Diffusion Length Scale Research Articles

Catalyst-derived hierarchy in 2D imine-based covalent organic frameworks.

Identifying facile strategies for hierarchically structuring crystalline porous materials is critical for realizing diffusion length scales suitable for broad applications. Here, we elucidate synthesis-structure-function relations governing how room temperature catalytic conditions can be exploited to tune covalent organic framework (COF) growth and thereby access unique hierarchical morphologies without the need to introduce secondary templates or structure directing molecules. Specifically, we demonstrate how scandium triflate, an efficient catalyst involved in the synthesis of imine-based COFs, can be exploited as an effective growth modifier capable of selectively titrating terminal amines on 2D COF layers to facilitate anisotropic crystal growth. We systematically map a compositional pseudo-phase space and uncover key mechanistic insights governing the catalyst-derived evolution of globular COFs with sub-micron diffusion length scales into unique rosette structures. Comprised of interconnected, high-aspect-ratio crystalline porous sheets of only several unit cells in thickness, the resulting COFs offer orders of magnitude reduction in diffusion length scales and several-fold increase in external surface area, enabling rapid uptake of bulky dyes. Generally, the resulting synthesis-structure-function relations hold promise for realizing unique control over COF mesostructure, morphology, and function.

Diffusive-length-scale adjustable phase field fracture model for large/small structures

In phase field fracture (PFF) method, the sharp crack is approximated by a phase field crack zone whose size is characterized by a diffusive length scale. Recently, the diffusive length scale is usually regarded as a constant material parameter determined by the fracture toughness, material strength, and young's modulus. As a result, the application of the PFF method poses challenges when dealing with structures whose sizes are much too large or small compared to the constant diffusive length scale. In details, for a large-scale structure, a significant computational burden is inevitable due to the limitation imposed by the constant diffusive crack length scale on the element size (i.e. the element size is generally at least smaller than half of the diffusive crack length scale to achieve the sufficient precision for the phase field process zone). For a small-scale structure, the crack patterns tend to be unclear and unrealistic due to the excessively large diffusive crack zone. To address these limitations, we propose a novel PFF method to make the relation between the diffusive length scale and the material parameters adjustable via modifying the energetic degradation functions. It is found that with the increase of the diffusive crack length scale, a transition from quasi-brittleness to brittleness is observed in the constitutive relationship curves, which coincides with the classical size effect on structural strength. Further, the Bažant's size effect in classical fracture mechanics can be reproduced by the present PFF method through scaling the size of a geometrically similar structure, i.e. a large-scale structure exhibits toughness-dominated fracture while a small-scale structure behavior strength-dominated fracture. The present PFF method efficiently addresses the mismatch of diffusive length scales for various material constituents by examining phase field crack growth in particle-reinforced composite plates. By adjusting the diffusive crack length scale based on structure size, it avoids high computational costs from large structures and unrealistic crack patterns from small ones. Moreover, it outperforms traditional PFF methods using a constant diffusive length scale in handling composite materials.

Effects of mass diffusion on Rayleigh–Taylor instability under a large gravity

Rayleigh–Taylor instabilities (RTI) play an important role in the evolution of inertial confinement fusion (ICF) processes, while analytical prediction of the RTI growth rate often fails to reach an agreement with the experimental and simulation results. Accurate analytical prediction of RTI growth is of great significance to the success of ICF schemes. In this paper, we study the effects of mass diffusion and exponential density distribution on RTI under a large gravity by solving the Rayleigh equation with a linear approximation to the density distribution of the mixing layer. The width of the mixing layer is assigned by evaluating the length scale of concentration diffusion and gravitational sedimentation. The latter term is missing in the former isobaric diffusion treatment and is supposed to change the structure of the mixing layer under the gravity. While both effects tend to dampen the instability growth, mass diffusion dominates the damping of perturbations of larger wavenumber and exponential density distribution dominates those of smaller wavenumber, resulting in a non-monotonicity of the density suppression factor of the instability growth rate over perturbation wavenumbers.

