Reaction-diffusion waves

Reaction-diffusion waves Homogeneous and inhomogeneous effects Barry R. Johnson, Stephen K. Scott and Annette F. Taylor School of Chemistry, University of L eeds, L eeds, UK L S2 9JT

Experimental evidence for the spontaneous formation of spiral waves and crossing wave patterns for the BelousovÈ Zhabotinsky reaction in solution are presented. Also observed are so-called “ lateral instabilities Ï with the spontaneous deformation of circular fronts in systems of low excitability. Lateral instabilities in resin-based systems, with the redox catalyst immobilised on ion-exchange beads, and the e†ect of the ageing of solutions, are also reported.

The existence of target and spiral waves in so-called “ excitable Ï media is reasonably widely known since the early observations reported by Zhabotinsky and Zaikin1 and by Winfree2 on the BelousovÈZhabotinsky (BZ) reaction. In homogeneous solutions, targets comprising concentric circular waves arise spontaneously from pacemaker sites, perhaps corresponding to the presence of local inhomogeneities (such as dust particles) where the local conditions give rise to a spontaneously oscillatory region.3 Spirals are not spontaneous structures, but can be induced by breaking a wave to produce broken ends that then evolve into a counter-rotating spiral pair. More recently, attention has focused on inhomogeneous media, such as layers of resin beads4,5 or gel matrices6 in which the BZ reaction can be carried out. Such systems allow the spontaneous evolution of spirals and several theories7h9 for the development of such structures have been proposed, typically based on local inhomogeneities in the excitability of the medium, either present initially or induced by the reaction through a conditioning wave. Additional responses, such as crossing wave patterns have also been reported for the BZ system10,11 in such inhomogeneous media. In this paper, we report the observation of some of these features in an apparently homogeneous solution of the BZ reagent under an experimental composition in which the mixture has a low excitability. We also report some observations on the stability of circular fronts in solution to lateral perturbations (i.e. the evolution to non-circular waves) and describe similar e†ects in resin-based systems.

Experimental Solutions of analytical grade potassium bromate, potassium bromide, malonic acid and sulfuric acid were prepared with doubly distilled, de-ionised water and Ðltered to remove dust particles. Ferroin was prepared from analytical grade iron(II) sulfate and 1,10-phenanthroline. For the solution-phase experiments reported here, the initial concentrations after mixing were [KBrO ] \ 0.12 M, [CH (CO H) ] \ 0.01 M, 3 0 2 2 2 0 [Br~] \ 1 ] 10~4 M, [Fe(phen) 2`] \ 1 ] 10~3 M. The 0 3 initial sulfuric acid concentration was varied between 0.1 and 0.3 M. The solutions were mixed in the order bromate, bromide, sulfuric acid, malonic acid and, after the Br had 2 been reacted, ferroin. The reaction mixture was stirred and then poured into an optically Ñat Petri dish of diameter 5.5 or 9.0 cm mounted in a thermostatted water jacket at 20 ¡C. The

dish was loosely covered with a glass lid to prevent dust falling on to the liquid surface and to reduce convective e†ects. No condensation on the glass surface was observed during the reaction. The system was illuminated from below with light of 490 ^5 nm and the evolution of the reaction monitored by colour video imaging (JVC camera) with framegrabbing and subsequent image analysis using the OPTIMAS5 software package. For the resin-based experiments, ferroin was pre-loaded onto Dowex-50W ion-exchange resin (4È400 mesh) at 1 ] 10~4 mol g~1 and these beads were then spread as a thin layer (nominally 1 mm thick) under a 1 cm deep solution of the remaining reagents at the above concentrations except that [H SO ] \ 0.2 M and [KBrO ] \ 2 4 0 3 0 0.1 M. The illuminating light was not Ðltered for the resin experiments.

