J. Phys. Chem. 1996, 100, 15137-15144
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Spatiotemporal Catalytic Patterns Due to Local Nonuniformities Moshe Sheintuch Department of Chemical Engineering, Technion, Israel Institute of Technology, Haifa, Israel 32000 ReceiVed: March 20, 1996; In Final Form: June 21, 1996X
Nonuniformities of properties, like activity and transport coefficients, are common to catalytic systems and were evident in studies of spatiotemporal patterns. Under such conditions it is usually difficult to conclusively differentiate between the effects of spontaneous symmetry breaking and those due to nonuniformity. We analyze the dynamics of one-dimensional systems with single-variable or two-variable kinetics and spacedependent properties and show that these systems may admit stable stationary fronts, oscillatory fronts, source points, and unidirectional pulses. Uniform systems, with similar properties, admit traveling front and pulse solutions. Patterns in nonuniform systems are quite similar to those in systems with a global interaction that induces symmetry breaking, and both can be classified by the sequence of phase plane spanned by the system. We also analyze the impact of global interaction that preserves the symmetry and show that it may destroy the inhomogeneity due to nonuniform properties. Uniform and globally interacting systems admit reflection symmetry, and patterns may appear as symmetric pairs. Although this property would be a most discriminatory test, certain difficulties may be encountered in its implementation in catalytic systems.
Introduction Multiple steady states and periodic or aperiodic timedependent solutions were demonstrated in numerous studies of catalytic and electrochemical reactions (see reviews in refs 1-4). While the interaction of diffusion and nonlinear reaction is known to induce a plethora of spatiotemporal patterns, the identification of patterns observed in heterogeneous reactors is still ambiguous in many cases, partly due to experimental difficulties and nonuniformity of the system properties (see ref 5 for a recent review). Nonuniformities of properties, like catalyst loading and transport coefficients, are common to catalytic systems. Such nonuniformities not only occur because of technical difficulties but also may arise naturally: The catalytic activity of edge and corner sites usually differs from that of regular surface sites, and defect sites may affect the behavior of the whole system. Recent observations of patterns under high-vaccum conditions, made with catalysts that expose several crystal planes, showed wave inception at the plane boundaries and their propagation through the planes; the system exhibited self-sustained oscillations while the corresponding single crystal always reached a steady state.6 Furthermore, boundary conditions that apply to many heterogeneous reactors are different than the no-flux or fixed steady-state conditions commonly employed in models of reaction-diffusion patterns. While this difference may not be significant for patterns with a large wavenumber, most reported patterns in catalytic systems typically show one or few fronts. Recent observations of spatiotemporal patterns in catalytic systems indicate the existence of certain nonuniformities of properties (see below). Under such conditions it is usually difficult to conclusively differentiate between the effects of spontaneous symmetry breaking and those due to nonuniformity. Reproducing the experiment on another sample of catalyst (a wire or ribbon) does not always resolve this ambiguity since the new catalyst will usually demonstrate a catalytic activity that is somewhat different than the former. The nonuniformity may also be due to edge or end effects. Uniform systems admit reflection symmetry, and patterns may appear as symmetric X
Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)00857-X CCC: $12.00
pairs. Although this property would be a most discriminatory test, the transition from a certain pattern to its mirror image requires local perturbations that are not easily induced experimentally in catalytic systems. The purpose of this article is to classify the main patterns that may emerge in a one-dimensional system with nonuniform (space-dependent) properties, in a single-variable bistable system and in a two-variable oscillatory system, and to contrast them with the plethora of patterns known to exist in systems with local or global interaction. The reaction-diffusion model we use incorporates a diffusing activator and a localized inhibitor, which is the typical situation in high- or low-pressure catalytic systems. Inhomogeneous solutions in such a system emerge due to nonuniform properties or due to global interaction. We show that patterns in systems with symmetry-breaking global interaction (to be defined below) are quite similar to those due to nonuniformity; the latter lack, of course, the symmetry property. This similarity makes the experimental discrimination between these two mechanisms of pattern formation even more challenging. We also consider the effect of global coupling that preserves the symmetry and show that it may destroy the patterns due to nonuniformity. This will account for the observation of synchronous oscillatory behavior, even in the presence of large nonuniformity of properties. We are interested in the robust patterns, those likely to be detected experimentally, and ignore the fine structure of patterns that appear over a narrow range of parameters. Our analysis is aimed at accounting for some of the experimentally observed patterns (see below) and at aiding the experimentalist in designing and interpreting experiments. Pattern formation in high-pressure catalytic systems is distinctly different from that in liquid-phase oscillators (e.g., the Belousov Zhabotinski reaction) or from the classical diffusion-reaction models: In high-pressure reactors thermal effects provide positive feedback as well as the mechanism for the long-range communication. Thermal patterns usually emerge due to global interaction, the interaction of the catalyst with a global variable like the gas-phase concentration or an external-control variable. Our current understanding of catalytic oscillators suggests that they are governed by the interaction of © 1996 American Chemical Society
