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On some electrochemical oscillators at the mercury ∣ water interface
Journal of Electroanalytical Chemistry 552 (2003) 223 /229 www.elsevier.com/locate/jelechem
On some electrochemical oscillators at the mercury j water interface Robert de Levie * Bowdoin College, Department of Chemistry, Brunswick, ME 04011, USA Received 18 September 2002; received in revised form 12 November 2002; accepted 19 November 2002 In honor of the 70th birthday of Prof. Damaskin, who has contributed so much to our understanding of the electrochemical properties of the mercury j water interface.
Abstract The principles underlying electrochemical oscillators on mercury have been known for three decades. They are (1) the existence of a negative charge transfer resistance, (2) the general coupling between interfacial kinetics and mass transport, and (3) the inclusion of the effects of additional impedances, such as the double layer capacitance and any series (film, solution, and external) resistances. The charge transfer resistance needs to be defined in general rather than within the confines of a specific model, such as that of Erdey-Gru´z and Volmer. The effects of the double layer capacitance and of series resistance can be incorporated by using general stability criteria for electrical circuits. The above effects suffice for a quantitative representation of several types of oscillations observed at the mercury j water interface. Electrochemical oscillations are typically non-linear, and closed-form analytical solutions are not available. Models in which diffusion is approximated in terms of Nernst diffusion layers are useful for experiments enforcing stationary mass transport conditions, as with rotating disk electrodes, but do not apply to measurements on hanging or dropping mercury electrodes. Digital simulations do not require such steady-state approximations. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Oscillator; Electrochemical oscillator; Negative charge transfer resistance
1. Introduction Almost from the first kinetic observations, oscillating reactions have intrigued chemists. There are two general classes of such oscillating reactions: homogeneous and heterogeneous. Oscillations observed at the electrode j solution interface, like those observed at the membrane j solution interface (including the electrochemical oscillations observed in signal transmission along nerves), are inherently interfacial, although they may involve bulk processes such as mass transport. The first description in the literature of an electrochemical oscillator was given in 1828 by Fechner [1], even before galvanometers had been developed to measure electrochemical currents. What Fechner reported were repetitive bursts of effervescence (release of gas bubbles) from iron immersed in nitric acid.
* Tel.: /1-207-725-3028; fax: /1-207-725-3017. E-mail address: [email protected] (R. de Levie).
Much later, Heathcote [2,3] and Lillie [4 /6] emphasized the similarities in the appearance of electrochemical oscillations such as are observable on a passivating iron wire (the ‘iron nerve’) and the nerve impulse. However, neither process was understood at the time, and their apparent analogy led only to a short-lived interest: one equation between two unknowns does not a solution yield. At any rate, we now know that the similarity was only superficial: different non-linear differential equations can yield similar-looking cyclic instabilities, but this implies neither a common mechanism nor even a common mathematical description. Many electrochemical oscillators are observed with solid electrodes, and involve interfacial films such as are encountered in metal passivation. Their description is complicated because not enough is known yet about the kinetics of formation and dissolution of such films. Fortunately, several electrochemical oscillations on mercury do not involve interfacial films, and can therefore be expected to be more amenable to quantitative interpretation. It is this small but more readily studied
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subgroup which will be considered in the present overview. For examples on solid electrodes the reader may want to consult more comprehensive reviews [7 /13].
2. What makes these oscillators tick? We list here the common features of such electrochemical oscillators [14]. First of all, they all involve a negative charge transfer resistance. Below we will define the charge transfer resistance more precisely, but for now it suffices to state that, in the necessarily limited range of potentials over which a charge transfer resistance can be negative, the net interfacial charge transfer rate becomes less oxidizing as the potential becomes more oxidizing, or less reducing as the potential becomes more reducing. In other words, the negative charge transfer resistance reflects an unusual sign of the first derivative of the net reaction rate with potential. Some possible reasons for such seemingly bizarre behavior will be discussed below. Secondly, the interfacial process apparently must be coupled to some time-dependent process, in our case diffusion, such that the negative charge transfer resistance induces an inductive phase shift. Thirdly, as these oscillators involve shifting charge from one ‘reservoir’ to another, they apparently require at least two phase-shifting phenomena, with phase shifts containing opposing components. In the examples discussed below, these two elements are provided by a pffiffiffi pffiffiffiffi negative diffusion pseudo-capacitance 2=(Rct RW v); see below, and a normal, i.e. positive double layer capacitance Cdl. Finally, the oscillations are observed only when the external control of the potential is less than rigid, a condition achieved here by the insertion of a resistance inside the control loop. With solid electrodes, such a series resistance is often supplied by a passivating film, thus obviating the need for an external resistance, but complicating the description when that resistance varies during the oscillations due to film formation or dissolution. Under galvanostatic conditions, the analog of an external resistance can be found in the output impedance of the control circuit.
3. Qualitative description We will first try to explain qualitatively how such a combination of factors can lead to oscillations. Even though electrochemical oscillations reflect dynamic, kinetic processes, it will be useful to use as our reference the stationary current /voltage curve, such as the envelope of a normal polarogram. A common, ‘reversible’ stationary current /voltage curve has the form of a
Fig. 1. Schematic steady-state current /voltage curves. (a) The typical curve of an interfacial oxidation, or for the corresponding reduction, in which case both current and voltage have the opposite signs. (b) The same when the charge transfer resistance is negative between Va and Vb. The decrease in the current as the applied potential becomes more strongly oxidizing (or the equivalent decrease in the absolute value of the reduction current as the potential becomes more strongly reducing) can be caused by a variety of mechanisms, such as catalysis of the electrode process by an adsorbed ion, coulombic repulsion between the electroactive species and the electrode charge, or interference of the electrode kinetics by an adsorbate.
