Spontaneous membrane formation and self-encapsulation of active rods in an inhomogeneous motility field

We study the collective dynamics of self-propelled rods in an inhomogeneous motility field. At the interface between two regions of constant but different motility, a smectic rod layer is spontaneously created through aligning interactions between the active rods, reminiscent of an artificial, semipermeable membrane. This “active membrane” engulfes rods which are locally trapped in low-motility regions and thereby further enhances the trapping efficiency by self-organization, an effect which we call “self-encapsulation.” Our results are gained by computer simulations of self-propelled rod models confined on a two-dimensional planar or spherical surface with a stepwise constant motility field, but the phenomenon should be observable in any geometry with sufficiently large spatial inhomogeneity. We also discuss possibilities to verify our predictions of active-membrane formation in experiments of self-propelled colloidal rods and vibrated granular matter.

Figure 1

(a) Sketch of the active-rod model. A light-gray background corresponds to a region with higher activity $F_1$, and dark-gray shading corresponds to a lower activity $F_2$. (b) Illustration of the coordinate system used for simulations on a spherical surface. The sphere is centered around the origin, and the region associated with the higher self-propulsion speed $F_1$ lies on the negative $x$ axis. The angle ${\phi }_i$ represents the angle between a rod's orientation vector ${\widehat{\mathbf{u}}}_i$ and the rotated meridian perpendicular to the boundary. (c) Average fraction of particles $N_1/N$ residing on the “fast” $x\le 0$ hemisphere as a function of packing fraction $\varphi$ for a system of $N=200$ rods with particle aspect ratio $a=10$ and activity ratio $F_1/F_2=10$. (d) Representative snapshots of steady-state configurations at various packing fractions, obtained after a total simulation time of $60000\tau$.

Figure 2

Representative snapshots of steady-state configurations for $N=800$ particles and activity ratio $F_1/F_2=10$ on a 2D surface with periodic boundary conditions. (a) Particle aspect ratio $a=16$ and packing fraction $\varphi =0.2$; (b) particle aspect ratio $a=1$ and packing fraction $\varphi =0.2$, with ${\mu }_{\parallel }={\mu }_R=1$ and $\mathrm{Pe}=100$; and (c) particle aspect ratio $a=1$ and packing fraction $\varphi =0.1$, with ${\mu }_{\parallel }={\mu }_R=1$ and $\mathrm{Pe}=100$. In the latter two cases, the particles are spherically shaped and experience no aligning torques during collision.

Figure 3

Structural order parameters for $N=800$ active rods with aspect ratio $a=16$ at packing fraction $\varphi =0.2$ and activity ratio $F_1/F_2=10$, calculated for [(a)–(c)] a spherical confining surface, and [(d)–(f)] a 2D surface with periodic boundary conditions. [(a) and (d)] Average density profile $\rho \left(x\right){\lambda }^2$, [(b) and (e)] polar order parameter $P\left(x\right)$, and [(c) and (f)] nematic order parameter $S\left(x\right)$. In all panels, the $x$ coordinate is normalized by the rod length $\ell =16\lambda$.

Figure 4

[(a)–(d)] Time-correlation functions calculated for $N=200$ rods with $a=10$ at different packing fractions $\varphi$. The top two panels show the incoherent $C_s\left(t\right)$ functions [Eq. (9)] for (a) inhomogeneous activity ratio $F_1/F_2=10$ and (b) homogeneous activity, i.e., $F_1/F_2=1$. Insets illustrate the motility field. The lower panels show the corresponding collective $C_c\left(t\right)$ functions [Eq. (11)] for (c) $F_1/F_2=10$ and (d) $F_1/F_2=1$. Time is given in units of the characteristic time ${\tau }_c$ in which a free particle will swim across one great circle of the sphere. All legends are as in panel (a). Panel (e) shows the average particle flux $J/\left(\tau \lambda \right)$ across the interface for $F_1/F_2=10$ as a function of packing fraction $\varphi$.

Figure 5

Representative snapshots for a system of $N=800$ rods with aspect ratio $a=16$ and packing fraction $\varphi =0.2$, with step locations of the motility field located at [(a) and (b)] $x_0=0.55R$, [(c) and (d)] $x_0=0.78R$, and [(e) and (f)] $x_0=0.97R$, as indicated by the white dashed lines. Left panels show the ($x,z$) plane and right panels the ($y,z$) plane.

Figure 6

Density profiles $\rho \left(x\right){\lambda }^2$ for a system of $N=800$ rods with aspect ratio $a=16$, packing fraction $\varphi =0.2$, and interface location $x_0=0$, for different activity ratios $F_1/F_2$ of (a) 10, (b) 2, and (c) 1.1.