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Chapter 7 Chaos at Different Levels of Heterogeneous Catalytic Systems
249
Chapter 7 CHAOS AT DIFFERENT LEVELS OF HETEROGENEOUS CATALYTIC SYSTEMS The oscillations discussed in Chapters 3 and 4 were mainly periodic. However, in experimental studies a lot of aperiodic or irregular oscillations have frequently been observed. To some extent there are even more observations of irregular oscillation patterns than of truly periodic oscillations. In the early studies of oscillations in heterogeneous catalytic systems the main attention was focused on the regular oscillations, with the aim of obtaining information about the system from the study of the dependence of period and amplitude upon the experimental parameters. Irregular oscillations were considered to be attributable to different external sources of noise. Later it was discovered that the irregular aperiodic signals also contain much information about the system if they represent the chaotic behaviour. The phenomena related to the occurence of randomness and unpredictability in completely deterministic systems have been called 'deterministic chaos' or simply chaos [I]. The observed chaotic behaviour in time is not due to external sources of noise, but arises from the nonlinear nature of the dynamic systems. The actual source of irregularity is the property of the nonlinear system of separating initially close by trajectories exponentially fast in a bounded region of phase space. The investigation of chaotic oscillations in different kinds of systems has developed very rapidly in recent years. The fundamental results and the main definitions in this field can be found in a number of monographs 11-41. This chapter will briefly discuss how irregular and aperiodic oscillations can be analysed, how one can prove that the measured signal is a chaotic one, and the information that can be obtained from an analysis of the chaotic signal. The observed chaotic oscillatory behaviour at different levels of a heterogeneous catalytic system will be presented. There are a number of measures that characterize deterministic chaos [4,6]. First of all the experimental data can be presented as the time dependence of a measurable dynamic variable, v(t), determined at sequential time intervals, t,=k(At), where k=1,2,...,n. This is the so-called 'time series'. The time series must be recorded in the computer and the dynamic properties of the observed complex behaviour can be analysed. The Fourier power spectra analysis of the dynamic regime can distinguish between a complex quasiperiodic behaviour and the chaotic regime. This method allows one to resolve the observed time series into the component frequencies, that
250
are included in the time series. Periodic and quasiperiodic behaviour can be identified through the presence of sharp peaks in the spectra at a few representative frequencies. The presence of broadened peaks of a large number of frequencies indicates a chaotic regime or broad-band noise in the system. Chaos and noise can be distinguished on the basis of the properties of deterministic chaos. Chaos cannot be equated simply with disorder and it has a kind of inner order, but without periodicity. The geometrical image of the chaotic movement is a strange attractor, which is a stable object as a whole, while the motion within the attractor is unstable. The latter is connected with one of the most important properties of the strange attractor known as 'sensitive dependence on initial conditions', i.e. the distance between two points on the strange attractor, initially separated by an infinitesimal distance, grows exponentially fast with time on the average. The stability of the motion on a strange attractor is characterized with the help of Lyapunov exponents, each of which measures the average spreading of trajectories within the attractor in one direction. Therefore there is one Lyapunov exponent for each dimension of the appropriate phase space. One of the Lyapunov characteristic exponents, corresponding to the direction of flow of the trajectory, is A zero. A second Laypunov exponent is always negative in a dissipative system [6]. chaotic system is characterized by an exponential divergence of trajectories in at least one dimension of the flow, i.e. at least one of the Lyapunov exponents must be positive. Therefore the strange attractor must be located in a phase space that is more than two dimensional. Table 1 Lyapunov exponents for different attractors Stable steady state Limit cycle Torus Strange attractor Strange attractor (hyperchaos)
h,O hl,h>O
V O
&
h3<0 h3<0 h3<0 h3<0 h4<0
Table 1 shows Laypunov exponents for different attractors. A saddle node or a focus is characterized by three negative exponents; a limit cycle has one zero Laypunov exponent. A torus, i.e. quasiperiodic behaviour, can be characterized by
251 two zero exponents and one negative one. If the attractor has more than one positive Lyapunov coefficient, then so-called hyperchaos may develop. The largest Lyapunov exponent can be used to distinguish between periodic, quasiperiodic and chaotic behaviour. Important information about the system can be obtained by constructing a strange attractor from experimental data. In principle the strange attractor must be constructed from the knowledge of trajectories of all variables, vi, in N-dimensional space. In experimental studies in many cases only one variable, e.g. the reaction rate, or the temperature of the catalyst is measured continuously and as a time series. Takens has demonstrated [7] that a multidimensional phase portrait can be constructed, even from the measurement of one variable using the time delay method, since each variable reflects information about the dynamic behaviour of the system. For any observable variable, V, and time delay, z, the n-dimensional vector is constructed from the vectors {V(tk),V(tk+z),...V(tk+ (n-l)T)}, k = 1,2,3,... where n is the embedding dimension. The strange attractor obtained may have many of the same properties as the original attractor, constructed from Vj variables (j=1,2,...,N) if the embedding dimension is larger than the number of independent variables: n>2N+1. In some cases it is more convenient to analyse the properties of the Poincare maps, introduced in Chapter 2, than the properties of the strange attractor. The next amplitude or next maximum maps can also be constructed from the time series. A next amplitude map represents one of the cases of the Poincare map and plots the amplitude of the (n+l)th peak against the amplitude (maximum) of the (n)-th. For regular periodic oscillations the next amplitude map will reveal a finite number of discrete points, while for a chaotic behaviour the next amplitude map will consist of an infinite number of different points. It must be noted, however, that twodimensional next amplitude maps reflect the essential properties of a strange attractor only if there is only one expanding direction within the attractor [4]. One of the most important properties of the strange attractor is its dimension. The dimensions of different types of attractors are shown in Table 2. The dimension of an attractor is smaller than the dimension N [5].In a 3dimensional space the dimension of the strange attractor is smaller than 3. It can be expected, that a strange attractor must have a non-integer or fractal dimension in the range 2 < D < 3 for a three-variable system. The fact that the dimension of the strange attractor is always less than the dimension of the space in which it is embedded provides a lower limit for the number of state variables needed for modelling the main dynamic properties of the system.
