converter. Amplifiers used: (1) AD 52
i
30 k i ? in series with 0.1 through 30 k c l . Data taken at 400 Hz.
-
pF cri?
50-
K; (2,4) AD 234 K ; (3)66 3329/03; (5)AD 40 J
Table I. Dummy Cell Measurements Testing the Frequency Response" frequency,
resistance, k ci
capacitance, nF
23.0
50.03
300.6
50.0 100.0
50.02 50.02
200
50.01
400 800
50.01 50.01
300.7 301.0 300.8 300.9
1600 3201
50.01 50.00
Hz
300.9 305.7 313.8
a Dummy cell used. 50 k i l in series with 300 n F ; standard ~ ~ _ _ used: _ _ _ _3 0- k_n _. ___-__-
computer keyboard, as can the drop age around which the song is to be centered. During the measurement interval, typically between a drop age of 3 to 6 s, the ac polarographic current is sampled by a 14-bit analog-tedigital converter and co-added in double-length (32-bit) words. Sampling of a response point and output of a song point are done on alternate clock pulses, and hence interdigitated. After the data have been acquired for the specified number of songs, the mercury droplet is dislodged and another bias voltage applied to the cell. While the next drop is growing, the data obtained on the preceding drop are analyzed by double wordlength integer fast Hadamard transformation and the resulting response at the frequencies of interest stored on a magnetic disk. For a 512-point transform, this procedure takes a total of 1.6 s, leaving enough time for the next measurement to be performed on the same drop; a 1024-point transform takes 3.3 s. If one wants to cover an even wider range of frequencies, one could use alternate drops for measurements and data processing - a tactic which might also be useful to avoid the depletion effect ( 17 , 18) in cases involving diffusion. RESULTS AND DISCUSSION Dummy cell measurements, see Table I, were made with the series combination of a precision metal film resistor (Sprague type 420 E, 0.1%) and two precision glass capacitors (Corning type CY70 CE154J, nominal value 150 n F each) with the capacitance short-circuited for the reference measurements. T h e resistance R was recovered to within 0.1%, the capacitance to within 0.3% except for the two highest frequencies, where the accuracy in C drops rapidly. Note that, a t 3.2 kHz, the impedance of the 300-nF capacitor is only 166 0, or less than 0.4% of the total cell impedance.
o L - ~ - I - ~ - . L - - . l ~ - 0
-1 3
.----- - ~ > I~
E
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-23
The capacitance of mercury in contact with 0.1 M aqueous KCI, measured at 232.9 Hz. Each point represents the result obtained on a single drop on which, simultaneously, the capacitance was measured at 7 other frequencies, ranging from 29 Hz to 3.7 kHz. Drop area at average drop age: 1.9133 mm2; temperature 25 OC. The resuA of premature drop fall can be seen in two places Figure 4.
A check of the linearity of the response is shown in Table 11, where a 30-kR resistance was used in series with 0.1 to 1.0 FF (in increments of 0.1 hF) from a General Radio 1412 BC decade capacitance box. Errors in R are again within 0.1 YO, those in C within about 0.2%; the manufacturer specifies the capacitance values as accurate to within 0.5%. When the dummy cell impedance is less than 1 kR, significantly larger errors appear, which we believe are caused by load-dependent phase shifts in the potentiostat and current-to-voltage amplifier but which have not yet been remedied. Figure 4 shows measurements made on a dropping mercury electrode in aqueous 0.1 M KC1 which agree with the data of Grahame e t al. (19) to well within their stated accuracy. 