19118
J. Phys. Chem. 1996, 100, 19118-19123
Oscillating Langmuir-Hinshelwood Mechanisms M. Eiswirth,* J. Bu1 rger, P. Strasser, and G. Ertl Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany ReceiVed: June 10, 1996; In Final Form: October 3, 1996X
Sufficient and necessary conditions for the occurrence of a Hopf bifurcation in chemical reaction mechanisms are presented using the formalism of stoichiometric networks. The conditions are applied to determine the mechanistic basis of chemical oscillations in isothermal surface reactions. Realistic examples are given for the different oscillatory mechanisms.
1. Introduction Dynamic instabilities in catalytic surface reactions, giving rise, e.g., to temporal oscillations and spatiotemporal pattern formation, have attracted considerable interest over the past decade.1-7 To elucidate the underlying chemical mechanisms, it turned out to be advantageous to use “idealized” catalytic systems, i.e., single crystal surfaces at very low pressures (typically below 10-3 mbar).6-8 This not only eliminates complications stemming from catalyst inhomogeneity but also allows to isolate the surface chemistry, because due to the low turnover thermal effects and changes in the gas phase can be neglected. Still on single-crystal surfaces a large variety of self-organization phenomena have been observed. At the same time, surface science techniques were applied in order to clarify the underlying elementary steps (which essentially include adsorption, desorption, and surface reaction). Just about all surface reactions (see however ref 9 for an exception) take place according to the classical Langmuir-Hinshelwood (LH) mechanism, i.e., both educts have to adsorb on the surface before reacting:
X + νX* a Xad
(1a,b)
Z + νZ* a Zad
(2a,b)
Xad + Zad f (νX + νZ)* + P
(3)
Here asterisks denote a vacant surface site, νX and νZ give the site requirements for adsorption of X and Z, respectively. The product P (≡XZ) generally has to desorb sufficiently fast in order to prevent self-poisoning of the catalyst (an example of delayed product desorption is treated below). It is assumed that the gases are pumped through the reactor at a high rate (compared to the turnover); consequently the partial pressures of X and Z can be assumed constant, and process 3 is practically irreversible (since P is removed from the reactor). The desorption processes 1b and 2b need to be included (even if small), since otherwise the reaction scheme would give rise to unphysical states and nongeneric bifurcations.10,11 It was shown by Feinberg and Terman12 that the above scheme in general gives rise to bistable behavior (bounded by two saddle-node (sn) bifurcations and terminating in a cusp). Only if the site requirements of X and Z are exactly the same will no instability occur. In experiments not only bistability has been observed in a number of surface reactions, but also oscillations occur under isothermal conditions in several systems. The aim of the present work is to clarify the mechanistic basis of such oscillations. X
Abstract published in AdVance ACS Abstracts, November 15, 1996.
S0022-3654(96)01688-7 CCC: $12.00
We first present sufficient and necessary conditions for the occurrence of a Hopf bifurcation (based on stoichiometric network analysis SNA13-15). In section 3 simple extensions of the basic LH mechanism are described which fulfill these conditions. Then some realistic examples are presented (section 4). 2. Hopf Bifurcation in Chemical Mechanisms We use the formalism of stoichiometric network analysis (SNA) developed by Clarke.13 SNA allows to write down the complete set of stationary states of a chemical mechanism ()stoichiometric network) for the complete parameter set (i.e., arbitrary nonnegative rates). Consequently also the stability analysis (in particular the test for local bifurcations) can be carried out for all stationary states. The use of complete parameter sets (rather than a specific set of rate constants) is advantageous not only because rate constants are often not accurately known but also because the effective rate of a process can usually be varied to a large extent by changing the concentration of a nonessential species.15,16 For example, the adsorption processes 1a and 2a are proportional to the respective partial pressures, which can be assumed constant and included in the effective rate constants, which therefore vary all the way from zero to very large values. The SNA formalism is briefly sketched here, for details see refs 13-15. The equation of motion of a chemically reacting system is in the absence of gradients given by
x˘ ) νv
(4)
where the vector x(t) describes the concentrations of all species Xi, the stoichiometric matrix ν contains the (positive or negative) stoichiometric change of each species (rows) in each reaction (columns). The nonnegative reaction velocity vector v is usually given by mass action (power law) kinetics:
Vj ) kj∏xiκij
(5)
i
