Mobility-induced instability and pattern formation in a reaction-diffusion system

Ions undergoing a reaction-diffusion process are susceptible to electric field. We show that a constant external field may induce a kind of instability on the state stabilized by diffusion in a reaction-diffusion system giving rise to formation of pattern even when the diffusion coefficients of the reactants are equal. The origin of the pattern is due to the difference in mobilities of the two species and is thus markedly different from that of deformed Turing pattern in presence of the field. While this differential flow instability had been shown earlier to result in traveling waves, we realize in the context of stationary pattern formation in a typical reaction-diffusion-advective system. Our analysis is based on a numerical simulation of a generic model on a two-dimensional domain.     

Turing pattern formation in coupled reaction-diffusion system with distributed delays

Turing pattern formation in coupled two-layer system with distributed delayed is investigated. Numerical simulations prove that, when the coupling is weak, it can apparently accelerate the formation process and enhance the spatial amplitude of the pattern. When it is strong, it will prolong the formation process or even inhibit the pattern and turn the whole system into bulk oscillatory state by its influence on the transient oscillatory state. If the coupling covers only part of the system, Turing pattern can be prominently oriented according to the shape of the coupling area at tiny coupling strength. However, if the coupling is too strong, the Turing pattern may also be destroyed. This means that in coupled systems, the delay effect in the cross-layer signal transfer may significantly influence the spatial character and/or the evolution dynamics in Turing pattern formation, even to destroy the pattern. This work is of practical significance in the study of Turing pattern in biosystems, where bilayer membranes or multilayer tissues are often found.     

Pattern formation in the Belousov-Zhabotinsky-PAMAM dendrimer system

The Belousov-Zhabotinsky reaction was studied under the influence of nanometric confinements induced by a complex polymer, the PAMAM-G4 dendrimers. They are well-defined in both molecular weight and architecture and are capable of molecular inclusion, making "unimolecular active micelles". The effect of such nanocompartments in the BZ reaction is analyzed by changing both the excitability and the concentration of the dendrimer, obtaining a wide range of behaviours, ranging from stationary Turing-like patterns to time dependent structures, such as jumping waves or packet waves.     

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.     

Turbulent states in a reaction-diffusion system with period-doubling bifurcations

We investigate the spatially extended Hastings–Powell model in one and two dimensions with constant diffusion coefficients and nonflux boundary conditions. Nowave zones, spirals and chaos are found. An absolute instability of the spirals produces a transition to chaos. A constant number of defects, linearly increasing with the bifurcation parameter of the system is found, i.e. there do not exist defect-creation or defect-destruction events. Defects behave as hard disks, with translational degrees of freedom, which result from a cooperative interaction between pairs of defects.     

Intermediate colloidal formation and the varying width of periodic precipitation bands in reaction-diffusion systems

The mechanism of rhythmic pattern formation in reaction-diffusion systems is investigated theoretically by introducing a new concept. The boundary that separates the two reacting species virtually migrates as the diffusion proceeds into the gelatinous medium. Based on this boundary migration scenario, all the well-established relations on Liesegang patterns could be proved, in a rather modified way. The idea of formation of intermediate colloidal haze prior to patterning along with the moving boundary model proved to be efficient in predicting the concentration dependence of the width of the spatiotemporal patterns. The experimental observations support the width law relation developed.     

Instability and pattern formation in reaction-diffusion systems: a higher order analysis

We analyze the condition for instability and pattern formation in reaction-diffusion systems beyond the usual linear regime. The approach is based on taking into account perturbations of higher orders. Our analysis reveals that nonlinearity present in the system can be instrumental in determining the stability of a system, even to the extent of destabilizing one in a linearly stable parameter regime. The analysis is also successful to account for the observed effect of additive noise in modifying the instability threshold of a system. The analytical study is corroborated by numerical simulation in a standard reaction-diffusion system.     

Pattern formation in draining thin film suspensions

We demonstrate the emergence of complexity from remarkably simple and ubiquitous systems: draining thin-film suspensions exhibiting a striking transition between two classes of self-organizing patterns. Vertical channels form when attractive forces lead to transient gelation, while horizontal bands result from granular mixtures. We propose an explanation whereby the generic physical mechanisms require only the existence of viscous and excluded-volume couplings among the particles, solvent, and substrate. System-specific, small inhomogeneities trigger large-scale pattern formation, through collective dynamics, where jamming plays a crucial role. Our results shed light on emergent complexity in bio- and geophysical processes and have implications for coatings and food industries.     

