Spatial bistability in a pH autocatalytic system: From long to short range activation

Article abstract of DOI:10.1021/jp0522922

The acid-auto-activated chlorite-tetrathionate reaction is studied in a one-side-fed spatial reactor. It was previously shown that in these conditions the unstirred reaction-diffusion system can generate oscillatory and excitable states even though under well-stirred nonequilibrium conditions only steady-state bistability is observed. Numerical simulations suggest that these temporal reaction-diffusion instabilities result from long-range activation by rapidly diffusing protons. We study here experimentally and numerically the effect of introducing into this reaction-diffusion system macromolecular carboxylate species that reduce the effective diffusivity of protons. Consistent with the original assumption, the introduction of such slow mobility proton-binding species quenches both oscillatory and excitability dynamics. Within the bistability domain the direction of the propagation of an interface between the two steady states depends on control parameter value. We elaborate on the fact that beyond a low critical concentration of macromolecular carboxylate species, the stability limit of the thermodynamic branch of spatial steady state does not depend on this concentration. Despite the relative simplicity of the kinetic model used in the numerical simulations, the results are in quasi-quantitative agreement with the experimental observations.

Full text of DOI:10.1021/jp0522922

J. Phys. Chem. A 2005, 109, 7843-7849
7843
ARTICLES
Spatial Bistability in a pH Autocatalytic System: From Long to Short Range Activation
Istva´n Szalai,* F. Gauffre, V. Labrot, J. Boissonade, and P. De Kepper
Centre de Recherche Paul Pascal, CNRS Bordeaux, AVennue Schweitzer, F-33600 Pessac, France
ReceiVed: May 3, 2005; In Final Form: July 6, 2005
The acid-auto-activated chlorite-tetrathionate reaction is studied in a one-side-fed spatial reactor. It was
previously shown that in these conditions the unstirred reaction-diffusion system can generate oscillatory
and excitable states even though under well-stirred nonequilibrium conditions only steady-state bistability is
observed. Numerical simulations suggest that these temporal reaction-diffusion instabilities result from long-
range activation by rapidly diffusing protons. We study here experimentally and numerically the effect of
introducing into this reaction-diffusion system macromolecular carboxylate species that reduce the effective
diffusivity of protons. Consistent with the original assumption, the introduction of such slow mobility proton-
binding species quenches both oscillatory and excitability dynamics. Within the bistability domain the direction
of the propagation of an interface between the two steady states depends on control parameter value. We
elaborate on the fact that beyond a low critical concentration of macromolecular carboxylate species, the
stability limit of the “thermodynamic” branch of spatial steady state does not depend on this concentration.
Despite the relative simplicity of the kinetic model used in the numerical simulations, the results are in quasi-
quantitative agreement with the experimental observations.
Introduction
diffusion of this species decreases, while the others are left
unchanged, in particular the inhibitors, which then may have a
Great interest is paid to pattern formation in chemical reacting
and diffusing systems. Both their theoretical and experimental
studies are a major field of research in nonlinear science.1-5
Self-organization phenomena in reaction-diffusion systems
encompass different types of oscillatory, traveling, and stationary
patterns. Reactions leading to such patterns include competing
chemical activation and inhibition kinetic pathways. Many
families of such reactions are now known.1,3,6 The temporal and
spatial instabilities that lead to the development of patterns
crucially depend on the time and space scales over which
activation and inhibition processes evolve. Oscillations and
travelling wave patterns are associated with time scale separation
and are typical of fast activation and slow inhibition systems,
while stationary patterns are usually associated with long-range
inhibition and short-range activation.4,5,7 In the simple standard
cases, the observation of stationary concentration patterns
requires that the activation species, or at least a species
controlling the activation kinetic pathway, should have a slower
effective diffusivity than the antagonist inhibitory species.
