Physica D 239 (2010) 776–784

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Physica D

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Pattern formation in the thiourea–iodate–sulfite system: Spatial bistability,waves, and stationary patterns

Judit Horváth a, István Szalai b, Patrick De Kepper a,∗

a Centre de Recherche Paul Pascal, CNRS, University of Bordeaux, 115, Avenue Schweitzer, F-33600 Pessac, Franceb Institute of Chemistry, Eötvös Loránd University, Laboratory of Nonlinear Chemical Dynamics, P.O. Box 32, H-1518 Budapest 112, Hungary

a r t i c l e i n f o

Article history:Available online 21 July 2009

Keywords:Pattern formationSymmetry breakingTuring patternLandolt reactionBistability

a b s t r a c t

We present a detailed study of the reaction–diffusion patterns observed in the thiourea–iodate–sulfite(TuIS) reaction, operated in open one-side-fed reactors. Besides spatial bistability and spatio-temporaloscillatory dynamics, this proton autoactivated reaction shows stationary patterns, as a result of two back-to-back Turing bifurcations, in the presence of a low-mobility proton binding agent (sodiumpolyacrylate).This is the third aqueous solution system to produce stationary patterns and the second to do this througha Turing bifurcation. The stationary pattern forming capacities of the reaction are explored through asystematic design method, which is applicable to other bistable and oscillatory reactions. The spatio-temporal dynamics of this reaction is compared with that of the previous ferrocyanide–iodate–sulfitemixed Landolt system.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

The study of non-equilibrium chemical patterns is a challengingarea of nonlinear dynamics [1–3]. In solutions, quite diversemechanisms can be at the origin of chemical patterns. They canbe driven by buoyancy instabilities [4], by surface tension [5],by nonlinear colloidal particle growth mechanism in periodicprecipitation reactions [6,7] (e.g. Liesegang patterns [8]), or bysingle phase kinetic nonlinearities and molecular diffusion in thereaction–diffusion (RD) patterns. This report is focused on thelatter type of patterning mechanism. Dr. Stefan Müller was amongthe early contributors to the experimental studies of RD [9] andother types of patterning mechanisms [10,11] in chemical andbiochemical [12] systems.Reaction–diffusion systems can produce a wide range of

nonlinear phenomena including smooth and turbulent wavepatterns, self-replicating spots, regular and irregular patterns asa result of different types of spatial and temporal bifurcations.It is known since the 1952 theoretical work by Alan Turing [13]that autoactivated reactions with associated long-range inhibitionprocesses can spontaneously lead to the formation of stationary

∗ Corresponding author. Tel.: +33 5 56 84 56 56.E-mail address: dekepper@crpp-bordeaux.cnrs.fr (P. De Kepper).

0167-2789/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2009.07.005

symmetry-breaking patterns. In two variable activator–inhibitormodel systems, this implies that the inhibitor diffuses fasterthan the activator. Real nonlinear chemical systems include amuch larger number of species and the requirement for thedevelopment of stationary patterns can be more involved but aspecies controlling or interfering with the activatory loop musthave a reduced mobility.The first experimental evidences of sustained stationary

chemical patterns were provided in the early 90s [14,15]. Theseobservations boosted the field. Interestingly these stationarypatterns developed through two different routes: one, the first,came through a Turing bifurcation [13,16], the other through frontpairing interactions [17,18]. These findings were made possibleby the development of a large variety of chemical oscillatingreactions in the 1980s, and the subsequent invention of openspatial reactors. Among the large variety of systems able to producemacroscopic patterns, the isothermal solution chemical systemsare often thought to be the best suited to serve as simplifiedexperimental models for a number of morphogenic and dynamicself-organization phenomena found in the living world.After the euphoria of the discovery of stationary patterns in

two different reactions (the chlorite–iodide–malonic acid (CIMA)and ferrocyanide–iodate–sulfite (FIS) reactions) [14,15,19–21], thediversification of chemical systems came to a stop. Two reasonsfor this: (i) the patterning capacities of the CIMA and FIS reactionsare rich enough to serve as test grounds for many theoreticalaspects of pattern developments; (ii) the discovery of these two

J. Horváth et al. / Physica D 239 (2010) 776–784 777

first chemical examples were the fruit of targeted research butnot of a fully comprehensive method. The incomplete theoreticalunderstanding of the actual complexity of real open spatialreactors also hindered this diversification. However, differenttypes of sustained stationary patterns were found in othersystems such as: on the surface of metal plates during gas phasecatalytic reactions [22], on the electrodes during electrochemicalreactions [23], in gas discharge systems [24], andmore recently in amicroemulsion version of the Belousov–Zhabotinsky reaction [25].One of the oldest known nonlinear chemical reactions is the

