Electrochemical Relaxation Techniques W. H. Reinmuth, Department of Chemistry, Columbia University, N e w York, N . Y. 70027
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past 20 years a wide variety of electroanalytical techniques have been newly developed and a number of previously conceived techniques have received r2newed attention as a result of electronic advances which facilitated their practical implementation. The purpose of the present review, rather than considering only the advances of the past tv.oyears, will be t o discuss the relative merits of a selection of these methods. No attempt has been made at completely surveying the extensive literature, and discussion is largely confined t o whttt may be termed relaxation techniques. Relaxation methods in general are those in which a perturbing influence, usually in the form of a controlled current or potential, is tipplied to a cell initially a t equilibrium, and the transient relaxation of the system in response to this disturbance (a c u e n t or voltage) is observed. The distinction between small and large amplitude techniques arises naturally on theoretical grounds. I n the mathematical treatment of the expected responses, 1 he current and concentrations of electroactive species are normally related through linear algebraic or differential equations, but the relation between concentration and potential (the Nernst equation) or current and potential (absolute rate expression) is an exponential one [cf. (66) for a recent revi1.w of theoretical methods]. When the excursions of potential from initial equilibrium are no more than a few millivolts, the exponentials can be expanded t o produce linear equations. Thus, small amplitudl: techniques can be made to yield useful theoretical predictions even for relatively complicated postulated read ion mechanisms. I n contrast, the large amplitude techniques, those in which potential excursions are too large to allow expansion of the exponentials with retention of only a few terms, tend t o yield intractable theories for all but the simplest cases.
SMALL AMPI.ITUDE RELAXATION TECHNIQUES
The present discussion is limited to qualitative aspects of these techniques. The reader desiring more quantitative and extensive treatment is urged to
consult the recent review by Delahay ($5).
Although the perturbance from equilibrium in a relaxation technique could in principle be nonelectrical and in fact supersonic waves (84) and ultraviolet and visible radiation (11 ) have been employed, most workers have used electrical disturbances in the form of controlled current or voltage waveforms. One of the simplest of the techniques is the voltostatic one studied in detail by Vielstich and Delahay (81). I n this case the perturbing influence is a step change in voltage applied t o the cell, and the current is observed, normally oscilloscopically. The technique is useful for studying the kinetics of chargetransfer reactions, but only relatively slow ones. The reason is that the step-change in voltage does not appear instantaneously across the solution-electrode interface but rather a t a rate limited by the R-C time constant associated with the charging of the capacitance of the electrical double layer through the ohmic resistance of the electrolytic cell. Because the capacitance is very large [about lpf for a dropping mercury electrode (DME) 1, even cell resistances of a few ohms produce relatively long time constants. The logical improvement of this technique, at the cost of increased experimental complexity, is the potentiostatic technique pioneered by Gerischer and Vielstich (41, 82) in which the potential of the electrode of interest is monitored by a reference electrode with Luggin capillary and controlled by an electronic feedback loop. The difficulties of this technique a t very short times again lie in the charging of the electrical double-layer, for changing the electrode potential in very short time requires a large current pulse through the ohmic cell resistance. As the time of interest becomes shorter and shorter, the current, voltage, and band pass requirements of the potentiostat become more and more extreme and at present the shortest useful times for such devices appear to be a few microseconds. A recent paper by Booman and Holbrook discusses the design of fast potentiostats in detail (13). The converse technique in which a current step rather than voltage or potential step is applied and potential is measured-the galvanostatic technique (12)-ultimately founders also on diffi-
culties associated with the double layer. Initially, the major fraction of the applied current is used for nonfaradaic charging and, much as in the case of the voltostatic technique, the important waveform (cell potential in the one case and faradaic current in the other) is rounded off, even though the applied wave form be an ideal step-function. Gerischer and Krause's ingenious method of circumventing the practical limitations imposed by the double layer on the galvanostatic and potentiostatic techniques is the double pulse galvanostatic method (40). The first current pulse contains sufficient charge to drive the electrode a few millivolts from equilibrium. During this pulse the potential changes more or less linearly as the double layer is charged. The current is then cut back to a lower value. If the decrease in current is too large, the potential drifts back toward the initial (undisturbed) value; if too small, the potential displacement becomes larger. However, if the level is properly adjusted the potential remains constant for a short period a t its value at the end of the first pulse. The simplest view is that the first pulse simply charges the double layer. The more detailed theoretical treatment by Matsuda, Oka, and Delahay (54) indicates that faradaic contribution during the first pulse is important for fast reactions. Although the technique was developed primarily for the purpose of extending the lower limit of time of observation beyond that available with the simple galvanostatic technique-and it does so by an order of magnitude-another of its important advantages lies in the fact that it is a null method. I n contrast to the simpler methods i t is not necessary to make accurate measurements of the relaxation wave-form, but the oscilloscope is used as a null device to determine when the ratio of the currents during first and second pulses has been properly adjusted. According t o Delahay (25) the fastest available times with this technique (1 p sec.) are limited by the effect of stray inductance in the electrolytic cell and its associated circuitry (producing spurious transients a t the end of the first pulse) and the effect of ohmic drop in the cell on the voltage detecting oscilloscope amplifier. Although the ohmic drop is constant and therefore has no diI VOL. 36, NO. 5, APRIL 1964
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rect effect on the null point, when it becomes large as compared with the few millivolt waveform of the relaxation, the detector amplifier can become saturated. The newest addition to the family of transient relaxation techniques is the coulostatic impulse method (11, $4, 36, 19, 65, 67) conceived originally by Barker (7, 31), In this technique a charge-impulse of duration as short as 0.1 psec. is applied to the cell t o charge the double layer so that the electrode potential jumps a few millivolts. This charge then “leaks off” through the faradaic impedance and the potential decays toward its initial value. The impulse can be delivered by charging or discharging a small condenser through the cell so that the circuitry in the simplest case consists of no more than battery, switch, and condenser, and of course, a potential detector, normally an oscilloscope. Variants of the technique employing double impulses are also possible in analogy to the double pulse galvanostatic technique -\ second impulse of half the size of the initial one and opposite sign is applied to the cell a t a later and adjustable time. TWOforms of null are possible, one in which the electrode potential is brought back to its initial undisturbed value by the second impulse, the second in which the slope of the potential-time curve after the second impulse is