Random Distribution Of Particles Research Articles

Surface plasmon resonance evaluation of colloidal silver aerogel filters

Aerogels containing silver nanoparticles were fabricated for gas catalysis applications. By applying the concept of an average or effective dielectric constant to the heterogeneous interlayer surrounding each particle, we extend the technique of immersion spectroscopy to heterogeneous or porous media. Specifically, we extend the predominant effective medium theories for the determination of the average fractional composition of each component in this inhomogeneous layer. Hence, the surface area of metal available for catalytic gas reaction is determined. The technique is satisfactory for statistically random metal particle distributions but needs modification for aggregated systems.

A general constitutive theory for linear and nonlinear particulate media with microstructure evolution

This work is concerned with the development of a constitutive theory for composite materials with particulate microstructures, which is capable of predicting, approximately, the evolution of the microstructure and its influence on the effective response of composites under general three-dimensional finitestrain loading conditions, such as those present in metal-forming operations. In its present form, the theory is general enough to be used for linearly viscous, nonlinearly viscous and perfectly plastic composites with randomly oriented and distributed ellipsoidal inclusions (or pores), which, in the most general case, can change size, shape and orientation. In addition, the “shape” and “orientation” of their center-to-center statistical distribution functions can also evolve with the deformation. To illustrate the key features of the new theory in the context of a simple example, an application is carried out for plane-strain loading of two-phase systems consisting of random distributions of aligned rigid particles in a power-law matrix phase. The results show that the evolution of the relevant microstructural variables, as well as the effective response, depend in a complex fashion on the initial state of the microstructure, as well as on the specific boundary conditions. In particular, it is found that the changes in orientation of the particles provide a mechanism analogous to “geometric softening” in ductile single crystals, which can lead to significant changes in the instantaneous hardening rate of the composite. This is shown to have important consequences for the possible onset of shear localization in the composite.

Variational bound finite element methods for three-dimensional creeping porous media and sedimentation flows

SUMMARY We present an analytico-computational methodology for the prediction of the effective properties of two types of three-dimensional particulate Stokes flows: porous media and sedimentation flows. In particular, we determine the permeability and average settling rate of media that consist of non-colloidal monodisperse solid spherical particles immersed in a highly viscous Newtonian fluid. Our methodology recasts the original problem into three scale-decoupled subproblems: the macro-, meso- and microscale subproblems. In the macroscale analysis the appropriate effective property is used to calculate the bulk quantity of interest. The mesoscale problem provides this effective property through the finite element solution of the transport equations in a periodic cell containing many particles distributed according to a prescribed joint probability density function. Finally, the microscale analysis allows us to accommodate mesoscale realizations in which two or more inclusions are in very close proximity; this geometrical stiffness is alleviated by introducing simple domain modifications that relax the mesh generation requirements while simultaneously yielding rigorous bounds for the effective property. Our methodology can treat random particle distributions as well as regular arrays; in the current paper we analyse only the latter. # 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids, 26: 145‐175 (1998)

Study of the diffusion of Al–Li alloys subjected to an electric field

An experimental investigation of the homogenization treatment of 2091 Al–Li alloy in the presence of an electric field, has revealed the phenomena of reduced volume fraction, small size, spherical shape and random distribution of second-phase particles, which bring about an increase in ductility. The results show that the dissolution of second-phase particles is promoted by means of the vacancy mechanism, because of the larger diffusion coefficient of solute atoms than the vacancy–solute complexes at the beginning of the homogenization treatment. By increasing the homogenization time and applying an electric field, the diffusion coefficient of vacancy-solute complexes is raised, whereas that of solute atoms is reduced, because of the decreased potential energy of the second phase at grain boundaries. Therefore, the non-equilibrium segregation of magnesium and copper elements is generated near the surface of the ingot by a complex mechanism. An experimental study of the solution treatment under an electric field, revealed that the lithium non-equilibrium segregation is induced at grain boundaries responsible for the δ precipitates. © 1998 Chapman & Hall

Structure and temperature dependence of resistance in very high resistivity thick films

