Reaction-diffusion Processes Research Articles

Controlling the Spatiotemporal Self-Organization of Stimuli-Responsive Nanocrystals under Out-of-Equilibrium Conditions.

Self-organization under out-of-equilibrium conditions is ubiquitous in natural systems for the generation of hierarchical solid-state patterns of complex structures with intricate properties. Efforts in applying this strategy to synthetic materials that mimic biological function have resulted in remarkable demonstrations of programmable self-healing and adaptive materials. However, the extension of these efforts to multifunctional stimuli-responsive solid-state materials across defined spatial distributions remains an unrealized technological opportunity. This paper describes the use of a nonequilibrium reaction-diffusion process to achieve the synthesis of a multifunctional stimuli-responsive electrically conductive metal-organic framework (cMOF) in a gelled medium with control over particle size and spatial periodicity on a macroscopic scale. Upon integration into chemiresistive devices, the resulting cMOF particles exhibit a size-dependent response toward hydrogen sulfide gas, as determined by their distinct surface-to-volume ratio, porosity, unique synthesis methodology, and unusual microcrystallite morphology compared to their counterparts obtained through bulk solution phase synthesis. Taken altogether, these achievements pave the way toward gaining access to functional nanomaterials with well-defined chemical composition, dimensions, and precisely tailored functions using far-from-equilibrium approaches.

Reaction-Diffusion and Crystallization Kinetics Modulation of Two-Step Deposited Tin-Based Perovskite Film via Reducing Atmosphere.

The two-step deposition method effectively mitigates the efficiency decline observed in tin-based perovskite solar cells (TPVSCs) with increasing cell area, stemming from film in-homogeneity. However, the high solubility of SnI2 in the conventionally used solvent isopropyl alcohol, coupled with the absence of effective modulation of reaction-diffusion process, results in inadequate film coverage and conversion. In this study, we introduce formic acid as the second-step solvent and introduce dithiothreitol (DTT) to regulate reaction-diffusion/crystallization kinetics meticulously. Moreover, this research underscores a fundamental principle that the suitable binding energy ranging from -1.38 to -10.10 kcal mol-1 between ligands and Sn2+ significantly enhances the effectiveness of two-step crystallization control. Notably, a uniform perovskite film is achieved on large-scale substrate, and TPVSCs processed with DTT exhibit the highest efficiencies of 12.68 % for 0.04 cm2 device and 11.30 % for 1 cm2 device among tin-based perovskite devices in two-step sequential deposition method, even in the absence of dimethyl sulfoxide. This study lays the groundwork for the potential scale-up development of lead-free perovskite solar cells.

Assessing the vulnerability of empirical infrastructure networks to natural catastrophes

Human-made infrastructures are complex systems continually exposed to events that threat their function, such as cascading failures, occurring when the flow of physical quantities is redistributed within the network as a consequence of localized disruptions. Nevertheless, the role played by exogenous catastrophic events and internal failures on the robustness of critical infrastructures is usually addressed independently, under simplifying assumptions and lacking a unified picture for realistic risk assessments. Here, we fill this gap by introducing the Operational-Affected-Dismantled (OAD) model that captures both local and nonlocal failure propagation mechanisms. The model combines reaction–diffusion processes for local spreading with a global field effect for long-range interactions, allowing us to quantitatively characterize the cascade dynamics in infrastructure networks. Moreover, we include information on external stressors to assess the robustness of empirical network infrastructures and build spatial risk maps. By using data from severe storms (2009–2016) and from earthquakes (2000–2023) as stressors of the North American power grid and the worldwide airline transportation system, respectively, we offer a quantitative way to rank events by their potential to trigger systemic effects. By analyzing the response of the European power grid to simulated severe storms, we find that it can show high levels of systemic risk. Uncertainty in global climate and the accelerating frequency of extreme events all over the globe call for novel strategies to quantify, adapt to and mitigate systemic risk. Our framework provides a suitable starting point to assess the robustness of empirical systems in realistic and what-if scenarios.

