Towards focusing of a swarm of magnetic micro/nanomotors

Magnetic micro- and nanoparticles propelled by a rotating magnetic field provide a new technology for targeted drug delivery. The therapeutic effect of the technology is achievable with the collective action of large groups (swarms) of the motors. Narrowing of the swarm width before it reaches the target or,vice versa, its expansion prior to deposition on the channel walls are of high importance. Here we show how such swarm tuning can be achieved using a combination of the in-plane rotating and static magnetic field acting in the same plane. Although for this asymmetric actuating field the steady synchronous solutions of the problem do not materialize, the developed approach admits an analytical consideration for the average motor orientation and propulsion. The unique property of an average motion in the actuation by an asymmetric magnetic field is the emergence of propeller's net drift in plane of the field rotation. The drift velocity can be comparable in magnitude to the propulsion velocity along the axis of the field rotation. The field-induced drift is studied in detail for achiral V-shaped and chiral helical propellers, depending on their magnetization. Finally, we suggest spatial configuration of the constant magnetic field capable of focusing/defocusing swarms of magnetic motors.