Stability study and categorization of electrochemical oscillators by impedance spectroscopy

ELSEVIER

Journal of Electroanalytical

Chemistry

409 (1996)

175

182

Stability study and categorization of electrochemical oscillators by impedance spectroscopy ’ Marc T.M. Koper Department

of Electrochemistry,

Uniuersiry

of Utrecht,

Received

Paduuhan

31 August

2 8, 3584 CH Utrecht.

Netherlands

1995

Abstract The frequency response or impedance spectrum of an electrochemical cell obtained under potentiostatic conditions can be used to predict the stability features of the electrochemical cell under different electrical conditions, such as galvanostatic control for example. It is shown that the particular condition for which the system is resonant with the perturbing signal implies the existence of a so-called Hopf bifurcation to sustained spontaneous oscillations in the time domain. The ability of impedance spectroscopy to make predictions about the occurrence of oscillations leads in a natural way to a classification of electrochemical oscillators into two different categories, depending on the kinetic nature of the negative faradaic impedance. The theory is illustrated by experimental and model examples. Keywords:

Electrochemical

oscillators;

Stability;

Impedance

spectroscopy

1. Introduction With the advent of “chaos theory”, oscillatory electrochemical systems have recently witnessed revived experimental and theoretical interest. In the past ten years, electrochemical oscillators have proved to be a marvellous playground for the chaotically inclined, and several review articles are now available [l-3]. Electrochemists have known for a long time that quite a number of electrochemical systems give rise to oscillations, a number that is perhaps greater than many would like to admit. However, knowledge about the origin of electrochemical oscillations is much less widespread than knowledge about their existence. This is somewhat surprising since several papers were published in the 1960s and 70s [4-91 which stressed the importance of the negative faradaic impedance and the role of the external circuitry in the generation of electrochemical oscillations. Recent work has clearly demonstrated the difference between systems that oscillate under either potentiostatic or galvanostatic conditions [lo- 131. Mechanistically, this

’ This paper is based on a presentation given during the Snowdonia Conference on Electrified Interfaces, Harlech, Wales, UK, 17-21 July 1995. * Present address: Abteilung Elektrochemie, Universitit Ulm, Oberer Eselsberg, D-89069 Ulm, Germany. 0022.0728/96/$15.00 0 1996 Elsevier SD1 0022-0728(95)04391-8

Science

S.A. All rights

reserved

difference manifests itself in the presence or absence of what has been called a “hidden” negative faradaic impedance [lo]. A system exhibits a hidden negative impedance when it possesses a negative real impedance for a range of non-zero perturbation frequencies, but its zerofrequency real impedance is strictly positive, implying a positive slope in the steady-state polarization curve. Such systems will (normally) oscillate under galvanostatic control, as well as under potentiostatic control in the presence of a sufficiently large ohmic drop, whereas systems that exhibit a negative faradaic impedance only in combination with a negative polarization slope will never oscillate under galvanostatic control. These conclusions were recently drawn on the basis of a study of an abstract generic model for electrocatalytic reactions [lo], but can in fact be given a more general basis by employing the Nyquist stability theorem from frequency response analysis. This paper discusses the usefulness of frequency response analysis or impedance spectroscopy in studying oscillatory electrochemical systems. The paper is certainly not unique in that sense as several authors, most notably de Levie [8] and Gabrielli [9], have done similar work before, and to some extent much of what follows can be found in their work as well. Our viewpoint is somewhat different, however, as it is based on bifurcation theory, and our goal is fully different, as we aim to categorize electrochemical oscillators, for which impedance spectroscopy can be used

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profitably. Bifurcations divide the parameter space (where a “parameter” is any externally controllable quantity, like applied potential or current, bulk concentrations, temperature, etc.) into regions with various kinds of qualitative dynamics. Such a bifurcation map is also called a “phase diagram” to underline the similarity with a liquid-gassolid phase diagram. Certain bifurcations, and in particular the so-called Hopf bifurcation [ 141 which describes the transition to spontaneous oscillations, can be identified easily by the impedance spectroscopy method, and therefore the most important curve in the phase diagram of an electrochemical cell can be predicted by straightforward extrapolation of impedance spectra [15]. An experimental example of this application will be discussed and this phase diagram, as obtained in the “frequency domain”, will be shown to match exactly that obtained in the “time domain”. Experimental examples of the hidden negative impedance of galvanostatic oscillators will also be discussed, and it will be shown that these systems possess phase diagrams that are qualitatively different from electrochemical oscillators that oscillate only under potentiostatic conditions (with ohmic drop). A simple reaction model will illustrate this as well, and will give us a hint about the typical equivalent cell circuits derivable from an impedance analysis for these two different types of electrochemical oscillator. Finally, this classification of electrochemical oscillators will be shown to be equivalent to an existing classification for homogeneous chemical oscillators.

