J. Phys. Chem. 1989, 93, 27 11-27 16 We have not tried to establish the bifurcation diagrams (as described in ref 12) either in the simple D = 0 case (which yields comparatively simple ODE)or in the more complicated nonzero delay case. One simple conclusion can be drawn, however, without doing any calculation. Suppose a certain point in the constraint space of an uncontrolled system is in a steady state. Suppose, further, that by introducing the control, this point is located beyond some bifurcation. Then it must be that the stable steady state will continue to exist too. If one starts at the old system and then
2711
introduces the control, the system must remain where it had been; on the other hand, if one starts on some other initial point, the trajectory of the controlled system may go to a limit cycle or some other state, which was nonexistent before.
Acknowledgment. J.W. and F.W.S. thank the Stiftung Volkswagenwerk and the Fonds der chemischen Ind. for financial support. Registry
No. BrOy,
15541-45-4;
Br-,
24959-67-9;
Ce,
7440-45-1,
Observed Properties of Trigger Waves Close to the Center of the Target Patterns in an Oscillating Belousov-Zhabotinsky Reagent C. Vidal* and A. Pagola Centre de recherche Paul Pascal, Universitd de Bordeaux I, 33405 Talence, France (Received: May 2, 1988)
The light transmitted in the neighborhood of the center of two target patterns, appearing spontaneously in an oscillating Belousov-Zhabotinsky reagent, is measured, and its variations are reproduced by a very simple model. The scenario is built on the trivial, but crucial, idea that the three dimensions of physical space must be taken into consideration for understanding observations made at distances from the center shorter than the layer depth. Experimental evidence is given on the apparently “homogeneous” character of one of the two centers, at least down to a few micrometers.
Introduction Expanding target patterns and rotating spiral waves in thin layers of unstirred BZ reagents have been known for about 20 years.’ Away from the core, the propagation of these so-called trigger waves is well understood and described. By introducing, in the familiar reaction-diffusion equation, a “plausible” kinetic model, one gets a good agreement between theoretical predictions,2 numerical simulations,’ and experimental data. Satisfactory results are provided by a model as crude as a version of the Oregonator reduced to two variables (the catalyst and HBr02). Moreover, it was shown recently that the dispersion relation measured4 and the curve calculated from such an oversimplified modelS fit each other for a set of scaling factors well within the estimated range (less than 2-fold) of rate constants. The main features of the propagation mechanism of trigger waves can thus be considered now as appropriately understood. However, two important facts have still to be elucidated before looking at this kind of chemical self-organization phenomena as a solved problem: (i) Why do centers spontaneously appear in oscillating media, whatever the experimental measures taken to clean the vessel and to prevent further contamination by d ~ s t ? ~ , ~ (ii) What exactly is happening within the core region, while the wave builds With regard to the first question, the dispute between “homogeneity” and ”heterogeneity” is well-known. The issue has been addressed several times, and both deterministiclo and sto(1) Vidal, C.; Hanusse, P . Int. Rev. Phys. Chem. 1986, 5, I . (2) Tyson, J. J. In Oscillations and Traveling Waves in Chemical Systems; Wiley: New York, 1985. ( 3 ) Reusser, E. J.; Field, R. J. J . Am. Chem. SOC.1979, 101, 1063. (4) Pagola, A.; Ross, J.; Vidal, C. J . Phys. Chem. 1988, 92, 163. (5) Keener, J. P.; Tyson, J. J. Physica D 1986, 21D, 307. Dockery, J. D.; Keener, J. P.; Tyson, J. J. Physica D 1988, 300, 177. (6) Winfree, A. In Oscillations and Traveling Waves in Chemical Systems; W h y : New York, 1985. (7) Vidal, C.;Pagola, A.; Bodet, J. M.; Hanusse, P.; Bastardie, E. J. Phys. 1986, 47, 1999. ( 8 ) Muller, S . C.; Plesser, T.; Hess,B. Physica D 1987, 2 4 0 , 71. (9) Pagola, A.; Vidal, C. J. Phys. Chem. 1987, 91, 501. (IO) Tyson, J. J. J. Chim. Phys. 1987.84, 1359.