zero. For relasations limited entirely by charge-transfer kinetics the delay times for slope and potential null coincide. As the masstransfer process becomes more important the slope null time becomes shorter and shorter and no slope null can be observed for a purely diffusion controlled reaction. I n addition to the advantages of accuracy normally associated with null techniques this method provides some indication of the important contributors to the relaxation process, I t also carries an additional benefit in convenience of esperimental application. The null condition depends not on the absolute magnitudes of the impulses but only on their ratio. Thus a train of double impulses with suitable rest period between can be applied to a D M E and, although the magnitude of the relaxation pattern diminishes slowly as the drop expands, it is not difficult to adjust delay times manually t o achieve balance while observing the repeated relaxation on an oscilloscope screen. This eliminates one of the major sources of inconvenience of the transient relasation techniques in general, namely the necessity of photographically recording the fast transient relaxations and then analyzing the data. This is particularly a problem with the double pulse galvanostatic method in which several such records may need to be made be2 12 R
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fore proper balancing of the pulse sizes in achieved. The coulostatic technique shares with the galvanostatic techniques the advantage of simpler instrumentation than that required for the potentiostatic technique. I n its more complicated variants it also shares the advantage of null detection of the galvanostatic technique. Because no net current flows during the relaxation process, no ohmic corrections need be made to the observed potentials nor is there danger of detector amplifier saturation during the relasation. The lower limit of observable times by this technique allpears to be a few tenths of a microsecond (39). This limit is due to difficulties associated with the detector amplifier. During the coulombic impulse i t receives a pulse which may be as large as several volts due to the passage of the impulse through the ohmic cell resistance. The time required for the amplifier to recover from this transient is presently the limiting factor. Incidentally, although presently limited by other factors, the potentiostatic technique would become limited by this difficulty, if available potentiostats functioned sufficiently rapidly, because the same coulombic transient occurs initially in both caseb. Theoretically all of these relaxation methods would be expected to yield equivalent results when applied to simple systems. Consequently, the choice between them can be made on practical grounds. As between the coulostatic and galvanostatic techniques, the advantages appear to lie all on the side of the former. The major factors in which they differ significantly are that the coulostatic technique appears to be useful a t shorter times and does not suffer from complications due t o net current flow during relaxation. As between the coulostatic and potentiostatic techniques, again the advantage of shorter working times, and in this case of simpler instrumentation as well, favor the coulostatic technique. However, it is possible t o foresee circumstances in which the potentiostatic technique would prove advantageous nonetheless. For example, in a mechanism involving independent parallel relaxation schemes of differing time constants, with the potentiostatic technique the observed relaxation is simply the sum of the two. In coulostatic technique the two schemes are in competition for the limited amount of charge supplied in the coulostatic impulse and the observed relaxation is a much more complicated function. The same complication occurs in the galvanostatic case. Thus, certainly in some complicated reaction schemes potentiostatic data would be expected t o be easier to interpret. Another advantage in principle is that the interpretation of po-
tentiostatic results (insofar as the experiment is truly potentiostatic) does not depend directly on knowledge of the double layer capacitance as it does in the galvanostatic and coulostatic cases. One of the difficulties of the small amplitude relaxation techniques is that the relaxation curves are relatively featureless monotonic curves which differ very little in form whatever the mechanism of relaxation may be. The unfortunate consequence is that whether the theoretical model chosen to describe a particular experimental system is the correct one or not its application to experimental data is likely to yield not implausible results. Because the potentiostatic and coulostatic modes of relaxation do differ fundamentally it might be hoped that application of both techniques to the same system would produce divergent results for kinetic parameters if the theoretical model nere incorrect. The same possibility is not t o be hoped for in comparison of the coulostatic and galvanostatic techniques; for almost any relaxation scheme, the relaxation curve for the coulostatic technique is simply the derivative of that for the galvanostatic technique so that both curves are affected in substantially the same way by any anomaly save artifact, Competitive with the transient relaxation techniques are those employing repetitive perturbations. Sinusoidal and square wave potentials and currents have been employed as well as alternating coulostatic impulses. For a complete study of the kinetics of reaction of a couple it is desirable to have relaxation information a t a variety of equilibrium potentials-i.e., ratios of oxidized to reduced forms-and at a variety of times for any given potential. Transient relaxation techniques are of advantage in the latter aspect, for relaxation as a function of time is automatically obtained in a single transient experiment. To obtain similar information with repetitive techniques requires observation of response as a function of frequency. On the other hand, the repetitive techniques are, in general, amenable to modifications in which response as a function of direct current potential is automatically recordede.g., alternating current polarographywhile gaining comparable data by transient methods requires point by point investigation. Thus in the matter of experimental convenience, the transient and repetitive techniques are somewhat complementary for kinetic studies. For analytical investigations, the time dependence of response is relatively unimportant as compared with the d.c. potential dependence. Consequently only repetitive waveform techniques have been employed, nor is it likely that even in special circumstancps
application of transient techniques will prove a t tractive. The most accurate of the simple relaxation methods is the faiadaic impedance method, in which the impedance of the cell t o a small alternating potential is determined using con1 entional impedance bridge methods. Saturally the balancing procedure is a relatively tedious one, and Smith (77) has recently described apparatus for the automatic recording of amplitudes and phase angles of alternating currents which appears to alleviate som: of this tedium at low frequencies at some loss in accuracy. The major difficulty with the faradaic impedance method stems from the necessity of making correction for the nonfaradaic current through the electrical double layer. At frquencies much above 100 kc. the signal (faradaic current) to noise (nonfwadaic current) ratio becomes so low that the method becomes exceedingly difficult t o apply, though it has been used even beyond 1 mc. (63). Even below this range, the analysis of data requires that the double layer capacitance be known accurately. I n most published work using this method i t has been assumed that this capacitance remains the same in the presence and absence of the electroactive couple. This assumption is normally justified simply by its expediency. Recently, however, Sluyters and coworkers (64, 7f-73) have developed a method of treating data from impedance measurements which allows deduction of the double layer capacitance from the data itself. Their analysis of thallium reduction i? chloride and nitrate media (74) indicates that anomalies previously ascribed to these systems are removed when proper account is taken of the change in double layer capacitance on addition of thallium. This type of treatmen; removes one of the major disadvantages of this technique as composed with the competitive transient relaxation niethods. I n the potentiostatic technique the double layer capacitance does not enter the simple treatment; in the coulostatic and galvanostatic techniques, although the capacitance must be known, it can be obtained directly by analysis of the relaxation data. A number of variants of alternating current polarography (15) have been proposed for circumventing the necessity of accurate knowledge of double layer capacitance. The two most practical of these appear t o be phase-selective alternating current polarography and square wave alternating current polarography. I n both cases the capacitive current through the double layer is rejected before readout-in the square wave method by gating the readout t o reject the current immediately after each potential step in the square wave,
