biological reaction systems cleary exhibit stationary patterns whether they .... Figure 2 sketches the many OSFR geometries used so far in theoretical and experi-.
Spatial Bistability: A Source of Complex Dynamics. From Spatio-Temporal Reaction-Diffusion Patterns to Chemomechanical Structures J. Boissonade, P. De Kepper Centre de recherche Paul Pascal (CNRS), 115 av. Dr. A. Schweitzer, F-33600, Pessac, France F. Gauffre IMRCP, Universit´e Paul Sabatier, 31062 Toulouse Cedex 09, France. I. Szalai Institute of Chemistry, L. E¨otv¨os University, P.O Box 32, H-1518 Budapest 112, Hungary. Abstract: We show experimentally and theoretically that reaction systems characterized by a slow induction period followed by a fast evolution to equilibrium can readily generate “spatial bistability” when operated in thin gel reactors diffusively fed from one side. This phenomenon which corresponds to the coexistence of two different stable steady states, not breaking the symmetry of the boundary conditions, can be at the origin of diverse reaction-diffusion instabilities. Using different chemical reactions, we show how stationary pulses, labyrinthine patterns or spatio-temporal oscillations can be generated. Beyond simple reaction-diffusion instabilities, we also demonstrate that the cross-coupling of spatial bistability with the size responsiveness of a chemosensitive gel can give rise to autonomous spatio-temporal shape patterns, referred to as chemomechanical structures. PACS: 82.40.Ck, 82.40.Bj, 82.33.Ln
1
The study, control and design of nonequilbrium reaction-diffusion patterns is a stimulating and challenging field of nonlinear dynamics. Though appropriate experimental open spatial reactors have been developed in the past two decades, very few isothermal reaction systems have been shown to produce non trivial sustained patterns. Up to now, only three [1–3] non biological reaction systems cleary exhibit stationary patterns whether they result from a Turing instability [4, 5] or from reactive front interactions in a bistable system. This may seem astonishing since a large number of reactions involving simple ionic species have been shown to exhibit oscillatory and bistability phenomena when operated in continuous stirred tank reactors (CSTR). By nature, all these reactions encompass kinetic “activator/inhibitor” or “activator/substrate depletion” mechanisms which are suitable for the development of sustained reaction-diffusion patterns when appropriately handled. In the 80’s, a great number of oscillatory systems were systematically designed on the basis of “temporal bistability” [6, 7]. Bistability is the simplest non trivial phenomenon that can be generated by an autoactivated reaction in an open reactor. It is theoretically well known that, in the frame of the comfortable “pool chemical” approximation, it can also play a crucial role for the development of reaction-diffusion patterns [8]. However, this approximation is unrealistic from an experimental point of view, since it would require that no input reactant be significantly consumed or that all these reactants be permanently fed at every space point without creating convective motions. It is only recently that bistability phenomena were systematically studied in real open spatial reactors. It was proved that in such reactors, several distributions of concentration can be stable, for the same boundary conditions [9]. These studies opened the way to new understanding of pattern development in “real” open spatial reactors, beyond this pool chemical approximation. They offer guidelines for the development of functional methods to discover non trivial reaction-diffusion patterns in many different chemical systems, as will be seen below. Differences in the mobility (diffusivity) of reacting species play a crucial role in the formation of symmetry breaking spatial and spatio-temporal patterns. In this respect, the spatial bistability studies allowed to discover a new source of oscillatory instability in open spatial reactors. Beyond plain reaction-diffusion instabilities the sensitivity of the spatial bistability to some geometrical size of the reactors also triggered the theoretical development and experimental study of new chemomechanical pattern concepts. In these systems the chemical activity and size changes of chemo-responsive materials cross-talk to uncover new sources for the formation of dissipative structures. In this feature article, we elaborate on the different aspects of pattern formation that can be linked to a common phenomenon: spatial bistability. 2
1
INTRODUCTION
Bistability is one of the fundamental phenomena that rule the dynamics of chemical dissipative structures [7,10]. Its role in the dynamics of homogeneous systems has been underlined very early. In reaction-diffusion systems, spatial bistability is defined as the existence of two different stable spatial distributions of concentrations, not breaking the symmetry of boundary conditions, for a same set of control parameters. Surprisingly, though this problem has long been considered from a mathematical point of view [11], it had only been mentioned occasionally in the context of chemical dissipative structures [12–14]. Its importance for the development and the understanding of the experimental spatial structures in open reactors was only recognized in recent years [9, 15, 16]. Since then, spatial bistability has been associated to various new phenomena such as long range activation [17], large amplitude spatial structures [18], or chemomechanics [19,20]. In this feature article, we provide a general survey of the theoretical and experimental basis of spatial bistability and its main present applications in the field of chemical dissipative structures. Spatial bistability can occur in a large variety of nonlinear systems. However, in chemistry, large differences between the time scales of the elementary reaction steps generally make possible convenient simplifications. Here, we limit ourselves to a simplified theory [9] applicable to the so-called “clock reactions”, actually the only reactions for which spatial bistability has been experimentally evidenced. For a less restrictive frame and a more systematic approach, yet limited to simple models, see the paper by Benyaich et al. [21]. A typical clock reaction is an autocatalytic reaction in which one of the products, say X, considerably increases the reaction rate when a small critical amount of X is reached, so that, in a closed system, after a more or less long induction time ti , during which the system remains almost “unreacted”, the chemical transformation is completely achieved within a short time tc . In most cases, tc ti , a condition that will be assumed in the following. In a continuous stirred tank reactor (CSTR), initially void of reactants, the residence time τ must be shorter than ti for the contents to be kept in a low extent of reaction state (hereafter referred as the “unreacted” or “flow” state F). But if a sufficient amount of X is already present, the reaction starts immediately so that the contents are driven in a “reacted” or “thermodynamic” state where it remains as long as τ tc . For shorter τ , the renewal of the reactants is too fast for the chemical transformation to be achieved. Thus both the “unreacted” and the “reacted” states are stable for tc τ ti and a hysteresis cycle can be traveled up and down when τ is varied (other control parameters, such as the concentrations of the feed or the temperature can be used). This is the well-known classical temporal bistability in homogeneous systems. Spatial bistability is a direct extension to spatial chemical systems which are fed with fresh reactants by diffusion from boundaries, a standard way to sustain nonequilibrium reaction-diffusion patterns [23]. In this case, to avoid convective motions, the
