Flexibility of Micro-magnetic Flagella in the Presence of an Oscillating Field.

Magnetic microbeads have recently been the subject of much study because of their potential applications in microfluidic systems which facilitate mixing [1], labeling [2], separation [3] and transport [4] in lab-on-achip devices. It has been proposed that rotating superparamagnetic beads chains can enhance the mixing efficiency in microfluidic devices [5] and the magnetic micro-bead chain that is subjected to an oscillating field can be used to design a micro-swimmer [6]. To manipulate the micro-beads chain swimming in the low-Reynolds number environment, various magnetic actuation methods have been employed to obtain the higher propulsive efficiency for the locomotion in the viscous fluid. A flexible flagellum is arguably the simplest mechanism to duplicate as it is a one-dimensional structure. However, a challenge to create the stable planar beating motion for propulsion generation is to fabricate the magnetic flagellum simultaneously flexible and stable structure at a microscale. In order to effectively use an oscillating magnetic flagellum as a tail of a microswimmer, the measurement of the bending rigidity for the flexible micro-chains is essential. In this study, the curvatures and the bending rigidity of distorted chains with different lengths subjected to an external field with various controlling parameters (such as the field intensity and frequency) were probed to evaluate the flexibility of the magnetic flagellum. The magnetic flagellum is obtained by combining the self-assembling ability of micro-sized superparamagnetic particles composed the iron oxide magnetite (Fe 3 O 4 ) embedded in polystyrene microspheres suspended in distilled water. Properties of the commercially available micro-beads are density ρ=1500 kg/m3, diameter d = 4.5 μm, initial magnetic susceptibility χ = 1.6, and without magnetic hysteresis or remanence, which could be magnetized under an applying external field and completely demagnetized when the field is removed. Firstly, the magnetic beads magnetized by a homogeneous static field $H_{d}$ tended to aggregate and form the linear flagellum, and a dynamical sinusoidal field $H_{y}$ with a maximum amplitude $H_{p}$ and an adjustable frequency $f$, that is, $H_{y} = H_{p} { sin(2}\pi {ft)}$, were then applied in a direction perpendicular to $H_{d}$. Thus, the planar beating motion is created to form the dynamic flexible flagellum. In order to accurately obtain the chain flexibility and bending rigidity, we model the waggling chains as continuous elastic flagella and analytically derive their curvatures as a function of magnetic field strength. The maximum dimensionless curvature (denoted as $C_{max})$ of the chain is given as $C_{max}=d/R, R$ represents the flagellum’s radius of curvature. The bending rigidity κ of the waggling chain is measured from the S-shape of the chain by using the equation given as [4]: $C_{max}= {(Bd/4)(}\pi {/3}\kappa\mu _{0})^{0.5}$ Where the $\mu _{0}$ represents the vacuum permeability, $B$ is the overall magnetic field strength. Figure 1 shows the most significant deformed shapes for the flexible flagella of various lengths subjected to the increasing field strength. It can be seen that all the chains bend to S-shape for all values of applying field strength, and the deformation increases slightly with an increase in the perpendicular dynamic field intensity H p for each flagellum. On the other hand, the influences of flagellum strengths are not consistent. By examining figure 1 carefully, we found the increase of the length doesn’t enhance the deformation significantly. On the contrary, the longer length as flagellum consisting of 16 particles (denoted as P16) behaves more insignificant S-shape than the counterpart of the P15 chain because of the more significant constraint from the stronger hydrodynamic drag acting on a longer chain. The further inspection of the interesting finding can be done by the examination of figure 2, which shows the geometrical relation between the flagellum’s radius of curvature $R$ and the diameter $d$ of the magnetic bead. It can be seen that for all lengths of flagella the maximum curvature of the flagellum increases linearly with the intensity of the applied field. In addition, the value of $C_{max}$ of the flagellum firstly gets higher then declines with the increase of the length, which indicates the magnetic flagellum subjected to an oscillating field would have the most flexible structure at the certain length. The further discussions are shown in full paper demonstrating the effects of the other controlling parameters on $C_{max}$ and the bending rigidity κ.