## 2 Method
### Field Transformation for Particle Rotation and Transportation  
According to our previous study [53], acoustic force potential that can produce a strong constraint boundary on the particle is crucial to improve the manipulation stability. For stripe-shaped particle, the surrounded contours of force potential are expected to resemble the particle shape, not only for position constraint, but also for orientation constraint. As the contours of force potential always follow that of acoustic pressure [55], acoustic field with elongated pressure nodes, as illustrated in Figure 2a, will be synthesized. The orientation and spatial position of the elongated pressure node can be changed by transforming the sound field with two steps: set the pressure of the control points from Equation (9), and then solve the excitation parameter of each transducer from Equation (14). Through this way, the stripe-shaped particle trapped at the pressure node can be rotated and transported by modulating the excitation signals electronically instead of varying the transducer arrangement mechanically. Figure 2b-d show the synthesized acoustic field within different regions, in which the pressure amplitude has been normalized. The blue color indicates the pressure minima and the red one represents the maxima. The pressure field shown in Figure 2b is derived with the truncation order \(M=[kr]=6\) and employing thirteen transducers. The same number of control points is equally distributed along a circle with \(0.8\lambda\) radius indicated by the dashed line. Although the truncated error is unavoidable, the pressure distribution within the circular control region is identical to the desired pressure field in Figure 2a. The elongated pressure node is formed and surrounded by four anti-nodes, which can produce a strong force gradient to trap the target. However, the co-existing of pressure nodes within the field limits its selectivity. In Figure 2c, a smaller range of target field is synthesized by thirteen control points that are located at the dashed circle with radius \(r=0.5\lambda\). The corresponding truncation order is decreased to \(M=4\). It is satisfactory to see that there is only one elongated pressure node within the field. In addition, the smaller field region means that fewer employed transducers have to be used, which may greatly reduce the device cost and complexity. Figure 2d shows the generated sound field in the same region by decreasing the control point and transducer number to \(Q=L=9\). The contours of pressure anti-node show some slight change, but the pressure node maintains the same shape. Figure 3 shows the rotational transformation of the desired pressure field by applying corresponding amplitude and phase settings to the fixed sound sources (see Supplementary Table S1). The single pressure node performs an anticlockwise rotation with an interval of 45\({}^{\circ}\). Theoretically, the rod that is trapped in the pressure node will rotate round its center with the same interval. The rotation direction is easily to be changed by inverting the modulation sequence of the excitation parameters. Spatial transformation of the sound field can also be realized. In Figure 4a-d, the circular sound field is shifted by an appointed distance 0.5\(\lambda\) along 45\({}^{\circ}\), 135\({}^{\circ}\), 225\({}^{\circ}\), and 315\({}^{\circ}\), respectively. Supplementary Table S2 gives the source parameters for this transformation. The micro-particle will follow the movement of the pressure node, and move the same theoretical distance. Figure 3: Anticlockwise rotation of the synthesized pressure field from 0\({}^{\circ}\) to 135\({}^{\circ}\) with 45\({}^{\circ}\) interval: (a) 0\({}^{\circ}\); (b) 45\({}^{\circ}\); (c) 90\({}^{\circ}\); (d) 135\({}^{\circ}\). Figure 2: (a) The desired acoustic pressure field for manipulating non-spherical micro-particle. The black dashed line indicates the boundary of acoustic field to be synthesized.