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Phase Wave between Two Oscillators in the Photosensitive Belousov-Zhabotinsky Reaction Depending on the Difference in the Illumination Time Satoshi Nakata,*,† Kenji Kashima,† Hiroyuki Kitahata,‡,§ and Yoshihito Mori| Graduate School of Science, Hiroshima UniVersity, Kagamiyama 1-3-1, Higashi-Hiroshima 739-8526, Japan, Department of Physics, Graduate School of Science, Chiba UniVersity, Yayoi-cho 1-33, Inage-ku, Chiba 263-8522, Japan, PRESTO, JST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan, and Graduate School of Humanities and Sciences, Ochanomizu UniVersity, 2-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan ReceiVed: June 7, 2010; ReVised Manuscript ReceiVed: July 23, 2010
To investigate the nature of the phase wave between two connected oscillators, the photosensitive Belousov-Zhabotinsky (BZ) reaction was examined for two connected circular reaction fields, which were drawn by using computer software and then projected on a filter paper soaked with BZ solution by using a liquid-crystal projector. The difference in the time at which illumination was terminated between the two circles (∆t0) was changed to control the time at which the phase wave was induced. When ∆t0 was small (0-3 s), the phase wave normally propagated on the two circles in one direction. In contrast, when ∆t0 was large (6-10 s), the velocity of the wave decreased near the intersection of the two circles. These different features are discussed in relation to the excitability of the circles and ∆t0. The experimental results were qualitatively reproduced by a numerical calculation based on the modified three-variable Oregonator model that included photosensitivity. Introduction Experimental and theoretical studies on spatiotemporally developing phenomena among coupled nonlinear elements may help us not only to understand signal processing in biological systems,1 for example, Ca2+ wave propagation through adjoining cells,2–4 but also to create novel methods for artificial processing, such as image processing5–8 and information processing,9–14 based on a reaction-diffusion system. The characteristic features of wave propagation and oscillation in the Belousov-Zhabotinsky (BZ) reaction have been experimentally and theoretically clarified as an excitable or oscillatory chemical system.15–17 As an application of the BZ system, many studies have examined the features of wave propagation in excitable fields with various geometries.18–25 A photosensitive experimental setup for the BZ reaction5,6,13,26,27 makes it easier to create excitable fields with various geometries, which are drawn by computer software and then projected by using a liquid-crystal projector on a filter paper soaked with BZ solution.14,28 In this case, light illumination produces bromide ion, which inhibits the oscillatory reaction; that is, the degree of excitability can be adjusted by changing the intensity of illumination. Therefore, excitation and oscillation can be regulated by the spatiotemporal change in illumination.14,29–31 Although there have been many studies on the coupled oscillators in the BZ system, for example, continuous stirred tank reactor (CSTR)32–35 and ion-exchange beads,36,37 there have not yet been any experimental studies on the control of the initial phase difference and the strength of spatial coupling. In the present study, we examined the characteristic features of the phase wave on two connected circular fields, that is, unidirec* To whom correspondence should be addressed. Tel.: +81-82-424-7409; E-mail:
[email protected]. † Hiroshima University. ‡ Chiba University. § PRESTO. | Ochanomizu University.
tional propagation of the phase wave with or without a decrease in velocity near the intersection of the circles, when the excitabilities of two fields were controlled by illumination. The characteristic features of the phase wave depending on the difference in the time at which illumination was terminated between two circles were discussed in relation to the degree of excitability of the reaction field and the effect of the boundaries of the two circles. The experimental results were qualitatively reproduced by a theoretical calculation based on the modified three-variable Oregonator model that included photosensitivity.38 These results suggest that the photosensitive BZ reaction may be useful for studying spatiotemporal development that depends on the geometry and excitability of the reaction fields. Experiments Ru(bpy)3Cl2, purchased from Sigma-Aldrich (St. Louis, MO), was used as a catalyst for the photosensitive BZ reaction. The BZ solution consisted of [NaBrO3] ) 0.5 M, [H2SO4] ) 0.3 M, [CH2(COOH)2] ) 0.16 M, [KBr] ) 0.01 M, and [Ru(bpy)3Cl2] ) 1.7 mM. A cellulose nitrate membrane filter (Advantec, A100A025A; diameter, 25 mm; thickness, 150 µm) with a pore size of 1 µm was homogeneously soaked in BZ solution (5 mL) for about 1 min. The soaked membrane filter was gently wiped with another pure filter paper to remove excess solution and placed on a glass plate (77 × 52 × 1.3 mm3). The surface of the membrane filter was completely covered with 0.7 mL silicone oil (Wako, WF-30) to prevent it from drying. The experiments were carried out in an air-conditioned room at 298 ( 1 K. The medium was illuminated from below as shown schematically in Figure 1. The high-pressure mercury bulb of a liquidcrystal projector (Mitsubishi, LPV-XL8) was used as a light source, the spatial intensity distribution was controlled with a personal computer, and a magnifying lens was used to adjust the focus. The black-and-white picture created by the liquidcrystal projector served as an illumination mask to create an
10.1021/jp105204n 2010 American Chemical Society Published on Web 08/09/2010
Phase Wave between Two Oscillators in the BZ Reaction
J. Phys. Chem. A, Vol. 114, No. 34, 2010 9125
Figure 1. Schematic illustration of (a) the experimental system based on the photosensitive BZ reaction, (b) the reaction field composed of two circles, and (c) the relationship between light intensity and the state on the two circles. d is the minimum distance between the two circles. After the illumination for 30 s except at the two circles, the left half of the reaction field was masked. After the left half was masked for ∆t0, the right half was masked at ∆t ) 0 s.