in the phase selective method by observing only the component of current in phase with the voltage and thus rejecting the capacitive component (90" out of phase). Both of these methods require accurate control of the potential waveform at the electrode solution interface. Practically, this means t h a t they suffer the same difficulties a t high frequencies as the transient potentiostatic techniques and require the same type of feedback circuitry except at relatively low frequencies. Published work with them has been largely limited t o low frequency quantitative analytical applications (8,15). It is tempting to suggest, because the double layer troubles which limit the applicability of the repetitive techniques at high frequencies seem unlikely to be overcome, that these techniques will become less important in the future. In fact, quite the opposite appears t o be the case. The application of the faster techniques is bringing t o light a great number of anomalous results due presumably t o reaction mechanisms more complicated than those assumed in the simplest theoretical treatments. I n cases of complicated mechanisms for which several kinetic parameters must be derived, the accuracy of the impedance technique makes it the method of choice. The recent book by Breyer and Bauer on alternating current polarography (16) attests to the fact that analytical work can be performed with this and other repetitive relaxation methods. Inherently, however, these techniques are a t a disadvantage by comparison with classical polarography and its variants. Because the response depends on relaxation t o a small perturbation from equilibrium, systems which are irreversible in direct current polarography normally show negligible or much attenuated responses in alternating current polarography. Even for systems which do show responses, their magnitude depends on the kinetics of the charge-transfer process. Thus, accuracy in analytical work is much more susceptible t o influence bv uncontrolled factors. Surfactants in solution, seemingly minor changes in the treatment of solid electrodes, or changes in supporting electrolyte can materially affect alternating currents in cases in which the limiting current in direct current polarography would be unaffected. The advantages of alternating current measurements most often cited are higher resolution, due to the fact that the polarogram is roughly the derivative in form of the direct current polarogram, and higher signal-to-noise ratio leading to higher sensitivity. The first of these advantages accrues merely from the form of readout; the same resolution can be achieved in direct current polarography by recording the derivative of the
normal direct current wave. I n fact, the direct current gives higher resolution in many cases because the alternating current waves are often kinetically broadened. Although derivative readout of direct current polarograms has been employed for many years, simple reliable circuitry has only recently become available with the introduction of operational amplifiers (60, 68). I n cases in which the reaction of a minor component follows that of a major one, alternating current can sometimes help. Even in this case, however, the benefits are often illusory. Even though oxygen, for example, produces an alternating current wave of negligible height, its reduction can change the p H at the electrode and produce peroxy or hydroxy complexes which alter the characteristics of the waves of subsequently reduced metal ions. The second advantage is a real one only when phase selective or gated square wave polarography is employed. Without these expedients, the capacitive current limits the sensitivity of alternating current and direct current polarography t o comparable levels. The systems for which alternating current methods seem t o be most advantageous are those containing surfactants. The so-called tensammetric waves [see (16, 35) for arguments about terminology], due to the desorption of these materials a t potentials sufficiently anodic and cathodic to the point of zero charge where their place in the double layer is taken by ions, can be used for analytical purposes. Because the kinetics of adsorptiondesorption processes are very rapid, these studies are not plagued by the kinetic factors which interfere in redox systems. On the other hand, the responses are relatively complicated functions of concentrations of surfactants in solution. The potentials of desorption depend on concentration in roughly logarithmic fashion and the form of the desorption peak depends on the structure of the double layer a t the potential of desorption both in the presence and in the absence of surfactant, so t h a t analysis of data is reduced t o a fairly empirical operation. I n addition, presence of two or more surfactants often leads apparently t o synergistic effects in the adsorption process so that potential pitfalls are many. Xonetheless, the method appears to be competitive with nonelectrochemical methods which are just as empirical. The limitations in applicability of most of the relaxation techniques t o kinetic studies a t high frequencies and analytical studies at low concentrations arise directly or indirectly because of the double layer capacitance and its relatively low impedance. A popular recent approach t o the circumvention of this problem lies in what might be VOL. 36, NO. 5 , APRIL 1964
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termed second order relaxation techniques. I n the theory of such techniques the simplifying expansion of the exponentials of potential occurring in the Nernst and absolute rate theory expressions is carried out to square terms. Practically, this means taking account of nonlinearity of the faradaic process. The second-order terms account for three familiar forms of distortion of electrical signals, namely: rectification, production of direct current voltages on application of alternating currents and vice versa; second harmonic distortion, production of second harmonic currents on application of pure fundamental frequency voltages or vice versa; intermodulation distortion, production of sum and difference frequency components when signals of two or more frequencies are applied simultaneously. The virtues of techniques based on faradaic nonlinearity lie in the fact that the double layer capacitance is much less nonlinear than the faradaic impedance and consequently interferes much less directly than in linear techniques. Just as the first-order relaxation techniques give responses a t sufficiently low frequencies which are in form derivatives of direct current polarograms, so the second-order techniques give responses which are in form second derivatives. Thus, for a diffusion controlled process, the faradaic rectification current rises t o a maximum a t a potential part way along the direct current polarogram, passes through zero a t the halfwave potential (the point of inflection), reaches a minimum further along and returns to zero a t potentials far removed on either side of the half-wave potential. At very high frequencies where the kinetics of charge-transfer become important, either the positive or the negative peak grows a t the expense of the other depending on whether the charge-transfer coefficient is less than or greater than one half. I n the high frequency limit the curve is a simple derivative of the direct current polarogram with the peak at the half-wave potential. Qualitatively the forms of second harmonic and intermQdulation polarograms show behavior similar t o that of the rectification current described above, in fact, as the two frequencies modulated in the intermodulation technique approach one another the behavior of the difference-frequency component approaches that of the faradaic rectification component and the sum-frequency component approaches that of the second harmonic. One can consider the results when potential is controlled and current measured as being composed of three factors: one related to the amplitude of the component of potential a t the fundamental frequency, a second related to the nonlinearity of the faradaic process 21 4 R