252
Table 2 Dimension of different attractors Attractor
Dimension
Dynamic behaviour
Stationary state Iimit cycle torus strange attractor
0 1 2
stationary process regular periodic oscillations quasiperiodic oscillations chaotic behaviour
>2
Fractal dimension requires a new concept of dimension [8]. There are a number of definitions of the dimension of a strange attractor: the Hausdorff ,,, [9]. dimension, D; the information dimension, D, and the correlation dimension, ,D The most widely known is the Hausdorff dimension [2,5]. Let N(E)be the number of small n-dimensional cubes of side length E that are required to cover a region of the state space. Then the fractal dimension or Hausdorff dimension of this region is defined as:
The information dimension, D, considers the probabilities of finding the trajectory in small cubes of side lenght E and is connected with the value of the information entropy. According to Kaplan and Yorke [lo], the Hausdorff dimension can be expressed in terms of Lyapunov coefficients for arbitrary strange attractors:
The integer j in this expression is defined in such a way that the sum of the first j Lyapunov exponents is positive, while the sum of the first j + l is negative:
253 i
i+l
i=l
(3)
i=l
The correlation dimension, ,D ,, can be directly calculated from experimental data using an algorithm by Grassberger and Procaccia [I 1,121. According to this algorithm ,D ,,, is defined as
where c(r) is the correlation integral:
If'
where r is a lenght interval, 9 the Heaviside function, the sum is equivalent to the
+ +
number of pairs ij of points on the attractor the distance I xi- x j
I
of which is smaller
than r. The relation between the three dimensions is:
Therefore ,D ,,, obtained from experimental data represents a lower limit to the Hausdorff dimension. Another important characteristic of chaotic behaviour is the Kolmogorov entropy. This characterizes the loss of information due to the exponential spreading of neighbouring trajectories within the strange attractor. To some extent the Kolmogorov entropy measures the degree of the chaos. This value is zero for limit cycle oscillations, finite and positive for strange attractors and tends towards infinity for a totally random system. Therefore the Kolmogorov entropy can be used to distinguish between chaos and noise in the system. Since there is a connection between the average loss of information and the spreading of trajectories, the Kolmogorov entropy is related to the Lyapunov exponents. This relation is expressed as:
where I is the number of positive Lyapunov exponents [13]. The rate at which information about the system is lost is equal to the sum of positive Lyapunov exponents. The simple equation (7) can be used in most cases. The other very distinct definition of deterministic chaos is the route by which it develops. As was already discussed in Chapter 2, there are bifurcations which lead to the loss of stability of the limit cycle and the appearance of chaotic behaviour. These bifurcations are characterized by the Floquet multipliers or characteristic multipliers (CM), introduced in Chapter 2 . If one of the real CM at the bifurcation point crosses the unit circle at -1 on the real axis, then a period doubling or flip bifurcation takes place. In this case the original limit cycle of the dynamic system loses its stability by creating a stable limit cycle with a doubled period. With the variation of the bifurcation parameter the cycle with the doubled period produces a new stable cycle with period 4 and so on. This is the period doubling route to chaos. The universal features of this scenario was discovered by Feigenbaum [14], who demonstrated that the sequence of parameters {p,, p2....}for which the perioddoubling bifurcations occur, constitutes a converging series lim p m = p a
m+m
When p = pm the limit cycle has an 'infinite period' and a strange attractor arises. It was demonstrated, that the rate of convergence of the infinite series of p, was determined by the universal Feigenbaum constant, which is equal to 4.6692. This allows an easy identification of this route to chaos, although it is very difficult in the experimental studies to follow more than two period doubling bifurcations. The second route to chaos is connected with the phenomenon of
51 proposed three mechanisms for the onset intermittency. Pomeau and Manneville [I of chaos, all of which are related to intermittency. These types are recognized and distinguished according to the changes in the CM at the bifurcation point [4]. The dynamic behaviour in all three cases represents the sequences of almost periodic behaviour, interrupted by bursts of aperiodic oscillations of finite duration. The three types of intermittency are closely connected with three ways in which stable limit cycle oscillations can lose their stability. The third route to chaos is the result of the breaking of a torus. This is known as the Ruelle-Takens-Newhouse scenario. In this case two CM form a complex pair and they leave the unit circle simultaneously. The sequence of bifurcations which is included in this scenario comprises the standard Andronov-Hopf bifurcation and one subsequent bifurcation to a two-frequency torus. The next bifurcation into a 3-torus
is not structurally stable and chaos arises, because the three-frequency movement is destroyed by arbitrary small perturbations. The experimentalist in this case observes a transition from regular to quasiperiodic and then to chaotic behaviour. The properties of the chaotic behaviour allow the experimentalist to distinguish between complex quasiperiodic behaviour, noise, and true chaotic behaviour and to treat these experimental data. Software packages such as Dynamical Software [I 61 were developed especially for the analysis of experimental time series. They include computational methods for reconstructing the attractor, estimating its dimension, and measuring the largest Lyapunov exponent from experimental time series. Some packages also allow the computation of correlation functions and Kolmogorov entropy. The most widely used computational algorithm (WSSV method) for the evaluation of the largest Lyapunov exponent was developed in ref. 17. A comparison of different algorithms can be found in refs. 18,19. There are many difficulties encountered during the analysis of experimental data. The main problem is connected with the external noise which is an unavoidable factor in laboratory experiments. The effect of the external noise must be taken into account in the construction of the strange attractor and during the computation of the positive Lyapunov exponent. The second difficulty is connected with the fact that, in order to obtain reliable values of the correlation dimension of a strange attractor and its properties, a sufficiently long experimental time series must be obtained. For analysing the properties of a three dimensional attractor about 50000-70000 data points are necessary [20]. This means that the chaotic behaviour must be measured for a sufficiently long time, while all experimental parameters must be kept stable during the whole experiment. The observation of chaotic behaviour has been carried out at different levels of heterogeneous catalytic systems. It is very complicated to study the phenomenon of chaos in these systems, because unpredictable changes of the catalytic activity can occur during the time required to obtain a sufficient amount of experimental points in the time series.
7.1 Single crystal surfaces Chaotic behaviour during single crystal studies was reported for the oxidation of CO on the Pt (110) surface [21]. The transition to chaos was observed at high temperatures, when starting with the CO covered surface. The temporal oscillations of the laterally averaged work function, AQ, were monitored with a Kelvin probe.
256
Typical chaotic time series are presented in Chapter 1, Fig. 7. The parameters for both chaotic series are summarized in Table 3. Fig. 1 demonstrates the transition to chaos observed in this system. The lowering of the CO partial pressure led to the transformation of regular oscillations via two period doublings to completely irregular behaviour. This is the period doubling route to chaos. The time series corresponding to the irregular behaviour were analysed and proved to represent deterministic chaos.
h
150
I50
f 100
I
e
a
100
d 50
LOO
C 50
LOO
50
3 3
10
20
30
40
50
70
60
t.
80
90
LOO
110
120
130
140
150
Isec)
Fig. 1 The transition from regular oscillations to chaos during the CO oxidation on Pt (110) observed at T=540 K and Po,=l.lO~ mbar upon decreasing Pc0 in small steps from 5.12 .
to 4.52. 10-5 mbar. (Reprinted from ref. 21 with permission.)
Table 3 Characteristics of chaotic time series observed in different catalytic systems Catalytic system
Ref.