9'ince our method samples over a finite time interval, we investigated the magnitude of the resulting sampling error. Theoretical predictions, which apply to fast Fourier transform methods as well, are reported in a companion communication (ZO), and can be summarized as follows: the sampling error in both R and C in a measurement of double layer capacitance plus electrolyte resistance on a dropping mercury electrode should be less than 0.1% as long as the sampling interval does not exceed 20% of the drop age, and less than 1% for sampling over the last 50% of drop life. Table I11 shows experimental results obtained on a dropping mercury electrode in aqueous 10 m M NaF, using different sampling intervals. For a sampling interval of 1.6 s centered around a drop age of 5 s (Le., sampling from 4.2 to 5.8 s in drop life) we indeed find less than 0.1% error in R and C; 1%errors are observed only for sampling from 3.0 to 7.0 s! (Note that the average cell current decreases upon increasing the sampling interval; this corre-
ANALYTICAL CHEMISTRY, VOL. 52, NO. 9, AUGUST 1980
Table 111. Effect of Variation of the Sampling Interval on a Dropping Mercury Electrode“ no. of songs 1 2 0
10
20 30 40 50 60 80 100
120 140 160 180 200
sampling interval, s
resistance,
0.040
3.004 3.004 3.002 3.004 3.004 3.006
capacitance, nF
kn
0.080 0.200 0.400 0.800 1.20 1.60 2.00
319.8 319.8 320.3 320.1 320.1 319.9 319.7 319.3 319.0 318.1 316.9 315.5 313.8 311.8 309.4 306.3
3.006
3.008 3.011
2.10 3.20 4.00 4.80 5.60 6.40 7.20
3.018
3.029 3.041 3.056 3.075 3.095 3.1 26
8.00
Data reported a t 200 Hz. Sampling interval ccntered around d r o p age of 5 s; sampling period in all cases preceded by 10 presongs. a
_
_
_
_
.
~
~
_
_
sponds with a larger value for R but a smaller one for C.) The present approach is a rather conservative one: we have adopted the precaution of filtering out the sudden transitions between successive points of the computer-generated song, by analogy with our earlier Fourier transform work (16),even though we have observed no need for such filtering over and above that caused by the small feedback capacitors used to ensure unconditional stability of operational amplifiers. We have used 10 presongs even though the results appear to be no different with only one presong; we have not used iR compensation although preliminary experiments indicate that at least partial iR compensation can be used without penalty. Our technique is a substitution method and therefore compensates automatically for amplitude and phase errors in the circuit insofar as these are independent of the cell impedance. By processing the data obtained on one drop before the next set of data is collected, we reduce the need for data storage: only the response at the frequencies of interest is stored, instead of a large array of intermediate data. The only operation done afterward is the normalization of the response using a standard impedance; the time involved here is rather trivial, and it allows us t o make the standard measurement afterward, so that there is no need to anticipate the amplifier settings to be used in the electrochemical experiment. With our present equipment, the highest frequency at which data can be taken is about 20 kHz, so that useful impedance data can be extracted only up to 5 kHz. The lower frequency limit is set by drop time and sampling error: for a 5-s drop the maximum sampling and a sampling error of a t most 0.170, period is about 1 s so that 1 Hz would be the lowest practical frequency. On a stationary impedance, as often encountered with solid electrodes, the low-frequency limit will be determined by the maximum experiment time one is willing to invest, since a t least one complete cycle of the signal must be sampled. We have experimented with various songs, especially by increasing the relative signal amplitude a t higher frequencies in order to counteract the decreasing impedance of the double layer capacitance a t higher frequencies. Such an approach has so far resulted in rather minor further improvements of the high-frequency response. Comparison with Existing Methods. There are three major alternative methods with which we will compare the present technique, viz., the use of an impedance bridge, of a lock-in amplifier, and of fast Fourier transformation. Of these.
