Dynamic Properties of Two-Dimensional Polydisperse Granular Gases

We propose a two-dimensional model of polydisperse granular mixtures with a power-law size distribution in the presence of stochastic driving. A fractal dimension D is introduced as a measurement of the inhomogeneity of the size distribution of particles. We define the global and partial granular temperatures of the multi-component mixture. By direct simulation Monte Carlo, we investigate how the inhomogeneity of the size distribution influences the dynamic properties of the mixture, focusing on the granular temperature, dissipated energy, velocity distribution, spatial clusterization, and collision time. We get the following results: a single granular temperature does not characterize a multi-component mixture and each species attains its own "granular temperature"; The velocity deviation from Gaussian distribution becomes more and more pronounced and the partial density of the assembly is more inhomogeneous with the increasing value of the fractal dimension D; The global granular temperature decreases and average dissipated energy per particle increases as the value of D augments.     

Spatial Density Distributions and Correlations in a Quasi-one-Dimensional Polydisperse Granular Gas

By Monte Carlo simulations, the effect of the dispersion of particle size distribution on the spatial density distributions and correlations of a quasi one-dimensional polydisperse granular gas with fractal size distribution is investigated in the same inelasticity. The dispersive degree of the particle size distribution can be measured by a fractal dimension dr, and the smooth particles are constrained to move along a circle of length L, colliding inelastically with each other and thermalized by a viscosity heat bath. When the typical relaxation time τ of the driving Brownian process is longer than the mean collision time To, the system can reach a nonequilibrium steady state. The average energy of the system decays exponentially with time towards a stable asymptotic value, and the energy relaxation time τB to the steady state becomes shorter with increasing values of df. In the steady state, the spatial density distribution becomes more clusterized as df increases, which can be quantitatively characterized by statistical entropy of the system. Furthermore, the spatial correlation functions of density and velocities are found to be a power-law form for small separation distance of particles, and both of the correlations become stronger with the increase of df. Also, tile density clusterization is explained from the correlations.     

Dynamic Properties of Two-Dimensional Polydisperse Granular Gases

We propose a two-dimensional model of polydisperse granular mixtures with a power-law size distribution in the presence of stochastic driving. A fractal dimension D is introduced as a measurement of the inhomogeneity of the size distribution of particles. We define the global and partial granular temperatures of the multi-component mixture. By direct simulation Monte Carlo, we investigate how the inhomogeneity of the size distribution influences the dynamic properties of the mixture, focusing on the granular temperature, dissipated energy, velocity distribution, spatial clusterization, and collision time. We get the following results： a single granular temperature does not characterize a multi-component mixture and each species attains its own ＂granular temperature＂; The velocity deviation from Gaussian distribution becomes more and more pronounced and the partial density of the assembly is more inhomogeneous with the increasing value of the fractal dimension D; The global granular temperature decreases and average dissipated energy per particle increases as the value olD augments.     

Non-Gaussian and Clustering Behavior in One-Dimensional Polydisperse Granular Gas System

We present a one-dimensional dynamic model of polydisperse granular mixture with the fractal characteristic of the particle size distribution, in which the particles are subject to inelastic mutual collisions and are driven by Gaussian white noise. The inhomogeneity of the particle size distribution is described by a fractal dimension D. The stationary state that the mixture reaches is the result of the balance between energy dissipation and energy injection. By molecular dynamics simulations, we have mainly studied how the inhomogeneity of the particle size distribution and the inelasticity of collisions influence the velocity distribution and distribution of interparticle spacing in the steady-state.The simulation results indicate that, in the inelasticity case, the velocity distribution strongly deviates from the Gaussian one and the system has a strong spatial clustering. Thus the inhomogeneity and the inelasticity have great effects on the velocity distribution and distribution of interparticle spacing. The quantitative information of the non-Gaussian velocity distribution and that of clustering are respectively represented.     

