J. Phys. Chem. 1996, 100, 11643-11648
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Frequency Dependent Chemical Patterns in Nonuniform Active Media Chikoo Oosawa and Kaoru Kometani* Department of Biochemical Engineering and Science, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, Fukuoka 820, Japan ReceiVed: February 2, 1996; In Final Form: April 30, 1996X
Spatial patterns are investigated in a nonuniform Belousov-Zhabotinsky (BZ) reaction medium, a thin layer of ferroin-loaded cation-exchange resin bathed in a BZ reaction mixture containing no catalyst. Two different nonuniform systems are investigated: (i) a system composed of two resin layers with different refractory periods (longer one, Tr) separated by a sharp boundary, and (ii) a system in which one of the two layers in system i is replaced by a layer of a mixture of two different resin beads. Wave patterns in each type of nonuniform systems are critically influenced by wave periods. In a system i, when periods of waves originating from either side of the boundary are longer than Tr, the waves with a shorter period grow over those with a longer one and cover the whole space in the medium. When the period of waves in one of the two layers is shorter than Tr, a 2:1 entrainment is established for the waves propagating in the same direction on either side of the boundary. The waves propagating in opposite directions form independent patterns on either side of the boundary. In a system ii, two sets of waves are observed, one propagating along the upper part and another along the lower part of the layer. When waves are initiated with a period shorter than Tr, they propagate initially only in the upper part of the layer. After the waves cover the whole space of the mixture layer, new waves appear in the lower part of the layer. The upper and lower waves propagate independently of each other. When waves with a period longer than Tr propagate in the mixture layer, the upper and lower waves show complete spatiotemporal entrainment.
Introduction Experimental and theoretical studies on pattern formation and wave propagation in the Belousov-Zhabotinsky (BZ) reaction1-4 have established the most basic properties of chemical waves in spatially uniform reaction-diffusion systems. Chemical waves in active media have a short excited wave front followed by a long refractory region and propagate with constant velocities. Because of the refractory region, chemical waves cancel each other under collision. Spatial patterns in the system arise from an interplay between the local dynamics of autocatalytic chemical reaction and the diffusive coupling of neighboring spatial domains. Recent studies on wave behavior in catalyst-immobilized systems5-11 have revealed new aspects of pattern dynamics resulting from transverse coupling.9,12-15 Winston et al.9 have investigated pattern dynamics of chemical waves on either side of a ferroin-loaded cation-exchange Nafion membrane. They demonstrated that transverse coupling of waves through the membrane leads to complete spatiotemporal entrainment after several distinct transient phases depending on the coupling strength. Zhabotinsky et al. have found that waves propagating in a thin gel layer open to air form a new wave pattern, which consists of two almost independent waves propagating along the top and the bottom of the layer.12-14 We have recently demonstrated16 that dispersion relations are different for waves propagating in ferroin-loaded cationexchange resin layers of different mesh ranges and/or different percentages of cross-linkage. We have also demonstrated that, in a system with a sharp boundary between two different resin layers, chemical waves propagating across the boundary exhibit refraction, reflection, and frequency change. In the present study, we report wave patterns resulting from an interaction * Author to whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, June 1, 1996.
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between waves originating from either side of the boundary. The resultant patterns are found to be critically influenced by wave periods. We also report pattern dynamics in layers made of a mixture of different resin beads. When a period of initial waves in a mixture layer is sufficiently short, independent patterns are formed in the upper and lower parts of the mixture layer after transient phases. Initial waves with a long period lead to complete spatiotemporal entrainment of the upper and lower waves in the mixture layer. Experimental Section Experimental procedures are similar to those in our previous paper16 and are briefly described here. A thermostated Petri dish (6 cm in diameter) with a glass lid maintained at 25 °C was utilized in all the experiments. A reaction medium was a thin layer of ferroin-loaded cation-exchange resin (1.0 × 10-5 mol of ferroin/g of beads) covered with a reaction mixture (0.25 M KBrO3, 0.2 M H2SO4, and 0.05 M malonic acid). Two different cation-exchange resin beads (Bio-Rad, analytical grade, 50W) were used in the experiments. One (resin A) was resin beads of mesh range >400 (bead diameter, 38-75 µm) and 4% cross-linkage. The other (resin B) was resin beads of mesh range 100-200 (bead diameter, 106-250 µm) and 8% crosslinkage. In the present experiments, the two resins were used separately and were also used as a mixture. When the resins were used as a mixture, a mixing ratio of A and B was taken as 1:4. The experiments were performed on two different nonuniform systems. One is a system which had a sharp boundary between two different resin layers. The two layers were made of resin A and B, and the boundary was extended through the center of the Petri dish. The other had the same structure as that of the first, but a layer of resin B was replaced by that of a resin mixture. The method to make these systems has been described in detail in our previous paper.16 The total amount of resin © 1996 American Chemical Society
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Figure 2. (a) Chemical pattern formed by waves with short periods in a system with the same structure as in Figure 1. Image taken at 3 h after the start of the reaction. (b) Two times enlargement of region 2 in part a. Scale bars: 1 cm. Figure 1. Chemical patterns in a system with two resin layers separated by a sharp boundary. Images taken at (a) 2 h and (b) 3 h after the start of the reaction. Arrows indicate the location of the boundary between the two layers: left side, resin A; right side, resin B. Scale bar: 1 cm.
