spatial and temporal patterns in catalytic oscillations

Physica A 188 (1992) 302-321 North-Holland PHYSICA iI

Spatial and temporal patterns in catalytic oscillations*

D . G . Vlachos , F. Schiith, R. Aris and L .D. Schmidt 1 Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA

The spatial and temporal evolutions of catalytic oscillations are modeled for two prototype systems, (1) a Monte Carlo simulation of a unimolecular reaction and (2) the thermal coupling of distributed oscillators. In the single oscillator problem attractive adsorbate- adsorbate interactions predict oscillations in the continuum mean field approach. Using the Monte Carlo method, the roles of metastability, gas pressure, surface diffusion and defects are examined and contrasted with the mean field model. In the multiple oscillators problem, the behavior of C O / N O reaction on supported Pd is simulated to examine the role of thermal coupling and the transition from synchrony to chaos through period doubling.

1. Introduction

Oscillations in reaction systems are very challenging problems, which have driven considerable research to elucidate possible reaction mechanisms that cause oscillations. Temporal patterns, which are more easily measured ex- perimentally than spatial patterns, can be very complicated and vary from periodic to quasiperiodic, to chaotic [1]. The routes which destroy synchrony and lead to chaos have stimulated numerous theoretical contributions [2].

Oscillations are observed in both homogeneous systems in aqueous solution, like the Belousov-Zhabotinskii reaction [3], and heterogeneous catalytic systems (for an extensive review see ref. [4]). Since the nature of and oscillation mechanisms for homogeneous and heterogeneous systems are quite different, in table I we contrast the most important characteristics of the two types of reaction systems.

Homogeneous oscillators usually involve complicated reactions with many elementary steps. Also, the system is dilute, and the reaction is slow. Homoge- neous reactions take place in a continuous three dimensional space (a solution) in well characterized reactors. In contrast, catalytic oscillators consist of simple

*This work was partially supported by the NSF under Grant No. CTS 9000117 and the Minnesota Supercomputer Institute.

Author to whom correspondence concerning this article should be addressed.

0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved

D.G. Vlachos et al. / Spatial and temporal patterns in catalytic oscillations

Table I Comparison between homogeneous and heterogeneous reaction systems.

303

Characteristic Homogeneous sys tems Heterogeneous systems

dimensionality three one (wires), two (single crystals and foils), three (supported catalysts)

space continuous lattice concentrations low high reaction mechanisms complicated simple reaction rate slow fast reactor simple complicated applications few many oscillation mechanism autocatalysis nonlinear rate constants

competition of sites autocatalysis

thermal behavior isothermal non-isothermal diffusion Fickian uphill in many cases control of mixing external stirring no external control spatial uniformity homogeneous usually spatially nonuniform

reaction mechanisms (usually one bimolecular reaction), the reaction is usually very fast, and the adsorbed layer is near saturation. Atoms are chemisorbed on localized crystal sites (minima of the crystal potential surface) and move from one site to another. The sites can therefore be represented by a lattice with defects. Catalytic reactors are often complicated but have many applications such as chemical reactors, catalytic combustors , electrochemical cells, flames,

and propellants. In homogeneous systems, the reaction mechanism involves usually an au-

tocatalytic step such as 2A + B - ~ 3 A , which is modeled by nonlinear but isothermal functions [5]. In heterogeneous catalytic systems, nonlinear rate coefficients such as k = k 0 e x p ( - a 0 ) [6-10] often result in oscillations. In addition, heterogeneous catalytic systems are often nonisothermal, as will be discussed in an example below. Tempera ture variation introduces a strong nonlinearity, e x p ( - 1 / T ) , in the governing equations which can lead to oscillations [11, 12]. Site competi t ion between reactants and between reactants and products is also frequently encountered and introduces strong nonlinearities

[131. A d s o r b a t e - a d s o r b a t e interactions [10, 14], adsorbate phase transition such

as 1 x 1 and hex phases of Pt (100) [15, 16], and surface deactivation caused by format ion of oxides [17, 18], can result in formation of spatial patterns on surfaces. In all these systems islands of a phase are produced, and nucleation may be necessary to form sufficiently large clusters, which subsequently grow to a new phase. All these phenomena result in strong spatial variation of species concentrations and /o r tempera ture , and these phase changes can drive oscillations, as shown for the single oscillator example below.

304 D.G. Vlachos et al. / Spatial and temporal patterns in catalytic oscillations

Diffusion of species in homogeneous systems is usually Fickian [19], i.e., fluxes flow from high to low concentrations. It has been shown that the nonuniform distribution of species in solution can strongly affect oscillations [20], and mixing of reactants with propellers is a typical control parameter. In contrast, in catalytic systems diffusion over the surface is in many cases uphill [21], i.e., from low concentration (single atoms) to high concentration (clusters). In addition, there is typically no external control parameter to homogenize the system. Surface diffusion depends on surface temperature but cannot directly be controlled. Thus, spatial nonuniformities become more important and less controllable in catalytic reaction systems compared to solution systems. Most macroscopic models which have been used to study heterogeneous catalytic systems usually assume a Fickian diffusion, a perfect single crystal, and a homogeneous surface with uniform distribution of species [4].

