Separation of binary granular mixtures under vibration and differential magnetic levitation force

The application of both a strong magnetic field and a magnetic field gradient to a diamagnetic or paramagnetic material can produce a vertical force that acts in concert with the force of gravity. We consider a binary granular mixture in which the two components have different magnetic susceptibilities and therefore experience different effective forces of gravity when subjected to an inhomogeneous magnetic field. Under vertical vibration, such a mixture may rapidly separate into regions almost pure in the two components. We investigate the conditions for this behavior, studying the speed and completeness of separation as a function of differential effective gravity and the frequency and amplitude of vibration. The influence of the cohesive magnetic dipole-dipole interactions on the separation process is also investigated. In our studies insight is gained through the use of a molecular dynamics simulation model.

Figure 1

Schematic diagram showing the magnet, the upper position of the sample in the magnet bore, and the vibration arrangements.

Figure 2

The separation of millet and insulating beads. (a) After vibration at $10\mathrm{Hz}$ for $300\mathrm{s}$ in zero magnetic field. Middle; (b) after vibration for $30\mathrm{s}$ with $BdB∕dz=-730{\mathrm{T}}^2{\mathrm{m}}^{-1}$; (c) after vibration for $30\mathrm{s}$ with $BdB∕dz=+1000{\mathrm{T}}^2{\mathrm{m}}^{-1}$.

Figure 3

Separation of a mixture of fine bronze and bismuth grains in vacuum. Vibration at $10\mathrm{Hz}$ with an amplitude of $3.7\mathrm{mm}$ and $BdB∕dz=-500{\mathrm{T}}^2{\mathrm{m}}^{-1}$ was used. The images shown are after 0, 20, 60, and $90\mathrm{s}$ of vibration.

Figure 4

Separation of bronze (light) and bismuth (dark) vibrated at $10\mathrm{Hz}$, $\Gamma =2.5$ for over $20\min$; (a) poor separation at $-BdB∕dz=180{\mathrm{T}}^2{\mathrm{m}}^{-1}$; (b) good separation at $-BdB∕dz=300{\mathrm{T}}^2{\mathrm{m}}^{-1}$; (c) excellent separation at $-BdB∕dz=500{\mathrm{T}}^2{\mathrm{m}}^{-1}$.

Figure 5

Measured concentration of impurities in the bronze (solid line) and in the bismuth (dashed line) layers after equilibrium has been reached, under vibration at $\Gamma =2.5$ and a frequency of $10\mathrm{Hz}$.

Figure 6

Schematic diagram showing the separation behavior of fine bronze and bismuth grains in vacuum as a function of $\Gamma$ and $\mid BdB∕dz\mid$ at $f=10\mathrm{Hz}$. The labeled areas represent regions of no separation (a), poor separation (b), good separation (c), excellent separation (d), and magnetic cohesion (e). The dashed line indicates the onset of collisions of the bed with the roof of the container.

Figure 7

Simulated separation of fine bronze (light) and bismuth (dark) particles. The images shown from top left to bottom right are after 0, 0.1, 0.5, and $2.0\mathrm{s}$ of vibration at $10\mathrm{Hz}$, and $\mid BdB∕dz\mid =600{\mathrm{T}}^2{\mathrm{m}}^{-1}$.

Figure 8

The height difference $\Delta h$ between the centers of mass of bismuth and bronze components, in units of particle diameters as a function of time at a frequency of $10\mathrm{Hz}$, a $\Gamma$ of 2.5 at several values of $-BdB∕dz$: $656{\mathrm{T}}^2{\mathrm{m}}^{-1}$ $(▵)$; $506{\mathrm{T}}^2{\mathrm{m}}^{-1}$ $(\circ )$; $365{\mathrm{T}}^2{\mathrm{m}}^{-1}$ $(◇)$; $206{\mathrm{T}}^2{\mathrm{m}}^{-1}$ $(+)$; $106{\mathrm{T}}^2{\mathrm{m}}^{-1}$ $(\times )$.