Diffusion in multiphase systems with nonstationary boundaries I. Solution of diffusion in the region with boundaries movig in one direction

In this work the problem of diffusion in a multiphase system whose both boundaries move in one direction according to the parabolic law with different velocities is analysed. The mathematical problem is solved exactly by means of thermal potentials of a double layer. The solution of the diffusion equation in the proximity to the boundary was derived and the concentration gradients on these boundaries were calculated. The numerical procedure of determining the diffusion characteristics from experimental concentration gradients on the phase boundaries was presented. As the zero approximation the result of calculations according to Vasileff and Smoluchowski, that can lead to considerable differences in the determined diffusion coefficients, was used. theory numerical calculations     

Computer simulation of interdiffusion in polycrystalline thin film couples

Computer simulation of superimposed lattice, grain boundary and surface diffusion, characteristic for polycrystalline thin film diffusion systems, was performed by way of discretisation of the nonlinear diffusion law. In order to give a vivid impression how such a complex process takes place under some typical conditions we have chosen pseudo three-dimensional computer plots of the spacial distribution of the concentration instead of the commonly used iso-concentration diagrams. The following cases are considered:

A myopic random walk on a finite chain

We solve analytically the problem of a biased random walk on a finite chain of ‘sites’ (1,2,…,N) in discrete time, with ‘myopic boundary conditions’—a walker at 1 (orN) at timen moves to 2 (orN − 1) with probability one at time (n + 1). The Markov chain has period two; there is no unique stationary distribution, and the moments of the displacement of the walker oscillate about certain mean values asn → ∞, with amplitudes proportional to 1/N. In the continuous-time limit, the oscillating behaviour of the probability distribution disappears, but the stationary distribution is depleted at the terminal sites owing to the boundary conditions. In the limit of continuous space as well, the problem becomes identical to that of diffusion on a line segment with the standard reflecting boundary conditions. The first passage time problem is also solved, and the differences between the walks with myopic and reflecting boundaries are brought out.     

Numerical Simulation of Grain-Boundary Grooving by Level Set Method

A numerical investigation of grain-boundary grooving by means of a level set method is carried out. An idealized polycrystalline interconnect which consists of grains separated by parallel grain boundaries aligned normal to the average orientation of the surface is considered. Initially, the surface diffusion is the only physical mechanism assumed. The surface diffusion is driven by surface-curvature gradients, while a fixed surface slope and zero atomic flux are assumed at the groove root. The corresponding mathematical system is an initial boundary value problem for a two-dimensional equation of Hamilton–Jacobi type. The results obtained are in good agreement with both Mullins analytical “small-slope” solution of the linearized problem (W. W. Mullins, 1957, j. Appl. Phys. 28, 333) (for the case of an isolated grain boundary) and with the solution for a periodic array of grain boundaries (S. A. Hackney, 1988, Scripta Metall. 22, 1731). Incorporation of an electric field changes the problem to one of electromigration. Preliminary results of electromigration drift velocity simulations in copper lines are presented and discussed.     

Mechanism of Fermi acceleration in dispersing billiards with time-dependent boundaries

The paper is devoted to the problem of Fermi acceleration in Lorentz-type dispersing billiards whose boundaries depend on time in a certain way. Two cases of boundary oscillations are considered: the stochastic case, when a boundary changes following a random function, and a regular case with a boundary varied according to a harmonic law. Analytic calculations show that the Fermi acceleration takes place in such systems. The first and second moments of the velocity increment of a billiard particle, alongside the mean velocity in a particle ensemble as a function of time and number of collisions, have been investigated. Velocity distributions of particles have been obtained. Analytic and numerical calculations have been compared. Zh. éksp. Teor. Fiz. 116, 1781–1797 (November 1999)     

Short-Circuit Diffusion in Ceramics

This article discusses grain boundary diffusion in ceramics. It gives a brief review of the experimental data available for ionic oxides and the problems of interpretation associated with it. The fundamental differences between grain boundary diffusion in metals and ceramics are noted. Calculations of the segregation of defects and impurities to grain boundaries are discussed together with methods of calculating diffusion coefficients in these boundaries. New results for alumina and chromia are presented. The problem of defining a grain boundary width is discussed with respect to new calculations on nickel oxide.     