beads in each experiment was equal to 2 g, and the amount of the reaction mixture was 32 mL. The layer thickness of resin beads was about 0.8 mm, and that of the reaction mixture about 1.1 cm. Thin layers of ferroin-loaded resin beads were illuminated from below by light with a wavelength 490 nm (bandwidth 10 nm). The light transmitted through the layer was monitored with a CCD video camera and recorded on a video tape. An image was processed into a 512 × 480 array of pixels with 256 digital unit intensity resolution on an image processor and a workstation as described before.16 Results and Discussion Before describing the results obtained in the present experiments, wave characteristics in layers of resin beads A and B are briefly described here. As in our previous report,16 dispersion relations in layers of ferroin-loaded cation-exchange resins are similar to those in homogeneous aqueous solutions.17,18 The relations in layers of resin beads are different in both the trigger wave velocity and the refractory period, when mesh size and/or percentages of cross-linkage of resin beads are different. The refractory period here means the smallest possible period of the waves in the medium. A layer of resin A exhibits a relation with a large trigger wave velocity (about 2.1 mm/min) and a short refractory period (TA, about 90 s),
while a layer of resin B exhibits a relation with a small trigger wave velocity (about 1.2 mm/min) and a long refractory period (TB, about 130 s). The results described below show that the final form of chemical patterns is critically influenced by the period (T) of initial waves. Chemical patterns resulting from waves with T < TB are significantly different from those with T > TB. Wave Patterns in Two Resin Layers Separated by a Sharp Boundary. Figure 1 shows patterns appearing 2 and 3 h after the start of the reaction in a system with a boundary between two layers of resins A and B. The left side of the boundary is the layer of resin A, and the right side the layer of resin B. In Figure 1a, the wave initiation site in the resin A layer (upper left, not seen) appeared spontaneously and initiated waves with a period of 240 s (>TB). The counterrotating spirals in the layer of resin B was created by breaking a wave with a piece of silver wire. The wave period from the spirals was 158 s (>TB). Because the periods of the waves originating from the two sites are longer than TB, the waves can propagate into the other side of the boundary without changing their periods. When the waves collide, they annihilate one another. As time proceeds, the waves with a shorter period grow over those with a longer one. As a result, the waves that originate from the layer of resin B cover the whole space in the medium (Figure 1b). The pattern dynamics is essentially the same as in a homogeneous aqueous solution, except that reflection of waves takes place at the boundary in the present system. The wavelengths in the resin A layer in Figure 1b are about 2 times longer than those in the resin B layer, showing about 2 times difference in wave
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Figure 3. Pattern dynamics of waves with a short period in a layer of resin mixture originating from a resin A layer. Images taken at (a) 3, (b) 4, (c) 7, and (d) 10 h after the start of the reaction. Arrows indicate the location of the boundary between two layers: left side, resin A; right side, resin mixture. Numbers outside of part d are given to mark initiation sites in the bottom part of the mixture layer. Scale bar: 1 cm.