Random attractors for fractional stochastic reaction–diffusion systems with fractional Brownian motion

Fractional stochastic reaction–diffusion systems involving fractional Brownian motion (fBm) provide a comprehensive modeling framework for studying complex physical and biological phenomena. In this study, we investigate the dynamic behavior of solutions to fractional stochastic reaction–diffusion systems with fBm defined on Rn. Firstly, we establish the existence and uniqueness of solutions to the fractional stochastic reaction–diffusion systems with fBm. We also obtain uniform estimates for the solutions on average. Furthermore, we construct a mean random dynamical system based on the derived solutions. Finally, we prove the existence and uniqueness of weak pullback mean random attractors. The results of our investigation contribute to a deeper understanding of the complexities involved in reaction–diffusion processes, specifically considering the effect of fBm with long-range dependence and self-similarity properties. Additionally, our findings have broader applications, particularly in areas such as analyzing the dynamic behavior of complex systems and biological models.

Catalyst pellets with Gaussian activity distribution under forced periodic operation for reactions with Langmuir-Hinshelwood kinetics

This research investigates how combining forced periodic operation with spatially distributed catalyst activity can enhance heterogeneous catalytic processes. It focuses on analyzing reaction-diffusion phenomena within porous catalyst pellets. These pellets exhibit a Gaussian distribution of active sites, and the study investigates how externally forced periodic variations in bulk reactant concentration and temperature affect the reaction process. The paper establishes a mathematical model for a non-isothermal reaction based on Langmuir-Hinshelwood kinetics. This model is then transformed into its dimensionless form for numerical analysis. Numerical simulations are employed to investigate the impact of various parameters on the concentration and temperature profiles within the pellet, as well as on the pellet productivity. These parameters include the position and width of the Gaussian distribution of active sites, the Thiele modulus, the mass and heat Biot numbers, the Arrhenius number for reaction, the energy generation function, the ratio of characteristic times for diffusion and heat conductivity, and frequencies and amplitudes of periodic variations. The simulations reveal complex relationships between the spatial and temporal profiles of concentration and temperature within the pellets. Using porous granules with a non-uniform catalyst activity profile alongside forced periodic operations for reaction-diffusion processes enables higher productivity compared to granules with a uniform activity profile and subject to the steady-state operation. This study demonstrates the potential for optimizing catalytic processes in porous pellets with non-uniform activity profiles under forced periodic operation, offering valuable insights into enhancing process efficiency.

Enzyme-Responsive DNA Condensates.

Membrane-less compartments and organelles are widely acknowledged for their role in regulating cellular processes, and there is an urgent need to harness their full potential as both structural and functional elements of synthetic cells. Despite rapid progress, synthetically recapitulating the nonequilibrium, spatially distributed responses of natural membrane-less organelles remains elusive. Here, we demonstrate that the activity of nucleic-acid cleaving enzymes can be localized within DNA-based membrane-less compartments by sequestering the respective DNA or RNA substrates. Reaction-diffusion processes lead to complex nonequilibrium patterns, dependent on enzyme concentration. By arresting similar dynamic patterns, we spatially organize different substrates in concentric subcompartments, which can be then selectively addressed by different enzymes, demonstrating spatial distribution of enzymatic activity. Besides expanding our ability to engineer advanced biomimetic functions in synthetic membrane-less organelles, our results may facilitate the deployment of DNA-based condensates as microbioreactors or platforms for the detection and quantitation of enzymes and nucleic acids.

Learning-based optimal boundary control for parabolic distributed parameter system with actuator dynamics

Considering actuator dynamics, we investigate a coupled system of parabolic partial differential equations (PDEs) and ordinary differential equations (ODEs), developing a data-driven boundary optimal controller based on iterative learning. Notably, the boundary input appears in the ODE-style actuator dynamics, making the boundary condition reduction quite difficult. Optimizing infinite-dimensional performance indexes in coupled Hilbert spaces is not a trivial task. This work is the first to solve the optimal control problem for a coupled PDE-ODE system with actuator dynamics under Neumann boundary conditions. We equivalently reformat the coupled PDE-ODE system into a system with homogeneous boundary conditions and then derive its singular perturbation form in an infinite-dimensional space. Subsequently, by constructing critic and actor networks, we design a novel model-free iterative learning optimal control algorithm where weighted residual techniques are used. The algorithm uses a rich set of arbitrary control policies rather than limiting to evaluation policies, enhancing the exploration capability of the learning algorithm and relaxing the requirement for persistent excitation conditions. Furthermore, the uniformly asymptotic stability of the closed-loop coupled system is demonstrated in the infinite-dimensional Hilbert space for each learning iteration, not only for the final one. Finally, the effectiveness of the proposed approach is verified by simulations on the diffusion-reaction process.

