J. Phys. Chem. 1994, 98, 12759-12767
12759
A Forward and Reverse U-Sequence of Cyclic Voltammograms Yuanhang Xu, Afshin Amini, and Mark Schell" Deparhnent of Chemistry, Southern Methodist University, Dallas, Texas 75275 Received: August 9, 1994; In Final Form: September 26, 1994@
Sustained voltammetric responses, with periods 2- 12 times longer than the period of the cycling potential, were found in application of cyclic voltammetry to the oxidation of methanol at a rotating platinum disk in alkaline solution. Experiments were conducted in which a control parameter (the upper potential limit of the potential cycle) was varied. The results demonstrate that the periodic states are ordered in the same way as dynamical states with the same symbolic representations are ordered in a combined forward and reverse U-sequence. Between ranges of the bifurcation parameter for which periodic states were observed, aperiodic behavior was found that possessed properties that are characteristic of deterministic chaos. High-order periodic responses were also obtained in experiments in which the system was allowed to relax under fixed conditions following the transfer of the Pt electrode to the methanol solution. These results provide evidence that the high-order periodic states do not arise from a slow aging process but, instead, are a consequence of the electrochemical mechanism for the voltammetric oxidation of methanol.
of PtOH with the intermediate PtC09 and the conversion of ROH to Pt oxides. Within the past 10 years there has been a renewed interest in In this paper we show that any member of an ordered set of the study of electrochemical instabilities. Applications of ideas periodic dynamical states can be obtained as a attractor, in the from nonlinear chemical dynamics reveal that, besides simple same sense as a limit-cycle attractor, in applications of cyclic periodic oscillations, more complex oscillatory behaviors, such voltammetry to methanol oxidation. These states are ordered as quasiperiodic oscillations, chaos, and mixed-mode oscillain the same way as are members of a mathematical sequence tions, occur in a large number of electrochemical p r o c e ~ s e s . ~ - ~ that is a simple combination of a forward and reverse UMany investigations on complex electrochemical instabilities sequence.1° Each state of the ordered set corresponds to an IIE were conducted under conditions in which either the current or curve that is obtained after transients disappear. Each IIE curve the potential of the working electrode is held constant. In this possesses a period, the time required for the curve to retrace paper we report the occurrence of instabilities in a different itself, that is 2- 12 times the period of the potential, the time it situation: Instabilities that occur in the application of cyclic takes the potential to pass from one turning point to the other voltammetry to the oxidation of methanol at a rotating platinum and back again. The relationship between the set of dynamical disk in an alkaline solution. states and the U-sequence is deduced by assigning a sequence In cyclic voltammetry, the current is measured while the of binary symbols to each state according to the order in which potential of the working electrode is cycled at a fixed rate values of the current are visited by the IIE curve at the upper between two reversal points. Often, after a finite number of potential limit of the potential cycle. potential cycles, the current vs potential curve (LEcurve) begins Between ranges of parameter values in which consecutive to repeat at the beginning of each new potential cycle. We have subsets of the U-sequence are stable, theory predicts the made the observation that the cycle repeatedly traced out by occurrence of deterministic chaos;" the state with the lowest the IIE curve after the decay of transients is equivalent to the period is related to all other states of the subset through projection of a limit cycle from state space.6 On varying a subharmonic bifurcations. This prediction is consistent with constraint, limit cycles can become unstable. An overall goal our experimental findings on the locations of different aperiodic of the present research is to determine whether instabilities that behaviors (not periodic). The bifurcations leading to aperiodic occur during applications of cyclic voltammetry can be directly behaviors and the observed banded structures of underlying related to changes in the chemical processes taking place on attractors are also consistent with the occurrence of deterministic the surface of electrode. Such a determination implies that chaos. However, because chaos exhibits some characteristics application of cyclic voltammetry as a mechanistic probe, when similar to characteristics of random processes (such as a largecombined with the study of instabilities, can be extended to amplitude broad background in power spectra), it may be more complex electrochemical processes where the ability to thought of as not amenable to tests of reproducibility. Charinterpret IIE curves is considered limited.7 acteristics of the aperiodic behaviors are reproducible, but we Previously, in this journal,6 it was shown that application of prefer to focus on the occurrence of the ordered set of periodic cyclic voltammetry to the oxidation of a formatelformic acid states that can easily be tested in other laboratories and which solution at a platinum electrode produced a bistable response. we find to be always reproducible. Two different IIE curves (limit cycles) exist as limiting behavior. Two experimental procedures were employed in this invesThe same instability was later found in applying cyclic tigation. In one set of experiments the Pt electrode was voltammetry to the oxidation of primary alcohols (ethanol, transferred to a methanol solution, and the potential was cycled 1-propanol, and l-butanol).8 The instability was explained with between two turning points until transients disappeared. In other a nonlinear feedback mechanism involving both the reaction experiments, the upper potential limit (upl) of the potential cycle was set at a value where limiting behavior corresponded to an IIE curve that repeated on every potential cycle. The up1 was Abstract published in Advance ACS Abstracts, November 1, 1994.