rounded step, as in Fig. 1a, with a shape described by the Heyrovsky´ /Ilkovicˇ equation for a polarographic wave [15] when (as will be assumed throughout the remainder of this discussion) diffusion of reagents and products towards and from the interface is the currentlimiting process at sufficiently extreme potentials. In this overview, we will be concerned with electrochemical reductions on mercury, and we will therefore use the polarographic convention of showing negative (reduction) currents and negative (reducing) voltages in the first quadrant. But for the sake of clarity we will first explain the phenomenon for oxidations, for which they can indeed be observed, as in the oxidation of Eu(II) / Eu(III) in dilute, acidified aqueous solutions [16]. When, in such a system, the applied voltage V is stepped at time t/0 from an initial value V0 (where no electrochemical reaction takes place) to a final value V1 (in the region of the limiting current), we can expect the
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resulting current I to be a transient reflecting the timedependence of diffusion. For sufficiently fast electrochemical reactions and planar diffusion, the current (except at very short times t) would be inversely proportional to t 1/2, as the system maintains the interfacial concentration c ? at zero. We now consider a system with a region of negative charge transfer resistance, with a stationary current / voltage curve such as that of Fig. 1b. Following a similar, stepwise change in applied voltage from V0 to V1 the current I will be much smaller than that in Fig. 1b. When we place a resistance R in series with the electrochemical cell, inside the feedback loop that controls the applied voltage V , then the potential E applied to the interface is given by E V IR
(1)
Immediately after the voltage step at t /0, the magnitude of the current I is large, so that E will be smaller than V . As the current I decays, and with it the corresponding voltage drop IR across the resistance, the interfacial potential E approaches the applied voltage V . So far nothing special. However, the system may now become unstable in the following way. As the current is much below its diffusion-limited value, diffusion will tend to increase the interfacial concentration to make it approach the bulk value c*. Now consider some fluctuation by which the interfacial potential temporarily becomes somewhat less negative. The corresponding rate of the electrode reaction is larger, so the current I increases. This, in turn, will increase the IR drop across the resistance, thereby reducing the interfacial potential E . But that only increases the rate of the electrochemical reaction, and hence the magnitude of the current I, and finally the amount of the voltage drop IR, further reducing E , etc. This, then, is a case of positive feedback, in which an arbitrarily small perturbance can start a move of the interfacial potential towards smaller values as the interfacial reaction rate increases. The process will stop eventually, e.g. when the potential reaches the region of the diffusion-limited current, where the current is independent of the interfacial potential E , and where the interfacial concentration c0 is maintained at zero. As the current decreases because of a gradual depletion of the electroactive species in the immediate neighborhood of the electrode, the magnitude of the IR drop decreases as well, and the current more or less retraces what happened immediately after the voltage step, although usually on a slightly shorter time scale. This sequence can repeat itself many times, until the surroundings of the electrode are sufficiently depleted of the electroactive material to end the oscillations. In the presence of forced mass transport and replenishment of the solution, the process
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could even be maintained indefinitely, which would be the electrochemical equivalent to using a continuously stirred reactor in studies of homogeneous chemical oscillations. The above, hand-waving explanation, though qualitatively correct, ignores the kinetic aspects of the electrochemical oscillations. Still, it allows us to recognize the major factors involved. The primary players are the negative charge transfer resistance (without which there would be no positive feedback during the initial part of the oscillatory excursion), and the resistance R needed to relax the control of the applied voltage V over the interfacial potential E . The diffusion coefficient is involved because it is diffusion that brings the oscillatory excursion to a halt by reducing the interfacial concentration of the electroactive reagent to zero, and at other times restarts the process again by supplying fresh electroactive material to the interfacial region. Finally, least obvious player is the interfacial capacitance C , which provides the opposing phase shift, and thereby influences how quickly the interfacial potential can change.
4. Quantitative considerations The charge transfer resistance Rct is most conveniently defined as [17,18] 1 1 @k @k c?red ox c?ox red Rct (2) nFA @E @E where n denotes the number of electrons involved in the net reaction, e.g. n/3 for the reduction of In(III) to In(0), and n/2 for the reduction of BrO4 to BrO3 or of Cu(II) /Cu(0). F is the Faraday constant, A the electrode area in contact with the solution, c ? red is the interfacial aqueous concentration of the reduced (and therefore oxidizable) form of the redox couple (in all our examples, c ? red will be zero), c ? ox the interfacial concentration of its oxidized (hence reducible) form (i.e. the interfacial concentrations of In(III), BrO4, or Cu(II), respectively), kox and kred the potential-dependent rate constants for oxidation and reduction, respectively, and E the interfacial potential. The constants n , F , and A in Eq. (2) are all positive, as are the interfacial concentrations c ? red and c ? ox, and the rate constants kox and kred. The sign of Rct is therefore determined exclusively by that of the derivatives of the rate constants k with respect to the potential E . Typically, the term @kox/@E will be positive, and @kred/@E negative, as it is in the usual model description of electrochemical kinetics of Erdey-Gru´z and Volmer [19], so that Rct will be positive. However, as we will see below, special conditions can be found under which Rct is negative over a limited range of potentials. Under the
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experimental conditions pertaining to the specific examples considered below, the oxidation reaction can be shown to be negligible, so that we can set the term c ? red (@kox/@E ) equal to zero. Under such conditions, the sign of @kred/@E will determine that of Rct. The coupling of mass transport to interfacial kinetics can be described in general terms [20]. Here, we will merely consider semi-infinite planar diffusion in the absence of any oxidation process, in which case the kinetic impedance Zk associated with the interfacial redox reactions can be shown [18] to be simply the product of the interfacial resistance Rct and the diffusional impedance which, for planar diffusion, can be written specifically as pffiffiffiffiffi (3) Zk Rct RW = jv pffiffiffiffiffi where RW = jv is called the Warburg impedance [21]. Consequently, for Rct B/0, the quadrature component of Zk has an inductive phase angle or, more precisely, has the frequency dependence of the above-mentioned negative pseudo-capacitance. This dependence of the sign of the phase angle on that of Rct had remained hidden as long one assumed a fixed model for electrode kinetics that implied Rct /0, such as that of Erdey-Gru´z and Volmer [19]. The interfacial (differential) double layer capacitance Cdl is usually a positive, thermodynamic quantity. In electrical circuit terms, the negative diffusion capacitance and the positive double layer capacitance provide two reservoirs between which the charge can slosh during an oscillation. Of course, not all current /voltage curves with ‘descending’ branches lead to electrochemical oscillations, and the particular kinetics of each specific system must be examined. For instance, the Fe(II)-catalyzed reduction of ClO2 studied by Gierst et al. [22] exhibits a kinetic impedance with a negative in-phase component and a positive quadrature component [23], and therefore is non-oscillatory. Current /time curves such as are obtained with linear sweep or cyclic voltammetry also exhibit decreases in the reduction (or oxidation) current with increasingly reductive (or oxidative) potentials, but these are purely diffusion-controlled phenomena although they have occasionally been misinterpreted in terms of negative interfacial impedance. The same applies to the currents in the decreasing branch of a polarographic maximum.