15138 J. Phys. Chem., Vol. 100, No. 37, 1996 a fast and long-range autocatalytic variable (typically, the temperature) with slow and localized changes in catalytic activity. Such systems are sometimes referred to as FitzHughNagumo type. Such one-dimensional systems are excitable and may exhibit transient spatiotemporal patterns like a traveling pulse;7 such systems are unlikely to exhibit sustained spatiotemporal patterns in the absence of global interaction. Patterns in such systems emerge due to the interaction of the fluid phase, through which the reactants are supplied, and the reactive solid phase. This interaction may produce patterns under conditions that, in the absence of such interaction, induce homogeneity in the catalyst phase. In a series of works on patterns that may emerge in a one-dimensional catalytic (wire, ribbon, or bed) system, interacting globally via the mixed or plug-flow gas phase or via an external integral control device,8-11 we analyzed these patterns in terms of the sequence of phase planes spanned by the system as the global variable is varied. Here we employ a similar attitude and classify the behavior of the system in terms of the phase planes spanned by the nonuniform variable. Since the sequences spanned in both cases are similar, we find similar patterns in systems interacting via global interaction and in nonuniform systems. The latter, however, do not admit the reflection symmetry of the uniform system and consequently do not admit the mirror-imaged solutions that exist in the former. Nonuniformities were evident in almost all studies of catalytic patterns at high pressures: A regular pattern in catalytic systems, in the form of two temperature pulses emanating from a source point and traveling in opposite directions, was presented by Cordonier et al.12 during ammonia oxidation on a Pt wire heated by a constant current. Spatial nonuniformity was evident by the difference in oscillation amplitude recorded by the six IRsensitive photodiodes, and the identification of the pulse pattern was based on the phase difference between these records. Antiphase oscillations were observed by Cordonier and Schmidt13 during the endothermic methylamine decomposition catalyzed by a 5 cm long Rh wire, again showing nonuniform character of the oscillations. In a similar study with a Pt or Ir wires all measured points were oscillating periodically or chaotically with different amplitude but without measurable lag time. This synchronization is rather surprising in light of the expected nonuniformity of activity. Philippou and Luss14 in a study of propylene oxidation on a 14.5 cm long Pt ribbon heated by constant electrical current, using IR thermography, observed aperiodic birth of fronts near the support and their propagation inside the ribbon. The front stopped at a certain position and moved backward before another front originated at the other support. With a uniform wire one would expect homogeneous oscillations, and the observed pattern is probably induced by end effects and the nonuniformity of the wire, as we show below. The same system exhibited temperature pulses moving back-and-forth along the ribbon, under constant resistance control.15 While the control accounts for the back-and-forth motion, the pulse did not sweep the whole wire, probably due to its low activity. Analysis of heat-generation time series revealed quasiperiodic oscillations when a single-pulse motion was observed locally and inception of a second pulse was documented as a route to chaotic dynamics. Nonuniform properties were evident also in patterns observed with two-dimensional catalytic systems, and most observed patterns did not conform with any known motion of reactiondiffusion systems. Investigators usually attribute the ambiguity to nonuniformity of surface properties. Several studies demonstrated the existence of one or more pacing centers that determine the motion of the whole surface. In several cases aperiodic solutions were shown to be associated with the
Sheintuch interaction between two or more such pacing centers. Brown et al.16 observed, using IR thermography, significant variation in surface temperature even when all parts of the catalyst belonged to the same steady-state branches. Studying selfsustained oscillations during ethylene oxidation on supported catalysts, Kellow and Wolf17observed that active hot zones around catalytic spots were contracting and expanding but in most cases were not changing their position. Chen et al.,18 in a study of ethylene oxidation on a wafer covered with Rh catalyst, observed hot spot flickering and hot spot wandering over the support. The authors suggest that solutions dominated by localized structures are very likely to be observed on catalysts with long-range nonuniformity. This conclusion suggests that complete classification of patterns will remain an elusive task, but it underscores the importance of spatially resolving techniques for analyzing these patterns, since localized measurements are inadequate measures of spatial and temporal organization in patterns dominated by few hot spots. Lobban and Luss19 used IR thermography to monitor thermokinetic oscillations during hydrogen oxidation on a 3.8 cm nickel disk. Ignition and extinction occured nonuniformly on the surface and then propagated through the entire disk. The review of experimental studies suggest that nonuniformities are indeed common to many catalytic systems. These observations motivate us to study the typical patterns that may emerge in a nonuniform system. We analyze the behavior of one-dimensional nonuniform system with bistable (singlevariable) kinetics or with oscillatory (two-variable) kinetics. In the latter case we assume wide separation of time scales, so that we can gain some insight from the analysis of motion in the phase plane. The main typical motions that two-variable systems exhibit are oscillations, bistability, and unistability of low or high activity. Part of the latter domain may be excitable. In the parameter plane the domains of existence of these motions are organized in an X-shaped diagram. To classify the nonuniformity, we denote the sequence of motions (phase planes) spanned by it and show that this classification provides the main information for predicting the emerging patterns. A similar argument was advanced in our studies of pattern formation due to global interaction, in the form of external control8,20 or as a mixed gas phase9 or due to self-generated longitudinal gradients of reactant concentrations in a packedbed reactor.11 Gradients of properties act very much like selfgenerated temporal or spatial gradients of global variables, and consequently the patterns produced seem to be similar; the main difference is that surface properties in the former case are independent of the motion and certain symmetries are destroyed, while in the latter case gradients are self-generated and symmetries are maintained. Surprisingly, the systems that exhibited synchronous oscillations did so even though they were apparently not uniform, as evident from the space-dependent amplitude of the oscillations. We believe that this synchronization is due to symmetrypreserving global interaction that occurs in many such systems. We consider the effect of nonuniformity under local interaction, due to diffusion or conduction, as well as under global, longranged interaction, due to gas-phase mixing. We show that indeed global communication can restore the synchronization to nonuniform systems. The effect of nonuniformity on pattern formation was investigated for systems that can be described by a phasediffusivity equation21 and for systems of a large number of oscillators interacting with each other. These descriptions, however, do not apply for the systems of interest, which admit widely separated time and length scales and can sustain the
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Figure 1. Typical fronts of single-variable (I, II, III) and two-variable (II) models in a system with constant properties and no-flux boundary conditions (I), in a system with nonuniform parameter R (I, IV), or in a system with fixed edge conditions (III). The front in I will typically travel out of the system, in II or IV it will become stationary or oscillatory, and in III the upper state may collapse.
