Self-Organized Density Relaxation by Tapping

Transcription

1 Self-Organized Density Relaxation by Tapping Anthony D. Rosato, Oleksandr Dybenko, David J. Horntrop *, Vishagan Ratnaswamy, and Lou Kondic * Granular Science Laboratory, Department of Mechanical Engineering *Department of Mathematical Sciences and Center for Applied Mathematics and Statistics New Jersey Institute of Technology, Newark, NJ Abstract The density relaxation phenomenon is modeled using both Monte Carlo and discrete element simulations to investigate the effects of regular taps applied to a vessel having a planar floor filled with monodisperse spheres. Results suggest the existence of a critical tap intensity which produces a maximum bulk solids fraction. We find that the mechanism responsible for the relaxation phenomenon is evolving quasi-ordered packing structure propagating upwards from the plane floor. PACS Numbers: n; b; Ns; Rm 1. Introduction Density relaxation describes the phenomenon in which granular solids undergo an increase in bulk density as a result of properly applied external loads. The ability of granular materials to undergo density changes is an inherent property that is not wellunderstood, and thus it remains a critical impediment in developing predictive models of flowing bulk solids. From a historical perspective, studies on density relaxation have their basis in the extensive literature on the packing of particles [1], where the concern was often in characterizing loose and dense random structures. 1

2 The focus of early studies on density relaxation was on how optimal packings could be produced through the use of continuous vibrations or discrete taps [2-4]. Recently, there has been a resurgence of interest (partly triggered by experiments [5-8]) in uncovering particle-level mechanisms responsible for observed macroscopic behavior. These experimental results have spurred theoretical studies involving free volume arguments [9], parking lot paradigms [10-13], and stroboscopic decay approaches [14, 15]. Computational investigations involving stochastic and deterministic simulations [16-22] have also been carried out to directly address the coupling between the detailed particle dynamics and bulk behavior. In this regard, we note the single particle and collective dynamics [17], jump and push filling mechanisms identified in [23], and the appearance of local structural order evidenced by a second peak in the radial distribution function [12]. Important progress related to density relaxation has also been made in characterizing glass and jamming transitions in granular materials [24]. The results reported in this paper provide significant new insight regarding the process of density relaxation in granular materials exposed to discrete taps imposed by the motion of a flat plane floor. The process is modeled through (1) hard-sphere Monte Carlo (MC) simulations, and (2) discrete element simulations (DE) using inelastic, frictional spheres. Although these particle-based methods are quite different (one purely stochastic and the other involving deterministic dynamics), both reveal a clear picture of the dynamical process responsible for density relaxation, namely, the upward progression of self-organized layers induced by the plane floor as the taps evolve. Indeed, its occurrence in both the MC and DE simulations suggests the universality of this mechanism in density relaxation which, to our knowledge, has not been previously 2

3 reported in the literature. We also unambiguously demonstrate the importance of the nature of the applied tap in the temporal evolution of these granular systems. In particular, we highlight the essential difference in behavior when the system is subjected to continuous vibrations as opposed to discrete taps. The remainder of this paper is organized as follows. We first describe our MC simulation model (reported in detail elsewhere [25]), which is based on a modification of the standard Metropolis scheme [26]. In this approach, the effect of taps applied to a containment vessel filled with spherical particles of diameter d is idealized by applying a normalized vertical lift Δ d γ of the particle assembly from the floor. We demonstrate the importance of incorporating this separation (as compared to employing a uniform spatial expansion) to dislodge metastable configurations in attaining an increase in density. We find evidence of a critical tap intensity that optimizes the evolution of packing density. Next, we discuss our DE model and simulations, the results of which reveal the same tap-induced ordering effect of the floor on the local microstructure, provided that the amplitude and frequency of the tap is large enough. 2. Monte Carlo Simulation Model The computational domain used for the simulations in this paper is a box having a solid plane floor, and periodic boundary conditions in the lateral directions (x and z; 12d x 12d). These periodic conditions are used to eliminate the ordering effect of the side walls. Spheres that are at the outset randomly placed in the box, settle under gravity to a loose assembly, which will be termed a poured assembly. (See Fig. 1a for a depiction). The poured assembly is then subjected to a series of taps of intensity γ. A typical assembly after the application of several taps is depicted in Fig. 1b. 3

