Mathematical modeling
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 (Fig. 2b).
Motion at low Reynolds numbers
The hydrodynamics of the microswimmer in low Reynolds number regime can be described by the Stokes equations:
$$\eta \nabla ^2{\mathbf {u}}-\nabla p=0$$
(1)
$$\begin{aligned} \nabla \cdot u=0 \end{aligned}$$
(2)
where p is pressure and u is moving velocity of the fluid. The relationship of external force \({\varvec{F}}\) together with torque \(\varvec{\tau }\) and translational velocity \(\varvec{V}\) together with angular velocity \(\varvec{\omega }\) is described as [36]:
$$\begin{aligned} \left[ \begin{array}{lll} \varvec{V} \\ \varvec{\omega } \end{array} \right] = \left[ \begin{array}{ccc} \varvec{K} &{} \varvec{C_o}\\ \varvec{C_o^T} &{}\varvec{\Omega _o} \end{array} \right] \left[ \begin{array}{ccc} \varvec{F} \\ \varvec{\tau } \end{array} \right] \end{aligned}$$
(3)
where \(\varvec{K}\) is the translation tensor and \(\varvec{\Omega _o}\) is the rotation tensor. \(\varvec{C_o}\) is the coupling tensor, representing coupling of translational and rotational motions of a microagent. For the microagent in Fig. 2a, the matrices \(\varvec{K}\), \(\varvec{\Omega _o}\) and \(\varvec{C_o}\) are given by
$$\begin{aligned} {\mathbf {K}} = \left[ \begin{array}{lll} K_1 &{} 0 &{} 0 \\ 0 &{} K_2 &{} 0 \\ 0 &{} 0 &{} K_3 \\ \end{array} \right] ,\quad \mathbf {\Omega _o} = \left[ \begin{array}{ccc} \Omega _1 &{} 0 &{} 0 \\ 0 &{} \Omega _2 &{} 0 \\ 0 &{} 0 &{} \Omega _3 \\ \end{array} \right] ,\quad \mathbf {C_o} = \left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} C_{23} \\ 0 &{} C_{32} &{} 0 \\ \end{array} \right] \end{aligned}$$
(4)
Magnetic actuation
The magnetic force and torque exerted on the microswimmer are given by:
$$\begin{aligned} \varvec{F}=(\varvec{m}\cdot \nabla )\varvec{B} \end{aligned}$$
(5)
$$\begin{aligned} \varvec{\tau }=\varvec{m} \times \varvec{B} \end{aligned}$$
(6)
where \(\varvec{m}\) is the induced magnetic dipole moment and \(\varvec{B}\) is the flux density of the magnetic field. Here we have \(\varvec{F}=0\) 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
$$\begin{aligned} \varvec{\omega }=\varvec{\Omega _o}(\varvec{m}\times \varvec{B}) \end{aligned}$$
(7)
$$\begin{aligned} \varvec{V}=\varvec{C_o}(\varvec{m}\times \varvec{B}) \end{aligned}$$
(8)
Two equations above indicate that if the coupling tensor \(\varvec{C_o}\) is nonzero, a rotating microswimmer can exhibit translational velocity.
Next, we show the two torques (i.e., drag torque and magnetic torque) counterbalanced with each other. When the pitch angle \(\alpha\) is 0, \(\varvec{m}\) and \(\varvec{B}\) are both in a plane perpendicular to the Z-axis (Fig. 3), making the angular velocity \(\varvec{\omega } = [0\quad 0 \quad \omega _z]^T\) and magnetic torque \(\varvec{\tau _m} = [0\quad 0 \quad \tau _{mz}]^T\). The induced magnetic torque can be treated as the torque exerted on a chain that consists of three spherical particles [37], as expressed by
$$\begin{aligned} \tau _{mz}=\frac{3}{4}\pi a^3 \mu _0 \chi ^2 B^2 \sin (2\theta ) \end{aligned}$$
(9)
where a is radius of the particle, \(\mu _0\) is the vacuum permeability, \(\chi\) is the particle susceptibility and \(\theta\) 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 \(\sin (2\theta )<1\) [37]. The drag torque \(\varvec{\tau _r}\) due to hydrodynamic interaction can be obtained by combining torque from each particle individually [38]. Similarly, we have \(\varvec{\tau _r} = [0\quad 0 \quad \tau _{rz}]^T\). For each particle, the drag torque is given by
$$\begin{aligned} \varvec{\tau _{rz,i}}=\varvec{d_i}\times \varvec{F_{d,i}} \end{aligned}$$
(10)
$$\begin{aligned} \varvec{F_{d,i}}=D_d\varvec{V_i}=D_d(\varvec{\omega _z}\times \varvec{d_i}) \end{aligned}$$
(11)
where \(\varvec{d_i}\) and \(\varvec{F_{d,i}}\) are vector position and drag force of the \(i-th\) microparticle, and \(D_d\) is the drag force coefficient, respectively. For spherical microparticles without any effects from boundary \(D_d=-6\pi \eta a\). The total drag torque is given by
$$\begin{aligned} \varvec{\tau _{rz}}=\sum _{n=1}^{3}\varvec{d_i}\times \varvec{F_{d,i}} = D_d \varvec{\omega _z}\sum _{n=1}^{3} {d_i^2} \end{aligned}$$
(12)
Eq.12 shows that larger \(\sum _{i=1}^{3}d^2_i\) 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 [37]
$$\begin{aligned} \sin (2\theta )=\frac{96\eta \omega }{\mu _0\chi ^2 B^2\ln (\frac{3}{2})} \end{aligned}$$
(13)
In order to balance the two torques, term \(\sum _{i=1}^{3}d^2_i\) 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 \(\gamma\). Since we only consider microswimmer with two perpendicular planes of symmetry, \(\gamma\) in our analysis is set to be \(\pi /3<\gamma <\pi\).