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reactions are performed within a porous medium, usually a chemically inert gel. The fresh reactants are provided by diffusion from faces of the gel that are kept in contact with large reservoirs of feeding solutions. In many cases, all the reactants are provided in a well mixed unique solution by means of a CSTR kept at a low extent of reaction by operating at a short enough residence time. Such a reactor, commonly referred to as an OSFR (“one side fed reactor”) is characterized by a size l, the shortest typical distance of the gel core to the feeding edge. This can be, for instance, the thickness w of a thin gel plate fed on one face, the difference w between the outer and the inner radius for a flat gel annulus fed along the outer rim, or the radius of a long cylindrical or a spherical piece of gel immersed in the CSTR. The different chemical states are characterized by different concentrations profiles within the gel. If the reaction is a clock reaction kept within the CSTR in the “unreacted” state F, the arguments of the former paragraph can be applied to the contents of the gel. The characteristic time tD ∼ l2 /D for the reactants to reach the core of the gel by diffusion, where D is the typical diffusion coefficient of these reactants, defines the time scale over which the contents of the gel are renewed. Thus, for tD ti there exists a stable almost uniform concentration profile√corresponding to an unreacted state. When tD ti or equivalently, when l > lsup ∼ D ti the core of the gel switches to a “reacted” state. It follows, under the action of diffusion and of the autocatalytic process, an extension of this state over nearly the whole size l of the gel. This switch also occurs when l < lsup if a sufficient amount of X is already initially present within the core. However, since continuity must be satisfied at the feeding boundary, there is a boundary layer within which the composition changes from an unreacted state at the CSTR boundary to a reacted state in the deep core of the gel. To distinguish this spatial profile from the flat reacted profile where the CSTR and the gel would both be in the reacted state, it is referred to as the “mixed state”. When the thickness δ of the boundary layer becomes of the same order as the size l, the reaction can no longer be achieved in the core. Thus, if δ l < lsup , the reacted state profile and the mixed state profile are stable for the same set of concentrations in the CSTR, which defines the spatial bistability range. In most cases, tc is short, so that the chemical transformation is localized in a very narrow region: the X profile exhibits a steep front at a point P located at a distance δ from the feed boundary B, taken along a direction normal to this boundary (Fig. 1). Let us now, consider a substrate A that is consumed during the reaction. Since all the chemical transformation is concentrated at point P, the system is almost purely diffusive between points B and P., i.e. within the boundary layer. Thus, at stationary state, the concentration profile is linear and the gradient of A is given by ∆A/δ where ∆A is the change of the substrate concentration between the CSTR (unreacted state) and the reacted state (Fig.1). Equating the flux of A through the feed boundary surface S to the rate Q˙ of consumption of A, one gets the value of the boundary layer thickness [9]:
δ≈
Di |∆A|S Q˙ 4
(1)
In spite of the strong approximations above, it turns out that this formula is accurate with most of the realistic models of the reactions that have been studied. As a matter of fact, due to large differences between the different time scales, these approximations remain valid in most practical situations. For a given reaction, the dynamical behavior and the nonequilibrium phase diagram of the CSTR/gel system can be obtained by solving numerically the system of equations: ∂cih (ci0 − cih ) = fi (ch ) + + Gi (2) ∂t τ ∂ci = fi (c) + Di ∇2 ci (3) ∂t where ci0 , cih and ci are the concentrations of species i respectively in the input flow, in the CSTR, and inside the gel, Di and fi are terms that respectively correspond to the diffusion coefficients and reaction rates of the species. In the right hand side of Eq. (2), the first term gives the contribution of the reactions. The second one represents the input and output flows of the species. It contains all the expandable control parameters of the system, namely, τ and the ci0 . The third one represents the feedback of the gel contents on the CSTR and is given by the diffusion flow of the species through the surface of contact, S. Thus: Z 1 (4) Gi = Di ∇ci .n dS V S where n is the unit vector orthogonal to the boundary and V the volume of the CSTR. In most experiments, the volume of the gel is very small in regard to V and these terms are negligible, but they can be kept in the computations, to account for marginal cases [24]. In the unreacted state, they are always negligible since [∇ci ]S ≈ 0. In practice, the control parameter that defines the bistability range is not the geometrical size l but the composition of the CSTR input flow or the residence time τ . When the Gi terms are not negligible, the concentrations in the CSTR slightly differ when the gel contents are either in the unreacted or the mixed state, and spatial bistability must only be considered in reference to the control parameters of the CSTR, rather than to the concentrations of the CSTR contents. In this article, we shall present several examples of spatial bistability. When the diffusion coefficients are similar, the dynamics only exhibits spatial bistability. When the autocalytic species diffuses much faster or slower than the other species, there is an ambiguity on the definition of the “typical” coefficient D. We shall see that, without turning down spatial bistability, this can lead to additional dynamical behaviors. Changing size l is unpractical in bench experiments. However, we shall also see that using a specific gel geometry with a continuous change of l, such as a conical shape, allows for the direct determination of spatial bistability parameters related to the size. Moreover, when the size of the gel is chemoresponsive, the size changes induced by the chemistry and the switch between the two states can exert a mutual feedback that can 5
eventually lead to spectacular nonstationary chemomechanical behaviors. This will be the subject of the last paragraph.
2
EXPERIMENTAL TECHNIQUES
Prior to the presentation and analysis of the diverse observations in the following sections, a brief description of experimental methods is necessary. Figure 2 sketches the many OSFR geometries used so far in theoretical and experimental studies of spatial bistability and other related reaction-diffusion patterns. The essential part of these reactors – thin disk (Fig.2a), flat annulus (Fig. 2b), long cylinder (Fig. 2c), sharp cone (Fig. 2d), sphere (Fig .2e) – is usually made of 2% agarose gel. This relatively tough hydrogel acts as a chemically inert and mechanically stable porous medium. The porous medium quenches convective fluid movements and thus enables the development of chemical patterns resulting from the sole interplay between reaction and diffusion. The pieces of gels are immersed in CSTRs which provide uniform feed composition at the gel/CSTR boundary. This composition directly depends on the chemical state of the CSTR which is controlled through the input flow concentrations ci0 and residence time τ . The CSTR to gel volume ratio is typically greater than 50 so that, in most cases, the chemical activity in the gel has little influence on the composition of the CSTR. The CSTRs are thermoregulated and input flows are controlled by precision syringe pumps (P500 from Pharmacia). The contents of the CSTRs are monitored by Pt or pH electrodes. The concentration patterns of colored species in the gel part are monitored by CCD cameras connected to a time lapse video recorder and a computer with an image grabber. The different gel reactor geometries were developed for specific observation needs. The first OSFRs [25, 26] were thin disks fed by one face, the other face being pressed against a transparent impermeable wall across which the chemical state of the gel contents was optically monitored (Figure 2a). This spatial reactor geometry was assumed to approximate uniformly constrained two-dimensional systems, a geometric condition commonly used in model simulations. However, it was shown [26] that the third dimension, the thickness, could be neglected only for very special reaction systems. These are reactions that exhibit long batch oscillatory dynamics where the initial major reagents are sparingly consumed during each oscillatory cycle, like for the BZ and CDIMA reactions, together with special boundary conditions. The very large majority of CSTR oscillatory and bistable reaction systems [7] only exhibit simple “clock” behavior in batch, a situation that usually excludes the simultaneous excess of all major initial reagents. Indeed, after the “tick” of the clock, at least one of the initial reagent is nearly completely consumed! The system switches to a reacted state, usually close to the thermodynamic equilibrium state. In this case, as seen in the introduction, different concentration profiles may develop in the thickness of an OSFR and an integrated light absorption in the feed 6
direction is not always unambiguous [27]. Thin flat annular OSFRs (Figure 2b) expose the concentration profiles in the direction orthogonal to the CSTR/gel feed surface so that the absorption by the colored species can be directly measured between the feed boundary and the deep core. This makes the different states easier to discriminate. A flat annular OSFR can be thought as a section across a thin disk OSFR with periodic conditions along the feed interface. Other OSFR geometries with a periodic feed boundary –cylinder, cone, sphere (Figure 2c-e)– have been designed for specific experiments. As mentioned above, the cone geometry is convenient to rapidly test the relative stability of the states as a function of the size l = r. In chemomechanical experiments (section 5), the cylinders of gels are made of a cross-linked copolymer of N-isopropylacrylamide and acrylic acid, a pH responsive gel. This suspended cylindrical geometry was used because it allows for unconstrained changes in size and shape along the axis. The spherical geometry is technically difficult to handle in experiments and was only investigated in numerical studies [19], up to now.