4 In our MC simulation, a single step in the algorithm consists of the random selection of a particle x = ( x, y, z ) i i i i, followed by its assignment to a trial position x = x + δ(1 2 î ), where î is a random vector with each component sampled from a i i uniform distribution on (0,1), and δ is a maximum allowed displacement. The trial position is accepted unconditionally (provided that an overlap does not occur) as the new location if the change in the system energy ΔE mg ( y i y ) 0. Otherwise, if the change in energy is positive ( ΔE > 0 ), the trial position is accepted with probability N i= 1 i E e βδ. For the macroscopic particles under study, β is very large so that the likelihood of an accepted upward displacement is small. Another particle is then selected at random, and the above procedure is repeated. As this settling process advances through many thousands of MC steps, the mean free path decreases resulting in a drastically slowed down rate of approach to a local equilibrium. Therefore, the parameter δ is modified every 10 4 MC steps in accordance to δ = δ if fewer than half of these steps are accepted. The effect of a single tap applied to a containment vessel filled with spherical particles of diameter d is idealized by lifting the entire particle assembly from the floor by an amount Δ, whose normalized value is γ Δ d. Our assumption here is that a tap is sufficiently energetic to cause a small separation of the assembly from the floor. That is, physically γ represents the lift that a granular mass will experience as it is energetically tapped. Within a single tap n of intensity γ, we monitored the bulk solids fraction ν ( n; γ) (i.e., the fraction of volume occupied by spheres) at MC step j every 10 6 steps. The j 4

5 settling process was terminated when the difference in the bulk solids fractions ν ( n ; γ ) ν ( n ; γ ) < 0.001, k = 0, 1, 2,. This protocol was validated by 6 6 ( k + 1)10 10 k enforcing the latter criterion twice to ensure that premature termination could not occur. 2.1 Monte Carlo Results In all cases reported in this letter, poured assemblies (consisting of 3,456 particles) with solids fractions in the range 0.56 to 0.58 filled the periodic box to a depth of approximately 22d. For each tap n, we computed the ensemble averaged bulk solids fraction over M realizations, and its standard deviation where.the equilibrium bulk solids fraction was found by increasing the number of realizations over a sufficient number of taps N T, until the condition was satisfied. Our MC tapping procedure differs from that used in [17], where vertical positiondependent displacements of the particles ( ) were applied in sync with random lateral perturbations. In an earlier paper [25] we reported that statistically equivalent solids fractions were obtained with or without application of these random lateral displacements. Furthermore, for the packings (~22d high) reported here, application of taps consisting of the vertical position-dependent displacements produced little or no increase in solids fraction in the upper portion of the bed, so that the bulk solids fraction remained relatively low. We attribute this behavior to an overly aggressive vertical displacement of the particles with distance from the floor, so that after each tap, the 5

6 system tends to lose memory of its previous microstructure. Physically, this corresponds to the situation in which vigorous taps or shakes are applied so that the system relaxes to nearly the same or lower bulk density. As further evidence of the importance of the choice of model for a tap in the Monte Carlo simulations, we considered a system in which exactly the same potential energy was introduced for both types of tapping. Fig. 2 compares these results; a significantly higher densification rate occurred when the entire particle assembly was raised from the floor, further demonstrating that the nature of the tap plays a crucial role in the MC simulation. In fact, extended studies carried out using the position-dependent tap model revealed that little or no significant increase in bulk density could be attained even after the application of many thousands of taps. In what follows, the remainder of the MC results will employ the lift model ( γ Δ d ) for a tap. The simulated mean coordination number (i.e., average number of contacts per particle) versus solids fraction is shown in Fig. 2. The behavior of these simulation results is in reasonably good agreement with the experimental measurements of Aste et al. [27], showing the same trend over a range of ν. Since the error of the experimental results was not reported, it is impossible to judge the significance of the slight shift of the two lines shown in Fig. 2. We found further corroboration (reported in [25]) of the simulation results in the fit of the solids fraction evolution data at low tap intensities to the inverse log phenomenological law reported in [5-8]. A series of case studies for was carried out, from which the corresponding bulk solids fractions were determined. Graphs of versus the number of taps N T at are shown in Fig. 4. The dependency of the densification rate on γ can be seen in the inset of the figure - for, the equilibrated 6

