Living Clusters and Crystals from Low-Density Suspensions of Active Colloids

Recent studies aimed at investigating artificial analogs of bacterial colonies have shown that low-density suspensions of self-propelled particles confined in two dimensions can assemble into finite aggregates that merge and split, but have a typical size that remains constant (living clusters). In this Letter, we address the problem of the formation of living clusters and crystals of active particles in three dimensions. We study two systems: self-propelled particles interacting via a generic attractive potential and colloids that can move toward each other as a result of active agents (e.g., by molecular motors). In both cases, fluidlike “living” clusters form. We explain this general feature in terms of the balance between active forces and regression to thermodynamic equilibrium. This balance can be quantified in terms of a dimensionless number that allows us to collapse the observed clustering behavior onto a universal curve. We also discuss how active motion affects the kinetics of crystal formation.

Figure 1

(a) SP particles (with propulsion force $F$) interacting via a Lennard-Jones potential. (b) SD particles moving towards each other over a distance $\Delta x_A$ with a rate $\nu$ if they are within a distance $R_A$. Particles also undergo diffusion ($D$). Depending on the values of $\Delta x_A$, $\nu$, and $D$, either clustering or collisions dominate.

Figure 2

Degree of clustering $\Theta$ versus aggregation propensity (see text for the definitions) for the SP (a) and the SD systems (b). Results for several propulsion forces $F$ (a) and diffusion coefficients $D$ (b) collapse onto two universal curves. The insets represent typical snapshots of living clusters, the cluster size distribution function (a), and the pair distribution function (b) at significant $P_{\mathrm{agg}}$ where finite clusters form. Particles belonging to different clusters in (a) are colored differently and, for clarity, monomers and small clusters are not shown. In the SP case, $\varphi =0.1$ and $F_{\mathrm{ref}}{\sigma }_p=10k_{B}T$, whereas in the SD case $d_{\mathrm{c}.\mathrm{m}.}/R_A=2$ ($\varphi =0.008$), $N_{\mathrm{part}}=1000$, and $D_{\mathrm{ref}}=0.01\sigma /{\tau }_{MC}$, ${\tau }_{MC}$ being the simulation time unit.

Figure 3

(a) Fraction of clusters in the SD suspension as a function of the processivity parameter $\Delta x_A/(R_A-\sigma )$ versus the aggregation propensity $P_{\mathrm{agg},\mathrm{SD}}$ and (b) as a function of their average distance $d_{\mathrm{c}.\mathrm{m}.}={\rho }^{-1/3}$ versus $P_{\mathrm{agg},\mathrm{SD}}$. In (a), $d_{\mathrm{c}.\mathrm{m}.}/R_A=2$ ($\varphi =0.008$).

Figure 4

Crystallization of active particles. Fraction of solidlike particles in the system versus the aggregation propensity in the SP (a) and SD (b) cases. The snapshots in (a) show the crystallization process of a single cluster ($\varphi =0.1$, $ϵ=60k_{B}T$, and $F{\sigma }_p=80k_{B}T$) during the course of a simulation. Fluidlike particles are depicted in blue, with solidlike particles in red. Crystals span the whole simulation box. (b) SD particles at low $\Delta x_A/\sigma$ form small faceted crystals that are replaced by arrested aggregates when $\Delta x_A$ increases (inset snapshots). In this case, $d_{\mathrm{c}.\mathrm{m}.}/R_A=2$ ($\varphi =0.008$).