Here kj is the (effective) rate constant of the jth reaction. The kinetic matrix K contains the kinetic exponents of all species in all reactions. The kinetic exponents are small integers for elementary steps in well-mixed dilute systems, but noninteger exponents and more complicated rate laws can result otherwise. For surface reactions at low pressure, surface diffusion is very fast compared to adsorption, desorption, and reaction so that a locally well-mixed adsorption layer can be assumed and the above rate law (corresponding to the first-order mean-field limit) remains applicable.8 Still it should be checked that the results are robust with respect to small variations of the kinetic © 1996 American Chemical Society
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J. Phys. Chem., Vol. 100, No. 49, 1996 19119
Figure 1. Diagrammatic representation of the Sel’kov model (a) and the Brusselator, which is a linear combination of (b) and (c). In this and the following figures the stoichiometric and kinetic coefficients νik and κik are encoded in the network diagrams as suggested by Clarke: 13 the (positive) stoichiometric coefficient of products are given by the number of barbs on the arrow, the (negative) ones of reactants by the total number of feathers, while the kinetic exponent of a reactant is symbolized by the number of feathers to the left. By convention no feathers are shown if both stoichiometric and kinetic coefficient are unity. For example the autocatalytic reaction 2X + Z f 3X in the above mechanisms is assumed to be of second order in X and first order in Z.
exponents (which in general need not be integers). Structurally unstable networks can easily be constructed (cf. ref 16) but are of little practical value and therefore not discussed here. Since the complete parameter set is to be considered, a mechanism of n species and r reactions is completely specified by two nxr matrixes ν and K. Rather than writing these down explicitly, they are usually encoded in so-called network diagrams DN (cf. ref 13 and Figure 1). If ν is of rank d < n, not all species are independent, i.e., there are conservation constraints, given by the vectors spanning the (n - d)-dimensional left nullspace of ν.14 The stationary state condition x˘ ) 0 in eq 4 is satisfied for the velocity vectors v which lie in the intersection of the right nullspace of ν (of dimension r - d) with the nonnegative orthant. This intersection (the set of all physically meaningful stationary states) forms a convex cone.14 Consequently all stationary states can be written as a linear combination with nonnegative coefficients of the vectors Ei pointing along the edges of this cone:
C ) {∑jiEi|ji ∈ R+ 0}
(6)
The stationary states are usually referred to as currents, the Ei as extreme currents. For the stability analysis we include the mixture of extreme currents to be considered in the stoichiometric matrix νc ) ν (diag Ej). Referring to the reciprocal steady-state concentrations of species Xi by hi, we define the matrixes S ) -νcKt and A ) S diag h and refer to their principal minors as βi and Ri, respectively. The matrix A corresponds to the Jacobian of the equation of motion; due to its special form the stability considerations can to a large extent be reduced by considering the signs of βi. As shown by Clarke,13 a sufficient condition for instability of a network (i.e., the existence of an unstable state) is that there exists a current with at least one negative principal minor of S (∃βi < 0). This condition by itself leads to a saddle point (generally giving rise to bistability), while for a Hopf bifurcation an additional condition needs to be fulfilled. 2.1. Hopf Bifurcation: Sufficient Condition. A network contains a Hopf bifurcation if there exists a species Xl with reciprocal concentration hl and a natural number k < d such that for some current (a) the sum over all Rk which depend on hl is negative and (b) the sum over all Rd which depend on hl remains positive. This condition can be tested using the βi. Criteria (a) and (b) lead to 2 inequalities for the hi (possibly including some ji). If these do not contradict each other, the existence of a Hopf bifurcation is guaranteed.
Criterion (a) actually implies that there is a positive feedback loop (autocatalysis) involving k species, which gives rise to an unstable steady state. Criterion (b) reflects the existence of a negative feedback loop involving at least one additional species (d > k) and thus (indirectly) also a different time scale. The above condition therefore just represents a precise formulation of the well-known argument that oscillations can be due to the interplay of a fast autocatalysis and a negative feedback loop occurring on a slower time scale. Condition 2.1 is applicable as long as the mechanism becomes unstable via an autocatalysis. A more general condition for the occurrence of a Hopf bifurcation can be obtained from the Routh-Hurwitz criterion (cf. ref 13). The characteristic equation of matrix A is of the form
f(λ) ) λn-d(a0λd + a1λd-1 + ... + ad)
(7)
with a0 ) 1 and ai ) ∑Ri. A modified Routh scheme can be defined as
D1,0 D2,0 D3,0 ...
