Dynamic Clustering and Chemotactic Collapse of Self-Phoretic Active Particles

Recent experiments with self-phoretic particles at low concentrations show a pronounced dynamic clustering [I. Theurkauff et al., Phys. Rev. Lett. 108, 268303 (2012)]. We model this situation by taking into account the translational and rotational diffusiophoretic motion, which the active particles perform in their self-generated chemical field. Our Brownian dynamics simulations show pronounced dynamic clustering only when these two phoretic contributions give rise to competing attractive and repulsive interactions, respectively. We identify two dynamic clustering states and characterize them by power-law-exponential distributions. In case of mere attraction a chemotactic collapse occurs directly from the gaslike into the collapsed state, which we also predict by mapping our Langevin dynamics on the Keller-Segel model for bacterial chemotaxis.

Figure 1

Snapshots of colloid configurations for increasing ${\zeta }_{\text{trans}}$ at ${\zeta }_{\mathrm{rot}}=-0.38$ and $\mathrm{Pe}=19$. Top left: gaslike state. Top right: dynamic clustering 1. Bottom left: dynamic clustering 2. Bottom right: collapsed state. $N_c$ is the mean cluster size.

Figure 2

State diagram: ${\zeta }_{\mathrm{tr}}$ versus ${\zeta }_{\mathrm{rot}}$ at $\mathrm{Pe}=19$. The mean cluster size $N_c$ is color coded.

Figure 3

Cluster-size distributions $P(n)$ for increasing translational phoretic parameter ${\zeta }_{\mathrm{tr}}$ at $\mathrm{Pe}=19$ and ${\zeta }_{\mathrm{rot}}=-0.38$. ${\zeta }_{\mathrm{tr}}$ assumes the values 0, 8.8, 14.5, 15.4, 16.3, and 18.2. The transition between dynamic clustering states 1 and 2 occurs between the red and green curves. Inset: mean cluster size $N_c$ versus ${\zeta }_{\mathrm{tr}}$. The transition is indicated.

Figure 4

State diagram plotting mean cluster size $N_c$ against Péclet number Pe and chemotactic control parameter ${\zeta }_{\mathrm{tr}}$. ${\zeta }_{\mathrm{rot}}=-0.38$. The color code indicates the mean number of particles $N_c$ in a dynamic cluster.

Figure 5

Mean cluster size $N_c$ versus Pe for different lines in the full parameter space defined via a parametrization with $x\in [0,1]$. We vary Pe as in experiments of Ref. [25], $\mathrm{Pe}=9.5+11.5x$, and choose ${\zeta }_{\mathrm{tr}}=4.8+16.6x$ and ${\zeta }_{\mathrm{rot}}=-0.16-{\zeta }_{0}x$, where the parameter ${\zeta }_0$ defines the different graphs. The transition between clustering states 1 and 2 roughly occurs at the intersection of the two straight lines.