Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with square spatial periodic forcing

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns to two-dimensional "square" spatial forcing, implemented as orthogonal sets of bright bands projected onto the reaction medium. Various resonant structures emerge in a broad range of forcing wavelengths and amplitudes, including square lattices and superlattices, one-dimensional stripe patterns and oblique rectangular patterns. Numerical simulations using a model that incorporates additive two-dimensional spatially periodic forcing reproduce well the experimental observations.     

Complex wave patterns in an effective reaction-diffusion model for chemical reactions in microemulsions

An effective medium theory is employed to derive a simple qualitative model of a pattern forming chemical reaction in a microemulsion. This spatially heterogeneous system is composed of water nanodroplets randomly distributed in oil. While some steps of the reaction are performed only inside the droplets, the transport through the extended medium occurs by diffusion of intermediate chemical reactants as well as by collisions of the droplets. We start to model the system with heterogeneous reaction-diffusion equations and then derive an equivalent effective spatially homogeneous reaction-diffusion model by using earlier results on homogenization in heterogeneous reaction-diffusion systems [S.Alonso, M.Ba?r, and R.Kapral, J. Chem. Phys. 134, 214102 (2009)]. We study the linear stability of the spatially homogeneous state in the resulting effective model and obtain a phase diagram of pattern formation, that is qualitatively similar to earlier experimental results for the Belousov-Zhabotinsky reaction in an aerosol OT (AOT)-water-in-oil microemulsion [V.K.Vanag and I.R.Epstein, Phys. Rev. Lett. 87, 228301 (2001)]. Moreover, we reproduce many patterns that have been observed in experiments with the Belousov-Zhabotinsky reaction in an AOT oil-in-water microemulsion by direct numerical simulations.     

Particle dynamics simulations of Turing patterns

The direct simulation Monte Carlo method is used to reproduce Turing patterns at the microscopic level in reaction-diffusion systems. In order to satisfy the basic condition for the development of such a spatial structure, we propose a model involving a solvent, which allows for disparate diffusivities of individual reactive species. One-dimensional structures are simulated in systems of various lengths. Simulation results agree with the macroscopic predictions obtained by integration of the reaction-diffusion equations. Additional effects due to internal fluctuations are observed, such as temporal transitions between structures of different wavelengths in a confined system. For a structure developing behind a propagating wave front, the fluctuations suppress the induction period and accelerate the formation of the Turing pattern. These results support the ability of reaction-diffusion models to robustly reproduce axial segmentation including the formation of early vertebrae or somites in noisy biological environments.     

Migration-driven instability in the chlorite-tetrathionate reaction

We have studied the lateral stability of planar reaction-diffusion fronts in an autocatalytic reaction between aqueous ions in an externally imposed electric field. In our experiments, migration drives the pattern formation leading to cellular structures where the sufficiently greater migrational flux of the reactant with respect to that of the autocatalyst is the driving force. The difference in electric field strength between the two sides of the thin reaction front results from the significant increase in conductivity during the reaction. The results of the theoretical analysis based on the empirical rate-law model of the reaction reproduce the behavior observed experimentally.     

Dancing waves in reaction-diffusion systems

The mechanism of pattern formation in reaction-diffusion systems is treated as an interesting subject, generally for understanding self-organization observed in living systems and natural phenomena. Several spatial patterns appear in the reaction-diffusion systems where an activator and an inhibitor coexist as an intermediate, as represented by a traveling wave, a stationary wave called a Turing structure, etc. Here, we show new kinds of waves in reaction-diffusion systems, which exhibit reciprocating motion without colliding into each other or blinking periodically. These patterns have never been observed in the conventional numerical models, although experimentally oscillating spots have been often observed. Our model demonstrates that other than the ratio of diffusion coefficients for both intermediates, the thickness of reaction media acts to generate inhibitory effect. The spatial factor of the medium contributes to new pattern formation in reaction-diffusion systems. For the design of new functional materials, the concept might be useful as a simple controlling method for pattern dynamics.     

Pattern formation in the iodate-sulfite-thiosulfate reaction-diffusion system

Sodium polyacrylate-induced pH pattern formation and starch-induced iodine pattern formation were investigated in the iodate-sulfite-thiosulfate (IST) reaction in a one-side fed disc gel reactor (OSFR). As binding agents of the autocatalyst of hydrogen ions or iodide ions, different content of sodium polyacrylate or starch has induced various types of pattern formation. We observed pH pulses, striped patterns, mixed spots and stripes, and hexagonal spots upon increasing the content of sodium polyacrylate and observed iodine pulses, branched patterns, and labyrinthine patterns upon increasing the starch content in the system. Coexistence of a pH front and an iodine front was also studied in a batch IST reaction-diffusion system. Both pH and iodine front instabilities were observed in the presence of sodium polyacrylate, i.e., cellular fronts and transient Turing structures resulting from the decrease in diffusion coefficients of activators. The mechanism of multiple feedback may explain the different patterns in the IST reaction-diffusion system.     