0022-3654/89/2093-2711$01.50/0
chastic” theories are available today. Whether a fluctuationnucleation mechanism might, or might not, be involved in the birth of at least some centers is still a matter of controversy. In this context, worthy of notice is the fact that the singular perturbation theory developed by Tyson’O does not require a “heterogeneous” pacemaker to account for the appearance of a center having a period nearly equal to that of the bulk oscillation;a local dephasing would suffice as well. Consequently, the homogeneous character of a center, even if demonstrated, is not a criterion allowing one to distinguish determinism from stochasticity in this matter. Only a detailed statistical study, of the kind previously ~ n d e r t a k e n , ~ seems to be capable of providing an answer. In the present article, we mainly deal with the second of the above-mentioned questions. Experimental data have been collected by observing with a microscope the core region of two “spontaneous” target patterns. Their analysis illuminates the way in which an oxidizing wavefront develops in the reduced medium. In particular, the three-dimensional (in physical space) character of the phenomenon cannot be disregarded at distances from the center smaller than or comparable to the depth of the layer. A serendipitous result is the experimental measurements carried out on a target whose center is shown to be free of particles, down to a size of a few micrometers.
Experimental Section The experimental setup and methodology used in this work have been reported in previous publications; we briefly recall here their main characteristics; the reader is referred to ref 7, 9, and 12 for more details. We use an inverse microscope, coupled to a video recorder device, to record the light transmitted at 475 nm by a layer of an oscillating BZ reagent. Its initial composition is the following: malonic acid 0.5 M; sulfuric acid 0.5 M; sodium bromate 0.31 M; ferroin 0.004 M. The solution is poured into a Petri dish, which is covered; the depth (about 0.65 mm) is small enough to prevent (1 1) Walgraef, D.; Dewel, G.; Borckmans, P. J . Chem. Phys. 1983, 78, 3043. (12) Pagola, A. These Universitt de Bordeaux 1, 1987
0 1989 American Chemical Society
2712 The Journal of Physical Chemistry, Vol. 93. No. 7, 1989
Pagola
m
il.Pl .*--
E
..@“
-
.E
Figure 1. Three-dimensional perspective representations of the light transmitted through the core of the heterogeneous target. Snapshots are taken at 1-s interval each. The sequence labeled a. ._.,h, reads from left to right, and from bottom to top.
-r
8a-” a 0-
0 ” 0
0.5
d (nml
1.0
Figure 3. Width (a. top) and relative amplitude (b, bottom) of a front versus distance to the border of the heterogeneity. Points rep-nt experimental data, and continuous curve values calculated with h, = h, h2 = 0.
Figure 2. Same representation as ~n Figure I in the ‘homogenmus” case. Note the difference in length scale (twice that used in Figure 1).
the onset of convection.” Under these conditions and at room temperature (Le., 21-22 “C), the reagent exhibits relaxation oscillations, more or less periodic, with a period close to 26 s. A video film is taken at a speed of 25 images per second. Later on, selected images are converted into a grid of 512 X 512 pixels, the signal of each element being digitized over a scale of 256 gray levels. At the magnification used, the observed area is a square of 2.85 X 2.85 mm2, scanned with a spatial resolution of 5.57 pm per pixel. As usual, subtraction of the background, smoothing of the signal, and noise reduction are performed using the standard techniques of image processing. In so doing, we eventually get three-dimensional perspective representations of light transmission, a sample of which is displayed in Figures 1 and 2.