Understanding the nonlinear reactive transport model in porous catalysts

This paper discusses the steady-state nonlinear reaction-diffusion processes in porous catalysts. This model is based on a nonlinear second-order differential equation that includes a nonlinear term associated with the Michaelis-Menten and non-Michaelis-Menten kinetics of the reaction. The nonlinear equations can be approximately solved using the Taylors series, modified Taylors series and Akabri-Ganji technique to obtain the concentration of dissolved species. The influence of the half-saturation parameter and the characteristic reaction rate on concentration is explored. Sensitivity analysis of parameters is discussed. Our analytical findings were compared with numerical solutions.

Dynamics behaviours of N-kink solitons in conformable Fisher–Kolmogorov–Petrovskii–Piskunov equation

PurposeThis manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.Design/methodology/approachThe generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.FindingsThe results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implicationsThe work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.Practical implicationsThe generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.Social implicationsPublic health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/valueThe use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

Applications of Optimal Transportation

The mathematical theory of optimal transportation is constantly expanding its range of application, while applications give impulses for new research directions in the field. This workshop was specifically devoted to applications of optimal transportation in the natural sciences, and to the recent developments of the theory that have been motivated by these. The bouquet of current applications includes mathematical models for large-scale air motion, dynamics of plasmas, material design, pattern formation in fluids, collective behaviour in biology, and many more. Related theoretical developments are in the analysis of the Hellinger-Kantorovich metric for modeling reaction–diffusion processes, and in efficient numerical methods for multi-marginal optimal transport, to name two of many examples encountered in this workshop.

Traveling waves reflecting various processes represented by reaction–diffusion equations

The aim of this paper is to discover analytically the interactional responses of populations in a dynamic region where the reaction–diffusion process with forcing effects takes place through traveling wave solutions. An expansion method is considered here to properly capture the responses for the first time. In order to profoundly analyze the physical and mathematical discussions, some illustrative behavioral results are exhibited for various values of physical parameters. Especially for the different values of diffusion coefficients in the model under consideration, their effects on the behavior of the solitary wave are discussed and observationally supported by considering various illustrations. It is also seen that the solutions representing the diffusion seen to be in the form of the behavior of hexagonal Turing patterns in different time periods. The application of this study in mathematical biology is to analyze the relationship between the population density of certain species in any local region and the specific population density with invasion characteristics. In addition, the formation of the extinction vortex of the invading population, depending on the characteristics of the solutions presented, is also descriptively discussed.

Anisotropic corrosion behavior of laser-based direct energy deposited H13 steel in molten ADC12 aluminum alloy

The immersion corrosion experiment of laser-based direct energy deposited H13 steel in ADC12 aluminum alloy melt was conducted to evaluate the demands for corrosion resistance after mold repair. The effect of the anisotropic microstructure of the deposited H13 steel on corrosion behavior was elucidated by characterizing the microstructure before and after corrosion. The results show that the microstructural heterogeneity affects the initiation, development and stabilization process of corrosion. Corrosion tends to initiate near the heat-affected zone due to the heterogeneous distribution of grain size, carbides, and dislocation density. Planes with weak microstructural heterogeneity are less likely to develop cracks in the intermetallic compounds (IMCs) layer during the corrosion development stage, thus exhibiting a slower corrosion rate. The IMCs layer of plane with weak microstructural heterogeneity stabilizes more quickly during the corrosion stabilization stage. The plane perpendicular to the build direction exhibits the highest corrosion resistance due to fewer corrosion initiation locations and the weakest microstructural heterogeneity. Furthermore, the composite structure of carbide core + Si phase shell, generated during corrosion, contributes to improving localized corrosion resistance. This study provides insights into understanding the influence of the heterogeneous microstructure from additive manufacturing on the reaction-diffusion process and offers guidance on direction selection when preparing components for applications in contact with aluminum melts using additive manufacturing techniques.

