Diffusion-reaction Models Research Articles

On the structural sensitivity of some diffusion–reaction models of population dynamics

In mathematical ecology, it is often assumed that properties of a mathematical model are robust to specific parameterization of functional responses, in particular preserving the bifurcation structure of the system, as long as different functions are qualitatively similar. This intuitive assumption has been challenged recently (Fussmann & Blasius, 2005). Having considered the prey–predator system as a paradigm of nonlinear population dynamics, it has been shown that in fact both the bifurcation structure and the structure of the phase space can be rather different even when the component functions are apparently close to each other. However, these observations have so far been largely limited to nonspatial systems described by ODEs. In this paper, our main interest is to investigate whether such structural sensitivity occurs in spatially explicit models of population dynamics, in particular those that are described by PDEs. We consider a prey–predator model described by a system of two nonlinear reaction–diffusion–advection equations where the predation term is parameterized by three different yet numerically close functions. Using some analytical tools along with numerical simulations, we show that the properties of spatiotemporal dynamics are rather different between the three cases, so that patterns observed for one parameterization may not occur for the other two ones.

Hybrid Deep Learning and Sensitivity Operator-Based Algorithm for Identification of Localized Emission Sources

Hybrid approaches combining machine learning with traditional inverse problem solution methods represent a promising direction for the further development of inverse modeling algorithms. The paper proposes an approach to emission source identification from measurement data for advection–diffusion–reaction models. The approach combines general-type source identification and post-processing refinement: first, emission source identification by measurement data is carried out by a sensitivity operator-based algorithm, and then refinement is done by incorporating a priori information about unknown sources. A general-type distributed emission source identified at the first stage is transformed into a localized source consisting of multiple point-wise sources. The second, refinement stage consists of two steps: point-wise source localization and emission rate estimation. Emission source localization is carried out using deep learning with convolutional neural networks. Training samples are generated using a sensitivity operator obtained at the source identification stage. The algorithm was tested in regional remote sensing emission source identification scenarios for the Lake Baikal region and was able to refine the emission source reconstruction results. Hence, the aggregates used in traditional inverse problem solution algorithms can be successfully applied within machine learning frameworks to produce hybrid algorithms.

Diffusion-reaction models for concrete exposed to chloride-sulfate attack based on porosity and water saturation

Reinforced concrete structures in an unsaturated state experiences significant degradation when exposed to combined chloride and sulfate attack in marine environments. For evaluating durability and predicting service life of reinforced structures, it is crucial to better understand ions diffusion, porosity variation and water saturation of concrete. Considering effect of porosity and water saturation degree on chloride diffusivity, a new diffusion-reaction model was established to investigate how concrete degrades exposed to the severe marine environment. The model was validated using experimental results from both hydrophobic and plain concrete that had undergone a 14-month exposure test at the Qingdao Wheat Island exposure station. The simulation results of the multi-ions model showed that both plain concrete and hydrophobic concrete experienced maximum reduction of chloride concentration of about 68% when compared to concrete exposed to the unsaturated condition.

Extricating the effect of sodium monovalent cation in the crystallization kinetics of bioactive glass and its influence on bioactivity

Sintering-induced crystallization phenomenon plays a major role in deciding a biomaterial's application. Among all, sodium cation still creates curiosity and controversy among researchers on their role in the bioactive glass network. The kinetic study of bioactive glasses (BG) with and without sodium cations was studied using TG-DTA analysis carried out under non-isothermal conditions at different heating rates. Kinetic triplets were calculated from both model-free (KAS) and model-fitting (CR) methods. It is found that traditional 45S BG obeys power law whereas sodium-free obeys third-order and diffusion-reaction models respectively with a lower pre-exponential factor. The variation in Eaof the system i.e., 11.690 kJ/mol for 45S BG and an average of 7.591 kJ/mol for 59S BG. Entropy (ΔS), Enthalpy (ΔH), and Gibb's free energy (ΔG) were also calculated. This present study is effectively note-worthy as it demonstrated important kinetic parameters to design biomaterial with erythrocyte and tissue compatibility for appropriate biomedical applications.