Measurements and Results The central zone of two target patterns, having spontaneously appeared in the same oscillatory reagent (but in two different experiments), was thoroughly observed. We hereafter call “heterogeneous” one of the two centers, simply because we see a “large” particle, shadowing about 145 pixels; that means a typical size of about 75 pm. On the contrary, the other center will be called “homogeneous”, because we cannot detect any extra ahsorption of light at any time. Beware not to misunderstand this word, whose meaning is strictly defined by the above-stated experimental facts. No conclusion at all can be drawn on the very homogeneity of the medium at a scale shorter than 6 pm. Another striking difference between these two targets concerns the emission (13)
Rcdriguez, J.: Vidal. C., submitted for publication in J. Phys. Chem.
period of their center, slightly shorter than the bulk period (equal to 26 s) for the heterogeneous center, only about one-fourth of it for the =homogeneous”one. The origin of such a large difference remains an open question, whose answer cannot be anticipated at the moment. Besides, we can hardly assert with precision at which phase of the oscillation each center had really become an “active” pacemaker. For both, the local dephasing reached a visible level while the surrounding solution lasts-in its reduced stage, in other words, during the 16 s preceding the burst of oxidation. Despite this lack of accuracy, there is no doubt however that we observed the birth of trigger waves, owing to their morphological and dynamical features. We now come to a more thorough examination of the qualitative similarities and quantitative differences exhibited by these two targets. In both cases, the oxidizing wavefront first develops, and then propagates, according to the same general scenario. At the beginning, the oxidation phase starts up within a small area around the center. As shown in Figures 1 and 2, which gather eight successive snapshots, each taken at I-s intervals, the radius of the area grows (linearly, in the range observable, that is, beyond 0.1 mm) in time, and so also does the intensity ofthe transmitted light, for a while during which a flat plateau of increasing height is observed (snapshots a to e). Then, the oxidized area keeps growing in size, always at a constant speed, hut now its central part begins to be depressed (snapshot f); a sort of “volcano” is thus formed when the reduction pr,ocess which follows the oxidation phase of the oscillation enters into play. As time goes on, the diameter of the volcano still grows linearly. Simultaneously, its crater becomes deeper and deeper, due to two effects: the height of the edge continues to increase, up to an asymptotic value characteristic of the medium, and the whole central part slowly goes hack toward the reduced level (snapshots g and h) At first glance, the two targets behave very similarly. However, there are several quantitative differences which cannot escape attention. For example, the radius at which the plateau phase ends (snapshot e) is c l w to 0.7 and 0.35 mm for the heterogeneous and the ‘homogeneous” target, respectively. The amount of transmitted light reaches its maximum (and asymptotic) value at about 1.1 and 0.6 mm from these two centers. In order to achieve this study on a quantitative basis, we measured how the amplitude and the width of the oxidizing wavefront varies with the distance from the center. These three quantities were defined as follows: froni amplitude, difference in grey levels (averaged over 3-5 pixels) between the.bottom and the top; front width, distance between. the “foot” of the front and the point corresponding to 75% of the amplitude; distance io the cenier, distance
The Journal of Physical Chemistry, Vol. 93, No. 7, 1989 2713
Trigger Waves in an Oscillating BZ Reagent
I
, @
I
t
OV
0
,
,
,
,
. 0.5
,
,
1 .o
,
I
d (mml
Figure 4. Width (a, top) and relative amplitude (b, bottom) of a front
versus distance to the "homogeneous" center. Same symbols as in Figure 3. This time h, = h2 = h / 2 .
L 0 5 10 t (I)
Figure 6. Temporal oscillation at three increasing distances from the border of heterogeneity: (a) the slowing down of the oxidation phase is striking near the surface (0.044 mm) of the particle; (b) only the last part of the oxidation process is still slow at 0.485 mm; (c) farther (0.769 mm), the oscillation has recovered a shape similar to that of the unperturbed
medium (Figure 5a).