Landolt reaction [26], that is, the autocatalytic oxidation of sulfiteions by iodate ions. This reaction is widely used as a classroomdemonstration of autocatalytic (or clock) reactions as the color ofiodine appears abruptly after a well-defined induction time. Thisis an acid and iodide activated reaction. Yet, the protons have beenshown to be the main autoactivatory species during the oxidationof sulfite [27]. A large variety of oscillatory reactionswere designedby adding a second substrate with antagonist action on theproton production when reacting with iodate (e.g., [Fe(CN)6]4−,S2O2−3 , SC(NH2)2) [28–30], and by operating thesemixed substrateLandolt reactions in continuously fed stirred tank reactors (CSTR).These types of oscillators are often referred to as ‘‘pH-oscillators’’,since the observed large amplitude pH changes are the drivingforces of the kinetic instability [31]. The mechanism, the kinetics,and the temporal dynamics of these systems have been studied indetail during the last decade [27,32–38].Refined studies of sustained RD patterns require the use of

open spatial reactors. Open spatial reactors [39] consist in a pieceof porous material (e.g. a hydrogel) permanently fed by freshreagents. The porousmedium quenches all hydrodynamicmotionsin the reacting mixture and enables the development of the pureRD patterns. The feeding is provided by diffusive exchanges withthe contents of a CSTR. The chemical state of the latter is controlledby the input flow concentration ([]0), the residence time (τ ), andtemperature of the thermostat (T ).In the case of the CIMA reaction, the long-range inhibition

required for the Turing patterns was realized by adding starchor PVA to the reacting mixture. These macromolecules, togetherwith iodine, reversibly bind the iodide ions, which are themajor activatory species of the reaction. The chemical andphysical mechanisms at play in the patterning processes of thissystem were clearly understood and controlled very soon afterthe breakthrough result [40–43]. Shortly after the discovery ofTuring patterns in the CIMA reaction, a rich variety of spatio-temporal patterns arising from planar front multiplicity and frontmorphological instabilities were observed in the FIS reaction[19–21]. In particular, stationary labyrinthine patterns, self-replicating spots, and bouncing waves were observed when thisreaction was operated in an open one-side-fed reactor (OSFR).However, for many years, the actual origin of the short-rangecharacter of the activatory process, necessary for the onsetof stationary patterns, was uncertain and made the originalobservations difficult to reproduce. Recently, we revisited thissystem and clarified this aspect [44,45]. This revised studywas used to develop an effective systematic design methodto discover stationary RD patterns in an OSFR operated withother targeted reactions. The major ingredients of the methodare: (i) the development of spatial bistability [46] and spatio-temporal oscillations in autoactivated reactions operated in anOSFR, (ii) the addition of a species enabling to control thenegative feedback loop independently from the positive one, and(iii) the introduction of a low-mobility reversible complexingagent to generate simultaneously appropriate time and space scaleseparations between the activatory and the inhibitory processes.The method, lately applied to the thiourea–iodate–sulfite

(TuIS) reaction, was confirmed by the discovery of the second

experimental example of sustained stationary patterns resultingfrom a Turing bifurcation [47]. Here, we provide a detaileddescription of the spatio-temporal dynamics observed in this newsystem. The similarities and differences between the TuIS and FISreactions are discussed.

2. Materials and methods

Two geometries of OSFRwere used in the present experiments:The one, hereafter named as the discOSFR, consists of a transparentthin disc (22 mm diameter and 0.75 mm thick) made of 4 wt%agarose gel (Fluka, BioChemika 05070 or 05077). One face of thedisc is in direct contact with the contents of a CSTR through acircular hole (18 or 20 mm diameter) in a mask which holdsthe gel tightly pressed against a flat observation window. Beyondthe contact surface, the disc extends (≈ 2 mm) under the mask(for details see Ref. [42]). Patterns are observed by transparencyand are monitored by a CCD video camera connected to a videotape recorder and to a black and white frame grabber board.To enhance the contrast of the transmitted light patterns, astabilized soft orange light illumination (using a longpass filterwith a transition at 515 ± 29 nm) was shined opposite to thecamera. The observations bring information on the integratedlight absorption of the patterns across the thickness of the discbut not on how these changes organize within the thickness.This complementary information is provided by the other OSFRgeometry, hereafter named as the annular OSFR: It consists in a flatannular gel reactor (w = 1 mm deep and 25 mm outer diameter)(for details see Ref. [48]). All the faces of the annulus are tightlypressed against impermeable walls except the outer edge which,as above, is in direct contact with the contents of a CSTR. Theobservations, in reflected light, are made in a direction orthogonalto the feed surface and thus allow resolving the color changes inthe feed direction. This geometry of OFSR is self-sustained. It isfree of boundary limit perturbations and offers periodic boundaryconditions. In this respect, it mimics an infinite boundary system.Note that both reactors are confined in the direction of the feedbut while the disc reactor is extended in the two other directions(apparent 2D system) the annular reactor is extended only inanother direction (apparent 1D system).All reported experiments were made at 30 ◦C with a resi-

dence time of the CSTRs of τ = 240 s. The chemical solutionswere stored in three separated reservoirs and injected by preci-sion pumps (Pharmacia P500) with equal flow rates at the baseof the CSTRs. They enter through a single port in the close vicin-ity of a turbine rotating at 1000 rpm. An overflow pipe maintainsthe reaction volume constant and at room pressure, with no airinterface. The respective reservoirs contain: (1) potassium iodate(Fluka or Riedel-deHaën); (2) sodium sulfite (Fluka); (3) sulfuricacid (Roth). Thiourea (SC(NH2)2, Fluka) and sodium poly(acrylate)(Aldrich,Mw ≈ 15 000Dalton) are added into the second reservoir.Bromocresol Green was used as a pH color indicator. It switchesfrom blue to yellow in the pH range 5.4–3.8. All the chemicalsare of analytical grade and solutions are prepared in deionizedwater. The feed concentration of potassium iodate ([KIO3]0) andsodium sulfite ([Na2SO3]0) were fixed at 75 mM and 89 mM,respectively. Here [ ]0 denotes the concentration that the inputspecies would have in the reactor after mixing and prior to anyreaction. The chemical state of the CSTR contents was monitoredby the mixed potential E measured by a bright Pt electrode ver-sus a Hg/Hg2SO4 (sat. K2SO4) reference electrode (Eref = 650mV).To explore the dynamics of the system, we varied the feed con-centration of thiourea ([Tu]0), sulfuric acid ([H2SO4]0), and sodiumpoly(acrylate) ([NaPAA]0) respectively in the ranges 0–6 mM,1–4 mM, and 0–12 mM. The value of [NaPAA]0 corresponds to the

