Influence of the long-range forces in non-Gaussian random-packing dynamics

In this paper, we perform molecular dynamics (MD) simulations to study the random packing of spheres with different particle size distributions. In particular, we deal with non-Gaussian distributions by means of the Lévy distributions. The initial positions as well as the radii of five thousand non-overlapping particles are assigned inside a confining rectangular box. After that, the system is allowed to settle under gravity towards the bottom of the box. Both the translational and rotational movements of each particle are considered in the simulations. In order to deal with interacting particles, we take into account both the contact and long-range cohesive forces. The normal viscoelastic force is calculated according to the nonlinear Hertz model, whereas the tangential force is calculated through an accurate nonlinear-spring model. Assuming a molecular approach, we account for the long-range cohesive forces using a Lennard-Jones (LJ)-like potential. The packing processes are studied assuming different long-range interaction strengths.     

高压电场内细颗粒堆积机理研究

细颗粒的堆积既是电厂尾部除尘系统的核心问题,也是航天领域空间环境内限制动力或光学元件性能的关键问题。电场力因具有长程有效性和强可操作性成为控制细颗粒堆积的主要手段。本文通过微观实验研究了高压平行板电场间细颗粒堆积的机理,观察到颗粒在预极化、预荷电、变外电场电压等工况下的堆积形貌,并发展了图像处理的方法统计堆积颗粒数。研究表明,偶极力是外电场下颗粒成链的主因,而颗粒的倒伏则是来自曳力的作用,颗粒链的极限高度主要受外电场场强和颗粒堆积结构的影响。     

Morphology of Nanostructured Films for Environmental Applications: Simulation of Simultaneous Sintering and Growth

A sequential Brownian dynamics approach was used to establish the morphological evolution of a nanostructured particle deposit accounting for random diffusion, particle–particle and particle–surface interactions through van der Waals forces, and sintering of deposited particles. Monodisperse (30nm radius) titanium dioxide particles were used in the simulations. A linear sintering law rate expression was used to account for the decrease in total surface area of the deposit. Characteristics such as packing thickness, total surface area, and fractal dimension are reported as a function of time during the deposition process. Sintering resulted in higher fractal dimensions (as defined) for the deposits, and elevated temperatures resulted in more compact deposits.     

粉末材料堆积的物理模型与仿真系统

研究了粉末材料堆积过程仿真的物理模型和系统,并探讨了适合多种不同粒径颗粒混合堆积过程仿真的高性能计算方法.在该仿真系统中,考虑了重力、接触力、阻尼力、摩擦力和范德瓦耳斯力等多种作用力的影响,集成了多种接触力模型和阻尼模型,使其适用于三维大规模粉末材料堆积过程的计算机仿真.利用该系统对粉末材料领域中的两个典型应用进行了模拟研究.模拟了两种相同密度不同粒径颗粒（粒径比为10）的混合堆积过程.当小颗粒数为大颗粒数的300倍时,得到最大的堆积密度（体积分数）为0.82.另外,还模拟了两种不同密度相同粒径颗粒的混合堆积过程.当堆积结束时,出现了明显的分离（segregation）现象和团聚现象.所研究的物理模型和仿真系统既可用于粉末材料堆积过程研究,亦可用于普通的球形物体堆积过程的模拟研究. 关键词： 粉末堆积 物理模型 仿真系统 离散元法     

Coupling of ferromagnetic nanoparticles through dipolar interactions

We consider two ferromagnetic nanoparticles coupled via long-range dipolar interactions. We model each particle by a three-dimensional array of classical spin vectors, with a central spin surrounded by a variable number of shells. Within each particle only ferromagnetic coupling between nearest neighbor spins is considered. The interaction between particles is of the dipolar type and the magnetic properties of the system is studied as a function of temperature and distance between the centers of the particles. We perform Monte Carlo simulations for particles with different number of shells, and the magnetic properties are calculated via two routes concerning the dipolar contribution: one assuming a mean-field like coupling between effective magnetic moments at the center of the particles, and other one, where we take into account interactions among all the pairs of spins, one in each particle. We show that the dipolar coupling between the particles enhances the critical temperature of the system relative to the case in which the particles are very far apart. The dipolar energy between the particles is smaller when the assumption of effective magnetic moment of the particles is used in the calculations.     