Introduction
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0022-365419412098-12759$04.50/0
0 1994 American Chemical Society
12760 J. Phys. Chem., Vol. 98, No. 48, 1994
then increased in variable increments. At each new value, the system was allowed to relax until transients disappeared. The latter procedure established that the set of dynamical states is ordered. However, the procedure is time-consuming, and the former procedure was necessary to establish that states did not arise from a slow aging process. Fundamental differences exist between the physical causes of the instabilities that lead to the dynamical states discussed in this paper and the physical causes of oscillatory instabilities obtained in constant-potential or constant-current experiments. The high-order, periodic responses presented here represent oscillations only in the same sense that a typical cyclic voltammogram represents oscillations. It is not necessary that the electrochemical system at constant potential supports sustained oscillations to exhibit instabilities when subjecting it to a cycling potential. The high-order periodic responses arise from the interaction between the cycling potential and the nonlinearities of the electrochemical process. When an electrochemical system exhibits sustained, autonomous, current oscillations, application of cyclic voltammetry typically leads to ZIE curves that possess several spikes.12-14 The responses reported in this paper do not exhibit this characteristic. The discussions of this paper are restricted to the existence of the ordered set of periodic states. The important issue of the relationship between the dynamical responses and the electrochemical mechanism for the voltammetric oxidation of methanol is briefly addressed. A more detailed mechanistic description of the instabilities requires the results of several additional experiments and is presented in a following paper.
Experimental Section Solutions, including base solutions (solutions without methanol), were prepared immediately before their use. High-purity water, obtained by passing distilled water through an Easypure Watersystem (Barnstead, Dubuque, IA), resistance 2 17.6 MQ/ cm, was used in all solutions. All methanol solutions contained OmniSolv Purge and Trap Grade methanol (E M Science, Gibbstown, NJ). Initially, both sodium hydroxide mesh beads (Aldrich, Milwaukee, WI) and sodium hydroxide monohydrate Suprapur Grade (EM Science) were used in making solutions. Solutions prepared from the two sources of sodium hydroxide revealed no detectable differences in results. Hence, most solutions were made using the less expensive mesh beans; [NaOH] = 0.01 M. Solutions were deaerated with a gas (nitrogen or argon) for at least 45 min before their use; a gas flow was maintained above the solution during experiments. Experiments were conducted in a three-compartment electrochemical cell located in a water bath possessing a temperature of 25.0 & 0.2 "C. The main compartment was a three-neck, 250 mL flask (Pine Instrument Co., Grove City, PA; Part No. RRF696170) that contained approximately 200 mL of solution. A compartment (Pine Instrument, ACOl142), separated from the main compartment by a glass frit, was placed in one of the side necks and contained the counter electrode, a platinum wire. The reference electrode was located in a third compartment that contained a solution with the same composition through a stopcock. Experiments were also conducted using a three-neck 500 mL flask with all three electrodes in the same solution; the latter cell was more convenient to use in experiments in which either the electrode was transferred between solutions or different amounts of methanol were repeatedly added to the cell. A rotating disk, 7.6 mm diameter of polycrystalline Pt, was employed as the working electrode. To check reproducibility, three separate electrodes of this type were used (Pine Instrument, two AFDD20Pt and one AFDDl158OPt). The roughness factor,
Xu et al.
V Figure 1. Cyclic voltammograms for solutions without methanol. Current is plotted against potential. Sweep rate = 0.1 V/s; rotation rate = 0.0 rpm. (a) 0.50 M H2S04;lower potential limit = -300 mV (AgCl, with NaZS04 electrolyte); upper potential limit = 900 mV. (b) 0.01 M NaOH; lower potential limit = -880 mV; upper potential limit = 400 mV. Zero current is denoted in all voltammograms with
horizontal lines. R, the ratio of the actual area to the geometric area of the electrode, was approximately the same for all three electrodes: Assuming a monlayer of hydrogen with a corresponding charge of 210 pC cm-*, R = 1.09 with a standard deviation of 0.08. The working electrode was attached to a Pine Instrument AFASR rotator. For most results presented in this paper the disk was rotated at 1000 & 10 rpm. Results are also reported from experiments in which the rotation rate was varied. A AgCl reference electrode in which the reference elements are surrounded by a NazS04 electrolyte (Fisher Scientific) and a saturated calomel electrode (Fisher Scientific) were used as reference electrodes. The potentials reported are with respect to the AgCl electrode. Potentials with respect to other electrodes can be determined from the voltammograms shown in Figure 1. A Model RDE4 potentiostat (Pine Instrument) was used to control the potential and measure the current. Current-potential curves were recorded on an x-y recorder (the 200 X-Y Recorder, The Recorder Co., San Marcos, TX). The working electrode was polished with water-based alumina before each set of experiments. A particle size of 0.05 pm was used in the final polishing. The electrode was then placed in either an acid or an alkaline solution, and the potential was cycled until either the CV in Figure l a (0.5 M HzSO4) or the CV in Figure l b (0.01 M NaOH) was achieved. For experiments that examined the effects of changing the upper potential limit (upl), the electrode was then transferred to an alkaline methanol solution. The results from these experiments were independent of whether the electrode was transferred from the acid solution or from the alkaline solution. For future reference in this and the following paper, it is convenient to review the features of the CV in Figure lb. In Figure 1, the two cathodic peaks (negative current) that occur close to the lower potential limit during the reverse sweep (decreasing potential) and the first set of peaks (three peaks in Figure l a and two peaks in Figure lb) that occur during the forward sweep (increasing potential) correspond respectively to the formation and oxidation of PtH, surface-bonded hydrogen.
