Steady-state Diffusion Research Articles

A theoretical investigation of steady-state concentration processes at a carrier-mediated transport model using Akbari-Ganji and differential transform methods

In this article, a carrier-mediated transport model is considered. The steady-state non-linear diffusion equations were solved by two effective and accessible analytical methods, Akbari-Ganji and the differential transform methods. Here, we present the generalized approximate analytical solution for the substrate, product, and reactant concentrations for the small experimental values of the kinetic and diffusion parameters. The numerical solution to this problem was derived using the Mat-lab/Scilab program, showing a satisfactory agreement between the analytical and numerical/computational outcomes. The simulated results can be compared to the relevant theories. This result through the physical parameters shows that the Akbari-Ganji method is better than the differential transform method. The effects of this study are applicable across the entire solution domain. Furthermore, the concentration of species, facilitation factor F, and the reaction equilibrium constant K are compared to the parameters are also obtained.

Migration behavior of water molecules and chloride ions at the nanopore scale based on molecular properties under temperature effect

The durability of cement-based materials is closely related to the transmission characteristics of water and chloride ions in the pores of calcium silicate hydrate (C-S-H) gel, especially in the climate conditions with significant temperature difference. The local structure and transport process of water molecules and chloride ions in gel pores are affected by interfacial effects. The unsaturated transport model of sodium chloride solution in C-S-H gel pores is established on the molecular scale, under different temperature and pore size conditions. Divide the regions in the C-S-H gel pore model into internal adsorption zone, external adsorption zone, and free zone. Taking into account the intrinsic mechanisms and molecular properties of water molecules and ions during transport in the C-S-H gel pores, the local structures and dynamic properties of water molecules and ions in the gel pores are quantitatively described. The electrical double layer and the double-layer water film overlap in the 1.5 nm gel pore, the water molecule and the chloride ion are closely arranged. Thus the diffusion coefficient is far lower than others, and is rarely affected by temperature. Considering the changes in the molecule properties of water and ion in different zones in molecular simulation results, a theoretical model of nanopore water and ion migration was established, revealing the migration behavior of water and ions. The results indicate that the steady-state diffusion fluxes of water and chloride ions are positively correlated with the effects of temperature and pore size, and the increasing amplitude of their diffusion fluxes is characterised by a similar phenomenon with that. This research provides a theoretical basis for the mechanism and durability of cement-based materials corroded by water and chloride ions in marine environments.

Water vapor sorption and transport in carbon molecular sieve membranes

Carbon Molecular Sieve (CMS) membranes have been extensively studied for gas separations. Many gas separation applications involve humidified streams, but reports on water vapor transport in CMS membranes are limited. In this study, water permeability, diffusivity, and solubility were determined as a function of water activity for CMS membranes. Water transport properties of membranes synthesized at different pyrolysis temperatures (550 °C and 800 °C) and with different polyimide precursors were examined. Water sorption followed Type V isotherms as previously observed for the adsorption of water in microporous carbons. Water permeability was much higher at all water activity values for CMS samples prepared at lower pyrolysis temperature. Water permeabilities as a function of water activity of the three different polyimides pyrolyzed at 800 °C were very similar. Water permeability of CMS membranes was high compared to many other polymeric materials, showing potential for dehydration applications. Finally, the steady-state water vapor diffusivity (Dss), estimated from steady state permeability and equilibrium sorption measurements, was compared to the experimental transient diffusivity (D), matching reasonably well.