3
STANDARD SPATIAL BISTABILITY IN THE CDI-PVA SYSTEM
The theoretical concept and gross properties of spatial bistability were initially tested [15] on a formal kinetic model inspired by the “minimal bromate-bromide oscillator” reaction [28,29]. The first experimental observations [9] were made on the oxidation reaction of iodide ions by chlorine dioxide in the presence of polyvinyl alcohol (CDI-PVA). The reaction was chosen for its prototypical “clock” behavior in batch conditions, the high color contrast between the reacted and unreacted states due to the formation of dark purple PVA-polyiodide complexes and the fact that excellent kinetic models were available [30, 31]. Also, it is connected to reaction systems that can exhibit stationary Turing patterns [1]. In the absence of PVA, the chlorine dioxide-iodide reaction operated in a CSTR exhibits a large domain of oscillations and a small domain of bistability between two branches of steady states. However, introducing PVA quenches oscillations and greatly increases the extent of the bistability domain. One steady state is a high iodide and high iodine branch, stable at high input iodide concentration [I− ]0 and at high flow rate. It corresponds to the flow branch F which is characterized by a low extent of reaction. The other steady state is a low iodide, low iodine branch which is stable at low iodide input feed and low flow rate. This is a branch of large extent of reaction referred to as the thermodynamic branch T. In the presence of PVA the F and T states of the CSTR are respectively colored dark purple and clear yellow. Inside or in the vicinity of the parameter domain of CSTR bistability, a flat annular OSFR fed with compositions of the F branch of the CSTR can exhibit two stable PVApolyiodide color profiles between the inner and outer rims, as illustrated in Fig. 3. Just as depicted in the introduction, one state is quasi uniform (Fig. 3a) and corresponds to a spatial state with a low extent of reaction, the F state of the gel. The other exhibits 7
a sharp color switch from dark to clear and corresponds to the mixed state M of the gel (Fig. 3b). The domains of stability of these two states were determined by decreasing or increasing [I− ]0 for OSFRs with different width w (Fig. 4A). As predicted by theory, the stability of these states in the gel, in particular the F state, sensitively depends on the width w of the annulus. At high and low [I− ]0 , the F and M states are respectively the only stable states of the gel. In the intermediate range of [I− ]0 , the domains of stability of the two states of the gel can overlap, for the same feed composition of the CSTR, and give rise to spatial bistability [9]. The extent of the spatial bistability decreases when w increases from 0.5 to 2 mm. At low [I− ]0 , the stability limit of the M state is determined by the stability limit of the F state of the CSTR (noted LF ). At high [I− ]0 , the M to F state transition is slightly shifted to higher [I− ]0 when w increases. The stability limit of the F state sharply varies with w (dashed line on Fig. 4A), from w = 0.5 mm where the gel and the F state limits LF of the CSTR cannot be distinguished, to w = 2 mm where there is almost no overlap with the M state. Numerical calculations based on the detailed scheme proposed by Lengyel et al. [31] and taking into account the actual experimental setting (i.e. a gel of thickness w diffusively connected to a CSTR) were performed [9]. The computed phase diagram presented in Fig. 4B is in quasi-quantitative agreement with the experimental results, Fig. 4A. In particular, the right domain of spatial bistability is obtained. In addition, calculations predict that below a critical value, w < winf = 0.039 mm, a size not accesible in our experiments, no mixed state can be observed, whatever the value of [I− ]0 , so that spatial bistability disappears. In the domain of parameters where both F and M are stable, it is possible to set different parts of the gel in the two different states and study the relative stability of the two states as a function of control parameter. Not surprisingly the stability of the F state is favored by increasing values of [I− ]0 (Fig. 3d), while that of the M state is favored by a decrease (Fig. 3c). This was the first experimental implementation of a tunable control of the relative stability between two stable states in a nonequilibrium bistable chemical system [9]. This aptitude will be used in section 3.1 to produce stationary interface pairing. Interestingly, the aspect ratio of this interface changes as a function of control parameter, since as illustrated Figs. 3c and 3d, the size δ of the boundary layer depends on the control parameter through Q˙ (Eq. 1). Similar changes in the shape of the interface are also found in model calculations [15]. Clearly, the system thickness cannot be discarded to account for the dynamics and the pattern mode selection mechanisms in true open spatial reactors. Note that, consistently with a no-flux boundary condition, the connection of the high and low iodide part of the gel becomes orthogonal to the impermeable wall at the inner rim of the annulus. This naturally introduces a curvature at the interface between state F and M.
8
4
PHENOMENA INTRODUCED BY DIFFERENTIAL DIFFUSION
In this section, we elaborate on the new phenomena that emerge in spatial bistable systems when the main activatory species diffuses significantly slower (short range activation) or faster (long range activation) than the species driving the inhibitory or moderating processes. In the standard inorganic chemistry used for bistable and oscillatory reactions all the species have approximatively the same mobility in aqueous solution within a factor 2, with the exception of the hydroxyl ions and the protons which have a mobility a factor 5 to 10 higher than the others. Thus, long range activation instabilities will be naturally found in pH driven self-activatory reactions. Two such examples will be provided. On the other hand, it is known since the first unquestionable discovery of Turing patterns in a chlorite-iodide-malonic acid reaction [1] and from the subsequent theoretical analysis [32, 33] that short range activation can be obtained by addition of uniformly distributed immobile (or of very reduced mobility) functional sites able to reversibly bind the activator of the reaction. The effective diffusion of the bound species decreases, while the others are left unchanged, leading to a situation where the inhibitor may have a higher diffusivity than the activator. Theoretical developments show [32–34] that, the rescaling of the diffusion coefficient D is given by Deff =D/σ with S0 (5) σ =1+ KS where S0 is the complexing agent concentration in large excess, and KS is the complexation equilibrium constant with the targeted autoactivatory species. In addition, if the bound form of the activator is kinetically inert, the introduction of such binding sites would also quench kinetically induced oscillatory instabilities. This property was used in the CDI-PVA system described in the previous section to extend the domain of CSTR bistability at the expense of the domain of oscillations.
4.1
Short range activation
It was envisioned that in spatial bistable system, it would be possible to create stationary pulses of one of the stationary states immersed in the other when two interfaces driven to head on collision have strong enough mutual repulsion to prevent their mutual annihilation. Though, in the previous CDI-PVA system some repulsion between colliding interfaces could be induced by the addition of PVA, all attempts to produce stationary pulses failed even using PVA concentrations as high as 30 g/dm3 . Remarkably, in the related monostable CDIMA reaction, stationary Turing patterns were observed for PVA concentrations as low as 0.5 g/dm3 [26]. In an endeavor to understand the possible connections or interactions between spatial bistability and Turing patterns, a combination of the CDI-PVA and CDIMA reactions was studied. Below, we describe in what conditions short range activation could lead to the formation of stationary pulses and labyrinthine patterns in the spatial bistable regime [18] of this 9
involved system, here on referred to as the CDIIMA reaction. We also show that these “large amplitude” patterns have no direct connection the previously observed Turing patterns. This reaction was studied as a function of the concentration of iodide and malonic acid in the feed flow. The feed concentration of chlorine dioxide and iodine are fixed. In the absence of malonic acid feed, the system is similar to the CDI-PVA reaction. At the high [PVA]0 values used in these experiments, the CSTR dynamics show bistability between a clear color state (low polyiodide-PVA complex), stable from [I− ]0 = 0 to [I− ]0 = 0.55 × 10−3 M, and a dark color state (high polyiodide-PVA complex), stable at high [I− ]0 and down to [I− ]0 = 0.85 × 10−3 M). Following our initial convention, we call T and F respectively the low iodide and the high iodide states. Introducing MA reduces the range of the bistability of the CSTR and for [MA]0 > 0.7 × 10−3 M, small amplitude Pt-potential oscillations are detected in a narrow region of parameter that does not overlap the spatial bistability region. At [I− ]0 = 0 M, in the presence of MA, the system is identical to the CDIMA reaction. No bistability is observed in the CSTR as a function of [MA]0 . A smooth increase in the polyiodide-PVA complex is observed with the increase of [MA]0 . In an OSFR, a finite domain of Turing patterns is observed. This domain is delineated in the right part of the phase diagram shown in Fig. 5. These patterns are observed when the contents of the CSTR does not belong to the F state branch (for a more detailed description see [18]). Below, we focus on the dynamics of this CDIIMA reaction in an annular OSFR when the contents of the CSTR belong to the F branch. Front interactions in an annular OSFR
When the CSTR contents belong to the F branch, two different stable color profiles can coexist over a finite range of parameters (Fig. 5). This is the standard spatial bistability phenomenon between F and M states. At low [MA]0 values, the two spatial states look much alike those observed above with the CDI-PVA system (Figs. 3a and 3b). However, when [MA]0 is increased the M state takes gradually new features. An additional dimer purple-blue stripe emerges at the inner rim of the annulus, as illustrated in Fig. 6a. This new feature is not due to a reaction-diffusion instability but to simple kinetics. The dark inner stripe corresponds to an iodide recovery process associated to the I 2 iodination reaction of malonic acid. Using an extended version of the Lengyel-Epstein model of the CDI reaction [31], it is possible to quasi-quantitatively account for the experimentally observed phase diagram (Fig. 5) and the different aspects taken by the M state [24], in particular at high [MA]0 as illustrated by the calculated [I− ] profiles in Fig. 6b. In all cases, the M state looses stability and the gel contents switches to the F state beyond a critical high value of [I− ]0 or of [MA]0 . Note that in this quite involved system, the domain of spatial bistability between the F and M states of the gel considerably exceeds that of the CSTR bistability domain, so that spatial bistability may be observed in conditions far from those where 10