DEM simulation of small particles clogging in the packing of large beads

The percolation of small particles through a periodic random loose packing of large beads is studied numerically with the Distinct Element Method. The representativity of periodic mono-sized sphere packing of varying system size was first studied by comparing their pore size distributions and tortuosities with those of a larger system, considered as an infinite medium. The results show that a periodic packing of size as low as 4-grain diameters gives a reasonable representation of the porous medium and allows reducing considerably the number of particles that has to be used in the simulations. The flow and clogging of small particles of varying concentrations and friction coefficients flowing through the former packing are then studied numerically. Results show that a steady state is rapidly reached where the mean velocity and mean vertical velocity of small particles are both constant. These mean velocities decrease with an increase in friction coefficient and in small particle concentration. The influence of the friction coefficient μ is much less marked for values of μ larger than or equal to 0.5. The distribution of small particles throughout the crossed packing becomes rapidly heterogeneous. Small particles concentrate in some pores where their velocity vanishes and where the density can reach values larger than the density of the random loose packing. The proportion of particles blocked in these pores varies linearly with concentration. Finally, the narrow throats of the porous medium responsible for blocking are identified and characterized for different values of the friction coefficient.     

Brownian dynamics of polydisperse colloidal hard spheres: Equilibrium structures and random close packings

Recently we presented a new technique for numerical simulations of colloidal hard-sphere systems and showed its high efficiency. Here, we extend our calculations to the treatment of both 2- and 3-dimensional monodisperse and 3-dimensional polydisperse systems (with sampled finite Gaussian size distribution of particle radii), focusing on equilibrium pair distribution functions and structure factors as well as volume fractions of random close packing (RCP). The latter were determined using in principle the same technique as Woodcock or Stillinger had used. Results for the monodisperse 3-dimensional system show very good agreement compared to both pair distribution and structure factor predicted by the Percus-Yevick approximation for the fluid state (volume fractions up to 0.50). We were not able to find crystalline 3d systems at volume fractions 0.50–0.58 as shown by former simulations of Reeet al. or experiments of Pusey and van Megen, due to the fact that we used random start configurations and no constraints of particle positions as in the cell model of Hoover and Ree, and effects of the overall entropy of the system, responsible for the melting and freezing phase transitions, are neglected in our calculations. Nevertheless, we obtained reasonable results concerning concentration-dependent long-time selfdiffusion coefficients (as shown before) and equilibrium structure of samples in the fluid state, and the determination of the volume fraction of random close packing (RCP, glassy state). As expected, polydispersity increases the respective volume fraction of RCP due to the decrease in free volume by the fraction of the smaller spheres which fill gaps between the larger particles.     

Role of interparticle forces in the formation of random loose packing

We present a physical and numerical study of the settling of uniform spheres in liquids and show that interparticle forces play a critical role in forming the so-called random loose packing (RLP). Different packing conditions give different interparticle forces and, hence, different RLP. Two types of interparticle forces are identified: process dependent and process independent. The van der Waals force, as the major cohesive force in the present study, plays a critical role in effecting the process-dependent forces such as drag and lift forces. An equation is formulated to describe the relationship between the macroscopic packing fraction and microscopic interparticle forces in a packing. We argue there is no lowest packing fraction for a mechanically stable RLP; hence, the packing fractions of RLP can range from 0 to 0.64 depending on the cohesive and frictional conditions between particles.     