Microstructures have been examined for superhigh-resistivity thick films having typical resistivities of 108–1011 Ω/□ and based on Sn0.9Sb0.1O2-Sb2O4 and Pb2Ru2O6 powders, amorphous glass, and crystallizing glass. The temperature dependence of the resistance has been measured at 77–870 K. Films based on Sn0.9Sb0.1O2-Sb2O4 mixtures have a chain structure, which consists of a network of particles of the conducting phase and groups of them, which are arranged around the particles of glass. There is a random distribution of the Pb2Ru2O6 particles in the glass matrix, and mutual diffusion occurs between the phases. The temperature dependence of the resistance is of activation type, which is due to carrier transport through the thin glass layers.

Writing transition noise due to random particle distribution in thin recording media

Fluctuations in the magnetization transition due to random distribution of position, particle size, anisotropy and misorientation of easy axis of a system of thermally activated noninteracting single domain particles (SDPs) in recording media under a recording head field has been analyzed, and the corresponding variance examined, by Monte Carlo simulation. Larger values of the thermal factor Kv p/ k B T give rise to steeper magnetization hysteresis curves, narrower transition widths and, correspondingly, sharper and narrower noise distributions. Heterogeneous media with an admixture of several different particle size distributions lead to a wider transition width at a lower noise level. Random anisotropy constant distribution imparts a far more severe effect to broaden the transition width while keeping the variance wider and shallower than that due to random particle size distribution. Similarly, random in-plane misorientation distribution of the easy axis is much less effective to broaden the transition width than that due to the out-of-plane misorientation. Variance in the read-out voltage becomes asymmetric when the head gap length is comparable to the transition width.

Application of an Effective Medium Theory to Composite Materials with Randomly Dispersed Particles of Specific Shapes

The effective medium theory used to describe the magnetic properties of randomized heterogeneous materials includes as parameters the intrinsic permeabilities μ i of the magnetic particles and an effective shape factor N. This paper compares the experimental variations of N to the theoretical variations predicted by au EMT. It is shown that the effective shape factor N cannot be seen as the mean shape factor of the composite components. The values of N belong to the range ]0,1/3[, and arc strongly correlated to the permeabilities μ i . In the low concentration range, random distributions of needle-like and flake-like particles both lead to low values for N. The result N=1/3 is obtained in the whole concentration range only in the case of randomly dipersed spheres. The values of the μ i might be kept constant up to a volumic fraction c in magnetic matter equal to 50%, according to previous results we got. The experimental results presented here arc well described by this theory up to c=50%.

Crossover from geometrical to stochastic fractal statistics for translationally invariant random distributions of independent particles in n-dimensional Euclidean space

The paper deals with the statistics of translationally invariant random distributions of independent particles in Euclidean space. The geometrical fractal model used in astrophysics and hydrodynamics assumes that the number of particles enclosed in a hypersphere of radius r obeys a Poisson law v N ( N!) −1exp(− v), where the average number of particles is a power function of radius r: v ∼ r d f and d f is the dimension of a fractal structure embedded in the Euclidean space considered. A statistical fractal distribution is introduced by assuming that the probability density of the distance between two nearest particles has a long tail of the inverse power law type φ 0( r) ∼ r −(1 + H r ) as r → ∞, where H r > 0 is a statistical fractal exponent. The distribution of the number of particles enclosed in a hypersphere of radius r is also Poissonian but the average number of points increases logarithmically rather than algebraically with the radius r: v∼ H r ln( r r 0 ) as r → ∞. The spatial distribution of points corresponding to this statistical fractal model is much rarer than in the geometrical fractal case. An alternative approach is derived by assuming that the probability density of the distance between two particles has a very broad logarithmic tail φ 0(r) dr ∼ d[ ln m( r r 0 )] [ ln m( r r 0 )] 1 + H , H > 0, r → ∞ where ln m( r r 0 ) = ln … ln( r r 0 ) is the mth iterated logarithm of r r 0 . For such a logarithmic statistical fractal the number of particles increases much more slowly with the radius r than in the ‘pure’ statistical fractal case: v ∼ H ln m + 1( r r 0 ) as r → ∞. By using a heuristic approach two general probabilistic models are derived which include both the geometrical and statistical fractal models as particular cases; these models predict power law dependences of the average number of particles for small systems and logarithmic dependences for large systems, respectively. The significance of the generalized models is elucidated by using the notion of a specific fractal hypervolume \\ ̃ gw which corresponds to a given particle. For the generalized model derived from the pure statistical fractal \\ ̃ gw is made up of two additive contributions: a constant one which determines the behavior of the model for small systems and a linearly increasing contribution with the total fractal hypervolume which is predominant for large systems. A similar structure of \\ ̃ gw exists for the generalized model leading to the logarithmic statistical fractal regime.