In the first problem, we consider at the molecular level a single oscillator describing a unimolecular reaction on catalyst surfaces. The roles of phase changes, metastability, gas pressure synchronization, surface diffusion, and defects on oscillations are examined to evaluate the validity of continuum models and the importance of spatial inhomogeneities on temporal behavior.

Catalytic systems consist of wires, single crystals and supported catalysts, which can be modeled as one-, two- and three-dimensional systems. All these systems can produce oscillations. Oscillations from large catalyst areas usually require a synchronization mechanism such as gas pressure, surface temperature (for nonisothermal systems), surface diffusion, phase transitions, potential (for electrochemical systems). In the second problem, we consider many randomly distributed oscillators as a prototype for oscillations on supported catalysts. The importance of thermal coupling on the complex spatial and temporal behavior and on the transition from synchrony to chaos is investigated.

2. Single oscillator for a unimolecular reaction

In this section, a simple oscillator which describes a unimolecular reaction on a (100) plane of a catalyst surface is examined. Four elementary processes are simulated: adsorption and desorption of species A from the gas Ag on a catalyst surface A~,

r a

Ag + S~-A~ , (1)

the unimolecular reaction to transform A s into product B, which instantly desorbs from the surface,

D.G. Vlachos et al. / Spatial and temporal patterns in catalytic oscillations 305

r R

A s ~ Bg + S, (2)

and finally migration or surface diffusion of A. Here , the subscripts s and g denote adsorbed and gaseous species respectively and S means a vacant site. Oscillations in this model are caused by the inclusion of attractive adsorba te - adsorbate ( A - A ) interactions which can produce clusters on the surface. The existence of attractive interactions results in a spatially nonuniform surface consisting of single atoms and islands of A [21, 22]. This problem is modeled at both the macroscopic level in section 2.1 and microscopic (molecular) level in

section 2.2.

2.1. Macroscop ic models

Macroscopically, the spatial and temporal evolutions of the single oscillator are described using either ordinary (ODEs) or partial (PDEs) differential equations. Below, these models are presented and their limitations are indi-

cated. The conservation of mass in the gas phase of an ideal continuous stirred tank

reactor (CSTR) shown in fig. l a is

dPA _ PAO-- PA

dt ~'R C ( r a - rd) , (3)

where PA0 is the inlet pressure of reactant A, PA is its partial pressure in the reactor , 7 R is the mean residence time in the reactor, and C is the surface capacity [10].

The surface coverage 0 A (fraction of filled surface sites) is assumed to be given by Langmui r -Hinsche lwood kinetics with the exception of the coverage dependent activation energy for desorption. In the presence of attractive adso rba t e -adso rba t e interactions, the conservation of mass on the catalyst surface then is

dO A dt - ra - r a - r a = k~PA(1 -- OA) -- kdO A exp(--aOA) -- kRO A , (4)

where ra, rd, r R are the adsorption, desorption and reaction rates, respectively, ka, k j , k R are the corresponding rate constants, and a is a dimensionless constant , a = ZsW/kT , which depends on the strength of interactions w, the surface t empera ture T, and the orientation of single crystal, i.e., the maximum number of nearest neighbors z S (for a square lattice we considered here z s = 4). The ratio of adsorption and desorption rate constants defines the equilibrium constant K: K =- k a / k d.


(a)

(c)

2 "P" • L

i • •

P | !

1.o (b)

i

0 A

0.5

0.0 0 1200 2400

t k PA

(d)

Fig. 1. Panel (a) illustrates a CSTR reactor with a catalyst surface consisting of a lattice. Panel (b) shows a dynamic path for the system obtained from the MC method. The metastable and stable solutions are indicated. The parameters used are w / k T = 2, k R - 0, K P A - 0.0195, and zero initial coverage. Panels (c) and (d) show snapshots of the surface a short time interval apart (corresponding to the arrows in (b)), which demonstrates the rapid growth of a supercritical cluster.

The assumpt ions of the mean field model are encoun te r ed in the desorp t ion

rate. Since it is assumed that t he s u r f a c e is h o m o g e n e o u s , the surface coverage

is un i fo rm across the surface and thus the desorp t ion probabi l i ty is the same for

all a toms. However , single a toms and atoms in clusters exper ience a different

n u m b e r of neares t ne ighbors and thus a different act ivat ion energy for desorp-

t ion. In the m e a n field model the act ivat ion energy is: E = E o + ( z~w )O A, where

E o is the act ivat ion energy for desorp t ion on a clean surface. The d e p e n d e n c e

of desorp t ion on the local e n v i r o n m e n t results in desorp t ion probabi l i t ies that

can differ by several orders of magni tude .

