Characterizing dynamic behaviors of three-particle paramagnetic microswimmer near a solid surface
Wang et al. Robot. Biomim.
Characterizing dynamic behaviors of three-particle paramagnetic microswimmer near a solid surface
Qianqian Wang 0
Lidong Yang 0
Jiangfan Yu 0
Li Zhang 0 1
0 Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong , Shatin, Hong Kong SAR , China
1 Chow Yuk Ho Technology Centre for Innovative Medicine, The Chinese University of Hong Kong
Particle-based magnetically actuated microswimmers have the potential to act as microrobotic tools for biomedical applications. In this paper, we report the dynamic behaviors of a three-particle paramagnetic microswimmer. Actuated by a rotating magnetic field with different frequencies, the microswimmer exhibits simple rotation and propulsion. When the input frequency is below 8 Hz, it exhibits simple rotation on the substrate, whereas it shows propulsion with varied poses when subjected to a frequency between 8 and 15 Hz. Furthermore, a solid surface that enhances swimming velocity was observed as the microswimmer is actuated near a solid surface. Our simulation results testify that the surface-enhanced swimming near a solid surface is because of the induced pressure difference in the surrounding fluid of the microagent.
Swimming microrobot; Magnetic actuation; Boundary effect; Low Reynolds number; Dynamic behavior
Microswimmers remotely actuated by magnetic fields
have been considered as promising microrobotic tools
because of their great potential in biomedical
], such as targeted therapy [
], drug delivery [
] and minimally invasive surgery . Various designs of
microswimmers combined with diverse magnetic
actuation strategies have been proposed [
]. Among them,
inspired by E. coil bacterial, helical microswimmer has
drawn attention of many researchers. For propulsion
of helical microswimmers, rotating magnetic fields are
widely used for the generation of corkscrew motion at
low Reynolds number. It was reported that, actuated by
a rotating magnetic field, the “artificial bacterial flagella”
(ABF) perform versatile swimming behaviors and can
act as effective tools for cargo transport and
micromanipulation tasks [
]. These ABF were fabricated using
self-scrolling technique [
], 3-D direct laser writing [
glancing angle deposition technique [
flagellar bundles , and so on. The dynamics of such
helical swimmers have been studied systematically
below, near and higher than the step-out frequency. For
instance, to perform corkscrew motion with continuous
rotation, usually a magnetic helical swimmer should be
actuated with an input frequency that is below its
stepout frequency, whereas the actuation with a frequency
that is higher than the step-out frequency will lead to a
so-called “jerky motion” [
], i.e., the combination of
a rotation with stops and backward motions , which
results in a decrease in its translational velocity [
]. Interestingly, Ghosh et al.  reported that a helical
microswimmer could exhibit bistable behaviors under an
external field near the step-out frequency, showing
random switch between two configurations, i.e., propulsion
or tumbling motion.
Unlike the propulsion of tiny structures with
chirality in low Reynolds number regime, it has been
demonstrated recently that randomly shaped microswimmers
can also be actuated effectively using a rotating magnetic
]. These microswimmers are obtained using
iron oxide nanoparticle aggregations with varied shapes
based on hydrothermal carbonization. Alternatively,
Cheang et al.  reported that achiral three-particle
microswimmers exhibit controlled swimming motion
under a rotating magnetic field. These microswimmers
consist of three polystyrene microparticles embedded
with paramagnetic or ferromagnetic nanoparticles, and
varied swimming behaviors are triggered because of their
different magnetic properties, despite the geometrical
It is notable that recent studies of three-particle
microswimmers focus on swimming behaviors in fluid with
negligible boundary effects [
]; however, their
swimming behaviors near a solid surface can be
significantly affected due to the boundary effect. Previously, the
boundary effects were reported on both natural
swimming organisms and artificial swimmers. The influence
of solid boundaries has been observed and analyzed for
E. coil bacteria [
], and spermatozoa self-organized
into dynamic vortices resembling quantized rotating
waves on a planar surface . A solid surface affects
swimming direction of ABF, resulting in drifting
], and wobbling motion of the ABF enhances the
sidewise drift due to wall effects [
]. Simulation results
indicate that microswimmer exhibits enhanced mobility
when swimming between inclined rigid boundaries [
and a surface can deform the induced streamlines of a
rotating microagent [
Here, we report the dynamic behaviors of a
paramagnetic three-particle microswimmer, which is actuated
near a solid surface using a rotating magnetic field. With
rotation axis of the magnetic field perpendicular to the
horizontal surface, the microswimmer exhibits simple
rotation when the input frequency is below 8 Hz, whereas
it shows propulsion when subjected to a frequency
between 8 and 15 Hz (Fig. 1). Furthermore, enhanced
swimming velocity can be achieved if the
microswimmer exhibits propulsion near the surface, because of the
induced pressure difference in the surrounding fluid of the
microswimmer. While with the rotation axis of the field
parallel to the surface, the microswimmer exhibits
lowfrequency tumbling (1–3 Hz) and wobbling (3–15 Hz).
