1D Reaction-diffusion Research Articles

Enhancing Neural Network Training For High Accuracy In Solving Singularly Perturbed 1D Reaction-Diffusion Equations

Enhancing Neural Network Training For High Accuracy In Solving Singularly Perturbed 1D Reaction-Diffusion Equations

Hydroxide-Source-Dependent Polymorphism and Phase Stability of Cobalt(II) Hydroxides in Diffusion-Driven Systems.

Hydroxides of cobalt(II) exist predominantly in two polymorphic forms, namely, the blue-green α-form [α-Co(OH)2] and reddish β-form [β-Co(OH)2]. These hydroxides have a layered structure with interlayer galleries of around 7 and 4 Å, respectively, for α- and β-Co(OH)2. In most of the previous studies, both the polymorphs were synthesized separately, and a few of them showed that the α-form gets converted to a thermodynamically more stable β-form via physical processes. In the present work, we have optimized the conditions for the simultaneous synthesis of both polymorphs under identical conditions in the same reactor using the 1D reaction-diffusion framework by employing different outer electrolytes. We found that the polymorph chemistry of Co(OH)2 depends on the source and concentration of OH- rather than other reaction conditions or later physical transformation. The products are characterized to confirm their morphology, structure, and chemical environment. We observed that the use of NaOH and NH4OH as the OH- precursor leads to α-Co(OH)2 only; however, with NaOH, a continuous precipitate is formed, and with NH4OH, periodic precipitation is formed. On the other hand, with hydrazine (HYZ) as the OH- source, Liesegang bands of α-Co(OH)2 and β-Co(OH)2 as granules are formed throughout the diffusion reactor. Another intriguing observation on the HYZ system is that at its high concentration, the bands of α-Co(OH)2 get converted to β-Co(OH)2. We articulate the reasons and mechanism of those observations.

Influence of nanocolumnar electrode geometry on electrochemical sensor performance

The concentration of glucose in sweat recently has been measured with high sensitivity, selectivity, and reproducibility by nanostructured NiO electrodes manufactured by the glancing angle deposition (GLAD) technique. The GLAD technique allows electrode morphological properties such as porosity and film thickness to be tightly controlled, providing ample opportunity to enhance the sensor performance. Currently, the selection of optimal GLAD parameters is determined experimentally by trial and error, at high costs in time and resources. Numerical simulation allows the effects of various parameters on sensor performance to be investigated at a much lower cost compared to experimental studies. In this work, a 2D reaction–diffusion model for the surface-catalyzed reactions in the nanostructured GLAD electrodes is developed, which are then solved using the finite element method. Parametric studies on the nanocolumn thickness and nanocolumn separation of the GLAD structures are then conducted to optimize GLAD-based electrode structures with different adsorption and catalytic rates. This research offers new guidance for rapidly designing highly effective sensors with higher sensitivity and lower limits of detection.

Investigating the coupling between transport and reaction within a catalyst pellet using operando magnetic resonance spectroscopic imaging

In heterogeneous catalysis, the distribution of reactant and product species within a catalyst pellet are of central importance to the catalyst performance, and ultimately to the overall conversion and selectivity achieved in commercial-scale reactors. Despite this importance, fundamental understanding of pellet scale behaviour at operando conditions is limited due to the coupled, multi-scale nature of the underlying transport processes, and the limited techniques capable of mapping the intra-pellet composition at industrially relevant length scales. Here, we demonstrate the coupling between transport and reaction at the pellet scale by using operando magnetic resonance spectroscopic imaging (MRSI) to map the liquid composition within Pd/γ-Al2O3 catalyst pellets at an isotropic in-plane spatial resolution of 86 µm during a styrene hydrogenation reaction. The results reveal that asymmetric liquid wetting at the pellet surface, typical of that occurring in trickle-bed reactors, causes significant composition heterogeneity across a catalyst pellet. Due to the wetting inhomogeneity, commonly used 1D reaction-diffusion models are inadequate for describing the observed composition heterogeneity across the pellet. The results reported here provide unique experimental insight into the coupling between transport and reaction at the pellet scale within an operating reactor.