" " " ' I
I " " '
.;:. 0
-;-*..+'
5
'0 t
(5)
Figure 5. Temporal oscillation of the bulk solution (a) and at a point
nearby (0.016 mm) the "homogeneous" center. between the point at half-width and either the "homogeneous" center of the border of the heterogeneity. The results are displayed in Figures 3 (heterogeneous case) and 4 ("homogeneous" case), where the symbols correspond to rough experimental data and the curves are the outcome of a calculation which will be presented in the next section. No internal monitor being available for absolute calibration of light transmission, we can only plot for each experiment the relative amplitude of the front, normalized to its asymptotic height (set equal to 100). The very noticeable differences between Figures 3a and 4a, 3b and 4b, respectively, need not be outlined. Perhaps the most striking fact is the maximum exhibited at 0.15 mm from the center by the front width in the "homogeneous" target (Figure 4a). This unanticipated observation remained very puzzling, until we developed the comprehensive phenomenological description presented in the next section. Of course, we measured the front velocity, and we checked its constancy too. The values are significantly different: 0.10 mms-' for the heterogeneous target and 0.15 m m d for the homogeneous one. This gap first appeared to us very puzzling. Indeed, in a given medium at a given temperature, the propagation velocity of trigger waves is known to have a typical value, not depending on other factors. Now, the difference falls beyond the 10% standard deviation which characterizes our experimental technique.' In order to understand what was going on, we also recorded the transmitted
light variations versus time at several locations. Figure 5a shows the oscillation in an unperturbed region of the layer, i.e., outside the target area. This "bulk" oscillation and the oscillation observed almost at the center of the "homogeneous" target (Figure 5b) look very much alike. In contrast, the oscillation is strongly modified near the heterogeneity; there, the oxidation phase lasts for 7 s,I4 which is about 4 times longer than in the bulk (Figure 6a). It is only far enough from the heterogeneity that the oscillation more or less recovers its "standard" shape (Figure 6c). In between, the oxidation takes place in two stages with different rates: the slow one is a witness of what occurs close to the surface of the heterogeneity (Figure 6b). Because the gradients, ahead of the oxidizing front, are less steep, due to the slowing down of the oxidation process by the particle, one expects the propagation to be less rapid in this case, as it actually is. Therefore, the difference in propagation speeds is qualitatively consistent with the observations on the rate of oxidation.
Discussion A rather trivial remark was the starting point of our attempt to understand the scenario depicted by Figures 1 and 2 and to account for the quantitative data plotted in Figures 3 and 4. As long as observations are performed at distances from the center comparable to the layer depth (say 1 mm, or less), the medium cannot be considered as (almost) two-dimensional. Obviously, all lengths having the same order of magnitude, the need to take into consideration the three dimensions of physical space is of course undisputable. With this in mind, one can derive a phenomenological description, relying essentially on the following heuristic assumption: we suppose that the asymptotic profile of a front, far from the core, reflects the way it has appeared at the center. We are thus enabled to compute the (relative) amount of light transmitted by the layer at any time. This calculation is given in the Appendix. A few approximations are introduced for the sake of convenience, (14) According to the FKN mechanismi5 of the Belousov-Zhabotinsky reaction, it seems likely that radicals are scavenged at the surface of the particle. (15) Field, R. J.; Koros, E.; Noyes, R. M. J . Am. Chem. SOC.1972, 94, 8649.
2714 The Journal of Physical Chemistry, Vol. 93, No. 7, 1989
Vidal and Pagola
0
length scrk
’&
Figure 7. Three recorded profiles (a) and the corresponding calculated ones (b) if the center is assumed to lie on a boundary of the layer. The central deep hole in (a) is due to the heterogeneity.
