Three phases of odd robotic active matter

Abstract

Nonreciprocal interactions in active matter are known to generate exotic mechanical behaviors such as odd elasticity and odd viscosity. However, these phenomena have largely been studied in isolation, raising a fundamental question: Is there a single system that embodies these distinct regimes of odd matter and can transition between phases, establishing a unified phase diagram for nonreciprocal active matter? To address this, we introduce a tunable robotic active matter platform, the Magnetomechanically Augmented Spinning roBotic (MASBot) collective, in which particle-level control of chirality, activity, and pairwise interactions enables access to distinct phases of odd matter. By continuously increasing repulsive forces relative to attractive and transverse forces, we experimentally map a transition from an odd elastic crystal to an odd viscous liquid, and then to a chiral active gas. We find that this latter phase forms a non-space-filling, nonreciprocal active gas stabilized by long-range hydrodynamic attractive forces, whose statistical signatures are consistent with those of a two-dimensional self-gravitating point vortex gas. Within these phases, adjusting spinning frequency and introducing spatially patterned activity allows us to fine-tune odd mechanical responses and tailor power spectra. Further polar and rotational symmetry breaking at the particle scale leads to novel emergent states such as phase separation and collective translation. Together, our system provides a fundamental experimental testbed for nonequilibrium physics and establishes a blueprint for treating robotic swarms as programmable states of matter, enabling functions that range from resilient structures to adaptive swarm reconfiguration.

Figures (5)

