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Edited by: Sanja Dogramadzi, University of the West of England, United Kingdom

Reviewed by: Bradley J. Nelson, ETH Zürich, Switzerland; Serhat Yesilyurt, Sabanci University, Turkey

This article was submitted to Biomedical Robotics, a section of the journal Frontiers in Robotics and AI

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Several microorganisms swim by a beating flagellum more rapidly in solutions with gel-like structure than they do in low-viscosity mediums. In this work, we aim to model and investigate this behavior in low Reynolds numbers viscous heterogeneous medium using soft microrobotic sperm samples. The microrobots are actuated using external magnetic fields and the influence of immersed obstacles on the flagellar propulsion is investigated. We use the resistive-force theory to predict the deformation of the beating flagellum, and the method of regularized Stokeslets for computing Stokes flows around the microrobot and the immersed obstacles. Our analysis and experiments show that obstacles in the medium improves the propulsion even when the Sperm number is not optimal (_{p} ≠ 2.1). Experimental results also show propulsion enhancement for concentration range of 0−5% at relatively low actuation frequencies owing to the pressure gradient created by obstacles in close proximity to the beating flagellum. At relatively high actuation frequency, speed reduction is observed with the concentration of the obstacles.

Efficient propulsion on the microscale is one of the main targets of micro- and nanorobotics research. Various propulsion mechanisms have been demonstrated in the last decades (Nelson et al.,

The ability to actuate artificial microswimmers in complex environments is likely to be an important advancement toward their translation into

A soft microrobotic sperm swims in a viscous heterogeneous medium under the influence of a periodic magnetic field. _{1}(_{2}(_{1}(

A soft microrobotic sperm swims by a beating flagellum in low Reynolds numbers. The flagellar wave propagation along its tail is achieved by exerting a periodic magnetic torque on the magnetic head of the microrobotic sperm. In the case of a medium with immersed particles, the elastic tail of the microrobotic sperm interacts with the surrounding fluid and the immersed particles. The resistive-force theory (RFT) is implemented to predict the influence of the particles on the deformation of the elastic tail and the flagellar propulsion.

The soft microrobotic sperm consists of a prolate spheroidal head of length 2_{t}. The microrobots are allowed to swim in a medium with viscosity μ, characterized by low Reynolds numbers hydrodynamics (_{x}(^{−5}), where ρ is the density of the medium and _{x} is the swimming speed. The medium contains randomly distributed and isotropic spherical particles with an average diameter 2_{p} and concentration _{p}^{2}

where _{1}(_{2}(_{1}(_{2}(_{1}(_{n} is the following normal drag coefficient (Leshansky,

where _{p}(α_{p}) is the modified Bessel function of degree

where

The concentration (φ) and size (_{p}) of the immersed obstacle influence the normal drag coefficient (_{n}) and the Sperm number (_{0}(α_{p}) and _{1}(α_{p}) based on (2). _{t} = 8 μm,

The tangential drag coefficient (_{t}) is also affected by the concentration and size of the immersed particles and is calculated using

The normal and tangential drag coefficients provide an anisotropic operator that relates the hydrodynamic drag force exerted on a segment (δ_{f} on δ

where _{y} is the transverse velocity component of the segment δ

In contrast to sperm cells and flagellated microorganisms, our soft microrobotic sperm depends on an external magnetic field with a sinusoidally varying orthogonal component to achieve flagellar propulsion. This magnetic field exerts a magnetic torque (_{1}(^{2}^{2} = 0 and ∂^{3}^{3} = 0. The magnetic field

The fluid surrounding the soft microrobotic sperm and the immersed particles are influenced by the beating tail. The governing fluid mechanics for a soft microrobotic sperm in low Reynolds numbers are given by the following Stokes equation:

where _{k} at a point _{k} along the flexible tail is approximated by (Cortez,

where _{k} = |_{k}| and ϵ is a parameter that describes the sharpness of a delta-function. This delta function approximates the forces exerted by the beating tail on the fluid. The velocity field, due to force _{k} at points _{k}, is given by

Equation (2.2) can be used to calculate the velocity field given the force exerted by the flexible tail on the surrounding fluid. It can also be used to calculate the necessary forces _{k} at Stokeslets points to initiate the given velocities _{k} at positions _{k}. _{y} = d_{t}. The sharpness of the delta function is ϵ = 0.25d

A soft microrobotic sperm achieves flagellar propulsion in a medium with immersed particles (black circles).

The deformation of the tail and the drag forces are determined using finite-difference discretization of (1). The tail of the soft microrobotic sperm is discretized into ^{−3} s. The tail deformation (

The boundary conditions provide _{1} = _{2} = _{1} + Δ_{2} = _{1} + tan α sin (ω(

Similarly, the

Finally, Equations (12) and (13) are arranged in a system of

where ℓ is given by

The initial configuration of the elastic tail is set to a straight line along the propulsion axis _{1}(

Forward speed (_{x}) of the soft microrobotic sperm is calculated vs. the concentration (φ) of the immersed particles for actuation frequency range of 1 ≤ _{t} = 8 μm,

Frequency response of soft microrobotic sperm samples is studied using an electromagnetic system under microscopic guidance, for various concentrations and random initial positions of the immersed particles.

