Research ArticleAPPLIED SCIENCES AND ENGINEERING

Flocking ferromagnetic colloids

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Science Advances  15 Feb 2017:
Vol. 3, no. 2, e1601469
DOI: 10.1126/sciadv.1601469
  • Fig. 1 Schematics of the experiment and particle trajectories.

    (A) Schematics of the experiment. A nonsmooth ferromagnetic particle with the magnetic moment μ is energized by an applied vertical ac magnetic field. Particle rotation leads to a self-propelled motion with the velocity V. (B) Trajectories of 25 particles observed over 100 periods of the ac magnetic field. Some particles escape the center of the lens, whereas other particles remain near the center of the lens most of the time. Scale bar, 1 mm.

  • Fig. 2 Main observed phases.

    (A to D) Snapshots of the individual particle velocities for the four major phases: gas (f = 20 Hz) (A), flocking (f = 30 Hz) (B), vortex (f = 40 Hz) (C), and reentrant flocking (f = 50 Hz) (D) (see also movies S1 to S8). Individual flocks in (B) and (D) are accented by light colors. (E to H) Magnitude of the corresponding coarse-grained velocity fields. Scale bars, 1 mm.

  • Fig. 3 Collective particle dynamics.

    (A) Spatial particle velocity correlation functions for different frequencies; here, a is the particle radius. Inset: Correlation length versus frequency f. When the correlation length becomes comparable to the system size for the frequency f ≈ 40 Hz, the particle motion is self-organized into a large vortex. The dashed line serves as a guide to the eyes. (B) Rotational order parameter φR for gas, flocking, and vortex states. (C) Vortex velocity profile versus distance from the center.

  • Fig. 4 Individual particle dynamics.

    (A) Mean square displacement (MSD) of individual particles at frequency f = 40 Hz and magnetic field magnitude H0 = 70 Oe; the dashed line is a linear law. For longer times, the curve saturates due to finite system size. Inset: Angular mean square displacement (〈δϕ2〉) for the same experimental conditions. (B) Particle velocity distribution function; the average is taken over 4 × 104 instantaneous velocity values of individual rollers at f = 40 Hz. (C) Typical particle velocity normalized by maximally attained velocity V0 = ωa, with ω = 2πf. The dashed line indicates the value of the slipping parameter αs used in simulations.

  • Fig. 5 Diagrams of the dynamic states.

    (A) Individual particle dynamic states in the absence of noise. Solid line shows the stability limit of the locked state (Eq. 7). Below the solid line, the particle is oriented along the field direction. The thin dashed line depicts the rotating state existence boundary from the condition μH0 = 2αrω. Chaotic regimes, characterized by spontaneous reversal of the rotation direction, exist above the black diamond line. The symbols show the locations of gas, flocking, and vortex states. (B) Diagram of the dynamic states spanning the magnitude H0 and the frequency f of the alternating external field obtained from numerical simulations for rotational diffusion DR = 1 s–1. Squares correspond to gas, triangles correspond to flocking, and circles correspond to vortex. The vortex state with maximal correlation length is shown in light blue circles. Data points inside the gray rectangle are also shown in (A). (C) Diagram of the dynamic states for the parameters of (B) but without noise (DR = 0). Flocking and gas phases for higher frequencies are replaced by a stationary crystal-like phase (× symbols). Inset: Stationary crystal phase observed for high frequencies without noise.

  • Fig. 6 Dynamic states.

    (A to D) Snapshots of the emerging dynamic states for H0 = 11 dimensionless units: gas (f = 6 Hz), flocking (f = 20 Hz), vortex (f = 47 Hz), and reentrant flocking (f = 88 Hz) (see also movies S9 to S12). Arrows indicate the velocity of each particle, and the flocks are highlighted by a green background. (E to H) Corresponding intensity plot of the coarse-grained velocity fields.

  • Fig. 7 Characterization of the dynamic states.

