Jan 9, 1989 - Centre de Recherche Paul Pascal, Université Bordeaux I, Domaine Universitaire, F-33405 ... irreversible, periodic precipitation patterns [3,8,9] -.
Volume 134, number 5
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9 January 1989
SUSTAINED REACTION-DIFFUSION STRUCTURES IN AN OPEN REACTOR
Q. OUYANG, J. BOISSONADE, J.C. ROUX and P. DE KEPPER Centre de Recherche Paul Pascal, Université Bordeaux I, Domaine Universitaire, F-33405 Talence, France Received 29 September 1988; accepted for publication 9 November 1988 Communicated by AR Fordy
Spontaneous emergence ofsustained spatial chemical dissipative structures resulting from only reaction and diffusion processes is a major yet incompletely solved problem in experimental nonlinear dynamics and pattern formation. We introduce a new type of open reactor where both chemical constraints and transport rates are controlled. A sequence of different spatiotemporal reaction—diffusion structures, and most interestingly the first nontrivial stationary pattern, have been obtained by Continuous variation of a control parameter with variants of the chlorite—iodide and the Belousov—Zhabotinskii reactions.
Chemical systems constrained by permanent nonequilibrium conditions can lose stability and spontaneously develop various temporal or spatial concentration patterns called “chemical dissipative structures” [1,21. When a system is maintained homogeneous by a vigourous stirring, one may only observe temporal organization phenomena such as multiple steady states, periodic [3] or chaotic [4] oscillations, originating in the nonlinear kinetics of the reaction processes. Their study has become standard in theoretical and experimental nonlinear dynamics. Still more exciting are the so-called reaction— diffusion structures resulting from the coupling of reaction and diffusion processes in unstirred systems [5]; it was immediately recognized that the possible spontaneous emergence of stationary concentration patterns (spatial structures) could be ofprimary importance in early stages of morphogenesis and other biological organization phenomena [61. Furthermore, the dynamics of nonstationary patterns (spatiotemporal structures), which can form in less stringent conditions, could become a paradigm in the growing field of the geometry of non-equilibrium [7]. In experiments on temporal chemical dissipative structures, the continuously stirred tank reactor (CSTR) provides an effective implementation of sustained and controlled constant nonequilibrium conditions. Unfortunately there has been no equivalent piece of apparatus to produce sustained, a for282
tiori stationary, genuine reaction—diffusion structures. Thus most previous experiments on these spatial structures have been limited to transient observations of wave patterns in batch conditions or irreversible, periodic precipitation patterns [3,8,9] The development of appropriate reactors has been checked both by technical difficulties (in particular in discarding any parasitic terms of convective form [101) and by experimentally inconvenient scales of space and time due to the intrinsic smallness of molecular diffusion coefficients. It is only recently that two different spatial open reactors have been developed in order to overcome the difficulties in controlling the chemical constraints [11,121. Although such reactors allow in principle the formation of stationary Turing type patterns [13,141 hitherto only sustained chemical waves have been obtained. Here, we propose a radically new type of reactor, namely the “Couette-flow reactor” where both chemical constraints and transport rates are controlled (fig. 1). We expect that it will play in spatial studies the role the CSTR did for temporal studies. With this device, various spatial and spatiotemporal reaction— diffusion structures have been obtained with variants of the chlorite—iodide and the Belousov—Zhabotinskii reactions. In this Letter, we limit ourselves to a specially representative series of results in order to illustrate the capabilities of this system. After a description of the basic principles and of the tech-
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stirrer chemicaL outlet
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Couette reactor
~STRI~5T~I chemical inLet
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Fig. 1. Cross-sectional view ofthe reactor. The cylinders are made of transparentglass. The inner cylinder radius is a = 10.5 mm and the 3. The rotation of the inner gap is e=2.l5 mm.byThe length andup volume of rpm the circular Couettetoflow are respectively 34cm and 49 cylinder is driven a step motor to 1500 corresponding a maximum Reynolds number Recm50R, where R~is the critical Reynolds number at which the laminar flow becomes unstable. The identical and CSTRs have a volume V= 26 cm3 and are vigorously mixed by means of six-paddle stirrers rotating at 600 rpm. Two different compositions of the reactants are pumped into each CSTR through a premixer by a series of high precision piston pumps. The input and output pumping rates are carefully balanced in order to avoid any net flow through the annular reactor, so that the transport only results from turbulent diffusion. The whole system is immersed in a thermostated bath. For control purposes, the electrochemical potential in each CSTR is monitored by a Pt electrode. The evolution of concentration patterns inside the Couette reactor is revealed by appropriate colour indicators for each reactionand recorded through a video camera. Light and colour intensity are ultimately processed on a NUMELEC Pericolor 2000 image analyser.
