J. Phys. Chem. 1995,99, 17403-17409
17403
Chaotic Behavior of Current Oscillations during Iron Electrodissolution in Sulfuric Acid Mauricio U. Kleinke Instituto de Fisica- UNICAMP, 13083-970 Campinas, SP,Brazil Received: July 13, 1 9 9 p
Current oscillations us time series were measured during iron electrodissolution in sulfuric acid solutions. The load line of the potentiostat was modified by an external resistance connected between the iron wire electrode and the potentiostat. The complex temporal behavior of these time series was diagnosed as deterministic chaos by the (i) reconstruction of the strange attractor, (ii) measurement of the GrassbergerRocaccia dimension (iii) calculation of the first Lyapunov exponent, and (iv) analysis of the power spectra. In the anodic scan, oscillations start and stop with high values of Lyapunov exponent (greater than 40 bitski). For the cathodic scan, the Lyapunov exponent increases as the potential increases. The calculated dimension varies between 1 and 3, for distinct values of potential and external resistances. The values of the dimension and the Lyapunov exponent decrease with the increasing of the external resistance. The route from periodicity (or quasi-periodicity) to chaos is probably related to the doubling at the period, with typical characteristics of a degenerated Hopf bifurcation.
Introduction
the Flade potential. The characteristic period of these oscillations depends on the electrode potential. The formation of Many electrochemical systems present an oscillatory behavior waves of birrefringent crystals of FeS04.7H20 was observed in their properties. For a large proportion of the cases, this during the negative slope in the current us time oscillations. behavior is related to the periodic formation and dissolution of More recently, Hudson et aL6 reported similar surface spatial anodic films.’ In recent years, a considerable effort has been oscillations during the electrodissolutionof static hemispherical devoted to the characterization of the complex oscillatory iron rod in sulfuric acid. The digitized image of the surface behavior as a deterministic chaos in chemical2 and electroshows a spacial wave of precipitated film associated to the chemical system^.^ decreasing of the current peak. Beni and Hackwood’ used Sustained current oscillations during the electrodissolution chaotic parameters to characterize the oscillations at the activeof iron were observed in two regions of fixed potentials: (i) at passive transition in the iron electrodelsulfuric acid system. He potentials in the vicinity of the Flade ~ o t e n t i a l ~(the - ~ potential described a period-doubling and intermittent turbulence route at which the active-passive transition occurs) and (ii) at to chaos in the current us time series oscillations and proposed potentials near the begining of the limiting current p l a t e a ~ . ’ ~ - ’ ~ a model where one of the control parameters is the resistance The fiist class of sustained current oscillations, near the Flade of the precipitated film and solution. Pagitsas et aL8 studied potential, presents a remarkable characteristic, shown in all of the current oscillations reponse of the same electrochemical the works: the low current in these current time series is close system when a sinusoidal perturbation was superimposed to the to zero, indicating a blocked (like passivated) electrode surface. applied potential, near the active-passive transition. In the same The frequencies of the oscillations are between 1 and 0.05 Hz. region of potential, Sazou et ale9analyzed the current response The second class of oscillations presents low values of the with respect to perturbation potential and proposed relations current around one-half of the maximum peak of current and between his experimental results and the Hopf bifurcations. frequencies in the range of 1-40 Hz. In general, a clear The second class of oscillations presents a more controversial distinction between the two types of sustained current oscilladescription in the limiting current plateau. Russel and Newtions is not made. manlo described the sustained current oscillations when the The first class of current oscillations is currently described potential is fixed within a certain range of this limiting current by the classical model of F r a n c k - F i t ~ H u g h , ~In ~ ,this ~ ~ model, plateau during the electrodissolution of iron in sulfuric acid. the electrode surface is blocked by a passivating film depending They proposed that a porous salt film, composed of hydrated on the pH to forming or dissolving the passivating layer. For iron sulfate, covers the surface of the electrode. iron electrodissolution in sulfuric acid, the passivating layer is Weihong Li and Ken Nobe” applied the deterministic chaos probably a porous sulfate layer and the sustained current technique to characterize the current oscillations during the iron oscillations (first class) are related to the cyclic between the electrodissolutionin aqueous NaCl solution, with a rotating disk active and passive states. electrode. The distinctive dynamical behavior of the potential Podesti and h i a 4 described regular oscillations on the oscillations was compared to the duplex film model composed current near the transition from active to passive region. The of an inner nonporous film and an outer porous film. oscillatory region was associated with the intrinsic negative The influence of viscosity in the oscillations observed during resistance of the Fe/H$304 system, observed in the stationary the iron electrodissolution in sulfuric acid, in the current plateau, potentiostatic polarization curve. For stationary iron electrodes was discussed by Ferreira et a1.