5. The thiocyanate-catalyzed reduction of indium(III) The reduction of In3 at the mercury j water interface is a rather slow process, but is greatly accelerated at higher pH-values where indium forms hydroxy species [24], and at low pH in the presence of a number of
adsorbable ligands [25]. Here we will consider one such adsorbable ligand, the pseudo-halide thiocyanate anion, SCN , which catalyzes the indium reduction through the complexation reaction of two adsorbed thiocyanate anions with the incoming In(III) species to form an adsorbed In(SCN)2 complex [17,26]. This complex can then be reduced, leading to a ‘reversible’ polarographic wave at potentials where, in the absence of such catalysis, virtually no reduction would take place. Eventually, at sufficiently negative potentials, hydrogen is reduced, and the resulting pH change at the electrode interface causes a resumption of the indium reduction, in this case catalyzed by OH . We will disregard this latter region of potentials, which has a positive charge transfer resistance, and is therefore of no particular interest in the present context. As the potential is made more negative, the electrode acquires a more negative charge density, and thiocyanate anions gradually desorb, thereby reducing the . Conseconcentration of adsorbed thiocyanate, SCNads quently the rate of the SCNads-catalyzed reduction falls, producing an example of the above-mentioned pathological decrease of the reduction rate as the potential becomes more strongly reducing. Experimentally, the reduction rate constant kred can be obtained readily as a function of potential E from the polarographic wave, using the ratio of the actual current to its diffusion-limited value [27], see Fig. 2. The polarographic experiment also yields the value of the diffusion coefficient D , while capacitance measurements can provide numerical values for the double layer capacitance Cdl which, like kred , is a function of E . As it turns out, in this case both ln kred and C are approximately linear functions of E over the range of potentials covered by the oscillations. Consequently, software simulations that parameterize kred and C at that level can represent the oscillations with surprising realism [28]1, see Fig. 3. The importance of this result lies in the demonstration that such a simulation apparently does not omit any significant effects. We note here that all experimental results were obtained on stationary (hanging mercury drop) electrodes, yet invariably show transient rather than steadystate oscillations. This reflects the transient nature of spherical diffusion; in none of these cases was a steadystate oscillation observed. (Unlike planar diffusion, which has no steady-state, spherical diffusion theoretically can eventually reach a non-zero steady-state. But in practice, by that time, poorly reproducible ‘natural’ convection will have taken over.) Likewise, the simulations produce transient rather than steady-state oscillations. The same applies to all the experimental data 1 The actual test function used in the simulation was the fourparameter function ln k /a sin(bE/c )/d .
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Fig. 2. (a) The polarographic currents at the end of the mechanically controlled drop time (open circles). The curve shown merely connects the data points. (b) The reduction rate constant k calculated from the polarographic currents in (a). As shown by the drawn line, the calculated data (open circles) in this particular case can be fitted quite well by an equation of the form ln k/a0/a1E .
reported below for the other chemical systems studied, which were all obtained on hanging mercury drop electrodes.
6. The reduction of perbromate In this case the mechanism causing the negative charge transfer resistance is the coulombic repulsion of
the reducible anion, BrO4. Similar oscillations can be observed with the reduction of peroxydisulfate S2O2 8 and many other anions studied by Frumkin et al. [29 / 31], as well as in the oxidation [16] of Eu2 to Eu3. We [32] have studied the oscillations in the reduction of BrO4 because this anion shows little tendency towards either adsorption or ion-pairing [33], two factors that often complicate the mathematical analysis. Unfortunately, in the dilute solutions necessary to obtain
Fig. 3. Comparison of the experimental and simulated current oscillations in the reduction of 1 mM In(NO3)3 in aqueous 2.5 M NaSCN at 18 8C and pH 3.15. Panels (a), (c), (e), (g) observed oscillations, at the indicated applied voltage, all with an added 20 kV resistance in series with the measurement cell (i.e. inside the potential control loop). Panels (b), (d), (f), (h): The current /time response simulated for these same conditions, using parameters determined from experiments in the absence of the external resistance, where the system is non-oscillatory and therefore allows the use of standard electrochemical methods to determine its kinetic and equilibrium parameters. Reprinted with permission from R. de Levie, J. Phys. Chem. 102 (1998) 4405, copyright 1998 American Chemical Society.
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significant repulsion between the anion and the electrode charge density, the interfacial capacitance has a pronounced minimum due to the minimum in the Gouy (diffuse) double layer capacitance at the potential of zero charge. As a result, the double layer capacitance Cdl is a highly non-linear function of potential E in the potential range of interest, and the agreement between experiment and our current simulation software (which is limited to a linear dependence of Cdl on E ) is consequently less satisfactory. Similar observations have been made for tetrathionate [34], where the simulation is similarly limited. Moreover, tetrathionate is specifically adsorbed [35], which further complicates the picture.