FINITE-SIZE SCALING FOR CELLULAR AUTOMATA WITH RANDOMIZED GRIDS AND FOR FRACTAL RANDOM FIELDS IN DISORDERED SYSTEMS

The finite-size effect on the statistics of independent random point processes is analyzed in connection with the theory of cellular automata with randomized grids. The space distribution of a large but finite number N0 of independent particles confined in a large region of ds-dimensional Euclidean space of size VΣ is investigated by using the technique of characteristic functionals. Exact formal expressions are derived for all many-body correlation functions of the positions of the particles as well as for all cumulants of the concentration field. These functions are made up of the contributions of the different negative powers of the total number N0 of particles from the system [Formula: see text] where n =1, 2, … are the orders of the correlation functions or of the cumulants of the concentration field. In the thermodynamic limit N0, VΣ → ∞ with N0 / VΣ = constant only the terms An(0) survive; the other terms An(m) with m > 0 express the finite-size effects. It is shown that, even though the particles are independent, for a finite size of the system a correlation effect different from zero exists among their positions and this correlation vanishes in the limit of an infinite size. The correlation among the positions of the different particles is a finite-size effect due to the conservation of the total number of particles which is similar to the correlation among ideal bosons or fermions at low absolute temperatures. The stochastic properties of an additive scalar field generated by a random distribution of independent particles are investigated. The approach can be applied to the study of stochastic gravitational fluctuations generated by a random distribution of stars or galaxies, of the short-range mean field generated by the particles making up a disordered medium, or of the distribution of the offspring number generated by a plant population randomly distributed in space. Special attention is paid to the finite-size scaling corrections to the long-range self-similar fractal fields. The computations lead to the surprising result that for large values of the resulting field the finite size of the system has practically no influence on the tails of the probability densities of the resulting field, which obey a statistical fractal scaling law of the negative power law type. This apparent paradoxical effect is due to the fact that the very large values of the field corresponding to the tails of the probability densities are generated by the closest neighbor of the test particle considered and are not influenced by the more distant particles.

Characterization of planar local arrangement by means of the Delaunay neighbourhood

A suitable procedure is presented to characterize the neighbourhood in disperse micro‐structures by measuring the Delaunay neighbour distances and neighbour numbers. Results are discussed from simulation experiments with three germ‐grain models, regular arrangements and a seam‐like model. A plot of variation coefficients is suggested to differentiate between various random distributions of particles.

Influence of homogenization treatment in an electric field on the workability of 1420 Al-Li alloy during hot rolling

In A1-Li alloys after casting, a significant amount of dendritic segregation is present, which leads to collapse during bot rolling. Antes [1] shows that a dramatic improvement in ductility is achieved by eliminating dendritic segregation. The effect of undissolved second phase on ductility has been investigated by Singh and Flemings [2]. Their results show that the ductility during hot rolling depends on not only the volume fraction but also the shape and the distribution of second phase particles. For a given amount of second phase, spherical shape and random distribution of second phase particles are beneficial for the improvement of ductility. Purdy and Kirkaldy [3] have drawn attention to the advantages to be gained by homogenization at temperamre above the melting point of segregated alloys. However, for A1-Li alloys, high temperature homogenization treatment results in local overheating or local underheating and a heavy loss of Li. The homogenization process invented by David [4] is to heat slowly at rates not exceeding 50°C/h to temperature of at least 530°C up to a final homogenization. But, due to the difficulties in controlling temperature, the total content of Li and Mg is not permitted to exceed 6%. Deformation of segregated alloys promotes the homogenization of alloys [5]. However, for the as-cast A1-Li alloys, great risk of collapse exists during deformation. The heat treatment of 2091 AI-Li alloy with an electric field has been investigated [6, 7]. The results show a significant effect of the electric field on the diffusion and phase transformation of 2091 A1-Li alloy. In this letter, the effect of an electric field on the workability of 1420 A1-Li alloy is reported. 1420 A1-Li alloys were melted under an argon atmosphere. Table I gives the chemical compositions of the alloy. Ingots were treated either with or without an electric field at the same homogenization time (32 h) and temperature (470 °C). The effect of homogenization time (16 h, 24 h, 32 h) in an electric field on the volume fraction, shape and distribution of second-