_
1509
the bridge method is the oldest and is still widely used. In principle, it should be the most precise since it uses a null method and should have the accuiracy of its impedance standards. Measurement.s on dummy cells will often be of impressive precision and accuracy i:ndeed but, for reasons which largely lie outside the bridge itself, it is questionable whether a comparable accuracy can be obtained with a dropping mercury electrode. In the first place, the measured components of the cell impedance, especially the capacitance, show a variation with frequency which is due t o the cell geometry used, and for which only approximate corrections exist. In particular, a solution film between the mercury thread and the glass walls of the capillary (21-25:1and, to a lesser extent, shielding of the top part of the drop by that capillary (26-28) are believed to be the major causes of the frequency dispersion, which typically makes the measured capacitance vary by about 1%if measured over a sufficiently wide range of frequencies. Also, conversion of measured interfacial capacitance to capacitance per unit area usually involyes the tacit assumption that the electrode area at fixed drop age is constant, Le., independent of applied potential. Thii; is not the case, because the mercury flow rate depends on the potential-dependent interfacial tension, through the so-called back-pressure (29, 30). Neglect to correct for this effect can result in an inaccuracy of several tenths of a percenl.. Finally, balancing of a bridge when the measured impedance varies with time almost always involves a tuned amplifier, and the resulting signal delay causes a systematic error. There may well be some fortuitous cancellation of errors here: the resulting signal delay depends on filter sharpness (“Q”) and frequency and usually decreases with increasing frequency; at lower frequencies, the measured capacitance actually pertains to an earlier moment in drop life, Le., to a smaller elect,rode surface. On the other hand, the earlier-mentioned frequency dispersion has an opposite effect: at higher frequencies a sinusoidal signal penetrates less deeply in less accessible lxuts of the cell, such as the solution film inside the capillary, hence the effective electrode area is larger a t lower frequencies. These effects are of consequence since it is not the capacitance per se, but the capacitance per unit electrode area, which is important. In view of the above, we estimate that the accuracy of bridge measurements on a dropping mercury electrode is seldom better than fl 7’0.Consequently, the exquisite bridge precision is not really relevant, and direct-reading methods of somewhat lower inherent precision need not yield results of less accuracy. Bridge measurements are tedious and, therefore, ususally discourage the user from making more than the minimum number of measurements. Especial.ly, effects of frequency dispersion will often go unnoticed s,ince measurements are seldom repeated at a number of widely separated frequencies. The present method is a t least an order of magnitude faster than bridge measurements a t a single frequency and two orders of magnitude faster if a similar frequency range were used, since we obtain the cell impedance at a number of frequencies during the life of a single drop of, typically, 4 to 8 s. Also, the well-defined sampling interval allows accurate estimation of the sampling error (20), so that the frequency dispersion inherent in the use of a dropping mercury electrode can be studied accurately. We will return to this more specialized aspect in a separate communication. The advantages of the present method of digital synchronous detection over its analog coun’terpart, lock-in amplification, lie mostly in measurement :speed and convenience, especially when data are to be obtained a t multiple frequencies. A similar result can, of course, be achieved by using multiple lock-in amplifiers, although isuch analog multiplexing ( 1 1 , 12) becomes rather awkward when more than a few frequencies are involved. The shorter sampling interval (of only
1510
ANALYTICAL CHEMISTRY, VOL. 52, NO. 9, AUGUST 1980
Table IV. Example of Eight-Point Hadamard Transform' a'=a+ b b'=a- b e'=c + d d'=c-d e'=e+ f f'=e-f g'=g+ h h'=g- h
b + c + d i e + f i g + h b + c -- d + e f + g - h c -- e + g " = a + b - c - d i e i f - g - h d"' = d " + h " = a - b - c + d + e - - f - g + h e"' -a "-e"=a+ b + c + d - e - f - g - h f ' " = b" - f " = a - b i c - d - e + f - g + h e''-g" = a + b - c - d - e - f + g + h = d " - h" a --- b - c + d - e + f + g - h a" b'" = b
ie " =
~
I '
+ f"
a
=a
+
-
~
I o
g,'f::=
a By successive additions and subtractions, 8 independent, linear combinations of the 8 data ( a through h ) are formed. For 2 3 data, 3 x 8 such operations are required; for 2" data, the number of operations is N X 2".
