Research ArticleAPPLIED SCIENCES AND ENGINEERING

Cargo capture and transport by colloidal swarms

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Science Advances  24 Jan 2020:
Vol. 6, no. 4, eaay7679
DOI: 10.1126/sciadv.aay7679
  • Fig. 1 Cargo capture and transport using multiagent feedback-controlled colloidal swarm.

    (A) Feedback control system consists of multiple self-propelled Janus particles and a cargo particle on a light or electrode array. At each control update period, imaging senses the system state including swarm particle positions and orientations and cargo position. On the basis of the system state and a given control objective, the policy determines each particle’s propulsion speed, which is actuated for each particle by a light array (or other arrays that mediate locally actuated transport mechanisms). (B) Control policy determines each swarm particle propulsion speed by (i) first assigning swarm particles to track target locations around cargo (represented by dashed lines) and (ii) then path planning. (C) Scheme for controlling swarm particles to navigate to target lattice to cage cargo (Algorithm 1), which is unmodified for transient and steady-state capture processes. (D) Following cargo capture, cargo transport is realized by maintaining steady-state capture while simultaneously identifying a subset of swarm particles oriented within a threshold of the desired direction to produce swarm and cargo translation without swarm breakup (Algorithm 2).

  • Fig. 2 Trajectories in demonstration of cargo capture and transport by 90 particle swarms.

    (A) Representative configurations for 90 particle swarms with 0kT attraction at t = 0 s (I), 30 s (II), 240 s (III), and 870 s (IV). (B) Cargo (black) and swarm particle (inset: spectrum time scale) trajectories (movies S7 to S9) for 90 particles with pair attraction of 0kT (left), 3.2kT (middle), and 5.3kT (right) during capture (<300 s) and transport (>300 s) using feedback control (Algorithms 1 and 2). Cargo starts at origin, and swarm is in initial circular configuration ~20a from cargo. (C) All swarm particle distances to their assigned targets, ΔdS, (gray lines) and their mean value (black line). (D) Cargo distance relative to the swarm center of mass, ΔdC. (E) Instantaneous power (kT per second) based on propulsion speeds for the entire swarm (red), swarm particles involved in capture (green), and swarm particles involved in transport (blue), where WS = WC + WT (red is on top of the green curve during initial capture process).

  • Fig. 3 Swarm structure and forces during steady-state capture.

    (A) Column 1: Two-dimensional density profile of 90 particle swarms at steady-state surrounding Brownian cargo under feedback control for pair attractions of 0kT (top), 3.2kT (middle), and 5.3kT (bottom) (coordinate system relative to swarm center of mass). Column 2: Magnitude of inward radial force component (units are N/a2) due to swarm particle self-propulsion. Swarm size and pair attraction dependence of (B) cargo mean displacement relative to swarm center of mass, (C) cargo diffusivity relative to free-space value, and (D) swarm mean displacement from targets.

  • Fig. 4 Swarm structure and forces during steady-state transport.

    (A) Column 1: Two-dimensional density profile of 90 particle swarms at steady-state surrounding Brownian cargo under feedback control for pair attractions of 0kT (top), 3kT (middle), and 5kT (bottom) (coordinate system relative to swarm center of mass). Magnitudes of force densities (units are N/a2) in (column 2) transport direction and (column 3) inward radial directions due to swarm particle self-propulsion. Swarm size and pair attraction dependence of (B) cargo mean displacement relative to swarm center of mass, (C) cargo diffusivity relative to free-space value, and (D) swarm mean displacement relative to target sites.

  • Fig. 5 Steady-state swarm speed, power, and efficiency versus swarm size and attraction.

    (A) Scheme to illustrate the interplay of self-propulsion assignments, pair attraction, and swarm size during transport. (B) Cargo transport mean speed and its SD as fraction of max single particle speed, vmax. Dashed line is theoretical maximum relative speed of ~(2π)−1 (Eq. 5). Gray triangles show locus of points for threshold combinations of swarm size and attraction to generate finite transport speeds. (C) Swarm capture power (where WS = WC, WT = 0) mean and its SD. Gray triangles show locus of points indicating lowest power at each level of attraction, which progressively changes from the largest swarm at no attraction to the smallest swarm at the highest attraction. (D) Ratio of transport power to total swarm power during cargo transport while simultaneously maintaining cargo capture (where WS = WC + WT). (E) Energy efficiency (fuel efficiency) during cargo transport. Gray triangles show locus of points indicating highest efficiency at each level of attraction. Inverted black triangles are theoretical maximum efficiency from Eq. 7 for each system size.

  • Table 1 Brownian dynamic simulation parameters.

    ParameterEquationValueParameterEquationValue
    a (nm)*81000Dt (m2/s)82.145 × 10−13
    B (a/kT)102.29Dr (rad2/s)§80.161
    κ−1 (nm)1050N86, 18, 36, 60, 90
    L#10200vmax (m/s)**85 × 10−6
    ΔΠ (10−8kT/nm3)††100, 3.8, 5.8, 8.8ΔtC (s)‡‡1 and 20.1

    *Particle radii.

