Reentrant phase behavior in active colloids with attraction

Motivated by recent experiments, we study a system of self-propelled colloids that experience short-range attractive interactions and are confined to a surface. Using simulations we find that the phase behavior for such a system is reentrant as a function of activity: phase-separated states exist in both the low- and high-activity regimes, with a homogeneous active fluid in between. To understand the physical origins of reentrance, we develop a kinetic model for the system's steady-state dynamics whose solution captures the main features of the phase behavior. We also describe the varied kinetics of phase separation, which range from the familiar nucleation and growth of clusters to the complex coarsening of active particle gels.

Figure 1

Top: Phase diagram illustrated by simulation snapshots at time $t=1000\tau$ with area fraction $\varphi =0.4$, as a function of interparticle attraction strength $U$ and propulsion strength $\text{Pe}$. The colors are a guide to the eye distinguishing near-equilibrium gel states (upper-left) from single-phase active fluids (center) and self-trapping-induced phase-separated states (right). Bottom: Detail of reentrant phase behavior at $U=4$ as $\text{Pe}$ is increased. At $\text{Pe}=4$ (left), the system is nearly thermal and forms a kinetically arrested attractive gel. Increasing $\text{Pe}$ to 20 (center) suppresses phase separation and produces a homogeneous fluid characterized by transient density fluctuations. Increasing $\text{Pe}$ further to 100 (right) results in athermal phase separation induced by self-trapping.

Figure 2

Left: Fraction of particles in clusters $f_{\mathrm{c}}$ measured from simulations as a function of $\text{Pe}$ and $U$. Right: $f_{\mathrm{c}}$ as predicted by our analytic theory [Eq. (3)], which reproduces the major features of the phase diagram, including the gel region for $\text{Pe}<U$, the self-trapping region for high $\text{Pe}$, and the low-$f_{\mathrm{c}}$ fluid regime in between. The values of the adjustable parameters are $\kappa =2$ and $f_{\text{max}}=1.7$, with the fit made by eye.

Figure 3

Schematic representation of our kinetic model. The high- and low-density phases fill the left and right half-spaces. A particle on the cluster surface (center) escapes only if its direction points within the “escape cone” defined by $\gamma v_{\mathrm{p}}(\widehat{\nu }·\widehat{\mathbf{n}})>F_{\text{max}}$, while particles in the gas land on the surface at a rate proportional to the gas density and propulsion speed.

Figure 4

Visual guide to phase-separation kinetics at fixed $U=30$. To make mixing visible, particles are labeled in two colors based on their positions at $t=1$. A passive system (top row) forms a space-spanning gel that gradually coarsens. The addition of activity (second row) greatly increases the rate at which the gel evolves. In both cases, particles remain “local” and largely retain the same set of neighbors. When $\text{Pe}$ exceeds $U$ (third row), activity is strong enough to break the gel filaments and a fluid of large mobile clusters results. While the instantaneous configurations appear structurally similar to the gels above, these systems quickly become well-mixed due to splitting and merging of clusters (see also Fig. 5). In the high-$\text{Pe}$ limit (bottom row), self-trapping drives the emergence of a single well-mixed cluster surrounded by a dilute gas.

Figure 5

Quantitative measurements of states with $U=30$. Left: Mean interaction energy per particle ($E$ represents the total system potential energy). Low-$\text{Pe}$ gels are strongly arrested and do not approach bulk phase-separation on our simulation timescales; however, higher-$\text{Pe}$ systems evolve much faster and nearly reach the bulk limit. When $\text{Pe}>U$ (dashed lines), coarsening is arrested as the gel breaks into a fluid of mobile clusters with a $\text{Pe}$-dependent characteristic size, leading to a plateau in the system potential energy. Right: Discrimination of gel from fluid states by mean-square displacement. Dotted lines have slope 1 and highlight the distinction at short times between gels (subdiffusive) and active fluids (superdiffusive).