Discharge of elongated grains from silo with rotating bottom.

We study the flow of elongated grains (wooden pegs of length L=20 mm with circular cross section of diameter d_{c}=6 and 8 mm) from a silo with a rotating bottom and a circular orifice of diameter D. In the small orifice range (D/d<5) clogs are mostly broken by the rotating base, and the flow is intermittent with avalanches and temporary clogs. Here d≡(3/2d_{c}^{2}L)^{1/3} is the effective grain diameter. Unlike for spherical grains, for rods the flow rate W clearly deviates from the power law dependence W∝(D-kd)^{2.5} at lower orifice sizes in the intermittent regime, where W is measured in between temporary clogs only. Instead, below about D/d<3 an exponential dependence W∝e^{κD} is detected. Here k and κ are constants of order unity. Even more importantly, rotating the silo base leads to a strong-more than 50%-decrease of the flow rate, which otherwise does not depend significantly on the value of ω in the continuous flow regime. In the intermittent regime, W(ω) appears to follow a nonmonotonic trend, although with considerable noise. A simple picture, in terms of the switching from funnel flow to mass flow and the alignment of the pegs due to rotation, is proposed to explain the observed difference between spherical and elongated grains. We also observe shear-induced orientational ordering of the pegs at the bottom such that their long axes in average are oriented at a small angle 〈θ〉≈15^{∘} to the motion of the bottom.

Granular flow from silos with rotating orifice.

For dry granular materials falling through a circular exit at the bottom of a silo, no continuous flow can be sustained when the diameter D of the exit is less than five times the characteristic size of the grains. If the bottom of the silo rotates horizontally with respect to the wall of the silo, finite flow rate can be sustained even at small D. We investigate the effect of bottom rotation to the flow rate of monodisperse plastic beads of d=6mm diameter from a cylindrical silo of 19 cm inner diameter. We find that the flow rate W follows Beverloo law down to D=1.3d and that W increases with the rotation speed ω in the small exit regime. If the exit is at an off-center distance R from the axis of the silo, W increases with the rate of area swept by the exit. On the other hand, when the exit diameter is large, W decreases with increasing ω at small ω but increases with ω at large ω. Such nonmonotonic dependence of flow rate on rotation speed may be explained as a gradual change from funnel flow to mass flow due to the shear at the bottom of the silo.

Effect of hopper angle on granular clogging.

We present experimental results of the effect of the hopper angle on the clogging of grains discharged from a two-dimensional silo under gravity action. We observe that the probability of clogging can be reduced by three orders of magnitude by increasing the hopper angle. In addition, we find that for very large hopper angles, the avalanche size (〈s〉) grows with the outlet size (D) stepwise, in contrast to the case of a flat-bottom silo for which 〈s〉 grows smoothly with D. This surprising effect is originated from the static equilibrium requirement imposed by the hopper geometry to the arch that arrests the flow. The hopper angle sets the bounds of the possible angles of the vectors connecting consecutive beads in the arch. As a consequence, only a small and specific portion of the arches that jam a flat-bottom silo can survive in hoppers.

Flow and clog in a silo with oscillating exit.

When grains flow out of a silo, flow rate W increases with exit size D. If D is too small, an arch may form and the flow may be blocked at the exit. To recover from clogging, the arch has to be destroyed. Here we construct a two-dimensional silo with movable exit and study the effects of exit oscillation (with amplitude A and frequency f) on flow rate, clogging, and unclogging of grains through the exit. We find that, if exit oscillates, W remains finite even when D (measured in unit of grain diameter) is only slightly larger than one. Surprisingly, while W increases with oscillation strength Γ≡4π^{2}Af^{2} as expected at small D, W decreases with Γ when D≥5 due to induced random motion of the grains at the exit. When D is small and oscillation speed v≡2πAf is slow, temporary clogging events cause the grains to flow intermittently. In this regime, W depends only on v-a feature consistent to a simple arch breaking mechanism, and the phase boundary of intermittent flow in the D-v plane is consistent to either a power law: D∝v^{-7} or an exponential form: D∝e^{-D/0.55}. Furthermore, the flow time statistic is Poissonian whereas the recovery time statistic follows a power-law distribution.

Coupled Leidenfrost states as a monodisperse granular clock.

Using an event-driven molecular dynamics simulation, we show that simple monodisperse granular beads confined in coupled columns may oscillate as a different type of granular clock. To trigger this oscillation, the system needs to be driven against gravity into a density-inverted state, with a high-density clustering phase supported from below by a gaslike low-density phase (Leidenfrost effect) in each column. Our analysis reveals that the density-inverted structure and the relaxation dynamics between the phases can amplify any small asymmetry between the columns, and lead to a giant oscillation. The oscillation occurs only for an intermediate range of the coupling strength, and the corresponding phase diagram can be universally described with a characteristic height of the density-inverted structure. A minimal two-phase model is proposed and a linear stability analysis shows that the triggering mechanism of the oscillation can be explained as a switchable two-parameter Andronov-Hopf bifurcation. Numerical solutions of the model also reproduce similar oscillatory dynamics to the simulation results.