7 value was achieved quite rapidly requiring only taps. Although we were unable to run to equilibrium for smaller γ ( ) because of the very large number of taps required, the inset suggests the existence of a critical intensity that produces an optimal dense packing. To further explore this hypothesis, the simulation process was reversed by starting with a hexagonal crystal structure ( ), which was subsequently tapped at the same intensities ( ) previously used. After only a few taps, the dense crystal structure is disrupted thereby causing a significant reduction in solids fraction. This is shown in the inset of Fig. 5a. After many thousands of taps, the solids fraction graphs flatten as shown for intensity values γ [0.2, 0.25] of Fig. 5a. For large intensities, equilibration is attained for less than 10 4 taps. The results of this study which are summarized in Fig. 5b reveal a trend very similar to that visible in Fig. 4. That is, the graph peaks at roughly the same range of intensity values as in Fig. 4. This finding regarding the existence of a critical intensity is also supported by experiments reported elsewhere [28], and by the conjecture [29] that bulk density is related to impact velocity. Fig. 6 illustrates the mechanism responsible for density relaxation in our Monte Carlo model namely, the upward advance of self-organized layers induced by the plane floor. The progression is quantified by the ensemble-averaged fraction of the total number of sphere centers n as function of the distance y/d from the floor. For the untapped poured system, n (Fig. 6a) is uniform except for a well-known ordering [30, 31] of the first few layers adjacent to the floor. As the tapping proceeds, there is an upward advance of the layering: after 50 and 250 taps (Figs. 6b,c), peaks in n have formed near the floor, while at 90,000 taps (Fig. 6d), the peaks appear throughout the depth indicating organization 7

8 present throughout the system. We remark that experimental evidence on the influence of the floor in promoting this ordering was reported in the Ph.D. dissertation of Philippe [32]. A growth in the magnitude of the peaks as the taps evolve is a manifestation of the gradual filling of voids. The locations of the first four peaks correspond exactly to a hexagonal close-packed structure. The increase from ν o = to < ν > = is accompanied by an (approximate) 2d reduction of the bed s effective height from its initial level. In Fig. 6d one can also see smaller secondary peaks in the upper side of the bed, showing existence of (yet) an imperfect structure there. These secondary peaks gradually vanish as the tapping progresses. While the results of the MC simulations are interesting and novel on their own right, we also carry out discrete element simulations, which could be argued to be closer to a realistic physical system. As we will see, the results of these studies exhibit the same qualitative behavior observed in the MC simulations, confirming universality of the results. 3. Discrete Element Model and Simulations The discrete element method is based on the numerical solution of Newton s laws for a set of discrete, interacting particles. A recent and detailed review on this method and applications can be found in [33, 34]. In our study, particles were inelastic, frictional, monodisperse spheres obeying binary, soft-sphere interactions [35] in which normal and tangential impulses are functions of a small overlap between colliding particles. Energy loss in the normal direction (i.e., along center line of contacting spheres) is produced by linear loading (K 1 ) and unloading springs (K 2 ), corresponding to a constant restitution 8