7. The reduction of copper(II) in the presence of tribenzylamine The polarographic reduction of copper has a rather positive half-wave potential, outside the range of adsorption of many organic neutral and cationic species. As the applied potential is made more negative, adsorption may start to interfere with the copper reduction, again leading to a negative charge transfer resistance. This is the case with tribenzylamine in acidic media [36,37]. Again, the system oscillates reproducibly, and ln kred is again an essentially linear function of E over most of the range of potentials involved in the oscillations [38]. In this case, quantitative agreement between the measurements and our simulation is again limited by the non-linear dependence of the capacitance on potential, which our present software cannot represent. Computationally this is not a trivial matter, and the computation is quite sensitive to the dependence of capacitance on potential because of the large swings in E during an electrochemical oscillations. Moreover, at potentials slightly more negative than the oscillatory behavior, tribenzylamine can undergo a phase transition (with the usual hysteresis) leading to the formation of a compact film [39]. Here, then, we encounter a system on mercury that has a complexity (including film formation) approaching that of electrochemical oscillators at solid electrodes.
8. Summary The good agreement between simulation and experiment in the case of the indium thiocyanate oscillator suggests that we fully understand the principles involved in such oscillators. This understanding, reached some 30 years ago but only recently confirmed by digital simulation, has remained the common filling of variously relabeled bottles.
The complications in simulating the details of the electrochemical oscillations in the case of perbromate, tetrathionate, and peroxydisulfate are due to a limitation of our present simulation software, which as yet cannot represent the highly non-linear dependence of capacitance on potential associated with the potential of zero charge in solutions of low ionic strength. Similarly, the strongly potential-dependent adsorption of tribenzylammonium leads to a highly non-linear interfacial capacitance. But these appear to be limitations in a relatively minor detail of the digital simulation, rather than in our understanding of what makes these systems oscillate. At least on mercury, where these effects can be studied most quantitatively, we suspect it to be only a matter of time and effort before all such systems can be simulated quantitatively to within their experimental reproducibility, as we have done with the indium thiocyanate oscillator. An encouraging start in that direction has recently been made by Jurczakowski and Orlik [40]. On solid electrodes the situation is quite different as long as one uses practical electrodes rather than welldefined single-crystal phases. And even on the latter one may have to incorporate other potentially time-dependent effects, such as interfacial reconstruction, which would further complicate mathematical analysis and simulation. Moreover, many electrochemical oscillations on solid electrodes involve the formation and dissolution of passive layers, processes for which a full theoretical description is still lacking. Even though the majority of reported electrochemical oscillators have been observed on solid electrodes, a quantitative description of electrochemical oscillations has so far been achieved only at the mercury j solution interface, which usually is physically and chemically homogeneous, as well as readily reproducible. The approach described here differs from that taken by Koper et al. [41,42], who has attempted to describe the oscillations in closed form by using approximate models. This is a valuable alternative approach, but (as with digital simulations) its success or failure depends on the appropriateness of the assumptions made. Koper and Sluyters first used a single Nernst diffusion layer in order to simplify the diffusion problem [41], and subsequently Koper and Gaspard refined this model by using two concentric Nernst diffusion layers [42], a modification that led to more realistic-looking oscillations. The noticeable effect of such a minor change in modeling on the simulation indicates the pivotal role of mass transport, and the need to represent it properly. While Koper’s analysis yields interesting results, it is perhaps not always realized that modeling diffusion in terms of single or double Nernst diffusion layers (which assume all solution concentrations to be constant beyond those layers) leads to essentially steady-state oscillations. As we have emphasized before, on a
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hanging or dropping mercury electrode, both the amplitude and frequency of the oscillations are clearly time-dependent. Koper’s approach is therefore more appropriately applied to experimental conditions that enforce a steady-state, such as can be obtained with a rotating disk electrode. Unfortunately, we are not aware of any reported observations of steady-state electrochemical oscillations on mercury.
Acknowledgements The simulation program used was kindly provided by Dr Manfred Rudolph of the Institu¨t fu¨r Anorganische und Analytische Chemie of the Friedrich-Schiller-Universita¨t in Jena, Germany. The measurements on indium and tetrathionate were performed by Dr Magdalena Hromadova and Ms Toyia N. James at Georgetown University, those on perbromate and copper by Mr Michael A. Prendergast and Ms Chutikarn Butkinaree at Bowdoin College. The initial research was supported financially by the Air Force Office of Scientific Research, and subsequently (until experiments on mercury were no longer fashionable) by the National Science Foundation.
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T.Z. Fahidı, Z.H. Gu, Mod. Asp. Electrochem. 27 (1995) 383. M.T.M. Koper, Adv. Chem. Phys. 92 (1996) 161. K. Krischer, Mod. Asp. Electrochem. 32 (1999) 1. R. de Levie, J. Electroanal. Chem. 25 (1970) 257. J. Heyrovsky´, D. Ilkovicˇ, Coll. Czech. Chem. Commun. 7 (1935) 198. L. Gierst, P. Cornelissen, Coll. Czech. Chem. Commun. 25 (1960) 3004. R. de Levie, A.A. Husovsky, J. Electroanal. Chem. 22 (1969) 29. R. de Levie, L. Pospı´sˇil, J. Electroanal. Chem. 22 (1969) 277. T. Erdey-Gru´z, M. Volmer, Z. Phys. Chem. A 150 (1930) 203. S.K. Rangarajan, J. Electroanal. Chem. 55 (1974) 297, 329. E. Warburg, Ann. Phys. Chem. (3) 67 (1899) 493. L. Gierst, L. Vandenberghen, E. Nicholas, J. Electroanal. Chem. 12 (1966) 462. R. de Levie, J.C. Kreuser, H. Moreira, Anal. Chem. 43 (1971) 784. A.J. Engel, J. Lawson, D.A. Aikens, Anal. Chem. 37 (1965) 203. (a) D. Cozzi, S. Vivarelli, Z. Elektrochem. 57 (1953) 408; (b) D. Cozzi, S. Vivarelli, Z. Elektrochem. 58 (1954) 907. R. de Levie, L. Pospı´sˇil, J. Electroanal. Chem. 25 (1970) 245. J. Koutecky´, Coll. Czech. Chem. Commun. 