Diffusionless crystal growth in a eutectic system during rapid solidification

Experiments on nonequilibrium rapid eutectic growth are surveyed. The applicability limits of the modern theoretical models describing rapid solidification of binary systems are assessed. A problem of rapid eutectic growth when the local equilibrium is violated in the solute diffusion field (in the bulk liquid and at the solid-liquid interface) is formulated. An analytical solution to the problem of rapid lamellar eutectic growth under local nonequilibrium conditions in the solute diffusion field is found. It is shown that the diffusion-limited growth of a eutectic pattern ceases as soon as a chemically homogeneous crystalline phase begins to grow when the critical point V=V _D is achieved (V is the solid-liquid interface velocity and V _D is the solute diffusion speed in the bulk liquid). At V ≥ V _D, eutectic decomposition is suppressed and the nascent homogeneous crystalline phase has the initial (nominal) chemical composition of the binary system.     

Sharp fronts in attracting-adatom monolayers

The problem of pattern formation by adsorbates undergoing attractive lateral interactions, is described by a parabolic integrodifferential equation having the scaled inverse temperature ? and the scaled pressure α of the vapor phase as parameters. A coexistence region of high- and low-coverage stable homogeneous states has been reported in the (?, α) plane. In the small interaction-range limit an effective diffusion coefficient can be defined, which becomes however negative for a coverage range in between the stable homogeneous ones. A novel free-energy-like Lyapunov functional is found here for this problem. When evaluated on the homogeneous states, it leads to a Maxwell-like construction which selects essentially the same value α(?) as the originally posited zero front-velocity condition. Moreover, its value on static fronts at this particular α(?) coincides with those of the homogeneous states. This article is dedicated to Prof. Helmut Brand with occasion of his 60th birthday.     

一种抑制合成孔径雷达图像相干斑的各向异性扩散滤波方法

本文提出了一种抑制合成孔径雷达图像乘性相干斑噪声的各向异性扩散滤波新方法.该方法将受自适应耦合函数控制的平均曲率运动嵌入到传统相干斑抑制的各向异性扩散方程中,形成了一种可有效抑制边缘区域相干斑与同质区块效应现象的各向异性扩散新方程,同时在新建的扩散方程中,引入了由改进Frost滤波与局部方向比率联合构建的一种带方向约束的新扩散函数,进一步削弱了块效应现象且明显改善了抑斑图像的边缘抖动扭曲问题.实验结果表明该方法在有效保护图像边缘的同时,能充分平滑同质区与边缘区域的相干斑,明显削弱块效应现象,有效改善抑斑图像边缘抖动扭曲问题,而抑斑图像无论视觉效果还是参数指标均比多种传统抑斑方法更具优势.     

Perturbed Lie Symmetry and Systems of Non-Linear Diffusion Equations

Abstract

The method of one parameter, point symmetric, approximate Lie group invariants is applied to the problem of determining solutions of systems of pure one-dimensional, diffusion equations. The equations are taken to be non-linear in the dependent variables but otherwise homogeneous. Moreover, the matrix of diffusion coefficients is taken to differ from a constant matrix by a linear perturbation with respect to an infinitesimal parameter. The conditions for approximate Lie invariance are developed and are applied to the coupled system. The corresponding prolongation operator is derived and it is shown that this places a power law and logarithmic constraints on the nature of the perturbed diffusion matrix. The method is used to derive an approximate solution of the perturbed diffusion equation corresponding to impulsive boundary conditions.     

Diffusionless crystal growth in rapidly solidifying eutectic systems

Using a local nonequilibrium model of solidification, experiments on rapid eutectic growth are analyzed. An analytical solution of a problem of rapid lamellar eutectic growth under local nonequilibrium conditions in the solute diffusion field is found. It is shown that solute diffusion-limited growth of a eutectic pattern is completely finished, and diffusionless growth of the chemically homogeneous crystalline phase begins to proceed at a critical point V = V(D), where V is the solid-liquid interface velocity and V(D) is the solute diffusion speed in bulk liquid. A suppression of eutectic decomposition occurs in the range V > or = V(D) that results in a growth of homogeneous crystal phase with the initial (nominal) chemical composition of the binary system.     

Time-dependent electromagnetic perturbations in a magnetized plasma with relaxing conductivity

The problem of one-dimensional propagation of a time-dependent electromagnetic perturbation in a homogeneous ionospheric plasma with relaxing conductivity in the presence of a horizontal geomagnetic field is considered. The approximate Green's function is constructed assuming that the dynamics of electromagnetic perturbations is determined by the diffusion mechanism if the diffusion is due to the sound wave motion. It is found that the relaxation of plasma conductivity may result in a solitary perturbation precursor preceding the arrival of the sound wave front.State University, St. Petersburg. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 7, pp. 668–677, July, 1995.     

