Observation of several different temporal patterns in the oxidation of formic acid at a rotating platinum-disk electrode in an acidic medium

151

J. Electroanal. Chem., 308 (1991) 151-173 Elsevier Sequoia S.A., Lausanne

Observation of several different temporal patterns oxidation of formic acid at a rotating platinum-disk electrode in an acidic medium

in the

F.N. Albahadily and Mark Schell * Department of Chemistry TX 75275 (USA) (Received

and Center for Nonequilibrium

19 June 1990; in revised form 24 January

Structures,

Southern Methodist

University, Dallas,

1991)

Abstract Results of an experimental study on the oxidation of formic acid at a rotating platinum-disk electrode are presented. Experiments were conducted under galvanostatic conditions using solutions containing sulfuric acid. By increasing the applied current slowly, the system was moved through sequences of oscillatory states. Most states exhibited temporal patterns consisting of different combinations of large and small oscillations and without a substantial range of intermediate amplitudes. The sequence found closest to the point where stationary behavior became unstable was ordered in such a way that the ratio of the number of small oscillations to the number of large oscillations increased with respect to increases in current. Each state that was identified contained one large oscillation and a number of small oscillations. Peak values in measured waveforms are consistent with the idea that a strongly adsorbed intermediate reacts with hydroxyl radicals. The first sequence was followed by a family of large-amplitude oscillations that exhibited periods with durations of up to several hours. Following the family of large-amplitude oscillations, a sequence of oscillatory states was found with an ordering reversed to that of the first sequence. This latter sequence was followed by a set of oscillatory states whose members could be identified as elements from a Farey sequence. The oxidation process also exhibited bistability: high-potential states and low-potential states were found within the same range of values of the current. Models are constructed from previously formulated kinetic steps and rate laws, and these models predict that potential oscillations accompany the oxidation process.

INTRODUCTION

During the past thirty years the electrocatalytic oxidation of formic acid has been the subject of numerous research studies. Parsons and VanderNoot [l] have written a review that covers electrochemical research on formic acid and other small organic

l

To whom correspondence

0022-0728/91/$03.50

should be addressed. 0 1991 - Elsevier Sequoia

S.A.

152

molecules from 1981 to the end of 1987. Several other reviews exist which cover previous years [2-61. Much of the motivation for research on the oxidation of formic acid originated because of potential applications in fuel cells. The reaction was also the focus of pioneering studies that employed surface spectroscopies. (For examples of the application of Electrochemically Modulated IR Spectroscopy see Beden et al. [7] and Kunimatsu [8]; for examples of IR Reflection-Absorption Spectroscopy see Kunimatsu and Kita [9], Beden et al. [lo] and Corrigan and Weaver [ll] and for an example of the application of Differential Electrochemical Mass Spectroscopy see Hartung et al. [12].) As a consequence of the large body of research, a considerable amount of both general and specific information is known concerning the mechanism for the oxidation of formic acid. It is widely accepted that the oxidation process follows at least two parallel paths. In one path, which we refer to as the direct path, reactive intermediates are produced that possess relatively short lifetimes. In a second path, intermediates are formed on the surface of the electrode that, at sufficiently low values of the working-electrode potential, remain essentially inert. Despite the large amount of information, many details of the kinetic mechanisms are still not known. Several variations of both the direct path and the path involving strongly adsorbed intermediates exist in the literature. In order for mechanisms to be considered plausible, they must be consistent with the occurrence of oscillations in the potential observed to accompany the oxidation of formic acid under galvanostatic conditions; see for examples, Buck and Griffith [13], Schell et al. [14] and Anastasijevic et al. [15]. The principal purpose of this paper is to demonstrate that the oscillatory behavior contains more information on the mechanism than realized. We present experimental results that show that by changing the current the system can be moved through different sequences of oscillatory states. Each state is qualitatively different. These results provide stringent tests for kinetic mechanisms. Furthermore, in principle, the peak potentials in a waveform, i.e., a plot of potential vs. time, contain information regarding the potential dependence of reactions that suddenly turn on. The optimum use of our results requires a model for the oxidation of formic acid that has a predictive capability as varied as the temporal structures that we find. Unfortunately, in this sense, our results are far ahead of any theory. In fact, although there have been several descriptive explanations of the occurrence of oscillations, we are unaware of any explicit kinetic model (one that includes mathematical equations describing the rates of change of chemical species and the potential with respect to time) which can be used to simulate potential oscillations. Therefore, in this paper, we also present models that are based on previously proposed approximations for the kinetics that accompany the oxidation of formic acid. Before presenting our experimental results, it is convenient to outline the current understanding of the mechanism for the oxidation of formic acid which contains an explanation for the occurrence of oscillations. The kinetic steps stated in the following will be referred to throughout the discussion on our results.

153

A mechanism for the direct also Pavese et al. [16]), consists HCOOH

path, as proposed by Kunimatsu of the following steps

-+ COOH + &j

and Kita

[9] (see

(1)

COOH-*CO,+H+e-

(2)

H-+H++e-

(3)

We have modified the scheme so that the intermediate I-j is shown. Although, for most of the potential range spanned by the oscillations, reaction (3) is expected to be fast, the fact that a site is necessary should in principle be important in modelling the process. A second point, which can, in principle, affect the predictions of a model, is whether to consider reaction (1) to consist of two steps, the first of which produces the intermediate HCOOH(a), i.e., HCOOH

+ HCOOH(a)

(4)

Reaction (4) is written in a number of places; see for example Sun et al. [17]. Without reference to the intermediate step, i.e., reaction (4) then either the next step must be considered as very fast or reaction (1) must be interpreted as a reaction involving reactants in the boundary layer surrounding the electrode. Among the many other possible reactions that might constitute part of the direct path, we also list the suggestion of Capon and Parsons [3] HCOOH

+ H --j COOH

+ H,

(5)