temporal bistability is observed in a CSTR. In the range of spatial bistability, a local perturbation with an acid solution of chlorite can create interfaces between the F and the M states. As already found with the CDI-PVA reaction, the direction of propagation of the interface sensitively depends on the actual value of [I− ]0 . In the major fraction of the spatial bistability domain, M propagates into F. The inversion values for the propagation direction are located close to the high [I− ]0 limit. Near the location of the inversion line for the direction of propagation of the interface (not shown in Fig. 5) and at the highest values of [MA]0 , stable front pairing phenomena between opposing repulsive interfaces can be obtained as shown in Fig. 7. One can observe the evolution of four interfaces. In the first snapshot, two opposing interfaces are far from each other while two others are very close. The sequence of snapshots shows that the pairs of interfaces located far apart move towards each other at ∼ 0.8 mm/hour, while those which are in a close together pair do not move. In the last snapshot, the distance between the two pairs of interfaces are similar and at halt. Depending on the experimental conditions, the stationary interface-pair distance ranges from 0.5 to 0.7 mm. Stable pulses of this type persist for as long as the control parameters are unchanged. Their distribution along the annulus is not periodic and depends on the initial conditions. The widths of these standing pulses are larger than the wavelength of the Turing patterns (∼ 0.2 − 0.3 mm) observed in this system. Moreover, as already mentioned the CSTR composition for the observation of Turing patterns is totally different from that leading to spatial bistability. There is no direct relation between the two types of standing structures. At lower values of [MA]0 the repulsion between interfaces is not strong enough to prevent the slow mutual annihilation of the colliding interfaces. The numerical simulations confirm that a small domain of stable pulses emerges at high enough [MA]0 [24]. In agreement with the experiments, simulations in a long rectangular area with Dirichlet boundary conditions along one long edge and no-flux boundary conditions along the other edges show that a M state propagating into a F state stops as the interface approaches the right side boundary (Fig. 8). The no-flux boundary condition on this wall acts as a mirror for the propagating interface which stops at ∼0.8 mm from its mirror image, in quantitative agreement with experimental observations (Fig. 8a). In the simulations, when the front is rigidly shifted to the right or to the left, it returns to the initial rest position, which demonstrates that this state is stable (Fig. 7). Transient Labyrinthine patterns in a disc OSFR
In a disk shaped OSFR, an “interface line” between M and F states can be initiated by brief appropriate back and forth jumps in [I− ]0 , when the gel contents are in state F. Typically a smooth interface line starts to develop from the edge of the disc. This interface can develop a transversal morphological instability and lead to a wavy interface. As before, the speed of propagation of the interface can be changed by tuning [I− ]0 . When the propagation of the interface is slowed, the wavelength of the wavy structure 11
increase and the amplitude decreases [27]. At high enough [MA]0 , it is possible to find conditions where the direction of propagation depends on the curvature sign of the interface, as illustrated in Fig. 9. In this situation, both the M and the F states are able to grow fingers into one another. When, after a small change in control parameter value, fingers start to grow in opposite directions, as in Fig. 9, the M state also becomes globally unstable. Large patches of F appear spontaneously inside the M regions. In Fig. 10, the first snapshot, taken several hours after the initial perturbation, shows islands of F in a gel that is mainly in the M state (darker regions). At this value of [I− ]0 , F propagates into M, so that the islands of F grow. The outburst of a new island is viewed in the region marked by an arrow in Figs. 10b and 10c. From the evolution of this new patch, one can see that there is a strong repulsion between the limits of adjacent islands of F and a strong dependence of the speed of the interface propagation on curvature. The curved part at the top of the patch moves faster than the flatter parts. Head to head interfaces lock for a while at a critical distance. However, the whole disc ultimately fills uniformly with the F state. In the region of parameters explored (Fig. 5), we found no set of values for which domain patterns would definitely stabilize although stable repulsive pulses were observed in the annular OSFR. The increased dimensionality seem to lower the stabilization capacity of patterns due to front curvatures in the additional dimension.
4.2
Long range activation
In this section, it is shown that reaction systems exhibiting only steady state bistability in a CSTR, can however lead to oscillatory and excitable states when operated in an annular OSFR if the activatory species of the reaction diffuses faster than any other substrate in the system. The emerging dynamic states are the result of a genuine reaction-diffusion instability and not from a direct kinetic instability as in the case of uniform oscillatory reactions. As mentioned above such phenomenon naturally develop in acid autocatalytic reactions as examplified in the chlorite–tetrathionate (CT) and iodate–sulfite (IS) reactions. Dynamics of the CT reaction
The CT reaction is an experimental system frequently used to investigate autocatalytic acid front propagation in unstirred batch conditions [35]. Although the detailed kinetic mechanism of the CT reaction is complex [36], the overall balance equation of the reaction is appropriately represented by: 2− 2− − + 7 ClO− 2 + 2 S4 O6 + 6 H2 O −→ 7 Cl + 8 SO4 + 12 H
(6)
and in a narrow range of stoichiometric conditions, fulfilled in the experiments pre-
12
sented here, the driving kinetics can be conveniently described by the following empirical rate law [37]: 1d − 2− + 2 [ClO− (7) vk = − 2 ] = k[ClO2 ][S4 O6 ][H ] 7 dt The CT reaction operated in a CSTR exhibits bistability between a branch F of alkaline states (pH∼8), and a branch T of acid states (pH∼ 2) [17, 38] as function of [NaClO2 ]0 and [K4 S4 O6 ]0 as well as the pH of the input feed. The ratio α between an acid and alkaline feed flow was used as control parameter for experimental convenience. The pH of the feed mixture decreases as α increases from 0 to 1. The actual feed concentrations associated to α are given in the caption of Fig. 11. In the conditions of the experiments reported in Fig. 11, the F state of the CSTR is stable from α = 0 to α = 0.83 while the T state is stable from α = 1 down to α = 0.18. As it is standard with spatial bistability phenomena, when the CSTR contents are maintained on the branch of F states, two different concentration profiles corresponding to the F and M states of the gel are observed in the gel and their domain of stability can overlap. A series of experiments was performed in annular OSFRs, with widthes w ranging from 0.5 to 3.0 mm (Fig. 11). The points correspond to experiments while the lines correspond to the numerical calculations detailed below. The chemical states of the gel are make visible by introducing bromophenol blue in the feed: This makes the acid part clear yellow and the alkaline part dark purple. The F state is then characterized by a quasi uniform dark color since the composition of the gel does not significantly differ from that of the CSTR. In contrast, the M state exhibits a typical sharp color front between an alkaline dark boundary layer and a clear acid core. On decreasing α, the radial extension of the alkaline boundary layer eventually starts to grow. At a critical value, the acid/base switch position δ becomes unstable and starts to oscillate, moving back and forth across the width of the annulus. The parameter domain over which these oscillations are observed is usually very small (Fig. 11). On decreasing α beyond this oscillatory region, the gel contents turns to the F state. In the region of parameters where the two states of the gel overlap for the same composition of the CSTR (Fig. 11), it is possible to prepare different parts of the gel in either state. Contrary to the previously described observations on CDI related systems, in the range of α corresponding to the spatial bistability domain, the relative stability of the M and F states does not depend on the value of the control parameter. The M state always takes over the F state, even at the very limit of the domain of stability of the M state at low α. In fact, even when the F state is the only asymptotically stable state, an acid perturbation can lead to a F/M state interface that propagates undamped into the F state. However, behind this interface the acid region survives only during a finite period of time. Ultimately, the F state totally recovers. As a result, a traveling pulse forms (Fig. 12). This property defines an excitable F state. In Fig. 11, the computed limits (full and dashed lines) of the bistability domain including the oscillation area and the excitable region of the F monostable domain were made by using a model based on a balance equation (Eq. 6) and the empirical 13