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at the fundamental frequency, a third related to the faradaic admittance a t the readout frequency. I t is the second of these factors and its variation with frequency and direct current potential which is of primary interest in kinetic studies. It is convenient in order to have close control of the first factor to work a t very low frequencies where the electrode impedance is primarily determined by the faradaic impedance or a t very high frequencies where i t is determined primarily by the double layer capacitance. The presence of the third factor renders the results from second harmonic polarography more difficult to interpret than those of the rectification and intermodulation techniques and thus makes it less attractive. The problem is that whereas in the rectification or intermodulation techniques the readout frequency can be a very low one a t which the faradaic impedance is free of kinetic complication, in the second harmonic technique it is of necessity higher so that not only the second factor but the third as well contains kinetic factors. In the matter of experimental simplicity as well, the second harmonic technique suffers, for the detector must be of narrow and variable band pass to reject the much larger first harmonic and allow studies a t a variety of frequencies; whereas, in the intermodulation technique the carrier frequencies can be varied over a wide range while their difference is held fixed. A single frequency of detection serves, and the band pass can be much wider because the carrier frequencies are far removed from the readout frequency. As with the linear relaxation techniques, both transient and repetitive approaches are possible. I n principle, the repetitive techniques are to be preferred because the basic advantage of transient techniques in linear conditions, that of allowing the observation of relaxation over an extended time range in a single experiment, is missing in the second order case since the characteristic time associated with the technique is not the observation time but the reciprocal of the fundamental frequency. Thus, repetitive techniques are the ones of choice in kinetic studies. It is not practical to employ the rectification techniques except as transient ones. For example, the application of a superimposed alternating current potential on the conventional direct current polarographic scan does lead t o a rectification current, but this is superimposed on the normal polarographic current [producing a Fournier polarogram (%$)I in which relatively large error would be made in attempting to separate the normal direct current and faradaic rectification components. Thus, the intermodulation method appears to be the one of choice for kinetic studies.
It is represented in the recent literature by Barker's radio-frequency polarography ( 7 ) . Although its inventor terms the technique one of faradaic rectification, experimentally it consists in applying to a cell superimposed on the normal direct current polarographic scan a small amplitude high frequency carrier (100 kc. to 6.4 mc.) square wave modulated at 225 C.P.S. In terms of Fourier analysis this is equivalent to adding side bands to the original carrier 225 c.p.s. removed (and, because the modulation is square wave, a t other frequencies as well). The detector then measures the component produced by the intermodulation of carrier and sidebands at the intermodulation frequency (225 c.p.s.) by means of a conventional square wave polarograph. Neeb (69) has also used intermodulation methods at much lower frequencies for analytical purposes. Despite the inherent advantages of the repetitive technique, it must be abandoned a t frequencies much above those employed by Barker. In order to produce potential drops of even a few millivolts across the extremely low impedance offered by the double layer at multimegacycle frequencies, very high currents must be passed and large amounts of power are dissipated in the ohmic resistance of the cell. The resulting temperature rise restricts the useful time of observation t o shorter and shorter intervals as the frequency increases. Delahay (26) estimates the interval for a 0.01' temperature rise to be 30 msec. in a 1 M strong acid solution a t 1 mc., and it is inversely proportional to the square of the frequency. I n practice two types of rectification techniques exist: one in which the direct current impedance of the cell is high so that application of the high frequency carrier produces a rectification voltage and the other in which the direct current impedance is low so that a rectification current is produced. Neglecting double layer effects, the magnitude of the current produced in the latter case is identical with that which would be observed in a potentiostatic experiment in which the potential step were of the magnitude of the rectification voltage in the former case. I n principle, then, both techniques lead to the same result. I n practice the cell potential does not immediately jump on application of the carrier, but relaxes to the new value a t a rate determined by the rate of charging of the electrical double layer by the faradaic process. This is the same process as that encountered in the coulostatic technique. Because such relaxation times can be relatively long by comparison with those required to produce substantial rise in temperature, Senda, Imai, and Delahay (70) were prompted to devise a double pulse method in
which a separate direct current pulse is applied t o charge the double layer to the rectification poteritial and then a high frequency pulse is applied t o maintain it at this value. Imai and Delahay successfully applied this technique with carrier frequencies as high as 50 mc. (46). I n this case, the double pulse method is adophed not with the aim of improving accuracy, but simply as a n expedient for getting the job done quickly. The rectification technique in which current is measured has also been adapted to a double pulse modification by Delahay and couorkers (47, 70). I n this case a potentioh tatic step is combined with the high ?equency carrier and the magnitude of this step is a d justed so that the net current, the sum of the opposing currents induced by the step and carrier, is zero. Techniques in which the rectification voltage is measured rely on high impedence a t low frequencies and are thus more difficult t o adapt to in situ polarographic generation of electroactive species than techniques in which current is measured or nulled. It would be pleasant t o be able to report that these techniques-at present, a n order of magnitude more rapid than the linear relar, ation methodshave produced relia d e measures of kinetic parameters for hitherto inaccessible systems. I n fact, however, they seem to have raised more questions than they have answered. Thus, Imai and Delahay (46) found a chargetransfer rate constant for the discharge of mercurous ion on mercury in 1M HCIOI forty fold larger than that observed by Matsuda, Oka, and Delahay (64) using the galvanos tatic double pulse method (1.3 cm./sec. us. 0.047 cm./sec.). Delahay, Senda, and Weis (31) concluded that the apparent charge-transfer coefficient for the reduction of titanium (IV) in acid taitrate was 0.25 or 0.46 depending on the way in which the data was analyzed. Barker’s analysis of a number of systems by the radiofrequency method discloses internal inconsistencies in many cases if simple charge transfer coupl3d with diffusion is assumed the mechanism of discharge. I n some of these cases plausible explanations in qualitative agreement with the observations have been put forth by invoking mechanisms involving adsorption or coupled homogeneous chemical reactions. Nonet heless, convincing proofs of the validity of these explanations remain for the future, and, in some cases, the explanations themselves have yet to be put forth. Imai (46) has proposed an interesting modification of the faradaic rectification technique. Because the form factor for the rectification current passes through zero a t some direct current potential for finite frequencies, the
