Modeling reaction-diffusion pattern formation in the Couette flow reactor

Transcription

1 Modeling reaction-diffusion pattern formation in the Couette flow reactor J. Elezgaraya) Center for Appied Mathematics, Cornell University, thaca, New York A. Arneodo ) Centerfor Nonlinear Dynamics and Department of Physics, The University of Texas, Austin, Texas (Received 3 December 199; accepted 26 February 1991) We report on a numerical and theoretical study of spatio-temporal pattern forming phenomena in a one-dimensional reaction-diffusion system with equal diffusion coefficients. When imposing a concentration gradient through the system, this model mimics the sustained stationary and periodically oscillating front structures observed in a recent experiment conducted in the Couette flow reactor. Conditions are also found under which oscillations of the nontrivial spatial patterns become chaotic. Singular perturbation techniques are used to study the existence and the linear stability of single-front and multi-front patterns. A nonlinear analysis of bifurcating patterns is carried out using a center manifold/normal form approach. The theoretical predictions of the normal form calculations are found in quantitative agreement with direct simulations of the Hopf bifurcation from steady to oscillating front patterns. The remarkable feature of these sustained spatio-temporal phenomena is the fact that they organize due to the interaction of the diffusion process with a chemical reaction which itself would proceed in a stationary manner if diffusion was negligible. This study clearly demonstrates that complex spatio-temporal patterns do not necessarily result from the coupling of oscillators or nonlinear transport.. NTRODUCTON n recent years there has been increasing interest in pattern forming phenomena in chemical systems. - n the early eighties a great deal of attention has been paid to the dynamics of homogeneous chemical reactions. n well-mixed media, the intrinsic nonlinear nature of chemical kinetics has provided a field of experimentation for the study of lowdimensional dynamical systems.5 When maintained far from thermodynamic equilibrium in a continuously stirred tank reactor (CSTR), chemical reactions have been shown to exhibit a transition from coherent (periodic) temporal patterns to chemical chaos. 3 Among these chemical oscillators, the Belousov-Zhabotinskii (BZ) reaction has revealed most of the well-known scenarios to chaos including period doubling, intermittency, frequency locking, collapse of tori and crisis phenomena. 5*16 n contrast, for years there has been only a little progress in the experimental research on sustained spatial and spatio-temporal chemical structures where, in addition, a diffusive transport process competes with the local chemical kinetics. Most experiments have been performed in closed systems where the system uncontrollably and irreversibly relaxed to thermodynamic equilibrium. Therefore the applicability of these experiments were limited to the study of transient patterns developing in a rather short time, in practice those resulting from excitability phenomena such as the so-called target patterns and spiral waves.5-7~1* ~ 7-23 n the past two years, however, there has been a rebirth of interest in the formation of dissipative structures in chemically reacting and diffusing systems. This interest has been mainly sparked by the development of open spatial reactors by groups in Texas2 3 and in Bordeaux.3-38 Basical- *) Permanent address: Centre de Recherche Paul Pascal, Universite de Bordeaux, Avenue Schweitzer, 336 Pessac, France. ly, two types of open reactors are currently operating: (i) the two-dimensional continuously fed unstirred reactors where the transport process is essentially natural molecular diffusion and where the feeding is either uniform (continuously fed unstirred reactorz6. ) or from the lateral boundaries ( linear3s-37 or annular24v25v38 gel reactors); and (ii) the Couette flow reactor29-34 which provides a practical implementation of an effectively one-dimensional reaction-diffusion system with well-defined boundary conditions and controllable diffusion process. These new available pieces of apparatus have already produced a wealth of genuine sustained chemical dissipative structures. The aim of the present study is to provide theoretical and numerical support for the recent observations of one-dimensional spatio-temporal patterns in the Couette flow reactor with variants of the chlorite-iodide reaction. 39 n this introduction, we briefly describe the corresponding experimental system with special emphasis on the characteristic properties of the observed patterns which require a specific mathematical description. n the conclusion, we will discuss the possibility of generalizing our theoretical approach to two-dimensional reactiondiffusion systems that model self-organization phenomena observed in the gel reactors. For technical details concerning this new generation of open spatial reactors, we refer the reader to the original publications2 39 and to the review article by Boissonade.4 Nonlinear reaction-diffusion equation models have been widely used to account for pattern forming phenomena in chemical systems maintained far from equilibrium. - 2* 8* 9 From a theoretical point of view, one may distinguish two types of reaction-diffusion structures: (i) global structures resulting from intrinsic symmetry-breaking instabili- ties, e.g., the Turing structures -3v4 742 and the phase-wave structures;4 and (ii) localized structures associated with fronts, i.e., steep spatial changes of concentration which ac- Downloaded J.Chem.Phys.Q5(1),1 9 Dec 23 to July1991 Redistribution /91/ $3. subject to AP license or copyright, 1991 see American nstitute of Physics 323

