Geometric constraints and optimization in externally driven propulsion

Micro/nanomachines capable of propulsion through fluidic environments provide diverse opportunities in important biomedical applications. In this paper, we present a theoretical study on micromotors steered through liquid by an external rotating magnetic field. A purely geometric tight upper bound on the propulsion speed normalized with field frequency, known as propulsion efficiency, δ, for an arbitrarily shaped object is derived. Using this bound, we estimate the maximum propulsion efficiency of previously reported random magnetic aggregates. We introduce a complementary definition of the propulsion efficiency, δ*, that ranks propellers according to their maximal speed in body lengths per unit time and that appears to be preferable over the standard definition in a search for fastest machines. Using a bead-based hydrodynamic model combined with genetic algorithms, we determine that δ*-optimal propeller deviates strongly from the bioinspired slim helix and has a surprising chubby skew-symmetric shape. It is also shown that optimized propellers with preprogrammed shape are substantially more efficient than random magnetic aggregates. We anticipate that the results of the present study will provide guidance toward prospective experimental design of more efficient magnetic micro/nanomachines.