A mechanism for a path involving the formation of strongly adsorbed intermediates is more difficult to formulate primarily because the identity of these intermediates remains controversial. Experimental evidence has been presented for several different candidates that include HCO and CO [l]. Experiments also indicate the formation of more than one strongly bound-intermediate [7-9,16-181. In order to present reasonably condensed models and descriptions, we consider only one strongly bound intermediate and take it to be linear carbon monoxide. Two reactions proposed for the formation of the intermediate [9] are represented by COOH+H+CO+H,O

(6)

and 2_COOH -+ CO + H,O + CO, Formation HCOOH(a)

of CO directly + CO + H,O

from HCOOH(a),

(7) i.e., (8)

has also been proposed; see for example Sun et al. [17]. In addition, Parsons and VanderNoot [l] proposed that the first step in poison formation involves cleavage of the C-OH bond in formic acid and formation of the relatively short lived intermediate HCO. The latter intermediate is oxidized to CO. This scheme, which is

154

similar to one proposed by Walter et al. [18], explains the detection of both HCO and CO. The above schemes do not include all the ingredients necessary for the prediction of oscillations. From time to time explanations for oscillations are given in terms of film formation, adsorption of anions or dissolution of metals. Arguments presented some time ago by Wojtowicz et al. [19] discount these processes as fundamental causes. Although these processes may accompany oscillations, the occurrence of oscillations must involve a nonlinear feedback mechanism; for a discussion of feedback mechanisms see Franck [20]. We know of only two such feedback mechanisms that have been formulated for the oxidation of formic acid/formate; see Wojtowicz et al. [19]. We modify one of these feedback mechanisms so that it is incorporated into a scheme that is consistent with more recent information conceming the mechanism for the oxidation of formic acid. The feedback mechanism is assumed to occur in a reaction between the poison and species formed in the process that leads to the formation of an oxide layer. At sufficiently high potentials, hydroxyl radicals are chemisorbed [17,21], H,O + QH + H++ This step formation.

eP

is often used to represent the initiation of the first The hydroxyl radicals in turn react with the poison:

QH+CO-,H++CO,+e-

(9) stage

of oxide

(10)

Wojtowicz et al. [19] have proposed that the rate of a reaction, which is similar to reaction (lo), is proportional to the surface density of vacant sites. (The paper cited considered reactions in which intermediates of a direct path reacted with QH.) If the rate of reaction (10) is proportional to the density of sites and the reaction also produces at least two vacant sites, the reaction rate increases. This property is characteristic of certain feedback mechanisms [20]. Such an explosive rate law seems to be consistent with spectroscopic data [ll] and, as will be noted, with features found in our measurements on oscillations. A physical explanation for the occurrence of oscillations under galvanostatic conditions can be given in terms of the stated kinetics: under appropriate conditions, reactions lead to the formation of a poison which partially blocks the direct path. Then, in order to satisfy the requirements of the applied current, the system responds with an increase in potential so that the rate of reactions in the direct path increase. At sufficiently high potentials, QH is formed, reaction (9) and subsequently reacts with the poison, reaction (10). This latter reaction cleans the electrode surface of hydroxyl radicals and poison and consequently, the potential drops. However, after a sufficient amount of poison is again formed on the surface of the electrode, the potential begins to rise and the cycle repeats. The above description is adequate to describe many features of oscillations. However, our study provides evidence that, in addition to the reaction of QH with CO, there exist other fundamental nonlinear feedback mechanisms coupled directly to that reaction. Measurements also indicate that the oxidation of formic acid is

1.55

coupled through additional nonlinear feedback mechanisms to the later stages of oxide-layer formation. EXPERIMENTAL

The electrochemical cell A three-neck, 500 ml flask, which was immersed in a bath held at 50.0 ’ C, served as the electrochemical cell. The cell contained a 400 ml solution of 2.0 M sodium formate and 1.0 M H,SO,. Prepurified nitrogen flowed through the solution during all experiments. Chemicals The sodium formate was obtained from Fisher Scientific. Reagent grade sulfuric acid (EM Science, Cherry Hill, NJ) and distilled water, which was further purified by processing through a millipore deionization unit, were used in all solutions. Instrumentation A PAR-273 potentiostat/galvanostat, which was controlled by an IBM-XT computer, was used in the experiments to measure the potential and set the current. Measurements were directed to an HP, model-9237 computer through an HP, model-3852A, data-acquisition unit and/or a strip-chart recorder. The data-acquisition unit was equipped with a model-44702A, HP, 13-bit voltmeter. The disk-electrode was connected to a PAR, model-616 rotator; the rotation rate was held fixed at 4000 rpm throughout all experiments. E~ectrode~~treatment of working electrode and methods A saturated calomel electrode (SCE) was employed as the reference electrode (note, this electrode was held at the temperature 50.0°C. All measured potentials reported in this paper are with respect to this electrode. A graphite rod was used as the counter electrode. A rotating, polycrystalline-platinum-disk, electrode (PAR part number 4666), with a diameter of 18 mm, 5.0 mm in platinum and 13 mm in teflon, served as the working electrode. Before each set of experiments, the working electrode was subjected to a cleaning treatment: first the electrode was polished with 0.25 pm diamond paste. After washing the electrode with distilled-deionized water, it was transferred to an electrochemical cell containing a solution of 1.0 M sulfuric acid. The working electrode was then subjected to a cycle in which the potential was held fixed at an upper limit of 2.0 V (SCE) for 40 s and at a lower limit of -0.40 V, for 10 s. Between switching from one limit to the other, the potential was held for 5 s at 0.0 V. This cycle was repeated 30 times. The electrode was tested by cyclic voltammetry using the potential range - 0.20 to 1.20 V. Cyclic voltammograms were obtained that were identical to those reported in the literature. The electrode was then washed, transferred to a cell containing a solution of the same composition used for measurements, and subjected to 10 cycles of the type described.