D1,1 D2,1 D3,1
D1,2 ... D2,2 ... D3,2 ... (8)
with D1,k ) a2k, D2,k ) a2k+1 and Dm+1,k ) Dm,0Dm-1,k+1 Dm-1,0Dm,k+1 (k ) 0, 1, ...). 2.2. Hopf Bifurcation: Necessary and Sufficient Condition. f(λ) has two roots (iω and (d - 2) roots with negative real part, if (a) ad > 0 and (b) Dk,0 > 0 for k ) 1, ..., d - 1, and (c) Dk,0 ) 0 for k ) d, d + 1. One also has to verify that the highest two Dk,0 change sign upon variation of a parameter (which is actually no problem in a complete parameter set). It suffices to check Dd,0, since Dd+1,0 has the same sign. Condition 2.2 is actually needed only for rare cases of nonautocatalytic oscillators,17 not treated here. On the other hand, it is also useful to make sure that no Hopf bifurcation can occur in a certain network. Oscillations can arise not only from a Hopf but also from other bifurcations. However, we are unaware of any oscillatory chemical model that does not exhibit a Hopf bifurcation in the complete parameter set. A mechanism is therefore referred to as oscillatory if it exhibits a Hopf bifurcation. An example of the application of the above criteria is treated in Appendix 1. 3. Abstract Models We first consider two minimal models of chemical oscillators, namely the Sel’kov model18 and the Brusselator19 (Figure 1). Both contain a strong (i.e., unstable) autocatalysis 2X + Z f 3X, such that β1(X) < 0. Z causes a negative feedback loop (i.e., X consumes Z which in turn produces X). The only difference is that in the Sel’kov model Z flows in, while in the Brusselator it is produced internally from X. Now consider the Brusselator without the in- and outflow of X (Figure 1c). The remaining network (Figure 1b) still contains an autocatalysis (in X) and a species consumed by it (Z), but since there are no in- nor outflows it exhibits a conservation constraint (X + Z ) constant). Therefore β2(X,Z) must vanish and condition 2.1(b) cannot be fulfilled. The effect of the flow in X is obviously to lift the constraint allowing for the additional degree of freedom required for oscillations. The situation is analogous in the standard LangmuirHinshelwood mechanism (eqs 1-3 and Figure 2). The species with the higher site requirements (called X) gives rise to an instability (β2(*,X) < 0), but β3 is zero because of the constraint
19120 J. Phys. Chem., Vol. 100, No. 49, 1996
Figure 2. Network diagram DN and extreme currents Ei of a Langmuir-Hinshelwood mechanism. * denotes a vacant site; X and Z are adsorbed species. The gaseous species (not shown) are nonessential. Note that it is also assumed that X requires two adsorption sites and that its adsorption kinetics is second order in the empty sites.
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Figure 4. Variations of a LH mechanism with the assumption that the initially adsorbed species can transform into another one before reacting. In (a) the primary species Z1 has to be less reactive than Z2, in (b) X1 has to be more reactive than X2 in order to allow for oscillatory behavior.
Figure 5. Vacancy model assuming that the surface reaction requires additional sites. In the example shown the reaction is formally X + Z + 2* f 4*.
Figure 3. Simple extensions of a LH mechanism. The reversible formation of an additional surface species A from * or X replaces the original constraint * + X + Z ) constant by * + X + Z + A ) constant, introducing the extra degree of freedom needed for oscillations.
