Programmable self-organization of heterogeneous microrobot collectives

Significance Microscale collectives composed of simple, locally reactive constituents can harness the effects of self-organization to enable diverse global behaviors. While phase separation of homogeneous collectives is well studied, heterogeneous collectives are relatively unexplored. This study focuses on a collective of magnetic microdisks of different sizes and examines how the group can self-organize into homogeneous subgroups using an external magnetic field. We find that heterogeneity enables collective behaviors including morphology reconfiguration, organized aggregation, dispersion, and locomotion, and caging and expulsion of external objects. Our work furthers insights into self-organization of heterogeneous microrobot collectives and may provide useful insights into the future of active matter.


This PDF file includes:
Appendix A Section 1. Pairwise Interaction Forces. Section 2. Physical Model. Section 3. Asymmetric Pairwise Interactions. Section 4. Calculation of entropy by neighbor distances.

Section 1. Pairwise Interaction Forces
This discussion reviews the pairwise interactions present in our microrobot collective system: the magnetic dipole-dipole force, the capillary force, and the hydrodynamic force.

Magnetic dipole force:
Each micro-disk is sputtered with a Cobalt thin film that make the disks ferromagnetic with a permanent magnetic dipole. The magnetic dipole-dipole force is attractive on average over one rotation and is caused by the magnetic dipole on each micro-disk. A rotating magnetic field exerts magnetic torque on each micro-disk; this causes each micro-disk's dipole to align with the instantaneous magnetic field vector and spin about its own axis. At high frequencies, in the step-out regime, micro-disks are unable to spin at the magnetic field vector's angular velocity; here, a micro-disk's response to the magnetic field is more erratic and in the right situations can lead to interesting emergent behaviors. The step-out frequency is dependent on several parameters, including the volume of the magnetic material on each micro-disk, the fluid medium's viscosity, and the magnitude of the external magnetic field.
Capillary force: Capillary interactions are weak since the micro-disks are flat; however, Supplementary Fig. 1 shows a flat 400 μm -diameter micro-disk at a fluid interface where small irregularities around its perimeter slightly deform the air-water interface; this enables weak capillary forces along those regions. Previous studies with a similar system show that micro-disks' edges can be patterned with corrugations to increase capillary attraction along certain orientations to form various lattice structures; two micro-disks experience attractive capillary forces when their corrugations are aligned and are repulsive when they are misaligned; at lower magnetic field frequencies, these micro-disks' capillary interactions dominate, and they are able to aggregate into specific types of structures. For the purposes of this study, micro-disks have no edge corrugations, and the numerical model assumes that there are no capillary interactions between agents.

Hydrodynamic force:
The third pairwise interaction is the repulsive hydrodynamic force which is modulated by micro-disks' rotation behavior in response to the rotating magnetic field. The hydrodynamic lift force is dependent on the size of the micro-disks and it can be non-symmetric if the sizes of the two micro-disks are different. The hydrodynamic force enables micro-disks to increase their neighbor distance; as the rotation frequency increases so does the repulsive force, which causes the collective to have a larger radius at a higher frequency. When the rotation frequency surpasses a microdisk's step-out frequency, its angular velocity becomes non-uniform and the hydrodynamic repulsion it exerts on neighbors is significantly reduced. Throughout most experiments, we keep the rotation frequency below the micro-disks' step-out frequency which enables us to use a numerical model to reproduce some of the self-organization behaviors we observe in experiments. In the last portion of our experimental selforganization, we demonstrate that the collective exhibits a different form of self-organization when part of the collective has stepped out.

Section 2. Physical model.
The model used for simulations was adapted from literature 20 and modified to include the different magnetic field profile: where and are the position vectors of micro-disks; = − is the vector pointing from the center of micro-disk to the center of micro-disks ; and are the orientations of micro-disks; is the angle of dipole moment with respect to . It is assumed to be the same for both micro-disks, as = = ; is the instantaneous spin speed of ℎ micro-disk; = | | is the magnetic field strength (10 mT); = arctan( / ) is the orientation of the external magnetic field; Ω and Ω are the oscillation frequencies of the x and y component of the external magnetic field respectively; is the radius of ℎ micro-disk; µ is the dynamic viscosity of water (10 -3 Pa·s); is the density of water (10 3 kg/m 3 ); is the magnetic dipole moment of the ℎ micro-disk (0.44 * 2 A·m 2 ); − , , and − , , are the magnetic dipole force on and off the center-to-center axis, respectively, and they are functions of and ; (see Ref. 20 or the details) − , , is the magnetic dipole torque, and it is a function of and ; , , is the capillary force, and it is a function of and and embeds the symmetry of a micro-disk; , , is the capillary torque, and it is a function of and and embeds the symmetry of a micro-disk; , ℎ , , and are the distances of a micro-disk to the four sides of the physical boundary; is the magnitude of the force due to curvature of the air-water interface and is set to be 1 × 10 −9 N; center is the position vector of the center of the arena; is the radius of the arena.
If the center-center distance < + , a repulsion term is added to the force equation, where is a small number (10 -10 µm/R); is set to be 10 -7 N.
: In a heterogeneous collective with two radii present in the system ( 1 and 2 ), there are four different possible sets of pairwise interactions between the micro-disks. A disk with radius 1 exerting forces on a disk with radius 2 ; a disk with 2 exerting forces on a disk with 1 ; a disk with 1 exerting forces on a disk with 1 ; a disk with 2 exerting forces on a disk with 2 . Each of these pairwise interactions result in a different total force ( ( )), which means that the exerted force will be unequal between two neighboring disks of differing radii and thus the calculated local spring constant ( =              Hz. Movie S13. Anisotropic Deformation Under Isotropic Compression. Self-organized collective deforms into an ellipse-like configuration after being compressed isotropically. Movie S14. Organized Collective Locomotion. A self-organized collective maintains its organized state as it moves. Movie S15. Caging and Expulsion of Passive Objects. Passive objects of different sizes are caged and expelled according to their size.