156

At the beginning of each set of experiments, the system was allowed to relax under open circuit conditions for approximately 150 min. For detailed study of the oscillations, the current was changed in small increments and the system was held at each value until a temporal pattern was sustained. Following the stated procedure, the sequences of behavior and the order in which they occurred were always reproduced. RESULTS AND DISCUSSION

B&ability Measured relationships between the current and potential responses are shown in Fig. 1. Data used for the plot was collected in an experiment that was conducted as described in the following: After cleaning the electrode and allowing the system to relax under open circuit conditions, the current was increased in increments G 5.0 x lop4 mA to a value of 0.24 mA. Each value of the current was held fixed for a period of 8 minutes; in almost all cases the change in the potential response over the final minute was less than 1%. The current was then decreased to a value of zero at a rate of 5.0 x 10m4 mA/min. Except for the last value at the low current end of the upper curve, all final potentials appeared to be constant within the accuracy of the experiment. The measurements demonstrate that the system supports an oscillatory response

1.40

I

I/mA

0.24

Fig. 1. (a) Measured potential response [E vs. SCE (50.0°C)] at different values of the applied current (I). Open squares represent data recorded after changing the current in the positive direction. Two open squares for one value of the current represent maximal and minimal values of an oscillating potential. Closed squares represent data recorded after changing the current in the negative direction. Inset: An enlargement of the low current region.

157

over a significant range of values for the applied current and that the system exhibits bistability, i.e., one of two different asymptotic behaviors can be realized under the same conditions. Bistability has been observed in several different types of electrochemical processes; see for examples Russell and Newman [22], Mark Jr. et al. [23] and Xu and Schell [24] and, in a previous study of formic acid, see Conway and Dzieciuch [25]. We first present a possible explanation for the occurrence of bistability that does not include coupling to oscillatory behavior. For this purpose, consider a hypothetical system that is identical to the one under study the oxidation of formic acid occurs only except for two imposed restrictions: through the direct path and no oxidation reactions occur in which the reactants include both the species that are involved in the oxidation of formic acid and the species that partake in the process by which an oxide layer is formed. It is well known that, if a large enough current is applied, the system will respond with a potential at which the evolution of oxygen occurs. This evolution is preceded by the formation of oxide layers, which occurs in stages and takes place at potentials lower than that of the evolution reaction. The explanation of bistability is based on the current understanding and observations of oxide formation [21,26].At small values of the applied current, the system meets the requirements imposed by the current through low-potential oxidation reactions of the direct path, e.g., eqns. (1) to (3). When the current is increased in small increments, the system responds with increases in the potential in order that the rates of reactions increase to values that meet the requirements of the current. Eventually, the potential becomes large enough that the initial stage of oxide layer formation occurs, eqn. (9). During some of the stages of oxide formation intermediates are produced which remain inert until higher values of the potential are achieved, and, in addition, some steps in the process of oxide-layer formation do not produce current [21,26]. Consequently, a feedback mechanism (which, of course, is different from the feedback mechanism that causes oscillatory behavior) is initiated that operates as described in the following; a potential is reached at which oxide formation occurs. Since oxides impede the oxidation of formic acid, the potential rises further, and in turn, the increase in potential increases the rate of oxide formation. Hence, once initiated, the oxide-feedback process continues until oxidation reactions, such as the evolution of oxygen, take place that match the requirements imposed by the applied current. Thus the oxide-feedback mechanism drives the system to a high-potential steady state. However, once on the upper branch of high-potential states, the system responds to small decreases in the current with small decreases in the potential (see Fig. 1) until the current is reduced to a value that is in the vicinity of a critical value. At this critical value of the current, high-potential oxidation reactions cannot take place at a rate that satisfies the requirements of the small current. Consequently, the oxide layer breaks up [27] and the oxidation of formic acid on vacant metal sites resumes. Consider now the modifications of the above description when both the formation of the “poison”, eqns. (7) and (8) and the complete coupling of the oxidation of formic acid with the oxide formation are allowed. Formation of the poison on the

158

electrode will cause substantial increases in the potential at a value of the applied current which is lower than that value at which equivalent increases would occur in the hypothetical system. (In a system without formic acid, the same rise in potential occurs at even small values of the current.) Eventually, a value of the current is reached at which the system responds with a value for the potential where the initial stage of oxide formation occurs. Species such as hydroxyl radicals then react with the poison and these reactions clean the surface of the electrode. However, formation of the poison causes the potential to increase, oxides form again, and the same events repeat. Hence, as described in the Introduction, this leads to the replacement of low-potential stationary states, i.e. stable behavior, by oscillations in the region of coexistence. The underlying cause of bistability remains the same. Oscillations The results of a detailed study of the oscillations is presented in the remainder of this section. Within the interval of oscillatory behavior, sequences of temporal patterns were found by increasing the applied current in small amounts. After each change, the current was held fixed until the response exhibited a sustained temporal pattern. During many of the initial experiments, it was often necessary to backtrack in order to ensure that an intermediate dynamical state had not been skipped. Three distinct sequences were discovered as well as a family of oscillations that possessed large amplitudes and extremely long periods. A sequence found near the low current end of the interval of oscillatory behavior is described first. A sequence containing principal oscillatory states Measured waveforms representing four different oscillatory states are shown in Fig. 2. The state corresponding to Fig. 2a consists of simple oscillations and was observed at a value of the current that was slightly larger than that value at which stationary behavior became unstable. Each remaining state represented in Fig. 2 exhibits a combination of large and very small oscillations. It is convenient to characterize a state according to the lengths of unbroken stretches of large and small oscillations that occur over a period. For this purpose we introduce notation for labelling states. The notation can be understood through the example, Lf’Lf’, in which the symbols represent a periodic state in which L, consecutive large oscillations are followed first by S, small oscillations, then by L, large oscillations and finally by S, small oscillations. Using this notation the states represented in Fig. 2 are denoted as lo, Fig. 2a, l’, Fig. 2b, 12, Fig. 2c and 13, Fig. 2d. It is noted that the states represented in Fig. 2 are treated as periodic in a symbolic sense; the small oscillations are too tiny to test for periodicity within the accuracy of the experiment. Oscillatory states that have either one large oscillation and a given number of small oscillations in one period or, one small oscillation and a given number of large oscillations per period, are usually the most prominent in regard to the size of the interval they occupy in local regions of constraint space (see Albahadily et al. [28],

159

a (b)

a

0

900

(c)

0

Time/s

900

0

Time/s

1200

Fig. 2. Potential, measured with respect to a saturated calomel electrode, plotted as a function of time. The range for the potential is 0 to 400 mV. (a) a lo state, I =1.20X 10m3 mA; (b) a 1’ state, 1=1.35~10-~ mA; (c) a l2 state, I=1.40X10U3 mA; (d) a l3 state, I=1.5OX1O-3 mA. Fig. 3. Measured waveforms. Potential range is the same as Fig. 2. (a) A l4 state, I =1.60X 10K3 mA; (b) An oscillatory state with a large number of small oscillations. I = 1.75 X 10W3 mA; (c) A state possessing a large period and no detectable small oscillations. Z = 2.75 X 10m3 mA.