rate law (Eq. 7) supplemented by the acid-base equilibria of water and hydrosulfate ions and the faster diffusivity of H+ and OH− (for the details of the numerical simulations see ref. [17]). The diagram is in good quantitative agreement with the experimental observations. Note that just as in the case of the CDI-PVA system, calculations predict a lower size w limit for the observation of the M state. But, contrary to the case of the CDI-PVA reaction, the spatial bistability persists at large value of w. This results from the extremely slowness of the reaction in the alkaline F state, leading to a quasi infinite induction time. Thus, one cannot switch spontaneously from the F state to the M state by changing w over reasonable experimental sizes; an external acid perturbation is necessary. Since the model predicts no oscillations in an homogeneous system, it is clear that the calculated oscillations and the related excitability properties can only result from the competition between the diffusive transport and the reactive processes. Actually, the long range activation due to the fast diffusion of the proton is responsible for the destabilization of the M state as it is clearly evidenced in various simple models [17,21, 22]. In particular, Benyaich et al. have performed a bifurcation analysis of the Hopf bifurcation leading to the oscillations in a case where some analytical approximations are possible [21]. Testing the effect of size on the characteristics of states and their relative stability is experimentally very tedious in systems with uniform size since this geometric parameter cannot be continuously tuned at will. For experimental convenience, conical shape reactors were developed. In this reactor geometry, it is possible to directly establish the critical coexistence size of the two spatial states at a given feed parameter and, conversely, to get a very good approximation the critical feed value for the stabilization of an F/M state interface at a given system thickness l = r [39, 40]. Dynamics of the IS reaction
The CT reaction presented above was the first system to exhibit spatiotemporal instabilities as a result of long range activation by protons. An interesting feature is that it can be operated under conditions where the reaction kinetics can be accounted in simple terms. However, the extremely long induction time of this reaction excludes that the large size limit of the spatial bistability domain be reached in practice. The reduction of iodate ions by sulfite ions is another long known autocatalytic reaction [41]. It is autocatalytic both with protons and iodide ions. For a typical initial composition it exhibits induction periods in the minute to tens of minutes range. The kinetics of this reaction is complex, but the dominant positive feedback process is the autocatalytic oxidation of hydrogen-sulfite which is enhanced by increasing acidity [42]: − 2− − + IO− 3 + 3 HSO3 −→ I + 3 SO4 + 3 H
(8)
while the iodide autocatalytic process become important only at the end of the oxida-
14
tion of sulfite when the pH drops below 5: − + IO− −→ 3 I2 + 3 H2 O 3 + 5 I + 6 H − + I2 + HSO3 + H2 O −→ 2 I− + SO2− 4 + 3 H
(9) (10)
According to Eq. 9 which corresponds a proton consuming process, the dynamics of 2− the IS reaction depends on the initial iodate/sulfite ratio q=[IO− 3 ]0 /[SO3 ]0 . In excess of sulfite ions (q ≤ 1/3), it acts similarly to the CT reaction. But, when q ≥ 1/3 (excess of iodate ions), in a closed system, the pH drops down from 7 to 2.5 (clock behavior) but then increases back to values up to 5.5, due to slow proton consumption through Eq. 9. In the range of feed concentrations of the phase diagram (Fig. 13), the IS reaction operated in a CSTR exhibits bistability between a F branch of states (pH∼ 7), and a T branch of states (pH∼ 3) as a function of the concentrations of the input reagents and acid in the feed. For experimental convenience and notation simplification h=[H2 SO4 ]0 /mM is used as an expendable control parameter. As in the case of the CT reaction the pH changes in the gel are made visible by introducing bromophenol blue as a color indicator. Figure 13 presents the experimentally established phase diagram in the (h, q) plane when the contents of the CSTR belong to the F branch. This section of phase diagram emphasizes the role of the stoichiometric iodate/sulfite ion ratio on the nature and relative stability of states. Spatial bistability, oscillations and excitability can be observed like in the case of the CT reaction. However, the domain of existence of the oscillatory state forms a bubble in the phase diagram (Fig. 13), connected in a typical “cross-shaped” topology [44] with the domain of the spatial bistability. In the domain of spatial bistability the stable stationary pH color indicator profiles are similar to those observed with the CT reaction. However, for q values above 1/3, the M state is characterized by a clear stripe sandwiched between two darker regions, for reasons analogous to what was already observed in the CDIIMA system (Fig. 7). According to Eq. 9, the pH increases again as the extent of reaction increases with the distance to the feed boundary. In the domain of the spatio-temporal oscillations, when the F state looses stability the inner part of the gel suddenly becomes acid. A standard front between an alkaline (dark) and an acid (clear) part forms, but this front moves back and forth in the depth of the gel (Fig. 14). The spatial amplitude of the oscillations varies as a function the control parameter. This amplitude can be as large as w, the width of the annulus. By increasing h the following sequence can be observed: stable state F −→ large amplitude periodic oscillations −→ large amplitude aperiodic oscillations −→ small amplitude oscillations −→ stable M state. Within our experimental accuracy, no hysteresis is observed between the oscillatory state and the stable F or M states. Next to the boundary of the oscillatory domain, in the vicinity of the “cross-point” of the phase diagram, a local acid perturbation of the F state can initiate an undamped traveling acid pulse, similar to the excitability waves observed with the CT reaction (Fig. 12). 15
If an interface between the F and the M states is created in the spatial bistability domain close to the “cross-point”, the M state always invades the F state. However, at high q, contrary to the CT reaction, the propagation direction of such interface can be reversed by changing h within the spatial bistability domain. At this stage, the origin of the spatio-temporal instabilities may appear ambiguous in the IS reaction. Among other sources, the complexity of this system comes from the fact that, in excess iodate, the overall proton production is decreased through Eq. 9 and that at large excess sulfite the pH is buffered by the protonation equilibrium of the hydrosulfate ions left over. Preliminary model calculations [45] with a simple kinetic mechanisms based on the competing overall processes (Eqs. 8-10) and the rate laws proposed by Rabai et al [42], well account for the standard spatial bistability phenomenon, but not for the oscillatory pocket at low q, even when the ion charge balance and the differential diffusivity of ions are taken into account. Though some oscillatory relaxation behavior was obtained, the development of a cross-shaped diagram seem to require more subtle models with refined description of the mobility of the different ionic and non ionic species. Yet, there are indications that here again the dynamical instabilities are promoted by the fast mobility of activatory protons [46]. As we shall see below the effective slowing down of the proton mobility quenches oscillations in both pH driven reactions.
4.3
Quenching long range activation by complexing agents with reduced mobility
As mentioned in section 4.1 (§short range activation), the introduction of a species of reduced mobility that would bind reversibly the activator reduces the effective diffusivity of this activator. In the case of protons, this can be obtained by introducing long polycarboxylic chains into the gel and in the feed solutions. It was shown experimentally and by simulations that even a small amount of such a binding agent can significantly decreases the diffusivity of the protons [47] in agreement with Eq. 5. The proton activated reactions offer a privileged experimental situation where it is possible to tune the relative activatory/inhibitory space scales from long-range to short-range activation.
Case of the CT reaction The effects of introducing sodium-polyacrylate (PA) (with a molecular mass of 20 000D) in the gel and in the feed solution of the CT reaction are summarized in the (α, [-COO− ]0 ) phase diagram (Fig. 15). The concentration [-COO− ]0 corresponds to the carboxylate function concentration actually introduced with PA. As seen in the previous paragraph, in the absence of PA, there is a narrow region where the boundary layer δ of the M state oscillates at the low α stability limit of this M state. Below this limit, the monostable F state is excitable. The introduction of increasing amounts 16
of [-COO− ]0 , makes the width of the domain of oscillations of the M state and of the domain of excitability of the F state rapidly decrease. Remarkably, the low α stability limit of the stationary state M does not depend upon [-COO− ]0 . However, the high α stability limit shifts out of the present experimental range with the rapid increase of stability of the F branch of the CSTR. Numerical calculations made with the CT model described in ref. [43] generate a phase diagram (Fig. 16) in remarkable agreement with experimental observations (Fig. 15). In particular, the quasi-invariant stability limit of the M state at low α is well accounted. This surprising behavior can be understood in the following way. In the introduction, we have seen on the basis of simple arguments that the M state looses stability when δ is close to l=w, the thickness of the gel. The value of δ (Eq. 1) directly depends on the concentration of the substrate at the feed boundary (in practice the concentration of the substrate in the feed-flow of the CSTR) and on the ratio of the diffusion coefficient D of the activatory species to the consumption rate Q˙ of the substrates which are driven by the activator. When a certain amount of complexing agent is added, the diffusion coefficient D and the consumption rate Q˙ of the autocatalytic reaction are both divided by the same factor σ (Eq. 5). Thus, according to Eq. 1 the critical value of δ (∼ l) is reached for the same value of the substrate in the feed of the CSTR. The low α stability limit of the M state is basically independent of the added amount of complexing agent. This invariance is similar to the result obtained by Lengyel and Epstein [32] and, Pearson and Bruno [33] for the threshold of Turing structures [26]. It is noteworthy that, for [-COO− ]0 ≥ 0.02 M and inside the spatial bistability domain, an interface between the F and M states can be made to reverse by tuning a control parameter, like it was the case for the CDI-PVA or CDIIMA systems. Using this newly acquired property, we searched for stable interface pairing at [-COO− ]0 = 0.1 M to produce stationary pulses in the same way we did with the CDIIMA system but the quest for stationary pulses remained unsuccessful. Even if repulsions were detected –i.e. interfaces slow down when they approach each other– the opposing interfaces would ultimately slowly merge. Though only a small domain of parameter space was explored with this goal in mind, we anticipate that producing such type of stationary pulse is very unlikely in such straightforward autocatalytic reaction system. We presently think, by analogy with the case of the CDIIMA reaction, that an additional proton consumption reaction is needed to compensate for the slow escape of protons linked to the introduction of the polycarboxylate molecules.
