Belousov-Zhabotinsky Reaction Research Articles

Emergent Synchronous Volumetric Oscillation in Hierarchically Structured Self-Oscillating Gel Clusters.

Emergent properties accompanying synchronization among oscillators are vital characteristics in biological systems. Belousov-Zhabotinsky (BZ) oscillators are an artificial model to study the emergence and synchronization in life. This research represents a self-oscillating gel system with clusterable properties to experimentally examine synchronous and emergent properties at a fundamental hierarchical level. Incorporating acrylic acid (AAc) moieties within the gel network facilitates cluster formation through hydrogen bonding in an acidic BZ substrate solution. Upon clustering, both homogeneous and heterogeneous gel assemblies─ranging from double to quadruple clusters─exhibit increased and synchronized periods and amplitudes during the BZ reaction. Notably, in heterogeneous clusters, gel units with initially short periods and small volumetric amplitudes display a significant increase, aligning with the lonfger periods and larger amplitudes of other elements within the cluster, an emergent property. This research can pave the way for a better understanding of synchronous and emergent properties in biological oscillators such as cardiomyocytes.

Periodical alternation between precipitation and dissolution of camphor solid film on an ethanol solution.

Rhythmic behaviors are generally observed in nonlinear chemical reactions such as the Belousov-Zhabotinsky reaction and enzymatic reactions. Similarly, a simple phase change can also lead to rhythmic behavior. It has been reported previously that camphor solid films alternate between generation and disappearance on ethanol (EtOH) solution, and a phenomenological mechanism has been suggested for this. The evaporation of EtOH decreases the temperature on the surface of the solution via vaporization heat and induces precipitation in the camphor solid film. At this time, the film prevents evaporation, and thus, the surface temperature increases due to thermal diffusion from the atmosphere, resulting in dissolution of the solid film. To verify the previously suggested phenomenological mechanism, we controlled the evaporation rate of EtOH using a porous plastic cover. As a result, the period of oscillation increased with decreasing pore diameter, and finally, the oscillation did not occur without pore in the cover, where the camphor solid film was not observed. Additionally, a new mathematical model was proposed, and the numerical calculations agreed well with experimental observations. Linear stability and bifurcation analyses revealed the detailed mechanism of this phenomenon, which agreed well with the phenomenological explanation mentioned above.

On phase correspondence between chemical oscillations in liquid Belousov-Zhabotinsky mixture filling catalyst-containing matrix and resulted gel's chemomechanical oscillations

We investigated phases and phase differences between coupled mechanical and chemical self-oscillations running in two kinds of newly synthesised chemomechanical gels. Processing of experimental data was model-free and it was based on spectral and wavelet approaches of the theory of dynamical systems. It was applied to samples of different sizes and compositions (Ru- and Fe-based gels). It was revealed that the Ru-based gels demonstrate a faster mechanical response to the redox changes of the Ru-moiety coupled to the Belousov-Zhabotinsky chemical reaction than the Fe-based gels. This demonstrated a significant phase difference between the redox changes of the Fe-moiety and the mechanical oscillations. Special attention was paid to the phase-coupling features for averaged higher harmonics, which are significant due to the high degree of nonlinearity of the chemomechanical oscillatory process.

Harmonic resonance and entrainment of propagating chemical waves by external mechanical stimulation in BZ self-oscillating hydrogels

Smart polymer materials that are nonliving yet exhibit complex "life-like" or biomimetic behaviors have been the focus of intensive research over the past decades, in the quest to broaden our understanding of how living systems function under nonequilibrium conditions. Identification of how chemical and mechanical coupling can generate resonance and entrainment with other cells or external environment is an important research question. We prepared Belousov-Zhabotinsky (BZ) self-oscillating hydrogels which convert chemical energy to mechanical oscillation. By cyclically applying external mechanical stimulation to the BZ hydrogels, we found that when the oscillation of a gel sample entered into harmonic resonance with the applied oscillation during stimulation, the system kept a "memory" of the resonant oscillation period and maintained it post stimulation, demonstrating an entrainment effect. More surprisingly, by systematically varying the cycle length of the external stimulation, we revealed the discrete nature of the stimulation-induced resonance and entrainment behaviors in chemical oscillations of BZ hydrogels, i.e., the hydrogels slow down their oscillation periods to the harmonics of the cycle length of the external mechanical stimulation. Our theoretical model calculations suggest the important roles of the delayed mechanical response caused by reactant diffusion and solvent migration in affecting the chemomechanical coupling in active hydrogels and consequently synchronizing their chemical oscillations with external mechanical oscillations.