Schell and Albahadily [29] and Ringland et al. [30]). Therefore, states like those in Fig. 2 are called principal states. An interesting feature of these oscillatory states is that they are part of an ordered set. The waveforms in Fig. 2 were measured at successively higher values of the applied current. As the current was increased beyond those values at which the l3 state was observed, the system continued to pass through a sequence of states in which the ratio of the number of large to the number of small oscillations decreased. A waveform representing the l4 state is shown in Fig. 3a. This state was the last member of the set under consideration that could be clearly identified in the direction of increasing current. Due to their size, the number of small oscillations could not be determined for states within this sequence that appeared at larger values of the current. An example of a measured waveform belonging to a state with several small oscillations is shown in Fig. 3b. Nearly homoclinic orbits On increasing the current from that value at which the waveform generated, the amplitudes of the small oscillations were observed

in Fig. 3b was to decrease. A

160

value of the current was finally reached at which small oscillations could not be detected. A waveform with this characteristic is shown in Fig. 3c. Notice that the period between the large oscillations increased to a relatively large value. On the application of further increases in the current, even larger periods between the large oscillations were observed. Eventually, the period became so long, and the slope on the top part of the waveform so small, that the system appeared as if it was approaching a stationary state. However, stationary behavior was not achieved. The largest observed period between oscillations was greater than seven hours. The interval of current values in which the period of time between large oscillations was longer than an hour was approximately 7.5 X 1O-4 mA. Figure 4 contains several

1

J

0

2 Time/h

I

I

8 Time/h I

Time/h

Cd) 0

2 Time/h

Fig. 4. Waveforms of oscillatory 3.75~10-~ mA; (b) 4.15~10~~

states with long periods. Potential range-100 mV to 600 mV. Current mA; (c) 4.35~10-~ mA; (d) 4.5~10-~ mA.

(a)

161

waveforms that exhibit large periods between the oscillations including examples on both sides of the the value of the current at which the largest period was observed. The results shown in Fig. 4 are consistent with the notion that the system was moved through an interval of current values that includes a value at which the underlying deterministic system possesses a cycle of infinite period. A cycle of infinite period is a concept that is well substantiated in dynamics theory [31] and is best illustrated in state space, a space in which the coordinates are all the dynamical variables of the system. In state space, a curve (trajectory, orbit) represents the evolution of the system. A periodic orbit of infinite period can arise in special cases due to the presence of an unstable stationary state, which is represented by a fixed point in state space. Local to the unstable stationary state are directions (eigenvectors) that can be classified as stable and unstable. The system evolves towards the stationary state along the stable directions and away from the stationary state along unstable directions. The formation of an infinite period occurs when the extension of an unstable direction (a curve in the state space} connects with a stable direction. Sketches illustrating the formation of a cycle with infinite period in state space are shown in Figs. 5a-c. A possible situation previous to the connection of the stable and unstable directions is shown in Fig. 5a. The location of the stationary state in the state space is labelled “fixed point”. The stable directions, local to the fixed point, are labelled with arrows pointing towards the fixed point and local unstable directions are labelled with arrows pointing away from the fixed point. The curve associated with the unstable direction that, after changing a parameter value to a critical value, becomes associated with a stable direction is labelled “unstable manifold”. For the situation illustrated in Fig. 5a, this unstable manifold approaches a limit cycle which is the representation in state space of a periodic state. The situation at the critical value of a parameter is shown in Fig. 5b, and a possible situation for a value of the parameter greater than its critical value is shown in Fig. 5c. The cycle in Fig. 5b is called a homoclinic orbit. According to the theory of differential equations, the system can only get closer and closer to the fixed point along the stable direction. It cannot reach the fixed point. It is this property that leads to an infinite period. As a constraint is changed, in our case the applied current, and the system approaches the point at which a homoclinic orbit occurs, cycles with increasing periods will be observed. The homoclinic orbit cannot be realized in practice because a small fluctuation will knock the system off the cycle. The dynamics associated with waveforms in Fig. 4 can be compared with the idea of a homoclinic orbit by constructing a phase portrait using the time-delay method. {A method that was developed for the purpose of constructing orbits in state space when only one observable can be measured; see Packard et al. 1321.)Such a portrait is shown in Fig. 5d embedded in the space, [I’(t), V(t + T), V(t + 27.), where 7 is the delay time. The trajectory is consistent with one that both approaches the neighborhood of a fixed point (outlined by fluctuations) along, but not on, a stable direction and is ejected away from the fixed point along an unstable direction. Within this interpretation the top part of the waveforms in Fig. 4 corresponds to “relaxation” towards the stations state along a stable direction, and the sudden

162

fixed

Cc)

,225 I i

Fig. 5. Sketches demonstrating formation of a homoclinic orbit in three-dimensional state space and the construction of phase plot from experimental data using the time delay method. (a) A possible situation prior to the formation of a homoclinic orbit. Stable directions are those with arrows pointed towards the fixed point; arrows pointed away from the fixed point denote unstable directions. The unstable manifold that also becomes identified with a stable manifold at a critical value of the bifurcation parameter is identified in the figure. The cycle shown in the figure is a limit cycle, a state-space representation of a period solution. (b) A homoclinic orbit. (c) Possible situation for a value of the bifurcation parameter that is slightly greater than the critical value. (d) Phase trajectory constructed from data used in Fig. 4(a). Delay time r = 4.5 s; Axes: 1= E(f), 2 = E(t + r), 3 = E(t +2r), where E represents the potential; angle =15O.

drop in the potential corresponds to moving away from the stationary state along an unstable direction. Evidence was obtained for the occurrence of nearly homoclinic orbits in other electrochemical processes by Bassett and Hudson [33], Albahadily and Schell [34] and Schell and Albahadily [29]. In these cases, the cycles were of a different nature; the system exhibited a spiral motion on moving away from the stationary state instead of the monotone motion shown in Fig. 5d. Region of a reverse sequence The observed behavior that follows the occurrence of oscillations with long periods appears to constitute part of an additional ordered set that resembles a reverse sequence of the type already discussed. For example, the ratio of large oscillations to small oscillations was found to increase with respect to increases in current. However, only two states could be clearly characterized in the sequence.