2 324 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation tually correspond to transitions between two chemical states (e.g., a reduced and a oxidized state) with fast kinetics, e.g., the traveling waves in excitable media. -7~18~ 9*4349 Several hindrances have delayed experimental researchs of global dissipative structures. According to theoretical works on model systems, Turing instability4 *42 from a homogeneous state to steady cellular patterns requires the diffusion coefficients of at least two different species to be significantly different. V3*5s5s These conditions are met in systems that are governed by the competition between an activator and an inhibitor, when the inhibitor diffuses faster than the activator. This is a common situation in biological systems,3 where many processes are activated by enzymes immobilized in a matrix. This is a generally unrealistic situation for chemical reactions, such as the much-studied BZ reaction,14 involving small size molecules in aqueous solutions with comparable diffusion coefficients D- 1-5 cm* s-. As pointed out by Pearson and Horsthemke,59 a way to overcome this difficulty consists in positioning the chemical system in the vicinity of a Takens-Bogdanov point6*6 where Turing patterns can still organize with nearly equal diffusion coefficients. 58 Unfortunately the research of multicritical points of oscillating chemical reactions turns out to be a very hard experimental task. Diffusive instabilities have also been proposed to account for propagating patterns in homogeneously oscillating systems.4 Near the oscillatory onset, the Ginzburg-Landa@ amplitude equation (Hopf normal form for extended systems) can be reduced to the Kuramoto-Sivashinsky equation,63*64 which describes slow spatial and temporal variations of the phase of the oscillators. This equation has been the center of increasing interest during the past few years.65 Besides regular cellular and propagating solutions, this equation was shown to exhibit chaotic solutions,65-67 a form of weak turbulence usually called phase turbulence. 4*63 But the Hopf bifurcations identified thus far in chemical systems are for the most part subcritical and lead to large amplitude relaxation oscillations5 Relatively few supercritical Hopf bifurcations have been observed experimentally.68 Therefore, wave patterns observed in chemical systems are not the paradigm for Kuramoto propagative structures as generally believed. Traveling waves in excitable media are the best known example of such propagative patterns, *49 they actually belong to the class of localized structures described just below. Localized front structures consist in spatial sequences of abrupt concentration jumps corresponding to rapid switches between steady or quasisteady states. They can originate in initially homogeneous media from a local perturbation giving rise to the well documented propagating waves in excit- able media. 5-7,1*11,17-23,4349 Stationary front patterns have been theoretically predicted69-73 but such patterns require the diffusion coefficients of different species to be controlled selectively. Actually they have been mainly observed in the presence of spatial concentration non uniformities. p7 77 Localized heterogeneous reacting sites were shown to induce local chemical structures. 78V79 A concentration gradient externally imposed from the boundaries can be used to sustain reaction-diffusion fronts in homogeneous systems.8c-8 Some of the recently developed open reactors were designed with feeding coming from the boundaries of the system.4 Originally, the basic idea was to localize all the significantly dynamical phenomena inside a narrow stationary reaction front in order to locate the region of strong chemical activity away from the boundaries where the perturbations associated to the feed may disturb the dynamics. Since instabilities can only develop inside the active region, these open reactors provide a very promising experimental support to the study of front pattern formation phenomena in reaction-diffusion systems. The results of preliminary experiments in the gel reactors 24*25*35-38 and the Couette flow reactor have confirmed the capability of these apparatus to produce and control sustained chemical front patterns. Because it mimics a one-dimensional reaction-diffusion system with externally adjustable concentration gradient and controllable diffusion rate, the Couette flow reactor is very likely to play a privileged role in the experimental approach of spatiotemporal phenomena in nonequilibrium systems. The Couette flow reactor29-34v39 consists of two CSTRs connected by a Couette-Taylor flow, with the inner cylinder rotating and the outer cylinder at rest. Chemicals injected in the CSTRs diffuse and react in the annular region between the two cylinders. At large Reynolds numbers, the Taylor vortices are turbulent enough for the fluid to be well mixed both in radial and azimuthal directions. Under these conditions, the mass transport along the cylinder s axis was shown to be diffusive over length scales larger than the vortex scale.86 Consequently, the Couette flow reactor can be modeled as a one-dimensional array of homogeneous cells, coupled by a diffusion process with a unique diffusion coefficient D for all chemical species. This diffusion coefficient is a tunable parameter which depends mainly on the rotation rate of the inner cylinder. The accessible D values range from 1 - * to 1 cm* s -, i.e., several orders of magnitude larger than molecular diffusion coefficients. This rather wide range of diffusion control makes possible a continuous variation of the structure length scale for a fixed geometry, changing progressively from small to extended system behavior. Let us remark that the number of pairs of vortices is rather low ( -5O-11, so that this system is neither a continuous sys- tem nor a low dimensional system. The role of the two CSTRs is to maintain nonequilibrium boundary conditions, e.g., by imposing a concentration gradient to the system (Bordeaux reactor) The Couette reactor can be fed as well by direct reactant flow (Texas reactor).29v3 Two different reactions have presently been studied in the Couette flow reactor, namely, the variants of the Belou- sov-zhabotinskii29-32v34 and chlorite-iodide3-34v39 reactions. The BZ reaction has revealed a rich variety of steady, periodic, quasiperiodic, frequency-locked, period-doubled, and chaotic spatio-temporal pattems,29*3 well described in terms of the diffusive coupling of oscillating reactor cells, the frequency of which changes continuously along the Couette reactor as the result of the imposed spatial gradient of constraints. This experimental observation has been successfully simulated with a schematic model of the BZ kineticss7 and the recorded bifurcation sequences of patterns resemble that obtained when coupling two nonlinear oscillators. Much more in the spirit of our prospective theoretical J. Chem. Phys., Vol. 95, No. 1,1 July 1991