Fractional Approach for Belousov-Zhabotinsky Reactions Model with Unified Technique

Fractional Approach for Belousov-Zhabotinsky Reactions Model with Unified Technique

A comparative analysis of Noyes-Field model for the non-linear Belousov–Zhabotinsky reaction using two reliable techniques

Within the domain of nonlinear dynamics, the Belousov-Zhabotinsky reaction system has consistently captivated researchers, sustaining its position as a vibrant and dynamic area of exploration. As an enduring subject of study, the Belousov-Zhabotinsky system provides continuous opportunities to unravel the foundational tenets of nonlinear dynamics within intricate systems. In our quest to deepen comprehension of this complex system, we introduce an innovative methodology for addressing the time fractional Belousov-Zhabotinsky system. This novel approach leverages both the Natural Transform Iterative Method (NTIM) and the Optimal Homotopy Asymptotic Method (OHAM), aiming to contribute novel insights and methodologies to the ongoing discourse surrounding this intriguing and influential area of research. We obtained a series solution to test the accuracy of the suggested approach. The proposed technique has the advantage of requiring few calculations while producing high-precision results. To help you better understand, 2D and 3D graphical representations are provided to demonstrate the model's behavior and how changing the fractional order derivative in Caputo's sense and time affects the solutions.

A programmable hybrid digital chemical information processor based on the Belousov-Zhabotinsky reaction

The exponential growth of the power of modern digital computers is based upon the miniaturization of vast nanoscale arrays of electronic switches, but this will be eventually constrained by fabrication limits and power dissipation. Chemical processes have the potential to scale beyond these limits by performing computations through chemical reactions, yet the lack of well-defined programmability limits their scalability and performance. Here, we present a hybrid digitally programmable chemical array as a probabilistic computational machine that uses chemical oscillators using Belousov-Zhabotinsky reaction partitioned in interconnected cells as a computational substrate. This hybrid architecture performs efficient computation by distributing information between chemical and digital domains together with inbuilt error correction logic. The efficiency is gained by combining digital logic with probabilistic chemical logic based on nearest neighbour interactions and hysteresis effects. We demonstrated the computational capabilities of our hybrid processor by implementing one- and two-dimensional Chemical Cellular Automata demonstrating emergent dynamics of life-like entities called Chemits. Additionally, we demonstrate hybrid probabilistic logic as a viable logic for solving combinatorial optimization problems.

Active Pulsatile Gels: From a Chemical Microreactor to a Polymeric Actuator.

We report on a synthesis protocol, experimental characterization, and theoretical modeling of active pulsatile Belousov-Zhabotinsky (BZ) hydrogels. Our two-step synthesis technique allows independent optimization of the geometry, the chemical, and the mechanical properties of BZ gels. We identify the role of the surrounding medium chemistry and gel radius for the occurrence of BZ gel oscillations, quantified by the Damköhler number, which is the ratio of chemical reaction to diffusion rates. Tuning the BZ gel size to maximize its chemomechanical oscillation amplitude, we find that its oscillatory strain amplitude is limited by the time scale of gel swelling relative to the chemical oscillation period. Our experimental findings are in good agreement with a Vanag-Epstein model of BZ chemistry and a Tanaka Fillmore theory of gel swelling dynamics.

Collective chemomechanical oscillations in active hydrogels

We report on the collective response of an assembly of chemomechanical Belousov-Zhabotinsky (BZ) hydrogel beads. We first demonstrate that a single isolated spherical BZ hydrogel bead with a radius below a critical value does not oscillate, whereas an assembly of the same BZ hydrogel beads presents chemical oscillation. A BZ chemical model with an additional flux of chemicals out of the BZ hydrogel captures the experimentally observed transition from oxidized nonoscillating to oscillating BZ hydrogels and shows this transition is due to a flux of inhibitors out of the BZ hydrogel. The model also captures the role of neighboring BZ hydrogel beads in decreasing the critical size for an assembly of BZ hydrogel beads to oscillate. We finally leverage the quorum sensing behavior of the collective to trigger their chemomechanical oscillation and discuss how this collective effect can be used to enhance the oscillatory strain of these active BZ hydrogels. These findings could help guide the eventual fabrication of a swarm of autonomous, communicating, and motile hydrogels.

