Effect of grain anisotropy on ordering, stability and dynamics in granular systems

C. J. OLSON,¹, C. REICHHARDT,¹, M. MCCLOSKEY² and R. J. ZIEVE^2
1Theoretical and Applied Physics Divisions, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
²Department of Physics, University of California - Davis, California 95616, USA

(received 31 August 2001; accepted in final form 2 January 2002)

PACS. 81.05.Rm - Porous materials; granular materials.
PACS. 45.70.-n - Granular systems.
PACS. 45.50.-j - Dynamics and kinematics of a particle and a system of particles.

Abstract. - With experiments and simulations, we show that monomer, dimer, and trimer beads in 2D represent an ideal system in which specific types of order, dynamics, and granular stability can be controlled. We determine the distinct types of ordering for increasing grain anisotropy. Dimers possess a quasi-ordered state in which translational but not orientational order exists, while trimers are disordered. We show that these different orderings give rise to specific types of flow. Monomers show a large-scale collective ordered shear motion, dimers show tumbling motion near the surface, and trimers show disorderly bulk tumbling.

A fundamental question for granular assemblies and materials is the relation between the mechanical response of the material and the structural ordering of the granular assembly or the shape of the individual grains [1,2]. Although it is well known that the degree of anisotropy can change the structural order, stability, and dynamical response of a granular packing, the exact relationship is not well understood. In addition, it is often difficult to control the properties of granular materials. For example, in the case of rice, dispersion in the shape and size of the individual particles is always present. Due to these issues it is of great interest to find a model system in which specific types of ordering can be controllably prepared and compared to the dynamical response.

In this work we numerically and experimentally study a simple system of granular assemblies confined in a 2D plane in which we demonstrate that specific types of ordering can be controllably produced by tuning the anisotropy of the grains. We also correlate the order of the granular packing with the angle of stability and specific types of dynamic responses. For monomers both translational and orientational order are observed while for dimers a novel quasi-ordered state occurs with translational but not orientational order. For trimers neither orientational nor translational order occur. Our system is relevant to other problems, such as the ordering of atomic dimers on surfaces, frustrated spin models, and the dynamic response of liquid crystals or elongated colloidal particles, as well as glassy and frustrated systems.

Most previous work on the effects of grain geometry has focused on angles of repose for grains that are roughly spherical, but with size dispersion [3], varying grain roughness [4-6], or grain nonsphericity [7-13]. Anisotropic or elongated grains have generally been considered in the context of the flow of feed grains through a hopper [14] or heaps [15]. Only one systematic numerical study has considered the effect of gradual elongation of the grains. In ref. [12], it was observed that as the grain is elongated from a monomer to a dimer, the angle of repose increases monotonically. The grain shape was continuously varied, so it would not be possible to form an ordered lattice with the same lattice constant for all the grain geometries. In addition the specific ordering of the dimers and monomers was not considered, nor was the dynamic response.

In this work we consider with experiments and simulations a system composed of grains that can always be packed into a triangular lattice, but which are individually composed of monomers, dimers, or trimers. Some previous simulation studies have focused on fully three-dimensional systems [16,17]. This introduces considerable complexity to the problem since there are many more degrees of freedom, and it is difficult to determine experimentally the behavior of the grains below the surface of the pile [18]. For this reason, we consider a system constrained to move in two dimensions. Here the microscopic behavior of individual grains can be directly observed experimentally and compared with computer simulations.

Experiment. - We consider a system of steel balls of radius 3.2 mm constrained to move in 2D by two Plexiglas sheets [19]. The steel balls are either used individually, or are spark welded together to form dimers or linear trimers (see inset to fig. 1). All three shapes can tile space in triangular arrays with the same lattice constant. Shapes longer than trimers were susceptible to breakage. The grains are poured into a vertical enclosed area 0.23 m wide by 0.36 m high, and the container is shaken vertically 20 times to prepare an ordered initial starting condition. Monomers and dimers form nearly perfect triangular lattices, but a large amount of disorder remains present in the trimers [19]. Longer shaking times do not produce further ordering. We require the initial configuration to have a horizontal surface level within three balls. The container is slowly tilted at 0.005 rad/s in the plane of the Plexiglas sheets to change the angle between the surface of the granular packing and gravity [3]. We record the angle at which the first grain motion occurs, giving us the maximum angle of stability for the system.