Pattern formation and self-organization in a simple precipitation system

Various types of pattern formation and self-organization phenomena can be observed in biological, chemical, and geochemical systems due to the interaction of reaction with diffusion. The appearance of static precipitation patterns was reported first by Liesegang in 1896. Traveling waves and dynamically changing patterns can also exist in reaction-diffusion systems: the Belousov-Zhabotinsky reaction provides a classical example for these phenomena. Until now, no experimental evidence had been found for the presence of such dynamical patterns in precipitation systems. Pattern formation phenomena, as a result of precipitation front coupling with traveling waves, are investigated in a new simple reaction-diffusion system that is based on the precipitation and complex formation of aluminum hydroxide. A unique kind of self-organization, the spontaneous appearance of traveling waves, and spiral formation inside a precipitation front is reported. The newly designed system is a simple one (we need just two inorganic reactants, and the experimental setup is simple), in which dynamically changing pattern formation can be observed. This work could show a new perspective in precipitation pattern formation and geochemical self-organization.     

Transient and steady state CO oxidation kinetics on nanolithographically prepared supported Pd model catalysts: experiments and simulations

Applying molecular-beam methods to a nanolithographically prepared planar PdSiO2 model catalyst, we have performed a detailed study of the kinetics of CO oxidation. The model catalyst was prepared by electron-beam lithography, allowing individual control of particle size and position. The sample was structurally characterized by atomic force microscopy and scanning electron microscopy before and after reaction. In the kinetic experiments, the O-rich and CO-rich regimes were investigated systematically with respect to their transient and steady-state behaviors, both under bistable and monostable reaction conditions. Separate molecular beams were used in order to supply the reactants, allowing individual control over the reactant fluxes. The desorbing CO2 was detected by both angle-resolved and angle-integrated mass spectrometries. The experimental results were analyzed using different types of microkinetic models, including a detailed reaction-diffusion model, which takes into account the structural parameters of the catalyst as well as scattering of the reactants from the support. The model quantitatively reproduces the results as a function of the reactant fluxes and the surface temperature. Various kinetic effects observed are discussed in detail on the basis of the model. Specifically, it is shown that under conditions of limited oxygen mobility, the switching behavior between the kinetic regimes is largely driven by the surface mobility of CO.     

Locking of Turing patterns in the chlorine dioxide-iodine-malonic acid reaction with one-dimensional spatial periodic forcing

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns with spatial periodic forcing. Wavenumber-locked stripe patterns are the typical resonant structures that labyrinthine patterns exhibit in response to one-dimensional forcing by illumination when images of stripes are projected on a working medium. Our experimental results reveal that segmented oblique, hexagonal and rectangular patterns can also be obtained. However, these two-dimensional resonant structures only develop in a relatively narrow range of forcing parameters, where the unforced stripe pattern is in close proximity to the domain of hexagonal patterns. Numerical simulations based on a model that incorporates the forcing by illumination using an additive term reproduce well the experimental observations. These findings confirm that additive one-dimensional forcing can generate a two-dimensional resonant response. However, such a response is considerably less robust than the effect of multiplicative forcing.     

Nanoscale changes induce microscale effects in Turing patterns

A water-in-oil microemulsion loaded with a reaction-diffusion chemical system (Belousov-Zhabotinsky reaction) is able to exhibit Turing patterns that are believed to be responsible for differentiation processes in Nature. Using polymers, such as polyethylene oxide, longer than the droplet size changes the distribution of droplets due to cluster formation. This difference in the nanoscale has relevant consequences in the observed the Turing pattern's wavelength, which is three orders of magnitude larger than the droplet size.     

Pattern formation induced by ion-selective surfaces: models and simulations

Simple inorganic reactions in gels, such as NaOH + CuCl(2), NaOH + AgNO(3), and CuCl(2) + K(3)[Fe(CN)(6)], can yield to various precipitation patterns. The first compound penetrates in a hydrogel by diffusion, and reacts with the second compound homogenized in the gel. The precipitate patterns formed in these reactions have got two kinds of bordering surfaces. Recent experimental results suggested that one of these surfaces has an ion-selective (semipermeable) character: It restrains the diffusion of the reacting ion contained by the reactant that diffuses into the gel. In this paper, we built the above experimental observation into a reaction-diffusion cellular-automata model of the pattern formation. Computer simulations showed that the model is able to reproduce the basic building elements of the patterns.     