X + Z + * ) constant (which simply reflects a constant area of the catalyst). In other words, while the LH mechanism contains in principle the mechanistic features required for oscillations (a strong autocatalysis and a species consumed by it), these cannot manifest themselves due to the constraint. The corresponding network can be decomposed into three extreme currents (Figure 2). E1 is unstable and gives rise to a saddle point, while E2 and E3 are stable and dominate the stable nodes, i.e., the mechanism exhibits bistability. Linear combinations of the unstable extreme current with one of the stable ones will at a certain point give rise to a zero eigenvalue (i.e., an sn bifurcation). Of course a flow of a surface species (which would eliminate the constraint) is not possible. However, another, fourth surface species may be involved in the reaction. The constraint would then manifest itself in a vanishing β4, allowing the β3’s to be different from zero. Such possibilities are described below. It will mostly be assumed that one of the reactants has higher site requirements than the other. For the examples shown the adsorption of X will be assumed to be of second order in the vacant sites, and that of Z of first order. Note that the results are robust with respect to variations of the reaction order (as long as the kinetic exponents in the adsorption terms do not become equal). 3.1. Inert Species. The simplest extension (formally analogous to a flow) of the basic LH mechanism is the assumption that one of the species reversibly forms another inert one (Figure 3). The mechanism in Figure 3a can actually be regarded as a system with a third adsorbing and desorbing gas A which is otherwise inert. Such a model was first suggested by Eigenberger almost 20 years ago.20 In Figure 3b it is assumed that the adsorbed species X can reversibly form another nonreactive species A. The third possibility (i.e., an equilibrium between Z and another species) does not give rise to oscillations, as can be checked using the conditions above.
3.2. Secondary Adsorbed Species. It is assumed that the primarily adsorbed species (Z1 respectively X1) can transform into another one (Z2 respectively X2) with different reactivity (Figure 4). Condition 2.1 predicts a Hopf bifurcation, provided that for the system of Figure 4a the secondary species Z2 is more reactive than the primary one Z1 (i.e., k2 > k1), whereas for Figure 4b the primary species X1 must react faster than X2 (k1 > k2). Physically the different species may correspond to a chemical change of the adsorbate itself or to its occupying different adsorption sites on the substrate, but also a transformation of the substrate could be responsible for a change in the reactivity of adsorbed particles (see section 4). 3.3. Vacancy Requirement of the Reaction. If the surface reaction can take place only when additional vacant sites are available, a mechanistic scheme as in Figure 5 is obtained (as suggested in refs 21 and 22). In such a case no fourth species is needed, provided the site requirements of the reaction are high enough to give rise to β1(*) < 0. This is the case if the order of the reaction in the free site is at least as high as the higher-order adsorption kinetics. In the case shown (corresponding to the model in ref 21) the adsorption kinetics are both first order, in which case the kinetic exponent of the reaction must be higher than 1 in order to lead to oscillations. 3.4. Delayed Product Desorption. Up to now it has been assumed that the reaction product P desorbs immediately. Actually a finite (but small) lifetime of P on the surface does not qualitatively change the dynamical behavior of the LH mechanism. The situation becomes different, however, if P can transform into another more tightly bound species Q which may form P again (Figure 6a) or desorb directly (Figure 6b). Both mechanisms are oscillatory since β3(*,X,P) < 0 while β4(*,X,P,Z) > 0. Note that the dynamics does not change qualitatively if a part of the vacant sites is already formed in the reaction to P (Figure 6c), which one would expect unless P is a very bulky molecule. 4. Realistic Mechanisms A number of binary surface reactions have been studied at low pressure under isothermal conditions and have been shown to exhibit bistability and oscillations.3,7 Here we present some examples in order to illustrate how the above abstract mechanisms can manifest themselves in real systems.
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J. Phys. Chem., Vol. 100, No. 49, 1996 19121
Figure 6. LH mechanism with delayed product desorption. The reaction product P can either desorb directly or form another species Q which desorbs directly (b) or via re-formation of P (a, c). All three variations are oscillatory.