Reactant mixure
Temperature [KI
Correlation Kolmogorov dimension entropy [s-l]
c 0 +0 2 / Pt(l10)
[23]
P ~ = 7 . 5 .1 0 - 5Torr
TK = 540
2.4 k 0.2
0.2k0.05
0.12 f 0.05
Pc0=3.4.10-~ pq=6.0.10-5 Torr
TK = 540
4.5 f 0.4
0.4 f 0.05
0.12 f 0.02
T K = 513
3.4
0.306
c0+02 I Pt foil
[27]
c0+02 I Pd zeolite CH30H+02/ Pd zeolite
[32]
c 0 +0 2 / Pt/A1203 catalyst bed
[34] [37]
Pc0=3.0.10-~ [CO]=l% [02]=20.7% [N2]=78.8% [CO]=O.5% in air [CH30H]=3.5% [02]=18% [N2]=72% [CO]=O.S% in air
Largest Lyaponov exponent Is-’]
T ~ = 4 4 0 2.1
0.013
T K = 337
0.01
2.4
T ~ = 4 4 0 2.1
-
0.01 0.06
The reconstructed strange attractors for both chaotic time series presented in Fig. 7, Capter 1 are shown in Fig. 2. At the same temperature of 540 K, but at lower oxygen concentrations, the shape of the reconstructed attractor was much more complicated (Fig. 2b). The analysis of the time series revealed that the correlation dimension and the lower bound of the Kolmogorov entropy were higher in this case. Two positive Lyapunov exponents were calculated for this case, which indicates the presence of hyperchaos in the system [23]. The origin of the chaotic behaviour could be elucidated by spatially-resolved PEEM measurements. No spatial nonuniformities had been observed during the transition to chaos via 2 period doublings, i.e. the total surface oscillated as a whole, strictly in phase [22]. Therefore this may be a case of intrinsic kinetic chaos, which can be described by the point model. However, there are some PEEM measurements that demonstrate the existence of ordered spatial structures in the form of standing waves, which oscillate strictly in phase during the limit cycle oscillations before the bifurcation to chaos. There is also some evidence that the first period doubling is still associated with ordered spatial structures and that turbulent, moving spatial structures on the surface can exist during the chaotic time series [24,25], indicating that even at the single crystal surface level spatiotemporal chaos can arise. The possibility of
258 different feedback mechanisms in this system together with the existence of spatial structures may be the reason for the detection of hyperchaos in this system [221. The same route to chaos, i.e. the transition via the period doubling route, was detected in the study of the NO + H, reaction on Pt(100) [26]. The correlation dimension of the reconstructed strange attractor was found to be 2.6 k 0.3 indicating that a minimum number of 3 variables were necessary to describe the dynamic behaviour of the system. (0)
Fig. 2. Strange attractors, reconstructed from the chaotic time serieqpresented in Fig. 7, Chapter 1. (Reprinted from ref. 22 with permission.)
7.2 Polycrystalline metals The first identification of chaotic behaviour at atmospheric pressure has been reported by Razon, Chang and Schmitz for the oxidation of CO on polycrystalline Pt ribbon [27]. Experiments were conducted in a gradientless reactor under isothermal conditions. The chaotic time series, represented in Fig. 3, were obtained at a catalyst temperature of 250°C in an excess of oxygen. A definite route to chaos was not identified in this study. The analysis of the time series presented in Fig. 3 proved the behaviour to be chaotic on the basis of the positive and finite value of the Kolmogorov entropy. The correlation dimension of the reconstructed strange attractor was found to be 3.4.
259 F A Fractions: C0.0010 Op ~ 0 . 2 0 6
x U
-
g
h B
e U
2 LL
Ng 10.784
0.2 1 -0 .
0
611
€51
691
731
771
o.3-Fi&aB 0.20.1.
- 0
I222 1262
I302 1342 TIME (rninl
1382
Fig. 3. Chaotic time series during oxidation of CO on Pt foil at atmospheric pressure. (Reprinted from ref. 27 with permission.)
No spatiotemporal observations were available in this study at atmospheric pressure, although the authors [27] excluded a temperature heterogeneity on the surface. The long term deactivation of the catalyst due to the diffusion of silicon from the bulk of the catalyst to the surface also makes it difficult to treat the experimental data in this system. Chaotic behaviour was also observed and identified during the oxidation of propylene on a Pt catalyst in the form of a ribbon. This was carried out for both kinds of experiments, when the ribbon was heated electrically either with a constant current [28] or controlled at a set average temperature [29]. As was discussed in Chapter 5, the metallic wire represents a striking example of a distributed system, where the temperature controller can stabilize different spatial structures. For such exothermic reactions as propylene oxidation they are produced by alternating zones of kinetic and diffusion regimes. Although significant differences were observed between the shape of oscillations in both kinds of experiments, the origin of the chaotic behaviour is nearly the same. A thermal imaging of the wire revealed that the back-and-forth movement of the high activity regime is responsible for the oscillations during the oxidation of propylene on the Pt wire. In the constant average temperature experiment in most cases only a fraction on the ribbon was in the ignited state. The origin of chaos was connected with the splitting of the ignited section (Fig. 4, profile a) into two waves (Fig. 4 , profile d), which moved along the ribbon with the same direction, but with different velocities. The deterministic chaos was identified with the help of the Fourier power spectral
260 analysis and the time delay method was used to construct the strange attractor [28]. The strange attractor, constructed from the time series measured for a mixture of 0.3% propylene and for an average temperature of 225"C, had a correlation dimension equal to 3.2. It increased with the decrease of propylene concentration and can be even larger than 5 for 0.2 % of propylene and an average temperature of 259°C. t
-
N
0
500
[seci 1000
1500
2000
I
I
E 0
I
I
400
I
I
400,
I
.33
j
I
- - - . _._ ,.b_
c
Y -
8
I
Fig. 4. Chaotic behaviour of the overall heat production and related temperature profiles of the electrically heated Pt ribbon during propylene oxidation. (Reprinted from ref. 28 with permission.) In the constant current experiments the ribbon remained fully ignited nearly all the time and the origin of the chaotic behaviour is due to the wide spread in the time intervals between successive wave movements of either the left or the right front of the diffusion regime. The chaotic behaviour identified in this case was found to be more regular, although the quantitative parameters of the reconstructed attractors were not reported [29].
26 1 The observation of the chaotic behaviour in the constant current experiments is rather surprising, because there is no temperature control, which can induce spatial reorganization and a behaviour of higher complexity [30]. However, in ref. 31 it was demonstrated that travelling waves of the high activity regime originated from a fixed point of the uncontrolled wire. Quasiperiodicity and aperiodic behaviour arose when a second point on the wire also started to emanate waves. The aperiodic time series were not treated for the identification of the deterministic chaos. However, it can be supposed that the surface heterogeneities and wave phenomena can lead to spatiotemporal chaos in such distributed catalytic systems like metallic wires.