2. Bifurcation

analysis by impedance

spectroscopy

Over the years many authors have addressed the problem of stability of electrochemical cells. From a theoretical point of view, stability can be defined as the ability of a system to suppress small fluctuations away from the stationary state. If one disposes of a kinetic model of the system, the stability is assessed by linearizing the kinetic (differential) equations about the stationary state, and subsequently evaluating the local time evolution of small perturbations by solving the linearized differential equations. Stability is then uniquely determined by the sign of the eigenvalues of the associated (Jacobian) matrix of coefficients: negative eigenvalues implying a damping out of fluctuations, i.e. stability, and one or more positive eigenvalues implying an exponential growth of fluctuations, i.e. instability. This is a standard and elementary technique known as linear stability analysis; readers unfamiliar with it may consult a well-known textbook such as Ref. [16]. But what if a kinetic model is missing? Is it possible to make predictions about the stability of an electrochemical cell without the need for a detailed mathematical (phenomenological) model of the system’s time evolution? As is well known in electrical engineering, the method for doing

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this is frequency response analysis 117,181. It is intuitively clear that if one determines experimentally a system’s frequency spectrum over a broad enough frequency range, one has essentially all the linear information of the system, with which it should be possible to make statements about the stability of the system. The theorem which establishes the relation between frequency response and stability is the Nyquist stability criterion [ 191 (de Levie [8], in his pioneering article, unjustly ascribes it to Llewelyn [20]). Unfortunately, the Nyquist criterion in its most fundamental form is not particularly enlightening or easy to use, and is based on rather abstract concepts of the theory of complex functions. A derivation with an eye to electrochemical systems can be found in Ref. [3]. However, in the majority of experimental cases we can do without the Nyquist criterion and simply make use of the fact that it is very easy to predict the possibility of bifurcations from a frequency spectrum. Recall that a bifurcation is a transition point where the system switches from one to the other type of qualitative dynamics. In the linear stability analysis as described above, a bifurcation occurs if one of the eigenvalues {h} has zero real part. One then distinguishes two cases [14]: a real eigenvalue being zero, h = 0 (a so-called saddle-node bifurcation), and a pair of complex eigenvalues having zero real part, A = f i w (a so-called Hopf bifurcation). By making use of more elaborate non-linear techniques, it is possible to prove universal features of these bifurcations (provided that a few side-conditions are satisfied which provide no serious obstacle in generic experimental circumstances) [ 141. If a saddle-node bifurcation exists, this always implies the existence of another stationary state, and if the system is not to explode then normally a third stationary state will exist as well. In the most typical situation there are always two saddle-node bifurcations bordering a hysteresis loop (see Fig. l(a). Note that this is not a kinetic hysteresis due to certain very slow or irreversible processes in the system, it is entirely due to the non-linearities in the kinetic equations, very much like that in the Van der Waals isotherm). If a Hopf

varmble

t

(a)

(b)

Fig. 1. Generic bifurcation diagrams near (a) saddle-node bifurcation and (b) Hopf bifurcation. Solid lines indicate the stability of the stationary state, dashed lines indicate instability. For the oscillatory state the minimum and maximum of the oscillation amplitude are shown.

