Piling and avalanches of magnetized particles

We performed computer simulations based on a two-dimensional distinct element method to study granular systems of magnetized spherical particles. We measured the angle of repose and the surface roughness of particle piles, and we studied the effect of magnetization on avalanching. We report linear dependence of both angle of repose and surface roughness on the ratio $f$ of the magnetic dipole interaction and the gravitational force (interparticle force ratio). There is a difference in avalanche formation at small and at large interparticle force ratios. The transition is at $f_c\approx 7$. For $f<f_c$ small vertical chains follow each other at short times (granular regime), while for $f>f_c$ the avalanches are typically formed by one single large particle- cluster (correlated regime). The transition is not sharp. We give plausible estimates for $f_c$ based on stability criteria.

Figure 1

Dipole-dipole arrangement. The ${\mathbf{m}}_1$ and ${\mathbf{m}}_2$ vectors denote the magnetic dipoles, ${\mathbf{r}}_{21}$ denotes their relative position, and ${\mathbf{F}}_{21}$ is the force acting on the second dipole as a result of the dipole-dipole interaction.

Figure 2

The Coulomb graph of the friction model used. The frictional force ${\mathbf{F}}_f$ is oriented oppositely to the relative translational velocity ${\mathbf{v}}_t$. For large velocities, $F_f=\mu F_n$, where $\mu$ is the friction coefficient and $F_n$ is the normal force. For small velocities, $F_f=\lambda v_t$, where $\lambda$ is a large viscous friction coefficient.

Figure 3

(Color online) Simulation setup. The particles are introduced with constant rate one by one at small random distances from the left wall. The external magnetic field is vertical. The particles can leave the system on the right side. The angle of repose is measured by fitting a straight line over the positions of the surface particles (marked with black). The figure also shows the normal contact forces. The thickness of the lines connecting the centers of the particles in contact is proportional to the normal contact force. The sample corresponds to $f=6$ interparticle force ratio.

Figure 4

Angle of repose (upper panel) and surface roughness (lower panel) at different magnetic interparticle force ratios. The angle of repose is measured in degrees. The surface roughness is measured in (average) particle diameters. We carried out three sets of simulations: In (a) and (c) the particles were fired into the pile, while in (b) the particles were placed gently on the pile. In (c) an artificial side wall effect was switched on (see text for details).

Figure 5

Distribution of particle avalanche sizes (upper panel) and avalanche durations (lower panel) at zero magnetization. We examined three different simulation setups (see text for details). At small avalanche size and duration linear binning, at large avalanche size and duration logarithmic binning was used. Firing the particles into the pile (i.e., dropping them from a given height) or placing them gently, and switching on or off the side wall effect gave no qualitative difference. Over our simulation data (for both avalanche sizes and avalanche durations) we could fit a stretched exponential with $\gamma =0.43$.

Figure 6

(Color online) Consecutive simulation snapshots. Each row corresponds to a different value of $f$. The particles are placed gently on the pile [see the (b) simulation set]. Particles leave the system, falling at the system’s boundary on the right side. Only the system’s lower right corner is shown. The external magnetic field is vertical.

Figure 7

Numerical results on cluster stability. At different chain lengths $\left(N\right)$ and cluster widths ($\nu$) the loss in magnetic energy and the gain in gravitational energy was compared when one chain is blocked while the rest of the cluster moves down by half a particle diameter (see inset). The dashed line shows the $\nu =1+f∕6$ function.

Figure 8

Dependence of average avalanche duration on avalanche size in the (b) simulation set. The connected points correspond to equal $f$ values. The error bars show standard deviation obtained with binning. In the granular regime the average duration of avalanches is proportional to their size (see upper panel). In the correlated regime the square of the average avalanche durations is proportional to the avalanche size, with no further dependence on $f$ (see lower panel). [Similar results were found for the (a) simulation set.]

Figure 9

Scaled avalanche size distribution in granular regime. A characteristic size can be observed. We examined three different simulation setups (see text for details). The avalanche size distribution at interparticle force ratios $1<f<7$ are scaled together using the ansatz $P(s,f)=f^{-1}Q(s∕f)$, where $s$ denotes avalanche sizes. We conclude that the characteristic avalanche sizes increase linearly with $f$.

Figure 10

Scaled avalanche duration distribution in correlated regime. A characteristic duration can be observed. We examined two simulation setups (see text for details). The avalanche duration distribution at interparticle force ratios $7<f⩽24$ are scaled together using the ansatz $P(\tau ,f)=f^{-1∕2}Q(\tau ∕f^{1∕2})$, where $\tau$ denotes avalanche durations. This justifies that the square of avalanche durations increase linearly with $f$.