Guiding Self-Assembly of Active Colloids by Temporal Modulation of Activity

Self-organization phenomena in ensembles of self-propelled particles open pathways to the synthesis of new dynamic states not accessible by traditional equilibrium processes. The challenge is to develop a set of principles that facilitate the control and manipulation of emergent active states. Here, we report that dielectric rolling colloids energized by a pulsating electric field self-organize into alternating square lattices with a lattice constant controlled by the parameters of the field. We combine experiments and simulations to examine spatiotemporal properties of the emergent collective patterns and investigate the underlying dynamics of the self-organization.We reveal the resistance of the dynamic lattices to compression and expansion stresses leading to a hysteretic behavior of the lattice constant. The general mechanism of pattern synthesis and control in active ensembles via temporal modulation of activity can be applied to other active colloidal systems.

Figure 1

Self-assembly of Quincke rollers into square lattices under a pulsating electric field. (a) A sketch of the experimental setup. (b) Temporal dependence of the electric field and the averaged particles’ velocity. $T={\tau }_{\mathrm{on}}+{\tau }_{\mathrm{off}}$ is the period of the signal. (c) A trajectory of an isolated roller energized by the pulsating electric field. (d) Mean square displacement curves of the rollers under periodic activity modulations. The bottom (purple) and the top (red) curves correspond to the motion of the rollers in gas and vortex phases, respectively. $E=2.7\mathrm{V}\mu {\mathrm{m}}^{-1}$ and $T=100\mathrm{ms}$ for all curves except the case of a constant field. Slopes marked as 1 and 2 indicate diffusive ($\sim \tau$) and ballistic $(\sim {\tau }^2)$ regimes, respectively. (e) An experimental snapshot of a dynamic square lattice captured between electric field pulses. The blue squares marks four unit cells with a lattices constant $a$. The same area is also shown in (f)–(h). The particle area fraction $\varphi =0.114$, $E=3.0\mathrm{V}\mu {\mathrm{m}}^{-1}$, $T=125\mathrm{ms}$, ${\tau }_{\mathrm{on}}=112.5\mathrm{ms}$. The scale bar is 0.5 mm. See also Supplemental Material, Video S1 [38]. (f) Overlayed square lattices formed after two consecutive cycles represented by blue and red circles. (g) Visualization of the particle trajectories over five periods of the field. For clarity only 10% of particle trajectories are shown. (h) Color-coded visualization of the time evolution of particles positions during one cycle. The color in each point of a particle trajectory corresponds to the time between zero (purple) and ${\tau }_{\mathrm{on}}$ (red) as shown by the color bar.

Figure 2

(a) Dynamic phases formed by rollers for different ${\tau }_{\mathrm{on}}$ and ${\tau }_{\mathrm{off}}$. $\varphi =0.120$, $E=2.7\mathrm{V}\mu {\mathrm{m}}^{-1}$. The system is confined by a cylindrical well with a diameter $D=1\mathrm{cm}$. The phases marked as “$*$” have been previously reported in Ref. [6]. See Video S3 in the Supplemental Material [38] for more details. (b) Lattices and gas formed at different ${\tau }_{\mathrm{on}}$ and $\varphi$ (${\tau }_{\mathrm{off}}\ge 40\mathrm{ms}$). The dashed line is a fit of the boundary between phases by a function $\varphi =A/{\tau }_{\mathrm{on}}$, see text for details. Inset: dynamic phase diagram obtained from simulations. $N$ is the number of particles in the system. (c),(d) Dependence of the lattice constant on ${\tau }_{\mathrm{on}}$ for different area fractions in (c) experiments and (d) simulations. $D=1.5\mathrm{mm}$. The dashed lines are linear fits for the lowest and highest area fractions. Insets in (c) and (d), experimental and simulation data with the lattice constant rescaled by a $\sqrt{\varphi }$.

Figure 3

Probability distribution functions of the local order parameters ${\mathrm{\Psi }}_4$ in a stable lattice formed at different ${\tau }_{\mathrm{on}}$ in (a) experiments and (b) simulations. Insets: temporal evolution of $⟨{\mathrm{\Psi }}_4⟩$. $E=3.0\mathrm{V}\mu {\mathrm{m}}^{-1}$, ${\tau }_{\mathrm{on}}=112.5\mathrm{ms}$, ${\tau }_{\mathrm{off}}=12.5\mathrm{ms}$, $\varphi =0.114$, $D=1\mathrm{cm}$. Also see Videos S4 and S5 in [38].

Figure 4

Hysteresis loops for different ramping rates of ${\tau }_{\mathrm{on}}$ in (a) experiments and (a) simulations. Arrows indicate the direction of the hysteresis loop. The starting point is ${\tau }_{\mathrm{on}}=80\mathrm{ms}$ (experiments) and 30 (simulations). See also Videos S6 and S7 in [38]. (c),(d) The evolution of the (c) lattice constant and (d) the local order parameter $⟨{\mathrm{\Psi }}_4⟩$ of the square lattices in response to the increase (solid symbols) or decrease (open symbols) of the ${\tau }_{\mathrm{on}}$ relative to the initial value of ${\tau }_{\mathrm{on}}=120\mathrm{ms}$. The data are averaged over five experimental realizations for each curve. Inset in (c) shows the absolute change of $a$ compared to the initial value $a_0$.