163

Beginning at a location where the system exhibited an oscillatory response with a long period and then increasing the current, small oscillations were eventually observed at the high potential part of the response. These small oscillations appeared somewhat similar to those which occurred in some of the oscillatory states in the sequence previously discussed. A waveform recorded near a value of the current where these small oscillations first appeared is shown in Fig. 6a. The first measured oscillatory state that exhibited an identifiable temporal pattern was a 1’ state. The waveform for this 1’ state is shown in Fig. 6b. Finally, after further increases, a value of the current was reached at which only one large oscillation was observed, a “lo ” state. See Fig. 6c. A sequence

of Farey states

Yet another sequence of oscillatory states was found on increasing the current beyond those values at which the state represented by the waveform in Fig. 6c occurred. Most of the states observed in this sequence consisted of a mixture of two

(b)

Time/s Fig. 6. Measured waveforms of mixed oscillations. 4.90~10-~ mA; (b) 5.00~10-~ n~4; (c) 5.30x10-*

Potential mA.

range:

-200

mV to 900 mV.

Z = (a)

164

types of oscillations; oscillations like those in Fig. 6c and oscillations with very large amplitudes. The states were ordered in such a way that, as the system passed from one state to the next in the direction of increasing current, the ratio of the number of oscillations with very large amplitudes to the number of oscillations like those in Fig. 6c increased. In labelling states in this new sequence, oscillations of the type shown in Fig. 6c play the role of small oscillations. Therefore, we treat the state in Fig. 6c as a 0’ state and, for the remainder of this section, refer to the corresponding type of oscillations as O’-like oscillations. (We note that the label O’, and the label l”, as used here, represent the same state.) A measured waveform that was collected near the beginning of the sequence is shown in Fig. 7a. For a fairly large segment, the waveform consists of the O’-like oscillations (Fig. 6~). The waveform in Fig. 7a also shows that these oscillations were interrupted by a relatively large-amplitude oscillation. On returning to the O’-like oscillations, the system evolved through an oscillation with a form similar to the oscillations shown in Fig. 6a and then through two oscillations that are similar to the oscillations in Fig. 6b. The occurrence of the last two types of oscillations persisted throughout the first part of the sequence, but were not observed in the latter part. For convenience, we treat these “complex oscillations” as if they are the O’-like oscillations when labelling states. Waveforms for states that occurred well into the sequence, a l4 and a l3 state, are shown in Figs. 7b and c. The l3 state was the last state in the sequence for which oscillations of the same form as those in Fig. 6b were observed. It is also interesting to note that, when the peak potential of a small oscillation was G 500 mV (an estimate), the small oscillations were complex

Time/s

240

Fig. 7. Measured waveforms of mixed oscillations. Potential range the same as Fig. 6. (a) I = 6.00 X lo-* mA. (b) a l4 state, I = 6.40 x lo-* mA. (c) a l3 state, I = 6.50 x lo-* mA.

165

and, when the peak potential was > 500 mV, the small oscillations were of the simpler type, i.e., the Or-like oscillations. A major characteristic of the sequence is revealed by the waveforms shown in Fig. 8. The states 12, Fi g. 8b, l’, Fig. 8d, and 2l, Fig. 8e, are by definition principal states, whereas the states 1213, Fig. 8a, 1’12, Fig. 8c were found intermediate to adjacent principal states. Waveforms of the intermediate states appear as simple combinations of the waveforms representing the adjacent principal states. A sequence of periodic states with this characteristic obeys Farey addition: for each state define a “firing number” F = S/( S + L), where S and L represent the

I

Time/s

240

Fig. 8. States from a Farey sequence. Potential range same as Fig. 6. (a) A 1213-state, mA; (b) a 12-state, Z = 7.40~ 10m3 mA; (c) a 1’12-state, = 8.20x 10m2 mA; (d) a l’-state, mA; (e) a 2’-state, Z = 0.1040 mA; (f) IO-state, Z = 0.1030 mA.

Z = 6.80 X 10m2 Z = 8.80~ 10m2

166

number of small and large oscillations, respectively. The Farey numbers is obtained by summing the numerator and denominator s,(s,+~,)+~*/(~*+~,)=(s,+~,)/(s,+~,+~,+~*)

sum of two firing separately, i.e., (11)

For example the two principal states, 1’ and l2 have firing numbers l/2 and 2/3, respectively. The Farey sum of these two firing numbers is 3/5; which is the firing number of the 1’1’ state, i.e., the state found between the 1’ and l2 states. We found only principal states and states that are associated with the Farey sum of the two firing numbers of adjacent principal states. These include states that were found between the 2i and lo states. For examples, the 2i3l and 3i4i states were observed (not shown). A sequence of Farey states was also obtained in a previous experimental study of the oxidation of formic acid [14] that was conducted under different conditions (0.5 M H,SO,, 3O’C). In a completely different electrochemical system, a sequence of states was characterized in which many additional Farey states were observed [28].