velocities in the two layers. Due to the difference in the velocities in the two layers, the waves from the resin B layer are reflected at the boundary.16,19 Figure 2a shows an image taken 3 h after the start of the reaction in the same system as in Figure 1. The difference of the pattern in Figure 2a from that in Figure 1 arises from the difference in the period of waves in the layer of resin A. A period, 90 s (TB), is not so different from that in Figure 1. Figure 2a shows that the waves from the two sites collide at the boundary and result in two different patterns. The pattern inside the square box 1 shows independent propagation of the waves on either side of the boundary. The interaction of the waves propagating in the opposite directions causes no apparent changes in wave form and in wave velocity. The pattern inside the square box 2, of which an enlargement is shown in Figure 2b, shows entrainment of the waves in a 2:1 fashion. Every second wave in the layer of resin A is coupled with the waves in the layer of resin B, and the others have free ends without coupling partners in the resin B layer. The waves with free ends propagate in the layer of resin A along the boundary without changing their shape. Since the periods of the waves in the two layers in Figure 2a are close to the refractory periods in the respective layers, regions between successive wave fronts are almost in the refractory state. Thus, the waves propagating in opposite directions along the boundary cannot cross through the boundary and result in the pattern shown inside the square box 1. The interaction between waves propagating in the same direction can cause a modulation of wave frequency. When waves with different frequencies interact, low-frequency waves are entrained to high-frequency ones. In Figure 2a, a frequency ratio of the waves in the two layers is 155:90, and high-frequency waves are the waves propagating in the resin A layer. Because the wave period of the high-frequency waves is shorter than the
refractory period of the waves in the resin B layer, the entrainment cannot occur in a 1:1 fashion. The period of the waves in the resin B layer are elongated to double the period of the waves in the resin A layer, and a 1:2 entrainment is established, as shown in Figure 2b. The waves in the resin A layer without entrained partners in the resin B layer have free ends and propagate along the boundary without changing their shape. When waves are broken, spiral cores emerge at the broken ends.2 Such spiral cores did not appear at the free ends of the waves in Figure 2b. This is due to the refractoriness in the neighboring regions of the free ends. Independent Patterns in a Mixture Layer. Figure 3 shows a temporal behavior of pattern formation observed in a system composed of a resin A layer (left side of the boundary) and a layer of a resin mixture (right side of the boundary). The counterrotating spirals in the layer of resin A give rise to nearly circular waves with a period of 97 s (TB) and 1.54 mm/min, respectively. It should be noted that the velocities of the waves in the two figures are larger than the trigger wave velocity in a layer of resin B. The two patterns in Figure 5 were quite stable, and new patterns as in Figures 3 and 4 did not appear during our observations, up to about 12 h after the start of the reaction, showing complete spatiotemporal entrainment of the waves in the upper and lower parts of the mixture layer. A phenomenon reminiscent of phase death as in Figures 3 and 4 was not observed for waves with periods longer than TB. The coupling strength of waves in the upper and lower parts of mixture layers in Figure 5 is not different from that in Figures 3 and 4, indicating that the appearance of the phenomenon reminiscent of phase death is critically influenced by wave periods in systems with the same coupling strength. The relations between the velocities and the periods of the waves in Figure 5 show that the waves follow the dispersion relation for waves in a resin A layer, and the velocity of the waves is larger than the trigger wave velocity propagating in resin B layers. The results indicate that the wave patterns are determined by the property of the resin A layer. The waves in the lower layer are in a state reminiscent of forced oscillation by the waves in the upper part of the mixture layer. Ga´spa´r et al. have shown by numerical simulations22 that when spirals with slightly different rotational periods are mutually entrained, the period of the entrained spiral lies between those of initial spirals. Though the situation of their simulations is a bit different from that of our experiments, our results suggest that
