Classical Diffusion Research Articles

The Existence of Solution of Diffusion Equation with the General Conformable Derivative

In this paper, the general conformable fractional derivative is used in the classical diffusion equations, and the corresponding maximum principle is obtained. By the maximum principle, this paper proves the uniqueness of the solution and the continuous dependence on source function and initial-boundary conditions of the solution. Furthermore, by employing the variable separation method, this paper obtains some existence results and the asymptotic behavior of the classical solution.

A new model for predicting the thickness of intermetallic compounds in friction stir welding

Abstract The thickness of intermetallic compounds (IMCs) is one of the main factors affecting the weld quality. In addition, the IMC thickness will affect the product functionality, e.g., the thermal or electrical conductivity of the IMC layer can be the limiting factor in the related applications. Modeling and prediction of the IMC thickness in the friction stir welding (FSW) process is an important role that has not been elaborated in the literature. Additionally, many studies used a model for the IMC thickness that is related to the classical diffusion phenomenon. That model is suitable for the pure (intrinsic) diffusion process and does not consider the main unique characteristic of FSW, i.e., the stirring and subsequent linear velocity of particles (that has an impact on the diffusion process). To address this research gap, first, we develop a new model for the IMC thickness that takes the velocity of particles into account. Second, we provide an analysis on the velocity of particles in FSW based on vortex dynamics. Third, we analyze the proposed IMC thickness model with experimental data from the literature, discuss the added values of our model, and finally examine a case study. Understanding various manufacturing processes can further lead to the improvement of the existing processes and the development of novel processes.

Mesoscopic wave physics in fish shoals

Ultrasound scattered by a dense shoal of fish undergoes mesoscopic interference, as is typical of low-temperature electrical transport in metals or light scattering in colloidal suspensions. Through large-scale measurements in open sea, we show a set of striking deviations from classical wave diffusion, making fish shoals good candidates to study mesoscopic wave phenomena. The very good agreement with theories enlightens the role of fish structure in such a strong scattering regime that features slow energy transport and brings acoustic waves close to the Anderson localization transition.

Global stability in a networked SIR epidemic model

A graph Laplacian reaction–diffusion system is introduced to a networked SIR epidemic model. By the means of Lyapunov function, we show that the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than 1, while the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is lower than 1. We extend the stability theory of SIR models with classical Laplacian diffusion to models with graph Laplacian.

On the convergence rate of the quasi- to stationary distribution for the Shiryaev-Roberts diffusion

For the classical Shiryaev-Roberts martingale diffusion considered on the interval where A > 0 is a given absorbing boundary, it is shown that the rate of convergence of the diffusion’s quasi-stationary cumulative distribution function (c.d.f.), to its stationary c.d.f., H(x), as is no worse than uniformly in The result is established explicitly by constructing new tight lower- and upper-bounds for using certain latest monotonicity properties of the modified Bessel K function involved in the exact closed-form formula for recently obtained by Polunchenko (2017b).

IMPLEMENTATION OF HIGH-ORDER, DISCONTINUOUS GALERKIN TIME STEPPING FOR FRACTIONAL DIFFUSION PROBLEMS

AbstractThe discontinuous Galerkin (DG) method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses coefficients defined by integrals that are not easily evaluated. We describe specialized quadrature techniques that efficiently maintain the overall accuracy of the DG method. In addition, we observe in numerical experiments that known superconvergence properties of DG time stepping for classical diffusion problems carry over in a modified form to the fractional-order setting.

Classical Advection-Diffusion in Heterogeneous Media

A method is proposed for solving the problem of impurity transport in heterogeneous medium due to the classical diffusion and advection. The case when advection is absent was analyzed separately. It focuses on distances from the impurity source, which are much larger than the main body of its localization, and the asymptotic approach developed by one of the authors (P.S.K.) is used. The problem is reduced to solving a differential equation of the first order, which determines the linear trajectory of the concentration signal from the source that arises with this approach to the point of observation. The result for concentration is expressed in terms of one-dimensional integrals along the concentration signal line. The solution of the transport problem in the presence of advection is obtained by transition into the coordinate system accompanying advection. The key elements entering the resulting concentration expression are the effective time and impurity displacement, both are caused by advection.