From crystal to amorphous: A novel route to unjamming in soft disk packings

Numerical studies on the unjamming packing fraction of bi- and polydisperse disk packings, which are generated through compression of a monodisperse crystal, are presented. In bidisperse systems, a fraction f ₊ = 0.400 up to 0.800 of the total number of particles has their radii increased by D \Delta R , while the rest has their radii decreased by the same amount. Polydisperse packings are prepared by changing all particle radii according to a uniform distribution in the range [- D \Delta R,D \Delta R] . The results indicate that the critical packing fraction is never larger than the value for the initial monodisperse crystal, f₀ \phi_{0}^{} = p \pi/?{12} \sqrt{{12}} , and that the lowest value achieved is approximately the one for random close packing. These results are seen as a consequence of the interplay between the increase in small-small particle contacts and the local crystalline order provided by the large-large particle contacts.     

双分散颗粒体系在临界堵塞态的结构特征

堵塞行为是颗粒体系中一种常见的现象,其力学性质与堆积结构的关联非常复杂.本文采用离散元法研究了由两种不同半径颗粒组成的二维双分散无摩擦球形颗粒体系在临界堵塞态所呈现的结构特征,讨论了大小颗粒粒径比与大颗粒百分比对临界堵塞态的影响.数值模拟结果表明,当粒径比小于1.4时,临界平均接触数与大颗粒百分比关系不大,当粒径比大于1.4时随着大颗粒百分比的增大临界平均接触数先减小再增大.而临界体积分数在粒径比小于1.8时随着大颗粒百分比的增加先减小后增大,大于1.8时又基本不随大颗粒百分比而变化.大颗粒百分比在接近0或1时,系统近似为单分散体系,临界平均接触数与体积分数基本不随半径比的增大而变化;在接近0.5时,临界平均接触数随着半径比的增大逐渐减小,而临界体积分数则是先减小后增大.文中对大-小颗粒这一接触类型的百分比也进行了探讨,其值随着大颗粒百分比的增大呈二次函数的变化趋势,粒径比对这一变化趋势只有较小的影响.     

非球体填充的组合球模型及松弛算法

现有的松弛算法由于仅用于球填充而只考虑颗粒的平动,故提出考虑非球体转动的改进松弛算法并采用组合球模型,使其能够模拟任意形状非球体的随机填充以及多种非球体的混合填充.用多个球体的外包络面近似一个非球体外形的组合球模型,将非球体之间的接触转化为球体之间的接触,从而简化并统一非球体接触判断算法.通过引入非球体的转矩和转角松弛机制,使改进松弛算法克服了"自锁"现象,并能生成非球体的随机密填充.算例表明,填充结果与现有的数值模拟及实验结果相符.     

Numerical Simulation of Random Close Packings in Particle Deformation from Spheres to Cubes

Variation of packing density in particle deforming from spheres to cubes is studied. A new model is presented to describe particle deformation between different particle shapes. Deformation is simulated by relative motion of component spheres in the sphere assembly model of a particle. Random close packings of particles in deformation form spheres to cubes are simulated with an improved relaxation algorithm. Packings in both 2D and 3D cases are simulated. With the simulations, we find that the packing density increases while the particle sphericity decreases in the deformation. Spheres and cubes give the minimum (0.6404) and maximum (0.7755) of packing density in the deformation respectively. In each deforming step, packings starting from a random configuration and from the final packing of last deforming step are both simulated. The packing density in the latter case is larger than the former in two dimensions, but is smaller in three dimensions. The deformation model can be applied to other particle shapes as well.     

Charged particles on a two-dimensional lattice subject to anisotropic Jahn-Teller interactions

The properties of a system of charged particles on a 2D lattice, subject to an anisotropic Jahn-Teller-type interaction and 3D Coulomb repulsion, are investigated. In the mean-field approximation without Coulomb interaction, the system displays a phase transition of first order. When the long-range Coulomb interaction is included, Monte Carlo simulations show that the system displays very diverse mesoscopic textures, ranging from spatially disordered pairs to ordered arrays of stripes, or charged clusters, depending only on the ratio of the two interactions (and the particle density). Remarkably, charged objects with an even number of particles are more stable than with an odd number of particles. We suggest that the diverse functional behavior-including superconductivity-observed in oxides can be thought to arise from the self-organization of this type.     

