Rolling at right angles: magnetic anisotropy enables dual-anisotropic active matter

TL;DR

This work investigates active matter in which motion is constrained to four cardinal directions by an in-plane magnetic field $B$, using magnetite-doped Quincke rollers driven by the Quincke instability. Experiments show isotropic circular motion at $B=0$, a perpendicular linearization at intermediate field strengths, and a second, parallel linearization at higher fields, indicating a dual-axis control mechanism. The authors propose and test an anisotropic magnetic susceptibility model $m = X B$ (with a rotating tensor $X$), which can sustain both modes and is described by a torque-balance relation written as $4\pi\epsilon_f P \times E + (4\pi/\mu)(X B) \times B = 8\pi\eta a^3 \omega$, contrasting with simple paramagnetism or a permanent dipole. Numerical simulations reveal a fixed-point attractor for the perpendicular mode and a limit-cycle-like wobble for the parallel mode, with phase diagrams showing basins of attraction and mode-switching events driven by interactions; this establishes a new class of dual-axis anisotropic active matter with potential for advanced control of single-particle and collective dynamics.

Abstract

We report on an experimental active matter system with motion restricted to four cardinal directions. Our particles are magnetite-doped colloidal spheres driven by the Quincke electrorotational instability. The absence of a magnetic field (|B| = 0) leads to circular trajectories interspersed with short spontaneous runs. Intermediate fields (|B| < 20 mT) linearize the motion along the axis perpendicular to B. At high magnetic fields, we observe the surprising emergence of a second, distinct linearization along the axis parallel to B. With numerical simulations, we show that this behavior can be explained by anisotropic magnetic susceptibility

Figures (3)

• Figure 1: (a) Intrinsically unstable dipole $\bm{P}$ induced in the particle by an applied electric field $\bm{E}$ when the condition $\frac{\epsilon_f}{\sigma_f} < \frac{\epsilon_p}{\sigma_p}$ is met lampa1906dielectric. For $\bm{E} > \bm{E}_c$ any small perturbation leads to persistent 'Quincke' rotation, or rolling in the vicinity of a surface. (b) Schematic of the experimental setup: particles lie on the bottom surface of our ITO-coated glass cell. A potential is applied between the glass electrodes to drive a DC electric field out of the plane of motion, while two electromagnets (not shown) drive a constant in-plane magnetic field $\bm{B}$. (c-e) The trajectories of an ensemble of rollers confined to a circular region (1 mm in diameter) at different magnetic field intensities and fixed potential $|\bm E| \approx 2E_c$ (scale bar is 200 µ m, see supplementary movies 1-3). The figures show a top-down view of the schematic shown in (b), with $\bm B$ aligned with the vertical axis and $\bm E$ coming out of the page. (c) $\bm{B}$ = 0 mT: isotropic motion consisting of tight circular orbits interspersed with runs. (d) $\bm{B}$ = 13 mT: first linearization of motion when a magnetic field is applied. A magnetic torque aligns the induced moment with the field, fixing the Quincke rotation axis to give rolling $\perp$ to $\bm{B}$. (e) $\bm{B}$ = 43 mT: emergence of the second linearization, with motion parallel to $\bm{B}$ above 20 mT. The magnetic moment must have a non-zero component in the vertical plane, to drive motion collinear with $\bm{B}$. (f-h) Normalized distributions of the angular displacements computed over 10-frame segments of the trajectories shown in (c-e). A 12° tolerance window is defined and highlighted in blue (red) for characterization of the perpendicular (parallel) mode, with remaining displacements colored in gray. (h) Homogeneous distribution in the absence of $\bm{B}$, corresponding to the trajectories in (c). (h) Collapse in the distribution of angular displacements of trajectories shown in (d), as roller motion is driven perpendicular to the magnetic field axis when a 13 mT magnetic field is applied. (i) The emergence of dual-anisotropy with the coexistence of the perpendicular and parallel modes at higher field intensities, evident from the trajectories in (e). (f) The probability of observing a roller in a given mode at a specific field strength. This is estimated by computing the fraction of time each roller spends in the perpendicular or parallel mode, and the ensemble average of this fraction is plotted as a function of the magnetic field intensity.
• Figure 2: (a-c) The trajectory of a single roller confined to a 0.25 mm circular region (scale bar is 50 µ m, see supplementary movies 4-6), under the same field conditions as in Fig. \ref{['fig:1']}, with $\bm B$ along the vertical axis and $\bm E$ coming out of the page. The trajectory is colored by time, with a circle (cross) denoting the start (end) point, while (d-f) show the corresponding angular distributions. (a), (d) Circular motion and a homogeneous distribution at 0 mT. (b) Motion perpendicular to the field dominates the trajectory, evident in the collapse of the angular distribution in (e) at intermediate field strengths. (c), (f) Emergence of motion parallel to the field at higher field strengths, with the roller switching spontaneously between parallel and perpendicular modes. This points to the dual mode capacity of rollers, rather than an intrinsic preference for a particular mode at high fields. (g-i) Stills from raw videos (see supplementary movie 7) capturing non-conservative interactions between rollers at $|\bm E| = 2E_c$ and $\bm B$ = 61 mT, demonstrate switching back and forth between perpendicular and parallel rolling. Sequential columns show the rollers before, during and after a collision respectively, as indicated by the annotated time. (g) Two rollers, initially in orthogonal modes (perpendicular (blue) and parallel (red)), interact and both emerge in the perpendicular mode. (h) A roller switches from perpendicular to parallel rolling, following a head-on collision with another perpendicular roller. (i) A collision between orthogonal rollers triggers a switch from perpendicular to parallel rolling. We emphasize the two-way switching between modes, despite the perpendicular motion being more stable and energetically favorable. Scale bar is 50 µ m.
• Figure 3: Results from dynamic numerical simulations of an isolated sphere with an anisotropic magnetic susceptibility under the influence of a driving electric and steering magnetic field. Simulations are carried out for a range of initial conditions given by $\phi$---the direction of motion---and $\theta$---the rotation of the soft axis in the xy-plane. (a) First 1250$\tau$ segment of a 5000$\tau$ trajectory resolving to perpendicular motion: $\phi$ = 25.5°, $\theta$ = 0°. (b) First 1250$\tau$ segment of a trajectory resolving to parallel motion: $\phi$ = 23.5°, $\theta$ = 0°. (c) Phase diagram showing the converged mode from a 5000$\tau$ sweep over pairwise values for $\phi$ and $\theta$ from -180° to 180°, with resolution 400$\times$400. Blue (red) denotes the roller converging to the perpendicular (parallel) mode. Enlarged region shows the proximity of the initial conditions corresponding to the trajectories shown in (a, b): '$\bm \times$' ('$\parallel$') is the perpendicular (parallel) trajectory. (d) The dynamics of the soft axis as a function of time, in the frame of the virtual roller for the final 250$\tau$ of the trajectory in (a). The soft axis spirals towards a stable fixed point, located at the spiral center. Every 100$^{\text{th}}$ point is plotted, as for a 250 $\tau$ segment at this scale, the spiral generated from the inclusion of every point would be overly dense. (e) The dynamics of the soft axis as function of time, in the frame of the roller body from $50-750\tau$ of the trajectory in (b), with every 100$^{\text{th}}$ point shown once again. The soft axis enters a steadily oscillating state, strongly reminiscent of a limit cycle in the xz-plane.