Hence, it is of fundamental interest to be able to control the
relative space and time scales over which these competing
chemical processes operate.
higher diffusivity than the activator. In addition, if the complex
is nonreactive, the apparent reactivity of the activator is reduced
and oscillatory states can be quenched.8,9 (ii) Second is the use
of dispersed multiphase systems where some species stay
trapped in droplets of the minority phase, while others diffuse
in the continuum phase. Presently, the only example of this type
is the so-called BZ-AOT system.10
At this time, most documented stationary symmetry breaking
chemical patterns rely on the presence of binding sites of
reduced mobility. It is also the case of interest to us in this
work. In this category, only two families of reactions have led
to stationary reaction-diffusion patterns in open spatial reactors.
One consists of chlorite-iodide driven systems11 where the
activator path is controlled by iodide ions. Using starch or poly-
(vinyl alcohol) to immobilize these ions in the form of
polyiodide complexes, these systems exhibit Turing patterns12-14
and standing pulses or transient domain patterns15 when the
system displays multiple steady states. The other is the ferro-
cyanide-iodate-sulfite (FIS) reaction which produces laby-
rinthine patterns and self-replicating and oscillating spots.16 The
reaction is primarily auto-activated by protons, but iodide ions
could also have a secondary auto-activation role.17 Beside the
above-mentioned reactions, only two other reactions have been
operated in open spatial gel-reactors: the popular Belousov-
Zhabotinsky (BZ)1,2 reaction under both constant constraints
(traveling waves, spiral core dynamics, spiral turbulence)18-20
or periodic forcing (stationary and traveling resonant patterns)21
and the chlorite-tetrathionate (CT) reaction.22,23
Two different approaches have been developed to selectively
control the effective diffusivity of species: (i) First is the
addition of uniformly distributed immobile (or reduced mobility)
functional sites able to reversibly bind a targeted species. If the
targeted species is the activator of the reaction, the effective
* Corresponding author. Permanent address: Department of Inorganic
and Analytical Chemistry, L. Eo¨tvo¨s University, P.O. Box 32, H-1518
Budapest 112, Hungary.
Among the different open spatial reactors developed in recent
years, the one-side-fed reactor (OSFR) is the most commonly
10.1021/jp0522922 CCC: $30.25 © 2005 American Chemical Society
Published on Web 08/17/2005

7844 J. Phys. Chem. A, Vol. 109, No. 35, 2005
Szalai et al.
This model is profitably used in the theoretical section of
this work. When operated in a CSTR, the system exhibits an
extended domain of bistability between two steady states
separated by large pH differences. In a OSFR, the reaction
exhibits a domain of spatial bistability, supplemented by
domains of sustained spatiotemporal oscillations and of excit-
ability. Interestingly, these dynamical phenomena do not seem
to originate from the sole kinetic mechanism, as observed in
other reactions, but are thought to be linked to the fact that the
proton, the driving activatory species, diffuses faster than the
reagents.22,23
Here, we focus on the effect of introducing long polymer
chains bearing carboxylate functions on the dynamics of the
CT reaction operated in an OSFR. The reversible protonation
of carboxylic groups with very reduced mobility can lead to a
decrease in the effective diffusivity of protons. Under batch
conditions, it is shown experimentally and by simulations that
even a small amount of binding agent significantly decreases
the diffusitivity of the protons.27,28 The CT reaction offers a
privileged experimental situation where it should be possible
to tune the relative activatory/inhibitory space scales from long-
range activation and short-range inhibition to short-range
activation and long-range inhibition. The addition of a macro-
molecular carboxylate enables a test of the previously sug-
gested23 mechanism for the observed oscillations. The experi-
mental results reported in the second section are supported by
numerical calculations in the third section. The implications of
our observations on colliding fronts are discussed in the last
section.
Figure 1. Sketch of the gel core of an annular one-side-fed reactor.
used.14,16,18 It typically consists of a thin disk of gel fed with
fresh reactants by diffusion from one face kept in contact with
the contents of a continuous stirred tank reactor (CSTR). The
other face of the gel is pressed against an impermeable wall.
The gel medium cuts down the hydrodynamic turbulence of
the CSTR and preserves diffusive molecular transport. The
CSTR ensures constant and uniform feed conditions along the
contact boundary with the piece of gel. Beside disk-shaped
OSFRs, other geometries have been developed for specific
purposes, like the flat annular shape that makes it possible to
visualize the chemical concentrations profiles across the depth
of the gel. In the present work, we mainly use this geometry of
reactor, schematically represented in Figure 1.