but they are not crucial. One point must, however, be emphasized: the derivation does not involve any arbitrary or adjustable parameter, since the two quantities w and p (seeAppendix and Figure 9 for definition of symbols) are supplied by the experiment. There is only one thing which is a matter of guess (and of conclusion): the height at which the center is located in the layer. For the two targets, we took the same value of w,namely 70 Mm, since the medium was the same. Indeed, the model implies that, far away from the center, the switching from the reduced to the oxidized stage (and the converse) takes place over a given distance, depending upon the medium only, whatever the center properties are. In other words, w and p should be, in principle, single-valued. Despite this conclusion, we were led to adopt two characteristic lengths, noticeably different: p(heterogeneous) = 0.55 mm; p(“homogeneous”) = 1.30 mm. As a matter of fact, we found very significant differences between the two profiles, even at the farthermost position of the front (not reproduced here) we used to determine the two parameters. Whereas a unique value of w is acceptable for the oxidation rise, the two relaxation parts cannot be fitted with the same value of p. One must assume that the influence of the particle is still strong enough to modify the reaction kinetics at the rather short distance where p could be measured. Figure 6 supports this idea. Since the particle slows down the oxidation process, it must speed up even more the reduction phase, the emission period of the center being necessarily smaller than the bulk period. Accordingly, due to this particle, one expects to observe a smaller value of p, as we do. A first qualitative, though very interesting, outcome of this approach is the prediction of a plateau phase (see Figure 12), always followed by a further increase of the front amplitude. When I!. is equal to 0.65 mm, this plateau phase is expected to end when the outer radius reaches 0.72 or 0.39 mm, depending on whether the center lies on a boundary or at the middle height of the layer. These two values meet pretty well those determined experimentally. A more thorough comparison between calculation and experiment deserves to be performed in two ways. First, it is appropriate to compute vertical cross sections of the profile (through the center) and to superimpose them to the measurements, for a whole set of external radii. An example of the result is given in Figures 7 and 8. In each, a sample of three cross sections of the measured (a) and calculated (b) profiles are put side-to-side, to get an easier reading. A striking resemblance is obtained in both cases. However, to reach it, we were led to choose once again the following values: h l = 0; h2 = 0.65 mm (or else the contrary) for the heterogeneous target (Figure 7); hl = h, = 0.325 mm for the “homogeneous” target (Figure 8). Otherwise, larger and larger differences are noticed as h l and h2 are moved from these two sets. The deep central hole in Figure 7a is due to light absorption by the heterogeneity. It is useless to reproduce it in the calculation, and this is the reason why Figure 7b is free of this extra absorption. A further step consists in computing the front width and amplitude already defined, and to plot them as functions of the distance from the center. For direct comparison with experimental data, the result obtained with the same set of conditions is plotted in Figures 3 and 4 (continuous curves). The main general trends
length rcrk
Figure 8. In the “homogeneous”case, the recorded profiles (a) never exhibit any extra absorption. Profiles in (b) are those calculated for a center staying just at middle height of the layer depth.
observed experimentally are reproduced for both targets. And so are, too, some typical differences between them, such as, for instance, the much more rapid increase in front amplitude noticed in the “homogeneous” case. Nevertheless, the most convincing fact is the appearance of a maximum in the curve width versus distance, not very far from the “homogeneous” center. As said before, we were wondering for weeks about this amazing and unanticipated observation, which was clearly outside the range of experimental uncertainties. The calculation shows that such a maximum is not at all mysterious. At the same time, this unexpected agreement points out the relevance of the description proposed. Owing to the various simplifications used and, above all, to the absence of any adjustable parameter (except, of course, h,), the deviation between the calculated and the recorded abscissa of this maximum is not a substantial drawback. It is far beyond any reasonable aim to achieve a perfect fit with so crude a modelling procedure. In summary, the whole set of experimental data is readily understandable in the framework of the simple description introduced. From a phenomenological point of view, a center may be considered as a pointlike defect,I6 radiating waves isotropically. The manner in which the observed properties of a front evolve at distances comparable to the layer depth depends noticeably on the direction of observation. When watching perpendicularly to a layer, for instance, the location (height) of the center is a crucial factor. Conversely, measuring the front properties over this range of distance provides a means to determine approximately where the center lies between the two limits: at a boundary or at the middle height. Finally, in the two cases under examination, we can assert that (i) the heterogeneity stayed on a layer boundary and (ii) the second center, where no absorption due to a particle could be detected (Figure 8a), was located around the middle of the layer. If we actually think about a ”particle”, then it must have had a very small size (a few micrometers) and, at the same time, a density (almost) equal to that of the solution. This last property precludes a gas bubble. It might have happened, as well, that this center was a genuinely homogeneous pacemaker. Conclusion A quite simple description of wave birth accounts well for all
the qualitative and quantitative observations made on the core region of two “spontaneous”‘target patterns. This description makes available a tool for locating the depth from which waves are radiated. In certain circumstances, especially when a center looks “homogeneous”, this is very useful, as we have shown. However, the most important conclusion of this work is a rather trivial statement: the approximation of a two-dimensional medium (16) In contrast to this, according to the smallness of the variations observed at the core of a spiral? it seems that a line of defects, extending across the layer, would be involved at the tip of a spiral. The formation of such a line obviously requires very peculiar conditions. One cannot expect them to be fulfilled spontaneously in a medium at rest. Here lies presumably the need for an external perturbation for creating spirals.