motion of fronts and pulses. Analysis of a single-variable (scalar) nonuniform reaction-diffusion model (Schutz et al.22 and references therein) shows that fronts can be pinned to positions with properties that correspond to a stationary front (we review this argument and relevant information below). Front motion in two-variable nonuniform systems was analyzed mainly for bistable kinetics: If time-scale separation is mild, then such a system with uniform properties behaves like a singlevariable system. A uniform system with wide separation of time scales will exhibit multiplicity of fronts in which either the upper or the lower state expands. A nonuniformity in properties may arrest this front and induce oscillations in front positions. The dynamics closed to a pinned-front solution was analyzed by Schutz et al.,22 the transmission or blocking of periodic wave trains in a nonuniform system was analyzed computationally by Bar et al.23 and the formation of stationary or breathing wave trains in a system with space-periodic properties was computed by Bangia et al.24 and analyzed by Hagberg et al.;25 the latter analysis employed a long-range inhibitor. Traveling wave blockage due to space-dependent diffusivity was analyzed by Ikeda and Mimura.26 Stationary and Moving Fronts (Bistable Kinetics) Theory. Fronts may be sustained in single-variable reactiondiffusion (or conduction) one-dimensional systems, like wires or ribbons, that catalyze a reaction with bistable kinetics. The source of multiplicity may be thermal acceleration in nonisothermal systems, reactant inhibition in an isothermal continuous porous layer, or even isothermal multiplicity in a surface that is free of any resistances. Single-variable systems that are generally described by
τuut - Lu2uzz ) f(u,R)
(1)
admit a moving front solution in an infinitely long system that connects the two stable solutions (u+,u-) of a source function (Figure 1, curve I). There is no analytical solution for front velocity in the general case, but it obeys the following properties in a uniform system: (i) The front is stationary when
∫uu (R(R)) f(u;R0) du ) 0 +
-
0
0
(2)
Figure 2. Heat generation curves (Qg(T;Cf)) and heat removal lines (T - Tf) in a single-variable model in a system with (a) constant properties and no-flux boundary conditions, (b) nonuniform parameter R, (c) fixed edge conditions, and (d) global interacion.
where R is a certain (say, kinetic) parameter and (u-,u+) are the two stable solutions of f(u,R0) ) 0. (ii) Stationary fronts are structurally unstable in one-variable systems, and their locus in the parameter space (R0) separates domains with an expanding upper or lower state. Typically, if the solution of f(u,R) ) 0 is a clockwise hysteresis curve, then the upper state is expanding for R > R0. The steady-state temperature (T) of an exothermic reaction is commonly presented as the intersection of the heat generation function Qg ) (-∆H)r/hav (i.e., the product of reaction enthalpy and reaction rate over heat-transfer coefficient per unit catalyst volume) and the heat removal lines Qr ) (T - Tf) (Figure 2a). Stationary solutions in a uniform long catalyst exist only for a certain set of parameters for which the areas bounded by these two lines are equal (i.e., when eq 2 is satisfied with u ) T and f ) Qg - Qr). To find the conditions (like the fluid temperature, Tf ) Tf0, Figure 2a) that correspond to a stationary system we adjust the ambient temperature, by displacing the heat removal line parallel to itself, to satisfy eq 2. In general, however, either the ignited (upper) or extinguished (lower) state will be more stable, and the system will eventually achieve homogeneity. Varying a parameter (say Tf ) will result then in a simple hysteresis curve bound by ignition and extinction (denoted by arrows, Figure 2a, at Tfi and Tfe, respectively). The stable branches are denoted by bold lines in Figure 2a. Nonuniform Systems. We consider below the effect of space-dependent parameters (e.g., catalytic activity) and of boundary conditions of the Newman form. We review first the argument that a nonuniformity in parameters can create structurally stable stationary fronts. (We use the term “nonuniformity” of parameters to distinguish it from inhomogeneity of the solution; by homogeneous solutions we imply that the whole system is at a state that belongs to the same, upper or lower, branch). Suppose that parameter R in eq 1 is space dependent, R ) R(z), but monotonic and that a stationary front exists for a certain R0, which is within the range spanned by R(z), and it occurs at a certain z0 (Figure 1, curve II). We choose R so that its increase will make the ignited state more stable; in catalytic systems R may represent surface activity, fluid temperature, current in resistive heating, or the inverse heat transfer coefficient. For these choices of R the front velocity is a monotonic increasing function of R. Now, we can place the front at z0 so that the parameter (R) and the variable (u) profiles show a similar (curve II) or opposite inclination (not shown). In the latter case, if the front is perturbed to a new position with R > R0, the upper state is more stable and it will expand until the
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front moves out of the system. In the former case, however, the front will move back toward its stationary position and the front is stable (Figure 1, curve II). To check the local stability of the stationary front, we can describe the front position (η) by
dη/dt ) -V(R(η))
(3)
where the front velocity (V) is assumed to depend only on the local activity, which in turn is space dependent; we use the convention that the velocity is positive when the active zone is expanding and assume that the front inclination is positive (as in Figure 1). At η ) z0, R ) R0 and V(R0) ) 0. The front is stable if
dV dV dR ) >0 dη dR dη
(4)