appropriate boundary. Two black circles (diameter, 4 mm; minimum distance, d; see I in Figure 1b) were located around the center of a larger circular field (Figure 1). The light intensity at the illuminated part was measured with a light intensity meter (As-one, LX-100). To create a difference in the time at which illumination was terminated between the two black circles, the following steps were taken. (1) At the initial stage, the light intensities, which were adjusted by changing the gray level of the computer
software, on the black circles and the other white region were 160 and 17500 lx, respectively. The reaction medium did not show a chemical wave or oscillation (Stage I in Figure 1b); that is, Stage I is the excitable state, as indicated in Figure 1c. (2) Thirty seconds after Stage I was started, the left half of the reaction medium was masked; that is, the light intensity of the left half was changed to 160 lx (Stage II in Figure 1b). In Stage II, no chemical wave and no oscillation were observed. (3) At ∆t0 after Stage II was started, the right half was also masked
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Figure 3. Experimental results of ∆t depending on ∆t0 for d ) 0 mm.
Figure 2. Experimental results of (1) a spatiotemporal plot and (2) typical snapshots of oscillation (top view) for ∆t0 ) (a) 2 s and (b) 7 s when d ) 0 mm. The space analyzed for the spatiotemporal plot is denoted by the white horizontal line on the two circles (a-1). ∆t ) tR - tL, where tR and tL are the times when the oscillations are generated on the left (t ) tL) and right (t ) tR) multiple marks at the centers of the two circles. The vertical dotted lines in panels a and b denote the location of the connection of the two circles.
(this moment was defined as the time origin, Stage III in Figure 1b). At least four examinations were performed for each experimental condition. The experiments were monitored from above with a digital video camera (Sony DCR-VX700) and recorded on videotape. A blue optical filter (Asahi Techno Glass, V-42) with a maximum transparency at 410 nm was used to enhance the images of the green-colored solution, which correspond to the oxidized state, [Ru(bpy)3]3+. The features of the oscillations on the two circles were analyzed by an image-processing system (ImageJ, National Institutes of Health, USA). Results Figure 2 shows the experimental results of (1) a spatiotemporal plot and (2) snapshots of the oscillation for different values of ∆t0 when the circles were in contact with one another, that is, d ) 0 mm. Here, the periods of individual oscillations in the two circles were 19 ( 2 s and independent of t0 and d. For ∆t0 ) 2 s, the phase wave was generated periodically along the two connected circles with ∆t ()tR - tL) ≈ 0.7 s, where tR and tL are the times at which oscillations were generated at the left (t ) tL) and right (t ) tR) circles (see crosses) in Figure 2a-1 (upper). The velocity of the phase wave was ∼7 mm s-1 except at the edges of the circles. For ∆t0 ) 7 s, the oscillation was generated at the left circle, the chemical wave propagated from the intersection of the two circles toward the right circle, and the oscillation was generated at the right circle except near the intersection of the two circles (see vertical dotted line). The velocity of the wave around the intersection was ∼0.18 mm s-1, and ∆t ≈ 5 s.
Figure 4. Experimental results of (a) a spatiotemporal plot (∆t ) 0 s) and (b) ∆t depending on ∆t0 for d ) 5 mm.