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bifurcation exists, then the Hopf theorem [14] guarantees the existence of a so-called limit cycle which gives rise to sustained periodic oscillations. If the Hopf bifurcation is supercritical, the oscillations are stable and thus observable (see Fig l(b)); if the Hopf is subcritical, the oscillations are unstable and thus not observable experimentally. The supercritical Hopf is the most typical situation, in which the amplitude of the oscillation grows continuously from zero when one moves away from the bifurcation. The subcritical Hopf may also give rise to the observation of stable oscillations, but always with a non-zero amplitude at the bifurcation point; this is due to the existence of another bifurcation that is global in nature and that cannot be inferred from a linear stability analysis of stationary states. However, it is possible that no oscillations are observed beyond a subcritical Hopf bifurcation. Therefore, a Hopf bifurcation does not guarantee that one will actually observe oscillations. However, if sufficient parameters exist, a subcritical Hopf can be made supercritical, and therefore in practice the hypothesis that a Hopf bifurcation must lead to observable oscillations somewhere in the phase diagram is a very reliable working horse. Since at a bifurcation small perturbations do not die out or grow (in the linear approximation), the solution of the linearized equations must be a sine wave, of finite frequency ou in the case of the Hopf bifurcation and of zero frequency in the case of a saddle-node bifurcation. If we now drive the system externally by a sine wave of exactly that critical frequency, a condition of resonance is encountered in which - in the ideal, linear case - the driving signal is blown up by the system without any phase delay. That is, the system has zero real and imaginary impedance. Therefore, a bifurcation is a situation for which the impedance spectrum intersects the origin of the complex impedance plane. If the intersection occurs for a non-zero frequency, the system exhibits a Hopf bifurcation; if the impedance spectrum terminates in the origin for zero frequency, the system exhibits a saddle-node bifurcation. Clearly in practice it is not feasible to measure the impedance spectrum close to a bifurcation, because if we measure on the “unstable side” of the bifurcation the stationary state is unstable (violating one of the basic conditions for impedance spectroscopy [21]), whereas on the stable side of the bifurcation the system is extremely sluggish due to critical slowing down (note that this does not mean that the impedance spectrum of an unstable system is not well defined; it is only not measurable). It is at this point that the role of the external circuit must be invoked. Since in the majority of cases electrochemical cells are stable under truly potentiostatic conditions (i.e. if one may really neglect the external ohmic potential drop) (the possible few exceptions will be discussed in the last section of the paper), and become unstable due to ohmic losses in the external circuit (i.e. a high ohmic drop due to a high electrolyte resistance or a large electrode surface area), and since the influence of the ohmic loss is nothing

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more than a simple horizontal shift in the complex impedance plane, it is possible to test graphically for the possibility of bifurcations from the impedance characteristics measured potentiostatically. This test includes the galvanostatic control, since this situation corresponds ideally to an infinite series resistance and an infinite applied potential, with ratio corresponding to the finite applied current. In order to have a bifurcation in the galvanostatic control, one must therefore have infinitely negative real impedance and zero imaginary impedance, which is equivalent to zero real and imaginary admittance. All this allows us to formulate the following conditions for detecting bifurcations. For a fixed applied potential (one may call this potentiostatic provided one realizes that there must also be considerable ohmic drop), an electrochemical cell will exhibit - a saddle-node bifurcation if Z(w) = 0, w = 0; - Hopf bifurcation if Z(w) = 0, w = on # 0. For a fixed applied current (galvanostatic), an electrochemical cell will exhibit - a saddle-node bifurcation if Y(o) = 0, o = 0; - a Hopf bifurcation if Y(w) = 0, w = ou # 0.

3. Experimental

examples

We will now illustrate the application of the theory of the previous section for two experimental examples: the reduction of In3+ on the HMDE from thiocyanate solution, a system that oscillates only under fixed applied potential conditions, and the oxidation of formaldehyde on a rhodium RDE, a system that oscillates under both fixed applied current and fixed applied potential conditions. The In3+ reduction on a mercury electrode from thiocyanate or chloride solution has long been known for its rather peculiar voltammetric characteristics. The polarogram exhibits a potential region of negative slope due to the desorption of the tbiocyanate or chloride ions from the mercury Jelectrolyte interface. These specifically adsorbed ions catalyze the otherwise irreversible reduction of the In3+. In the presence of an external ohmic resistor, de Levie [8] showed that the system may give rise to sustained periodic oscillations. Recently, the author and coworkers [22] have shown that much more complex oscillations, including deterministic chaos, may also occur, and that these different qualitative dynamics make up an intricate phase diagram whose details are well reproduced by a simple model that incorporates the most essential phenomenology of the system [23]. The impedance characteristics have also been measured, by de Levie and Pospisil [24] for a thiocyanate electrolyte, and by Sluyters-Rehbach et al. [25] for a chloride electrolyte. However, both measurements were carried out for a DME, which does not give rise to a well-defined stationary state and is therefore unsuitable for the kind of phenomena we are interested in.

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R/kD zo-

0’ -1 (bl

I

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-1.4 v/ (vs. SE)

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-1.8

Fig. 2. (a) Impedance diagram for the It?’ reduction on a HMDE from SCNsolution at E = - 1.075 V vs. SCE. 9 mM In(NO,), +5 M NaSCN, pH 2.9, T = 25°C. This particular steady state exhibits a Hopf bifurcation for R, = 4.2 kR and a corresponding applied potential V = E + IR, = - 1.15 V (vs. SCE), where I is the steady-state current. Indicated frequencies in Hertz. (b) Line of Hopf bifurcations determined by the impedance method as explained under (a). Dots represent the onset of oscillations as observed by insertion of an external resistor.

