Simulations of two-dimensional arrays of beads under external vibrations: Scaling behavior

Statistics of a granular cluster ensemble at a liquid-solid-like phase transition

We report on the construction of a granular network of particles to study the formation, evolution, and statistical properties of clusters of particles developing at the vicinity of a liquid-solid-like phase transition within a vertically vibrated quasi-two-dimensional granular system. Using the data of particle positions and local order from Castillo et al. [G. Castillo, N. Mujica, and R. Soto, Phys. Rev. Lett. 109, 095701 (2012)0031-900710.1103/PhysRevLett.109.095701], we extract granular clusters taken as communities of the granular network via modularity optimization. Each one of these communities is a patch of particles with a very well defined local orientational order embedded within an array of other patches forming a complex cluster network. The distributions of cluster sizes and lifespans for the cluster network depend on the distance to the liquid-solid-like phase transition of the quasi-two-dimensional granular system. Specifically, the cluster size distribution displays a scale-invariant behavior for at least a decade in cluster sizes, while cluster lifespans grow monotonically with each cluster size. We believe this systematic community analysis for clustering in granular systems can help to study and understand the spatiotemporal evolution of mesoscale structures in systems displaying out-of-equilibrium phase transitions.

Characterization of sand convective motions at a vertical wall subjected to long-term cyclic loading

By conducting a two-dimensional experimental study, this paper aims to enhance the understanding of the mechanism of sand convective motions in the vicinity of a wall subjected to long-term cyclic lateral loadings. The experimental tests were conducted in a rectangular sandbox with a transparent front-wall, through which the process of sand particle motions could be recorded by using a high-resolution digital camera. The images were processed with a high time-resolved PIV (Particle Image Velocimetry) system. Based on the experimental data, this work (1) presents the sand flow field in the convective zones; (2) provides means to describe the convection mechanism; (3) proposes the relationships between the loading conditions and dimensions of the region with intense sand movement; and (4) elaborates the similarity of the sand flow velocity structure within the sand convective zones.

Mechanical excitation of rodlike particles by a vibrating plate

The experimental realization and investigation of granular gases usually require an initial or permanent excitation of ensembles of particles, either mechanically or electromagnetically. One typical method is the energy supply by a vibrating plate or container wall. We study the efficiency of such an excitation of cylindrical particles by a sinusoidally oscillating wall and characterize the distribution of kinetic energies of excited particles over their degrees of freedom. The influences of excitation frequency and amplitude are analyzed.

Softening of stressed granular packings with resonant sound waves

We perform numerical simulations of a two-dimensional bidisperse granular packing subjected to both a static confining pressure and a sinusoidal dynamic forcing applied by a wall on one edge of the packing. We measure the response experienced by a wall on the opposite edge of the packing and obtain the resonant frequency of the packing as the static or dynamic pressures are varied. Under increasing static pressure, the resonant frequency increases, indicating a velocity increase of elastic waves propagating through the packing. In contrast, when the dynamic amplitude is increased for fixed static pressure, the resonant frequency decreases, indicating a decrease in the wave velocity. This occurs both for compressional and for shear dynamic forcing and is in agreement with experimental results. We find that the average contact number Zc at the resonant frequency decreases with increasing dynamic amplitude, indicating that the elastic softening of the packing is associated with a reduced number of grain-grain contacts through which the elastic waves can travel. We image the excitations created in the packing and show that there are localized disturbances or soft spots that become more prevalent with increasing dynamic amplitude. Our results are in agreement with experiments on glass bead packings and earth materials such as sandstone and granite and may be relevant to the decrease in elastic wave velocities that has been observed to occur near fault zones after strong earthquakes, in surficial sediments during strong ground motion, and in structures during earthquake excitation.

Maximizing energy transfer in vibrofluidized granular systems

Using discrete particle simulations validated by experimental data acquired using the positron emission particle tracking technique, we study the efficiency of energy transfer from a vibrating wall to a system of discrete, macroscopic particles. We demonstrate that even for a fixed input energy from the wall, energy conveyed to the granular system under excitation may vary significantly dependent on the frequency and amplitude of the driving oscillations. We investigate the manner in which the efficiency with which energy is transferred to the system depends on the system variables and determine the key control parameters governing the optimization of this energy transfer. A mechanism capable of explaining our results is proposed, and the implications of our findings in the research field of granular dynamics as well as their possible utilization in industrial applications are discussed.