The main contributions of this work include the
following two aspects. First, a mathematical model is proposed
for the analysis of dynamic poses under different input
frequencies. Second, simulation results show that the
induced pressure near a surface can enhance swimming
velocity of a three-particle microswimmer, which are
validated by experimental results.
The remaining parts of this paper are structured as
follows. Mathematical modeling and simulations of the
microswimmer are presented in Methods section. Then,
in section Results and discussion, we discuss the dynamic
behaviors of this microswimmer, and the experimental
results are analyzed as well. Finally, Conclusions are given
in the last section.
The three-particle microswimmer is treated as a rigid
structure with two perpendicular planes of
symmetry, forms an achiral structure (Fig. 2a). It is placed on a
solid surface and actuated by a rotating magnetic field
Motion at low Reynolds numbers
The hydrodynamics of the microswimmer in low
Reynolds number regime can be described by the Stokes
η∇2u − ∇p = 0
∇ · u = 0
where p is pressure and u is moving velocity of the fluid.
The relationship of external force F together with torque
τ and translational velocity V together with angular
velocity ω is described as [
where K is the translation tensor and o is the rotation
tensor. Co is the coupling tensor, representing coupling of
translational and rotational motions of a microagent. For
the microagent in Fig. 2a, the matrices K , o and Co are
Co = 00 C032 C023
Two planes of symmetry
Fig. 2 Structure of the microswimmer and the applied magnetic field. a Microswimmer is treated as a rigid structure with two mutually
perpendicular planes of symmetry. b Schematic of the rotating magnetic field with constant flux density. Blue dashed line and arrow refer to the normal
line and rotation direction of the magnetic field, respectively. Pitch angle α is between the normal line and Z-axis, and yaw angle β is between X-axis
and projection of the normal line in the XY-plane
The magnetic force and torque exerted on the
microswimmer are given by:
F = (m · ∇)B
τ = m × B
o(m × B)
V = Co(m × B)
where m is the induced magnetic dipole moment and B is
the flux density of the magnetic field. Here we have F = 0
because the applied magnetic field has uniform flux
density. The angular and translational velocity of the
microswimmer due to induced magnetic torque is denoted by
Two equations above indicate that if the coupling
tensor Co is nonzero, a rotating microswimmer can exhibit
Next, we show the two torques (i.e., drag torque and
magnetic torque) counterbalanced with each other.