It has been shown that reaction systems exhibiting bistability
in a CSTR can lead to a phenomenon called “spatial bistabil-
ity”24 when they are operated in an OSFR. This phenomenon
corresponds to the coexistence of two different concentration
profiles across the depth of the gel reactor, for the same fixed
composition of the CSTR contents. The phenomenon has been
clearly analyzed both from the theoretical and experimental
viewpoint for the chlorine dioxide-iodide24,25 and for the
chlorite-tetrathionate22,23 systems, and is implicit in the FIS
reaction experiments.16
Experimental Conditions
The core of our spatial reactor consists of a flat annular piece
of gel tightly inserted into a groove in a polished transparent
Plexiglas cylinder immersed in the CSTR contents. The gel
annulus, made of a 2% agarose (Fluka 05070) network, has an
outer radius r ) 2.5 cm, a width (the difference between the
external and internal radius) w ) 1.0 mm, and a height h )
0.25 mm. Only the outer rim of the annulus is in contact with
the CSTR contents. This design makes it possible to observe
the color changes across the 1.00 mm width of the flat annulus
(Figure 1). The volume of the CSTR is V ) 25 cm3. The
residence time τ ) 600 s and the CSTR temperature T ) 25 °C
were kept constant. The feed is provided by three separate
streams of chemicals, injected by precision pumps (Pharmacia
P 500), that enter the CSTR by a single inlet port. Two streams
respectively contain fixed concentrations of sodium chlorite
(Prolabo, 96% purity) and potassium tetrathionate (Fluka)
stabilized in 3 × 10-4 M NaOH solutions. Variable amounts
of perchloric acid and sodium hydroxide are added through the
third stream to control the pH of the input solutions. This is
done by pumping different volume ratios of acid and base,
keeping the overall flow rate constant. For experimental and
graphic convenience, this relative distribution is characterized
by a control parameter R on which the acid and the base flow
concentrations depend linearly: respectively as [HClO4]0 ) R
× [HClO4]res, [NaOH]0 ) (1 - R) × [NaOH]res, where
[HClO4]res and [NaOH]res are the acid and base feed concentra-
tions from the reservoirs which were respectively fixed to 0.33
× 10-2 and 1.67 × 10-2 mol/dm3. The pH of the feed mixture
decreases as R increases from 0 to 1. To reduce the effective
diffusivity of protons, controlled amounts of sodium poly-
(acrylate) (PA) (Polyscience, 20000D) are also added through
this third stream. The solutions contain bromophenol blue, a
pH color indicator that changes from purple-blue (basic) to
The present work focuses on the chlorite-tetrathionate (CT)
reaction. Though the detailed kinetic mechanism of this reaction
is complex, the overall balance equation of the reaction is
appropriately represented by
7ClO2- + 2S4O62- + 6H2O f 7Cl- + 8SO42- + 12H+
(1)
and in a narrow range of stoichiometric conditions, fulfilled in
this work, the driving kinetics can be conveniently described
by the following empirical rate law:26
1 d
7 dt
-
-
Vk ) -
[ClO2 ] ) k[ClO2 ][S4O62-][H+]2
(2)
Since in our experiments, the pH of the solution can change
from basic to acid, it was shown that to express correctly the
overall kinetic process it is also necessary to take into account
the fast dissociation equilibria:22,23
H+ + OH- h H2O
H+ + SO42- h HSO4
(3)
(4)
-
Their rate laws are respectively given by
Ve ) ke+ - ke- [H+][OH-]
(5)
(6)
-
2-
Va ) k+a [HSO4 ] - k-a [H+][SO4
]

Spatial Bistability in a pH Autocatalytic System
J. Phys. Chem. A, Vol. 109, No. 35, 2005 7845
Figure 2. Illustration of an interface between the stable B state (left)
and the stable M state (right) in an annular gel OSFR (between white
dash lines). For experimental conditions, see text.
yellow-orange (acid) around pH ) 3.8. In the following, the
bracketed terms [X]0 are the concentration that the species X
would have in a collective feed-stream, prior to any reaction.