U-Sequence of Cyclic Voltammograms
To shorten the time of experiment, unless stated otherwise, the lower potential was fixed at -660 mV, the approximate location of the smaller cathodic hydrogen peak. We state results that show varying the lower potential limit within the hydrogen region has only a small effect on the phenomena investigated. During the forward sweep, the structures in the currentpotential curve that appear after the hydrogen peaks are due to the initial stages of the process by which an oxide layer is formed. The three peaks labeled 01,011, and Om in Figure l b are considered to correspond to the discharge of hydroxide ions and formation of PtOH on three different sublattices. [This is analogous to how the three peaks, 0 1 , 02,and 0 3 , were identified for the acid solution15 (Figure la). Later, it was shown anions affect the way OH is deposited in acid solution, and the discharge of anion-water complexes may be responsible for some of the structure in cyclic v ~ l t a m m o g r a m s . ~ ~ExperiJ~] mental evidence also indicates that some PtOH forms immediately on leaving the hydrogen region. The broad region that follows the peak O m corresponds to formation of Pt oxides, e.g., PtO. The broad cathodic peak for the reverse sweep corresponds to the reduction of the oxygen-containing species, much of which can be of a different nature from that of the species originally formed. l5
J. Phys. Chem., Vol. 98, No. 48, 1994 12761
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Results High-Order Periodic Voltammograms Obtained Immediately following Transfer to Methanol Solutions. Cyclic voltammograms (CVs) shown in Figure 2 provide representations of the general phenomena considered in this paper. These voltammograms were recorded during experiments in which the working electrode was first placed in the base solution (0.01 M NaOH) and the potential cycled between two limits until the achievement of a stable response. Next, the electrode was transferred from the base solution to a methanol solution, and the potential was again cycled until a stable CV was obtained. The working electrode was then returned to the base solution to examine whether the original stable voltammogram achieved in that solution could be reproduced. Figure 2a shows the response to the f i s t 10 potential cycles after transfer to a 0.25 M methanol solution; the upper potential limit (upl) was equal to 1600 mV. A pattern became clear during subsequent cycles as can be seen in Figure 2b where the response to the seven cycles that followed the 10th cycle of Figure 2a is shown. The system appeared to alternate between two types of behavior. During every other cycle the IIE curve exhibited a relatively large loop (Le., a large separation between the current of the forward and the reverse parts of the potential cycle) as well as a small peak at low values of the potential during the reverse sweep. During a cycle that immediately followed a cycle with the stated response, the current-potential curve exhibited a substantially smaller separation between the current of the forward and reverse parts of the potential cycle, a larger maximum current value, and no current peak during the reverse part of the cycle. Finally, after approximately 20 cycles from the time the electrode was placed in the methanol solution, the current-potential curve would begin to repeat after every other potential cycle; see Figure 2c. We call this type of response a period-two cyclic voltammogram. A cyclic voltammogram recorded in the base solution, with all other conditions the same as those used in obtaining Figure 2a-c, is shown in Figure 2d. The sharp rise in the current, in this case, is due to the onset of oxygen evolution. A comparison of parts d and c of Figure 2 indicates that oxygen evolution is
I
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Potential (mV) Figure 2. Voltammetric responses of methanol solutions and base solution. Current is plotted against potential. (a) The response to the first 10 potential cycles; the first two are labeled 1 and 2. Upper potential limit (upl) = 1600 mV; [CHsOH] = 0.25 M; rotation rate = lo00 rpm; sweep rate = 0.10 V/s. (b) The response to seven potential cycles that followed those in (a). (c) The stable cyclic voltammogram (CV), a period-two CV. (d) The stable CV recorded for the base solution, 0.01 M NaOH, using the same conditions as those in (a) to (c). (e) A stable CV for the base solution recorded before transfer to the methanol solution. up1 = 500 mV; rotation rate = zero. (f) The same as in (e) except the CV was recorded after the measurements in (a) to (c) were made. (g) A period-five CV. [CH30H] = 0.50 M; other conditions the same as in (a) to (c). (h) A period-two CV. [CH3OH] = 0.20 M; up1 = 700 mV; other conditions are the same as in (€9.
greatly diminished in the presence of methanol. Otherwise, a contribution to the current on the order of that in Figure 2d would cause a noticeable increase in the current at a potential around 500 mV. A cyclic voltammogram recorded for the base solution before the working electrode was transferred to the methanol solution (Figure 2e) and a voltammogram recorded after the electrode was transferred back from the methanol solution (Figure 2f) are also shown. Both voltammograms are essentially identical. Several experiments were conducted following the same procedures but using different methanol concentrations and different up1 values. Stable cyclic voltammograms (recorded after the disappearance of transient behavior) are shown in Figure 2g,h. The conditions used to obtain Figure 2g were the same as those used to obtain the period-two voltammogram in Figure 2c, except that the methanol concentration was 0.5 M. The result was period-five cyclic voltammogram; the currentpotential curve would begin to repeat after every fifth cycle. A period-two voltammogram, similar in form to that of Figure 2c, was obtained using the 0.5 M methanol solution with the upper potential limit set at 1800 mV. The result indicates that the potential limits in which period-two voltammograms exist vary with concentration; for a fixed lower potential limit, an increase in the concentration causes the period-two response to move to a region of larger values for the upper potential limit. Further evidence for this trend is provided by the voltammogram
12762 J. Phys. Chem., Vol. 98, No. 48, 1994
Xu et al. A
Potential (mv) Figure 4. Voltammetric responses of a 1.0 M methanol solution: sweep rate = 0.10 Vls; rotation rate = lo00 rpm. (a) The response recorded during the loth potential cycle; up1 = -200 mV. (b) The same as (a) but recorded after 90 min. (c) A period-one CV up1 = 100 mV. (d) A period-two CV; up1 = 108 mV.