778 J. Horváth et al. / Physica D 239 (2010) 776–784

concentration of carboxylic groups in the feed solution introducedby the poly(acrylate) polymer chains. These polyacid chains slowlydiffuse into the agarose gel where their asymptotic concentrationis expected to be close to that of the CSTR solution. For each newNaPAA feed concentration, the gel parts of the reactors were left toimpregnate in a solution with the targeted value for at least 24 h.Note that our agarose (Fluka BioChemika 05077) 4 wt% hydrogelcontains≈ 0.8 mM residual acid functional groups as determinedby conductometric titration of an agarose suspension.

3. Results

3.1. CSTR dynamics

The TuIS system is a mixed Landolt type pH-oscillator [32].The kinetic mechanisms of the composed reactions are detailedin the literature [27–30,32–38]. It can be grossly accounted bytwo overall processes,where the hydrogen ion driven autocatalyticoxidation of hydrogen sulfite ((R1) positive feedback) is coupledwith a hydrogen ion consuming reaction, in the present case withthe oxidation of thiourea ((R2) negative feedback):

IO−3 + 3 HSO−

3 −→ 3 SO2−4 + I−+ 3 H+ (R1)

IO−3 + 6 H++ 6 Tu −→ I− + 3 Tu2+2 + 3 H2O. (R2)

Although reaction (R1) is autocatalytic both for protons and iodideions, Rábai and coworkers assert that the kinetic contribution ofthe latter ones is negligible in unbuffered systems [27]. The mainactivation is provided by the concentration of free protons whichfavors the protonation of sulfite ions to their more reactive form,i.e., HSO−3 . In batch, the total oxidation of Tu by iodate goes throughseveral organic sulfur and iodine containing intermediates, andthe end products are ammonium sulfate, carbon dioxide, andiodine [30]. However, in open systems with residence timestypically below 2000 s, the proton consuming reaction betweeniodate and Tu can be satisfactorily described by (R2). Taking intoaccount the rate constants of the consecutive steps only the firststep, that is, the oxidation (R2) to the dimer, formamidine disulfideion (Tu2+2 ) is relevant. It is known, that the iodate–sulfite reaction(R1) shows bistability in a CSTR [28,48]: the contents of the reactorcan be either in a high pH (7.0–6.0), high redox potential, lowextent of reaction or in a low pH (≈ 3), low redox potential,high extent of reaction steady state. They, respectively, correspondto the so-called ‘‘flow’’ and ‘‘thermodynamic’’ branches [28]. Thestability limits of these two branches sensitively depend on thesulfuric acid feed concentration ([H2SO4]0) which is convenientlyused as the major tunable control parameter throughout theexperiments. In the absence of Tu, the stability domains of thesetwo steady-state branches overlap for 0.5 mM ≤ [H2SO4]0 ≤4.5 mM (Fig. 1(a)).Introducing thiourea decreases the range of steady-state bista-

bility in the CSTR as the stability limit of the low pH branch shiftsfaster to higher acid feed values than that of the high pH branch(Fig. 1(a)). Nevertheless, the potential difference between the twobranches remains large throughout the domain. It exceeds 400mVeven at [Tu]0 = 5.7 mM, as it can be seen in Fig. 2. Above a crit-ical feed concentration of Tu ([Tu]0 ≥ 5.7 mM) the flow branchloses stability to produce large amplitude oscillations prior to thetransition to the stationary thermodynamic branch. In the param-eter range explored, the oscillatory domain enlarges as [Tu]0 is in-creased but coexists with a stable domain of the thermodynamicbranch. This behavior of the system is better understood by exam-ining the potential response as a function of [H2SO4]0 at a constantvalue of Tu. This is shown in Fig. 3(a) for the case [Tu]0 = 10 mM.The flow state is stable up to [H2SO4]0 ≤ 8.5 mM, beyond it isfollowed by large amplitude oscillations (≈ 350 mV). The system

a

b

Fig. 1. CSTR phase diagram of the TuIS reaction in the ([H2SO4]0; [Tu]0) plane at[NaPAA]0 = 0mM (a) and 12mM (b). T = 30 ◦C; τ = 240 s. Symbols: N flow state(zone I); H thermodynamic state (zone II); � flow/thermodynamic state bistability(zone III);� oscillatory/thermodynamic state bistability (zone IV). Gray areas indi-cate the CSTR feed conditions where RD patterns were observed in the disc OSFR.

a b

Fig. 2. Pt potential response of the CSTR contents. Feed conditions: [Tu]0 =5.71 mM and [NaPAA]0 = 0 mM (a) and 12 mM (b). The vertical segment denotesthe amplitude of the oscillations. Other symbols and zones as in Fig. 1.