The fixed values of [NaClO2]0 and [K2S4O6]0 are respectively
1.9 × 10-2 and 0.5 × 10-2 mol/dm3. In experiments with PA,
the gel reactors are set in contact with solutions of appropriate
poly(acrylate) concentrations for a day before starting the
experiment to let the macromolecule diffuse into the agarose
matrix. The chemical state of the CSTR is monitored by the
redox potential of a bright platinum electrode. The color profiles
of stationary states and wave patterns in the gel are monitored
by a CCD camera and the dynamics are recorded on a time-
lapse VCR. A frame grabber digitizes the images for further
processing.
Figure 3. Experimental phase diagram in the ([OH-]0, [-COO-]0)
plane (-COO- functions from poly(acrylate)), observed in the annular
OSFR. The symbols correspond the experimental points and are
attributed to states of the gel: (9) monostable (nonexcitable B state);
([) monostable (excitable B state); (2) bistable (stationary B/M state);
(b) bistable (stationary B state/oscillatory M state); (/) CSTR contents
switches to the thermodynamic state.
Experimental Results
Under our experimental conditions, the CT reaction operated
in a CSTR exhibits bistability between a branch B of basic states
(pH ∼ 10), and a branch A of acid states (pH ∼ 2).22,23 The
CSTR contents are maintained on the branch of B-states.
Different states can be observed in the gel depending on R and
on the initial pH conditions of the gel, as shown in previous
publications.22,23 If the gel is initially acid, and if R is slightly
lower than the value corresponding to the limit of stability of
the CSTR B branch, the inner part of the gel remains acid while
the part in contact with the CSTR contents becomes basic. A
sharp front forms parallel to the gel/CSTR contact surface
between the basic boundary layer and the acid core. This state
of the gel is referred to as the mixed state or state M (Figure
2). On decreasing R, state M remains stable over some range
of this parameter. However, the radial extension of the basic
peripheral zone eventually starts to grow. At a critical value,
the acid/base front becomes unstable and starts to oscillate. In
this oscillatory regime, the front between the basic and the acid
part moves back and forth in the radial direction. The parameter
domain over which these surprising oscillations are observed
is usually very small (a few percent of the parameter domain
over which the stable M state is observed). On decreasing R
beyond this oscillatory region, the gel turns completely basic.
It was shown22,23 that in this case the composition of the gel
does not significantly differ from that of the CSTR and so, by
extension, we call this the B state of the gel (Figure 2). A further
decrease of R brings no qualitative change in the state of the
gel. Now, on increasing R the B state of the gel remains stable
until the branch of CSTR basic states loses stability. Thus, there
is a region of parameter space over which states B and M coexist
in the gel. This defines the domain of spatial bistability. In this
domain, it is possible to prepare29 different parts of the gel in
either state. It is then possible to test their relative stability and
to study the geometric properties of the M-B interface. Figure
2 is an illustration of such an interface between state B (the
quasi-uniform dark-gray left portion of the gel) and state M
(the right portion of the gel with a sharp switch from dark to
pale gray). Note that, consistent with a no-flux boundary
condition, the connection of the acid/base switch is orthogonal
to the impermeable wall at the bottom of the groove. Thus, the
acid region which makes the difference between state M and
state B naturally develops a curvature at the interface.
In the range of R corresponding to the spatial-bistability
domain, one would expect that one of the stationary states would
expand at the expense of the other, the direction of expansion
depending on the value of R. We have systematically probed
this relative stability of states, as a function of R and for different
concentrations of poly(carboxylic acid) in the feed solution. In
the absence of carboxylated polymer chains, the M state always
takes over the B state, even at the very limit of the domain of
stability of the M state at low R. In fact, even when the B state
is the only asymptotically stable state, an acid perturbation of
this state can lead to a M-B-state-like interface that propagates
undamped into the B state. However, behind this interface the
acid region survives only a short period of time. Ultimately,
the B state totally recovers. A traveling pulse forms. This
property defines an excitable B state. Excitability of the B state
extends some distance beyond the limit of the spatial bistability
domain.