W h e n w / k T > 4 / z ~ (strong at tract ive in teract ions and low tempera tu re ) , the

i so therm (r~, - r o = 0) is mul t iple valued. Three solut ions exist, as shown in fig.

2a, and it has been shown [10] that two of them are stable and the in te rmedia te


Mean field Monte Carlo 1.0 (a) 1.0 (c)

0 A

0.5

0.0

KP~x 10 ~

1.0 (b)

0 A

0.5

0.0

K~ x 102

6

I

i

0^

0.5

0£ 1.6 1.9

K P^ x l0 a

°Af 0.5

0.0

x 102

1.9

4 10 1.6

i

2.2

' 3 . 5 ' t/x R

Fig. 2. Pane l s (a) and (c) show i s o t h e r m s and pane l s (b) and (d) show t ime ser ies for coverage and

r e a c t a n t p r e s su re o b t a i n e d for the M F and M C mode l respect ive ly . The p a r a m e t e r s used in the M F m o d e l a re w / k T = 1.5, KPA, = 0 . 1 0 0 5 , K C = 0.03, kd'C R = 1000, and kR/k d =0 .00333 . In the M C m o d e l the p a r a m e t e r s are w / k T = 2, KPAo = 0.024, KC = 0.00144, k d r R = 6.58 X 104, kR/k d = 10 4,

and la t t ice size 30 x 30.

one unstable. Under certain conditions, Hopf bifurcation can be found [10], and both the surface coverage and gas pressure oscillate periodically with time,

as shown in fig. 2b. In the mean field model, an infinite downhill diffusion is implicitly assumed

which homogenizes the distribution of species on the adsorbed layer so that V0 A = 0. The presence of attractive adsorbate-adsorbate interactions results in the formation of adsorbed layers that consist of single atoms and islands of

reactants so that species concentrations are strongly spatially nonuniform.


Phase transitions, for example, from a dilute phase (gas-like phase) to a condensed phase (liquid-like phase) of large islands usually include a nucleation barrier. Fluctuations, which can play a crucial role on the formation of islands of a phase and subsequent growth to the other phase, are not incorporated in this model. Surface defects and structural changes of the surface [14] result in energy inhomogeneities of the catalyst surface, Which are also not included in this model. Consequently, the mean field model is an oversimplification regarding the structures and dynamics of adsorbed layers.

In an attempt to describe the spatial and temporal evolution on surfaces, catalytic reaction systems can in principle be modeled using generalized Fickian partial differential equations (PDEs) of the form

00 A - V(DVOA) + r~ - r d - r R , (5)

Ot

where the first term on the right-hand side shows the spatial variation of surface coverage and the other terms account for the fact that the system is open to mass transfer, eq. (4).

Fick's second law predicts that an initial accumulation of mass somewhere on a surface will be spread out by Fickian or downhill diffusion so that a homogeneous solution will finally be achieved. However, in order to predict the formation of islands on surfaces caused by attractive interactions, transfer of material from low concentration (single atoms) to high concentration (clusters) is necessary. This requires an uphill diffusion of atoms. Mathemati- cally, a negat ive e f f ec t i ve d i f fus iv i t y will be required in eq. (5), which is physically unrealistic. This is analogous to spinodal decomposition problem in alloys where Cahn and Hilliard [23] have proposed the classical theory for separation of phases by including higher terms of the free energy. However, MC simulations of Ising reversible systems have shown that the Cahn-Hil l iard theory is only approximate and does not correctly predict the kinetics of phase separation [24]. Since irreversible systems are outside statistical thermodynamics, MC simulations [24] are unique tools to study reaction-diffusion systems. Thus, the above model, eq. (5) or its modifications, will not be further considered here.

In summary, both mean field models (ODEs) and Fickian type of models (PDEs) are structurally incorrect, do not incorporate properly the physics, and consequently cannot predict the spatial structures of adsorbed layers and temporal behavior of these systems. Thus, in the next section we shall model the single oscillator using the Monte Carlo method which correctly captures the microscopic physics of the problem.


2.2. Microscopic simulations of single oscillator

The four elementary processes modeled and their transition probabilities in the MC method are summarized in table II. The relation between MC trials and real time has been developed [21] to relate results with experimental data. The gas phase is taken to be homogeneous (no spatial variation of gas pressure), i.e., a CSTR reactor is assumed, eq. (3), as shown in fig. la.

