Modeling and Characterization of the Passive Bending Stiffness of Nanoparticle‐Coated Sperm Cells using Magnetic Excitation

Of all the various locomotion strategies in low‐Re$Re$, traveling‐wave propulsion methods with an elastic tail are preferred because they can be developed using simple designs and fabrication procedures. The only intrinsic property of the elastic tail that governs the form and rate of wave propagation along its length is the bending stiffness. Such traveling wave motion is performed by spermatozoa, which possess a tail that is characterized by intrinsic variable stiffness along its length. In this paper, the passive bending stiffness of the magnetic nanoparticle‐coated flagella of bull sperm cells is measured using a contactless electromagnetic‐based excitation method. Numerical elasto‐hydrodynamic models are first developed to predict the magnetic excitation and relaxation of nanoparticle‐coated nonuniform flagella. Then solutions are provided for various groups of nonuniform flagella with disparate nanoparticle coatings that relate their bending stiffness to their decay rate after the magnetic field is removed and the flagellum restores its original configuration. The numerical models are verified experimentally, and capture the effect of the nanoparticle coating on the bending stiffness. It is also shown that electrostatic self‐assembly enables arbitrarily magnetizable cellular segments with variable stiffness along the flagellum. The bending stiffness is found to depend on the number and location of the magnetized cellular segments. A contactless electromagnetic‐based excitation method is presented to estimate the apparent bending stiffness of soft bio‐hybrid microrobots and nanoparticle‐coated flagella of bull sperm cells. The method works by dynamically exciting the magnetizable sperm cells using a controlled magnetic field and measuring the relaxation time after the removal of the field.