Preliminary in Vitro Investigations on The Inhibitory Activity of The Original Dietary Supplement Oxidal® on Pathogenic Bacterial Strains

The antimicrobial action of the dietary supplement Oxidal® was tested using the classic Bauer and Kirby agar-gel diffusion method. Clinical and reference strains of Staphylococcus aureus and Escherichia coli were used in the studies. The tested dietary supplement showed a well-pronounced inhibitory effect against the microbial strains commensurable with that of the broad-spectrum chemotherapeutic agent Enrofloxacin and showed even higher activity than the broad spectrum antibiotic Thiamphenicol. The proven inhibitory effect of the tested dietary supplement against the examined pathogenic bacteria is in accordance with the established clinical effectiveness standards for antimicrobial agents.

Hypoplasia of cerebellar afferent networks in Down syndrome revealed by DTI-driven tensor based morphometry

Quantitative magnetic resonance imaging (MRI) investigations of brain anatomy in children and young adults with Down syndrome (DS) are limited, with no diffusion tensor imaging (DTI) studies covering that age range. We used DTI-driven tensor based morphometry (DTBM), a novel technique that extracts morphometric information from diffusion data, to investigate brain anatomy in 15 participants with DS and 15 age- and sex-matched typically developing (TD) controls, ages 6–24 years (mean age ~17 years). DTBM revealed marked hypoplasia of cerebellar afferent systems in DS, including fronto-pontine (middle cerebellar peduncle) and olivo-cerebellar (inferior cerebellar peduncle) connections. Prominent gray matter hypoplasia was observed in medial frontal regions, the inferior olives, and the cerebellum. Very few abnormalities were detected by classical diffusion MRI metrics, such as fractional anisotropy and mean diffusivity. Our results highlight the potential importance of cerebro-cerebellar networks in the clinical manifestations of DS and suggest a role for DTBM in the investigation of other brain disorders involving white matter hypoplasia or atrophy.

Vertex correction and conductivity in 2D Dirac-like systems with non-conserving spin disorder

We determine the classical diffusion of two dimensional Dirac-like quasiparticles, in the presence of conserving spin disorder (scattering off electric impurities) and non-conserving spin disorder (scattering off magnetic impurities). We use the Kubo formula for the conductivity tensor and employ diagrammatic perturbation theory to calculate the vertex correction and the renormalisation of the current operator for both electric and magnetic scattering. Scattering off electric impurities is isotropic and the current operator renormalised to two times the bare current operator irrespective of the direction of the dynamics, as usual for Dirac-like fermions. For magnetic scattering the renormalisation of the current operator depends on the direction of the dynamics and on the polarisation of the magnetic impurities, making the system anisotropic. We calculate the anisotropic magnetoresistance (AMR) and analyse it as a function of the ratio of the strength of the electric to the magnetic scattering potentials, for short range Gaussian correlation.

Growth of intermetallic compounds in solder joints based on strongly coupled thermo–mechano–electro–diffusional theory

The growth of intermetallic compounds (IMCs) is one of the key problems that restrict the reliability of solder joints. A phase field model based on strongly coupled thermo–mechano–electro–diffusional theory is proposed to investigate the growth of IMCs and formation mechanism. Of which, the diffusion coefficients and effective charge numbers of Cu and Sn were regarded as a continuous function of concentration on the basis of experimental data. The effect of back stress can be determined to be equivalent to that of volumetric strain through the analogy of the diffusion equation in the strongly coupled theory and the classical diffusion equation. On this account, the evolution of back stress can be evaluated during the entire process.Simulation results demonstrate that the growth curves are well consistent with the experiment values at low current densities when the coupled phase field model is adopted. The reasons for the poor coincidence at high current density are investigated and explained. And then, the thermodynamic driving forces in the entire growth process of IMCs are quantitative assessed in detail. The result shows that the evolution of back stress is agree with the sum of Cu and Sn concentrations in the IMCs, which indicates that the analogy is practical. Lastly, the concentration factor of the total driving force is defined to characterise the effect of current crowding on the growth of IMCs in solder joints.

