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

https://doi.org/10.1016/j.physd.2009.07.005Get rights and content

Abstract

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-temporal oscillatory 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 (sodium polyacrylate). This is the third aqueous solution system to produce stationary patterns and the second to do this through a Turing bifurcation. The stationary pattern forming capacities of the reaction are explored through a systematic 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–sulfite mixed Landolt system.

Introduction

The study of non-equilibrium chemical patterns is a challenging area of nonlinear dynamics [1], [2], [3]. In solutions, quite diverse mechanisms can be at the origin of chemical patterns. They can be driven by buoyancy instabilities [4], by surface tension [5], by nonlinear colloidal particle growth mechanism in periodic precipitation reactions [6], [7] (e.g. Liesegang patterns [8]), or by single phase kinetic nonlinearities and molecular diffusion in the reaction–diffusion (RD) patterns. This report is focused on the latter type of patterning mechanism. Dr. Stefan Müller was among the early contributors to the experimental studies of RD [9] and other types of patterning mechanisms [10], [11] in chemical and biochemical [12] systems.

Reaction–diffusion systems can produce a wide range of nonlinear phenomena including smooth and turbulent wave patterns, self-replicating spots, regular and irregular patterns as a 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 inhibition processes can spontaneously lead to the formation of stationary symmetry-breaking patterns. In two variable activator–inhibitor model systems, this implies that the inhibitor diffuses faster than the activator. Real nonlinear chemical systems include a much larger number of species and the requirement for the development of stationary patterns can be more involved but a species controlling or interfering with the activatory loop must have a reduced mobility.

The first experimental evidences of sustained stationary chemical patterns were provided in the early 90s [14], [15]. These observations boosted the field. Interestingly these stationary patterns developed through two different routes: one, the first, came through a Turing bifurcation [13], [16], the other through front pairing interactions [17], [18]. These findings were made possible by the development of a large variety of chemical oscillating reactions in the 1980s, and the subsequent invention of open spatial reactors. Among the large variety of systems able to produce macroscopic patterns, the isothermal solution chemical systems are often thought to be the best suited to serve as simplified experimental models for a number of morphogenic and dynamic self-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], [20], [21], the diversification of chemical systems came to a stop. Two reasons for this: (i) the patterning capacities of the CIMA and FIS reactions are rich enough to serve as test grounds for many theoretical aspects of pattern developments; (ii) the discovery of these two first chemical examples were the fruit of targeted research but not of a fully comprehensive method. The incomplete theoretical understanding of the actual complexity of real open spatial reactors also hindered this diversification. However, different types of sustained stationary patterns were found in other systems such as: on the surface of metal plates during gas phase catalytic reactions [22], on the electrodes during electrochemical reactions [23], in gas discharge systems [24], and more recently in a microemulsion 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 sulfite ions by iodate ions. This reaction is widely used as a classroom demonstration of autocatalytic (or clock) reactions as the color of iodine appears abruptly after a well-defined induction time. This is an acid and iodide activated reaction. Yet, the protons have been shown to be the main autoactivatory species during the oxidation of sulfite [27]. A large variety of oscillatory reactions were designed by adding a second substrate with antagonist action on the proton production when reacting with iodate (e.g., [Fe(CN)6]4−, S2O32−, SC(NH2)2) [28], [29], [30], and by operating these mixed substrate Landolt 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 driving forces of the kinetic instability [31]. The mechanism, the kinetics, and the temporal dynamics of these systems have been studied in detail during the last decade [27], [32], [33], [34], [35], [36], [37], [38].

Refined studies of sustained RD patterns require the use of open spatial reactors. Open spatial reactors [39] consist in a piece of porous material (e.g. a hydrogel) permanently fed by fresh reagents. The porous medium quenches all hydrodynamic motions in the reacting mixture and enables the development of the pure RD patterns. The feeding is provided by diffusive exchanges with the contents of a CSTR. The chemical state of the latter is controlled by the input flow concentration ([]0), the residence time (τ), and temperature of the thermostat (T).

