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J. Phys. Chem. 1992,96, 6713-6116
Spatial Bistabiiity of Two-Dimensional Turing Patterns in a Reaction-Diffusion System Q.Ouyang,t Z. Noszticzius~~* and Harry L.Swinney*qt Centerfor Nonlinear Dynamics and Department of Physics, The University of Texas, Austin, Texas 78712, and Department of Chemical Physics, Technical University of Budapest, H-I 521 Budapest, Hungary (Received: February 28, 1992)
A Turing bifurcation from an uniform state to a striped patterned state was observed in experiments conducted in a single-phase spatial open gel reactor with the chlorite-iodide-malonic acid-starch (CIMA) reaction; previous experiments had revealed a bifurcation from a uniform state to hexagons rather than stripes. A modiied reactor is used to demonstrate that the hexagonal and striped patterns are quasi-two-dimensional; this is further confirmed by a direct measurement of the third dimension of patterns with a camera of high resolution in depth of field. For some range of chemical concentrations the hexagonal and striped patterns are bistable; this is the first evidence of spatial bistability between different Turing structures.
Introduction A Turing' (diffusion-induced) instability arises as the result of the interplay of diffusion and chemical reaction. Turing patterns-tationary periodic concentration structures-have been extensively studied by theoretical physical chemists2and theoretical biologi~ts~-~ for more than two decades but were only recently observed in laboratory experiments. Experiments with the chlorite-iodide-malonic acidstarch (CIMA) reaction in Bordeaux- and Austh~~,'~ in a single-phase reaction-diffusion system revealed clear evidence of the Turing patterns and Turing bifurcations (transitions, as a control parameter changes, from a uniform state to a patterned state). Experiments in Bordeaux indicated that the observed spatial concentration patterns can be three-dimensional with, e.g., a body-centered cubic (bcc) structure,'~~ while in our experiments the question of dimensionality remained pen.^,'^ Our camera had insufficient depth resolution to resolve any possible structure in the direction normal to the pattern. In this paper we present results obtained in a modified spatial open reactor that was developed to distinguish two-dimensional from threedimensional spatial patterns. Starting from an uniform state and decreasing the malonic acid in the CIMA reaction, we observed first a Turing bifurcation to a quasi-two-dimensional striped pattern. With further decrease in malonic acid, there was a transition to hexagons. The transition from stripes to hexagons was hystereticthere was a range in malonic acid in which either stripes or hexagons can be stable. Finally, our experimental observations of Turing bifurcations and hysteresis are compared with the results of recent numerical simulations conducted by Dufiet and Boissonade with an activatorsubstrate depleted model. I *
Experimental Section Materipla Analytical grade potassium iodide (Baker), sodium chlorite (Eastman Kodak), malonic acid (Sigma), and Thiodhe (a triiodide indicator, Prolabo)12were used without further purification. Sodium hydroxide was added to the chlorite solution to keep it from decomposing. The stock solution of potassium iodide was refrigerated and prepared frequently because of possible oxidation of iodide with air. The polyacrylamidegels, which have about 95% void space and 80-A average pore size, were prepared using the same recipe and procedure described by Castets et al.5 No chemical and physical changes were found in the gel after continuous experiments for 6 weeks; however, since the indicator faded slightly during this long period, the gel was replaced every 3 weeks. Appnnhra The experiments were conducted in a spatial open reactor like that described previously,9v10except that the reaction medium consists of three gel disks rather than one: a thin (0.2The University of Texas. 'Technical University of Budapest.