‘* A viscosity profile from the in sulfuric acid solutions, characterizedby voltammograms with electrode interface to the bulk of solution was proposed to low values of scan rate, Froment et ~ 1 described . ~ oscillations explain the results of electrohydrodynamics impedance. on the current in the plateau of iron electrodissolution, close to Teschke et aZ.I3 observed that the turbulence at the precipitated fildsolution interface promotes a breakdown in the precipitated film with consequent current oscillations. A Abstract published in Advance ACS Absrracrs, November 1, 1995. @
0022-365419512099-17403$09.0010
0 1995 American Chemical Society
17404 J. Phys. Chem., Vol. 99, No. 48, 1995
positive resistance polarization device was developed to observe these current oscillations at the two ends of the active current ~1ateau.I~ At the start of the anodic scan, before the precipitated film formation, the Fe Fez+ 2e- dissolution process defines the reaction line (like a Tafel curve). At the instability region, there is a partial removal of the electrode coating. The removal is accompanied by an abrupt increase of the current. This current now generates an iron sulfate coating that partially blocks the current. The iron sulfate coating seems to display two apparently contradictory behaviors: it insulates the metallic surface and generates turbulence which leads to its removal and to electrode e x p ~ s u r e . ’ ~ The relation between the negative resistance and the oscillatory behavior has been known for a long time: first electronic relaxation oscillators were projected with negative-resistance devices such as unijunction transistors and neon bulbs.” The effect of negative resistance in the oscillatory electrochemical systems was studied by many authors, in experimental and theoretical works. Epelboin et al.I8 developed a polarization device known as a negative impedance converter to trace in a stable manner the Z vs E curve of the active-passive transition. This device provides a load line with a variable slope. One option of these device is a resistance connected between the working electrode and the ground (or the potentiostat). In the potentiostatic case, it is possible to distort the load line by adding an external resistance (R) between the working electrode and the potentiostat. The electrochemical Co/H3P04 system is similar to the Fe/HzS04 with respect to the activepassive tran~ition.’~ The addition of an external resistance in this system promotes a current oscillation in the plateau of the active current.Ig This effect is related to a change in the “load line”. Working on the theory of the electrochemical oscillations, Kelzer20 characterized the negative-resistance region detected in the steady-state solutions (I us F) as preferential oscillations region. These results also suggest that the existence, amplitude, and period of the oscillations are a function of the external resistance in the circuit. Koper et a1.I6 proposed that a system with negative resistance characteristics needs an ohmic series resistance to give rise effectively to instabilities. By adding a ohmic drop to the model of Franck and FitzHugh it can be shown that the oscillatory current time series can change from periodic solution to complex oscillations or chaotic solutions. One of the possibilities to observe the electrochemical oscillations in the electroreduction of indium is to find the range of potentials in which this reaction exhibits a negative charge resistance. This can be accomplished by adding an external resistance to the system.21 In this work we report the observation of sustained current oscillations during the static iron cylindrical electrode electrodissolution in sulfuric acid, observed in potentiodynamic sweep curves. These current series were obtained through a change in the load line of a classical potentiostat. The load line was modified by the addition of an external resistance between the iron electrode and the potentiostat. These current series were sampled at various static potentials in the region of the limiting current plateau, for distinct values of external resistance. The complex temporal behavior of these time series was diagnosed as deterministic chaos by applying the following methods: the reconstruction of the system dynamics,22measurement of the Grassberger-Procaccia or correlational dimension, CD,23calculation of the first Lyapunov exponent, and analysis of the power spectra.2.22