We have studied the stability domains of the B and M states
in the (R, [PA]0) plane and systematically tested the response
of the B state to local acid perturbations. The results are gathered
in a phase diagram (Figure 3). Excitability of the state B rapidly
decreases as the polycarboxylate ion concentration increases in
the feed. For concentrations of carboxylic functions above 0.02
mol/dm3 no traveling acid pulse is observed in the monostability
domain of state B. In the present experimental conditions and
within our experimental accuracy, the oscillatory M state is not
detected when 0.01 mol/dm3 of carboxylate functions are
introduced in the feed. Remarkably, the stability limit of the
stationary state M, at low R, does not depend on the concentra-
tion of proton-complexing sites (i.e., carboxylate functions). This
limit ranges between 0.6 and 0.62. However, the high-R stability
limit shifts out of our experimental range with the rapid increase
of stability of state B with increased poly(acrylate) in the feed.

7846 J. Phys. Chem. A, Vol. 109, No. 35, 2005
Szalai et al.
Figure 4. Propagation velocity of the M-B states interface, in the
annular OSFR, as a function of R: (0) in the absence of carboxylic
groups (left “y” scale); (b) in the presence of [-COO-]0 ) 0.1 mol/
dm3 (from poly(acrylate))(right “y” scale).
Figure 5. Sketch of the gel core of a cylindrical OSFR.
these slow interfaces are very fuzzy and difficult to follow,
especially when the optical path across the piece of gel is only
a few tenths of millimeters.
Figure 4 illustrates the qualitative and quantitative changes
induced by the introduction of PA on the relative stability of
states. In the absence of poly(acrylate), the M state always
propagates into the B state, when R is within the spatial
bistability range. We count as positive this direction of interface
propagation. The interface velocity decreases with decreasing
R but never changes sign. In the domain where the B state is
excitable, the velocity dependence of an acid pulse on R is in
perfect continuation (no curve break) with the values obtained
in the spatial bistability domain, except that the gel contents
now ultimately returns to the B state. This indicates that, in
first approximation, the front velocity is affected only by the
concentrations ahead of the front and not by the recovery
process, except maybe very close to the pulse propagation limit
R = 0.37. Below this limit, any acid perturbation decays.
In the presence of poly(acrylate) the velocity of propagation
dramatically slows down. At 0.1 mol/dm3 of carboxylate groups
the velocity is 100 times slower than in the absence of this
proton-binding agent (Figure 4). More interestingly, within the
spatial bistability range, the velocity of the interface changes
sign as a function of R.
The velocity of the M-B interface drops to values close to
zero at the limit of stability of the M state when the concentra-
tion of acrylate functions reaches approximatively 0.02 mol/
dm3. The introduction of a reversible binding agent for protons
quenches the excitability properties of the B state. The standard
qualitative behavior of spatial bistable systems is recovered with
the addition of a high enough quantity of PA: in the bistability
domain the direction of the propagation of an interface between
the two steady states changes for a critical parameter value. This
result supports the assumption that the oscillatory and excit-
ability properties of the CT reaction operated in an OSFR are
linked to long-range activation by migrating protons and are
not the direct result of the reaction kinetic mechanism.
Previous experimental observations25 and theoretical calcula-
tions24 have shown that, in standard spatial bistable conditions,
the critical control parameter value at which the relative stability
of spatial steady states changes can sensitively depend on the
width of the OSFR. In connection with the quest for chemo-
mechanical instabilities,30 the study of the relative stability of
states as a function of the width w of the OSFR is of
fundamental interest. However, in our annular OSFR with a
height h ) 0.25 mm, negative propagation velocities of the
M-B state interface are difficult to detect. The critical switching
value as a function of R remains very close to that of the limit
of stability of state M, and the interface velocities remain very
small. Furthermore, because bromophenol blue is slowly
bleached by chlorine-dioxide produced in the acid/base front,31
To solve the above interface-detection problem and to test
the sensitivity of the relative stability of states B and M on the
width of an OSFR, additional experiments were performed in
a cylindrical-gel OSFR (Figure 5). Cylindrical, conical,32,33 and
spherical34 pieces of gel uniformly fed at their surface can also
be considered as one-side-fed-reactors with the difference that
the curvature of the feed surface must be taken into account.
