Oscillatory Belousov-Zhabotinsky Reaction Research Articles

A nonlinear continuum framework for constitutive modeling of active polymer gels

Chemo-mechanical transduction is one of the key mechanisms that has formed the basis for designing bio-inspired self-driven synthetic systems from soft materials. Polymer hydrogels that use Belousov–Zhabotinsky (BZ) reaction are a unique class of dynamical reaction–diffusion (RD) systems that can continuously transduce internal chemical energy, from the reaction, to produce sustained mechanical work. In particular, BZ gels represent a complex nonlinear chemo-mechanical system, wherein, the autocatalytic oscillatory BZ reaction drives the rhythmic mechanical deformations through polymer-solvent inter–diffusion. The objective of our work is to develop a standardized finite element (FE) framework for chemically driven active hydrogels that captures nonlinear elastic deformations with limited chain extensibility. The distinguishing feature of our approach is that, unlike other approaches, it combines reaction kinetics, solvent transport, elastodynamics of the polymeric network, and polymer-solvent friction under a unified FE framework. Moreover, we adapt our approach to a specific case of BZ gels and capture their swelling-deswelling characteristics. We first implement our FE framework in MATLAB that subsequently, forms the basis for constructing a three-dimensional user element subroutine (3D-UEL) in ABAQUS. Ultimately, through our simulations, we are able to capture all the essential features of BZ gels that includes chemically driven mechanical deformations. In addition, we also demonstrate that our 3D-UEL efficiently captures the chemo-mechanical response of “stent-shaped” BZ gels–a non-standard 3D geometry. In essence, our FE approach not only allows us to simulate BZ gels but also provides a template for other active, dynamical, RD-based systems, driven by chemo-mechanical transduction, irrespective of internal or external mechanisms.

Lie point symmetry infinitesimals, optimal system, power series solution, and modulational gain spectrum to the mathematical Noyes–Field model of nonlinear homogeneous oscillatory Belousov–Zhabotinsky reaction

Introduction:The chemical oscillators are identified as open system that demonstrate periodic changes in the concentration of some reaction species as a result of intricate physico-chemical mechanisms which can lead to bi-stability, the occurrence of limit cycle attractors, the emergence of spiral waves and turing patterns, and finally, deterministic chaos. Objectives:The main objective of this paper is to analyze the simple Noyes–Field governing system of differential equations for the nonlinear Belousov–Zhabotinsky reaction which delineates the non-linear oscillatory behavior of chemical systems that occurs in the homogeneous media. Methodology:The Lie symmetry invariance analysis performed to extract the symmetries infinitesimal generators and the adjoint representation carried out to develop optimal system for the obtained Lie vectors. The significant power series approach applied to obtain the analytical solution. The modulation instability criteria ensured the stability of nonlinear oscillatory Belousov–Zhabotinsky reaction process. Results:The one-dimensional Lie symmetry generators algebra of the mathematical Noyes–Field governing system for oscillatory reaction is established. Furthermore, similarity reductions are carried out as well as the development of an optimal system of the sub-algebras. The similarity transformation technique converted the controlling system to ordinary differential equations and generates the large quantity of analytical traveling wave solutions. Moreover, the closed-form analytical solution for the proposed homogeneous nonlinear oscillatory chemical process is secured. The (MI) gain spectrum graphically visualized with the suitable choice of arbitrary parameters. Conclusion:The graphical performance of the Noyes–Field model solution at various settings reveals new perspectives and fascinating model phenomena. The attained outcomes have significant applications and have opened up innovative development areas for research across numerous scientific fields.

Information Processing Using Networks of Chemical Oscillators.

I believe the computing potential of systems with chemical reactions has not yet been fully explored. The most common approach to chemical computing is based on implementation of logic gates. However, it does not seem practical because the lifetime of such gates is short, and communication between gates requires precise adjustment. The maximum computational efficiency of a chemical medium is achieved if the information is processed in parallel by different parts of it. In this paper, I review the idea of computing with coupled chemical oscillators and give arguments for the efficiency of such an approach. I discuss how to input information and how to read out the result of network computation. I describe the idea of top-down optimization of computing networks. As an example, I consider a small network of three coupled chemical oscillators designed to differentiate the white from the red points of the Japanese flag. My results are based on computer simulations with the standard two-variable Oregonator model of the oscillatory Belousov–Zhabotinsky reaction. An optimized network of three interacting oscillators can recognize the color of a randomly selected point with >98% accuracy. The presented ideas can be helpful for the experimental realization of fully functional chemical computing networks.