-
+
,1,24925
Kleinke The time delay, z, is characteristic time after which the system has lost an essential part of information. If t is too small, Z(t) and Z(t z) become indistinguishableand are linearly correlated; if z is too small, Z(t) and Z(t z) become indistinguishable and are linearly correlated; it z is too large the points are uncorrelated.26 Consequently, to find z we apply to the current time series the standard autocorrelation function22
+
+
where Z(f,) is the value of current sampled at time 4. z is estimated as the value of time when the standard autocorrelation function crosses zero, i.e., Scr(z)= O.*6 The first Lyapunov exponent is then the measurement of the deviation of the fiducial trajectory after the evolving time (Zevolv) and is evaluated by the model proposed by W0lP4
where m is the number of replacements steps, L’u-1) is the initial distance (in the Euclidian sense) between two neighbor points in the attractor, and LO) is the distance between the same points separated by zevOlv.In this work, zevolv is equal to 1 3 , m = 50, and the embedding dimension, ED, is equal to 3 or 10. The correlation dimension, CD, is calculated as suggested by Grassberger-Pro~accia,~~ where CD is the slope of a loglog plot of C(r/)us r/,
(3) and C(r,) is the correlation integral defined by N
k=j
(4)
where H(x) is the Heaviside function (1 for x > 0 and 0 for x I0). The norm I l ( f k ) - Z(tj)l is the Euclidean norm, and the correlational integral was calculated using 20 values for rl, between the greater and the minor distance in the attractor. The CD and the associated error were estimated by the least-squares technique. Experimental Section
The working electrode (WE) is an iron wire (4 = 0.25 cm) embedded in an insulating cylinder made of silicone rubber except for 20 mm at one of its ends. The electrode surface is initially abraded with emery paper. A polishing cloth is then used with 50 p m diamond pastes. The glass cell dimensions are 15 x 5 x 5 cm3. The depth of the electrolyte is ca. 4 cm. The reference electrode is a saturated calomel electrode (WE) and the counter electrode is a platinum mesh with 15 cm2. The distance between the SCE and the WE was 0.5 f 0.3 cm. The distance between the SCE and the WE to start oscillations in the same system without external resistance was around 7 cm. Nitrogen was bubbled through the electrolyte for at least l/2 h before the beginning of the experiments. The cell was thermostated at 25 f 1 “C. The experiments were carried out using a potentiostat/galvanostat PAR 273A with the slope of the load line modified by a resistance, R, equal to 1, 2, or 4 S2. I-E plots were recorded on the potentiostat connected to a PC386-
J. Phys. Chem., Vol. 99, No. 48, 1995 17405
Iron Electrodissolution in Sulfuric Acid
it
\ -0.6
(V vs SCE)
-0.3
0.0
0.3
0.6
-0.3
0.0
0.3
0.6
(V vs SCE)
Figure 1. Voltammograms of current vs working electrode potential measured in the anodic scan at 1 mV/s for iron electrodes in 1 M sulfuric acid, with distinct external resistances: (a) R = 0 R; (b) R = 1 Q; (c) R = 2 52; and (d) R = 4 Q.
Figure 2. Voltammograms of current us working electrode potential measured in the cathodic scan at 1 mV/s for iron electrodes in 1 M sulfuric acid, with distinct external resistances: (a) R = 0 Q; (b) R = 1 R; (c) R = 2 Q; and (d) R = 4 Q.
like HP Vectra with a GPIB interface bus. These series were sampled (20 000 points and frequency of 400 Hz, HP 7090A measurement plotting system) at various static potentials, when the scanning potentials was stopped for a few seconds. The working electrode potential, EWE,is referenced to the SCE. In the classical potentiostat control, the potential of the working electrode is only a function of time. With the addition of R, the applied potential of the potentiostat, EA^^, can be expressed as
the current oscillation region, the current decreases when the potential increases, a characteristic behavior of negative resistance systems. This negative resistance is associated with the coverage of the surface by a precipitate. After this negative pulse, the current oscillations start. This result shows the effect of the negative resistance on the instability of electrochemical systems. The cathodic scans are shown in Figure 2. Figure 2a shows a voltammogram without external resistance, with the typical current spikes at the passive-active transition. After the spikes the surface electrode is passivated. In the cathodic scan, the current oscillations start at distinct potentials. The initial potential of oscillations is a composition between the effects of the external resistance and the region of oscillations in the anodic scan. For all resistances, the range of oscillations in the cathodic scan are greater than in the anodic one. Current