Uniqueness,reciprocity theorem,and plane waves in thermoelastic diffusion with a fractional order derivative

In this work, a theory of thermoelasticity with diffusion is taken into consideration by using the methodology of fractional calculus. The governing equations for particle motion in a homogeneous anisotropic fractional order generalized thermoelastic diffusive medium are presented. Uniqueness and reciprocity theorems are proved. The plane wave propagation in the homogeneous transversely isotropic thermoelastic diffusive medium with fractional order derivative is studied. For the two-dimensional problem, there exist a quasi-longitudinal wave, a quasi-transverse wave, a quasi-mass diffusion wave, and a quasi-thermal wave. From the obtained results, the different characteristics of waves, like phase velocity, attenuation coefficient, specific loss, and penetration depth, are computed numerically and presented graphically. Some special cases are also discussed.     

Quasi-linear stage of synchrotron instability I. The limit of low initial radiation energy density

We consider the initial problem of quasi-linear relaxation of synchrotron instability in a spatially homogeneous system of relativistic electrons embedded in a cold plasma. We analyze evolution of the energy spectrum of electrons and the frequency spectrum of radiation at the limit of low initial radiation energy density. At this limit, the problem is reduced to the solution of a diffusion equation with a stationary diffusion coefficient for the electron distribution over momentum which does not depend on the initial radiation energy density. The radiation frequency spectrum is approximated by the Gaussian profile, and its parameters are expressed via the electron energy density at the given and initial instants of time. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 11, pp. 964–976, November 2006.     

Comparison between fixed and Gaussian steplength in Monte Carlo simulations for diffusion processes

We analyze the different degrees of accuracy of two Monte Carlo methods for the simulation of one-dimensional diffusion processes with homogeneous or spatial dependent diffusion coefficient that we assume correctly described by a differential equation. The methods analyzed correspond to fixed and Gaussian steplengths. For a homogeneous diffusion coefficient it is known that the Gaussian steplength generates exact results at fixed time steps Δt. For spatial dependent diffusion coefficients the symmetric character of the Gaussian distribution introduces an error that increases with time. As an example, we consider a diffusion coefficient with constant gradient and show that the error is not present for fixed steplength with appropriate asymmetric jump probabilities.     

Continuum description of anomalous diffusion on a comb structure

Anomalous diffusion on a comb structure consisting of a one-dimensional backbone and lateral branches (teeth) of random length is considered. A well-defined classification of the trajectories of random walks reduces the original problem to an analysis of classical diffusion on the backbone, where, however, the time of this process is a random quantity. Its distribution is dictated by the properties of the random walks of the diffusing particles on the teeth. The feasibility of applying mean-field theory in such a model is demonstrated, and the equation for the Green’s function with a partial derivative of fractional order is obtained. The characteristic features of the propagation of particles on a comb structure are analyzed. We obtain a model of an effective homogeneous medium in which diffusion is described by an equation with a fractional derivative with respect to time and an initial condition that is an integral of fractional order. Zh. éksp. Teor. Fiz. 114, 1284–1312 (October 1998)     

Problems of homogeneous irradiation of Si ingots more than 200 mm in diameter under neutron doping

The problem of homogeneous irradiation of Si ingots of large diameter (i.e., comparable with the thermal neutron diffusion length in silicon—20 cm) by thermal neutrons is considered. The effect of neutron absorption on the irradiation homogeneity is discussed. It is established for some usual cases of neutron doping that the neutron distribution in the irradiation zone affects the irradiation homogeneity. The results obtained can be useful for choosing and designing the irradiation zone for neutron doping of silicon.     

Spatial moments for the nonstationary one-velocity problem of transport theory with isotropic scattering. I. Point instantaneous source

Equations for the moments of spatial particle distribution in a homogeneous medium with isotropic scattering are derived for the nonstationary one-velocity problem of transport theory. Exact analytical and numerical solutions are found for five even moments (from the second to the tenth one) by the Laplace transform method, and an algorithm for calculating the moments of arbitrary order is described. Convergence to the corresponding moments is investigated in the diffusion approximation for t→∞, and its nonuniform character is established: higher moments differ significantly from the corresponding diffusion moments at any t. The physical causes for such behavior of the moments are discussed. Ul'yanovsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 88–94, February, 2000.     