Verification methods for drift–diffusion reaction models for plasma simulations

Compared to other computational physics areas such as codes for general computational fluid dynamics, the documentation of verification methods for plasma fluid codes remains under developed. Current analytical solutions for plasma are often highly limited in terms of testing highly coupled physics, due to the harsh assumptions needed to derive even simple plasma equations. This work highlights these limitations, suggesting the method of manufactured solutions (MMSs) as a potential option for future verification efforts. To demonstrate the flexibility of MMS in verifying these highly coupled systems, the Multiphysics Object-Oriented Simulation Environment (MOOSE) framework was utilized. Thanks to the MOOSE framework’s robustness and modularity, as well as to its physics module capabilities and ecosystem applications (i.e. Zapdos and the chemical reaction network) developed for plasma physics modeling and simulation, this report lays the groundwork for a structured method of conducting plasma fluid code verification.

Kinetic study of uranium (VI) extraction with tributyl-phosphate in a stratified flow microchannel

Solvent extraction kinetics are of considerable importance in process and apparatus design, especially when the mass transfer is coupled with a chemical reaction. In such circumstances, the mass transfer can either be diffusion limited, reaction limited, or both in a mixed regime. Conventional extraction techniques lead to acquire an overall mass transfer coefficient for fast chemical systems. The latter brings together diffusional and reactional parts that are difficult to distinguish independently. This problem is addressed using a high velocity parallel flow microfluidic setup for the investigation of the extraction kinetics of the fast chemical system U(VI):HNO3/TBP:TPH. Experimental data were analyzed using a combination of overall mass transfer resistance calculations and advection–diffusion-reaction models. Results indicate that a chemically limited regime was obtained at a high initial concentration (119 g.l−1) of uranium (VI) due to Marangoni convections. Whereas, for low initial concentrations (10–30 g.l−1) and short residence times in the microchannel (≤13 ms), both diffusion and reaction impact the transfer rate. The obtained value of the apparent rate constant under current operating conditions ([HNO3] = 3 M, % TBP = 30 % (v/v)) was of (6.5 ± 1.7) × 10−4 m.s−1. This value is much higher than what conventional extraction techniques predicted, indicating the better mass transfer efficiency in the microfluidic setup due to the decrease of the diffusional limitation.

Simulation of conformality of ALD growth inside lateral channels: comparison between a diffusion-reaction model and a ballistic transport-reaction model.

Atomic layer deposition (ALD) has found significant use in the coating of high-aspect-ratio (HAR) structures. Approaches to model ALD film conformality in HAR structures can generally be classified into diffusion-reaction (DR) models, ballistic transport-reaction (BTR) models and Monte Carlo simulations. This work compares saturation profiles obtained using a DR model and a BTR model. The saturation profiles were compared qualitatively and quantitatively in terms of half-coverage penetration depth, slope at half-coverage penetration depth and adsorption front broadness. The results showed qualitative agreement between the models, except for a section of elevated surface coverage at the end of the structure, 'trunk', observed in the BTR model. Quantitatively, the BTR model produced deeper penetration into the structure, lower absolute values of the slope at half-coverage penetration depth and broader adsorption fronts compared to the DR model. These differences affect the values obtained when extracting kinetic parameters from the saturation profiles.