    †Translational diffusivity.

    ‡Electrostatic prefactor.

    §Rotational diffusivity.

    ‖Debye length.

    ¶Swarm size.

    #Depletant radius.

    **Maximum propulsion speed.

    ††Osmotic pressure.

    ‡‡Control update time.

    Supplementary Materials

    • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/4/eaay7679/DC1

      Section S1. Nomenclature

      Section S2. Supplemental Methods and Results

      Fig. S1. Swarm particle pair potentials with different amounts of attraction.

      Fig. S2. Examples illustrating the optimal assignment solution.

      Fig. S3. Optimal control policy for single particles–single targets, multiple particles–multiple targets, and cargo capture.

      Fig. S4. Osmotic pressure from sedimentation equilibrium and swarm configurations.

      Fig. S5. Steady-state density and force distribution during the capture process for N = 60.

      Fig. S6. Steady-state density and force distribution during the transport process for N = 60.

      Fig. S7. MSD analysis for cargo and swarm particles during steady-state capture.

      Fig. S8. MSD analysis for cargo and swarm particles during steady-state transport.

      Fig. S9. Steady-state energy consumption rate for swarm particles during cargo transport.

      Fig. S10. Example extensions of colloidal swarm functions.

      Movie S1. High-quality three-dimensional rendering of cargo capture by N = 60 swarms for UM = 5.3kT.

      Movie S2. Cargo capture by N = 90 swarms for UM = 0kT.

      Movie S3. Cargo capture by N = 90 swarms for UM = 3.2kT.

      Movie S4. Cargo capture by N = 90 swarms for UM = 5.3kT.

      Movie S5. Cargo capture by N = 6 (0kT), 18 (5.3kT), 36 (5.3kT), and 60 (3.2kT) swarms.

      Movie S6. Crystalline swarm melting without capture control for N = 18, 36, 60, and 90 and UM = 5.3kT.

      Movie S7. Cargo transport by N = 90 swarms for UM = 0kT.

      Movie S8. Cargo transport by N = 90 swarms for UM = 3.2kT.

      Movie S9. Cargo transport by N = 90 swarms for UM = 5.3kT.

      Movie S10. Cargo capture and transport by N = 36 (3.2kT), 36 (5.3kT), 60 (3.2kT), and 60 (5.3kT) swarms.

      Movie S11. Cargo transport without capture control for N = 18 (8.7kT), 36 (5.3kT), 60 (5.3kT), and 90 (5.3kT) swarms.

    • Supplementary Materials

      The PDFset includes:

      • Section S1. Nomenclature
      • Section S2. Supplemental Methods and Results
      • Fig. S1. Swarm particle pair potentials with different amounts of attraction.
      • Fig. S2. Examples illustrating the optimal assignment solution.
      • Fig. S3. Optimal control policy for single particles–single targets, multiple particles–multiple targets, and cargo capture.
      • Fig. S4. Osmotic pressure from sedimentation equilibrium and swarm configurations.
      • Fig. S5. Steady-state density and force distribution during the capture process for N = 60.
      • Fig. S6. Steady-state density and force distribution during the transport process for N = 60.
      • Fig. S7. MSD analysis for cargo and swarm particles during steady-state capture.
      • Fig. S8. MSD analysis for cargo and swarm particles during steady-state transport.
      • Fig. S9. Steady-state energy consumption rate for swarm particles during cargo transport.
      • Fig. S10. Example extensions of colloidal swarm functions.
      • Legends for movies S1 to S11

      Download PDF

      Other Supplementary Material for this manuscript includes the following:

      • Movie S1 (.avi format). High-quality three-dimensional rendering of cargo capture by N = 60 swarms for UM = 5.3kT.
      • Movie S2 (.avi format). Cargo capture by N = 90 swarms for UM = 0kT.
      • Movie S3 (.avi format). Cargo capture by N = 90 swarms for UM = 3.2kT.
      • Movie S4 (.avi format). Cargo capture by N = 90 swarms for UM = 5.3kT.
      • Movie S5 (.avi format). Cargo capture by N = 6 (0kT), 18 (5.3kT), 36 (5.3kT), and 60 (3.2kT) swarms.
      • Movie S6 (.avi format). Crystalline swarm melting without capture control for N = 18, 36, 60, and 90 and UM = 5.3kT.
      • Movie S7 (.avi format). Cargo transport by N = 90 swarms for UM = 0kT.
      • Movie S8 (.avi format). Cargo transport by N = 90 swarms for UM = 3.2kT.
      • Movie S9 (.avi format). Cargo transport by N = 90 swarms for UM = 5.3kT.
      • Movie S10 (.avi format). Cargo capture and transport by N = 36 (3.2kT), 36 (5.3kT), 60 (3.2kT), and 60 (5.3kT) swarms.
      • Movie S11 (.avi format). Cargo transport without capture control for N = 18 (8.7kT), 36 (5.3kT), 60 (5.3kT), and 90 (5.3kT) swarms.

      Files in this Data Supplement:

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