3 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation 325 study, the chlorite-iodide reaction provides a remarkable illustration that spatiotemporal patterns can organize in a chemical system from the diffusive coupling (with equal diffusion coefficients for the different chemical species) of (nonoscillating) steady states reactor cells A detailed report of the patterns, state diagrams and details on the chemistry, feed compositions and experimental procedures can be found in Ref 39. The chlorite-iodide reaction88*89 is a bistable reaction, i.e., the reacting medium can be in either a reduced or an oxidized state. Taking advantage of this bistability, stationary nonhomogeneous spatial patterns can be obtained when feeding unsymmetrically at the two ends of the Couette flow reactor, e.g., the left-end (respectively, right-end) CSTR in a reduced (respectively, oxidized) state. Under these conditions, a stationary single-front pattern appears in the Couette reactor: Two rather homogeneous regions corresponding to the reduced and oxidized states, respectively, are spatially separated by a sharp transition front. This somewhat trivial pattern is due to the existence of a switching process in the kinetics of the reaction, which induces a spatial transition between these two states. When varying the chemical input concentrations or the transport rate D as control parameters, bifurcation sequences of patterns have been observed.39 For example, when tuning the chlorite concentration in the feed flow of the right-end CSTR, the stationary single-front pattern bifurcates to a time-dependent state where the position of the front oscillates periodically over a finite spatial region in the Couette reactor. When further increasing the chlorite concentration, this oscillating front sweeps a larger and larger domain in the Couette reactor until a new (oscillating) band of oxidized state comes off the original oxidized region resulting in a periodic alternation of a single-front and a three-front pattern. Ultimately, this oscillating pattern stabilizes to produce a multi-front stationary pattern with three spatial switchings from the reduced to the oxidized state. Historically, these multi-peaked spatial concentration profiles were the first experimental evidence for genuine sustained stationary chemical patterns in an isothermal and homogeneous continuous reaction-diffusion system3 (without external field). Since then, the Bordeaux group has reported the observation of a symmetry-breaking instability leading to a stationary Turing structure in the linear gel reactor35-37 as commented in the conclusion. A very rich variety of spatiotemporal patterns has been observed in the Couette flow reactor under either nonsymmetric or symmetric feeding conditions.39 An example of patterns obtained with symmetric feeding are the bursting patterns, where a burst of oxidized state appears periodically in a reduced region imposed from the boundaries. Despite the observation of rather irregular displacements of front patterns, no definite experimental evidence for chaotic spatio-temporal behavior has been obtained thus far. Our goal is to demonstrate that the experimental spatiotemporal patterns observed in the Couette flow reactor can be described by a reaction-diffusion process and to show that the observations are characteristic of a wide class of systems. More generally, we wish to identify the main ingredients required for pattern formation and to develop a theo- retical analysis of the bifurcations that produced those dissipative front structures. n Sec. we define our reactiondiffusion system model; this model is a two-variable Van der Pol-like system with equal diffusion coefficients. n Sec. we report numerical simulations of this reaction-diffusion model under concentration gradient imposed by either Dirichlet or CSTR boundary conditions. A comparative study of the numerical and experimental spatio-temporal patterns is carried out for both asymmetric and symmetric feedings. Section V is devoted to the theoretical study of the existence and stability of single and multi-front patterns. Our approach of these localized structures is essentially based on singular perturbation techniques. 9*71.72 Exact analytical results are derived when considering a piecewise linear slow manifold. n Sec. V we perform a nonlinear analysis of bifurcating patterns using center manifold/normal form techniques. 6*6 Special attention is paid to the Hopf bifurcation from steady to oscillating front patterns. We compare the theoretical predictions of the normal form calculations with the results of direct simulations. We conclude in Sec. V with a discussion of the possible generalization of this theoretical study to sustained front patterns recently observed in annular and linear gel reactors.. A REACTON-DFFUSON SYSTEM MODEL WTH EQUAL DFFUSON COEFFCENTS n most theoretical studies of chemical systems, pattern forming phenomena as well as the concept of chemical turbulence have been addressed in terms of the linear coupling of spatially distributed nonlinear oscillators.4 According to this prerequisite, the analysis of the partial differential equations which model realistic reaction-diffusion systems has been commonly simplified to the investigation of coupled nonlinear oscillators,4.9s94 coupled nonlinear maps95,96 and cellular automata.97 The aim of the present study is to emphasize that nontrivial regular and irregular spatio-temporal regimes can be attained when coupling (nonoscillatory) steady state reactors with equal diffusion coefficients for the different chemical species, provided a spatial concentration gradient, e.g., a nonhomogeneous feed, is imposed to the system As pointed out in the ntroduction, the spatially extended open Couette flow reactor29-34,39 provides a practical implementation of an effectively one-dimensional reactiondiffusion system with an external concentration gradient imposed from the boundaries. With the specific motivation to provide theoretical and numerical support for the recent experimental observations of sustained dissipative structures in the Couette flow reactor, we will consider the standard reaction-diffusion equation, d,c = R(C) + DAC, (1) where C is a concentration vector, D the diffusion matrix, and R( C!) models the reaction process. A faithful modeling of the experimental situation would consist in considering a reaction-diffusion system which meets the experimental conditions and the specific requirements of the chemical kinetics laws of the chlorite-iodide reaction. Here we will adopt a strategy which is much more in the spirit of the Downloaded 9 Dec 23 to J. Redistribution Chem. Phys., Vol. subject 95, No. to AP 1,l July license 1991 or copyright, see