Collision Statistics of Driven Polydisperse Granular Gases

We present a dynamical model of two-dimensional polydisperse granular gases with fractal size distribution, in which the disks are subject to inelastic mutual collisions and driven by standard white noise. The inhomogeneity of the disk size distribution can be measured by a fractal dimension df. By Monte Carlo simulations, we have mainly investigated the effect of the inhomogeneity on the statistical properties of the system in the same inelasticity case. Some novel results are found that the average energy of the system decays exponentially with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state after a long evolution time. Furthermore, the inhomogeneity has great influence on the steady-state statistical properties. With the increase of the fractal dimension df, the distributions of path lengths and free times between collisions deviate more obviously from expected theoretical forms for elastic spheres and have an overpopulation of short distances and time bins. The collision rate increases with df, but it is independent of time. Meanwhile, the velocity distribution deviates more strongly from the Gaussian one, but does not demonstrate any apparent universal behavior.     

Steady State Properties of One-Dimensional Non-uniform Granular Gases Subjected to Gaussian White Noise Driving

We present a model of non-uniform granular gases in one-dimensional case, whose granularity distribution has the fractal characteristic. We have studied the nonequilibrium properties of the system by means of Monte Carlo method. When the typical relaxation time T of the Brownian process is greater than the mean collision time To, the energy evolution of the system exponentially decays, with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state in which the velocity distribution strongly deviates from the Gaussian one. Three other aspects have also been studied for the steady state： the visualized change of the particle density, the entropy of the system and the correlations in the velocity of particles. And the results of simulations indicate that the system has strong spatial clustering; Furthermore, the influence of the inelasticity and inhomogeneity on dynamic behaviors have also been extensively investigated, especially the dependence of the entropy and the correlations in the velocity of particles on the restitute coefficient e and the fractal dimension D.     

Steady-State Properties of Driven Polydisperse Granular Mixtures

We represent a two-dimensional model of polydisperse granular mixtures with a power-law size distribution. The model consists of smooth hard disks in a rectangular box with inelastic collisions, driven by a homogeneous heat bath at zero gravity. The width of particle size distribution is characterized by the only
parameter, namely, the fractal dimension D. The energy dissipation of the mixture is increased as D increases or as e decreases. Furthermore, it is found that the steady-state properties of the mixture such as the collision rate, granular temperature, kinetic pressure and velocity distribution depend sensitively on size distribution parameter D.     

Dynamic behavior of non-uniform granular gases system with the fractal characteristic in one dimension

A one-dimensional dynamic model of polydisperse granular mixture with a power-law size distribution is presented, in which the particles are subject to inelastic mutual collisions and driven by Gaussian white noise. The particle size distribution of the mixture has the fractal characteristic, and a fractal dimension D as a measurement of the inhomogeneity of the particle size distribution is introduced. We define the global granular temperature and the kinetic pressure of the mixture, and obtain their expressions. By molecular dynamics simulations, we have mainly investigated how the inhomogeneity of the particle size distribution and the inelasticity of collisions influence the steady-state dynamic properties of the system, focusing on the global granular temperature, kinetic pressure, velocity distribution and distribution of interparticle spacing. Some novel results are found that, with the increase of the fractal dimension D, the global granular temperature and the kinetic pressure decrease, the velocity distribution deviates more obviously from the Gaussian one and the particles cluster more pronouncedly at the same value of the restitution coefficient e (0<e<1). On the other hand, as the restitution coefficient e decreases, the dynamic behavior has the similar evolution as above at the fixed fractal dimension D. The dynamic behavior changing with e and D is, respectively, presented.     

Effects of Continuous Size Distributions on Pressures of Granular Gases

Direct Monte Carlo simulations are employed to investigate the granular pressures in granular materials with a power-law particle size distribution. Specifically, smooth circular discs of uniform material density are engaged in a two-dimensional rectangular box, colliding inelastically with each other and driven by a homogeneous heat bath at zero gravity. The resulting pressures are found to decrease as the widths of particle size distribution are increased. Moreover, the granular pressures in power-law systems are found to be unequally distributed among the various sizes of particles, with large particles possessing more pressure than their smaller counterparts. The width-dependent nature of the total pressures is induced by the more dispersion of smaller particles in the system as the particle size distribution is widened.     

Temperature Properties in Polydisperse Granular Mixtures

Numerical simulations are employed to consider the problem of determining the granular temperatures of the species of a homogeneous heated granular mixture with a power-law size distribution. The partial granular temperature ratios are studied as functions of the fractal dimension D, the restitution coefficient e, the rescaled viscosity time, the average occupied area fraction φ, the total particle number N and the number fraction. Different species of particles in a power-law system typically do not have the same mean kinetic energy, namely the granular temperature. It is found that the extent of nonequipartition of kinetic energy is determined by the fractal dimension D, the restitution coefficient e and the rescaled viscosity time, while is insensitive to the total particle number N , the area fraction φ and the number fraction.     

