Research ArticleCOLLECTIVE BEHAVIOR

Magnetic quadrupole assemblies with arbitrary shapes and magnetizations

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Science Robotics  30 Oct 2019:
Vol. 4, Issue 35, eaax8977
DOI: 10.1126/scirobotics.aax8977
  • Fig. 1 Quadrupole magnetic modules with a tunable dipole moment.

    (A) Structure of a dipole magnet and a quadrupole module. There are four alternating poles on each side of the quadrupole module structure. (B) The simulated magnetic fields around the quadrupole modules with different angles θ between the two magnets. A “pure” dipole (θ = 180°) and a pure quadrupole module (θ = 0°) can be achieved. In this work, θ = 20° was selected for a dominant quadrupole structure with a balanced dipole moment. (C) Analytical solution of the magnetic potential evaluated at x = 0 and y = −3 mm contributed from dipole and quadrupole terms of a type A module (see “Multipole expansion of quadrupole module” section). (D) Magnetic potential energy map of two quadrupole modules (white type B module moves around fixed gray type A module). There are four local minimal energy regions (purple), which allow four stable assembly positions on each side due to the dominant quadrupole symmetry. (E) The magnetic potential energy in the line from (A) to (E), which is the central position of type B module. The relative positions between two modules are shown. (F and G) Actual assembly between two quadrupole modules. When two quadrupoles are brought close together, they will automatically assemble (movie S1). Quadrupole assemblies between the same type and different types are demonstrated. Scale bars, 5 mm.

  • Fig. 2 Combinatorial design process for quadrupole magnetic assemblies.

    (A) First, we digitalized the target structure in a 2D square lattice; each module represents a quadrupole module to be filled. The checkerboard pattern made of quadrupoles with different orientations automatically ensures the matching of north poles with south poles. This allows one to design arbitrary shape assemblies without internal magnetic frustrations. (B) Second, we chose the magnetization of each module using the hidden dipole inside each module. On the basis of the selection map, the internal dipole moment can align in four directions (up, down, left, and right) for each quadrupole module for both types of quadrupole orientations in the checkerboard. (C) The calculated assembly structures with the experimental realization (movie S1). Scale bar, 4 mm.

  • Fig. 3 Arbitrary shapes and arbitrary magnetizations of quadrupole assemblies and alignment in an external magnetic field.

    (A) Examples of complex quadrupole assemblies in 2D using the shapes of emoji pixel art. Scale bar, 1 cm. (B) The magnetization space for a given assembly structure (“Small Rocket”). The Small Rocket composed of nine quadrupole modules, and possible magnetizations were reachable for all black dots and yellow dots in the map. Cases with large magnetization were selected (yellow dots on the map) to experimentally demonstrate the alignment in an external magnetic field (10 mT pointing upward, C to N). Scale bar, 5 mm.

  • Fig. 4 Actuated soft material with programmable deformations.

    A soft segment with two quadrupole modules was proposed in this experiment. The quadrupole modules (in white) on both ends of the soft segment have no dipole moment. The deformation is completely dependent on the dipole moment of quadrupoles (in dark blue) attached to it. This allowed us to program the magnetization direction of each rigid part and to create programmable motions of structured material under uniform magnetic fields. We demonstrate a star material (A and B) and a square material (C to E) with programmable motion. The deformation could be directly controlled with externally generated magnetic fields (movie S2).

Supplementary Materials

  • robotics.sciencemag.org/cgi/content/full/4/35/eaax8977/DC1

    Fig. S1. Magnetic assembly stage for quadrupole modules.

    Fig. S2. Assemble process of a “pi” shape using the quadrupole modules in the assembly stage.

    Fig. S3. Multipole expansion of the quadrupole structures.

    Fig. S4. Strong poles and weak poles.

    Fig. S5. Magnetic potential energy map of two quadrupole modules with varying angle θ.

    Fig. S6. Ising model of magnetic quadrupole assemblies.

    Fig. S7. Design space of the magnetization of a “small rocket”.

    Fig. S8. Dynamic performance of the auxetic material assembled by quadrupole modules.

    Fig. S9. Single-segment bending motion and collapse of soft metamaterials under large magnetic fields.

    Fig. S10. Stability of quadrupole assemblies with mechanical disturbances.

    Movie S1. Assembly process of magnetic quadrupole modules.

    Movie S2. Actuation of programmable soft metamaterials assembled by magnetic quadrupole modules.

  • Supplementary Materials

    The PDF file includes:

    • Fig. S1. Magnetic assembly stage for quadrupole modules.
    • Fig. S2. Assemble process of a “pi” shape using the quadrupole modules in the assembly stage.
    • Fig. S3. Multipole expansion of the quadrupole structures.
    • Fig. S4. Strong poles and weak poles.
    • Fig. S5. Magnetic potential energy map of two quadrupole modules with varying angle θ.
    • Fig. S6. Ising model of magnetic quadrupole assemblies.
    • Fig. S7. Design space of the magnetization of a “small rocket”.
    • Fig. S8. Dynamic performance of the auxetic material assembled by quadrupole modules.
    • Fig. S9. Single-segment bending motion and collapse of soft metamaterials under large magnetic fields.
    • Fig. S10. Stability of quadrupole assemblies with mechanical disturbances.

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    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.mp4 format). Assembly process of magnetic quadrupole modules.
    • Movie S2 (.mp4 format). Actuation of programmable soft metamaterials assembled by magnetic quadrupole modules.

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