18 (1953) 597. M. Rudolph, M. Hromadova, R. de Levie, J. Phys. Chem. 102 (1998) 4405. (a) A.Ya. Gokhsthein, A.N. Frumkin, Dokl. Akad. Nauk SSSR 132 (1960) 388; (b) A.Ya. Gokhsthein, A.N. Frumkin, Dokl. Akad. Nauk SSSR 144 (1962) 821. A.N. Frumkin, O.A. Petri, N.V. Nikolaeva-Fedorovitch, Dokl. Akad. Nauk SSSR 136 (1961) 1158. (a) A.Ya. Gokhsthein, Dokl. Aakd. Nauk SSSR 140 (1961) 1114; (b) A.Ya. Gokhsthein, Dokl. Aakd. Nauk SSSR 148 (1963) 136; (c) A.Ya. Gokhsthein, Dokl. Aakd. Nauk SSSR 149 (1963) 880. M.A. Prendergast, R. de Levie, unpublished results. R. de Levie, M. Nemes, J. Electroanal. Chem. 58 (1975) 123. T.N. James, R. de Levie, unpublished results. T. Ohsaka, R. de Levie, J. Electroanal. Chem. 99 (1979) 255. H. Jehring, U. Ku¨rschner, J. Electroanal. Chem. 75 (1977) 799. H.-D. Do¨rfler, E. Mu¨ller, J. Electroanal. Chem. 135 (1982) 37. C. Butkinaree, R. de Levie, unpublished results. R. de Levie, Chem. Rev. 88 (1988) 599. (a) R. Jurczakowski, M. Orlik, J. Electroanal. Chem. 478 (1999) 118; (b) R. Jurczakowski, M. Orlik, J. Electroanal. Chem. 486 (2000) 65. M.T.M. Koper, J.H. Sluyters, J. Electroanal. Chem. 303 (1991) 73. (a) M.T.M. Koper, P. Gaspard, J. Phys. Chem. 95 (1991) 4945; (b) M.T.M. Koper, P. Gaspard, J. Phys. Chem. 96 (1992) 7797.
On some electrochemical oscillators at the mercury j water interface Robert de Levie * Bowdoin College, Department of Chemistry, Brunswick, ME 04011, USA Received 18 September 2002; received in revised form 12 November 2002; accepted 19 November 2002 In honor of the 70th birthday of Prof. Damaskin, who has contributed so much to our understanding of the electrochemical properties of the mercury j water interface.
Abstract The principles underlying electrochemical oscillators on mercury have been known for three decades. They are (1) the existence of a negative charge transfer resistance, (2) the general coupling between interfacial kinetics and mass transport, and (3) the inclusion of the effects of additional impedances, such as the double layer capacitance and any series (film, solution, and external) resistances. The charge transfer resistance needs to be defined in general rather than within the confines of a specific model, such as that of Erdey-Gru´z and Volmer. The effects of the double layer capacitance and of series resistance can be incorporated by using general stability criteria for electrical circuits. The above effects suffice for a quantitative representation of several types of oscillations observed at the mercury j water interface. Electrochemical oscillations are typically non-linear, and closed-form analytical solutions are not available. Models in which diffusion is approximated in terms of Nernst diffusion layers are useful for experiments enforcing stationary mass transport conditions, as with rotating disk electrodes, but do not apply to measurements on hanging or dropping mercury electrodes. Digital simulations do not require such steady-state approximations. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Oscillator; Electrochemical oscillator; Negative charge transfer resistance
1. Introduction Almost from the first kinetic observations, oscillating reactions have intrigued chemists. There are two general classes of such oscillating reactions: homogeneous and heterogeneous. Oscillations observed at the electrode j solution interface, like those observed at the membrane j solution interface (including the electrochemical oscillations observed in signal transmission along nerves), are inherently interfacial, although they may involve bulk processes such as mass transport. The first description in the literature of an electrochemical oscillator was given in 1828 by Fechner [1], even before galvanometers had been developed to measure electrochemical currents. What Fechner reported were repetitive bursts of effervescence (release of gas bubbles) from iron immersed in nitric acid.
* Tel.: /1-207-725-3028; fax: /1-207-725-3017. E-mail address: [email protected] (R. de Levie).
Much later, Heathcote [2,3] and Lillie [4 /6] emphasized the similarities in the appearance of electrochemical oscillations such as are observable on a passivating iron wire (the ‘iron nerve’) and the nerve impulse. However, neither process was understood at the time, and their apparent analogy led only to a short-lived interest: one equation between two unknowns does not a solution yield. At any rate, we now know that the similarity was only superficial: different non-linear differential equations can yield similar-looking cyclic instabilities, but this implies neither a common mechanism nor even a common mathematical description. Many electrochemical oscillators are observed with solid electrodes, and involve interfacial films such as are encountered in metal passivation. Their description is complicated because not enough is known yet about the kinetics of formation and dissolution of such films. Fortunately, several electrochemical oscillations on mercury do not involve interfacial films, and can therefore be expected to be more amenable to quantitative interpretation. It is this small but more readily studied
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subgroup which will be considered in the present overview. For examples on solid electrodes the reader may want to consult more comprehensive reviews [7 /13].
2. What makes these oscillators tick? We list here the common features of such electrochemical oscillators [14]. First of all, they all involve a negative charge transfer resistance. Below we will define the charge transfer resistance more precisely, but for now it suffices to state that, in the necessarily limited range of potentials over which a charge transfer resistance can be negative, the net interfacial charge transfer rate becomes less oxidizing as the potential becomes more oxidizing, or less reducing as the potential becomes more reducing. In other words, the negative charge transfer resistance reflects an unusual sign of the first derivative of the net reaction rate with potential. Some possible reasons for such seemingly bizarre behavior will be discussed below. Secondly, the interfacial process apparently must be coupled to some time-dependent process, in our case diffusion, such that the negative charge transfer resistance induces an inductive phase shift. Thirdly, as these oscillators involve shifting charge from one ‘reservoir’ to another, they apparently require at least two phase-shifting phenomena, with phase shifts containing opposing components. In the examples discussed below, these two elements are provided by a pffiffiffi pffiffiffiffi negative diffusion pseudo-capacitance 2=(Rct RW v); see below, and a normal, i.e. positive double layer capacitance Cdl. Finally, the oscillations are observed only when the external control of the potential is less than rigid, a condition achieved here by the insertion of a resistance inside the control loop. With solid electrodes, such a series resistance is often supplied by a passivating film, thus obviating the need for an external resistance, but complicating the description when that resistance varies during the oscillations due to film formation or dissolution. Under galvanostatic conditions, the analog of an external resistance can be found in the output impedance of the control circuit.
