8677
J . Phys. Chem. 1990, 94, 8677-8682
Experimental Study of Spiral Waves in the Ce-Catalyzed Belousov-Zhabotinskii Reaction Zs. Nagy-Ungvarai,**+vgJ. J. Tyson,*-! S. C. Muller: L. T. Watson,! and B. Hess' Max- Planck- Institut fur Ernahrungsphysiologie, Rheinlanddamm 201, D-4600 Dortmund I, FRG, and Department of Biology, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 (Received: April 6 , 1990; In Final Form: May 30, 1990)
Systematic measurements of spiral wave characteristics (rotation period, normal velocity, wavelength, and wave form) in the cerium-catalyzed BZ reaction were carried out over a wide range of reagent concentrations. These data provide a stringent test for theoretical models of the reaction. Numerical calculations of spiral wave solutions to modified Oregonator equations are in satisfactory agreement with the experimental results.
Introduction The Belousov-Zhabotinskii (BZ) reaction' is an important example for studying nonlinear phenomena because experimentally it can be handled very easily. Systematic studies describe the properties of temporal oscillations in homogeneous, stirred systems' and also various properties of concentric trigger waves in thin layers of the reaction mixture as a function of the chemical composition of the system.24 Spiral waves in thin layers of the BZ reaction were discovered by Zhabotinskii and Zaikin5 and Winfree6 and since then they have been the subject of several specific investigations, both and experimentally.g-' Most of the experimental data have been measured in particular systems with a given set of selected concentrations.6 However, no systematic work has been published about the properties of spiral waves measured under controlled variation of the chemical conditions. Such a study, though time-consuming, can both serve as a test for comparison of models with experiments and contribute to the understanding of differences between experimental data measured in different laboratories under different conditions. In this paper we report on such systematic measurement of various properties of spiral waves, in continuation of our previous work on velocity and wave profiles in the Ce-catalyzed BZ r e a ~ t i o n , ~ , ~ in a wide concentration range as a function of the initial composition of the system.
TABLE I: Composition of the Reactant Solutions after Completion of the Bromination Reaction of Malonic Acid (MA) to Bromomalonic Acid (&MA) According to the Equation Br03- 2Br- + 3MA + 3H+ = 3BrMA + 3H,O rcactants concn range, M standard conc, V 0.3 NaBr03 0.12-0.57 2so4 0.16-0.9 1 0.4 1 MA 0.0 1-0.366 0.03 BrMA 0.09 0.09 Ce(S04)2 0.003-0.01 2 0.006
+
Experimental Procedure The reagent solutions (analytical reagent grade) were filtered through 0.45-pm Millipore filters, thermostated to 25.0 f 0.1 OC, and mixed to give the initial concentration as summarized in Table I . All the experiments were carried out at 25.0 f 0.1 "C. The reacting BZ solutions were placed in an optically flat Petri dish. The layer thickness was varied from 0.2 to 0.6 mm. Circular waves in the excitable solution layer were triggered by a Ag wire. Spiral waves were obtained by disruption of the circular waves with a gentle blast of air ejected from a pipet. Immediately afterwards the dish was covered with a quartz plate either without or with a IO-" air gap above the solution. The light absorption was measured by a UV-sensitive 2D spectrophotometer9 at 344 nm. On the images, dark areas represent the oxidized waves as the absorption of the oxidized catalyst Ce( IV) was measured. Computerized storage of images, evaluation of intensity data, their
* Authors to whom correspondence should be addressed. Max-Planck-lnstitut fur Ernahrungsphysiologie. *Virginia Polytechnic Institute and State University. 9 On leave from the Institute of Inorganic and Analytical Chemistry, University L. Eotviis, Budapest, Hungary. ( 1 ) Oscillations and Traveling Waves in Chemical Systems; Field, R. J., Burger, M., Eds.; Wiley-lnterscience: New York, 1985. (2) Field, R. J.; Noyes, R. M. J. Am. Chem. Soc. 1974, 96, 2001. (3) Nagy-Ungvarai, Zs.; Tyson. J. J.; Hess, B. J. Phys. Chem. 1989, 93, 707 and references therein.