We have measured the impedance spectra of the In3+ reduction on a HMDE from thiocyanate solution in order to test the theory outlined above. A typical impedance spectrum as measured on the branch of negative slope in the current-voltage curve is shown in Fig. 2(a). Although the lowest frequencies were not measured, the spectrum should qualitatively have the shape of the dashed line, terminating on the negative real impedance axis at a value equal to the slope of the steady-state polarization curve (the steady state is well-defined because diffusion is towards a stationary spherical electrode). It is seen that this particular steady state with electrode potential E = - 1.075 V (vs. SCE) is expected to exhibit a Hopf bifurcation if an external resistance R, of ca. 4.2 k0 is connected in series with the working electrode, since then the spectrum will intersect the origin for a frequency of ca. 30 Hz. The corresponding applied potential V is E + IR, = - 1.15 V. Therefore the point (V, R,) = (-- 1.15 V, 4.2 kfi) lies on the curve of the Hopf bifurcation in the phase diagram spanned by V and R,. The entire curve, as extrapolated from spectra for 15 different electrode potentials, is given by the solid line in Fig. 2(b). The dots in Fig. 2(b) represent the onset of oscillations as actually observed by

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409 (19961 175-182

inserting an adjustable ohmic resistor in series with the cell. Clearly, the Hopf bifurcation as observed in the “time domain” matches exactly the curve as extrapolated from the frequency spectra. This is of course only to be expected, but we are not aware of such a detailed experimental test of the equivalence of frequency-domain and time-domain phase diagrams in the literature. The phase diagram has the familiar fish-shape that is predicted by simple models [26] Oscillations are found inside the head of the fish. However, inside the tail of the fish no oscillations are observed because the Hopf bifurcation is subcritical [26]. Instead, this region is characterized by a hysteresis as in Fig. l(a). Here we see a clear advantage of the impedance method: it is able to identify a Hopf bifurcation that is difficult to identify by a time-domain method. It may be mentioned that in the real phase diagram there are many more bifurcation curves that the impedance method is not able to identify [22]. The Hopf bifurcation remains the most important, however, as it separates the oscillatory region from the non-oscillatory region. The spectrum in Fig. 2(a) leads to an unstable steady state in the galvanostatic control also, but the instability is not oscillatory. This is because, as we increase the external resistance, the Hopf bifurcation is followed by a saddlenode bifurcation which changes the local time evolution around the steady state so as to prohibit the possibility of a limit cycle. (Actually to prove this, one has to employ the Nyquist stability criterion.) Beyond this critical R, the steady state no longer changes its stability properties, so it must be the same in the galvanostatic limit. Therefore, systems with impedance characteristics like that of Fig. 2(a) do not oscillate under galvanostatic conditions. It seems therefore that if we want a system to oscillate under galvanostatic conditions, the Hopf bifurcation must be the only bifurcation to be encountered as we increase the external resistor from small (zero) to large (infinite) value. This can only be achieved if for the lowest frequencies the real impedance becomes positive again. This then implies that the steady-state polarization curve must have a positive slope. The negative impedance is in some sense “hidden” because it cannot be inferred from the steadystate current-voltage curve. An experimental example of such behavior is given in Fig. 3, which shows experimental results obtained with the oxidation of formaldehyde on a rhodium rotating disk electrode, a system that was shown by Hachkar et al. [27] to give rise to quite easily reproducible oscillations. In the absence of an external resistor, this reaction system gives a cyclic voltammogram (Fig. 3(a)) without oscillations if care is taken that the ohmic potential drop in the electrolyte is small enough (in this particular case the formaldehyde concentration was lowered until the oscillations disappeared). If we now carry out the experiment under controlled current, and make an “amperogram”, oscillations are observed as also shown in Fig. 3(a). These galvanostatic potential oscillations indeed

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0.6 I/rnA 0.4

(a) -6000

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0

z/n

(b)

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x -350

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(cl

)3

0 y/n-'

Fig. 3. (a) Voltammogram of 0.1 M HCHO in 0. I M NaOH RDE for 0, 1000 and 1500 L8 external resistance (internal ca. 95 0). Scan rate 10 mV s-‘, 3000 rev min- ‘, room Amperogram 0.01 mA s- ‘. (b) Impedance diagrams at -0.45 V, 0; and -0.35 V, A. Indicated frequencies Admittance plane representation of the same data as in (b).