Table of results from experiment and simulation. The angle of stability was obtained by tipping the container and identifying the first motion of the flat granular surface. P_d is the percentage of individual grain elements that do not have six neighbors (0 indicates perfect order).

In fig. 1 we show the appearance of a system of dimers after an avalanche has occurred. The grains remained motionless until the container had been tipped by 60^o. A significant amount of order remains in the system even after the avalanche. The maximum angles of stability measured in this way for grains of differing anisotropy in 18 experiments are listed in Table I. We find the rms lowest angle =43^o3^o for the disordered trimer system. The well-ordered monomers have a slightly higher value of =45^o1^o, which is in reasonable agreement with previous experiments on spherical or nearly spherical grains [3-6]. In contrast dimers display a much larger angle of stability of =62^o10^o. To better understand the stability of the dimer shape, we turn to a simulation.

Simulation. - We consider a system of N_g=705 grains constrained to move in a 2D plane in a system of size 10 x 18 to 60 x 18 cm, with 25 x 28 monomers. We use a granular dynamics method similar to the one employed in [20-22] to integrate the equations of motion for each grain, given by: Here, f_g = -0.0025 is the force of gravity and f_fric=0.3 is the friction between the grain and the plane in which the grains move (representing the Plexiglas plate). Two grains interact only when their relative distance is smaller than the sum of their radii, r_g=0.4 cm. The elastic restoration force between two grains is where k_g=20 is the strength of the restoring spring, 2r_g is the grain diameter, and r_ij=r_j-r_i, the distance between grains located at r_i and r_j. The dissipation force due to the inelasticity of the collision is where =2.4 is a phenomenological dissipation coefficient, m_i=1 g is the grain mass, and v_ij=v_i-v_j is the relative velocity. The shear friction force mimicking solid friction is where =1.2 is the shear friction coefficient and t_ij=(-r_ij^y,r_ij^x) is the vector r_ij rotated by 90^o. This corresponds to the limit = of Coulomb friction [23-26]. Time is given in units of ; velocity in . The behavior we observe is not sensitive to changes of , , or the relative strengths of f_g, f_fric, and k_g. In some cases, we constrain the grains to form linear dimers, trimers, or quadrimers by rigidly fixing the grains together. The walls and floor are evenly lined with immobile grains of the same size as the simulated monomers. Such a rough floor is necessary to produce a finite angle of repose in the case of monomers; without it, the monomers continue to slump until their height is reduced to a single grain. Dimers and trimers, however, produce a finite angle of repose even on a smooth floor due to their tendency to interlock.

The system is prepared in one of two ways. In studies of , the grains are prepared in a box of size 10 x 18 such that the top surface of the grains is roughly flat, either by dropping individual grains from above, or by placing them into an ordered arrangement directly. In studies of the dynamics of collapsing grains, the grains are first introduced into a box that is 1/6 as wide as the full simulation area of 60 x 18, either by dropping or by direct placement. The right wall is then instantaneously moved out to the edge of the system, and the grains collapse to the right. This method has been previously employed in refs. [27,28] to study angles of repose.

When the grains are dropped slowly from above, we find that monomers and dimers fall into well-ordered arrangements but trimers are disordered. The positions of the grains are illustrated in fig. 2(a,c,e), and the Delaunay triangulations of the lattice are shown in fig. 2(b,d,f). The monomers of fig. 2(a,b) are perfectly ordered, with no dislocations, as indicated in table I. For dimers as in fig. 2(c,d) the positional order is nearly complete but there is a slight chance for a vacancy to occur. Note that although the individual elements composing the dimers are ordered, the bond angles have no overall order. For trimers, shown in fig. 2(e,f), there is some local ordering of the bond angles, but no overall order, and there is a large amount of positional disorder.

There is a simple geometric reason for the difference in packing of the three grain shapes. If a layer of previously dropped grains is present, and there is a single monomer-sized vacancy in the layer, then an additional monomer or dimer dropped over the vacancy will fall into the vacancy and fill it. In contrast, the trimer is long enough to span the vacancy and block it from being filled. Additionally, a trimer is long enough to assume a continuum of angles if dropped near a second trimer lying flat, depending on the exact relative spacing of the two trimers, whereas a dimer will assume only the angles allowed by a triangular lattice. Thus with trimers long-range order cannot be achieved through random dropping. We have also checked the behavior of longer, quadrimer grains, and find that it resembles that of the trimers. The ordering behavior of the shapes is specific to 2D, and different behavior is possible in a three-dimensional system. Further, if smooth rods are used instead of dimers, the unique ordering behavior found for the dimers is not expected to occur.