Temperature and orientation dependences of the grain-boundary diffusion coefficient of antimony in copper bicrystals

An investigation is made of the diffusion of antimony through the bulk and along grain boundaries in copper bicrystals containing a symmetric 〈100〉 misorientation boundary with misorientation angles from 20 to 37.2°. The bicrystals are grown by the method of horizontal zone recrystallization. The temperature range for these studies is 480–580 °C, where the solubility of Sb in Cu is about 6 atomic % and practically temperature-independent. The concentration profiles are obtained by x-ray spectral microanalysis, and the grain-boundary diffusion parameters are computed by the method of Whipple and Suzuoka. The orientation dependence of the triple product P=sδD _b (where s is the segregation coefficient, δ the width of the grain boundary, and D _b the grain-boundary diffusion coefficient) is nonmonotonic, with a maximum for the special ∑5 misorientation boundary (36.9°). The effective activation energy for grain-boundary diffusion ranges from ∼70 kJ/mol for ∑5 to140 kJ/mol for general boundaries. Fiz. Tverd. Tela (St. Petersburg) 39, 1153–1157 (July 1997)     

Magnetic Properties of Grain Boundaries of Nanocrystalline Ni and of Ni Precipitates in Nanocrystalline NiCu Alloys

Perturbed γγ-angular correlation spectroscopy (PAC) was used to investigate nanocrystalline Ni and NiCu alloys, which are prepared by pulsed electrodeposition (PED). Using diffusion for doping nanocrystalline Ni with ¹¹¹In four different ordered grain boundary structures are observed, which are characterized by unique electric field gradients. The incorporation of ¹¹¹In on substitutional bulk sites of Ni is caused by moving grain boundaries below 1000 K and by volume diffusion above 1000 K. The nanocrystalline NiCu alloys prepared by PED are microscopically inhomogeneous as observed by PAC. In contrast, this inhomogeneity cannot be detected by X-ray diffraction. The influence of the temperature of the electrolyte, the current density during deposition, and the optional addition of saccharin to the electrolyte on the homogeneity of nanocrystalline NiCu alloys was investigated. This revised version was published online in September 2006 with corrections to the Cover Date.     

Spectroscopy of GaAs/AlGaAs microstructures with submicron spatial resolution using a near-field scanning optical microscope

A near-field scanning optical microscope (NSOM), which we built, is used to investigate 1–5-μm wide stripes with a 10-nm thick layer—a quantum well — on a GaAs surface. A map of the photoluminescence intensity is obtained synchronously with the topographic profile of the structures. The measured spatial distribution of the photoluminescence intensity is described satisfactorily in a model that takes into account carrier diffusion in the layer and the existence of a region with a short carrier lifetime near the side boundaries of the layer. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 7, 523–527 (10 April 1996)     

Equivalent network parameters of millimeter wave E-plane septum discontinuities with finite conductivity

The equivalent T—network parameters of millimeter wave E—plane discontinuities of bilateral metal septa with finite conductivity are calculated by the method of lines. An efficient approach is presented to extend this numerical method to EM boundary value problems with imperfectly conducting metal boundaries parallel to the discretization lines. Numerical results of both equivalent reactances and resistances in the T—network are obtained. The insertion losses of a Ka—band filter are calculated based on the lossy equivalent circuit. Numerical results are in agreement with the published experimental results.     

Short-Range Ordering in Amorphous Structures

A numerical investigation is made of some details of the short-range order in model amorphous structures with free boundaries and a number of different interaction potentials. Five- and seven-member rings — typical features of amorphous structures — are found out in all two-dimensional cross-sections of the generated clusters. Histograms of distribution of the nearest-neighbor numbers are constructed and an analysis of the positions of atoms with different coordination numbers within a model cluster volume is performed. An evaluation is made of the effects of the boundary on the coordination number distribution. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 42–45, September, 2005.     

Theoretical study of hydrogen absorption near symmetric tilt grain boundaries in Pd and TiFe

The results of calculations of the atomic and electron structure of Pd and TiFe with symmetrical Σ5 tilt grain boundaries obtained using the methods of electron density functional theory are reported. Hydrogen sorption at tilt grain boundaries and corresponding surfaces is considered. It is shown that the hydrogen absorption energy increases in magnitude by ∼0.2 eV at the Pd Σ5(210) grain boundary and by ∼0.5 eV in B2-TiFe with the Σ5(310) grain boundary. The binding energy of hydrogen in palladium, as well as in TiFe, in the most preferred positions at the surface is higher than near grain boundaries. It is found that, as in the case of a defect-free material, the following tendency is observed at a symmetrical tilt grain boundary: the strong bond of the impurity at the grain boundary in the metal or alloy matrix reduces the sorption energy of hydrogen.     

