Many electrochemical systems are known to exhibit complex non-linear ... We
discuss the origin of electrochemical oscillations and argue that most of the ...
Non-linear phenomena in electrochemical systems Marc T. M. Koper L aboratory of Inorganic Chemistry and Catalysis, Eindhoven University of T echnology, 5600 MB Eindhoven, T he Netherlands
Many electrochemical systems are known to exhibit complex non-linear behaviour such as spontaneous oscillations of current or potential, complex and chaotic oscillations and associated bifurcation scenarios and spatial pattern formation when maintained far from thermodynamic equilibrium. We discuss the origin of electrochemical oscillations and argue that most of the currently known electrochemical oscillators belong to one of two di†erent classes. These classes are distinguished by the di†erent types of electrical control under which oscillations are observable and are characterized by qualitatively di†erent impedance spectra. The possibility of observing electrochemical oscillations under truly potentiostatic conditions is discussed brieÑy, as well as the role of global coupling, due to the electroneutrality condition in the formation of spatiotemporal patterns on the electrode surface.
1 Introduction Spontaneous oscillations of electric current or potential are a widespread phenomenon in electrochemistry. Observations of periodically varying current or potential during electrochemical reactions date back to the nineteenth century.1 Even though there were many publications on electrochemical oscillations long before, in the 1970s, chemical oscillations were recognized as an interesting subject of study, a coherent understanding of electrochemical oscillations and their various manifestations has emerged only during the last couple of years. This development is certainly related to the interest people from the non-linear chemical kinetics Ðeld started to develop in electrochemical systems. This interest has initiated the use of methods from the theory of dynamical systems and nonlinear science and has helped to unravel more systematically the origin of various non-linear dynamical phenomena of electrochemical systems. There are several recent reviews on electrochemical oscillations and spatiotemporal pattern formation,2h4 and even only a quick comparison of these papers with the much quoted 1973 review of Wojtowicz,5 reveals the substantial advances that have been made. For one thing, non-linear dynamists have exploited the marvelous opportunities electrochemistry o†ers to study non-linear systems. This has provided us with some of the most clear-cut experimental examples of important concepts in dynamical systems theory, such as di†erent routes to chaos, bifurcation diagrams, the e†ect of global coupling in spatially extended systems, and experimental chaos control. Furthermore, using the tools of bifurcation theory and non-linear science, electrochemists now have more precise and quantitative explanations for oscillations, spatiotemporal patterns and waves in electrochemical systems. Here, I want to give a brief and non-comprehensive overview of what has been achieved recently in the study of nonlinear phenomena in electrochemistry. I have attempted to write this article in such a way that it should be comprehensible to electrochemists with no previous knowledge of the concepts of bifurcation theory. My main reason for this is that, among electrochemists, there still seems to be a relatively widespread belief that oscillations are some kind of artefact related to an improper electrical control of the electrochemical cell. Especially in the older literature, one sometimes comes across such obfuscating statements as that oscillations are caused by the external circuitry. It must be emphasized that, although there is certainly an essential role of the external cir-
cuitry in electrochemical oscillations, or, more generally, of the way the electrochemical cell is controlled electrically, this is not equivalent to saying that the external circuitry is causing the oscillations. Such statements are not paying proper credit to the importance of the interfacial electrode kinetics. In fact, the heart of the unstable behaviour lies in the negative impedance characteristics of the faradaic processes at the electrode, but whether or not this gives rise to instabilities or oscillations depends crucially on the way the electrode is coupled electrically to the rest of the system. After a short outline of some important concepts of bifurcation theory, the relationship between bifurcation theory and a technique familiar to many electrochemists, namely impedance spectroscopy, is discussed. Impedance spectroscopy is a useful tool for classifying electrochemical oscillators, as will be shown in the next sections. We will discuss the origin of oscillations under conditions of a Ðxed applied potential for which the ohmic drop cannot be neglected, under galvanostatic conditions and, Ðnally, under truly potentiostatic conditions, for which any role of a residual uncompensated ohmic resistance can be ruled out. Finally, the spatial extension of the electrode will be taken into account to treat spatiotemporal phenomena and to specify the conditions under which a “ lumped Ï treatment of the interfacial electrochemical dynamics is justiÐed.
2 Stability analysis and bifurcations For a thorough understanding of the occurrence of spontaneous oscillations in physico-chemical systems far from thermodynamic equilibrium, some knowledge of the mathematical description of dynamical systems is indispensable. In this section, I will brieÑy discuss the main concepts that are necessary for a comprehension of the subsequent sections. My treatment will be incomplete and, at times inaccurate, and will mainly involve the introduction of jargon. To start with, it is useful to be very accurate about our usage of the terms variable and parameter. A parameter is a quantity that is controlled by the experimenter, and whose value can be changed at will. Examples are the externally applied potential, temperature, bulk concentration of a certain species, such as bulk pH, etc. A variable is a quantity that is “ chosen Ï by the system, and its value usually depends on the parameters. Examples are the electric current in a potentiostatic experiment, or the concentration of surface species when a surface reaction is taking place. Note that, in a galvanostatic experiment, electric current is no longer a variable, but a parameter. J. Chem. Soc., Faraday T rans., 1998, 94(10), 1369È1378