Figure 3. Voltammetric responses of methanol solutions and base solution. Conditions for (a) to (c): [CH30H] = 0.20 M, up1 = 225 mV; other conditions the same as Figure 2a. (a) The response to the first 18 potential cycles. (b) The response to potential cycles numbered 40-52. (c) The stable response, a period-four CV. (d) A stable CV recorded in the base solution before transfer to the methanol solution. Conditions the same as for (a) to (c). (e) The same as in (d) except it was recorded after the transfer from the methanol solution use in (a) to (c). (f) A period-three CV. up1 = 150 mV; other conditions the same as those in (a) to (c).
in Figure 2h, a period-two voltammogram. The concentration and up1 were respectively 0.2 M and 700 mV. When up1 values were considerably less than those listed in Figure 2, a larger number of potential cycles were usually required before transient behavior disappeared. The response to the f i s t 18 potential cycles is shown in Figure 3a for a 0.2 M methanol solution and a up1 value of 225 mV. The response to 12 later potential cycles, beginning at the 40th cycle, is shown in Figure 3b. A stable response was achieved approximately 35 min after the transfer of the working electrode to the methanol solution. It was a period-four CV (Figure 3c). Cyclic voltammograms for the base solution, one recorded before the transfer to the methanol solution used in Figure 3a-c and one recorded after the transfer from the methanol solution, are shown respectively in Figure 3, d and e. A period-three CV (Figure 3 0 was the stable response when a value of 150 mV was used for the upl. This stable CV was not achieved until approximately 60 min had elapsed. Drastic Changes in the Cyclic Voltammetric Response with Respect to Variations in a Control Parameter. Highorder periodic voltammetric responses were also obtained following an experimental procedure in which, immediately following the placement of the working electrode into the methanol solution, the potential was cycled between -660 and -200 mV for a period between 90 and 120 min. The up1 was then increased in variable increments. At each value of the up1 the potential was cycled until a stable periodic response was achieved or until it was assumed that the response was aperiodic.
The results from these experiments demonstrate that periodic states, including those of Figures 2 and 3, are ordered in a particular sequence. At the low value of the up1 (-200 mV), the initial response was one in which the current, beginning at a potential of approximately -430 mV, underwent a rapid increase during the forward sweep; see Figure 4a. The response was characterized by a relatively small separation between the current of the forward and reverse parts of the cycle. Although the form of the response was maintained during the cycling process, the current slowly decreased. Compare the 10th cycle (Figure 4a) with the response obtained after 90 min (Figure 4b). Similar current-potential curves accompanied with similar decreases in current are observed under other conditions for the oxidation of methanol (see, for example Figure 4 in ref 19>, as well as for the oxidation of other oxygenated organics so long as the up1 is not too large.20 The decreases in amplitude are attributed to the accumulation of surface-bonded C0.19 Increasing the up1 in small increments caused the stable CV to change from the form possessed by the CV in Figure 4b to a form possessed by the CV shown in Figure 4c. The characteristics of the latter CV have been observed in the voltammetric oxidation of a large number of different oxygenated organic compounds under a wide range of conditions.6,8,21,22 They are generally interpreted as follows: the increase in current during the forward sweep is due to the oxidation of the organic, and the decrease in current is attributed to the formation of platinum oxides that inhibit the oxidation of organics; the sharp increase in current during the reverse sweep corresponds to the reduction of the inhibiting oxides and the resumption of the oxidation of the organic.22 Upon further increases in the upl, a drastic change occurred in the response: the period-one CV was replaced, as stable behavior, by a period-two CV. It was established that, close to the bifurcation point, the two segments of the period-two CV were close in proximity. An example of a CV exhibiting this proximity, which was recorded at a up1 value 8 mV larger than a value at which a stable period-one CV was observed, is shown in Figure 4d. This observation is evidence that the period-two CV arises through a subharmonic bifurcation. In a subharmonic bifurcation, the two segments of the response would be superimposed at the bifurcation point. The inner loop of the period-two response in Figure 4d rapidly decreased in size with respect to increases in the up1 and then
J. Phys. Chem., Vol. 98, No. 48, 1994 12763
U-Sequence of Cyclic Voltammograms 0.66
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Potential (mv) Figure 5. High-order periodic cyclic voltammograms: (a) a periodtwo CV, up1 = 120 mV; (b) a period-four CV, up1 = 125 mV; (c) a period-three CV, up1 = 130 mV; (d) a period-six, CV up1 = 150 mV.