spikes in the passive-active transition region shown in the Figure 2a (cathodic scan) are related to the passive film breakdown and the formation of a precipitated film over the electrode surface. In this case it is possible to observe the repassivation of the surface near the Hade potential. The potential attained after the passive-active transition pulse decreases with the increasing of R ; with R = 4 Q, the end of the pulse of potential associated with the transition from the passive to active state is inside the oscillation region. The most important parameter to confirm the presence of a chaotic behavior is the Lyapunov exponent. When 1 is greater than zero, the system is chaotic. The value of 1 reflects the time scale on which system dynamics becomes unpredictable. Figure 3 shows the values of 1 calculated from the sustained current time series obtained during the interruption of the cathodic (square symbols and solid line) and anodic (circle symbols and dashed line) scan as a function of EWE. The values of , Iwere calculated for global dimension equal to 3 (filled symbols) and 10 (open symbols), for all measured current series. The presented value of 1 is the average of 10 values calculated at different first I($), with 50 steps in the fiducial trajectory inner the attractor. This procedure was adopted to minimize the local effects of the attractor, since the wire iron electrode/ sulfuric acid is a dissipative system, and the experimental conditions may change quickly. The values of 1 calculated with ED equal to 10 (Figure 3) are lower than these obtained with ED equal to 3. All values of 1 decrease with R and increase with EWE. In the anodic scan, values of 1 (circle and dotted lines in Figure 4)present great starting values, with an abrupt transition from stability to chaos. These high values of 1
EApp(t) = E,
+ at = EWE@) + (R, + R)Z(t)
(5)
where EO is the initial potential, a is the scan, t is the scan time, EWEis the potential of the working electrode, Rs is the resistance of the solution (resistance between the WE and the SCE), and I is the total current. For high conductivity solutions and reference electrode close to the WE, the potential of the working electrode, once corrected for the imposed external resistance, can be expressed as
=
- Rz(t)
(6)
For all voltammograms, the WE was initially polarized at -0.6 V us SCE applied potential during 10 min before the anodic scan, with a scan rate of 1.0 mV/s.
Results and Discussion A typical anodic scan voltammogram of iron electrode in sulfuric acid is shown in the Figure la. Here, there is no external resistance and the load line is equal to zero. For a potential around -0.3 V us SCE, the current increases monotonically as the potential increases. After attaining its maximum, the current decreases due to the film formation: presenting now a negative pulse, followed by a steady-state current region (between 0.0 and 0.35 V us SCE). At potentials around 0.35 V us SCE the iron electrode is passivated and the current take very low values. Anodic scan voltammograms obtained with R equal to 1, 2, and 4 52 are shown in the Figure lb, c, and d, respectively. In these voltammograms, the new load line resulting from the addition of the external resistance is clearly observed at the active-passive transition region. The value of the cot(@ is equal to the value of the resistance of the load line, where 8 is the angle between the current and the potential of the working electrode in the active-passive transition. The current oscillations in the anodic scan, in the three voltammograms (Figure 2a-c), start at EWEaround 0.4 V us SCE. Before
17406 J. Phys. Chem., Vol. 99, No. 48, 1995
Kleinke
.-8 3.0 2o Fi
e,
B
c
.e
n
I
- 1
.-8
-E
I
2.5 2.0
Y
m
u
1.5 1.0 0.5
2
4
6
8
10
Embedding Dimension
Figure 5. Correlation dimension
us embedding dimension for characteristic current-time series of cathodic (open symbol) and anodic (filled symbol) scans and distinct values of external resistance: (i) 1 R, 0 for EWE= -30 and 0 for EWE= 108 mV us SCE; (ii) 2 S2, 0 for E = -158 and W for EWE= 68 mV us SCE; (iii) 4 52, 0 for EWE = -160 and for EWE= 540 m V us SCE.
-0.2
-0.1
0.0
E,,
0.1
0.2
(V vs. SCE)
Figure 3. First Lyapunov exponent us EWE for cathodic ,(square symbols and solid line) and anodic (circle symbols and dashed line) scans and distinct values of R. The values of I were calculated for global dimension equal to 3 (filled symbols) and 10 (open symbols).
in the cathodic scan. Cathodic scan series, for lower values of EWEand R equal to 2 and 4 Q, present results for A close to or even less than zero. The analysis of the route to chaos requires low or negative values of A to define the critical potential of the working Le., the value for EWEcorresponding to 13. is electrode, EWEC; equal to (or crossing) zero. This value of electrode potential is associated with a stable solution of a Hopf bifurcation.22 The bifurcation parameter control, p, is defined as p = (EWE EWEC).
00.