Competition between geometrically induced and density-driven segregation mechanisms in vibrofluidized granular systems

The behaviors of granular systems are sensitive to a wide variety of particle properties, including size, density, elasticity, and shape. Differences in any of these properties between particles in a granular mixture may lead to segregation, or "demixing," a process of great industrial relevance. Despite the known influence of particle geometry in granular systems, a considerable fraction of research into these systems concerns only uniformly spherical particles. We address, for the case of vertically vibrated granular systems, the important question of whether the introduction of differing particle geometries entirely invalidates our existing knowledge based on purely spherical granulates, or whether current models may simply be adapted to account for the effects of particle shape. We demonstrate that while shape effects can indeed influence the dynamical and segregative behaviors of a granular system, the segregative mechanisms associated with particle geometry are decidedly secondary to those related to particle density. The relevant control parameters determining the extent of geometrically induced segregation are established. Finally, a manner in which shape effects may be accounted for in simulations utilizing purely spherical particles is proposed.

Center of mass scaling in three-dimensional binary granular systems

Using a combination of experimental results acquired through positron emission particle tracking and simulational results obtained via the discrete particle method, we determine a scaling relationship for the center of mass height of a vibrofluidized three-dimensional, bidisperse granular system. We find the scaling to be dependent on the characteristic velocity with which the system is driven, the depth of the granular bed, and the elasticities of the particles involved, as well as the degree of segregation exhibited by the system and the ratio of masses between particle species. The scaling is observed to be robust over a significant range of system parameters.

Effects of packing density on the segregative behaviors of granular systems

We present results concerning the important role of system packing in the processes of density- and inelasticity-induced segregation in vibrofluidized binary granular beds. Data are acquired through a combination of experimental results acquired from positron emission particle tracking and simulations performed using the discrete particle method. It is found that segregation due to inelasticity differences between particle species is most pronounced in moderately dense systems, yet still exerts a significant effect in all but the highest density systems. Results concerning segregation due to disparities in particles' material densities show that the maximal degree to which a system can achieve segregation is directly related to the density of the system, while the rate at which segregation occurs shows an inverse relation. Based on this observation, a method of minimizing the time and energy requirements associated with producing a fully segregated system is proposed.

Influence of thermal convection on density segregation in a vibrated binary granular system

Using a combination of experimental results and discrete particle method simulations, the role of buoyancy-driven convection in the segregative behavior of a three-dimensional, binary granular system is investigated. A relationship between convective motion and segregation intensity is presented, and a qualitative explanation for this behavior is proposed. This study also provides an insight into the role of diffusive behavior in the segregation of a granular bed in the convective regime. The results of this work strongly imply the possibility that, for an adequately fluidized granular bed, the degree of segregation may be indirectly controlled through the adjustment of the system's driving parameters, or the dissipative properties of the system's side-boundaries.

Hydrodynamics of granular gases with a two-peak distribution

Vibrating walls, frequently employed to maintain the temperature (i.e., average velocity) in a granular gas, modify the system strongly, rendering it dissimilar to a molecular one in various aspects. As evidenced by microgravity experiments employing a quasi-two-dimensional (quasi-2D) rectangular box and by 2D simulations, the one-peak velocity distribution is split into two, rendering the stress both nonuniform and anisotropic-without a shear flow and in the absence of gravitation. To account for this, granular hydrodynamics (as first proposed by Haff and later derived employing the kinetic theory) is generalized by introducing two additional variables, with one accounting for the distance between the two peaks and a second for the difference between the average velocities along different directions. The hydrodynamic theory thus generalized relates the velocity distribution to the stress, yielding results that agree with experiments and simulations.

Bottlenecks in granular flow: when does an obstacle increase the flow rate in an hourglass?