Photoinduced Nonlinear Dynamics of Strongly Correlated Systems with Spin Crossover: Autocatalytic Spin Transition

Nonlinear phenomena similar to the Belousov–Zhabotinsky reaction (autocatalytic oscillations of the population of high-spin and low-spin multielectron states of a transition metal ion) in open systems with spin crossover near bistability are considered. The conditions for possible experimental observation of autocatalytic oscillations of the magnetization in magnetically ordered systems with spin crossover are analyzed.

Bistable Wavefronts in the Delayed Belousov–Zhabotinsky Reaction

Bistable Wavefronts in the Delayed Belousov–Zhabotinsky Reaction

Neuromorphic Engineering in Wetware: Discriminating Acoustic Frequencies through Their Effects on Chemical Waves.

Some features of the human nervous system can be mimicked not only through software or hardware but also through liquid solutions of chemical systems maintained under out-of-equilibrium conditions. We describe the possibility of exploiting a thin layer of the Belousov-Zhabotinsky (BZ) reaction as a surrogate for the cochlea for sensing acoustic frequencies. Experiments and simulations demonstrate that, as in the human ear where the cochlea transduces the mechanical energy of the acoustic frequencies into the electrochemical energy of neural action potentials and the basilar membrane originates topographic representations of sounds, our bioinspired chemoacoustic system, based on the BZ reaction, gives rise to spatiotemporal patterns as the representation of distinct acoustic bands through transduction of mechanical energy into chemical energy. Acoustic frequencies in the range 10-2000 Hz are partitioned into seven distinct bands based on three attributes of the emerging spatiotemporal patterns: (1) the types and frequencies of the chemical waves, (2) their velocities, and (3) the Faraday waves' wavelengths.

Analytical and numerical computation of self-oscillating gels driven by the Belousov-Zhabotinsky reaction

Self-oscillating gel is a class of deformable polymers driven by Belousov-Zhabotinsky (BZ) reactions, which can form periodic deformations without any external stimuli, and are widely used in the research of micro actuators, AI sensors, drug release carriers or biomimetic materials. However, quantitative study on formation of the self-oscillating gel is limited especially from the perspective of energy conservation. This work adopts frequency domain analysis to the chemo-mechanical model, and the basic frequency is obtained to evaluate the maintenance energy of the deformable gel. For accurate computation, boundary value problem with unknown period is formulated; then, continuation algorithm based on technique of perturbation is performed to obtain the periodic trajectories with varying model parameters. The results could be implemented to design self-oscillating gels with prescribed periodicity.

Chemistry does general relativity: reaction-diffusion waves can model gravitational lensing

Gravitational lensing is a general relativistic (GR) phenomenon where a massive object redirects light, deflecting, magnifying, and sometimes multiplying its source. We use reaction-diffusion (RD) Belousov-Zhabotinsky (BZ) chemistry to study this astronomical effect in a table-top experiment. We experimentally observe BZ waves passing through non-planar, quasi-two-dimensional molds and reproduce the waveforms in computer simulations using planar RD waves propagating with variable diffusion. We tune the variable diffusion to match the Schwarzschild-coordinate light speed near a spherical mass so the RD propagation approximates Einstein’s famous light deflection relation. We discuss varying the diffusion or reaction rates with a gel matrix or with illumination, electric field, or temperature gradients.

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.

Anisotropically self-oscillating gels by spatially patterned interpenetrating polymer network.

Here we introduce sub-millimeter self-oscillating gels that undergo the Belousov-Zhabotinsky (BZ) reaction and can anisotropically oscillate like cardiomyocytes. The anisotropically self-oscillating gels in this study were realized by spatially patterning an acrylic acid-based interpenetrating network (AA-IPN). We found that the patterned AA-IPN regions, locally introduced at both ends of the gels through UV photolithography, can constrain the horizontal gel shape deformation during the BZ reaction. In other words, the two AA-IPN regions could act as a physical barrier to prevent isotropic deformation. Furthermore, we controlled the anisotropic deformation behavior during the BZ reaction by varying the concentration of acrylic acid used in the patterning process of the AA-IPN. As a result, a specific directional deformation behavior (66% horizontal/vertical amplitude ratio) was fulfilled, similar to that of cardiomyocytes. Our study can provide a promising insight to fabricating robust gel systems for cardiomyocyte modeling or designing novel autonomous microscale soft actuators.