Computation of quantum‐corrected noise in graphene‐SiC‐based impact avalanche transit time diode

AbstractHere, we have performed the noise analysis in graphene‐SiC‐based double drift region impact avalanche transit time diode (IMPATT). For computation, the noise model taken is based on drift diffusion with quantum corrections. Though noise computation in IMPATT with classical drift diffusion model are widely followed in earlier years, noise analysis through classical drift diffusion model is independent of confinement and tunneling due to quantum effects. Noise computed through quantum‐corrected noise model is more accurate to measured noise. Therefore, by considering quantum effects in addition to classical noise model, we have computed the noise in graphene‐SiC IMPATT. The noise performance of graphene‐SiC IMPATT is evaluated at Ka band which ranges from 26.5 to 40 GHz. The obtained noise from graphene‐SiC IMPATT is in good agreement with earlier reported noise behavior in IMPATT. We observed that the graphene‐SiC IMPATT is giving the better noise performance in terms of lower computed noise and noise spectral density as compared to other materials reported so far.

Effect of initial crystallization temperature and surface diffusion on formation of GaAs multiple concentric nanoring structures by droplet epitaxy**Project supported by the National Natural Science Foundation of China (Grant Nos. 61564002 and 11664005), the Science and Technology Foundation of Guizhou Province, China (Grant No. QKH-[2017]1055), and Guizhou University Talent Foundation (Grant No. GDJHZ-[2015]23).

GaAs multiple concentric nano-ring structures (CNRs) are prepared with multistep crystallization procedures by droplets epitaxy on GaAs (001) to explore the influence of different initial crystallization temperatures on CNRs morphology. Atomic force microscope (AFM) images show that GaAs nanostructures are more likely to form elliptical rings due to diffusion anisotropy. Meanwhile, with the increase of initial crystallization temperature, the inner ring height and density of CNRs are increased, and outer rings are harder to form. In addition, the mechanism of formation of CNRs is discussed by classical nucleation theory and diffusion theory. The method can be used to calculate the diffusion activation energy of gallium atoms (0.7±0.1 eV) on the GaAs (001) surface conveniently.

New Findings on an Old Question: Can Defect‐Free Graphene Monolayers be Superior Metal‐Ion Battery Anodes?

AbstractGraphene has numerous extraordinary properties and applications, ranging from water purification to energy storage. Previous studies identified that defect‐free graphene monolayers cannot be adopted as lithium‐ion battery anodes. Approaches, such as functionalization and introducing defects in graphene, have been employed to improve its performance in energy storage. However, defects in graphene boost dendrite growth, resulting in safety issues and fast power losses in batteries. Herein, by adopting state‐of‐the‐art first‐principles calculations, orbital theory, and transition state theory, it is found that defect‐free graphene monolayers are a promising potassium‐ion battery (KIB) anodes, however, they cannot be employed as lithium‐/sodium‐/magnesium‐/calcium‐/zinc‐/aluminum‐ion battery anodes. The evidence of adsorption energy and electron transfer indicates that van der Waals interactions dominate the adsorption of K atoms while delocalized electrons neutralize repulsion between K ions. The KIB anode delivers a high capacity of 1487.7 mA h g−1 and ultrafast charging/discharging rates. Transition state theory results demonstrate that its ultrahigh classic diffusion constants achieve 2.36 × 1011/1.55 × 109 s−1, its Wigner zero‐point‐energy‐/tunneling‐corrected diffusion constants approach 2.38 × 1011/1.56 × 109 s−1, and its low quantum‐mechanical tunneling effects are 0.67%/0.72% under the low/high concentration diffusion of K ions at room temperature. This work offers a more comprehensive understanding of defect‐free graphene.