Physics-informed neural networks for the reaction-diffusion Brusselator model

In this work, we are interesting in solving the 1D and 2D nonlinear stiff reaction-diffusion Brusselator system using a machine learning technique called Physics-Informed Neural Networks (PINNs). PINN has been successful in a variety of science and engineering disciplines due to its ability of encoding physical laws, given by the PDE, into the neural network loss function in a way where the network must not only conform to the measurements, initial and boundary conditions, but also satisfy the governing equations. The utilization of PINN for Brusselator system is still in its infancy, with many questions to resolve. Performance of the framework is tested by solving some one and two dimensional problems with comparable numerical or analytical results. Validation of the results is investigated in terms of absolute error. The results showed that our PINN has well performed by producing a good accuracy on the given problems.

Adaptive learning of effective dynamics for online modeling of complex systems

Predictive simulations are essential for applications ranging from weather forecasting to material design. The veracity of these simulations hinges on their capacity to capture the effective system dynamics. Massively parallel simulations predict the systems dynamics by resolving all spatiotemporal scales, often at a cost that prevents experimentation. On the other hand, reduced order models are fast but often limited by the linearization of the system dynamics and the adopted heuristic closures. We propose a novel systematic framework that bridges large scale simulations and reduced order models to extract and forecast adaptively the effective dynamics (AdaLED) of multiscale systems. AdaLED employs an autoencoder to identify reduced-order representations of the system dynamics and an ensemble of probabilistic recurrent neural networks (RNNs) as the latent time-stepper. The framework alternates between the computational solver and the surrogate, accelerating learned dynamics while leaving yet-to-be-learned dynamics regimes to the original solver. AdaLED continuously adapts the surrogate to the new dynamics through online training. The transitions between the surrogate and the computational solver are determined by monitoring the prediction accuracy and uncertainty of the surrogate. The effectiveness of AdaLED is demonstrated on three different systems - a Van der Pol oscillator, a 2D reaction–diffusion equation, and a 2D Navier–Stokes flow past a cylinder for varying Reynolds numbers (400 up to 1200), showcasing its ability to learn effective dynamics online, detect unseen dynamics regimes, and provide net speed-ups. To the best of our knowledge, AdaLED is the first framework that couples a surrogate model with a computational solver to achieve online adaptive learning of effective dynamics. It constitutes a potent tool for applications requiring many computationally expensive simulations.

A compact ADI finite difference method for 2D reaction–diffusion equations with variable diffusion coefficients

Reaction–diffusion systems on a spatially heterogeneous domain have been widely used to model various biological applications. However, solving such partial differential equations (PDEs) analytically is rarely possible. Therefore, efficient and accurate numerical methods for solving such PDEs are desired. In this paper, we apply the well-known Padé approximation-based operator splitting techniques and develop a fourth-order compact alternative directional implicit (ADI) scheme. The new scheme is compact and fourth-order accurate in space. Combined with the Richardson extrapolation, the method can be improved to fourth-order accuracy in time. Stability analysis shows that the method is unconditionally stable; thus, a large time step can be used to improve the overall computational efficiency. Numerical examples have also demonstrated the new scheme’s high efficiency and high order accuracy.

Dynamic transitions and bifurcations of 1D reaction-diffusion equations: The non-self-adjoint case

The main goal of this paper is to classify the first transitions of a class of 1D second order reaction diffusion equation with a non-self-adjoint linear part and semilinear nonlinearity on a bounded interval subject to homogeneous boundary conditions. The emphasis is on the interaction of the nonlinear terms, the boundary conditions and the strength of the first order derivative term which makes the linear operator non-self-adjoint. The equations admit a trivial steady state solution which loses stability as a control parameter exceeds a threshold. According to our results, the first transitions of the system are either continuous, jump or mixed type. We compare our results with a recent study where only self adjoint linear operator was considered. In the Dirichlet, Neumann and periodic boundary condition cases, the first transition occurs via a transcritical/pitchfork bifurcation and the dynamics remain unchanged between the self-adjoint and non-self-adjoint cases. However, for the periodic case, with imposed zero mean condition, in the non-self adjoint case, the first transition is always accompanied by a Hopf bifurcation with a stable/unstable bifurcated limit cycle which is different than the self-adjoint case. Finally, we apply our results to determine the first transitions of some well-known reaction diffusion equations such as the Kolmogorov-Fisher, Chaffee-Infante equation and the Burger's equation.

Design principles for a nanoconfined enzyme cascade electrode via reaction-diffusion modelling.

