Mechanisms in the size segregation of a binary granular mixture

A granular mixture of particles of two sizes that is shaken vertically will in most cases segregate. If the larger particles accumulate at the top of the sample, this is called the Brazil-nut effect (BNE); if they accumulate at the bottom, it is called the reverse Brazil-nut effect (RBNE). While this process is of great industrial importance in the handling of bulk solids, it is not well understood. In recent years ten different mechanisms have been suggested to explain when each type of segregation is observed. However, the dependence of the mechanisms on driving conditions and material parameters and hence their relative importance is largely unknown. In this paper we present experiments and simulations where both types of particles are made from the same material and shaken under low air pressure, which reduces the number of mechanisms to be considered to seven. We observe both BNE and RBNE by varying systematically the driving frequency and amplitude, diameter ratio, ratio of total volume of small to large particles, and overall sample volume. All our results can be explained by a combination of three mechanisms: a geometrical mechanism called void filling, transport of particles in sidewall-driven convection rolls, and thermal diffusion, a mechanism predicted by kinetic theory.

Figure 1

Left: Experimental setup. The particles are shaken vertically in an evacuated square glass tube. Pictures are taken from the top and the bottom (using a mirror) of the granular sample. Right: An image of the bottom plate. All large particles detected by the image processing are marked with black circles.

Figure 2

Side view of the shaken sample while it is most expanded. The labels give the shaking frequency and amplitude, corresponding to $\Gamma =20$, 1.7, 2.5, and 7, respectively. The container is filled with 128 large and approximately 1040 small brass spheres (corresponding to equal volumes); their diameter ratio is 2. The elongated appearance of the particles in the horizontal direction is a consequence of the illumination. The height of the images is one-eighth of the overall tube length; this height is large enough so that the spheres do not collide with the top of the container.

Figure 3

(Color online) Top row: Test of the reproducibility in three experimental runs, each starting with washed particles as described in Sec. 4A. Open symbols correspond to the average number of large particles detected at the bottom; closed symbols to the number at the top. Each set of points represents experiments with identical shaking acceleration. Bottom row: Comparison of the average of the experimental results in the top row (squares) and the molecular dynamics simulations (triangles). The sample contains 64 large and 1540 small brass spheres with a diameter ratio of 2.

Figure 4

Relaxation into a steady state exhibiting the BNE. The shaking frequency in the experiment is $15\mathrm{Hz}$. The container is filled with 128 large and approximately 1040 small brass spheres (corresponding to equal volumes); their diameter ratio is 2.

Figure 5

(Color online) Experimental segregation results as a function of the shaking acceleration. The different symbols correspond to different shaking frequencies; the open version of a symbol represents the average number of large particles visible at the bottom, the closed version the number at the top. These experiments exhibit the BNE except for those with $f=15\mathrm{Hz}$ and $\Gamma >4$. The container is filled with 128 large and approximately 1040 small brass spheres (corresponding to equal volumes); their diameter ratio is 2. All data points for $\Gamma >2.5$ are averaged over 19 independent experiments. For the measurements at $1.7<\Gamma <2.5$ we used relaxation curves of the type shown in Fig. 4, where the initial shaking time necessary to reach a steady state (up to $15\min$) was discarded.

Figure 6

(Color online) Experimental segregation results as a function of the shaking amplitude. The data presented are the same as in Fig. 5. ($p_{\mathrm{mix}}=0.523$ [62]).

Figure 7

(Color online) At $15\mathrm{Hz}$ (to the right of the vertical dotted line): simulations with frictional sidewalls (엯) agree with experiment (●), while simulations with periodic boundary conditions (◻) differ. This shows the importance of sidewall-driven convection rolls at this frequency. At $80\mathrm{Hz}$ (to the left of the vertical dotted line): the simulations are independent of the boundary conditions. The experimental results are the same as in Fig. 6. All simulation results are averaged over 10 runs starting from different initial conditions.

Figure 8

Convection becomes important as the shaking amplitude increases. These measurements, performed using high-speed images as in Fig. 2, show the downward velocity of the particles at the sidewalls of the container. The sample is the same as in Fig. 5. The velocities are averaged over all phases of $6\text{cycles}$ $\left(15\mathrm{Hz}\right)$ or $30\text{cycles}$ $\left(80\mathrm{Hz}\right)$.

Figure 9

(Color online) Number density and granular temperature in vertical and horizontal directions in simulations with periodic boundaries. As predicted by thermal diffusion, the larger particles accumulate preferentially at the minimum of the granular temperature. The sample contains one monolayer (measured in units of $d_{\mathrm{L}}$) each of small (엯) and large (▴) particles with a diameter ratio of two. The RBNE observed in the experiment at $15\mathrm{Hz}$ is in accord with the relative positions of the arrows, which show the height of the centers of mass for the small and large particles. The granular temperatures were only computed for heights where there was in average at least one particle in a height interval of $d_{\mathrm{L}}$ of the given type at a time; their values are scaled by the energy needed to raise a large particle its own diameter. Zero height corresponds to the lowest position of the vibrating plate. Data are averaged at a single phase of the cycle for 1200 $\left(80\mathrm{Hz}\right)$ or 434 $\left(15\mathrm{Hz}\right)$ cycles. At $15\mathrm{Hz}$ this phase corresponds to the most expanded sample at the upper turning point; at $80\mathrm{Hz}$ no dependence on the phase was observed.

Figure 10

(Color online) Dependence of the observed segregation on the diameter ratio. The samples in the upper plot contain a single large particle in two layers (measured in $d_{\mathrm{L}}$) of small particles. The samples in the lower plot consist of equal volumes of each 1 monolayer of small and large particles. All data points are averaged over 19 independent experiments.

Figure 11

(Color online) Dependence of the segregation on the diameter ratio with and without convection. The closed circles (●) correspond to the experimental results for $f=15\mathrm{Hz}$, $A=3.25d_{\mathrm{L}}$ and equal volumes of small and large particles (Fig. 10). The open circles (엯) are the results of the MD simulations using frictional sidewalls. The open squares (◻) represent the simulation results if periodic boundary conditions are used to suppress convection. All simulation results are averaged over 10 runs starting from different initial conditions.

Figure 12

(Color online) Segregation due to sidewall driven convection increases with diameter ratio. The mean vertical velocity for small and large particles was measured in experiment at the sidewall of the container for diameter ratios of 1.5 (▾) and 3 (●). Small and large symbols represent the respective particles. Both samples contain equal volumes of small and large particles. The velocities are averaged over all phases of $6\text{cycles}$ $\left(15\mathrm{Hz}\right)$ or $30\text{cycles}$ $\left(80\mathrm{Hz}\right)$.

Figure 13

(Color online) Dependence of the observed segregation on the volume ratio between small and large particles. All samples have a total volume of two layers (measured in $d_{\mathrm{L}}$). The leftmost data points in each frame correspond to a single large intruder. All results are averaged over at least 18 independent experiments.

Figure 14

(Color online) Driving parameters adequate for the condensation mechanism were searched in simulations (with periodic boundary conditions) that yielded this plot of the mean square displacement. The diameter ratio is 2; the symbol size corresponds to the particle type. The sample size is two layers with a volume ratio of large to small spheres of 1.5. All data points are averaged over 200 time steps.

Figure 15

(Color online) Dependence of the observed segregation results on the total sample height, either two or four layers (measured in $d_{\mathrm{L}}$). All samples contain equal volumes of small and large particles. All data points are averaged over at least 12 independent experiments.