6773
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
i
1
J
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.