Pattern propagation in nonlinear dissipative systems

We discuss the problem of pattern selection in situations where a stable, nonuniform state of a nonlinear dissipative system propagates into an initially unstable, homogeneous region. Our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known; and we point out that these known properties are consistent with a marginal-stability hypothesis that has been suggested in the theory of dendritic crystal growth. We then describe a more general interpretation of the marginal-stability hypothesis and, finally, present numerical evidence for its validity from three different pattern-forming models.     

Joint diffusion of rays in a homogeneous medium with random inhomogeneities

Fluctuations of the direction and amplitude of two rays propagating in a homogeneous, on aver-age, medium with random inhomogeneities of dielectric permittivity are considered. The solution to the problem is based on the method of ray diffusion, which enabled us to obtain the probability densities of the distribution of the relative amplitude and the relative direction of the rays.     

Microdiffusion: A microscopic technique for study of the diffusion of quinoline in PVC powder

Observation of the diffusion of quinoline inside PVC grains by microscopy represents a very sensitive method of PVC powder differentiation. The kinetics of microdiffusion is measured as a function of time and temperature (exponential function) by the fraction of impregnated grains of the powder which varies, on a broad range, as a function of resin properties, which are mainly influenced by the conversion factor. Gaussologarithmic representation of the phenomenon gives a Galton's straight line, the slope of which is related, more or less, to the homogeneous character of microdiffusion in the powder. That aspect is described in relation to the problem of formation of “fish eyes.”     

Equations for the distributions of functionals of a random-walk trajectory in an inhomogeneous medium

Based on the random-trap model and using the mean-field approximation, we derive an equation that allows the distribution of a functional of the trajectory of a particle making random walks over inhomogeneous-lattice site to be calculated. The derived equation is a generalization of the Feynman-Kac equation to an inhomogeneous medium. We also derive a backward equation in which not the final position of the particle but its position at the initial time is used as an independent variable. As an example of applying the derived equations, we consider the one-dimensional problem of calculating the first-passage time distribution. We show that the average first-passage times for homogeneous and inhomogeneous media with identical diffusion coefficients coincide, but the variance of the distribution for an inhomogeneous medium can be many times larger than that for a homogeneous one.     

Positron diffusion in solids and the reconstruction of inhomogeneous defect distributions from lifetime measurements

The time dependent diffusion trapping equations for positrons implanted into inhomogeneous solids are analyzed. This problem is of central importance in the study of polycrystalline materials and for the application of pulsed positron beams to defect studies in materials research. The main problem in previous investigations was the necessity to solve the time-dependent diffusion equation. It prevented analytical treatment in all but the simplest applications. For the first time this difficulty is eliminated by invoking a new concept, the observable local annihilation characteristics for local implantation of positrons into the thermalized ensemble. It will be shown that the local annihilation characteristics are governed by field equations which reduce to the well known quantities of the standard trapping model in the case of homogeneous defect distributions. Furthermore, inhomogeneous defect distributions are uniquely determined from the field equations provided the local annihilation characteristics are known. Analytical solutions are derived and applied successfully to recent experimental results for a selection of simple, but realistic problems. The formal procedure includes internal drift fields and could be extended to cover also the epithermal period of positron thermalization, if necessary.     

The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds

It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem is still open. The standard way to deal with homogeneous geodesics in the pseudo-Riemannian case is to use the so-called “Geodesic Lemma”, which is a formula involving the inner product. We shall use a different approach: namely, we imbed the class of all homogeneous pseudo-Riemannian manifolds into the broader class of all homogeneous affine manifolds (possibly with torsion) and we apply a new, purely affine method to the existence problem. In dimension 2, it was solved positively in a previous article by three authors. Our main result says that any homogeneous affine manifold admits at least one homogeneous geodesic through each point. As an immediate corollary, we prove the same result for the subclass of all homogeneous pseudo-Riemannian manifolds.     

Kinetic models for diffusion generated by an external force

Two kinetic models are used to study the homogeneous color diffusion problem in a dilute binary mixture. Both kinetic models incorporate a temperature dependence in the collision frequencies, which allows for the consideration of a general repulsive molecular interaction. The main transport properties as well as the velocity distribution functions are explicitly obtaines in terms of the field strength and the parameters characterizing the mixture. The results are illustrated for the two extreme cases of Maxwell molecules and hard spheres. A comparison between both models and with previous results derived from the Boltzmann equation for Maxwell molecules is carried out.