Towards the understanding of convective dissolution in confined porous media: thin bead pack experiments, two-dimensional direct numerical simulations and physical models

We consider the process of convective dissolution in a homogeneous and isotropic porous medium. The flow is unstable due to the presence of a solute that induces a density difference responsible for driving the flow. The mixing dynamics is thus driven by a Rayleigh–Taylor instability at the pore scale. We investigate the flow at the scale of the pores using Hele-Shaw type experiment with bead packs, two-dimensional direct numerical simulations and physical models. Experiments and simulations have been specifically designed to mimic the same flow conditions, namely matching porosities, high Schmidt numbers and linear dependency of fluid density with solute concentration. In addition, the solid obstacles of the medium are impermeable to fluid and solute. We characterise the evolution of the flow via the mixing length, which quantifies the extension of the mixing region and grows linearly in time. The flow structure, analysed via the centreline mean wavelength, is observed to grow in agreement with theoretical predictions. Finally, we analyse the dissolution dynamics of the system, quantified through the mean scalar dissipation, and three mixing regimes are observed. Initially, the evolution is controlled by diffusion, which produces solute mixing across the initial horizontal interface. Then, when the interfacial diffusive layer is sufficiently thick, it becomes unstable, forming finger-like structures and driving the system into a convection-dominated phase. Finally, when the fingers have grown sufficiently to touch the horizontal boundaries of the domain, the mixing reduces dramatically due to the absence of fresh unmixed fluid. With the aid of simple physical models, we explain the physics of the results obtained numerically and experimentally. The solute evolution presents a self-similar behaviour, and it is controlled by different length scales in each stage of the mixing process, namely the length scale of diffusion, the pore size and the domain height.

Identifying diffusive length scales in one-dimensional Bose gases

In the hydrodynamics of integrable models, diffusion is a subleading correction to ballistic propagation. Here we quantify the diffusive contribution for one-dimensional Bose gases and find it most influential in the crossover between the main thermodynamic regimes of the gas. Analysing the experimentally measured dynamics of a single density mode, we find diffusion to be relevant only for high wavelength excitations. Instead, the observed relaxation is solely caused by a ballistically driven dephasing process, whose time scale is related to the phonon lifetime of the system and is thus useful to evaluate the applicability of the phonon bases typically used in quantum field simulators.

A Thermodynamically Based Modified Cam‐Clay Model for Post‐Bifurcation Behavior of Deformation Bands

AbstractCompaction bands are a type of localized deformation that can occur as diffuse or discrete bands in porous rocks. While modeling of shear bands can replicate discrete and diffusive bands, numerical models of compaction have so far only been able to describe the formation of discrete compaction bands. In this study, we present a new thermodynamic approach to model compaction bands that is able to capture both discrete and diffuse compaction band growth. The approach is based on a reaction‐diffusion formalism that includes an additional entropy flux. This entropic velocity regularizes the solution, by introducing a characteristic diffusion length scale and controlling the mode change from discrete to diffusive post‐localisation growth. The approach is used to model compaction band growth in highly porous carbonates. The model can replicate the areas of material damage exhibiting reduced porosity which are often observed as nuclei for the growth of compaction bands in experiments. The model also has the versatility to predict the formation of diffuse compaction bands, which is a significant advance in the field of compaction band modeling. The method can potentially be used for investigating the effect of material heterogeneities on compaction band growth and is heuristic for developing new methodologies for forecasting compaction band formation.

Binary bilayer simulations for partitioning within membranes.

Membrane proteins (MPs) often show preference for one phase over the other, which is characterized by the partition coefficient, Kp. The physical mechanisms underlying Kp have been only inferred indirectly from experiments due to the unavailability of detailed structures and compositions of ordered phases. Molecular dynamics (MD) simulations can complement these details and thus, in principle, provide further insights into the partitioning of MPs between two phases. However, the application of MD has remained difficult due to long time scales required for equilibration and large system size for the phase stability, which have not been fully resolved even in free energy simulations. This chapter describes the recently developed binary bilayer simulation method, where the membrane is composed of two laterally attached membrane patches. The binary bilayer system (BBS) is designed to preserve the lateral packing of both phases in a significantly smaller size compared to that required for macroscopic phase separation. These characteristics are advantageous in partitioning simulations, as the length scale for diffusion across the system can be significantly smaller. Hence the BBS can be efficiently employed in both conventional MD and free energy simulations, though sampling in ordered phases remains difficult due to slow diffusion. Development of efficient lipid swapping methods and its combination with the BBS would be a useful approach for partitioning in coexisting phases.