Case of the IS reaction In this other reaction, the addition of polyacrylate to concentrations of carboxylate functions [-COO− ]0 ≥10−3 M quenches the oscillatory domain which is then replaced by a narrow extension of the spatial bistability domain that eventually resumes at low q 17
values. This indicates that the fast diffusion of the protons plays a significant role in the mechanism of these oscillations. As in the case of the CT reaction, no standing pulse was observed with the IS reaction, in an OSFR. However, it is interesting to remind that labyrinthine patterns were observed in a disc OSFR with the FIS reaction [2, 48]. This latter reaction differs from the IS reaction by the addition of ferrocyanide which competes with sulfite to reduces iodate but the redox steps with ferrocyanide consumes protons. This proton scavenger species could then reinforce the depletion of protons in the core of the OSFR and would stabilize localized domains of F state in a M state background. This is just the type of mechanism that has helped the stabilization of a standing pulse the CDIIMA system (see §4.1).
5
CHEMOMECHANICAL INSTABILITIES
Gels are polymer networks swollen by a solution. In some cases, the amount of solution uptake, and consequently the volume of the gel, strongly depends on the composition (pH, nature of solvated ions, ionic strength,...) of the solution. Such gels are called “chemically-responsive gels” [49, 50]. Whereas non-responsive gels have been widely used as porous media to observe various reaction-diffusion patterns, the recent introduction of chemically responsive gels in the field of nonlinear chemistry has brought the prospect of creating stationary or dynamical shape patterns of gels [51–53]. This is particularly interesting in the case of spatial bistable systems since, in this case, the selection of one steady state can be controlled by the width of the gel through the parameter l (section 1). In addition, the two states exhibit very different concentrations profiles except within the thin boundary layer of width δ, so that the gel can shrink or swell almost as a whole, inducing large volume changes. Thus, appropriate chemically responsive gels can dynamically adapt themselves to the establishment of a chemical “steady” state by a volume change, that might in turn destabilize this steady state. The emergence of new instabilities resulting from the cross-coupling of spatially bistable systems and size changes was recently evidenced numerically and experimentally [20, 56]. These instabilities have been called “chemomechanical” instabilities. They are expected to give rise to a variety of stationary and dynamical shape changes in the gels. In the present section, we will first describe a simple heuristic model for the production of chemomechanical instabilities based on spatial bistability. Then, we will present numerical results illustrating this model and eventually experimental observations when performing the CT reaction in an OSFR with a responsive gel. Heuristic model. A recent theoretical and numerical study demonstrates that volume oscillations of a spherical piece of chemically responsive gel can be obtained through coupling with an autocatalytic chemical reaction [19, 57]. Consider such a reaction in an OSFR made of 18
a sphere of nonresponsive gel of radius R immersed in a CSTR with the contents kept on the unreacted F branch of state. Suppose that in the gel, as for the CT reaction, large amounts of autocatalytic species x are produced in the core of the gel. In this simple case, x = xc is highest at the center of the gel and remains high in a large part of the core and then decreases to reach the value (x = xh ) at the gel/CSTR boundary. Conversely, in the unreacted F state, x remains very low everywhere within the gel. As displayed in Fig. 17, two quite different compositions may coexist in the center of the sphere as a function of R. Rinf and Rsup are respectively the limits of stability of the M and F states. Now, suppose that the sphere is made of a chemically responsive gel that can swell up to a radius Rswoll when x is low and shrink down to Rshrunk when x is high. Obviously, the sphere will be respectively in a swollen or skrunken state when the chemical composition in the gel corresponds to the F or M state. If, when the gel is in the shrunken state, the sphere radius Rshrunk is shorter than Rinf , the M state cannot be sustained in that sphere. This causes a switch of the sphere contents to a F state (low x) and the sphere slowly swells. If Rswoll > Rsup , this process is not brought to completion. Instead, when the radius goes beyond the value Rsup (point D in Fig. 17), the system rapidly switches to the reacted state, at point A, causing the sphere to shrink again and follow the high x branch until R = Rinf (point B) where x suddenly decreases to the lower branch, at point C. This process is repeated indefinitely, producing relaxation oscillations. The resulting chemical cycle is given by the sequence D → A → B → C → D → . . . in Fig. 17. Figure 18 shows the corresponding time evolution of xc and the radius R of a sphere computed with the model described below. Quantitative numerical analysis. A comprehensive numerical model was developed to support the chemomechanical oscillatory mechanism described above. This model uses a two-variable autocatalytic chemical system with the following reaction-diffusion equations : ∂x 12 2 2 = x a + ∇2 x ∂t 7
∂a = −x2 a2 + ∇2 a, ∂t
(11)
where x and a are respectively the concentrations of the autocatalytic species X and of a substrate A. Note that in the present equations, the two diffusion coefficients that appear in Eq. (11) were set equals (Dx = Dy = 1). In such conditions, the system can asymptotically be reduced to a one variable system and therefore cannot develop oscillations when R is fixed [17]. It only exhibits spatial bistability as represented in Fig. 17. The size dependence of the chemically responsive gel upon the chemical state is introduced in a phenomenological way, using the classical Flory description of polymer gels [54, 55]. The generalized chemical potential describing equilibrium properties of the gel is the sum of two independent terms ∆µ = ∆µmix + ∆µel , which represents 19
respectively the contributions of the mixing and of the elastic forces in the generalized chemical potential: ∆µmix = RT (φ + ln(1 − φ) + χφ2 ), ∆µel = Knet RT [(
φ 1/3 1 φ ) − ( )], φ0 2 φ0
(12) (13)
The constant Knet is characteristic of the structure of the polymer network, φ is the volume fraction of polymer within the gel and φ0 the volume fraction in a reference state that depends on the initial preparation of the gel. The Flory parameter χ represents the energy of interaction between the polymer network and the solution and plays a major role in the swelling properties. In the model, it is assumed that the equilibrium swelling state of the gel is completely described by the dependence χ(x). The function χ(x) is taken as a sigmoid to represent at best the swelling equilibrium states of the responsive gel. Modeling the swelling dynamics of the polymer network is less straightforward [58] than the equilibrium swelling properties. However in the case of a sphere of gel with only moderate isotropic deformations and no tangential instabilities, one can use an approach based on a Maxwell-Stefan description. It is assumed that each point of the network is submitted to a driving force caused by the gradient of the chemical potential: Fdr = −∇(∆µ)
(14)
that is balanced by a friction force resulting from the drag exerted by the creeping flow of the solvent against the network: Ffr = φG(φ)(ξS − ξP )
(15)
The function G(φ) describes the dependence of friction with polymer density, ξS and ξP are the velocities of the solvent and of the polymer respectively. Details on the functions used in this model, on its limitations and on quantitative computations are reported in [57]. The data used in Fig.17 correspond to this model and to the parameters that define the sigmoidal shape of function χ. The computed oscillations of the sphere radius and of the concentration xc at the center of the sphere are reported in Fig.18. The different stages of the oscillations are labelled with letter points in correspondance with their location on the hysteresis cycle of Fig.17.
