Distribution Of Particle Displacements Research Articles

On entropy estimates of contact forces in static granular assemblies

As an extension of previous work by others, this articles deals with maximum-entropy estimates for the statistics of static contact forces in assemblies of nearly rigid grains. As found in the previous works, the constraint of mean stress leads provisionally to a distribution of contact force that is exponential at large force. This behavior, found in various experiments and numerical simulations, does not depend on special models of force propagation postulated in the contemporary physics literature. Following K. Bagi [Behringer, R. Jenkins, J.T. (Eds.), Powders and Grains, Balkema, 1997, p. 251] consideration is also given to entropy maximization under the constraint of constant mean strain, which leads to a similar exponential tail in the distribution of particle displacements. In contrast to the previous works, it is emphasized that the exact form of the probability density depends on the statistical weight (a priori probability) assigned to elementary volumes in the state-space of contact forces or displacements. This leads to the conclusion that the large-force exponential is a general representation of maximum-entropy statistics arising from global constraints, whereas the state-space measure is dictated by local mechanics. A few examples of state-space measure are considered, and the resulting distributions are compared to previous experiment and simulation. A striking analogy is revealed between the force distribution in a static sphere assembly and the Maxwell–Boltzmann velocity distribution for gases. Based on the methods of statistical thermodynamics, a virtual-thermodynamic formalism is presented for complementary strain energies in granular statics. This involves no direct appeal to the concept of (static) granular temperature favored in certain statistical-physics literature. As a possible test of the general validity of the entropy principle in granular statics, the question is raised as to whether it can describe the heterogeneous two-phase structure found in photoelastic experiments and numerical simulations of granular assemblies.

Supersonic Detonation in Rock Mass - Part 2: Particle Displacements and Velocity Fields for Single and Multiple Non-Delayed and Delayed Detonating Blastholes

In Part 1 of this contribution [1] constant speed supersonic detonation of an infinitely long column charge and the resulting stress wave propagating from the blasthole were addressed and an analytical solution for the stress field was presented. This publication (Part 2) deals with the associated distributions of particle displacements and velocities using previously obtained results. The wave field in the immediate vicinity of multiple non-delayed and delayed detonating blastholes is discussed. Results show that the wave fronts decay with different decay coefficients for the P- and S-wave and these decay coefficients depend primarily on the velocity of detonation. Regions of large concentrations of stress and particle velocity could be identified in the regions adjacent to the blastholes as well as between the blastholes. These regions are indicative of material crushing and tensile failure. The wave field scenario is discussed for two, three and four delayed blastholes.

Development of particle dispersion characteristics from arbitrary initial conditions in isotropic turbulence

Discrete particles released into turbulent carrier flows respond to that turbulence over periods of time which depend on the particle inertia and the initial state of the particle. Due to the turbulence, the velocity of the particles becomes random and they will become dispersed throughout the carrier flow. The level of randomness of the particle velocities can be quantified by the particle kinetic stress and the spreading rate is characterized by the dispersion coefficient. In the idealized case of stationary isotropic turbulence, the kinetic stress and dispersion coefficients approach limiting values long after release. Previous analysis by the author has provided development times for dispersion coefficients and kinetic stress in cases where particles were released from a point source: (i) from an initial state of rest, and (ii) with kinetic stress identical to that found in the steady state. This paper generalizes the analysis by allowing for arbitrary initial particle velocity and displacement distributions. As with the previous analysis, the primary focus is on high-inertia particles. Unlike the previous work, however, the present analysis is applicable to the common practice in direct numerical simulations of setting initial particle velocities equal to local fluid velocities. Several different estimators of the particle dispersion coefficient are considered. Whereas each estimator leads to the same long-time value, the time taken to approach this value can vary dramatically. Expressions for development times are given for the particle kinetic stress and dispersion coefficients. Consequences of the analysis for experimental and computational studies are discussed.