switches to the thermodynamic branch at [H2SO4]0 ≥ 10.5 mM.The potentials on the flow and thermodynamic branches, respec-tively, do not differ much from the values measured in the bista-bility region (compare with Fig. 2). However, on decreasing now[H2SO4]0, we do not return to the oscillatory domain again, butthe thermodynamic branch remains stable down to [H2SO4]0 ≈5.5mM. In domain IV, the oscillatory state and the stationary ther-modynamic state are both stable. In domain III, the two stationarystates, the thermodynamic state and the flow state, are stable. Onestationary state is stable in each of the domains I and II. This meansthat the overall topology of the phase diagram is not that of thestandard cross shape [49] as observed for the analog reaction withferrocyanide [28]. This is a signature of the more complex kineticmechanism between iodate and Tu.In our pattern formation experiments we use NaPAA for the re-

versible binding of protons, thus, we also explored the effect ofNaPAA on the CSTR dynamics. The addition of NaPAA induces ageneral shift of the domain stability limits to higher [H2SO4]0 val-ues without changing qualitatively the topology of the diagram

J. Horváth et al. / Physica D 239 (2010) 776–784 779

a

b

Fig. 3. Pt potential response of the CSTR contents. Feed conditions: [Tu]0 = 10mMand [NaPAA]0 = 0 mM (a) and 12 mM (b). The vertical segments denote theamplitude of the oscillations. Other symbols and zones as in Fig. 1.

(see Fig. 1(b)). This shift is understandable as we supply the weakpolyacid in the form of its stoichiometric sodium salt (NaPAA)which hydrolyzes basically. The stability limit of the thermody-namic branch is more affected by NaPAA than that of the flowbranch. Although, this way, thewidth of the bistability domain (III)is reduced to less than 50% at [Tu]0 = 5.7 mM (Fig. 2(b)), the po-tential difference between the two branches remains nearly unaf-fected, it is still mostly larger than 300 mV. On the contrary, al-most no shift occurs in the [Tu]0 value where oscillations appear([Tu]0 ≥ 5.7 mM). Noteworthy, the oscillatory domain is widerthan in the absence of NaPAA. This widening by more than 70%along the [H2SO4]0 coordinate can be observed at [Tu]0 = 10 mMin Fig. 3(b). At the same time, the amplitude of the oscillations is re-duced to≈ 200mV. This is the side-effect of NaPAAwhich, in addi-tion to its role in the kinetics, acts also as a pH buffer on the system.A sudden decrease to 40–10 mV occurs in the amplitude directlypreceding the end of the oscillatory state. The amplitude tends tozero as if an additional stationary state existed over a tiny con-centration range directly before the system switches to the ther-modynamic state. In this vanishingly stable stationary state no I2precipitate is present yet but the system is already in the acidicstate. The same phenomenon was observed in the presence of6 mM NaPAA, but not in its absence.The CSTR feed conditions for which patterns were observed in

the OSFR aremarked as gray areas in Fig. 1(a) and (b). The contentsof the CSTRwere alwaysmaintained on the stationary state branchwith the lowest extent of reaction, i.e., the high pH flow branch.This ensures far from equilibrium conditions at the boundary ofthe gel. In all cases, these patterning conditions remain far awayfrom the stability limit of this branch, and also from the oscillatorydomain, at any Tu or NaPAA feed concentration.

3.2. Dynamics in annular OSFR

An annular OSFR can be conveniently used to determine therange of parameters at which the phenomena of interest can beobserved – e.g., spatial bistability and oscillations. In Fig. 4, wepresent a section of phase diagram in the ([Tu]0; [H2SO4]0) plane

Fig. 4. Experimental annular OSFR phase diagramof the TuIS reaction. The symbolscorrespond to the state of the gel: H stable M-state; N stable F-state; • oscillatorystate. The solid lines indicate the approximative limits of existence of the steady F-and M-states.

for the annular OSFR. The phase diagram was established by step-wise changes of [H2SO4]0 at different fixed values of the [Tu]0.Different spatial states can be observed in the gel depending on[H2SO4]0 and on the initial state of the gel contents. In the absenceof Tu, at [H2SO4]0 = 0mM, the gel is in a quasi-uniform dark colorstate (exemplified by the right part of Fig. 5(a)), the F-state of theOSFR. The F-state of the gel is stable up to [H2SO4]0 = 2.38 mM.A further increase of [H2SO4]0 to 2.68 mM makes the inmost partof the annulus suddenly turn clear. A stable pH-indicator colorswitch settles parallel to the rim of the annulus, as exemplifiedby the left part of Fig. 5(a); this is the mixed state M of theOSFR. Now, decreasing [H2SO4]0, the M-state remains stable downto 1.68 mM. Further lowering [H2SO4]0 to 1.04 mM makes thecontents of the annulus return to the uniformly dark F-state. Thestability range of the F- and theM-states of the gel overlap between[H2SO4]0 = 1.68 and 2.38mM, this is spatial bistability [46]. In therange of spatial bistability, an appropriate perturbation can createinterfaces between the M- and the F-states. Fig. 5(a) illustratessuch an interface. The direction of the propagation of the interfacedepends on the actual value of [H2SO4]0.The extent of the domain of spatial bistability decreases as the

input concentration of the thiourea is increased and vanishes ata critical value ([Tu]0 ≈ 2.5 mM). Let us describe the changesof states above this limit, e.g., at [Tu]0 = 3 mM. Starting from[H2SO4]0 = 0 the content of the gel is in the F-state until [H2SO4]0= 2.9 mM. A further increase to [H2SO4]0 = 3.0 mM makes theinmost part of the gel suddenly become acidic. As above, a mixedstate appears but now the switching position of the color indica-tor is no longer stable but moves back and forth across the widthw of the annulus (Fig. 5(b)). It manifests as periodic rapid expan-sion of the low pH core followed by its slow decay. The domainof existence of this oscillatory state interchanges with the domainof spatial bistability through a typical cross-shaped phase diagramtopology [49] (Fig. 4). The oscillation of the color switch positionhas a spatial amplitude less than 0.2mm in an annular reactorwithw = 1mm, and decreases as the feed concentration of the Tu is in-creased. The period of the spatio-temporal oscillations is in 10minrange (Fig. 5(b)).