faradaic rectification current (or potential) does likewise. By measuring the direct current potential of rectification null rather than the magnitude of the current or potential directly, necessity of accurate knowledge of the amplitude of the carrier frequency voltage is avoided. Paynter and Reinmuth (63) have used a n analogous technique employing intermodulation rather than rectification nulls. Alas, their conclusions are much the same as those of other workers, namely, that results are often inconsistent with the simplest discharge mechanisms though plausible explanations can be devised for the discrepancies. The forerunner of these null techniques is one devised by van Cakenberghe (18) and applied by Bauer and Elving (10) using second-harmonic nulls. Unfortunately, van Cakenberghe’s theoretical treatment is incorrect and a more careful treatment leads t o the conclusion, verified by experiment (sa),that second-harmonic measurements, in contrast t o the intermodulation and rectification techniques, does not produce exact nulls except when the charge-transfer coefficient is one-half or at very low frequencies. For kinetic studies, the second-order relaxation techniques seem, a t the present state of the art, t o be complementary to the linear relaxation techniques because they are useful at very high frequencies at which the linear techniques fail. For relatively slow relasation processes, the complication of their instrumentation and of their theory makes them much less attractive than the linear techniques. Both intermodulation and secondharmonic polarography are feasible as analytical techniques. It would seem that in this case the advantage lies with the second-harmonic technique. T o avoid, insofar as possible, the kinetic complications which plague small amplitude relaxation techniques in general in analytical applications, it is desirable t o work at fundamental frequencies as low as possible. At the dropping mercury electrode both fundamental and readout frequencies are limited a t the low end by the necessity that they be high enough t o avoid complications due t o the rate of drop growth (>--20 c.P.s.). These considerations combined with the lack of necessity for variable carrier frequency in analytical studies make second-harmonic polarography easier t o instrument than intermodulation polarography, though the difference is slight. The same disadvantages with regard to susceptibility t o kinetic factors apply to second-order techniques as t o the first order ones. Their virtues lie in the high signal-to-noise ratio because of substantially lessened interference from the nonfaradaic double layer current. Several workers in harmonic (9, 60, 78)
and intermodulation (6, 69) techniques have noted this. The enhancement of sensitivity on this account is of the same order of magnitude as that associated with gated square wave snd phase selective alternating methods. Third- and fourth- and presumably even higher order nonlinearities of the faradaic process can be utilized for the development of relaxation techniques. Paynter and Reinmuth (63) have briefly described results with thirdand fourth-harmonic polarography. Because the magnitudes of the responses decrease with the order of the nonlinearity while the major virtue, minimization of double layer response, is achieved with the second harmonic, these techniques seem little more than curiosities. Although, as noted above, theoretical treatments become quite cumbersome when the amplitudes of potentid excursions are not restricted to small values, Smit and Wijnen (76, 83) have concerned themselves with. repetitive square wave current and potential techniques a t somewhat larger amplitudes. By appropriate choice of experimental conditions, the theory can be much simplified from the more formidable general caqes, and kinetic parameters are relatively easy to deduce. The principle is much the same as for rectification or intermodulation nulls. The difficulty with the approach lies, of course, not in applying these techniques to the simplest kinetic schemes, but in modifying the theory for cases of more involved mechanisms, and of diagnosing these schemes experimentally. Because the small amplitude relaxation techniques rely on small displacements from equilibrium, they are not strictly applicable to highly irreversible reactions such as those encountered often in fuel cell studies and the like. For the most part, such systems are studied by stationary state techniques in which direct current is measured as a function of direct current potential under conditions of steady state convective mass transfer. Variants of the relaxation techniques emplojing perturbations from the stationary state rather than perturbations from equilibrium may then prove useful. Current interrupter techniques have been common for many years (44). The analogs of many of the equilibrium relaxation techniques are feasible. Perhal)i: the “fanciest” is Barker’s application of intermodulation methods t o the hydrogen evolution reaction ( 7 ) . When the reaction schemes are simple ones, little is t o be gained from relaxation techniques that cannot be obtained by care ul analysis of data from stationary state current voltage curves. For more complex reaction schemes involving transient intermediates and adsorption phenomena, possible benefits accrue. VOL. 36, NO. 5 , APRIL 1964
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As the discussion above indicates, the most interesting applications of the small amplitude techniques lie in the determination of kinetic parameters of charge-transfer processes, while their most obvious disadvantages lie in their lack of ability to provide qualitative insight into mechanisms, with the result that the significance of parameters so deduced often remains open to question. In the determination of kinetic parameters these techniques are competitive with the stationary state methods primarily Levich’s rotating disk (37) and Jordan’s hydrodynamic (48) voltammetry. The rotating disk technique appears particularly attractive in that it gives very reproducible convective mass transfer conditions at the cost of a minimum in experimental complexity. Instrumentation, theory, and practical analysis of experimental data are much simpler than with relaxation methods. The disadvantages of this technique lie in its difficulty of implementation with mercury, the lack of reproducibility of solid electrode surfaces, and difficulties with situations in which the stationary state characteristics of the reaction are not the same as the transient ones- g., cases of surface active reactants or products. With regard to irreproducibility, for example, in the application of a transient relasation technique a t a DME, the impurity levels which can be tolerated are determined by the amounts which can diffuse to the electrode during drop-life. At a rotating disk, because the basis of the technique lies in the pressure put on the charge-transfer proceqs by a very high rate of convective mass transfer, tolerable impurity levels are orders of magnitude lower when such impurities can accumulate a t the electrode. At the present the competition appears to be a standoff; convective stationary state techniques are becoming of increasingly greater interest to workers at solid electrodes, while workers concerned with mercury tend to choose relaxation methods for kinetic studies.