The surface is t reated at the molecular level by simulating the four elementary processes described in table II. The local environment of each atom is explicitly taken into account in the desorption rate in contrast to the mean field approximation. In addition, uphill surface diffusion can properly be included in the model (see below), fluctuations and nucleation are explicitly incorporated, and the role of defects on the dynamics can be studied. In this way, the roles of attractive interatomic interactions, phase transition and surface inhomogeneity on spatial and temporal patterns of oscillations can be examined.

MC simulations are extremely expensive, especially close to critical points. Typical runs for a few periods of oscillations on a 30 × 30 lattice require about

109 trials and several CPU hours on the Cray 2 for a single computer experiment and a bifurcation diagram requires many of these data points. However , these simulations are the only rigorous way to examine the role of surface inhomogeneities on the oscillations.

2.2.1. Metastability in a differential reactor without surface diffusion We first examine multiple solutions and their stability using the MC method

for a fixed pressure (differential) reactor where only the corresponding master equation [22] for the surface is solved.

Fig. lb shows the surface coverage (an average variable determined from the microconfigurations generated) as a function of time for a 30 × 30 lattice. The surface coverage fluctuates about an average value. Atoms aggregate and form

Table II Elementary processes and transition probabilities used in the MC simulations, cj is the occupation variable for the j th site which is 0 or 1 if the site is empty or filled respectively. The ith site is a nearest neighbor to the j th site. For diffusion, the ith site is randomly chosen among the four nearest neighbors, x J a . is the number of atomic jumps of each atom before it desorbs. Details of the simulations are given elsewhere [21, 22].

Elementary process Microscopic transition probability

adsorption k , P A (1 - cj) desorption k d e x p ( - n jw / k T ) c j reaction k~c ~ migration (x~/a~):k d e x p ( - n j w / k T ) cj(1 - c~)


islands in the presence of attractive interactions to reduce the free energy of the system. Peaks of high coverage indicate the formation of clusters which are subsequently destroyed. The adsorbed layer consists of single atoms and clusters and it is highly nonuniform. When a sufficiently large cluster is formed through fluctuations (a supercritical cluster), it can persist and grow further. An abrupt transition is thus observed from the low to the high concentration state in a very short time interval, as illustrated in figs. l b -d .

From dynamic runs such as that shown in fig. lb, the isotherm is constructed, as shown in fig. 2c. Starting with a low concentration (with fixed pressure) the system stays in the low concentration state for some time until a sufficiently large perturbation occurs which drives the system to the high concentration phase. Hysteresis is found when the pressure is gradually decreased from high values. Thus, two solutions are found by the MC method, if w / k T > 1.7625 [25], one of which is metastable (a local minimum of free energy in which the system stays for some time) and the other stable (a global minimum of free energy in which the system stays for an infinite time once the transition from the metastable to the stable state occurs). No unstable solutions are found as in laboratory experiments.

2.2.2. Oscillations in a CSTR reactor and gas pressure synchronization In this section, the CSTR model of eq. (3) for the pressure PA and the

master equation for the surface (which provides the coverage 0A) are solved simultaneously. We have observed oscillations in parameter space, as shown in fig. 2d, synchronized by the gas pressure. These are relaxation oscillations because the system stays in the low or high concentration state for a long time compared to the abrupt transition from one state to the other. Nucleation of a supercritical cluster through thermal fluctuations is essential in the phase transition from the low to the high 0 A state. However, the surface reaction is also essential in the transition from the high to the low 0 A state. In the limit of k R--~ 0, no oscillations are found [14], indicating that the oscillations are driven by the interplay of nucleation and chemical reaction.

Fourier spectrum analysis of the time series shows the existence of a characteristic frequency. Furthermore, the correlation integral method gives a constant correlation exponent as the embedding dimension increases, and a size effect analysis demonstrates [14] that the amplitude of oscillations does not decrease with 1/~/(system size) [25]. This implies that the oscillations correspond to a real attractor and are not statistical noise.

2.2.3. Surface diffusion and synchronization In the limit of kR----~0 , a dynamic equilibrium is established at the surface

where atoms desorb to the gas phase and readsorb onto the surface. Through


adsorpt ion and desorption, clusters develop on the surface, as discussed above. Fluctuations in the average coverage shown in fig. lb are caused by the randomness in number of atoms which adsorb and desorb. Desorpt ion of an a tom and readsorption can occur at a random site a long distance from the initial site, so that t ransport over long distances can effectively occur through the gas phase without surface diffusion.

Surface diffusion, on the other hand, allows atoms to migrate from a site to an adjacent site, if the site is empty, and thus results in formation of clusters in the presence of attractive interactions. Atomic jumps of one lattice constant are allowed, i.e., t ransport over short distance occurs. However , since it is usually expected that xs/a o >> 1, an a tom jumps many times before a desorption event takes place. Surface diffusion will then be the dominant mechanism for cluster format ion at short times.