Localized pattern formation: semi-strong interaction asymptotic analysis for three components model

We investigate a three-component system involving the Belousov–Zhabotinsky reaction in water-in-oil microemulsions. Our goal is to investigate the connection between homoclinic snaking and semi-strength interaction in a three-variable reaction–diffusion system. A two-parameter bifurcation diagram of homogeneous, periodic and localized patterns is obtained numerically, and a natural asymptotic scaling for semi-strong interaction theory is found where an activator source term a = O ( δ 1 ) and b = O ( δ 1 ) , with δ 1 ≪ 1 being the diffusion ratio. Under this regime, singular perturbation techniques are used to construct localized steady-state patterns, and new types of non-local eigenvalue problems (NLEP) are derived to determine the stability of these patterns to O ( 1 ) time-scale instabilities. We extend the scope of the NLEP by considering a general scenario where both time-scaling parameters are non-zero. All analytical results are found to agree with numerics. Further numerical results are presented on the location of various types of breathing Hopf instability for localized patterns.

Coevolution Analysis of the Sustainable Development Systems of Energy Firms From a Strategic Philanthropy Perspective

As the objectives of the “dual carbon goal”—reaching peak carbon emissions by 2030 and achieving carbon neutrality by 2060—continue to be promoted, the fulfilment of social responsibilities is critical in enabling energy firms to achieve sustainable development goals. However, due to the frequent occurrence of nonstrategic charitable behaviors, energy firms cannot effectively manage the outcomes of charity. This affects the sustainable development processes of energy firms. While previous studies have mainly focused on the assessment of the economic, social, and environmental dimensions of the static evaluation of sustainable development, few studies focus on the coevolution analysis of the triple combination. To explore the coevolution of the sustainable development systems of energy firms from the perspective of strategic philanthropy, based on triangular fuzzy numbers, a multiattribute evaluation system has been used. This system integrates strategy features, sustainability features, adaptability features, and matching degrees. It has been used to improve the traditional Belousov–Zhabotinsky (B-Z) reaction model, otherwise known as complex system theory. The coevolution model of sustainable development systems stemmed from the perspective of strategic philanthropy. The feasibility of the dynamic model was verified using data from Chinese energy enterprises (Firm-G and Firm-S) gathered between 2011 and 2020. The improved “B-Z” model proposed in this study is able to dynamically simulate the sustainable coevolution track of energy firms and the action mechanism of strategic philanthropy, which provides new decision-making ideas and methods to enable energy firms to effectively manage the behaviors and effects of strategic philanthropy.

Energy and power characteristics of nanocatalyzed Belousov-Zhabotinsky reactions via bifurcation analyses.

Active stimuli-responsive materials, intrinsically powered by chemical reactions, have immense capabilities that can be harnessed for designing synthetic systems for a variety of biomimetic applications. It goes without saying that the key aspect involved in the designing of such systems is to accurately estimate the amount of energy and power available for transduction through various mechanisms. Belousov-Zhabotinsky (BZ) reactions are dynamical systems, which exhibit self-sustained nonlinear chemical oscillations, as their catalyst undergoes periodic redox cycles in the presence of reagents. The unique feature of BZ reaction based active systems is that they can continuously perform mechanical work by transducing energy from sustained chemical oscillations. The objective of our work is to use bifurcation analyses to identify oscillatory regimes and quantify energy-power characteristics of the BZ reaction based on nanocatalyst activity and BZ reaction formulations. Our approach involves not only the computation of higher order Lyapunov and frequency coefficients but also Hamiltonian functions, through normal form reduction of the kinetic model of the BZ reaction. Ultimately, using these calculations, we determine amplitude, frequency, and energy-power densities, as a function of the nanocatalysts' activity and BZ formulations. As normal form representations are applicable to any dynamical system, we believe that our framework can be extended to other self-sustained active systems, including systems based on stimuli-responsive materials.

Insights into time fractional dynamics in the Belousov-Zhabotinsky system through singular and non-singular kernels

In the realm of nonlinear dynamics, the Belousov-Zhabotinsky reaction system has long held the fascination of researchers. The Belousov-Zhabotinsky system continues to be an active area of research, offering insights into the fundamental principles of nonlinear dynamics in complex systems. To deepen our understanding of this intricate system, we introduce a pioneering approach to tackle the time fractional Belousov-Zhabotinsky system, employing the Caputo and Atangana-Baleanu Caputo fractional derivatives with the double Laplace method. The solution we obtained is in the form of series which helps in investigating the accuracy of the proposed method. The primary advantage of the proposed technique lies in the low amount of calculations required and produce high degree of precision in the solutions. Furthermore, the existence and uniqueness of the solution are investigated thereby enhancing the overall credibility of our study. To visually represent our results, we present a series of 2D and 3D graphical representations that vividly illustrate the behavior of the model and the impact of changing the fractional order derivative and the time on the obtained solutions.