4 326 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation normal form approach :6*61 We will simply consider a formal reaction-diffusion model that will be shown to retain the minimal ingredients necessary to reproduce most of the phenomena associated with the observed front patterns. A. Reaction process For most of the oscillating reactions, the knowledge of the reaction mechanism and rate constants is generally very sketchy. Since the details of a particular kinetic model are not relevant close to bifurcation conditions, we will consider the simplest ordinary differential equation model which accounts for the characteristic features of the chlorite-iodide reaction,88.89 namely, bistability, excitability, and relaxation oscillations. Our model of the reaction term is a two-variable Van der Pol-like system98 99 [C = ( U,U) 1, -=g du dt - (v -f(u) 1, -= dv ---+, (2) dt where E is a small positive parameter and a a free parameter the role of which will be explained shortly. These equations ensure the existence of a pleated slow manifold v =f( u), on which all trajectories are attracted in a time - ( 6). The S shape of this manifold accounts for the excitable character of the dynamics. The only steady state of the reaction term [u, = a, v, =f( a) ] is necessarily located on the slow manifold; an elementary linear analysis shows that this steady state is stable for a < al (lower branch), or a > au (upper branch) while it is unstable for CYLE [ all,a,] as sketched in Fig. 1. The critical values au and al correspond to Hopf bifurcations leading to oscillatory behavior. According to the specific shapef( U) of the slow manifold, this bifurcation can be either supercritical or subcritical.99 When adding a flux term to this Van der Pol-like equation, bistability can also be recovered. Despite the fact that model (2) does not have all the properties required in a chemical scheme, u and u FG. 1. Sketch of the slow manifold u =f( U) = u2 - u + us. The unique steadystateofeq.(2):u=cx,u=f(a),isafocus(f)fora; <Ada,+ or a,- < cr < ao+, and a node (N) elsewhere. A solide line indicates a stable steady state (a<~, or cz> a,); a dashed line an unstable steady state (a,<a<a,). play the role of concentration variables and we will refer to the upper and lower branches of the slow manifold as the analogues of the reduced and oxidized state branches of the chlorite-iodide reaction, respectively. Let us remark that a change of variables of the form ii = u + udr t, = v + vd can be used to ensure these concentration variables to be positive, at the expense of a slight modification of the exact form OfEq. (2). B. Diffusion process When taking into account the diffusive transport process, the reaction-diffusion model reads -= au at!?!!=dd-u++ -, (3) at ax where XE[, 1 ] is the single space variable; the spatial length of the Couette reactor is resealed to unity for convenience. The cross diffusion terms between the two species u and v are neglected and the diffusion coefficients D, = D, = D are set equal in order to mimic the turbulent mass transport that drives pattern formation in the Couette flow reactor. C. Boundary conditions 1. Dirichet boundary conditions n most experimental runs, the volume and feeding flows of the two CSTRs at both ends of the Couette reactor were large enough for their internal state not to be significantly influenced by the dynamics inside the Couette reactor.39 This corresponds mathematically to imposing Dirichlet boundary conditions to our model reaction-diffusion system ( 3). n most of the simulations reported in this paper (Sets. A 1 and A 3), the Couette flow reactor is unsymmetrically fed, with the left-end CSTR (x = ) in a (reduced) upper-branch state while the right-end CSTR (x = 1) is maintained in an (oxidized) lower-branch state, v(x=o) =f(u,) with u. =u(x=o)>a,, v(x = 1) =f(u,) with u, = u(x = 1) <a,. (4) Symmetric feeding (u. = u, ) will also be considered in Sec. A 2. For the sake of simplicity, the value of a in the system (3) is set independent ofx, a > au, so that when switching off the diffusion process, all the intermediate cell points evolve asymptotically to the same stable reduced steady state on the upper branch of the slow manifold. A more realistic model should probably take into account a spatial dependence of a, so that a(x = ) = u. and a(x = 1) = ul. n related models29*87 of the Couette flow reactor experiments conducted with the BZ system by the Texas group, the role of a is played by a third variable which corresponds to a set of reactants whose concentrations can be considered as being time independent during the experiment, and which act as an effective nonequilibrium constraint at each point of the Couette reactor. t is assumed that their spatial profile is a linear concentration gradient, which explains the linear spatial dependence of a in these models. The situation is not so clear for the chlorite-iodide reaction since such a set of reac- J. Chem. Phys., Vol. 95, No. 1,l July 1991