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Variables are usually time-dependent quantities. If we perturb the system by suddenly changing one of the parameters to a new value, variables usually undergo transients until they settle down onto a new state. Usually, this state is a steady state, or stationary state, in which the variables no longer depend on time. However, such a (steady) state can only be reached if it is stable. By stability we mean the systemÏs insensivity to small (thermal) Ñuctuations : if a small perturbation takes the system away from the steady state, the system will return to the stable state after the perturbation has died out. Instability therefore implies that any Ñuctuation, no matter how small, will make the system leave the steady state. Hence, an unstable state can never be observed experimentally. Apart from the steady state, another state is possible in dynamical systems : the oscillatory state. An oscillatory state can also be stable or unstable, in the sense that it recovers from small Ñuctuations or not. In general, spontaneous oscillations in a physico-chemical system are related to the fact that the stationary state is unstable. Therefore, the question of why stable spontaneous oscillations occur in a given system must always also involve the question of why the stationary state is not stable. The transition from a stable state to an unstable state is usually accompanied by a bifurcation. Hence, a bifurcation in a nonequilibrium system is very much like a phase transition in an equilibrium system : under the inÑuence of a certain control parameter, the system switches from one to the other type of qualitative behaviour. The analogy between the phase theory of equilibrium systems and the bifurcation theory of dynamical (non-equilibrium) systems is useful for a Ðrst grasp of the understanding of what bifurcation theory entails, but is not, of course, entirely accurate. It is, however, very helpful to treat a state of a dynamical system on the same footing as a phase of an equilibrium system. As we have solid and liquid phases in equilibrium systems, we have stationary and oscillatory states in a dynamical system. However, one possible misconception must be cleared out of the way. In the equilibrium phase theory a metastable phase is sometimes considered unstable. However, a metastable state is unstable only with respect to large Ñuctuations but is stable with respect to small Ñuctuations and, hence, considered stable in terms of bifurcation theory. A typical example of an unstable state is the transition state which separates the less stable (metastable) state from the more stable state. Realizing this, it will be clear to every physical chemist that, although an unstable state cannot be observed experimentally, it plays an immensely important role in the overall behaviour of the dynamical system. Bifurcation theory is a mathematical machinery which enables us to predict if and when a certain bifurcation can occur in a set of di†erential equations which is assumed to model some kind of dynamical system, such as an electrochemical system. Bifurcation theory also tells us about the properties of a certain bifurcation, properties which are often universal, enabling us to identify unequivocally the occurrence of that particular bifurcation in an experimental system. Good introductions to the application of bifurcation theory to chemical kinetics can be found in the books of Nicolis6 and Gray and Scott,7,8 the in-depth mathematical details can be found in the books of Guckenheimer and Holmes9 and Wiggins.10 Three bifurcations are of particular importance in the classiÐcation of electrochemical oscillators that will be discussed in the ensuing sections. Two of these bifurcations, the socalled saddle-node and Hopf bifurcations, are local in nature and can be assessed from a linear stability analysis of the stationary state. Linear stability analysis is a standard and elementary technique which studies the stability of stationary states by linearizing the systemÏs di†erential equations about the stationary state. Since stability is deÐned in terms of recovering from small Ñuctuations, the non-linear terms are 1370
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small enough to be ignored. Solving the linearized equations amounts to calculating the eigenvalues and eigenvectors of the associated (Jacobian) matrix of coefficients. Stability is then uniquely determined by the sign of the eigenvalues : negative eigenvalues imply a damping out of the Ñuctuations, i.e. stability ; one or more positive eigenvalues imply an exponential growth of the Ñuctuations, i.e. instability. Since linear stability analysis looks at the behaviour of the system in the immediate neighborhood of the stationary state, it derives only local properties. The third bifurcation, the so-called homoclinic bifurcation, is global in nature and requires a more involved non-linear stability analysis. There exists a complicated way to predict if a given model of a system contains a homoclinic bifurcation but, in practice, its existence will be assessed from a numerical integration of the di†erential equations. The stability of a given stationary state may change under the inÑuence of a control parameter k. As mentioned, typical control parameters in electrochemical systems are the applied potential, applied current (in the galvanostatic mode), bulk concentrations, pH, temperature, etc. For a critical value of the control parameter, for which one of the eigenvalues becomes exactly zero, a bifurcation occurs : A saddle-node bifurcation occurs when one of the eigenvalues passes through zero. Two steady states merge in a saddle-node bifurcation, as shown in the bifurcation diagram of Fig. 1(a). When k is driven beyond the saddle node, a sudden jump in the systemÏs response usually occur. The linear stability analysis does not allow one to say what will be the new behaviour of the system : usually it settles down into another steady state [as in the example shown in Fig. 1(a)] or onto a periodic solution (i.e. oscillatory state) of the system [not shown in Fig. 1(a)]. Note that saddle-node bifurcations, in practice, always come in pairs and lead to hysteresis. Hysteresis occurs when a cyclic sweep is made through the bifurcation diagram of Fig. 1(a), as illustrated by the dashed line, which depicts the response of the variable to the parameter sweep. In between the two saddle-node bifurcations, the system can be in one of two stable steady states. Which one is chosen depends on where one comes from, or, more generally, on the initial conditions. This situation is known as bistability. A Hopf bifurcation occurs when the real part of a pair of complex conjugate eigenvalues passes through zero. In a Hopf bifurcation, the stationary state becomes unstable and an oscillatory state is born. If the Hopf bifurcation is supercritical, the oscillatory state is stable and oscillations will always be observed. If the Hopf bifurcation is subcritical, the oscillatory state is unstable and oscillations may or may not be observed. The bifurcation diagram of a supercritical Hopf bifurcation is shown in Fig. 1(b). Typical for a supercritical Hopf bifurcation is that, close to the bifurcation, the oscillations are small-amplitude, almost sinusoidal oscillations.
Fig. 1 Generic bifurcation diagrams near (a) a saddle-node bifurcation and (b) a Hopf bifurcation. Solid lines indicate stable steady states, dashed indicate instability. For the oscillatory state the maximum and minimum of the oscillation are shown. In (a) the thin dashed line illustrates the path a cyclic parameter sweep would follow, leading to hysteresis in the systemÏs response.