Shear deformation of binary mixtures of dry particulate solids

Abstract The shear deformation behavior of dry granular materials is of interest in the analysis of flowability, mixing, segregation and degradation. Quite often the particulate systems are mixtures of materials with a wide size distribution. In spite of a large amount of work reported in the literature, the shear behavior of poly-disperse systems has not been widely addressed. In this paper experimental work using an annular shear cell and numerical simulations by the distinct element method (DEM) have been carried out to quantify the normal and shear stress behavior of assemblies made of binary mixtures. The experimental results on mixtures of different sizes of glass beads show a linear relationship between the normal and shear stresses with a slope which depends on the shear rate and particle size ratio. This is an indication of flow in the intermediate regime between the quasi-static and rapid shear flows. The results also indicate a highly non-linear particle displacement distribution across the shear layer. Numerical simulations using the DEM agree qualitatively well with the experimental results for the normal-shear stress behavior.

Freezing transition and correlated motion in a quasi-two-dimensional colloid suspension.

Recent experiments have demonstrated that the deviation of the single-particle displacement distribution from Gaussian form in a dense quasi-two-dimensional colloid suspension is a result of heterogenous dynamics that involves cooperative motions of neighboring colloid particles [J. Chem. Phys. 47, 9142 (2001)]. In this paper, we report the results of molecular dynamics (MD) simulations of a quasi-two-dimensional assembly of nearly hard-sphere colloid particles. The colloid-colloid interaction we use is short ranged and everywhere repulsive; it is related to the Marcus-Rice (MR) and modified MR interactions used in a previous study [Phys. Rev. E 58, 7529 (1998)]. As is the case for those systems, the one we study supports liquid, hexatic, and solid phases. Our calculations show that the deviation of the single-particle displacement distribution from Gaussian form is present in the liquid phase, and that a sharp increase in its magnitude occurs at the liquidus density and extends into the crystalline phase. For densities greater than the liquidus density we find three dynamical relaxation processes that include, at intermediate times, a slowing down in the rate of growth of the diffusive displacement of a particle due to the cage effect. As the density increases toward the solidus density, the dependence of the mean squared displacement on time, at intermediate times, changes from sublinear to zero. The onset of the long-time relaxation mode corresponds to the time at which the deviation of the particle displacement distribution from Gaussian form is a maximum. At this time, which increases exponentially with the density, the self-part of the van Hove function exhibits multiple maxima with respect to r while the distinct part of the van Hove function is a maximum at the origin, thereby signaling jump dynamics. At long times the particle mean square displacement has diffusive character at all densities including solid phase densities. A remarkable feature of our findings is the continuity of character of the particle displacement from the liquid phase through the hexatic phase and into the solid phase. Cooperative jumps that lead to diffusive process in crystals can be explained by a mechanism that involves many such correlated hops in random locations and random directions (but along the crystallographic axes) thereby generating effective random walk behavior. We argue that the collective motion we have found is generated by superpositions of instantaneous normal mode vibrations along diffusive paths. The diffusive paths are along the directions with strong bond orientation correlation, and start to grow in amplitude rapidly on entry into the hexatic phase.

Reaction-controlled diffusion: Monte Carlo simulations.

We study the coupled two-species nonequilibrium reaction-controlled diffusion model introduced by Trimper et al. [Phys. Rev. E 62, 6071 (2000)] by means of detailed Monte Carlo simulations in one and two dimensions. Particles of type A may independently hop to an adjacent lattice site, provided it is occupied by at least one B particle. The B particle species undergoes diffusion-limited reactions. In an active state with nonzero, essentially homogeneous B particle saturation density, the A species displays normal diffusion. In an inactive, absorbing phase with exponentially decaying B density, the A particles become localized. In situations with algebraic decay rho(B)(t) approximately t(-alpha(B)), as occurring either at a nonequilibrium continuous phase transition separating active and absorbing states, or in a power-law inactive phase, the A particles propagate subdiffusively with mean-square displacement <x-->(t)(2)(A)> approximately t(1-alpha(A)). We find that within the accuracy of our simulation data, alpha(A) approximately alpha(B) as predicted by a simple mean-field approach. This remains true even in the presence of strong spatiotemporal fluctuations of the B density. However, in contrast with the mean-field results, our data yield a distinctly non-Gaussian A particle displacement distribution n(A)(x-->,t) that obeys dynamic scaling and looks remarkably similar for the different processes investigated here. Fluctuations of effective diffusion rates cause a marked enhancement of n(A)(x-->,t) at low displacements /x-->/, indicating a considerable fraction of practically localized A particles, as well as at large traversed distances.