A transport theory of pulse propagation in a strongly forward scattering random medium

The scalar time-dependent equation of radiative transfer is used to develop a theory of pulse propagation in a discrete random medium whose scatter function (phase function) consists of a strong, narrow forward lobe superimposed over an isotropic background. The situation analyzed is that of a periodic sequence of plane-wave pulses, incident from an air half-space, that impinges normally upon the planar boundary surface of a random medium half-space; the medium consists of a random distribution of particles that scatter (and absorb) radiation in accordance with the aforementioned phase function. After splitting the specific intensity into the reduced incident and diffuse intensities, the solution of the transport equation in the random medium half-space is obtained by expanding the angular dependence of both the scatter function and the diffuse intensity in terms of Legendre polynomials, and by using a point matching procedure to satisfy the boundary condition that the forward travelling diffuse intensity be zero at the interface. Curves of received power show that, at small penetration depths, the coherent (reduced incident) intensity dominates, whereas at large depths, the incoherent (diffuse) intensity is the strongest and causes the pulses to broaden and distort. The motivation for this study was to complement a test series, on mm-wave pulse propagation in vegetation, by a theory that provides understanding of overall trends and assistance in the interpretation of measured results. In the mm-wave region, all scatter objects in a forest have large dimensions compared to a wavelength and, therefore, produce strong forward scattering and a phase function of the type assumed in this paper.

A microscopic investigation of the precipitation phenomena observed during the pearlite reaction in vanadium alloyed carbon steels

During the isothermal pearlitic transformation in vanadium alloyed low and medium carbon steels the formation of proeutectoid ferrite precedes that of pearlitic formation. Within this proeutectoid ferrite interphase precipitation of vanadium carbide (VC) occurs in a form of linear uniformly sized and shaped arrays of particles. When pearlite subsequently forms, two distinctive pearlitic morphologies exist. Lamellar and non-lamellar spheroidised types of pearlite. Within the pearlitic constituent of both of these pearlitic microstructures VC interphase precipitation occurs. However, in the lamellar type of pearlite curved arrays of uniformly sized and shaped VC particles whose direction is perpendicular to the cementite long axis are observed while in the non-lamellar type of pearlite apparently random distributions of VC particles exist. It was consistently found that VC interphase precipitation always occurred on the moving austenite/ferrite interphase boundaries at all temperature of isothermal pearlitic transformation.

A theoretical model for composite electrolytes—II. Percolation model for ionic conductivity enhancement

A novel method of a combination of a percolation model with the space charge layer model is derived. For the first time, this gives an analytical solution for the ionic-conductivity enhancement of composite electrolytes. The conductivity enhancement is directly related to the size of second phase particles and matrix particles, the density of the defects as charge carriers and temperature. The theoretical optimum-content of the second phase with an ideal random distribution is solved as 40 vol%. At the optimum-content, the ionic conductivity of the composite electrolytes reaches the maximum. With the introduction of the non-ideal random distribution of matrix particles or second phase particles, an analytical solution based on the percolation model was derived. The theoretical results fit the experimental data quantitatively very well for LiI/Al 2O 3 and CuCl/Al 2O 3 systems in the whole range of the second phase content admixed in the matrix. The shift effect of optimum-content away from the theoretical optimumcontent of 40 vol%, the reason for which is unknown so far, is caused by the non-uniform distribution of either matrix particles or second phase particles and this non-uniform distribution always degenerates the conductivity enhancement seriously.

