Dynamics and interactions of Quincke roller clusters: From orbits and flips to excited states

Abstract

Active matter systems may be characterized by the conversion of energy into active motion, e.g., the self-propulsion of microorganisms. Artificial active colloids form models that exhibit essential properties of more complex biological systems but are amenable to laboratory experiments. While most experimental models consist of spheres, active particles of different shapes are less understood. Furthermore, interactions between these anisotropic active colloids are even less explored. Here, we investigate the motion of active colloidal clusters and the interactions between them. We focus on self-assembled dumbbells and trimers powered by an external dc electric field. For dumbbells, we observe an activity-dependent behavior of spinning, circular, and orbital motions. Moreover, collisions between dumbbells lead to the hierarchical self-assembly of tetramers and hexamers, both of which form rotational excited states. On the other hand, trimers exhibit flipping motion that leads to trajectories reminiscent of a honeycomb lattice.

Fig. 1.. Spinning, DO, and OO motions of dumbbells.

(A) Scanning electron microscopy (SEM) micrograph of a Quincke dumbbell. Scale bar, 5 μm. (B) Representation of a dumbbell body frame. Perpendicular ⊥ and longitudinal ∥ orientations $\hat{\mathbf{n}}$ with respect to the bond between the two spheres are shown. In addition, the velocity v is given by the displacement of the center of mass r. The angles θ_i, corresponding either to the velocity or to the orientation, are defined with respect to the reference axis. (C) Field-dependent behavior of dumbbells. Passive dumbbells become active spinners above E_Q and then circular rollers above E_dis. (D and E) Schematics of incoming electrohydrodynamic flow field as a result of the field application, whose angular velocity leads to rotation in the horizontal plane (D) and elevated side view (E). Flow fields are obtained with eq. S2. (F to H) Representative trajectories of the three states. (F) A spinning dumbbell at low E. Solid blue line represents the displacement of the center of mass r, and the dashed line is from the motion of one of the sites as the dumbbell spins. (G) DO motion and (H) OO motion. Solid lines indicate the displacement of the center of mass, and arrows correspond to the orientation ${\hat{\mathbf{n}}}_{\perp }$. Insets in (F) to (H) show the time evolution of the angles θ_v (dashed lines) and ${\theta }_{\hat{\mathbf{n}}}$ (solid lines).

Fig. 2.. Dynamics of dumbbells.

(A) Angular velocity ω as a function of the field strength E. Two regimes are identified: spin motion (S; shaded region) appears with low values of E, whereas DO and OO orbital trajectories emerge with increased E. (B) Trajectory radius R for the different regimes obtained with E. (C) MSDs measured at different amplitudes of E. Symbols are from experiments and solid lines are fits to Eq. 4. (D) Self-propulsion speed of dumbbells versus the field amplitude. (E) Mean displacement of the trajectory central point r_c. (F) Rotational diffusion coefficient D_r obtained from fits in (C) using Eq. 4.

Fig. 3.. Formation of tetramers and hexamers.

(A) Active dumbbells performing DO motion may collide with a consequent change in their trajectory. We take the angle ϕ made between the orientations ${\hat{\mathbf{n}}}_{ij}$ and velocities v_ij to characterize the collisions. (B) When aligned, two colliding dumbbells form spinning tetramers whose motion results from the dynamical frustration exerted by one dumbbell on the other. (C) The formation of hexamers is possible when a third dumbbell collides with a previously formed tetramer. The resulting spinning motion of hexamers is also attributed to the dynamical frustration of single circular trajectories. (D and E) Schematic representation of the Quincke rotation of dumbbells. Hydrodynamic coupling is schematically illustrated with the black arrows. (F) Formation sequence of a hexamer. A tetramer is previously formed by two dumbbells. A third dumbbell approaches with its orientation ${\hat{\mathbf{n}}}_k$ pointing toward the tetramer. Upon collision, the dumbbells rearrange to form a triangular shape as indicated by the orientations ${\hat{\mathbf{n}}}_{ijk}$. Scale bar, 10 μm. (G) For dumbbells forming tetramers, the distributions of ϕ indicate that the process is dominated by the dumbbell orientation rather than the velocity. Inset shows the distribution of the orientation angles ϕ for successful and unsuccessful formation of tetramers. (H and I) Spinning angular velocities ω for (H) tetramers and (I) hexamers as function of E. Inset in (H) is the evolution of the orientation angle ${\theta }_{\hat{\mathbf{n}}}$ as the tetramer spins. Inset in (I) shows the mean angular displacement 〈Δθ(t)²〉 of a spinning hexamer. E = 3.1 V μm⁻¹.

Fig. 4.. Quincke trimers.

(A) Top: SEM micrograph of a Quincke trimer. Scale bar, 5 μm. Bottom: Trimer body frame. The orientation $\hat{\mathbf{n}}$ of each vertex is given by an angle φ formed with respect to a reference axis and each vertex position. (B) Schematic representation of the flip motion performed by active trimers. Every jump corresponds to a leapfrogging mechanism of one vertex over the opposite side of the trimer, which takes the trimer out of the plane close to the substrate. (C) Distribution of flip angles α for a trajectory E ≈ 2 V μm⁻¹, shown at the inset. (D) A trimer trajectory dominated by flips at E ≈ 3.33 V μm⁻¹ (inset) shows a strong distribution of α → 0. (E) Flip rate as function of the electric field strength E. (F) Effective translational diffusion coefficients D_t obtained from filtered trajectories. Inset shows diffusive MSDs. Arrow indicates increase in E. (G) Reorientational time correlation functions for six different ranks m as defined in the main text. Symbols are obtained from experimental trajectories at E ≈ 2 V μm⁻¹, and dashed lines are fittings from Eq. 7. Inset in (H) displays the collapsed scaled functions for the same data shown in (G). Solid line is a fitting using the mean rotational diffusion coefficient D_r extracted from the fits in (G). (H) Effective rotational diffusion coefficients D_r versus the applied electric field strengths.