Magnesium recovery from ferrochrome slag: kinetics and possible use in a circular economy

The ever-increasing demand for ferrochrome alloys has resulted in a substantial accumulation of ferrochrome slag by-products in many mining areas. On the other hand, the ferrochrome slag has been identified as one waste material that is rich in magnesium (Mg) and has not been effectively exploited. Beneficiating of ferrochrome slag (FCS) waste material is envisaged as a means of achieving sustainable recovery of Mg. Previous studies have used sulphuric acid as a lixiviant for leaching FCS at moderate temperatures to recover Mg. In this study, the recovery of Mg from ferrochrome slag was investigated using hydrochloric acid (HCl) as the lixiviant at low temperatures. Previous studies have shown that various metal oxides have been proven to be more amenable to leaching using HCl. This study examined the effects of acid content, leaching temperature, and reaction time on the recovery of Mg from FCS. Kinetic and thermodynamic analysis of the leaching process were also investigated as these are critical factors for maximum extraction of the Mg. The results showed that the highest recovery of Mg of 88.2% was obtained from FCS using 5 M HCl with a solid to liquid ratio of 1:10, mixing intensity of 250 rpm, reaction time and temperature of 150 min and 70 °C, respectively. The shrinking core model (SCM) was used in kinetic analysis to find the experimental data's best fit. A linear relationship was obtained with the coefficient of determination for the chemical reaction model (Kc) of >0.9 which indicates a good fit. The activation energy obtained for the diffusion and chemical reaction models were 95.44 and 41.45 kJ/mol, respectively, demonstrating that the rate-limiting phase is the one involving the chemical reaction.

Reaction-diffusion models in weighted and directed connectomes.

Connectomes represent comprehensive descriptions of neural connections in a nervous system to better understand and model central brain function and peripheral processing of afferent and efferent neural signals. Connectomes can be considered as a distinctive and necessary structural component alongside glial, vascular, neurochemical, and metabolic networks of the nervous systems of higher organisms that are required for the control of body functions and interaction with the environment. They are carriers of functional phenomena such as planning behavior and cognition, which are based on the processing of highly dynamic neural signaling patterns. In this study, we examine more detailed connectomes with edge weighting and orientation properties, in which reciprocal neuronal connections are also considered. Diffusion processes are a further necessary condition for generating dynamic bioelectric patterns in connectomes. Based on our precise connectome data, we investigate different diffusion-reaction models to study the propagation of dynamic concentration patterns in control and lesioned connectomes. Therefore, differential equations for modeling diffusion were combined with well-known reaction terms to allow the use of connection weights, connectivity orientation and spatial distances. Three reaction-diffusion systems Gray-Scott, Gierer-Meinhardt and Mimura-Murray were investigated. For this purpose, implicit solvers were implemented in a numerically stable reaction-diffusion system within the framework of neuroVIISAS. The implemented reaction-diffusion systems were applied to a subconnectome which shapes the mechanosensitive pathway that is strongly affected in the multiple sclerosis demyelination disease. It was found that demyelination modeling by connectivity weight modulation changes the oscillations of the target region, i.e. the primary somatosensory cortex, of the mechanosensitive pathway. In conclusion, a new application of reaction-diffusion systems to weighted and directed connectomes has been realized. Because the implementation was realized in the neuroVIISAS framework many possibilities for the study of dynamic reaction-diffusion processes in empirical connectomes as well as specific randomized network models are available now.

Towards the modeling of the interplay between radiation induced segregation and sink microstructure

Excess point defects created by irradiation in metallic alloys diffuse and annihilate at sinks available in the microstructure, such as grain boundaries, dislocations, or point defect clusters. Fluxes of defects create fluxes of alloying elements, leading to local changes of composition near the sinks and to a modification of the properties of the materials. The direction and the amplitude of this radiation-induced segregation, its tendency to produce an enrichment or a depletion of solute, depend on a set of transport coefficients that are very difficult to measure experimentally. The understanding of radiation-induced segregation phenomena has, however, made significant progress in recent years, thanks to the modeling at different scales of diffusion and segregation mechanisms. We review here these different advances and try to identify the key scientific issues that limit the development of predictive models, applicable to real alloys. The review addresses three main issues: the calculation of the transport coefficients from ab initio calculations, the modeling of segregation kinetics at static point defects sinks—mainly by kinetic Monte Carlo or diffusion-reaction models—and the more challenging task of modeling the dynamic interplay between radiation-induced segregation and sink microstructure evolution, especially when this evolution results from annihilation of point defects. From this overview of the current state-of-the-art in this field, we discuss still-open questions and guidelines for what constitutes, in our opinion, the desirable future works on this topic.