To elucidate the role of surface diffusion on oscillations, two different compute r experiments were performed. In the first the activation energy for leaving a cluster is larger than the activation energy for joining the cluster because an a tom at the cluster 's edge had additional bonds to break, as shown in fig. 3a. Under these circumstances, atoms aggregate and islands develop on

(a)

(b) 1

0A ~ , / . a 20=

0

1

e^

01.6 1.9 K P ^ x 1 0 •

(c)

(d) 1

0A ~ x,l%=0

eA

x,/ao=O.1 2.2 01.6 ' 2.0 2.4

KPAX 102 Fig. 3. One-dimensional potential diagram for migration of atoms on a single crystal in uphill diffusion (a) and Fickian diffusion (b). Panels (c) and (d) show limit cycles as the uphill and Fickian diffusion increase. The parameters used are the same as in fig. 2.


surfaces by surface diffusion. Uphill migration is expected to operate in the presence of attractive interactions, which results in formation of islands as the adsorp t ion-desorp t ion mechanism does. Fig. 3b shows that, as the surface diffusion increases, the width of the limit cycle is not strongly affected and the oscillations are not strongly altered.

In the second computer experiment the activation energies for joining and leaving the cluster are equal, as shown in fig. 3c, i.e., there is no preference of a toms to form islands. This is a downhill or Fickian diffusion, which is expected to destroy the clusters developed by adsorpt ion-desorpt ion if it is sufficiently fast. A toms at edges of clusters migrate away from clusters and the surface is homogenized.

Fig. 3d shows that as downhill diffusion increases the width of the limit cycle is reduced and above a small value of atomic jumps xs/a o the oscillations disappear. Downhill diffusion kills the nucleation of clusters which is essential for the transition from the low to the high coverage state. This is perhaps surprising because one expects that in the limit of an infinite downhill diffusion the surface will be randomized and the results of the mean field model will be obtained. However , it turns out that even in the limit o f an infinite downhi l l

di f fusion the mean f ield mode l is not adequate to describe the discrete nature of active sites, the fluctuations affecting species distribution, and the cooperative

phenomenon between atoms.

2.2.4. Ef fec t o f surface defects on oscillations It is usually assumed that catalyst surfaces are homogeneous and that all

surface sites are equivalent. However , catalyst surfaces always contain defects such as adsorbed atoms, impurities, vacancies (point defects), and also disloca- tions, steps and grain boundaries (line defects). Atoms are frequently bound more strongly on defects than on perfect surfaces [26], and therefore defects make surfaces inhomogeneous and act as nucleation sites. Such nucleation sites have been thought to be sources which generate waves and other exotic spatial pat terns on surfaces. Here we examine the role of defects on oscillations by including sites where atoms are irreversibly bound. Atoms adsorbed on these surface sites cannot be removed by any process such as desorption or diffusion, i.e., they form a permanent island. These atoms of reactant A interact with other A atoms through first nearest neighbors, as atoms on regular sites do.

Figs. 4a -c show snapshots of the adsorbed overlayer after long time. Sites in black correspond to defect sites, sites in gray to atoms adsorbed on regular sites, and the white region to empty sites. The concentration of defects is 4% for all three snapshots. However , f rom (a) to (b) the defects are dispersed more on the lattice so that the role of distribution of defects on oscillations can be studied.

D.G. Vlachos et al. / Spatial and temporal patterns in catalytic oscillations 3 1 3

(a) []

[] []

[]

m

fl

1 (d)

OA

(b) (c)

=d~= ~ ,~"~i~

0 0 , i

l(e_)

OA

1 (f)

0A

0 0 i i I i i i i I i I

' ' 6 3 6 3 t/~R t/XR t/XR

Fig. 4. Panels (a)-(c) are snapshots of surfaces with defects (in black) and panels (d)-(f) show the surface coverage versus time for these imperfect surfaces. In the three panels the defect concentration is 4% but from (a) to (c) the defects are spread out on the surface. The defect dispersion causes a transition from quite periodic to chaotic regime and finally to disappearance of oscillations.

Fig. 4d shows the surface coverage 0 A as a funct ion of t ime for an isolated

pa tch of defects depic ted in fig. 4a. It is seen f rom fig. 4d that an isolated

defec ted spot does not s trongly affect the oscillations and the oscillations are quite regular as on a single (100) crystal surface. The nucleat ion site at the

cen te r of the surface forces nucleat ion to start at the center ra ther than a

r a n d o m posi t ion of the surface. Fig. 4e shows the surface coverage 0 A versus t ime for a surface where the

defects are regularly spread into nine nucleat ion centers, as shown in fig. 4b.