The study of enzymes by direct electrochemistry has been extended to enzyme cascades, with a key development being the 'electrochemical leaf': an electroactive enzyme is immobilized within a porous electrode, providing in situ cofactor (NADP(H)) regeneration for a co-immobilized downstream enzyme. This system has been further developed to include multiple downstream enzymes, and it has become an important tool in biocatalysis, however, the local environment within the porous electrode has not been investigated in detail. Here, we constructed a 1D reaction-diffusion model, comprising the porous electrode with 2 kinds of enzymes immobilized, and an enzyme-free electrolyte diffusion layer. The modelling results show that the rate of the downstream enzyme is a key parameter, and that substrate transport within the porous electrode is not a main limiting factor. The insights obtained from this model can guide future rational design and improvement of these electrodes and immobilized enzyme cascade systems.

Fractional approximate solutions of 2D reaction–diffusion Brusselator model using the novel Laplace-optimized decomposition approach

The dynamical Brusselator reaction–diffusion system of time-fractional is used to describe chemical models and chemical processes with nonlinear oscillation. In this study, the Laplace optimized decomposition scheme is proposed for approximating solutions of three applications of the two-dimensional (2D) reaction–diffusion Brusselator model with the noninteger derivative proposed in the Caputo approach. Complete descriptions of the scheme and solution steps are utilized and mentioned. By applying the procedures of the Laplace inversion operator and truncating the optimized series, the approximate solutions are drawn, tabulated and sketched. Numerical results show the efficiency, reliability and accuracy of the technique for the nonlinear systems of partial differential equations of noninteger-different order derivatives. Finally, focused notes and futures planning works are mentioned with the most-used references.

Optimizing Immunofunctionalization and Cell Capture on Micromolded Hydrogels via Controlled Oxygen-Inhibited Photopolymerization.

With circulating tumor cells (CTCs) playing a critical role in cancer metastasis, the quantitation and characterization of CTCs promise to provide precise diagnostic and prognostic information in service of personalized therapies. However, as CTCs are extremely rare, high yield, high purity strategies are required to target and isolate CTCs from patient samples. Recently, we demonstrated the selective capture of CTCs upon antibody-functionalized polyethylene glycol diacrylate (PEGDA) hydrogels photopolymerized within polydimethylsiloxane (PDMS) microfluidic molds. Isolated CTC purity was subsequently enriched by selectively releasing desired cells from photodegradable hydrogel capture surfaces. However, the fabrication of these acrylate-based hydrogels by photopolymerization is subject to oxygen inhibition, which dramatically affects the physical and chemical properties of hydrogel interfaces formed in proximity to PDMS boundaries. To evaluate how antibody conjugation density and cell capture is impacted by fabrication parameters affected by oxygen inhibition, PEGDA hydrogel features were polymerized within PDMS micromolds under different UV exposure conditions and linker (acrylate-PEG-biotin) concentrations. Predictions of acrylate conversion throughout the hydrogel feature were performed using a 1D reaction-diffusion model that describes oxygen-inhibited photopolymerization. The functional consequences of photopolymerization parameters and solution stoichiometry on CTC capture were experimentally quantified and evaluated. Results show that hydrogel surfaces polymerized under shorter exposure times and with higher linker concentrations display superior functionalization and higher CTC capture efficiency. Conversely, highly cross-linked hydrogel surfaces polymerized under longer exposure times are insensitive to functionalization and display poor capture, regardless of linker concentration. By highlighting the importance of oxygen-inhibited photopolymerization, these findings provide guidelines to design micromolded hydrogels with controlled ligand expression. In addition to enhancing the selective cell capture capacity of immunofunctional hydrogels, the ability to quantifiably design hydrogel interfaces described here will improve the sensitivity of hydrogel biosensors, provide a platform to finely screen cell-matrix interactions, and generally enhance the fidelity of micromolded hydrogel features.