3.3. Dynamics in disc OSFR

Under the feed conditions that correspond to the oscillatorydomain in the annular OSFR ([Tu]0 = 3–6 mM and [NaPAA]0 =

780 J. Horváth et al. / Physica D 239 (2010) 776–784

Fig. 5. Illustration of the spatial states in the annular gel OSFR: (a) A sectorof the annular gel with an interface between the M- and F-states in the spatialbistability domain. On the left, the sharp color switch (pH drop) from dark to lightgray characterizes the M-state. On the right, the quasi-uniformly dark gray partcorresponds to the F-state. Feed conditions: [Tu]0 = 1.0 mM and [H2SO4]0 =2.0 mM. (b) Time-space plot of the spatio-temporal oscillatory state at [Tu]0 =3.0 mM and [H2SO4]0 = 3.0 mM.

Fig. 6. Time-space plot of spatio-temporal patterns in the disc OSFR: (a) quasi-uniform oscillations, (b) traveling waves, (c) fragmentedwaves. Time goes from leftto right, each image spans 20 min. Experimental conditions: (a) [Tu]0 = 5 mM,[NaPAA]0 = 0 mM, [H2SO4]0 = 3.54 mM, diameter of the disc 18 mm; (b)[Tu]0 = 6 mM, [NaPAA]0 = 3 mM, [H2SO4]0 = 3.66 mM, diameter of the disc20 mm; (c) [Tu]0 = 5 mM, [NaPAA]0 = 3 mM, [H2SO4]0 = 3.49 mM, diameter ofthe disc 20 mm.

0 mM), quasi-uniform oscillations and traveling waves areobserved in the disc OSFR. The time-space plot that is analogouswith Fig. 5(b), obtained in the annular OSFR, is shown in Fig. 6(a)for the disc OSFR. It demonstrates immediately that the oscillationsarewith good approximation in-phase throughout the gel disc. Theasymmetry in the amplitude, the burst expansion of the indicatorswitching position and its slow draw back, can be observed inthis representation, as well. Now, the amplitude is transformedto grayscale as we detect the integrated light intensity acrossthe thickness of the gel. We see a very bright (left) edge at thebeginning of each appearance of the light color of the indicator thatfades away gradually on the right-hand side.In the same region of spatio-temporal oscillations, we also

observe propagating low pH waves at slightly lower [H2SO4]0(Fig. 7 top row). According to the grayscale, we can concludeto a sharp front and a less steep back in the direction ofpropagation. A small amount (3 mM) of NaPAA is not enoughto stop traveling waves, moreover, they remain very regular(Fig. 6(b)). Nevertheless, the effect of NaPAA can appear as thedestabilization and fragmentation of the wave front, and thesewaves often leave temporarily surviving low pH spots behind(Fig. 7 bottom row). They are also slower as compared to the case

in the absence of NaPAA. A more peculiar behavior is that thewaves are not only segmented but their propagation appears tobe discontinuous in the direction of propagation (Fig. 6(c)). Thewave front is more or less regularly interrupted and apparentlyresets at a small but finite distance further thanwhere the previousone died. These observations show some similarities with thediscontinuous wave propagation reported recently in the BZ-AOTsystem [50].The region in the ([Tu]0; [NaPAA]0) plane, where traveling or

fragmented waves exist, extends only up to [NaPAA]0 ≈ 3 mMand for 3 mM . [Tu]0 . 6 mM, as is shown in Fig. 8. If we movefurther upwards in the diagram, we notice that [NaPAA]0 = 6 mMis sufficient to quench any oscillations, and stationary patternsdevelop instead. However, the region of [H2SO4]0 and [Tu]0 wherepattern formation occurs shrinks to the right half of the previousoscillatory domain. [Tu]0 ≈ 2.5 mM, i.e., around the cross-pointof the OSFR phase diagram without NaPAA (Fig. 4), sets thelower limit of existence of stationary symmetry-breaking patterns.Between concentrations [Tu]0 = 2.5–4.0 mM, irregular highlybranched stationary patterns develop (Fig. 9). These ‘‘branched’’structures do not appear ‘‘as is’’, but are results of a complexprocess involving the coalescence of low pH patches and fingeringgrowth.On the contrary, very regular stationary patterns emerge be-