Biased Brownian motion in narrow channels with asymmetry and anisotropy.

We study Brownian motion of a single millimeter size bead confined in a quasi-two-dimensional horizontal channel with built-in anisotropy and asymmetry. Channel asymmetry is implemented by ratchet walls while anisotropy is introduced using a channel base that is grooved along the channel axis so that a bead can acquire a horizontal impulse perpendicular to the longitudinal direction when it collides with the base. When energy is injected to the channel by vertical vibration, the combination of asymmetric walls and anisotropic base induces an effective force which drives the bead into biased diffusive motion along the channel axis with diffusivity and drift velocity increase with vibration strength. The magnitude of this driving force, which can be measured in experiments on a tilted channel, is found to be consistent with those obtained from dynamic mobility and position probability distribution measurements. These results are explained by a simple collision model that suggests the random kinetic energy transfer between different translational degrees of freedom may be turned into useful work in the presence of asymmetry and anisotropy.

Torque generation through the random movement of an asymmetric rotor: A potential rotational mechanism of the γ subunit of F(1)-ATPase.

The rotation of the γ subunit of F(1)-ATPase is stochastic, processive, unidirectional, reversible through an external torque, and stepwise with a slow rotation. We propose a mechanism that can explain these properties of the rotary molecular motor, and that can determine the direction of rotation. The asymmetric structures of the γ subunit, both at the tip of the shaft (C and N termini) and at the part (ε subunit) protruding from the α(3)ß(3) subunits, are critical. The torque required for stochastic rotation is generated from the impulsive reactive force due to the random collisions between the γ subunit and the quasihexagonal α(3)ß(3) subunits. The rotation is the result of the random motion of the confined asymmetric γ subunit. The steps originate from the chemical reactions of the γ subunit and physical interaction between the γ subunit and the flexible protrusions of the α(3)ß(3) subunits. An external torque as well as a configurational modification in the γ subunit (the central rotor) can reverse the rotational direction. We demonstrate the applicability of the mechanism to a macroscopic simulation system, which has the essential ingredients of the F(1)-ATPase structure, by reproducing the dynamic properties of the rotation.

Boltzmann distribution in a nonequilibrium steady state: measuring local potential by granular Brownian particles.

We investigate experimentally the steady state motion of a millimeter-sized granular polyhedral object on vertically vibrating platforms of flat, conical, and parabolic surfaces. We find that the position distribution of the granular object is related to the shape of the platform, just like that of a Brownian particle trapped in a potential at equilibrium, even though the granular object is intrinsically not at equilibrium due to inelastic collisions with the platform. From the collision dynamics, we derive the Langevin equation which describes the motion of the object under an effective potential that equals the gravitational potential along the platform surface. The potential energy is found to agree with the equilibrium equipartition theorem while the kinetic energy does not. Furthermore, the granular temperature is found to be higher than the effective temperature associated with the average potential energy, suggesting the presence of heat transfer from the kinetic part to the potential part of the granular object.

Entropic force on granular chains self-extracting from one-dimensional confinement.

The entropic forces on the self-retracting granular chains, which are confined in channels with different widths, are determined. The time dependence of the length of chain remaining in the channel Lin(t) is measured. The entropic force is treated as the only parameter in fitting the solution of the nonlinear equation of motion of Lin(t) to the experimental data. The dependence of the entropic force on the width of the confining channel can be expressed as a power-law with an exponent of 1.3, which is consistent with the previous theoretical predictions for the entropy loss due to confinement.

Force generation by granular chains moving randomly on periodic ratchet plates.

A variation of the Brownian ratchet mechanism for the force generated by the combination of the random motion and the ratchet structure is proposed and simulated with granular chains moving randomly on periodic ratchet plates. The present mechanism differs from the flashing ratchet model of the kinesin-microtubule molecular motor. When the bead chain bounces against the periodic ratchet, the chain as a whole will gain an impulse in the direction of the long side (the side with smaller slope). The observed behaviors of the simulating system, including (i) the force-velocity relation, (ii) the stall force as a function of the number of chains, (iii) the increase of velocity with the excitation, and (iv) the appearance of steps at low velocity and its distribution function, are similar to the corresponding ones of the kinesin-microtubule system.

Collapse kinetics of vibrated granular chains.