0 -0 2
-0 1
.e
00
E,,
01
0 -
02
(V vs. SCE)
Figure 4. Correlational dimension us EWEfor cathodic (square symbols and solid line) and anodic (circle symbols and dashed line) scan and distinct values of R. The values of CD were calculated for embedding dimension equal to 2 (filled symbols) or 10 (open symbols).
indicate disorder in the oscillations, probably related to interfacial turbulence in the precipitated film.I4 The dispersion of 13. values is greater in the anodic scan than in the cathodic one. When the cathodic and anodic scan present current oscillations in the same range of EWEand with the same R, the values of 13. and CD for the anodic and cathodic scan are very close. For low values of EWEin the cathodic scan, 13. decreases with respect to EWEwith a rate around (30 bits/s)/O.l V. The origin of this effect is probably the proximity between the iron electrode electrodissolution reaction line and the current ~1ateau.I~ Anodic scan series start and stop with values greater then those measured
The correlational dimension often can be interpreted as the lower number of essential variables needed to model the dynamics.27 Figure 4 shows CD results for the sustained current oscillations sampled at the interruption of the cathodic (square symbols and solid line) and anodic (circle symbols and dashed line) scan as a function of EWE.The values were calculated for global dimension equal to 2 (filled symbols) or 10 (open symbols). For the cathodic scan and ED equal to 10, the values of CD decreases with R, with values laying in the range 2-3. For ED equal 2 the values of CD are closed to 1 in the anodic as in the cathodic scan. For lower values of EWEin the cathodic scan, and R equal to 2 and 4, the values of CD for ED = 10 are lower or close than the values of CD for ED = 2. Figure 5 shows the correlational dimension as a function of the embedding dimension, for typical current series in the cathodic (hollow symbols and continuous line) and anodic (filled symbols and dashed line) scans. For the anodic scan, the values of CD increases continuously with ED. This effect is possibly related to a random noise around the attractor.** Attractors for great values of A (Figures 9-1 IC)present characteristics of noise around the preferential trajectory of the system in the phase space (clearly observed in the Figures 9-11, a and b also suggesting the existence of noise around the measured values. In the anodic scan with R = 1 52, CD presents a linearlike increasing behavior with respect to ED. For R = 2 and R = 4 Q the curves of CD us ED do not present a behavior suitable to attain a dimension limit. The values of CD for the cathodic scan present an asymptotic trend to equilibrium with respect to the ED. For R = 4 S2 the CD presents a maximum (1.06 f 0.01) for ED = 3. This is the maximum CD for this system. The asymptotical trend to attain the maximum is more accentuated for R = 2 Q than for R = 1 Q. Values of CD measured in this work were compatible with those obtained in rotating ring-disk electrode," in the range 2-3. Diem and Hudson29 measured a correlational dimension up to 6 for the iron
J. Phys. Chem., Vol. 99, No. 48, 1995 17407
Iron Electrodissolution in Sulfuric Acid 160
12
120 68
80
3 I60
2 80
v
v
-
-E E
E
120
6 E 80
10
8 6 0
200
100
1-50
80 I20 60
80 0
1
2
3
0
10
20
30
40
Time (s) Frequency (Hz) Figure 6. Current us time series and power spectra us frequency for R = 1 52, in the cathodic scan. Current series shown examples of (a) periodicity, (b) quasi-periodicity, and (c) chaotic behavior. The and CD are shown in Table 1. corresponding values of EWE,p , ,I, 82 80 78
3
-8 E
v
E
90
80 70
U 100
80
0
1
2
3
0
10
20
30
40
Time (s)