0022-3654/90/2094-8677$02.50/0
Figure 1. Spiral wave in a BZ solution with :I high [ M A ] of 0.366 M and standard concentration of the other components (no. 7 in Table 11). Image area "** Layer thicknessOe3 *"
conversion into Ce(1V) concentration, extraction of intensity and concentration profiles, their superposition, determination Of spiral parameters, and other numerical operations, such as measurement (4) Nagy-Ungvarai, Zs.; Muller, S. C.; Tyson, J. J.; Hcss, R. J . Ph.v.7. Chem. 1989, 93, 2760. ( 5 ) Zhabotinskii, A. M.; Zaikin, A. N. Oscillatory Processes in Bioloxicnl and Chemical Systems; Nauka: Puschino, USSR, 1971: p 279. ( 6 ) Winfree, A. T. Science 1973, 175, 634. (7) Keener, J. P.; Tyson, J. J. Physica 1986. 2 I D , 307. (8) Mikhailov, A. S.; Krinskii, V. I. Physica 1983, 9D, 346. (9) Muller, S. C.; Plesser, Th.; Hess, B. Physica 1987, 24D, 71, 87. ( I 0) Agladze, K. 1.; Krinskii, V. I. Nature 1982, 296, 424. ( 1 I ) Jahnke, W.; Skaggs, W. E.; Winfree, A. T. J. Phys. Chem. 1989. 93. 740.
0 1990 American Chemical Society
8678 The Journal of Physical Chemistry, Vol. 94, No. 24, 1990
Nagy-Ungvarai et al.
TABLE 11: Observed Properties of Spiral Waves and Concentric Waves concentric wave
sDiral wave initial concentrations [Br03-], no.
M
1 2 3 4 5 6 7 8 9
0.57 0.12 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.57
IO
[Ce(lV)], M 0.006 0.006 0.012 0.003 0.006 0.006 0.006 0.006 0.006 0.006
[H'], M 0.41 0.41 0.41 0.41 0.91 0.16 0.41 0.41 0.41 0.91
[MA], M 0.03 0.03 0.03 0.03 0.03 0.03 0.366 0.01 0.03 0.31
amp1 [BrMA],
M 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09
[ce(rv)l- m M max min A 2.5 2.2 0.3 1.2 0.9 0.3 2.3 1.5 0.8 0.9 0.65 0.25 2.8 2.5 0.3 0.3 1.4 0.9 0.5 0.4 0.4 0.3
wavelength, mm 1.08 1.70 1.28 1.32 1.02 1.90 0.98 1.47 1.31 0.71
rotatn time, s 17.1 45.2 21.6 29.9 15.0 40.3 15.0 24.6 26.0 8.5
speed, pm s-I 63.3 37.6 59.3 44.1 68.0 47.1 65.3 59.8 50.4 83.5
max
min
A
2.50 1.30 2.50 0.90 3.15
1.70 0.90 1.35 0.50 2.00
1.40 2.15
0.80 1.50
0.80 0.40 1.15 0.40 1.15 0.3 1 0.60 0.65
speed, pm s-' 135.0 73.3 138.3 96.7 133.4 58.3 115.0 108.4 1 1 1.7 135.5
Figure 2. Spiral waves in BZ solutions with (a) a low [MA] of 0.01 M (no. 8); (b) a high [Br03-] of 0.57 M (no. 1); ( c ) a high [H'] of 0.91 M (no. 5); (d) [BrO,-] = 0.58 M, [H+] = 0.91 M, [MA] = 0.31 M (no. 10) and in all cases standard concentrations of the other components. Image area 5 X 5 mm2. Laye: thickness 0.3 mm.
of intensity distribution as a function of time, were performed as described earlier.3.4.9The center of rotation of regular spirals was
dctcrmined by an overlay t e ~ h n i q u e .Concentration ~ profiles of spirals were exkracted along a line passing through this center.
The Journal of Physical Chemistry, Vol. 94, No. 24, 1990 8679
Waves in the Ce-Catalyzed BZ Reaction
1
0
2
3
5
4
X["I
Figure 3. Concentration profile passing through the center of the spiral extracted from (a) Figure I and (b) Figure 2c.
I
a
1.41.2-
r
1.0-
.
E
1 5 E
C
SI
0
1
2
3
4
5
X["I
Figure 4. Overlapped concentration profiles, passing through the center of the spiral, derived during the revolution of the spiral in three characteristic cases: (a) from Figure 1; (b) from the standard case (no. 9 in Table 11); and (c) from Figure 2d.