A diffuse interface–lattice Boltzmann model for conjugate heat transfer with imperfect interface

A diffuse interface-lattice Boltzmann model is proposed for modeling conjugate heat transfer problems with imperfect interfaces. According to the continuity condition of heat flux and jump condition of temperature across the imperfect interface, a unified sharp interface equation of the temperature field with a singular source term in the composite media domain is established. Then a diffuse interface equation is obtained by making a smoothed approximation of the unified sharp interface equation. The novel diffuse interface equation is restructured into a second-order heat diffusion equation with an anisotropic diffusion coefficient tensor. To numerically solve the second-order anisotropic diffusion equation, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) scheme is employed. Several numerical simulations of benchmark test examples with analytical solutions are carried out to demonstrate the accuracy of the present model, including the steady-state heat diffusion in a two-layer medium with a flat interface, steady heat diffusion in a square domain with a curved interface and unsteady-state temperature diffusion with two-layer media. The numerical results show that the phenomenon of temperature jump across the imperfect interface can be successfully and correctly captured. Moreover, as an example of practical application, the new proposed diffuse interface model is utilized to predict the effective thermal conductivity of porous media with thermal contact resistance at pore scale. The variation law of the effective thermal conductivity is obtained for each parameter by studying the effects of thermal conductivity ratios of the dual-component material, the shape and volume fraction of blocks in the material, the interfacial thermal conductance coefficient, and the number of blocks on the effective thermal conductivity. Both uniform and random pore structures are considered. The previous theoretical equations strongly support the validity of the present model.

A Quadrature Scheme for Steady-State Diffusion Equations Involving Fractional Power of Regularly Accretive Operator

A Quadrature Scheme for Steady-State Diffusion Equations Involving Fractional Power of Regularly Accretive Operator

Steady-state inhomogeneous diffusion with generalized oblique boundary conditions

We consider the elliptic diffusion (steady-state heat conduction) equation with spacedependent conductivity and inhomogeneous source subject to a generalized oblique boundary condition on a part of the boundary and Dirichlet or Neumann boundary conditions on the remaining part. The oblique boundary condition represents a linear combination between the dependent variable and its normal and tangential derivatives at the boundary. We first prove the well-posedness of the continuous problems. We then develop new finite volume schemes for these problems and prove rigorously the stability and convergence of these schemes.We also address an application to the inverse corrosion problem concerning the reconstruction of the coefficients present in the generalized oblique boundary condition that is prescribed over a portion Γ0 of the boundary ∂Ω from Cauchy data on the complementary portion Γ1 = ∂Ω\Γ0.

Label-free learning of elliptic partial differential equation solvers with generalizability across boundary value problems

Traditional numerical discretization-based solvers of partial differential equations (PDEs) are fundamentally agnostic to domains, boundary conditions and coefficients. In contrast, machine learnt solvers have a limited generalizability across these elements of boundary value problems. This is strongly true in the case of surrogate models that are typically trained on direct numerical simulations of PDEs applied to one specific boundary value problem. In a departure from this direct approach, the label-free machine learning of solvers is centered on a loss function that does not use computed field solutions as labels. Instead, the PDE and boundary conditions are directly incorporated in residual form to express the loss function during training. However, their generalization across boundary conditions is limited and they remain strongly domain-dependent. Here, we present a framework that generalizes across domains, boundary conditions and coefficients while simultaneously learning the PDE in weak form. Our work explores the ability of convolutional neural network (CNN)-based encoder–decoder architectures to learn to solve a PDE in greater generality than its restriction to a particular boundary value problem. In this first Communication, we take the canonical path through elliptic PDEs and focus on steady-state diffusion, linear and nonlinear elasticity. Importantly, the learning happens independently of any labeled field data from either experiments or direct numerical solutions. We develop probabilistic CNNs in the Bayesian setting using variational inference. Extensive results for these problem classes demonstrate the framework’s ability to learn PDE solvers that generalize across hundreds of thousands of domains, boundary conditions and coefficients, including extrapolation beyond the learning regime. Once trained, the machine learning solvers are orders of magnitude faster than discretization-based solvers. They therefore have relevance to high-throughput solutions of PDEs on varied domains, boundary conditions and coefficients, such as for inverse modeling, optimization, design and decision-making. We place our work in the context of other recent machine learning solvers of PDEs including continuous operator learning frameworks, as well as make comparisons of performance where possible. Finally, we note extensions to transfer learning, active learning and reinforcement learning.

