Energy input and scaling laws for a single particle vibrating in one dimension

Motion behaviour of ellipsoidal granular system under vertical vibration and airflow

We studied the motion behaviour of ellipsoid particles under vertical vibration and airflow. Three typical convection patterns were observed when submitted to vertical vibration with frequency (f) from 20 Hz to 80 Hz and dimensionless vibration acceleration (Γ) from one to six. We studied the effects of f and Γ on the change of convection patterns. We quantitatively studied the effects of f, Γ, airflow direction, airflow velocity, and particle shape on the convection area and intensity using the area fraction λ and average velocity vz characterizing the convection area and intensity, respectively. Results showed that the convection first occured occurred in the upper part of the granular system. Increasing f and A can both increase the convection area and strengthen the convection intensity. A had a greater influence than f at the same Γ. The wheat particles were more likely to enter the global convection state under the action of the airflow in the opposite direction of gravity. The maximum convection intensity of wheat particles under the airflow in the opposite direction of gravity was approximately 30-35% of the value measured under the airflow along the direction of gravity. The convection area and maximum convection intensity of the spherical particles were approximately 85% and 93% of the measured values for the ellipsoidal particles, respectively. We also analysed the effects of f, Γ, airflow direction, airflow velocity, and particle shape on the convection area on the basis of energy dissipation.

Protocol for assembling micro- and nanoparticles in a viscous liquid above a vibrating plate

In this protocol, we demonstrate the use of a vibrating plate to drive the assembly of micro- and nanoparticles as an approach to high-throughput, large-scale directed assembly in a viscous liquid. Vibration drives the assembly of glass bead microparticles and iron oxide nanoparticles in contact with water over an area of 6400 mm². We use a scaling analysis to show that there is a competition between acoustic radiation force and vibration-generated fluid flow in a viscous medium, which determines particle transport characteristics. For assembly in a viscous liquid, we find close agreement between the observed experimental results when compared to a numerical solution of the 2D wave equation that describes plate displacement. This model indicates that microparticles migrate along displacement gradients towards displacement anti-nodes where the magnitude of displacement is maximum. We also observe that nanoparticles migrate toward displacement nodes where the magnitude of displacement is zero.

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Cost-effective directed assembly technique without the need for microfabrication facilities

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Large-scale assembly produces heterogeneously ordered structures on a vibrating substrate

Response of jammed packings to thermal fluctuations

We focus on the response of mechanically stable (MS) packings of frictionless, bidisperse disks to thermal fluctuations, with the aim of quantifying how nonlinearities affect system properties at finite temperature. In contrast, numerous prior studies characterized the structural and mechanical properties of MS packings of frictionless spherical particles at zero temperature. Packings of disks with purely repulsive contact interactions possess two main types of nonlinearities, one from the form of the interaction potential (e.g., either linear or Hertzian spring interactions) and one from the breaking (or forming) of interparticle contacts. To identify the temperature regime at which the contact-breaking nonlinearities begin to contribute, we first calculated the minimum temperatures T_{cb} required to break a single contact in the MS packing for both single- and multiple-eigenmode perturbations of the T=0 MS packing. We find that the temperature required to break a single contact for equal velocity-amplitude perturbations involving all eigenmodes approaches the minimum value obtained for a perturbation in the direction connecting disk pairs with the smallest overlap. We then studied deviations in the constant volume specific heat C[over ¯]_{V} and deviations of the average disk positions Δr from their T=0 values in the temperature regime T_{C[over ¯]_{V}}<T<T_{r}, where T_{r} is the temperature beyond which the system samples the basin of a new MS packing. We find that the deviation in the specific heat per particle ΔC[over ¯]_{V}^{0}/C[over ¯]_{V}^{0} relative to the zero-temperature value C[over ¯]_{V}^{0} can grow rapidly above T_{cb}; however, the deviation ΔC[over ¯]_{V}^{0}/C[over ¯]_{V}^{0} decreases as N^{-1} with increasing system size. To characterize the relative strength of contact-breaking versus form nonlinearities, we measured the ratio of the average position deviations Δr^{ss}/Δr^{ds} for single- and double-sided linear and nonlinear spring interactions. We find that Δr^{ss}/Δr^{ds}>100 for linear spring interactions is independent of system size. This result emphasizes that contact-breaking nonlinearities are dominant over form nonlinearities in the low-temperature range T_{cb}<T<T_{r} for model jammed systems.