0 .oc13 on a rhodium cell resistance temperalure. -0.50 V, W ; in Hertz. (c)

occur around a branch in the potentiostatic polarization curve with positive slope. Clearly, this branch can also be de-stabilized by adding an external resistor to the circuit, leading to current oscillations as exemplified in Fig. 3(a) for two different values of R,. If the impedance spectrum is measured (potentiostatically), it is found that in the unstable region there is a hidden negative impedance, as expected, since the real impedance is negative for a finite range of non-zero frequencies but becomes positive for the

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lowest frequencies, in accordance with the polarization slope (Fig. 3(b)). If the data are plotted in the admittance plane for the three different potentials, it is observed that for E = ca. - 0.45 V the plot almost intersects the origin (Fig. 3(c)). Th’s1 is indeed the potential for which we observe the development of small oscillations in the amperogram, thereby confirming the condition for the observation of a galvanostatic Hopf bifurcation. A few comments are in order about the results displayed in Fig. 3. First, a very strong hysteresis is observed in the cyclic voltammogram. This is a kinetic hysteresis, not a hysteresis of the type described in the previous section (Fig. l(a)). It is probably due to the formation of CO during the oxidation of formaldehyde, which tends to absorb irreversibly onto the electrode surface. The poison can only be removed by going to very positive potentials where it may react electrochemically with adsorbed OH. Second, the process most likely to be responsible for the negative impedance characteristics is the formation of a (hydr)oxide layer, since the potential region of negative impedance coincides with the oxide-formation wave in an electrolyte without electroactive species. Third, there must be an additional, as yet unidentified process responsible for the impedance being positive at the lowest frequencies in the oscillatory potential region. Clearly, since this process is only observed at low frequencies it is necessarily a slow process. It is likely, although this still needs to be shown unambiguously, that it is somehow related to the potential dependence of the formaldehyde adsorption. The origin of this potential dependence can be manifold, and may be related to other (slow) adsorption processes of species that are not electroactive, or do not give rise to a steady-state current. A last point concerns the way the oscillations cease in the amperograrn for increasing applied current. Note that for I = 0.4 mA the oscillations suddenly terminate and the potential sets off for very positive potentials, where finally an electrode potential is reached where the rhodium and/or water oxidation is favorable enough to satisfy the pre-set current. It is no coincidence that this sudden transition occurs when the anodic oscillation amplitude meets the (galvanostatically unstable) down-going part of the potentiostatic cyclic voltammogram. What we witness here is a so-called homoclinic bifurcation [14], in which the limit cycle collides with an unstable steady state. This is a global bifurcation that one can never foresee from an impedance spectrum. Such bifurcations are in fact typical for galvanostatic electrochemical oscillators. We will return to the topic of typical bifurcations and phase diagrams for the two types of oscillator at the end of the paper.

4. A simple kinetic model

In this section we will briefly consider a very simple abstract model [lo] to illustrate how the theory described

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above relates to an explicit electrochemical mechanism and to derive equivalent circuits that correspond to the two different kinds of electrochemical oscillator described. The models are not claimed to represent the correct kinetic interpretation of the experimental systems considered above, but are arguably not too far off either. We consider an electrocatalytic mechanism for which a single species X first adsorbs onto the electrode, where it is subsequently oxidized (or reduced): ka kr -P+nex-xadsk, where k,, k,, and k, are the rate constants for adsorption, desorption and reaction respectively. They may all depend on the electrode potential E. Mass transport is assumed to be fast so that it may be left out of consideration. The differential equation governing the time evolution of the surface coverage 8 of X is given by

Chemistry

k, + k,

dk, -k, + k, + k, W X

k,+:,+k,(;+:)

k,+k,+k,+jw

where the RHS should be preceded by a minus sign in the case of a reduction reaction. Assuming, for simplicity, that the adsorption process does not contribute significantly to the charging of the electrode, the total interfacial impedance is Zi,[ = l/(j,C, + Z, ‘>, and therefore the total impedance, including any uncompensated ohmic resistance, is Z,,, = Zi,, + R,. Assume first that the adsorption and desorption rates are independent of potential, ak,/aE = ak,/aE = 0. It is easy to show that in this case the frequency response is that of the following equivalent cell circuit: Z, = R,, +