The amplitude grows with the square of the distance of k from the critical k value. A Hopf bifurcation is a very generic way by which oscillations may appear or disappear in a dynamical system. If one disposes of a model which is to be investigated for the existence of sustained oscillatory behaviour, it is common practice to Ðrst linearize the systemÏs equations and to check for the existence of a Hopf bifurcation. In a model with a sufficient number of control parameters, this is usually a very reliable test whether oscillatory solutions can occur or not. A homoclinic bifurcation occurs if an oscillatory state collides with a stationary state of the saddle type. This is illustrated in the phase plane picture of Fig. 2(a). The phase plane is the plane spanned by the two variables X and Y , say the interfacial electrode potential and the surface concentration of a certain species. Note that an oscillatory solution is a closed orbit in the phase plane, also known as a limit cycle. A stationary state is a point in the XÈY phase plane. Two steady states are shown in Fig. 2(a) : the unstable steady state around which the limit cycle has developed, and a saddle-type steady state. At the critical parameter value k , the limit cycle conc nects to the saddle in a homoclinic orbit. The homoclinic orbit is, therefore, an oscillatory state of inÐnite period. When approaching k , the period of the oscillation will grow c logarithmically ; beyond the bifurcation, oscillations no longer occur. Fig. 2(b) clariÐes the situation in a bifurcation diagram, where the steady-state value or the extrema of the oscillation of the variable X are plotted vs. k. In this picture we assumed that the oscillatory state was born earlier in a Hopf bifurcation. It is also clear how this oscillatory state touches a saddle-type steady state in a homoclinic bifurcation. The saddle-type unstable steady state is related to the existence of a saddle-node bifurcation at yet another value of k. In Fig. 3 we show an example of a the current response as a result of slow voltage sweep (10 mV s~1) during the electrodissolution of nickel in a 1 M sulfuric acid solution. In this Ðgure all three above-mentioned bifurcations take place. Here, the applied voltage V is the bifurcation parameter ; in series with the working electrode there is an ohmic resistance of 12 k). At low potential, the dissolution current is low and in a stable steady state. When scanning the potential in the posi-
Fig. 3 Cyclic voltammogram of the dissolution of nickel wire in 1 M sulfuric acid. Scan rate 10 mV s~1. Inset shows the period of the oscillation as the voltage approaches the critical value V . In series with 2 resistor. Referthe nickel working electrode there was a 12 k) ohmic ence electrode : SCE.
tive direction, a critical voltage is passed, where the current starts oscillating spontaneously. The oscillations develop smoothly from the steady state : the system undergoes a Hopf bifurcation. The oscillations are stable over quite a wide voltage range, until, at V they suddenly terminate by drop2 ping to a lower current value where the system is in a steady state again. This bifurcation is a homoclinic bifurcation, as is evidenced by the fact that the period of oscillation indeed varies logarithmically with the distance from the bifurcation point, as is shown in the inset of Fig. 3. After the potential reversal, the system remains on the steady-state branch. Apparently, in between the voltages V and V the system can 1 2 be in one of two possible stable behaviours (bistability) : a stable oscillation at high currents, and a steady state at lower current. Which behaviour is chosen by the system depends on the initial conditions ; in fact, by a suitably chosen perturbation, one may switch from one to the other type of behaviour, at one and the same externally applied voltage. The two stable behaviours are separated by an unstable steady state of the saddle type (or, more precisely, by the stable manifold of the saddle9) ; it is, in fact, this saddle which is involved in the homoclinic bifurcation at V . At the voltage V , during the 2 1 return scan, a new transition is observed : the system suddenly switches to the higher-current oscillating branch of the forward scan. This is due to a saddle-node bifurcation. Note that, during the return scan, the oscillations do not become extinct at the same potential as they started during the forward scan. This is a dynamic delay e†ect due to the Ðnite scan rate.11,12
3 Bifurcation theory and impedance spectroscopy
Fig. 2 (a) Phase plane representation of a limit cycle bifurcating in a homoclinic orbit and disappearing. (b) Typical bifurcation diagram belonging to the situation illustrated in (a).
The idea of linearizing the systemÏs equations about a stationary reference state will appear familiar to many electrochemists : it lies at the heart of impedance spectroscopy, a popular electrochemical measuring technique.13 It is intuitively clear that, if one determines experimentally the systemÏs linearized response over a wide enough frequency range, one essentially has all the linear information of the system and, hence, it must be possible to make statements about the systemÏs stability. In an impedance spectroscopy experiment, one drives the electrochemical systems by a small-amplitude sinusoidal voltage modulation superimposed on a constant potential, and measures the resulting small amplitude sinusoidal current oscillations. The amplitude and the phase shift of the current modulation with respect to the voltage modulation together determine the complex impedance of the system. The real J. Chem. Soc., Faraday T rans., 1998, V ol. 94
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impedance, Z@, is given by (*E/*I) cos /, and the imaginary impedance, ZA, by (*E/*I) sin /, where *E and *I are the amplitudes of the voltage and current modulation and / is the phase shift. By plotting ZA vs. Z@ as a function of the modulation frequency u, one obtains an impedance spectrum. The theorem which establishes the relation between a systemÏs frequency response or impedance spectrum and stability is the Nyquist stability criterion.14 In its most rigorous and formal form, the Nyquist criterion is not particularly enlightening or easy to use, as it makes use of the theory of complex functions and conformal mappings. I prefer a simpler test which is entirely equivalent and which is based on the fact that it is very easy and intuitively clear to recognize the possibility of bifurcations from an impedance spectrum.3,15 Recall that a bifurcation is a transition point where the system switches from one type of qualitative dynamics to the other. Linear stability analysis predicts that, at a bifurcation, small perturbations do not die out or