tween [Tu]0 = 4.5–6.0 mM. Three different planforms are ob-served (Fig. 10): two inverted hexagonal arrays of spots and onearray of parallel stripes. They represent the three standard 2D pat-tern planforms expected from a Turing bifurcation. The realizationof one or the other depends on [H2SO4]0 at a given [Tu]0. Start-ing from the uniform F-state, the smallest sufficient increment in[H2SO4]0 initiates the formation of a low pH spot pattern followinga hexagonal symmetry. The spots appear one by one in the vicinityof previous spots, starting from the borders of the disc. The bound-ary of the disc OSFR is at a slightly higher extent of reaction thanthe center because the gel under themask is always in theM-state.The spots appear at their proper places on the array followed byonly a very slight rearrangement. Deviances from the exact hexag-onal symmetry are the consequence of the mismatch between thepattern symmetry and the circular symmetry of the boundary, andalso of small local defects (dust particle or air bubble) in the gel.The Turing stripe pattern in Fig. 10(b) emerges when [H2SO4]0 isset slightly higher. Similarly, stripes appear one after the other, atdefined distance from each other, starting from the border, whena superthreshold parameter change is made directly from the uni-form state. A further increment in [H2SO4]0 results in the devel-opment of a hexagonal array of ‘‘inversed’’ color spots (Fig. 10(c)).These three types of Turing structures can be transformed into eachother by varying [H2SO4]0: On increasing [H2SO4]0, the hexagonallow pH spots merge into evenly spaced stripes, and further, theclear stripes fuse together and leave an array of high pH spots withhexagonal symmetry in a low pH background. The conversion isreversible in both directions without hysteresis within the accu-racy of our minimal steps, i.e., 0.02 mM [H2SO4]0. The coexistenceof the hexagonal high pH dark spots and the stripes in Fig. 10(c)is a consequence of the narrowness of the domain of stability ofeach planform. It is sensitive to small local shifts in parameter con-ditions, e.g., to a tiny gradient in the gel thickness or differencesin the feed exchange at the surface. Around [Tu]0 = 5.5–6.0 mMthe contrast of the Turing patterns becomes weak and above thisconcentration range only uniform spatial states, gradually chang-ing from F to M type states, are observed as a function of [H2SO4]0.Furthermore, we tested the effect of gel thickness in the disc

OSFR at w = 0.50 mm and 1.00 mm. Decreasing the gelthickness shifts the stability limit of the Turing structures tohigher [Tu]0 and higher [H2SO4]0 values, while their wavelengthsshowed no major changes. The extent of the shift was found to

J. Horváth et al. / Physica D 239 (2010) 776–784 781

Fig. 7. Snapshots of traveling waves in the disc OSFR at [Tu]0 = 5 mM and two different NaPAA feed concentrations. Waves move from the bottom to the top. Top row:traveling wave at [NaPAA]0 = 0 mM and [H2SO4]0 = 3.39 mM; sampling interval 3 min, period of waves 8 min. Bottom row: fragmented wave at [NaPAA]0 = 3 mM and[H2SO4]0 = 3.49 mM; sampling interval 6 min, period of waves 10 min. White pointer indicates the position of the vertical section at which the time-space plot in Fig. 6(c)was constructed.

Fig. 8. Patterning regions in the disc OSFR at gel thickness w = 0.75 mm. Aprojection onto the ([Tu]0;[NaPAA]0) plane along the [H2SO4]0 axis. Letters denotethe types of patterns observed in the gel: W = waves; Wfr = fragmented waves;B= branched patterns; T= Turing patterns; X= uniform states.

be around ∆[Tu]0/∆w ≈ −1 mM/0.25 mm at [NaPAA]0 =6 mM. Remarkably, the patterns appear more contrasted for w =0.50 mm than at thicker disc values.Doubling of [NaPAA]0 to 12 mM causes no qualitative changes

except for a very slight shift to lower [Tu]0 by ≈ 0.5 mM (Fig. 8).Contrary to the stability domains of the different states of theCSTR, the range of [H2SO4]0 over which patterns are observed inthe OSFR is little sensitive to the amount of NaPAA introducedin the feed. This can be seen in Fig. 1(a) and (b) where thesmall area tinted in gray indicate the domain of parameterswhere patterns develop in the OSFR. The quasi-invariance of thepatterning region with the increased amounts of [NaPAA]0 isconsistent with the theoretical predictions that the addition of a

low-mobility complexing agent for the activator of an oscillatorysystem only shifts the Hopf bifurcation limit while leaving theTuring bifurcation limit unchanged as a function of the mainchemical reagent concentrations [40]. The further doubling of[NaPAA]0 to 24 mM reduced the color contrast of the patterns andconsiderably slowed down all processes. The latter results from thereduced apparent reactivity of the protons probably enhanced bythe increased viscosity of the solution.

4. Discussion

Let us recall briefly the spatio-temporal dynamics of theFIS reaction and compare it with the observations in the TuISsystem. Both systems produce large amplitude spatio-temporalpH oscillations/waves in the absence of NaPAA (Fig. 11(a)). Inthe FIS reaction, one can clearly distinguish two types of sharpfronts: an acid producing (+)front (transition from F- to M-state)and an acid consuming (−)front (transition from M- to F-state).In contrast, in the TuIS reaction, the (−)front is smooth andalways lags behind the (+)front. A special common feature ofthe (+)fronts in these two systems is their tendency to drawaway from the edge of the circular mask. When a (+)frontapproaches the mask, it slows down, stops, and reverses direction(‘‘bounces’’) in the FIS system while it ‘‘dies’’ in the TuIS system.The extent of reaction is naturally higher under the mask thanin the part of the gel directly in contact with the contents of theCSTR. The composition under the mask turns acidic before therest of the gel. However, in the patterning parameter domains,the M-state never starts from under the mask but develops ata distance from the circular rim. In the absence of NaPAA, thedynamics of spatio-temporal patterns are quite different in thetwo systems: While in the FIS reaction, the (+) and (−)frontsexhibit complex interactions (e.g. inversions in the sign of a front:under certain form of mutual or boundary interactions and criticalcurvatures) reminiscent of Bloch fronts [17]. In the TuIS reaction,more standard relaxation oscillations and waves are observed.In the presence of NaPAA, the development of the stationary

