5674
J. Phys. Chem. 1992, 96, 5674-5675
A One-Parameter Bifurcatlon Analysis of the IndiumRhiocyanate Electrochemical Oscillator Marc T.M. Koper,*.+Pierre Gaspard,*and J. H.Sluyterst Department of Electrochemistry, University of Utrecht. Padualaan 8, 3584 CH, Utrecht, The Netherlands, and Service de Chimie Physique, UniversitE Libre de Bruxelles, Campus Plaine CP 231, Boulevard du Triomphe, 1050 Brussels, Belgium (Received: March 4, 1992; In Final Form: May 12, 1992)
We present a short experimental study of the indium/thiocyanate electrochemical oscillator that is similar to a theoretical one-parameter bifurcation analysis we published earlier. The qualitative agreement between the present experiment and the predictions of the model is remarkable.
Introduction In an earlier Letter for this journal,’ we reported a one-parameter bifurcation analysis of a simple threevariable model that should qualitatively describe the current oscillations accompanying the indium(II1) reduction at a mercury electrode in a concentrated sodium thiocyanate solution. This reaction has been known to release (highly nonlinear) osciUating currents since 1970: but more complex behaviors like mixed-mode and chaotic oscillations have been reported only recently.) Comparable behaviors have been observed in a wide variety of electrochemical systems of different nature. These include the electrodissolution of copper in phosphoric acid4 or hydrochloric acid,5 the oxidation of hydrogen at platinum: the oxidation of formic acid at platinum,’ and the reduction of copper(I1) at mercury in the presence of triben~ylamine.~ Our present investigations aim at reexamining experimentally the indium/thiocyanate system in a more distinct relation with the detailed predictions of our threevariable modelas In the present Letter, we describe some preliminary experimental results by performing a simple and short one-parameter bifurcation analysis whose interest lies mainly in that it is qualitatively similar to the theoretical prediction published in our previous Letter. As it turns out, the agreement is remarkable. More detailed studies of the dynamic behavior as a function of two control parameters are left for a second paper. Experimental Section The electrochemical cell consisted of an EG&G Parc 303 static mercury drop electrode that served as the negative working electrode and a large mercury pool employed as the positive counter and reference electrode. (The solubility of the mercury thiocyanate being sufficiently low, the large mercury pool will also function as an internal reference electrode.) The described experiment applies to an aqueous electrolyte solution containing 7.3 mM In(N03)3.5H20 (Aldrich, 99.99%) and 5.0 M NaSCN (Merck, >98.5%). Its pH was adjusted to 3.2 by the addition of nitric acid (Merck, 65%) in order to prevent the formation of indium hydroxides. The size of the drop was approximately 0.05 cm2. The cell was thermostated at 5.0 OC (this appeared to give more stable signals than at room temperature), and oxygen was removed from the electrolyte solution by flowing argon through it before the experiment. In series with the cell an adjustable Ohmic resistance was connected. Potential control across the circuit (i.e., cell plus series resistance) was performed by an Ecochemie Autolab PSTAT 10 in its two-electrode configuration. The Autolab was connected to a Hewlett-Packard 54504A digitizing oscilloscope to follow the time evolution of the current and to an Olivetti PCS 368SX personal computer that was used for data acquisition and processing. *Towhom correspondence should be addressed. ‘University of Utrecht. f Universite Libre de Bruxelles.
0022-3654/92/2096-5674S03.00/0 I
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Results and Discussion The results described below apply to an experiment with fixed circuit potential V = -0,906 V (vs Hg/HgSCN). This potential lies in the region of negative faradaic impedance of the current-voltage characteristic of the system under the conditions described above. We study the system’s current response to varying the external resistance R,from a low to high value. Low-Current Stationary State. Initially, at low R, ( ~ 0 - 8kQ) the system is in a stable steady state of low current. While increasing R,, we observe that at R,= 8.4 kO the current starts oscillating with very small amplitude (Figure la) after a (supercritical) Hopf bifurcation. Low-Resisbaee Mixed-Mode Oscillatiorw The small-amplitude oscillation becomes marginally stable, and at R, = 8.5 kQ it changes into a high-period mixed-mode oscillation (MMO), depicted in Figure Ib, where a large number of growing small oscillations is interspersed with a fast and large relaxation spike. It is convenient to characterize such dynamical states with a symbolic notation Ls, L giving the number of large spikes and S the number of small-amplitudeoscillations in one periodic entity. Perfectly repetitive and stable states of the one depicted in Figure 1b are difficult to obtain because of the high sensitivity to system parameters. After rescaling the current axis of Figure 1b, we count 18 small oscillations in the first periodic entity and 17 in the second. Increasing the series resistance decreases the number of small oscillations until they eventually disappear (R,z 10.2 kQ). A typical 1 state is shown in Figure IC. Model simulations predict that, in between two principal MMO states Ls and LS-l,concatenations of these two states should exist which follow an incomplete Farey a r i t h m e t i ~ . ~Basically, ~~ such states are