Frustrated crystallization of a monolayer of magnetized beads under geometrical confinement

We present a systematic experimental study of the confinement effect on the crystallization of a monolayer of magnetized beads. The particles are millimeter-scale grains interacting through the short range magnetic dipole-dipole potential induced by an external magnetic field. The grains are confined by repulsing walls and are homogeneously distributed inside the cell. A two-dimensional (2d) Brownian motion is induced by horizontal mechanical vibrations. Therefore, the balance between magnetic interaction and agitation allows investigating 2d phases through direct visualization. The effect of both confinement size and shape on the grains' organization in the low-energy state has been investigated. Concerning the confinement shape, triangular, square, pentagonal, hexagonal, heptagonal, and circular geometries have been considered. The grain organization was analyzed after a slow cooling process. Through the measurement of the averaged bond order parameter for the different confinement geometries, it has been shown that cell geometry strongly affects the ordering of the system. Moreover, many kinds of defects, whose observation rate is linked to the geometry, have been observed: disclinations, dislocations, defects chain, and also more exotic defects such as a rosette. Finally, the influence of confinement size has been investigated and we point out that no finite-size effect occurs for a hexagonal cell, but the finite-size effect changes from one geometry to another.

Figure 1

Evolution of the global orientational order ${\mathrm{\Psi }}_6$ during the 200 second cooling process for $N=331$ beads. For each geometry of the cell, the points correspond to an average over three independent experiments. Typical error bars corresponding to the standard deviation are indicated.

Figure 2

Global bond order parameter ${\mathrm{\Psi }}_6$ at the end of the cooling process as a function of the number $n$ of edges of the cell. The error bars correspond to the standard deviation of ${\psi }_6$ over three independent experiments.

Figure 3

Stack of Voronoi diagrams and Delaunay triangulations for each geometry with $N=331$ beads. Beads with respectively fivefold, sixfold, and sevenfold coordination are colored in red, yellow, and green in the Voronoi diagram. Voronoi cells for beads at the boundary are not shown for clarity.

Figure 4

(Left) Rosette defects in a pentagonal confinement. (Right) Six topologically charged defects near the boundary drawing a hexagon with the same orientation as the lattice.

Figure 5

Cumulated radial distribution of defects $N_D\left(r\right)$ for the final state of the system for square, pentagonal, heptagonal, and circular geometries for $N=331$ beads. In the insert is the cumulated radial distribution of the particle number $N\left(r\right)$; the black curve corresponds to a homogeneous distribution. The distance $r$ from the center is normalized by the mean distance $a$ between two beads in the system.

Figure 6

Final value of ${\mathrm{\Psi }}_6$ for pentagonal, hexagonal. and circular geometries for $N=169$, 331, and 721 beads at constant filling fraction. The lines correspond to a linear fit.