Gel discs loaded with
thin (0.2 mm) gel disc loaded with much starch (15g/L)
I -t
25.4 nun
Vycor porous
I
d
reaction medium
commrtment B
(b)
Figure 1. Schematicdiagram of the open gel reactor: (a) reaction media; (b) reaction system.
mm-thick) gel disk loaded with 15.0 g/L of Thiodbne is sandwiched between two identical wedge-shaped gel disks loaded with 0.5 g/L Thiodbne; the wedge-shaped disks increase linearly in thickness from zero at one edge to 2.00 mm at the opposite edge (Figure la). The diameter of the gel disks is 25.4 mm. The sandwich of three gel disks form a disk of uniform thickness, 2.0 mm. This reaction medium is sandwiched between two 0.4"thick porous glass disks (Vycor glass, Coming), which have 25% void space and 1OO-A average pore size. The outer surface of each porous glass disk is in turn in contact with a chemical reservoir, where reactant concentrations are kept constant and uniform by mixiig and a continuous flow of fresh reagents (see Figure lb). The reactor is immersed in a temperature-controlled water bath. Chemical amcentration gradients were imposed in the direction normal to the plane of the gel: chlorite is only in compartment A and malonic acid is only in compartment B (see Figure 1b); the other chemical species are contained in equal amounts in both reservoirs, except for sulfuric acid, which is more concentrated in compartment B than in Compartment A. The distribution of the reactants in each compartment is chosen in such a way that the chemical solutions of neither compartment are separately reactive. The chemicals diffuse through the porous glass disks into the gel where the reaction occurs. The gel, loaded with the starch indicator,changes in color from yellow to blue with increases in concentration of 13-during the reaction; thus the behavior of the patterns can be monitored in transmitted light (580 nm, the wavelength of high absorption of Ic-starch complex) using a video
0022-3654/92/2096-6173$03.00/00 1992 American Chemical Society
Ouyang et al.
6774 The Journal of Physical Chemistry, Vol. 96, No. 16, 1992
10 m m
0.0
2
0.0 1 .o
0.0
Figure 2. (a) Two-dimensional and (b) three-dimensional patterns. The wavelengths of the patterns are respectively (a) 0.33 and (b) 0.20 mm. Image (a) was obtained using a 0.2-mm-thick polyacrylamide gel as the thin disk (see text); image (b) was obtained using a 0.2-mm-thick 10% poly(viny1 alcohol) (PVA) gel as the thin disk. The concentrations in compartments A and B were [I-]oA (=[I-IoB), [C102-]oA, [CH,(COOH)2]oB(in mM) in (a), 3.5, 16.0, 13.0; in (b), 3.0, 20.0, 9.0. Other control parameters were held fixed: [Na2S0410A= [Na2S0410B= 4.5 mM, [H2S0410A= 0.5 mM, [H2S0410B= 4.0 mM, T = 5.6 "C.
2.0
Z("> Figure 3. Measurement of amplitude as a function of the depth 2 for (a) a 0.01-mm wire (which serves as a 6 function in 2 direction) and (b) a hexagonal pattern. The deconvolution of (b) from (a) is shown (c). The amplitude is evaluated by calculating the root-mean-squaredeviation of the measured optical intensity and averaging this deviation over a few wavelengths. The conditions for (b) are the same as for Figure 2a.
camera. Digitized black and white images were processed and analyzed on a Silicon Graphics work station. Rt?!3ults Dimerrsionality. For certain conditions two-dimensional Turing patterns were observed, while for other conditions the patterns were three-dimensional. The two-dimensional patterns contain only a single thin patterned layer, which is parallel to the plane of gel, while three-dimensional structures extend over a depth that is larger than the wavelength. In our reactor the height of the indicator varies across the width of the gel, thus probing the structure as a function of distance normal to the plane of the gel. Patterns can be clearly seen only where a patterned layer and the thin diagonal gel disk intersect. Outside of the intersection regions, patterns are faint since the indicator concentration is much less. As a result, a two-dimensional pattern has only one intersection region with the thin gel disk; this yields a single narrow patterned band when the reactor is viewed normal to the plane of the disk. In contrast, a three-dimensional structure (e.g., bcc) would have multiple intersection regions with the thin gel disk From the width of the patterned band ( W ) and the ratio between the thickness of thin gel disk (h = 0.2 mm) and that of reaction medium (H = 2.0 mm), one can calculate the thickness d of the patterned layer: d = Wh/H, which can be compared with the wavelength of the pattern (A) obtained from a spatial Fourier transform in order to distinguish a two-dimensional pattern (d S A) from a three-dimensional pattern (d > A). Figure 2a is an example of singlelayer hexagonal patterns. The width Wof the intense pattern is 3.6 mm. Thus the thickness of the patterned layer is
d = W h / H = 3.6 mm
X
0.2 mm/2.0 mm = 0.36 mm
which is approximately the same as the wavelength of the pattern (0.33 mm); hence this is a two-dimensional pattern.