The ferroin-immobilized cation-exchange resin system for the BZ reaction has a great potential in the study of pattern dynamics in nonuniform systems. Nonuniform systems with different structures can be easily constructed by combining resin beads with different mesh size and/or different percentages of crosslinkage. Studies of pattern dynamics in nonuniform systems with different forms may provide insights into important features of biological excitable media. In the present study, pattern dynamics is investigated in two types of nonuniform systems. One type is a system with two layers of different resin beads separated by a sharp boundary, and the other type a system with two layers being stratified. In each type of system, the final form of chemical patterns is critically influenced by periods of initial waves. The final patterns resulting from waves with short periods are quite different from those with long periods. The border between “short” and “long” lies in the refractory period of the waves in the medium. In a system of two layers of different resin beads separated by a sharp boundary, pattern dynamics arising from initial waves with long periods is essentially the same as that in a homogeneous aqueous solution, except that waves from a low-velocity medium are refracted at the boundary. Initial waves with short periods result in independent and entrained patterns depending on the directions of the waves propagating on either side of the boundary. The waves propagating in opposite directions form independent patterns and those in the same direction established an entrainment. The transverse coupling of chemical waves has been investigated experimentally9,12-14 and by numerical simulations.14,20,22 The investigations9,22 of pattern dynamics resulting from transverse coupling of waves have shown that complete spatiotemporal entrainment is achieved after several distinct transient phases, and the transient behavior and the final form of the entrainment are critically influenced by the coupling strength and periods of the waves. Waves propagating in a mixture layer in the present experiments exhibit transverse coupling, and the study of pattern formation in the mixture layer reveals the effects of wave periods on the final form of chemical patterns resulting from transverse coupling. The period of initial waves is shown to play a critical role in the final form of chemical patterns in systems with the same strength of transverse coupling. Entrainment of waves is achieved for waves with long periods, while waves with short periods form independent patterns after transient phases reminiscent of phase death. References and Notes (1) Zaikin, A. N.; Zhabotinsky, A. M. Nature 1970, 225, 535. (2) Winfree, A. T. Science 1972, 175, 634. (3) Field, R. J.; Burger, M., Eds. Oscillations and TraVeling WaVes in Chemical Systems; Wiley: New York, 1984. (4) Ross, J.; Mu¨ller, S. C.; Vidal, C. Science 1988, 240, 460. (5) Maselko, J.; Reckley, J. S.; Showalter, K. J. Phys. Chem. 1989, 93, 2774.
11648 J. Phys. Chem., Vol. 100, No. 28, 1996 (6) Agladge, K. I.; Krinsky, V. I.; Panfilov, A. V.; Linde, H.; Kuhnert, L. Physica D 1989, 39, 38. (7) Swinney, H. L., Krinsky, V. I., Eds. WaVes and Patterns in Chemical and Biological Media; Physica D 1991, 49, Nos. 1&2. (8) Yamaguchi, T.; Kuhnert, L.; Nagy-Ungvarai, Zs.; Mu¨ller, S. C.; Hess, B. J. Phys. Chem. 1991, 95, 5831. (9) Winston, D.; Arora, M.; Maselko, J.; Ga´spa´r, V.; Showalter, K. Nature 1991, 351, 132. (10) Perez-Mun˜uzuri, V.; Aliev, R.; Vasiev, B.; Perez-Villar, V.; Krinsky, V. I. Nature 1991, 353, 740. (11) Agladge, K.; Keener, J. P.; Mu¨ller, S. C.; Panfilov, A. Science 1994, 264, 1746. (12) Zhabotinsky, A. M.; Mu¨ller, S. C.; Hess, B. Chem. Phys. Lett. 1990, 172, 445. (13) Zhabotinsky, A. M.; Mu¨ller, S. C.; Hess, B. Physica D 1991, 49, 47.
Oosawa and Kometani (14) Zhabotinsky, A. M.; Gyo¨rgyi, L.; Dolnik, M.; Epstein, I. R. J. Phys. Chem. 1994, 98, 7981. (15) Linde, H.; Engel, H. Physica D 1991, 49, 13. (16) Oosawa, C.; Fukuta, Y.; Natsume, K.; Kometani, K. J. Phys. Chem. 1996, 100, 1043. (17) Pagola, A.; Ross, J.; Vidal, C. J. J. Phys. Chem. 1988, 92, 163. (18) Sevcikova, H.; Marek, M. Physica D 1989, 39, 15. (19) Zhabotinsky, A. M.; Eager, M. D.; Epstein, I. R. Phys. ReV. Lett. 1993, 71, 1526. (20) Bugrim, A. E.; Zhabotinsky, A. M.; Epstein, I. R. Phys. ReV. Lett. 1995, 75, 1206. (21) Tyson, J. J. Ann. N.Y. Acad. Sci. 1979, 316, 279. (22) Ga´spa´r, V.; Maselko, J.; Showalter, K. Chaos 1991, 1, 435. (23) Crowley, M. F.; Epstein, I. R. J. Phys. Chem. 1989, 93, 2496.
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