7.3 Zeolite supported catalysts The transition from regular periodic behaviour to chaos has also been observed for the oxidation of CO on a Pd zeolite catalyst [32]. The details of experiment and the properties of the regular oscillations have been discussed in Chapter 3, Section 2. The complex dynamic behaviour, which was observed at CO concentrations higher than 0.57%CO at 220"C, was identified as deterministic chaos, with the characteristics shown in Table 3. The transition from regular to chaotic behaviour was found to be related to an increase of the catalyst activity. This could be achieved either by an increase of the CO concentration or by the reduction of the catalyst with pulses of hydrogen as demonstrated in Fig. 5. This indicated that the origin of the chaotic behaviour may be connected with the appearance of internal diffusion limitations when the catalyst activity exceeded a critical level. In the presence of a CO concentration gradient within the supporting zeolite crystal, the Pd clusters oscillate under different conditions. The coupling between different oscillators can lead to chaotic regimes outside the entrainment band. The possibility of diffusion chaos can also not be excluded in this case. The fact that intracrystalline transport processes within the zeolite can cause complicated dynamic behaviour was demonstrated by Jaeger, Plath and Svensson during the investigation of the methanol oxidation on zeolite supported Pd [33]. They observed toroidal oscillations with Pd embedded within a Faujasite NaX zeolite matrix and noted that this phenomenon cannot be observed using an amorphous support. Oscillations occuring in the oxidation of methanol are nonisothermal. In addition, the oxidation-reduction of Pd particles can take place. Together with the complex transport processes in this system these feedback mechanisms lead to the
262 appearance of chaotic behaviour [34]. The parameters of the chaotic motion are presented in Table 3.
Fig. 5.The transition from regular to aperiodic oscillations on Pd zeolite catalyst: (a) after the addition of O.O2%CO in the inlet mixture; (b) after the third H, pulse treatment of the catalyst surface. (Reprinted from ref. 32 with permission.) Two routes to chaos have been discovered for this catalytic system. The first one, via period doubling, was detected on moving from the high activity state, characterized by the large temperature difference between the catalyst and the gas phase temperature. The second route was less common and led via the destruction of a torus and type-ll intermittency to homoclinic chaos. A characteristic of type-ll intermittency is the beat-like modulation of the limit cycle oscillations with increasing modulation amplitude (Fig. 6). Homoclinic chaos includes homoclinic structures characterized by the presence of large bursts (Fig. 7).
263 T CKI
10 m i n
time
,
Fig. 6. Type-ll intermittency dynamic behaviour during the oxidation of methanol on Pd zeolite catalyst. (Reprinted from ref. 34 with permission.)
10
0 20 I
1
10
0
-
10 m i n
time ------+
Fig. 7. Homoclinic chaos observed during the oxidation of methanol on Pd zeolite catalyst. (Reprinted from ref. 34 with permission.)
7.4 Catalyst bed level The catalyst bed with many oscillating pellets represents a system of a higher level of complexity. However, the same route to chaos as has been observed in single crystal studies, i.e. period-doubling transition, was reported for the oxidation of CO on Pt/AI,O, [35].Two factors render the interpretation of the results difficult. The first one is that the catalyst activity was the bifurcation parameter in the experiment. This is the reason for the irreproducibility of the observed phenomena. The second is that the appearance of regular osciilations was detected from the
264
steady state with nearly complete conversion and inhomogeneities can arise in this case both on the surface and in the gas phase. The complete identification of a chaotic time series, observed during the oxidation of carbon monoxide on supported Pt catalyst in an adiabatic packed bed reactor had been carried out by Wicke and Onken [36,37]. The characteristics of the observed chaos can be found in Table 3. Together with the chaotic behaviour of the C02 production rate the authors [36,37] observed the appearance and the evolution of temperature waves in the catalyst bed. The origin of the temperature waves in the catalyst bed was demonstrated to be due to the high sensivity of the system to external perturbations in the bifurcation region, where a transition from one to three steady states in the catalyst bed of the reactor takes place. This state of high sensitivity to external perturbations may be realized under the conditions when the line of convectional heat removal, Q, and the line of reaction rate heat production, Q,, coincide with a common straight line within a certain range of reaction temperatures on the heat balance diagram, presented in Fig. l c , Chapter 1. In this case there is no definite intersection and hence no definite steady state [38,39]. For the nonoscillatory, exothermic oxidation of ethane in the same adiabatic reactor, small temperature oscillations introduced in the inlet of the catalyst bed increased by a factor of 50 at the exit of the catalyst bed. The same phenomena could be also observed in the bifurcation region, when the ethane concentration was modulated in the feed. Deterministic chaos was detected for the oscillating CO oxidation reaction. In this case self-sustained oscillations of the reaction rate at single catalyst pellets in the entrance section of the packed bed turned out to be the source of the reaction waves. The local small amplitude of kinetic nonisothermal oscillations from the catalyst pellets at the entrance of the catalyst bed initiate the growth of reaction waves that propagate through the catalyst bed causing large fluctuations of temperature and conversion in the exit section (Fig. 8 ). The larger the number of pellets that can oscillate in the entrance cross section of the catalyst bed, the denser is the sequence of reaction waves and the more chaotic the behaviour of the observed overall reaction rate [37]. The possibility of the formation of chaotic patterns, produced by the process of selective amplification of fluctuations induced by some weak external noise, has been discussed in ref. 4. In the bifurcation region, due to the high sensitivity, the system responds strongly to noise provided by the environment. It enhances one of the spatial modes present in the noise and suppresses the others, resulting in chaotic spatial patterns.
265
A
01
-
I 10 min
-I
Fig. 8.Chaotic time series during the oxidation of CO in the adiabatic packed bed tubular reactor, which is accompanied with the evolution and movement of temperature peaks along the catalyst bed, 1% CO in air, To=431 K. (Reprinted from ref. 36 with permission.) The chaos originating in the catalyst bed during the oxidation of CO is even more deterministic, because the fluctuations originate in the system itself. The experimental data presented in this chapter demonstrate that chaotic behaviour is widely spread in heterogeneous catalytic systems and that it has been identified on different levels. It is possible to say that it is easier to observe complex aperiodic or chaotic behaviour than regular periodic oscillations. The reason is that heterogeneous catalytic systems are inhomogeneous in space and, in most cases, the chaotic behaviour arises due to the complex structure or wave phenomena on the catalytic surface. The 'lumped' character of chaos is doubtful, even for single crystals. The study of chaotic behaviour gives the experimentalist additional information about the system. The similarity of the transitions to chaos for different levels of heterogeneous catalytic system proves the similarity of organization in time and space at different levels of the system. The origin of the diffusion chaos in a heterogeneous system is one of the most interesting problems which must be solved in the nearest future.
