Field-Induced Self-Assembly of Suspended Colloidal Membranes

We report experiments that probe the self-assembly of micrometer-size colloids into one-particle-thick, robust, and self-healing membranes. In a magic-angle precessing magnetic field, superparamagnetic spheres experience isotropic pair attraction similar to the van der Waals force between atoms. But the many-body polarization interactions among them steer an ordered aggregation pathway consisting of growth of short chains, cross-linking and network formation, network coarsening, and consolidation of membrane patches. This generic aggregation scenario can be induced in any particles of large enough susceptibility.

Figure 1

Colloids in precessing magnetic field: The vertical coils generate the static component and the two horizontal pairs produce the rotating field (a). The cell is located in the center and the laser tweezers’ beam enters the setup from below. The field cone opening angle ${\theta }_m$ depends on the relative magnitude of the static and the rotating component (b). To lowest order, the induced magnetic moments of the colloids ${\mathbf{m}}_i$ follow the external field. The pair potential $U_2(r)$ (c) was measured by analyzing the motion of two colloids in a microfluidic chamber $5.5\mu \mathrm{m}$ in diameter and $1.1\mu \mathrm{m}$ in height (inset). By comparing their spatial distribution in zero field to that with the field on 26, we verified that $U_2(r)\propto r^{-6}$ (red curve) and determined the susceptibility of the particles $\chi =0.89$, which is consistent with earlier reports [27].

Figure 2

Snapshots of the evolution of a dilute suspension (0.15 particles per $\mu {\mathrm{m}}^3$) after the field has been switched on. The virtually instantaneous appearance of short chains (0.5 s) is followed by their growth, branching, and formation of a loose network (4.5 s). Most remaining unconnected clusters are rapidly captured and the network is subsequently coarsened such that the membrane patches grow at the expense of the chainlike sections (10.8 s and 23.8 s). The distance between adjacent vertices of the network does not exceed the branching length of about 8 colloids. Shown here is a small part of the field-of-view where the majority of the colloids confined to a $100\mu \mathrm{m}$-thick cell are in the focal plane.

Figure 3

Computed potential of a test colloid near a straight 4-colloid chain in a plane that includes chain axis (a): The colloid is drawn to the ends so as to make the chain grow lengthwise, whereas a sideway approach is prohibited by the potential barrier whose magnitude is largest in the normal plane through chain center. With increasing chain length, the lateral barrier along the midline decreases; panel (b) shows the potential of a 16-colloid chain and panel (c)  depicts the profile of the midline lateral barrier of 4-, 8-, 16-, and 48-colloid chain. Potential of a membrane patch (d) is attractive along the perimeter and repulsive normal to the patch. Energy per particle in chain, membrane, and crystallite in a field of 2.5 mT vs cluster size (e): The red (light gray) curve is the fit described in the text and the thin black lines are the guide to the eye. The data correspond to rhombic close-packed membranes and bipyramidal close-packed crystallites; except in small clusters, the more isotropic shapes produce very similar results.

Figure 4

$z$ scan through a colloidal membrane partly lying on the bottom slide (a) and schematic interpretative image of the membrane (b). The four micrographs obtained by moving the focal plane from top to bottom disclose the vertical, the curved cylindrical, and the flat horizontal part of the membrane. Time-sequence micrographs of the passage of a $9\mu \mathrm{m}$ colloid through a membrane (c). As the large colloid held by laser tweezers and marked by a red cross is pushed against it, the membrane bends until it breaks so that the colloid can pass through it; after that the membrane is self-healed spontaneously.