LARGE AMPLITUDE RELAXATION TECHNIQUES
The large amplitude techniques, as well as the small. can be delineated experimentally by whether current or potential is controlled, and further by whether transient or repetitive waveforms are employed. I t will be convenient, however, to divide the discussion below into three parts based on the three basic types of applications for the techniques are normally employed. namely: quantitative kinetic studies, qualitative kinetic studiesi.e., mechanism elucidation-and analytical $tudies. 216 R
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Although the small amplitude relaxation methods are useful for the study of charge-transfer kinetics, they are less useful for studies of coupled chemical reactions. For example, in a mechanism of the type Y (not electroactive) 0; 0 ne + R, the small amplitude relaxation proceeds in three steps in which (a) the concentrations of 0 and R a t the electrode readjust by charge transfer, (b) in response to this diffusional mass transfer of 0 and R occurs in the bulk of solution, and (c) in response to this the chemical interconversion of 0 and Y occurs. The time constants for all of these processes influence the observed results, and a d justing conditions so that the third is the important one may be difficult or impossible. However, by driving the potential sufficiently far from equilibrium with a large amplitude technique it is possible t o make the rate constant for charge-transfer sufficiently large that complications due to charge-transfer kinetics are absent. These objections notwithstanding, Smith recently (76) and others before him have used alternating current polarography and other small amplitude relaxation techniques for quantitative study of chemical kinetic complications, and indeed such study would be mandatory in cases in which the chemical reaction occurred sufficiently rapidly that the double layer thickness was comparable with the homogeneous reaction layer thickness so that potential dependence of the rate of the homogeneous reaction might be expected. From a theoretical viewpoint, the easiest large amplitude techniques to treat are those employing potentiostatic, galvanostatic, or coulostatic steps. The instrumentation is simpler for the coulostatic and galvanostatic techniques than for the potentiostatic one. On the other hand, they have a number of disadvantages. The least attractive technique would be the coulostatic one which has as yet not been applied to kinetic studies. In both the coulostatic and potentiostatic cases the system initially a t equilibrium a t a point corresponding to the foot of a polarographic wave is driven essentially instantaneously to a position corresponding to the plateau region. In both cases exactly the same relaxation process occurs so long as the system remains on the plateau, with exactly the same faradaic current flowing in each situation. In the potentiostatic case, this current is measured directly. I n the coulostatic case no net current flows; the faradaic current is balanced by an equal and opposite capacitive current through the double layer. The current associated with the faradaic process is deduced indirectly by observation of the change of electrode potential with time due to discharge of
+
--)I
the double layer capacitance. Not only does this require knowledge of the double layer capacitance as a function of direct current potential, and greatly complicate data analysis when the capacitance varies over the potential range of the relaxation, but, in addition, it places the results at the mercy of factors which affect the double layer capacitance--e.g., surface active impurities. It would appear that these difficulties might be alleviated by application of a double pulse modification, the second pulse to return the system t o its initial condition. Such a technique would remove the dependence of the observed quantity (delay time between first and second impulses necessary to bring the potential back to its undisturbed value or alternatively ratio of the impulse magnitude at fixed time) on the double layer capacitance except in so far as this changed a t the null potential as a result of electrolysis (formation of surface active products). To the extent that the direct coulostatic method measures this change in charge on the double layer capacitance as a result of faradaic reaction, it is very similar in principle to the integral potentiostatic technique introduced some vears ago by Booman, Morgan, and Crittenden (14). These workers measured the integral of the relaxation current on application of a potentiostatic step, rather than the current directly. Fundamentally this technique differs from the coulostatic one in that it stores charge-Le., integrates current-on an external condensor rather than the cell double layer capacitance. The virtue that the external capacitance is less subject to uncontrolled variation than the double layer, is paid for in increased complexity of instrumentation (electronic integrator and potentiostat). The choice between the galvanostatic (or chronopotentiometric) and potentiostatic techniques is less obvious. When the times of interest become very short, analysis of chronopotentiometric data is complicated by the fact that a substantial fraction of the applied current is used to charge the double layer. ?Jot only does the deduction of transition times become a highly empirical art because of distortion of the potentialtime curves, but correction of them for double laver effects is a problem which has not yet been solved in a generally satisfactory manner. The potentiostatic results (insofar as the potentiostat functions) do not have this complication. Superficially, the treatment of chronopotentiometric data appears simpler (when double layer effects do not interfere) than that of potentiostatic data. This and the simpler instrumentation probably account for the relative popularity of the former technique. However, detailed anal-
ysis of a single poteitiostatic relaxation curve can yield information analogous to that obtained from chronopotentiometric transif ion time measurements at several current densities, so that ease of analysii of data must be weighed against the necessity of having more of it. As computer methods of data analysis become more important, this consideration u ill weigh more heavily in favor of potentiostatic studies. Two excellent and relatively recent studies by this technique are those of Delahay, OLa, and Matsuda (30)and of Alberts and Shain (1). I n large amplitucle potentiostatic studies many types o madsorption phenomena are unobse-vable. A reactant adsorbed on the electrode before the potentiostatic step reacts essentially instantaneously when the step is applied and is lost in the double layer charging transient. Such cases are in principle amenable to study by chronopotentiometry and much use of the technique has been made recently (3, 43, 51, 52, 57, 79) Unfortunately, several difficulties attend them. First, reactions of adsorbed species are quite similar in principle to double layer charging; in both cases charge is stored a t the electrode surface, in the one case with and in the other case without actual trar sfer across the hypothetical plane clividing electrode and solution. This means that in analysis of data the two phenomena are not a priori distinguishable. Practically, the amounts of charge associated with the two phen3mena are comparable and adsorption in systems of greatest interest must be studied under conditions in which double layer phenomena are importani-i.e., short transition times. T o add further complication, even without the interference of double layer phentlmena, accurate analysis of chronopotentiometric data in cases with both adsorption and diffusion requires some assumption or independent knowledge of the relative fractions of the applied current devoted t o reaction of diffusing; and adsorbing species as function of time through the experiment. Presently seniiquan t itative studies a t best are possible except in ideal cases. A study by Bard (5) gives an idea of the difficulties of interpreting complicated reaction schemes unambiguously. The large amplitude techniques are adaptable to modifications which have no useful counterparts in small amplitude studies, namely double step methods a hich allow generation of unstable intermediates and subsequent study of their chemical decays. Chronopotentiometry n i t h current reversal is the most prominent of these techniques a t present but potentiostatic and coulostatic analogs are possible. The considerations which gcvern the relative