Structure of the attraction zones of the final states in the presence of dynamical period doubling bifurcations

The structure of the attraction zones of the final states associated with dynamical period doubling bifurcations is investigated. It is found that on the “initial value—transition rate” plane the attraction zones of the two possible final states alternate with each other and that a subdivision of the attraction regions occurs with a decrease in the transition rate. It is shown that the boundaries of the attraction zones are smeared out because of the effect of noise and in this situation the fine structure of the attraction zones is destroyed. As analytical and numerical calculations have shown, the critical value of the noise variance, corresponding to the boundary between the dynamical (or predictable) and stochastic (or unpredictable) modes, has a power-law dependence on the transition rate with a typical exponent value of one. The existence of “noise” invariants is also observed: the integrated (over all initial values) probability of achieving the final state is invariant with respect to the noise level. Zh. éksp. Teor. Fiz. 113, 369–380 (January 1998)     

Accurate Solutions of the Navier-Stokes Equations Despite Unknown Outflow Boundary Data

A very common procedure when constructing boundary conditions for the time-dependent Navier-Stokes equations at artificial boundaries is to extrapolate the solution from grid points near the boundary to the boundary itself. For supersonic outflow, where all the characteristic variables leave the computational domain, this leads to accurate results. In the case of subsonic outflow, where one characteristic variable enters the computational domain, one cannot in general expect accurate solutions by this procedure. The problem with outflow boundary operators of extrapolation type at artificial boundaries with errors in the boundary data of order one will be investigated. Both the problem when the artificial outflow boundary is located in essentially uniform flow and the situation when the artificial outflow boundary is located in a flow field with large gradients are discussed. It will be shown, that in the special case when there are large gradients tangential to the boundary, extrapolation methods can be used even in the subsonic case.     

Diffusion through a polycrystalline thin film

Analytical solutions to Fick’s second law of diffusion have been simultaneously derived without the restrictions of parabolic profiles along the x-axis in grain boundaries and expressed in a series for both grain interior and grain boundary diffusing through a polycrystalline thin film. The analysis takes segregation of diffusion species at grain boundaries into account. The analytic solutions lead to the concentration profiles in the grain interior and in the grain boundary, to the average-integrated amount of diffusion species at the exit surface, and to the time lag, which can be technologically used for depth profile studies and kinetic accumulation measurements.     

Importance of exactb-tensor calculation for quantitative diffusion tensor imaging and tracking of neuronal fiber bundles

Quantitative diffusion tensor imaging (DTI) is a novel method of magnetic resonance (MR) imaging providing information on the brain’s microstructure in vivo. DTI can be effectively measured with modern clinical MR scanners. However, imaging sequence details required for accurateb matrix calculation and for following DTI quantification are normally unknown to the user. In this work, we investigated the accuracy ofb value approximation if theb matrix is calculated without taking into account the effect of imaging gradients. It was found that an error of more than 4% in DTI estimation arises for a quite typical brain imaging protocol. The errors in mean diffusivity and fractional anisotropy index depend on diffusion tensor shape and eigenvectors orientation and exceed noise level in DTI quantification. These errors however have a strong impact on fiber tracking — up to 30% difference was found between the fiber tracks corresponding to exact and approximate calculated DTI data. Since these errors are dependent on imaging parameters and sequence implementation, accurateb matrix calculations are important for adequate comparison between data acquired on different MR scanners and also for data measured with the different imaging protocols.     

Homogeneous averaging of the stochastic diffusion equation in chaotic porous media

The diffusion of solution components in the pore space of a chaotic porous medium is treated. The problem of homogeneous averaging of the diffusion boundary problem is developed. In the end a system of homogeneous equations is obtained on whose basis one can treat particular transport processes. Russian University of Friendship for the People. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 52–58, September, 1996.     

Rhythmic pattern formations in gels and Matalon—Packter law: A fresh perspective

The periodic precipitation pattern formation in gelatinous media is interpreted as a moving boundary problem. The time law, spacing law and width law are revisited on the basis of the new scenario. The explicit dependence of the geometric structure on the initial concentrations of the reactants is derived. Matalon—Packter law, which relates the spacing coefficient with the initial concentrations is reformulated removing many ambiguities and impractical parameters. Experimental results are discussed to establish the significance of moving boundary concept in the diffusion controlled pattern forming systems