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

TL;DR

This study analyzes a simple two-link flexible nano-swimmer actuated by a rotating magnetic field, emphasizing the transition between planar tumbling, spatial helical propulsion, and asynchronous motion as the actuation frequency changes. By introducing a phase lag $\beta=\phi-\Omega t$ and reducing the full 7-DOF dynamics to a time-invariant 4-DOF system, the authors derive explicit analytic solutions for synchronous planar and spatial motions and perform a comprehensive stability and bifurcation analysis, including a Hopf bifurcation at the step-out frequency. They also quantify the forward speed and pitch, and perform parametric optimizations over geometry, elasticity, and magnetic field properties to maximize propulsion, providing practical guidelines for nano-swimmer design. The results illuminate the nonlinear dynamics governing magnetically actuated nano-swimmers and offer optimization strategies relevant to biomedical applications, while noting model limitations and avenues for extending to more complex hinge mechanics, hydrodynamics, and swarms control.

Abstract

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.

Figures (27)

• Figure 1: Schematic sketch of the two-link nano-swimmer model. p is the center of the head link, J is the pointed revolute joint point that connects the head and tail link. The joint's axis of rotation is $\mathbf{{z'}}$, whereas the swimmers' links lie within $\mathbf{{x'}}-\mathbf{{y'}}$ plane.
• Figure 2: Numerical simulations under rotating magnetic field of f$=\Omega/2\pi=4[Hz]$, giving planar tumbling motion in $\hat{x}-\hat{y}$ plane.
• Figure 3: Numerical simulations under rotating magnetic field of f$=\Omega/2\pi=20[Hz]$, giving synchronous motion of spatial helical trajectory.
• Figure 4: Numerical simulations under rotating magnetic field of f$=\Omega/2\pi=40[Hz]$, resulting in asynchronous motion.
• Figure 5: The average swimming speed $V_z$ as function of frequency. Solid line - calculated from numerical simulations; black dots are from experimental data wu2021helical.