merits of the single transient methods would appear t o be applicable to their double step modifications as well. Triple steps ad infinitum are also possible, but i t appears to the reviewer that the usefulness of the multiple step techniques varies more or less inversely as the square of the number of steps insofar as quantitative kinetic work is concerned. The only repetitive waveform technique easily derivable from the transient ones is Kalousek’s large amplitude square wave potential polarographic one which is a logical extension of the transient potentiostatic technique. I t remains a puzzle to the reviener why this technique has not received more attention. I t appears to have some advantages in the way of experimental convenience over the transient techniques for studies a t relatively low (say audio) frequencies; though for very high frequencies this might well be outweighed by instrumental complexity, as is the case with the small amplitude analog. Coulostatic and galvanostatic techniques a t large amplitude are not generally suited to repetitive extension in its simplest form because rectification by the electrode process tends t o drive the system into potential regions of solvent or electrode decomposition. For example, in chronopotentiometry with only oxidized form initially present, the first reverse halfcycle can only be one third the length of the first forward half-cycle if oxidation of species other than the one of interest is t o be avoided. If potential excursions are limited, steady states are ultimately reached in which forward and reverse cycles become equal in length (4Z),but the virtues of techniques based upon the idea remain t o be shown, and applicability is limited because they depend on the stoichiometric reversibility of the electrode process. Techniques for qualitative elucidation of mechanism of electrode reactions depend for their virtues on being able (a) to provide ready criteria to distinguish between various modes of mass transfer: adsorption diffusion and chemical kinetics, (b) to distinguish between various modes of charge-transfer: reversible or irreversible, and (c) to detect and characterize transient intermediates and unstable products. For a survey investigation, controlled potential techniques are much better suited than controlled current ones because in principle the qualitative characteristics of a couple are determined by the potential a t which it reacts and not the current. Repetitive techniques are better suited than transient ones because they allow more convenient study of the effect of changes in variables, and, again because they are more adaptable t o repetitive modifications, con-
trolled potential techniques are best suited. Probably the most popular technique at present is linear potential scan, in either single sweep or repetitive (triangular wave) modifications. I t is readily applicable to slow times with recorder readout or relatively rapid ones (to 100 or 1000 c.p.s.) with oscilloscopic display. Although quantitative analysis of data can become complicated because forms of theoretical results even for simplest cases make computer calculations almost mandatory, qualitative detection of intermediates requires simply counting bumps on the polarogram, deducing reversibility is simply a matter of judging symmetry of cathodic and anodic peaks and of judging wave shapes (reversible waves are more sharply peaked than irreversible ones and do not shift in potential with frequency, kinetically limited peaks are flat topped and do not increase in height with frequency, for example). Sinusoidal potential scans (67A j can be employed rather than triangular ones but suffer the disadvantage that the rate of potential scan varies over the display. Since peak heights and shapes (not to mention double layer c-harging current) depend on this f a d o r , the display is somewhat more complicated to interpret-though whether the added complication is enough to justify purchase or construction of a triangular wave generator is something each individual must decide. Despite the reviewer’s reservations, sinusoidal controlled current techniques have long been employed successfully by Eastern European workers led by Heyrovsk? (49). On each full cycle the driven (at a mercury electrode) from mercury oxidation to solvent or supporting electrolyte reduction and back, with potential holdups corresponding to each electroactive species in between. The fact that’ for an appreciable fraction of each full cycle the system rests in regions in which the estraneous solvent and electrode reactions occur is the major disadvantage of the technique because products of these reactions diffusing away from the electrode can interact with the species of interest t o produce unintended results. The lengths of time associated with these holdups have roughly the same characteristics as chronopotentiometric transition times. Thi$ technique has the simplest instrumental requirements of any of the repetitive ones requiring merely a step-down transformer and large series resistor for power source. Rather than varying frequency as is necessary in controlled potential techniques, variation of current density (within limits) can be employed for the same purpose. In this simplest form it is not useful a t the DhIE because the high current densities necessary VOL. 36, NO. 5 , APRIL 1964
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a t full drop-life would damage the capillary when the drop detached. Although the large amplitude repetitive waveform techniques seem best suited at present to general qualitative survey work, in cases in which specific questions are t o be asked about a specific system, large amplitude transient techniques or small amplitude techniques, insofar as they are more readily adaptable to quantitative measurements and theoretical derivations for various mechanisms are more readily derivable, are also very useful. It is doubtless true that kinetic studies could be made using any peculiar waveform at controlled potential or current or combinations of the two. The kinetic researcher finds little merit in such fancification because it ordinarily increases the cost and complexity and decreases the reliability of his instrumentation and complicates theoretical interpretation of his data without offering promise of giving him answers he cannot get with simpler techniques. I n analytical applications, on the other hand, an obscure technique] however limited its generality] may solve a specific problem which must be faced each day in routine analysis much better than any other. The most elegant technique available will prove of little value if it requires the attention of a Ph.D. continuously in its most menial applications Highly empirical methods have no disadvantage if they produce the right answers inexpensively and reliably. Because of the seemingly infinite variety of such techniques and the fact that practically none seem totally without merit, we restrict ourselves below to a brief discussion of general purpose techniques and criteria for choosing them. The uncontested champion of electroanalytical techniques is polarography. Not only does it have very wide applicability and give a convenient form of readout, but instrumentation is easy to acquire or construct and by choosing it one automatically avails himself of 40 years of literature experience in both theory and practice. When polarography with the D M E is impossible because of the available potential range, stationary state voltammetry with solid electrodes, particularly rotated disks, is often an adequate substitute and the most desirable one where applicable. From an electrochemical viewpoint, the features of these techniques which make them advantageous are that data are obtained in the region in which the least number of factors influence the resultLe., on the limiting current plateau where mass transfer rather than charge transfer or a combination governs the magnitude of response. Secondly, in both cases the mass transfer process can he made to occur in a reproducible 218 R