9 coefficient e = K1 K2. This model has been shown to reproduce the nearly linear loading behavior for a spherical surface that experiences plastic deformation of the order of 1% of a particle diameter. In the tangential direction, a Mindlin-like model [36] is used in which tangential stiffness, given by K T [ 1 ( ) µ ] / [ 1 ( ) µ + ] 1/ 3 K T T o N T, increasing T = 1 3, (where K o = 0.8K1 ), decreases with Ko T T N T, decreasing T tangential displacement until full sliding occurs at the friction limit µ [37]. The computational domain is again a laterally periodic box (12d x 12d) in which particles which are randomly placed at t = 0 within the box settle under gravity to a fill height of approximately 22 particle diameters. We note that the number of particles in these studies was the same as for the MC simulations. By tuning the value of the friction coefficient µ, it is possible to obtain different poured solids fractions. This is shown in the simulation results summarized in Fig. 7, which were obtained by averaging twenty realizations. For large friction coefficients, less dense systems are formed after pouring due to the formation of bridges and arches within the structure. For smooth particles (µ = 0), the bulk solids fractions obtained are in good agreement with the value ( ± ) normally attributed to a random packing of spheres [38]. For the remainder of the discussion, we restrict attention to spheres for which e = 0.9 µ = 0.1 and ρ = 1200 kg/m 3. These properties are close to those measured by Louge [39] for acrylic spheres (i.e., µ = ± , e = ± ). It is understood that these material properties may also have a significant effect on system behavior. This will be the subject of a future paper. 9

10 3.1 Ensemble-Average Bulk Density Evolution We find that for poured configurations which are statistically indistinguishable with regard to their initial bulk solids fractions and distributions of nearest neighbors and free volume, significant variations in the solids fraction evolutions were observed. This is not surprising since intuition suggests that it is the details of the microstructure that have a dominant influence on how the system advances to a dense configuration. We demonstrate this with an ensemble of 20 poured realizations, For each realization, the Voronoi diagram was constructed, from which its distribution of nearest neighbors (i.e., coordination number distribution) was obtained. The distributions for all realizations were individually fit ( R ) to a normal form 2 1 n a f = exp, from which a = ± and b = ± b π b Although not shown here, each individual distribution was statistically indistinguishable from that of the ensemble-averaged coordination number (R 2 = , b = 7.271, and c = 1.773) given in Fig. 8. The free volume is defined as the ratio of the volume of a sphere to that of the encapsulating polyhedron in the Voronoi diagram. Although not shown in this paper, we also observed very little visual difference between the free volume distributions of the ensemble of 20 realizations, which were quite similar (both qualitatively and quantitatively) to the ensemble-averaged distribution given in Fig. 9. Thus, from the perspective of bulk density, coordination number and free volume distribution, each realization in the ensemble was very similar. And yet, as will be demonstrated, the bulk solids fraction trajectories of the realizations were rather different. 10

11 We applied taps to each realization for the same time duration using identical vibration parameters, i.e., a / d = and f = 7.5 Hz, corresponding to Γ = 2. The ensembleaveraged trajectory of the bulk solids fraction is given in Fig. 10a, where the inset shows the standard deviation. In some of the trajectories, rather dramatic jumps or steep gradients in the solids fraction (see Fig. 10b) occurred, which at first glance appeared to be spurious. However, further consideration of these trajectories suggested that the jumps were caused by collective reorganizations of the microstructure that took place over small time durations relative to the time of tapping. The later study demonstrates the sensitive dependence of the trajectories on the details of the microstructure, and the importance of using ensemble averages in these simulations rather than only single realizations. 3.2 Dependence on Tap Amplitude and Frequency After a poured assembly is obtained, the system is tapped by applying harmonic displacement oscillations to the floor consisting of a half-sine wave (amplitude a and period τ), followed by a relaxation time t r of sufficient duration to ensure that upon collapse, a quiescent state of minimal kinetic energy is attained. The occurrence of densification depends on the choice of the tap amplitude as can be seen (Fig. 11) in striking difference between the evolution of bulk solids fraction for Γ = 0.5 (f = 7.5 Hz, a/d = ) and Γ = 2.0 (f = 7.5 Hz, a/d = ). In the later case, the qualitative behavior of n (the number of sphere centers a given distance from the floor) shown in Fig. 12 is comparable to the MC results (Fig. 6); that is, as the taps progress, one observes the formation of a front, separating a saturated relatively dense and quasiordered region (originating at the floor), from the one that is evolving to a higher density. 11