Belousov–Zhabotinsky reaction: an open-source approach

The time-dependent change in concentration of reactants in the oscillatory Belousov–Zhabotinsky reaction defined by non-linear ordinary coupled differential equations has been derived by applying normalized non-dimensional computational methods and presented graphically. The accurate normalisation procedure eschews complex mathematical steps; however, choosing the appropriate references to define the dimensionless variables necessitates a deep physical understanding of the problem. The dimensionless groups are then derived to provide the functional dependences of the unknowns of interest. Finally, to cross-check all the mentioned dependences, a computational simulation has been carried out using domain-specific to general-purpose programming languages. Although there are countless programming languages present to carry out computer simulations, a flexible, portable, and open-source language like python proves fruitful.

Pattern formation in a four-ring reaction-diffusion network with heterogeneity.

In networks of nonlinear oscillators, symmetries place hard constraints on the system that can be exploited to predict universal dynamical features and steady states, providing a rare generic organizing principle for far-from-equilibrium systems. However, the robustness of this class of theories to symmetry-disrupting imperfections is untested in free-running (i.e., non-computer-controlled) systems. Here, we develop a model experimental reaction-diffusion network of chemical oscillators to test applications of the theory of dynamical systems with symmeries in the context of self-organizing systems relevant to biology and soft robotics. The network is a ring of four microreactors containing the oscillatory Belousov-Zhabotinsky reaction coupled to nearest neighbors via diffusion. Assuming homogeneity across the oscillators, theory predicts four categories of stable spatiotemporal phase-locked periodic states and four categories of invariant manifolds that guide and structure transitions between phase-locked states. In our experiments, we observed that three of the four phase-locked states were displaced from their idealized positions and, in the ensemble of measurements, appeared as clusters of different shapes and sizes, and that one of the predicted states was absent. We also observed the predicted symmetry-derived synchronous clustered transients that occur when the dynamical trajectories coincide with invariant manifolds. Quantitative agreement between experiment and numerical simulations is found by accounting for the small amount of experimentally determined heterogeneity in intrinsic frequency. We further elucidate how different patterns of heterogeneity impact each attractor differently through a bifurcation analysis. We show that examining bifurcations along invariant manifolds provides a general framework for developing intuition about how chemical-specific dynamics interact with topology in the presence of heterogeneity that can be applied to other oscillators in other topologies.

PH Oscillating System for Molecular Computation as a Chemical Turing Machine.

It has previously been demonstrated that native chemical Turing machines can be constructed by exploiting the nonlinear dynamics of the homogeneous oscillating Belousov–Zhabotinsky reaction. These Turing machines can perform word recognition of a Chomsky type 1 context sensitive language (CSL), demonstrating their high computing power. Here, we report on a chemical Turing machine that has been developed using the H2O2–H2SO4–SO32––CO32– pH oscillating system. pH oscillators are different to bromate oscillators in two key ways: the proton is the autocatalytic agent, and at least one of the reductants is always fully consumed in each turnover—meaning the system has to be operated as a flow reactor. Through careful design, we establish a system that can also perform Chomsky type 1 CSL word recognition and demonstrate its power through the testing of a series of in-language and out-of-language words.

Oscillatory microcells connected on a ring by chemical waves.

The dynamics of four coupled microcells with the oscillatory Belousov-Zhabotinsky (BZ) reaction in them is analyzed with the aid of partial differential equations. Identical BZ microcells are coupled in a circle via identical narrow channels containing all the components of the BZ reaction, which is in the stationary excitable state in the channels. Spikes in the BZ microcells generate unidirectional chemical waves in the channels. A thin filter is put in between the end of the channel and the cell. To make coupling between neighboring cells of the inhibitory type, hydrophobic filters are used, which let only Br2 molecules, the inhibitor of the BZ reaction, go through the filter. To simulate excitatory coupling, we use a hypothetical filter that let only HBrO2 molecules, the activator of the BZ reaction, go through it. New dynamic modes found in the described system are compared with the "old" dynamic modes found earlier in the analogous system of the "single point" BZ oscillators coupled in a circle by pulses with time delay. The "new" and "old" dynamic modes found for inhibitory coupling match well, the only difference being much broader regions of multi-rhythmicity in the "new" dynamic modes. For the excitatory type of coupling, in addition to four symmetrical modes of the "old" type, many new asymmetrical modes coexisting with the symmetrical ones have been found. Asymmetrical modes are characterized by the spikes occurring any time within some finite time intervals.