Mechanochemically synthesized Al–Fe (oxide) composite with superior reductive performance: Solid-state kinetic processes during ball milling

Recently, mechanical ball milling (BM), a simple and green powder processing method, has been successfully applied to improve the performance of zero-valent metals (ZVMs) for efficient water treatment. However, until now BM is still regarded as a “black box” in which the processes of the solid-state reaction during activation remain unclear. In this paper, firstly, FeSO4·7H2O crystal was used to activate and modify inert microscale zero-valent aluminum (mZVAl) by BM to synthesize Al–Fe (oxide)bm composite that showed superior reactivity in reductive removal of various contaminants and excellent reusability, which may be mainly ascribed to the newly formed iron oxide layer on mZVAl by mechanochemical reaction. At the same time, the formation of iron oxides on mZVAl was closely related to BM parameters. Further kinematic analysis revealed that the occurrence of mechanochemical reaction depended on the impact energy and input energy, which BM speed and BM time were two main factors determining reaction extent on the premise that the precursors were full dose. Moreover, kinetic fitting uncovered the solid-state reaction mechanism between mZVAl and FeSO4·7H2O conformed to three-dimensional diffusion and phase boundary reaction models. This study ponders deeply upon the mechanochemical process and solid reaction mechanism during the preparation of Al–Fe (oxide)bm composite, which deepens comprehensions of material synthesis procedures by BM and promotes applications of ZVM-based composite in polluted water or wastewater treatment.

Mild sulphuric acid pre-treatment for metals removal from biosolids and the fate of metals in the treated biosolids derived biochar

Biosolids are contaminated with heavy metals (HMs) and alkali and alkaline earth metals (AAEMs). These metals limit the suitability of biosolids for land application as well as their pyrolytic conversion to high-quality products. In this work, a mild sulphuric acid pre-treatment of biosolids was carried out at different stirring speeds (300–900 rpm), temperatures (25–100 °C), extraction time (0–180 min), and acid concentration (1–5% v/v) to reduce the metals load in biosolids and their biochar derived from pyrolysis. The metal leaching process was very rapid and reached equilibrium in less than 30 min. The optimum conditions removed about 75–95% HMs and 80–95% AAEMs except Ca due to the formation of CaSO4 hydrates. Temperature was the driving parameter for Cd and Ni extraction, whereas temperature and acid concentration played the leading roles in Cu extraction. The shrinking core product layer diffusion and surface chemical reaction models described the extraction kinetics of Ni, Cu and Cd. A leaching activation energy of 10.02 kJ/mol and 7.37 kJ/mol was estimated for Ni and Cd, respectively. FTIR, SEM and XRD characterisation of the treated biosolids jointly indicated that the leaching mechanism was dominated by acid dissolution of metal-containing components followed by ion exchange of metal species with protons from H2SO4. Treatment at 25 °C and 3% H2SO4 lowered the biosolids ash content by 50% and preserved the physicochemical attributes, which enhanced the pyrolysis upcycling of the treated biosolids. Pre-treatment influenced the migration characteristics of the metals during pyrolysis and the produced biochar had several folds lower HMs and AAEMs contents than the raw biosolids-derived biochar.

Millisecond timescale reactions observed via X-ray spectroscopy in a 3D microfabricated fused silica mixer.

