Zero-flux Condition Research Articles

Unified theory of nucleation and coarsening in the solid state

ABSTRACT Despite numerous contributions made to the modelling of precipitation kinetics, no theory seems to have unified all the concepts required to simulate the nucleation, growth, and ripening processes as one continuous process. In the present work, a novel approach was used to reshape the master equation calculating the evolution of the size distribution, so that the zero-flux condition can represent a steady-state evolution. This modification was made by introducing explicitly the probability for clusters to grow or to dissolve in each size class. The expression found to calculate this probability allowed the calculation of the entire evolution of the size distribution from the first step of nucleation to the final stages of the ripening process. The winners-losers concept also allowed for a better definition of the average condensation and evaporation rates, which were found by applying the thermodynamic extremum principle and assuming that a population of clusters of a given size behaves on the average as one cluster of the same size and submitted to the same driving forces. Furthermore, the thermodynamic extremum principle allowed us to show that the subcritical growth regime must be interface controlled to maximise the dissipation rate of energy. The computational scheme based on the new theory was applied to calculate the size evolution of precipitates in the Al-Cu system. The influence of aging temperature on the evolution of the number density of precipitates during homogeneous nucleation was found to agree with experimental estimations of number densities made on uniformly distributed θ′-Al2Cu precipitates.

A Diffuse-Interface Approach for Solid-State Dewetting with Anisotropic Surface Energies

We present a diffuse-interface model for the solid-state dewetting problem with anisotropic surface energies in {mathbb {R}}^d for din {2,3}. The introduced model consists of the anisotropic Cahn–Hilliard equation, with either a smooth or a double-obstacle potential, together with a degenerate mobility function and appropriate boundary conditions on the wall. Upon regularizing the introduced diffuse-interface model, and with the help of suitable asymptotic expansions, we recover as the sharp-interface limit the anisotropic surface diffusion flow for the interface together with an anisotropic Young’s law and a zero-flux condition at the contact line of the interface with a fixed external boundary. Furthermore, we show the existence of weak solutions for the regularized model, for both smooth and obstacle potential. Numerical results based on an appropriate finite element approximation are presented to demonstrate the excellent agreement between the proposed diffuse-interface model and its sharp-interface limit.

Exact solutions and asymptotic behaviors for the reflected Wiener, Ornstein-Uhlenbeck and Feller diffusion processes.

We analyze the transition probability density functions in the presence of a zero-flux condition in the zero-state and their asymptotic behaviors for the Wiener, Ornstein Uhlenbeck and Feller diffusion processes. Particular attention is paid to the time-inhomogeneous proportional cases and to the time-homogeneous cases. A detailed study of the moments of first-passage time and of their asymptotic behaviors is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions for the restricted Wiener, Ornstein-Uhlenbeck and Feller processes are proved. Specific applications of the results to queueing systems are provided.

Rival interpretations of hydride precipitation-dissolution hysteresis in zirconium: Rebuttal to Critical Comments to the Letter to the Editor of G.A. McRae and C.E. Coleman published in Journal of Nuclear Materials 556 (2021) 153169

Two models of precipitation-dissolution hysteresis in zirconium have been discussed recently. In one an elastic model for the hydrogen flux is written as the superposition of a diffusion current and a drift current. The solvus, TSS, is defined when the currents sum to zero at a hydride interface. Stable hydrides are surrounded by a cloud of hydrogen terminating at concentration C- calculated from another zero-flux condition. The temperature difference between TSS and C- agrees with the hysteresis. The second ‘traditional’ model invokes plastic deformation in zirconium caused by differences between the crystal axes in zirconium and those of the hydride.

Time-Inhomogeneous Feller-Type Diffusion Process in Population Dynamics

The time-inhomogeneous Feller-type diffusion process, having infinitesimal drift α(t)x+β(t) and infinitesimal variance 2r(t)x, with a zero-flux condition in the zero-state, is considered. This process is obtained as a continuous approximation of a birth-death process with immigration. The transition probability density function and the related conditional moments, with their asymptotic behaviors, are determined. Special attention is paid to the cases in which the intensity functions α(t), β(t), r(t) exhibit some kind of periodicity due to seasonal immigration, regular environmental cycles or random fluctuations. Various numerical computations are performed to illustrate the role played by the periodic functions.