Growth of Massive Molecular Cloud Filament by Accretion Flows. I. Slow-shock Instability versus Ambipolar Diffusion

The Herschel Gould Belt Survey showed that stars form in dense filaments in nearby molecular clouds. Recent studies suggest that massive filaments are bound by the slow shocks caused by accretion flows onto the filaments. The slow shocks are known to be unstable to corrugation deformation of the shock front. Corrugation instability could convert the accretion flow's ram pressure into turbulent pressure that influences the width of the filament, which, according to theory, determines the self-gravitational fragmentation scale and core mass. In spite of its importance, the effect of slow-shock instability on star-forming filaments has not been investigated. In addition, the linear dispersion relation obtained from ideal magnetohydrodynamics (MHD) analysis shows that the most unstable wavelength of shock corrugation is infinitesimally small. In the scale of dense filaments, the effect of ambipolar diffusion can suppress the instability at small scales. This study investigates the influence of ambipolar diffusion on the instability of the slow shock. We perform two-dimensional MHD simulations to examine the linear growth of the slow-shock instability, considering the effect of ambipolar diffusion. The results demonstrate that the most unstable scale of slow-shock instability is approximately 5 times the length scale of ambipolar diffusion ℓ AD calculated using post-shock variables, where ℓ AD corresponds to the scale where the magnetic Reynolds number for ambipolar diffusivity is unity.

Spatial confinement of Co-N-C catalyst in carbon nanocuboid for water Decontamination: High Performance, mechanism and biotoxicity assessment

Encapsulating free radical species with super-short lifetime and organic pollutants inside the chemical theoretical diffusion length scale by spatial confinement could offer a strategy with great promise to resolve the mass transfer restrictions in heterogeneous Fenton-like systems. Herein, we reported the spatial confinement of Co, N co-doped carbon (Co-N-C) catalyst in carbon nanocuboids (Co-CNCs), via encapsulation of Co-N-C catalyst into CNCs, for activating CaSO3 to efficiently degrade sulfamethoxazole. The detailed characterizations confirmed ZIF-9 was successfully transformed into N-doped carbon nanocuboids at 850 °C calcination. Co-N-C catalyst was successfully confined into carbon nanocuboids, is beneficial to prevent the aggregation of Co nanoparticles, thereby facilitating its stability and catalyst activity in sulfamethoxazole (SMX) removal via CaSO3 activation. Compared with other sulfites (including Na2SO3 and NaHSO3), CaSO3 (82%) presented much higher degradation efficiency than those of NaSO3 (18%) and NaHSO3 (41%) in SMX removal, emphasizing the critical role of CaSO3 in water decontamination. Because CaSO3 could discharge SO32- persistently and slowly, preventing from the consumption of the formed SO4•- at a high concentration of aqueous SO32-. Quenching test and mechanism study had verified that •OH and SO4•- (SO3•-) were the major reactive species in Co-CNC-850/CaSO3 system. In addition, Zebrafish toxicology assays and seed germination experiments had revealed the biotoxicity of SMX to zebrafish and wheat seeds was significantly reduced after degradation via Co-CNC-850/CaSO3 system.