Bottlenecks occur in a wide range of situations from pedestrians, ants, cattle, and traffic flow to the transport of granular materials. We examine granular flow across a bottleneck using simulations of monodisperse disks. Contrary to expectations but consistent with previous work, we find that the flow rate across a bottleneck actually increases if an obstacle is optimally placed before it. Using the hourglass theory and a velocity-density relation, we show that the peak flow rate corresponds to a transition from free flow to congested flow, similar to the phase transition in traffic flow.

Horizontal segregation of mono-layer granules coordinated by vertical motion

We experimentally investigate the segregation of a binary mixture of spherical beads confined between two horizontal vertically vibrating plates. The two kinds of beads are of equal diameter and mass but have different restitution coefficients. Segregation occurs in particular ranges of vibration amplitude and frequency. We find that the collisions between beads at an angle to the horizontal plane induce an effective horizontal repulsive force. When one or both bead types bounce up and down in synchronization, the effective repulsive force between the two types of beads is likely to be larger than that found within a single bead type, resulting in the mixture segregating. Non-horizontal collisions also play a role in stabilizing the segregation state by transferring the horizontal kinetic energy back into vertical motion.

Scaling of granular temperature in a vibrated granular bed

Granular temperature underpins the kinetic theory of granular flows as well as models for heat transfer, segregation, erosion, attrition, and aggregation in various granular systems. It is generally thought that granular temperature in vibrated granular systems scales with the square of the peak vibrational velocity. However, careful diffusing wave spectroscopy experiments and statistical analysis of data obtained from these for a three-dimensional vibrated bed of monodisperse glass particles reveals that the granular temperature is also significantly correlated with other vibrational parameters. Reexamination of previously published data obtained by others using alternative methods further supports our thus far unremarked upon observation reported here.

Depletion forces drive polymer-like self-assembly in vibrofluidized granular materials

Ranging from nano- to granular-scales, control of particle assembly can be achieved by limiting the available free space, for example by increasing the concentration of particles ("crowding") or through their restriction to 2D environments. It is unclear, however, if self-assembly principles governing thermally-equilibrated molecules can also apply to mechanically-excited macroscopic particles in non-equilibrium steady-state. Here we show that low densities of vibrofluidized steel rods, when crowded by high densities of spheres and confined to quasi-2D planes, can self-assemble into linear polymer-like structures. Our 2D Monte Carlo simulations show similar finite sized aggregates in thermally equilibrated binary mixtures. Using theory and simulations, we demonstrate how depletion interactions create oriented "binding" forces between rigid rods to form these "living polymers." Unlike rod-sphere mixtures in 3D that can demonstrate well-defined equilibrium phases, our mixtures confined to 2D lack these transitions because lower dimensionality favors the formation of linear aggregates, thus suppressing a true phase transition. The qualitative and quantitative agreement between equilibrium and granular patterning for these mixtures suggests that entropy maximization is the determining driving force for bundling. Furthermore, this study uncovers a previously unknown patterning behavior at both the granular and nanoscales, and may provide insights into the role of crowding at interfaces in molecular assembly.

Spiral trajectory in the horizontal Brazil nut effect

An intruder to a group of identical beads contained in a circular plate which is subjected to a circular vibration will trace approximately a cyclic spiral. The trajectory is a result of both migration and rotation. The intruder migrates in the radial direction while rotating with a constant speed with respect to the center of mass of the whole group of beads. The rotation velocity is due to friction between the beads and the container wall and determined by the vibration amplitude and the number of beads. The migration direction is dependent on the size ratio and mass ratio of the intruder to the background beads. The migration speed is constant for the outward migration, but decreases gradually when the intruder migrates toward the center of mass of the whole cluster for the inward case.

Event-driven Brownian dynamics for hard spheres

Brownian dynamics algorithms integrate Langevin equations numerically and allow to probe long time scales in simulations. A common requirement for such algorithms is that interactions in the system should vary little during an integration time step; therefore, computational efficiency worsens as the interactions become steeper. In the extreme case of hard-body interactions, standard numerical integrators become ill defined. Several approximate schemes have been invented to handle such cases, but little emphasis has been placed on testing the correctness of the integration scheme. Starting from the two-body Smoluchowski equation, the authors discuss a general method for the overdamped Brownian dynamics of hard spheres, recently developed by one of the authors. They test the accuracy of the algorithm and demonstrate its convergence for a number of analytically tractable test cases.