_
I
I
0
-
a 1 Y
a
b-b,
F v 4. Transition from a uniform state ([CH2(COOH)2]oB> 27 mM)
to a striped pattern ( [CH2(COOH)2]oB< 27 mM) and hysteresis between states of stripes ( 0 )and hexagons (A): (a) Present laboratory experiments, (b) numerical simulation by Dufiet and Boissonade (Figure 3 of ref 11) for the Schnackenberg model23in a two-dimensional reaction-diffusion system. In the experiments the amplitude was obtained from a two-dimensional fast Fourier transform of a band near the spatial frequency 3.0 cm-I. All concentrations other than [CH2(COOH)2]oB were held fixed at the values given in Figure 2a.
For the control parameter range explored in this experiment, we did not observe regular three-dimensional patterns. However, when we used poly(viny1 alcohol) gel13instead of polyacrylamide gel as the thin gel disk, patterns such as that in Figure 2b appeared for certain chemical conditions. There are at least four different patterned layers, each with a thickness comparable to the
Two-Dimensional Turing Patterns
The Journal of Physical Chemistry, VoI. 96. No. 16. 1992 6775
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Figme 5. Observed bistability between hexagonal and striped patterns as function of [CH2(COOH)2]oR.The control variable ([CH2(C0OH),loBin m M ) was (a) 13.0, (b) 21.0, (c) 25.0, (d) 21.0, (e) 14.0, and (0 13.0. Other conditions were held fixed at the values given for Figure 2a.
wavelength of the patterns (0.2 mm). Each layer has its own morphology: honeycomb, stripes, mixture of dots and stripes, and dots. Although these patterns show some three-dimensional features, they are different from a regular three-dimensional structure such as body-centered cubic. The different patterns possibly arise because imposed concentration gradients normal to the plane of the gel lead to a variation of the chemical concentrations with height. In this experiment, we also measured directly the pattern intensity in the normal (2)direction for an ordinary reaction medium (one piece of polyacrylamide gel disk loaded with 15 g/L starch). Figure 3a shows that the camera resolution in the 2 direction is 0.2 mm, achieved using a 20-mm macro lens, aperture F = 2.0, with a 112-mm macro bellows. Figure 3b shows the amplitude as a function of depth 2 for a hexagonal pattern with chemical conditions the same as those in Figure 2a. After de-
convolution of the signal (Figure 3b) from the camera’s response function (Figure 3a), we get a single peak as a function of depth 2 (Figure 3c), indicating that the measured pattern has no three-dimensional structure. Moreover, the depth of the pattern given by the half-width at half-height of the measurement in Figure 3c is in agreement with the pattern thickness determined with the diagonal gel indicator. Spatial Bistability. The patterns ohserved in the following series of experiments are all two-dimensional stationary spatial patterns. Figure 4a presents the main results of this experiment. Starting from an uniform state, we observed a transition to a striped pattern as the malonic acid concentration in compartment B was decreased. Previous experiments with temperature as the control parameter had yielded bifurcations from a uniform state only to hexagon^.^.'^ No hysteresis was observed around the transition = 27 mM); thus the Turing bifurcation point ( [CH2(COOH)2]oB
6776 The Journal of Physical Chemistry, Vol. 96, No. 16, 1992
is supercritical within our resolution. The striped pattern increases in amplitude as [CH2(COOH)z]oe decreases, while its wavelength remains the same. If [CH2(COOH)2]oBis decreased below 17 mM, the striped state becomes unstable and the system jumps to a state of hexagons, which has the same wavelength as the stripes. From this point if [CHz(C00H)2]oBis increased, the hexagonal pattern persists until the concentration reaches 24 mM, where the system goes back to the striped state. Thus there is a large domain of spatial bistability as a function of the feed concentration of malonic acid. Figure 5 shows the exchange of stabilities of two-dimensional hexagonal and striped patterns as function of the concentration of malonic acid in compartment B. Parts b and d of Figure 5 illustrate that both hexagons and stripes can exist for the same conditions ([CH2(COOH)2]oB = 21 mM), depending on the initial conditions. Figure 5e is a transient state observed during the transition from a striped state to a hexagonal state; this picture was taken about 70 h after the change of control parameter. Near the transition point, the motion is very slow: the typical invading speed of a hexagonal region into a striped region is about 2 lattice sites/day, but with a small further decrease in the feed concentration of malonic acid, the striped pattern disappeared within a few hours.
