Rolling microswarms along acoustic virtual walls

Rolling is a ubiquitous transport mode utilized by living organisms and engineered systems. However, rolling at the microscale has been constrained by the requirement of a physical boundary to break the spatial homogeneity of surrounding mediums, which limits its prospects for navigation to locations with no boundaries. Here, in the absence of real boundaries, we show that microswarms can execute rolling along virtual walls in liquids, impelled by a combination of magnetic and acoustic fields. A rotational magnetic field causes individual particles to self-assemble and rotate, while the pressure nodes of an acoustic standing wave field serve as virtual walls. The acoustic radiation force pushes the microswarms towards a virtual wall and provides the reaction force needed to break their fore-aft motion symmetry and induce rolling along arbitrary trajectories. The concept of reconfigurable virtual walls overcomes the fundamental limitation of a physical boundary being required for universal rolling movements.


Supplementary Notes
Supplementary Note 1. Estimation of the Reynolds number of the microchain. The Reynolds number of the microchain, Re = ρv s L/µ, was estimated as Re = 5 × 10 −3 , where ρ = 1000 kg · m −3 , the density of deionized water; L = 100 µm, the maximum characteristic length of the microchain; v s = 50 µm · s −1 , the maximum translational velocity of the microchain; and µ = 10 −3 Pa · s, the fluid viscosity.
The Reynolds number of the particle at the endpoint of a microchain was estimated as Re = 2.45×10 −4 , where L = 1.63 µm, the diameter of the particle; and v s = 150 µm · s −1 , the maximum linear velocity of the particle.
Supplementary Note 2. Acoustic contrast factor. The acoustic contrast factor ϕ dictates the minimum position of the Gor'kov potential 1 . If the density-related ration (5 ρ − 2)/(2 ρ + 1) is bigger than the compressibility ratio κ, ϕ is positive and the microparticle will be pushed towards the pressure node of the wave field. Otherwise, the microparticle will be pushed toward the pressure antinode. The density of the deionized water is 1000 kg · m −3 . The compressibility of the deionized water, which equals to k 0 = 1(ρ 0 c 2 0 ), is estimated to be 4.4 × 10 −10 Pa −1 , based on the sound speed in water c 0 = 1500 m · s −1 2 . The Commercially available magnetic particles of diameter 1.63 µm (COMPEL, Bangs Laboratories) have the density of 1580 kg · m −3 (provided in the technical data sheet by the manufacturer). The compressibility of the polymer/iron composite, can be obtained by the expression where V F e , V P S is the volume fraction of iron and polystyrene, which is 2% ∼ 6% and <16%, respectively 3 ; K F e , K P S is the compressibility of iron and polystyrene, which is ∼ 5.88×10 −7 Pa −1 and ∼ 220×10 −6 Pa −1 respectively 4 . Therefore, the acoustic contrast factor ϕ of the magnetic particles is positive and was estimated to be ∼0.29, which suggests that the particles will be pushed to the pressure nodal lines of the one-dimensional acoustic standing wave field. Supplementary Fig. 4. Control experiment in a narrow capillary. Three rows of rolling microchains were observed, among which microchains on both sides rolled along the capillary boundary, the microchains in the middle were identified as rolling along the virtual wall. The outer and inner diameters of the capillary are 1500 µm and 1000 µm, respectively. The distance between the two side rows is nearly 989 µm, implying the virtual wall is developed in the middle and far away from the bottom boundary of the circular capillary. The green curved arrow, the yellow arrow, and the blue dotted line denote the magnetic rotational direction, the rolling direction, and the acoustic virtual wall, respectively. The acoustic excitation voltage and frequency were 20 V PP and 1.55 MHz, respectively. The magnetic rotational direction was counterclockwise, and the magnetic rotational velocity and intensity were 24 rpm and 21 mT, respectively.  1) The translational velocity increases with the increasing acoustic frequency. These results are due to a higher excitation frequency can produce a higher ARF. Subsequently, the higher ARF can cause a larger off-center rotation to increase the translational velocity. (2) The translational velocity is also increased by increasing the intensity of the magnetic field due to the stronger dipole-dipole interactions make the self-assembly microswarms more stable by reducing the possibility to break into short chains during rolling. Each data point represents the average translational velocity analysed from 3-5 microchains (Source Data). Error bars represent the standard deviation (s.d.) of data. The spread of the data points can be explained as follows: (1) the data was captured from five different microchains from different acoustic pressure nodal lines, which exhibit a relatively large variety of lengths; (2) the dynamic acoustic pressure nodal line was not perfectly straight. Additionally, it is worth noting that the translational velocity of microchains in the capillary is bigger than that in the acoustic chamber, which can be explained as: (1) as the distance between two opposite transducers decreases, the intensity of acoustic field decays less; (2) the hydrodynamic interaction in the narrow capillary boundary may accelerate the translational velocity (which is beyond the scope of this study and will be explored in future work).

