Colloidal vortex rings

In this section, we examine some more complex phenomena that arise under a more restricted set of conditions, and which provide new insights into the interplay of electrohydrodynamic and other forces in macroionic systems.

Applying a positive bias in our cell tends to drive negatively charged silica spheres upward into the bulk electrolyte. The field is uniform in our parallel plate geometry, so that the upward electrophoretic force on spheres of constant charge therefore should be independent of height $h$ in the cell. Under such conditions, spheres dense enough to sediment under gravity should remain near the lower electrode at low biases, and should rise directly to the upper electrode at higher biases where electrohydrodynamics forces exceed gravity.

This effect should be accentuated in charge-regulating silica spheres because the magnitude of their surface charge increases with pH, and therefore should increase with $h$. Indeed, individual isolated silica spheres exhibit hysteresis in their promotion to the upper electrode with increasing bias, and relaxation to the lower electrode with decreasing bias, consistent with these considerations. More concentrated dispersions, on the other hand, exhibit substantially more interesting collective behavior.

At biases between 2.2 and 4 V, which only slightly exceed the threshold for hydrolysis, the background electrolyte is stable against convective instabilities. Nevertheless, initially quiescent collections of spheres develop several classes of striking many-body convective patterns that highlight a previously unsuspected role for cooperativity in charge-stabilized dispersions subjected to electric fields. Unlike the electrohydrodynamically stabilized surface crystals described in previous reports, these quasi-steady-state patterns form in the bulk of the electrolyte and reflect the interplay of electrohydrodynamic forces and gravity.

Figure 6: Typical patterns formed in dispersions of $3.0~\ensuremath{\unit{\mu m}}\xspace$ diameter silica spheres at $H=200~\ensuremath{\unit{\mu m}}\xspace$. (a) $V = 0$: Equilibrium monolayer, $\phi \approx 100\%$. (b) $V = 2.6~\unit{V}$: Diffuse tumbling clouds. (c) $V = 3.0~\unit{V}$: Circulating flower-like cluster levitated to $h = 40~\ensuremath{\unit{\mu m}}\xspace$. (d) $V = 3.8~\unit{V}$: Compact circulating cluster, also at $h = 40~\ensuremath{\unit{\mu m}}\xspace$.

Figure 6 shows the typical sequence of dynamic patterns that forms when an upward bias is applied to dispersions of 3.0  $\unit{\mu m}$ diameter silica spheres in water. Positively biasing the upper electrode by as much as $0.8~\unit{V}$ causes no discernible out-of-plane motion. Abruptly applying a larger bias in the range $0.8~\unit{V} \lesssim V \lesssim 2.5~\unit{V}$ causes a transient in which spheres jump off the lower surface and then settle back to the bottom. This transient reflects the establishment of static ionic gradients within the water that screen out the field.

The spheres' collective behavior passes through distinct regimes as they are driven out of equilibrium. Just above threshold, the monolayer of spheres breaks into diffuse billowing clouds occupying the lower half of the sample cell, as shown in Fig. 6(b). Pushing the system further from equilibrium might be expected to yield increasingly chaotic behavior; but quite the opposite occurs. Increasing the bias beyond 2.6 V coalesces the itinerant clouds into extraordinary flower-like clusters such as the example in Fig. 6(c), all floating with their midplanes at $h = 40~\ensuremath{\unit{\mu m}}\xspace$ above the lower wall. Each cluster forms around a rapidly circulating toroid whose spheres travel downward along the inner surface and return upward along the outside, completing one cycle in a few seconds. The sense of this circulation is indicated schematically in Fig. 1. Because these clusters' structure and motion are reminiscent of hydrodynamic vortex rings, we will refer to them as colloidal vortex rings.

Colloidal vortex rings formed at higher biases tend to consist of fewer clusters, each containing a larger number of spheres at higher density. The dense yet vigorously circulating cluster in Fig. 6(d) was observed at $V = 3.8~\unit{V}$. Increasing the bias once clusters have already formed causes each to shrink and its circulation to accelerate. Such over-driven clusters merge with their neighbors until they achieve the size appropriate to the final bias. The clusters' height, $h$, does not change appreciably with voltage.

By around $V = 4~\unit{V}$, some clusters' density increases enough that they become jammed and stop circulating. These compactified clusters still dissociate immediately once the bias is removed.

At higher voltages, the electrolyte itself becomes unstable against electroconvection. The resulting patterns span the electrochemical cell, with the colloid acting principally as passive tracers. We will discuss this distinct range of conditions elsewhere.

Most often, a densely packed colloidal vortex ring is surrounded by a diffuse circulating corona that extends outward for tens of micrometers. Fig. 7 shows alternate forms consisting of compact circulating toroidal clusters without coronas. Such bare colloidal vortex rings form most often at slightly lower voltages, 2.6 V versus 2.8 V, and can coexist with fully dressed colloidal vortex rings. These clusters also are levitated to $h \approx 40~\ensuremath{\unit{\mu m}}\xspace$ in a system of thickness $H=200~\ensuremath{\unit{\mu m}}\xspace$.