5 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation 327 tants does not seem to exist.lm For the sake of simplicity we will assume in the present study that the nonequilibrium constraint a is constant (zero-order approximation) all the way along the reactor. At this point, let us mention that the spatio-temporal patterns reported in the numerical study in Sec. are robust against (smooth) spatial perturbations of the a = cst working hypothesis. 2. CSTR boundary conditions For some feeding conditions, the dynamics inside the Couette reactor has been observed to introduce some feedback in the ending CSTRs. 34*39 n such conditions the two CSTRs cannot be considered anymore as maintained in a steady state. n order to account for this phenomenon we will also consider the following set of boundary conditions: d,c=r(c) fk,(c, -C) +Td,C, for x=, d,c=r(c) +k,(c, -C) -$d,c, for x= 1, where R(C) is the reaction term; the inlet concentrations C, = (u,,v, ) and C, = (u,,u, ) are chosen such that without coupling with the Couette reactor, both CSTRs are set in a stable steady state on either the upper or lower branch of the slow manifold. is the relative size of the CSTRs with respect to the overall size of the one-dimensional Couette reactor. k, mimics the input flow rate of the two CSTRs. Equation (5) ensures flux conservation at x = and x = 1, respectively. Let us remark that the Dirichlet boundary conditions are recovered in the limit k, + CO. ll. FRONT PATTERNS N ONE-DMENSONAL REACTON-DFFUSON SYSTEMS UNDER CONCENTRATON GRADENT The spatio-temporal patterns observed when performing numerical simulations of the reaction-diffusion model (3) are in many respects very similar to those observed experimentally in the Couette flow reactor with the chloriteiodide reaction This reaction and its variants provide a remarkable illustration that stationary and oscillating front patterns can organize in a chemical system from the diffusive coupling of steady state reactor cells. The aim of this section is to detail some specific transitions leading to spat&temporal patterns which seem to be generic in both the experiments and the simulations.8-85 ntuitive arguments will be given explaining the pattern formation phenomena through bifurcation mechanisms. A theoretical understanding of these bifurcations based on singular perturbation techniques and center manifold/normal form calculationsay6 will be reported in Sets. V and V. The partial-differential equations (3) are solved numerically through finite difference approximation for the spatial derivatives and the method of line for time advancement. The model medium is represented by a discretized line with a resolution from 5 up to 2 points. The resulting set of ordinary differential equations is integrated with a stiff ODE solver. Care is taken to vary the spat&temporal (5) resolution in order to check the reliability of the reported phenomena. A. Dirichlet boundary conditions 1. Asymmetric feeding Let us consider first the situation where the values of the two concentration variables of our reaction-diffusion model, u and U, are kept fixed at the two boundaries x = (reduced upper-branch state) and x = 1 (oxidized lowerbranch state) according to Eq. (4). As long as the fronts are located far enough from the two CSTRs at both ends of the Couette reactor, this seems to be a rather good approximation of the experimental situation with asymmetric feeding. 33Y34,39 Let us suppose that E is kept fixed to a (small) positive value. For D$E-, the diffusion term is predominant, and all the trajectories of the system converge asymptotically to a unique stable steady state. This solution is merely a linear spatial concentration profile linking the two ending concentrations. When D is decreased and becomes - O( 1 ), the reaction term eventually becomes of the order of the diffusion term: the former diffusion-like solution is still stable but it develops a sharp front which corresponds to a spatial switching between the two attracting branches of the slow manifold. n other words, in addition to the characteristic size of the system (which has been chosen to be equal to l), a smaller length scale -n (the width of the front) comes into play. When D is further decreased, the transition front becomes sharper and sharper until this single-front solution loses its stability; in parallel, an increasing number of multifront solutions actually appear, either stable or unstable. This evolution turns out to be generic independently of the specific S shape of the slow manifold. The limit D -+ can be identified to the limit of an extended system: in fact, the effective number of degrees of freedom increases when the diffusion is decreased. 2s 3 n this limit, we may expect to observe a very rich variety of dynamical behavior. But this limit is far from (i) the current experimental conditions: it would require low rotation rates of the inner cylinder of the Couette reactor for which the transport process could no longer be considered as diffusive; and (ii) the resolution of current numerical simulations. n the present numerical study, we mainly focus on the early bifurcations of front patterns observed on the way to this extended system limit and discuss the existence of up to three-front pattern solutions.* The numerical patterns shown in Figs. 2 and 3 have been obtained with the following form of the slow manifold: f(u) = u* - u3 + u5. (6) t is easy to check that a =, a, = - 1, a; , a; = , a; = , a;d ao+ = This choice makes the Hopf bifurcation in the reaction term subcritical,99 as is the case in most experimental situations. The following model parameters are kept unchanged: E =.1 and the feed concentration of the right-end CSTR U, = - 1 S. The feed concentration u. of the left-end CSTR, the diffusion coefficient D and the J. Chem. Phys., Vol. 95, No. 1,l July 1991

6 328 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation FG. 2. A perspective plot of the spatiotemporal variation of the variable u(x.r) as computed with the reaction-diffusion system (3), with Dirichlet boundary conditions (4). the slow manifold (6) and the model parameters: U, = - 1.5,~ =.1. (a) Stationary single-front pattern (u,, = 1.1, D=O.l, a=o.ol); (b) periodically oscillating single-front pattern (u,, = 1.1, D =.45, a =.1); (c) periodic alternation of a single-front and a three-front pattern via colliding fronts (u. = 1.1, D=O.Ol, a=o.ol); (d) stationary three-front pattern (uo =.5, D =.8, a =.2); (e) periodically oscillating three-front pattern (u, =.5, D =.6, a =.2); (f) periodic alternation ofa single-front and a three-front pattem(u=.5,d=.2,a=.2). parameter a are taken as control parameters. n Fig. 2, we characteristic length of the front decreases until use a three-dimensional space-time representation which il- the steady front solution eventually becomes unstable and lustrates the time evolution of the spatial concentration pro- starts oscillating periodically in time, as shown in Figs 2 (b) file of the u species. The same patterns are illustrated in Fig. and 3(b). To gain some understanding of these oscizzating 3 under a concentration coding similar