R;’

+ jwC a

with R,, = nF

k,

%

k, + k, + k, %?’

k,R,,

R, = k, + k, ’

Note that the sign of all three elements is the same as the sign of ak,/aE, i.e. the potential dependence of the electron transfer rate. This implies that if R,, is negative, the real part of the faradaic impedance is negative for all frequencies. As we have seen above, this may lead (and

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will lead) to oscillations under fixed applied potential provided that R, is sufficiently large, but excludes the possibility of galvanostatic potential oscillations. Galvanostatic potential oscillations can be obtained if we allow for a potential-dependent adsorption or desorption. Considering, for simplicity, only potential-dependent adsorption, the following equivalent circuit can be derived: 1 -xc ZF

1

1 iT+ct

R, + joL

with R,‘=nF

R,=

ka

%

k, + k, + k, aE ’ (k

+k,

k,( k, + k,) 2 L=

where the faradaic current density flowing is j, = & nFk,B. Hence, for the faradaic impedance one obtains

409 (1996)

+kJ2 - k,k,G

’

Ra k, + k, + k,

provided that

is This last inequality will be satisfied when ak,/aE negative, or when ak,/aE is about to change sign, or, more generally, when adsorption exhibits a stronger potential dependence than electron transfer. A negative R,, will give rise to a negative impedance over a certain frequency range, but it is now possible that the zero-frequency impedance is positive, as required for galvanostatic oscillations. The condition for this follows from l/Z, = l/R,, + 1/R, > 0, i.e.

$!$ > -k;z This condition was already derived in Ref. [lo] by another, conceptually similar method. This analysis illustrates that galvanostatic oscillators are accompanied by inductive behavior in their impedance spectra, especially in the case where R,,-’ is small and positive and about to change sign. From these elementary calculations we can already appreciate a fundamental difference between potentiostatic and galvanostatic oscillators. For a system that oscillates only under potentiostatic conditions, the only potential-dependent process is electron transfer. For a system that also oscillates under galvanostatic conditions, an additional potential-dependent process must be active, to break the symmetry between the potential-dependence of the charge balance of the electrified interface and the mass balance of the electroactive species. This process should therefore either be a non-electrochemical process (like adsorption) or any other potential-dependent process that affects the mass balance in a way different from the charge balance.

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In these terms it also becomes clear from a more physical point of view why the indium reduction at the HMDE as described above does not give rise to galvanostatic oscillations: there is only one potential-dependent process, the electron-transfer rate. The process that replenishes the interface with indium is diffusion, which is independent of potential.

5. A classification

of electrochemical

oscillators

On the basis of our findings above, we suggest the following categorization of electrochemical oscillators. We begin with a category that we have not yet considered, because it cannot be treated with the formalism described in the previous section. 5.1. Class 1. Oscillations potentiostatic conditions

(of the current)

occur under truly

It is certainly conceivable that oscillations occur in an electrochemical system in which the feedback onto the external circuit is not essentialfor the oscillation mechanism. Such oscillations will still occur under truly potentiostatic conditions, where the ohmic drop can be safely neglected. As we have pointed out before [lo], the instability mechanismfor such systemsmust be purely chemical, and may for instance be related to an autocatalytic surface chemistry, to adsorbate-inducedsurface phase transitions, to non-ideal adsorption isotherms,etc. There are very few, if any, experimental examplesof this class.In the author’s belief, practically all systems that have been assumedto belong to this class actually belong to class 2 or 3, described below. Whenever current oscillations are observed, one must always make absolutely sure that the effect of ohmic drop is minimized, by minimizing the electrolyte resistivity, the electrode surface area, etc. If this is done, and the oscillations disappear,they do not belong to class 1. For instance, there have been several (more or less implicit) claims in the literature that the oxidation of formic acid on (single-crystal) platinum oscillates under truly potentiostatic conditions, but it seemsthat in none of the cases has the utmost been done to minimize ohmic drop. In fact, the results described above for the presumably related oxidation of formaldehyde on rhodium lend support to the hypothesis that the oxidation of small organic molecules does not give rise to “class 1 oscillations”. There are, however, a few systemsthat may be related to class 1. An example of a system that is unstable under truly potentiostatic conditions, but doesnot oscillate, is the electrocrystallization of zinc in a so-called Leclanche’cell, as studied by Epelboin et al. [28]. This system exhibits an S-shaped polarization curve, and the middle branch is unstablepotentiostatically. The peculiar shapeof the polarization curve must indeed be interpreted by invoking a