Results For low acid concentration, [H SO ] \ 0.15 M, the above 2 4 0 compositions give rise to a system that is initially in a reduced, excitable state. At higher [H SO ] , the system 2 4 0 becomes spontaneously oscillatory under well-stirred batch conditions. In both cases, the long time state is the oxidised state due to the high [BrO ~] : [CH (CO H) ] ratio. For 3 2 2 2 0 the reaction in the unstirred reactor, the solution volume was adjusted to provide average layer depths of 1 or 2 mm. It should be noted, however, that the solution depth is not constant across the dish diameter : the meniscus e†ects produce a deeper solution at the edges of the dish and a thinner solution near the centre. Thin layer : spontaneous spiral formation The evolution of a system with a nominal depth of 1 mm and with [H SO ] \ 0.16 M is shown in Plate 1. In this Ðgure, part 2 4 of the edge of the Petri dish can just be seen in the top righthand corner. The solution is not illuminated over the whole of the dish, just in a small circular area, giving rise to a dark circular section between the illuminated area and the edge of the dish which can also be seen in the Ðgure. The initial mixture is allowed to undergo one oscillation before the solution is poured into the Petri dish. Plate 1(a) shows a region of the dish close to the edge 23 min after pouring. Previous to this snap-shot, a wave initiated spontaneously at a point at the rim ; this wave propagated a small distance into the illuminated area before failing. This is a characteristic feature of the early behaviour in this system ; waves are initiated but fail to propagate. A second initiation has now developed further around the rim and the oxidation front can be seen developing from this site into the illuminated area. In Plate 1(b), this wave has propagated away from the rim and, in addition, a second wave has been initiated and propagated away from the original pacemaker site, the two waves coalescing. The uppermost segment of the wave in the Ðgure is entering a region of solution in which no previous reactiondi†usion structure has entered. In this region, the wave is again beginning to fail. Further along the wave (lower in the J. Chem. Soc., Faraday T rans., 1997, 93(20), 3733È3736

3733

Plate 1 Contrast-enhanced images showing the development of spiral wave structures in solution with [H SO ] \ 0.16 M : (a) 23 min after the 2 4 0 start of the reaction, an oxidation wave enters the illuminated region. Also indicated by the dashed curve lower in the Ðgure is the extent propagated by an earlier, failing wave ; (b) 5 min later, a wave break develops in the upper part of the diagram, with the lower segment propagating successfully into the wake of the previous wave ; (c) and (d) 3 and 5 min later respectively, the broken end develops into a spiral. The image area corresponds to 3 cm vertically and 1.8 cm horizontally.

Ðgure), the wave is propagating into the solution in which the earlier event propagated. Here the new wave appears to be able to propagate successfully. This leads to a break in the wave front and the subsequent images show the typical spiral development from such a broken end. The spiral cores that develop in this system typically show signiÐcant meandering. As the initial acid concentration is increased, so the frequency of wave breaking increases and multiple spirals form in the solution. A slightly di†erent scenario is illustrated in Plate 2, for a system with [H SO ] \ 0.15 M. Plate 2(a) corresponds to a 2 4 0 time 31 min after the introduction of the solution into the Petri dish. The remnants of a failed wave are evident in the lower left-hand corner of the Ðgure. A new pacemaker site has developed at the rim in the top left-hand corner and is just entering the illuminated area. The subsequent images show

how the developing wave from this site is forced to evolve “ around Ï the failed wave and so develops into a spiral as that region recovers. The behaviour of a system with yet lower acidity, [H SO ] \ 0.1 M, is shown in Plate 3. The waves developing 2 4 0 initially from pacemaker sites have the usual smooth circular form at low radius. As the radius of the wave increases, however, this circular aspect can be lost and a wiggly front with a lateral or transverse mode develops. In this case, the wave does not deform sufficiently so as to break and so no spirals develop. Thicker layers : crossing wave patterns In thicker layers of solution, nominally 2 mm depth, signiÐcant three-dimensional e†ects may develop. Under some con-

Plate 2 Contrast-enhanced images showing the development of a spiral wave around the refractory region created by a previous wave in solution with [H SO ] \ 0.15 M : (a) 29 min after start of reaction, the extent of the previous wave propagation before failure indicated by 2 a new 4 0 wave entering the illuminated region ; (b)È(e) 36, 39, 49 and 63 min after the start of the reaction, respectively, Showing dashed curve, with the development of a spiral. The image area corresponds to 1.6 cm vertically and 2.0 cm horizontally.

Plate 3 Evolution from an initially circular wave due to lateral instability in solution with [H SO ] \ 0.1 M : (a) ca. 20 min after start of 4 0 non-circular waves. The image area reaction, circular wave (subsequent images at 2 min intervals) ; (b) approximate onset of instability2 ; (c)È(d) corresponds to 2.2 cm vertically and 3.2 cm horizontally.