to the one used to single-frontpatterns, one may consider a spatially discretized visualize the changes of color in the experimental study. Fig- version of our continuous reaction-diffusion system (3). A ure 3 has to be compared with Fig. 2 in Ref. 31 and Fig. 9 in straightforward linear calculation shows that, among the Ref. 39. one-dimensional array of coupled elementary reactor cells, For values of u. )a and values of D- ( 1 ), one ob- the ones that are located at the front zone are driven by serves only steady single-frontpatterns; left to the front, the diffusion to an attracting limit cycle because of the presence solution is confined to the upper branch of the slow mani- of a steep concentration gradient. (The physical mechanism fold; right to the front the solution belongs to the lower underlying this oscillatory instability has been identified in branch [Figs. 2(a) and 3(a). When D is decreased, the the direct diffusive coupling of two unsymmetrically fed J. Chem. Phys., Vol. 95, No. 1, 1 July 1991

7 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation a -- d 329 ~ 1 FG. 3. The spat&temporal variation of the variable u(x,t) is coded in order to mimic the spatial color profiles observed in the Couette flow reactor (Refs. 33, 34, and 39); 32 shades are used from the left-end upper branch (reduced) state (black) to the rightend lower branch (oxidized) state (white). The numerical spat&temporal patterns in (a)-(f) are the same as in Fig. 2. space - space CSTRs in Refs. 14 and 15.) When the diffusive coupling is weak enough, these reactor cells are no longer stabilized by the stable steady state cells located close to the boundaries and the whole system starts oscillating. t is then clear that the amplitude of oscillation is much larger for the cell points located at the front zone: The instability originates in the active region at the front zone and is propagated by the diffusive coupling to the other cell points of the reactor. One may wonder whether this instability persists in the continuous limit. The soundness of an extrapolation to the continuous limit is supported by the analytical study reported in Sec. V. When further lowering D, the spatial amplitude of the oscillation of the front pattern increases until a qualitative change occurs in the spatio-temporal evolution of the system as indicated by the reentrance phenomenon shown in the space-time representation in Figs. 2(c) and 3(c). A three-front profile alternates periodically with a single-front profile. The three-front profile proceeds from the periodic appearance of two traveling fronts. The single-front profile is recovered from the periodic coalescence of one of these two traveling fronts with the originally oscillating front. A similar pattern where the period is about twice the period of the previous one is shown in Figs. 2 ( f) and 3 ( f). These colliding front patterns have also their experimental counter part.3-34*39 They give hints that steady multi-front patterns are very likely to exist. For u. -a, we have succeeded in freezing asteady threefront pattern involving three spatial switches between the two branches of the slow manifold [Figs. 2 (d) and 3 (d) 1. Again a decrease of the diffusion coefficient induces a transition to a periodically oscillating three-front pattern [Figs. 2(e) and 3 (e) 1. n our numerical simulations, this stafion- J. Chem. Phys., Vol. 95, No. 1, 1 July 1991

8 33 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation ary multipeaked structure coexists with the oscillating single-front pattern, in contrast with the experimental situation39 where these two patterns apparently take place in two different regions of the constraint space. However, let us emphasize the striking similarities between the patterns shown in Figs. 2 and 3, and the corresponding patterns identified thus far in the experiments (Fig. 2 in Ref. 31 and Fig. 9 in Ref. 39). As discussed in Sec. V, we have strong indications that the three-front solution originates from a saddlenode bifurcation, i.e., at some distance in phase space from the natural single-front solution. As pointed out in Refs. 33 and 39, despite the fact that no hysteresis has been observed thus far, one cannot exclude the presence of multiple stable states in the experiments. 2. Symmetric feeding n order to mimic sustained patterns observed in the Couette flow reactor with symmetric feeding,39 let us now consider symmetric Dirichlet boundary conditions: u. and U, are for example located on the (oxidized) lower branch of the slow manifold, while a still belongs to the (reduced) upper branch. Eventhough there is no asymmetry in the feeding, there still exists a concentration gradient in the system close to the two boundaries. When D+ CO, the diffusive transport process dominates and all the trajectories converge asymptotically to a homogeneous solution imposed from the boundary conditions: All the reactor cells are uniformly constrained to the lower branch u(x) = u = u,. When D is decreased, the homogeneous solution becomes unstable, and a new stationary solution pops up [Figs. 4(a) and 4(e)]; this new solution displays a spatial profile which involves two fronts separating a central region of reduced states from the two regions of oxidized states close to the two boundaries. As previously observed for the single-front patterns, the steady two-frontpattern solution undergoes a Hopf bifurcation when the diffusion D is further decreased, leading to a periodically breathingpattern as illustrated in Figs. 4(b) and 4(f). When lowering D, the amplitude of oscillation of the two fronts increases as shown in Figs. 4(c) and 4( g ), and the breathing pattern transforms into a burstingpattern as observed in the experiments.39 This bursting phenomenon corresponds to the periodic appearance and coalescence of the two oscillating fronts, i.e., a burst of (reduced) upper-branch state emerges periodically in the Couette reactor from a rather uniform (oxidized) lower-branch state induced by the boundary conditions. Special attention has to be paid, however, to identify unambiguously this bursting phenomenon since the concentration coding used in Fig. 3, as well as in the experiments, can make the distinction between breathing and bursting patterns quite confusing. At this point let us mention that subsequent