## 2 Method  
### Mechanical Effects on Micorparticles in a Sound Field  
It has been well demonstrated that micro-particles suspended in a sound field will be stably trapped at certain positions. A time-averaged acoustic radiation force, which arises from the second-order effects of the incident and scattered waves, constrains the target in the force potential minimum. For less compressible micro-particle that is adopted in this paper, the stable force potential minimums always correspond to the pressure nodes. Correspondingly, the particle position is related to the spatial location of pressure node  
\[(x_{c},y_{c})=(x_{s},y_{s})\Big{|}_{\min(P)}\,, \tag{1}\]  
where \((x_{s},y_{s})\Big{|}_{\min(P)}\) indicates the position with minimum sound pressure. Since the sound pressure \(P(x_{s},y_{s})\) is also a function of the source frequency, amplitude and phase, the node position is only determined when these parameters are set. Then the trajectories of the micro-particles can be easily predicted by the movement of the acoustic pressure nodes.  
When considering the non-spherical particles, even if the sound field does not contain any acoustic vortices, an additional torque may be induced due to the asymmetric distribution of the acoustic radiation pressure on the particle surface. Theoretical calculation of the acoustic radiation torque on a rigid rectangular disk was first proposed by [PERSON], in which the general expression was derived by an approach analogous to the radiation force [36]. Nowadays, the theory has been further developed with the influence of particle shape and material [37; 38; 39; 40; 44; 48].  
When the dissipation in the wave propagating process can be neglected, the acoustic radiation torque on an irregularly shaped scatterer can be expressed by the time-averaged flux of angular momentum through any surface enclosing the particle [48], as shown in Figure 1a,  
\[T_{i}=\left\langle-\iint\limits_{\Sigma}\epsilon_{ijk}r_{j}\Pi_{kl}n_{l}dS\right\rangle, \tag{2}\]  
where the angle brackets indicate the time-averaged operation, \(i\), \(j\), and \(k\) denote the Cartesian axes, \(\Sigma\) is the integral surface, \(n_{l}\) is the outward normal of \(\Sigma\), \(\epsilon_{ijk}\) is the permutation tensor, \(r_{j}\) is a position vector from the integral unit to the coordinate origin, and \(\Pi_{kl}\) is the tensor of momentum flux density. Unlike the torque from the orbital angular momentum that is associated with acoustic vortex, this acoustic torque on the scatterer is dependent on the angle position [49]. Accordingly, the scatterer will be rotated and finally stopped at the certain angular position where the acoustic radiation torque is zero.  
The acoustic medium (water) in this study can be regarded as an ideal fluid since its viscous boundary layer \(\xi=\sqrt{\
u/\pi f}\approx 0.3\)\(\mu\)m is much smaller than the object (\(90\)\(\mu\)m in length and \(15\)\(\mu\)m in diameter \(r\)) [50; 51; 52]. The particle to be manipulated has the stiffness and density larger than the medium. As proposed by [PERSON] et al. [38; 44] and [PERSON] et al. [53], a stripe-shaped particle satisfying the above conditions would be trapped at the pressure node and rotated to a specific angular position. If the contours of acoustic pressure around the particle also have the stripe shape, the major axis of the stripe-shaped particle will coincide with that of the contours around the pressure node. Thus, the particle will always follow the rotation of the sound field and the rotation angle \(\phi\) of target particle equals to that of the sound field  
\[\phi=\delta, \tag{3}\]  
where \(\delta\) is the rotation angle of the sound field as illustrated in Figure 1a. Theoretically, the particle can be controlled with any orientation compared with its initial angle.