A method for determining electrophoretic and electroosmotic mobilities using AC and DC electric field particle displacements

We have developed a method for measuring the electrophoretic mobility of submicrometer, fluorescently labeled particles and the electroosmotic mobility of a microchannel. We derive explicit expressions for the unknown electrophoretic and the electroosmotic mobilities as a function of particle displacements resulting from alternating current (AC) and direct current (DC) applied electric fields. Images of particle displacements are captured using an epifluorescent microscope and a CCD camera. A custom image-processing code was developed to determine image streak lengths associated with AC measurements, and a custom particle tracking velocimetry (PTV) code was devised to determine DC particle displacements. Statistical analysis was applied to relate mobility estimates to measured particle displacement distributions.

The influence of interface layer characteristics on Lamb waves in layered anisot ropic media

In this paper, the spring model for the anisotropic interface layer is built and introduced to the global matrix technique that is popularly used for analyzing layered composites. In introducing the spring interface condition, the philosophy adopted is to incorporate it with a minimum disruption to the program structur e, particularly the assembly of the global matrix. The spring interface has ther efore been introduced as a "material layer". This layer has material constants w hich govern the stiffness across it and has zero thickness. The layer can be bui lt into the appropriate location in the global matrix without making any changes in the other layers of the system. Numerical computations for the Lamb wave dis persion curves in two-layered anisotropic composite media with different interfa ce conditions including rigid connection, slip connection, and complete disbond are made，and plots of the through-thickness particle displacement distributions of Lamb modes for the rigid bond interface and slip bond interface are compared .

Deterministic Advection-Diffusion Model Based on Markov Processes

A new deterministic numerical formulation named DisPar-k based on particle displacement probability distribution for Markov processes was developed to solve advection-diffusion problems in a one-dimensional discrete spatial grid. DisPar-k is an extension of DisPar, and the major difference is the possibility of establishing a number of consecutive particle destination nodes. This was achieved by solving an algebraic linear system where the particle displacement distribution moments are known parameters taken from the Gaussian distribution. The average was evaluated by an analogy between the Fokker-Planck and the transport equations, being the variance Fickian. The particle displacement distribution is used to predict deterministic mass transfers between domain nodes. Mass conservation was guaranteed by the distribution concept. It was shown that, for linear conditions, the accuracy order is proportional to the number of particle destination nodes. DisPar-k showed to be very sensible to physical discontinu...

Particle motion in gas-fluidized granular systems by pulsed-field gradient nuclear magnetic resonance.

Nuclear magnetic resonance k-space and q-space imaging methods were applied for the first time to monitor the time-averaged density variations and the random motion of granular particles in the presence of gas flow of variable velocity in a model gas-fluidized bed reactor. The transitions toward fluidization, bubbling phase, and slugging phase are visible as sudden changes in the density and the root mean-squared displacement. The onset of these transitions is shown to depend on the particle sizes for a given cell geometry. The effective diffusion coefficients are measured and the distributions of particle displacements, or propagators, are presented.

Geometric approach to the dynamic glass transition.

We numerically study the potential energy landscape of a fragile glassy system and find that the dynamic crossover corresponding to the glass transition is actually the effect of an underlying geometric transition caused by the vanishing of the instability index of saddle points of the potential energy. Furthermore, we show that the potential energy barriers connecting local glassy minima increase with decreasing energy of the minima, and we relate this behavior to the fragility of the system. Finally, we analyze the real space structure of activated processes by studying the distribution of particle displacements for local minima connected by simple saddles.

Investigating the microenvironments of inhomogeneous soft materials with multiple particle tracking.

