Accelerated Dynamics in Active Media: From Turing Patterns toSparkling WavesJorge Carballido-Landeira* and Alberto P. Muñuzuri

Group of Nonlinear Physics, Universidade de Santiago de Compostela, Santiago de Compostela E-15782, Spain

*S Supporting Information

ABSTRACT: We report the destabilization of stationary Turing patterns and thesubsequent emergence of fast spatiotemporal dynamics due to reactant consumption.The localized hexagonal Turing spots switch from a stationary regime to a dynamicsstate by exhibiting spatial oscillations with two characteristic wavelengths and onerepresentative temporal period. These oscillatory Turing spots are not temporally stableand evolve into traveling spiral tips that, in addition to the unexpected birth of spots,rapidly transform into target patterns and originate multiple collisions and wavebreakups due to their proximity, degenerating into a chaotic scenario.

■ INTRODUCTIONPattern formation in chemical oscillators1,2 has been thoroughlystudied since the first traveling waves reported in theBelousov−Zhabotinsky (BZ) reaction,3,4 which consists of themetal-ion-catalyzed oxidation of an organic compound (usuallymalonic acid) in an acidic solution by bromate ions. The BZreaction has been frequently used as a prototype of anonequilibrium system that exhibits nonlinear spatiotemporalbehavior with the reaction−diffusion (RD) structures similar tothose self-organized patterns that appear in living organisms.5

The possibility to generalize the results obtained fromnonlinear oscillators to understand complex real phenomenahas prompted the scientific community to develop newchemical reactors. Thus, the BZ reaction was proposed to beconfined inside water-in-oil (w/o) microemulsions using AOT(sodium bis(2-ethylhexyl) sulfosuccinate). The relevance of theBZ-AOT system lies in the fact that the main RD mechanismstake place in nanometer-sized compartments. Meanwhile, theresultant spatiotemporal pattern is several orders of magnitudelarger.6

The wide variety of patterns exhibited by the BZ-AOTsystem has been related to the quite significant cross-diffusioncoefficients obtained for the BZ reagents confined in the waternanodroplets7,8 while several nonpolar intermediates (such asbromine, the inhibitor of the reaction) are able to diffusethrough the oil phase even faster than those BZ reagentsresiding in the AOT nanodroplets.9 These differences in thediffusion coefficients in addition to the high concentration ofinhibitor in the oil phase are required for the appearance ofstationary structures associated with the Turing instability.10

These structures occupy the entire reactor with a characteristicwavelength, λT (usually measured through 2D FFT), remainstationary for almost 1 h, and fade away slowly after that time.6

However, for volume droplet fractions of the dispersed phase

(water plus surfactant, Φd) larger than the percolationthreshold (Φd ≫ Φp), AOT microemulsions transform fromspherical droplets to elongated clusters that eventuallyinterconnect the entire system.11,12 In this regime, it is plausibleto assume that the diffusion coefficients of the nonpolarintermediates and the aqueous BZ reagents become similar as aresult of the fast diffusion of the water molecules into theelongated clusters.13 Under these conditions, the BZ-AOTsystem typically exhibits bulk temporal oscillations, which arehomogeneous in space, with a characteristic frequency as aconsequence of the prevalent Hopf instability.14

Previous results across the percolation transition have shownthat modifications in the volume fraction generate considerablechanges in the kinetic observables of the BZ-AOT system, suchas the induction period and the period of oscillations.15−17 Ourstudy is focused in the emergence of new spatiotemporaldynamics that arise under the interaction of the Turing andHopf modes during the onset of percolation (Φd ≈ Φp), i.e.,when spherical droplets start to convert into elongatedclusters.18

■ EXPERIMENTAL SECTIONIn a typical sample preparation, a solution of 1.5 M AOT in octane wasemployed to prepare two AOT microemulsions (MEs) with differentaqueous contents. The first microemulsion (ME1) was composed ofthe organic compound (malonic acid, MA) and H2SO4, and thesecond one (ME2) was loaded with sodium bromate and the catalyst(Ferroin). Both stock microemulsions ME1 and ME2 were preparedwith the same droplet size, obtained through the relation ω ≡ [H2O]/[AOT] = 18, and the same droplet fraction, Φd = Φw(1 + 21.6/ω) =0.75 (where Φw is the volume fraction of the water phase).9 To obtain

