Emergent hydrodynamic bound states between magnetically powered micropropellers

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Science Advances  26 Jan 2018:
Vol. 4, no. 1, eaap9379
DOI: 10.1126/sciadv.aap9379
  • Fig. 1 Propulsion of the hematite micropropellers.

    (A) Schematic showing two hematite particles subjected to a rotating magnetic field circularly polarized in the (x, z) plane. (B) Scanning electron microscope image of the hematite particles, where a and b are the long and short axes, respectively. (C) Orientational angle θ between the particle long axis and an applied field H. The particle moment is obtained by balancing the magnetic and viscous torque during reorientation (see Materials and Methods). (D) Average speed 〈vx〉 versus driving frequency ω of one micropropeller subjected to a rotating field with amplitude of H0 = 4400 A m−1. Orange and blue fits denote synchronous and asynchronous regimes. (E) Microscope images showing a pair of propelling particles at frequencies ω = 188.5 rad s−1 (left) and ω = 502.6 rad s−1 (right). Scale bar, 10 μm. (see movie S1). Bottom: The evolution with time of the relative positional angle θ.

  • Fig. 2 Observation of the bound states.

    (A to C) Sequence of graphs illustrating (A) the time evolution of the angle ϑ, (B) particle velocities v1, v2, and (C) normalized relative velocity squared α = (vrel/vcv)2 for a pair of propellers driven by a rotating field with amplitude of H0 = 4400 A m−1 and at two different frequencies. The first column refers to ω = 125.7 rad s−1 and second and third columns to ω = 502.6 rad s−1. The increase in speed of the pair of propellers is shown in movie S2. The time t = 0 corresponds to an arbitrary starting point. (D and E) Probabilities P(τ) of lifetimes τ, where the pair of propellers have α < 0.01 for frequencies ω = 125.7 rad s−1 (D) and ω = 502.6 rad s−1 (E). The continuous red lines are fits to the data of an algebraic function as a guide to the eye.

  • Fig. 3 Self-assembly due to sole HI.

    (A) Sequence of images showing the formation of a chain of five propellers under a rotating field with amplitude of H0 = 4400 A m−1 and frequency of ω = 502.6 rad s−1. Scale bar, 10 μm. The time t = 0 corresponds to an arbitrary staring point. The corresponding video is movie S3. (B) Evolution with time of the average positional angle ϑ (top), 〈vcv〉 in blue and 〈vrel〉 in orange (bottom). The shaded red region in the graph denotes the assembly stage. Inset illustrates the flow velocity ux calculated at the same elevation of a chain composed of five propellers.

  • Fig. 4 Hydrodynamic bound states.

    (A and B) 3D trajectories of two particles (one in blue and the other in red) in a hydrodynamic bound state. The data are obtained from numerical simulations of the equations in the Supplementary Materials, with an initial angle ϑ = 45°, a distance of d = 2.67a, being a the particle radius for Fg/h = 0 (A) and Fg/h = 10−1 (B). Fg/h denotes the ratio between the gravitational Fg and vicous Fh forces (see movie S4). (C and D) Distance (C) and angle (D) between the two particles measured in term of the particle radius a versus period of the driving field for different values of Fg/h.

  • Fig. 5 Average center of velocity of the pair.

    (A and B) Components of the velocity 〈vcv〉 along the Embedded Image (A) and Embedded Image (B) directions versus angle ϑ. The pair of propellers forming the bound states is driven by a rotating field with amplitude H0 = 4400 A m−1 and frequency ω = 502.6 rad s−1. 〈v0〉 = 10.3 μm s−1 denotes the speed of a single propeller driven by the same field. r is the center-to-center distance. Scattered symbols denote experimental data; continuous lines are fits following the model developed in the text.

Supplementary Materials

  • Supplementary Materials

    This PDF file includes:

    • Theoretical Model

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    Other Supplementary Material for this manuscript includes the following:

    • movie S1 (.avi format). Movie corresponding to Fig. 1E.
    • movie S2 (.avi format). Movie corresponding to Fig. 2B.
    • movie S3 (.avi format). Movie corresponding to Fig. 3A.
    • movie S4 (.avi format). Movie corresponding to Fig. 4A.

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