simulations have revealed more complicated periodic as well as intermittent (chaotic) burstings. Moreover, these phenomena seem to be robust with respect to the choice of the boundary conditions; in particular they can be observed in the more surprising situation where the two CSTRs and the Couette reactor steady state are located on the same branch of the slow manifold. A detailed study of bursting patterns in the reaction-diffusion model ( 3 ) will be reported elsewhere. lo6 Actually there are two different types of oscillating modes of the two-front pattern (see Sec. V). The breathing pattern shown in Figs. 4(b) and 4(f) corresponds to an inphase (or symmetric) oscillating mode: The reactor cells located at the two front zones oscillate in phase. The wigglingpattern shown in Figs. 4(d) and 4(h) corresponds to a out-of-phase (or antisymmetric) oscillating mode: The reactor cells at the two-front zone oscillate out of phase. Because the instability of the in-phase mode occurs generally prior to the instability of the out-of-phase mode, the wiggling patterns are usually observed as transient phenomena to either the steady two-front pattern (before the oscillatory instability threshold) or more or less complicated breathing patterns (beyond the oscillatory instability threshold). 3. Diffusion-induced spatio-temporal chaos The whole zoology of patterns reported in Sets. A 1 and A 2 are robust, in the sense that they can generically be observed in any reaction-diffusion system with a S- shaped slow manifold. n particular, we have reproduced the patterns reported in Figs. 2-4 with the following slow manifolds: -U O<u<l-s f(u) = -fc -u) c = -is- (1 -U)4f~S- (l -U) &;u(l +a* 1 u-2 u>l +s (7) n the limit 6-, this one-parameter family of slow manifolds reduces to a piecewise linear function f which will be used in Sec. V to derive analytical results. With the slow manifolds (7)) we have found conditions where the oscillating single-front pattern undergoes secondary instabilities leading to more complicated spatio-temporal behavior.* n Fig. 5 (a), we show a chaotically oscillating front structure computed with 6 = 1e2 in Eq. (7). The phase portrait reconstructed from the temporal evolution of the variables u and u recorded at an intermediate spatial cell point is shown in Fig. 5 (b). The corresponding Poincare map and 1D map are illustrated in Figs. 5 (c) and 5 (d), respectively. The fact that the Poincare map is not a scattering of points but that all the points lie to a good approximation along a smooth curve indicates that the trajectories lie approximately on a (multifolded) two-dimensional sheet in the phase space. The welldefined single humped shape of this 1 D map is a clear signature of the low dimensional chaotic nature of these oscillations. t is somewhat puzzling that the phase portraits obtained in our simulations [ Fig. 5 (b) ] are strikingly simi- J. Chem. Phys., Vol. 95, No.,1 July 1991

9 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation 331 mace FG. 4. A perspective plot of the spatio-temporal variation of the variable u(x,t) as computed with the reaction-diffusion system (3) with Dirichlet boundary conditions, the slow manifold (6), and the model parameters: a,,=~, = -11.5,a=.2,~=.1.(a)Stationary two-front pattern (D =.5); (b) two in-phase periodically oscillating fronts (D =.3); Cc) bursting pattern (D =.55); (d) two out of phase periodically oscillating fronts (D =.2). Figures (e)- (h) represent the same spatio-temporal patterns as in (a)-(d), respectively, using 32 shades from the lower branch (oxidized ) state (white) to the upper branch (reduced) state (black). lar to the strange attractors observed in the BZ reaction when conducted in a CSTR S16 Moreover, as in the homogeneous BZ reaction, period-doubling bifurcations are observed as precursors to this macroscopic chaos. Let us mention that these chaotic spat&temporal patterns have been obtained when using a spatial discretization ( - 1 intermediate reactor cells) compatible with the number of characteristic diffusion lengths in the Couette flow reactor. We have checked that these chaotic patterns are preserved when increasing spatial resolution. The generality of the observed transition to spatio-temporal chaos extends to a rather large range of values of 6, As pointed out in Ref. 39, be- Downloaded 9 Dec 23 to Redistribution J. Chem. Phys., subject Vol. 95, to No. AP 1,1 license July 1991 or copyright, see

10 332 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation -3 GQ-yy-J FG. 5. Diffusion-induced chaos obtained when integrating the reactiondiffusion system (3) with Dirichlet boundary conditions (4) and the slow manifold (7) (6 = O- ). The model parameters are u = 2.5, U, = - 3.1, a = 1.1, E =.1, D =.5. (a) Spatio-temporal variation of the variable u(x,t) coded as in Fig. 3; (b) phase portrait; (c) Poincart map; (d) 1Dmap. cause of the difficulties in controlling experimental conditions during sufficiently long period of time, there has been thus far no clear demonstration of the chaotic character of the nonperiodic time series recorded in the Couette flow reactor. 6. CSTR boundary conditions n some experiments performed with some variants of the chlorite-iodide reaction, the oscillating front patterns have been observed to invade one of the ending CSTRS. ~*~~ Henceforth the two CSTRs cannot be considered in a steady state during the experimental run as before. n order to account for the interplay of the dynamics inside the Couette reactor and in the CSTRs, we have performed subsequent numerical simulationse3 of our reaction-diffusion model (3) with the CSTR boundary conditions defined in Eq. (5). We give here a short description of the patterns observed when considering the slow manifold (6). The following parameters are kept fixed: e = 1-2, a =.5, u,, = 2, zll = - 4, ui =f( ui ), i =,l. D = k, is our control parameter. As compared to the feeding concentrations used thus far, the left (x = ) and right (X = 1) end CSTRs are fed on the (reduced) upper and (oxidized) lower branches of the slow manifold, respectively, but the values of u. and 11, considered here induce a very strong gradient of concentration through the reactor. Under Dirichlet boundary conditions, the single-front solution would become unstable at very low values ( - 1w3) of the diffusion coefficient. The