3. Qualitative description We will first try to explain qualitatively how such a combination of factors can lead to oscillations. Even though electrochemical oscillations reflect dynamic, kinetic processes, it will be useful to use as our reference the stationary current /voltage curve, such as the envelope of a normal polarogram. A common, ‘reversible’ stationary current /voltage curve has the form of a
Fig. 1. Schematic steady-state current /voltage curves. (a) The typical curve of an interfacial oxidation, or for the corresponding reduction, in which case both current and voltage have the opposite signs. (b) The same when the charge transfer resistance is negative between Va and Vb. The decrease in the current as the applied potential becomes more strongly oxidizing (or the equivalent decrease in the absolute value of the reduction current as the potential becomes more strongly reducing) can be caused by a variety of mechanisms, such as catalysis of the electrode process by an adsorbed ion, coulombic repulsion between the electroactive species and the electrode charge, or interference of the electrode kinetics by an adsorbate.
rounded step, as in Fig. 1a, with a shape described by the Heyrovsky´ /Ilkovicˇ equation for a polarographic wave [15] when (as will be assumed throughout the remainder of this discussion) diffusion of reagents and products towards and from the interface is the currentlimiting process at sufficiently extreme potentials. In this overview, we will be concerned with electrochemical reductions on mercury, and we will therefore use the polarographic convention of showing negative (reduction) currents and negative (reducing) voltages in the first quadrant. But for the sake of clarity we will first explain the phenomenon for oxidations, for which they can indeed be observed, as in the oxidation of Eu(II) / Eu(III) in dilute, acidified aqueous solutions [16]. When, in such a system, the applied voltage V is stepped at time t/0 from an initial value V0 (where no electrochemical reaction takes place) to a final value V1 (in the region of the limiting current), we can expect the
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resulting current I to be a transient reflecting the timedependence of diffusion. For sufficiently fast electrochemical reactions and planar diffusion, the current (except at very short times t) would be inversely proportional to t 1/2, as the system maintains the interfacial concentration c ? at zero. We now consider a system with a region of negative charge transfer resistance, with a stationary current / voltage curve such as that of Fig. 1b. Following a similar, stepwise change in applied voltage from V0 to V1 the current I will be much smaller than that in Fig. 1b. When we place a resistance R in series with the electrochemical cell, inside the feedback loop that controls the applied voltage V , then the potential E applied to the interface is given by E V IR
(1)
Immediately after the voltage step at t /0, the magnitude of the current I is large, so that E will be smaller than V . As the current I decays, and with it the corresponding voltage drop IR across the resistance, the interfacial potential E approaches the applied voltage V . So far nothing special. However, the system may now become unstable in the following way. As the current is much below its diffusion-limited value, diffusion will tend to increase the interfacial concentration to make it approach the bulk value c*. Now consider some fluctuation by which the interfacial potential temporarily becomes somewhat less negative. The corresponding rate of the electrode reaction is larger, so the current I increases. This, in turn, will increase the IR drop across the resistance, thereby reducing the interfacial potential E . But that only increases the rate of the electrochemical reaction, and hence the magnitude of the current I, and finally the amount of the voltage drop IR, further reducing E , etc. This, then, is a case of positive feedback, in which an arbitrarily small perturbance can start a move of the interfacial potential towards smaller values as the interfacial reaction rate increases. The process will stop eventually, e.g. when the potential reaches the region of the diffusion-limited current, where the current is independent of the interfacial potential E , and where the interfacial concentration c0 is maintained at zero. As the current decreases because of a gradual depletion of the electroactive species in the immediate neighborhood of the electrode, the magnitude of the IR drop decreases as well, and the current more or less retraces what happened immediately after the voltage step, although usually on a slightly shorter time scale. This sequence can repeat itself many times, until the surroundings of the electrode are sufficiently depleted of the electroactive material to end the oscillations. In the presence of forced mass transport and replenishment of the solution, the process
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could even be maintained indefinitely, which would be the electrochemical equivalent to using a continuously stirred reactor in studies of homogeneous chemical oscillations. The above, hand-waving explanation, though qualitatively correct, ignores the kinetic aspects of the electrochemical oscillations. Still, it allows us to recognize the major factors involved. The primary players are the negative charge transfer resistance (without which there would be no positive feedback during the initial part of the oscillatory excursion), and the resistance R needed to relax the control of the applied voltage V over the interfacial potential E . The diffusion coefficient is involved because it is diffusion that brings the oscillatory excursion to a halt by reducing the interfacial concentration of the electroactive reagent to zero, and at other times restarts the process again by supplying fresh electroactive material to the interfacial region. Finally, least obvious player is the interfacial capacitance C , which provides the opposing phase shift, and thereby influences how quickly the interfacial potential can change.