266
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25 K.Krischer, M.Eiswirth and G.Ertl, J.Chem.Phys. 96 (1992) 9161. 26 P.D.Cobden, J.Siera and B.E.Nieuwenhuys, J. Vac. Sci. Technol. (A), 10 (1992) 2487. 27 L.F.Razon, S.-M.Chang and R.A.Schmitz, Chem.Eng.Sci., 41 (1986) 1561. 28 G.Philippou, F.Schultz and D.Luss, J.Phys.Chem., 95 (1991) 3224. 29 G.PhiIippou and D.Luss, J.Phys.Chem., 96 (1992) 6651. 30 M.Sheintuch, J.Phys.Chem., 94 (1990) 5889 31 G.A.Cordonier and L.D.Schmidt, Chem.Eng.Sci., 44 (1989) 1983. 32 M.M.Slinko, N.I.Jaeger and P.Svensson, in: Yu.Sh,Matros (Editor), 'Unsteady State Processes in Catalysis', VSP, Utrecht, 1990, p.415. 33 N.I.Jaeger, P.J.Plath and P.Svensson, in: L.Rensing and N.I.Jaeger (Editor), 'Temporal Order', Springer Series in Synergetics, Vo1.29, Springer-Verlag, Berlin, 1985, p.101. 34 H.Herzel, P.Plath and P.Svensson, Physica D 48 (1991) 340.
35 J.Kapicka and M.Marek, Surf.Sci., 222 (1989) L885.
Chapter 7 CHAOS AT DIFFERENT LEVELS OF HETEROGENEOUS CATALYTIC SYSTEMS The oscillations discussed in Chapters 3 and 4 were mainly periodic. However, in experimental studies a lot of aperiodic or irregular oscillations have frequently been observed. To some extent there are even more observations of irregular oscillation patterns than of truly periodic oscillations. In the early studies of oscillations in heterogeneous catalytic systems the main attention was focused on the regular oscillations, with the aim of obtaining information about the system from the study of the dependence of period and amplitude upon the experimental parameters. Irregular oscillations were considered to be attributable to different external sources of noise. Later it was discovered that the irregular aperiodic signals also contain much information about the system if they represent the chaotic behaviour. The phenomena related to the occurence of randomness and unpredictability in completely deterministic systems have been called 'deterministic chaos' or simply chaos [I]. The observed chaotic behaviour in time is not due to external sources of noise, but arises from the nonlinear nature of the dynamic systems. The actual source of irregularity is the property of the nonlinear system of separating initially close by trajectories exponentially fast in a bounded region of phase space. The investigation of chaotic oscillations in different kinds of systems has developed very rapidly in recent years. The fundamental results and the main definitions in this field can be found in a number of monographs 11-41. This chapter will briefly discuss how irregular and aperiodic oscillations can be analysed, how one can prove that the measured signal is a chaotic one, and the information that can be obtained from an analysis of the chaotic signal. The observed chaotic oscillatory behaviour at different levels of a heterogeneous catalytic system will be presented. There are a number of measures that characterize deterministic chaos [4,6]. First of all the experimental data can be presented as the time dependence of a measurable dynamic variable, v(t), determined at sequential time intervals, t,=k(At), where k=1,2,...,n. This is the so-called 'time series'. The time series must be recorded in the computer and the dynamic properties of the observed complex behaviour can be analysed. The Fourier power spectra analysis of the dynamic regime can distinguish between a complex quasiperiodic behaviour and the chaotic regime. This method allows one to resolve the observed time series into the component frequencies, that
250
are included in the time series. Periodic and quasiperiodic behaviour can be identified through the presence of sharp peaks in the spectra at a few representative frequencies. The presence of broadened peaks of a large number of frequencies indicates a chaotic regime or broad-band noise in the system. Chaos and noise can be distinguished on the basis of the properties of deterministic chaos. Chaos cannot be equated simply with disorder and it has a kind of inner order, but without periodicity. The geometrical image of the chaotic movement is a strange attractor, which is a stable object as a whole, while the motion within the attractor is unstable. The latter is connected with one of the most important properties of the strange attractor known as 'sensitive dependence on initial conditions', i.e. the distance between two points on the strange attractor, initially separated by an infinitesimal distance, grows exponentially fast with time on the average. The stability of the motion on a strange attractor is characterized with the help of Lyapunov exponents, each of which measures the average spreading of trajectories within the attractor in one direction. Therefore there is one Lyapunov exponent for each dimension of the appropriate phase space. One of the Lyapunov characteristic exponents, corresponding to the direction of flow of the trajectory, is A zero. A second Laypunov exponent is always negative in a dissipative system [6]. chaotic system is characterized by an exponential divergence of trajectories in at least one dimension of the flow, i.e. at least one of the Lyapunov exponents must be positive. Therefore the strange attractor must be located in a phase space that is more than two dimensional. Table 1 Lyapunov exponents for different attractors Stable steady state Limit cycle Torus Strange attractor Strange attractor (hyperchaos)
h,
V O
&
h3<0 h3<0 h3<0 h3<0 h4<0
Table 1 shows Laypunov exponents for different attractors. A saddle node or a focus is characterized by three negative exponents; a limit cycle has one zero Laypunov exponent. A torus, i.e. quasiperiodic behaviour, can be characterized by
251 two zero exponents and one negative one. If the attractor has more than one positive Lyapunov coefficient, then so-called hyperchaos may develop. The largest Lyapunov exponent can be used to distinguish between periodic, quasiperiodic and chaotic behaviour. Important information about the system can be obtained by constructing a strange attractor from experimental data. In principle the strange attractor must be constructed from the knowledge of trajectories of all variables, vi, in N-dimensional space. In experimental studies in many cases only one variable, e.g. the reaction rate, or the temperature of the catalyst is measured continuously and as a time series. Takens has demonstrated [7] that a multidimensional phase portrait can be constructed, even from the measurement of one variable using the time delay method, since each variable reflects information about the dynamic behaviour of the system. For any observable variable, V, and time delay, z, the n-dimensional vector is constructed from the vectors {V(tk),V(tk+z),...V(tk+ (n-l)T)}, k = 1,2,3,... where n is the embedding dimension. The strange attractor obtained may have many of the same properties as the original attractor, constructed from Vj variables (j=1,2,...,N) if the embedding dimension is larger than the number of independent variables: n>2N+1. In some cases it is more convenient to analyse the properties of the Poincare maps, introduced in Chapter 2, than the properties of the strange attractor. The next amplitude or next maximum maps can also be constructed from the time series. A next amplitude map represents one of the cases of the Poincare map and plots the amplitude of the (n+l)th peak against the amplitude (maximum) of the (n)-th. For regular periodic oscillations the next amplitude map will reveal a finite number of discrete points, while for a chaotic behaviour the next amplitude map will consist of an infinite number of different points. It must be noted, however, that twodimensional next amplitude maps reflect the essential properties of a strange attractor only if there is only one expanding direction within the attractor [4]. One of the most important properties of the strange attractor is its dimension. The dimensions of different types of attractors are shown in Table 2. The dimension of an attractor is smaller than the dimension N [5].In a 3dimensional space the dimension of the strange attractor is smaller than 3. It can be expected, that a strange attractor must have a non-integer or fractal dimension in the range 2 < D < 3 for a three-variable system. The fact that the dimension of the strange attractor is always less than the dimension of the space in which it is embedded provides a lower limit for the number of state variables needed for modelling the main dynamic properties of the system.