When the pitch angle α is 0, m and B are both in a plane
perpendicular to the Z-axis (Fig. 3), making the
angular velocity ω = [0 0 ωz]T and magnetic torque
τm = [0 0 τmz]T. The induced magnetic torque can be
treated as the torque exerted on a chain that consists of
three spherical particles [
], as expressed by
where a is radius of the particle, µ 0 is the vacuum
permeability, χ is the particle susceptibility and θ is the phase
lag between external field and induced dipole moment. If
the microswimmer is actuated with steady rotation, the
phase lag must satisfy the condition sin(2θ ) < 1 [
drag torque τr due to hydrodynamic interaction can be
obtained by combining torque from each particle
]. Similarly, we have τr = [0 0 τrz]T. For each
particle, the drag torque is given by
τrz,i = di × Fd,i
Fd,i = Dd Vi = Dd (ωz × di)
where di and Fd,i are vector position and drag force of
the i − th microparticle, and Dd is the drag force
coefficient, respectively. For spherical microparticles without
any effects from boundary Dd = −6π ηa. The total drag
torque is given by
view, the rotation axis is a dot with coordinate (xr , yr ).
From geometrical perspective, we have
di2 = (xi − xr )2 + (yi − yr )2
where (xi, yi) is the coordinate of the i − th particle’s
center. The minimal value of i3=1 di2 exists if the rotation
axis passes through the centroid of the simplified
isosceles triangle, given by
(xc, yc) =
n=1 xi ,
Substituting Eq. 15 into Eq. 14 yields
di2min = 32 L2(2 + sin γ − cos γ )
dimin ∈ (1.58L2, 2.28L2)
Then, we assume the microswimmer is actuated with
propulsion as shown in Fig. 4b. In this scenario, the
minimal value exists if the rotation axis is parallel to the
longest side of the triangle, which is the side respected
to angle γ since γ > π/3. Calculation results show that
the minimal value exists when the rotation axis passes
through the point p, a point of trisection of the height
with respect to the longest side. Similarly, we have
Eq.12 shows that larger i3=1 di2 leads to larger drag
torque under the same input frequency.
Pose change frequency
The microswimmer undergoes constant magnetic
torque due to uniform magnetic flux density. However,
the drag torque is in dependence on the input frequency
of magnetic field and rotation pose of the
microswimmer. Next, from the torque-balance perspective, we
show how swimming behaviors of our microswimmer
varied by increasing the input frequency. The phase lag
for a given input frequency and magnetic flux density is
sin(2θ ) =
µ 0χ 2B2 ln( 32 )
In order to balance the two torques, term i3=1 di2 in
Eq. 12 must change its value corresponding to
different input frequencies, which results in different rotation
poses of the microswimmer. However, the adjustable
range of this term has limitation. Let us consider two
cases of the microswimmer under actuation, i.e., simple
rotation and propulsion. We simplify the
microswimmer as an isosceles triangle with two sides of identical
length L and the included angle γ. Since we only
consider microswimmer with two perpendicular planes of
symmetry, γ in our analysis is set to be π/3 < γ < π.
First, we assume that the microswimmer is actuated
with simple rotation as shown in Fig. 4a. From the top
The analysis results above, in particular Eqs. 17 and 19,
show that with the same input frequency, drag torque
becomes smaller if the microswimmer exhibits
propulsion rather than simple rotation. Finally, let us consider
a specific case. We increase the input frequency
continuously, at first, the microswimmer exhibits simple
rotation, and then, it tends to change its actuation
behaviors toward reducing the drag torque. The only
feasible method is to reduce the distance (term i3=1 di2 in
Eq. 12) between each microparticle and the rotation axis.
Therefore, the microswimmer has to change from
simple rotation to propulsion when the input frequency is
higher than a certain value ωc, and here we name ωc as
the pose-change frequency. When the angular velocity is
high enough, the propulsion force of the microswimmer
is larger than the combination of gravitational force and
buoyancy, so that it will swim. A switch from simple
rotation to propulsion can be realized by increasing the input
frequencies with a value higher than ωc. For example in
Fig. 1, ω1 is below ωc while ω2, ω3, ω4 are higher than ωc.