Bottom pressure scaling of vibro-fluidized granular matter

Vibrated granular beds show various interesting phenomena such as convection, segregation, and so on. However, its fundamental physical properties (e.g., internal pressure structure) have not yet been understood well. Thus, in this study, the bottom wall pressure in a vertically vibrated granular column is experimentally measured and used to reveal the nature of granular fluidization. The scaling method allows us to elucidate the fluidization (softening) degree of a vibrated granular column. The peak value of the bottom pressure pm is scaled as [formula in text], where pJ, d, g, ω, H, and Γ are the Janssen pressure, grain diameter, gravitational acceleration, angular frequency, height of the column, and dimensionless vibrational acceleration, respectively. This scaling implies that the pressure of vibrated granular matter is quite different from the classical pressure forms: static and dynamic pressures. This scaling represents the importance of geometric factors for discussing the behavior of vibro-fluidized granular matter. The scaling is also useful to evaluate the dissipation degree in vibro-fluidized granular matter.

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.

Nuclear magnetic resonance measurements of velocity distributions in an ultrasonically vibrated granular bed

We report the results of nuclear magnetic resonance imaging experiments on granular beds of mustard grains fluidized by vertical vibration at ultrasonic frequencies. The variation of both granular temperature and packing fraction with height was measured within the three-dimensional cell for a range of vibration frequencies, amplitudes and numbers of grains. Small increases in vibration frequency were found--contrary to the predictions of classical 'hard-sphere' expressions for the energy flux through a vibrating boundary--to result in dramatic reductions in granular temperature. Numerical simulations of the grain-wall interactions, using experimentally determined Hertzian contact stiffness coefficients, showed that energy flux drops significantly as the vibration period approaches the grain-wall contact time. The experiments thus demonstrate the need for new models for 'soft-sphere' boundary conditions at ultrasonic frequencies.

Driven inelastic-particle systems with drag

We study steady states of the motion of a large number of particles in a closed box that are excited by a vibrating boundary and experience a linear drag force from the interstitial fluid. The dissipation in such systems arises from two main sources: Inelasticity in particle collisions and the effects of interstitial fluid on the particles. In many applications, order of magnitude estimates suggest that the dissipation due to interstitial fluid effects may greatly exceed that due to inelasticity and one is naturally led to neglect inelastic effects. In this study, we show that, if one adopts a linear drag force and inelastic effects are neglected, a steady state only exists when the vibration speed of the boundary is below a critical value. For vibration speeds above this critical value, no steady state exists since the kinetic energy of the particles grows without bound. We show that, for vibration speeds above the critical value, inelastic effects must be included to obtain a steady state even if order of magnitude estimates suggest they are negligible. Numerical simulations confirm these theoretical predictions. We also show that inclusion of apparently small nonlinear drag terms can also play a similar role in preventing the kinetic energy of the particles growing without bound.

Bouncing ball problem: stability of the periodic modes

Exploring all its ramifications, we give an overview of the simple yet fundamental bouncing ball problem, which consists of a ball bouncing vertically on a sinusoidally vibrating table under the action of gravity. The dynamics is modeled on the basis of a discrete map of difference equations, which numerically solved fully reveals a rich variety of nonlinear behaviors, encompassing irregular nonperiodic orbits, subharmonic and chaotic motions, chattering mechanisms, and also unbounded nonperiodic orbits. For periodic motions, the corresponding conditions for stability and bifurcation are determined from analytical considerations of a reduced map. Through numerical examples, it is shown that a slight change in the initial conditions makes the ball motion switch from periodic to chaotic orbits bounded by a velocity strip v=+/-Gamma(1-epsilon) , where Gamma is the nondimensionalized shaking acceleration and epsilon the coefficient of restitution which quantifies the amount of energy lost in the ball-table collision.

Inelastic collapse of a ball bouncing on a randomly vibrating platform

A theoretical study is undertaken of the dynamics of a ball which is bouncing inelastically on a randomly vibrating platform. Of interest are the distributions of the number of flights nf and the total time tauc until the ball has effectively "collapsed," i.e., coalesced with the platform. In the strictly elastic case both distributions have power law tails characterized by exponents that are universal, i.e., independent of the detail of the platform noise distribution. However, in the inelastic case both distributions have exponential tails: P(nf) approximately exp[-theta1nf] and P(tauc) approximately exp[-theta2tauc]. The decay exponents theta1 and theta2 depend continuously on the coefficient of restitution and are nonuniversal; however, as one approaches the elastic limit, they vanish in a manner which turns out to be universal. An explicit expression for theta1 is provided for a particular case of the platform noise distribution.