252
Table 2 Dimension of different attractors Attractor
Dimension
Dynamic behaviour
Stationary state Iimit cycle torus strange attractor
0 1 2
stationary process regular periodic oscillations quasiperiodic oscillations chaotic behaviour
>2
Fractal dimension requires a new concept of dimension [8]. There are a number of definitions of the dimension of a strange attractor: the Hausdorff ,,, [9]. dimension, D; the information dimension, D, and the correlation dimension, ,D The most widely known is the Hausdorff dimension [2,5]. Let N(E)be the number of small n-dimensional cubes of side length E that are required to cover a region of the state space. Then the fractal dimension or Hausdorff dimension of this region is defined as:
The information dimension, D, considers the probabilities of finding the trajectory in small cubes of side lenght E and is connected with the value of the information entropy. According to Kaplan and Yorke [lo], the Hausdorff dimension can be expressed in terms of Lyapunov coefficients for arbitrary strange attractors:
The integer j in this expression is defined in such a way that the sum of the first j Lyapunov exponents is positive, while the sum of the first j + l is negative:
253 i
i+l
i=l
(3)
i=l
The correlation dimension, ,D ,, can be directly calculated from experimental data using an algorithm by Grassberger and Procaccia [I 1,121. According to this algorithm ,D ,,, is defined as
where c(r) is the correlation integral:
If'
where r is a lenght interval, 9 the Heaviside function, the sum is equivalent to the
+ +
number of pairs ij of points on the attractor the distance I xi- x j
I
of which is smaller
than r. The relation between the three dimensions is:
Therefore ,D ,,, obtained from experimental data represents a lower limit to the Hausdorff dimension. Another important characteristic of chaotic behaviour is the Kolmogorov entropy. This characterizes the loss of information due to the exponential spreading of neighbouring trajectories within the strange attractor. To some extent the Kolmogorov entropy measures the degree of the chaos. This value is zero for limit cycle oscillations, finite and positive for strange attractors and tends towards infinity for a totally random system. Therefore the Kolmogorov entropy can be used to distinguish between chaos and noise in the system. Since there is a connection between the average loss of information and the spreading of trajectories, the Kolmogorov entropy is related to the Lyapunov exponents. This relation is expressed as:
where I is the number of positive Lyapunov exponents [13]. The rate at which information about the system is lost is equal to the sum of positive Lyapunov exponents. The simple equation (7) can be used in most cases. The other very distinct definition of deterministic chaos is the route by which it develops. As was already discussed in Chapter 2, there are bifurcations which lead to the loss of stability of the limit cycle and the appearance of chaotic behaviour. These bifurcations are characterized by the Floquet multipliers or characteristic multipliers (CM), introduced in Chapter 2 . If one of the real CM at the bifurcation point crosses the unit circle at -1 on the real axis, then a period doubling or flip bifurcation takes place. In this case the original limit cycle of the dynamic system loses its stability by creating a stable limit cycle with a doubled period. With the variation of the bifurcation parameter the cycle with the doubled period produces a new stable cycle with period 4 and so on. This is the period doubling route to chaos. The universal features of this scenario was discovered by Feigenbaum [14], who demonstrated that the sequence of parameters {p,, p2....}for which the perioddoubling bifurcations occur, constitutes a converging series lim p m = p a
m+m
When p = pm the limit cycle has an 'infinite period' and a strange attractor arises. It was demonstrated, that the rate of convergence of the infinite series of p, was determined by the universal Feigenbaum constant, which is equal to 4.6692. This allows an easy identification of this route to chaos, although it is very difficult in the experimental studies to follow more than two period doubling bifurcations. The second route to chaos is connected with the phenomenon of
51 proposed three mechanisms for the onset intermittency. Pomeau and Manneville [I of chaos, all of which are related to intermittency. These types are recognized and distinguished according to the changes in the CM at the bifurcation point [4]. The dynamic behaviour in all three cases represents the sequences of almost periodic behaviour, interrupted by bursts of aperiodic oscillations of finite duration. The three types of intermittency are closely connected with three ways in which stable limit cycle oscillations can lose their stability. The third route to chaos is the result of the breaking of a torus. This is known as the Ruelle-Takens-Newhouse scenario. In this case two CM form a complex pair and they leave the unit circle simultaneously. The sequence of bifurcations which is included in this scenario comprises the standard Andronov-Hopf bifurcation and one subsequent bifurcation to a two-frequency torus. The next bifurcation into a 3-torus
is not structurally stable and chaos arises, because the three-frequency movement is destroyed by arbitrary small perturbations. The experimentalist in this case observes a transition from regular to quasiperiodic and then to chaotic behaviour. The properties of the chaotic behaviour allow the experimentalist to distinguish between complex quasiperiodic behaviour, noise, and true chaotic behaviour and to treat these experimental data. Software packages such as Dynamical Software [I 61 were developed especially for the analysis of experimental time series. They include computational methods for reconstructing the attractor, estimating its dimension, and measuring the largest Lyapunov exponent from experimental time series. Some packages also allow the computation of correlation functions and Kolmogorov entropy. The most widely used computational algorithm (WSSV method) for the evaluation of the largest Lyapunov exponent was developed in ref. 17. A comparison of different algorithms can be found in refs. 18,19. There are many difficulties encountered during the analysis of experimental data. The main problem is connected with the external noise which is an unavoidable factor in laboratory experiments. The effect of the external noise must be taken into account in the construction of the strange attractor and during the computation of the positive Lyapunov exponent. The second difficulty is connected with the fact that, in order to obtain reliable values of the correlation dimension of a strange attractor and its properties, a sufficiently long experimental time series must be obtained. For analysing the properties of a three dimensional attractor about 50000-70000 data points are necessary [20]. This means that the chaotic behaviour must be measured for a sufficiently long time, while all experimental parameters must be kept stable during the whole experiment. The observation of chaotic behaviour has been carried out at different levels of heterogeneous catalytic systems. It is very complicated to study the phenomenon of chaos in these systems, because unpredictable changes of the catalytic activity can occur during the time required to obtain a sufficient amount of experimental points in the time series.
7.1 Single crystal surfaces Chaotic behaviour during single crystal studies was reported for the oxidation of CO on the Pt (110) surface [21]. The transition to chaos was observed at high temperatures, when starting with the CO covered surface. The temporal oscillations of the laterally averaged work function, AQ, were monitored with a Kelvin probe.
256
Typical chaotic time series are presented in Chapter 1, Fig. 7. The parameters for both chaotic series are summarized in Table 3. Fig. 1 demonstrates the transition to chaos observed in this system. The lowering of the CO partial pressure led to the transformation of regular oscillations via two period doublings to completely irregular behaviour. This is the period doubling route to chaos. The time series corresponding to the irregular behaviour were analysed and proved to represent deterministic chaos.
h
150
I50
f 100
I
e
a
100
d 50
LOO
C 50
LOO
50
3 3
10
20
30
40
50
70
60
t.
80
90
LOO
110
120
130
140
150
Isec)
Fig. 1 The transition from regular oscillations to chaos during the CO oxidation on Pt (110) observed at T=540 K and Po,=l.lO~ mbar upon decreasing Pc0 in small steps from 5.12 .
to 4.52. 10-5 mbar. (Reprinted from ref. 21 with permission.)