The most extensively studied system is the CO oxidation on platinum group metals:
O2 + 2* a 2Oad
(9a,b)
CO + * a COad
(10a,b)
COad + Oad f 2* + CO2
(11)
The well-known asymmetric inhibition of adsorption (i.e., preadsorbed CO blocks oxygen adsorption but not vice versa) can be taken into account by an additional reaction step:
Oad + CO f [Oad + COad] f * + CO2
(12)
As before, the gaseous species are nonessential. Moreover the short coexistence of Oad and COad in process 12 can be neglected, such that it can be treated as a pseudoreaction Oad f *. Since this reactive removal of Oad is much faster than oxygen desorption for the conditions in question, the latter can be neglected. The resulting scheme is obviously bistable, since the adsorption site requirements for CO and O2 are different due to the dissociative adsorption of the latter. On certain surfaces oxygen can go below the surface of the catalyst (subsurface oxygen), resulting in a mechanism as depicted in Figure 7a, which is oscillatory (cf. Figure 3b) and actually applies for the CO oxidation on Pd single-crystal surfaces.23,24 Experimentally oscillations were observed only for conditions where the formation of subsurface oxygen takes place. Although oscillations were predicted for coadsorption of an inert gas long ago,20 no corresponding experiments have been carried out yet. A different mechanism operates for the CO oxidation on Pt surfaces. While Pt(111) exhibits only bistability, Pt(100) and Pt(110) show oscillations in addition. The decisive difference is that the densely packed (111) oriented surface does not undergo any change, whereas the clean (100) surface is reconstructed to a quasihexagonal arrangement (“hex”25). This hex phase hardly adsorbs oxygen and is therefore practically
Figure 7. Real examples of oscillating LH mechanisms as determined in experimental studies: (a) CO oxidation with subsurface oxygen formation, found on Pd single crystal surfaces; (b) reconstruction model of oscillatory CO oxidation, Pt(100) and Pt(110); (c) oxide model of CO oxidation, supported Pt/SiO2 catalyst; (d) vacancy model for the CO/NO reaction, Pt(100) (under conditions where the phase transition is not involved); (e) H2/O2 reaction on a Pt field-emitter tip; H2Of represents field-adsorbed water, i.e., a product that does not desorb immediately.
unreactive. However CO does adsorb and lifts the reconstruction, such that the 1 × 1 phase forms which has a higher sticking coefficient for oxygen and the reaction can take place. The corresponding reaction scheme is shown in Figure 7b. It corresponds to the type shown in Figure 4a, since CO-1 × 1 is (in part) formed via the unreactive species hex and CO-hex. Basically, the same mechanism has been shown to be responsible for oscillations on Pt(110), though the difference in reactivity of the two surface structures (1 × 2 missing row and 1 × 1) is much smaller.10 A model in which adsorbed oxygen forms another less reactive species (namely an oxide Ox) has been suggested by Sales et al.;26 see Figure 7c. The oxide formed is only slowly reduced by CO. Experimental support for this mechanism was found on a supported Pt/SiO2 catalyst using the chemical shift of O in the X-ray photoemission spectra.27 The experiments were carried out at near-atmospheric pressure and were not strictly isothermal. No evidence for oxide has been found at low pressure (where platinum oxides are thermodynamically not stable and would certainly not be expected to form). Figure 7d represents a simple model of the CO/NO reaction on Pt. Both molecules have virtually the same adsorption requirements, but NO has to decompose into O and N (which
19122 J. Phys. Chem., Vol. 100, No. 49, 1996 desorbs right away) before the reaction takes place.28,29 Since the NO dissociation requires empty sites, the model is of the vacancy type (section 3.3). The model predicts oscillations only if the dissociation kinetics is of order higher than 1 in the vacant sites. Experimentally it was found that NO dissociation is very slow if there are few vacant sites and rises steeply as the number of sites increases,30 as is characteristic of higher-order kinetics. The H2/O2 reaction on Pt single-crystal surfaces exhibits bistability, but no oscillations have been observed.31 In contrast, sustained oscillations have been obtained in field-ionization studies using a Pt emitter tip.31-33 Under these conditions the water formed in the surface reaction need not desorb directly but can also form a field-adsorbed state (denoted H2Of) with a longer lifetime. The mechanism shown in Figure 7e is simplified in order to visualize its crucial features (cf. ref 32). Note that it is assumed that O2 has higher site requirements than H2. The intermediate species OH has been omitted. Fieldadsorbed H2Of desorbs via H3O+ formation with Had or in a second order process (branching in the latter is neglected here).34 Although not all details of the mechanism have been clarified yet, this reaction represents a likely candidate for an oscillatory mechanism with delayed product desorption. In particular the fact that oscillations cease in the absence of field effects suggests that the formation of field-adsorbed water is a crucial step in the oscillatory cycle. 