Now, dV/dR > 0, by our definition of R, and stability is assured if dR/dη > 0, i.e., if the activity profile is synclinal to the profile of the state variable (as in Figure 1, II). For a moderate nonuniformity an expression for front velocity was developed by Schutz et al.,22 which requires knowledge of the front solution (u(z+Vt;R)) in a system with uniform properties; such solutions exist, however, only for a cubic f(u) expression. Close to z0 the front velocity is linear with position. If we change now the operating condition (say, Tf or Cf) and try to obtain a bifurcation diagram, we find that the front will move to a new position so that its activity satisfies the stationary system condition. Analysis of such a system shows that the wider the domain of nonuniformity, the larger is the domain over which stationary fronts exist; to show that let us denote the Qg curves with the largest (R+) and the smallest (R-) activities (Figure 2b). Obviously, an ignited or extinguished state cannot exist unless (T - Tf) intersects the upper (lower) branch of all Qg(T,R) curves with R ∈ (R-,R+). When that is not possible (as in Figure 2b), as one edge is ignited and another is extinguished, a front must be formed; once formed, this front will travel until it becomes stationary at z0 where R ) R0. End Effects. The analysis above was limited to zero-flux boundary conditions. In an actual catalytic wire, which is connected to two inactive ports at its edges, the boundary conditions should account for some heat loss through the ports. With high heat loss the edge temperature should approach the ambient one. Suppose now that the activity is uniform but the edge conditions specify that T ) T-. If the system lies at an ignited state, outside the domain of steady state multiplicity, the edge effect will be limited to a narrow transition zone. In the domain where bistability is possible but the ignited state is more stable, the front connecting T- and T+ cannot propagate inward and again will lie close to the edge. However, when the lower steady state is more stable this front will conquer the whole surface (Figure 1, III); thus, under these conditions an ignited homogeneous state cannot be sustained, while it can be sustained in a wire with no-flux boundary conditions. The resulting diagram is shown in Figure 2c, which denotes the stable branches by bold lines; note that the upper branch is unstable for Tfe < Tf < Tf0. Similar results are expected when the edge temperature lies even below T-. Global Interaction. Several mechanisms of global coupling have been recently discussed in the literature. Controlling the average temperature (i.e. the resistance) of a wire or a ribbon, subject to resistive heating, has been shown to induce symmetry breaking when the reaction admits local bistability. Global interaction can also result from interaction of a distributed catalyst (ribbon or disk) with a mixed gas phase. We define
the interaction between the phases to be symmetry breaking (symmetry preserving) if local ignition or extinction at one point inhibits (accelerates) the same process at another point. Consider a catalyst that exhibits local bistability, so that its temperature can differ from that of the surrounding, placed in a mixed reactor. Two counteracting effects are important in nonisothermal systems, and the outcome depends on the thermal regime: As an ignition is induced locally at a certain section of the catalyst, the fluid temperature ascends while the fluid concentration of a reactant descends. Both effects are important in an adiabatic reactor, but the thermal effect more than compensates for reactant depletion, and the interaction preserves the symmetry. In an isothermal-fluid reactor, where the fluid temperature is kept constant by external heating or cooling but temperature gradients between the phases are not negligible, mixing induces symmetry breaking in reactions with positiveorder kinetics: When ignition occurs locally in an isothermalfluid reactor, it inhibits subsequent ignition of another spot due to declining concentration. This isothermal-fluid model is a limiting asymptote of the general nonisothermal heterogeneous model of a catalytic reactor. Under both mechanisms of global interaction, due to control or due to mixing, a uniform catalyst is described by eq 1 with f(u,R0, λ), where, in turn, the global variable obeys λ ) B(us 〈u〉): In controlled systems the sensitivity factor (B) is the gain and is typically very large, so that the space averaged value obeys 〈u〉 ) us; in gas-phase interaction B reflects the ratio between reaction rate and convective flow. In both cases we ignore the interaction dynamics, assuming an instantaneous control or a small reactor volume. The steady state is the intersection of Qg(Cf) and (T - Tf) (Figure 2d). Now, when the system admits local bistability, there exists a certain λ0 (either Cf or Tf) for which a stationary front exists (recall that R is assumed constant now). To check its stability we may apply local (in the phase space) analysis and consider the response of λ to perturbation in front position. The stationary front may become globally stable if neither an ignited nor an extinguished state exists. Local stability of the front position requires that dV/dη ) (dV/dλ)(dλ/dη) > 0 (for the inclination in Figure 1, I). A simple analysis will verify the stability of the stationary front in the external-control and the isothermal fluid cases, while in the adiabatic reactor the front is unstable: Increasing the resistive current will render the ignited state more stable (dV/ dλ > 0), but expansion of the ignited zone will cause the control to lower the current (dλ/dη > 0). To study whether a “homogeneous” solution exists, we plot in Figure 2d three heat generation curves (Qg(Cf), with Cf1 < Cf2 < Cf3) and three removal lines (Tf1 < Tf2 < Tf3), where the intermediate set corresponds to a stationary system. In the case of a constant-resistance control of a wire exposed to a stream of fixed properties, we fix Cf and Qg but allow the current to vary so that the space-averaged temperature meets a certain set point, Ts. Since the resistive heating compensates for heat loss from the wire, varying the current amounts to varying the effective Tf. (This is approximately so, since the resistance is temperature dependent, but for practical problems it is a good approximation; see ref 27.) Evidently if the homogeneous state T ) Ts is unstable (intermediate solution), then the system cannot satisfy the set point with another homogeneous solution, whether active or inactive, and a stationary front must exist. In the adiabatic mixed reactor, in which the catalyst is placed, both Qg(Cf) and (T - Tf) curves may shift while in the isothermalfluid reactor Tf is fixed but Cf may vary. In the adiabatic case the system must attain either the cold (the intersection of Tf1 and Qg(Cf3)) or the hot homogeneous solution, since high Cf