782 J. Horváth et al. / Physica D 239 (2010) 776–784

Fig. 9. Development of a branched pattern at [Tu]0 = 3 mM, [NaPAA]0 = 12 mM, [H2SO4]0 = 3.36 mM. The sampling interval of the snapshots from left to right and downis 20 min except between the last two which is 3 h. The wavelength in the final image is λ = 1.55 ± 0.07 mm.

Fig. 10. Planforms of Turing structures in the disc OSFR: (a) Hexagonal array of low pH spots at [Tu]0 = 5.0 mM, [H2SO4]0 = 3.57 mM, and [NaPAA]0 = 6 mM,λ = 1.85 ± 0.09 mm; (b) Stripe pattern at [Tu]0 = 4.5 mM, [H2SO4]0 = 3.60 mM, and [NaPAA]0 = 6 mM, λ = 1.55 ± 0.07 mm; (c) Coexistence of spots andstripes at [Tu]0 = 4.5 mM, [H2SO4]0 = 3.63 mM, [NaPAA]0 = 6 mM, λ = 1.86 ± 0.09 mm.

Fig. 11. Oscillations and labyrinthine patterns observed in the FIS reaction at [K4Fe(CN)6]0 = 20 mM: (a) double spiral wave pattern at [NaPAA]0 = 0 mM and[H2SO4]0 = 2.92 mM; (b) stationary pattern at [NaPAA]0 = 4.0 mM and [H2SO4]0 = 2.92 mM; (c) stationary pattern at [NaPAA]0 = 1.0 mM and [H2SO4]0 = 3.04 mM. Fordetailed description see Ref. [44].

patterns is also quite different between the two systems. In the FIS,(−)front interactions produce stableM-state filamentous domainswhich lead to frozen labyrinthine patterns through tip growth andbudding (Fig. 11(b) and (c)). In the TuIS system, at high [Tu]0, theonset of stationary patterns is characteristic of two back-to-backTuring bifurcations. One corresponds to the local growth of lowpH-spikes at well-defined distance from each other in a uniformhigh pH state, when [H2SO4]0 is increased. Inversely, the other

corresponds to the individual growth of a hexagonal array of highpH spots in the uniform M-state when [H2SO4]0 is decreased.Consistently, the two regular spot arrays are separated by a domainof stripe pattern, the third standard 2D planform expected from aTuring bifurcation. We searched for such pattern bifurcation in theFIS reaction with no success. In the TuIS reaction, at low [Tu]0, theirregular branched patterns essentially result from the coalescenceof M-state spots and spot growth.

J. Horváth et al. / Physica D 239 (2010) 776–784 783

Fig. 12. Labyrinthine patterns in the FIS system. (a) M-state pattern viewedby the polyiodide indicator (3.3 g/L PVA). The dark color corresponds to theI−3 –PVA complex, the light background corresponds to the F-state. (b) M-statepattern viewed by a mixture of the polyiodide indicator (3.3 g/L PVA) and the pHindicator (0.6 g/L Bromocresol Green). Thin dark gray structure: I−3 –PVA complex;clear regions around: extent of the acid form of the pH indicator; medium graybackground: basic form of the pH indicator (F-state). Other feed conditions:[K4Fe(CN)6]0 = 20 mM; [NaPAA]0 = 4 mM; [H2SO4]0 = 3.10 mM (a) and2.86 mM (b). Diameter of the disc 18 mm.

From the chemical mechanism viewpoint, the TuIS and FIS re-actions are closely related to each other. The positive feedbackreaction is identical in both systems: it is the acid autoactivatedoxidation of HSO−3 by IO

−

3 . The second substrate, that is respon-sible for the negative feedback, is exchanged from [Fe(CN)6]4− toSC(NH2)2 in the TuIS system. This exchange introduces clear dif-ferences in the detailed kinetics and also in pattern development.As written in (R2), this negative feedback reaction is stoichio-

metrically identical to the analogous reaction in FIS. However, theoxidation product in (R2), Tu2+2 , is actually a double protonatedspecies with dissociation constants pKa = 5.49 and 7.66 [51]. Thereactant, Tu, is not protonated even down to pH = 0 in aque-ous medium [52]. In consequence, the stoichiometry of the protonconsumption in (R2) holds only if Tu2+2 is fully protonated, that is,only when pH< 4.5. The proton consumption in (R2) falls to halfat pH = 6.5 where the dimer is only monoprotonated. This leadsto some kind of ‘‘buffering’’ that hinders the increase of pH. Thestoichiometry is the same as in the FIS reaction when we are in theacid state but less protons are consumed at higher pH when lessprotons are available. This more moderated proton consumptionis in accordance with the higher amount of NaPAA that is neededin the TuIS system to stabilize stationary patterns. To sufficientlyreduce the effective diffusivity of the protons, [NaPAA]0 ≥ 6 mMwas needed in the TuIS system while the lower limit was only[NaPAA]0 ≥ 2 mM in the FIS system [45].Regarding the kinetics of the feedback reactions in the TuIS and