grow and, hence, the solution of the linearized equations must be a sine wave, of Ðnite frequency u in the case of a Hopf bifurcation, and of H zero frequency in the case of a saddle-node bifurcation. If the system is driven externally by a sine wave of exactly that critical frequency, a condition of resonance is encountered for which, in the ideal, linear case, the driving signal is blown up by the system without any phase delay. Hence, a bifurcation is a situation for which the impedance spectrum intersects the origin of the complex impedance plane. If the intersection occurs for 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, an impedance spectrum is measurable only if the system is stable. It is at this point that the role of the external circuit must be invoked. As we will see in the remainder of this paper, the majority of electrochemical oscillators lose their stability due to ohmic losses in the external circuit. Since the inÑuence of an external ohmic resistance on the impedance spectrum amounts to a simple horizontal shift in the complex impedance plane, it is easy to test graphically for the possibility of a Hopf or saddle-node bifurcation from the impedance spectrum measured potentiostatically in the absence of an appreciable external ohmic drop or with the potentiostatÏs automatic IR compensation turned on. This test includes the galvanostatic control, since this situation corresponds ideally to an inÐnite series resistance and an inÐnite applied potential, with their ratio equalling the Ðxed applied current. Hence, one must have inÐnitely negative real impedance and zero imaginary impedance in order to have a bifurcation under galvanostatic control, which is equivalent to zero real and imaginary admittance. To summarize, we have the following conditions for detecting bifurcations from impedance characteristics (with Z denoting impedance and Y admittance) : For Ðxed applied potential (one may call this potentiostatic, provided it is realized that there must be a sizable ohmic drop), an electrochemical cell will exhibit : A saddle-node bifurcation if Z(u) \ 0, u \ 0. A Hopf bifurcation if Z(u) \ 0, u \ u D 0. H For Ðxed applied current (galvanostatic), an electrochemical cell with exhibit : A saddle-node bifurcation if Y (u) \ 0, u \ 0. A Hopf bifurcation if Y (u) \ 0, u \ u D 0. H
thiocyanate or chloride solution at a hanging mercury drop electrode (HMDE). The oscillations in this system were Ðrst investigated by Tamamushi16 and de Levie.17 Fig. 4 shows the polarogram of the indium reduction at the HMDE. A region of negative polarization slope is clearly recognized, the origin of which will be discussed further below. In the potential region of negative slope, a typical impedance spectrum looks like that shown in Fig. 5(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. 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 k) is connected in series with the s working electrode, since then the impedance spectrum will intersect the origin for a frequency of ca. 30 Hz. The corresponding applied potential V is E ] IR \ [1.15 V, where s I \ [17.8 lA is the steady-state current at E \ [1.075 V. Therefore the point (V , R ) \ ([1.15 V, 4.2 k)) lies on the s Hopf bifurcation curve in the phase diagram spanned by V and R . The entire curve, as extrapolated from spectra for 15 s di†erent electrode potentials, is given by the solid line in Fig. 5(b). The dots in Fig. 5(b) represent the onset of oscillations as actually observed by inserting an adjustable ohmic resistor in series with the cell. Clearly, the Hopf bifurcation, as measured in the time domain, matches exactly the curve measured in the frequency domain. This conÐrms the theory of the previous section and shows that impedance spectroscopy is an easy and accurate way to perform an experimental linear stability analysis of electrochemical systems. The phase diagram has the familiar Ðsh-shape that is predicted by simple models.18 Oscillations occur inside the head of the Ðsh. A voltametric scan through this region is shown in Fig. 6(a). Inside the tail of the Ðsh, oscillations do not occur, rather, this region is characterized by a bistability of two steady states. This is illustrated by the scan in Fig. 6(b), where the hysteresis typical for this kind of bistability is observed. In this region of the V ÈR parameter plane the Hopf bifurcation s is subcritical and does not give rise to stable, observable oscillations. Of course, impedance spectroscopy allows only a detection of local bifurcations such as Hopf and saddle-node. In reality, much more complicated bifurcations also occur and Fig. 7 shows a more complete phase diagram of the various possible
4 Oscillations at Ðxed applied potential 4.1 Experiment As an example of current oscillations at Ðxed applied potential, we will discuss the reduction of In3` from a concentrated 1372
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Fig. 4 Polarogram and steady-state currentÈvoltage curve for a solution of 11.3 mM In3` in 5.0 M NaSCN at an HMDE. No external resistor. Reference electrode : mercury pool in the same solution.
Fig. 5 (a) Impedance diagram for a 9 mM In3` solution in 5.0 M NaSCN at E \ [1.075 V at the HMDE. This particular steady state exhibits a Hopf bifurcation for R \ 4.2 k) and a corresponding s applied potential V \ E ] IR \ [1.15 V vs. SCE, with I \ [17.8 s lA the steady-state current. Indicated frequencies in Hz. (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. Hatched area shows the parameter region where sustained oscillations are observed. Reference electrode : SCE.
oscillatory waveforms that can occur during the In3` reduction at an HMDE.19 Most conspicuous are the so-called mixed-mode oscillations, waveforms consisting of alternating small and large amplitude excursions. These oscillations are recurrent in many oscillating reactions, not only electrochemical reactions.20,21 Mixed-mode oscillations follow complicated but highly regular bifurcation sequences, and we owe some of the most beautiful experimental examples of these bifurcation sequences to electrochemistry, such as the copper electropolishing in phosphoric acid, as studied by Schell and co-workers,22 and the indium reduction at the HMDE.19 These sequences also contain chaotic oscillations, a subject reviewed in greater detail in ref. 3. An example of the so-called strange attractor of a chaotic oscillation observed experimentally during the In3` reduction at the HMDE from thiocyanate solution is shown on the front cover of this issue. 4.2 Mechanistic interpretation The occurrence of spontaneous oscillations during the indium reduction from thiocyanate solution at the HMDE can be rationalized by the following simple model.23,24 First, we have to give an explanation for the negative polarization slope that is observed in Fig. 4. Pospisil and de Levie25 proposed that the thiocyanate, which is speciÐcally adsorbed at the mercury/ solution interface, acts as a catalyst for the otherwise slow indium reduction : In3` ] 2 SCN ~ ] In(SCN) ` ads 2, ads In(SCN) ` ] 3e~ ] In(Hg) ] 2 SCN ~ 2, ads ads