In the case of the CIMA reaction, the long-range inhibition required for the Turing patterns was realized by adding starch or PVA to the reacting mixture. These macromolecules, together with iodine, reversibly bind the iodide ions, which are the major activatory species of the reaction. The chemical and physical mechanisms at play in the patterning processes of this system were clearly understood and controlled very soon after the breakthrough result [40], [41], [42], [43]. Shortly after the discovery of Turing patterns in the CIMA reaction, a rich variety of spatio-temporal patterns arising from planar front multiplicity and front morphological instabilities were observed in the FIS reaction [19], [20], [21]. In particular, stationary labyrinthine patterns, self-replicating spots, and bouncing waves were observed when this reaction was operated in an open one-side-fed reactor (OSFR). However, for many years, the actual origin of the short-range character of the activatory process, necessary for the onset of stationary patterns, was uncertain and made the original observations difficult to reproduce. Recently, we revisited this system and clarified this aspect [44], [45]. This revised study was used to develop an effective systematic design method to discover stationary RD patterns in an OSFR operated with other targeted reactions. The major ingredients of the method are: (i) the development of spatial bistability [46] and spatio-temporal oscillations in autoactivated reactions operated in an OSFR, (ii) the addition of a species enabling to control the negative feedback loop independently from the positive one, and (iii) the introduction of a low-mobility reversible complexing agent to generate simultaneously appropriate time and space scale separations 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 resulting from a Turing bifurcation [47]. Here, we provide a detailed description of the spatio-temporal dynamics observed in this new system. The similarities and differences between the TuIS and FIS reactions are discussed.

Section snippets

Materials and methods

Two geometries of OSFR were used in the present experiments: The one, hereafter named as the disc OSFR, consists of a transparent thin disc (22 mm diameter and 0.75 mm thick) made of 4 wt% agarose gel (Fluka, BioChemika 05070 or 05077). One face of the disc is in direct contact with the contents of a CSTR through a circular hole (18 or 20 mm diameter) in a mask which holds the gel tightly pressed against a flat observation window. Beyond the contact surface, the disc extends (≈2 mm) under the

CSTR dynamics

The TuIS system is a mixed Landolt type pH-oscillator [32]. The kinetic mechanisms of the composed reactions are detailed in the literature [27], [28], [29], [30], [32], [33], [34], [35], [36], [37], [38]. It can be grossly accounted by two overall processes, where the hydrogen ion driven autocatalytic oxidation of hydrogen sulfite ((R1) positive feedback) is coupled with a hydrogen ion consuming reaction, in the present case with the oxidation of thiourea ((R2) negative feedback): IO3+3HSO33

Discussion

Let us recall briefly the spatio-temporal dynamics of the FIS reaction and compare it with the observations in the TuIS system. Both systems produce large amplitude spatio-temporal pH oscillations/waves in the absence of NaPAA (Fig. 11(a)). In the FIS reaction, one can clearly distinguish two types of sharp fronts: 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

Acknowledgements

We acknowledge the support from the French Agence National de la Recherche, the French-Hungarian CNRS-MTA collaboration program (21420), and the Hungarian funds OTKA (77986, 67701). I.S. thanks the support of the Bolyai Fellowship. We thank Jacques Boissonade and Pierre Borckmans for fruitful discussions.

References (54)

  • D.S. Chernavskii et al.

    Physica D

    (1991)
  • J.J. Perraud et al.

    Physica A

    (1992)
  • J.D. Murray

    Mathematical Biology

    (2004)
  • I.R. Epstein et al.

    An Introduction to Nonlinear Chemical Dynamics

    (1998)
  • J. D’Hernoncourt et al.

    Chaos

    (2007)
  • L. Rongy et al.

    Phys. Rev. E.

    (2008)
  • P. Ortoleva

    Geochemical Self-Organization

    (1994)
  • S.C. Müller et al.

    J. Phys. Chem. A

    (2003)
  • R.E. Liesegang

    Geologische Diffusionen

    (1913)
  • S.C. Müller et al.