Pattern formation in phase separating binary mixtures

We experimentally investigate the interplay of thermodynamics with hydrodynamics during phase separation of (quasi-) binary mixtures. Well defined patterns emerge while slowly crossing the cloud point curve. Depending on the material parameters of the experimental system, two distinct scenarios are observed. In quasi-binary mixtures of methanol-hexane patterns appear before macroscopic phase separation sets in. In course of time the patterns turn faint while the overall turbidity of the sample increases until the mixtures become completely turbid. We attribute this pattern formation to a latent heat induced instability resembling a Rayleigh-Bénard instability. This is confirmed by calorimetric data and an estimate of its Rayleigh number. Mixtures of C(4)E(1)-water doped with decane phase separate under heating. After passing the cloud point curve these mixtures first become homogenously turbid. While clearing up, pattern formation is observed. We attribute this type of pattern formation to an interfacial tension induced Bénard-Marangoni instability. The occurrence of the two scenarios is supported by the relevant dimensionless numbers.     

Amplitude equations for breathing spiral waves in a forced reaction-diffusion system

Based on a multiple scale analysis of a forced reaction-diffusion system leading to amplitude equations, we explain the existence of spiral wave and its photo-induced spatiotemporal behavior in chlorine dioxide-iodine-malonic acid system. When the photo-illumination intensity is modulated, breathing of spiral is observed in which the period of breathing is identical to the period of forcing. We have also derived the condition for breakup and suppression of spiral wave by periodic illumination. The numerical simulations agree well with our analytical treatment.     

Turing pattern amplitude equation for a model glycolytic reaction-diffusion system

For a reaction-diffusion system of glycolytic oscillations containing analytical steady state solution in complicated algebraic form, Turing instability condition and the critical wavenumber at the Turing bifurcation point, have been derived by a linear stability analysis. In the framework of a weakly nonlinear theory, these relations have been subsequently used to derive an amplitude equation, which interprets the structural transitions and stability of various forms of Turing structures. Amplitude equation also conforms to the expectation that time-invariant amplitudes are independent of complexing reaction with the activator species.     

Spatiotemporal chaos arising from standing waves in a reaction-diffusion system with cross-diffusion

We show that quasi-standing wave patterns appear in the two-variable Oregonator model of the Belousov-Zhabotinsky reaction when a cross-diffusion term is added, no wave instability is required in this case. These standing waves have a frequency that is half the frequency of bulk oscillations displayed in the absence of diffusive coupling. The standing wave patterns show a dependence on the systems size. Regular standing waves can be observed for small systems, when the system size is an integer multiple of half the wavelength. For intermediate sizes, irregular patterns are observed. For large sizes, the system shows an irregular state of spatiotemporal chaos, where standing waves drift, merge, and split, and also phase slips may occur.     

Locally implicit solution of a reaction-diffusion system with stiff kinetics

The Tyson-Fife reaction-diffusion equations are solved numerically using a locally implicit approach. Since the variables evolve at very different time scales, the resulting system of equations is stiff. The reaction term is responsible for the stiffness and the time step is increased by using an implicit method. The diffusion operator is evaluated explicitly and the system of implicit nonlinear equations is decoupled. The method is particularly useful for parameter values in which the equations are very stiff, such as the values obtained directly from the experimental reaction rate constants. Previous efforts modified the parameters on the equations to avoid stiffness. The equations then become a simplified model of excitable media and, for those cases, the locally implicit method gives a faster although less accurate solution. Nevertheless, since the modified equations no longer represent a particular chemical system an accurate solution is not as important. The algorithm is applied to observe the transition from simple motion to compound motion of a spiral tip.     

Jumping solitary waves in an autonomous reaction-diffusion system with subcritical wave instability

We describe a new type of solitary waves, which propagate in such a manner that the pulse periodically disappears from its original position and reemerges at a fixed distance. We find such jumping waves as solutions to a reaction-diffusion system with a subcritical short-wavelength instability. We demonstrate closely related solitary wave solutions in the quintic complex Ginzburg-Landau equation. We study the characteristics of and interactions between these solitary waves and the dynamics of related wave trains and standing waves.     