Recognition of spatiotemporal patterns of the periodically precipitating 2D reaction-diffusion system by determination of precise band location: Implications on the Matalon-Packter law

Recently focus of studies on the periodic precipitation in gels is shifting from evaluation of its spatio-temporal characteristics of the self-organized patterns to that of as a novel and viable method of synthesis of hierarchical monodispersed micro and nanomaterials in a single reactor. One of the parameters that profoundly affect the morphology, shape, size, and self-organization in the chemical systems is the supersaturation of the participating species. The Matalon-Packter (MP) law correlates the effect of the concentration of invading electrolytes to the spatial patterning in the Liesegang system. Although concentration and supersaturation have quantitative relations, the implications of supersaturation on the spatial arrangement of the band are not entirely understood. The present paper deals with studies on the periodically precipitating Co(OH) 2 in agar gel with respect to supersaturation of the participating reactants during the pattern formation. Varying inner and outer electrolytes concentrations were employed to determine spatial trends. Pattern image analysis was used to determine the precise location of the Liesegang bands. The supersaturation values showed a decreasing trend as the diffusion progressed outward. The present study led to the correlation between the supersaturation and the spacing coefficient.

Critical Role of Layer Thickness in Frontal Polymerization.

Thermal frontal polymerization (FP) is a chemical process during which a cold monomer-initiator mixture is converted into a hot polymer as a polymerization front propagates in the system due to the interplay between heat diffusion and the exothermicity of the reaction. The theoretical description of FP generally focuses on one-dimensional (1D) reaction-diffusion (RD) models where the effect of heat losses is encoded into an effective parameter in the heat equation. We show here the limits of such 1D models to describe FP under nonadiabatic conditions. To do so, the propagation of a polymerization front is analyzed both analytically and numerically in a rectangular two-dimensional (2D) layer. The layer thickness is shown to control the dynamics of the front and to determine its very existence. We find that for given heat losses, a minimum thickness is required for front propagation as recently observed in FP experiments of 2D thin films on wood. Moreover, when the thickness exceeds a critical value, the front is observed to survive independently of the rate of heat losses. This result cannot be predicted with 1D models where front extinction is always possible. A scaling analysis is proposed to highlight the physical interpretation of such a front survival. The influence of dimensionality on thermal instabilities is also analyzed, with a focus on the differences with the 1D predictions.

Deuterium retention and removal in liquid lithium determined by in situ NRA in Magnum-PSI

In this work, Li-filled 3D-printed porous tungsten samples were exposed to deuterium (D) plasma in Magnum-PSI with a wide ion flux from 4 × 1022 to 1.5 × 1024 m−2 s−1 and with a corresponding wide temperature range from below Li melting point (180.5 °C) to above Li deuteride (LiD) melting point (∼690 °C). The formation, decomposition and melting of LiD have been directly observed in the experiment via infra-red thermometry and visually post-mortem while still in vacuo, and correlated to the D retained content. The LiD formation was characterized by a solid precipitate layer formed on the surface with high emissivity (0.6–0.9) characterized by a blue or dark blue color after exposure. The melting of Li–LiD layer was found to occur close to the temperature predicted by Li–LiD phase diagram. In situ nuclear reaction analysis (NRA) was applied to perform the measurement of D retained in Li samples immediately after exposure without breaking the vacuum. D depth profiles were determined by NRA, in which the highest D concentration (15–45 at.%) was found in the top several micrometers and decreases with depth to low levels (<5%) within 5–30 μm. No pure LiD layer was found on the sample surfaces, however a D concentration close to 50 at.% was observed on a Li-D co-deposited layer on the clamping ring in some cases. The experiments also indicate that the D retained increases with increasing temperature until ∼500 °C. At temperatures beyond ∼500 °C the dissociation of LiD starts to dominate and the deuterium retention started to decrease. Overall, D retained fraction for all cases was found to be below ∼2%, which is significantly different from literatures where full uptake has been suggested. A 1D reaction–diffusion (RD) model based on D diffusion and chemical reactions with Li has been built. D depth profiles from the RD modelling can roughly match that from NRA measurement and a low D retained fraction below ∼2% was also indicated by the model. The model can also help explain the relationship between D retained and the surface temperature and fluence. After D plasma exposure, either helium or H plasma was utilized to remove the retained D in Li and both were proved to be effective and the removal efficiency can be as high as 96% above 420 °C.