Received: January 28, 2015Revised: February 28, 2015Published: March 1, 2015

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the reactive microemulsion, we mixed equal volumes of both stockmicroemulsions (ME1 and ME2) and diluted with octane to acquirethe desired droplet concentration. The control parameters in thisstudy are, on the one hand, the concentration of droplets (in the rangeof 0.45 ≤ Φd ≤ 0.75) and, on the other hand, the excitability of thereaction, calculated as the ratio of concentrations in the aqueous phase[H2SO4]0[NaBrO3]0/[MA]0, where we employed [MA]0 = 0.2−0.25M, [H2SO4]0 = 0.18−0.225 M, [NaBrO3]0 = 0.16−0.18 M, and[ferroin]0 = 4 mM. The spatially extended experiments were carriedout at room temperature (T = 20 °C), the temperature at which theBZ-AOT microemulsion with Φd = 0.45 remains below thepercolation threshold.19,20

To image the spatiotemporal patterns, a drop of the resultantmicroemulsion was sandwiched between two transparent windowsseparated by a Teflon spacer (Zefluor membrane) of 80 μm.21 Tovisualize the formation of patterns, we used a camera (Guppy AVT 64fps) with an achromatic objective (DIN 4× Edmund Optics)connected to a personal computer. Spectrophotometric measurementswere made inside 3.5 mL continuously stirred cuvettes with a Vernierspectrometer (Ocean Optics) connected to a computer. Spectropho-tometric data were visualized and analyzed via Logger Pro software.Conductivity experiments were carried out at room temperature byusing an inLab 730 measuring cell and a Mettler Toledo SevenEasyconductivy meter. Dynamic light scattering (DLS) measurements wereperformed at a scattering angle of 90° with a device equipped withvertically polarized incident light (λ = 488 nm, 2 W argon ion laser,Coherent Inc., Santa Clara, CA) coupled to a digital correlator (ALV5000E, ALV GmbH, Germany). DLS experiments were conductedwith microemulsions containing all BZ reagents except ferroin (toavoid solution coloration) at a constant w = 18.3 and varying thevolume droplet fraction. The samples were filtered with 0.2 μm PTFEmembranes (Millex, Millipore Inc., USA), and the CONTIN methodwas used to obtain distributions of decay rates.22,23 The hydrodynamicradius (rh) was calculated from the measured diffusion coefficient usingthe Stokes−Einstein equation (D = kBT/6πηrh), where D is thetranslational diffusion coefficient at zero concentration, kB is theBoltzmann constant, T is the absolute temperature, and η is thedynamic viscosity of octane. DLS experiments were all carried out atthe same temperature as for spatially extended systems (T = 20 °C) toavoid temperature-induced percolation transitions.19,20

■ RESULTS AND DISCUSSIONAt the percolation edge, the dynamics of the system differsfrom those explained before. Initially, Turing structures

displayed as hexagonal arrangements of spots remain stationaryonly for a short period of time (around 20−30 min). Thistransition state gives way to a more complex dynamics (calledoscillatory Turing patterns, OTP): each individual spot (Figure1a) starts to move toward one of its closest neighbors until theyreach a gap equal to half of the initial wavelength, λT/2 ≈ 0.13mm (Figure 1b). At this particular moment, both spots stopand turn back, moving away each other as far as 3λT/2, theinstant at which they repeat the same displacement cycle. (Seemovie 001 in the Supporting Information.) By performing aspace-time plot of the described motion (Figure 1c), weobserved that the system demonstrates a frequency ofoscillation and also two characteristic wavelengths coexistingsimultaneously. OTP also demonstrate that spots diameterremain unaltered the whole cycle (Figure 1c), in contrast withother reported oscillatory structures (breathing modes andoscillons) where the spots or stripes periodically fluctuate theirsize.10,21,24

The presence of multiple wavelengths has been proventhrough a 2D fast Fourier transform (FFT) at a given time.Thus, it exhibits two well-defined wavelengths (Figure 1d),simultaneously demonstrating that one spot may be separatedλT/2 from some neighbors while 3λT/2 from others (result alsoevidenced through the space−time plot in Figure 1c), recallingthat the effective distance among three spatially alignedneighbors in the OTP stage is constant and equal to 2λT (asin the early Turing phase). However, a more representativeFFT reveals a bandwidth spectrum (Figure 1e) indicating that asingle spot can be concurrently separated via multiple gapsfrom the neighbors but always inside the interval of accessiblevalues (λT/2 < λ < 3λT/2). The oscillatory demeanor of thesystem has been confirmed by spectrophotometric measure-ments of the same BZ-AOT microemulsion under continuouslystirred conditions (Figure 4a) by monitoring the opticalabsorption maxima of ferroin (510 nm).Oscillatory Turing spots can be easily understood in 1D

systems, where an array of spots is periodically oscillating inspace and time. A spatial alteration in the activator domain inrelation to the inhibitor region makes each individual spotasymmetric, inducing their spatial propagation to itsneighbors.25,26 However, in the usual 2D experiments, Turing