Frequency (Hz) Figure 7. Current us time series and power spectra us frequency for R = 2 Q, in the cathodic scan. Corresponding values are shown in Table 1. electrodissolution. The comparison with our results indicates the use of low values of embedding dimension (ED 5 10) or noise in the case of the cathodic scan. Time series were selected for a tight description of the route to chaos, in the cathodic scan, and the corresponding Fourier transform are shown in the Figures 6, 7, and 8 for external resistance equal to 1, 2, and 4 P, respectively. Figures 6-8 show the original periodic regime (a), the transition to quasiperiodicity (b), and their evolution to chaos (c). Their respective reconstructed attractors are shown in Figures 9- 11. The values of EWE,p, d, and CD for these series and the reconstructed attractors are in Table 1. These time series show increasing values of Eweand d from the top of the bottom. The current series presented in the Figure 6a ( R = 1 52) shows a clearly periodic pattern (composed by two peaks). With the increase EWE(Figure 6b) the period of the oscillations also increases presenting three peaks. Figure 6c shows a current series with very poor pattern repetition and a rather chaoticlike behavior. For R = 2 Q, the current series are more interesting: Figure 7a presents a series with a negative Lyapunov exponent (see Table l), with the characteristic
0
1
2
3
0
10
20
30
40
Time (s) Frequency (Hz) Figure 8. Current us time series and power spectra us frequency for R = 4 Q, in the cathodic scan. Corresponding values are shown in Table 1. shutting down of the current amplitude and sinusoidal-like pattern. A small increase in the potential (Figure 7b) doubles the period, with periodic pattern composed by two intercalated pulses, with a clear characteristic of quasi-periodicity.22 For greater potential values (Figure 7c), the current signal becomes chaotic. The sinusoidal characteristics, the decreasing of the amplitude, and the period-doubling are important parameters to characterize the route to chaos. For R = 4 Q, the results are similar to others previously observed. Figure 8a shows a periodic pattern, with a Lyapunov exponent close to zero. Figure 8b presents chaotic-like characteristics and the pattern were truncated. The chaotic series is shown in Figure 8c. The Fourier transform spectrum shows a great peak of dominant frequency, characteristic of periodic behavior in all figures with values of p close to or lower than zero (Figures 6-8a). The peak amplitude is 2 orders of magnitude greater than the background in these curves. With the increase of p, the power spectrum contains quasi-periodical characteristics, showing various peaks of frequency and the base line of the power spectra starting to increase (Figures 5-7b). For the series with high values of p (Figures 5-7c) the background of power spectrum increases and broadens, and the peaks of characteristic frequencies disappear, a typical chaotic behavior. The attractor is reconstructed following a standard procedure based on the time-delayed coordinates for the state space.2s22 Figures 9- 11 show attractors reconstructed from the current series of Figures 6-8, respectively. For R equal to 1 52, the attractors present a deviation from the simple cyclic trajectory, like a painting brush (Figure 9a). The Figure 9b is reconstructed from the current series shown in the Figure 6b. The attractor presents two cyclic orbits, related to the distinct groups of pulses. The discussions of the route from stability to chaos were restricted to the cathodic scan at low potentials, because in this range the values of d are close to zero. The critical potential EWECdefines a circular orbit in the phase space, associated with the stable limit circle of Hopf bifurcation. The experimental result similar or closer to the stable limit circle condition is the current series shown in the Figure 8a with the corresponding attractor in the Figure 1la. The attractor shows a clear periodic structure in the phase space. The experimental description of a Hopf bifurcation is very For values of p < p , (consequently p , is equal to zero) the perburbations decay back to the stationary state in a damped oscillatory manner. The decrease of the oscillators is associated with a characteristic
17408 J. Phys. Chem., Vol. 99, No. 48, 1995
Kleinke
150
100
Figure 9. Reconstructed attractor with the time delay technique, for the time series of R = 1 Q, present in the figure 6. Characteristic values are in Table 1.
80 78
Figure 10. Reconstructed attractor with the time delay technique, for the time series of R = 2 Q, present in the figure 7. Characteristic values are in Table 1.
Fieure 11. Reconstructed attractor with the time delav techniaue. for the time series of R = 4 Q, present in the figure 8. Characteristic values are inyable 1.