The oscillations generated from the centers of the two circles developed to the outside regions across their boundaries. These oscillatory phenomena were maintained for a few minutes (5-8 cycles). After 5-8 cycles of these oscillations, the propagating waves (velocity ≈ 0.09 mm s-1), which were generated from the boundary of the circle (A and B), disturbed the oscillations on the two circles. Figure 3 shows the experimental results for ∆t depending on ∆t0 for d ) 0 mm. When d ) 0 mm, ∆t was almost lower than ∆t0 at ∆t0 < 6 s, and ∆t was constant at ca. 4 s at 6 s e ∆t0 e 10 s. In this experiment, it was difficult to examine the conditions at ∆t0 > 10 s because the oscillation at the left circle often started before the right side was masked. Figure 4 shows the experimental results of (a) a spatiotemporal plot at ∆t0 )0 s and (b) ∆t depending on ∆t0 for d ) 5 mm. Phase waves, which were independently generated from the two circles, collided and disappeared between the outside of the two circles. ∆t was almost equal to ∆t0. Figure 5 shows the experimental results for ∆t depending on d when ∆t0 ) 7 s. ∆t was ∼4 s for d e 2 mm, but ∼7 s for 4 mm e d. Discussion On the basis of the experimental results and related studies, we discuss the nature of the phase wave on the two circles with a difference in the time at which illumination was terminated by using the suggested scheme, as indicated in Figure 6. Because light illumination produces Br- as an inhibitor because of the photochemical reactions of the catalyst,5 the concentration of Br-, [Br-], in the reaction field under a bright background is higher than that under a dark background, and thus light illumination inhibits the BZ reaction (State I in Figure 6).7,27,29 When both the two circles and the outer area were not illuminated, several chemical waves were generated; that is, the system was considered to be in an oscillatory state, but no
Phase Wave between Two Oscillators in the BZ Reaction
Figure 5. Experimental results for ∆t depending on d at ∆t0 ) 7 s. The horizontal dotted line (∆t0 ) ∆t) indicates that ∆t is determined by ∆t0. If ∆t < ∆t0, the phase of the oscillation at the right circle is accelerated by that at the left circle.
Figure 6. Suggested mechanism for the features of the phase wave on two circles (d ) 0 mm and ∆t0 g 7 s). The phase wave is repeated by the cycling through States III-1, 2, 3, and 4.
chemical waves were observed when the outer area was illuminated but the two circles were not. This suggests that the excitabilities on the circles may be weakened by light scattering from the periphery of the circles, as indicated in Figure 1c. Actually, the velocities of the phase wave around both ends of the circles were lower than those around their centers, as seen in Figures 2a1 and 2b1. Because the left half of the reaction field is initially masked (State II in Figure 6) and then the right half is masked, [Br-] at the left circle is lower than that at the right circle at t ) 0 (State III-1 in Figure 6). Therefore, the left circle reaches an oscillatory state before the right circle because of the difference in [Br-] between the right and left circles, which is caused by the difference in the illumination time, ∆t0.31 The phase wave with a constant and rapid velocity from left to right at small ∆t0 (1-3 s) may be caused by a small difference in [Br-] between them; that is, the oscillation at the right circle is rapidly induced by the oscillation at the left circle. Thus, the increase in the concentration of HBrO2 (activator) due to the oscillation at the left circle induces oscillation at the right circle as propagation of the phase wave (State III-3 in Figure 6). In contrast, the decrease in the velocity of the wave near the intersection of the connected circles at large ∆t0 (see Figure
J. Phys. Chem. A, Vol. 114, No. 34, 2010 9127 2b) may be caused by higher [Br-] near the intersection due to hysteresis of the light illumination (State III-2 in Figure 6). The concentration of the activator does not reach the threshold value for oscillation (or starting of the autocatalytic reaction) at the right circle due to the high [Br-] even if oscillation is generated at the left circle, and therefore, the slower chemical wave rather than the phase wave is generated from left to right near the intersection. The constant value of ∆t ≈ 4 s for ∆t0 ) 6-10 s suggests that the induction time of oscillation at the right circle after the right half of the reaction field is masked is shortened because of wave propagation from the left circle (State III-3 in Figure 6). Thus, the concentration of the activator around the left end of the right