Inhibitory effect of La3+ on hydrogen damage of 27CrMo44S/1 steel in H2S-containing acidic solution

The effect of La3+ on the hydrogen permeation behavior of 27CrMo44S/1 steel in acidic environments containing H2S is investigated through the measurement of steady-state permeation current density (i∞), hydrogen diffusivity (D), and subsurface hydrogen concentration (C0). Results indicate that the addition of La3+ can remarkably reduce i∞. The corrosion products with La3+ involved in the film formation process in the H2S-free environment have a good hydrogen barrier effect, but the addition of La3+ hardly influence the D value of 27CrMo44S/1 steel in the H2S-containing environment. However, the presence of La3+ substantially reduces the C0 on metal surface regardless of the presence of H2S in the solution. Static immersion and tensile experiments illustrate that the addition of La3+ can improve the corrosion resistence and plasticity loss of the material in the H2S environment.

A 3D face interpolated discretisation method for simulating anisotropic diffusive processes on meshes coming from wood morphology

In this paper, a new 3D numerical discretisation method for solving the anisotropic, steady-state diffusion problem is developed and analysed. The scheme is constructed on hexahedral meshes using the geometrical properties of the cells. Indeed, each cell provides a local basis formed using the centres of its lateral faces. The discrete cell gradient approximation is then obtained by using three discrete directional derivatives resolved in terms of this basis, and by invoking the consistent relationships between opposite faces of the same cell. The face degrees of freedom are interpolated to reduce the complexity of the numerical scheme, resulting in the main unknowns being entirely nodal based. The scheme is unconditionally coercive and admits a unique solution. Various numerical experiments are performed to highlight the accuracy and the robustness of the method with respect to the mesh and anisotropy. An important outcome is that second order convergence is observed for all problems considered, even for highly deformed meshes. After this validation process, the method is applied to the prediction of the effective thermal conductivity of wood from its real 3D morphology. The property is estimated in the radial tangential and longitudinal directions. The solver is robust, efficient and yields to similar results compared to recent contributions.

Positional influence on cellular transcriptional identity revealed through spatially segmented single-cell transcriptomics.

Single-cell RNA sequencing (scRNA-seq) is a powerful technique for describing cell states. Identifying the spatial arrangement of these states in tissues remains challenging, with the existing methods requiring niche methodologies and expertise. Here, we describe segmentation by exogenous perfusion (SEEP), a rapid and integrated method to link surface proximity and environment accessibility to transcriptional identity within three-dimensional (3D) disease models. The method utilizes the steady-state diffusion kinetics of a fluorescent dye to establish a gradient along the radial axis of disease models. Classification of sample layers based on dye accessibility enables dissociated and sorted cells to be characterized by transcriptomic and regional identities. Using SEEP, we analyze spheroid, organoid, and invivo tumor models of high-grade serous ovarian cancer (HGSOC). The results validate long-standing beliefs about the relationship between cell state and position while revealing new concepts regarding how spatially unique microenvironments influence the identity of individual cells within tumors.

Inertial active ratchet: Simulation versus theory.

We present the inertial active dynamics of an Ornstein-Uhlenbeck particle in a piecewise sawtooth ratchet potential. Using the Langevin simulation and matrix continued fraction method (MCFM), the particle transport, steady-state diffusion, and coherence in transport are investigated in different parameter regimes of the model. Spatial asymmetry is found to be a key criterion for the possibility of directed transport in the ratchet. The MCFM results for net particle current of overdamped dynamics of the particle agree well with the simulation results. The simulated particle trajectories for the inertial dynamics and the corresponding position and velocity distribution functions reveal that the system passes through an activity-induced transition in the transport from the running phase to the locked phase of the dynamics. This is further corroborated by the mean square displacement (MSD) calculations, where the MSD gets suppressed with increase in the persistent duration of activity or self-propulsion in the medium and finally approaches zero for a very large value of self propulsion time. The nonmonotonic behavior of the particle current and Péclet number with self-propulsion time confirms that the particle transport and its coherence can be enhanced or reduced by fine tuning the persistent duration of activity. Moreover, for intermediate ranges of self-propulsion time as well as mass of the particle, even though the particle current shows a pronounced unusual maximum with mass, there is no enhancement in the Péclet number, instead the Péclet number decreases with mass, confirming the degradation of coherence in transport.