nical arrangement, we report a sequence of different spatiotemporal structures obtained by continuous variations of a control parameter, which ultimately lead to a stationary pattern. The core ofthe reactor is a cylindrical Couette flow, where the fluid fills the annular gap comprised between an outer fixed cylinder and a coaxial rotating cylinder. The ends of this reactor open into two identical CSTRs which perform the feeding functions. When the system is operated at large Reynolds numbers (typically Re>> l5R~)the flow inside the Couette reactor becomes turbulent enough for the fluid to be well mixed both in the radial and azimuthal directions; in these conditions the transport along the cylinder’s axis was previously shown to behave as a onedimensional diffusion process with a unique diffusion coefficient D for all species [15]. Unidimensional concentration patterns, revealed by appropriate colour indicators, can develop in the axial direction. The key features of the Couette reactor are the following: (a) Chemical dynamics are ruled by a reaction—diffusion process. Experimental studies of simple acid—base and oxidation reactions performed in a closed continuously rotated cellular
reactor [16], an elementary form of a Couette reactor, support this point. (b) The axial diffusion coefficient D is tunable by a simple change of the rotation rate. (c) It ranges from 7x l0—~to approximately 1 cm2/s, very high values in regard to those of natural molecular diffusion (-~l0—~cm2/s). Most known liquid phase reactions are expected to produce spatial dissipative structures that evolve on a time scale r of the order of a few minutes. Thus the order of magnitude of the expected structure wavelength). \/~, should range on a laboratory scale, ideal for visual or photographic analysis. (d) Nonequilibrium conditions are sustained at the boundaries through the CSTRs. In most of our experiments, the volume and feeding flows of each CSTR were large enough for their internal state not to be significantly modified by the dynamics inside the Couette reactor. This corresponds from a mathematical point of view to Dirichiet boundary conditions. (e) Since D is identical for all species, structures cannot origmate in the differences of transport delays [171, but in the Dirichlet conditions (ref. [11, p. 102), in a response to a finite perturbation beyond a secondary bifurcation [181 or more likely in the nonuniformity of the basic steady state [19—211.Thus, in order to 283
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increase this nonuniformity and favour self-organization, the system was fed with different compositions at each boundary, inducing a natural gradient of concentrations.A spatial dissipative structure must then be distinguished from this trivial pattern caused by the asymmetric constraints and should correspond to a multipeaked or a nonstationary concentration profile along the reactor. We have investigated two different reactions known to exhibit wave patterns [3,321, a Belousov— Zhabotinskii type reaction and the chlorite—iodide— malonic acid reaction. We report here selected observations with the latter, which was also previously shown to produce an elementary spatiotemporal structure in a system of two coupled-cell reactors [23]. The reacting medium can be in a reduced (high [I ]) or oxidized (low [I — 1) state, respectively characterized in presence of starch by blue or yellow colour. In each sequence of experiments, performed at a constant rotation rate, CSTR I and II were respectively fed with compositions which should correspond to reduced and oxidized states in the isolated CSTR but, whereas composition I was kept constant, composition II was varied in a stepwise manner and the system left to relax to an asymptotic state after every change. Different sequences of sustained spatial and spatiotemporal patterns were readily obtamed, leading ultimately to nontrivial stationary patterns. We present here a selected representative sequence, postponing to a forthcoming publication more complex situations and extensive phase diagrams where both chemical input concentration and the transport rate D are used as control parameters. In order to visualize the changes of colours as a function of time, we use the spacetime representation of fig. 2. On the first pattern (fig. 2a), the yellow and blue regions are separated by a sharp fixed front. The strong localization in the colour change reveals a switching process in the kinetics. Increasing gradually the chlorite concentration in the feed flow of reactor II induces self-organization: above some critical value, the front position becomes unstable and achieves periodic oscillations in the Couette flow, whereas the concentrations in the CSTRs are found to remain constant (fig. 2b). This is the spatiotemporal equivalent of temporal periodic oscillations. Increasing further the chlorite concentration gener284
