Quincke Oscillations of Colloids at Planar Electrodes

Dielectric particles in weakly conducting fluids rotate spontaneously when subject to strong electric fields. Such Quincke rotation near a plane electrode leads to particle translation that enables physical models of active matter. In this Letter, we show that Quincke rollers can also exhibit oscillatory dynamics, whereby particles move back and forth about a fixed location. We explain how oscillations arise for micron-scale particles commensurate with the thickness of a field-induced boundary layer in the nonpolar electrolyte. This work enables the design of colloidal oscillators.

Figure 1

(a) Schematic illustration of the experimental setup. The inset shows the ionization of AOT micelles, which is balanced by recombination in the bulk and by field-induced migration in boundary regions of thickness $\ell$. (b) Time-lapse microscopy images showing the three observed particle behaviors: stationary, rolling, and oscillating. Here, the particle radius is $a=5\mu \mathrm{m}$, the AOT concentration is $[\mathrm{AOT}]=150\mathrm{mM}$, and the electrode separation is $L=150\mu \mathrm{m}$. Scale bars are $40\mu \mathrm{m}$. See also Supplementary Material [11], Video 1. (c) Time-averaged particle speed vs external field strength $E_{\mathrm{e}}$. For each 20-ms trajectory, we compute the mean and standard deviation of the particle speed. Markers denote the median of these mean speeds for ca. 1000 trajectories; error bars denote the median of the corresponding standard deviations. The plotted data are colored based on probability assignments of the Bayesian classifier. The solid curve is a fit of the form $U=(\kappa a/{\tau }_{\mathrm{mw}})[(E_e/E_o{)}^2-1{]}^{1/2}$ with $\kappa =0.40$ and $E_o=2.3\mathrm{V}/\mu \mathrm{m}$; the Maxwell-Wagner time is ${\tau }_{\mathrm{mw}}=0.70\mathrm{ms}$ from independent conductivity measurements (Supplemental Material [11], Sec. 1). Note that the fitted value of the field strength $E_o$ differs from that predicted by Eq. (1) for an unbounded sphere, $E_c=0.91\mathrm{V}/\mu \mathrm{m}$.

Figure 2

Phase diagram showing the observed dynamics as a function of two dimensionless parameters: $a/\ell$, the ratio of the particle radius and the boundary layer thickness; $E_e/E_c$, the ratio of the external field strength and the critical field of Eq. (1). Plotted data correspond to experiments on five different particle sizes $a=0.5,1.5,2.5,5,25\mu \mathrm{m}$ (for $[\mathrm{AOT}]=150\mathrm{mM}$) and three different AOT concentrations $[\mathrm{AOT}]=50$, 100, 150 mM (for $a=5\mu \mathrm{m}$). Markers are colored based on probability assignments of the Bayesian classifier (Supplemental Material [11], Sec. 2).

Figure 3

(a) Particle position $r$ vs time $t$ for PS spheres (radius $a=5\mu \mathrm{m}$) in different AOT-hexadecane solutions. The applied field is $E_e/E_c=5.4$, which corresponds to $E_e=2.2$, 3.7, and $4.8\mathrm{V}/\mu \mathrm{m}$ for $[\mathrm{AOT}]=50$, 100, and 150 mM, respectively. (b) Oscillation frequency $\omega$ vs external field strength $E_e$ for different AOT concentrations. Markers denote the mean frequencies within populations of particle trajectories of equal duration; error bars denote standard deviations of these populations. (c) Peak-to-peak oscillation amplitude $2A$ vs external field strength $E_e$ for the three AOT concentrations in (b). Markers denote the mean frequencies; error bars denote standard deviations.

Figure 4

(a),(b) Simulated electric field around a stationary sphere in a model electrolyte above a plane electrode; color map shows the charge density [11]. The radii of the large (a) and small (b) spheres are $a/\ell =3.5$ and $a/\ell =0.14$, respectively. Other parameters include the Debye length ${\lambda }_D/\ell =0.028$, the surface separation $\delta /a=0.1$, the particle permittivity ${ϵ}_p/{ϵ}_f=1.2$, and the recombination rate constant $k_r{ϵ}_f/e^2\mu =2$. (c) Time-averaged angular speed $\mathrm{\Omega }$ scaled by ${\tau }_{\mathrm{mw}}^{-1}$ vs external field strength $E_e$ scaled by $E_c$ for three variations of the leaky dielectric model: an unbounded sphere, a sphere at a plane electrode with constant fluid conductivity, and a sphere at an electrode with a conductivity gradient. The particle permittivity is ${ϵ}_p/{ϵ}_f=1.5$; the surface separation is $\delta /a=0.1$; the conductivity gradient is ${\sigma }_f/(2a+\delta )$; the resistance coefficient is $R/8\pi \eta a^3=1.45$. The shaded region denotes one standard deviation about the average speed. (d) Angular position $\theta$ vs oscillation phase $\omega t$ for $E_e=5.3E_c$. (e) Oscillation frequency $\omega$ scaled by ${\tau }_{\mathrm{mw}}^{-1}$ vs external field strength $E_e$ scaled by $E_c$.