The Effect of Herbal Penetration Enhancers on the Skin Permeability of Mefenamic Acid Through Rat Skin.

Mefenamic acid (MA) is a strong non-steroidal anti-inflammatory drug, but because of its limited oral bioavailability and the side effects that come with taking it systemically, it is better to apply it topically. The major goal of this study was to see how certain permeation enhancers affected MA is in vitro skin permeability. In manufactured Franz diffusion cells, MA permeability tests using rat skin pretreatment with several permeation enhancers such as corn oil, olive oil, clove oil, eucalyptus oil, and menthol were conducted and compared to hydrate rat skin as a control. The steady-state flux (Jss), permeability coefficient (Kp), and diffusion coefficient are among the permeability metrics studied. The permeability enhancement mechanisms of the penetration enhancer were investigated using fourier transform infrared spectroscopy (FTIR) to compare changes in peak position and intensities of asymmetric and symmetric C-H stretching, C=O stretching, C=O stretching (amide I), and C-N stretching of keratin (amide II) absorbance, as well as differential scanning calorimetry (DSC) to compare mean transition temperature and their enthalpies. Clove oil, olive oil, and eucalyptus oil were the most effective enhancers, increasing flux by 7.91, 3.32, and 2.6 times, as well as diffusion coefficient by 3.25, 1.34, and 1.25, respectively, when compared to moist skin. FTIR and DSC data show that permeation enhancers caused lipid fluidization, extraction, disruption of lipid structures in the SC layer of skin, and long-term dehydration of proteins in this area of the skin. According to the findings, the permeation enhancers used improved drug permeability through excised rat skin. The most plausible mechanisms for greater ERflux, ERD, and ERP ratios were lipid fluidization, disruption of the lipid structure, and intracellular keratin irreversible denaturation in the SC by eucalyptus oil, menthol, corn oil, olive oil, and clove oil.

A Statistical Finite Element Method Integrating a Plurigaussian Random Field Generator for Multi-scale Modelling of Solute Transport in Concrete

A new model for the multi-scale simulation of solute transport in concrete is presented. The model employs plurigaussian simulations to generate stochastic representations of concrete micro- and meso-structures. These are idealised as two-phase medium comprising mortar matrix and pores for the micro-structure, and mortar and large aggregate particles for the meso-structure. The generated micro- and meso-structures are employed in a finite element analysis for the simulation of steady-state diffusion of solutes. The results of the simulations are used to calculate effective diffusion coefficients of the two-phase micro- and meso-structures, and in turn, the effective diffusion coefficient at the macro-scale at which the concrete material is considered homogenous. Multiple micro- and meso-structures are generated to account for uncertainty at the macro-scale. In addition, the level of uncertainty in the calculated effective diffusion coefficients is quantified through a statistical analysis. The numerical predictions are validated against experimental observations concerning the diffusion of chloride through a concrete specimen, suggesting that the generated structures are representative of the pore-space and coarse aggregate seen at the micro- and meso-scales, respectively. The method also has a clear advantage over many other structural generation methods, such as packing algorithms, due to its low computational expense. The stochastic generation method has the ability to represent many complex phenomena in particulate materials, the characteristics of which may be controlled through the careful choice of intrinsic field parameters and lithotype rules.