First, we assume that the microswimmer is actuated with simple rotation as shown in Fig. 4a. From the top view, the rotation axis is a dot with coordinate \((x_r,y_r)\). From geometrical perspective, we have
$$\begin{aligned} \sum _{n=1}^{3} {d_i^2}=(x_i-x_r)^2 + (y_i-y_r)^2 \end{aligned}$$
(14)
where \((x_i,y_i)\) is the coordinate of the \(i-th\) particle’s center. The minimal value of \(\sum _{i=1}^{3}d^2_i\) exists if the rotation axis passes through the centroid of the simplified isosceles triangle, given by
$$\begin{aligned} (x_c,y_c)=\left( \frac{\sum _{n=1}^{3}x_i}{3},\frac{\sum _{n=1}^{3}y_i}{3}\right) \end{aligned}$$
(15)
Substituting Eq. 15 into Eq. 14 yields
$$\sum _{n=1}^{3}d^2_{i{\rm min}} = \frac{2}{3}L^2 (2+\sin \gamma -\cos \gamma)$$
(16)
$$\begin{aligned}&\quad \sum _{n=1}^{3}d^2_{i{\rm min}}\in (1.58 L^2, 2.28 L^2) \end{aligned}$$
(17)
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 \(\gamma\) since \(\gamma >\pi /3\). 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
$$\sum _{n=1}^{3}d^2_{i{\rm min}} = \frac{2}{3} L^2 \cos ^2\frac{\gamma }{2}$$
(18)
$$\begin{aligned}&\quad \sum _{n=1}^{3}d^2_{i{\rm min}} \in \left( 0,\frac{1}{2}L^2\right) \end{aligned}$$
(19)
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 \(\sum _{i=1}^{3}d^2_i\) 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 \(\omega _{c}\), and here we name \(\omega _{c}\) 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 \(\omega _{c}\). For example in Fig. 1, \(\omega _1\) is below \(\omega _{c}\) while \(\omega _2\), \(\omega _3\), \(\omega _4\) are higher than \(\omega _{c}\).
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 \(\varphi\) with X-axis and the distance between the vertex and rotation axis is \(d_m\). These two parameters are able to represent the propulsion pose with rotation axis. Here the \(\sum _{i=1}^{3}d^2_i\) is calculated as
$$\begin{aligned} \sum _{i=1}^{3}d^2_i = 3d_m^2 + L^2[\cos ^2(\varphi )+\cos ^2(\varphi -\gamma )]-2Ld_m[\cos (\varphi )+\cos (\varphi -\gamma )] \end{aligned}$$
(20)
For a given microswimmer \(\gamma =\pi /2\), if we define \(d_m=\sigma L\) (\(0 \le \sigma \le 1\)) and Eq. 20 can be simplified as
$$\begin{aligned} \sum _{i=1}^{3}d^2_i = L^2[3\sigma ^2 + 1 -2\sigma (\sin \varphi + \cos \varphi )] \end{aligned}$$
(21)
The range of \(\varphi\) is set to be \(0\le \varphi \le \pi /4\), while \(\pi /4 \le \varphi \le \pi /2\) results in the same value of \(\sum _{i=1}^{3}d^2_i\) 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 \(\sigma = 1\) and \(\varphi =0\), corresponding to the pose with angular velocity \(\omega _2\) in Fig. 1. Interestingly, the minimum value is \(\sum _{i=1}^{3}d^2_i=L^2 /3\) when \(\sigma = 0.47\) and \(\varphi = \pi /4\), which also proves that minimal value exists when the rotation axis is parallel to the longest side of the simplified triangle model.
Simulations
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 \(\upmu \hbox {m}\) and an angle \(\gamma = \pi /2\), 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 \(\upmu \hbox {m}\)) the surface, and Fig. 6d–f are results with it farther away (20.75 \(\upmu \hbox {m}\)) 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 flagellum [15, 39]. The fluid impacts on the substrate, resulting in enhanced pressure [40]. 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 \(\upmu \hbox {m}\) 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 \(10^1\)
\(\upmu \hbox {m}^2\) 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 \(\hbox {g}/\hbox {cm}^3\) and a diameter of 4–5 \(\upmu \hbox {m}\) 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 (\(\beta\)) and pitch angles (\(\alpha\)) can be adjusted through this program. Schematic of the magnetic field is shown in Fig. 2b. A swimming tank (\(21 \hbox {mm} \times 21 \hbox {mm} \times 3 \hbox {mm}\)) 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 50 fps.