Other conditions the same as in Figure 4. took on a form that exhibits little hysteresis. An example of a period-two CV exhibiting little hysteresis during every second potential cycle is shown in Figure 5a. Eventually, the periodtwo CV was replaced, as stable behavior, by a period-four CV (Figure 5b). At up1 values larger than those values for which this periodfour CV was stable, aperiodic responses were found. Aperiodic behavior was intempted by large intervals in which periodic responses were found. The fist periodic response observed after the onset of aperiodic behavior was a period-three state. A period-three CV is shown in Figure 5c. Between the interval of up1 values in which the period-three CV was observed as stable behavior and the return to aperiodic behavior, an interval was found in which a period-six CV was observed; see Figure 5d. To show the characteristics of additional states from a different viewpoint, we use a representation commonly employed in studies of chemical instabilities, a time-series representation. To make easier the correspondence between the potential-current curves and time-series representations, the time-series representation of a period-one CV and the timeseries representation of the CVs in Figure 5 are shown in Figure 6. An interesting feature of these time series is that they provide simple examples of the fact that the period of a given state does not necessarily correspond to the number of peaks within that period. The extra peaks arise from the sharp increase in the current during the reverse part of some of the potential cycles. Time-series representations of a period-four state and its subharmonic, as well as a period-five state and its subharmonic, are shown in Figure 7. The period-four state was the first periodic state found after the period-six state in Figure 6e. Only aperiodic behavior was found between them. Aperiodic responses were also the only behaviors found between the intervals of stability for the subharmonic of the period-four state (Figure 7b) and the period-five state (Figure 7c). The period-four state
0 0
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Time (s) Figure 6. Time-seriesrepresentations. Current is plotted against time. (a) A period-one CV; up1 = 75 mV. (b)-(e) Times-series representations of the CVs shown in Figure 5 . 1.05
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Time(s) Figure 7. Time-series representations of high-order periodic responses. (a) A period-four state; up1 = 160 mV. (b) The first subharmonic of the period-four state in (a); up1 = 170 mV. (c) A period-five state; up1 = 190 mV. (d) The subharmonic of the period-five state; up1 = 225 mV. All other conditions the same as in Figure 4.
in Figure 7a is different from the period-four state shown in Figures 5b and 6c. The latter state is the first subharmonic of the period-two state. Symbolic Representation of Dynamical States. To distinguish periodic states, we assign a symbolic notation to each state. This is done by comparing the values of the current at the up1 between sequential potential cycles. The symbol R is assigned to a cycle
Xu et al.
12764 J. Phys. Chem., Vol. 98, No. 48, 1994 R P2 = (Ply I
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if the value of the current at the up1 is greater than the corresponding value of the previous cycle. Otherwise, the symbol L is assigned to the cycle. The assignments are made sequentially and begin with the cycle that follows the cycle with the smallest value of the current at the upl. The last symbol is omitted as it is understood that it is always an L. It is also possible in many cases to deduce the symbol sequence from the time series: Begin with the largest peak and assign an R; ignore all peaks that arise from the reduction of oxides during the reverse sweep; assign an L to all peaks for which the preceding peak is higher, etc. This latter method, although applicable to the time series that we show, is not always unique due to the fact that a time series is a projection of the dynamics. Using either construction, the period-two state is represented by the symbol R and its subharmonic, the period-four state shown in Figure 5b, is represented by the symbolic sequence RLR. The symbolic sequences representing the period-three state (Figure 5c) and its subharmonic (Figure 5d) are respectively RL and RLLRL. The symbolic sequence representing the periodfour state in Figure 7a is RLL. Periodic states, their symbolic representation, and the approximate location and length of their stability regions along the up1 axis are depicted in Figure 8. Up to the value of 400 mV, the periodic states are arranged in the same order in which dynamical states with the same symbolic representation appear in the U-sequence.lo Previously, dynamical states measured in a study of the Belousov-Zhabotinski reaction were shown to be consistent with the occurrence of a U - ~ e q u e n c e . In ~ ~our results, only states with the symbolic sequence RL" and their
360
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Time (s) Figure 9. Time-series representation of periodic states with the symbol sequence RL"R2, measured at large values of the upl: (a) a periodtwelve (P12), up1 = 600 mV; (b) P l l , up1 = 650 mV; (c) P10, up1 = 700 mV; (d) P9, up1 = 840 mV; (e) P8, up1 = 900 mV.
subharmonics were found for up1 values less than 400 mV. Many other states belonging to the U-sequence were not found. (In the U-sequence, 27 states with periods of 11 or less exist between the period-two and period-three states.lO) In studies on the oxidation of ethanol24and the oxidation of l - p r ~ p a n o l , ~ ~ RL" states were observed. The CVs in these studies are of the same from as those presented here (compare Figure 5a with Figure l b in ref 24), indicating that the same chemical processes are responsible for the occurrence of these states in the three oxidation processes. At values of the up1 greater than 400 mV, states were found that possessed the same symbolic sequence as states observed at lower up1 values. However, the former set of states appeared in reverse order, Le., a "reverse U-sequence". In addition to finding states with the symbolic sequence RL" and some of their subharmonics, states with the symbolic sequence RLnR2(Figure 9) and states with the symbolic sequence RL"R (Figure 10) were also found. Eventually, at a potential greater than 1800 mV, the system returned to period-one behavior. Time-series representations and CVs of RL" states and a CV of a periodone state found at large values of the up1 are shown in Figure 11. From Figure 11 it can be deduced that not only are the symbol sequences the same as states found at lower values of the up1 but also the form of the ZIE curves are of the same topology. This observation provides evidence that, although the rates are different, the same mechanism is responsible for the occurrence of the high-order periodic states at both the smaller and larger values of the upl. Reproducibility and Changes Induced by Varying Other Control Parameters. All the RL" states, with n 5 4, and their subharmonics listed in Figure 8, and the order in which they appear, would always be reproduced in one experimental setting: the incremental increases in the up1 were 50 mV for a
. I . Phys. Chem., Vol. 98, No. 48, 1994 12765
U-Sequence of Cyclic Voltammograms
2.48 0 -6
Potential (mv)
5
Potential (mv) Figure 12. (a) CV and phase portrait for a stationary electrode. [CHsOH] = 1.0 M; up1 = 200 mV; time delay (t)= 3.3 s; current range 0-1540 PA. Bars across the cycles denote where they cross the plane at [(t) = 551 PA. (b) CV and phase portrait for a rotating working electrode (lo00 rpm). up1 = 185 mV; t = 2.6 s; current range 0-1470 PA; plane is at I(t) = 538 PA; other conditions the same as (a).