strongly dominant periodic windows in the chaotic regions of parameter space. The most easily observed periodic window is the chief concatenation between 1 and lo, i.e., the 1 l o or 2’ state, shown in Figure Id. Relaxation (kcitions. From R, = 10.2 to -24.0 kO the system exhibits its dominant behavior, being a simple spike-type relaxation oscillation (labeled High-Resistance Mixed-Mode Oscillations. On further increasing the external series resistance R,, the electrode potential effectively decreases and the system enters its diffusion-limited regime. However, the steady state associated with the diffusion-limited current is still unstable, and although the system stays in the neighborhood of the steady state most of the time, it repetitively oscillates out of it and thereupon returns again. This results in the mixed-mode oscillations of Figure 2, a and b, labeled l 4 and Is. They are preceded, at lower R,, by the l i , 12,and l 3 states. It is known that such states should make part of highly regular periodic-chaotic sequences.’ For the present potential, at the transition from one principal MMO to the other, we observed periodic concatenations like those of Figure Id, which are, as mentioned, the periodic windows of the chaotic region. The chaos in these sequences is not very robust on account of the strong i0).293
0 1992 American Chemical Society
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Letters
The Journal of Physical Chemistry, Vol. 96, No. 14, 1992 5675
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Conclusion The simple experiment described above-a oneparameter cross section of the control parameter spacecompares favorably with the results predicted by a numerical simulation of a three-variable model that we reported on previously.’ The qualitative agreement is more than satisfying, especially as regards the sequence of dynamical states that is observed with increasing the external series resistance. The three-variable model is based on a reaction mechanism proposed by Pospisil and de Levie’O in which the indium(II1) reduction is catalyzed by the thiocyanate ions. The thiocyanate ions are specifically adsorbed onto the mercury electrode surface and desorb from it at more negative electrode potentials, as a result of Coulombic repulsion. This effect lies at the heart of the observed negative faradaic impedance. A second important feature of the model, which appears essential to explain the complex dynamical behaviors, is the incorporation of a diffusion relaxation that is a kind of first-order correction for the deviation from diffusion-layer-like behavior under non-steady-state oscillatory conditions.8 It is also noteworthy that the bifurcation scenario described above is very similar to a typical one-parameter scenario for the copper electrodissolution in phosphoric acid, as studied by Schell and co-~orkers.~As we argued earlier,s this suggests that its explanation could be mechanistically equivalent to the one for the indium/thiocyanate system, that is, the coupling of a simple negative faradaic impedance with a diffusion relaxation caused by the non-steady-state behavior of the diffusion (or mass transport) layer. More details concerning the dynamic behavior with respect to the Farey arithmetic, the deterministicchaos, and a tweparameter bifurcation diagram will be presented in a future publication.
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dissipation in the time evolution of the electrode potential. (A closer look at Figure 2a shows, however, that the first couple of periods are slightly period doubled.) In mathematical terms, the capacity of the electrical double layer acts as a kind of smallness parameter that causes the system to stay close to its steady-state (faradaic) current value most of the time. At lower potentials, where the dissipation is weaker, one observes chaotic states which are more persistent, stable, and deterministic, as predicted by the modeL8 SlllalGAmplitUde oscitiom and High-curreat ShrtiONUy state. At about R, = 25.7 kQ, the system is close to the transition from the larger-amplitude MMOs to small-amplitudeoscillations. The small-amplitude oscillations exhibit chaotic transients but more easily observable are time series like in Figure 2c (R, = 25.8 k0) which is essentially of period two. At R, = 25.9 k0, this has become a simple monoperiodic small-amplitude oscillation, and at still higher R, the system is eventually allowed to reach-via a Hopf bifurcation-a stable steady state again, which is of high (diffusion limited) current.
Acknowledgment. P.G.is “Chercheur Qualifit” of the National Fund for Scientific Research (Belgium).
References and Notes (1) Koper, M. T. M.; Gaspard, P. J . Phys. Chem. 1991, 95,4945. (2) de Levie, R. J . Elecrroanal. Chem. 1970, 25, 257. (3) Koper, M. T. M.; Sluyters, J. H. J. Elecrroanal. Chem. 1991, 303,65. (4) Albahadily, F. N.; Ringland, J.; Schell, M. J . Chem. Phys. 1989, 90, 813. Schell, M.; Albahadily, F. N. J . Chem. Phys. 1989, 90, 822. (5) Bassett, M.R.;Hudson, J. L. J . Phys. Chem. 1988,92,6963. Bassett, M. R.; Hudson, J. L. J . Phys. Chem. 1989, 93, 2731. (6) Krischer, K.; LIibke, M.; Wolf, W.; Eiswirth, M.; Ertl, G. Ber. Bunsen-Ges. Phys. Chem. 1991, 95, 820. (7) Albabadily, F. N.; Schell, M. J. Elecrroanal. Chem. 1991, 308, 151. (8) Koper, M. T. M.; Gaspard, P. J . Chem. Phys. 1992, 96, 7707. (9) Glendinning, P.; Sparrow, C. J . Srat. Phys. 1984, 35,645. Gaspard, P.; Kapral, R.; Nicolis, G. J . Star. Phys. 1984, 35, 697. (10) Pospisil, L.; de Levie, R. J. Electroanal. Chem. 1970, 25, 245.