5. Discussion We have applied matrix stability theory to stoichiometric networks in order to explicitly formulate conditions for the occurrence of instability and oscillations in chemical mechanisms. For not too large networks these criteria allow a reasonably quick analytical check whether a given network exhibits a Hopf bifurcation (see Appendix 1). Though the procedure was applied to a specific class of systems (oscillating LH mechanisms), it is perfectly general and may be useful for other types of chemical reactions. Stoichiometric network analysis has been used before in order to classify chemical oscillators according to mechanistic features.16,35 The prototype models constructed in this context did not involve any special assumptions such as conservation constraints. In contrast, surface reactions always exhibit a constraint, namely, that the sum over all surface species (corresponding to the sum over all essential species) remains constant. Thus the considerations from refs 16 and 35 do not apply directly. In the nomenclature of ref 35 the basic LH mechanism contains all ingredients for an oscillator of type 2C, but the constraint makes things slightly more complicated. The classical Langmuir-Hinshelwood mechanism is always autocatalytic in the vacant sites. While this generically leads to bistability,12 the constraint prevents the occurrence of oscillations. However, it was shown that the introduction of a single additional species is sufficient to allow oscillatory behavior. The additional species can be connected to the LH skeleton in different ways; as shown in section 3 almost all simple possibilities lead to oscillations. In other words it suffices to somewhat “loosen” the constraint (by extending it over more species). Thus it is not surprising that oscillations have been detected in quite a few binary surface reactions under isothermal experimental conditions.7 For ternary systems (involving three gases) oscillations become generic; more precisely, the corresponding mechanism is oscillatory provided one of the reacting species has higher site requirements than the others (the simplest such case corresponds to Figure 3a). A system exhibiting bistability and oscillations in adjacent parameter regions is generally also excitable in the vicinity.
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Figure 8. Current diagram DC and current matrix diagram DCM of a current of the network shown in Figure 3a used to prove the existence of a Hopf bifurcation. The principal minors of S, called βi, can be obtained by summation over the respective feedback loops in the DCM.13
Consequently the models described form the mechanistic basis for bistability, oscillations, and excitability in surface reactions, and therefore for the large variety of chemical wave phenomena (such as fronts, spirals, and turbulence) observed on catalytically active surfaces at low pressure.6-8 For higher pressures adsorption becomes comparable to or even faster than surface diffusion, and the mean-field limit as well as the simple power law kinetics are not applicable any more.36 Moreover the catalyst can no longer be kept strictly isothermal so that thermokinetic effects have to be taken into account. The experimental investigation and appropriate modeling of nonlinear phenomena in catalysis at higher pressures are a field of active current research.37-39 Acknowledgment. The authors are indebted to S. Kro¨mker for helpful discussions. Appendix 1 The application of the conditions for the occurrence of a Hopf bifurcation is demonstrated by way of an example. Consider the subsurface oxygen model of Figure 7a. The network can be decomposed into four extreme currents; three are analogous to those in Figure 2, the fourth is just the reversible reaction O S Osub. We choose the current shown as DC in Figure 8, where the fourth extreme current carries a weight j. The principal minors βi are most conveniently obtained diagrammatically from the so-called current-matrix diagram DCM (Figure 8), as described in refs 13 and 15. The only negative minor not of maximum order is β2(*,O). According to condition 2.1 we compute the sum over all R2 and R3 which depend on *:
(-8 + 10j)hO + 12hCO + 10jhOsub < 0
(a)
12hOhCO - 8hOhOsub + 12hCOhOsub > 0
(b)
(Note that not all βi are needed.) Both conditions can readily
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J. Phys. Chem., Vol. 100, No. 49, 1996 19123
be fulfilled provided hO is sufficiently large, j is small, and hOsub is not too large. Therefore oscillations are expected as long as * and O are fast variables and the formation of Osub is slow. For illustration we also show the application of condition 2.2 (though the above consideration is sufficient). Scheme 8 becomes
1
∑R1 ∑R1∑R2 - ∑R3 (∑R1∑R2 - ∑R3)∑R3
∑R2 ∑R3
0 0
0 0
The last two entries in the first column change sign simultaneously if ∑R1∑R2 - ∑R3 changes sign. ∑R3 must be positive but cannot lead to a sign change of this expression because all its terms are compensated by terms stemming from ∑R1∑R2 (see the list of βi in Figure 8). ∑R1 is always positive, and thus ∑R1∑R2 - ∑R3 will definitely have changed sign as soon as ΣR2 does, which is the case when hO and h* become large. References and Notes (1) Razon, L. F.; Schmitz, R. A. Catal. ReV. Sci. Eng. 1986, 28, 89. (2) Ertl, G. AdV. Catal. 1990, 37, 214. (3) Ertl, G. Science 1991, 254, 1756. (4) Imbihl, R. Prog. Surf. Sci. 1993, 44, 185. (5) Eiswirth, M. In Chaos in Chemistry and Biochemistry; Field, R. J., Gyo¨rgyi, L., Eds.; World Scientific: Singapore, 1993. (6) Eiswirth, M.; Ertl, G. In Chemical WaVes and Patterns; Kapral, R., Showalter, K., Eds.; Kluwer: Dordrecht, 1994. (7) Imbihl, R.; Ertl, G. Chem. ReV. 1995, 95, 697. (8) Eiswirth, M.; Ba¨r, M.; Rotermund, H. H. Physica D 1995, 84, 40. (9) Rettner, C. T.; Auerbach, D. J. Science 1994, 263, 365. (10) Krischer, K.; Eiswirth, M.; Ertl, G. J. Chem. Phys. 1992, 96, 9161.