T h e (s) Figure 10. Periodic states with the symbol sequence RL"R: (a) P9, up1 = 800 mV; (b) P8, up1 = 855 mV; (c) €7,up1 = 930 mV; (d) P6, up1 = 1100 mV; (e) P5, up1 = 1400 mV.
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Potential (mv) Figure 11. Time-series representations and CVs of periodic states with the symbol sequence RL" and a CV of a period-one state with a large up1 value: (a) P5, up1 = 950 mV; (b) P4, up1 = 1300 mV; (c) P3, up1 = 1450 mV; (d) P3 CV, up1 = 1450 mV; (e) P2 CV, up1 = 1650 mV; (f) P1 CV, up1 = 1900 mV.
up1 e100 mV, 20 mV for 100 mV up1 e300 mV, and 100 mV for a up1 > 300 mV. Sometimes back-tracking was required. Locating a specific state among the remaining states shown in Figure 8 sometimes required long searches that involved first determining its approximate location by finding nearby states
and then by changing the up1 by variable increments or decrements. The location with respect to nearby states was always consistent with the representation of the sequence in Figure 8. To examine the reproducibility of a quantitative result, the value of the up1 at which the system was observed to make the transition to period-two was measured several times. A 1.0 M methanol solution was prepared for each measurement. Instead of allowing transient behavior to disappear after each change in the upl, the following procedure was implemented. The up1 was held at -200 mV for 2 h. Next, the up1 was increased 25 mV every 10 min until the value of 25 mV was reached. At this point the time the up1 was held fixed was changed to 20 min, and the incremental increase was changed to 10 mV. When a value of the up1 was reached that was less than 20 mV from the transition point obtained in the preceding experiment, the incremental increase in the up1 was changed to 2.0 or 1.0 mV. The average value for the up1 at which period-two was first observed is 105.5 mV. The standard deviation is 7.52 mV (range 95-1 16 mV, eight experiments). Effects of Rotation Rate and Sweep Rate. Along with the rotation rate of 1000 rpm, experiments in which the up1 was varied were conducted using the rotation rates 0, 200,500, and 2000 rpm. It was found that as the rotation rate was decreased, the value of the up1 at which the period-one CV lost its stability decreased. The variation was small. The point on the up1 axis at which the transition from period-one to period-two occurred decreased approximately 20 mV on changing the rotation rate from 2000 to 200 rpm. At zero rotation rate, no high-order periodic states were observed. Incremental increases in the upper potential limit led to a direct transition from a period-one CV to aperiodic behavior. The measured aperiodic behavior at zero rotation rate appears to be of the same nature as the aperiodic behavior obtained using a rotating electrode. In Figure 12 we show, along with CVs, phase portraits constructed using the time delay methodz6from data in which a stationary electrode was used and from data in which a rotating electrode was used. The two phase portraits possess similar structures. The results are consistent with the possibility that the periodic windows become rapidly smaller as zero rotation rate is approached. Unfortunately, we cannot
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Potential (mv) Figure 13. Additional results. (a) Plot of peak current (forward sweep) up1 = of period-one CVs (Ip)vs the square root of rotation rate (d2): 300 mV; sweep rate = 100 mVls; [CHjOH] = 0.06 M (circles) and 0.02 M (rectangles). (b) Stable cyclic voltammograms obtained in an experiment in which the sweep rate (S)was increased (up1 = 125 mV). Other conditions the same as Figures 4-7. (c) A period-three CV obtained using a lower potential limit of -800 mV (up1 = 130 mV). Other conditions the same as Figures 4-7. follow the fate of the periodic windows as the limit of zero rotation rate is approached; at small rotation rates, the rotators possess an axial vibration which affects the dynamics. We can demononstrate that the singular limit in the dynamics is consistent with other limits that probably arise from the fact that the diffusion boundary layer is the whole reaction cell at zero rotation rate. In Figure 13a we show plots of the peak current obtained during the forward sweep of period-one CVs vs the square root of rotation rate. The methanol concentration was sufficiently small that period-one CVs were stable for most conditions. At finite rotation rates the data fit a straight line. The peak current decreases with respect to increases in rotation rate. At zero rotation rate, the peak current does not fall on that line. Usually, the peak current measured at zero rotation rate is larger than that at finite rotation rates; see data represented by rectangles. Under conditions near bifurcation points, a different result is possible. The data represented by circles reveal that the peak current at zero rotation rate is smaller than that at finite rotation rate. In this case, a period-one CV at zero rotation rate was not stabilized; the response remained aperiodic. All curves were very close together, and an average was easily obtained. The response at 200 rpm, peak current not shown, was a periodtwo CV. We point out that in switching from finite to zero rotation rate the system almost immediately changed to a response that is close to that of the CV obtained after transients disappeared. Also, the changes shown in Figure 13a with respect to varying the rotation rate are reversible: the points obtained increasing the rotation rate are approximately (within 5 % ) the same as those obtained on decreasing the rotation rate. The decrease in peak current with respect to increases in rotation rate can also be achieved by increasing the concentration of NaOH. (PtOH is requried for the oxidation of methanol, but it also inhibits the reaction by occupying sites and by being transformed to PtO.) Thus, there is evidence that increases in rotation rate increases the amount of ion complexes in the electrode boundary layer. The results shown Figure 13a are reported for the purpose of indicating that they are consistent with the singular limit obtained for the ordered sequence of high-order periodic states at zero rotation rate. The important result is that changes in