(I) (II)
Fig. 6 Voltammograms of the In3` reduction at the HMDE in the presence of an external resistor R ; cell composition as in Fig. 5. (a) s R \ 10 k). (b) R \ 30 k). s s
where the Ðrst step is rate determining. Because of its negative charge, the thiocyanate ion is repelled from the mercury with progressively negative potential, and this implies that, in the potential region where the thiocyanate desorbs from the surface, the catalysing action of the thiocyanate is lost. Hence, in this potential region, the rate of catalysis decreases with increasing polarization, and this explains the negative polarization slope observed experimentally. A similar e†ect is seen for the indium reduction from chloride solution, or for the nickel reduction from thiocyanate solution. The oscillations can be understood by coupling the charge balance of the electrochemical cell to the mass balance of the reactive species. The total current Ñowing through the cell is equal to the sum of the double-layer charging current and the faradaic current. It is also equal to the potential drop across any external resistance, divided by that resistance. Hence : V [E dE \ I ] I \ AC ] AJ (1) C F d F R dt s where I denotes current, J current density, A electrode surface area and C the double-layer capacity. Rewritten explicitly as d a di†erential equation for the electrode potential E, one has I\
dE V [ E C \ [J d dt F AR s The faradaic current is given by FaradayÏs law :
(2)
J \ [nFk(E)c (3) F s where n and F have their usual meaning, k(E) is the heterogeneous rate constant, i.e. the rate of the rate-determining reaction, and c is the surface concentration of the In3`. Note s J. Chem. Soc., Faraday T rans., 1998, V ol. 94
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Fig. 7 Bifurcation phase diagram for the In3`/SCN~ system where regions of stationary, oscillatory and complex oscillatory behaviour are mapped onto the V ÈR plane. Cell conditions as in Fig. 4. s
that, for the In3` reduction from thiocyanate solution, the rate constant k(E) must lead to a negative dJ (E)/dE in a F certain potential region. A simple equation for the time dependence of the surface concentration c can be obtained by making a di†usion-layer s approximation. Within the di†usion layer of thickness d (which in fact equals the radius of the stationary mercury sphere) the concentration proÐle is, at all times, linear. Taking into account the reaction at the surface and the mass transport through the di†usion layer, one Ðnds the following equation for dc /dt23 s D dc 2 s\ [k(E)c ] (c [c ) (4) s d bulk s dt d
C
D
The combination of eqn. (2)È(4) may be shown to reproduce qualitatively the phase diagram of Fig. 5(b). The existence of a Hopf bifurcation can easily be established analytically, and the existence of a limit cycle can also be proved by a so-called PoincareÈBendixson analysis.3 Clearly, the model also qualitatively reproduces the impedance spectrum of Fig. 5(a). The di†usion-layer approximation which led to eqn. (4) is only rigorously correct under steady-state conditions. Eqn. (4) essentially entails that the concentration proÐle in the electrolyte always immediately assumes its steady-state proÐle, even under oscillatory conditions. However, under time-dependent conditions, the correct approach would be to solve FickÏs second law with the appropriate boundary conditions. Numerically, this would mean a substantial increase in computational e†ort as the solution side of the interface would have to be divided into a large number of compartments to model the partial di†erential equation. Koper and Gaspard26 have suggested a simpler model which divides the di†usion layer into only two compartments, where the second compartment, i.e. the one between the surface and the bulk, approximately takes into account the delay with which the 1374
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concentration proÐle will settle down into its steady state. It turns out that this simpliÐed approach is already sufficient to reproduce the entire phase diagram of Fig. 7, including the mixed-mode and chaotic oscillations, in qualitative detail. The remarkable fact about oscillatory reduction reactions at mercury is that they all show qualitatively exactly the same kind of phase diagram as in Fig. 7, including the mixed-mode oscillations.24 This suggests that the above model not only applies to In3` reduction from thiocyanate solution, but also to the In3` reduction from chloride solution, the Ni2` reduction from thiocyanate solution, the Cu2` reduction from a solution containing tribenzylamine, and the PtCl 2~ 4 reduction from an unsupported solution. All these systems show a region of negative polarization slope which, in the presence of a sufficiently large external resistance, gives rise to spontaneous current oscillations. All the systems exhibit these oscillations in a potential region where there is mixed kinetic and di†usion control. Although the mechanistic origin of the negative impedance is quite di†erent for various systems (desorption of a catalyst, adsorption of an inhibitor, doublelayer repulsion), they are all described by essentially the same mathematical model whose properties are clearly insensitive to the precise chemical origin of the negative dJ (E)/dE. This F strongly suggests that it is indeed the combination of negative impedance, external resistance, and relatively slow mass transport which accounts not only for the simple oscillations, but also for the complex mixed-mode oscillations. A very similar story also applies to simple reduction reactions at gold and platinum rotating disk electrodes (RDE). Wolf et al.27 have studied the peroxodisulfate S O 2~ 2 8 reduction at gold and platinum RDEs from an unsupported or poorly supported electrolyte solution, a system which possesses a region of negative polarization slope due to the double-layer repulsion of the reactive anion. The phase diagram they Ðnd is completely similar to that shown in Fig. 5(b). Koper28 has reported that, under appropriate conditions, these systems can also give rise to mixed-mode oscillations, which behave very similarly to those observed in the indium reduction at the HMDE. This again suggests that a qualitatively similar model should apply to both systems. A Ðnal word about the external resistance. The values of the external resistance, in the order of k)s, needed for oscillations in e.g. Fig. 7 may seem very high and may give a rather artiÐcial impression. However, this is largely due to the fact that the current in this system is low, in the order of lAs. In many practical systems, currents are rather of the order of mAs, and hence the external resistance needed to destabilize a negative impedance in such a system would be roughly three orders of magnitude smaller, i.e. the order of several (tens of) )s. This is very close to a typical uncompensated ohmic cell resistance. Several recent studies have shown that it is in fact the uncompensated ohmic cell resistance which destabilizes the negative impedance of the activeÈpassive transition in many metal electrodissolution systems and leads to current oscillations.29,30