The Journal of Physical Chemistry, Vol. 93, No. 7, 1989 2715
Trigger Waves in an Oscillating BZ Reagent
/
U E
,
o
w
r
-
?
/-
t
Figure 11. Decomposition of the layer in two slices, when the center stays somewhere between the two boundaries.
Figure 9. Sketch of the approximate transmission profile (along a radius) used in the calculation.
I
:
ccnler
Figure 12. End of the plateau phase, just prior to crater formation.
fz
X
r,
r,- w
If ro is the distance reached by a front at time t (Le., ro = ut), then the “local” transmission at a point located at altitude z may be determined by a rather elementary calculus, involving the conditions mentioned above. Setting, for instance, the origin of a front on a boundary of the layer (Figure lo), it follows that 0
4Ib) =
Figure 10. Definition of the reference frame, the center being on a boundary. The continuous and dashed lines correspond to the edge of
ro - (x2 + W
the front and to the local maximum of light transmission respectively. Here, and in the next two figures, the geometric sketch is drawn at a scale such that the range over which a wavefront interacts with the impermeable boundaries is negligible.
between the two circles of radius ro and ro - w, and
does not apply to domains whose extension is not large compared to the depth! Such a sentence is known as a truism. Nevertheless, it reminds us of the absolute need to take into consideration the three dimensions of physical space when dealing with the central region of a pattern (whatever this pattern, target, spiral,”-’* or anything else) especially because, so far, experiments were all carried out in layers very thick (a few tenths of a millimeter, at least) with respect to the characteristic length of chemical waves (say 10 pm). Accordingly, how could we improve significantly our knowledge of the early stages of spontaneous self-organization phenomena, unless we give up the usual two-dimensional approximation? Topology tells us that switching from two to three dimensions has drastic consequences.
inside the inner circle (1x1 Iro; 0 Iz Ih). The amount of light A ( x ) transmitted at a distance x from the center, in a direction perpendicular to the layer, is obtained by integrating these expressions along a vertical. Time is taken into consideration through ro. Besides the trivial condition 1x1 > ro, which corresponds to A ( x ) 0, three different situations may be encountered ro I 1x1 I ro - w
Appendix Determination of the Light Transmitted Perpendicularly to a Layer through an Expanding Spherical Front. Let us suppose that along any radius the transmission profile consists of a linear part, ranging from 0 to w,followed by an exponential decay with characteristic length p, as sketched in Figure 9. Putting the origin at the bottom of the front, the equations giving the transmission at a distance r are
(
41dZ) = exp -
(ro - w ) - (x2 + z2)I/2 P
)
A ( x ) = Jz’41(z) dz; zI = min (h,(ro2- x2)i/2) 0
ro - w > 1x1 L either 0 or ((ro- w ) -~h2)’/2whenever t is such that ro - w I h
zII = ((ro- w)’ - x2)1/2
((ro - w ) -~h2)’l2> 1x1 I 0 (which implies ro - w
> h)
Whereas the integral of 411(z) needs to be determined numerically, the other one has a simple analytical expression
( .aw)
4 ( r ) = exp --
; r> w
The variation amplitude being arbitrary, we set the maximum a t 1 and the origin at 0 for the sake of simplicity. Here we are merely choosing a set of equations that accounts for the salient features of the profile actually observed far from the center. Accordingly, the parameters w and p are provided by measurements. We now introduce two other simplifying hypotheses: (i) A front propagates throughout the medium at the same speed u in all directions. (ii) The shape of the profile along a radius is the same at any time, starting from the birth of the front. We thus ignore any curvature effect. (17) Agladze, K. I.; Panfilov, A. V.;Rudenko, A. N. Physica D 1988,29D, 409. (18) Muller, S.C.; Plesser, Th.;Hess, B. Physica D 1987, 240, 71.