We develop a multiple particle tracking technique for making precise, localized measurements of the mechanical microenvironments of inhomogeneous materials. Using video microscopy, we simultaneously measure the Brownian dynamics of roughly one hundred fluorescent tracer particles embedded in a complex medium and interpret their motions in terms of local viscoelastic response. To help overcome the inherent statistical limitations due to the finite imaging volume and limited imaging times, we develop statistical techniques and analyze the distribution of particle displacements in order to make meaningful comparisons of individual particles and thus characterize the diversity and properties of the microenvironments. The ability to perform many local measurements simultaneously allows more precise measurements even in systems that evolve in time. We show several examples of inhomogeneous materials to demonstrate the flexibility of the technique and learn new details of the mechanics of the microenvironments that small particles explore. This technique extends other microrheological methods to allow simultaneous measurements of large numbers of probe particles, enabling heterogeneous samples to be studied more effectively.

Extraction of all coefficients of coupled-mode equations for natural, single-phase, unidirectional SAW transducers from dispersion characteristics computed by hybrid finite element method.

A numerical method is presented for determining all of the coefficients of coupled-mode equations for natural, single-phase, unidirectional SAW transducers (NSP-UDTs). Substituting numerical results computed by the hybrid finite element method for infinite NSPUDTs with shorted and open electric ports into several relations derived from the coupled-mode theory, we can determine all of the coefficients. Specifically, the edge frequencies of a stop-band and the static capacitances yield the self-coupling coefficients and the amplitudes of mutual-coupling and transduction coefficients, and the electric potential standing wave on the substrate surface, which can be derived from the projection of the standing wave distributions of particle displacements and electric potential in the whole substrate onto the set of those predicted by the coupled-mode theory, yields the phases of mutual-coupling and transduction coefficients. NSPUDTs on ST-25 degrees X quartz, Y-51.25 degrees Z LiTaO3, and 50 degrees Y-25 degrees X La3Ga5SiO14 substrates are investigated. Our results agree well with the earlier experimental ones.

Excitation and propagation of non-axisymmetric guided waves in a hollow cylinder.

Excitation and propagation of non-axisymmetric guided waves in a hollow cylinder is studied by using the normal mode expansion method (NME). Different sources such as angle beam, tube end excitation with normal beam, and comb transducer possibilities are discussed based on the derivations of the NME method. Numerical calculations are focused on the case of angle beam partial loading. Based on the NME method, the amplitude coefficients for all of the harmonic modes are obtained. Due to the difference of phase velocities for different modes, the superimposed total field varies with propagating distances and hence makes particle displacement distribution patterns (angular profile) change with distance. This varying non-axisymmetric angular profile of guided waves represents a nonuniform energy distribution around the hollow cylinder and thus has an impact on the inspection ability of guided waves. The angular profiles of an angle beam source are predicted by theory and then verified by experiments. The predicted angular profiles also provide information for determining the transducer location to find defects in a certain position on the hollow cylinder.

Pore-scale network modelling of flow propagators derived from pulsed magnetic field gradient spin echo NMR measurements in porous media

Pulsed magnetic field gradient spin echo (PGSE) NMR experiments carried out on porous media where fluid flow is occurring, may be analysed to give the propagator, P( ξ, t), This quantity is the ensemble density distribution of particle (nuclei) displacements in a chosen direction, ξ, in a given time interval, t. These displacements arise as a result of both the convection and diffusion of molecules in the flowing fluid. The propagator can be derived for various displacement times, t, and hence these gradually probe a wider domain of the pore-scale velocity field within the porous medium. Such measurements can be performed separately on the oil and water phases in a two-phase flowing system. The interpretation and modelling of these single- and two-phase propagators in terms of the pore-scale flow field within the porous medium presents a difficult and interesting scientific challenge. In this paper, we model the main qualitative features of the experimentally measured propagators for both single- and two-phase flow using connected 3D pore network models of porous media. The calculated flow field within such models shows some non-trivial and qualitatively correct predictions about real flow fields in porous media. The propagator is modelled directly by incorporating transport due to both convection and diffusion for large numbers of marker particles (the nuclei) for both single- and two-phase flow. In the latter case, the transport within each of the two immiscible phases (oil and water) has been modelled in their separate pore occupancy networks. The network model captures most of the qualitative features for both single- and two-phase propagators thus giving us the capability to clearly interpret the respective flowing and non-flowing fractions of the oil (non-wetting) and water (wetting) phases in two-phase experiments. Therefore, our findings offer a powerful approach — PGSE NMR experiments and associated pore-scale modelling — for understanding and characterising two-phase flow through porous media.

