E. Dulos

Turing-type chemical patterns in the chlorite-iodide-malonic acid reaction

Abstract We describe experimental observations of symmetry breaking stationary patterns. These patterns are interpreted as the first unambiguous evidence of Turing-type structures in a single-phase isothermal chemical reaction system. Experiments are conducted with the versatile chlorite-iodide-malonic acid reaction in open spatial reactors filled with hydrogel. A phase diagram gathering the domain of existence of symmetry breaking and no-symmetry breaking standing patterns is discussed.

EXPERIMENTAL STUDY OF STATIONARY TURING PATTERNS AND THEIR INTERACTION WITH TRAVELING WAVES IN A CHEMICAL SYSTEM

We give a brief review of recent observations of Turing patterns in an isothermal single-phase chemical system. The basic principles of open spatial reactors used in the experiments are described. Different types of one-, two- and three-dimensional symmetry breaking reaction-diffusion patterns are discussed in relation with the geometric dimensions of the reactors and with the localization of the patterned regions in the concentration ramps. We also present a set of new spatiotemporal structures resulting from the interaction of the Turing (spatial) and Hopf (temporal) instabilities. Among other things, these interactions lead to antisymmetric wave sources in quasi-one-dimensional systems and to spatiotemporal intermittency in quasi-two-dimensional systems. We also report on a “cell splitting” growth mechanism of stationary patterns after a supercritical change in parameter value beyond the onset of the Turing instability.

Experimental study of synchronization phenomena under periodic light irradiation of a nonlinear chemical system.

A photosensitive chemical oscillating reaction, i.e., the Briggs-Rauscher (B.R.) reaction, exhibiting a wealth of nonlinear behavior, when performed in a continuous-flow stirred-tank reactor, and subjected to periodic light irradiation, is studied as an experimental example of entrainment phenomena observable in biological systems. The adaptation patterns under periodic light irradiation are elucidated by means of the response of the system to continuous and single-pulse light irradiation. It is shown that self-oscillating states, excitable steady states and bistable systems can exhibit the same types of synchronization patterns when submitted to periodic external forces with appropriate amplitude and time scale conditions.

DIFFUSIVE INSTABILITIES AND CHEMICAL REACTIONS

Diffusive instabilities provide the engine for an ever increasing number of dissipative structures. In this class autocatalytic chemical systems are prone to generate temporal and spatial self-organization phenomena. The development of open spatial reactors and the subsequent discovery in 1989 of the stationary reaction–diffusion patterns predicted by Turing [1952] have triggered a large amount of research. This review aims at a comparison between theoretical predictions and experimental results obtained with various type of reactors in use. The differences arising from the use of reactions exhibiting either bistability of homogeneous steady states or a single one in a CSTR are emphasized.

Pattern selection and localized structures in reaction-diffusion systems

We present some theoretical concepts that have been used in the study of chemical disspative structures together with a brief description of recent experimental work on Turing patterns and their interaction with travelling waves.

Turing Patterns: From Myth to Reality

Besides classical equilibrium structures, such as solid state crystals, nature exhibits a number of dissipative structures in systems kept far from equilibrium by permanent driving forces. These structures result from a symmetry breaking instability of the basic thermodynamic state induced by nonlinearities and competition between antagonistic processes [1, 2]. Their archetype is the family of convective instabilities in hydrodynamics [3–4]. Other well-known examples are the homogeneous isothermal chemical systems fed with a permanent flow of fresh reactants which can exhibit oscillating phenomena, provided they encompass appropriate antagonistic catalytic and inhibitory steps [1, 5]. It seems to follow from common sense that introducing molecular diffusion — a transport process which tends to damp any inhomogeinety — should not promote the spontaneous formation of a spatial pattern. However, this naive statement is actually false because, when several species have different diffusion rates, the responses of the antagonistic processes to a local perturbation do not spread at the same rate. As a result, the subtle balance between these processes can break in a nonhomogeneous way. In these conditions, a spatial instability leading eventually to a stationary spatial structure can take place. Although this idea can be tracked down to Rashevsky [6], its modern formulation, published in 1952, is more commonly attributed to Turing [7]. Turing structures were further theorized from the late sixties, in particular by the Brussels group [1, 8, 9], with a progressive introduction of bifurcation theory. Since the basic ingredients — permanent feed, reactions with antagonistic feedbacks, large differences in diffusion coefficients — are common in biological media, the concept has become very popular among a small community of biologists and biomathematicians as a promoter of the early stages of morphogenesis and has initiated a large amount of work in this direction [10–12].

From quasi-2D to 3D Turing patterns in ramped systems

Abstract We elaborate on the transition from quasi-two-dimensional to three-dimensional Turing patterns in a chemical reaction-diffusion system confined in gradients of chemicals between two feed boundaries. This transition is observed in open spatial reactors specially designed to make possible the unfolding of a pattern sequence in one direction of the plane of observation. In this direction, the confinement of the structure is progressively relaxed. Complementary observations from two reactor geometries allow the dimensionality of the structure to be elucidated: quasi-two-dimensional and three-dimensional patterns, respectively, correspond to patterns developing in monolayers and in bilayers. Beyond the now classical hexagonal and stripe patterns, various new stable planforms are shown to result from the coupling of these two classical pattern modes which develop in two adjacent layers, with well-defined phase relations between the two pattern modes.

Stationary Turing patterns versus time-dependent structures in the chlorite-iodide-malonic acid reaction

Abstract The standing concentration patterns recently discovered in open gel-filled reactors, with the chlorite-iodide-malonic acid (CIMA) oscillating reaction in the presence of starch, were ascribed to a Turing-type reaction-diffusion symmetry breaking instability. Here we extend the investigations to other regions of parameters, with a particular emphasis to the role played by the chemical nature of the gel matrix and by the starch concentration on the onset of stationary patterns. Stationary Turing patterns are shown to develop in gel-free systems. Transitions between stationary Turing structures and wave patterns are presented. The first evidence of an anti-symmetric “homogeneous” wave source is presented.

Standard and nonstandard Turing patterns and waves in the CIMA reaction

We describe experimental evidence of stable triangular and hexagon-band mixed mode nonstandard patterns, in a three-dimensional chemical reaction-diffusion system with steep gradients of chemical constraints. These gradients confine the structures in a more or less thick stratum of the system. At onset, patterns develop in monolayers which approximate two-dimensional systems; but beyond onset, three-dimensional aspects have to be considered. We show that the nonstandard pattern symmetries result from the coupling of standard hexagonal and striped pattern modes which develop at adjacent positions, due to the differences in parameter values along the direction of the gradients. We evidence a Turing-Hopf codimension-2 point and show that some mixed mode chaotic dynamics, reminiscent of spatio-temporal intermittency combining the Turing and the Hopf modes, are also a consequence of the three-dimensional aspect of the structure. The relations between these observations and the theoretical studies performed in genuine two-dimensional systems are still open to discussion.

Spatial bistability and waves in a reaction with acid autocatalysis

The phenomenon of spatial bistability has recently been proposed for a comprehensive understanding of a number of chemical patterns observed in open spatial reactors consisting of thin films of gel diffusively fed from one side. We study experimentally and numerically this phenomenon in the tetrathionate-chlorite reaction characterized by an acid superautocatalysis. We focus on the similarities and differences with previous studies on the chlorine dioxide-iodide reaction. In addition, we show that this reaction, which is only bistable in a continuous stirred tank reactor, can exhibit oscillatory and traveling waves when diffusion comes into play. Our computations suggest that the nonstationary behaviour originates from differential diffusive transport.