5 Oscillations at Ðxed applied current 5.1 Experiment The indium reduction discussed in the previous section gives rise to current oscillations under potentiostatic control in the presence of a sufficiently large uncompensated series resistance. In contrast, at Ðxed applied current, i.e. galvanostatic conditions, the indium reduction at mercury does not exhibit oscillations, only bistability of a low and high steady polarization state. This can be seen from the polarogram in Fig. 4 : drawing a horizontal line through the currentÈvoltage curve can result in three intersections : one at less negative potential, one at more negative potential, and one at intermediate
potential. The latter is unstable, the former two are the two bistable polarization states that the system can choose between at this particular current value. There are, however, many examples of systems that exhibit spontaneous potential oscillations under galvanostatic control. As we will see, these systems also oscillate under potentiostatic control in the presence of a series resistance, though not in quite the same way as the systems discussed in the previous section. It is, therefore, not entirely accurate to classify these two systems according to the type of electrical control under which they may exhibit oscillations. After we have discussed the origin of the oscillations in a typical “ galvanostatic oscillator Ï in this section, we will be able to formulate a more precise criterion to distinguish between the two classes of electrochemical oscillators. A typical example of a system which exhibits potential oscillations under galvanostatic control is the electrocatalytic oxidation of small organic molecules, such as methanol, formaldehyde and formic acid, at transition metals such as platinum, rhodium or iridium.31h37 Strasser et al.38 have recently formulated a simple model for formic acid oxidation at platinum which seems to capture all the essential features of the system. It seems reasonable to assume that this model should also provide the basic mechanism for the oscillations in the other systems. Let us Ðrst look at the typical experimental observations. We will do this for the formaldehyde oxidation at a rhodium RDE,31,36 as some good impedance spectra are available for this system. Fig. 8(a) gives the cyclic voltammogram of 0.1 M HCHO in 0.1 M NaOH at rhodium. The “ 0 ) Ï curve gives the stable cyclic voltammogram in the absence of an external
resistor. With the insertion of an external resistor, current oscillations appear on the forward scan (though there is a non-zero electrolyte resistance, it is not sufficiently large to induce oscillations), as is illustrated by the curves labelled “ 1000 ) Ï and “ 1500 ) Ï. Under galvanostatic control, potential oscillations are observed. What is noticeable is that the oscillations occur about a branch of the polarization curve with positive slope. If one measures the impedance spectra (under potentiostatic, stable, conditions) in that potential region, one observes the curves shown in Fig. 8(b). The curve with the open circles, taken at V \ [0.45 V, is typical for this type of oscillator. It is seen that the real part of the interfacial impedance is negative for a range of non-zero frequencies, but is positive for the lowest frequencies, in accordance with the positive polarization slope. From the theory described in a previous section on stability and impedance spectroscopy, one can deduce that such a spectrum will lead to an oscillatory instability under galvanostatic control. The negative impedance is in some sense “ hidden Ï,39 as it cannot be detected from the polarization curve, but does lead to the oscillations observed in the presence of an external resistor or at Ðxed applied current. The spectrum should be contrasted with the spectrum shown in Fig. 5(a), which is typical for a system which oscillates only under potentiostatic conditions with an external resistor but never under galvanostatic conditions. Note that the curves labelled “ 1000 ) Ï and “ 1500 ) Ï are very similar to the curve in Fig. 3 for the nickel electrodissolution. In fact, the sudden drop in current which accompanies the extinction of the oscillations during the forward scans, is a homoclinic bifurcation. The homoclinic bifurcation also occurs during the galvanostatic scan where a sudden transition to very positive potentials occurs. It is no coincidence that this occurs when the amplitude of the potential oscillations touches on the down-going branch of the “ 0 ) Ï cyclic voltammogram, as this branch is the galvanostatically unstable saddle to which the homoclinic orbit occurs. This homoclinic bifurcation is in fact typical for systems with a hidden negative impedance, and does not occur in the systems discussed in the previous section.
5.2 Mechanistic interpretation Strasser et al.38 have suggested a model for the oscillations during the formic acid oxidation on platinum single-crystal electrodes, a system which shows behaviour very similar to the formaldehyde oxidation on rhodium. A well accepted mechanism in the literature is that the formic acid oxidation follows a dual-path mechanism.40,41 In the direct path, the formic acid is oxidized via a reactive intermediate : HCOOH ] * ] ~COOH ] H` ] e~ (III) surf ads ~COOH ] CO ] H` ] e~ ] * (IV) ads 2 where “ * Ï denotes a free surface site. In the indirect path, the formic acid decomposes on the surface to the unreactive CO adsorbate : HCOOH ] * ] CO ] H O (V) surf ads 2 The CO is said to poison the surface, as it blocks surface sites that would otherwise be available to the direct oxidation path. The CO can be removed from the surface at higher potentials by OH adsorbates that are formed through the electrochemical water dissociation. The reactions are : Fig. 8 (a) Voltammogram of 0.1 M HCHO in 0.1 M NaOH on a rhodium RDE for a 0, 1000 and 1500 ) external resistance (internal cell resistance ca. 95 )). Scan rate 10 mV s~1, 3000 rev min~1. Amperogram taken at 0.01 mA s~1. (b) Impedance diagrams taken at [0.50 V (=), [0.45 V (L), [0.35 V (|). Indicated frequencies in Hz. Reference electrode : SCE.