circle is increased by oscillation at the left circle. State III-4 returns to State III-1, and the spatiotemporal oscillation is generated by the repetition of States III-1, 2, 3 and 4. Figure 5 suggests that there are two parts, that is, (i) d e 2 mm at ∆t ≈ 4 s and (ii) 4 mm e d at ∆t ) ∆t0. In the case of (i), the phase wave at the right circle starts before it is affected by the phase wave from the left circle. If we consider that the velocity of a phase wave is 7 mm s-1 and that the radius of the circle is 2 mm, the velocity of the phase wave from the left circle decreases at the intersection of the circles, and it takes about 4 s to go ahead from one circle to another. Thus, the critical length of phase-wave propagation between the two circles is 2 mm at ∆t0 ) 7 s. In the case of (ii), the time dependency is constant at 7 s. Before the phase wave reaches the right circle, oscillation is independently generated at the right circle. Therefore, the two oscillatory waves collide and then disappear outside of the two circles. In order to confirm the above-mentioned mechanism, we made a numerical calculation based on the modified threevariable Oregonator model describing the photosensitve BZ reaction:38
∂u 1 ) [qV - uV + u(1 - u) + p2φ] + Du∇2u ∂t ε
(1)
∂V 1 ) [-qV - uV + fw + p1φ] + DV∇2V ∂t ε′
(2)
(
)
p1 ∂w )u-w+ + p2 φ + Dw∇2w ∂t 2
(3)
where u, V, and w are dimensionless variables that correspond to the concentrations of the activator (HBrO2), bromide ion (Br-), and oxidized catalyst ([Ru(bpy)3]3+), respectively. f, ε, ε’, and q are positive parameters that determine the nature of the BZ reaction. p1 and p2 are positive parameters related to photosensitive chemical reaction. Du, DV, and Dw are the diffusion constants for u, V, and w, respectively. φ corresponds to the light intensity. On the basis of eqs 1-3, numerical simulations in a twodimensional space were performed. The light intensity was changed on the basis Figure 1c. We set four parts in calculation field (just as A, B, C, and D in Figure 1c). At first, light intensities were set to φ1 in A and B and φ2 in C and D. Then, at t ) -∆t0 the light intensities in A and C were instantaneously changed to φ3. At t ) 0, the light intensities in B and D were also changed to φ3. We set the values of φ1, φ2, and φ3 so that the system was suppressed for φ1, excitable for φ2, and oscillatory for φ3. The radius of the circles of A and B were set as 32, and the minimum distance between two circles, d, was
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Figure 7. Numerical results represented by (a) the plots of ∆t against ∆t0 for (a-1) d ) 0 and (a-2) 80 and (b) that of ∆t against d for ∆t ) 4, based on eqs 1-3. In panel a, the broken line corresponds to ∆t ) ∆t0. ∆t were calculated by averaging the time differences for the first three peaks. The parameters are f ) 1.2, ε ) 0.06347, ε’ ) 0.00021164, q ) 0.000095238, p1 ) 0.0176, p2 ) 1.13018, and Du ) DV ) Dw ) 5.0.38 The variables for the light intensities were set as φ1 ) 0.01, φ2 ) 0.003, and φ3 ) 0.0.
varied. The spatial and temporal grid sizes were set as 1 and 10-4, respectively. The size of the calculation field was 384 × 96. The Neumann boundary condition was adopted at the edges of the calculated field. Figure 7a shows the plots for ∆t vs ∆t0 for different d. When d ) 0, ∆t is almost equal to ∆t0 for small ∆t0 (∆t0 < 2), whereas ∆t becomes smaller than ∆t0 for large ∆t0 (∆t0 > 3). Here, the intrinsic period of the oscillation without any interaction is 11.6. In contrast, when d ) 80, ∆t is almost equal to ∆t0, which means the phase waves on two circles do not interact. These results correspond well to the experimental results as shown in Figures 3 and 4b. In Figure 7b, ∆t is plotted against d for ∆t0 ) 4. Those results qualitatively correspond to the experimental ones, as shown in Figures 3-5. The numerical results on the spatiotemporal plots and snapshots based on eqs 1-3 are shown for d ) 0 and ∆t0 ) 1 (Figure 8a), for d ) 0 and ∆t0 ) 4 (Figure 8b), and for d ) 80 and ∆t0 ) 1 (Figure 8c). The results shown in Figure 8a,b,c correspond to the experimental results as shown in Figures 2a,b and 4a, respectively.
Figure 8. (a) The spatiotemporal plot and snapshots for (a) every one time unit achieved by the numerical calculation for d ) 0 and ∆t0 ) 1, (b) every two time units achieved by the numerical calculation for d ) 0 and ∆t0 ) 4, and (c) every one time unit achieved by the numerical calculation for d ) 80 and ∆t0 ) 1. Both in the spatiotemporal plots and snapshots, only the field near the circles are shown. The parameters for numerical calculation are the same as those in Figure 7.