Besides the two specific scenarios shown in Fig. 4a,
b, other dynamic behaviors can be realized as well. As
shown in Fig. 4c, we define the simplified triangular has
an angle ϕ with X-axis and the distance between the
vertex and rotation axis is dm. These two parameters are able
to represent the propulsion pose with rotation axis. Here
the i3=1 di2 is calculated as
di2 = 3dm2 + L2[cos2(ϕ) + cos2(ϕ − γ )]
− 2Ldm[cos(ϕ) + cos(ϕ − γ )]
For a given microswimmer γ = π/2, if we define
dm = σ L (0 ≤ σ ≤ 1) and Eq. 20 can be simplified as
di2 = L2[3σ 2 + 1 − 2σ (sin ϕ + cos ϕ)]
The range of ϕ is set to be 0 ≤ ϕ ≤ π/4, while
π/4 ≤ ϕ ≤ π/2 results in the same value of i3=1 di2
because of the symmetry of the model. We use MATLAB
to calculate the distribution of the value in Eq. 21, and
the results are shown in Fig. 5. The maximum value exists
when σ = 1 and ϕ = 0, corresponding to the pose with
angular velocity ω2 in Fig. 1. Interestingly, the minimum
value is i3=1 di2 = L2/3 when σ = 0.47 and ϕ = π/4,
which also proves that minimal value exists when the
rotation axis is parallel to the longest side of the
simplified triangle model.
To simulate and understand how a solid surface affects
swimming behaviors, two finite element method (FEM)
models are established using COMSOL Multiphysics
(two insets in Fig. 6a, d) to investigate the induced fluid
flows (Fig. 6a, b) and pressure (Fig. 6b, c, e, f ) by the
rotating microswimmer. The microswimmer is
modeled as three spheres with a diameter of 4.5 μm and
an angle γ = π/2, and set to be actuated in water at
a frequency of 10 Hz. A solid surface is modeled as a
no-slip wall at the bottom. Simulations consist of two
cases: Fig. 6a–c are simulation results with
microswimmer near (0.75 μm) the surface, and Fig. 6d–f are
results with it farther away (20.75 μm) from the
surface. After over ten full rotations, the induced pressure
distribution and streamlines of the surrounding fluid
are calculated. Figure 6a, b show that the
microswimmer induces a net flow of fluid along the direction of
the rotation axis, similar to the propulsion of a helical
]. The fluid impacts on the substrate,
resulting in enhanced pressure . For the case of
rotation near the surface, the induced pressure
difference between the area near the top and bottom space of
the microswimmer is observed in Fig. 6b, c. However,
such difference becomes negligible when the
microswimmer 20.75 μm above the surface (Fig. 6e, f ). The
largest pressure difference around each particle is in
the order of 10−2 Pa (Fig. 7). The affected area on the
microswimmer is in the order of 101 μm2 and the net
force along Z-axis works out in piconewton range due
to the pressure difference.
The microswimmer and experimental setup
In our experiments, the microswimmer was obtained by
direct sediment of paramagnetic microparticles colloidal
suspensions (Spherotech PMS-40-10) in DI water. These
microparticles have a density of 1.27 g/cm3 and a
diameter of 4–5 μm with a smooth surface. Sediment
introduces randomness to the process, resulting in different
structures. Nonetheless, the three-particle structures can
be easily obtained and directly used in our experiments.
During the magnetic actuation, we did not observe
deformation of the swimmer by turning on and off the field,
which indicates the link between two microparticles is
fixed and stable.