H O ] * ½ OH ] H` ] e~ (VI) 2 ads CO ] OH ] CO ] H` ] e~ ] 2* (VI) ads ads 2 Note that the adsorbed OH also blocks surface sites for the direct path. Assuming that reaction (IV) is fast, the current J. Chem. Soc., Faraday T rans., 1998, V ol. 94
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Ñowing due to the direct path can be calculated by a steadystate approximation for the ~CO H adsorbate : 2 J \ FK(E, c )(1 [ h [ h ) (5) direct HCOOH CO OH where F is the Faraday constant and K is some expression depending on the potential and the formic acid concentration. Since the OH coverage increases with more positive potential, dJ /dE may become negative in a certain potential region direct and this process is held responsible for the negative faradaic impedance observed experimentally. This fact, in itself, could explain current oscillations under potentiostatic conditions (with ohmic drop), but it cannot explain the hidden negative impedance feature and the galvanostatic potential oscillations. To this end, we must invoke the role of adsorbed CO. Its coverage is determined by two di†erent potential-dependent processes : OH adsorption and CO oxidation reaction (VI). If this latter reaction would not occur, the surface would always be completely blocked and very little current would Ñow. However, owing to reaction (VI), a potential region exists where the CO coverage drops more quickly than the OH coverage rises and, hence, the overall dJ /dE is positive. direct However, since CO poisoning is a much slower process than OH poisoning, dJ /dE is, in fact, a frequency-dependent direct quantity. For high frequencies, only OH can respond and dJ /dE is negative ; lowering the frequency leads to a direct stronger response of CO and, for the lowest frequencies, dJ /dE becomes positive. This is exactly the chemical direct explanation of the hidden negative impedance shown in Fig. 8(b). Of course, this reasoning only applies to a certain potential region. For low potentials, dJ /dE is always positive direct due to the K(E), for high potentials, OH completely blocks the surface and dJ /dE will be negative for all frequencies direct (except for such high frequencies that even OH can no longer respond to the potential modulation). Strasser et al.38 have written the di†erential equations for E, h and h and have calculated cyclic voltammograms CO OH numerically. Some typical results from their model are shown in Fig. 9. The current oscillations are indeed found to occur about a branch with positive polarization slope, and it is also remarkable that, during the positive scan, the oscillations abruptly disappear owing to a homoclinic bifurcation. There have been several other suggestions, and models, for formic acid oxidation published in the literature, but the model of Strasser et al. stands alone as the model which does not include any assumption which is not commonly accepted in the literature. To conclude, systems which oscillate under both Ðxedapplied potential and Ðxed-applied current conditions require a hidden negative impedance. A hidden negative impedance
implies the coupling of two processes : a fast “ poisoning Ï process which governs the negative impedance, and a second potential-dependent process, which must be inherently slower, and which leads to a positive impedance at the lowest frequencies and, as a consequence, to a positive polarization slope. Another noteworthy example of a hidden negative impedance oscillator, that has also been studied in quite some detail by the Berlin group, is hydrogen oxidation on platinum in the presence of copper cations and chloride anions.42h44 Nickel electrodissolution in sulfuric acid also belongs to this class,45 as well as the hydrogen peroxide reduction on various semiconductors.46,47
6 Truly potentiostatic oscillations In the previous sections we have seen that, for electrochemical oscillations to develop, it is apparently necessary that the interfacial electrode potential is given a certain freedom to choose its own value, either by the presence of an uncompensated series resistance, or by driving the cell at constant applied current. What if the electrode potential is held strictly constant, by ruling out any residual uncompensated resistance, or by using automatic IR compensation ? Are oscillations possible under such truly potentiostatic conditions, where the electrode potential is a parameter rather than a variable, as in the previous two sections ? Current oscillations under truly potentiostatic conditions are certainly conceivable,39 as they are comparable to oscillations in heterogeneously catalysed reactions in the gas phase, of which many examples exist.48 Remarkably, however, there are no really convincing examples of such electrochemical oscillations in the literature. There are one or two systems that seem to completely fall outside the categories described in the two previous sections, silicon dissolution in Ñuoridecontaining electrolyte being the most prominent, but whether this system is really a truly potentiostatic oscillator remains somewhat unclear. Decaying oscillatory transients are observed in this system under conditions where the role of the series resistance can be reasonably neglected,49 but a series resistor is still needed to observe sustained oscillations.50 Chazalviel et al.51 have presented a model which describes the system as a collection of local oscillators which can be synchronized by the presence of an external resistor. What would be causing the local oscillations, however, is as yet not clear. If truly potentiostatic oscillations occur, then their explanation must be quite di†erent from those discussed in the previous two sections. From heterogeneously catalysed reactions in the gas phase, it is known that oscillations can occur on account of autocatalytic surface chemistry, deviations from ideal Langmuir adsorption due to lateral interactions, and adsorbate-induced substrate transformations.
7 Spatiotemporal phenomena
Fig. 9 Calculated voltammograms in the StrasserÈEiswirthÈErtl model.37 Parameter values are as in Fig. 4 of the original paper, but for di†erent values of the external resistor, as indicated in the Ðgure.
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In the previous sections it was assumed that the spatial extension of the electrode did not play a role in the dynamics. However, it is known since the early work of Ostwald52 and Franck53 that the electrode cannot always be considered as spatially homogeneous. It is, of course, well known that chemical instabilities may lead to spectacular spatiotemporal phenomena, such as spiral waves and other concentration patterns.6 These patterns result from the coupling of autocatalytic reactions and di†usion. In electrochemical systems, the situation is di†erent, as potential gradients that lead to migration of charged species may also exist. As migration has a very strong tendency to maintain electroneutrality at every location in the electrolyte and, hence, the potential distribution will, to a good approx-
imation, be governed by the Laplace equation, local disturbances tend to make their occurrence felt at large distances. This e†ect is known as global coupling.54 Flatgen and Krischer54h56 have studied the spatiotemporal phenomena that accompany the transition between two stable states or the oscillations during the peroxodisulfate reduction at silver ring or disk electrodes, a system that shows current oscillations at Ðxed applied potential (see Section 4). They Ðnd that the transition from one state to the other is mediated by fronts at the electrode. These fronts can be made visible by local potential measurements55 or by an elegant technique making use of surface plasmon spectroscopy.56 (During metal electrodissolution, where these fronts are also often observed, they can simply be observed by optical microscopy.)57 On the ring electrode, these fronts do not have a constant velocity, but accelerate during their travel.55 Such accelerating fronts have also been observed during cobalt electrodissolution.57 A model which couples the local charge balance of all the inÐnitesimally small elements of the electrode to the strict electroneutrality of the electrolyte solution, reproduces this front acceleration.54 The acceleration can be shown to originate from the global coupling constraint mentioned above. On the disk electrode, Flatgen et al.56 observed that, during the transition from a low-current to a high-current state, a front often starts from the rim of the electrode. This is understandable, as the current density at the rim is higher and therefore the “ work Ï to create a high-current nucleus is smaller at the rim. The relaxation back from the high-current to the low-current state seems, rather, to be accompanied by radially symmetric waves starting from the rim. Mazouz et al.58,59 have further studied the FlatgenÈ Krischer model to assess in more detail several manifestations of the global coupling. They Ðnd that the range of the coupling depends crucially on the length scales of the system i.e. the areal size of the working electrode relative to the distance between the working electrode and the equipotential plane provided by the counter or reference electrode, and that the strength of the coupling is proportional to the conductivity of the electrolyte. In particular, the range of the coupling has interesting e†ects. Long-ranged coupling brought about by a small electrode area or a long distance to the reference electrode, has the tendency to synchronize the events on the electrode surface. An ohmic resistor, connected in series with the working electrode, has the same synchronizing e†ect. Under these conditions the “ lumped Ï description of the previous sections becomes very accurate. Very short-ranged coupling results in “ ordinary Ï reactionÈdi†usion behaviour, as described in standard textbooks,7 and leads to waves of constant velocity. The intermediate range of “ non-local Ï coupling leads to the accelerating fronts mentioned in the previous paragraph. In the oscillatory regime, non-local coupling may lead to destabilization of the homogeneous oscillation mode and torus-like or irregular oscillations of the overall current may develop owing to the superposition of the homogeneous oscillation and various types of standing waves on the surface. As it is difficult to tune experimentally the range of the coupling without changing other properties of the system, it is as yet difficult to conÐrm these results experimentally. However, they do point towards the importance of considering spatial inhomogeneities in the explanation of complex or irregular oscillation patterns.
