Simulations of the Spot Model – Introduction

For some states of matter, a microscopic picture of the component particles exists, from which it is possible to derive macroscopic physical laws. For example, in gases, the model of individual molecules undergoing randomized collisions led to Boltzmann's kinetic theory. Similarly, in crystals, where particles are held in a dense, ordered packing, diffusion and flow can be thought of as mediated by defects, such as vacancies and dislocations. However, for granular materials, where particles are held in a dense, amorphous packing, it is unclear what, if any, is the correct microscopic picture.

Based on experimental work, our group showed that in dense granular flows, the motion of a particle is spatially correlated with its neighbors, typicially on a mesoscopic length scale of three to five particle diameters [1]. Intuitively, this makes sense: in dense granular materials, a particle is often fixed in place due to packing constraints with its neighbors. It cannot move independently, and it must flow co-operatively with its neighbors.

Spot model microscopic mechanism

(a) (b) (c)
Initial spot displacement Elastic relaxation step Overall motion

Our group suggested that this behavior could be modeled by the motion of “spots” [2], which represent a small amount of free space spread across several particle diameters, as shown by the blue circle in (a). When the spot moves according to the blue arrow, it induces a small, correlated motion of all particles within range. This simple mechanism captures many results in granular drainage, such as particle diffusion and spatial velocity correlations.

However, while this basic model remains simple enough for mathematical analysis, it is clear that it does not explicitly enforce packing constraints of the particles. In order to preserve valid packings, a second step has been proposed [3]. After the block motion has been carried out, a small relaxation step is applied, during which the particles and their nearest neighbors experience a soft-core repulsion with each other, as shown in (b). The net effect, as shown in (c), is therefore a co-operative local deformation, whose mean is roughly the original block motion.

Schematic diagram
A single spot carrying out a random walk.

Spot simulations of granular drainage

The above mechanism could potentially be applied to flow in a wide variety of situations. However, we concentrated on the case of drainage from a silo. We introduced spots at the silo exit following an exponential waiting time distribution to match the outflow of particles. Motivated by previous work on the Kinematic Model [4] and Void Model [5,6], we then proposed that these spots move upwards through the packing following a random walk, moving up-left or up-right with equal probability, and causing a corresponding downwards motion in the particles. Each spot moves according to an exponential waiting time distribution, and once it reaches the top of the packing, it is removed from the simulation.

Computational testing and implementation

To test the above algorithm, we decided to carry out a systematic comparison of the spot model to a Discrete Element Method (DEM) simulation of 55,000 spheres in a cuboidal container. The DEM simulation was carried out using the code using the LAMMPS code developed at Sandia National Laboratories: this is a brute-force simulation technique whereby every particle DEM simulation was first run, and then five parameters for the spot model are calibrated off the DEM data:

The algorithm was implemented in C++. Since the influence of a single spot is local, the algorithm can be very efficiently constructed. We take the initial packing of particles from the DEM data. We divide the simulation into cuboidal regions, each of which keeps track of the particles within it. Each time we apply the a spot's influence, we only need to test the regions which it overlaps with.

Comparison between Spot Model and DEM

The frames below show a comparison between the spot model and DEM.

DEM Simulation (3 days of computation on 24 processors)
Montage of snapshots from the DEM Simulation
Spot Model (8 hours of computation on one processor)
Montage of snapshots from the spot model

It is clear that the spot model provides a good description of the mean flow, and also captures the correct amount of mixing at the interface between the two colors of particles. Carrying out a detailed analysis of the two simulations shows a high degree of quantitative agreement, which suprising given the small number of parameters that were calibrated. See also a comparison movie between the spot model (left) and DEM (right).

Comparison of particle statistics between the spot at DEM simulations

Comparison of packing statistics between the DEM and spot simulations (click to enlarge).

Perhaps the most exciting result of the simulation is that the spot model accurately tracks the microscopic structure of the amorphous packing seen in DEM. The graphs to the right show the radial distribution function and the bond angle distribution function for the initial packing, the flowing DEM state, and the flowing spot state. These two functions are frequently used as way of characterizing the microscopic structure of a particle packing. We see that not only does the spot simulation create a valid packing, but it tracks the minuscule changes to these functions that are seen in the DEM simulation during flow.

Algorithm stability

The match between the packing statistics in the DEM and spot simulations lasts for a long period of time, during which the flow undergoes considerable rearrangement. However, after a long time, the radial distribution function of packings in the spot model starts to be more peaked than that from DEM, suggesting that the packings generated have slightly more local ordering. To investigate this further we carried out some much longer simulations in taller silos, to examine the algorithm's stabilty. We also tried many alterations to the model to investigate the effects on the amount of deviation in packing statistics. We found that reducing the step size of the random walks would result in a progressive improvement in the spot model's stability, to the point where we could create an accurate match in packing statistics for an entire simulation. We therefore view the random walk step as analogous to a step size in a numerical method, whereby decreasing the size results in a better match, but at a higher computational cost.


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