Experiments. For technical reasons, it is difficult to run clean experiments on small spherical gels. Thus, experiments have been conducted using long cylinders of gels suspended by one 20
end in a CSTR [20]. In these experiments pH sensitive poly(N-isopropylacrylamide-coacrylic acid) gels were used and associated with the acid autocatalytic CT reactiondiffusion system described in section 4.2. These gels swell when pH increases and shrink when pH decreases. Sharp changes of volume occur at pH∼4.5. Thus the system swell and shrink by a large amount when it switches to the F (alkaline) or M (acid) state respectively. In addition, the carboxylic acid functions polymerized within the polymer network quench the long range activation process and insure that no oscillatory instability can occur through reaction-diffusion only [43]. As mentioned before, even in experimentally reasonable large pieces of gel, the CT reaction-diffusion system never switches spontaneously to the reacted state because the induction time of the reaction is too long. In other words, we can consider that Rsup shifts to quasi-infinity so that the right half of the curves cannot be reached for reasonable gel sizes since one has always Rsup > Rswoll . As a consequence, chemomechanical oscillations such as described in the model above are not possible. However, if starting from a swollen gel in the unreacted state (e.g. point S in Fig. 17), a supercritical perturbation ∆x (acid here) shifts the composition to some S1 value in the gel, the autocatalytic process will drive the composition all the way to the reacted state branch at S2 . From there, the associated shrinking of R can shift the system to B on the hysteresis loop (Fig. 17) where it jumps to C and swells back. However, the process stops at the initial position S. A self-oscillatory cycle cannot be completed. This sequence of events S forced to S1 , followed by the spontaneous transformation sequence (S1 → S2 → B → C → S) characterizes an excitable behavior of chemomechanical origin. A typical sequence of experimentally observed stationary and dynamic properties of the states of the gel is displayed in Fig.19 when the pH responsive cylinder of gel is immersed in a CSTR fed with the CT reaction. The contents of the CSTR are on the unreacted F state. Whatever the value of the control parameter, if the gel is initially alkaline or neutral, it settles in a stable transparent and swollen state. However, when a strong acid perturbation is applied at the free end of the cylinder, the gel can either switch back to the initial swollen state or settle in a stable collapsed turbid state, depending on the value of the control parameter. These states correspond to the previous F and M states of the gel but, here, in addition to the differences in chemical composition, they are associated to changes in the degree of swelling and turbidity. As before, the stability domains of these states continue to overlap over a finite range of parameter (concentration of OH− in the total input flow). Interestingly, in the whole bistability domain, when an interface is created between the shrunken and swollen states, the shrunken state always propagates into the swollen state. A stationary interface, or an interface moving in the opposite direction was never observed, eventhough, the concentration of carboxylic functions on the gel polymer network are much higher than those established in fixed size agarose gels loaded with polyacrylic acid (§4.3). Furthermore, in a very narrow range of parameter values (Fig. 19) at the limit of the spatial bistability domain very large amplitude complex oscillatory gel contraction waves are observed. For a detailed description of these waves and tentative 21
explanation see reference [20]. Just beyond the spatial bistability limit (Fig. 19) in the domain of the monostable swollen F state (Fig. 19), the acid perturbation initiates a transient contraction wave that propagates undamped into the unperturbed part of the cylinder, as illustrated in the series of snapshots in Fig. 20. The contraction locally reaches up to 50% of the diameter in the swollen state. Behind the contraction front, the gel slowly relaxes back to its initial swollen state. In the absence of long range activation mechanism, the propagation of polymer density waves can be accounted for by the chemomechanical excitability mechanism described above by the (S→ S1 → S2 → B→ C→ S) transformation sequence (Fig. 17) extrapolated to the extended cylindrical geometry. Indeed, let us consider slices of the cylinder of responsive gel. If a supercritical acid perturbation is applied to one of these slices, the local acid part of the gel (core of the M state) that transiently develops can contaminate by diffusion the neighboring unperturbated slices in the swollen unreacted state and make them also cross the excitability threshold. A large amplitude contraction of the gel follows the development of the acid core. In return the contraction makes the gel locally switch back to the F state and a slow recovery process follows. As a result, a large amplitude contraction wave travels undamped along the main axis of the cylinder, with a sharp gradient of radius ahead and a recovery tail as observed in Fig. 20 .
6
CONCLUSIONS AND PERSPECTIVES
The systematic use of the CSTR in the 80’s have made possible a significative qualitative and quantitative jump in the control, the phenomenological analysis, and the discovery of new oscillatory reaction systems. The phenomenon of temporal bistability is directly linked to the material exchange fluxes with the environment. It has played an essential role in the formidable diversification of oscillatory reactions. We have shown above that spatial bistability in one side fed spatial reactors could play an analog key role in a systematic quest for new spatio-temporal patterns in a potentially large diversity of reaction systems. We anticipate that all known oscillatory or of course simply bistable systems in a CSTR can exhibit spatial bistability. However, since the spatial extension introduces an infinite number of degrees of freedom and defines a boundary value problem, the domains of respectively temporal and spatial bistability do not match. As shown in our experimental observations and model calculations, spatial bistability can actually be observed for large set of parameters for which no temporal bistability is observed in a CSTR (and vice versa). As expected, diffusion coefficient differences can play a determinant role in the development of spatial symmetry breaking structures. In particular, as inferred by Turing more than 50 years ago, stationary patterns are promoted by short range activation. We have shown that stationary pulses could be made by design, using the possibility to control the rate and direction of propagation of interfaces between steady spatial states 22
and to selectively reduce the effective diffusivity of an activatory species by introducing macromolecular complexing agents. However, this work further revealed the complexity of developing sustained boundary symmetry breaking patterns in real open spatial reactors. Not only the full dimensionality of the system has to be taken into account but also some contradictory requirements between the slowing down of the activatory species and their necessity to be fed or evacuated through the OSFR/CSTR exchange boundary. Though there is no definite theoretical proof yet, we infer that to obtain stationary patterns, the production or consumption of the activatory species cannot be just counterbalanced by the escape or feed flows of this species at the boundary of the system. Other chemical processes acting as independent sinks or sources must be introduced. The case of long range activation instabilities had not been predicted theoretically prior to the first observation in the proton autoactivated CT reaction. The phenomenon was naturally first discovered in such a reaction since protons have the highest diffusivity coefficient in aqueous solutions. However, one could imagine that similar instabilities could also be obtained in systems activated by other species if, contrary to the previous case, one would selectively slow down the diffusivity of a dominant inhibitory species or of a substrate instead of the activatory species. In both cases, to be able to generalize the studies of reaction-diffusion instabilities to a wider variety of reaction systems, the limitations will come from the posibility of finding suitable complexing agents. In the case of iodide ions and protons simple selective complexing agents as starch or PVA and polycarboxylic macromolecules could be used. This “trick” would be more difficult to play for species like HBrO2 or Br− in the BZ reaction. The studies of chemomechanical structures are still in their infancies and we think that much is still to come both from the theoretical and experimental view point. In these studies, one needs to further explore the potentialities of materials with different elastic properties. Could it be possible by long range deformation to produce not only spatio-temporal shape patterns but also standing spatial patterns? We anticipate that systems under mechanical “tension” (e.g. by imposing a fixed size to the chemoresponsive pieces of gel) can lead to global coupling phenomena that would induce the self-locking of a single traveling interface connecting, for exemple, the swollen and skrunken states in the spatial bistability domain. The shrinking of one part of the system could induce an effective elongation of the swollen part and thus a decrease in the diameter of the cylinder. This diameter could drop below the critical value for which the skrunken state can propagate into the swollen part. An externally increased tension could make the shrunken part recede. Conversely if the tension is relaxed the shrunken state would extend again until the critical tension is reached once more. This could lead to self-regulation of a mechanical tension by a chemical process. Our studies and the envisioned extensions emphasize the richness of nonequilibrium and chemomechanical patterns. Their potential impact on numerous fields is obvious, ranging from morphogenesis or mechanical processes driven by chemistry in biological systems, to autonomous devices in microfluidics or microrobotics. Spatial bistability 23
appears as a basic engine for a number of patterns. This should boost further interest for this phenomenon in the near future. Aknowledgments: This work has been supported by CNRS, by the Marie Curie Program of European Union (I.S.), by the French-Hungarian Balaton program, and the french Agence Nationale de la Recherche (J.B., P.De K.).
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27
.
28
FIGURES
29
X
A
B
δ
P
Figure 1: Schematic representation of concentration profiles of a substrate A and of an autocatalytic species X in the mixed state. The CSTR/gel boundary is located at position B. Point P corresponds to the location of the steep concentration front in X.
Figure 2: Sketches representing OSFRs with different geometries. The gel parts are in blue while the CSTR parts, associated to a symbolic propeller, are in grey. The white and black parts represent respectively transparent and non transparent impermeable walls supporting the gel parts.
30
Figure 3: Experimental spatial bistability states and aspects of interfaces between these states in the CDI-PVA system operated in a flat annular OSFR.Common parameters: τ =500 s, T=22 ◦ C, [ClO2 ]0 =3.0 × 10−4 M, w=2.0 mm. a) Quasi-uniform dark purpleblue state F (high concentration of polyiodide-PVA complex); b) Sharp color switch between a dark blue (high concentration of polyiodide-PVA complex) region and a clear yellow (very low concentration of polyiodide-PVA complex) region: M state. c) and d) Interfaces between the F and M states in the bistability domain. c) State M takes over the F state: [I− ]0 = 1.1 × 10−3 M; d) State F takes over the M state: [I− ]0 = 1.2 × 10−3 M.
31
A
B
Figure 4: Sections of the spatial bistability phase diagram of the CDI-PVA reaction in the ([I− ]0 , w) plane; A) experimental, B) numerical: The symbols correspond to the actual experimental points and are attributed to the different stationary states of the gel: ▲ F state; ● M state; : limit of stability of M state; : low − [I ]0 limit of stability of the F state. LF and LT respectively point out the stability limits of the F and T state of the CSTR.