Pattern formation in excitable media with concentration-dependent diffusivities

We study a model of pattern formation in an excitable medium with concentration-dependent diffusivities. The reaction terms correspond to a two-variable Gray-Scott model in which the system has only one stable steady state. The diffusion coefficients of the two species are assumed to have a functional relationship with the concentration of the autocatalyst. A transition from self-replicating behavior to stationary spots is observed as the influence of the local autocatalyst concentration on the diffusion process increases. Notably, the transition occurs even though there is no change in the relative diffusivities of the activator and inhibitor. The observed time-independent patterns exhibit an unusual dependence on the size and geometry of an initial perturbation. Initial perturbations with a large spatial size, for example, sometimes revert to the homogeneous equilibrium state, whereas perturbations of smaller spatial extent develop into stable spots at the same parameter values.     

Pattern formation during electrodeposition of alloys

The increase in the content of the alloying element in electrodeposited alloys reflects in the changes of their phase composition, when the saturation limit of the lattice of the basic metal is reached. At higher percentages, the excess amount of the alloying element forms one or more new, richer in this element phases. The coatings become multi-phase, heterogeneous and their physical–mechanical properties change. Sometimes an ordered distribution of the different phases of the heterogeneous alloy coating could be observed. Examples of self-organization phenomena during electrodeposition of different alloy systems, such as Ag-Sb, Ag-Bi, Ag-In, Ag-Sn, Ag-Cd, Cu-Sb and In-Co, resulting in pattern formation and formation of spatio-temporal structures on the surface of the obtained coatings are presented and compared. Instabilities resulting in potential or current oscillations are registered in most of the investigated systems. The phase composition of the alloy coatings and especially of the observed pattern is determined and some similarities in the structure of the phases forming the pattern are registered. The pattern formation is registered on the cathode not only in cyanide silver alloys electrolytes, but also during deposition of other alloy systems in acidic electrolytes like Cu-Sb and In-Co. The effect of the natural convection in non-agitated electrolytes on the pattern formation is discussed. The possibility of formation of periodic structured coatings without applying external electrical pulses which could result in appropriate modification of some properties of the electrodeposited alloys is demonstrated. The hypothesis that similar pattern formation could be observed in agitated electrolytes at different hydrodynamic and electrolysis conditions, when the same percentage or the same phase composition of the alloy is reached was examined for Ag-Cd, In-Co and Ag-Sb alloys in jet-plating experiments.     

Pattern formation and morphology evolution in Langmuir monolayers

We present a study of how patterns formed by Langmuir monolayer domains of a stable phase, usually solid or liquid condensed, propagate into a metastable one, usually liquid expanded. During this propagation, the interface between the two phases moves as the metastable phase is transformed into the more stable one. The interface becomes unstable and forms patterns as a result of the competition between a chemical potential gradient that destabilizes the interface on one hand and line tension that stabilizes the interface on the other. During domain growth, we found a morphology transition from tip splitting to side branching; doublons were also found. These morphological features were observed with Brewster angle microscopy in three different monolayers at the water/air interface: dioctadecylamine, ethyl palmitate, and ethyl stearate. In addition, we observed the onset of the instability in round domains when an abrupt lateral pressure jump is made on the monolayer. Frequency histograms of unstable wavelengths are consistent with the linear-instability dispersion relation of classical free-boundary models. For the case of dendritic morphologies, we measured the radius of the dendrite tip as a function of the dendrite length as well as the spacing of the side branches along a dendrite. Finally, a possible explanation of why Langmuir monolayers present this kind of nonequilibrium growth patterns is presented. In the steady state, the growth behavior is determined by Laplace's equation in the particle density with specific boundary conditions. These equations are equivalent to those used in the theory of morphology diagrams for two-dimensional diffusional growth, where morphological transitions of the kind observed here have been predicted.     

Pattern formation and fluctuation-induced transitions in protein crystallization

A kinetic model of protein crystallization accounting for the nucleation stage, the growth and competition of solid particles and the formation of macroscopic patterns is developed. Different versions are considered corresponding successively, to a continuous one-dimensional crystallization reactor, a coarse grained two-box model and a model describing the evolution of the space averaged values of fluid and solid material. The analysis brings out the high multiplicity of the patterns. It provides information on their stability as well as on the kinetics of transitions between different states under the influence of the fluctuations.