3734

J. Chem. Soc., Faraday T rans., 1997, V ol. 93

Plate 4 Crossing wave patterns in solution of nominal 2 mm depth with [H SO ] \ 0.2 M : (a) early wave propagating throughout the whole 2 4 0 depth of the solution ; (b)È(d) image of the same region of the solution at 20 min intervals showing the development from bottom-propagation waves through a crossing wave pattern to di†use spirals ; (e) independent bottom- and surface-propagating waves. The image area in (a)È(d) corresponds to 3 cm vertically and 1.5 cm horizontally ; in (e) to 3 cm by 4.3 cm.

ditions, a wave may propagate throughout the whole solution depth. Such a case is shown in Plate 4(a). The leading edge of this wave is propagating along the bottom of the solution but just behind this there is a brighter band corresponding to a region in which the wave also breaks through to the surface of the solution. Such a wave is typically observed early in a reaction, which the pacemaker site has a low frequency and the wave has a relatively high velocity, in this case the wave speed is 3 mm min~1. As the solution ages, so typically the pacemaker frequency increases. For these higher frequency sites, the resulting waves propagate more slowly and initially only along the bottom of the solution, Plate 4(b), in which the wave speed is 1.1 mm min~1. Eventually, the wave also reaches the surface layer, but, as indicated in Plate 4(c), this may not occur over the whole length of the wave front. In this particular example, the wave does not break through to the surface in the region closer to the edge of the Petri dish (not visible in the Ðgure) where the solution is somewhat deeper due to the meniscus e†ect. This creates a series of broken ends in the surface layer that, naturally, tend to form spirals. These nascent spirals produce a connection from one wave to the next, giving rise to a staircase pattern as shown in the Ðgure. This simple scenario is similar to that described by Bugrim et al.11 in a gel matrix. However, more complex situations also arise and the breaking of the waves does not always seem to

be associated with the deepest regions of the solution. In general, it is clear that this system exhibits a great sensitivity to extremely small variations in the liquid depth. (An attempt to measure any variation of depth across the concentration region of the petri dish spectrophotometrically using the BeerÈLambert law on an unreacting ferroin solution was made but any such variation is below that which can be detected with the precision of this method.) Fig. 4(d) shows an example of a structure that develops following the staircase pattern in this system, with di†use spirals evolving. In some cases, waves can be initiated at the surface of the liquid layer : if the frequency of the pacemaker is sufficiently high, the resulting waves propagate only in the surface layer. The whole reaction domain may thus develop a series of wave trains, some propagating in the bottom layer and some propagating in the surface layer, essentially independent of each other, as shown in Plate 4(e). Lateral instabilities in resin systems Maselko et al.5 have examined the evolution of wave structures in system for which the redox catalyst is immobilised by loading onto a cation-exchange resin. Following their work, we have examined a similar system, but with a reactant concentration range corresponding more closely to that used for

Plate 5 Transition from circular waves on increasing radius due to lateral instability in resin-based system : (a) approximately 1 h after the start of the reaction ; subsequent images (b), (c) at 5 min intervals. The image area corresponds to 3 cm vertically and 4 cm horizontally.

Plate 6 Loss of lateral instability at long times : (a) 30 min after the start of the reaction ; subsequent images show waves passing the same region of observation at approximately 6 h intervals. The image area corresponds to 1.5 cm vertically and 3.0 cm horizontally.

J. Chem. Soc., Faraday T rans., 1997, V ol. 93

3735

the solution-phase experiments described above. As shown in Plate 5, pacemaker sites arise in these systems too and give rise to familiar target structures over a relatively wide range of compositions. With the low excitability employed in our experiments, however, the expanding circular fronts can display an instability and develop the transverse modes described previously for solutions. The departure from circular evolution is particularly clear in this sequence, with the wave being e†ectively circular at low radius but developing the additional modes beyond a critical radius. This behaviour is typical of freshly mixed solutions early on in this batch reaction. At longer times, as the reactant concentrations fall, the transverse instability is lost and only simple circular fronts are observed. A sequence of waves in an ageing system is shown in Plate 6.

Discussion The various phenomena presented here are relatively robust and reproducible between experiments. Although novel in the present contexts, each has counterparts already observed in other systems. Nagy-Ungvarai and MuŽller12 present an overview of experimental observations and some theoretical interpretations of wave instabilities which can, at least in part, be drawn on here. Lateral instabilities have been observed previously in the BZ system in thick silica gels (14% SiO ). In 2 more dilute gels, Horvath and Showalter13 report the phenomenon in the iodateÈarsenite reaction. In the latter case, the instability is related to the di†erence in di†usion coefficients for the feedback and reactant or recovery species. This instability mechanism was predicted theoretically14 and has been investigated in some detail. For circular fronts, the theory predicts15 a critical radius beyond which the non-circular wave structure can develop, consistent with the observations here. It is, however, not clear that selective di†usion can be operating in the BZ system : Ðrst, in solution-phase the di†usion coefficients for HBrO and ferriin are not likely to be 2 signiÐcantly di†erent and in the resin system, the recovery species is immobilised and so cannot have the higher mobility. Secondly, theory shows that, for simple autocatalytic systems at least, a cubic feedback is required whereas the autocatalysis is of the quadratic type for the BZ system. It may be added, however, that the waves in the BZ system are not simple fronts, but are pulses and have a di†erent mathematical structure. The additional inhibitory role of Br~ also enhances the non-linearity in the BZ system so that its combination with the quadratic autocatalysis is able to support limit cycle oscillations. Current interpretations for the BZ system focus more on the role of low excitability. Nagy-Ungvarai et al.16h18 report the breaking of waves under such conditions, but the broken ends were then typically unable to propagate and develop into spirals in their experiments. In our experiments, there is some evidence of “ co-operative e†ects Ï between di†erent pacemaker sites producing the instabilities. The origin of crossing wave patterns has been interpreted by Bugrim et al.11 as involving concentration gradients in O 2 and Br as functions of depth, with [Br ] being lower at the 2 2 surface than at the bottom of the gel due to evaporation of the vapour and [O ] being lower at the bottom due to reaction. It 2 is possible that this mechanism operates in solution here