Rayleigh-Taylor instabilities in reaction-diffusion systems inside Hele-Shaw cell modified by the action of temperature

The influence of temperature in the buoyancy driven Rayleigh-Taylor instability of reaction-diffusion fronts is investigated experimentally in Hele-Shaw cells. The acid autocatalysis of chlorite-tetrathionate reaction coupled to molecular diffusion yields exothermic planar reaction-diffusion fronts separating two miscible reactant and product solutions. The resulting chemical front moves downwards invading the fresh reactants, leaving the products of the reaction behind it. The density of the product solution is higher than the reactant solution; hence, the traveling front is buoyantly unstable and develops density fingers in time (Rayleigh-Taylor instability) when the products are above the reactants. The kinetic constant of a chemical reaction varies due to thermal effects. This may stabilize the exothermic descending front when temperature is increased, so that the mixing zone decreases, modifying the fingering patterns, until it almost disappears. The authors study the influence of the temperature variation on the instability pattern figure observed in the chlorite-tetrathionate reaction for long times, corresponding to the nonlinear regime.     

Sustained self-organizing pH patterns in hydrogen peroxide driven aqueous redox systems

Many pattern developments in nature are believed to result from the interplay between self-activated (bio)chemical processes and the diffusive transport of constituents. Though the details are difficult to work out, the relevance of reaction-diffusion processes is widely accepted in many aspects of biological development. Due to their easier manipulation and control, aqueous phase chemical reactions are commonly preferred to probe the patterning capacity of reaction-diffusion processes. Nonetheless, sustained patterns of such a type were observed only in reactions involving oxyhalogen compounds. We report on halogen free solution chemistry systems which lead to stationary or oscillatory spatiotemporal pH patterns. They are based on the acid autocatalytic oxidation of sulfite ions by hydrogen peroxide in combination with two significantly different proton consuming feedback reactions. Besides the chemical novelty, yet experimentally and even theoretically undocumented pattern dynamics are uncovered. This success, based on a well-defined method, further paves the way to the discovery of stationary patterns in delicate biochemical reactions.     

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.     

Spatial bifurcations of fixed points and limit cycles during the electrochemical oxidation of H2 on Pt ring-electrodes

Pattern formation during the oscillatory oxidation of H2 on Pt ring-electrodes in the presence of electrosorbing ions was studied under potentiostatic control for three different positions of the reference electrode (RE). The position of the RE crucially affects the degree of the global feedback which is imposed by the potentiostatic operation mode, and the three configurations selected corresponded to zero, maximum and intermediate global coupling. In the absence of global coupling, 'communication' among different positions occurs exclusively through migration coupling (the electrochemical counterpart to diffusion in reaction-diffusion systems). In this case, spatially inhomogeneous oscillations that were attributed to a spatial bifurcation of the homogeneous limit cycle were observed throughout. This implies that the system is Benjamin-Feir unstable. For the strongest global coupling adjustable, travelling pulses were found that emerged in a wave bifurcation with n = 1 from the homogeneous steady state. The pulses exhibited modulations in velocity and width that most likely resulted from the interaction between inhomogeneities of the catalytic surface and the nonlinear reaction dynamics. In the case of an intermediate global coupling strength, a diversity of spatio-temporal motions was observed. The dynamics ranged from pulses over target patterns and so-called asymmetric target patterns to mixed states where two or three of these states alternate. For some parameters these mixed states were in addition separated by bursts of the system to a nearly homogeneous unreactive state.     

Subcritical wave instability in reaction-diffusion systems

We report an example of subcritical wave instability in a model of a reaction-diffusion system and discuss the potential implications for localized patterns found in experiments on the Belousov-Zhabotinsky reaction in a microemulsion.     

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

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.     

Neutron reflectivity characterization of the photoacid reaction-diffusion latent and developed images of molecular resists for extreme ultraviolet lithography

Lithographic feature size requirements have approached a few radius of gyration of photoresist polymers used in thin-film patterning. Furthermore, the feature dimensions are commensurate with the photoacid diffusion length that defines the underlying latent image. Smaller imaging building blocks may enable reduced feature sizes; however, resolution limits are also dependent upon the spatial extent of the photoacid-catalyzed reaction diffusion front and subsequent dissolution mechanism. The reaction-diffusion front was characterized by neutron reflectivity for ccc stereoisomer-purified, deuterium-labeled tert-butoxycarbonyloxy calix[4]resorcinarene molecular resists. The spatial extent of the reaction front exceeds the size of the molecular resist with an effective diffusion constant of (0.13 ± 0.06) nm(2)/s for reaction times longer than 60 s, with the maximum at shorter times. Comparison to a mean-field reaction-diffusion model shows that a photoacid trapping process provides bounds to the spatial and extent of reaction via a reaction-limited mechanism whereas the ratio of the reaction rate to trapping rate constants recovers the effective diffusion peak. Under the ideal step-exposure conditions, surface roughness was observed after either positive- or negative-tone development. However, negative-tone development follows a surface-restructuring mechanism rather than etch-like dissolution in positive-tone development.