8 Conclusion We have discussed the origin of spontaneous oscillations in electro-chemical systems and the various types of non-linear phenomena this may give rise to. We found that the majority of electrochemical oscillators can be divided into two classes :3,4,15
1. Systems with a negative impedance, like that shown in Fig. 5(a), which always occur concurrently with a negative slope in the polarization curve. In the presence of a sufficiently large external resistor (which may be the internal cell resistance if the current is high enough) these systems give rise to current oscillations around the branch with negative slope. For very high external resistance, and in the galvanostatic limit, these systems show bistability and never oscillations. Typical examples are various reduction reactions at mercury, notably the In3` reduction from thiocyanate solution at the HMDE, the peroxodisulfate reduction at platinum and gold RDE, and various metal electrodissolutions at the activeÈ passive transition. 2. Systems with a hidden negative impedance, like that shown in Fig. 8(b), which occurs concurrently with a positive slope in the polarization curve. In the presence of a sufficiently large external resistor these systems give rise to current oscillations around the branch with positive slope. In the galvanostatic control, potential oscillations occur around the branch of positive slope. Typical examples are formic acid oxidation on platinum, formaldehyde oxidation on rhodium and platinum, hydrogen oxidation on platinum in the presence of copper cations and chloride anions, hydrogen peroxide reduction at GaAs semiconductor electrodes, and nickel electrodissolution in sulfuric acid. These two classes exhibit di†erent bifurcation diagrams ; the most conspicuous di†erence is the occurrence of a homoclinic bifurcation in systems with a hidden negative impedance, whereas this bifurcation is not expected to occur in the other class. It is possible to devise a more complete classiÐcation60 which includes classes that are conceivable but for which convincing examples are still missing ; the truly potentiostatic oscillations are an example of such a class. Furthermore, the systems with a hidden negative impedance can be further subdivided into two categories ;60,61 again, only for one of them are there convincing experimental examples. As for the formation of spatiotemporal patterns, it has become apparent that the global coupling constraint provided by the strong tendency to maintain local electroneutrality in the electrolyte plays an important role in electrochemical systems. The global coupling causes synchronization of the events at the electrode if the working electrode is small or the reference electrode is far away, but as the electrode area increases with respect to the distance from the equipotential plane, global coupling starts to break down, leading to accelerated wave fronts and complex oscillations as the spatially homogeneous oscillation becomes unstable. This latter phenomenon still awaits experimental conÐrmation. In this article, the focus was on the physico-chemical origin of oscillations and non-linear phenomena in electrochemical systems. It seems that the general picture now seems fairly well established, though many individual systems may of course have their proper peculiarities. From the electrochemical kinetics point of view, the biggest challenge seems to be to Ðnd a system that oscillates under truly potentiostatic conditions, without any role of an uncompensated ohmic resistance. Furthermore, there is also the modelling of complex oscillations that have been observed experimentally in many electrochemical systems. Complex oscillations may be due to the complexities of the underlying interfacial chemistry, or to spatial non-uniformities, although for the latter there is still no conclusive experimental evidence. Mixed-mode and chaotic oscillations occurring in the spatially uniform indium reduction at mercury were mentioned in Section 4. However, mixed-mode oscillations occur in many systems and are a type of complex oscillation that is very common and can be observed in most models that have at least three variables. More interesting are complex bursting oscillations such as have been observed during formic acid oxidation on platinum,62 iron electrodissolution in sulfuric acid in the presence J. Chem. Soc., Faraday T rans., 1998, V ol. 94
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of halides,63 and hydrogen peroxide reduction on platinum.64 They exhibit complex oscillation patterns that are really unique to these systems and, therefore, they have something to tell us about the underlying chemical kinetics. The main future direction of electrochemical oscillations and spatial pattern formation, however, seems to be to serve as a playground for the experimental application of concepts developed in the rapidly growing Ðeld of non-linear dynamics. The e†ect of global coupling on spatiotemporal pattern formation has already been mentioned. Furthermore, electrochemistry has already provided us with some of the richest and most clear-cut experimental examples of complex bifurcation scenarios and di†erent routes to chaos in real-world systems, and will continue to do so in the future. Another interesting application is chaos control, or control theory in general.61,65,66 In these respects, electrochemistry has some important advantages over other Ðelds of chemical kinetics :67 oscillations usually occur on a much more favourable timescale than, for instance, in heterogeneous catalysis or in liquidphase reactions, high-precision measurements and control are easily achieved, and the required instrumentation is relatively cheap. It is a pleasure to thank all my colleagues who have worked with me on the subject of electrochemical oscillations. I would like to mention two people in particular : Professor Jan Sluyters and Dr. Pierre Gaspard, whose e†orts, stimulation and support have been a great help. I also gratefully acknowledge the Royal Netherlands Academy of Arts and Sciences (KNAW) for supporting my research at Eindhoven University. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
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Paper 7/08897C ; Received 10th December, 1997