one period of the lowest measurement frequency used, see Table 111) is also an advantage of the present method. On the negative side, we note the obvious increase in instrument cost and complexity and the limitation to measurements a t rather low frequencies imposed by the time required to digitize signals to sufficient precision; this limitation can be circumvented with specially-constructed excitation sources (31)but only a t the expense of increased instrument complexity. The relative advantages of the present method over the use of lock-in amplifiers are most obvious when comparison is made with those instruments in which lock-in amplifiers feed into dedicated computers. Such a measurement approach seems to be popular in electrochemistry (32,33) and in other fields of physicochemical instrumentation, such as spectroscopy, because of the increased availability of dedicated computers, and of anticipated gains in data precision and ease of analysis. In this case, we can add to the list of advantages of our digital approach, apart from frequency multiplexing and the use of shorter sampling intervals, the instrumental simplification, since the lock-in amplifier can often merely be deleted. Finally, comparison with the fast Fourier transform method ( I 3, 14) shows advantages and disadvantages, both relatively minor: the speed of the present method is higher, but it is more restrictive in the choice of measurement frequencies which can be multiplexed. (In fact, it uses exactly those frequency combinations which Smith had carefully avoided in his Fourier transform work. For measurements of the double layer capacitance, which is only weakly dependent on potential and hence does not generate strong harmonics, this is of little consequence; it may be more important for redox and, especially, desad phenomena.) The fast Hadamard transform algorithm provides a fast and convenient method of digital synchronous detection, but it may not always be the method of choice. For example, if one measures the impedance of one or more batteries (e.g., to monitor state-of-charge) a t low frequencies, there may be ample time between data points to perform the straightforward digital synchronous rectification. The advantage here is that the result is ready as soon as the last measurement has been taken and processed. Although the total analysis time may be larger, t h e analysis can be interspersed with data taking whereas the fast Hadamard transform requires a complete data set before analysis. Also, even though the memory needed for the fast Hadamard transform is small since it is a n in-place algorithm (i.e., i t stores intermediate and final results over original data, without requiring more space), straightforward digital synchronous detection needs even less memory: it requires one register each for the in-phase and quadrature response a t a given frequency, because it does not store individual data (analogous to the way most pocket calculators handle a linear regression). Simple digital synchronous detection may also be all that is needed when the perturbation contains only one frequency, as in hydrodynamic voltammetry (34)or in optical experiments using chopped light beams; in that case, the present method may still have an advantage over lock-in amplification because of its shorter
dc
a + b t c +d+e+f+g+h
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Figure 5. The 8 simultaneous equations in a through h of Table I V can be regarded as the original data modulated by switching functions which assume values of 1 arid -1 only. Among those Walsh switching functions, shown on the right, are the very waveshapes used in synchronous detection. They are denoted here by their frequency and by their phase relationship ( i = in-phase, q = quadrature). Some of them are not square waves and are not used here ( * )
+
sampling interval, which allows its use for time-dependent phenomena with time c0nstant.s intermediate between that of the measurement frequency and that of the low-pass output filter required in analog synchronous detection.
APPENDIX In this appendix we will summarize, for the benefit of readers unfamiliar with the Hadamard transform, its most relevant aspects. The Hadamard transform is based on the fact that two independent linear combinations of two numbers contain t h e same information as the original numbers. Therefore, as long as the protocol to form the independent linear combinations is agreed upon, there is a unique relation between the original numbers, say a and b, and their linear combinations, a ' m d b ', such that a'and b'are unambiguously defined by a and b, and vice versa. The independent linear combinations involved in the Hadamard transform are simply the sum and difference, i.e., a'=
Q
+b
b'=a-b and the inverse operation then is a = l / ( a ' + b?
b = y 2 ( a ' - b') In Table IV we show the Hadamard transform of 23 = 8 data, a, b, c, . , ., h. In the first set of operations, we combine a and b t o form a'= Q + b, b ' = a - b, and do the same for c and d , e and f , and g and h. In the second set, a'and c'are combined, as are b ' a n d d', etc., leading to two sets of four linear combinations, one of a through d , the other of e through h. In the third operation, 8 independent linear combinations of all eight input data are obtained; these form the Hadamard transform of the original data set. In Figure 5 we depict the operations which led from the original data to their transform. Except for the first one, which represents the simple addition of all data a through h, the
Anal. Chem. 1980, 52, 1511-1515
operations can be looked at as functions switching between +1 and -1. Several of these Walsh functions (35) are square waves, and these have been indicated with their respective frequency as w , Zw, etc. We note that these are the very switching functions used in synchronous detection; in fact, both the in-phase and quadrature switching functions are present, as indicated by the subscripts i and q respectively. Some Walsh functions, identified by an asterisk, are not square waves and are not useful to our purposes. Clearly, application of the Hadamard transform accomplishes the task of synchronous detection in a digital way, without the need for physical switches as in a lock-in amplifier. A number of simple “fast” Hadamard transform programs are available in the literature (36-38).