2D reaction-diffusion model of quorum sensing characteristics during all phases of bacterial growth

2D reaction-diffusion model of quorum sensing characteristics during all phases of bacterial growth

Dynamic transitions and bifurcations of 1D reaction–diffusion equations: The self‐adjoint case

This paper deals with the classification of transition phenomena in the most basic dissipative system possible, namely, the 1D reaction–diffusion equation. The emphasis is on the relation between the linear and nonlinear terms and the effect of the boundaries which influence the first transitions. We consider the cases where the linear part is self‐adjoint with second‐order and fourth‐order derivatives which is the case which most often arises in applications. We assume that the nonlinear term depends on the unknown function and its first derivative which is basically the semilinear case for the second‐order reaction–diffusion system. As for the boundary conditions, we consider the typical Dirichlet, Neumann, and periodic boundary settings. In all the cases, the equations admit a trivial steady state which loses stability at a critical parameter. We aim to classify all possible transitions and bifurcations that take place. Our analysis shows that these systems display all three types of transitions: continuous, jump and mixed. Moreover they exhibit transcritical, supercritical bifurcations with bifurcated states such as finitely many equilibria, circle of equilibria, and slowly rotating limit cycle. Many applications found in the literature are basically corollaries of our main results. We apply our results to classify the first transitions of the Chaffee–Infante equation, the Fisher‐KPP equation, the Kuramoto–Sivashinsky equation, and the Swift–Hohenberg equation.

Multiscale Modelling of Nanostructured Foil Anode for Next Generation Batteries

Interdigitated Eutectic Alloy anode (IdEA) is a promising new candidate for next generation batteries due to its high specific capacity (250-300 mAh/g) [1]. Multiscale modeling approaches are essential to study such new battery chemistries. They help examine the underlying electrochemical properties and phase transitions, including the transport, kinetics and reaction schemes. In this study, we present a multiscale modeling analysis, of a Li-ion cell with a ZTB (Zn-Sn-Bi) IdEA, consisting of DFT simulation and continuum level electrochemical engineering modeling aided by experimental measurements. The DFT simulation is conducted to obtain diffusivity of the alloy anode undergoing phase changes. The electrochemical engineering model is developed based on a 1D reaction-diffusion equation and it models the kinetics and transport in the alloy anode-based battery system. Using this 1D model, a moving phase front has been observed when the eutectic alloy undergoes phase transition during charge/discharge. The cell level voltage response obtained from this model is validated with the experimental data. The practical relevance can be maximized by formulating a techno-economic model. Heligman, B.T., Kreder III, K.J. and Manthiram, A., 2019. Zn-Sn interdigitated eutectic alloy anodes with high volumetric capacity for lithium-ion batteries. Joule, 3(4), pp.1051-1063.

Reaction-diffusion in a growing 3D domain of skin scales generates a discrete cellular automaton

We previously showed that the adult ocellated lizard skin colour pattern is effectively generated by a stochastic cellular automaton (CA) of skin scales. We additionally suggested that the canonical continuous 2D reaction-diffusion (RD) process of colour pattern development is transformed into this discrete CA by reduced diffusion coefficients at the borders of scales (justified by the corresponding thinning of the skin). Here, we use RD numerical simulations in 3D on realistic lizard skin geometries and demonstrate that skin thickness variation on its own is sufficient to cause scale-by-scale coloration and CA dynamics during RD patterning. In addition, we show that this phenomenon is robust to RD model variation. Finally, using dimensionality-reduction approaches on large networks of skin scales, we show that animal growth affects the scale-colour flipping dynamics by causing a substantial decrease of the relative length scale of the labyrinthine colour pattern of the lizard skin.

Implementation of Modified Cubic UAT Tension B-spline DQM for Numerical Approximation of 1D and 2D Reaction-Diffusion System

In present paper, a new approach, “modified cubic UAT tension B-spline DQM” has been developed to solve the 1D and 2D Reaction-Diffusion system numerically. The modified cubic UAT tension B-spline is used as basis function, to find the required weighting coefficients. The resulting system of ODE has been solved by SSP-RK43 scheme. The proposed scheme is checked by five test examples.

Numerical approximation of 1D and 2D reaction diffusion system with modified cubic UAH tension B-spline DQM

In this paper, a new numerical approach “Modified cubic UAH tension B-spline DQM” is projected to find the numerical approximation of 1D and 2D Reaction-Diffusion system. The modified cubic UAH tension B-spline is used in space to discretize the partial derivatives. The obtained system of ODE is dealt with SSP-RK43 scheme. To check the adaptability and efficiency of the proposed scheme, five numerical examples are discussed. The present method is easy to implement and economical as compared to the existing approaches available in literature for different types of linear and non-linear PDEs.