Table 3 Characteristics of chaotic time series observed in different catalytic systems Catalytic system
Ref.
Reactant mixure
Temperature [KI
Correlation Kolmogorov dimension entropy [s-l]
c 0 +0 2 / Pt(l10)
[23]
P ~ = 7 . 5 .1 0 - 5Torr
TK = 540
2.4 k 0.2
0.2k0.05
0.12 f 0.05
Pc0=3.4.10-~ pq=6.0.10-5 Torr
TK = 540
4.5 f 0.4
0.4 f 0.05
0.12 f 0.02
T K = 513
3.4
0.306
c0+02 I Pt foil
[27]
c0+02 I Pd zeolite CH30H+02/ Pd zeolite
[32]
c 0 +0 2 / Pt/A1203 catalyst bed
[34] [37]
Pc0=3.0.10-~ [CO]=l% [02]=20.7% [N2]=78.8% [CO]=O.5% in air [CH30H]=3.5% [02]=18% [N2]=72% [CO]=O.S% in air
Largest Lyaponov exponent Is-’]
T ~ = 4 4 0 2.1
0.013
T K = 337
0.01
2.4
T ~ = 4 4 0 2.1
-
0.01 0.06
The reconstructed strange attractors for both chaotic time series presented in Fig. 7, Capter 1 are shown in Fig. 2. At the same temperature of 540 K, but at lower oxygen concentrations, the shape of the reconstructed attractor was much more complicated (Fig. 2b). The analysis of the time series revealed that the correlation dimension and the lower bound of the Kolmogorov entropy were higher in this case. Two positive Lyapunov exponents were calculated for this case, which indicates the presence of hyperchaos in the system [23]. The origin of the chaotic behaviour could be elucidated by spatially-resolved PEEM measurements. No spatial nonuniformities had been observed during the transition to chaos via 2 period doublings, i.e. the total surface oscillated as a whole, strictly in phase [22]. Therefore this may be a case of intrinsic kinetic chaos, which can be described by the point model. However, there are some PEEM measurements that demonstrate the existence of ordered spatial structures in the form of standing waves, which oscillate strictly in phase during the limit cycle oscillations before the bifurcation to chaos. There is also some evidence that the first period doubling is still associated with ordered spatial structures and that turbulent, moving spatial structures on the surface can exist during the chaotic time series [24,25], indicating that even at the single crystal surface level spatiotemporal chaos can arise. The possibility of
258 different feedback mechanisms in this system together with the existence of spatial structures may be the reason for the detection of hyperchaos in this system [221. The same route to chaos, i.e. the transition via the period doubling route, was detected in the study of the NO + H, reaction on Pt(100) [26]. The correlation dimension of the reconstructed strange attractor was found to be 2.6 k 0.3 indicating that a minimum number of 3 variables were necessary to describe the dynamic behaviour of the system. (0)
Fig. 2. Strange attractors, reconstructed from the chaotic time serieqpresented in Fig. 7, Chapter 1. (Reprinted from ref. 22 with permission.)
7.2 Polycrystalline metals The first identification of chaotic behaviour at atmospheric pressure has been reported by Razon, Chang and Schmitz for the oxidation of CO on polycrystalline Pt ribbon [27]. Experiments were conducted in a gradientless reactor under isothermal conditions. The chaotic time series, represented in Fig. 3, were obtained at a catalyst temperature of 250°C in an excess of oxygen. A definite route to chaos was not identified in this study. The analysis of the time series presented in Fig. 3 proved the behaviour to be chaotic on the basis of the positive and finite value of the Kolmogorov entropy. The correlation dimension of the reconstructed strange attractor was found to be 3.4.
259 F A Fractions: C0.0010 Op ~ 0 . 2 0 6
x U
-
g
h B
e U
2 LL
Ng 10.784
0.2 1 -0 .
0
611
€51
691
731
771
o.3-Fi&aB 0.20.1.
- 0
I222 1262
I302 1342 TIME (rninl
1382
Fig. 3. Chaotic time series during oxidation of CO on Pt foil at atmospheric pressure. (Reprinted from ref. 27 with permission.)
No spatiotemporal observations were available in this study at atmospheric pressure, although the authors [27] excluded a temperature heterogeneity on the surface. The long term deactivation of the catalyst due to the diffusion of silicon from the bulk of the catalyst to the surface also makes it difficult to treat the experimental data in this system. Chaotic behaviour was also observed and identified during the oxidation of propylene on a Pt catalyst in the form of a ribbon. This was carried out for both kinds of experiments, when the ribbon was heated electrically either with a constant current [28] or controlled at a set average temperature [29]. As was discussed in Chapter 5, the metallic wire represents a striking example of a distributed system, where the temperature controller can stabilize different spatial structures. For such exothermic reactions as propylene oxidation they are produced by alternating zones of kinetic and diffusion regimes. Although significant differences were observed between the shape of oscillations in both kinds of experiments, the origin of the chaotic behaviour is nearly the same. A thermal imaging of the wire revealed that the back-and-forth movement of the high activity regime is responsible for the oscillations during the oxidation of propylene on the Pt wire. In the constant average temperature experiment in most cases only a fraction on the ribbon was in the ignited state. The origin of chaos was connected with the splitting of the ignited section (Fig. 4, profile a) into two waves (Fig. 4 , profile d), which moved along the ribbon with the same direction, but with different velocities. The deterministic chaos was identified with the help of the Fourier power spectral
260 analysis and the time delay method was used to construct the strange attractor [28]. The strange attractor, constructed from the time series measured for a mixture of 0.3% propylene and for an average temperature of 225"C, had a correlation dimension equal to 3.2. It increased with the decrease of propylene concentration and can be even larger than 5 for 0.2 % of propylene and an average temperature of 259°C. t
-
N
0
500
[seci 1000
1500
2000
I
I
E 0
I
I
400
I
I
400,
I
.33
j
I
- - - . _._ ,.b_
c
Y -
8
I
Fig. 4. Chaotic behaviour of the overall heat production and related temperature profiles of the electrically heated Pt ribbon during propylene oxidation. (Reprinted from ref. 28 with permission.) In the constant current experiments the ribbon remained fully ignited nearly all the time and the origin of the chaotic behaviour is due to the wide spread in the time intervals between successive wave movements of either the left or the right front of the diffusion regime. The chaotic behaviour identified in this case was found to be more regular, although the quantitative parameters of the reconstructed attractors were not reported [29].
26 1 The observation of the chaotic behaviour in the constant current experiments is rather surprising, because there is no temperature control, which can induce spatial reorganization and a behaviour of higher complexity [30]. However, in ref. 31 it was demonstrated that travelling waves of the high activity regime originated from a fixed point of the uncontrolled wire. Quasiperiodicity and aperiodic behaviour arose when a second point on the wire also started to emanate waves. The aperiodic time series were not treated for the identification of the deterministic chaos. However, it can be supposed that the surface heterogeneities and wave phenomena can lead to spatiotemporal chaos in such distributed catalytic systems like metallic wires.