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ates a continuous succession of radically new patterns, the continuity of which is demonstrated in the spatiotemporal representation in figs. 2b—2f. The amplitude ofthe front oscillations increases (fig. 2c), then a band of oxidized state periodically splits the reduced blue region in two separated parts (fig. 2d) and finally becomes permanent: in fig. 2e, this band is still delimited by two oscillating fronts, but at a larger chlorite flow rate all fronts ultimately stabilize to produce the stationary spatial pattern shown in fig. 2f. The alternation of blue and yellow regions cannot be explained by a simple gradient of input reactants. The length scale of the pattern is much smaller than the reactor length and no equivalent structure was obtained in experiments with symmetric feeds. This rules out a trivial structure that would result from the sole Dirichiet boundary conditions. The pattern readily originates in the interplay between the highly nonlinear kinetics and the diffusive process and constitutes a genuine reaction— diffusion structure which can be sustained indefinitely. Experiments performed with a bubble Belousov— Zhabolinskii free reaction [24] give more complicated patterns, some of them presenting a larger number of simultaneously moving fronts between oxidized and reduced regions. We have actually succeeded in stabilizing the first three fronts and have hints that additional fronts could be stabilized to produce a multipeaked stationary structure but at much lower diffusion rates. With the present geometric characteristics of our Couette reactor, this would require low rotation rates for which the transport can no longer be considered as diffusional. Other patterns exhibit oscillations which are not strictly periodic, but no definite evidence of a chaotic behavjour has yet been obtained. Existence of periodic or chaotic oscillations of sharp fronts in a one-dimensional reaction—diffusion system submitted to fixed boundary conditions agrees with recent theoretical predictions [21] where the nonlinear kinetic term has not an intrinsic oscillating character. Only the bistability properties, used in the choice of the boundary conditions in formal agreement with our experiment, are necessary. This differs slightly from parallel experiments, similar to ours, where the boundaries are left free to oscillate
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.1 ~Ii I
o
io
Ill 20
(cm)0
10
20
(cm)
Fig. 2. Sustained spatial and spatiotemporal patterns. Each figure represents the spatial colour profile (abscissa) in the Couette flow as a function of time (ordinate). Contrast and brilliance ofcolours are amplified by image processing. Reactants are chlorite, iodite, malonic acid, sulfuric acid and a starch-type colour indicator (PROLABO “Thiodêne”). The parameters common to the whole sequence are: 2/s experimentally determined), T= 26°C,input and output flows in both CSTRs 0.167 cm3/s, rotation rate = 840 rpm (D 0.32 cm compositions of input flows, [I]~=[I]~j =3.SxlO3M, [H 3M, [MA]~= [MA]W=2xl03M, [ClOfl~=0. (a) Stationary front ([ClOflh/=0.8xl02M). (b)2SO4Th=[H2SO4]~i Oscillating front ([ClOfl~/=0.9xl02M). =5X10 (c) ([ClO~]~/= 1.1 X l02M). (d) ([ClOfl~/= l.2x l02M). (e) ([ClOflU= l.4X lO2M). (f) Stationary structure ([ClO~]U= l.5x102M).
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and are well accountedby a set ofcoupled oscillators with a frequency gradient [251. We have thus constructed a reliable piece of apparatus to produce and study sustained reaction—diffusion structure with simultaneous control of the chemical constraints and the transport rate. We have given proof of principle that various sustained nontrivial spatiotemporal patterns can readily be obtamed. Moreover we exhibit the first genuine sustained chemical stationary pattern. This experimental system allows a systematic study of bifurcation between spatial or spatiotemporal states by continuous variation of one or several control parameters, much the same way the CSTR does for homogeneous states. We thus expect a large development of these studies in the more general frame of nonlinear dynamics and pattern formation. In particular, this type of reactor c~uldprovide convenient solutions to the problem of the transition from a low dimensional system to extended ones [211. This work is part of a joint project with the Nonlinear Dynamics group of Texas University (who performed the experiments of refs. [11,12,15,25] with support of the BP Venture Research Unit.