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implies low Tf, etc. In the isothermal-fluid case these solutions are not possible as the active state implies low Cf, but the lower Qg curve may acquire only a cold solution. The system must attain then an inhomogeneous solution. The situation may be more involved with other kinetics, and under some conditions the stationary front solution may coexist with a homogeneous state, but this simple analysis verifies the broad classification suggested above into symmetry-preserving or -breaking interactions. Nonuniform Systems with Global Interaction. The discussion above demonstrates that stationary fronts may emerge in nonuniform systems or in systems with symmetry-breaking global interaction. In the latter case two mirror-imaged inhomogeneous solutions must exist. We show now that in systems with symmetry-preserving interaction (such as an adiabatic reactor), stationary fronts pinned by nonuniformities may become unstable. The system is described by eq 1 with f(u,R(z),λ(us - 〈u〉)). Their stability depends on the term
dV ∂V dR ∂V dλ ) + dη ∂R dη ∂λ dη
(5)
which incorporates now two counteracting contributions. The first term is positive as before (using the conventions employed before; see profile in Figure 1, II), but the second is negative in an adiabatic reactor (λ ) Tf): ∂V/∂Tf > 0, since the ignited state becomes more stable with increasing fluid temperature, and dTf/dη < 0, since as the front moves to the right the hot zone shrinks (see Figure 1) and the fluid temperature, which reflects the average reaction rate, declines. The first term in eq 5 depends on the slope of nonuniformity, while the second is determined by the sensitivity factor B (recall that λ ) B(us 〈u〉); if B is sufficiently large, the nonuniform system will obtain an almost-homogeneous solution, in which all points of the system belong to the same branch of solutions. Oscillatory Kinetics (Two-Variable Model) We consider the motion of systems of the form
ut - uzz ) f(u,V,λ;R(z)) 1 V ) g(u,V,β(z)) t
(6)
where , the ratio of time scales, is usually a small parameter and f(u,V) ) 0 is a multivalued curve. This model applies to most high-pressure exothermic catalytic systems: the catalyst temperature is the activator (u) and its length-scale is larger than that of the inhibitor (V), which is believed to be a slowly changing surface concentration (see ref 5 for further discussion). Similar models have been suggested for isothermal low-pressure catalytic oscillations.3 For fixed λ and space-independent parameters the system exhibits homogeneous oscillations, when the phase plane is oscillatory (let us denote it by O), it exhibits a propagating pulse when it is perturbed from a unique but excitable state (U), while multiple solutions or propagating fronts are possible when it is bistable (M). The nature of the phase planes, in this problem with widely separated time scales, and with S-shaped f ) 0 and monotonic g ) 0 null curves of the type shown in Figure 3, can be readily determined from the number and location of the steady states. For the limiting case of ) 0 the two Hopf lines typically form the common X-shaped diagram that divides the (λ, R) plane into four domains with U phase planes at top and bottom sections and O and M motions at the left and right sections. The system reaches a uniform solution in the U or M domain. In the excitable part
Figure 3. Phase-plane analysis (upper row) and the resulting spatiotemporal t-z patterns (lower row, u > 0 in the dark domain) in a two-variable system with nonuniform properties that span the oscillatory regime (a, b), the O-U regime (c), the U-O-U regime (d), and the U-M-U regime (e) and in an oscillatory nonuniform system with symmetry-preserving global interaction (f). The phase planes show the f ) 0 (solid lines) and g ) 0 (dashed) null curves at the left and right edges (superscript L, R) and at certain intermediate situations.
of the unique-solution domain the homogeneous rest state is linearly stable but is susceptible to finite perturbations. A local perturbation may send a solitary wave that propagates at constant speed and shape; unlike solitons, these waves annihilate each other in face-to-face collision. Similarly, a local perturbation in the M domain will send a propagating front that will switch the system from one homogeneous state to another. Unlike in the case of a single-variable model, there are two stable traveling fronts now, and the velocity dependence on a parameter exhibits a hysteresis or pitchfork behavior.22 These fronts and waves, however, are transient features. The excitable domain may sustain a propagating wave train if subject to continuous oscillatory perturbation of its edges. Nonuniform Systems. We allow now for nonuniformities in parameters that affect either the u or V kinetics (eq 6). The motions in a nonuniform system differ from those in a uniform medium since the local state of the catalyst (i.e., the corresponding phase plane) may vary. We try to understand and classify the possible motions in terms of the phase planes spanned by the system. Thus, we consider the effect of nonuniformities on the null curves f ) 0 and g ) 0, and in that sense we can study changes in R in the same framework as changes in β. Excitable domains may sustain a passage of waves but will not generate new ones. Waves will be incited by a local perturbation or by a self-sustained oscillatory domain.