Fluidization of a vertically vibrated two-dimensional hard sphere packing: a granular meltdown

We report measurements of the fluidization process in vertically vibrated two-dimensional granular packings. An initially close packed granular bed is exposed to sinusoidal container oscillations with gradually increasing amplitude. At first the particles close to the free surface become mobile. When a critical value of the forcing strength is reached the remaining crystal suddenly breaks up and the bed fluidizes completely. This transition leads to discontinuous changes in the density distribution and in the root mean square displacement of the individual particles. Likewise the vertical center of mass coordinate increases by leaps and bounds at the transition. It turns out that the maximum container velocity v0 is the crucial driving parameter determining the state of a fully fluidized system. For particles of various sizes the transition to full fluidization occurs at the same value of v 2 0/gd, where d is the particle diameter and g is the gravitational acceleration. A discontinuous fluidization transition is only observed when the particles are highly elastic.

Collision frequencies and energy flux in a dilute granular gas

Recent experimental study of a granular gas fluidized by vibrations in a low gravity environment has reported that the collision frequency nu(rho) of the particles with the container boundary scales roughly like N(alpha) with alpha=0.6 +/- 0.1, where N is the number of particles. Using numerical simulations, we show that this scaling is observed on a wide range of N, both for nu(rho) and for the particle-particle collision frequency nu(c). Simple scaling arguments show that this behavior is related to the energy flux in the granular gas, from injection by the moving boundary to dissipation by inelastic collisions. We predict in the dilute limit that the collision frequencies scale such as (sqrt)N are in fair agreement with experimental measurements.

Granular Leidenfrost effect: experiment and theory of floating particle clusters

Granular material is vertically vibrated in a 2D container: above a critical shaking strength, and for a sufficient number of beads, a crystalline cluster is elevated and supported by a dilute gaseous layer of fast beads underneath. We call this phenomenon the granular Leidenfrost effect. The experimental observations are explained by a hydrodynamic model featuring three dimensionless control parameters: the energy input S, the number of particle layers F, and the inelasticity of the particle collisions epsilon. The (S,F) phase diagram, in which the Leidenfrost state lies between the purely solid and gas phases, shows accurate agreement between experiment and theory.

Fermi-like behavior of weakly vibrated granular matter

Vertical movement of zirconia-yttria stabilized 2 mm balls is measured by a laser facility at the surface of a vibrated 3D granular matter under gravity. Realizations z(t) are measured from the top of the container by tuning the fluidized gap with a 1D measurement window in the direction of the gravity. The statistics obeys a Fermi-like configurational approach which is tested by the relation between the dispersions in amplitude and velocity. We introduce a generalized equipartition law to characterize the ensemble of particles which cannot be described in terms of a Brownian motion. The relation between global granular temperature and the external excitation frequency is established.

Simulations of vibrated granular medium with impact-velocity-dependent restitution coefficient

We report numerical simulations of strongly vibrated granular materials designed to mimic recent experiments performed in both the presence and the absence of gravity. The coefficient of restitution used here depends on the impact velocity by taking into account both the viscoelastic and plastic deformations of particles, occurring at low and high velocities, respectively. We show that this model with impact-velocity-dependent restitution coefficient reproduces results that agree with experiments. We measure the scaling exponents of the granular temperature, collision frequency, impulse, and pressure with the vibrating piston velocity as the particle number increases. As the system changes from a homogeneous gas state at low density to a clustered state at high density, these exponents are all found to decrease continuously with increasing particle number. All these results differ significantly from classical inelastic hard sphere kinetic theory and previous simulations, both based on a constant restitution coefficient.