4. Quantitative considerations The charge transfer resistance Rct is most conveniently defined as [17,18] 1 1 @k @k c?red ox c?ox red Rct (2) nFA @E @E where n denotes the number of electrons involved in the net reaction, e.g. n/3 for the reduction of In(III) to In(0), and n/2 for the reduction of BrO4 to BrO3 or of Cu(II) /Cu(0). F is the Faraday constant, A the electrode area in contact with the solution, c ? red is the interfacial aqueous concentration of the reduced (and therefore oxidizable) form of the redox couple (in all our examples, c ? red will be zero), c ? ox the interfacial concentration of its oxidized (hence reducible) form (i.e. the interfacial concentrations of In(III), BrO4, or Cu(II), respectively), kox and kred the potential-dependent rate constants for oxidation and reduction, respectively, and E the interfacial potential. The constants n , F , and A in Eq. (2) are all positive, as are the interfacial concentrations c ? red and c ? ox, and the rate constants kox and kred. The sign of Rct is therefore determined exclusively by that of the derivatives of the rate constants k with respect to the potential E . Typically, the term @kox/@E will be positive, and @kred/@E negative, as it is in the usual model description of electrochemical kinetics of Erdey-Gru´z and Volmer [19], so that Rct will be positive. However, as we will see below, special conditions can be found under which Rct is negative over a limited range of potentials. Under the
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experimental conditions pertaining to the specific examples considered below, the oxidation reaction can be shown to be negligible, so that we can set the term c ? red (@kox/@E ) equal to zero. Under such conditions, the sign of @kred/@E will determine that of Rct. The coupling of mass transport to interfacial kinetics can be described in general terms [20]. Here, we will merely consider semi-infinite planar diffusion in the absence of any oxidation process, in which case the kinetic impedance Zk associated with the interfacial redox reactions can be shown [18] to be simply the product of the interfacial resistance Rct and the diffusional impedance which, for planar diffusion, can be written specifically as pffiffiffiffiffi (3) Zk Rct RW = jv pffiffiffiffiffi where RW = jv is called the Warburg impedance [21]. Consequently, for Rct B/0, the quadrature component of Zk has an inductive phase angle or, more precisely, has the frequency dependence of the above-mentioned negative pseudo-capacitance. This dependence of the sign of the phase angle on that of Rct had remained hidden as long one assumed a fixed model for electrode kinetics that implied Rct /0, such as that of Erdey-Gru´z and Volmer [19]. The interfacial (differential) double layer capacitance Cdl is usually a positive, thermodynamic quantity. In electrical circuit terms, the negative diffusion capacitance and the positive double layer capacitance provide two reservoirs between which the charge can slosh during an oscillation. Of course, not all current /voltage curves with ‘descending’ branches lead to electrochemical oscillations, and the particular kinetics of each specific system must be examined. For instance, the Fe(II)-catalyzed reduction of ClO2 studied by Gierst et al. [22] exhibits a kinetic impedance with a negative in-phase component and a positive quadrature component [23], and therefore is non-oscillatory. Current /time curves such as are obtained with linear sweep or cyclic voltammetry also exhibit decreases in the reduction (or oxidation) current with increasingly reductive (or oxidative) potentials, but these are purely diffusion-controlled phenomena although they have occasionally been misinterpreted in terms of negative interfacial impedance. The same applies to the currents in the decreasing branch of a polarographic maximum.
5. The thiocyanate-catalyzed reduction of indium(III) The reduction of In3 at the mercury j water interface is a rather slow process, but is greatly accelerated at higher pH-values where indium forms hydroxy species [24], and at low pH in the presence of a number of
adsorbable ligands [25]. Here we will consider one such adsorbable ligand, the pseudo-halide thiocyanate anion, SCN , which catalyzes the indium reduction through the complexation reaction of two adsorbed thiocyanate anions with the incoming In(III) species to form an adsorbed In(SCN)2 complex [17,26]. This complex can then be reduced, leading to a ‘reversible’ polarographic wave at potentials where, in the absence of such catalysis, virtually no reduction would take place. Eventually, at sufficiently negative potentials, hydrogen is reduced, and the resulting pH change at the electrode interface causes a resumption of the indium reduction, in this case catalyzed by OH . We will disregard this latter region of potentials, which has a positive charge transfer resistance, and is therefore of no particular interest in the present context. As the potential is made more negative, the electrode acquires a more negative charge density, and thiocyanate anions gradually desorb, thereby reducing the . Conseconcentration of adsorbed thiocyanate, SCNads quently the rate of the SCNads-catalyzed reduction falls, producing an example of the above-mentioned pathological decrease of the reduction rate as the potential becomes more strongly reducing. Experimentally, the reduction rate constant kred can be obtained readily as a function of potential E from the polarographic wave, using the ratio of the actual current to its diffusion-limited value [27], see Fig. 2. The polarographic experiment also yields the value of the diffusion coefficient D , while capacitance measurements can provide numerical values for the double layer capacitance Cdl which, like kred , is a function of E . As it turns out, in this case both ln kred and C are approximately linear functions of E over the range of potentials covered by the oscillations. Consequently, software simulations that parameterize kred and C at that level can represent the oscillations with surprising realism [28]1, see Fig. 3. The importance of this result lies in the demonstration that such a simulation apparently does not omit any significant effects. We note here that all experimental results were obtained on stationary (hanging mercury drop) electrodes, yet invariably show transient rather than steadystate oscillations. This reflects the transient nature of spherical diffusion; in none of these cases was a steadystate oscillation observed. (Unlike planar diffusion, which has no steady-state, spherical diffusion theoretically can eventually reach a non-zero steady-state. But in practice, by that time, poorly reproducible ‘natural’ convection will have taken over.) Likewise, the simulations produce transient rather than steady-state oscillations. The same applies to all the experimental data 1 The actual test function used in the simulation was the fourparameter function ln k /a sin(bE/c )/d .
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Fig. 2. (a) The polarographic currents at the end of the mechanically controlled drop time (open circles). The curve shown merely connects the data points. (b) The reduction rate constant k calculated from the polarographic currents in (a). As shown by the drawn line, the calculated data (open circles) in this particular case can be fitted quite well by an equation of the form ln k/a0/a1E .
reported below for the other chemical systems studied, which were all obtained on hanging mercury drop electrodes.