A scaling procedure for designing thermochemical energy storage system

Thermochemical energy storage (TCES) technology offers a promising avenue for achieving prolonged thermal energy storage through reversible gas-solid reactions. In the present work, a scaling analysis approach is presented to derive the relevant length and time scales for heat charging and discharging processes occurring inside a porous reactor bed of a TCES system. The processes involved are vapour flow through the porous bed, coupled heat and mass transfer, and heat and mass absorption/release in conjunction with reaction kinetics, for the case of water vapour and potassium carbonate as the gas-solid pair. This analysis employs an order-of-magnitude approach on mass, momentum, and energy conservation equations. The scaling analysis of continuity and momentum equations derives the scale for critical vapour penetration depth. It signifies the length scale beyond which pressure drop becomes significant, leading to a significant reduction in the reaction rate. Similarly, a critical thermal diffusion length scale is derived from the scaling analysis of the energy equation. This scale signifies the length beyond which the reaction rate drops significantly due to limited thermal diffusion. The derived scales characterize the performance of the energy storage bed and help in determining bed dimensions for an effective flow of heat and water vapour through the bed.

Mass transfer in atmospheric water harvesting systems

In this work, the rate-limiting diffusion mechanisms of MOF-303, MOF-333, and a multivariate (MTV) version of these metal–organic frameworks (MOFs), where the organic linkers are present in a 50/50 ratio, are identified and quantified using concentration swing frequency response (CSFR). The data show that the single-atom precision of MOFs allows for precise tuning of the diffusion rate that is not easily achieved in traditional adsorbent materials. The Maxwell-Stefan diffusivity as a function of loading was calculated to decouple the influence of molecular mobility and equilibrium effects. To further understand the diffusion process in these MOFs, samples with different crystal sizes were synthesized and diffusion rates were measured. The results show that the controlling diffusion length scale is similar between the small and large crystal samples, as evidenced by similar diffusion rate constants. The MOFs were then incorporated into a film using a binder system and the mass transfer mechanisms were identified using CSFR. When placed in this particular binder system, the macropore diffusion behavior dominates over the MOF micropore diffusion. To illustrate how these diffusion parameters govern the adsorption rates and dynamics of water-harvesting systems, a model of the MOF-coated tube was developed. The results show that, with only milligrams of adsorbent, CSFR can quantify the diffusion rate needed to predict the adsorption times in a water-harvesting system. More broadly, the results illustrate the connectivity between atomically-precise reticular chemistry and water-harvesting system performance.

Virtual Volumetric Additive Manufacturing (VirtualVAM)

AbstractTomographic volumetric additive manufacturing (VAM) produces arbitrary 3D geometries by exposure of a rotating volume of photopolymer resin to tomographically‐patterned illumination. This enables high speed, layer‐less printing of parts from a wide range of photopolymers not amenable to layer‐by‐layer processes. Since the entire geometry is produced at once over the course of a few seconds to minutes, molecular diffusion length scales become significant to the printing process. Understanding these molecular reaction and diffusion processes is imperative for advancing VAM to a usable technology. These processes are experimentally very difficult to monitor and measure. Herein, VirtualVAM ‐ a simulation framework for modeling the tomographic VAM process, is developed and experimentally validated. VirtualVAM simulates reaction, diffusion, and heat generation processes over the course of a print with single‐voxel resolution. From a few experimentally‐determined input parameters and a set of images for projection, VirtualVAM is able to generate a large spatio‐temporal data set for any given tomographic VAM print. Using VirtualVAM, a number of experimentally‐unattainable aspects of the VAM process are investigated such as single‐voxel conversion profiles, effect of molecular oxygen, and stopping time determination. VirtualVAM also enables the optimization of exposure patterns to further improve contrast between in‐part and out‐of‐part delivered dose.

Magma mingling and ascent in the minutes to hours before an explosive eruption as recorded by banded pumice

High-threat explosive silicic eruptions commonly contain banded pumice, reflecting magma mingling in the conduit prior to or during eruption. Heterogeneities in tuffs have been attributed to the draw-up of compositionally distinct magmas, in which low-viscosity magmas ascend more quickly than high-viscosity magmas. The Rattlesnake Tuff of the High Lava Plains in Oregon (northwestern United States) represents a zoned magma reservoir where at least five different rhyolite compositions are preserved in banded pumice samples in variable mingled combinations. Geochemical gradients recorded across band boundaries in pumice were modeled using a Monte Carlo least-square minimization procedure to find the complementary error function that best fit observed Si and Ba diffusion profiles by iteratively varying the concentration of each plateau (i.e., the concentration on either side of the band boundary), the center and spacing of the diffusion profile, diffusion length scale, and temperature. Modeling indicates maximum time scales between mingling and conduit ascent from minutes to hours. Viscosity calculations for each rhyolite composition confirm that highly viscous rhyolites have longer ascent times than low-viscosity magmas, strongly supporting a model of sequential tapping of a zoned chamber controlled by viscosity.