Local equation of state and velocity distributions of a driven granular gas

We present event-driven simulations of a granular gas of inelastic hard disks with incomplete normal restitution in two dimensions between vibrating walls (without gravity). We measure hydrodynamic quantities such as the stress tensor, density and temperature profiles, as well as velocity distributions. Relating the local pressure to the local temperature and local density, we construct a local constitutive equation. For strong inelasticities the local constitutive relation depends on global system parameters, like the volume fraction and the aspect ratio. For moderate inelasticities the constitutive relation is approximately independent of the system parameters and can hence be regarded as a local equation of state, even though the system is highly inhomogeneous with heterogeneous temperature and density profiles arising as a consequence of energy injection. With respect to local velocity distributions we find that they do not scale with the square root of the local granular temperature. Moreover the high-velocity tails are different for the distribution of the x and the y components of the velocity, and even depend on the position in the sample, the global volume fraction, and the coefficient of restitution.

Anisotropic energy distribution in three-dimensional vibrofluidized granular systems

We examine the energy flows in a three-dimensional model of a granular system consisting of N inelastic hard spheres contained in an open cylinder of radius R under the influence of gravity. Energy is supplied to the system in the vertical direction by a vibrating base and is transferred to the perpendicular directions through particle-particle collisions. We examine how the local and global dissipation of energy by particle-particle and particle-wall collisions depends on the number of particles, the velocity of the vibrating base, and the restitution coefficients.

NMR experiments on a three-dimensional vibrofluidized granular medium

A three-dimensional granular system fluidized by vertical container vibrations was studied using pulsed field gradient NMR coupled with one-dimensional magnetic resonance imaging. The system consisted of mustard seeds vibrated vertically at 50 Hz, and the number of layers N(l)<or=4 was sufficiently low to achieve a nearly time-independent granular fluid. Using NMR, the vertical profiles of density and granular temperature were directly measured, along with the distributions of vertical and horizontal grain velocities. The velocity distributions showed modest deviations from Maxwell-Boltzmann statistics, except for the vertical velocity distribution near the sample bottom, which was highly skewed and non-Gaussian. Data taken for three values of N(l) and two dimensionless accelerations Gamma=15,18 were fitted to a hydrodynamic theory, which successfully models the density and temperature profiles away from the vibrating container bottom. A temperature inversion near the free upper surface is observed, in agreement with predictions based on the hydrodynamic parameter micro which is nonzero only in inelastic systems.

Energy of a single bead bouncing on a vibrating plate: experiments and numerical simulations

The energy of a single bead bouncing on a vibrating plate is determined in simulations and experiments by tracking the bead-plate collision times. The plate oscillates sinusoidally along the vertical with the dimensionless peak acceleration Gamma, and the bead-plate collisions are characterized by the velocity restitution coefficient epsilon. Above the threshold dimensionless peak acceleration Gamma(s) approximately 0.85, which does not depend on the restitution coefficient, the bead energy is shown to initially increase linearly with the vibration amplitude A, whereas it is found to scale like v(2)(p)/(1-epsilon), where v(p) is the peak velocity of the plate, only in the limit Gamma>>Gamma(s). The threshold Gamma(s) is shown to decrease when the bead is subjected, in simulations, to additional nondissipative collisions occurring with the typical frequency nu(c). As a consequence, the bead energy scales like v(2)(p)/(1-epsilon) for all vibration strengths in the limit nu(c)>>nu(*)(c). From the experimental and numerical findings, an analytical expression of the bead energy as a function of the experimental parameters is proposed.

Ratchet-induced segregation and transport of nonspherical grains

We consider through simulations the behavior of elongated grains on a vibrating ratchet-shaped base. We observe differences in layer velocity profile and in net grain velocity for grains that are composed of one, two, or three collinear spheres. In the case of mixtures of different species of grains, we demonstrate layer-by-layer variation in the average velocity as well as layer segregation of species, and show that horizontal separation of the species can be achieved using this geometry. We also find that the addition of a small number of shorter grains to a sample of long grains provides a lubrication effect that increases the velocity of the long grains.

Bifurcations of a driven granular system under gravity

The molecular dynamics study on the granular bifurcation in a simple model is presented. The model consists of hard disks, which undergo inelastic collisions; the system is under the uniform external gravity and is driven by the heat bath. The competition between the two effects, namely, the gravitational force and the heat bath, is carefully studied. We found that the system shows three phases, namely, the condensed phase, the locally fluidized phase, and the granular turbulent phase, upon increasing the external control parameter. We conclude that the transition from the condensed phase to the locally fluidized phase is distinguished by the existence of fluidized holes, and the transition from the locally fluidized phase to the granular turbulent phase is understood by the destabilization transition of the fluidized holes due to mutual interference.

