Structural characterization of two-dimensional granular ... - Springer

494 S. Živković etal.event-driven molecular-dynamics algorithm. We employedthe Walton model [23,24] that captures the major featuresof granular interactions. The most realistic simulations usinge.g. soft particle methods [25], allowing the determination offorces and inter-particle contacts, require much more computationalpower when applied to dense granular systems. Itwas shown that the compaction dynamics strongly dependson the material properties of the grains. The organization ofgrains at local level was studied by analyzing the time evolutionof connectivity numbers and through the distribution ofthe Delaunay “free” volumes. Pores have a distribution witha long tail which progressively reduces while the packinggets more compact.In this paper we pay attention to the numerical study ofthe behavior of a granular system exposed to discrete taps.We explore the effects of several factors, e.g., the drivingacceleration and dissipative properties of the grains, on thelocal organization of particles. Analysis at the “microscopic”scale is based on the Voronoï tessellation, which allows usto unambiguously decompose any arbitrary arrangement ofdisks (spheres in 3D) into space-filling set of cells. Our aim isto characterize the structure of disordered disk packings andto quantify the structural changes associated with differentdensities. Voronoï diagrams are used to visualize the structuralorder in packing structures during the compaction process.Quantitative information on the type of order inducedcan be obtained by calculating the shape factor of the Voronoïcells. The shape factor and its distribution was introducedin [26] for tracking the change in structure as a liquid-likesystem approaches a disordered jammed state. The shape factoris a dimensionless measure of deviation of the Voronoïcells from circularity. Distribution of the shape factor clearlyindicates the presence of different underlying substructures(domains) in the packing.The phenomenology of the vibrated granular materialsis very rich, showing many characteristic glassy behaviors.Here we will focus on one of them, namely on the responseof the granular system to sudden perturbations of the “effectivetemperature” given by the vibration intensity. It has beenshown that, during the compaction, the response in the evolutionof the density to a change in the vibration intensity orshear amplitude at a given time, is not determined by the densityat that time [27,28]. Our goal is to study the link betweenthe macroscopic dynamics and the microscopic structure ofthe packing during the short-term memory effect.Finally, we use the simple experimental set-up [29] tostudy the arrangement of granular particles in an experimentallyrealized packings in two dimensions. We image a rectangularregion containing more than 4,500 metallic disks witha resolution of each disk position to better than 0.01 diskdiameters. To validate the simulation results, we measurethe probability distribution of the shape factor experimentallyfor various packing fractions. A quite good agreementbetween the experiment and the simulation is observed whenthe grains are modeled with the values of material parameterscorresponding to the values for hard metal surfaces.In Sec. 2 we summarize the most important features andtechnical details that are relevant to our numerical simulation.A more detailed description of our model can be found inRef. [22]. The results of numerical simulation are presentedin Sec. 3. In Sec. 4 we describe the experimental proceduresand compare the simulation results to the experimental ones.In the last section, we draw some conclusions.2 Simulation methodWe present the event-driven molecular dynamics simulationsin two dimensions on a model system of N = 1,000 monodisperse,frictional hard disks of diameter d and mass m,under the influence of gravitational acceleration g. The systemis spatially periodic in the xy plane (gravitational fieldacts along the negative y direction), with a unit cell of widthL = 1, and is bounded in the y direction by a rough bed atthe bottom and an open top.One “shake cycle” of the granular assembly, which modelsthe effects of a vibration cycle in a real powder is decomposedin two stages. First, the effect of a single tap applied toa disk packing is idealized by lifting the entire disk assemblyfrom the floor proportionally to factor ξ. A disk at height y israised to a new height y ′ = (1 + ξ)y. The parameter ξ in ourmodel plays a similar role as the shaking acceleration Ɣ inreal experiments (Ɣ is defined as the ratio of the peak accelerationof the tap to the gravitational acceleration g). Philippeand Bideau [19] have suggested that the expansion factor ξshould be proportional to the square of the experimentallycontrolled parameter Ɣ.At the beginning of the second phase of the shakecycle, we give a random initial velocity to each disk, insuch a way that the total momentum is equal to zero. Let〈E p 〉 denote the average increment of the potential energyper disk during the vertical expansion of the disk packing.Initial linear and angular velocities are chosen randomlyfrom a uniform distribution in the intervals [−2v T , 2v T ] and[−2ω T , 2ω T ], respectively, where v T = [2〈E p 〉/m] 1/2 ,ω T =[2〈E p 〉/I ] 1/2 , and I is the moment of inertia aboutthe center of disk.The second part of the shake cycle is the formation ofstatic granular pack in the presence of gravity. The packingis compressed and stabilized in a uniaxial external field, usingan event-driven algorithm [30]. In the event driven methodthe disks follow an undisturbed motion, under the influenceof gravity, until either the collision of two disks or the collisionof one disk with the wall occurs. In order to proceedfrom one collision to the next, a search of all upcoming collisions,and the time required for collisions to occur, is made.123