Results
In our measurements one of the reactant concentrations was varied over the range as shown in Table I, while all other concentrations were maintained constant. The most important paramctcrs of spiral waves are collected in Table 11, where they are compared with the respective data of concentric waves measured undcr thc same conditions. All solutions except no. 2 and 6 in Table I1 are amply excitable. I n such solutions open ends of wave fronts curl up to regular (rigidly rotating Archimedian) spirals which have a well-defined wavelength and speed. Depending on the reactant concentrations, even regular spirals show a variety in their concentration profiles. The cross-sectional shape of spiral fronts depends on the ratio of the initial concentrations of the reducing and oxidizing agents in
Figure 5. Irregular spiral in a BZ solution with [ Br03-] = 0.12 M and standard conccntration of the other components (no. 2 in Table I I ) , sandwiched between two glass plates. Image area 4 X 4 mm2. Layer thickness 0.3 mm.
thc BZ solution. At high MA concentrations we have the well-known sharply shaped spiral form (maximum intensity at the leading edge of the front) in a quite broad reduced background (Figure I ) . On the other hand, at low MA concentrations or at high Br03- and/or H+concentrations, broad spirals (maximum intcnsity in the middle of the front) are formed with a narrow reduced line between them (Figure 2). Concentration profiles of the spirals clearly show this change from asymmetric, relaxation shape to an almost symmetric form, as is represented in Figure 3. The center of rotation of regular spirals, as determined by the overlay technique, is a small site with quasi-stationary singular properties, for example, catalyst concentration. On changing the initial reactant concentrations so as to get more symmetric concentration profiles, the concentration of the catalyst in the center of the spiral is shifted to the more oxidized region. The well-known narrow sharply shaped spirals give results similar to those published for the fcrroin-catalyzed case:9 in the center of the spiral, the concentration of the catalyst lies in the more reduced half of the total change in catalyst concentration (Figure 4a). At our standard concentrations, for example, it lies in the middle of the total catalyst change (Figure 4b). In case of very broad, almost symmetric spiral fronts, the catalyst concentration in the spiral core lies in the more oxidized region (Figure 4c). Solutions no. 2 and 6 represent cases of low excitability. I n general, when the initial concentration of Br03- and/or H+ is dccrcascd, the excitability of the solution will be lowered. In such solutions the tip of the spiral waves is meandering during rotation, that is, it moves on loops resembling a cycloid, as was described for fcrroin-catalyzed solution^.^^-'^ The spirals show an irregular non-Archimedian form and do not have a unique wavelength (Figure 5). For such irregular spirals we give the minimum wavelength in Table I I . At extremely low excitabilities, broken ends of wave fronts do not form spirals; they rather keep their elongated shape and contract in time. Such a process is shown in the series of Figure 6. The spiral waves described above were measured in freshly prepared solutions during the first 5-10 min of the reaction. During the whole lifetime of spirals, their properties show a (12) Agladze, K. I.; Panfilov, A. V.; Rudenko, A. N. Physica 1988, 290. 409. ( 1 3) Plesser, Th.; Muller, S. C.; Hess, B. To be submitted for publication.
Nagy-Ungvarai et al.
8680 The Journal of Physical Chemistry, Vol. 94, No. 24, 1990
Figure 6. Contraction of a broken wave front in a BZ solution with [BrO, ] = 0.1 2 M, [ H + ] = 0.16 M, and standard concentration of the other components, 30 min after mixing the reagents. Image area 6.5 X 5 mm2. Layer thickness 0.3 mm.
Figure 7. Spiral waves in solution no. 8, as shown a t the beginning of the reaction in Figure 2, (a) I5 min, (b) 30 min, and (c) 40 min later.
considcrablc change in time. Their wavelength and rotation time are increasing (Figure 7), and the velocity of spirals is decreasing in time. The amplitude, which is the difference between maximum and minimum concentration of the catalyst, is decreasing as well. Howcvcr, its position on the total catalyst concentration scale is shiftcd during the course of the reaction either to more oxidized or to morc rcduccd states of the catalyst, depending on the ratio of oxidizing and reducing agents in the initial solution. In Figure 8 wc show thc timc evolution of the intensity distribution of the wholc spiral image, a mcasurc of the overall oxidation statc of the catalyst, for two characteristic cases. At low [ Br03-] [H+]/[MA] [BrMA] ratios the average intensity is incrcasing and the wholc solution ends up in the oxidized state of the catalyst (Figurc 8a). Figure 8b shows the opposite case in a solution with high [BrO,-][H+]/[MA] + [BrMA] ratio. At a laycr thickness 10.4 mm, in solution layers which are 'covcrcd with a glass plate leaving a IO-" air gap between layer surface and cover, dark oxidized patches can be observed on the continuously broadening spiral patterns (Figure 9a). The patches causc a notcworthy cffcct also in the corresponding superposed conccntration profilcs (Figurc 9b) when compared with profiles mcasurcd in thinncr laycrs of the same solution as shown in Figure 4b. In thin Iaycrs which arc closed without an air gap or react
+
undcr a N, atmosphere no such patterns can be observed.