12 Furthermore, after application of hundreds of taps, the effective height of the system is reduced by approximately two particle diameters (analogous to what took place in the MC simulations). In order to further illustrate the influence of a reduced amplitude (at constant frequency) on the development of ordered layers and the eventual increase in bulk density of the system with increasing number of taps, we carefully monitored the complete evolution of particle centers as a function of distance y/d from the floor for two example cases already discussed, i.e., f = 7.5 Hz, a/d = , Γ = 0.5 and (b) f = 7.5 Hz, a/d = Γ = 2.0. The evolution of ensemble averages (taken over 20 realizations) for the full time interval over which the simulations were run are presented in Fig. 13 a,b. (Note that Fig. 12 are slices taken as discrete times from Fig. 13b). The vertical axis ( N i / Nmax ) in the figures represents the number of particle centers N i as a function of the position above the floor, normalized by the maximum number of particles N max that can occupy a layer (i.e., a planar triangular packing structure). We observed that the location of the peaks of approximately the first 8 10 layers compared well with those for an hexagonal closed-packed structure. For the larger amplitude, there is a clear evolution of structure, as opposed to the case with the smaller amplitude where, no definitive organization is present upon completion of thousands of taps. We again emphasize that this behavior supports the finding in the MC simulations of a critical value of the tap amplitude that optimizes the formation of a quasi-ordered microstructure. However, within the context of the DEM simulations, we interpret this amplitude as being related to a peak dilation of the system. Indeed, one can imagine two extremes that engender no microstructure development or appreciable increase in bulk density: (i) an 12

13 aggressive, energetic tap that greatly expands the system so that any existing microstructure is destroyed and upon contraction and, (ii) a low energy tap that merely transmits a stress wave through the contact network, but does not dislodge or disturb the structure to any appreciable degree. Thus, we hypothesize that the densification process is regulated by the degree of dilation, and that ordering of the microstructure occurs within some narrow range of values of dilatations. The results of our current study to explore this conjecture will be reported in a future paper. Of particular relevance to the results reported here are the studies of Zhu et al. [40], who report on experiments and discrete element simulations of the density relaxation process through the application of continuous vibrations. In particular, they employ a procedure in which very dense structures (solids fraction ~ 0.7 and higher) are built up by vibrating sequentially deposited thin poured layers (of approximately one diameter). This procedure differs from our approach, in which the entire system is exposed to taps, resulting its evolution to a dense structure ( ) as in Fig. 10a. Their findings are essentially in agreement with our earlier reported discrete element investigation [28] in that high solids fractions (greater than ~ 0.66) were not found in relatively deep systems through the application of continuous vibrations. This can be seen in the phase portrait of Fig. 14 (redrawn from [28]) for a system of 8000 particles (e = 0.9 and µ = 0.1) within a laterally periodic box (aspect ratio L/d = 25), in which the improvement in solids fraction is mapped as a function of the vibration amplitude ( ) and frequency ( Hz). Distinct regions are visible according to the color scale corresponding to various improvement levels, with a maximum value of only approximately 5%. Although not shown here, results from this 13

14 study also indicated a correlation between these regions with depth profiles of granular temperature, solids fraction and ratio of the lateral to vertical components of the kinetic energy. Fig. 14 may represent a typical phase portrait associated with the density relaxation process, albeit quantitative differences are expected based on the mode of energy input (tapping as opposed to continuous vibrations), and on material properties, aspect ratios and mass overburdens. 4. Summary and Conclusions In this paper, the results of our MC and DE investigations of tapped density relaxation were reported. Good agreement of the MC-generated coordination number versus solids fraction with reported experiments was found. We find compelling evidence of a critical tap intensity that optimizes the evolution of packing density. Both our stochastic (MC) and deterministic (DE) models revealed the same dynamical process responsible for density relaxation, namely, the progression of self-organized layers induced by the plane floor as the taps evolve. Furthermore, our results suggest that the evolution of bulk density is highly dependent on the microstructure and contact network. We expect that this prediction will be of interest to numerous scientific and engineering problems involving self-organization and its relation to the approach to equilibrium. In future work, we will carry out a broad systematic DE study of dilation behavior and its relationship to the relaxed state, including the dependence of the relaxation process on material parameters. 5. Acknowledgements 14

