Single Particle Behavior in Flowing Fine Powders: Augmenting Experiments with Simulations

The flow of granular matter has been studied in great detail for many years. However, some important questions remain unanswered because flowing granular matter is hard to characterize experimentally on a single particle level. Here, we explore this domain for dense, slow flows. We will present experiments in a miniature shear cell capable of tracking each individual particle within the cell. We will also explore the limits of simulating our experiments on a single particle level using LIGGGHTS [1]. The direct comparison of our simulations and experiments demonstrates a high degree of agreement on the individual particle level between experiment and simulation.

The experiment uses a strain-controlled shear cell composed of a fixed bottom and a movable upper plate [2]. The shear cell is laterally open, i.e., contains no lateral walls. The model system is composed of polydisperse spherical silica particles with an average diameter of 7 µm suspended in an aqueous electrolyte solution. The shear cell is approximately ten particle diameters high. We use a laser scanning confocal microscope [3,4] developed in-house to record simultaneously position and size of each individual particle within our field-of-view. Due to number of voxel used to characterize each particle, position and size resolution are significantly better than the optical resolution of the confocal microscope, i.e., below the Abbe limit. The samples are imaged continuously in 3D during externally applied large amplitude oscillatory shear. We follow the trajectories of each individual particle during the experiment using the approach pioneered by Crocker and Grier [4] and adapted to our experimental system [2,3]. Typically, trajectories are available for all about 1400 particles are in the field-of-view for each 3D image, with particle trajectories available for all particles visible in the field of view. During a typical experiment, a total shear amplitude in the order of two shear units (140 µm absolute travel) is applied to the system, however, including between two to five shear reversals. We assume that kinetic effects can be excluded, because the experiments were run at very low speeds in the quasi-static limit. Due to the viscosity of the carrier liquid our model system is overdamped. Nevertheless, we took extra care to also exclude mechanical noise coming from the microscope or other external sources.

The knowledge of experimental particle trajectories for all observed particles allows us exploring the initial formation of shear bands, wall effects and layering within the sample. We also characterized density gradients induced by shear as a measure for the bulk behaviour. In the experiments, the shear band width decreases with shear amplitude and does not recover at shear reversals. The shear band width only depends on the absolute shear amplitude in these experiments.

Flow inside shear bands is non-affine, looking chaotic in nature. Simulations often produce the same general trends and features at medium or large length scales (and on average) as seen in the experiments, but fail to reproduce the exactly same trajectories as measured experimentally. This leads to the fundamental question how well simulations can reproduce real experiments at small length scales.

Our experiments provide the necessary reference to compare simulations and experiments on a single particle level. We initialise our simulations with the experimental particle positions and sizes. This allows us to directly compare the simulated particle trajectories during shear to the experimentally measured ones. We show that both the non-affine and the chaotic nature of the flow manifests itself in a divergence between particle trajectories in simulation and experiment. More precisely, for short times up to 0.01 shear units, the median deviation between experiment and simulation is close to the experimental resolution. On longer timescales, up to 0.04 shear units, the median deviation increases up to 5% of the particle radius. This increase in deviation shows the characteristic exponential divergence expected for a chaotic system, suggesting a positive Lyapunov exponent. On this basis, we discuss the limits of predictability of these systems.

We also studied the match between experiment and simulation for large scale features, e.g., the formation of layered structures close to walls. Layering, enhanced by shear near walls, can be characterized via the density variations along the shear cell height. We correlate the experimental and simulated profile over time and found slowly decreasing, but reasonable correlation up to 0.1 shear units. This hints at a second, longer timescale on which our experiments and simulations follow the same general trend, even if the individual particle trajectories already to deviate.

In conclusion, our combination of experiments and simulations demonstrates the power of simulations to enhance the analysis of experimental data on granular flow. Using high resolution input data and elaborate simulation setups, it is possible to predict individual particle trajectories within a few percent of the particle diameter for short time scales. For long times the chaotic nature of the particle trajectories induces an exponential divergence between experimental and simulated trajectories. Still, studying larger bulk features is even possible over longer timescales.

References

[1] Kloss C., Goniva C., Hager A., Amberger S., Pirker S., Progress in Computational Fluid Dynamics, 12, 140 (2012).

[2] Wenzl, J., Seto, R., Roth, M., Butt, H.-J., Auernhammer, G., Granul. Matter, 15, 391 (2012).

[3] Roth, M., Schilde, C., Lellig, P., Kwade, A., Auernhammer, G., Eur. Phys. J. E, 35, 124 (2012)

[4] Crocker, J.C., Grier, D.G, J. Colloid Interface Sci. 179, 298 (1996).