Electric-field assisted cascade reactions to create alginate/carboxymethyl chitosan composite hydrogels with gradient architecture and reconfigurable mechanical properties

Rational designs of polysaccharide-based hydrogels with organ-like three-dimensional architecture provide a great possibility for addressing the shortages of allograft tissues and organs. However, spatial-temporal control over structure in bulk hydrogel and acquire satisfied mechanical properties remain an intrinsic challenge to achieve. Here, we show how electric-field assisted molecular self-assembly can be coupled to a directional reaction-diffusion (RD) process to produce macroscopic hydrogel in a controllable manner. The electrical energy input was not only to generate complex molecule gradients and initiate the molecular self-assembly, but also to guide/facilitate the RD processes for the gel rapid growth via a cascade construction interaction. The hydrogel mechanical properties can be tuned and enhanced by using an interpenetrating biopolymer network and multiple ionic crosslinkers, leading to a wide-range of mechanical modulus to match with biological organs or tissues. We demonstrate diverse 3D macroscopic hydrogels can be easily prepared via field-assisted directional reaction-diffusion and specific joint interactions. The humility-triggered dissipation of functional gradients and antibacterial performance confirm that the hydrogels can serve as an optically variable soft device for wound management. Therefore, this work provides a general approach toward the rational fabrication of soft hydrogels with controlled architectures and functionality for advanced biomedical systems.

Biological invasions and epidemics with nonlocal diffusion along a line.

The goal of this work is to understand and quantify how a line with nonlocal diffusion given by an integral enhances a reaction-diffusion process occurring in the surrounding plane. This is part of a long term programme where we aim at modelling, in a mathematically rigorous way, the effect of transportation networks on the speed of biological invasions or propagation of epidemics. We prove the existence of a global propagation speed and characterise in terms of the parameters of the system the situations where such a speed is boosted by the presence of the line. In the course of the study we also uncover unexpected regularity properties of the model. On the quantitative side, the two main parameters are the intensity of the diffusion kernel and the characteristic size of its support. One outcome of this work is that the propagation speed will significantly be enhanced even if only one of the two is large, thus broadening the picture that we have already drawn in our previous works on the subject, with local diffusion modelled by a standard Laplacian. We further investigate the role of the other parameters, enlightening some subtle effects due to the interplay between the diffusion in the half plane and that on the line. Lastly, in the context of propagation of epidemics, we also discuss the model where, instead of a diffusion, displacement on the line comes from a pure transport term.

Gradient-Based Monte Carlo Methods for Relaxation Approximations of Hyperbolic Conservation Laws

Particle methods based on evolving the spatial derivatives of the solution were originally introduced to simulate reaction-diffusion processes, inspired by vortex methods for the Navier–Stokes equations. Such methods, referred to as gradient random walk methods, were extensively studied in the ’90s and have several interesting features, such as being grid-free, automatically adapting to the solution by concentrating elements where the gradient is large, and significantly reducing the variance of the standard random walk approach. In this work, we revive these ideas by showing how to generalize the approach to a larger class of partial differential equations, including hyperbolic systems of conservation laws. To achieve this goal, we first extend the classical Monte Carlo method to relaxation approximation of systems of conservation laws, and subsequently consider a novel particle dynamics based on the spatial derivatives of the solution. The methodology, combined with asymptotic-preserving splitting discretization, yields a way to construct a new class of gradient-based Monte Carlo methods for hyperbolic systems of conservation laws. Several results in one spatial dimension for scalar equations and systems of conservation laws show that the new methods are very promising and yield remarkable improvements compared to standard Monte Carlo approaches, either in terms of variance reduction as well as in describing the shock structure.

A study using a semi-analytical approach to the reaction kinetics that describes the behavior of heterogeneous reaction–diffusion process with a glyoxyl-agarose immobilized enzyme (penicillin G acylase) in a batch reactor

This research article examines the reaction–diffusion process in a batch reactor by means of immobilized penicillin G acylase (PGA) with a glyoxyl-agarose matrix support. This study employs the Akbari-Ganji’s Method (AGM) when the system is under the internal (intra-particle) diffusional restrictions, and the external (film) diffusional restrictions is negligible. The semi-analytical approximations by the AGM technique are employed for computing the dimensionless steady-state solutions for a system of nonlinear differential equations to examine the impact of internal diffusional restrictions (IDR) as a function of catalyst enzyme loading and particle size. The model entails highly nonlinear components that adhere to the standard Michaelis-Menten kinetics. In addition to the proposed model, the dynamics for the mean integrated effectiveness factor (MIEF) of the dimensionless substrate concentration phenylglycine methyl ester (PGME) is presented. In light of this, there is a satisfactory level of agreement on the comparison of the semi-analytical result with the simulated numerical results across the whole concentration range where the dimensionless initial substrate and product concentrations are S1b=30,S2b=15, and P1b=60,P2b=100. The behavior of concentration profiles under the steady/unsteady state conditions for the effect of IDR on the overall reaction rates by immobilized biocatalyst PGA were examined. A sensitivity analysis is carried out to identify the effective parameter which can influence the diffusional restrictions on MIEF. An error analysis is performed on the dimensionless concentration equations to assess the amount of accuracy and the convergence of a solution.