• Figure 1: Magnetomechanically Augmented Spinning roBotic (MASBot) collectives.a-c. Inspired by starfish embryos that self-assemble into living chiral crystals (a-b), we build MASBot composed of a cylindrical base with a rotating propeller (c). Scale bars are $100~\mathrm{\mu m}$ and $200~\mathrm{\mu m}$ in (a) and (b), respectively. d-f. Single MASBot shows vortical flow in the vertical plane (d), clockwise rotational flow in the top horizontal plane (e), and inward attractive flow at the mid-horizontal plane (f). Scale bars in (c-f) are $2~\mathrm{cm}$. g. Rotational flow exerts nonreciprocal transverse and attractive interactions, and magnets induce short-range magnetic repulsion on neighboring MASBots. h. Activity and pairwise interactions can be tuned at single MASBot level, including translational self-propulsion. i-l MASBot collectives exhibit dynamic reorganization, forming clusters with hexagonal symmetry due to hydrodynamic attraction (i). With increasing short-range magnetic repulsion, the collective undergoes phase transitions from solid to gas (j). By tuning the rotation speed, MASBot collectives show phase-separation-like behaviors (k). Introducing a translational flow director to a standard MASBot breaks polar symmetry and enables the MASBots to actively translate (l). Scale bars in (j-l) are one MASBot diameter ($7.5~\mathrm{cm}$).
• Figure 2: Phase transitions in chiral active matter.a-f. Experimental analysis of phase transitions due to increasing magnetic repulsion. (a) Snapshots of MASBot collectives, from the solid to gas state, with correspondingly increasingly stronger magnetic repulsive force (left to right, i-v). Time averaged 2D pair distribution function $G(\textbf{r})$ (b) and the local bond orientational order parameter $\psi_6$ (c) for the corresponding area in (a). Scale bars in (a-c) are $7.5~\mathrm{cm}$. Mean squared pairwise displacement (MSPD) as a function of lag time $\tau$ for different states from solid (purple) to gas (yellow) (d). The inset is the oscillation seen in the solid states (linear scale). (e) Mean $\psi_6$ as a function of magnetic repulsion. (f) Radial distribution function $g(r)$ for liquid ($0.023~\mathrm{N}$), liquid-gas ($0.031~\mathrm{N}$), and gas ($0.041~\mathrm{N}$) states. g-j. Numerical simulations of phase transitions. Phase diagrams of $|\psi_6|$ (g) as functions of spinning frequency $\omega$ and repulsive strength $E_M$. (h) Time averaged $G(\textbf{r})$, shown from left to right for the solid ($E_M=0$), liquid ($E_M=80$), and gas states ($E_M=640$) marked in (g) where $\omega = 2$. (i) $\psi_6$ corresponding to (h). (j) MSPD as a function of $\tau$ for the states in (h, i). Scale bars in (h, i) are 2.5 MASBot diameters.
• Figure 3: Odd viscosity in chiral robotic matter extending to active chiral gas enabled by inertia.a-b. Schematic (a) and scattering angle distribution (b) of collision experiment, indicating microscopic parity violation. c-d. The temporal velocity correlation of gas state $C(\tau)$ in experiment (c) and simulation (d). The gray lines in (c) are single particle $C(\tau)$ and the black line is the averaged $C(\tau))$. e-f. The separation conditioned velocity correlation field $\mathbf{V}(\mathbf{r})$ in experiment (e) and simulation (f). The arrows indicate the direction and magnitude of the velocity. The color map in the background shows the magnitude of the angular component of velocity $V_{\theta}$. The black circle in the center indicates the particle size. g-h. Schematic of the boundary $R(\phi,t)$ (Black boundary) of MASBot collectives (g), and the angular-time kymograph (h). i-j. Power spectrum of angular-time kymograph shows spinning frequency dependent parity breaking in experiment for full frequency $\omega_0$ (g) and half frequency $\omega_0/2$ (h). The red dashed line (guide for the eye) highlights the peak in the spectrum. k-l. Power spectrum of angular-time kymograph shows spinning frequency dependent parity breaking in simulation for full frequency $\omega_0$ (i) and half frequency $\omega_0/2$ (j). The red dashed line (guide for the eye) highlights the peak in the spectrum. The parameters in simulation are $\omega = 20, E_M = 160$ for (d) and (f) (gas phase), and $\omega_0 = 6, E_M = 80$ for (h) (liquid phase). m. Spontaneous droplet fission driven by odd instabilities in a MASBot cluster with effective magnetic repulsion $0.016~\mathrm{N}$ (liquid phase, also see SI Video 9). Scale bar: $10~\mathrm{cm}$.
• Figure 4: Odd waves and cycling states in nonreciprocal active solid.a-c. Spontaneous strain waves and odd elastic engine cycle (Methods 2.10-2.11). A snapshot of the MASBot collective decomposed into the four principal strain components (a). Space-time kymographs of strain waves along the boundary (counterclockwise) of a MASBot solid (b). Probability density current in the dilation–rotation strain (top) and shear 1–shear 2 strain (bottom) shows counterclockwise strain cycle (c). d-g. Strain power spectra of homogeneous (d-e) and patterned (f-g) active solid. Active solid with two layered spinning frequencies (f, $\omega_0=2.63~\mathrm{Hz}$) exhibits secondary peaks (g) in the power spectra where the peaks are marked with black lines and defined as $60\%$ of the highest value. h-j. Excitable frequencies modulated by rotation speed (orange arrows) and p-atic symmetry of repulsion (green circles). Tuning the particle-level interactions (h) excites distinct vibrational frequencies (i-j). Case (ii) corresponds to the case in Fig. \ref{['fig:activesolids']}d, while cases (i) and (iii) correspond to the cases in Fig. \ref{['fig:phase']}d with magnetic repulsion of $0~\mathrm{N}$ and $0.01~\mathrm{N}$ respectively.
• Figure 5: Tunable interactions through broken symmetries.a. A pair of MASBot particles shows orbital motion under hydrodynamic attraction and transverse interaction. b. Shallow water introduces short range repulsion, $r$, resulting in oscillatory nonlinear dynamics. c. By attaching a slat to redirect fluid flow, MASBot particle breaks polar symmetry, resulting in helical path and neighbor alignment. d. Two MASBot particles with opposite chirality results in net directed motion. Scale bars in (a-d) and (g): $7.5~\mathrm{cm}.$e-f. Collective of self-propelled MASBot particles shows dynamic reorganization and eventually settle into a rigid body when a $+1$ topological defect (red dot in (f), blue arrow: the propelling direction) emerge at center of cluster. The scale bar in (f): $12.5~\mathrm{cm}$. g-h. A faster spinning MASBot can robustly escape an ensemble of slower spinning particles.