Global Pressure of One-Dimensional Polydisperse Granular Gases Driven by Gaussian White Noise

We study the global pressure of a one-dimensional polydisperse granular gases system for the first time,in which the size distribution of particles has the fractal characteristic and the inhomogeneity is described by a fractal dimension D. The particles are driven by Gaussian white noise and subject to inelastic mutual collisions. We define the global pressure P of the system as the impulse transferred across a surface in a unit of time, which has two contributions,one from the translational motion of particles and the other from the collisions. Explicit expression for the global pressure in the steady state is derived. By molecular dynamics simulations, we investigate how the inelasticity of collisions and the inhomogeneity of the particles influence the global pressure. The simulation results indicate that the restitution coefficient e and the fractal dimension D have significant effect on the pressure.     

Global Pressure of One-Dimensional Polydisperse Granular Gases Driven by Gaussian White Noise

We study the global pressure of a one-dimensional polydisperse granular gases system for the first time, in which the size distribution of particles has the fractal characteristic and the inhomogeneity is described by a fractal dimension D. The particles are driven by Gaussian white noise and subject to inelastic mutual collisions. We define the global pressure P of the system as the impulse transferred across a surface in a unit of time, which has two contributions, one from the translational motion of particles and the other from the collisions. Explicit expression for the global pressure in the steady state is derived. By molecular dynamics simulations, we investigate how the inelasticity of collisions and the inhomogeneity of the particles influence the global pressure. The simulation results indicate that the restitution coefficient e and the fractal dimension D have significant effect on the pressure.     

Fractal Analysis of Surface Roughness of Particles in Porous Media

A fractal dimension for roughness height （RH） is introduced to characterize the degree of roughness or disorder of particle surface characters which significantly influence physical-chimerical processes in porous media. An analytical expression for the fractal dimension of RH on statistically self-similar fractal surfaces is derived and is expressed as a function of roughness parameters. The specific surface area （SSA） of porous materials with spherical particles is also derived, and the proposed fractal model for the SSA of particles with rough surfaces is expressed as a function of fractal dimension for RH and fractal dimension for particle size distribution, relative roughness of particle surface, and ratio of the minimum to the maximum particle diameters of spherical particles.     

Velocity Profiles of Slow Blood Flow in a Narrow Tube

A fractal model is introduced into the slow blood motion. When blood flows slowly in a narrow tube, red cell aggregation results in the formation of an approximately cylindrical core of red cells. By introducing the fractal model and using the power law relation between area fraction φ and distance from tube axis ρ, rigorous velocity profiles of the fluid ia and outside the aggregated core and of the core itself are obtained analytically for different fractal dimensions. It shows a blunted velocity distribution for a relatively large fractal dimension (D～2), which can be observed in normal blood; a pathological velocity profile for moderate dimension (D = 1), which is similar to the Segre-Silberberg effect; and a parabolic profile for negligible red cell concentration (D = 0), which likes in the Poiseuille flow.     

The Thermal Conductivity Theory of Non-uniform Granular Flow and the Mechanism Analysis

According to the fractal characteristics appearing in non-uniform granular system, we found the fractalmodel to study the effective thermal conductivity in the mixed system. Considering the quasi-equilibrium, we bringforward the fractal velocity probability distribution function. The equipartition of energy is employed to the non-uniform granular system, and the granular temperature is derived. We investigate the thermal conductivity in granularflow due to the movement of the particles, namely the heat transfer induced by the streaming mode only. The thermalconductivity in the mixed system changes with the fractal parameters such as the solid fraction v, structural characterparameter η, and fractal dimension D of size distribution. These parameters depict the characteristics of the thermalconductivity in the actual complex granular system. Comparing our conclusion with the correlative experimental dataand the theoretical conclusion of binary mixture of granular materials, the results can qualitatively confirm the generalityof our prediction on the granular system.     