Relationship to mechanism The upper portion of the waveforms in Figs. 2 to 4 spans a potential range within which OH is expected to form on the electrode. It has been claimed that the initiation of oxide formation on a platinum electrode begins at 170 mV (SCE), ref. 6, p. 81. By comparing voltammograms and galvanostatic charging curves, it was also deduced that oxides form first on a platinum electrode at potentials G 330 mV (SCE) in solutions containing formic acid and sulfuric acid [35]. We have verified that the characteristic oxide arrest [35] in a voltammogram measured with a sweep rate of 5 mV/s and under the same conditions as those used in the constant current experiments occurs at 410 mV (an upper bound for the value at which oxide formation is initiated). This result is consistent with more recent work using spectroscopic methods [ll] and the proposal that reactions involving OH, or a closely related species, remove the poison. The rate law proposed by Wojtowicz et al. [19] is consistent, as seen for example, in Fig. 4, with the sudden drop in potential that follows its rise to a relatively high value. In an attempt to examine this point in more detail, we constructed models, which are based on the proposed elementary kinetic steps given in the Introduction, that describe the temporal evolution of the system. The formulation of the models is presented in the Appendix. Potential oscillations, which were calculated using these models, are shown in Fig. 9. The waveforms in Fig. 9 resemble waveforms obtained in experiments; a sudden drop in potential is predicted to follow a relatively slow increase. In Fig. 9c, the calculated surface concentrations of CO and OH (&o and 0OH, respectively) are plotted as a function of time. The phase difference between the two concentrations follows behavior expected when the underlying fundamental cause of oscillatory behavior is the autocatalytic step proposed by Wojtowicz et al. [19]; i.e., the maximum in @co precedes that of So,. Other models were examined that contained different variations of the direct path and/or mechanism for the formation of the poison. They also predict oscillations as long as the autocatalytic

167

0.0

0.0

100

200

TIME/s Fig. 9. Calculations of potential and surface concentrations as a function of models and the parameter values used in the calculation are given in calculated from model 1; see Appendix. (b) Potential calculated from model concentrations of CO (upper waveform) and OH calculated using model concentration of OH is terminated so that the shape of an oscillation in clearly.

time. The formulation of the the Appendix. (a) Potential 2. (c) Dimensionless surface 2. The time series for the the upper waveform is seen

step is included. Sequences of mixed oscillations, like those observed in the experiments, were not simulated. However, a complete analysis of models with this level of complexity is almost impossible. In other types of chemical processes, the occurrence of mixed oscillations, like those in Figs. 2 and 3, are explained by the coupling of two instabilities [36]. See also Barkley [37] and references therein. Perhaps the simplest conjecture here is that two forms of strongly bound CO partake in similar nonlinear reactions but with a slightly different potential dependence. In addition to those at low currents, the occurrence of mixed oscillations at higher currents implies other nonlinear couplings. It is important to note that the value of the largest peak potentials in Figs. 7 and 8 are over 200 mV larger than the value of the peak potentials in Figs. 2-4. These results provide evidence that the oxidation of the fuel couples to a later stage of oxide formation. Evidently, after a number of small oscillations (see Fig. 8) the surface concentration of primary oxides passes through a threshold value that cannot be completely consumed in reactions with the poison and, consequently, the potential rises to larger values.

168

However, the system does not evolve to the upper branch of stationary states; see Fig. 1. Instead, reactions occur that remove at least partially the species that form in the late stages of oxide-layer formation. Consequently, these reactions allow the potential to drop to small values. SUMMARY

The sequences of temporal behavior exhibited by the oxidation of formic acid, are summarized in Fig. 10. The results show that a large variety of dynamical behaviors is possible which can provide stringent tests for mechanisms. The observation of oscillations implies a nonlinear feedback mechanism. The proposal that the reaction between OH and CO is strongly nonlinear and autocatalytic provides an explanation for the occurrence of oscillations and some features in the measured waveforms. However, the observation of mixed oscillations at low values of the current indicates the existence of at least two feedback mechanisms. At higher current values, the occurrence of mixed oscillations with large peak potentials provides evidence of additional nonlinear feedback mechanisms involving removal of species associated with later stages of oxide-layer formation. Spectroscopic methods should provide additional details of the different surface chemistries associated with the different types of sustained dynamical behavior. The measurewill provide additional ment of a phase difference between two concentrations

I I

1.

A Sequence of Mixed Oscillations. The principal states LO, 11, 12, 13, and 14 were observed; see Figs. 2 and 3. The ratio of the number of small to the number of large oscillations increased.

2.

A Family of Large-amplitude Long periods of time between See Fig. 4.

3.

A Sequence of Mixed Oscillations. The ratio of the number of small to the number of large oscillations decreased. See Fig. 6.

4.

A Sequence of Farey States. Between two intervals of the current in which the principal states, lM+l and lM, were found, the Farey intermediate, LMlM+L, was observed. See Fig. 8.

Oscillations. oscillations.

Fig. 10. Summary of the oscillatory behavior observed in the oxidation of formic acid. Behaviors are listed in the order that they were found.

169

details of kinetic mechanisms (see Fig. SC). In related work, Anastasijevic et al. [15] have demonstrated that differential mass spectroscopy can be used to study dynamical behavior. In their investigation of the oxidation of formic acid, oscillations in both the production of CO, and the potential that followed an anodic polarization were correlated to the recharging of the double layer and the oxidation of the electrode surface. APPENDIX

Formulation of model 1; reaction directly from boundary layer and explicit temporal evolution of surface hydrogen In this appendix we present the formulation of models that describe the oxidation of formic acid. The models are based on the kinetics presented in the Introduction and include the formation of only one strongly adsorbed intermediate. Concentrations of all species in the electrode boundary layer are assumed constant; the behavior was not affected by the rotation rate except very close to bifurcation points. In this first model, model 1, which was used to calculate the time series in Fig. 9a, we assume that the reactions in eqns. (1) to (3) and eqn. (5) constitute the direct path and that the reactions represented by eqns. (6) and (7) account for the formation of CO. In the differential equations for the model, surface concentrations are scaled by the surface concentration of possible sites on the electrode, S,, and time is scaled by l/(&k,,), where k,O is the rate coefficient of the reaction written in eqn. (1). This coefficient is assumed to be potential independent. The potential is scaled by an arbitrary potential, E,. After scaling the differential equations, the factors S, and k,, can be adsorbed into the rate coefficients in such a way as to render these coefficients dimensionless. All rate laws will be written in the corresponding dimensionless form. In the following, the dimensionless surface concentrations COOH, CO, QH, _H and Q will be represented by 13,, r3,,, r&,, @,, ando,. We assume that two vacant sites are required for the first dehydrogenation reaction, eqn. (1). This assumption leads to the following scaled rate law. r, =S2