r0 J64,(z) dz = -(b - a) W
2w
b(x2
x 2 + b2)i/2 + b2)i/2- a(x2 + a z ) l / z+ x2 In a6 ++ ((x2 + a2)i/2
1
When the center is not located on a boundary, one has simply to sum the contribution of two sides, each having its own width, say hl and hz, as indicated in Figure 11. Considering the particular case hl = h2 = h / 2 (Le., a particle centered in midlayer), let us point out the following. Of course, the calculus yields a picture very much resembling that computed for hi = h and h2 = 0, since there is simply a scaling factor of 2. Now the quantitative difference between the two cases lies precisely in this factor, which thus allows to distinguish between them. For instance, the radius at which the plateau phase will end is obviously one-half in the former situation. In Figure 12 we draw the geometrical condition which, in this approach, corresponds to the end of the plateau phase. Once again, we put, in this example, the center on a boundary. The value of
J . Phys. Chem. 1989, 93, 2716-2718
2716
the associated radius r, is equal to w + h in that case. It would be equal to w h/2 with a center staying at middle height. A further increase in the light transmitted through the front takes place beyond this condition. The maximum amplitude is reached asymptotically, while ro goes to infinity. However, the change in amplitude is no longer measurable as soon as the front curvature has become negligible, simply due to the projection from three
+
to two dimensions. It is not difficult to estimate that, if w is much smaller than h, this happens for ro of the order of one-half to one-eighth of the ratio h2/w. Farther, the profile gets its asymptotic shape, schematically depicted in Figure 9. Registry No. Malonic acid, 141-82-2; bromate, 15541-45-4; ferroin, 14708-99-7.
Stationary Patterns in a Discrete Beiousov-Zhabotinsky Medium with Small Catalyst Diffusibility A. B. Rovinsky Institute of Biological Physics of the Academy of Sciences, Puschino, Moscow Region, 142292, USSR (Received: May 19, 1988; In Final Form: September 6, 1988)
Diffusive instability of spatially homogeneous stable stationary states in models of the Belousov-Zhabotinsky reaction leading to symmetry breaking can only occur when the diffusion coefficient of the catalyst exceeds that of the other active species. However, stable inhomogeneousstructures can be found in a system of diffusively coupled cells even if the diffusion coefficient of the catalyst is smaller than that of other dynamic species (which is more likely in reality). Formation of such structures requires that the symmetry be broken by appropriate initial conditions.