Solution of Fourth Order Diffusion Equations and Analysis Using the Second Moment

The classical concept of diffusion characterized by Fick’s law is well suited for describing a wide class of practical problems of interest. Nevertheless, it has been observed that it is not enough to properly represent other relevant applications of practical interest. When in a system of particles their spreading is slower or faster than predicted by the classical diffusion model, such a phenomenon is referred to as anomalous diffusion. Time fractional, space fractional and even space-time fractional equations are widely used to model phenomena such as solute transport in porous media, financial modelling and cancer tumor behavior. Considering the effects of partial and temporary retention in dispersion processes a new analytical formulation was derived to simulate anomalous diffusion. The new approach leads to a fourth-order partial differential equation (PDE) and assumes the existence of two concomitant fluxes. This work investigates the behavior of the bi-flux approach in one dimensional (1D) medium evaluating the mean square displacement for different cases in order to classify the diffusion process in normal, sub-diffusive or super-diffusive.

Evaluation of the impact of multi-phase flow on reservoir signatures in the Wolfcamp shale

The dominant transient flow regime for multi-fractured horizontal wells (MFHWs) producing from low-permeability and shale (unconventional) reservoirs has historically been interpreted to be transient linear flow (TLF) with analysis performed in the framework of classical diffusion (CD). Recently, observed deviations away from this classical behavior for Permian Basin Wolfcamp shale (oil) wells have been attributed to reservoir heterogeneity and anomalous diffusion (AD). However, some wells in the basin also produce below bubble point pressure and exhibit characteristics of multi-phase flow. The objective of the current study is therefore to investigate if multi-phase flow may also be contributing to observed deviations from classic TLF behavior.The conventional log-log diagnostics used to identify flow regimes do not account for reservoir complexities such as multi-phase flow and reservoir heterogeneity. Failure to correct for these effects when they are occurring may result in misdiagnosis of flow regimes. A new workflow is therefore introduced herein to improve flow regime identification when reservoir complexities are exhibited, with a specific focus on multi-phase flow. The workflow involves the correction of log-log diagnostics for multi-phase flow through the use of modified pseudo-variables (pseudo-pressure and pseudo-time) once characteristics of multi-phase flow are identified (e.g. rapid increase of gas-oil ratio after bubble point pressure is reached). Although reservoir heterogeneity and AD are accepted causes of deviations from TLF, the impact of multi-phase flow has not been investigated in detail. Therefore, corrections to pseudo-variables for multi-phase flow, a known reservoir complexity exhibited by Wolfcamp shale wells, are presented. Pressure-dependent permeability is also accounted for in pseudo-variable calculations, although its impact is demonstrated to be relatively minor in this study.Application of the new workflow to a simulated case and a Wolfcamp shale field case demonstrates the following: 1) multi-phase flow, and in particular the appearance of a mobile gas phase after two-phase oil and water production, results in deviations from classical TLF behavior when data is analyzed using conventional (uncorrected) diagnostics; 2) application of the modified diagnostics to a simulated case that includes multi-phase flow results in the “true” flow regime signature of TLF being observed; 3) application of the modified diagnostics to a field case exhibiting evidence of multi-phase flow reduces the deviation from TLF.This study suggests that multi-phase flow may be impairing our ability to identify flow regimes using conventional flow regime identification methods.

Developing an Inkjet Printer I: RGB Image to CMY Ink Amounts - Image Processing and Color Management

In this paper, we propose a novel error diffusion halftoning algorithm for the conversion of cyan, magenta and yellow (CMY) ink amounts to a multi-bit halftone image. We adopt the idea of allowing multiple drops of colorants in order to achieve print results with better saturation, which is implemented by modifying the classic Floyd-Steinberg error diffusion algorithm. For better halftone texture and more efficient use of colorants, we add a dot-off-dot feature to the classic Floyd-Steinberg error diffusion. Blending-in the original input image with its DBS screened multilevel halftone image as a preprocessing step to dot-off-dot error diffusion is proposed as a measure to reduce halftone artifacts. Digitally simulated halftone images will be presented to illustrate the better halftone texture that can be achieved by applying the proposed algorithm.