Observations of Nearbed Turbulence over Mobile Bedforms in Combined, Collinear Wave-Current Flows

Collinear wave-current shear interactions are often assumed to be the same for currents following or opposing the direction of regular wave propagation; with momentum and mass exchanges restricted to the thin oscillating boundary layer (zero-flux condition) and enhanced but equal wave-averaged bed shear stresses. To examine these assumptions, a prototype-scale experiment investigated the nature of turbulent exchanges in flows with currents aligned to, and opposing, wave propagation over a mobile sandy bed. Estimated mean and maximum stresses from measurements above the bed exceeded predictions by models of bed shear stress subscribing to the assumptions above, suggesting the combined boundary layer is larger than predicted by theory. The core flow experiences upward turbulent fluxes in aligned flows, coupled with sediment entrainment by vortex shedding at flow reversal, whilst downward fluxes of eddies generated by the core flow, and strong adverse shear can enhance near-bed mass transport, in opposing currents. Current-aligned coherent structures contribute significantly to the stress and energy dissipation, and display characteristics of wall-attached eddies formed by the pairing of counter-rotating vortices. These preliminary findings suggest a notable difference in wave-following and wave-opposing wave-current interactions, and highlight the need to account for intermittent momentum-exchanges in predicting stress, boundary layer thickness and sediment transport.

Digital fluxgate current sensor based on second harmonic detection

This paper presents a new digital fluxgate current sensor based on second harmonic detection for DC and AC measurement. The sensor utilizes a feedback loop to obtain an almost zero-flux condition, i.e., a balance between the magnetic flux of the primary current and the feedback current, in which way the feedback current is proportional to the primary current. The AC magnetic flux is detected with an induction coil, and the DC zero-flux condition is realized by magnetic saturation effect method, where the magnetic core is periodically magnetized and then the second harmonic of the magnetization current is calculated as an indication of the DC magnetic flux. After theoretical derivation, the operating principle of the sensor was investigated using a numerical simulation model built with Simulink of MATLAB. In addition, a prototype sensor was developed and tested. The experiment results demonstrate that the current sensor works properly for DC and AC measurement. The average error is about 0.06% for DC measurement.

Threshold dynamics of a delayed nonlocal reaction-diffusion HIV infection model with both cell-free and cell-to-cell transmissions

In this paper, we derive and analyze an HIV infection model containing a fixed latent period and both cell-free and cell-to-cell transmissions. The model is described by a spatially non-local reaction-diffusion system with the zero-flux condition on the boundary. We first show that the model admits global solutions and possesses a global attractor. Following the definition of the next infection operator, we obtain the expression of the basic reproduction number ℜ0, which is shown as a threshold parameter for the model dynamics. Our results indicate that the clearance/persistence of HIV is governed by the sign of ℜ0−1. In particular, an explicit formula of ℜ0 is given in the homogeneous case.

Improved Modeling of Sediment Oxygen Kinetics and Fluxes in Lakes and Reservoirs.

To understand water quality degradation during hypoxia, we need to understand sediment oxygen fluxes, the main oxygen sink in shallow hypolimnia. Kinetic models, which integrate diffusion and consumption of dissolved oxygen (DO) in sediments, usually assume a downward flux of DO from the sediment-water interface (SWI) with a zero-flux condition at the lower boundary of the oxic sediment layer. In this paper, we separately account for the oxidation of an upward flux of reduced compounds by introducing a negative flux of DO as a lower boundary condition. Using in situ measurements in two lakes, kinetic models were fit to DO microprofiles using zero-order and first-order kinetics with both zero and non-zero lower boundary conditions. Based on visual inspection and goodness-of-fit criteria, the negative-flux lower boundary condition, -0.25 g O2 m-2 d-1, was found to more accurately describe DO consumption kinetics. Fitted zero-order rate constants ranged from 50 to 510 mg L-1 d-1, and first-order rate constants ranged from 60 to 400 d-1, which agree well with prior laboratory studies. DO fluxes at the SWI calculated from the simulated profiles with the negative-flux lower boundary condition also showed better agreement with the observed DO fluxes than the simulated profiles with the zero-flux lower boundary condition.

The general setting for the zero-flux condition: The lagrangian and zero-flux conditions that give the heisenberg equation of motion.

Generalizing our recent work on relativistic generalizations of the quantum theory of atoms in molecules, we present the general setting under which the principle of stationary action for a region leads to open quantum subsystems. The approach presented here is general and works for any Hamiltonian, and when a reasonable Lagrangian is selected, it often leads to the integral of the Laplacian of the electron density on the region vanishing as a necessary condition for the zero-flux surface. Alternatively, with this method, one can design a Lagrangian that leads to a surface of interest (though this Lagrangian may not be, and indeed probably will not be, "reasonable"). For any reasonable Lagrangian for the electronic wave function and any two-component method (related by integration by parts to the Hamiltonian) considered, the Bader definition of an atom is recaptured. © 2018 Wiley Periodicals, Inc.