Evolving radon diffusion through earthen barriers at uranium waste disposal sites

Field measurements of Rn-222 fluxes from the tops and bottoms of compacted clay radon barriers were used to calculate effective Rn diffusion coefficients (DRn) at four uranium waste disposal sites in the western United States to assess cover performance after more than 20 years of service. Values of DRn ranged from 7.4 × 10−7 to 6.0 × 10−9 m2/s, averaging 1.42 × 10−7. Water saturation (SW) from soil cores indicated that there was relatively little control of DRn by SW, especially at higher moisture levels, in contrast to estimates from most steady-state diffusion models. This is attributed to preferential pathways intrinsic to construction of the barriers or to natural process that have developed over time including desiccation cracks, root channels, and insect burrows in the engineered earthen barriers. A modification to some models in which fast and slow pathway DRn values are partitioned appears to give a good representation of the data; 4% of the fast pathway was needed to fit the data regression. For locations with high Sw and highest DRn (and fluxes) at each site, the proportion of fast pathway ranged from 1.7% to 34%, but for many locations with lower fluxes, little if any fast pathway was needed.

A variational nodal formulation for multi-dimensional unstructured neutron diffusion problems

A variational nodal method (VNM) with unstructured-mesh is presented for solving steady-state and dynamic neutron diffusion equations. Orthogonal polynomials are employed for spatial discretization, and the stiffness confinement method (SCM) is implemented for temporal discretization. Coordinate transformation relations are derived to map unstructured triangular nodes to a standard node. Methods for constructing triangular prism space trial functions and identifying unique nodes are elaborated. Additionally, the partitioned matrix (PM) and generalized partitioned matrix (GPM) methods are proposed to accelerate the within-group and power iterations.Neutron diffusion problems with different fuel assembly geometries validate the method. With less than 5 pcm eigenvalue (keff) error and 1% relative power error, the accuracy is comparable to reference methods. In addition, a test case based on the kilowatt heat pipe reactor, KRUSTY, is created, simulated, and evaluated to illustrate the method's precision and geometrical flexibility. The Dodds problem with a step transient perturbation proves that the SCM allows for sufficiently accurate power predictions even with a large time-step of approximately 0.1 s. In addition, combining the PM and GPM results in a speedup ratio of 2–3.

Development and validation of multiphysics PWR core simulator KANT

KANT (KAIST Advanced Nuclear Tachygraphy) is a PWR core simulator recently developed at Korea Advance Institute of Science and Technology, which solves three-dimensional steady-state and transient multigroup neutron diffusion equations under Cartesian geometries alongside the incorporation of thermal-hydraulics feedback effect for multi-physics calculation. It utilizes the standard Nodal Expansion Method (NEM) accelerated with various Coarse Mesh Finite Difference (CMFD) methods for neutronics calculation. For thermal-hydraulics (TH) calculation, a single-phase flow model and a one-dimensional cylindrical fuel rod heat conduction model are employed. The time-dependent neutronics and TH calculations are numerically solved through an implicit Euler scheme, where a detailed coupling strategy is presented in this paper alongside a description of nodal equivalence, macroscopic depletion, and pin power reconstruction. For validation of the steady, transient, and depletion calculation with pin power reconstruction capacity of KANT, solutions for various benchmark problems are presented. The IAEA 3-D PWR and 4-group KOEBERG problems were considered for the steady-state reactor benchmark problem. For transient calculations, LMW (Lagenbuch, Maurer and Werner) LWR and NEACRP 3-D PWR benchmarks were solved, where the latter problem includes thermal-hydraulics feedback. For macroscopic depletion with pin power reconstruction, a small PWR problem modified with KAIST benchmark model was solved. For validation of the multi-physics analysis capability of KANT concerning large-sized PWRs, the BEAVRS Cycle1 benchmark has been considered. It was found that KANT solutions are accurate and consistent compared to other published works.