Temperature- and pressure-dependent gas diffusion in coal particles: Numerical model and experiments

The diffusion coefficient is a key parameter related to gas migration in coalbed methane extraction and CO2 geologic sequestration applications. Isothermal assumptions are usually used in diffusion models and experimental tests, in which the thermal effect and its impact on gas adsorption/desorption are neglected. Coal powder gas diffusion tests are not exactly reflective of an isothermal process, in fact, due to gas-desorption-induced decreases in temperature. In this study, a coupled thermal-diffused-mechanical model for gas desorption–diffusion under non-isothermal conditions is developed in which the effects of gas pressure, temperature, and coal porosity on the gas diffusion coefficient are considered. The effect of gas-desorption-induced temperature variation is also included. Gas desorption–diffusion tests are conducted on coal powder under isothermal and adiabatic conditions, respectively. The developed model is then used to simulate the gas desorption–diffusion process under adiabatic conditions and the results are compared against the test data. The proposed non-isothermal model with varying diffusion coefficient, which can be regarded as an improvement of the classical unipore diffusion model, is shown to replicate the whole desorption–diffusion process and back-calculate a relatively consistent diffusion coefficient value range for all scenarios. The results also show that the assumption of isothermal conditions in a diffusion model significantly overestimates the accumulated gas diffusion content throughout the gas desorption–diffusion process. The assumption of constant diffusion coefficient partially underestimates the gas diffusion rate and accumulated gas diffusion content in the later stages.

The effect of initial values on extinction or persistence in degenerate diffusion competition systems.

When the asymptotic spreading for classical monostable Lotka-Volterra competition diffusion systems is concerned, extinction or persistence of the two competitive species is completely determined by the dynamics of the corresponding kinetic systems, while the size of initial values does not affect the final states. The purpose of this paper is to demonstrate the rich dynamics induced by the initial values in a class of degenerate competition diffusion systems with weak Allee effect. We present various extinction or persistence results by selecting different initial values although the corresponding kinetic system is fixed, which also implies the existence of balance between degenerate nonlinear reaction and diffusion. For example, even if the positive steady state of the corresponding kinetic system is globally asymptotically stable, we observe four different spreading-vanishing phenomena by selecting different initial values. In addition, the interspecific competition of one species may be harmful to the persistence of the other species by taking proper initial values. Our results show that the superior competitor in the sense of the corresponding kinetic system is not always unbeatable, it can be wiped out by the inferior competitor in the sense of the corresponding kinetic system depending on the size of initial habitats as well as the intensity of Allee effect.

In-situ observation and analysis of solid-state diffusion and liquid migration in a crystal growth system: A segregation-driven diffusion couple

Energy-resolved neutron imaging is employed for in-situ measurements of dopant transport in a simple experiment performed before the crystal growth of the scintillator BaBrCl:5%Eu via a vertical gradient freeze technique. During a stabilization period preceding growth, we observed the diffusion of Eu from the solid phase into the melt over a period of approximately 4 h. Comparing the measured centerline concentration profile with a mathematical model for the system, we estimate the solid-state diffusivity of Eu in BaBrCl as D1=1.9×10−10 m2/s and an upper limit for the liquid-phase diffusivity of Eu in the melt as D2*=2.5×10−10 m2/s, at temperatures near the melting point. We compare this experiment, where diffusion is driven by a concentration discontinuity arising from segregation, to the classical diffusion couple technique. Suggestions are offered on how this segregation-driven couple might be improved as a tool for measuring diffusion coefficients, and we draw attention to the great promise of neutron imaging for in-situ measurements of the distribution of elements, with sufficiently high neutron attenuation coefficients, in difficult environments.