A chemical micropump actuated by self-oscillating polymer gel

Abstract We demonstrated a self-driven chemical micropump powered by self-oscillating motions of synthetic polymer gel converting chemical energy into displacements of a thin polydimethylsiloxane (PDMS) sheet to produce a directional flow via micro check valves. The gel oscillations induced by the oscillatory Belousov-Zhabotinsky (BZ) reaction generate large volume change. To effectively exploit this volume change, gel bars were used as drivers. They were sandwiched by a fixed cover glass and a push-bar to communicate the expanding (pushing) force of the gel bars to the pumping chamber. At 25 °C, this gel micropump generated a 0.07 mN stroke force and a 0.28 μL/min positive net flow in the microchannel. Additionally, the flow rate of this gel micropump was increased as the temperature was raised from 20 to 30 °C. Compared to existing chemical micropumps, this gel-based micropump provides advantages of self-oscillating property, easy refueling and thermal controllability. This gel-based soft micropump would be used as wearable or medical devices exploiting the flexibility and quietness.

Multimode Self‐Oscillating Vesicle Transformers

AbstractEngineering synthetic materials that mimic the complex rhythmic oscillatory behavior of living cells is a fundamental challenge in science and technology. Up to now, the reported synthetic model system still cannot compete with nature in oscillatory modes and amplitudes. Presented here is a novel alternating copolymer vesicle that exhibits drastic and multimode shape oscillations in real time, which are controlled by polymer concentrations and driven by the Belousov—Zhabotinsky oscillatory reaction, including swelling/deswelling, twisting/detwisting, stretching/shrinking, fusion/fission, and multiple division. Some of them, especially the fission oscillation, have not been observed before. In addition, the oscillation magnitude with regard to diameter is much larger than that of previously reported self‐oscillating vesicles. Such a self‐oscillating vesicle transformer would extend the complexity and capacity of membrane deformations in synthetic systems, approaching those of natural cells.

Multimode Self-Oscillating Vesicle Transformers.

Engineering synthetic materials that mimic the complex rhythmic oscillatory behavior of living cells is a fundamental challenge in science and technology. Up to now, the reported synthetic model system still cannot compete with nature in oscillatory modes and amplitudes. Presented here is a novel alternating copolymer vesicle that exhibits drastic and multimode shape oscillations in real time, which are controlled by polymer concentrations and driven by the Belousov-Zhabotinsky oscillatory reaction, including swelling/deswelling, twisting/detwisting, stretching/shrinking, fusion/fission, and multiple division. Some of them, especially the fission oscillation, have not been observed before. In addition, the oscillation magnitude with regard to diameter is much larger than that of previously reported self-oscillating vesicles. Such a self-oscillating vesicle transformer would extend the complexity and capacity of membrane deformations in synthetic systems, approaching those of natural cells.

0D-2D heterostructures as nanocatalysts for self-oscillating reactions: an investigation into chemical kinetics.