TABLE 1: Calculated Values of p, Lyapunov Exponent, and Grassberger-Procaccia Dimension for the Current Series (See Figures 6-8) and Reconstructed Attractors (See Figures 9-11) 1 (bitsh) CD ED ED ED ED EWE p fig
6-9a 6-9b
R ( Q ) (mV) (mV) 1
6-9c
7-10a 7-lob 7-10c 8-lla 8-llb 8- 1I C
2 4
-30 -10 105 -168 -163 -133 -160 -152 -90
20 40 155 -2 3 33 -1 7
69
=3
=10
6.4 6 0 . 2 2.4 f O . l 20.8 f 0 . 7 4.0f0.1 42.8 f 0 . 3 7.8 f 0 . 3 -0.5 f O . 1 -0.7 f 0 . 1 0.4 f O . 1 0.5 f 0 . 2 18.0f0.2 4.0 f O . l -0.13 f O . l O.OfO.l 3.3 f O . l 1.8 f 0 . 2 23.6 f 1.0 5.2 i.0.4
effect is observable in the current series of Figures 7a,b and 8a, when the sinusoidal-like waveform is present. The amplitude of oscillations in Hopf bifurcation, p, can be expressed as
= 2 = 10 1.1 1.1 1.1 0.4 1.3 1.2 1.0 1.2 1.0
1.7 2.0 2.6 0.8 1.0 1.8 1.0 1.8 2.0
relaxation time z of the oscillatory envelope of the transients to the asymptotically stable steady state which diverges when p approaching pC3'
For values of p < 0, the oscillation amplitude decreases with the time, like a stable sink (see current series of Figure 7a and the attractor Figure loa). For positive values of p, close to the Hopf bifurcation point, the theoretical oscillations have small amplitude and sinusoidal the experimental related
where the exponent p, for a supercritical Hopf bifurcation, is equal to 0.5. If the higher-order terms are added to the system of equations, it is possible to obtain a value for p = I l 4 , and this value is associated to a degenerate Hopf b i f u r ~ a t i o n . ~ ~ Figure 12 shows a log-log plot of the experimental values of the mean current oscillation amplitude us p. For distinct external resistances, the curves present a good fitting with the law of amplitude in the model of Hopf bifurcations. The measured values of B are equal to 0.36 f 0.01, 0.34 f 0.01, and 0.31 f 0.01 for R equal to 1, 2, and 4 Q, respectively. p presents the same functional behavior with respect to the resistances as the Lyapunov exponent. As the EWEincreases, the evolution of the system modifies the values of the mean current in the plateau. To minimize this effect the mean current amplitudes were normalized. Figure 13 shows the results of normalized amplitude value (ratio between the amplitude of oscillations and the mean current in the plateau) for R = 2 and 4 Q. For these two resistances, /3 = 0.25 f 0.01, a value characteristic of a Hopf degenerated bifur~ation.~
Iron Electrodissolution in Sulfuric Acid
J. Phys. Chem., Vol. 99,No. 48, 1995 17409
II
IO'
'
'
"'
0.01
0.10
I
EwE-EwEc (")
Figure 12. log-log plot of mean cument amplitude of series us p, for low values of EWEin the cathodic scan, for R equal to 1, 2, and 4 9.
The values of the first Lyapunov exponents in the current oscillations vs time series are greater in the anodic scan than the cathodic scan. The mean current amplitude obey a scaling law of the current amplitude characteristic of the Hopf model of bifurcation, p = ,up, with values of /3 equal to 0.36 & 0.01, 0.34 f 0.01, and 0.31 f 0.01 for R equal to 1, 2, or 4 8,respectively. The normalized mean current amplitude gives values of /3 equal to 0.25 f 0.01, a characteristic of a Hopf degenerated bifurcation for R equal to 2 or 4 S2. The chaotic parameters in the cathodic and anodic scan, for the same external resistance and potential, are close to each other. These results reflect the possibility of characterization of an electrochemical dynamic system by chaotic parameters. Acknowledgment. The author is grateful for the financial of FAF'ESP 93-2412/9 and CNPq.
SUPPOI?
References and Notes
0.01
P (V)
0.10
Figure 13. log-log
plot of the ratio between the amplitude of oscillations and the mean current in the plateau us p, for low values of EWEin the cathodic scan, for R equal to 2 and 4 Q.
The great values of Lyapunov exponents and the current time series with sharp transitions from low to high values of current for R = 1 S2 (Figure 6a) are far from the necessary conditions (sinusoidal-like curves and A closed to zero) to adjust the results to the Hopf model. Other possible explanation for the deviation from the Hopf degenerated bifurcation is the greater noise in the series with R equal to 1 S2 (see Figure 6 ) , which should characterize a randomic system rather than a chaotic system. Conclusions
As we have seen, the addition of an external resistance between the working electrode and the potentiostat changes the load line and can promote oscillations in the active current plateau during static iron electrodissolution in sulfuric acid. For an iron electrodissolution in sulfuric acid, the current oscillation time series present a chaotic behavior, characterized by a positive Lyapunov exponent, the Grassberger-Procaccia dimension, and the power spectrum. The dimension and the Lyapunov exponent are strongly related to the external resistance: both parameters decrease with the increasing of the external resistance. The correlational dimension and the Lyapunov exponent as a function of the working electrode potential. The values of Grassberger-Procaccia dimension characterize a strange attractor. Current-time series present a correlational dimension in the range of 2-3 for the anodic and 1-2 for the cathodic scan (the embedding dimension is equal to 10).
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