The soft microrobotic sperm samples are prepared by electrospinning a solution of polystyrene (168 N, BASF AG) in dimethylformamide (DMF) and magnetic particles with average diameter of 30 μm. The polymer concentration is 25 wt % in DMF and the weight ratio of the iron to polystyrene is 1:2. The solution is injected using a syringe pump at flow rate of 20 μl/min under the influence of an applied electric potential with electric gradient of 100 kV/m. This electric potential is applied between the syringe and a collector and beaded-fibers are fabricated and cut to provide soft microrobotic sperm samples, as shown in

The frequency response of the soft microrobotic sperm samples is characterized in the absence and presence of immersed spherical particles with average diameter of 30 μm. The area concentration is varied between 0 and 10%. In each trial, the soft microrobotic sperm is allowed to achieve flagellar propulsion for the mentioned concentration range and under the influence of oscillating magnetic fields with frequency range between 1 and 5 Hz. The frequency response is limited to this range owing to the step-out frequency of the microrobotic sperm samples (above 5 Hz). The initial position of the immersed spherical particles is influenced after each trial due to the induced flow-field by the microrobotic sperm. Therefore, the average forward speed is measured vs. the average concentration of the immersed particles. In each experiment, the soft microrobotic sperm is allowed to swim in the absence of particles, as shown in

Image sequence of a soft microrobotic sperm demonstrating flagellar propulsion inside a viscous medium with and without spherical particles. The forward swimming speed (_{x}) is measured under the influence of actuation frequency of 1 Hz. 2_{x} = 103 μm/s. _{x} = 115.8 μm/s.

Image sequence of a soft microrobotic sperm demonstrating flagellar propulsion inside a viscous medium with spherical particles at actuation frequencies of 1 and 2 Hz. The forward swimming speed (_{x}) is measured at concentration (φ) of 7%. 2_{x} = 124.1 μm/s. _{x} = 149.8 μm/s.

The forward speed (_{x}) of a soft microrobotic sperm sample is characterized vs. the concentration (φ) of spherical particles and actuation frequency (_{t} = 8 μm, _{x} for _{x} for _{x} for _{x} for _{x} for _{p} is calculated using (3).

The forward speed (_{x}) of a soft microrobotic sperm sample is characterized vs. the concentration (φ) of spherical particles and actuation frequency (_{t} = 8 μm, _{x} for _{x} for _{x} for _{x} for _{x} for _{p} is calculated using (3).

In another set of experimental results, the frequency response of a soft microrobotic sperm with relatively longer flexible tail is characterized, as shown in _{p} ≈ 2.1). Nevertheless, propulsion is enhanced for 0 < φ < 2.5% and 0 < φ < 4% at

_{n}(α) > _{t}(α) based on (6). Equations (2) and (4) indicate that the concentration of the immersed particles influences the ratio between the normal and tangential drag coefficients. This dependency provides additional explanation to the behavior of the soft microrobotic sperm samples in a medium with immersed particles. Our experimental results and simulations suggest that planar flagellar propulsion of soft artificial swimmers is enhanced at relatively low actuation frequencies. This behavior implies that the swimming velocity of these artificial swimmers is less likely to be affected in complex and crowded environments at relatively low frequencies of the beating flagellum.

The forward speed (_{x}) of a soft microrobotic sperm sample is measured (in μm/s) vs. the concentration (φ) of spherical particles and the actuation frequency (

1.2 ≤ |
1.3 ≤ |
1.1 ≤ |
1.1 ≤ |
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106 ± 17 | 105 ± 21 | 45 ± 5 | 46 ± 10 | 56 ± 8 | 82 ± 8 | 73 ± 4 | 85 ± 10 | |

1.4 ≤ |
1.6 ≤ |
1.3 ≤ |
1.3 ≤ |
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228 ± 20 | 212 ± 44 | 93 ± 30 | 124 ± 27 | 51 ± 10 | 68 ± 7 | 157 ± 10 | 142 ± 28 | |

1.6 ≤ |
1.8 ≤ |
1.4 ≤ |
1.5 ≤ |
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259 ± 33 | 369 ± 59 | 151 ± 34 | 178 ± 44 | 58 ± 5 | 81 ± 13 | 175 ± 55 | 193 ± 35 | |

1.7 ≤ |
2 ≤ |
1.5 ≤ |
1.6 ≤ |
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334 ± 104 | 435 ± 30 | 149 ± 11 | 203 ± 26 | 88 ± 7 | 110 ± 7 | 160 ± 101 | 260 ± 17 | |

1.8 ≤ |
2.1 ≤ |
1.6 ≤ |
1.7 ≤ |
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394 ± 43 | 520 ± 65 | 198 ± 5 | 268 ± 48 | 65 ± 8 | 82 ± 8 | 198 ± 61 | 246 ± 26 |

Like various microorganisms, the propulsion of soft microrobotic sperm samples is enhanced with the concentration of the immersed particles in a viscous heterogenous medium. A hydrodynamic model of the microrobotic sperm is developed based on the RFT to predict the deformation of its tail and the swimming velocity for various concentrations and actuation frequencies. Our simulation results and experiments show that the pressure field created in close proximity to the beating tail is greater than that near to the head at relatively low actuation frequencies (

IK wrote the paper, conceived the experiments, and analyzed the data. AK designed the simulation results. YH fabricated the robots and conducted the experiments. VM wrote the paper and analyzed the data. MT conducted the experiments. SM participated in drafting the paper and revising it critically.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

VM thanks the Zukunftskonzept of the TU Dresden, an excellence Initiative of the German Federal and State Government for funding.