Sedimented interfacial patterns

Figure 12: Sedimented interfacial patterns. (a) Islands of 3.0  $\unit{\mu m}$ diameter silica spheres in $H = 90~\ensuremath{\unit{\mu m}}\xspace$ cell at 2.6 V. (b) The same sample, after increasing the bias to 2.8 V. (c) Labyrinths of 3.0  $\unit{\mu m}$ diameter silica spheres in $H=200~\ensuremath{\unit{\mu m}}\xspace$ cell at 2.6 V. (d) The same sample, after increasing the bias to 3 V.

Although the patterns described in the previous Section form reproducibly, they occasionally are replaced by qualitatively distinct labyrinthine and island-like patterns such as those shown in Fig. 12. Unlike the vigorously circulating free-floating colloidal vortices, these slow-moving clusters form near the bottom electrode under the same conditions. These patterns also are distinct from the previously reported interfacial crystals, which form on the upper electrode under positive bias.

Once sedimented interfacial clusters nucleate, the choice between forming islands, as in Fig. 12(a) and 12(b), or labyrinths, such as Fig. 12(c) and 12(d), depends principally on the concentration of particles. Two-dimensional islands form at surface coverages of $10\% \lesssim \phi \lesssim 60\%$, and two-dimensional labyrinths at $60\% \lesssim \phi \lesssim 80\%$. Three-dimensional labyrinths form at higher concentrations. Islands and labyrinths can coexist at intermediate sphere concentrations.

The spheres in islands and labyrinths usually flow fluidly as the patterns form, in a manner that resembles the circulation in bulk colloidal vortices. Unlike bulk colloidal vortices, whose circulation retains its sense over time, labyrinthine domains sometimes flip circulation direction as they coarsen. Unlike Rayleigh-Bénard convection or bulk electroconvection, moreover, neighboring domains' circulation directions generally are not correlated. Larger labyrinthine domains, however, develop double-roll structures, with particles rising along the edges and sinking along the center line.

After a few minutes of circulation, islands and labyrinths often become jammed. Increasing the bias can fluidize them again, and coarsen their features. This can be seen in the transitions from Fig. 12(a) to Fig. 12(b) and from Fig. 12(c) to Fig. 12(d).

Figure 13: Sedimented interfacial vortex opening out into a smoke ring cluster. 3.0  $\unit{\mu m}$ spheres in an $H=200~\ensuremath{\unit{\mu m}}\xspace$ cell. (a) A typical vortex ring at 3.4 V. (b) The same cluster at 2.8 V.

Increasing the bias still further causes coarsened blobs such as those in Fig. 12(d) to circulate faster and develop into interfacial toroidal vortex rings, an example of which appears in Fig. 13(a). These surface-hugging clusters differ from the free-floating vortex rings of the previous section in that they respond irreversibly to changes in bias. Reducing the bias on circulating blobs causes them to open out into structures resembling smoke rings, Fig. 13(b). This morphology can be formed only by first increasing and then decreasing the bias, and clearly inherits its topology from the progenitor toroidal cluster.

The islands and labyrinths we have observed somewhat resemble the interfacial crystals described in Refs. (17) and (18), even though the electric field is reversed. We conjecture that gravity maintains the particles on the lower electrode, while the in-plane electrohydrodynamic attraction discovered in Refs. (17) and (18) draws them together into islands (Fig. 12(a)) and labyrinths (Fig. 12(c)). Once formed, such close-packed clusters would have substantially reduced hydrodynamic drag coefficients (29), and so would remain sedimented at biases that would levitate a single sphere. Applying the bias rapidly appears to favor the formation of levitated clusters, presumably by disrupting this stabilizing mechanism, while slowly increasing the bias improves the chances to form interfacial clusters.

We have observed free-floating colloidal vortex rings coexisting with interfacial labyrinths and islands. This demonstrates that the two types of structures can be generated under the same conditions, and therefore are not distinguished by variations in the experimental procedure.