15 The authors thank D. Blackmore, P. Singh and J. T. Jenkins (Cornell University) for discussions of this work. DJH acknowledge partial support from the grant NSF-DMS and LK from the grants NSF-DMS and NSF-DMS Partial computational support was provided by NSF-DMS In addition, a portion of this research was done using resources of the Open Science Grid, which is supported by the National Science Foundation and the U.S. Department of Energy's Office of Science. 6. References [1] D. J. Cumberland, and R. J. Crawford, The Packing of Particles (Elsevier, 1987), Vol. 6. [2] J. E. Ayer, and F. E. Soppet, J. Am. Ceram. Soc. 48, 180 (1965). [3] P. E. Evans, and R. S. Millman, Powder Met. 7, 50 (1964). [4] M. Takahashi, and S. Suzuki, Ceramic Bulletin 65, 1587 (1986). [5] J. B. Knight et al., Phys. Rev. E 51, 3957 (1995). [6] E. R. Nowak et al., Phys. Rev. E 63, (R) (2001). [7] E. R. Nowak et al., Phys. Rev. E. 57, 1971 (1998). [8] E. R. Nowak et al., Powder Technology 94, 79 (1997). [9] T. Boutreux, and P. G. DeGennes, Physica A 244, 59 (1997). [10] K. L. Gavrilov, Phys. Rev. E 58, 2107 (1998). [11] A. J. Kolan, E. R. Nowak, and A. V. Tkachenko, Phys. Rev. E 59, 3094 (1999). [12] J. Talbot, G. Tarjus, and P. Viot, Eur. Phys. J. E 5, 445 (2001). [13] M. Wackenhut, and H. J. Herrmann, Phys. Rev. E 68, (2003). [14] S. J. Linz, Phys. Rev. E 54, 2925 (1996). 15

16 [15] S. Linz, and A. Dohle, Phys. Rev. E 60, 5737 (1999). [16] D. Arsenovic et al., PRE 74, (2006). [17] G. C. Barker, and A. Mehta, Phys. Rev. A 45, 3435 (1992). [18] G. Liu, and K. E. Thompson, Powder Tech 113, 185 (2000). [19] L. F. Liu, Z. P.Zhang, and A. B. Yu, Physica A 268, 433 (1999). [20] M. Nicodemi, and A. Coniglio, Phys. Rev. Letters 82, 916 (1999). [21] M. Nicodemi, A. Coniglio, and H. J. Herrmann, PRE 59, 6830 (1999). [22] S. Remond, and J. L. Galias, Physica A 369, 545 (2006). [23] X. Z. An et al., Phys. Rev. Lett. 95, (2005). [24] M. Nakagawa, and S. Luding, Proceedings of the 6th International Conference on Micromechanics of Granular Media (AIP, Golden, CO, 2009), Vol [25] O. Dybenko, A. D. Rosato, and D. Horntrop, Kona, Powder and Particle 25, 133 (2007). [26] N. Metropolis et al., J. Chem. Phys. 21, 1087 (1953). [27] T. Aste, M. Saadatfar, and T. J. Senden, Phys. Rev. E 71, (2005). [28] N. Zhang, and A. D. Rosato, KONA 24, 93 (2006). [29] J. C. Macrae, W. A. Gray, and P. C. Finlason, Nature 179, 1365 (1957). [30] E. M. Kelly, Powder Technology 4, 56 (1970). [31] T. G. Owe Berg, R. L. McDonald, and R. J. Trainor, Powder Technology 3, 183 (1969/1970). [32] P. Philippe, and D. Bideau, Physical Review Letters 91, (2003). [33] H. P. Zhu et al., Chemical Engineering Science 62, 3378 (2007). [34] H. P. Zhu et al., Chemical Engineering Science 63, 5728 (2008). 16