Modeling
In the first two parts of our studies about waves in the Cecatalyzed BZ r e a ~ t i o nwe ~ , ~showed that a simple but realistic mathcmatical model, based on the Oregonator equations,I4 could account in quantitative detail for velocities and profiles of expanding circular waves. In this paper we press the model further by calculating spiral wave solutions to the reaction-diffusion cquations: ax 1 = eV2x + ;f(x.z) dt dZ = tV2z + g ( x , z ) at
(1)
In eq 1, V2 is the Laplacian operator in two spatial dimensions (sI+ s,). x and z are scaled concentrations of bromous acid and Cc(lV), respectively, t is a small parameter ( e = 0.01) reflecting thc faster time scale for changes in [HBr02] compared to changes ~~
(14) Field, R. J.; Noyes, R. M. J . Chem. Phys. 1974, 60, 1877.
The Journal of Physical Chemistry, Vol. 94, No. 24, 1990 8681
Spiral Waves in the Ce-Catalyzed BZ Reaction
i
250
a
I'
99%
a
200 150
loo
1
I -
b
I b 04 0
1
10
20
30 40 T I M E ( min )
50
60
70
Figure 8. Temporal evolution of the spatial average and maximum (99%) and minimum ( 1 % ) intensity distribution (a) in solution no. 7 and (b) in solution no. 8 of Table 11. 0
in [Cc( IV)], and f ( x . 2 ) and g(x,z) are functions describing the kinetics of local chemical reactions. For the modified Oregonator modcl wc arc using f(X,z)
= X( I - X) - hz(x - 9 ) / ( X
2x(2 u = u(x,z) =
(C - Z ) -k [(C - 2)'
+ 9) -
K-5U2
+ K-62) + 8 ~ - , ~ (+2 K+jZ)]"2
(2)
(4)
For the dimensionless parameters appearing in eqs 2-4, the following values are a p p r ~ p r i a t e : ~ . ~ h = 1.5, 4 = 0.0002, K-5 = 0.01, K+ = 300, c = [CeItotal/O.125 M The dimensionless variables in eq 1 are given by3*4 x = [HBr02]/1.2 mM,
z = [Ce(IV)]/125 mM, t = time/21 s, s = space/l.8 mm
To solve eq 1 numerically over a square spatial domain, we introduce a uniform grid of M points and approximate the Laplacian operator by finite differences. (We assume no flux of x or z through the boundaries of the domain.) This procedure yields a set of 2N2 ordinary differential equations (ODEs) that arc difficult to solve numerically because of the small parameter in the equations. We have taken two approaches to solving these ODEs. The more conservative approach is to use an implicit backwards-diffcrcnce formula of variable order. There are several well-tested versions of such algorithms; we used DEBDF (from the DEPAC libraryi5) which is an implementation of LSODE. The finitc-diffcrcnce approximation to the Laplacian operator introduces a sparsc banded matrix that must be inverted frequently by the algorithm. We tried several approaches to handling this matrix, looking for a reasonable trade-off between speed and accuracy. An advantage of these commercial programs is that they include error-controlling features, so that the user can have some confidence in the results. However, in our experience with cq I , DEBDF is quite expensive to run in terms of CPU time. For ( I 5 ) Shampine, L. F.; Watts, H . A. Technical Report SAND 79-2374 (Sandia Laboratories, Albuquerque NM).
1
3
2
4
5
X [ " J
Figure 9. Spiral wave in a 0.6 mm thick standard solution layer, covered with a glass plate leaving a IO-" air gap between layer surface and cover. (a) Image area 9 X 9 mm2. (b) Overlapped concentration profiles from a spiral in the same solution.
a 56 X 56 grid (a small grid by usual standards), DEBDF required ovcr 90 min of CPU time (on a VAXstation 3200) to advancc the solution of eq 1 by 1 dimensionless time unit. To compute two rotations of a spiral wave (3 time units) on a 280 X 280 grid (morc rcalistic) would take over 100 h CPU time! The second, riskicr but faster, approach to numerical solution of thc 2M ODEs is to use a straightforward, explicit, first-order Eulcr approximation." This approach turns out for our problem to bc about 10 times faster than the more sophisticated DERDF routine. In the several cases where we have compared output from thc Eulcr algorithm and the DEBDF algorithm, there arc no significant diffcrcnccs between the calculations with respect to thc frcqucncy of rotation and speed of propagation of spiral wavcs. Thcrcforc, wc uscd thc Euler algorithm for all calculations reported hcrc. Our first calculations were done for the standard concentration valucs (no. 9 in Tablc 11). In this case, c = 0.048 in eq 4. We took a squarc domain of cdgc length 4.4 space units (7.9 mm). For the temporal and spatial step sizes we chose dt = 0.001 and dh = 0.0125 (grid = 353 X 353). Under these conditions the calculation rcquircs 1 1 h CPU time (VAXstation 3200) pcr revolution of thc spiral. To check the accuracy of the calculation, wc rcduccd dt and dh in steps down to dt = 0.00035 and dh = 0.004 17 (6.3 h CPU time per revolution, on IBM 3090 Vector Processor, about 40 times faster than the VAXstation 3200 for this problem). Wave speed and period of the spiral changcd somewhat with dt and dh, but finally settled on values close to thosc computed at dt = 0.001, dh = 0.0125. So we used these largcr step sizcs in all further calculations, In Table I I I we report characteristics of the computcd spiral wavcs for conccntration values corresponding to three cases in Tablc I I . Though thc agreement is not exact, there is certainly qualitative (and some degree of quantitative) consistency bctwccn calculations and observations. It should be noted that we have no free parameters to adjust to get a better fit between the model and experiment.