Figure 1. Oscillatory Turing spots. (a) Snapshot of initial Turing patterns (b) 250 s after (a). Frame size is 1.7 mm × 0.7 mm. (c) Space−time plotof oscillatory Turing patterns for the cross section shown in panels a and b. Frame size = 1.2 mm × 1000 s. Time is on the vertical axis evolvingupward as indicated by the arrow. (d, e) Two-dimensional FFT exhibiting the two characteristic wavelengths (for snapshot a) and a bandwidthspectrum between them (calculated for a picture 4 times greater than snapshot a). Initial reagent concentrations: [MA]0 = 0.21 M, [H2SO4]0 = 0.2M, [NaBrO3]0 = 0.17 M, and [ferroin]0 = 4 mM, with Φd = 0.59.

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structures appear, displaying a hexagonal configuration, whichmeans that each spot is typically surrounded by six neighborsand, consequently, has three different directions in which topropagate. Experiments showed that spots typically randomlychose one route to oscillate, presenting periodic cycles in thisdirection without experiencing the other alternatives (as can beobserved by the space−time plot in Figure 1). However, moreintricate excursions are also possible, and we even observedspots that were able to follow circular trajectories with time(Figure 2a−d and movie 002 in the Supporting Information).

In this case, after the spot reached the minimum gap λT/2 withone neighbor, it did not return but moved closer to another“fellow” located in a different angular phase; the process isrepeated until completing an entire rotation (anticlockwise inFigure 2a−d). The spot repeated the same gyrationconsecutively as can be observed by tracking its centroid(Figure 2e). The time required to complete the full cycle is 450± 50 s (Figure 2f), and this result is in agreement with the casewhere the spot oscillates only along one direction. Thecomplexity of this phenomenon lies in the fact that thepropagation direction of one spot is delimited by its neighbors,making the BZ-AOT system composed of multiple localizedindividuals, which are able to decide their sense of movementwithout any external control, in contrast to spots moving in aglobally synchronized manner.We remark that the destabilization of Turing patterns has

been previously examined by various groups.25,27−37 De Witand coworkers used reaction−diffusion (RD) models todemonstrate the OTP spatiotemporal signature. This mixedstate arises in the vicinity of a codimensional-two Turing−Hopfbifurcation point, specifically under subharmonic Turinginstabilities, where the dynamics exhibits two wavenumbersand one frequency of oscillation.28,29 They also reported theemergence of other mixed modes, experimentally demonstratedby De Kepper et al. using oscillatory redox reactions operatedin open spatial reactors. Among the variety of spatiotemporalpatterns is the superimposition of traveling waves overhexagonal spots. In this case, the wave motion does not alterthe Turing mode as a result of weak interactions between bothinstabilities.29,38,39 Analogous dynamics were obtained byEpstein’s group in their studies of the symmetry-breakinginstability interaction.33−37 We highlight their evidence ofsuperlattice structures switching cyclically between differentspatial−periodic patterns. The double periodicity (both in timeand space) of the superlattice and the shift in wavelength eachhalf period resemble the OTP dynamics with the exception thattheir spatial structures change their shape with time whereasOTP does not.33 Furthermore, Liu et al. accounted for thepossibility that more complex spatiotemporal oscillatingsolutions could appear by considering the nonlinearity termsin the activator−inhibitor reactions under a pure Turingunstable domain.30

Bearing in mind that the BZ-AOT microemulsion is a closedreactor (i.e., there is not an income of fresh reagents), the

Figure 2. Turing spot describing a circular trajectory. (a) Centroid(green dot) of the spot case of study (circled in green). (b−d)Centroid spot localization at consecutive times separated by 125 s inthe sequence of colors (green, red, blue, and purple). Images displaythe actual and former spot’s centroid position in movie 002 in theSupporting Information. Frame size = 0.7 mm × 0.55 mm. (e) Journeyof the spot’s centroid and (f) temporal excursion calculated throughthe Otsu algorithm.40,41 Initial reagent concentrations: [MA]0 = 0.21M, [H2SO4]0 = 0.2 M, [NaBrO3]0 = 0.17 M, and [ferroin]0 = 4 mM,with Φd = 0.59.