Particle segregation in vibrofluidized beds due to buoyant forces

We utilize two-dimensional discrete element computer simulations to investigate the equilibrium position of an impurity in a vibrofluidized bed of otherwise homogeneous particles. The steady state equilibrium height of the impurity increases with increasing vibration velocity amplitude and decreases with the impurity to the surrounding particle density ratio. A simple model whereby the impurity weight is balanced by a "buoyant" force due to the surrounding particle impacts makes a good prediction of the impurity position.

Stratified horizontal flow in vertically vibrated granular layers

A layer of granular material on a vertically vibrating sawtooth-shaped base exhibits horizontal flow whose speed and direction depend on the parameters specifying the system in a complex manner. Discrete-particle simulations reveal that the induced flow rate varies with height within the granular layer and oppositely directed flows can occur at different levels. The behavior of the overall flow is readily understood once this feature is taken into account.

Granular temperature profiles in three-dimensional vibrofluidized granular beds

The motion of grains in a three-dimensional vibrofluidized granular bed has been measured using the technique of positron emission particle tracking, to provide three-dimensional packing fraction and granular temperature distributions. The mean square fluctuation velocity about the mean was calculated through analysis of the short time mean squared displacement behavior, allowing measurement of the granular temperature at packing fractions of up to eta approximately 0.15. The scaling relationship between the granular temperature, the number of layers of grains, and the base velocity was determined. Deviations between the observed scaling exponents and those predicted by recent theories are attributed to the influence of dissipative grain-sidewall collisions.

Dissipative properties and scaling law for a layer of granular material on a vibrating plate

The dissipative properties and scaling law for a layer of vertically vibrated granular materials were investigated by means of a dynamical model of a single sphere colliding completely inelastically with a massive, oscillating plate. A relationship is presented of how the temperature of the layer scales with the acceleration of the plate and the restitution coefficient of the grains. The numerical calculation shows the existence of an "energy well" and a "temperature well," which could be used to explain the existence of a f/2 flat (where f is the external driving frequency) state in an experiment on vibrated granular material.

Experimental study of solid-liquid-type transitions in vibrated granular layers and the relation with surface waves

From pressure and surface dilation measurements, we show that a solid-liquid-type transition occurs at low excitation frequencies in vertically vibrated granular layers. This transition precedes subharmonic bifurcations from flat surface to standing wave patterns, indicating that these waves are in fact associated with the fluidlike behavior of the layer. At higher frequencies we show that another kind of subharmonic waves can be distinguished. These waves do not involve any lateral transfer of grains within the layer, and correspond to excitations for which the layer slightly bends alternately in time and space. These bending waves have very low amplitude, and we observe them in a vibrated two-dimensional layer of photoelastic particles.

Pattern formation in a vibrated granular layer: the pattern selection issue

We present a numerical study of a surface instability occurring in a bidimensional vibrated granular layer. The driving mechanism for the formation of stationary waves is closely followed. Two regimes of wavelength selection are identified: a dispersive regime where the wavelength decreases with increasing frequency and a saturation regime where the value of the wavelength is a constant depending on the number of grains in the vertical direction. For the dispersive regime an empirical relation is proposed, based on dimensional arguments involving transport properties in the layer. A comparison is made with existing experimental results in two and three dimensions. For the saturation regime, a connection is established between the pattern formation and an intrinsic instability occurring spontaneously in dissipative gases. The observed dependence on the layer height is linked to a detailed dissipation mechanism for the collisions between grains.

Hydrodynamic theory for granular gases

A granular gas subjected to a permanent injection of energy is described by means of hydrodynamic equations derived from a moment expansion method. The method uses as reference function not a Maxwellian distribution f(M) but a distribution f(0)=Phif(M), such that Phi adds a fourth cumulant kappa to the velocity distribution. The formalism is applied to a stationary conductive case showing that the theory fits extraordinarily well the results coming from our Newtonian molecular dynamic simulations once we determine kappa as a function of the inelasticity of the particle-particle collisions. The shape of kappa is independent of the size N of the system.