rotation rate only have a small effect on the sequence of periodic states over the range 200-2000 rpm. Decreasing the sweep rate also caused the periodic states and the bifurcation from period-one to period-two to shift to lower values of the upl. The effects of changing the sweep rate can be seen in Figure 13b. The three CVs were obtained in an experiment in which the sweep rate was changed and the up1 held fixed. A period-two CV was stabilized with the sweep rate fixed at 100 mV/s, the first CV in Figure 13b. The sweep rate was then changed to 200 mV/s. The stable response was again a period-two CV. A period-two CV was also obtained after changing the sweep rate to 300 mV/s. However, it is clear that the latter CV is close to the period-one to period-two bifurcation point. We note that the process of changing sweep rate was reversible; the original CV was obtained on changing the sweep rate from 300 mVls back to 100 mV/s. Effects of Different Lower Potential Limits. Besides the value -660 mV, values of -600, -720, -780, -800, and -880 mV were employed for the lower potential limit. In Figure 13c we show a period-three CV for which the lower potential limit was set at -800 mV. Comparisons with the other high-order periodic CVs reveal that decreasing the lower potential limit extends to lower potentials the part of the ZIE curve where the current is close to zero. The transition point for the onset of period-two behavior decreased approximately 50 mV on changing the lower potential limit from -600 to -800 mV. Using the value of -880 mV, 80 mV less than the lowest potential cathodic hydrogen peak in Figure lb, no high-order periodic states were found; the system appeared to make a direct transition into aperiodic behavior on increasing the upl.
Summary and Discussion In applying cyclic voltammetry to the oxidation of methanol at a rotating platinum disk, high-order, periodic responses and aperiodic responses were found under a wide range of conditions. Varying the upper potential limit revealed that the periodic states appear in the same order as states appear in a forward and reverse U-sequence. Because of the existence of these dynamical behaviors, short-time scale experiments that accompany cyclic voltammetry can yield a wide range of measured results that will not necessarily fit a Gaussian-type distribution. Even measurements involving large time scales can yield variable results. Given initial conditions close to the underlying period-two attractor of the cyclic voltammetry experiment, two, single, forward, sweep experiments (linear voltammetry) can yield two very different results. The statistical distribution for the total charge transferred during the sweep will be bimodal. It was determined that end points exist in intervals for values of a constraint where the set of ordered periodic states exists. The set of states was not found when the lower potential limit of the potential cycle was set approximately 80 mV less than the value of the lowest potential cathodic hydrogen peak (Figure lb). Instead, a direct transition to aperiodic behavior was observed. The set of states was observed when the value of the lower potential was from a range including all hydrogen peaks. Changing the sweep rate had only a small effect on the sequence of ordered states (Figure 13b). The sequence of ordered states was observed when the rotation rate was chosen from the range 200-2000 rpm. Within this range, changes in the rotation rate had little effect on measurements. At zero rotation rate, incremental increases in the upper potential limit
U-Sequence of Cyclic Voltammograms led to a direct transition into aperiodic behavior. No high-order periodic states were observed at zero rotation rate. We present here a brief explanation for the occurrence of the set of ordered states. That the set exists for a range of rotation rates spanning at least an order of magnitude implies their primary cause is due to surface reactions. Reaction of surface-bonded carbon monoxide, which was claimed as an identified intermediate in a study that combined spectroscopy and cyclic ~ o l t a m m e t r yis, ~taken ~ as the rate-determining step.19 Surface-bonded carbon monoxide reacts with PtOH.18,27,28(See discussion of Figure l b in the Experimental Section.) Besides reacting with PtCO, PtOH is also transformed to Pt oxides, e.g., PtO. The Pt oxides show very little reactivity with methanol in alkaline solution.27 Following Buck and Griffith,22 the characteristics of a decreasing current during the forward sweep and a sharp increase in current during the reverse sweep, like those in the CVs shown in Figures 4b,5, and 11, are taken to correspond to formation and reduction of platinum oxides that inhibit the oxidation of methanol. When there is no large decrease in current during the forward sweep, and no sudden increase in current during the reverse sweep, essentially no inhibiting oxides are formed during the potential cycle. The CVs in Figure 4a provide evidence for the latter statement: the upper potential limit for this CV is too low for Pt oxides to form so that any PtOH present will react with PtCO. Except near bifurcation points, the high-order periodic states consist of at least one cycle that exhibits the characteristics associated with formation and reduction of inhibiting oxides. Such cycles are followed by one or more cycles in which the characteristics of oxide formation and reduction are not exhibited. Therefore, we conclude that high-order periodic states originate because of the dual role played by PtOH. During one or more cycles, most of the PtOH that forms reacts with PtCO (cycles without the characteristics of oxide formation and reduction). These cycles are followed by a cycle in which a substantial amount of PtOH is transformed to PtO, which does not react with methanol. A more detailed explanation for instability is presented in a following paper.
