Partial derivatives of acoustic radiation force and dynamic equilibrium stability of acoustophoresis.

The theory of using transducer arrays for the contactless and stable manipulation of Rayleigh objects (ka≪1, where k is the wave number and a is the averaged radius of object) is well established, whereas retrieving the transducer parameters required for the stable and dynamic manipulation of Mie objects (1<ka<10) remains a significant challenge. Here, we develop an analytical model for the stable trapping and transportation of Mie (axisymmetric) objects by directly presetting the desired radiation force and torque, with appropriate stability constraints incorporated to ensure dynamical stability. Specifically, acoustic contributions from multiple transducers are superimposed using the translation addition theorem. The geometric features of the object are mapped onto a sphere through a conformal transformation, allowing the acoustic radiation force and torque to be expressed in terms of partial-wave expansions. The radiation force is further reformulated as a position-dependent function with the help of the addition theorem, enabling an analytical derivation of its partial derivatives. These derivatives are then used to express the Lyapunov stability conditions in closed form. Finally, a system of nonlinear inequality governing equations, incorporating the radiation force, torque, and stability constraints, is constructed to retrieve a set of transducer parameters capable of stably trapping a Mie object at specified spatial positions. At these positions, the partial derivatives of the radiation forces (i.e., the stability) are negative, implying the presence of a restoring mechanism that stabilizes the object at the target positions. The analytical predictions for both the radiation force and its partial derivatives show well agreement with the finite-element simulations. In practice, by switching among the retrieved sets of transducer parameters corresponding to different trapping positions, we demonstrate that a Mie object with a radius of 10mm (with ka≈7.39) can be stably transported over a distance equivalent to one transducer radius, i.e., 5mm with kd/2≈3.69.