641)

where S represents the dimensionless electrode and satisfies the equality, s = 1 - ( e,,+

surface

concentration

of vacant

sites on the

e,, + e,, + 8, + e,)

The rate law for the second r, = k,SB,,

reaction

(AZ) in the direct path, eqn. (2) is written

as (A3)

where k, is given by k, = k2, exp[P(E

- -%)I

(A4)

In eqn. (A4), E is the scaled potential difference and /3 is equal to aFE,/RT, where R is the gas constant, (Y is the symmetry factor, F is Faraday’s constant and T is

170

the temperature. We approximate the direct path, eqn. (5), as r3 =

the rate law for an additional

reaction

included

k304

in

(A9

The rate laws for the formation

of CO, eqns. (6) and (7) are given, respectively,

r4 = k4&L4,

as (A6)

and r5 = &a(%

(A7)

These expressions, eqns. (A6) and (A7), are approximated as potential independent. The rate law for the chemisorption of OH, eqn. (9) is assumed to be r, = k,S - k_,&,

648)

where k,=k,,

exp[P(E-&)I

(A9)

and k-,

= k_,,

-/3(1 - or)(E - E,)/a]

exp[

For the autocatalytic 17 =

k7(S

+

reaction

involving

(AlO) OH and CO, eqn. (9) we write

knVU’o~

WV

where k7=kT0

exp[D(E-&)I

(Al2)

and k,, is a constant. By setting k 72 equal to zero, eqn. (All) reduces to the rate law of Wojtowicz et al. A finite k,* takes into account the finite probability for reaction when all sites are occupied. We consider the ionization of hydrogen, which is represented by eqn. (3) to be reversible and use the expression rs = k&l, - k_,S

W)

for the rate law, where k8 = k8, exp[P(E

- Et,)]

(14)

and k-,

= k-,,

exp[ -/3(1-

ol)(E - E,)/a]

In the first model, we also include

the surface

(AI5) reaction

QH-+H++_O+e-

(Al6)

and the following

approximation

for the evolution

of oxygen

H,O+G+1/20,+2H++2e-+Q For the reaction

represented

r, = k,O,, - k-9t?o

(AI7) in eqn. (A15), we employ

the following

rate expression (A18)

171

where k,=&aexp[fl(E-E,)]

(AI9)

and k-,

= kP,,

exp[

-P(l

- a)( E - &)/a]

The rate law for the “evolution

of oxygen”

(‘420) is given as

rio = k,o%

(A21)

where kio = kioo exp[2P(E-

-G)I

The temporal evolution of the surface concentrations following set of differential equation: d8,,/dt d8,,/d

= r, - r, + r, - r, - 2r, t = r, + r, - r,

WV and the potential

satisfy the

(A23) (A24

d B,,/d t = r, - r, - r,

(‘425)

d8,/dt

= r, - r3 - r, - r,

(A26)

dt?,/dt

= r,

(A27)

dE/dt

= I’ - y( r, + r, + r, + r, + r, + 2r,,)

(A281

The symbols

in eqn. (A28) that remain undefined are explained in the following: I’ = l~,,,/(k,,S,CEo), where Iapp is the applied current density and C is the capacitance, which is approximated as a constant; and y = FS,/(CE,). The above equations were integrated on an HP, model-9237 computer using a semi-implicit method (the trapezoid rule) and the exact expression for the Jacobian. The values used for the parameters in the calculation shown in Fig. 9a are listed in the following: k,, = 0.40695,, Ikk,, = 4.1209 X 10p2, k, = 4.1209 X 10e2, k,, = 1.6484, k,, = 0.4632867, k_,, = 9.1928 x 10-2, k,, = 0.41418, k,, = 1.0 x 10-3, k,, = 1.5385, k_,, = 0.15385, k,, = 1.5385, k_,, = 1.5385, k,, = 15.385, E2 = 0.0, E,=O.3, E, =0.25, E, = -0.23, E,=O.65, E,,=0.80, I’=4.615 10-2, y = 0.39, /S!= 18.0, (Y= 0.5. The real time and potential scales in Fig. 9a are obtained by setting S,klo to 6.5 s-l and E, to 1.0 V. The approximations, eqns. (A15) and (A16), for representing oxide growth and subsequent oxygen evolution [3X] are crude. These reactions do not play a role in the calculations of the small potential oscillations, Fig. 9a. The concentration, f?,,, remains tiny even at the peaks of the oscillations. Model 2: Explicit account of the adsorption of formic acid The second model, model 2, which was used to calculate the waveforms in Fig. 9b and 9c, explicitly includes the adsorption of formic acid, eqn. (4). For this reaction we employ the rate law r, = S

(~29)

172

for the adsorption of formic acid, does not appear The rate coefficient, k,, explicitly in eqn. (A28) because in this model we scale time by the inverse of that rate coefficient. We now also require a rate law for the reaction HCOOH(a)

-+ COOH f H++ e-

(A30)

which we take to be (A31)

r11 = kJ&, where &, is the scaled surface concentration k,, = kIIO exp[P@-

En)]

of HCOOH(a)

and (~32)

We assume that the poison is formed only through the reaction represented by eqn. (8) and use the following expression for the corresponding rate law r12 =

km41

(A331

Instead of using eqn. (A16), we use a reaction by which oxide can be reduced [27] _O+ HCOOH + H,O + CO,

(A34)

which is given the rate law r13 =

~,3*%

(A351

Although eqn. (A34) by itself is a crude representation of the kinetics at high potentials, it is sufficient to illustrate a physical point. The reaction, (A34) is necessary to maintain oscillations. Upon lowering the value of k,,, by a factor of five from that used in the calculation of Fig. lb, the electrode surface becomes saturated with GH and 0. This result provides evidence, which is in addition to the experimental evidence, that the oxidation of formic acid is also strongly coupled to a later stage of oxide formation. The other reactions in model 2 are also in model 1. The complete temporal evolution of the system is given by the following set of differential equations: d&,/dt = r, - r,, - r12