Long thin cylindrical-gel OSFRs are easier to produce and
handle than flat annular gels. In addition, in the cylindrical
geometry, the absorbance of the dye is integrated over the
diameter of the cylinder which typically ranges from 1.0 to 2.0
mm, to compare to the 0.25 mm in the flat annular OSFR. Thus,
in the axial direction, the color contrast between the two spatial
states is much improved. However, the advantages of the
periodic boundary conditions of annular gels are lost.
For the cylindrical-gel OSFR experiments, the 2% agarose
gel is initially cast in silicon tubes of different diameters. After
being removed from their mold, each cylinder is then soaked
for at least 24 h in a 0.7% sodium poly(acrylate) aqueous
solution (corresponding to a concentration of 7.65 × 10-2 mol/
dm3 carboxylate groups) before being glued onto a stopper and
introduced at the top of a tall cylindrical CSTR designed for
fast recirculation of the contents from bottom to top (Figure 5).
The gel cylinders are totally immersed in the solution. The new
CSTR, made of polished transparent Plexiglas, has a volume
of 46 cm3, a residence time of 600 s, and is thermoregulated at
25 or 35 °Csas appropriately indicated in the captions. The
feed concentrations of the reagents are the same as above and
a fixed concentration of PA corresponding to 7.65 × 10-2 mol/
dm3 of carboxylate groups is injected in the CSTR. However,
to fit with the experimental conditions of parallel studies on
size-responsive gels, no acid + base mixtures are injected
through the third feed channel. Only variable additional amounts
of NaOH are injected through this channel and are used as the
control parameter. Note that, an increase in [NaOH]0 corre-
sponds to a decrease in R. Also, for practical reasons, the pH
indicator is changed to methyl-red (Aldrich), which turns from
clear yellow to dark-red when the solution switches from basic
to acid. Thus, the dark acid states in the cylinder of gel are
more easily observable across the clear alkaline contents of the
CSTR. The state of the CSTR is accurately monitored by a pH
electrode.
The snapshots in Figure 6 show the propagation of the M
state, characterized by an acid core (dark gray), into the alkaline
(clearer) B state, respectively in the positive (a-c) and the
negative (d-f) directions inside a OSFR with a radius r ) 0.5
mm. Note that for positive values (M state propagating into the

Spatial Bistability in a pH Autocatalytic System
J. Phys. Chem. A, Vol. 109, No. 35, 2005 7847
Figure 8. Computed stability diagram in the plane (R,[S0]). A zoom
on the oscillatory domain is given in Figure 9.
associated rate laws (eqs 2, 5, and 6). The following values
were used for the kinetic constants: k ) 5 × 106 M-3 s-1, ke-
) 1.4 × 1011 M s-1, and k+e ) Ke k- with Ke ) 10-14, ka-
)
e
1011 M s-1, and ka+ ) Ka ka- with pKa ) - log(Ka) ) 1.94. The
chemical feeds and other essential parameters were taken to fit
the experimental conditions of the annular OSFR. The residence
time of the CSTR is τ ) 600 s. The ratio of the volume of the
gel to the volume of the CSTR is 1.75 × 10-3 for a 1 mm
width. The concentrations in the input flow are [ClO2-]0 ) 2
× 10-2 M and [S4O62-]0 ) 5 × 10-3 M, and R is defined as in
the experiments. To account for the fast diffusion of H+ and
OH-, their diffusion coefficients were fixed to Dfast ) 3.4 ×
10-5 cm2 s-1 and all the other coefficients to a standard value
D ) 10-5 cm2 s-1. The value of Dfast is an effective coefficient
that was set by a previous22,23 fit of a few experimental data.
This effective value accounts both for the high diffusion
coefficient of these ions and for the electrostatic effect of the
other ions on the fast charged species. To include the effects of
a complexing agent, we introduce an additional fast equilibrium
with a species S-:
Figure 6. Sequences of snapshots illustrating the propagation of M-B
interfaces in the cylindrical OSFR (top of the cylinder on the left and
bottom on the right): (a-c) [OH-]0 ) 1.3 × 10-3 mol/dm3, interface
velocity +5.1 mm/h; (d-f) [OH-]0 ) 2.3 × 10-3 mol/dm3, interface
velocity -2.4 mm/h. Conditions: diameter of the gel cylinder 1 mm,
temperature 25 °C, and [-COO-]0 ) 7.65 × 10-2 mol/dm3 (from poly-
(acrylate)); for other parameters, see text. The gel boundary, not visible
in the snapshot, is schematically drawn in snapshot a.
