Geometric control of motility-induced phase separation

Abstract

Curvature fundamentally alters the collective properties of soft, active, and biological materials. Here we study motility-induced phase separation (MIPS), a canonical non-equilibrium transition, and demonstrate that even weak and slowly varying curvature provides robust geometric control over the dense MIPS phase. This includes dictating both the location and morphology of the MIPS cluster, even in regimes where the effect on the overall phase boundaries is minimal. Focusing on active Brownian particles confined to the surface of a torus, we show that varying the aspect ratio drives a structural transition of the dense cluster from a disk localized at the outer equator to a band wrapping the minor circumference. We then discuss how the curved geometry provides a platform for comparing different theoretical frameworks for the MIPS phase: by analyzing the geometries of the cluster boundaries, we compare the structures predicted by thermodynamic and kinetic pictures. Our results establish curved space not only as a tool to shape and guide non-equilibrium dynamics, but as a uniquely sensitive arena for probing the fundamental mechanisms of active matter.

Figures (5)

• Figure 1: (a) Snapshot of a steady-state configuration for a torus with aspect ratio $\xi=1.5$ and $N=20,000$ particles. Here the dense phase forms a disk. (b) Steady-state snapshot for $\xi=3.0$ and $N=5000$. Here the dense phase forms a band that wraps around the minor circumference of the torus. (c-d) Corresponding time-averaged density profiles for particles in the dense phase. Cluster locations are centered in the $\theta$ direction before averaging. Dashed lines show the contour enclosing the average dense phase area fraction $\phi_d^*=0.314$. The outer equator has $\psi=0$.
• Figure 2: (a) Probability density distribution of the area fraction $\phi_d$ of the torus covered by the dense phase for different aspect ratios $\xi$. The color scale is shown in panel c. (b) Variation in the mean area fraction with aspect ratio for different numbers of particles $N$. (c) Probability density distribution of the perimeter $P$ of the dense phase, scaled by the perimeter of an ideal band $P_b$. (d) Peak value of the perimeter distribution for different $\xi$ and $N$. (e) Fraction of snapshots $f_b$ that form a band as $\xi$ is varied. The solid lines are fits to $f_b = \frac{1}{2}\tanh{\frac{\xi-\xi_c}{h}}+\frac{1}{2}$ for each $N$, which give $(\xi_c, h) = (2.01, 0.3)$ for $N=5000$ and $(2.07,0.2)$ for $N=20,000$. The $\xi_c$ values are indicated by the dotted lines while the turquoise and purple dashes respectively show predictions for $\xi_c$ for disks of constant geodesic curvature ($\xi_c=1.68$) and radius ($\xi_c=2.10$). (f) Probability density distribution of the $\psi$ coordinate center of the dense phase for $\xi=1.5$ for different $N$. The outer equator of the torus has $\psi=0$. Distributions in panels a, c, and f are histograms constructed with a Gaussian smoothing kernel. The color scale in panel b also applies to panels d--f.
• Figure 3: (a) Variation in the perimeter $P$ of a constant geodesic curvature $\kappa_g$ disk centered at the outer equator, scaled by the perimeter $P_b$ of an ideal banded solution as the aspect ratio $\xi$ is varied. Curves correspond to different values of $\phi_d$, the area fraction covered by the disk. The solid turquoise line indicates $\phi_d^*=0.314$, the average dense phase area fraction in the simulations, and the corresponding vertical dashed line shows the predicted transition value $\xi_c = 1.68$, above which a banded solution has a lower perimeter than a disk. The torus inset shows the shape of the disk solution at the predicted $\xi_c$. (b) Variation in the minimized perimeter $P_{\psi_0}$ of a disk with fixed area fraction $\phi_d=0.1$ as the center position $\psi_0$ of the disk is varied for different values of $\xi$. Perimeter values are scaled by the perimeter $P_0$ of a disk centered at $\psi_0=0$. Solid markers show solutions obtained via numerical optimization while open markers indicate values for which the solution is a curve of constant $\kappa_g$. Note that for $\xi=1.1$, the numerical approach fails to find a valid solution near the inner equator. At the inner equator, the disk solution self-intersects slightly, indicating that the solution would instead be a cylinder spanning the inner equator. Inset tori show solutions for $\xi=2.1$ for $\psi_0=0,\frac{\pi}{2},\pi$.
• Figure 4: (a--b) Comparison between a contour of constant density, taken from the density plot in \ref{['fig:schematics-density']}(c) and area matched curves of constant geodesic curvature $\kappa_g$ and geodesic radius $r_g$ centered at the outer equator, shown (a) on the torus surface, and (b) in torus coordinate space. The area of the disks is taken as the average area fraction of the dense phase cluster $\phi_d^*=0.314$. The color scale for the constant $\kappa_g$ and $r_g$ curves indicated in panel (b) applies to all figure panels. (c) Variation in the perimeter $P$ of these different curves as the area fraction $\phi_d$ is varied. Results are shown for contours taken from the $N=5000$ samples (black) and $N=20,000$ samples (gray). The dashed vertical line indicates $\phi_d^*$. (d) Histogram of the perimeter deviation $w$ between individual disk-shaped clusters with centers within $\psi_{\text{Max}}$ of the outer equator and area-matched curves of constant $\kappa_g$ and $r_g$ centered at the outer equator. Results are shown scaled by the particle diameter $\sigma$. Solid lines are for $\xi=1.5$, $N=5000$, dashed lines are for $\xi=3.0$, $N=5000$ and lighter shades are for $\xi=1.5$, $N=20,000$. (inset) Mean deviation $\bar{w}/\sigma$ for the $\xi=1.5$ distributions as the maximum allowed deviation of the cluster center from the outer equator is varied. Solid markers are for $N=5000$, open markers are for $N=20,000$. Error bars, which are typically smaller than the marker size, indicate the standard error across independent trajectories.
• Figure 5: (a) Snapshot of the system on the hourglass surface with the coordinate directions marked. The shaded region represents the optimal isoperimetric solution on the surface for the expected dense phase area. (b) Time-dependence of the $z$-coordinate center of the dense phase cluster for two representative trajectories. In one trajectory, the dense cluster crosses between the lower and upper spheres; in the second, it remains centered on the lower sphere for the entire time sampled. Horizontal dashed lines indicate the boundaries of the neck region. (c) Probability distributions for the dense phase area fraction for clusters centered in different regions of the surface.