Dynamical Clustering and Phase Separation in Suspensions of Self-Propelled Colloidal Particles

We study experimentally and numerically a (quasi-)two-dimensional colloidal suspension of self-propelled spherical particles. The particles are carbon-coated Janus particles, which are propelled due to diffusiophoresis in a near-critical water-lutidine mixture. At low densities, we find that the driving stabilizes small clusters. At higher densities, the suspension undergoes a phase separation into large clusters and a dilute gas phase. The same qualitative behavior is observed in simulations of a minimal model for repulsive self-propelled particles lacking any alignment interactions. The observed behavior is rationalized in terms of a dynamical instability due to the self-trapping of self-propelled particles.

Figure 1

(a) Dynamical clustering of self-propelled colloidal particles at low densities ($\varphi \simeq 0.1$ and $v\simeq 0.65\mu \mathrm{m}/\mathrm{s}$). Shown is the formation and breaking up of one cluster. Every particle that at one time has been a member of the cluster is colored differently. (b) The mean cluster size increases linearly as a function of speed $v$. The dashed line is the fit $1.1+1.1v$. Error bars indicate the statistical uncertainty. (c) At higher densities ($\varphi \simeq 0.27$ and $v\simeq 1.63\mu \mathrm{m}/\mathrm{s}$), phase separation into a few big clusters and a dilute phase occurs. The aggregation is completely reversible: The snapshots show how the clusters dissolve after the illumination has been turned off.

Figure 2

(a) Structure of the passive suspension (without illumination): Experimental pair distribution function $g(r)$ at packing fraction $\varphi \simeq 0.37$ and simulation results employing the potential Eq. (3) for $\lambda =0$ (repulsive) and $\lambda =0.5k_{B}T$ (slight attractions). (b) Corresponding pair potentials.

Figure 3

Simulation results for the clustering of purely repulsive discs at area fraction $\varphi =0.3$: (a) Mean cluster size as a function of swimming speed $v$. (b) Snapshot for speed $\mathrm{Pe}=140$ (corresponding to $v\simeq 0.95\mu \mathrm{m}/\mathrm{s}$). Clusters with same relative size have the same color. For comparison: snapshots (c) and (d) show the experimental suspension for speeds (c) $v\simeq 0.36\mu \mathrm{m}/\mathrm{s}$ and (d) $v\simeq 1.51\mu \mathrm{m}/\mathrm{s}$.

Figure 4

Phase separation: (a) Relative mean size $P$ of the largest cluster as a function of swimming speed $v$. Shown are experimental results (open symbols) and simulation results (closed symbols). (b) Simulation snapshot of the separated system at $\varphi =0.5$ and speed $\mathrm{Pe}=100$. (c) Experimental snapshot at $\varphi \simeq 0.25$ and $v\simeq 1.45\mu \mathrm{m}/\mathrm{s}$.

Figure 5

(a) Consecutive close-ups of a cluster, where we resolve the projected orientations (arrows) of the caps. Particles along the rim mostly point inwards. The snapshots show how the indicated particle towards the bottom (left) leaves the cluster (center) and is replaced by another particle (right). (b) Sketch of the self-trapping mechanism: for colliding particles to become free, they have to wait for their orientations to change due to rotational diffusion and to point outwards.