H+ + S- h HS
(7)
with rate law:
Vs ) k+s [HS] - k-s [H+][S-]
(8)
Figure 7. Propagation velocities of the M-B interface as a function
of [NaOH]0 in cylindrical OSFRs of different diameters: (4) 1 mm;
(0) 1.5 mm; (+) 2 mm. Conditions: [-COO-]0 ) 7.65 × 10-2 mol/
dm3 (from poly(acrylate)) and a temperature of 35 °C; for other
parameter values, see text.
with k-s ) 1011 M s-1 and ks+ ) Ks k-s and pKs ) - log(Ks) )
5.5.
Diagrams of stability and boundaries of oscillatory domains
were computed in 1-D simulations, while the excitability
domains were determined in 2-D simulations. More details on
the numerical procedures are found in refs 23 and 33.
The computed phase diagram in the plane (R, S0), where S0
is the total amount of available immobile complexing sites for
protons, is given in Figure 8. Figure 9, is a zoom of the small
domain of oscillatory M state. The double arrows in Figure 8
delimit the domain of excitable B state for a few values of S0.
At S0 ) 2 × 10-2 M, this state is no longer found to be excitable.
Despite the simplicity of the model, the results are in good,
almost quantitative, agreement with the experimental results.
Except for small values of S0 where the B f M transition cannot
be distinguished from the B f A transition of the CSTR
contents, the former transition occurs before the latter. When
S0 increases, this B f A transition is shifted to higher values
of R, but, since the experimental procedure restricts the range
to R e 1, this limit is out of reach beyond S0 ∼ 2 × 10-2 M.
Let us now focus on the stability limit of the M state at low R.
B state) the acid core is significantly broader than for the
negative values (B state propagating into the M state). These
changes of state characteristics in the radial direction are in
agreement with previous observations in other systems.24,25 In
this range of radius (or width), the high [NaOH]0 (low R) limits
of the spatial bistability strongly depend on the radial size of
the OSFR,23 and as illustrated (Figure 7), the value of [NaOH]0
at which the M-B interface velocity changes direction decreases
with the decrease of the radius of the cylinder of gel.
Numerical Results and Discussion
Numerical simulations based on the kinetic model presented
earlier were performed in complement to experiments. It was
previously shown22,23 that, in the absence of complexing agent,
the dynamical properties of the CT reaction in a OSFR can be
accounted for almost quantitatively by eqs 1, 3, and 4 with the

7848 J. Phys. Chem. A, Vol. 109, No. 35, 2005
Szalai et al.
However, it was shown in the experiments that, for high enough
values of S0, the propagation velocity can be reversed in the
vicinity of the bistability limit. To check this point without
undertaking too lengthly calculations, we have used a smaller
width w ) 0.3 mm to decrease the relaxation times, and a value
S0 ) 10-1 M. With these parameters the bistability limit is
located at R = 0.715. The shift to higher R (lower [OH-]0)
with decreasing width follows the experimental observations
(Figure 7). In the bistability domain, at R ) 0.73, state M still
invades state B with a propagation velocity equal to 1.27 ×
10-4 mm/s. However closer to the bistability limit the situation
is reversed, for R ) 0.72, state B invades state M at a velocity
equal to 6.1 × 10-5 mm/s.
Figure 9. Detail of the stability diagram given in Figure 8: limits of
the oscillatory domain.
Conclusion
This report confirms that the oscillatory and the excitability
properties of the CT reaction operated in an OSFR are induced
by long-range-activation instabilities. The interface dynamic of
standard bistable systems is recovered when the diffusivity of
the protonssthe activatory speciessis selectively and suf-
ficiently slowed. The addition of large enough amounts of poly-
(acrylate) ions can even make the system short range activated.