Discussion The use of our reactor design to probe three-dimensional structure assumes that the starch in the thin diagonal layer is simply an indicator. However, Lengyel and Epstein14J5point out that starch can play a key role in pattern formation in the CIMA reaction. Since starch is immobile in a gel environment, it decreases the effective diffusion coefficient of iodide in the system through the formation of a IT-starch complex. In their modelle16 this decrease in the &ion rate of iodide is a necessary condition for Turing bifurcation. This prediction has been supported by experiments of Agladze et a1.* in a gel-free system, where a transition from standing (time-independent) to wave (time-dependent) patterns was observed as the starch concentration decreased. However, Agladze et al. found that varying the starch concentration had no effect on Turing patterns in a polyacrylamide gel medium. Moreover, Lee et al.’ discovered that, even in the absence of starch, a Turing pattern could be observed in a polyacrylamide gel. The reason is that the polyacrylamide gel is not a totally inert medium for the CIMA reaction. Binding sites in the gel can react with iodide or triiodide and decrease the diffusion rate of iodide. If starch concentration is low, the main cause of the decrease in diffusion rate of iodide is the binding sites of polyacrylamide gel; in this limit the pattern is unaffected by variations in starch concentration. Our experiments showed that a state of hexagons did not change when starch (Thiodhe) concentration was changed from 0.5 to 20 g/L. Hence we conclude that our method of identifying the dimensionality of spatial patterns is valid. Pattern formation in reaction-diffusion systems has been extensively studied both in the analysis of amplitude equations (normal f ~ r m ~ )and ~ *in- numerical ~~ The general scenario near the onset of a Turing bifurcation in a two-dimensional system was summarized by Dufiet and Boissonade:II “Hexagons should appear first via a subcritical bifurcation, while stripes arise supercritically but are unstable, b m i n g stable only at larger values of the control parameter. Hexagons become unstable at still higher values; there is a region of bistability where both patterns are stable.” If the control parameter is driven even further from the onset of Turing bifurcation point, according to their simulation,another type of hexagonal pattern (H2, a phase shift of r of the first one, HI) becomes stable; H2 patterns can
Ouyang et al. coexist with the striped patterns. In the present experiments we did not observe the transition from a uniform state to the a state of HI-hexagons,or the bistability between the HI-hexagonsand stripes. However, we did observe a large domain of bistability between stripes and H2-hexagons, and the bifurcation diagrams obtained in the simulation (Figure 4b) and experiment (Figure 4a) are in qualitative agreement. In practice H1-hexagonsand the hysteresis between HI-hexagons and stripes may occur in only a very small part of the control parameter space. In fact, in the simulations the regions of Hl-hexagons and hysteresis between H,-hexagons and stripes are contained within 0.1%of the control parameter, too small to observe with our experimental resolution. In summary, our experiment demonstrates that the stationary spatial patterns that were observed in earlier experiments are basically two-dimensional. A hysteresis between striped and hexagonal patterns was observed as the concentration of malonic acid was increased or decreased around a certain value. This bistability agrees qualitatively with that found in recent numerical simulations by M i e t and Boissonade.” Further experiments will use the gel reactor with the indicator varying in height to study the threedimensional patterns, which exist in other regions of the control parameters.