Global dynamics of a West Nile virus model in a spatially variable habitat

In this paper, we investigate a mathematical model that describes the transmission dynamics of West Nile virus (WNV) associated with the mosquito–bird population in a continuous bounded habitat. Our model is given by a spatial reaction–diffusion system with the zero-flux condition on the boundary, which is motivated by the models in the previous works of Wonham et al. (2004) and Lewis et al. (2006). By using the comparison theorem and the theory of uniform persistence, we show that the global dynamics of the model can be determined by two indices, mosquito reproduction number and infection invasion threshold.

The series solution to the modified mild-slope equation for wave scattering by Homma islands

Based on a newly derived explicit form of the modified mild-slope equation (MMSE), a series solution is constructed for wave scattering by Homma island and the convergence of the solution is analyzed. It is found that an intuitive and commonly adopted depth-averaged zero-flux condition on the cylinder surface is wrong and an alternative Robin-type condition is derived. To validate the present solution, comparisons are conducted against various known solutions to the long-wave equation, Helmholtz equation, the mild-slope equation (MSE) and time-dependent MSE. It is shown that the present approach can model wave scattering from very long waves to very short waves. Comparison between the present MMSE solution and its degenerated MSE version is also conducted. Significant contribution of the mass-conserving matching conditions in improving solution accuracy revealed by Porter and Staziker is confirmed for this particular island, which mainly occurs for intermediate waves. By using the present solution, wave amplification is investigated for original Homma island and a modified Homma island. Finally, contour lines of wave amplification in the vicinity around a modified Homma island are presented.

A Numerical Well-Balanced Scheme for One-Dimensional Heat Transfer in Longitudinal Triangular Fins

The temperature profile for fins with temperature-dependent thermal conductivity and heat transfer coefficients will be considered. Assuming such forms for these coefficients leads to a highly nonlinear partial differential equation (PDE) which cannot easily be solved analytically. We establish a numerical balance rule which can assist in getting a well-balanced numerical scheme. When coupled with the zero-flux condition, this scheme can be used to solve this nonlinear partial differential equation (PDE) modelling the temperature distribution in a one-dimensional longitudinal triangular fin without requiring any additional assumptions or simplifications of the fin profile.

Applications of the kinetic formulation for scalar conservation laws with a zero-flux type boundary condition

The authors are concerned with a zero-flux type initial boundary value problem for scalar conservation laws. Firstly, a kinetic formulation of entropy solutions is established. Secondly, by using the kinetic formulation and kinetic techniques, the uniqueness of entropy solutions is obtained. Finally, the parabolic approximation is studied and an error estimate of order \(\eta ^{\tfrac{1} {3}}\) between the entropy solution and the viscous approximate solutions is established by using kinetic techniques, where η is the size of artificial viscosity.

Threshold dynamics of an infective disease model with a fixed latent period and non-local infections

In this paper, we derive and analyze an infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain. The model is given by a spatially non-local reaction-diffusion system carrying a discrete delay associated with the zero-flux condition on the boundary. By applying some existing abstract results in dynamical systems theory, we prove the existence of a global attractor for the model system. By appealing to the theory of monotone dynamical systems and uniform persistence, we show that the model has the global threshold dynamics which can be described either by the principal eigenvalue of a linear non-local scalar reaction diffusion equation or equivalently by the basic reproduction number R₀ for the model. Such threshold dynamics predicts whether the disease will die out or persist. We identify the next generation operator, the spectral radius of which defines basic reproduction number. When all model parameters are constants, we are able to find explicitly the principal eigenvalue and R₀. In addition to computing the spectral radius of the next generation operator, we also discuss an alternative way to compute R₀.