7.3 Zeolite supported catalysts The transition from regular periodic behaviour to chaos has also been observed for the oxidation of CO on a Pd zeolite catalyst [32]. The details of experiment and the properties of the regular oscillations have been discussed in Chapter 3, Section 2. The complex dynamic behaviour, which was observed at CO concentrations higher than 0.57%CO at 220"C, was identified as deterministic chaos, with the characteristics shown in Table 3. The transition from regular to chaotic behaviour was found to be related to an increase of the catalyst activity. This could be achieved either by an increase of the CO concentration or by the reduction of the catalyst with pulses of hydrogen as demonstrated in Fig. 5. This indicated that the origin of the chaotic behaviour may be connected with the appearance of internal diffusion limitations when the catalyst activity exceeded a critical level. In the presence of a CO concentration gradient within the supporting zeolite crystal, the Pd clusters oscillate under different conditions. The coupling between different oscillators can lead to chaotic regimes outside the entrainment band. The possibility of diffusion chaos can also not be excluded in this case. The fact that intracrystalline transport processes within the zeolite can cause complicated dynamic behaviour was demonstrated by Jaeger, Plath and Svensson during the investigation of the methanol oxidation on zeolite supported Pd [33]. They observed toroidal oscillations with Pd embedded within a Faujasite NaX zeolite matrix and noted that this phenomenon cannot be observed using an amorphous support. Oscillations occuring in the oxidation of methanol are nonisothermal. In addition, the oxidation-reduction of Pd particles can take place. Together with the complex transport processes in this system these feedback mechanisms lead to the
262 appearance of chaotic behaviour [34]. The parameters of the chaotic motion are presented in Table 3.
Fig. 5.The transition from regular to aperiodic oscillations on Pd zeolite catalyst: (a) after the addition of O.O2%CO in the inlet mixture; (b) after the third H, pulse treatment of the catalyst surface. (Reprinted from ref. 32 with permission.) Two routes to chaos have been discovered for this catalytic system. The first one, via period doubling, was detected on moving from the high activity state, characterized by the large temperature difference between the catalyst and the gas phase temperature. The second route was less common and led via the destruction of a torus and type-ll intermittency to homoclinic chaos. A characteristic of type-ll intermittency is the beat-like modulation of the limit cycle oscillations with increasing modulation amplitude (Fig. 6). Homoclinic chaos includes homoclinic structures characterized by the presence of large bursts (Fig. 7).
263 T CKI
10 m i n
time
,
Fig. 6. Type-ll intermittency dynamic behaviour during the oxidation of methanol on Pd zeolite catalyst. (Reprinted from ref. 34 with permission.)
10
0 20 I
1
10
0
-
10 m i n
time ------+
Fig. 7. Homoclinic chaos observed during the oxidation of methanol on Pd zeolite catalyst. (Reprinted from ref. 34 with permission.)
7.4 Catalyst bed level The catalyst bed with many oscillating pellets represents a system of a higher level of complexity. However, the same route to chaos as has been observed in single crystal studies, i.e. period-doubling transition, was reported for the oxidation of CO on Pt/AI,O, [35].Two factors render the interpretation of the results difficult. The first one is that the catalyst activity was the bifurcation parameter in the experiment. This is the reason for the irreproducibility of the observed phenomena. The second is that the appearance of regular osciilations was detected from the
264
steady state with nearly complete conversion and inhomogeneities can arise in this case both on the surface and in the gas phase. The complete identification of a chaotic time series, observed during the oxidation of carbon monoxide on supported Pt catalyst in an adiabatic packed bed reactor had been carried out by Wicke and Onken [36,37]. The characteristics of the observed chaos can be found in Table 3. Together with the chaotic behaviour of the C02 production rate the authors [36,37] observed the appearance and the evolution of temperature waves in the catalyst bed. The origin of the temperature waves in the catalyst bed was demonstrated to be due to the high sensivity of the system to external perturbations in the bifurcation region, where a transition from one to three steady states in the catalyst bed of the reactor takes place. This state of high sensitivity to external perturbations may be realized under the conditions when the line of convectional heat removal, Q, and the line of reaction rate heat production, Q,, coincide with a common straight line within a certain range of reaction temperatures on the heat balance diagram, presented in Fig. l c , Chapter 1. In this case there is no definite intersection and hence no definite steady state [38,39]. For the nonoscillatory, exothermic oxidation of ethane in the same adiabatic reactor, small temperature oscillations introduced in the inlet of the catalyst bed increased by a factor of 50 at the exit of the catalyst bed. The same phenomena could be also observed in the bifurcation region, when the ethane concentration was modulated in the feed. Deterministic chaos was detected for the oscillating CO oxidation reaction. In this case self-sustained oscillations of the reaction rate at single catalyst pellets in the entrance section of the packed bed turned out to be the source of the reaction waves. The local small amplitude of kinetic nonisothermal oscillations from the catalyst pellets at the entrance of the catalyst bed initiate the growth of reaction waves that propagate through the catalyst bed causing large fluctuations of temperature and conversion in the exit section (Fig. 8 ). The larger the number of pellets that can oscillate in the entrance cross section of the catalyst bed, the denser is the sequence of reaction waves and the more chaotic the behaviour of the observed overall reaction rate [37]. The possibility of the formation of chaotic patterns, produced by the process of selective amplification of fluctuations induced by some weak external noise, has been discussed in ref. 4. In the bifurcation region, due to the high sensitivity, the system responds strongly to noise provided by the environment. It enhances one of the spatial modes present in the noise and suppresses the others, resulting in chaotic spatial patterns.
265
A
01
-
I 10 min
-I
Fig. 8.Chaotic time series during the oxidation of CO in the adiabatic packed bed tubular reactor, which is accompanied with the evolution and movement of temperature peaks along the catalyst bed, 1% CO in air, To=431 K. (Reprinted from ref. 36 with permission.) The chaos originating in the catalyst bed during the oxidation of CO is even more deterministic, because the fluctuations originate in the system itself. The experimental data presented in this chapter demonstrate that chaotic behaviour is widely spread in heterogeneous catalytic systems and that it has been identified on different levels. It is possible to say that it is easier to observe complex aperiodic or chaotic behaviour than regular periodic oscillations. The reason is that heterogeneous catalytic systems are inhomogeneous in space and, in most cases, the chaotic behaviour arises due to the complex structure or wave phenomena on the catalytic surface. The 'lumped' character of chaos is doubtful, even for single crystals. The study of chaotic behaviour gives the experimentalist additional information about the system. The similarity of the transitions to chaos for different levels of heterogeneous catalytic system proves the similarity of organization in time and space at different levels of the system. The origin of the diffusion chaos in a heterogeneous system is one of the most interesting problems which must be solved in the nearest future.
266
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