(A36)

dB,,/dt = r,, - r2

(A37)

df?,,/dt = r12 - r,

W8)

d&,/d t = r, - r, - r,

(A39)

d@,/dt = r, - r13

fA40)

dE/dt

= I’ - y(r2 + r, + r, + r, + rll)

(A411

We note, however, that the dimensionless concentration of vacant sites, S, now satisfies the equality S = 1 - (19,~+ 8, + f3,, + @,, + @J, and that the dimensionless rate coefficients, which are common to both models, are formally identical to those of the first model only after multiplication by the factor k,/(Srk,,). The proportionality factor arises because we scaled the time differently in each model. For the same reason, I’ = I,,,/( k&Z), in model 2. The values of the parameters used in the

173

calculation of Figs. 9b and 9c are listed in the following: k,,, = 3.3333, k,,, = 1.0025, k_,, = 0.13278, k,, = 0.68716, k,, = 1.0 x 10-3, k,, = 2.2222, k_,, = 2.2222, k,,, = 3.3333, k,,, = 0.43333, k,,, = 8.8889 x 10-2, V, = 0.04, V, = 0.4, V, = 0.35, V, = 0.60, V,, = 0.03, I’ = 6.6667 10P2, y = 0.39, /3 = 18.0, (Y= 0.5. The real time and potential scales in Figs. 9b and 9c are obtained by setting k, to 4.5 s-l and E, to 1.0 v. REFERENCES 1 R. Parsons and T. VanderNoot, J. Electroanal. Chem., 257 (1988) 9. 2 J. O’M. Bockris, B.E. Conway, E. Yeager and R.E. White (Eds.), Comprehensive Treatise of Electrochemistry, Vol. 3, Plenum Press, New York, 1981. 3 A. Capon and R. Parsons, J. Electroanal. Chem., 45 (1973) 205. 4 A. Capon and R. Parsons, J. Electroanal. Chem., 44 (1973) 1. 5 A. Capon and R. Parsons, J. Electroanal. Chem., 41 (1973) 239. 6 W. Vielstich, Fuel Cells, Wiley, New York, 1970. 7 B. Beden, M.C. Morris, F. Hahn and C. Lamy, J. Electroanal. Chem., 229 (1987) 119. 8 K. Kunimatsu, J. Electroanal. Chem., 213 (1986) 149. 9 K. Kunimatsu, and H. Kita, J. Electroanal. Chem., 218 (1987) 155. 10 B. Beden, A. Bewick, M. Razaq and J. Weber, J. Electroanal. Chem., 139 (1982) 203. Chem., 241 (1988) 143. 11 D.S. Conigan and M.J. Weaver, J. Electroanal., Chem., 205 (1986) 135. 12 T. Hartung, J. Willsau and J. Heitbaum, J. Electroanal. Sot., 109 (1962) 1005. 13 R.P. Buck and L.R. Griffith, J. Electrochem. 14 M. Schell, F.N. Albahadily, J. Safar and Y. Xu, J. Phys. Chem., 93 (1989) 4806. H. Baltruschat and J. Heitbaum, J. Electroanal. Chem. 272 (1989) 89. 15 N.A. Anastasijevic, Chem., 245 (1988) 145. 16 A. Pavese, V. Solis and M.C. Giordano, J. Electroanal. Chem., 240 (1988) 147. 17 S.G. Sun, J. Clavilier and A. Bewick, J. Electroanal. Sot., 132 (1985) 1635. 18 0. Walter, J. Sillsau and J. Heitbaum, J. Electrochem. 19 J. Wojtowicz, N. Marincic and B.E. Conway, J. Chem. Phys., 48 (1968) 4333. Berlin, 1985, pp. 20 U.F. Franck, in L. Rensing and N.I. Jaeger (Eds.), Temporal Order, Springer-Verlag, 2-12. 21 H. Angerstein-Kozlowska, B.E. Conway and W.B.A. Sharp, J. Electroanal. Chem., 43 (1973) 9. 22 P.P. Russell and J. Newman, J. Electrochem. Sot., 130 (1983) 547. Studies of 23 H.B. Mark Jr., T.M. Kenyhercz and P.T. Kissinger, in D.T. Sawyer (Ed.), Electrochemical Biological Systems, ACS Symposium Series, American Chemical Society, Washington, DC, 1977, p. 1. 24 Y. Xu and M. Schell, J. Phys. Chem., 94 (1990) 7137. 25 B.E. Conway and M. Dzieciuch, Can. J. Chem., 41 (1963) 38. 26 J.S. Hammond and N. Winograd, J. Electrochem. Sot., 78 (1977) 55. 27 A. Hoffman and A.T. Kuhn, Electrochim. Acta, 9 (1964) 835. 28 F.N. Albahadily, J. Ringland and M. Schell, J. Chem. Phys., 90 (1989) 813. 29 M. Schell and F.N. Albahadily, J. Chem. Phys., 90 (1989) 822. 30 J. Ringland, N. Issa and M. Schell, Phys. Rev. A, 41 (1990) 4233. 31 D.R.J. Chillingworth, Differential Topology with a View towards Applications, Pitman, London, 1976. 32 N.H. Packard, J.P. Crutchfield, J.D. Farmer and R.S. Shaw, Phys. Rev. Lett., 45 (1980) 712. 33 M.R. Bassett and J.L. Hudson, J. Phys. Chem., 92 (1988) 6963. 34 F.N. Albahadily and M. Schell, J. Chem. Phys., 89 (1988) 4312. 35 V.S. Bagotzky and Yu.B. Vasilyev, Electrochim. Acta, 9 (1964) 869. 36 M. Schell and J. Ross, J. Chem. Phys., 85 (1986) 6489. 37 D. Barkley, J. Chem. Phys., 89 (1988) 5547. 38 J.P. Hoare, J. Electrochem. Sot., 132 (1985) 301.