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Figure 4. Simulations of eq 4 with f ) -u3 + u + V + R(z) and g ) -βu - V, with R(z) that varies linearly and spans the O-U domains (RL ) 0.15 and RR ) 0.25 (a) or RL ) 0.17 and RR ) 0.21 (b)) or the O domain (RL ) -0.1 and RR ) 0.1 (c); in a-c L ) 20, β ) 1, ) 0.03, and the corresponding Hopf bifurcation at f 0 occurs at R ) (0.19) or with R that spans the U-M-U domains (RL ) -0.15 ) -RR, L ) 50, β ) 0.5; the boundaries of the domain with two stable states occur at R ) (0.096 as f 0) and ) 0.3 (f), ) 0.1 (e), and ) 0.03 (d).
A transition from one branch of f(x,y) ) 0 to another is induced either by a slow-moving trigger wave or by a phase wave in which the transition occurs spontaneously. If a section of the reactor spans the oscillatory domain, then it will serve as a pacing center for the whole reactor, sending fronts that will move through the excitable domains. The reactor point that will serve as a pacing center is expected to be the one with the highest frequency. In a patterned state a front (in u) will travel according the rules outlines for the single-variable problem, but its velocity depends now on the the local V as well as of R. Nonuniformity in R will affect the front velocity, but in the O or U phase planes the inhibitor (V) will vary slowly and the front will not become stationary. The inhibitor may reach a steady solution in the M phase plane, and we explain this situation below. If the nonuniformity does not cross bifurcation boundaries, then within a U or M region the system attains an almost homogeneous state, in which the whole system lies on the same (upper or lower) branch of the null curve f ) 0. Within the oscillatory domain a nonuniformity in R may express a shift of the f ) 0 curve, or of its shape, while variation in β implies that the g ) 0 null curve is translated and rotated as we move along the surface. In either case the period of natural oscillation varies along the surface, and specifically we can compute (for very small and for every z) the travel time between the two limit points, along the upper and lower branches of f ) 0. These functions, τ+(z) and τ-(z), are space dependent, and their sumsthe oscillations periodsis space dependent as well. To find a periodic solution, we need to compose a series of local oscillations, composed of slow motions along the stable f(u,V;R(z)) ) 0 branches and jumps between them that obey the local kinetics, with jumps triggered either by meeting a limit point or by the arrival of a front. If the whole system lies within the O region we may consider two limiting situations: In a
relatively short system, so that the front travel time (of order L) is much shorter than the oscillation period (of order 1/), and with mild nonuniformity the phase difference along the surface will tend to be minimal and synchronization of the period will occur by phase waves. For longer systems the transitions will be mediated by trigger fronts. In typical situations, where τ+(z) increases with z while τ-(z) decreases (or vice versa), we may find that activation and deactivation fronts originate at different positions or at opposing edges. The expected spatiotemporal pattern then is like that in Figure 3a: The figure presents the f ) 0 null curves corresponding to the right and left edges (fR ) 0 and fL ) 0; solid lines) and the g ) 0 null curve, which is assumed to be space independent (dashed line). The disposition indicates that motion on the lower (upper) f ) 0 branch is faster (slower) on the right than on the left edge. The local oscillatory behavior on the two edges is denoted by ABCD and A′B′C′D′ in the figure. When the transition AB is completed, the right edge sends a front that travels at varying speed (determined by the local V) until it triggers the A′B′ transition at the other edge. If the latter edge will undergo a spontaneous transition (C′D′), before the similar transition on the right, we will find another front moving from left to right. The local τ- and τ+ will be determined by the arrival time of the front, but every point of the surface should obey a trajectory so that the local period is constant (τ- + τ+ ) τ-L + τ+L). If a certain point admits the shortest τ+ and shortest τ-, then it will become the pace center and send fronts that will mediate the periodicity (Figure 3b). The spatiotemporal pattern is presented, in the t-z plane, in the lower part of Figure 3 (light and dark shades correspond to negative and positive u values). To demonstrate some of these claims we simulated eq 6 with f ) -u3 + u + V + R(z) and g ) -βu - V, where R(z) is assumed to vary linearly with z. Figure 4c presents a spatiotemporal contour map for L ) 50, β ) 1, ) 0.03, and R
Spatiotemporal Patterns varying between RL ) -0.1 (left edge) and RR ) 0.1. To find the phase planes spanned by the system we note that the Hopf bifurcation for β ) 1 and f 0 occurs at R ) (0.19. (The boundary will shift with increasing and other phase planes may develop near the transition, but these changes are expected to be small for the small value employed here). Thus, only oscillatory motions are spanned by the system simulated in Figure 4c and it shows fast-moving (phase) transitions; note that a certain intermediate point sends successive up fronts and another one sends down fronts. If the nonuniformity spans an oscillatory and an excitable domain, then the shortest period oscillator is still the pace center, sending successive fronts as before, that travel through and excite the excitable domain. This situation is demonstrated in Figure 3c by a g ) 0 null curve that shifts with position: for the disposition shown, the right edge is oscillatory while the left edge is excitable. Note that τ+ in the excitable domain is infinite. Consequently, the pace center is within the oscillatory domain, sending (phase) fronts that move fast through the oscillatory domain but turn to slow trigger fronts as they travel through the excitable domain. This pattern is verified by simulating the cubic kinetics described above with RL ) 0.15 and RR ) 0.25 (Figure 4a) or RL ) 0.17 and RR ) 0.21 (Figure 4b; other parameters as in Figure 4c; recall that Hopf bifurcation occurs at RΗ ) 0.19; with β ) 1 the system is excitable for RH < R < 1). When the nonuniformity is wide, so that both edges are excitable but the intermediate domain is oscillatory (Figure 3d), then the pace center will serve as a source point that sends pairs of pulses in opposite directions (Figure 3d). A nonuniformity that spans only the excitable or the bistable domain, or stretches between two such regions, will attain an almost homogeneous system in which all points belong to the upper or lower branch of f ) 0; u is space dependent due to the nonuniformity, but its gradients are small so that diffusion terms are typically negligible. To find the steady-state solution, we solve g(u,V) ) 0 and substitute into f ) 0 so that u(z) is approximately the solution of F(u,z) ) f(u,V(u)g)0;R(z)) ) 0. Within the M domain there also exists a certain R0 for which a stationary front exists. For reasons that we explain below this solution is unstable for small , and the front position oscillates and may exit the system. In a uniform system with similar characteristics, the u and V profiles are anticlinal (Figure 1, IV): A small implies that the front velocity is fast in comparison with the period of oscillations. Consequently, upon a small perturbation of the u front (thin line) from its stationary position, it will meet a lower V and will start moving and accelerating as it meets even lower Vs. Eventually the V profile will adjust to the new situation in the wake of the moving front, but this system is too slow to arrest the front. In a nonuniform system we should consider two situations: when R(z) is anticlinal with u(z) than the nonuniformity will cause even further acceleration of the front, and the system will reach almost-homogeneity. This conclusion is similar to the outcome of the single-variable front. If R(z) is synclinal, however (as in Figure 1, IV), the escaping front will slow down as it meets the high-activity domain. Eventually, the front may be arrested, thus giving ample time for the V profile to adjust. Once that happens, the front will reverse direction and move toward the other edge until again it meets a similar fate. A nonuniform system that spans the U-M-U phase planes (like the situation shown in Figure 3e, in which the solution is unique at both edges and bistable at intermediate positions) will exhibit a stationary front for large ; it will undergo a Hopf bifurcation upon decreasing and an oscillatory front will emerge (Figure 3e). The unfolding of the behavior of pinned front, close to a
J. Phys. Chem., Vol. 100, No. 37, 1996 15143 localized maximum in property or in the vicinity of a maximum and a minimum, was analyzed by Schutz et al.22 This transition portrayed above is verified in Figure 4d,e) in which the parameters span the U-M-U domains (RL ) -0.15 ) -RR, L ) 50, β ) 0.5; for these values the boundaries of the domain with two stable states occur at R ) (0.096 as f 0). For large ( ) 0.3, Figure 4f), the situation is similar to that of a single-variable problem and the stationary front is stable. For intermediate values ( ) 0.1, Figure 6e), the front is oscillating, while for sufficiently low values ( ) 0.03, Figure 6d), the front sweeps the whole surface and the motion is similar to that of an oscillatory system. Nonuniform Systems with Global Interaction. The effect of global interaction on oscillatory systems like eq 6 has been extensively investigated, 8,9 and a large plethora of patterns has been revealed for systems with symmetry-breaking interaction. With symmetry-preserving interaction the system typically exhibits either homogeneous oscillations or bistability of homogeneous solutions. We show now that the effect of symmetry-preserving interaction on a nonuniform system will be largely to eliminate inhomogeneities. To simplify the analysis we assume that the system can be portrayed by four f ) 0 null curves (Figure 3f): fR- and fL- are the null curves of the two edges (bold line), when the other edge is at the lower state, while when the other edge is ignited the null curves (fR+ and fL+) are shifted to the left by global interaction. Suppose that the left edge is about to be ignited (the state is denoted by point A on f ) 0, Figure 3f) while the right edge state is still away from the limit point on the lower branch. Ignition of the left edge will form a front that as it travels it will immediately affect the right edge, by global interaction (say, by raising the fluid temperature), and shift the null curve so that ignition is likely to be induced on the right edge (if not in the present cycle, then in a later one). Similar arguments apply to the extinction process, and with a moderate inhomegeneity such systems will reach synchronous transitions and almost homogeneous oscillations (local temperature may vary somewhat). Concluding Remarks Nonuniformities of properties, like activity and transport coefficients, are common to catalytic systems and were evident in studies of spatiotemporal patterns. Under such conditions it is usually difficult to conclusively differentiate between the effects of spontaneous symmetry breaking and those due to nonuniformity. We classified the main patterns that may emerge in a one-dimensional system with nonuniform (space-dependent) properties. We compare them now with the plethora of patterns known to exist in systems with local or global interaction: Single-variable models with bistable kinetics admit travelingwave solutions in a uniform system; a stationary front emerges due to global symmetry-breaking interaction or due to nonuniformity of properties, when its spatial inclination is similar to that of the variable, or due to special boundary conditions. The bifurcation diagrams obtained, upon varying a parameter, in systems with global interaction are quite similar to those with nonuniform properties. Two-variable models with excitable, oscillatory, or bistable kinetics admit moving-pulse and moving-front solutions in a uniform system. Nonuniformity may induce unidirectional pulses (Figures 3b, 4a,b), which move from the active edge to the other one, source points from which two pulses emerge and move toward the boundaries (Figure 4d), and stationary or oscillatory fronts (Figures 3e, 4e,f). Nonuniformity may induce also a sequence of fronts emitted from alternating edges (Figures 3a, 4d).