Discrete Particle Distribution Model for Advection-Diffusion Transport

A new methodology, named DisPar, based on a discrete probability distribution for a particle displacement, was developed to solve 1D advection-diffusion transport problems in water bodies. The discrete probability distribution for the particle displacement was developed as an average and variance function. These probabilities were used to predict the deterministic mass transfer between cells in one time step, and therefore the particle concentration in each cell was considered the state variable. The state equation was found to be similar to an explicit finite-difference formulation with a Eulerian grid. The model stability, positivity, and mass conservation are guaranteed by the probability distribution concept. DisPar produces solutions without numerical dispersion for constant velocity, diffusion coefficient, and cross-sectional area. In these conditions, DisPar was also developed as a function of space and time for an instantaneous mass spill. When the stability and positivity restrictions were respected, the model produced excellent results when compared to analytical solutions and other methods. The discrete particle displacement distribution concept differs from other numerical formulations, and therefore it represents a new modeling technique.

Suppression of bulk wave radiation from leaky surface acoustic waves by loading with thin dielectric films

The bulk wave, which is radiated into the substrate from the leaky surface acoustic wave (LSAW), can be suppressed by loading the substrate with a thin film of dielectric such as tantalum pentoxide (Ta2O5) and silicon dioxide (SiO2). Rotated Y-cut X-propagating lithium niobate (LiNbO3) is used for the substrate. For a certain range of the rotation angle, the attenuation of the LSAW can be suppressed by choosing the appropriate material and thickness for the film. For the free surface of 64° Y−X LiNbO3 with Ta2O5 film, and the metallized surface of 41° Y−X LiNbO3 with SiO2 film, the measured propagation losses decreased approximately to one-fourth to half of that of the sample without the thin film. It is shown that, even if the propagation path is a free surface, the particle displacement distribution in the LSAW is concentrated near the substrate surface by the loading with the thin film, as for the metallized surface. This result suggests that the bulk wave radiation in the LSAW excitation can be suppressed. The bulk wave radiation loss for the 41° Y−X LiNbO3 with the Ta2O5 film decreased approximately to one-sixth of that of the sample without the thin film.

Single-file diffusion of colloids in one-dimensional channels

Single-file diffusion, prevalent in many processes, refers to the restricted motion of interacting particles in narrow micropores with the mutual passage excluded. A single-filing system was developed by confining colloidal spheres in one-dimensional circular channels of micrometer scale. Optical video microscopy study shows evidence that the particle self-diffusion is non-Fickian for long periods of time. In particular, the distribution of particle displacement is a Gaussian function.

Direct observation of dynamical heterogeneities in colloidal hard-sphere suspensions

The real-space dynamics in a model system of colloidal hard spheres was studied by means of time-resolved fluorescence confocal scanning microscopy. Direct experimental evidence for the presence of dynamical heterogeneities in a dense liquid was obtained from an analysis of particle trajectories in two-dimensional slices of the bulk sample. These heterogeneities manifest themselves as a non-Gaussian probability distribution of particle displacements and also affect the onset of long-time diffusive behavior.

Suppression of bulk wave radiation of leaky surface acoustic waves by proton exchange

This paper shows that the bulk wave that is radiated to the inside of the substrate in the excitation and propagation of a leaky surface acoustic wave (LSAW) can be suppressed by forming a proton-exchanged layer on the surface of the substrate. It is theoretically shown that, even if the electrical condition of the propagation path is a free surface, the particle displacement distribution of the LSAW particle is concentrated at the substrate surface by proton exchange, as in the case of a metallized surface. This result suggests that the bulk wave radiation in LSAW excitation can be suppressed. The bulk wave radiation loss is examined by measuring the insertion loss between the input and the output IDT on the 41° Y-X LiNbO3 substrate, and it is shown that the bulk wave can be reduced to zero by proton exchange. It is shown experimentally that the leak loss in the propagation due to the bulk wave radiation can be reduced to approximately one-third by the proton exchange. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 83(1): 44–52, 2000