Figure 3. Sparkling waves stage. (a−d) Consecutive snapshots taken every 5 s for two spots (highlighted by red arrows), increasing their size in awavelike shape and posteriorly colliding with each other. Frame size = 0.6 × 0.5 mm2. (e, f) Sequential images of sparkling waves’ spontaneousgeneration taken every 2 s. (Green circles help to visualize the location of interest). Frame size = 2.5 × 1.75 mm2. (i) Space−time composition (timeincreasing upward) highlighting the generation of unprompted waves (encircled in green) and the collision of wavelike moving spots (red arrow) asa summary of the sparkling wave stage. Initial reagent concentrations: [MA]0 = 0.21 M, [H2SO4]0 = 0.2 M, [NaBrO3]0 = 0.17 M, and [ferroin]0 = 4mM, with Φd = 0.59.

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chemical concentrations changed during the course of thereaction. The reactants’ consumption is the main reason for thetransient state of oscillatory Turing spots which, after remaining20−30 min (the larger the volume fraction of the droplets, theshorter the time), evolve to another dynamics systemhenceforth called the sparkling waves stage (SW) until the“death” of the reaction. The spatiotemporal behavior of thisstage can be summarized as follows: each individual spotexpands and converts into an extended propagating wave byincreasing its size, resembling pairs of spiral tips (Figure 3a−cand movie 003 in the Supporting Information), a phenomenonalso manifested in an active network of biological single-celledorganisms.42 In an attempt to produce the spatial oscillationpreviously described, the recently generated tips overcome theminimum stable gap between them (λT/2), colliding with eachother (Figure 3d). The generation of multiple spirals, separatedinitially by a small distance (λT), produces via collisionsnumerous breakups close to the rotational tip of the spirals.This phenomenon, also known in the literature as “corebreakup”,43,44 involves dynamics faster than those displayedpreviously in the system (Figure 4c) and constitutes one of thepossible routes to a spatiotemporal chaotic-like scenario from

outwardly rotating spiral patterns.45 Earlier, Vanag et al.observed this chaotic transition by working under differentinitial conditions where spiral waves continuously break intosegments.46 Thus, the BZ-AOT system represents experimentalevidence of multiple and natural breakups, a phenomenonenhanced during the last decades because it is considered to bethe mechanism underlying heart fibrillation and is systemati-cally analyzed by numerical stability analysis.47,48 Concurrentwith core breakup, unprompted spots arise interstitially, fillingthe empty gaps by diffusing quickly as circular target patterns(Figure 3e−h). We called them sparkling waves due to thebrightness of their spontaneous appearance (Figure 3h and themovie 004 in the Supporting Information) and the rapidity oftheir spatial expansion (Figure 3i). SWs appear in randomplaces and require, according to experimental observances,blank spaces larger than λT. The impossibility to characterizespatial features between SWs makes us think that they originatefrom the underlying oscillatory behavior associated with theHopf mode (Figure 4a). The unpredictable location of theseinterstitial pulsatile patterns contrasts with spots rhythmicallyoscillating 2π/3 out of phase with their closest neighbor inarrangements of intertwined hexagonal lattices. These patterns,

Figure 4. (a) Temporal evolution of the BZ-AOT system at 510 nm (wavelength in the emission spectrum of ferroin) in a continuously stirred tankreactor at room temperature for a microemulsion loaded with [MA]0 = 0.225 M, [H2SO4]0 = 0.2 M, [NaBrO3]0 = 0.16 M, and [ferroin]0 = 4 mM,with Φd = 0.58. (b) Conductivity measurements of the active microemulsion as a function of the volume droplet fraction. Dashed lines delimit thevolume fraction regime 0.575 < Φd < 0.64, where the new dynamical behaviors emerge (here labeled as region II). Regions I and III correspond tothe regimes of Turing patterns and bulk oscillations, respectively. (c) Velocity in the three different regimes, Turing patterns (TP), oscillatory Turingpatterns (OTP), and sparkling waves (SW), here separated by dashed lines. Measurements are obtained by averaging the speed of the spots in eachstage for the microemulsion described in a. (d) Dynamic laser scattering measurement of an active microemulsion in the absence of catalyst at Φd =0.24 (solid line) and at Φd = 0.58 (dashed line).