3736

J. Chem. Soc., Faraday T rans., 1997, V ol. 93

although it may seem likely that such gradients will be less severe than in gels. If so, the results presented here serve to further testify to the remarkable sensitivity of oscillatory and non-linear reactions to the operating conditions. Finally, the possible role of convective e†ects should be addressed. We have repeated our experiments with solutions degassed with N and run under an N atmosphere but other2 2 wise with identical conditions. The various instabilities disappear under such conditions. Crossing wave patterns and lateral instabilities are also absent if the reaction is performed without a free solution/air interface, sandwiched between two glass plates. These features conÐrm the importance of O in 2 these reactions and at least suggest that the system excitability, rather than convective e†ects, is the main driving force. More evidence for the greater importance of low excitability can be drawn from the experiments of Markus et al.19,20 who exploit the light-sensitive [Ru(bipy) ]2` catalyst which 3 reduces the excitability under appropriate illumination and who observed transverse instability along spiral waves. B.R.J. is grateful to the University of Leeds and A.F.T. is grateful to the EPSRC for Ðnancial support. References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

A. M. Zhabotinsky and A. N. Zaikin, in Oscillating Processes in Biological and Chemical Systems II, ed. E. E. SelÏkov, Nauka, Puschino, 1971, p. 279. A. T. Winfree, Science, 1972, 175, 634. J. J. Tyson and J. P. Keener, Physica D, 1988, 32, 327. J. Maselko, J. S. Reckley and K. Showalter, J. Phys. Chem., 1989, 93, 2774. J. Maselko and K. Showalter, Physica D, 1991, 49, 21. Z. Noszticzius, W. Horsthemke, W. D. McCormick and H. L. Swinney, Nature (L ondon), 1987, 329, 619. K. I. Agladze, J. Keener, S. C. MuŽller and A. PanÐlov, Science, 1993, 264, 1746. M. Gomez-Gesteria, G. Fernandez-Garcia, A. P. Munuzuri, V. Perez-Munuzuri, V. I. Krinsky, C. F. Starmer and V. PerezVillar, Physica D, 1994, 76, 359. J. H. Merkin, A. J. Poole and S. K. Scott, J. Chem. Soc., Faraday T rans., 1997, 93, 1741. Z. M. Zhabotinsky, L. Gyorgyi, M. Dolnik and I. R. Epstein, J. Phys. Chem., 1994, 98, 7981. A. E. Bugrim, A. M. Zhabotinsky and I. R. Epstein, J. Phys. Chem., 1995, 99, 15930. Zs. Nagy-Ungvarai and S. C. MuŽller, Int. J. Bifurcation Chaos, 1994, 4, 1257. D. Horvath and K. Showalter, J. Chem. Phys., 1995, 102, 2471. D. Horvath, V. Petrov, S. K. Scott and K. Showalter, J. Chem. Phys., 1993, 98, 6332. R. A. Milton and S. K. Scott, J. Chem. Phys., 1995, 102, 5271. Zs. Nagy-Ungvarai, J. Ungvarai and S. C. MuŽller, Chaos, 1993, 3, 15. Zs. Nagy-Ungvarai, J. Ungvarai, S. C. MuŽller and B. Hess, J. Chem. Phys., 1992, 97, 1004. Zs. Nagy-Ungvarai, A. Pertsov, B. Hess and S. C. MuŽller, Physica D, 1992, 61, 205. M. Markus, G. Kloss and I. Kusch, Nature (L ondon), 1994, 371, 402. M. Markus, T. Schulte and Z. Czajka, Phys. Rev. A : Gen. Phys., 1997, 56, R1È4.

Communication 7/04616B ; Received 1st July, 1997