Excitable or oscillatory Belousov-Zhabotinsky systems present excellent examples of self-organization phenomena. In a stirred reactor, a number of phenomena have been found such as single-pulse generation, bistability, regular quasi-sinusoidal and relaxation oscillations, bursts of oscillations, multiperiodic, and chaotic oscillations.’,2 In a spatially extended reactor such as Petri dish, solitary traveling circular and spiral waves have been observed.IV2 Stationary mosaic patterns also have been found, but their origin and nature are not ~ l e a r . ~Formation .~ of stationary patterns is one of the general ways to s e l f - ~ r g a n i z a t i o n . ~ ~ ~ As Turing showed,’ stationary spatial structures may arise when the spatially homogeneous steady state of a uniform, active medium composed of stable local elements loses its stability because of the interplay of reaction and diffusion. Recent theoretical works has shown that in the ferroin-catalyzed Belousov-Zhabotinsky system the Turing bifurcation is possible and stationary structures may be formed. Stationary patterns also have been found by Becker and Field9 in the Oregonator model. Rovinsky8 has shown that the Turing bifurcation can occur in that system only if the diffusion coefficient of the catalyst exceeds that of bromous acid, which is an active intermediate in the reaction. The stable stationary patterns in ref 8 and 9 were found only under this condition. Although the diffusion coefficients of neither ferriin nor bromous acid are known exactly, one could hardly expect the diffusion coefficient of ferriin ion to be greater than that of bromous acid because ferriin is a rather large, charged complex ion with organic ligands, while HBr02 is a much smaller molecular species. If so, the Turing instability is impossible in that system. Reference 8 shows, however, that the Turing instability is not a necessary prerequisite for formation of inhomogeneous structures. Such structures may be found in a system with a stable homo(1) Oscillations and Traveling Waves in Chemical Systems; Field, R. J., Burger, M., Eds.; Wiley-Interscience: New York, 1985. (2) Non-Equilibrium Dynamics in Chemical Systems; Vidal, C., Pacault, A,, Eds.; Springer-Verlag: West Berlin, 1984. (3) Zhabotinsky, A. M.; Zaikin, A. N. J . Theor. Biol. 1973, 40, 45-60. (4) Showalter, K. J . Chem. Phys. 1980, 73, 3735. ( 5 ) Nicolis, G.; Prigogine, 1. Se[fOrganization in Non-Equilibrium Sysrems; Wiley-Interscience: New York, 1977. ( 6 ) Haken, H. Advanced Synergetics; Springer Series in Synergetics; Springer-Verlag: West Berlin, 1983. (7) Turing, A. Philos. Trans. R. Soc., London, Sei. B 1952, 2378, 31-72. (8) Rovinsky, A. B. J . Phys. Chem. 1987, 91, 4606-4613. (9) Becker, P. K.; Field, R. J. J . Phys. Chem. 1985, 89, 118-128.
0022-365418912093-27 16$01.50/0 , ,
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geneous state. In those systems, however, the stationary patterns cannot arise spontaneously but are formed when an appropriate perturbation is imposed on the system. The question then can be raised whether stable stationary reaction-diffusion structures can exist in this case if the diffusion coefficient of ferriin ion is smaller than diffusion coefficient of bromous acid. A positive answer to this question is important not only for this particular system but also for a broader class of “inhibitor-activator” (or “predator-prey”, or “stabilizer-destabilizer”) systems extensively studied in physics, biology, and ecol~gy.~-’JO-~~ In all models of these systems, stable stationary patterns were studied only for diffusibility of an inhibitor (predator) greater than that of an “activator” (prey).5-7,10-’5From the point of view of such models, the ferriin ion plays the role of a formal inhibitor (predator), despite the fact that it is really a catalyst of the reaction. Ferriin ion can be considered as an inhibitor because it inhibits the growth of HBR02 by producing Br-. The bromide ion rapidly eliminates HBrO2.I6 For qualitative explanations of pattern formation when the diffusion coefficient of an inhibitor exceeds that of an activator, see, e.g., ref 10 and 15. Nonhomogeneous stationary solutions were indeed found in the model of the ferroin-catalyzed BZ medium for D ( F e ( ~ h e n ) ~ ~ + ) < D(HBr02),but they were unstable.s Calculations are presented here based on a discrete model of a Belousov-Zhabotinsky medium. The system is a line of stirred vessels each connected to its neighbors by permeable walls. In such a system an inhomogeneous distribution of reactant concentrations in various vessels can be stable even for D ( F e ( ~ h e n ) , ~ +