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so-called “twinkling eyes”, were obtained by modeling RDsystems with two coupled layers and were explained in terms ofthe interaction between subharmonic Turing and Hopfmodes.34,35

This particular dynamical evolution has been observed onlyclose to the percolation threshold (Φd ≈ Φp), where DLSdemonstrates changes in the distribution of nanodroplets.Thus, experiments carried out at Φd = 0.24, (Φd ≪ Φp whereonly Turing patterns appear) exhibit a main peak centered ataround 3 to 4 nm (solid line in Figure 4d). However, as wereach the percolation (0.575 < Φd < 0.64) the distributionprofile differs, displaying a bimodal distribution of nanodroplets(dashed line in Figure 4d). These two subsystems representsingle droplets (a small subpopulation with the same size in thecase of Φd ≪ Φp, 3 to 4 nm) coexisting together with a largenumber of nanodroplet clusters, one order of magnitude greater(peak centered around 20−25 nm). Bearing in mind thatautocatalytic rate constants depend on droplet size,49 clusterscan be considered a chemical subpopulation slightly differentthan droplets. Moreover, conductivity measurements aroundthe percolation transition show values 10 times greater than inthe nanodroplets regime (Figure 4b), although still lower incomparison to those obtained for volume fractions at Φd ≫Φp.

18 The increase in conductivity suggests, according toKataoka et al.,50 an increment in the interchange of the water-soluble components between slowly diffusing clusters.The presence of nanodroplet clusters in addition to the fast

exchange of mass is not favorable to the ratio of effectivediffusion coefficients required for the Turing instability9,10 andtherefore are assumed to be the main destabilizing factor ofTuring patterns.

■ CONCLUSIONSIn this article, we present a system that is able to display theinteraction between spatial and time-breaking symmetries byjust adjusting the dilution of the nanocompartments where theinitial reagents are encapsulated. Moreover, the rich variety ofspatiotemporal behaviors expected in a bifurcation point isconsiderably increased by considering the consumption of thechemicals in a batch reactor. Thus, by increasing the volumedroplet fraction of the BZ-AOT microemulsion, the systemtransits from stationary Turing structures to oscillations that arehomogeneous in space, with both states corresponding tosamples below and above the percolation transition. However,our experiments, performed at the onset of percolation,demonstrated more striking dynamics, such as the emergenceof oscillatory Turing spots and sparkling waves.The spatiotemporal oscillations of hexagonal structures can

be understood as a consequence of the mixed state obtained inthe bifurcation onset, such as the patterns exhibiting thewavelength corresponding to the Turing instability andoscillating with a frequency according to the Hopf mode.This behavior contrasts with that obtained at the end of theexperiments, when individual spots transform into tips of spiralspropagating along the system. The proximity between the spot-converted curls produces continuous collisions of the wavesand spontaneous sparkling patterns in the remaining emptygaps. This dynamical stage is much faster than those observedin the previous steps and gives way to spatiotemporal chaoticpatterns through the core-breakup mechanism.The possibility to analyze the interaction between two

different instabilities by just tuning the physical properties ofthe microemulsion confinement represents a new tool in the

achievement of complex spatiotemporal dynamics that mayemerge as a consequence of mixed modes, enhancing theemployment of the BZ-AOT system for such a purpose.

■ ASSOCIATED CONTENT*S Supporting InformationMovies of the experimental results explained in the article. (File001) Movie displaying oscillatory Turing spots (Figure 1). (File002) Spot describing a circular trajectory (Figure 2). (File 003)Spot increasing in size and transforming into a tip of a wave(Figure 3a−d). (File 004) Appearance of spontaneous waves(Figure 3e−h). This material is available free of charge via theInternet at http://pubs.acs.org.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: jorge.carballido@gmail.com.Present Address(J.C.-L.) Nonlinear Physical Chemistry Unit, Service de ChimiePhysique et Biologie Theorique, Faculte ́ des Sciences,Universite ́ Libre de Bruxelles (ULB), CP231, 1050 Brussels,Belgium.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe acknowledge funding from Xunta de Galicia throughresearch project CN2012/315. J.C.-L also thanks FRS-FNRSfor financial support.

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Langmuir Article

DOI: 10.1021/acs.langmuir.5b00357Langmuir 2015, 31, 3021−3026

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