situation is quite different with the CSTR boundary conditions. At low values of D, almost all the reactor cell points are in a reduced state u = a, except a small region located near x = 1 where the reactor cells are driven in an oxidized state by the right-end CSTR. As shown in Fig. 6, when D is increased, the diffusion process carries further the influence of the right-end CSTR and the steady front moves to the left: n turn more and more reactors switch from the upper to the lower branch of the slow manifold. The displacement of the steady front is found to depend linearly on D. The boundary conditions become much more Dirichlet like when the front is located in the central region ofthe Couette reactor; in this stationary situation, the gradient term JC/& does not play any important role in Rq. (5), so that, for reasonable values of k,, one can consider that the two ending CSTRs evolve according to the equation k=r(c) +k,(ci -C), i=o,l. When the front approaches the left-end CSTR (x = ), it becomes unstable through a Hopf bifurcation, and starts oscillating periodically. At this point, the intuitive picture we gave for the case of Dirichlet boundary conditions breaks down: the steady single-front solution is then distabilized when increasing (instead of decreasing) the diffusion coefficient. This is a direct consequence of the specific type of boundary conditions we are imposing. We suspect that, in this case, the oscillatory instability is not only governed by the interplay of the reaction and diffusion processes at the front zone but is also driven by the dynamics of the left-end CSTR. This is clearly illustrated in Fig. 7; when further increasing D the spatial amplitude of oscillation increases until the front periodically disappears in the left-end CSTR. The system undergoes secondary instabilities, and the behavior of the front inside the Couette reactor become more or less regular as illustrated in Figs. 7 and 8. When looking at the periodic oscillations recorded at different spatial points, the corresponding temporal patterns shown in Figs. 7 (b), 7 (c) and 8 (b), 8 (c) are strikingly similar to the time series obtained in the pioneering experiments on the homogeneous BZ reaction.5* 5V 6. 7 Recently, the alternating periodicchaotic sequences observed in these experiments have been understood in terms of the frequency-locked and chaotic 2.,..,..1 *,,,.,..,.,. *. u t- D D -.5 D =. 1 space FG. 6. Displacement of a stationary front when increasing the diffusion coefficient D in the reaction-diffusion model ( 3 ). The boundary conditions are of CSTR type [ Eq. ( 5 ) 1, the slow manifold is given by Fq ( 6) and the model parameters are e = O-, a =.5, u, = 2, u, = - 4. i J. Chem. Phys., Vol. 95, No. 1, 1 July 1991

11 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation ~ _, (c) J = s, 1 : t (! ;,i,k,(b) 1 2 JO 4 fll*. LO f,l,ll FG. 7. (a) A periodically invading front obtained with the reaction-diffusion model (3) with the CSTR boundary conditions (5); the parameter valuesare: l=2.13 O-, c= 1 2, D =.1, a =.5, U = 2, 11, = - 4 and the slow manifold is given by Eq. (6); (b) and (c) illustrate the periodic oscillations recorded in two distinct intermediate reactor cells visited by the front. FG. 8. (a) An intermittent front obtained with the reaction-diffusion model (3) with the CSTR boundary conditions (5); the parameter values are E = 1, D =.99, a =.5, u(, = 2, U, = - 4 and the slow manifold is given by Eq. (6); (b) and (c) illustrate the oscillations recorded in two distinct intermediate reactor cells visited by the front. states issued from the breaking of a T torus.68* 8 But this chaos is a small-scale chaos 5P 6, 8 which turns out to be extremely difficult to identify as compared to the large-scale chaos encountered in nearly homoclinic conditions. 9 1 Therefore, in contrast to the macroscopic chaos observed with Dirichlet boundary conditions (Fig. 5)) very high accuracy numerical simulations are required to distinguish between periodically and chaotically oscillating fronts in our reaction-diffusion model (3) with CSTR boundary conditions. Figure 8(a) illustrates an intermittent front which shows up and disappears apparently in an erratic manner during our finite-time numerical experiment. n a local cell, the possible chaotic nature of the temporal pattern is contained in the small amplitude oscillations: Their number may differ from one temporal motif to the next while their amplitude may also fluctuate. We refer the reader to Refs. 68, 18, and 111 for a detailed experimental, numerical, and theoretical study of similar small-scale chaos in the BZ reaction conducted in a CSTR. V. EXSTENCE AND STABLTY OF FRONT PATTERNS n this section, we study the existence and stability ofthe stationary solutions of our reaction-diffusion model (3). Our goal is not to give a full description of all the possible cases, but rather to emphasize some general properties which are common to the class of reaction-diffusion systems given by a set of equations similar to Eq. (3 ) with Dirichlet boundary conditions. 2 n Sec. V A, we summarize the perturbative theory for the existence of stationary solutions proposed by Fife in Ref We then apply this approach to the particular case of a piecewise linear slow manifold 69, 4*115 and we give some estimates of the number of statllonary solutions. n Sec. V B, we study the linear stability of these solutions. Analytical calculations of some bifurcation diagrams are detailed in Sec. V C for a piecewise linear slow manifold. A. Existence of stationary front patterns The problem of the existence of stationary solutions for the system (3) consists in solving the stationary problem: o=d++u-f(u), dx* O=Dfi-u+a. (8) dx2 The singular perturbation analysis developed below is a heuristic version of the rigorous results derived by Fife. 13 According to this work, the original stationary problem [ Eq. ( 8) ] can in principle be reduced to a more tractable equation. For the specific case of a piecewise linear slow manifold, we will solve the reduced system analytically. 1. Perturbafive theory A key role in the treatment of the stationary problem (8) is played by the small parameter 7 = m. n Eq. (8a), Downloaded 9 Dec 23 to J. Redistribution Chem. Phys., Vol. subject 95, No. to AP 1,l July license 1991 or copyright, see