Stochastic dynamics of a rod bouncing upon a vibrating surface

We describe the behavior of a rod bouncing upon a horizontal surface which is undergoing sinusoidal vertical vibration. The predictions of computer simulations are compared with experiments in which a stainless-steel rod bounces upon a metal-coated glass surface. We find that, as the dimensionless acceleration parameter Gamma is increased appreciably above unity, the motion of a long rod passes from periodic or near-periodic motion into stochastic dynamics. Within this stochastic regime the statistics of the times between impacts follow distributions with tails of approximately Gaussian form while the probability distributions of the angles at impact have tails that are close to exponential. We determine the dependence of each distribution upon the length of the rod, upon frequency, and on Gamma. The statistics of the total energy and of the translational and rotational components each approximately follow a Boltzmann distribution in their tails, the translational and rotational energy components being strongly correlated. The time-averaged mean vertical translational energy is significantly larger than the mean rotational energy, and both are considerably larger than the energy associated with horizontal motion.

Dissipative area-preserving one-dimensional Fermi accelerator model

The influence of dissipation on the simplified Fermi-Ulam accelerator model (SFUM) is investigated. The model is described in terms of a two-dimensional nonlinear mapping obtained from differential equations. It is shown that a dissipative SFUM possesses regions of phase space characterized by the property of area preservation.

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 systems on a vibrating wall: the kinetic boundary condition

Dense granular media, fluidized by a vibrating wall, is studied in the high-vibrating frequency limit, where the plate vibration frequency is much greater than the collision frequency, and the plate acceleration is much greater than gravity. Using kinetic theory, it is shown that, regardless of the fluid density, external field, or restitution coefficient between particles, there is an asymptotic scaling for saying that if Aomega is kept constant, then different amplitudes A (with the corresponding frequencies omega ) produce the same macroscopic result. Furthermore, it is found that in the limit of high frequencies, the boundary condition associated with the vibrating wall can be replaced by a stationary heat source. The value of the heat flux depends linearly with density even for dense fluids. Numerical comparisons with molecular dynamics simulations confirm these predictions and show that the substitution of the vibrating wall by a thermal one gives poor results, while the substitution by a heat source is very accurate.

Collision statistics of driven granular materials

We present an experimental investigation of the statistical properties of spherical granular particles on an inclined plane that are excited by an oscillating side wall. The data is obtained by high-speed imaging and particle tracking techniques. We identify all particles in the system and link their positions to form trajectories over long times. Thus, we identify particle collisions to measure the effective coefficient of restitution and find a broad distribution of values for the same impact angles. We find that the energy inelasticity can take on values greater than one, which implies that the rotational degrees of freedom play an important role in energy transfer. We also measure the distance and the time between collision events in order to directly determine the distribution of path lengths and the free times. These distributions are shown to deviate from expected theoretical forms for elastic spheres, demonstrating the inherent clustering in this system. We describe the data with a two-parameter fitting function and use it to calculate the mean free path and collision time. We find that the ratio of these values is consistent with the average velocity. The velocity distributions are observed to be strongly non-Gaussian and do not demonstrate any apparent universal behavior. We report the scaling of the second moment, which corresponds to the granular temperature, and higher order moments as a function of distance from the driving wall. Additionally, we measure long-time correlation functions in both space and in the velocities to probe diffusion in a dissipative gas.

van der Waals-like transition in fluidized granular matter

A phase separation of fluidized granular matter is presented. Molecular dynamics simulations of a system of grains in two spatial dimensions, with a vibrating wall and without gravity, exhibit the appearance, coalescence, and disappearance of bubbles. By identifying the mechanism responsible for the phase separation, we show that the phenomenon is analogous to the spinodal decomposition of the gas-liquid transition of the van der Waals model. We have deduced a macroscopic model for the onset of phase separation which agrees quite well with molecular dynamics simulations.

Velocity correlations in dense granular gases

We report the statistical properties of spherical steel particles rolling on an inclined surface being driven by an oscillating wall. Strong dissipation occurs due to collisions between the particles and can be tuned by changing the number density. The velocities of the particles are observed to be correlated over large distances comparable to the system size. The distribution of velocities deviates strongly from a Gaussian. The degree of the deviation, as measured by the kurtosis of the distribution, is observed to be as much as four times the value corresponding to a Gaussian, signaling a significant breakdown of the assumption of negligible velocity correlations in a granular system.

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.

Non-gaussian velocity distributions in excited granular matter in the absence of clustering

The velocity distribution of spheres rolling on a slightly tilted rectangular two-dimensional surface is obtained by high speed imaging. The particles are excited by periodic forcing of one of the side walls. Our data suggests that strongly non-Gaussian velocity distributions can occur in dilute granular materials even in the absence of significant density correlations or clustering. When the surface on which the particles roll is tilted further to introduce stronger gravitation, the collision frequency with the driving wall increases and the velocity component distributions approach Gaussian distributions of different widths.