15144 J. Phys. Chem., Vol. 100, No. 37, 1996 Global symmetry-preserving interaction may destroy stationary fronts or patterns due to nonuniformity in both singlevariable and two-variable kinetics. Patterns in nonuniform systems are quite similar to those in systems with global interaction, but the former lack, of course, the symmetry property of the latter. To classify the nonuniformity, we denoted the sequence of phase planes spanned by it and show that this classification provides the main information for predicting the emerging patterns. Gradients of properties act very much like self-generated gradients due to global interaction, and consequently the patterns produced seem to be similar. It is necessary to discriminate between these two mechanisms in modeling a system that exhibits pattern formation. Reproducing the experiment on another sample (wire or ribbon) of catalyst does not always resolve this ambiguity since the new catalyst will usually demonstrate catalytic activity that is somewhat different than those of the former. The nonuniformity may also be due to edge or end effects. Uniform systems admit reflection symmetry, and patterns may appear as symmetric pairs. Although this property would be a most discriminatory test, the transition from a certain pattern to its mirror image requires local perturbations that are not easily induced experimentally in catalytic systems. These results can be extended to more complex profiles of properties, provided the distance between fronts is kept sufficiently large: A spatially periodic pattern of R will sustain a stationary wave in the single-variable case. With two-variable dynamics, in the bistable domain the systems will exhibit a stationary wave train for large , which will undergo a Hopf bifurcation with declining to form a breathing wave train; as its amplitude increases, the train will be destroyed to form almost homogeneous oscillations, in analogy with Figure 4d-f. Such a transition was simulated by Bangia et al.24 for a model of Pt(111)-catalyzed CO oxidation at low pressures. Acknowledgment. This work was supported by the U.S.Israel Binational Science Foundation.
Sheintuch References and Notes (1) Hudson, J. L.; Tsotsis, T. T. Chem. Eng. Sci. 1994, 49, 14931572. (2) Schuth, F.; Henry, B. E.; Schmidt, L. D. AdV. Catal. 1993, 39, 51. (3) Ertl, G.; Imbihl, R. Chem. ReV. 1995, 95, 697. (4) Slinko, M. M.; Jaeger, N. J. Stud. Surf. Sci. Catal. 1994, 86. (5) Sheintuch, M.; Shvartsman, S. AIChE J. 1996, 42, 1041. (6) Gorodetskii, V.; Lauterbach, J.; Rotermund, H.-H.; Block, J. H.; Ertl, G. Nature 1994, 370, 276. (7) Hagberg, A.; Meron, E. Nonlinearity 1994, 7, 805. (8) Middya, U.; Sheintuch, M.; Graham, M. D.; Luss, D. Physica D 1993, 63, 393-409. (9) Middya, U.; Luss, D.; Sheintuch, M. J. Chem. Phys. 1994, 100, 3568. (10) Barto, M.; Sheintuch, M. AIChE J. 1994, 40, 120. (11) Shvartsman, S.; Sheintuch, M. J. Chem. Phys. 1994, 101, 9573. (12) Cordonier, G. A.; Schuth, F.; Schmidt, L. D. J. Chem.Phys. 1989, 91, 5374-5386. (13) Cordonier, G. A.; Schmidt, L. D. Chem. Eng.Sci. 1989, 44, 19831993. (14) Philippou, G.; Luss, D. J. Phys. Chem. 1992, 96, 6651. (15) Philippou, G.; Shultz, F.; Luss, D. J. Phys. Chem. 1991, 95, 3224. (16) Brown, J. R.; D’Netto, G. A.; Schmitz, R. A. In Temporal Order; Rensing, L., Jaeger, N., Eds.; Springer-Verlag: Berlin, 1985; p 86. (17) Kellow, J. C.; Wolf, E. E. AIChE J. 1991, 37, 1844. (18) Chen, C. C.; Wolf, E. E.; Chang, H.-C. J. Phys. Chem. 1993, 97, 5, 1055. (19) Lobban, L.; Luss, D. J. Phys. Chem. 1989, 93, 6530. (20) Sheintuch, M. Chem. Eng. Sci. 1989, 44, 1081-1089. (21) Kuramoto, Y. Chemical Oscillations, WaVes and Turbulence; Springer-Verlag: Berlin, 1984. (22) Schutz, P.; Bode, M.; Purwins, H.-G. Physica D H-G 1995, 82, 382-397. (23) Bar, M.; Kevrekidis, I. G.; Rotermund, H. H.; Ertl, G. Phys. ReV. E 1995, 52, 5739-5742. (24) Bangia, A. K.; Bar, M.; Kevrekidis, I. G.; Graham, M. D.; Rotermund, H. H.; Ertl, G. Chem. Eng. Sci. 1996, 51, 1757-1765. (25) Hagberg, A.; Meron, E.; Rubinstein, I.; Zaltzman, B. Phys. ReV. Lett. 1996, 76, 427. (26) Ikeda, H.; Mimura, M. SIAM J. Appl. Math. 1989, 49, 515-538. (27) Sheintuch, M.; Schmidt, J. Chem. Eng. Commun. 1986, 44, 35-52.
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