ACKNOWLEDGMENT T h e authors are grateful to Tetsuya Osaka for calling our attention to the work of Hayakawa et al., to W. C. Craig for building the high-speed clock and for outfitting our computer with a pause instruction, and to M. Krishnan and C. Chang for technical assistance.
LITERATURE CITED (1) Grahame, D. C. J . Am. Chem. Soc. 1941, 6 3 , 1207. (2) MacAleavy. C. Belgian Patent 443003, 1941. (3) Breyer, B.; Gutmann, F. Trans. faraday Soc. 1946, 42, 645. (4) Jessop, G. British Patent 640 768, 1950. (5) Jessop, G. British Patent 776 543, 1957. (6) Evilia, R. F.; Diefenderfer, A . J. Anal. Chem. 1967, 3 9 , 1885. (7) Smith, D. E. Anal. Chem. 1963, 3 5 , 1811. (8) Hayes, S.W.; Reilley. C. N. Anal. Chem. 1965, 3 7 , 1322. (9) de Levie, R.; Husovsky, A. A. J . Electroanal. Chem. 1969, 20, 181. (10) de Levie, R.; Kreuser, J. C. J , Nectroanal. Chem. 1969, 2 7 , 221. (11) Glover, D. E.; Smith, D. E. Anal. Chem. 1972, 44, 1140.
1511
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RECEIVED for review March 7, 1980. Accepted May 5 , 1980. Work supported by the Air Force Office of Scientific Research under grant AFOSR 76-3027.
Gel Permeation Chromatography of Coal-Derived Products with On-Line Infrared Detection R. S. Brown, D. W. Hausler, and
L. T.
Taylor‘
Depatfment of Chemistry, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 2406 1
The compiextty of solvent refined coal (SRC) enables one to gain only negligible structural and chemical information with conventional liquid chromatographic detectors. Infrared spectrometric detection has the potential to readily provide a great deal of information regarding the presence of various chemical functionalities. The high selectivity of liquid chromatographic infrared detection with a conventional single beam spectrometer has been demonstrated in the gel permeation chromatographic separation of a seven-component synthetic mixture. Application of infrared detection to the size separation of hexane sdubie SRC and various fractions of SRC which had previously been separated on a silica column was achieved employing conventional infrared spectrometric detection. Numerous characteristic infrared absorptions were observed in these samples and tentative functionality assignments have been made for the coal-derived fractlons.
Solvent refined coal (SRC) is one of a number of synfuel processes which would allow our coal resources to be better utilized (cleaner power generation, petrochemicals, etc.). T o better understand coal conversion processes and to maximize their utility, analytical techniques are needed to more fully characterize products such as SRC as well at its predecessor coals. In previous work ( I , 2 ) , we have developed with refractive index detection preparative liquid chromatographic (LC) techniques which separate SRC into rough molecular 0003-2700/80/0352-151 l$Ol.OO/O
“sized” fractions (GPC). Analysis of these isolated gel permeation chromatography (GPC) fractions (3)employing I3C and ‘H nuclear magnetic resonance (NMR) resulted in negligible structural and chemical information; yet such measurements provided a clue as to the complexity of GPC fractions to SRC. Liquid chromatographic detection via refractive index and ultraviolet spectrometry of course provides even less information regardii3g speciation. Combination approaches involving SRC such as separation via s u e followed by functionality and vice versa have been met with similar results ( 4 ) . A much finer characterization of the total SRC product is desirable. Two options present themselves. T h e chromatography can be improved and optimized for a better separation, or a more selective detector system can be employed which responds to only specific functionalities. Since heteroatom content and heteroatom distribution in SRC products are important measures of SRC character, a n inexpensive highly selective detector which is especially sensitive to various heteroatom functionalities appeared to be the better option. A logical choice in this regard is a variable wavelength infrared detector system since many of the heteroatom containing functionalities (e.g., phenols, ethers, acids, aldehydes, etc.) are infrared (IR) active. Infrared spectrometry is also a well proven method for both qualitative and quantitative organic analysis. A limited number of scattered reports employing conventional single-beam or double-beam IR spectrometry as an LC 1980 American Chemical Society