Self-oscillating chemical reactions are dynamical reaction-diffusion systems that show immense potential in the design of synthetic soft materials with biomimetic functionalities. The Belousov-Zhabotinsky (BZ) reaction is one such reaction, where the periodic change in the redox state of the metal ion catalyst drives the rhythmic chemical oscillations. Inspired by the exceptional properties of graphene, specifically its catalytic activity for redox reactions, we investigate the effect of graphene-based nanocomposites on the dynamics of the BZ reaction. In particular, we synthesized catalytic mats by decorating ceria nanoparticles (CeNPs) on graphene-based nanosheets, thereby creating 0D-2D heterostructures and subsequently, incorporate these catalytic mats into the BZ reaction. Our investigations reveal that CeNP decorated nanocomposites significantly enhance the oscillating frequency of the BZ reaction, not only compared to the traditional solution-based catalysts but also compared to the bare graphene-based nanosheets. From our experiments at various temperatures and concentrations, together with modelling and simulations, we determine the apparent rate constant for different CeNP decorated graphene nanocomposites. Ultimately, we determine the apparent rate and estimate various kinetic parameters, including activation energy and reaction order. In short, we demonstrate that CeNP decorated nanomats are excellent catalysts and elucidate that the kinetics of the BZ reaction can be simulated using the Oregonator model with our kinetic parameters. We envisage that our findings can be utilized to harness multiscale interactions to design a variety of multifunctional stimuli responsive materials.

Flow chemistry controls self-assembly and cargo in Belousov-Zhabotinsky driven polymerization-induced self-assembly

Amphiphilic block-copolymer vesicles are increasingly used for medical and chemical applications, and a novel method for their transient self-assembly orchestrated by periodically generated radicals during the oscillatory Belousov-Zhabotinsky (BZ) reaction was recently developed. Here we report how combining this one pot polymerization-induced self-assembly (PISA) method with a continuously stirred tank reactor (CSTR) strategy allows for continuous and reproducible control of both the PISA process and the chemical features (e.g. the radical generation and oscillation) of the entrapped cargo. By appropriately tuning the residence time (τ), target degree of polymerization (DP) and the BZ reactants, intermediate self-assembly structures are also obtained (micelles, worms and nano-sized vesicles). Simultaneously, the chemical properties of the cargo at encapsulation are known and tunable, a key advantage over batch operation. Finally, we also show that BZ-driven polymerization in CSTR additionally supports more non-periodic dynamics such as bursting.

Experimental Investigation of the Dynamical Modes of Four Pulse‐Coupled Chemical Micro‐Oscillators

We present an experimental system of four identical microreactors (MRs) in which the photosensitive oscillatory Belousov-Zhabotinsky (BZ) reaction occurs. The inhibitory coupling of these BZ MRs is organized via pulses of light coming to each MR from a computer projector. These pulses are induced by spike(s) in other MR(s) of the same network. Time delay between the spike in one BZ MR and the pulsed perturbation of the other BZ MR(s), the amplitude of light pulses, their duration, and the connectivity of the MRs are controlled by the LabVIEW software. Recording the dynamics of the BZ reaction in the MRs via a microscope equipped with a CCD camera, we observe all the main dynamical modes of our network of MRs, which are the IP (in-phase), AP (anti-phase), W (walk), and WR (walk reverse) for the unidirectional coupling, and the IP, two-cluster, three-cluster, and splay modes for the all-to-all coupling. Our software detects all the modes of the network automatically and makes it possible to switch between them on demand using a few special "switching" pulses. As the result of the present work, the experimental implementation of the adaptive behaviour of the pulse-coupled chemical micro-oscillator networks becomes available.

Designing polymer gels and composites that undergo bio-inspired phototactic reconfiguration and motion

Inspired by the adaptive behavior of photo-responsive biological organisms, we develop analytical and computational models to design polymer gels and composites that can be dynamically reconfigured and driven to move with the application of light. We focus on gels formed from poly(N-isopropylacrylamide) and functionalized with spirobenzopyran (SP) chromophores, which become hydrophobic under blue light in acidic aqueous solution. Using our modeling approaches, we irradiate the gels through photomasks and demonstrate that the shapes of the samples can be reversibly and remotely ‘remolded’ by varying the apertures in the masks. By simulating the effect of repeatedly moving the light across the sample, we also show that the gel can undergo directed motion. We then examine gels that contain both SP chromophores and the ruthenium catalysts that drive the oscillatory Belousov–Zhabotinsky reaction. These dual-functionalized gels undergo spontaneous, self-sustained motion even when the lights are held stationary. We also simulate the behavior of composites formed from SP-functionalized fibers embedded in the poly(N-isopropylacrylamide) gel. With the SP-functionalization confined to the fibers, light and heat act as orthogonal stimuli and thus the composites display distinctly different modes of movement when the different cues are applied to the samples. Overall, our findings provide guidelines for using light to controllably reconfigure the shape and drive the movement of gel-based materials and thus, tailor the material to display different functionalities.

