Competing active and passive interactions drive amoebalike crystallites and ordered bands in active colloids

Swimmers and self-propelled particles are physical models for the collective behavior and motility of a wide variety of living systems, such as bacteria colonies, bird flocks, and fish schools. Such artificial active materials are amenable to physical models which reveal the microscopic mechanisms underlying the collective behavior. Here we study colloids in a dc electric field. Our quasi-two-dimensional system of electrically driven particles exhibits a rich and exotic phase behavior exhibiting passive crystallites, motile crystallites, an active gas, and banding. Amongst these are two mesophases, reminiscent of systems with competing interactions. At low field strengths activity suppresses demixing, leading to motile crystallites. Meanwhile, at high field strengths, activity drives partial demixing to traveling bands. We parametrize a particulate simulation model which reproduces the experimentally observed phases.

Figure 1

Phase diagram of Quincke rollers as a function of activity. (a) Schematic representation of mechanism of Quincke rolling. The charge distribution around the sphere forms a dipole oriented inversely to the field direction, and any fluctuation in the dipole orientation leads to particle rotation with a constant angular speed $\mathrm{\Omega }$. The field-dependent activity is translated to the Péclet number, as described in the text. (b) Experimental setup. Colloidal particles are suspended within a sample cell made of conductive glass slides. Colloids are confined by an electrokinetic flow to the region of interest and they become active (blue particles). The induced electrohydrodynamic flow is represented by the solid lines in the amplified illustration. This flow field leads to long-ranged electrohydrodynamic interactions between the particles [72]. (c) Phase behavior in the area fraction: Péclet number plane. In the low activity regime, electrohydrodynamic interactions due to flow fields [see inset in (b)] result in (passive) crystallite formation at sufficient area fraction. On increasing the activity, e.g., $\mathrm{Pe}$ = 2 ($E=E_Q$, with $E_Q\approx 8\times {10}^5\mathrm{V}{m}^{-1}$), the particles self-propel sufficiently that the dynamics change markedly [see Supplemental Material, Fig. S2(b) and Movie 1 67], and the active crystals split and coalesce with one another. With a further increase of activity, the crystallites melt and we find polar bands propagating through active gas. Black and white symbols represent experimental and numerical data, respectively. Symbols: $⬡$ represent passive crystals (X), $⊚$ active gas (G), $▵$ active crystals and gas ($\text{A}+\text{G}$), $\square$ bands and gas ($\text{B}+\text{G}$). Solid lines are drawn guides. Experimental snapshots for every phase in the diagram are indicated by arrows. Particles are colored according to their hexagonal order parameter ${\psi }_6$, whose magnitude is indicated by the color bar. Scale bars represent 10 $\mu \mathrm{m}$.

Figure 2

Changes in local structure as a function of field strength for Quincke rollers. (a) Local order determined with the bond-orientational parameter ${\psi }_6$ upon increasing $\mathrm{Pe}$ values. (b) Fluctuations of the bond-orientational parameter ${\chi }_6$ as defined in the text. Inset displays experimental measurements where a peak develops at the transition between the crystallites and the amoeba phase. Symbols and lines in (a) represent experimental and numerical data respectively for both (a) and (b). (c) Orientational correlation functions $g_6\left(r\right)$ for $\mathrm{Pe}$ as indicated in the color bar. Data obtained from simulations with $\varphi =0.15$. In (a) and (b) and in subsequent figures, the phases are denoted as X (crystallites), A (amoebae), B $+$ G [(active) gas and banding].

Figure 3

Dynamics of the Quincke rollers across various phases. (a) Dynamical overlap function $Q\left(t\right)$, using Eq. (5). Symbols represent experimental data for $\varphi =0.11$. Solid lines are exponential fits as described in the text, where the stretching exponent $b$ is constrained to 1 for both experiment and simulation. Color bar indicates the correspondent $\mathrm{Pe}$ for each line. Data are scaled by the Brownian time $\tau$. (b) Relaxation time ${\tau }_{\alpha }$ from stretched-exponential fitting to symbols in (a). Symbols represent experiments, solid line is obtained from simulations, and dashed line is a guide. The phases are denoted as X (crystallites), A (amoebae), B $+$ G [(active) gas and banding].

Figure 4

Characteristics of the clusters formed by the Quincke rollers. (a) Size of clusters as a function of activity. Rapid increase is observed as crystallites form. Symbols represent experiments and lines are from numerical analysis. (b) Polar order for different $\mathrm{Pe}$ values. Symbols are data obtained from many velocity measurements over different regions in the sampling cell. As above, the phases are denoted as X (crystallites), A (amoebae), B $+$ G [(active) gas and banding].