Trapping and sorting active particles: Motility-induced condensation and smectic defects

We present an experimental realization of the collective trapping phase transition [Kaiser et al., Phys. Rev. Lett. 108, 268307 (2012)], using motile polar granular rods in the presence of a V-shaped obstacle. We offer a theory of this transition based on the interplay of motility-induced condensation and liquid-crystalline ordering and show that trapping occurs when persistent influx overcomes the collective expulsion of smectic defect structures. In agreement with the theory, our experiments find that a trap fills to the brim when the trap angle $\theta$ is below a threshold ${\theta }_c$, while all particles escape for $\theta >{\theta }_c$. Our simulations support a further prediction, that ${\theta }_c$ goes down with increasing rotational noise. We exploit the sensitivity of trapping to the persistence of directed motion to sort particles based on the statistical properties of their activity.

Figure 1

(a)–(d) Typical trapped and untrapped states in experiment and simulation. The angles are mentioned in yellow. In terms of rod length, the diameter of the outer circle of the flower-shaped geometry in experiments is 20, and the dimension of the periodic simulation box is 39. (e) A sequence of images showing separation of polar particles based on their activity with only the red particle getting trapped.

Figure 2

Trapping efficiency shows a sudden jump at $\theta ={120}^{\circ }$ for (a) experiment and (b) simulation, indicating a trapping to detrapping transition. The critical transition angle decreases monotonically with angular noise as shown in the inset to (b). Close to the transition angle, we see repeated attempts where rods tend to form metastable structures inside the trap which become increasingly rare as we move away from the transition angle, for both experiment (c) and simulation (d).

Figure 3

(a) Idealized schematic representation of smectic ordering inside a wedge, with a tilt boundary line down the center. (b) Ordering of particles inside a wedge as seen in simulation ($\varepsilon =0.2$ ${\mathrm{cm}}^{-1}$).

Figure 4

Active polar particles at number density $n$ and self-propelled speed $v$ produce a flux of order $nv$ at the mouth of the trap. Admission to the trap (red dashed lines) is primarily for particles captured when they venture within a persistence length $v\tau$ of the arms. Particles entering far from the arms typically escape (green dashed line) by rotational diffusion of their axis, which is also the direction of their self-propelled velocity. An aggregate of length $r$ and width $\sim r\theta$ forms in the trap.

Figure 5

(a) A photograph of the R&T particle along with its $x$ component of the in-plane velocity $V_x$ and orientation $\theta$. (b) A typical trajectory of the particle showing run and tumble events with time. Plot of $V_x$ as function of time and its probability distribution in the inset. (c) Steady states of mixture of A and R&T for constant ${\epsilon }_{\text{A}}$ and different ${\epsilon }_{\mathrm{R}\&T}$, showing insensitivity of R&T particles to added angular noise. (d) Steady states of binary mixtures of type A particles distinguished only by angular noise strength; sorting sets in with increasing noise contrast.