Settling of particles studied by a hydrodynamic cellular automation for granular media

We implemented a hydrodynamic lattice gas cellular automation with rest particles on bonds to study the behaviour of granular media in a two-dimensional closed container under the influence of dissipation, static friction and gravity. The problems posed by rest particles on bonds are described and our solutions (rules and algorithm) are discussed. Starting our simulations with a random distribution of particles on the lattice we study the motion of the centre of mass for different sets of dissipation and gravity parameters. The roughness of the surface is measured for different system sizes (after all particles have stopped moving).

Numerical analysis of near—wall channelling in a packed bed

Regularities imposed by an external wall on the random distribution of particles in a packed bed lead to anisotropic wall effects of the bed. The resulting deviations of near—wall porosity from the average bed—porosity markedly affect the average velocity profile, not only near the wall but also some distance into the bed itself. The effect of wall channelling due to such variations in porosity near external boundaries is predicted by means of two different numerical solution methods for a unified model for granular porous media. The results are shown to compare favourably to experimental and numerical results reported in the literature.

The application of recursive aggregate T‐matrix algorithm in the Monte Carlo simulations of the extinction rate of random distribution of particles

The recursive T ‐matrix algorithm is applied to Monte Carlo simulations of multiple scattering by random distribution of dielectric spheres. The method is a fast algorithm for calculating the exact solution of Maxwell's equations for a large number of scattering objects. The extinction rate is calculated by averaging over many realizations. The computed results are compared with those from analytic approximations, namely, the quasi‐crystalline approximation.

Effect of high-temperature exposure on the structure of inertia-friction welds in RS/PM AL-8.5FE-1.3V-1.7SI

Isothermal elevated-temperature exposure of the inertia-friction welded AA-8009 showed negligible dispersoid coarsening in the base metal and weld HDZ and an absence of dispersoid transformation at 425°C for exposure times up to 500 h. Hardness of the base metal was unaffected by thermal exposure at this temperature. Although hardness of the central HDZ decreased slightly with increased exposure time, it remained higher than that of the base metal. Exposure at 500 and 575°C promoted dispersoid coarsening and the formation of acicular Al 13(Fe,V) 4 intermetallic particles, primarily along RS particulate boundaries in the base metal and weld outer HDZ. The random distribution of these intermetallic particles in the central HDZ resulted from the redistribution during welding of the coarse base metal dispersoids from which these particles transform. Thermal exposure to 575°C also resulted in a significant decrease in base metal and weld zone hardness.

The effects of conducting particle distribution on the behaviour of anisotropic conducting adhesives: non-uniform conductivity and shorting between connections

This paper presents mathematical models that are intended to assist in the design of anisotropic conducting materials. Such materials are usually random distributions of hard conducting spheres in epoxy resin matrices. Conducting joints are constructed in soft conducting materials by the plastic deformation of the conductor on the printed circuit boards. An earlier paper (Williams et al., 1993) describes some of the average behaviour of the materials during joint assembly and after assembly. This work explores the effects of the random distribution of particles in the material on resistivity variation and the probability of shorting between fine-pitch interconnections.

Monte Carlo calculations of the modulation transfer function of an optical system operating in a turbid medium

Using a Monte Carlo method, we investigate the effect of a turbid medium on image transmission by means of the modulation transfer function approach. We present results that refer to a medium that consists of a random distribution of water spherical particles in air. We analyze the effect of geometric conditions (medium width and position) and source characteristics (Lambertian, beam emission). We present results for small spheres (Rayleigh scattering) and spheres (1.0-microm diameter) that are not small in comparison with the wavelength lambda = 0.6328 microm. Numerical data show a large modulation transfer function dependence on the source emission aperture and a substantial independence of the medium width for a fixed value of the optical depth. In accordance with reciprocity principles, we test an inverse scheme of Monte Carlo calculation, the advantage of this scheme being a substantial reduction in calculation time.