Figure 7: Bare colloidal vortex ring with and without central core. 3.0  $\unit{\mu m}$ diameter silica spheres in $H=200~\ensuremath{\unit{\mu m}}\xspace$ cell. (a) Compact toroidal cluster enclosing a domain of two-dimensional colloid crystal at $V = 2.8~\unit{V}$. The toroid completes one cycle of circulation in about 1.5 sec. (b) Compact toroidal cluster with empty core.

Although colloidal vortex rings somewhat resemble conventional laminar vortex rings (27), they are driven by quite different mechanisms (28) and also have other distinctive features. Perhaps their most remarkable characteristic is the presence of close-packed colloidal crystals at their centers. The example in Fig. 7(a) is particularly noteworthy because much of its core consists of a static crystalline monolayer. These close-packed spheres are not flocculated, and disperse immediately once the driving field is turned off. Consequently, the crystals' formation and stability suggests the existence of a stagnation plane along each cluster's midplane, quite unlike the streaming flow within a purely hydrodynamic vortex ring.

The stagnation plane must end at the edge of the circulating toroid, and indeed particles from the core sometimes are swept into the circulating flow. While the crystalline domains tend to be stable for the duration of an experimental run (up to ten minutes at $V = 3~\unit{V}$), some are incorporated into the surrounding toroid after a few minutes, leaving hollow circulating rings such as that in Fig. 7(b).

In most cases, the crystalline monolayer core evolves from a layered fluid-like state as shown in Fig. 8. As the ring's diameter decreases with increasing bias, the concentration of spheres in the core increases to the freezing point, and a static crystal results.

Figure 8: Central crystal formed by compression in a dispersion of 3.0  $\unit{\mu m}$ diameter silica spheres in a layer of thickness $H=200~\ensuremath{\unit{\mu m}}\xspace$. The sequence of photographs show a cluster shrinking as the bias increases from 2.4 V to 4.0 V. An AC component (0.7 V square wave at 10 Hz) is applied to further stabilize the crystal. The circulation period of the toroid is about 1.5 sec. (a): 0 sec. (b): 0.5 sec. (c): 1.0 sec. (d): 1.5 sec. (e): 2.0 sec. (f): 3.0 sec.

Individual colloidal vortex rings sometimes develop breathing-mode instabilities with periods of a few seconds, with one cycle of a typical structural oscillation appearing in Fig. 9. The usual circulation continues throughout the oscillation. Remarkably, the cluster's crystalline core is destroyed and reformed with each cycle. Neighboring clusters' oscillations do not become phase locked, nor do they necessarily pulsate at the same frequency. Indeed, oscillating clusters can coexist with steadily circulating or compactified clusters.

Figure 9: This sequence of images, separated by 0.5 sec, shows one period of a vortex ring's breathing mode oscillation. 3.0  $\unit{\mu m}$ silica spheres in $H=200~\ensuremath{\unit{\mu m}}\xspace$ cell at 2.8 V constant bias. The cluster's core remains crystalline throughout the cycle.

Within the range of biases for which flower-like colloidal vortex rings reproducibly form (about 2.6 to 3.4 V), we also observe a variety of variant vortex ring structures that either coexist with or even replace the morphologies represented in Figs. 6 and 7. Some of these are shown in Fig. 10.

Figure 10: Variations on the theme of colloidal vortex rings. 3.0  $\unit{\mu m}$ diameter silica spheres in $H=200~\ensuremath{\unit{\mu m}}\xspace$ cells. (a) 3.0 V: Colloidal vortex rings with sharply defined coronas. (b) 2.6 V: Sharply bounded vortex ring without central core. (c) 2.6 V: Central crystal stabilized by a 0.1 V square wave at 1.0 Hz develops scalloped boundary. (d) 3.0 V: Rare variant in which the central core also is convective, $\phi \approx 150\%$.

Figure 10(a) shows colloidal vortex rings whose coronas have sharply defined boundaries, rather than the diffuse coronas evident in Fig. 6(c). These clusters also sometimes form without a crystalline core as can be seen in Fig. 10(b), and the two types can coexist. Unlike the diffusely bounded clusters that undergo breathing mode instabilities the corona in the diffuse morphology can undulate like a jellyfish. Figure 10(c) shows an extreme example of this circumferential instability.

In all of these cases, the vigorously circulating vortex ring appears to be virtually decoupled from the central crystalline core. Figure 10(d) shows a relatively rare case in which some of the particles in the central core are swept into the vortex ring's circulation.

In roughly half of our experiments, we observed vortex rings coexisting with two-dimensional surface crystals (17,18) that had nucleated on the upper electrode. The spheres in these crystals are driven to the upper electrode by the transient ionic current burst when the bias is initially applied. Particles that reach the upper electrode during these transients tend to stay there and subsequently form crystalline clusters through mechanisms that have been previously reported (17,18). Indeed, such surface crystals begin to form at biases below the onset for vortex ring formation. Above 2.6 V, clusters hovering in the bulk coexist with crystals trapped on the upper surface. Decreasing the bias to 1.4 V, which is the threshold for forming crystals on the upper electrode, causes all particles in crystals to sink to the bottom.