This is clearly shown to occur for concentrations of quasi
immobile carboxylic functions above 2 × 10-2 M. Note that
the different steps taken above: the characterization of a spatial
bistability domain, the search for conditions where the direction
of propagation of interfaces between the two spatial steady states
changes sign, and the development of effective short-range
activation are the first steps in a recently proposed systematic
method to produce stationary pulse patterns of one state
immersed into an other state (domain patterns). The final step
of the method is to study head-on collisions of spatial interfaces
in a range of parameters near to the values where the direction
of propagation of the spatial interface changes sign and search
for conditions where nonvanishing front pairing would form.
With this possibility in mind, we studied the collision dynamics
of spatial interfaces having slow positive velocities for a
concentration of acrylate function equal to 0.1 mol/dm3; the goal
was to observe stationary pulses of the B state localized in the
M state. Up to now, this quest is not successful. Beside the fact
that we have explored only a small range of possible feed
parameters, the bleaching of the color dye in the acid/base front,
makes the observation of a narrow region of the B state difficult.
Furthermore, a recent kinetic investigation31 shows that chloride
ions can play a secondary activatory role in the CT reaction.
The diffusivity of this species is not affected by the introduction
of poly(acrylate) and the slow head-on collisions of the spatial
state interfaces could not be prevented in these conditions. We
now continue the quest for stationary patterns in other pH-
activated systems, like the iodate-sulfite reaction, where we
expect to avoid the above difficulties.
At S0 ) 0, on decreasing R, the M state develops a small
domain of oscillations just before disappearing (Figure 9). In
the model, these oscillations were shown to result from the long-
range activation linked by the fast diffusion of H+ and have
been extensively discussed in ref 23. For the same reasons,
outside the bistability domain, the B state remains excitable over
a large domain of parameters (0.475 < R < 0.69). When the
complexing agent is introduced, the effective diffusion of H+
is divided by a factor σ ) 1 + S0/Ks.8 As a consequence, the
extent of the domains of oscillations and of excitability decrease
when S0 increases. These behaviors have totally vanished at S0
) 2 × 10-2 M. Numerical results are quantitatively consistent
with experimental determinations. In both cases, for S0 g 2 ×
10-2 M, the stability limit of the M state exhibits a remarkable
invariance as a function of S0. This surprising behavior can be
understood in the following way. On the basis of simple
arguments, it was previously shown24 that if the front of the M
state is sharp and a substrate (here OH-) is almost completely
transformed, the distance δ of this front to the CSTR boundary
is approximately given by the following relation:
DX
Q˙
δ
(9)
X is the concentration of the substrate at the feed-boundary, D
the diffusion coefficient of the driving species (here H+), and
Q˙ is the rate of consumption of the substrate. This distance does
not depend on the width w of the system. State M loses its
stability when w becomes of the same order as δ. When a certain
amount of complexing agent is added, the effective diffusion
D and the reaction rate Q˙ of the autocatalytic reaction are both
divided by the same factor σ. Thus, according to eq 9, δ remains
unchanged. It was checked during the computations that for a
given R the distance δ is independent of S0 and w provided
that S0 is large enough to avoid long-range activation effects.
Because δ only depends on R (through X) and for a given w M
loses stability when δ reaches a critical value (of order w), the
transition point does not depend on S0. This invariance is similar
to the result obtained by Pearson and Bruno9 for the threshold
of Turing structures. On the same basis, they show that this
threshold is independent of the concentration of the complexing
agent used to slow the diffusion of the activator. Obviously,
this does not infer that S0 has no effect on the dynamics.
Although the position of the transition is unchanged, the system
dynamics can be dramatically slowed and the concentration
profiles of the two different stable states are modified.
Acknowledgment. This research has been supported by a
Marie Curie Fellowship of the European Community program
Improving Human Potential under contract number HPMF-
CT-2002-011771. We thanks the support of the ESF “REAC-
TOR” program. I.S. expresses thanks for the support of the
Hungarian Academy of Sciences (HAS) (F049666, T0437473).
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Within the bistable domain and for low concentrations of the
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are in contact; i.e., the M state is more stable than the B state.

Spatial Bistability in a pH Autocatalytic System
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