6. The reduction of perbromate In this case the mechanism causing the negative charge transfer resistance is the coulombic repulsion of
the reducible anion, BrO4. Similar oscillations can be observed with the reduction of peroxydisulfate S2O2 8 and many other anions studied by Frumkin et al. [29 / 31], as well as in the oxidation [16] of Eu2 to Eu3. We [32] have studied the oscillations in the reduction of BrO4 because this anion shows little tendency towards either adsorption or ion-pairing [33], two factors that often complicate the mathematical analysis. Unfortunately, in the dilute solutions necessary to obtain
Fig. 3. Comparison of the experimental and simulated current oscillations in the reduction of 1 mM In(NO3)3 in aqueous 2.5 M NaSCN at 18 8C and pH 3.15. Panels (a), (c), (e), (g) observed oscillations, at the indicated applied voltage, all with an added 20 kV resistance in series with the measurement cell (i.e. inside the potential control loop). Panels (b), (d), (f), (h): The current /time response simulated for these same conditions, using parameters determined from experiments in the absence of the external resistance, where the system is non-oscillatory and therefore allows the use of standard electrochemical methods to determine its kinetic and equilibrium parameters. Reprinted with permission from R. de Levie, J. Phys. Chem. 102 (1998) 4405, copyright 1998 American Chemical Society.
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significant repulsion between the anion and the electrode charge density, the interfacial capacitance has a pronounced minimum due to the minimum in the Gouy (diffuse) double layer capacitance at the potential of zero charge. As a result, the double layer capacitance Cdl is a highly non-linear function of potential E in the potential range of interest, and the agreement between experiment and our current simulation software (which is limited to a linear dependence of Cdl on E ) is consequently less satisfactory. Similar observations have been made for tetrathionate [34], where the simulation is similarly limited. Moreover, tetrathionate is specifically adsorbed [35], which further complicates the picture.
7. The reduction of copper(II) in the presence of tribenzylamine The polarographic reduction of copper has a rather positive half-wave potential, outside the range of adsorption of many organic neutral and cationic species. As the applied potential is made more negative, adsorption may start to interfere with the copper reduction, again leading to a negative charge transfer resistance. This is the case with tribenzylamine in acidic media [36,37]. Again, the system oscillates reproducibly, and ln kred is again an essentially linear function of E over most of the range of potentials involved in the oscillations [38]. In this case, quantitative agreement between the measurements and our simulation is again limited by the non-linear dependence of the capacitance on potential, which our present software cannot represent. Computationally this is not a trivial matter, and the computation is quite sensitive to the dependence of capacitance on potential because of the large swings in E during an electrochemical oscillations. Moreover, at potentials slightly more negative than the oscillatory behavior, tribenzylamine can undergo a phase transition (with the usual hysteresis) leading to the formation of a compact film [39]. Here, then, we encounter a system on mercury that has a complexity (including film formation) approaching that of electrochemical oscillators at solid electrodes.
8. Summary The good agreement between simulation and experiment in the case of the indium thiocyanate oscillator suggests that we fully understand the principles involved in such oscillators. This understanding, reached some 30 years ago but only recently confirmed by digital simulation, has remained the common filling of variously relabeled bottles.
The complications in simulating the details of the electrochemical oscillations in the case of perbromate, tetrathionate, and peroxydisulfate are due to a limitation of our present simulation software, which as yet cannot represent the highly non-linear dependence of capacitance on potential associated with the potential of zero charge in solutions of low ionic strength. Similarly, the strongly potential-dependent adsorption of tribenzylammonium leads to a highly non-linear interfacial capacitance. But these appear to be limitations in a relatively minor detail of the digital simulation, rather than in our understanding of what makes these systems oscillate. At least on mercury, where these effects can be studied most quantitatively, we suspect it to be only a matter of time and effort before all such systems can be simulated quantitatively to within their experimental reproducibility, as we have done with the indium thiocyanate oscillator. An encouraging start in that direction has recently been made by Jurczakowski and Orlik [40]. On solid electrodes the situation is quite different as long as one uses practical electrodes rather than welldefined single-crystal phases. And even on the latter one may have to incorporate other potentially time-dependent effects, such as interfacial reconstruction, which would further complicate mathematical analysis and simulation. Moreover, many electrochemical oscillations on solid electrodes involve the formation and dissolution of passive layers, processes for which a full theoretical description is still lacking. Even though the majority of reported electrochemical oscillators have been observed on solid electrodes, a quantitative description of electrochemical oscillations has so far been achieved only at the mercury j solution interface, which usually is physically and chemically homogeneous, as well as readily reproducible. The approach described here differs from that taken by Koper et al. [41,42], who has attempted to describe the oscillations in closed form by using approximate models. This is a valuable alternative approach, but (as with digital simulations) its success or failure depends on the appropriateness of the assumptions made. Koper and Sluyters first used a single Nernst diffusion layer in order to simplify the diffusion problem [41], and subsequently Koper and Gaspard refined this model by using two concentric Nernst diffusion layers [42], a modification that led to more realistic-looking oscillations. The noticeable effect of such a minor change in modeling on the simulation indicates the pivotal role of mass transport, and the need to represent it properly. While Koper’s analysis yields interesting results, it is perhaps not always realized that modeling diffusion in terms of single or double Nernst diffusion layers (which assume all solution concentrations to be constant beyond those layers) leads to essentially steady-state oscillations. As we have emphasized before, on a
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hanging or dropping mercury electrode, both the amplitude and frequency of the oscillations are clearly time-dependent. Koper’s approach is therefore more appropriately applied to experimental conditions that enforce a steady-state, such as can be obtained with a rotating disk electrode. Unfortunately, we are not aware of any reported observations of steady-state electrochemical oscillations on mercury.
Acknowledgements The simulation program used was kindly provided by Dr Manfred Rudolph of the Institu¨t fu¨r Anorganische und Analytische Chemie of the Friedrich-Schiller-Universita¨t in Jena, Germany. The measurements on indium and tetrathionate were performed by Dr Magdalena Hromadova and Ms Toyia N. James at Georgetown University, those on perbromate and copper by Mr Michael A. Prendergast and Ms Chutikarn Butkinaree at Bowdoin College. The initial research was supported financially by the Air Force Office of Scientific Research, and subsequently (until experiments on mercury were no longer fashionable) by the National Science Foundation.
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