Our electromagnetic coils setup consists of three
orthogonally placed Helmholtz coil pairs, a swimming
tank containing a Si substrate inside and a light
microscope with a recording camera on the top. Rotating
magnetic field is generated by the coil system (Fig. 8) actuated
by three servo amplifiers (ADS 50/5 4-Q-DC, Maxon
Inc.). The amplifiers are controlled by a LabVIEW
program through an Analog and Digital I/O card (Model
826, Sensoray Inc.), frequency, field strength as well as
yaw (β) and pitch angles (α) can be adjusted through this
program. Schematic of the magnetic field is shown in
Fig. 2b. A swimming tank (21mm × 21mm × 3mm) filled
with DI water is placed in the middle of the coils, and the
Si substrate inside provides a solid surface. The top
camera records the motion of the microswimmer at a rate of
Results and discussion
The microrobot swims away from the solid surface (α = 0◦)
The microswimmer has been actuated at a frequency
range from 1 to 16 Hz on a Si substrate in the tank. The
flux density of the magnetic field maintains 9 mT
during the experiments. When the input frequency is below
8 Hz, the microswimmer exhibits simple rotation and
no translational velocity is observed (Fig. 9a), whereas it
exhibits propulsion with varied poses when the
input frequency is higher than 8 Hz (Fig. 9b). Experiment results
show that 8 Hz is the pose-change frequency ωc. When
the input frequency is below ωc (8 Hz), the
microswimmer exhibits simple rotation and the drag torque is small
enough to be balanced by the magnetic torque. The
projection of rotation axis in XY-plane is closing to the
centroid of the simplified triangle gradually with increasing
the input frequency, in order to reduce the drag torque
(Fig. 4a). Such adjustment of rotation axis cannot affect
the actuation pose (simple rotation) of the
microswimmer. Equations 14–17 have shown the limitation of this
adjustment method, which also explains why the
microswimmer cannot maintain simple rotation with input
frequency higher than ωc. While when the input frequency
is higher than ωc (8 Hz), the drag torque is affected by
both the pose angle ϕ and the distance dm (Fig. 4c).
Different input frequencies of the magnetic field change the
drag torque, and dynamic behaviors of the
microswimmer are governed by the interplay of magnetic and
resistant torques. The dynamic behaviors appear when turning
on the magnetic field or changing the input frequency
(see Additional file 1). As shown in the experimental
results (Fig. 9b), after turning on the magnetic field the
dynamic behaviors of the microswimmer last less than
2 s (0–2 s). After that the microswimmer exhibits steady
rotation and propulsion (2–29 s).
Swimming velocity of the microswimmer along the
Z-axis is measured by a fixed distance z divided by
time t. It follows three steps. First, the focal plane of the
microscope is set on the substrate, followed by
recording and turning the magnetic field on. This step aims to
record the starting time. Then, the focal plane is adjusted
to 20 μm above the substrate. The microswimmer
swimming across the focal plane is observed as it became in
focus gradually and then out of focus. Finally, we find
the best focusing frame from the recorded video to
count the time t. Using this method, swimming
velocity of the microswimmer in the space 0 to 20 μm above
the substrate (bottom space) is measured. The velocity
in the space 20–40 μm (upper space) above the
substrate is measured using the same method. After turning
off the magnetic field, the microswimmer will gradually
sink onto the substrate due to gravitational force. The
Three-axis Helmholtz coils
swimming velocity against frequency in the bottom and
upper space is depicted in Fig. 10.
Next, we show magnetic steering of the
microswimmer. It swims along the direction of +Z-axis after
exerting field at a frequency of 10 Hz with α = 0◦. After lifting
25 μm from the substrate, it can stay in focus by
adjusting pitch angle γ to 80◦, showing negligible displacement
along Z-direction (see Additiona file 1). The propulsive
force has the same direction with the normal line of the
applied magnetic field. In this scenario, gravitational
force and buoyancy are balanced by the component
of propulsive force on Z-axis. Steering can be done by
adjusting the yaw angle β of the field from 0◦ to 360◦ as
shown in Fig. 11. The microswimmer did not show
visible sidewise drift because of the absence of the boundary
]. The propulsive force is measured based on the
equilibrium of forces, which contains gravitational force,
buoyancy and propulsive force. The gravitational force
and buoyancy, respectively, are 1.82 and 1.43 pN, and the
propulsive force is calculated to be 1.14 pN. Based on the
simulation results, the net force generated by pressure
difference is in the order of piconewton as well, which
implies this net force can enhance swimming velocity.