(11) Ba¨r, M.; Zu¨licke, C.; Eiswirth, M.; Ertl, G. J. Chem. Phys. 1992, 96, 8595. (12) Feinberg, M.; Terman, D. Arch. Rational Mech. Anal. 1991, 116, 35. (13) Clarke, B. L. AdV. Chem. Phys. 1980, 43, 1; Cell Biophys. 1988, 12, 237. (14) Clarke, B. L. J. Chem. Phys. 1981, 75, 4970. (15) Eiswirth, M. Suri Kagaku 1994, 372, 59. (16) Eiswirth, M.; Freund, A.; Ross, J. AdV. Chem. Phys. 1991, 80, 127; J. Phys. Chem. 1991, 95, 1294. (17) Tyson, J. J. J. Chem. Phys. 1975, 62, 1010. (18) Sel’kov, E. E. Eur. J. Biochem. 1968, 4, 79. (19) Tyson, J. J. J. Chem. Phys. 1972, 58, 3919. (20) Eigenberger, G. Chem Eng. Sci. 1978, 33, 1263. (21) Takoudis, C. G.; Schmidt, L. D.; Aris, R. Surf. Sci. 1981, 105, 325. (22) McKarnin, M. A.; Aris, R.; Schmidt, L. D. Proc. R. Soc. London A 1988, 415, 363. (23) Ladas, S.; Imbihl, R.; Ertl, G. Surf. Sci. 1989, 219, 88. (24) Bassett, M. R.; Imbihl, R. J. Chem. Phys. 1990, 93, 811. (25) Heilmann, P.; Heinz, K.; Mu¨ller, K. Surf. Sci. 1979, 83, 4867. (26) Sales, B. C.; Turner, J. E.; Maple, M. B. Surf. Sci. 1982, 114, 381. (27) Hartmann, N.; Imbihl, R.; Vogel, W. Catal. Lett. 1994, 28, 373. (28) Schwartz, S. B.; Schmidt, L. D. Surf. Sci. 1988, 206, 169. (29) Fink, T.; Dath, J.-P.; Imbihl, R.; Ertl, G. J. Chem. Phys. 1991, 95, 2109. (30) Imbihl, R.; Fink, T.; Krischer, K. J. Chem. Phys. 1992, 96, 6236. (31) Gorodetskii, V.; Lauterbach, J.; Rotermund, H. H.; Block, J. H.; Ertl, G. Nature 1994, 370, 277. (32) Ernst, N.; Bozdech, G.; Gorodetskii, V.; Kreuzer, H. J.; Wang, R. L. C.; Block, J. H. Surf. Sci. 1994, 318, L1211. (33) Gorodetskii, V.; Block, J. H.; Drachsel, W. Appl. Surf. Sci. 1994, 76/77, 129. (34) Sieben, B.; Bozdech, G.; Ernst, N.; Block, J. H. Surf. Sci., in press. (35) Schreiber, I.; Ross, J., preprint. (36) Zel’dovich, Ya. B.; Mikhailov, A. S. SoV. Phys. Usp 1987, 30, 977. (37) Rotermund, H. H.; Haas, G.; Franz, R. U.; Tromp, R. M.; Ertl, G. Science 1995, 270, 608. (38) Mikhailov, A.; Ertl, G. Chem. Phys. Lett. 1995, 238, 104. (39) Hildebrand, M.; Mikhailov, A. S., preprint.
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