Algorithm for particle packing based on Aristotelian mechanics

A dynamic algorithm is proposed for three-dimensional packing of spherical solid particles. The particles are deposited within a specified region with a fixed rigid boundary. The velocity of each particle is proportional to its weight and forces due to contact of the particle with the boundary and neighbor particles. Dimensional analysis of the equations of particle motion is performed. The average density and coordination number distribution for an equilibrium packing are calculated. The dependence of these characteristics on viscosity, granulometric composition, and representation of initial conditions (numerical analogue of material pouring into a specified volume) is studied.     

Lattice Boltzmann simulation of the two-dimensional flow of a magnetic suspension between two parallel walls under a non-uniform magnetic field

We have developed a lattice Boltzmann method based on fluctuation hydrodynamics that is applicable to the flow problem of a particle suspension. In this method, we have introduced the viscosity-modifying method, rather than the velocity-scaling method, in which a modified viscosity is used for generating random forces in lattice Boltzmann simulations. The viscosity-modifying method is found to be applicable to the simulation of a magnetic particle suspension. We have applied this method to the two-dimensional Poiseuille flow of a magnetic suspension between two parallel walls in order to investigate the behavior of magnetic particles in a non-uniform applied magnetic field. From the results of the snapshots, the pair correlation function between the magnetic pole and the magnetic particles and the averaged local particle velocity and magnetization distributions, it was observed that the behavior of the magnetic particles changes significantly depending upon which factor dominates the phenomenon in the balance between the magnetic particle–particle interaction, the non-uniform applied magnetic field and the translational and rotational Brownian motion.     

Influence of dispersive long-range interactions on properties of vapour–liquid equilibria and interfaces of binary Lennard-Jones mixtures

The influence of dispersive long-range interactions on properties of vapour–liquid equilibria and interfaces of six binary Lennard-Jones (LJ) mixtures was studied by molecular dynamics (MD) simulations and density gradient theory (DGT). The mixtures were investigated at a constant temperature T, at which the low-boiling component, which is the same in all mixtures, is subcritical. Two different high-boiling components were considered: one is subcritical, the other is supercritical at T. Furthermore, the unlike dispersive interaction was varied such that mixtures with three different types of phase behaviour were obtained: ideal, low-boiling azeotrope, and high-boiling azeotrope. In a first series of simulations, the full LJ potential was used to describe these mixtures. To assess the influence of the long-range interactions, these results were compared with simulations carried out with the LJ truncated and shifted (LJTS) potential applying the corresponding states principle. The dispersive long-range interactions have a significant influence on the surface tension and the interfacial thickness of the studied mixtures, whereas the relative adsorption and the enrichment are hardly affected. Furthermore, the influence of the long-range interactions on Henry's law constants and the phase envelopes of the vapour–liquid equilibrium was investigated. The long-range interactions have practically no influence on the composition dependency of the investigated mixture properties.     

Dipolar Order in Molecular Fluids: II. Molecular Influence

The influence on the short-range packing in dipolar fluids by molecular shape and by additional higher order electrostatic moments has been investigated by molecular dynamic simulations. The dipole polarization was found to decrease as the particles were elongated parallel to the dipole and to increase for elongation perpendicular to the dipole, eventually forming a nematic order. The addition of a quadrupole lead to a reduction of the polarization, and the influence of an axial octupole was weaker and more complex. Both a decrease and an increase of the polarization is possible depending on the relative dipole–dipole and octupole–octupole interaction strengths and the relative direction of the symmetry axes of the moments. These observations were attributed to the different parity of a dipole and a quadrupole and the same parity of a dipole and an axial octupole under reflection. In addition, further insights into the formation of dipole polarization were obtained. Short polar and long equatorial radii and strong dipole–dipole interaction are particle properties that promote a fluid with a high dipole polarization.     