The ampl i tude of oscillations has been considerably decreased c o m p a r e d to fig. 4d and the oscillations have become less regular and more chaotic. This

behav io r is even more p r o n o u n c e d for single, regularly distr ibuted defect sites

where the oscillations vanish, as shown in fig. 4f. Similar behavior is also found for a r a n d o m distr ibution of defects [14]. For single defect sites which are

closely separa ted , there are s trong interactions between adjacent defects and the oscillations disappear. The global behavior of the system is caused by an

unsynchronous ignition of nucleat ion centers. Nucleat ion of many clusters close in space at different t imes results in more chaotic oscillations and in a


non-effective synchronization, at least when gas pressure is the dominant synchronization mechanism.

Defect sites act as nucleation centers, which are required for the transition from the low to the high coverage phase. In this way, the value of gas pressure required for the phase transition is decreased and the limit cycle shrinks. Thus, as the number of nucleation centers increases, the metastability (multiplicity) regime shown in fig. 2c is reduced [21] and the oscillations are gradually altered from quite periodic, to chaotic, to a fixed point with a little noise. When the concentration of defects is sufficiently large and the distance between defects is small, the gas pressure cannot synchronize the surface.

Aside from the concentration of defects, our simulations suggest that their spatial distribution on the surface is a crucial parameter which affects spatial patterns and temporal behavior. It is expected that impurity atoms with different interaction energies might also strongly alter the dynamics of heterogeneous catalytic systems.

3. Thermal coupling among distributed oscillators

The experimental temporal evolution of catalytic systems is often very complex, as shown in fig. 5a. Small change of parameters can result in change of the temporal behavior from periodic, to quasiperiodic, to beat structures, to chaos. Infrared spectroscopic experiments demonstrated that supported catalysts can contain many oscillating regimes which are not in phase and complex behavior has been found [27, 28]. Spatially resolving experiments using a setup of photodiodes in the unimolecular methylamine decomposition showed complicated spatial patterns on catalytic wires [11]. Recent photoelectron spec- troscopy revealed exotic spatial and temporal patterns, such as waves, spirals and turbulence [29].

Complex non-periodic oscillatory behavior in catalytic systems can be caused by a lack of synchronization of different regimes or crystallites of the system which are oscillating out of phase. Here, we examine whether complex macroscopic behavior has a prerequisite of complicated chemical kinetics on a microscopic level or whether this can be caused by coupling alone. Hence, relatively simple kinetics are used to avoid occurrence of complex behavior from the kinetics of the single oscillator and the thermal coupling among many oscillating regimes (cells) is investigated. The transition from synchrony to chaos and possible routes to chaos are examined as parameters are varied.

3.1. Mode l ing o f the N O ~ C O reaction sys tem on suppor ted Pd

We model a supported catalyst as a 10 x 10 two-dimensional array of cells


a 1.2-

1.1

I~ 1.0

~'~.. 635-

E I, ~ 630

0 I I

800 1600

b 10.0-

8.0-

0 O 6.0-

4 .0 -

2.0-

700-

640-

~. 580- E #.

520- 0.0 860 16100 24i00 ' ' 3200 4000

Time [s] Fig. 5. Panel (a) shows exper imental t ime series (CO 2 conversion and thermocouple temperature) for Pco = 84 Torr , PNo = 77 Torr , T = 570 K. Panel (b) shows simulated time series (CO 2 conversion and tempera ture of a cell) for Pco = 70 Torr , PNo = 70 Torr , T = 540 K. Only some peaks of the conversion appear in the thermocouple reading.

where 10 randomly chosen cells are made to be active and the other 90 are inactive but conduct heat to neighboring cells. These 10 active oscillators are coupled with both the gas phase and their nearest neighbors through heat transfer. We use periodic boundary conditions so in a sense we are considering 10 particles on a "spherical" support.

Each active cell (oscillator) describes the N O / C O reaction on supported Pd. Two basic steps in the reaction mechanism involve blocking of surface sites S by N atoms according to [30]

N O g + 2S--+ N~ + O s (6)

and liberation of surface sites (deblocking of surface sites) by desorption of N 2

according to


N s q- N~---> S2g + 2 S . (7)

Neglecting competit ive adsorption, the change of N coverage is then

dON - kb , , , c kPNo(1 -- 0N) 2 - kdeblock 0 2 (8) dt

where kb~,,~, k and kdebloc k a r e the rate constants of reactions (6) and (7), and 0 N is the surface coverage of N atoms for each oscillator.

The coverage of N atoms affects the reaction rate for CO 2 formation, which

is approximated by [30]

rR = kR(1 _ ON ) PNO Pco " (9)

The energy balance for each cell is

d T - - A H R dt - r R - - K~(T-- Tg) - K 2 E ( T - T , , ) , (10)

m c a t C c a t nn

where T is the cell temperature , AH R is the heat of reaction, Tg is the gas phase tempera ture (assumed to be homogeneous) , Tn, is the tempera ture of a nearest neighbor cell, rnc~ , is the catalyst mass, cca , is the specific heat, K~ is the heat transfer coefficient to the gas phase, and K 2 is the heat transfer coefficient

between cells. The summation is over all nearest neighbor cells. Energy communicat ion exists between all active and inactive cells.