Bénard instabilities in a binary-liquid layer evaporating into an inert gas: Stability of quasi-stationary and time-dependent reference profiles

This study treats an evaporating horizontal binary-liquid layer in contact with the air with an imposed transfer distance. The liquid is an aqueous solution of ethanol (10% wt). Due to evaporation, the ethanol mass fraction can change and a cooling occurs at the liquid-gas interface. This can trigger solutal and thermal Rayleigh-Benard-Marangoni instabilities in the system, the modes of which corresponding to an undeformable interface form the subject of the present work. The decrease of the liquid-layer thickness is assumed to be slow on the diffusive time scales (quasi-stationarity). First we analyse the stability of quasi-stationary reference profiles for a model case within which the mass fraction of ethanol is assumed to be fixed at the bottom of the liquid. Then this consideration is generalized by letting the diffusive reference profile for the mass fraction in the liquid be transient (starting from a uniform state), while following the frozen-time approach for perturbations. The critical liquid thickness below which the system is stable at all times quite expectedly corresponds to the one obtained for the quasi-stationary profile. As a next step, a more realistic, zero-flux condition is used at the bottom in lieu of the fixed-concentration one. The critical thickness is found not to change much between these two cases. At larger thicknesses, the critical time at which the instability first appears proves, as can be expected, to be independent of the type of the concentration condition at the bottom. It is shown that solvent (water) evaporation plays a stabilizing role as compared to the case of a non-volatile solvent. At last, an effective approximate Pearson-like model is invoked making use in particular of the fact that the solutal Marangoni is by far the strongest as an instability mechanism here.

Exact time-dependent solutions for the thin accretion disc equation: boundary conditions at finite radius

We discuss Green's-function solutions of the equation for a geometrically thin, axisymmetric Keplerian accretion disc with a viscosity prescription "\nu ~ R^n". The mathematical problem was solved by Lynden-Bell & Pringle (1974) for the special cases with boundary conditions of zero viscous torque and zero mass flow at the disc center. While it has been widely established that the observational appearance of astrophysical discs depend on the physical size of the central object(s), exact time-dependent solutions with boundary conditions imposed at finite radius have not been published for a general value of the power-law index "n". We derive exact Green's-function solutions that satisfy either a zero-torque or a zero-flux condition at a nonzero inner boundary R_{in}>0, for an arbitrary initial surface density profile. Whereas the viscously dissipated power diverges at the disc center for the previously known solutions with R_{in}=0, the new solutions with R_{in}>0 have finite expressions for the disc luminosity that agree, in the limit t=>infinity, with standard expressions for steady-state disc luminosities. The new solutions are applicable to the evolution of the innermost regions of thin accretion discs.

Quantifying steric effect with experimental electron density

Using experimental electron densities, the recent effort of quantifying steric effect within the framework of density functional theory is continued. In this work, steric potential, steric field, and steric charge distributions are systematically examines for diamond and boron nitride crystals. Bader's zero-flux condition has been employed to discuss the atomic contributions of these quantities. Two new concepts, characteristic radius r(s) of steric field and atomic steric charge q(s), are introduced in this work, which are intrinsic properties of a system and thus can be used to characterize atomic properties in a molecule or crystal. We anticipate that these steric effect related quantities together with the new concepts introduced in this work can be applied to characterize variety categories of the chemical bonds or weak interactions and provide in-depth insights to a wide range of organic, inorganic, and biological systems.

Functional Differential Equations Arising in Cell-growth

Non-local differential equations are notoriously difficult to solve. Cell-growth models for population growth of a cohort structured by size, simultaneously growing and dividing, give rise to a class of non-local eigenvalue problems, whose principal” eigenvalue is the time-constant for growth/decay. These and other novel non-local problems are described and solved in special cases in this paper. Keywords: Non-local differential equations; non-local eigenvalue problems; delay equation; pantograph equation; zero-flux condition; cohort of cells; Green’s function

Resonance Structures of the Amide Bond: The Advantages of Planarity

Delocalization indexes based on magnitudes derived from electron-pair densities are demonstrated to be useful indicators of electron resonance in amides. These indexes, based on the integration of the two-electron density matrix over the atomic basins defined through the zero-flux condition, have been calculated for a series of amides at the B3LYP/6-31+G* level of theory. These quantities, which can be viewed as a measure of the sharing of electrons between atoms, behave in concordance with the traditional resonance model, even though they are integrated in Bader atomic basins. Thus, the use of these quantities overcomes contradictory results from analyses of atomic charges, yet keeps the theoretical appeal of using nonarbitrary atomic partitions and unambiguously defined functions such as densities and pair densities. Moreover, for a large data set consisting of 24 amides plus their corresponding rotational transition states, a linear relation was found between the rotational barrier for the amide and the delocalization index between the nitrogen and oxygen atoms, indicating that this parameter can be used as an ideal physical-chemical indicator of the electron resonance in amides.