Segregation in a monolayer of magnetic spheres

Segregation and pattern formation is investigated for binary mixtures of granular magnetic spheres in a vertically vibrated monolayer. The spheres, all of equal mass and size, have a maximum surface magnetic field $B$ induced by encased cylindrical magnetic cores of length $l$. For binary mixtures of particles with equal $l$ but different $B$, we find that the particles spontaneously segregate when driven. For fixed vibration frequency, the segregation rate increases roughly linearly with driving acceleration over the amplitudes investigated. For systems of fixed particle number density, the rate of segregation also decreases as the volume fraction of “strong” (high $B$) particles increases. We find that segregation also occurs in binary mixtures of particles with equal $B$, but different $l$. Finally, using a simple model of spheres with dipolar and higher magnetic moments, we show that the observed segregation phenomena occur in conjunction with a decrease in magnetic energy.

Figure 1

(Color online) A general schematic of the magnetic particles used, together with a qualitative illustration of their respective magnetic field lines (obtained using VIZIMAG software).

Figure 2

(Color online) (a) The mixed initial state of a system of 88 weak (white/yellow, light/light in gray scale) and 23 strong (blue/green, dark/dark in gray scale) particles. (b) The final segregated state of the 88 weak and 23 strong particles after being shaken at $9.0g$ for 300 s. Note that there are two dense clumps of the strong particles, while the weak particles form a loose network pattern.

Figure 3

(Color online) (a) The mixed initial state of a system of 51 strong (blue/green, dark/dark in gray scale) and 59 long (yellow/green, light/dark in gray scale) particles. (b) The final state of the 51 strong and 59 weak particles after being shaken at $9.6g$ for 300 s. Note that the strong particles are largely in clusters and segregated from the long particles, which form long unbranching chains.

Figure 4

(a) The segregation $S$ versus time for the system of 88 weak and 23 strong particles shown in Fig. 2. Note that $S$ increases significantly over time, and it is greater than 1 (the $S$ of a randomly mixed system) at the end of the experiment. (b) $S$ versus time for the system of 51 strong and 59 long particles shown in Fig. 3. Note that $S$ again increases significantly over time, and it is greater than 1 at the end of the experiment.

Figure 5

(a) $S_f$ versus the strong particle mixture fraction for strong/weak systems of 111 particles accelerated at $9.0g$. Note that $S_f$ decreases as the number of strong particles in the mixture increases. (b) $S_f$ versus acceleration $a∕g$ for systems of 111 particles. Mixtures of 44 strong and 67 weak particles are shown as filled circles, while mixtures of 67 strong and 44 long particles are shown as filled triangles. Note that, for both mixtures, $S_f$ increases roughly linearly with acceleration.

Figure 6

(a) A schematic of our general magnetic particle model. (b) For the weak and strong particles, the model in (a) reduces to a two-point-dipole model, with the dipoles separated by a distance $w=0.31D$, where $D$ is the particle diameter. (c) For the long particles, the model in (a) reduces to a two-charge model, with the charges separated by a distance $h=0.69D$.

Figure 7

(a) The magnetic energy $U_{ex}$ versus time for the system of 88 weak and 23 strong particles shown in Fig. 2 is plotted as filled cirles. The magnetic energy of the randomly mixed state $U_r$ is also plotted as open circles for reference. Note that $U_{ex}$ decreases quickly within the first ten seconds, then decreases more gradually over time, while $U_r$ remains roughly constant. (b) A plot of $U_{ex}-U_r$ versus time. $U_{ex}-U_r$ decreases steadily after the first few seconds.