    Science

    (1985)
  • K. Matthiessen et al.

    Phys. Rev E

    (1995)
  • S.C. Müller et al.

    Development

    (1990)
  • A. Turing

    Phil. Trans. R. Soc.

    (1952)
  • V. Castets et al.

    Phys. Rev. Lett.

    (1990)
  • Q. Ouyang et al.

    Nature

    (1991)
  • P. Borckmans et al.

    J. Stat. Phys.

    (1987)
  • A. Hagberg et al.

    Chaos

    (1994)
  • S. Ponce-Dawson et al.

    Phys. Lett. A

    (2000)
  • K.J. Lee et al.

    Science

    (1993)
  • K.J. Lee et al.

    Phys. Rev. E

    (1995)
  • G. Li et al.

    J. Chem. Phys.

    (1996)
  • R.M. Eiswirth et al.

    Appl. Phys. A

    (1990)
  • Y. Li et al.

    Science

    (2001)
  • Y. Astrov et al.

    Phys. Lett. A

    (2001)
  • V. Vanag et al.

    Science

    (2001)
  • H. Landolt

    Ber. Dtsch. Chem. Ges.

    (1886)
  • Cited by (29)

    • Peristaltic waves in a responsive gel sustained by a halogen-free non-oscillatory chemical reaction

      2015, Polymer
      Citation Excerpt :

      The appearance of a maximum thickness is a new precondition in the experimental assembly of synergistic oscillators. Yet, this tendency had been experienced when reaction-diffusion patterns had been developed in idle agarose discs by different pH-oscillatory reactions: The thickness of the gel disc had to be set thinner (0.5 mm vs. 0.75 mm) in certain cases when the proton producing nonlinear subset was the HPS reaction compared to the oscillatory reactions where the IS subset was used [42,47]. A cylindrical gel with a considerably lower thickness was feasible by gluing a five times heavier glass weight to the bottom of a gel cylinder, 0.17 g compared to the 0.034 g used in the experiments of Figs. 7–9.

    • The effect of chloride on spatiotemporal dynamics in the electro-oxidation of sulfide on platinum

      2013, Electrochimica Acta
      Citation Excerpt :

      Moreover, the addition of strongly adsorbing species such as chloride adds a feedback loop and causes a shift in the position of the negative differential resistance region, thus changing the parameter region in which the positive feedback loop occurs. Experimental studies of spatiotemporal patterns on a single chemical system with multi-feedback loops [37] are still rare [38,39], and results in this direction pose interesting fundamental questions for future investigations. Besides deepening our understanding of the rich spatiotemporal dynamics of sulfur deposition and dissolution on platinum, the findings presented here reveal the impact exerted by surface adsorbed species on surface activity waves.

    • Tuning density fingering by changing stoichiometry in the chlorite-tetrathionate reaction

      2013, Chemical Physics Letters
      Citation Excerpt :

      Spatial bistability, excitable waves, or stationary patterns are observed experimentally depending on the initial concentrations of the reactants. In the ferrocyanide–iodate–sulfite and the thiourea–iodate–sulfite systems, the cross-shaped phase diagrams that map the dynamic behavior have been determined by De Kepper and co-workers [5,6]. Simple autocatalytic systems, like the iodate–arsenous acid reaction, may yield different products when the chemical composition is changed under batch conditions.

    • Simulation of reaction-diffusion processes in three dimensions using CUDA

      2011, Chemometrics and Intelligent Laboratory Systems
      Citation Excerpt :

      These kernels are not optimized because their job is very small compared to the Laplacian computation, they contribute approximately 2% to the computational time. Our second simulation example is a very extensively studied problem, both theoretically [42–44] and experimentally [4,5,45], in reaction–diffusion systems. Turing pattern formation occurs in case of sustained nonequilibrium conditions, where spatial patterns arise from an instability in a uniform medium.

    • Preface

      2010, Physica D: Nonlinear Phenomena
    View all citing articles on Scopus
    View full text