Acknowledgment. This research was sponsored by the Robert A. Welch Foundation. Grant N-1096.
J. Phys. Chem., Vol. 98, No. 48, 1994 12161 References and Notes (1) Hudson, J. L.; Tsotsis, T. T. Chem. Eng. Sci. 1994, 49, 1493. Hudson, J. L:; Bassett, M. R. Rev. Chem. Eng. 1991, 7, 109. (2) Koper, M. T. M.; Gaspard, P. J . Phys. Chem. 1991, 95, 4945. Koper, M. T. M.; Gaspard, P.; Sluyters, J. H. J . Chem. Phys. 1992, 97, 8250. (3) Dewald, H. D.; Parmananda, P.; Rollins, R. W. J . Electroanal. Chem. 1991, 306, 297. (4) Eiswirth, M.; Luebke, M.; Krischer, K.; Wolf, W.; Hudson, J. L.; E d , G. Chem. Phys. Lett. 1992, 192, 254. (5) Gu, Z. H.; Chen, J.; Fahidy, T. Z. Electrochim. Acta. 1992, 37, 2637. Karantonis, A.; Pagitsas, M.; Sazou, D. Chaos 1993, 3, 243. (6) Parida, G. R.; Schell, M. J . Phys. Chem. 1991, 95, 2356. (7) Compton, R. G.; Hillman, A. R. Chem. Br. 1986, 22, 1088. (8) Cai, X.; Schell, M. Electrochim. Acta 1992, 37, 673. (9) Parsons, R.; VanderNoot, T. J . Electroanal. Chem. 1988, 257, 9. (10) Metropolis, N.; Stein, M. L.; Stein, P. R. J. Combinatorial Theory 1973, 15, 1973. (1 1) Manneville, P. Dissipative Structures and Weak Turbulence; Academic Press: New York, 1990; pp 215-217. (12) Markovic, N.; Ross, P. N. J . Phys. Chem. 1993, 97, 9771. (13) Raspel, F.; Eiswirth, M., preprint. (14) Sazou, D.; Pagitsas, M. J . Electroanal. Chem. 1992, 323, 247. (15) Angerstein-Kozlowska, H.; Conway, B. E.; Sharp, W. B. A. J . Electroanal. Chem. 1973, 43, 9. Tilak, B. V.; Conway, B. E.; AngersteinKozlowska, H. J . Electroanal. Chem. 1973,48, 1. Conway, B. E.; Bamett, B.; Angerstein-Kozlowska,H.; Tilak, B. V. J . Chem. Phys. 1990,93,8361. (16) Angerstein-Kozlowska,H.; Conway, B. E.; Hamelin, A,; Stoicoviciu, L. J . Electroanal. Chem. 1987, 228, 429. (17) Wagner, F. T.; Ross, P. N. J. Electroanal. Chem. 1988, 250, 301. (18) Santos, E.; Giordano, M. C. J . Electroanal. Chem. 1984,172, 201. (19) Gasteiger, H. A.; Markovic, N.; Ross, P. N.; Cairns, E. J. J . Phys. Chem. 1993, 97, 9771. (20) Enyo, M. J . Electroanal. Chem. 1985, 186, 155. Burke, L. D.; O’Leary, W. A. In Proceedings of the Second Symposium on Electrode Materials and Processes for Enerav Conversion and Storaae; Srinivasan, S.; Wagner, S.; Wroblowa, H.; Ed;; Electrochemical Society: Pennington, NJ, 1987; Vol. 87-12, UP 378-393. (21) Anatasijevic, N: A.; Baltruschat, H.; Heitbaum, J. J . Electroanal. Chem. 1989,272, 89. Leung, W. H.; Weaver, M. Langmuir 1990,6,323. (22) Buck, R. P.; Griffith, L. R. J. Electrochem. SOC. 1962,109, 1005. (23) Coffman, K. G.; McCormick, W. D.; Noszticzius, Z.; Simoyi, R.; Swinney, H. L. J . Chem. Phys. 1987, 86, 119. (24) Schell, M.; Cai, X. J . Chem. Soc., Faraday Trans. 1991,87,2255. (25) Schell, M.; Cai, X. Electrochim. Acta 1993, 38, 519. (26) Packard, N. H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S . Phys. Rev. Lett. 1980, 45, 712. (27) Caram, J. A.; Gutierrez, C. J . Electroanal. Chem. 1992, 323, 213. (28) Also compare the CVs in ref 27 with the CVs for the oxidation of CO in ref 18.