Dynamics of a 1D array of inhibitory coupled chemical oscillators in microdroplets with global negative feedback.

We have investigated the effect of global negative feedback (GNF) on the dynamics of a 1D array of water microdroplets (MDs) filled with the reagents of the photosensitive oscillatory Belousov-Zhabotinsky (BZ) reaction. GNF is established by homogeneous illumination of the 1D array with the light intensity proportional to the number of BZ droplets in the oxidized state with the coefficient of proportionality ge. MDs are immersed in the continuous oil phase and diffusively coupled with the neighboring droplets via inhibitor Br2 which is soluble in the oil phase. At chosen concentrations of the BZ reactants, illumination suppresses the BZ oscillators. Without GNF, or at a very small ge < 0.29, local inhibitory coupling leads to out-of-phase oscillations of the neighboring BZ droplets with an almost constant phase shift Δφ between them, which makes a space-time plot of the BZ MDs look like a staircase. At 0.3 < ge < 0.6, regular oscillatory clusters consisting of distant BZ MDs (mostly 5-6 phase clusters) emerge. At 0.6 ≤ ge ≤ 1.0, chaotic clusters are observed. At 1.2 < ge < 1.8, regular (mostly 3-4-phase) clusters emerge again. At 1.8 < ge < (3-4), complex clusters with different (but multiple) periods of oscillation are observed. At the same time, some droplets stop oscillating. At large enough ge (>4), in the region of two-phase clusters (with several suppressed BZ MDs), final patterns seem to resemble the initial patterns. Intensive computer simulations with the ordinary differential equations support experimental results.

Chemical communication and dynamics of droplet emulsions in networks of Belousov-Zhabotinsky micro-oscillators produced by microfluidics.

Chemical communication leading to synchronization and collective behaviour of dynamic elements, such as cell colonies, is a widespread phenomenon with biological, physical and chemical importance. Such synchronization between elements proceeds via chemical communication by emmision, interdiffusion and reception of specific messenger molecules. On a lab scale, these phenomena can be modeled by encapsulating an oscillating chemical reaction, which serves as a signal (information) sender/receiver element, inside microcompartments such as droplet emulsions, liposomes and polymersomes. Droplets can thus be regarded as single units, able to generate chemical messengers that diffuse in the environment and hence can interact with other compartments. The Belousov-Zhabotinsky (BZ) reaction is a well-known chemical oscillator largely used as a model for complex nonlinear phenomena, including chemical, physical and biological examples. When the BZ-reaction is encapsulated inside microcompartments, its chemical intermediates can serve as messengers by diffusing among different microcompartments, to trigger specific reactions leading to a collective behavior between the elements. The geometry and constitution of the diffusion pathways play an important role in governing the collective behaviour of the system. In this context, microfluidics is not only a versatile tool for mastering the encapsulation process of the BZ-reaction in monodisperse microcompartments, but also for creating geometries and networks with well defined boundaries. The individual compartments can be engineered with selected properties using different surfactants in the case of simple emulsions, or with specific membrane properties in the case of liposomes. Furthermore, it enables the arrangement of these microcompartments in various geometric configurations, where the diffusive coupling pathways between individual compartments are both spatially and chemically well-defined. In this tutorial paper, we review a number of articles reporting various approaches to generate networks of compartmentalized Belousov-Zhabotinsky (BZ) chemical oscillators using microfluidics. In contrast to biological cellular networks, the dynamical characteristics of the BZ-reaction is well-known and, when confined in microcompartments arranged in different configurations with a pure interdiffusive coupling, these communicative microreactors can serve to mimic various types of bio-physical networks, aiding to comprehend the concept of chemical communication.