The kinetics of the collapse of the coil state into condensed states is studied with vibrated granular chain composed of N metal beads partially immersed in water. The radius of gyration of the chain, R(g) is measured. For short chains (N < 140), disk-like condensed state is formed and R(g) decreases with time such that the function ΔR(g)(2) (≡ R(g)(2) - R(g)(2)(∞)) = A e(-t/τ), where the relaxation time τ follows a power-law dependence on the chain length N with an exponent γ = 1.9 ± 0.2. For the chains with length N ≥ 300, rod-like clusters are observed during the initial stage of collapse and R(g)(2) = R(g)(2)(0) - Bt(ß), with ß = 0.6 ± 0.1. In the coarsening stage, the exponential dependence of ΔR(g)(2) on time still holds, however, the relaxation time τ fluctuates and has no simple dependence on N. Furthermore, the time dependence of the averaged radius of gyration of the individual clusters, R(g,cl) can be described by the theory of Lifshitz and Slyozov. A peak in the structure function of long chains is observed in the initial stage of the collapse transition. The collapse transition in the bead chains is a first order phase transition. However, features of the spinodal decomposition are also observed.

Granular dynamics of a slurry in a rotating drum.

We study the effects of interstitial fluid viscosity on the rates of dynamical processes in a thin rotating drum half-filled with monodisperse glass beads. The rotating speed is fixed at the rolling regime such that a continuously flowing layer of beads persists at the free surface. While the characteristic speed of a bead in the flowing layer decreases with the fluid viscosity µ, the mixing rate of the beads is found to increase with µ. These findings are consistent to a simple model related to the thickness of the flowing layer. In addition, our results indicate a possible transition from the inertial limit regime to the viscous limit regime (reported previously by S. Courrech du Pont [Phys. Rev. Lett. 90, 044301 (2003)]) when the Stokes number is reduced.

Traveling shock front in quasi-two-dimensional granular flows.

While the density profile of a granular shock front can be obtained by the conventional treatment of supersonic fluids, its temperature profile is very different from that in ordinary shocks. We study the density and temperature profiles of a traveling granular shock generated by piling up metal spheres in a closed bottom quasi-two-dimensional channel. We successfully account for the temperature profile in the granular shock using a simple kinetic theory in terms of energy transfer from the mean flow direction to the transverse direction. Contrary to ordinary fluids and previous granular shock experiments, the granular shock width is found to increase with the inflow rate.

Unusual diffusion in a quasi-two-dimensional granular gas.

We have studied diffusion in a quasi-two-dimensional granular gas composed of plastic balls confined in a vertically vibrating thin box. The horizontal motion of the balls in the box is found to follow the Langevin equation with the top and bottom of the box acting on the balls with a viscous drag like that in a fluid. Surprisingly, we find that both the granular temperature and the diffusion constant increase with the number of balls (N) in the box for small N . The unusual diffusion can be explained by a two-state model, in which a ball is in contact with two effective temperature baths due to collisions with the top or bottom of the box and collisions with other balls.

Jamming transition in two-dimensional hoppers and silos.

Jamming of monodisperse metal disks flowing through two-dimensional hoppers and silos is studied experimentally. Repeating the flow experiment M times in a hopper or silo (HS) of exit size d, we measure the histograms h(n) of the number of disks n through the HS before jamming. By treating the states of the HS as a Markov chain, we find that the jamming probability J(d), which is defined as the probability that jamming occurs in a HS containing m disks, is related to the distribution function F(n) is identical with (1/M) sigma(s=n) to (s=infinity) h(s) by J(d) = 1 - F(m) = 1 - e(-(alpha(m - n(o))). The decay rate alpha, as a function of d, is found to be the same for both hoppers and silos with different widths. The average number of disks N is identical with 1/alpha = [n] passing through the HS can be fitted to N = A e(Bd2), N = A e(B/(d(c) - d))), or N = A (d(c) - d)(-gamma). The implications of these three forms for N to the stability of dense flow are discussed.

Jamming pattern in a two-dimensional hopper.

We perform granular flow experiments using metal disks falling through a two-dimensional hopper. When the opening of the hopper d is small, jamming occurs due to formation of an arch at the hopper opening. We study the statistical properties of the horizontal component X and the vertical component Y of the arch vector that is defined as the displacement vector from the center of the first disk to the center of the last disk in the arch. As d increases, the distribution function of X changes from a steplike function to a smooth function while that of Y remains symmetrical and peaked at Y=0. When the arch vectors are classified according to the number of disk n in the arch, the mean value is found to increase with d. In addition, the horizontal component X(n) and the absolute value of the vertical component /Y(n)/ in each class have mean values increasing with n. Regarding the arch as a trajectory of a restricted random walker, we derive an expression for the probability density function a(n)(X) of forming an n-disk arch. The statistics (,, and the fraction g(d)(n) of n-disk arches) of the arches generated by a(n)(X) agree with those found in the experiment.