Large-deviation quantification of boundary conditions on the Brazil nut effect

We present a discrete element method study of the uprising of an intruder immersed in a granular media under vibration, also known as the Brazil Nut Effect. Besides confirming granular ratcheting and convection as leading mechanisms to this odd behavior, we evince the role of the resonance on the rising of the intruder by using periodic boundary conditions (pbc) in the horizontal direction to avoid wall-induced convection. As a result, we obtain a resonance-qualitylike curve of the intruder ascent rate as a function of the external frequency, which is verified for different values of the inverse normalized gravity $\mathrm{\Gamma }$, as well as the system size. In addition, we introduce a large deviation function analysis which displays a remarkable difference for systems with walls or pbc.

Figure 1

(a) A typical snapshot of the system studied. Grains are represented by discs: hollow discs ($\circ$) are moving downward while filled ones ($•$) are moving upward (against gravity). Color level shown at left side indicates the magnitude of the speed: red indicates no movement while blue indicates the maximum value for the speed. It is worth noting the inertia of the intruder accelerating the grains above him (bluish grains). It is also possible to observe the transition between the grains rising and falling in the reddish region around the intruder, a characteristic responsible for the granular ratchet effect. While the intruder easily displaces the grains on the way up, he is unable to move the grains that descend due to the steric exclusion. (b) Sinusoidal displacement imposed to the substrate. This snapshot was taken after the minimum of the oscillation of the substrate, marked as a blue dot in this plot.

Figure 2

(a) Sequential positions of the intruder $(x,y)$ taken at constant time intervals (1000 molecular dynamics steps), for a system with periodic boundaries in the horizontal direction. $y=0$ indicates the box floor and the unities are in unities of the mean grain diameter. The change in colors indicates the time elapsed in the simulation, the red corresponds to the beginning of the simulation and the blue to the end. As the color changes, it is possible to verify that the intruder crosses the periodic boundary several times. Middle panels: time series for intruder horizontal position at different $\mathrm{\Gamma }$ for system with (b) fw or (c) with pbc. In (b), simulations run until $2000\sqrt{d/g}$, while in (c) simulations run until the intruder rises beyond the highest grain. (d, e) Corresponding probability distribution functions (PDFs) of the intruder speed. Note the asymmetry features of PDFs. In the panels, $d$ is the average grain diameter, $t$ is the time, $g$ is the gravity, $v$ is the vertical intruder velocity, $\omega$ is the external frequency, and $\mathrm{\Gamma }$ is the normalized magnitude of the imposed acceleration. Legend presented in panel (b) applies to panels (b), (c), (d), and (e) equally.

Figure 3

Intruder ascent rate $v$ curves for various normalized external accelerations $\mathrm{\Gamma }$ for the pbc case. $\mathrm{\Gamma }=1.15$ and 2500 grains is in $■$, $\mathrm{\Gamma }=1.25$ and 2500 grains is in $•$, $\mathrm{\Gamma }=1.40$ and 2500 grains is in $▲$, $\mathrm{\Gamma }=1.50$ and 2500 grains is in $▼$, $\mathrm{\Gamma }=1.25$ and 1875 grains is in $◆$, all with $37.5d$ wide.

Figure 4

Left: Intruder position in function of time for $\mathrm{\Gamma }=1.25$ for pbc and different imposed frequencies exhibiting a nonmonotonic behavior as the frequency increases. Right: Ascent rate curves for two different system sizes with pbc. 1875 grains in $\square$, 2500 grains in $\circ$. The normalization was made by dividing the abscissa corresponding to the larger system by the ratio of system sizes. These results were taken from a single sample.

Figure 5

Inset: $P\left(W_{\tau }\right)$ for $\mathrm{\Gamma }=1.25$ and several values of $\tau$ as shown in the legends, for both systems: with frictional walls at left and pbc at right. Main panels: Collapse of the RF function calculated for $\mathrm{\Gamma }=1.25$, and several values of $\tau$. Data collapse was performed by dividing the ratio between upward and downward probabilities by the integration time $\tau$. For the range of integration times considered, a remarkable collapse is observed. The frictional walls' case is shown at left and pbc at right panel. Observe that distributions for pbc are much more symmetric than those for fw, which lean strongly to the positive side.

Figure 6

Collapse of the measured values. The data points are collapsed for all simulations with pbc. The fit can cover the ascent rate for different parameters, like stiffness $k_n$, normalized acceleration $\mathrm{\Gamma }$, and the column above the intruder $h$.

Figure 7

Fit of the parameters using Eq. (2). In (a), the fitted natural frequency ${\omega }_0$ shows little dependence with $\mathrm{\Gamma }$. In (b), the dissipation parameter shows agreement within the error bars to a constant value.

Figure 8

Temporal profile, according to stiffness. Rising trajectories for the intruder. In (a), for lower hardness ($k_n=1000$). we observe that the intruder can rise suddenly and fast. In (b), for a much higher hardness ($k_n=10000$), the intruder only rises for a limited range of frequencies, and it clearly jumps each layer after some waiting time, with the possible exception of the case for the fastest rising rate ($\omega =0.225\sqrt{g/d}$) where the waiting time is minimal.

Figure 9

In (a) the simulation was set to fw, width of 37.5d, shaken frequency of $0.256\sqrt{g/d}$, 2500 grains, and we varied the intruder density, compared to the medium. In (b), a comparison between fw and flw and a change in frequency.

Figure 10

In (a) 25d wide in pbc. Increase $\mathrm{\Gamma }$ leads to increase ascent rate. In (b) 37.5 wide in pbc.