Length-scales for efficient CFL conditions in high-order methods with distorted meshes: Application to local-timestepping for [formula omitted]-multigrid

We propose a strategy to estimate the maximum stable time-steps for explicit time-stepping methods for hyperbolic systems in a high-order flux reconstruction framework. The strategy is derived through a von-Neumann analysis (VNA) framework for the advection–diffusion equation on skewed two- and three-dimensional meshes. It directly incorporates the spatial polynomial- and mesh-discretization in estimating the convective and diffusive length-scales. The strategy is extended to the density-based Navier–Stokes system of equations, taking into account the omnidirectionality of the speed of sound.We compare the performance of this strategy with three other popular choices of length-scales across a wide range of polynomial-orders, meshes of drastically varying cell-quality, and flow-physics. The proposed strategy shows robust behavior across all test-scenarios with limited variation of the maximum stable CFL-number (0.1 to 1) for polynomial-orders 1 through 10, unlike other strategies where the CFL-number varies sharply. Finally, we show the advantage of the proposed methodology for local-timestepping for p-multigrid through a RANS-modeled steady-state turbulent flow case, on a mesh with large disparity of mesh elements and aspect ratios.

Physical interactions in non-ideal fluids promote Turing patterns.

Turing's mechanism is often invoked to explain periodic patterns in nature, although direct experimental support is scarce. Turing patterns form in reaction-diffusion systems when the activating species diffuse much slower than the inhibiting species, and the involved reactions are highly nonlinear. Such reactions can originate from cooperativity, whose physical interactions should also affect diffusion. We here take direct interactions into account and show that they strongly affect Turing patterns. We find that weak repulsion between the activator and inhibitor can substantially lower the required differential diffusivity and reaction nonlinearity. By contrast, strong interactions can induce phase separation, but the resulting length scale is still typically governed by the fundamental reaction-diffusion length scale. Taken together, our theory connects traditional Turing patterns with chemically active phase separation, thus describing a wider range of systems. Moreover, we demonstrate that even weak interactions affect patterns substantially, so they should be incorporated when modelling realistic systems.

Fundamental solution of diffusion equationfor Kappa gas: Diffusion length for suprathermal electrons in solar wind.

A recent numerical treatment of data obtained by the Parker Solar Probe spacecraft describes the electron concentration in solar wind as a function of the heliocentric distance based on a Kappa distribution with spectral index κ=5. In this work, we derive and, subsequently, solve an entirely different class of nonlinear partial differential equationsdescribing the one-dimensional diffusion of a suprathermal gas. The theory is applied to describe the aforementioned data and we find a spectral index κ≳1.5 providing the widely acknowledged identification of Kappa electrons in solar wind. We also find that suprathermal effects increase the length scale of classical diffusion by one order of magnitude. Such a result does not depend on the microscopic details of the diffusion coefficient since our theory is based on a macroscopic formulation. Forthcoming extensions of our theory by including magnetic fields and relating our formulation to nonextensive statistics are briefly addressed.