In our model, El is an effective heat transfer coefficient which expresses the strength of thermal coupling between each oscillator and the gas phase. K~ is an effective heat transfer coefficient across the surface, which is determined by the thermal conductivity of the material connecting the different particles, the distance between particles, and the surface area over which energy is ex- changed. The ratio of K~/K 2 indicates the relative importance of gas phase (external bath) to internal (among particles) coupling, and it varies from 0 (thermally insulated system from the gas phase) to oo ( independent oscillators coupled with a thermal bath). Estimates of the heat transfer coefficients for Pd indicate that the ratio K~/K 2 must be close to 1. The value of K 2 used here is K 2 = 0.4 S ~ whereas g~ has been treated as a bifurcation parameter .

The resulting system consists of 110 nonlinear O D E s (100 energy balances plus 10 mass balances). Simulations were performed on the Cray 2 of the Minnesota Supercomputer Institute using the L S O D E package. The Gear method of integration was used because it exhibits very good stability and convergence. A typical single time series of 10000 seconds required - 2 minutes of CPU time. Details of the simulations are reported elsewhere [30].


3.2. Spatial and temporal evolution o f coupled oscillators

In this model a single (isolated) cell oscillates periodically under certain conditions but cannot lead to complex behavior. We have checked that increase of t empera ture results in increasing frequency and decreasing amplitude and change f rom relaxation to sinusoidal oscillations, in agreement with exper iments [31]. Fixing the parameters under which a single cell oscillates, as described elsewhere [30], the behavior of thermally coupled cells is then examined. From the single to the multiple oscillators problem, the heat t ransfer coefficient to the gas phase K~ was however reduced, since in the coupling problem heat loss is also possible from inactive cells. We restrict the discussion below to the influence of thermal coupling on the appearance of complex spatial and temporal evolution.

For each active cell, the coverage and reaction rate are determined as functions of t ime whereas the tempera ture of all (active and inert) particles is obtained. From these time series, the global behavior of the entire system is examined.

Fig. 6 shows the Lyapunov exponent as a function of the thermal conductivity to the gas phase K 1 . The inset shows the total 100 cells and the 10 active cells in black, which are randomly distributed and whose reactivity is described by identical pa ramete r values. As K~ increases from low values the period of the oscillations doubles and a chaotic regime is found for larger values of Kl. At

@

@ I= ,-i

.~_

.1

._~

0.10 - 2 4 8

0.08 11

0.06 []

0 .04 ~ "

0.02 [ ]

0.00 • . |

0.12 0.14

I I I I I I I I I I I

2 1

/ l

0.16 0.18 0.20 0.22

K [l/s] 1

Fig. 6. Lyapunov exponent versus heat transfer coefficient K~ for the cell distribution shown in the inset. The arrows indicate the onset of certain periodicity as K, increases.


t ime [s I Frequency [Hz]

3000 0008 001B 0024 0032 0040

Frequency [Hz]

800 1600 2400 3200 4000 '0 000 0 0O4 0 008 0 012 0 0 t6 0 020

t ime [s] Frequency [Hz]

o

o~.

600 1600 2400 3200 4000 t ime [s]

o

8° = o t=

2-

o

aoo ~ Boo ~4oo 32oo 4000 o ooo o ooB o oi s o 024 o 032 o 040

t ime Is] Frequency [Hz]

17oo ~ - oooo o'oos o'o~s o'o24 o'o3~ ooao Boo 600 24~ 32~ 4000

t ime Is] Frequency [Hz] o

~ '09 a ,o

lU ~u I, ,~ . 80O TSO0 24O0 32OO 4OOO '0000 eioos O~01S 0'024 0032 0040

t ime {s] Frequency [Hz]

Fig. 7. Simulation time series and their Fourier spectra for different points of fig. 6. In (A) K~=0.125S ~,(B) K~=0.135S -1,(C) K] =0.1475s-' , (D) K, =0.175s 1,(E) K 1=0.19s l and(F) K~=0.21s 1.


even higher values of El, periodic temporal patterns are formed. At low values of gas-particle heat transfer coefficient where the oscillations are apparently periodic, the positive Lyapunov exponents are probably caused by computa- tional limitations in the construction of the attractor.

Fig. 7 shows time series of CO 2 formation and the corresponding Fourier spectrum of these time series as functions of thermal conductivity K1. A transition from seemingly periodic oscillations, to chaotic, to periodic (completely synchronized) behavior is found before the oscillations disappear. The three regions are separated by period doubling. It is surprising that even though the oscillators are identical, they do not necessarily synchronize. For the cell distribution shown in fig. 6, the transition with period doubling seems to reflect a Feigenbaum route to chaos. As K~ decreases from high values (completely periodic regime) towards the chaotic regime, no Feigenbaum sequence was observed. We have found that other random distributions of cells exhibit multiple periodicity but do not satisfy the Feigenbaum criteria, i.e., the route to chaos depends sensitively on the distribution of active cells. However, slight changes in the description of each cell does not strongly affect the basic behavior of the system indicating that the thermal coupling dominates the dynamics of the system.