References [1] G. Nicolis and I. Pngogine, Self-organization in nonequilibrium chemical systems (Wiley, New York, 1977). [2] H. Haken, Synergetics, an introduction, 2nd Ed. (Springer, Berlin, 1979). [31D. Field andM. Burger, eds. Oscillations andtraveling waves in chemical systems (Wiley, New York, 1985). [4] F. Argoul, A. Arneodo, P. Richetti, J.C. Roux and H.L. Swinney, Accounts Chem. Res. 20 (1987) 436. ES] G. Nicolis, T. Erneux and M. Herschkowitz-Kaufman, Adv. Chem. Phys. 88 (1978) 263. [6] A. Turing, Philos, Trans. R. Soc. B 327 (1952) 37.
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[7] P. Coullet and P. Huerre, eds., New trends in nonlinear dynamicsand pattern forming phenomena: the geometry of nonequilibrium (Plenum, New York), to be published. [8] G. Vidal and A. Pacault, in: Evolution of order and chaos, ed. H. Haken (Springer, Berlin, 1982) p. 74. [9] S.C. Muller, in: From chemical to biological organization, eds. M. Markus, S.C. Muller and G. Nicolis (Springer, Berlin, 1988) p. 83. [10] J.C. Micheau, M. Gimenez, P. Borkmans and G. Dewel, Nature 305 (1983) 43. [11] Z. Noszticzius, W. Horsthemke, W.D. McCormick, H.L. Swinney and W.Y. Tam, Nature 329 (1987) 619. [12] W.Y. Tam, W. Horsthemke, Z. Noszticzius and H.L. Swinney, J. Chem. Phys. 88 (1987) 3395. [13] G. Dewel, D. Walgraef and P. Borckmans, J. Chim. Phys. (Paris) 84 (1987) 1335. [14] J. Boissonade, J. Phys. (Paris) 49 (1988)541. [151W.Y. Tam and H.L. Swinney, Phys. Rev. A 36 (1987) 1374. [16] J.B. Grutzner, E.A. Patrick, P.J. Pellechia and M. Vera, J. Am. Chem. Soc. 1988 (1987) 726. [171 H. Meinhardt, Model of biological pattern formation (Academic Press, New York, 1982) ch. 3. [18]J.A. Vastano, J.E. Pearson, W. Horsthemke and H.L. Swinney, Phys. Lett. A 124 (1987) 320. [19] L. GlassandR. Perez, J. Chem. Phys. 61(1975)5242. [201 M. Boukalouch,J. Elezgaray, A. Arneodo, J. Boissonadeand P. De Kepper,J. Phys. Chem. 91(1987) 5843. [211 A. Arneodo and J. Elezgaray, in: Proc. Conf. on Spatial inhomogeneities and transient behaviour in chemical kinetics, Brussels, 1987 (Manchester Univ. Press, Manchester), to bepublished; J. Elezgaray and A. Arneodo, in: New trends in nonlinear dynamics and pattern forming phenomena: the geometry of nonequilibrium, eds. P. Coullet and P. Huerre (Plenum, New York), to be published. [221 P. De Kepper, I. Epstein, K. Kustin and M. Urban, J. Phys. Chem. 86(1982)170. [23] M. Boukalouch, J. Boissonade and P. Dc Kepper, J. Chim. Phys. (Paris) 84 (1987) 1354; J. Boissonade, M. Boukalouch and P. De Kepper, in: Proc. Conf. on Spatial inhomogeneities and transient behaviour in chemical kinetics, Brussels, 1987, ch. 30 (Manchester Univ. Press, Manchester), to be published. [24] Q. Ouyang, W.Y. Tam, P. De Kepper, W.D. McCormick, Z. Noszticzius and H.L. Swinney, J. Phys. Chem. 91(1987) 2181. [25] W.Y. Tam, J.A. Vastano, H.L. Swinney and W. Horsthemke, submitted for publication (1988).