Chaos in the Belousov–Zhabotinsky reaction

The dynamics of reacting chemical systems is governed by typically polynomial differential equations that may contain nonlinear terms and/or embedded feedback loops. Thus the dynamics of such systems may exhibit features associated with nonlinear dynamical systems, including (among others): temporal oscillations, excitability, multistability, reaction-diffusion-driven formation of spatial patterns, and deterministic chaos. These behaviors are exhibited in the concentrations of intermediate chemical species. Bifurcations occur between particular dynamic behaviors as system parameters are varied. The governing differential equations of reacting chemical systems have as variables the concentrations of all chemical species involved, as well as controllable parameters, including temperature, the initial concentrations of all chemical species, and fixed reaction-rate constants. A discussion is presented of the kinetics of chemical reactions as well as some thermodynamic considerations important to the appearance of temporal oscillations and other nonlinear dynamic behaviors, e.g., deterministic chaos. The behavior, chemical details, and mechanism of the oscillatory Belousov–Zhabotinsky Reaction (BZR) are described. Furthermore, experimental and mathematical evidence is presented that the BZR does indeed exhibit deterministic chaos when run in a flow reactor. The origin of this chaos seems to be in toroidal dynamics in which flow-driven oscillations in the control species bromomalonic acid couple with the BZR limit cycle.

Modeling the entrainment of self-oscillating gels to periodic mechanical deformation.

Polymer gels undergoing the oscillatory Belousov-Zhabotinsky (BZ) reaction are one of the few synthetic materials that exhibit biomimetic mechano-chemical transduction, converting mechanical input into chemical energy. Here, we consider self-oscillating BZ gels that are subjected to periodic mechanical forcing, and model the entrainment of the oscillatory gel dynamics to this external stimulus. The gel size is assumed to be sufficiently small that the chemo-mechanical oscillations are spatially uniform. The behavior of the system is captured by equations describing the kinetics of the oscillatory BZ reaction in the gel coupled to equations for the variations in gel size due to the inherent reaction and imposed force. We employ the phase dynamics approach for analyzing the entrainment of the BZ gel to force- and strain-controlled compressive deformations. The phase response curves are obtained using Malkin's method, and time-averaging is applied to extract the slow phase dynamics caused by the periodic forcing. We demonstrate that the entrainment of the self-oscillating BZ gel is sensitive to the chemo-mechanical coupling in gel, the mode of deformation, and the level of static compression. Kuramoto's model of phase oscillators is shown to be applicable if the external forcing is purely harmonic.

Designing Dual-functionalized Gels for Self-reconfiguration and Autonomous Motion.

Human motion is enabled by the concerted expansion and contraction of interconnected muscles that are powered by inherent biochemical reactions. One of the challenges in the field of biomimicry is eliciting this form of motion from purely synthetic materials, which typically do not generate internalized reactions to drive mechanical action. Moreover, for practical applications, this bio-inspired motion must be readily controllable. Herein, we develop a computational model to design a new class of polymer gels where structural reconfigurations and internalized reactions are intimately linked to produce autonomous motion, which can be directed with light. These gels contain both spirobenzopyran (SP) chromophores and the ruthenium catalysts that drive the oscillatory Belousov-Zhabotinsky (BZ) reaction. Importantly, both the SP moieties and the BZ reaction are photosensitive. When these dual-functionalized gels are exposed to non-uniform illumination, the localized contraction of the gel (due to the SP moieties) in the presence of traveling chemical waves (due to the BZ reaction) leads to new forms of spontaneous, self-sustained movement, which cannot be achieved by either of the mono-functionalized networks.

Excitability Modulation of Oscillating Media in 3D-Printed Structures.

Excitation and oscillation are central to living systems. For excitable systems, which can be brought into oscillation by an external stimulus, the excitation threshold is a crucial parameter. This is evident for neurons, which only generate an action potential when exposed to a sufficiently high concentration of excitatory neurotransmitters, which may only be achieved when multiple presynaptic axons deliver their action potential simultaneously to the synaptic cleft. Dynamic systems composed of relatively simple chemicals are of interest because they can serve as a model for physiological processes or can be exploited to implement chemical computing. With these applications in mind, we have studied the properties of the oscillatory Belousov-Zhabotinsky (BZ) reaction in 3D-printed reaction vessels with open channels of different dimensions. It is demonstrated that the channel geometry can be used to modulate the excitability of the BZ medium, switching a continuously oscillating medium to an excitable medium. Because large networks of channel-connected reaction wells of different depth can easily be fabricated by 3D printing, local excitability modulation could be built into the structure of the reaction vessel itself, opening the way to more extensive experimentation with networks of chemical oscillators.