8682 The Journal of Physical Chemistry, Vol. 94, No. 24, 1990
Nagy-Ungvarai et al.
TABLE 111: Calculated Properties of Spiral Waves and Solitary Waves no.
ICe(lVl1. M
C
a m d . mM
0.012 0.003 0.006
0.096 0.024 0.048
0.94 (0.8) 0.31 (0.25) 0.54 (0.4)
~~
3 4 9
spiral wavel.. m m 2.2 (1.3) 2.5 (1.3) 2.2 (1.3)
solitary period, s 31 (22) 37 (30) 32 (26)
speed, r m / s
ampl, mM
speed, w n / s
72 (59) 64 (44) 68 (50)
1.21 (1.15) 0.34 (0.40) 0.67
128 (138) 96 (97) 116 (112)
"The numbers in parentheses are the observed values, from Table 11.
1.51
0
1
2
4
3
5
X["I Figure 10. Concentration profiles of concentric waves in solution no. I of Table I I .
(e..)
and spiral (-)
Discussion Rotating spiral waves must satisfy not only the dispersion relation, which is the dependence of wave velocity on initiation frcquency, but also the relationship between propagation velocity and curvature.' Therefore, spiral waves have a characteristic speed, much less than that of an isolated circular wave in the same medium, and a characteristic wavelength as can be verified in the data of Table 1. The relationship between the shape of circular wavcs and spirals is evident in Figure 10 where we have drawn thc profile of circular and spiral waves in the same picture. Spirals are curling up close to the preceding wave fronts. Thus, especially in case of broad wave fronts, the concentration of the catalyst (and othcr variables, especially Br-) cannot be completely recovered before the next front arrives, as in the case of concentric waves. The velocity of spirals is therefore smaller. The thickness of the oxidized region in a spiral is generally larger than in an isolated circular wave and also the amplitude of a spiral is generally smaller
than that of an isolated circular wave. Our data agree well with sporadic data measured in BZ systems with particular choices of '-I6 conc~ntrations.~,' Agladze, Panfilov, and RudenkoI2 attributed irregular spiral wave forms in ferroin-catalyzed BZ systems to nonuniform saturation of the solution layers with oxygen. In our experiments (no. 2 and 6 in Table II), though less-pronounced, non-Archimedian spirals occurred even in solution layers with a 10 mm high N, gap above them and moreover in solution layers sandwiched between two parallel glass plates. Therefore, non-Archimedian symmetry can be an intrinsic property of solutions with sufficiently low excitability. This observation is in good agreement with the results of measurements in the ferroin-catalyzed BZ reacti~n.'~ Oxidized patches occurring on spiral waves were thought to be an aging phenomenon caused by convective proce~ses.~In agreement with measurements in silica gels" we think that those patches are 3D phenomena caused by the gradient in saturation with oxygen throughout the solution depth. The chemical propcrties of the upper layers containing more oxygen differ from the lower ones and, as the two sheets of solution interact, this can give risc to transversally moving structures superposed on the usual spiral pictures. This behavior is even more evident in discrete gel systems,'* where this interaction may lead to new types of wave gcomctrics.
Acknowledgment. The technical assistance of I. Beyer and A . Rohde is gratefully acknowledged. The work of J.J.T. is supported by N S F Grant DMS-8810456. Registry
No. MA, 141-82-2; Br03-, 15541-45-4; Ce, 7440-45-1.
(16) Maselko, J.; Showalter, K. Nature 1989, 339, 609. (17) Agladze, K. 1.; Krinskii, V. I.; Panfilov, A. V.; Linde, H.; Kuhnert, L. Physica 1989, 390, 38. (18) Zhabotinskii, A. M.; Muller, S. C.; Hess, B. To be submitted for publication.