    (A) Rotational order parameter φR for gas, flocking, and vortex states as a function of time (normalized by the filed period). (B) Tangential velocity profile for the vortex state as a function of distance from the center. The velocity is scaled by the reduced single-particle velocity αsV0. (C) Spatial particle velocity correlation functions for the different dynamic states; dashed lines correspond to the results including hydrodynamic interactions. Data are shown for the following frequencies: gas (f = 6 Hz), flocking (f = 20 Hz), and vortex (f = 47 Hz). (D) Correlation length Cd as a function of frequency; open symbols denote the results considering hydrodynamic interactions.

  • Fig. 8 Particle orientation distributions.

    (A to D) Stroboscopic plots of orientation angles θi between the particle magnetic moments and the vertical direction. Angles versus time for all particles in the system is shown in (B) and (C) and only for one particle in (A) and (D) for clarity. Colors mark individual particles, time is measured in the field periods 2π/ω, angles are extracted at the moments of maximal field, and sin(ωt) = 1. Gas phase for f = 6 Hz (A), low-frequency flocking for f = 20 Hz (B), vortex for f = 47 Hz (C), and high-frequency (reentrant) flocking for f = 88 Hz (D). (E) Probability distribution functions for the angles θi for different frequencies.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/2/e1601469/DC1

    movie S1. Experimental movie for the gas-like state at frequency f = 20 Hz.

    movie S2. Experimental movie for the flocking state at frequency f = 30 Hz.

    movie S3. Animation of the experimentally obtained flocking state at frequency f = 30 Hz, where the direction of motion is indicated by the color code.

    movie S4. Experimental movie for the vortex state at frequency f = 40 Hz.

    movie S5. Animation of the experimentally obtained vortex state at frequency f = 40 Hz, where the direction of motion is indicated by the color code.

    movie S6. Experimental movie for the reentrant flocking state at frequency f = 50 Hz.

    movie S7. Animation of the experimentally obtained reentrant flocking state at frequency f = 50 Hz, where the direction of motion is indicated by the color code.

    movie S8. Experimental movie for the gas-like state at frequency f = 60 Hz.

    movie S9. Numerically obtained gas-like state at frequency f = 6 Hz, indicating the direction of motion by the color code.

    movie S10. Numerically obtained flocking state at frequency f = 20 Hz, indicating the direction of motion by the color code.

    movie S11. Numerically obtained vortex state at frequency f = 47 Hz, indicating the direction of motion by the color code.

    movie S12. Numerically obtained reentrant flocking state at frequency f = 88 Hz, indicating the direction of motion by the color code.

    movie S13. Numerically obtained vortex state, considering hydrodynamic interactions, at frequency f = 47 Hz, indicating the direction of motion by the color code.

  • Supplementary Materials

    Other Supplementary Material for this manuscript includes the following:

    • movie S1 (.avi format). Experimental movie for the gas-like state at frequency f = 20 Hz.
    • movie S2 (.avi format). Experimental movie for the flocking state at frequency f = 30 Hz.
    • movie S3 (.avi format). Animation of the experimentally obtained flocking state at frequency f = 30 Hz, where the direction of motion is indicated by the color code.
    • movie S4 (.avi format). Experimental movie for the vortex state at frequency f = 40 Hz.
    • movie S5 (.avi format). Animation of the experimentally obtained vortex state at frequency f = 40 Hz, where the direction of motion is indicated by the color code.
    • movie S6 (.avi format). Experimental movie for the reentrant flocking state at frequency f = 50 Hz.
    • movie S7 (.avi format). Animation of the experimentally obtained reentrant flocking state at frequency f = 50 Hz, where the direction of motion is indicated by the color code.
    • movie S8 (.avi format). Experimental movie for the gas-like state at frequency f = 60 Hz.
    • movie S9 (.avi format). Numerically obtained gas-like state at frequency f = 6 Hz, indicating the direction of motion by the color code.
    • movie S10 (.avi format). Numerically obtained flocking state at frequency f = 20 Hz, indicating the direction of motion by the color code.
    • movie S11 (.avi format). Numerically obtained vortex state at frequency f = 47 Hz, indicating the direction of motion by the color code.
    • movie S12 (.avi format). Numerically obtained reentrant flocking state at frequency f = 88 Hz, indicating the direction of motion by the color code.
    • movie S13 (.avi format). Numerically obtained vortex state, considering hydrodynamic interactions, at frequency f = 47 Hz, indicating the direction of motion by the color code.

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