17 [35] O. R. Walton, and R. L. Braun, Acta Mechanica 63, 73 (1986). [36] R. D. Mindlin, and H. Deresiewicz, J. Appl. Mech. 20, 327 (1953). [37] O. R. Walton, Mechanics of Materials 16, 239 (1993). [38] G. D. Scott, and D. M. Kilgour, British Journal of Applied Physics 2, 863 (1969). [39] M. Y. Louge, in (Cornell University, Ithaca, NY, 1999). [40] A. B. Yu et al., Phys. Rev. Lett. 97 (2006). 17

18 Figure Captions Fig. 1. Snapshot of a system of hard spheres within the periodic box for: (a) an initial, random state and (b) a configuration of the ensemble tapped at for which. Fig. 2. Solids fraction versus tap number for two tap modes: γ = 0.25 and λ = 1.025, each producing the same increase in system potential energy. Fig. 3. Mean coordination number versus solids fraction ν from the MC simulation ( ) and experimental measurements ( ) of Aste et al. [27]. The straight lines are linear regression fits to show the trends of experimental (dashed) and computational (solid) results. Fig. 4. Bulk solids fraction versus the number of taps N T at. The inset shows for. Fig. 5. Bulk solids fraction versus the number of taps N T at, for which the initial state was an hexagonal close-packed structure ( ). The evolution of for is shown in the inset (b). Graph of versus tap intensity for. Fig. 6. MC ensemble-averaged fraction of the total number of sphere centers n as a function of y/d for tap intensity : (a) poured system, ; (b) 50 taps, ; (c) 250 taps, ; (d) 90,000 taps,. 18

19 Fig. 7. A wide range of bulk densities can be obtained by changing particle friction properties. Each point on the graph represents an average taken over 20 discrete element realizations, with the vertical lines representing the deviation from the average. Fig. 8. The points are ensemble-averaged coordination numbers for 20 poured realizations. The solid line is a fit (R 2 = ) to, where. Fig. 9. Ensemble-averaged free volume distribution for 20 poured discrete element realizations. Fig. 10. (a) Evolution of the ensemble-averaged bulk solids fraction over 20 discrete element simulations in which f = 7.5 Hz, a/d = , Γ =2. The inset shows the standard deviation. (b) Several trajectories from the ensemble of 20 realizations. Fig. 11. Evolution of the ensemble-averaged bulk solids fraction (over 20 realizations) for (f = 7.5 Hz, a/d = ) and (f = 7.5 Hz, a/d = ). Fig. 12. Ensemble-averaged (over 20 realizations) distributions of the particle centers measured from the floor (y/d = 0) to the top surface for a/d = , f = 7.5 Hz ( ) at t = 0s, t = 149.3s, t = 299.3s and t = 599.3s. Fig. 13. Evolution of the ensembled-averaged distribution of particle centers for (a) (f = 7.5 Hz, a/d = ) and (b) (f = 7.5 Hz, a/d = ). Fig. 14. The color scale shows the improvement in bulk solids fraction as a function of amplitude ( ) and frequency ( ) for continuously sinusoidal vibrations applied the floor (taken from [28]). 19

20 Figure 1 20

21 Figure 2 21

22 Figure 3 22

23 Figure 4 23

24 Figure 5a 24

25 Figure 5b 25

26 Figure 6 26

27 Figure 7 27

28 Figure 8 28

29 Figure 9 29

30 Figure 10a 30

31 ν = o ν = f ν ν o f = = ν ν o f = = ν = o ν = f Figure 10b 31

32 Figure 11 32

33 t = 0 s Tap 223 t = s Tap 448 t = s t = s Figure 12 33

34 Figure 13a (to be replaced by ensemble-averaged result) 34

35 Figure 13b 35

36 Amplitude (inch) Frequency (Hz) Figure 14 36