Variational multiscale method stabilization parameter calculated from the strain-rate tensor

The stabilization parameters of the methods like the Streamline-Upwind/Petrov–Galerkin, Pressure-Stabilizing/Petrov–Galerkin, and the Variational Multiscale method typically involve two local length scales. They are the advection and diffusion length scales, appearing in the expressions for the advective and diffusive limits of the stabilization parameter. The advection length scale has always been in the flow direction. The diffusion length scales in use have mostly been just element-geometry-dependent, but there is good justification for also making that direction-dependent, so that the spatial variation of the solution is taken into account somehow. The length scale in the solution-gradient direction, which was introduced in 2001, was intended for making sure that near solid surfaces, the element length in the surface-normal direction is selected even if that is not the minimum element length. It was also intended for making sure that in a 2D computation with a 3D mesh, there would be no dependence on the element length in the third direction. With the same objectives, and with better invariance properties, we are now introducing the direction-dependent diffusion length scale calculated from the strain-rate tensor. We accomplish those objectives, get invariance with respect to switching to a different inertial reference frame, and the element length in the surface-normal direction, even when the surface is undergoing rotation, is selected as the diffusion length scale.

Position dependent microparticle charge in a spatiotemporal afterglow plasma

In the growing field of dusty afterglow plasma physics, the key parameter is the residual charge of dust particles. However, the particle (de)-charging process in afterglow plasmas is still far from fully understood and further development of a governing theoretical framework requires experimental data. In this work, the influence of the location of a microparticle in a spatiotemporal afterglow plasma, at the moment when the plasma was terminated, on its residual charge is investigated. It is found that the measured charge depends strongly on the local characteristic diffusion length scale of the system, while the plasma power prior to the start of the temporal afterglow phase is of much less influence. Our results contribute to an improved understanding of particle (de)-charging in afterglow plasmas and are highly relevant to the design of applications in which afterglow plasmas are present and where the charge of dust particles needs be controlled for the sake of (nano)contamination control.

Relaxation-Diffusion Spectrum Imaging for Probing Tissue Microarchitecture.

Brain tissue microarchitecture is characterized by heterogeneous degrees of diffusivity and rates of transverse relaxation. Unlike standard diffusion MRI with a single echo time (TE), which provides information primarily on diffusivity, relaxation-diffusion MRI involves multiple TEs and multiple diffusion-weighting strengths for probing tissue-specific coupling between relaxation and diffusivity. Here, we introduce a relaxation-diffusion model that characterizes tissue apparent relaxation coefficients for a spectrum of diffusion length scales and at the same time factors out the effects of intra-voxel orientation heterogeneity. We examined the model with an in vivo dataset, acquired using a clinical scanner, involving different health conditions. Experimental results indicate that our model caters to heterogeneous tissue microstructure and can distinguish fiber bundles with similar diffusivities but different relaxation rates. Code with sample data is available at https://github.com/dryewu/RDSI.

Travelling wave solutions of the cubic nonlocal Fisher-KPP equation: I. General theory and the near local limit

We study non-negative travelling wave solutions, u ≡ U(x − ct) with constant wavespeed c > 0, of the cubic nonlocal Fisher-KPP equation in one spatial dimension, namely, for , where u(x, t) is the population density. Here ϕ(y) is a prescribed, piecewise continuous, symmetric, nonnegative and nontrivial, integrable kernel, which is nonincreasing for y > 0, has a finite derivative as y → 0+ and is normalised so that . The parameter λ is the ratio of the lengthscale of the kernel to the diffusion lengthscale. The quadratic version of the equation, with reaction term u(1 − ϕ*u), has a unique travelling wave solution (up to translation) for all c ⩾ c min = 2. This minimum wavespeed is determined locally in the region where u ≪ 1, (Berestycki et al 2009 Nonlinearity 22 2813–44). For the cubic equation, we find that a minimum wavespeed also exists, but that the numerical value of the minimum wavespeed is determined globally, just as it is for the local version of the equation, (Billingham and Needham 1991 Dynam. Stabil. Syst. 6 33–49). We also consider the asymptotic solution in the limit of a spatially-localised kernel, λ ≪ 1, for which the travelling wave solutions are close to those of the cubic Fisher-KPP equation, u t = u xx + u 2(1 − u). We find that when ϕ = o(y −3) as y → ∞, the minimum wavespeed is , but that when ϕ = O(y −n ) with 1 < n ⩽ 3, the minimum wavespeed is . In each case we determine the correction terms. We also compare these asymptotic solutions to numerical solutions and find excellent agreement for some specific choices of kernel.