Emergent Microrobotic Behavior of Active Flexicles in Complex Environments

TL;DR

This work introduces flexicles, vesicle-encapsulated active matter consisting of self-propelled rods that spontaneously form a polar cluster at the membrane and propel the whole composite. Through large-scale molecular dynamics simulations, the authors show that internal rod rearrangements create a functional bifurcation into latchers and navigators, driving geometry-dependent behaviors such as latching to curved obstacles, orbiting on surfaces, and even climbing over walls or stairs. The study reveals that membrane bending rigidity $\kappa_H$ and rod propulsion (Péclet number $\text{Pe}$) jointly control locomotion, obstacle interaction, and the transition between deformable and rigid flexicle dynamics, enabling transport of objects and cooperative escape. These findings establish flexicles as a versatile, autonomous platform for programmable, geometry-sensitive microrobotics with potential paths toward collective colloidal robotics and soft robotic systems at the microscale.

Abstract

Collections of simple, self-propelled colloidal particles exhibit complex, emergent dynamical behavior, with promising applications in microrobotics. When confined within a deformable vesicle, self-propelled rods cluster and align, propelling the vesicle and inducing changes in the vesicle shape. We explore potential microrobotic capabilities of such vesicle-encapsulated particles, which form a composite particle system termed a `flexicle'. Using molecular dynamics simulations, we demonstrate that the alignment of rods enables flexicles to locomote and respond adaptively to their physical environment. When encountering solid boundaries or obstacles, the rods reorient at the interface, triggering novel emergent behaviors such as crawling, corner-preferencing, wall climbing, and object-latching. These interactions and accompanying internal rod re-arrangement lead to spontaneous, temporary differentiation of the rods into `latchers' and `navigators'. This division of labor among the rods enables coordinated locomotion and environmental response. Our findings establish flexicles as a versatile platform for programmable, geometry-sensitive microrobotic behavior, offering a step toward autonomous colloidal robotics.

Figures (13)

• Figure 1: Illustration of the flexicle model considered in this paper. The top left snapshot depicts a self-propelled rod particle. The spherically capped rod is modeled as five closely overlapping spheres with diameter $\sigma$ rigidly connected along their diameters. The top right image shows a flexicle with 153 rod particles enclosed within a deformable membrane, which is represented by vertices connected by a triangulated mesh. The three images on the bottom represent the different external geometries that flexicles encounter in our study: a sphere, a cylinder, and a square box with walls.
• Figure 2: (a) Collision and latching behavior of a flexicle ($\kappa_H=100~k_B\,T$) interacting with a spherical obstacle ($R_\text{sph}=1.3~R_\text{flex}$). Left: Five snapshots show the latching process. Middle: Three snapshots illustrate steady-state circulation around the sphere. Right: Internal rod organization highlights two roles—latchers (blue) and navigators (yellow). (b) Four snapshots showing a rigid flexicle ($\kappa_H=10000~k_B\,T$) transporting a movable sphere in a straight line. Inset (far right): Rod alignment at the initial time point. (c) Trajectory over time of a flexicle moving on a static sphere. Top: Flexible membrane ($\kappa_H = 100~k_B\,T$). Bottom: Rigid membrane ($\kappa_H = 10000~k_B\,T$). (d) Latching probability as a function of obstacle radius.(e) Alignment distribution of rods, measured by the angle between each rod's axis ($n_i$) and the surface normal ($\hat{p}_i$). Solid and dotted lines show results for flexible and rigid membranes. Peaks are color-coded to match rod roles in (a). (f) Instantaneous velocity of flexicles on static spheres across varying membrane rigidities. All simulations were performed at Péclet number $\mathrm{Pe} = 100$.
• Figure 3: (a) Normalized tangential component of the net active force from the internal rod cluster as a function of membrane bending rigidity $\kappa_H$: $\frac{\mu_{\text{tan}}}{\sqrt{\mu_{\text{rad}}^2 + \mu_{\text{tan}}^2}}$, where $\mu_{\text{rad}}$ and $\mu_{\text{tan}}$ are the radial and tangential components of the net active force relative to the inward surface normal $\hat{\mathbf{n}}$. Solid lines represent results for spherical obstacles, and dotted lines for cylindrical ones. Insets show representative snapshots of flexicles interacting with the obstacles at $\kappa_H = 100~k_B\,T$ (top row) and $\kappa_H = 5000~k_B\,T$ (bottom row), with arrows indicating the direction of the net force. (b) Obstacle radius (red) and effective latching radius (blue) corresponding to a 50% flexicle attachment probability, plotted against $\kappa_H$. Values are extracted from error-function fits to the latching probability data shown in Fig.\ref{['fig:results1']}(d) for spheres and Fig.\ref{['fig:results3']}(b) for cylinders. Solid and dotted lines follow the same notation as in (a). All simulations were performed at a Péclet number of $\mathrm{Pe} = 100$.
• Figure 4: (a) Left: Snapshots showing a flexicle spiraling upward along a cylindrical surface. Right: Internal rod configuration, highlighting two functional roles—latchers (blue) and navigators (yellow). (b) Latching probability of flexicles on cylindrical obstacles. (c) Rod alignment distribution, comparing the rod's axis ($n_i$) with the local surface normal ($\hat{p}_i$). Solid and dotted lines represent flexible ($\kappa_H = 10~k_B\,T$) and rigid ($\kappa_H = 10000~k_B\,T$) membranes, respectively. Colored segments of the solid line correspond to latchers and navigators shown in (a). All simulations were performed at Péclet number $\mathrm{Pe} = 100$.
• Figure 5: (a) Snapshots showing a flexicle escaping from a square enclosure. Key time points: $\tau_0$ (center), $\tau_2$ (wall), $\tau_5$ (corner), and $\tau_6$ (escaped). Top-right inset shows a time-colored trajectory from a top view. Bottom-left inset shows rod alignment toward the corner at $\tau_5$. (b) Time evolution of rod orientation angles relative to the wall’s y-axis. Top: Full timeline. Middle: Zoom-ins on three key stages—(I) wall contact, (II) corner encounter, and (III) escape. Bottom: Angle distributions at specific times, comparing before, during and after each stage ($\tau_0$–$\tau_6$), with time points highlighted in the middle row. (c) Snapshots of a flexicle ($\kappa_H = 100~k_B\,T$) climbing a staircase (step height = $R_{\text{flex}}$) under a constant upward force $F_G = 3\,\epsilon/\sigma$. Color indicates progression in time. (d) Snapshots of two cooperative escape scenarios involving multiple flexicles. Each row shows a time series of four snapshots, with arrows indicating the progression over time. Top row – Cooperative Pushing Mechanism: Multiple flexicles work together to "help" one flexicle (blue) climb a tall wall ($h_{\text{wall}} = 1.75~R_{\text{flex}}$). Yellow and green flexicles push from behind, compressing the blue flexicle. This compression enables internal rods in the blue flexicle to align upward and grow taller, allowing it to climb over the barrier. Bottom row – Cooperative Leg-up Mechanism: A group of flexicles collaborates to overcome a wall ($h_{\text{wall}} = 1.5~R_{\text{flex}}$). The yellow flexicle approaches the group, the orange flexicle exits, and the blue flexicle climbs over the wall by riding atop the green one. The rod alignment within the blue flexicle adjusts to support the climb.