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manner from run t o run and day to day. Some of the small and large amplitude relaxation techniques improve on the sensitivity of polarography--e.g., alternating current polarography and linear voltage scan, but they do so a t the cost of making the observed response potentially a function of charge-transfer parameters, and, to this extent, become less generally applicable and of lower inherent accuracy. Ideally the types of relaxation methods which should be employed to improve sensitivity are adaptations of the large amplitude coulostatic or potentiostatic techniques in which the system is restricted to the plateau region during relaxation to avoid charge-transfer complications. When the system is restricted to the plateau region and mass-transfer is diffusion controlled, the faradaic current is inversely proportional to the square root of time. Thus, the shorter the time in which a measurement is made, the larger the response. The transient potentiostatic technique has been applied directly by Oka (61). Pulse polarography (8)is an adaptation which makes use of this idea by applying a step at a D M E at a fixed time in drop life and measuring current shortly thereafter. A second type of approach is t o amplify the signal simply by integrating it for long periods of time. Stripping analysis accomplishes this end by storing the product of reaction in or on the electrode. The coulostatic method (22, 27, 28) does the same thing by observing change in double layer charge (corresponding to the integral of faradaic current). Combination of both approaches in coulostatic stripping analysis ( 2 , 25) carries these approaches to their logical limit. One of the major limitations in the quest for high sensitivity a t the D M E by direct as opposed t o integral or pseudo integral measurements is the phenomenon of capillary noise (6, 20). Barker suggests that second-order relaxation techniques, because they appear to be less influenced by this factor offer a promising approach. The instrumental complexity of the technique, however, and its essential dependence on charge-transfer kinetics produce other disadvantage.. . Few, if any, of the general limitations of classical polarography are likely to be circumvented by relaxation techniques. As noted earlier, its resolution when readout is subjected to electronic differentiation is inherently as good as g r better than that of the relaxation techniques; its accuracy is ultimately limited by the reproducibility of the mass-transfer process, common t o all the relaxation techniques. Because the relaxation techniques can often be applied under conditions in which kinetic factors assume varying importance, in principle their selectivity is higher than
that of polarography. I n practice it is usually far more convenient to effect selectivity by chemical treatment of the sample prior t o analysis. The basic limitations of polarography, and the relaxation techniques as well, stem from the dependence of the results on mass-transfer processes. Accuracy and sensitivity would be improved if complete electrolysis took place. Conventional electrolysis techniques are relatively time consuming, but recent papers concerned with electrolysis in small volumes (19) and with large electrodes with high rates of stirring ( d ) , indicate that these techniques do hold promise of achieving high sensitivity and accuracy with relatively simple instrumentation and in short times. Insofar as the promise is fulfilled] they may well supplant polarography. Even a few years ago the myriad of available techniques all with possible special advantages presented an excrutiating choice to the researcher with limited time and funds. Fortunately] the introduction of operational amplifier based electronics to electronanalytical instrumentation has materially improved his lot. For a cost not exceeding that of a commercial recording polarograph he can construct an instrument which will perform any conceivable relaxation technique with a time-base up to 1000 C.P.S. The variety of applications discussed at a recent symposium attests to this (IS, ir, SS,66,58,69, 77,80). We seem t o be on the threshhold of a new era. Gone is the day when the polarographer cranked up his trusty instrument, gathered all the data he could, and speculated helplessly when he reached the limitations of the technique. With operational amplifiers, if polarography cannot solve the problem, switching a few wires determines quickly whether alternating current polarography, controlled potential electrolysis, chronopotentiometry, or some spur-ofthe-moment innovation offers greater hope. A particularly elegant study and perhaps a fitting keynote to the new era are provided by the researches of Adams and coworkers on the oxidation of N,N-dimethylaniline (36, 58, 55). These workers applied stationary state rotating disk studies at Pt and carbon paste electrodes, ring-disk studies of intermediates, Pt wire voltammetry, triangular potential scan relaxation techniques] tritium tracer studies to allow study of products formed a t low concentrations, fluorescence and absorption spectrophotometry, and electron spin resonance spectroscopy. Xot the least important conclusion to be drawn from this study is that, despite the wealth of relaxation techniques presently a t the disposal of the electro-
chemist, he need not be restricted t o them or to electrochemical methods in general. The techniques are tools for solving chemical problems, and not ends unto themselves,
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(2) Aramata, A,, Dehhay, I?., ANAL CHEM.35, 1117 (1963). (3) Bard, A. J., ANAL.CHEM. 35, 340 (1963). (4) Ibid., p. 1120. (5) Ibid., p. 1602. ( 6 ) Barker, G. C. in Charlot, G., ed. “Modern Electroanalytical Methods,” Elsevier, 1958, (71, Barker, G. tAE’.‘ieager, E., Ed., Symposium on Elecltrode Processes,” Philadelphia, 1959, Wiley, N. Y., 1961, p. 325. (8) Barker, G. C., in Zuman, P., Wolthoff, I. M., eds. “Progress in Polarography,” Interscience, 1962, V d . 11, p. 411. (9) Bauer, H. H., J . .Vlectroanal. Chem. 1, 256 (1960). (10) Bauer, H. H., Elving, P. J., ANAL. CHEM.30, 341 (1958 1. ( I 1 ) Berg, H., Naturciissenschaften 47, 320 (1960); 49, 11 (1962). (12) Berpins, T., Delahay, P., J . Am. (’hem. Soc. 77, 6448 (1955). (13) Rooman, G. L., .Holbrook, W. R., ANAL.CHEM.35, 1793 (1963). 14) Booman, G. L., Morgan, E., Crittenden, A. L., J . Am. C k m . Soc. 18, 5533 (1956). 15) Breyer, B., Bauer, H. H., “Alternat,; ing Polarography and Tenssmmetry, Interscience, N. Y., 1963, p. 101 ff. 16) Breyer, R., Hacobian, S., J . Electroanal. Chem. 3 , 45 (1962). 17) Buck, R. P., Eldridge, R. W., ANAL. CHEM.35, 1829 (1963). 18) Cakenberghe, J. van, Bull. soc. chim. bdyes 60, 3 (1951). 19) Christensen, C. It., Anson, F. C., ANAL.CHEM.35,205 (1963). 20) Cooke, W. D., Kelley, M. T., Fisher, I). J., ANAL.CHEM.33, 1209
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