Conclusion The nature of a phase wave in the photosensitive BZ reaction via the connection of two circular fields was examined both experimentally and numerically. When ∆t0 was small (0-3 s), the phase wave propagated unidirectionally, and the difference in time (∆t) when the phase wave was generated at the centers of the two circles was almost constant at around 1 s. In contrast, when ∆t0 was large (6-10 s), the velocity of the phase wave was decreased near the intersection of the two circles, and the time difference (∆t) was almost constant at 5 s. The experimental results can be qualitatively reproduced by a numerical calculation based on the three-variable Oregonator modified for the photosensitive BZ reaction. However, several problems remain in the present paper. That is, for 0 < d < 2 mm, ∆t seems to be constant at ∼4 s in Figure 5, but such a constant value of ∆t is not reproduced for a small d in Figure 7b. In addition, ∆t < ∆t0 at ∆t0 < 5 s in the experimental results of Figure 3, but ∆t ≈ ∆t0 at ∆t0 < 2 in the numerical calculation
in Figure 7a-1. These discrepancies between the experimental and numerical results may be due to the relaxation process of the concentrations of chemical substances between two circles after the mask of the reaction medium. It suggests that the relaxation time is very short for a small d or a small ∆t0 in the experimental system. In the present study, the reproducibility of these characteristics has not yet succeeded in the numerical calculation. Further optimization of the used parameters and development of the mathematical model is necessary to obtain the reproducibility of the experimental results for 0 mm < d < 2 mm and at ∆t0 < 5 s in the future work. The photosensitive BZ reaction may be useful for studying spatiotemporal development that depends on the geometric condition of excitable fields, because the shape of the reaction field can be easily varied by using a personal computer. To investigate reaction fields with further complex shapes or to experimentally reproduce theoretical predictions, the develop-
Phase Wave between Two Oscillators in the BZ Reaction ment of an experimental system in place of an immersed filter paper will be necessary to maintain and control the features of wave propagation and oscillation at a mesoscopic scale. In this study, the behaviors of chemical waves on the photosensitive BZ reaction might be affected not only by the instant light intensity but also by the nature of the temporal change and/or the spatial gradient of light intensity. It would be interesting to study the hysteresis effect in the photosensitive BZ reaction under spatiotemporal stimulation with light. Acknowledgment. The authors would like to thank Professor Takashi Amemiya (Yokohama National University, Japan) for his kind advice. This work was supported in part by a Grantin-Aid for Scientific Research (No. 20550124), by a grant from the Asahi Glass Foundation, and by a grant from Shiseido to S.N. References and Notes (1) Winfree, A. T. The Geometry of Biological Time; Springer: Berlin, 1980. (2) Korkiama¨ki, T.; Yla¨-Outinen, H.; Koivunen, J.; Karvonen, S.-L.; Peltonen, J. Am. J. Pathol. 2002, 160, 1981. (3) Frame, M. K.; de Feijter, A. W. Exp. Cell Res. 1997, 230, 197. (4) Denda, M.; Denda, S. Skin Res.Tech. 2007, 13, 195. (5) Kuhnert, L. Nature 1986, 319, 393. (6) Kuhnert, L.; Agladze, K. I.; Krinsky, V. I. Nature 1989, 337, 244. (7) Rambidi, N. G.; Shamayaev, K. E.; Peshkov, G. Y. Phys. Lett. A 2002, 298, 375. (8) Sakurai, T.; Mihaliuk, E.; Chirila, F.; Showalter, K. Science 2002, 296, 2009. (9) To´th, A.; Showalter, K. J. Chem. Phys. 1995, 103, 2058. (10) Steinbock, O.; Kettunen, P.; Showalter, K. J. Phys. Chem. 1996, 100, 18970. (11) Motoike, I.; Yoshikawa, K. Phys. ReV. E 1999, 59, 5354. (12) Ichino, T.; Igarashi, Y.; Motoike, N. I.; Yoshikawa, K. J. Chem. Phys. 2003, 118, 8185. (13) Gorecka, J.; Gorecki, J. Phys. ReV. E 2003, 67, 067203. (14) Nagahara, H.; Ichino, T.; Yoshikawa, K. Phys. ReV. E 2004, 70, 036221.
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