On hydraulic permeability of random packs of monodisperse spheres: Direct flow simulations versus correlations

Hydraulic permeability is studied in porous media consisting of randomly distributed monodisperse spheres by means of computational fluid dynamics (CFD) simulations. The packing of spheres is generated by inserting a certain number of nonoverlapping spherical particles inside a cubic box at both low and high packing fractions using proper algorithms. Fluid flow simulations are performed within the interparticulate porous space by solving Navier-Stokes equations in a low-Reynolds laminar flow regime. The hydraulic permeability is calculated from the Darcy equation once the mean values of velocity and pressure gradient are calculated across the particle packing. The simulation results for the pressure drop across the packing are verified by the Ergun equation for the lower range of porosities (<0.75), and the Stokes equation for higher porosities (∼1). Using the results of simulations, the effects of porosity and particle diameters on the hydraulic permeability are investigated. Simulations precisely specified the range of applicability of empirical or semi-empirical correlations for hydraulic permeability, namely the Carman-Kozeny, Rumpf-Gupte, and Howells-Hinch formulas. The number of spheres in the model is gradually decreased from 2000 to 20 to discover the finite-size effect of pores on the hydraulic permeability of spherical packing, which has not been clearly addressed in the literature. In addition, the scale dependence of hydraulic permeability is studied via simulations of the packing of spheres shrunk to lower scales. The results of this work not only reveal the validity range of the aforementioned correlations, but also show the finite-size effect of pores and the scale-independence of direct CFD simulations for hydraulic permeability.     

Two-dimensional small particles coupled by dipolar interactions

We study the effect of the dipolar coupling on the magnetic properties of two small interacting ferromagnetic particles. Each particle is a two-dimensional array of Ising spins with a central spin surrounded by a variable number of shells. The coupling between spins inside each particle is ferromagnetic and the dipolar interaction between the particles is determined as a function of the number of shells, temperature, and distance between their centers. We investigate the system by mean-field approximation and Monte Carlo simulations. The dipolar interaction is calculated in two ways, one assuming effective spins in the centers of the particles, and the other directly computing the interactions among all the pairs of spins, one in each particle. We show that the difference in the corresponding dipolar energies is a power law on the distance with exponent 5. We calculate the magnetization and susceptibility as a function of temperature, number of shells and distance between the particles’ centers. We show that the critical temperature increases with the number of spins in each particle, and it is more noticeable in the mean-field calculations than in the Monte Carlo simulations.     

Nonequilibrium statistical mechanics of systems with long-range interactions

Systems with long-range (LR) forces, for which the interaction potential decays with the interparticle distance with an exponent smaller than the dimensionality of the embedding space, remain an outstanding challenge to statistical physics. The internal energy of such systems lacks extensivity and additivity. Although the extensivity can be restored by scaling the interaction potential with the number of particles, the non-additivity still remains. Lack of additivity leads to inequivalence of statistical ensembles. Before relaxing to thermodynamic equilibrium, isolated systems with LR forces become trapped in out-of-equilibrium quasi-stationary states (qSSs), the lifetime of which diverges with the number of particles. Therefore, in the thermodynamic limit LR systems will not relax to equilibrium. The qSSs are attained through the process of collisionless relaxation. Density oscillations lead to particle–wave interactions and excitation of parametric resonances. The resonant particles escape from the main cluster to form a tenuous halo. Simultaneously, this cools down the core of the distribution and dampens out the oscillations. When all the oscillations die out the ergodicity is broken and a qSS is born. In this report, we will review a theory which allows us to quantitatively predict the particle distribution in the qSS. The theory is applied to various LR interacting systems, ranging from plasmas to self-gravitating clusters and kinetic spin models.