Supplementary Figures
Supplementary Fig. 7. Microswarms rolling along the acoustic virtual wall at different suspended planes. (a) Schematic of chain-shaped microswarms rolling at different suspended planes. The yellow curved arrow and the pink arrow denote the magnetic rotational direction and the translational direction, respectively. (b) Control experiment demonstrates microchains synchronously performed rolling along the acoustic virtual wall at different suspended planes. In contrast, no noticeable motion was observed of the nonmagnetic 15 µm polystyrene microbead laying below. The fuzzier the particle, the farther away it was from the polystyrene bead layer. The green curve arrow, yellow arrow, and pink dotted line denote the magnetic rotational direction, net translational direction, and displacement, respectively. Scale bar, 100 µm. During experiments nonmagnetic polystyrene beads were injected into the liquid first. Due to their larger mass, the polystyrene microbeads sediment rapidly to the bottom substrate, allowing them to act as a reference point. Then, magnetic particles were injected into the acoustic chamber. Following, a 1D acoustic standing wave field was introduced in the chamber, generating acoustic pressure nodal lines at multiple suspended planes, i.e. across different heights along the z-axis. The two types of particles were trapped along those pressure nodal lines. The vertical component of the ARF keeps magnetic particles suspended in the liquid [5][6][7][8] . We estimated the respective distances between the substrate and magnetic microparticles trapped at various suspended planes as: ∼15 µm, ∼35 µm, ∼70 µm, ∼110 µm, ∼400 µm, and even ∼1200 µm. These heights were measured by tuning and calibrating the focus knob of the inverted microscope. Tuning the focus knob allows the viewer to change the z-position that the microscope is focused on. We first moved the objective so that the 15 µm polystyrene microbeads on the glass substrate was in focus and marked the vertical position as Z 0 . Then we moved the objective to bring the suspended planes into focus in sequence and marked their respective vertical positions as Z 1 , Z 2 , Z 3 …. Finally, by subtraction (H = Z i − Z 0 ), we obtained the height of each suspended plane. The magnetic rotational direction was counterclockwise, and the magnetic rotational velocity and intensity were 12 rpm and 21 mT, respectively. (b) Measurement method of the distance between rotational center and geometry center. The distance P 1 C is defined as l 1 and the distance P 1 A is defined as l 2 (denoted by the orange lines). Thus, the off-center distance l(t) = l 2l 1 (denoted by the yellow line). When l 2 < l 1 , the sign of l(t) is defined as minus; When l 2 > l 1 , the sign of l(t) is defined as plus. The white line denotes the rotating microchain and the light blue lines are the auxiliary positioning lines. Scale bar, 50 µm. Supplementary Fig. 10. Tracked velocities of the microchain's two endpoints (P 1 and P 2 ) against the rolling frames (a) Linear velocities (Source Data). (b) Velocity components in the x and y directions, respectively (Source Data).
Supplementary Fig. 11. Acoustic radiation force estimation and analysis. (a) Acoustic radiation force and acoustic wave pressure estimation of a single magnetic particle versus the distance to the acoustic pressure node. The ARF shows sinusoidal dependence on the particle's position in the acoustic standing wave field, with maximum ARF being achieved when the particle is sited at one-eighth of the acoustic wavelength from the pressure node. As the particle moves towards the pressure node, the ARF decreases to zero (Source Data). The acoustic excitation frequency was 1.55 MHz. (b) Acoustic radiation force analysis of the microchain at different orientations. over one cycle of rolling, the ARF acting on the microchain always points to the pressure nodal line, but its magnitude varies as the microchain rotates. When the microchain aligns along the pressure nodal line (phase 1), it experiences the minimum ARF. Then, as it rotates away from the pressure nodal line in the counterclockwise direction, the ARF increases gradually. In phase 2, the ARF tends to impede the microchain's rotation, as the acoustic radiation torque opposes the magnetic torque. When the microchain is perpendicular to the pressure nodal line (phase 3), it experiences the maximum ARF and the ARF gradually decreases, as it rotates towards the pressure nodal line. In phase 4, the ARF tends to accelerate the rotation of the microchain on account of the acoustic radiation torque having a similar direction to the magnetic torque. The blue dotted line denotes the acoustic pressure nodal line. The red arrow, pink arrow, wathet arrow, and black curved arrows denote the acoustic radiation force, propulsion force, drag force, and magnetic torque, respectively. The black arrow along the pressure nodal line denotes the translational direction. During the development of the pressure node, the microchain is accelerated first and then slows down to 0 due to the reducing radiation force. The microchain has a stable translational velocity when rolling along the horizontal and vertical pressure nodal line at ∼10.63 µm · s −1 . The acoustic excitation voltage and frequency were 20 V PP and 1.55 MHz, respectively. The magnetic rotational direction was clockwise, and the magnetic rotational velocity and intensity were 24 rpm and 21 mT, respectively (Source Data). Insets show the movements of the tracked microchain at different rolling time points. The switching operation is distinguished from the rolling motion, and the resulting motion is highly related to the operating voltage and speed. The green box shows the tracked microchain. The blue dotted line and the dotted circle denote the acoustic pressure nodal line and the pressure node, respectively. Scale bar, 50 µm. Supplementary Fig. 16. Step-out frequencies of the microswarm and a single particle. (a) Variable self-assembly shapes of magnetic particles with different magnetic rotational frequencies in the x − y rotational magnetic field of 21 mT. (b) Rolling velocity of a single particle versus the magnetic rotational frequency within the x−z rotational magnetic field of 21 mT. The inset schematic shows the experimental rolling motion. Within the 21 mT rotational magnetic field, B(t) = B 0 cosωte x + B 0 sinωte z (the rotational axis is y-axis), the translational velocity of such a single particle increases almost linearly with the frequency till 40 Hz after which it decreases upon further increasing the frequency. Thus, the step-out frequency of the used single particle is around 40 Hz. Each data point represents the average translational velocity analysed from 3-5 microparticles (Source Data). Error bars represent the standard deviation (s.d.) of data.