Single-particle behavior

To best highlight the role of collective behavior in electrohydrodynamic pattern formation, we studied the simplest case: a single silica sphere in an electrolysis cell. A single charged colloidal sphere of buoyant mass $m$ and charge $q$ in a constant electric field not only experiences a downward gravitational force $mg$ and an electrostatic force $q \vec{E}$ but also more complicated electrohydrodynamic and osmotic forces due to the electrolyte's response to the field. When the lower electrode is negative, the positive screening cloud of the sphere flows downward. These flowing ions, including $\mathrm{OH^-}$ and $\mathrm{H_3O^+}$ created by hydrolysis, entrain a flow of water. The friction between this electrohydrodynamic flow and the sphere drags the sphere downward. The sphere's cloud of screening ions is deformed not only by the electric field, but also by the hydronium flux. This downward osmotic force and the downward electrohydrodynamic drag are second-order responses to the electric field, and therefore should be smaller than the electrostatic levitating force $q \vec{E}$.

Straightforward considerations allow us to estimate the bias needed to levitate a particle against gravity. A single sphere sediments through water at speed $v_g = mgb = \sigma^2 (\rho - \rho_0) g /(18 \eta)$, where $\rho_0 = 1~\unit{gm/cm^3}$ is the density of water and $\eta = 1~\unit{cP}$ is its viscosity. For 3  $\unit{\mu m}$ diameter silica spheres, $v_g \approx 5~\unit{\ensuremath{\unit{\mu m}}\xspace /sec}$. The same particle's electrophoretic velocity saturates at $v_e \approx e_0 \Sigma \, (V - V_t)/(H \kappa \eta)$, where $\kappa^{-1}$ is the electrolyte's Debye-Hückel screening length, and $e_0$ is the proton charge (1). If we choose the surface charge density $\Sigma_{pH=5}\sim 5000~\unit{\ensuremath{\unit{\mu m}}\xspace ^{-2}}$ (23) and $\kappa^{-1}=0.1~\ensuremath{\unit{\mu m}}\xspace$, these two velocity scales would be equal at about $V - V_t = 0.1~\unit{V}$. Thus, considering the neglected electrohydrodynamic and osmotic drags, the threshold for levitation should be at least 0.1 V higher than the water decomposition voltage. This is consistent with our experiments.

Figure 17: Schematic representation of the gravitational and electrostatic forces on a silica sphere. (a) Dissociation of water during hydrolysis establishes a vertical pH gradient. (b) The charge $q$ on a sphere tracks variations in pH. (c) The electric field $E$ is non-vanishing but substantially independent of height outside of the Debye-Hückel screening layers near the electrodes. (d) Even so, the charge-regulating spheres experience an upward force that increases with height in the cell. This force opposes the force due to gravity.

In fact, the pH is not uniform across the cell. Fig. 17 schematically shows how the profiles of pH, sphere charge $q$, electric field $E$, and electric force $qE$ vary with height in the cell. Even though the electric field is uniform in the bulk, the electrokinetic force levitating a charge-regulating silica sphere increases with height. When this force competes with gravity, therefore, it is possible to obtain two equilibrium heights, one near the lower electrode when gravity dominates, and another near the upper electrode where the electrokinetic force dominates.

The situation is more complicated for two spheres because hydrodynamic forces between the spheres and bounding surfaces are long-ranged. Reference (28) shows the numerically calculated streamlines for one or two electrophoretically levitated charged spheres. Two particles sinking towards the lower electrode create a back flow which tends to drive them apart (30). Conversely, two electrically levitated particles will attract each other. This electrohydrodynamic interaction at least heuristically explains why an initially homogeneous colloidal fluid breaks up into discrete clusters as it is driven away from the lower electrode.

We observed that the threshold for levitating a single sphere is lower than that for levitating an entire monolayer. This most likely reflects the reduced hydrodynamic drag coefficient for many spheres in a monolayer, as compared to that for a single sphere (29). The increased hydrodynamic attraction generated by the intensified ionic fluxes in the more concentrated dispersions would tend to draw these sedimented spheres together into clusters. This, however, tends to reduce their overall hydrodynamic drag (29), causing them to rise less than they otherwise might have.

To compare our system's behavior with these predictions, we performed experiments on 3  $\unit{\mu m}$ diameter silica spheres from a monolayer with areal density below $10^{-4}~\unit{\ensuremath{\unit{\mu m}}\xspace ^{-2}}$, i.e. fewer than three particles per frame. Abruptly applying a bias causes these isolated spheres to jump off the lower electrode during the transient in which vertical ionic concentration gradients are established. Below a threshold voltage, the particles eventually return to the lower wall. Above this threshold, they rise to the top wall and remain suspended. Fig. 18 shows how the single sphere's height changes with applied voltage as the voltage is first increased and then decreased. The point at which a single sphere is electrolevitated to the upper wall is hysteretic: levitated particles only return to the lower wall when the voltage is lowered substantially. The explanation for this hysteresis also can be found in Fig. 17. At the critical bias, $\vert q \vec{E}(z)\vert \gtrsim mg$ for all $z$ in the bulk. By contrast, a particle can only fall from the upper electrode, when the field falls below threshold at the upper electrode $\vert q \vec{E}(H)\vert \lesssim mg$. Therefore, the two thresholds are different. The inset to Fig. 18 shows this hysteresis more clearly at three different salt concentrations. Such hysteresis supports the contention that the levitating force acting on silica spheres increases with the height $h$.

For salt concentration above 1 mM, a single particle can never be forced to the upper electrode. This is consistent with the observation that particles move less vigorously and cooperative structures are harder to form at high ionic concentrations.

Figure 18: Height attained by a single 3  $\unit{\mu m}$ diameter in a $H=200~\ensuremath{\unit{\mu m}}\xspace$ cell as a function of applied bias. The voltage is increased and then decreased step by step. Inset: The threshold voltage required to levitate spheres.

Attractions between particles are detectable only at relatively low particle concentrations (e.g. $\phi \simeq 10^{-4}$) and high applied voltages (e.g. 2 V). In such cases, particles attract each other as they levitate and so form tumbling clouds. The typical inter-particle separation in the most diffuse clouds is about 10  $\unit{\mu m}$ and decreases with increasing bias. These field-induced attractions are surprisingly long-ranged, extending to roughly 50  $\unit{\mu m}$. At biases above roughly 5 V, even isolated spheres travel in circular orbits whose diameters range from 50 to 150  $\unit{\mu m}$ in the vertical plane. This motion is due to electroconvection in the underlying electrolyte. It almost certainly plays no role in forming the highly organized and substantially smaller-scale patterns discussed above.

Generally speaking, single particles are not stably levitated into the bulk of the electrolyte, and they certainly do not trace out the complex trajectories characterizing microscopic patterns such as those in Figs. 6 - 16. These observations help to confirm that such patterns do not arise from electroconvection in the electrolyte alone. We are left to conclude that the spheres play an active role in dynamical pattern selection.