We next consider , measured with the same procedure followed in the experiment. After dropping particles into the box and ensuring that the upper surface of the grains is horizontal, we simulate tipping the box slowly by rotating the direction of gravity, and record the angle at which the first avalanche occurs. As listed in table I, we find the same trend that was found in experiment: low values of =27^o for ordered monomers and =16^o for disordered trimers or quadrimers, and a much larger angle of =40^o for the dimers. The value of is not strongly affected by the size of our sample, as indicated in the inset to fig. 4 below. It is clear that the different types of order in these three geometries lead to the different angles of stability. The disordered trimers are very unstable to rearrangements since there are many dislocations in the granular packing where motion can occur [29]. The ordered monomers are more stable due to the absence of dislocations, but are still unstable to shearing motion along the lattice directions. In contrast, the dimers are stabilized by the absence of dislocations like the monomers, but shearing motion is suppressed due to the randomly arranged bonds between the dimers, which lead to interlocking. The larger we observe for dimers compared to monomers agrees with the results of ref. [12]. Our study of the even more anisotropic trimers indicates, however, that the increase in with anisotropy found in ref. [12] does not continue indefinitely but instead decreases again as the grains become longer than dimers.

To further explore how the structure of the granular packing affects its stability, we directly examine the dynamics of collapsing grains when one wall is removed, and compare the three geometries in fig. 3. We see a striking difference in the dynamics. Collapsing monomers move in large collective motions involving much of the bead pack, which follow the 30^o slip planes of the lattice. The motion is very orderly and occurs in the form of large pulses, during which triangular wedges of grains displace along their lattice vectors. This wedge-like motion is similar to motion seen in experiments with flowing steel balls [18,30].

In collapsing dimers, the motion is limited to the surface of the pile due to the interlocking of the dimers which prevents shearing motion along the lattice vectors. The motion that does occur is tumbling in nature and very disorderly. A relatively small portion of the bead pack is moving at any given time.

In collapsing trimers, the large number of dislocations present in the system serve as "lubrication" to the motion, allowing easy disorderly sliding of a large portion of the bead pack. There is no significant ordering present on a scale of more than a few grains, and no well-defined slip planes, so coherent motion is impossible. Motion occurs through a combination of gliding along the dislocated area, and tumbling down the topmost portion of the pile. Although the motion is widespread, it occurs more slowly than the collective motion seen in the monomers. This is due to the fact that a considerable amount of rearrangement within the bulk must occur as the movement continues, unlike the case of monomers when no rearrangement was necessary and the movement could follow the lattice vectors.

The different dynamics can also be observed by measuring the net velocity in the horizontal x-direction, , of the grains over time, as shown in fig. 4. Monomers produce extremely large pulses as the collective motion occurs, whereas dimers and trimers produce much smaller amplitude signals with their more disperse responses.

Summary. - We have shown through experiments and simulations that granular assemblies consisting of rigidly connected beads represent an ideal system in which specific types of ordering and dynamics can be controllably realized. Monomers order into a triangular lattice, while dimers show a novel quasi-ordered state in which only translational but not bond-orientational order exists. Trimers do not possess translational or orientational order. We also show that specific types of granular flow are associated with the grain anisotropy. Monomers flow during collapse by large-scale collective shear motions. The dimers, which do not allow for easy shear motion due to the lack of orientational ordering, show a surface tumbling motion. Trimers show a slower tumbling motion that occurs in the bulk as well as on the surface. Our system represents an ideal granular system in which specific types of ordering and dynamics can be controlled. We believe that the model can be used to study a rich variety of behavior such as stress network formation, jamming, and shear, as well as glassy dynamics in granular media and glassy systems in general.

* * *

We thank E. BEN-NAIM, N. GRONBECH-JENSEN, M. HASTINGS, H. JAEGER, and J. KAKALIOS for helpful discussions. This work was supported in part by CLC and CULAR (LANL/UC) and by NSF, DMR-9733898.

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