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special kind of chemistry, and Epelboin et al. [28] have suggesteda model which includes an autocatalytic step to reproduce their data. A second example of this type is electrodissolution of iron in nitric acid, studied by the same group of Gabrielli, Epelboin and coworkers [29,30]. The reduction of nitric acid, which acts as a strong oxidizing agent for the iron, has often been argued to be autocatalytic. Another system that we tentatively range in class 1 is the oscillatory electrodissolution of silicon (p-type in the dark, n-type under illumination) in fluoride media, far in the electropolishing potential region. This is basically because the impedancedata for this system as measuredby Ozanam et al. [31] are so complicated and unusualthat it is impossible to range this system in one of the other categories. Chazalviel and Ozanam [32] have presented a model of incoherently oscillating microdomainsthat reproduces their data impressively, although the physical mechanismfor thesepotentiostatically oscillating domains is not understood. 5.2. Class 2. Oscillations only under potentiostatic tions with suficiently large ohmic drop

condi-

These systemsmust show a region of negative slope in their steady-state current-voltage curve, and the current oscillations occur exactly around this branch in the current-voltage curve when a sufficiently large ohmic potential drop is present. Under galvanostatic conditions, these systemsdisplay bistability but never oscillations. The negative polarization slope must clearly be related to some kind of inhibition of the electron-transfer process,owing to passivation, electrostatic repulsion, etc. There are many examples of class 2 electrochemical oscillators, the In3+ reduction as described in Section 3 being only one of them. 5.3. Class 3. Oscillations and under potentiostatic ohmic drop

under galvanostatic conditions conditions with suficiently large

These systemsmust have a hidden negative impedance, i.e. their frequency-dependent impedance must have a negative real part for a range of non-zero frequencies, but the zero-frequency impedanceis positive, reflecting a positive slope in the steady-state current-voltage curve. Such characteristics require the coupling of at least two potential-dependent processes:a relatively fast process giving rise to the negative impedance, and a slower processthat has a positive impedance and dominates the potential dependenceat steady state. Such systemsdo not necessarily require (although they usually have) a region of negative polarization slope; oscillations may still occur in spite of the absenceof a negative impedancein the steady-state current-voltage curve. Examples include formaldehyde oxidation on rhodium, hydrogen oxidation on platinum in the presence of elec-

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trosorbing cations and anions [ 1l- 131, electrodissolution of nickel in the transpassiveregion [3], and reduction of hydrogen peroxide on various semiconductors[3]. Electrochemical oscillators of class 2 or 3 are also distinguished by qualitatively different bifurcation diagrams. It was already mentioned at the end of Section 3 that a homoclinic bifurcation is typical for class3 electrochemical oscillators. Such bifurcations also occur for class 2 electrochemical oscillators, but in that casethey can only involve unstable limit cycles, so one will never be able to observe them. In two parameters, the typical phase diagram for class 2 electrochemical oscillators is the crossshapedor fish-shapeddiagram, as studied in Refs. [26] and [33]. For class 3 electrochemical oscillators, the phase diagram in two parameters is rather different and, as mentioned above, will always involve a homoclinic bifurcation as one of the boundary curves to the observable oscillatory region. Examples of such phase diagrams are given in Refs. [lo], [ 121and [13]. They are related to the existence of a highly degeneratebifurcation point (a kind of triple point) that cannot occur in systemswith a crossshapedphasediagram [34,35]. There is an interesting correspondence between the classification of electrochemical oscillators into systems that oscillate only under potentiostatic conditions, and systems that oscillate under both potentiostatic and galvanostatic conditions, and the classification that has been suggestedby Eiswirth et al. [36] for homogoneouschemical oscillators. There one distinguishesbetween systemsin which the de-stabilizing (autocatalytic) and the stabilizing (“exit reaction”) processeshave the same stoichiometry (“a critical current cycle”), and systems in which the de-stabilizing process is of a higher stoichiometric order than the stabilizing process (“a strong current cycle”). The former leadsto cross-shapedphasediagrams,the latter to phasediagramswith a homoclinic bifurcation of a stable limit cycle. Apparently, the symmetry that exists in the stoichiometric order of the stabilizing and de-stabilizing processesplays a crucial role. In Section 4 it was argued that a similar symmetry - in the potential dependenceof the massand charge balances- determinesthe classification of electrochemical oscillators.