Fig. 5b shows the temperature of an active cell (representing reading of a thermocouple) and the overall conversion during oscillations. The simulations show that only part of the oscillatory behavior in CO 2 formation rate are found in the temperature of a local region. Our results are in agreement with experimental results shown in fig. 5a, suggesting that coupling among oscillators can result in complex and richer dynamics than a single oscillator.

4. Summary

We have examined two quite different models related to spatial and temporal patterns in catalytic oscillations: a single oscillator for a unimolecular reaction with the MC method and the thermal coupling among oscillating particles on a support.

For the single oscillator the interplay between the phase transformation and chemical reaction drives the oscillations in a stirred tank reactor. Since species concentrations are frequently not uniform and diffusion is not Fickian, macroscopic models (either ODEs or PDEs) are inadequate to correctly describe the spatial and temporal behavior of these catalytic systems. Fluctuations are crucial in the formation of clusters, nucleation and dynamics close to bifurcation points, such as cusp or Hopf bifurcation points. Metastability and nucleation are essential to form oscillations in this system where the variation of gas


pressure synchronizes the surface. Surface diffusion is an important process; a

downhill diffusion (Fickian) randomizes the surface, destroys clusters and

blocks the nucleation mechanism, which is important for oscillations, whereas

an uphill diffusion does not strongly affect oscillations. Finally, we find that

defects have a profound influence on oscillations. When their concentration is large and their distance is short, the oscillations disappear and fluctuations are damped.

The second problem addresses the thermal coupling between oscillators and

the complex spatial and temporal evolution caused by thermal communication.

Even though the model used a simple oscillator which can exhibit only periodic oscillations, the coupled system of identical oscillators can show very complex

behavior with periodic oscillations, period doubling cascades and chaos. It is

possible that the complex behavior in many experimental systems is caused in

some cases by coupling of many oscillating regimes, and does not reflect the existence of a complicated microscopic reaction mechanism.

These results should be regarded more as "computer experiments" rather than models. In the first problem, MC simulations incorporate exactly the

physics of the simple system chosen. In the second problem, the results are obtained from the solution of 110 ODEs and cannot be foreseen without

actually performing the "computer experiment".

The two models examined here suggest that temporal oscillations and the

observation of complex behavior in heterogeneous systems are often associated

with spatial inhomogeneities caused by defects, islands, fluctuations, or some

type of coupling across the surface. Thus, the use of distributed systems at the molecular or macroscopic level is frequently required to describe accurately the

dynamics and patterns of catalytic systems.

References

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[3] A.M. Zhabotinskii, Dokl. Akad. Nauk SSSR 157 (1964) 392. [4] F. Sch/ith, B.E. Henry and L.D. Schmidt, Advances in Catalysis, m press. [5] R.J. Field and M. Burger, eds., Oscillations and Traveling Waves in Chemical Systems

(Wiley, New York, 1985). [6] C.A. Pikios and D. Luss, Chem. Eng. Sci. 32 (1977) 191. [71 E.A. Ivanov, G.A. Chumakov, M.G. Slinko, D.D. Bruns and D. Luss, Chem. Eng. Sci. 35

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[11] G.A. Gordonier, F. Schiith and L.D. Schmidt, J. Chem. Phys. 91 (1989) 5374. [12] W.H. Ray and S.P. Hastings, Chem. Eng. Sci. 35 (1980) 589. [13] C.G. Takoudis, L.D. Schmidt and R. Aris, Surf. Sci. 105 (1981) 325. [14] D.G. Vlachos, L.D. Schmidt and R. Aris, J. Chem. Phys. 93 (1990) 8306. [15] R. Imbihl, M.P. Cox, G. Ertl, H. Muller and W. Brenig, J. Chem. Phys. 83 (1985) 1578. [16] M.P. Cox, G. Ertl, R. Imbihl and J. Rustig, Surf. Sci. 134 (1983) L517. [17] B.C. Sales, J.E. Turner and M.B. Maple, Surf. Sci. 114 (1982) 381. [18] T.H. Lindstrom and T.T. Tsotsis, Surf. Sci. 150 (1985) 487. [19] J. Crank, The Mathematics of Diffusion (Oxford Univ. Press, New York, 1986). [20] P. De Kepper, M. Boukalouch and J. Boissonade, in: Spatial Inhomogeneities and Transient

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spatial and temporal patterns in catalytic oscillations

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