Potential energy in a three-dimensional vibrated granular medium measured by NMR imaging

Fast NMR imaging was used to measure the density profile of a three-dimensional granular medium fluidized by vertical vibrations of the container. For container acceleration much larger than gravity, the rise in center of mass of the granular medium is found to scale as v(alpha)(0)/N(beta)(l) with alpha = 1.0+/-0.2 and beta = 0.5+/-0.1, where v(0) is the vibration velocity, and N(l) is the number of layers of grains in the container. This value for alpha is significantly less than found previously for experiments and simulations in one dimension ( alpha = 2) and two dimensions ( alpha = 1.3-1.5).

Onset of fluidization in vertically shaken granular material

When granular material is shaken vertically one observes convection, surface fluidization, spontaneous heap formation, and other effects. There is a controversial discussion in the literature as to whether there exists a threshold for the Froude number Gamma=A(0)omega(2)(0)/g, below which these effects cannot be observed anymore. By means of theoretical analysis and computer simulation we find that there is no such single threshold. Instead, we propose a modified criterion that coincides with the critical Froude number Gamma(c)=1 for small driving frequency omega(0).

Self-diffusion of grains in a two-dimensional vibrofluidized bed

The analogy of vibrofluidized granular beds with a thermal gas of hard discs has been tested. Analysis of the mean squared displacement behavior of the grains allowed comparison of the measured diffusion with the predicted value at a particular combination of granular temperature and packing fraction. High speed photography, image analysis, and particle tracking software enabled accurate location of the grains. Appropriate analysis of the three mean squared displacement regimes, ballistic, diffusive, and crossover between the two extremes, allowed both the diffusion coefficient and the granular temperature to be measured at the same packing fraction. Broad agreement between Chapman-Enskog theory relating temperature to self-diffusion and observation was observed up to packing fractions of eta approximately equal to 0.7. At higher packing fractions the grains showed evidence of caging and jump diffusion, with the observed diffusion rapidly diverging from that predicted by theory. Measurement of self-diffusion coefficients and subsequent use of kinetic theory was found to be an accurate method to determine the granular temperature for intermediate packing fractions (eta=0.4-0.6), and would be particularly suitable for those situations where the time resolution of the experimental facility is insufficient to resolve the speed of the grain between collisions.

Velocity statistics in excited granular media

We present an experimental study of velocity statistics for a partial layer of inelastic colliding beads driven by a vertically oscillating boundary. Over a wide range of parameters (accelerations 3-8 times the gravitational acceleration), the probability distribution P(v) deviates measurably from a Gaussian for the two horizontal velocity components. It can be described by P(v) approximately exp(-mid R:v/v(c)mid R:(1.5)), in agreement with a recent theory. The characteristic velocity v(c) is proportional to the peak velocity of the boundary. The granular temperature, defined as the mean square particle velocity, varies with particle density and exhibits a maximum at intermediate densities. On the other hand, for free cooling in the absence of excitation, we find an exponential velocity distribution. Finally, we examine the sharing of energy between particles of different mass. The more massive particles are found to have greater kinetic energy. (c) 1999 American Institute of Physics.

Temperature scaling in a dense vibrofluidized granular material

The leading order "temperature" of a dense two-dimensional granular material fluidized by external vibrations is determined. The grain interactions are characterized by inelastic collisions, but the coefficient of restitution is considered to be close to 1, so that the dissipation of energy during a collision is small compared to the average energy of a particle. An asymptotic solution is obtained where the particles are considered to be elastic in the leading approximation. The velocity distribution is a Maxwell-Boltzmann distribution in the leading approximation. The density profile is determined by solving the momentum balance equation in the vertical direction, where the relation between the pressure and density is provided by the virial equation of state. The temperature is determined by relating the source of energy due to the vibrating surface and the energy dissipation due to inelastic collisions. The predictions of the present analysis show good agreement with simulation results at higher densities where theories for a dilute vibrated granular material, with the pressure-density relation provided by the ideal gas law, are in error.