Figure 10 shows that the microswimmer has a higher
swimming velocity in the bottom space (0–20 μm above
the surface), which validates our calculation and
Actuation of the microrobot near the solid surface
(α = 90◦)
The microswimmer has been actuated with pitch angle
α = 90◦ (i.e. its rotation axis is parallel to the
horizontal substrate) above the Si substrate. It shows
frequencydependent motion regimes, that is, tumbling (1–3 Hz)
and wobbling (3–20 Hz). The plot of velocity verses
frequency is depicted in Fig. 12. When input frequency
is below 3 Hz, the microswimmer exhibits tumbling
motion with 90◦ precession angle. After increasing the
input frequency to be higher than 3 Hz, the
microswimmer exhibits wobbling motion and the precession angle
decreases continually under higher frequency. Previous
studies show that dynamic regimes for the
tumblingto-wobbling transition of a magnetic microswimmer
depend not only on the frequency of the magnetic field,
but also on the geometry and easy axis orientation of
the microswimmer [
]. In our experiments, when
the microswimmer is actuated at a frequency range of
1–3 Hz, both the easy axis and the induced magnetic
moment are oriented along the field direction. The easy
axis and the induced magnetic moment rotate with a
phase lag behind the magnetic field, resulting in
tumbling motion of the microswimmer. After increasing the
input frequency to be higher than 3 Hz, the drag torque
increases and its interplay with the magnetic torque
results in wobbling regime. During the experiments, the
precession angle of the microswimmer decreases with
increasing input frequency of the magnetic field, same as
the theoretical prediction [
]. The swimming velocity
reaches maximum under magnetic field at a frequency of
10 Hz, similar to the results shown in Fig. 10.
Drifting of the microswimmer occurs due to the
boundary effects. The drag coefficient is constant for
a certain sphere particle in bulk fluid, but the presence
of a solid surface increases the drag on a body, which
decreases with a growing distance between the
microswimmer and the surface [
]. To be specific, a segment
of the microswimmer closer to the surface exhibits larger
drag than that farther away the surface, which causes the
microswimmer drift sidewise, perpendicular to the
rotation axis. Figure 12 also indicates that unlike the ABF
], the drift velocity of the microswimmer is not
increasing linearly with the input frequency.
In this paper, we demonstrate dynamic behaviors of
a three-particle paramagnetic microswimmer near a
solid surface. These dynamic behaviors are dependent
on the input frequency of the rotating magnetic field,
and varied actuation poses can be switched by
adjusting the frequency. Simulations of the microswimmer
near (0.75 μm) and far farther away (20.75 μm) from a
solid surface are investigated, which are in good
agreement with the experimental results. Finally, the effects of
a solid surface on swimming behaviors are proposed, i.e.,
enhancing swimming velocity when the microswimmer
exhibits propulsion perpendicular to the horizontal
surface and causing sidewise drift when it is actuated
parallel to the surface. Future studies will focus on the motion
control of the microswimmer in biofluids with different
Additional file 1. This video demonstrates the microswimmer actuated
under a rotating magnetic field at a frequency of 7 Hz and 9 Hz, and
magnetic steering of the microswimmer.
QW designed experiments, built the analytical model and simulation, as
well as drafted the manuscript. QW and JY performed the experiments. LY
designed the magnetic actuated system. LZ supervised the project and made
contributions to the revision of the draft. Part of the work will be presented
at the 2017 IEEE International Conference on Robotics and Biomimetics (IEEE
ROBIO2017). All authors read and approved the final manuscript
Shatin, Hong Kong SAR, China. 3 Shenzhen Research Institute, The Chinese
University of Hong Kong, Shenzhen 518172, China.
The authors thank D. D. Jin (Chinese University of Hong Kong) for the fruitful
The authors declare that they have no competing interests.
This paper is supported by the Early Career Scheme (ECS) grant with Project
No. 439113, the General Research Fund (GRF) with Project No. 14209514,
14203715 and 14218516 from the Research Grants Council (RGC), the ITF
project with Project No. ITS/231/15 funded by the HKSAR Innovation and
Technology Commission (ITC) and the National Natural Science Funds of
China for Young Scholar with Project No. 61305124.
Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.
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