Determination of electronic structures during chemical reactions remains challenging in studies which involve reactions in the millisecond timescale, toxic chemicals, and/or anaerobic conditions. In this study, a three-dimensionally (3D) microfabricated microfluidic mixer platform that is compatible with time-resolved X-ray absorption and emission spectroscopy (XAS and XES, respectively) is presented. This platform, to initiate reactions and study their progression, mixes a high flow rate (0.50-1.5 ml min-1) sheath stream with a low-flow-rate (5-90 µl min-1) sample stream within a monolithic fused silica chip. The chip geometry enables hydrodynamic focusing of the sample stream in3D and sample widths as small as 5 µm. The chip is also connected to a polyimide capillary downstream to enable sample stream deceleration, expansion, and X-ray detection. In this capillary, sample widths of 50 µm are demonstrated. Further, convection-diffusion-reaction models of the mixer are presented. The models are experimentally validated using confocal epifluorescence microscopy and XAS/XES measurements of a ferricyanide and ascorbic acid reaction. The models additionally enable prediction of the residence time and residence time uncertainty of reactive species as well as mixing times. Residence times (from initiation of mixing to the point of X-ray detection) during sample stream expansion as small as 2.1 ± 0.3 ms are also demonstrated. Importantly, an exploration of the mixer operational space reveals a theoretical minimum mixing time of 0.91 ms. The proposed platform is applicable to the determination of the electronic structure of conventionally inaccessible reaction intermediates.

On a Robust and Efficient Numerical Scheme for the Simulation of Stationary 3-Component Systems with Non-Negative Species-Concentration with an Application to the Cu Deposition from a Cu-(β-alanine)-Electrolyte

Three-component systems of diffusion–reaction equations play a central role in the modelling and simulation of chemical processes in engineering, electro-chemistry, physical chemistry, biology, population dynamics, etc. A major question in the simulation of three-component systems is how to guarantee non-negative species distributions in the model and how to calculate them effectively. Current numerical methods to enforce non-negative species distributions tend to be cost-intensive in terms of computation time and they are not robust for big rate constants of the considered reaction. In this article, a method, as a combination of homotopy methods, modern augmented Lagrangian methods, and adaptive FEMs is outlined to obtain a robust and efficient method to simulate diffusion–reaction models with non-negative concentrations. Although in this paper the convergence analysis is not described rigorously, multiple numerical examples as well as an application to elctro-deposition from an aqueous Cu2+-(β-alanine) electrolyte are presented.

Parameter Identification and Sensitivity Analysis for Zero-Dimensional Physics-Based Lithium-Sulfur Battery Models

Abstract This paper examines parameter estimation for Lithium-Sulfur (Li-S) battery models from experimental data. Li-S batteries are attractive compared to traditional Lithium-ion batteries, thanks largely to their potential to achieve higher energy densities. The literature presents a number of Li-S battery models with varying fidelity and complexity levels. This includes both high-fidelity diffusion-reaction models as well as zero-dimensional models that neglect diffusion dynamics while capturing the underlying reduction-oxidation reaction physics. This paper focuses on four zero-dimensional models, representing different possible sets of redox reactions. There is a growing need for using experimental data sets to both parameterize and compare these models. To address this, Li-S coin cells were fabricated and tested. In parallel, a sensitivity analysis of key model parameters was conducted. Using this analysis, a subset of model parameters was selected for identification and estimation in all four Li-S battery models.

The Thermal-Oxidation Behavior of Pristine and Doped Magnéli Phase Titanium Oxides

Magnéli phase titanium oxides (Ti n O2n−1, 4 ≤ n ≤ 10) are important materials for solid state and electrochemical technologies such as memristors, batteries, fuel cells, and electrochemical devices for water treatment. Developing an understanding of transitions between Ti n O2n−1 and its product of oxidation, titanium(IV) oxide (TiO2), as well as strategies such as doping to modulate the conditions for such changes will enable the development of more effective devices. To elucidate a mechanism for their thermal oxidation and investigate the influence of doping, the thermal-oxidation behavior in air of Ti4O7 doped with vanadium, chromium, and iron were investigated by thermogravimetric analysis (TGA). These powders prepared by high-temperature H2 reduction of dopant-containing TiO2 were characterized by scanning electron microscopy (SEM), gas adsorption analysis, X-ray fluorescence (XRF), energy-dispersive X-ray (EDX) spectroscopy, X-ray photoelectron spectroscopy (XPS), and powder X-ray diffraction (PXRD). V- and Fe-‍doping improved the thermal stability of Ti4O7 as evidenced by higher onset temperatures in their thermograms. Three-dimensional diffusion reaction models adequately describe the solid-state kinetics of thermal oxidation of Ti4O7 in air as demonstrated by linear model-fitting. Doping shows a mixed influence on the kinetics for thermal oxidation in air reducing both the Arrhenius pre-exponential factor and the activation energy.

Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

How relevant is anisotropy in bimolecular electron transfer reactions in liquid crystals?

This work aims to describe anisotropy effects on chemical reactions imposed by the structure of liquid crystals. Photo-induced intermolecular electron transfer is studied by means of fluorescence quenching. The charge shift and charge separation reactions are investigated in two different chemical systems. Steady state and time resolved measurements are performed to track the reaction kinetics. In order to extract the electron transfer parameters which are independent of the solvent, the reactions are also conducted in isotropic media at different temperatures and viscosities. A set of diffusion-reaction models with different levels of anisotropy has been developed as well as a numerical method to solve the corresponding partial differential equations. Only the model which introduces anisotropic diffusion and reactivity gives physically meaningful values for all parameters entering the reactivity. It is therefore shown how to adapt diffusion-reaction models able to accommodate any sort of reactivity to complex environments.

The strategies for the modelling of the passive mass transport through porous membranes: Applicability to transdermal delivery systems.

The paper concerns the modelling of the passive solute transport through porous membranes. A general scheme for the mass transport has been developed upon the mixed diffusion-advection-reaction model. The passive advection has been introduced as a certain simplification of the Navier-Stokes problem, involving a pressure gradient-induced creeping flow of an incompressible Newtonian fluid. Nine scenarios for the drug transport process have been tested versus two experimental datasets acquired earlier (photoacoustic depth-profiling and contact angle surface wettability techniques) for the characterization of bulk and interfacial processes in a model pharmaceutical system: the synthetic dodecanol-collodion porous membrane in contact with a photodegradable pigment dithranol. The scenarios considered include three mass transport models (the diffusion-advection-reaction, diffusion-advection and diffusion-reaction models) under three distinct types of the lower (the donor/acceptor interface) boundary conditions: the Dirichlet-type instantaneous source, the Dirichlet-type interface relaxation, and the Neumann-type concentration gradient. The results obtained indicate a considerable agreement between the experimental data and predictions of the diffusion-reaction and the general models for long times, however, some deviations were exhibited at the initial stages of the permeation process. It is considered, that the discrepancies originate from a specific penetrant behaviour at the interfaces, which violates boundary transfer schemes classically employed for the mass transport phenomena quantification. Moreover, an additional mixing process taking place close to the interface related to the liquid flow driven by the surface tension gradients (so-called classic and thermal Marangoni effect) could play a still underestimated role in the trans-interfacial mass transport.

Mechanism based diffusion-reaction modelling for predicting the influence of SARA composition and ageing stage on spurt completion time and diffusivity in bitumen

Diffusion-reaction models derived within the frame of continuum thermodynamics of irreversible processes for reacting mixtures enjoy a number of advantages over purely phenomenological models. This paper explores the potential and challenges of this approach by means of a particular model for spurt oxidation in bitumen.Film ageing experiments are simulated considering different bitumen compositions in terms of SARA fractions. The predicted spurt completion times are contrasted with parameters characterizing the initial composition. The results demonstrate the consistency of the model. Furthermore, the trend regarding spurt completion time predicted by the model is in line with experimental results from the Strategic Highway Research Program (SHRP).In addition, the change in diffusivity due to oxidation is discussed for different compositions. The model consistently predicts a significant decrease in diffusivity due to oxidation in the absence of out-diffusion of reaction products.