32
■
■ ■
2
■ ■ ▲ ■
[MA] 0 /mM
M
■ ▲ ❋
▲
■ ▲
■ ▲
M/F ■ ■ ▲
F ▲ ▲
■ ▲
■ ▲
1
■ ▲ ❋
Turing states
Spatial bistability
0 0
0.4
■ ▲ ■ ▲▲
■ ■ ▲ ▲
■ ▲▲
■ ▲
[I −] 0 /mM
0.8
■ ❋▲
▲
■ ▲
▲
1.2
Figure 5: Experimental phase diagram in the ([I− ]0 , [MA]0 ) plane, for the CDIIMA reaction in an annular OSFR. The symbols correspond the experimental points. ▲: F state; ■: M state; ❋ labels the point in the vicinity of which the stability of interface pairing phenemon was tested. : upper and lower stability limits of the − M state; : low [I ]0 stability limit of the F state; q q q q q q q q : limit of the Turing states region. Fixed parameters: τ = 500 s, T= 5 ◦ C, [I2 ]0 =3.3 × 10−4 M, [ClO2 ]0 = 2 × 10−4 M, [PVA]0 = 10 g/dm−3 , w= 1.0 mm.
33
b
Figure 6: Mixed state M at high [I2 ]0 and high [MA]0 : a)Experimental observation of the polyiodide-PVA color complex in an annular OSFR. b) [I− ] profile across the width w, computed with the model in [24].
34
Figure 7: Dynamics of standing pulses (F state) immersed in the stable M state of the CDIIMA reaction in an annular reactor. Snapshots begin two hours after the initial series of local perturbations. Experimental conditions as in Fig. 5 with [I− ]0 =1.0 × 10−3 M, [MA]0 =5.0 × 10−4 M, w=1 mm.
35
a
b
Figure 8: Calculated interface propagation at high [MA]0 and high [I− ]0 in a rectangular OSFR with fixed boundary values along the bottom and no flux boundary conditions on the three other sides. Scale bar=0.2 mm. a) Interface at halt close to the impermeable right boundary. Grey scale: dark and clear color correspond respectively to high and low concentrations of the polyiodide-PVA complex. b) Time evolution of the distance y from interface to the impermeable boundary (right). At t=7×104 s this boundary was moved closer by 0.3 mm. The interface returns to y= 0.41 mm.
36
Figure 9: Growth of F/M interface fingers in a disc OSFR. The white dashed line indicates the original shape of the interface. The two front positions correspond to an interval of 10 minutes. Experimental conditions as in Fig. 5 with [I− ]0 =0.77 × 10−3 M and [MA]0 =2.1 × 10−3 M, w=1mm.
Figure 10: Development of transient patterns obtained after initial perturbation of state M in the disc reactor (disc diameter: 20 mm, gel thickness: 1.0 mm ). Experimental conditions as in Fig. 5 with [I− ]0 =0.8 × 10−3 M, [MA]0 =2.1 × 10−3 M, and [PVA]0 = 10 g/dm3 .
37
Figure 11: Nonequilibrium phase diagram of the CT reaction in the (α, w) plane. Symbols correspond to experimental points: : monostable non excitable F state; 4 : monostable excitable F state; •: bistability between the F state and the oscillatory M state; : bistability between the stationary F and M states; ❋ : CSTR switches to the T state . Experimental conditions: τ = 600 s, T = 25◦ C, [HClO4 ]0 = α × 0.33 × 10−2 M, [NaOH]0 = (1 − α)× 1.67 × 10−2 M, [NaClO2 ]0 = 1.9 × 10−2 M and [K2 S4 O6 ]0 = 0.5 × 10−2 M. Lines are calculated state limits with the extended CT model in [17]). : limit of M state; : limit of oscillatory M state; : low α limit of the excitable F state.
Figure 12: Traveling acid pulse in the monostable F state. pH changes are made visible by introducing bromophenol blue. Experimental conditions as in Fig 11 with w = 3.00 mm and α = 0.48.
38
0.36 ◆◆ ▼ ▼ ▼▼
0.34▼
◆ ◆ ◆ ▼
◆ ◆
◆ ◆ ◆ ▼
◆
Sp. Bist.
◆ ▼ ▼▼
◆
F/M
0.32▼ ▼
◆ ◆◆ ▼▼▼
▼
q 0.30▼ 0.28▼
▼
▼
● ● ◆ ▼ ▼
◆
▼
▼
▼
▼
◆
◆
Osc.
monostable F state
0.26▼ 0
◆
monostable M state
▼
◆● ● ◆ ◆ ▼
▼
▼
4
h
◆ ▼
◆
◆ ◆
◆
◆
8
Figure 13: Experimental phase diagram in the (q, h) plane of the IS reaction operated in an flat annular OSFR. Symbols: ▼: stable F state; ◆: stable mixed state M; ●: oscillatory mixed state. :limits of spatial bistability region; q q q q q q q q q q : limit of stability of the F state of the : limits of oscillation region; CSTR. Experimental conditions: τ = 500 s, T = 30 ◦ C, q=0.30 ([IO− 3 ]0 = 0.018 M and [SO−2 ] = 0.06 M), w = 1.0 mm. 3 0
x/mm
0
1 t/min 10 min
Figure 14: Spatio-temporal plot of large amplitude oscillations in the annular OSFR at q = 0.30. Sampling time 2 min. Grey scale: dark and clear correspond respectively to the alkaline and acid forms of the pH color indicator. Experimental conditions as in Fig. 13 with h = 5.3. limit
39
0.05 ■
■
■
■
■■
▲ ▲
[-COO-]/M
Monostability
■
■
■
■
■
■ ◆
◆
■
0.4
■
▲
▲
▲ ▲
Bistability
F
0.025
0
▲
F/M ■
■ ■ ▲ ▲
■
■
◆◆◆▲ ▲
■
◆ ◆◆
▲
▲
▲ ▲
▲ ▲
▲
▲
▲
▲ ▲
▲ ▲
▲
▲
▲
◆◆●▲ ▲
▲ ▲
❋
❋
T ❋
Exc. F Osc. M ◆
▲
0.6
0.8
1
α
Figure 15: Experimental section of the phase diagram of the CT reaction in the plane (α, [-COO− ]0 ). The symbols correspond to the states of the gel: ■: monostable nonexcitable F state;limit ◆: monostable excitable F state; ▲: bistability between the stationary F and M states; ●: bistability between the stationary F state and the oscillatory M state; ❋ : CSTR contents switches to the thermodynamic state branch. : limits of (F,M) bistability domain; : limit of the monostable excitable F state, : low α limit of the oscillatory M state, same experimental conditions as in Fig.11
40
Figure 16: Theoretical phase diagram in the (α, S0 ) plane calculated with the extended CT model in ref. [43]. S0 has the same signification as [-COO− ]0 in the experiments (Fig. 15). Regions of states F, M, bistability (F/M), thermodynamic state T, excitable F and oscillatory M are labelled in the figure.
Figure 17: Bistability in a sphere of radius R. Stable states are characterized by the concentration xc at the center of the sphere (full lines). The sequences (A-BC-D-A) and (S1 -S2 -B-C-S) in the (R, xc ) phase space correspond respectively to the trajectories during an chemomechanical oscillation and a chemomechanical excitation in a responsive gel. For the definition of these trajectories and parameters Rinf , Rsup , Rshrunk , Rswell , see text.
41
Figure 18: Chemomechanical oscillations of a chemoresponsive sphere: Radius R of the sphere (full curve), xc concentration at the center of the sphere (broken curve) as a function of time (calculated from the chemomechanical model in [19, 57]). Letters label the different states of the system along the hysteresis loop in Fig.17.
42
Figure 19: Chemomechanical phase diagram. Dynamics of a pH responsive cylindrical OSFR immersed in a CSTR fed with the CT reaction as a function of [OH− ]0 . Gel prepared by free radical copolymerisation of N-isopropylacrylamide, acrylic acid and N,N0 -methylene-bisacrylamide. After swelling, the polymer to solvant (water) ratio is ∼ 10% and the concentration of the carboxylate function on the polymer network is −2 −2 ∼ 4.5×10−2 M. CSTR feed conditions: [ClO− M; [S4 O−2 M; 2 ]0 = 6.32×10 6 ]0 = 1.8×10 ◦ τ = 18.5 min, T= 35 C.
43
Figure 20: Chemomechanical excitability: Series of shadowgraph snapshots of a contraction wave traveling in a pH responsive cylindrical OSFR with same feed conditions as in the excitable domain of the diagram in Fig. 19. The wave is triggered by a gentle acid perturbation made at the bottom end of a cylinder of gel initially in the alkaline swollen state. Time interval between snapshots, from left to right: 21, 30, 20 min; Scale bar=4.0 mm; Wave velocity ∼5.6 mm/h.
44
