Spatial dynamics of flexible nano-swimmers under a rotating magnetic field

Micro-nano-robotic swimmers have promising potential for future biomedical tasks such as targeted drug delivery and minimally-invasive diagnosis. An efficient method for controlled actuation of such nano-swimmers is applying a rotating external magnetic field, resulting in helical corkscrew-like locomotion. In previous joint work, we presented fabrication and actuation of a simple magnetic nano-swimmer composed of two nano-rods connected by a short elastic hinge. Experiments under different actuation frequencies result in different motion regimes. At low frequencies, in-plane tumbling; at higher frequencies, moving forward in a spatial helical path in synchrony with the rotating magnetic field; in further frequency increase, asynchronous swimming is obtained. In this work, we present mathematical analysis of this nano-swimmer motion. We consider a simple two-link model and explicitly formulate and analyze its nonlinear dynamic equations, and reduce them to a simpler time-invariant system. For the first time, we obtain explicit analytic solutions of synchronous motion under simplifying assumptions, for both solutions of in-plane tumbling and spatial helical swimming. We conduct stability analysis of the solutions, presenting stability transitions and bifurcations for the different solution branches. Furthermore, we present analysis of the influence of additional effects, as well as parametric optimization of the swimmer's speed. The results of our theoretical study are essential for understanding the nonlinear dynamics of experimental magnetic nano-swimmers for biomedical applications, and conducting practical optimization of their performance.