Effect of self-propulsion on equilibrium clustering

In equilibrium, colloidal suspensions governed by short-range attractive and long-range repulsive interactions form thermodynamically stable clusters. Using Brownian dynamics computer simulations, we investigate how this equilibrium clustering is affected when such particles are self-propelled. We find that the clustering process is stable under self-propulsion. For the range of interaction parameters studied and at low particle density, the cluster size increases with the speed of self-propulsion (activity) and for higher activity the cluster size decreases, showing a nonmonotonic variation of cluster size with activity. This clustering behavior is distinct from the pure kinetic (or motility-induced) clustering of self-propelling particles which is observed at significantly higher activities and densities. We present an equilibrium model incorporating the effect of activity as activity-induced attraction and repulsion by imposing that the strength of these interactions depend on activity superlinearly. The model explains the cluster size dependence of activity obtained from simulations semiquantitatively. Our predictions are verifiable in experiments on interacting synthetic colloidal microswimmers.

Figure 1

Overall interaction potential $u\left(r\right)$ showing competing attractive and repulsive interactions for ${\epsilon }_a=25k_{B}T$ from Eq. (1). The inset shows the potential near contact.

Figure 2

Effect of activity on the average cluster size $\langle n\rangle$ for various values of ${\epsilon }_a$. The lines serve as a guide to the eye.

Figure 3

Representative snapshots of clusters for $\text{Pe}=0$ (a) and $\text{Pe}=6$ (b) for ${\epsilon }_a=25k_{B}T$. The insets show the cluster size distribution $P\left(n\right)$ in the steady state. The scale bars correspond to $10\sigma$.

Figure 4

Effect of activity on the reduced variance of cluster size for various ${\epsilon }_a$ values. The lines serve as a guide to the eye.

Figure 5

Comparison between predictions of the model [Eq. (14)] (lines) with simulation data (points). Legends for the data points are same as in Fig. 2.

Figure 6

Representative configurations of various states of active particles with competing interactions for varied Péclet number $\text{Pe}$ and reduced number density $\rho {\sigma }^2$ for ${\epsilon }_a=25k_{B}T$.

Figure 7

State diagram of active particles with competing interactions in the plane spanned by Péclet number $\text{Pe}$ and reduced density $\rho {\sigma }^2$ for ${\epsilon }_a=25k_{B}T$.

Figure 8

Cluster size distribution for $\text{Pe}=100$, density $\rho {\sigma }^2=0.13$, and ${\epsilon }_a=25k_{B}T$.