References [ll J.L. Hudson [2] T.Z. Fahidy

and T.T. Tsotsis, Chem. Eng. Sci., 49 (1994) and Z.H. Gu, in R.E. White et al. (I%.),

1493. Modem

Chemistry

409 f 1996) 175-182

Aspects of Electrochemistry, Vol. 27, Plenum Press, New York, 1995 p. 383. Koper, Adv. Chem. Phys., 92(1996) 161. [31 M.T.M. Dokl. Akad. Nat&. SSSR, 133 (1960) 745. [41 Yu.A. Chizmadzev, El W.Schenk, Electrochim. Acta, 5 (1961) 301. 161J. Osterwald, Electrochim. Acta, 7 (1962) 523. [71 H. Degn, Trans. Faraday Sot., 64 (1968) 1348. @I R. de Levie, J. Electroanal. Chem., 25 (1970) 257. Corr. Ind., [91 C. Gabrielli, These, Universite de Paris-VI, 1973; M&IX 573 (1973) 171; 574 (1973) 223; 577 (1973) 309; 578 (1973) 356. [lOI M.T.M. Koper and J.H. Sluyters, J. Electroanal. Chem., 371 (1994) 149. [I 11W. Wolf, Thesis, Freie Universidt Berlin, 1994. W. Wolf, K. Krischer, M. Liibke, M. Eiswirth and G. Ertl, J. 1121 Electroanal. Chem., 385 (1995) 85. Koper, K. Krischer, M. Eiswirth and 1131 W. Wolf, M. Liibke, M.T.M. G. Ertl, J. Electroanal. Chem., 399 (1995) 185. and P. Holmes, Nonlinear Oscillations, Dynamical 1141 I. Guckenheimer Systems, and Bifurcations of Vector Fields, Springer, Berlin, 1986. 1151M.T.M. Koper, Thesis, Universiteit Utrecht, 1994. [161 E. Kreyszig, Advanced Engineering Mathematics, Wiley, New York, 1988. Analysis and Feedback Amplifier Design. D. [I71 H.W. Bode, Network Van Nostrand, Princeton, NJ, 1945. Reading, 1181R.C. Dorf, Modem Control Systems, Addison-Wesley, MA, 1981. [191 H. Nyquist, Bell. Syst. Tech. J., 11 (1932) 126. 1201F.B. Llewelyn, Proc. IRE, 21 (1933) 1532. 1211J.R. Macdonald (Ed.), Impedance Spectroscopy, Wiley, New York, 1987. 1221M.T.M. Koper, P. Gaspard and J.H. Sluyters, J. Phys. Chem., 96 (1992) 5674; J. Chem. Phys., 97 (1992) 8250. [231M.T.M. Koper and P. Gaspard, J. Phys. Chem., 95 (1991) 4945; J. Chem. Phys., 96 (1992) 7797. [241R. de Levie and L. Pospisil, J. Electroanal. Chem., 22 (1969) 277. B. Timmer and J.H. Sluyters, Z. Phys. Chem., [=I M. Sluyters-Rehbach, NF 52 (1967) 1. I261M.T.M. Koper, Electrochim. Acta, 37 (1992) 177 I. I271M. Hachkar, B. Beden and C. Lamy, J. Electroanal. Chem., 287 (1990) 81. 1281I. Epelboin. M. Ksomi, E. Lejay and R. Wiart, Electrochim. Acta, 20 ( 1975) 603. and H. Takenouti, [291C. Gabrielli, M. Keddam, E. Stupinek-Lissac Electrochim. Acta, 21 (1976) 757. 1301I. Epelboin, C. Gabrielli and M. Keddam, in J.O’M. Bockris, B.E. Conway, E. Yeager and R.E. White (Eds.), Comprehensive Treatise of Electrochemistry, Vol. 4, Plenum Press, New York, 1984 p. 151. J.-N. Chazalviel, A. Radi and M. Etman, J. Elec[311 F. Ozanam, trochem. Sot., 139 (1992) 2491. [=I J.-N. Chazalviel and F. Ozanam, J. Electrochem. Sot., 139 (1992) 2501. Ber. [331 W. Wolf, J. Ye, M. Purgand, M. Eiswirth and K. Doblhofer, Bunsenges. Phys. Chem., 96 (1992) 1797. Physica D, 20 (1986) I. 1341 J. Guckenheimer, Koper, J. Chem. Phys., 102 (1995) 5278. 1351 M.T.M. [%I M. Eiswirth, A. Freund and J. Ross, Adv. Chem. Phys., 80 (1991) 127.