KDP晶体单点金刚石车削表面形貌分形分析

 分别使用2维和3维分形方法对单点金刚石车削加工的KDP晶体表面形貌进行了分析，并对表面的3维分形维数和3维粗糙度表征参数进行了比较，分析了二者对表面形貌表征的差异。使用2维轮廓分形方法计算了KDP晶体表面圆周各方向上的分形维数。通过分析得出：3维分形维数与表面粗糙度值成反比关系；使用单点金刚石车削方法加工KDP晶体会形成各向异性特征明显的已加工表面，在一定程度上容易形成小尺度波纹；已加工表面是否具有明显的小尺度波纹特征与表面粗糙度值并无直接关系，但与其表面轮廓分形状态分布密切相关；KDP晶体表面2维功率谱密度与其分形状态具有相近的方向性特征。     

The Thermal Conductivity Theory of Non-uniform Granular Flow and the Mechanism Analysis

According to the fractal characteristics appearing in non-uniform granular system, we found the fractal model to study the effective thermal conductivity in the mixed system. Considering the quasi-equilibrium, we bring forward the fractal velocity probability distribution function. The equipartition of energy is employed to the non-uniform granular system, and the granular temperature is derived. We investigate the thermal conductivity in granular flow due to the movement of the particles, namely the heat transfer induced by the streaming mode only. The thermal conductivity in the mixed system changes with the fractal parameters such as the solid fraction v, structural character parameter η, and fractal dimension D of size distribution. These parameters depict the characteristics of the thermal conductivity in the actual complex granular system. Comparing our conclusion with the correlative experimental data and the theoretical conclusion of binary mixture of granular materials, the results can qualitatively confirm the generality of our prediction on the granular system.     

Dielectric constant of porous ultra low-k dielectrics by fractal-Monte Carlo simulations

The effective dielectric constant of porous ultra low-k dielectrics is simulated by applying the fractal geometry and Monte Carlo technique in this work. Based on the fractal character of pore size distribution in porous media, the probability models for pore diameter and for effective dielectric constant are derived. The proposed model for the effective dielectric constant is expressed as a function of the dielectric coefficient of base medium and the volume fractions of pores and base medium, fractal dimension for pores, the pore size, as well as random number. The Monte Carlo simulations combined with the fractal geometry are performed. The predictions by the present simulations are shown in good accord with the available experimental data. The proposed technique may have the potential in analyzing other properties such as electrical conductivity and thermal conductivity in porous ultra low-k dielectrics.     

Flow, diffusion, and thermal convection in percolation clusters: NMR experiments and numerical FEM/FVM simulations.

Percolation objects were fabricated based on computer-generated, two- or three-dimensional templates. Random-site, semi-continuous swiss cheese, and semi-continuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by NMR imaging and, in the presence of a pressure gradient, NMR velocity mapping. The fractal dimension, the correlation length, and the percolation probability were evaluated both from the computer-generated templates and the corresponding NMR spin density maps. Based on velocity maps, the percolation backbones were determined. The fractal dimension of the backbones turned out to be smaller than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law the exponent of which was not yet considered in the theoretical literature. The experimental results favorably compare to computer simulations based on the finite-element method (FEM) or the finite-volume method (FVM). Percolation theory suggests a relationship between the anomalous diffusion exponent and the fractal dimension of the cluster, i.e., between a dynamic and a structural parameter. We examined interdiffusion between two compartments initially filled with H2O and D2O, respectively, by proton imaging. The results confirm the theoretical expectation. As a third transport mechanism, thermal convection in percolation clusters of different porosities was studied with the aid of NMR velocity mapping. The velocity distribution is related to the convection roll size distribution. Corresponding histograms consist of a power law part representing localized rolls, and a high-velocity cut-off for cluster-spanning rolls. The maximum velocity as a function of the porosity clearly visualizes the percolation transition.     

Grayscale projection of finite size fractal clusters

Microscopy techniques are suitable to obtain structural information of colloidal clusters with high resolution, but yield only a two dimensional projection of the objects. When imaging finite size objects with fractal properties, such as clusters of colloidal particles, this projection process has to be taken into account for the calculation of the fractal dimension. In this paper we present a technique to calculate the fractal dimension of finite size clusters with fractal properties using grayscale projections such as images obtained by X-ray microscopy. The grayscales are interpreted as different occupation counts within a projection. It is shown, that the radial distribution of these occupation counts varies with the fractal dimension d of the cluster. Using the radius of maximum occupation probability the fractal dimension up to 2.2 of finite size clusters can be calculated. The theoretical predictions are verified by test calculations employing numerically generated clusters.