Tuning the Random Walk of Active Colloids: From Individual Run-and-Tumble to Dynamic Clustering

Active particles such as swimming bacteria or self-propelled colloids spontaneously self-organize into large-scale dynamic structures. The emergence of these collective states from the motility pattern of the individual particles, typically a random walk, is yet to be probed in a well-defined synthetic system. Here, we report the experimental realization of tunable colloidal motion that reproduces run-and-tumble and Lévy trajectories. We utilize the Quincke effect to achieve controlled sequences of repeated particle runs and random reorientations. We find that a population of these random walkers exhibit behaviors reminiscent of bacterial suspensions such as dynamic clusters and mesoscale turbulentlike flows.

Figure 1

(a) Quincke random walker: In a uniform direct current electric field $E$, free charges brought by conduction accumulate at the particle surface. For $E$ values above the threshold for Quincke rotation, a spontaneous symmetry breaking of the charge distribution gives rise to a net torque, and the sphere rolls about a randomly chosen axis in the plane perpendicular to the applied field direction. If a square-wave electric field is applied with a period between the pulses longer than the time needed for the sphere to depolarize (Maxwell-Wagner time), the sphere executes a random walk. (b)–(d) Quincke walker trajectories at different ${\tau }_T/{\tau }_{\mathrm{MW}}$ ratios, with ${\tau }_R=0.15\mathrm{s}$. Particle stops are marked with red circles. Insets: (Left) tumbling angle distribution, (right) log-log plot of time-averaged experimental (symbol) and theoretical (solid line) mean-squared displacement (crossover marked by the vertical dashed line at $t={\tau }_R$). (e) persistence index $\alpha$ for different ${\tau }_T/{\tau }_{\mathrm{MW}}$ ratios. (f) Run velocity depends solely on the amplitude of the applied electric field. Symbols: Measured velocity at different amplitude and frequencies $1/({\tau }_R+{\tau }_T)$ of the applied pulsed signal. Solid lines: Velocity measured in dc fields. (g) Particle velocity for 1 s duration of the pulsed signal ${\tau }_T/{\tau }_{\mathrm{MW}}=20$, ${\tau }_R=0.15\mathrm{s}$. $E=1.66\mathrm{MV}/\mathrm{m}$.

Figure 2

Top row: Run and tumble. Bottom row: Lévy walk. (a)–(f) Distribution of run-time ${\tau }_R$ drawn from an exponential and power-law probability density function. (b),(g) Trajectory of the Quincke walker performing a run-and-tumble and Lévy walk. (c),(h) Time-averaged mean-squared displacement (in ${\text{m}}^2$). Symbols, experiment; solid line, theory; vertical dashed line marks $\overline{\tau }$. (d),(i) Normalized velocity autocorrelation function. Symbols, experiment; solid line, theoretical exponential decay; dashed line, theoretical power-law decay. Inset: Same in log-log scale plot. (e),(j) Run length vs the corresponding run-time ${\tau }_R$. Symbols, experiment; solid line, linear fit showing a constant run velocity of $V=1.73\mathrm{mm}/\mathrm{s}$ for run-and-tumble and $V=0.84\mathrm{m}/\mathrm{s}$ for Lévy walk. $E=1.83\mathrm{MV}/\mathrm{m}$ for run-and-tumble and $E=1.5\mathrm{MV}/\mathrm{m}$ for Lévy walks. For both run-and-tumble and Lévy walk: $\overline{\tau }=0.075\mathrm{s}$, ${\tau }_T/{\tau }_{\mathrm{MW}}=20$. $\gamma =1.7$ for Lévy walk.

Figure 3

(a) Collective states formed by Quincke random walkers with different run and turn times (${\tau }_R$, ${\tau }_T$). Symbols indicate the experimentally observed ⊲ static clusters, △ swarm (SW), ∘ rotating clusters (RC), □ polar clusters (PC), * disordered clusters (DC). (b) Snapshots of the SW (${\tau }_T/{\tau }_{\mathrm{MW}}=0.5$, ${\tau }_R=4\mathrm{ms}$), RC (${\tau }_T/{\tau }_{\mathrm{MW}}=1.5$, ${\tau }_R=5\mathrm{ms}$), PC (${\tau }_T/{\tau }_{\mathrm{MW}}=3$, ${\tau }_R=8\mathrm{ms}$), and DC (${\tau }_T/{\tau }_{\mathrm{MW}}=10$, ${\tau }_R=40\mathrm{ms}$) phases from the experiments. Scale bar is 1 mm. Velocity vectors are superimposed to the particles. Particle area fraction and pulse amplitude are constant in all experiments and are equal to $\varphi \approx 0.15$ and $E=2.08\mathrm{MV}/\mathrm{m}$, respectively.

Figure 4

(a) Angular-averaged normalized two-point correlation vs normalized radial distance $r$. Top inset: In 2D, showing the spatial periodicity of rotating cluster. Bottom inset: Characteristic length of $S_2$. Angular-averaged velocity autocorrelation vs normalized radial distance. Top inset: In 2D, showing the periodicity of velocity correlation of rotating and polar clusters in space. Some degree of anisotropy in the velocity autocorrelation of the polar cluster is due to insufficient number of clusters in the field of view. Bottom inset: Characteristic length of $C_{vv}$. (c) Polar order parameter vs $n$, the number of particles in a cluster. Inset: Global order parameter; shaded areas indicate polar ordering higher than 75% and lower than 15%. (d) Probability distribution of clusters with n number of particles. (e) VACF of a disordered cluster with ${\tau }_R=100\mathrm{ms}$ evaluated at different run steps. Inset: Angular-averaged energy spectrum of disordered clusters. Symbols denote different phases: □ polar cluster (PC) (${\tau }_T/{\tau }_{\mathrm{MW}}=3$, ${\tau }_R=8\mathrm{ms}$), * disordered cluster (DC) (${\tau }_T/{\tau }_{\mathrm{MW}}=10$, ${\tau }_R=40\mathrm{ms}$), ▿ disordered cluster (${\tau }_T/{\tau }_{\mathrm{MW}}=10$, ${\tau }_R=100\mathrm{ms}$), $\star$ disordered cluster (${\tau }_T/{\tau }_{\mathrm{MW}}=10$, ${\tau }_R=150\mathrm{ms}$), △ swarm (SW) (${\tau }_T/{\tau }_{\mathrm{MW}}=1$, ${\tau }_R=4\mathrm{ms}$.