Bioturbation in sediment cores from the Clarion-Clipperton Zone in the northeast Pacific: Evidence from excess 210Pb

Radioactivity levels of 210Pb and 226Ra were measured to evaluate the bioturbation coefficients and mixing depths in one sediment core collected from the polymetallic nodule area inside the Clarion-Clipperton Zone (CCZ), a contract area of the China Ocean Mineral Resources Association (COMR) in the eastern Pacific Ocean. With a steady-state diffusion mode, the bioturbation coefficient and mixing depth of the sediment core were estimated to be 81.5 cm2/a and 51.3 cm, respectively, higher than that of the sediment core from the previous results from abyssal or global areas, which may be attributed to the distribution of total organic carbon (TOC) abundance. In addition, the transport flux of TOC from the surface layer to the deep layer were evaluated to be 1.15 mmol/(cm2·a), resulting in 83,200 tons of carbon buried annually into the sediment in the CCZ, indicating that a potential carbon sink will be disturbed with future exploration.

Crystallization by single diffusion through agar gel.

Crystallization is an important process in nature, various biological systems, and in industrial processes ranging from pharmaceuticals to material synthesis. In the industrial crystallization processes, common methods of crystallization include evaporative and cooling crystallization, and less frequently varying pH and addition of anti-solvents. The crucial driving force in all these techniques involves bringing the concentration of solutions to supersaturation to trigger the initial nucleation and growth of crystals. These conventional modes may not be applicable in systems where localized nucleation is required as these methods cause supersaturation throughout the bulk of solution which makes it challenging to affect crystallization at a specific desired location. In addition, these modes of reaching supersaturation cannot be used to explain or influence the crystallization in certain scenarios e.g., biological systems and in cases where the scope of temperature and a systemic pH change is very minimal. Another existing mode of crystallization involves the separation of ions into two different chambers and double diffusion through gels which is also not widely practical. Here we propose a mechanism in which we place salts in different solubility conditions on two sides of a permeable gel medium held at a steady state. Our steady state diffusion calculations predict that in certain pairings of different solubility conditions, the concentration of ions in the gel will always be above saturation concentration despite the absolute concentration decreasing along the length of the medium. We perform experiments in systems with variable solubilities in two different scenarios (pH and solvent mixtures) and we observe that the crystallization takes place in the gel indicating the supersaturation of the ions. This method offers an alternative way to get supersaturated solutions without evaporating solvents, changing temperatures, or separating the constituent ions into two different chambers across the diffusive medium.

Steady-state diffusion with the smooth step dependence of the diffusion coefficient on concentration

Steady-state diffusion with the smooth step dependence of the diffusion coefficient on concentration

Alleviation of Self-Heating Effect in Top-Gated Ultrathin In2O3 FETs Using a Thermal Adhesion Layer

In this work, a thin layer of hexagonal boron nitride (h-BN) or HfO2 serves as an adhesion layer between the ultrathin atomic-layer-deposited (ALD) indium oxide (In2O3) channel and a sapphire substrate to enhance the thermal interfacial conductance. A thermo-reflectance (TR) measurement system with high spatial resolution is introduced to experimentally demonstrates the improvement. With the thin h-BN or HfO2 interlayer, the temperature elevation ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta {T}$ </tex-math></inline-formula> ) induced by self-heating effect (SHE) is decreased by roughly 9% or 27%, respectively. To quantify the improvement of the interfacial heat transfer, a steady-state thermal diffusion model with a finite-element method is combined with the experimental TR observation to extract the effective thermal boundary conductance (TBC) values in each case. It is shown that the effective TBC is ameliorated by a factor of 2 or 7 with the h-BN or HfO2 interlayer, which is responsible for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta {T}$ </tex-math></inline-formula> reduction. Furthermore, phonon density of states (PDOS) distribution mismatch implies that the intersection over union (IOU) ratio in the acoustic phonon region of In2O3 with h-BN or HfO2 is roughly 3 or 11 times higher than that directly with sapphire, which is responsible for the profound TBC enhancement. Based on this, ultrahigh maximum drain current of 2.4 mA/ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> is achieved with 2.1-nm-thick In2O3 channel on a sapphire substrate with an HfO2 interlayer in between due to the alleviated SHE.