Efficient and scalable prediction of stochastic reaction–diffusion processes using graph neural networks

The dynamics of locally interacting particles that are distributed in space give rise to a multitude of complex behaviours. However the simulation of reaction–diffusion processes which model such systems is highly computationally expensive, the cost increasing rapidly with the size of space. Here, we devise a graph neural network based approach that uses cheap Monte Carlo simulations of reaction–diffusion processes in a small space to cast predictions of the dynamics of the same processes in a much larger and complex space, including spaces modelled by networks with heterogeneous topology. By applying the method to two biological examples, we show that it leads to accurate results in a small fraction of the computation time of standard stochastic simulation methods. The scalability and accuracy of the method suggest it is a promising approach for studying reaction–diffusion processes in complex spatial domains such as those modelling biochemical reactions, population evolution and epidemic spreading.

Emergence of power-law distributions in self-segregation reaction-diffusion processes.

Many natural or human-made systems encompassing local reactions and diffusion processes exhibit spatially distributed patterns of some relevant dynamical variable. These interactions, through self-organization and critical phenomena, give rise to power-law distributions, where emergent patterns and structures become visible across vastly different scales. Recent observations reveal power-law distributions in the spatial organization of, e.g., tree clusters and forest patch sizes. Crucially, these patterns do not follow a spatially periodic order but rather a statistical one. Unlike the spatially periodic patterns elucidated by the Turing mechanism, the statistical order of these particular vegetation patterns suggests an incomplete understanding of the underlying mechanisms. Here, we present a self-segregation mechanism, driving the emergence of power-law scalings in pattern-forming systems. The model incorporates an Allee-logistic reaction term, responsible for the local growth, and a nonlinear diffusion process accounting for positive interactions and limited resources. According to a self-organized criticality (SOC) principle, after an initial decrease, the system mass reaches an analytically predictable threshold, beyond which it self-segregates into distinct clusters, due to local positive interactions that promote cooperation. Numerical investigations show that the distribution of cluster sizes obeys a power law with an exponential cutoff.

Preparation of surgical meshes using self-regulating technology based on reaction-diffusion processes

While reaction-diffusion processes are utilized in multiple scientific fields, these phenomena have seen limited practical application in the polymer industry. Although self-regulating processes driven by parallel reaction and diffusion can lead to patterned structures, most polymeric products with repeating subunits are still prepared by methods that require complex and expensive instrumentation. A notable, high-added-value example is surgical mesh, which is often manufactured by weaving or knitting. In our present work, we demonstrate how the polymer and the biomedical industry can benefit from the pattern-forming capabilities of reaction-diffusion. We would like to propose a self-regulating method that facilitates the creation of surgical meshes from biocompatible polymers. Since the control of the process assumes a thorough understanding of the underlying phenomena, the theoretical background, as well as a mathematical model that can accurately describe the empirical data, is also introduced and explained. Our method offers the benefits of conventional techniques while introducing additional advantages not attainable with them. Most importantly, the method proposed in this paper enables the rapid creation of meshes with an average pore size that can be adjusted easily and tailored to fit the intended area of application.Graphical abstract

A Naturally Inspired Extrusion-Based Microfluidic Approach for Manufacturing Tailorable Magnetic Soft Continuum Microrobotic Devices.

Soft materials play a crucial role in small-scale robotic applications by closely mimicking the complex motion and morphing behavior of organisms. However, conventional fabrication methods face challenges in creating highly integrated small-scale soft devices. In this study, microfluidics is leveraged to precisely control reaction-diffusion (RD) processes to generate multifunctional and compartmentalized calcium-cross-linkable alginate-based microfibers. Under RD conditions, sophisticated alginate-based fibers are produced for magnetic soft continuum robotics applications with customizable features, such as geometry (compact or hollow), degree of cross-linking, and the precise localization of magnetic nanoparticles (inside the core, surrounding the fiber, or on one side). This fine control allows for tuning the stiffness and magnetic responsiveness of the microfibers. Additionally, chemically cleavable regions within the fibers enable disassembly into smaller robotic units or roll-up structures under a rotating magnetic field. These findings demonstrate the versatility of microfluidics in processing highly integrated small-scale devices.