Acknowledgment. We thank V. Dufiet and J. Boissonade for providing us their numerical results and for their comments; K. J. Lee, W. D. McConnick, and R. D. Vigil for daily enlightening discussions; and A. Arneodo, P. Borckmans, P. De Kepper, G. Dewel, A. De Wit, E. Dulos, and D. Walgraef for helpful discussions. This work is supported by the U.S.Department of Energy Office of Basic Energy Sciences. Regbtry No. Malonic acid, 141-82-2.
Referemces .ad Notes (1) Turing, A. Philos. Trans. R . Soc. London 1952, B237, 31. (2) Nicolis, G.; Prigogine, I. Self-Organization in Nonequilibrium Sysrems; Wiley: New York, 1977. (3) Meinhardt, H. Models of Biological Pattern Formation; Academic Pres: New York, 1982. (4) Murray, J. D. Mathematical Biology; Springer, Berlin, 1989. (5) Castets, V.; Dulos, E.; Boissonade, J.; De Kepper, P. Phys. Rev. Lett. 1990,61, 2953. (6) Boissonade, J.; Castets, V.; Dulos, E.; De Kepper, P. Inr. Ser. Numerical Marh. 1991, 97, 61. (7) De Kepper, P.; Castets, V.; Dulos, E.; Boissonade, J. Physica D 1991, 49, 161. (8) Agladzc, K.; Dulos, E.; De Kepper, P.; J. Phys. Chem. 1992,96,2400. (9) Ouyang, Q.; Swinney, H. L. Nature (London) 1991, 352, 610. (10) Ouyang, Q.; Swinney, H. L. Chaos 1991, I , 41 1 . (11) Dufiet, V.; Boiesonade, J. J . Chem. Phys. 1992, 96, 664. (12) A recent study shows that Thiodhe is composed of 93-94% (weight) of urea and 6-7% (weight) of soluble starch. See: Noszticzius, Z.; Ouyang, Q.;McCormick, W. D.; Swinney H. L. J. Phys. Chem. 1992, 96, 6302. (13) The polyvinyl alcohol gels are prepared by mixing 5 mL of 10% (weight) poly(viny1 alcohol) solution, 0.2 mL of 37% HC1, and 2 drop of 25% glutaraldehyde. A thin uniformly flat layer of the solution is left to polymerize at room temperature for 20 min. The resulting sheet of gel is then throughly washed and cut into the diak shape. (14) Lengyel, I.; Eptein, I. R. Science 1990, 251, 650. (15) Lengyel, I.; Epstein, I. R. Proc. Narl. Acad. Sci. U.S.A. 1992, 89. 3977. (16) Lengyel, I.; Rabai, G.; Epstein, I. R. J . Am. Chem. SOC.1990, 112, 4606. (17) Lee, K. J.; McCormick, W. D.; Noszticzius, Z.; Swinney, H. L. J . G e m . Phys. 1992, 96, 4048. (18) Pismcn, L. M.J . Chem. Phys. 1980, 72, 1900. (19) Walgraef, D.; Dewel, G.; Borckmans, P. Phys. Rev. A 1980,21,397. (20) Walgraef, D.; Dewel, G.; Borckmans, P. Adu. Chem. Phys. 1982.49, 311. (21) Lacalli, T.C. Philos. Trom. R. Soc. London 1981, 294, 547. (22) Lyons, M. J.; Harrison, L. G . Chem. Phys. Leu. 1991, 183, 158. (23) De Wit, A.; Dewel, G.; Borckmans, P.; Walgraef, D. Physica D,in press. (24) Schnackenberg, J. J . Theor. Eiol. 1979, 81, 389.