FIS systems, respectively, the oxidation of Tu to Tu2+2 by I2 is at leastfive times faster than the oxidation of [Fe(CN)6]4− by I2 [27,30].This causes that the stability limit of the thermodynamic stateshifts faster to higher [H2SO4]0 in the TuIS than in the FIS system(compare Fig. 4 in this work and Fig. 2 in Ref. [45]) because lessTu than [Fe(CN)6]4− is necessary to operate the negative feedbackprocess at the same rate. Also, the cross point of the OSFR phasediagramwhich can be considered as ameasure of the effectivenessof the feedback process occurs at lower concentration of the secondsubstrate in the TuIS system than in the FIS system. It is at [Tu]0 ≈2.5 mM in the TuIS system and at [K4Fe(CN)6]0 ≈ 9 mM in theFIS system. The difference by almost a factor of four approximatelymatches the ratio of the rate constants of the reactions of Tu andK4Fe(CN)6 with I2.The faster I2 consumption in the TuIS reaction is accompanied

by the consumption of less protons. To get some information on theI2 level in the two systems, we tried to visualize the distribution ofpolyiodide (I−3 ) in the patterns with the use of an I

−

3 color indicator,poly(vinyl alcohol) – PVA – (Aldrich, Mw = 9 000–10 000 Dalton,

80% hydrolyzed). This product is commonly used to detect I−3patterns in other systems [14]. Note thatwhen used as a polyiodidecolor indicator, PVA must contain acetate ester functional groups.Pure PVA gives no color complex with I−3 . The above commercialproduct is strictly speaking not poly(vinyl alcohol) but a copolymerof 80 mol% vinyl alcohol and 20 mol% vinyl acetate [53]. Acetatecontaining PVAs were found to be even more effective iodometricindicators than the conventional starch [54]. The I−3 –PVA complexis red with an absorption maximum at 490 nm but the colordiminishes ondecreasing the acetate content of the copolymer. Thechain length has no effect on color intensity [54]. The polymers inthe 10 000 Dalton range were found to be optimal for their not tooslow diffusion in the gel matrix.In the FIS reaction, a labyrinthine pattern could be visualized

by this I−3 color indicator alone (Fig. 12(a)). The pattern contrastsare greatly improved compared to the use of pH color indicator.Note that the grayscale is ‘‘inverted’’ when using PVA instead ofBromocresol Green as color indicator. The high contrast is due tothe fact that in the absence of pH color indicator the flow stateof the reactor is much clearer than in the presence of suitableamounts of pH indicator. Using a combination of the two types ofindicators, we can compare the relative spatial distributions of H+

and I−3 (Fig. 12(b)). It is observed that the color switches of thetwo indicators do not occur exactly at the same position in thepattern front. The boundaries of the M-state spread wider as seenby pH indicator than by I−3 indicator. We clearly see the light colorof the acid form of the pH indicator around the dark gray thinnercurves corresponding to the I−3 –PVA complex. Both types of edgeare sharp, whichmust result from high concentration gradients, soit is not probable that the displacement between the two fronts isa mere matter of color switch threshold.Despite the advantageous properties of the PVA indicator, we

failed to detect I−3 in the TuIS patterns. This is consistent with Tubeing a more efficient I2 consumer than [Fe(CN)6]4−. However,the I−3 -PVA complex becomes clearly visible for [Tu]0 ≤ 2 mMwhen the stable M-state invades the disc. At these low [Tu]0, theacid front precedes in space the I−3 front in this system, as well. At[Tu]0 = 2.5 mM, this I−3 front is no more detected but the acidfront remains. We can use the negative result of detectability togive an upper estimation for the I−3 concentration in the stationarypatterns. PVA is needed in a concentration ≥ 5 g/L to get maxi-mum color intensity in the case of 0.020 mM I−3 [54]. This meansan absorbance as high as 0.8 in 1 cm path length. The 3.3 g/L PVAconcentration used in our experiments would be sufficient to ob-tain a not much lower absorbance value, 0.7, under the same con-ditions. This should be detectable even at a path length of 0.2 mm(1/50 cm) that is the typical ‘‘thickness’’ of the patterns (the po-sition of the pH indicator color switching position from the back)according to the annular OSFR observations (Fig. 5). From this, wecan conclude to a I−3 concentration� 0.02 mM in the patterns orin theM-state at [Tu]0 > 2mM in the TuIS system. It is noteworthythat PVA has no effect on pattern development in the presence orabsence of NaPAA, neither in the TuIS nor in the FIS reaction. Thisimplies that I−, a secondary activator of the Landolt reaction, playsno role in pattern formation.After revisiting the FIS reaction, the present successful discov-

ery of stationary patterns in another double substrate Landolt re-action is a significant result of our recently developed systematicdesign method [47]. This method should be applicable to a largenumber of oscillatory reactions and one can envision that in a nearfuture the number of different reactions, including reactions of bi-ological significance, will be shown to produce in vitro stationarysymmetry-breaking patterns.

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Acknowledgements

We acknowledge the support from the French Agence Nationalde la Recherche, the French-Hungarian CNRS-MTA collaborationprogram (21420), and the Hungarian funds OTKA (77986, 67701).I.S. thanks the support of the Bolyai Fellowship. We thank JacquesBoissonade and Pierre Borckmans for fruitful discussions.

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