† Corresponding author. E-mail:
Project supported by the National Key R & D Program, China (Grant No. 2016YFF0203000), the National Natural Science Foundation of China (Grant Nos. 11774167 and 61571222), Fundamental Research Funds for the Central Universities, China (Grant No. 020414380001), State Key Laboratory of Acoustics, Chinese Academy of Sciences (Grant No. SKLA201609), and AQSIQ Technology R & D Program, China (Grant No. 2017QK125).
Based on the angular spectrum decomposition and partial-wave series expansion methods, we investigate the radiation force functions of two Airy–Gaussian (AiG) beams on a cylindrical particle and the motion trajectory of the particle. The simulations show that the particle can be pulled or propelled into either the positive or negative transverse direction by turning the phase difference between the two AiG beams appropriately; and the larger the beam widths of the two AiG beams are, the bigger the radiation force can be obtained to control the particle. In addition, the direction of the accelerated particle can be controlled while the dimensionless frequency bandwidth changes. The results indicate that the phase plays an important role in controlling the direction of the particle, which may provide a theoretical basis for the design of acoustical tweezers and the development of drug delivery.
Many scholars have investigated the acoustic radiation force, and the common beams, such as plane wave,[1,2] Gaussian beam,[3,4] focused beam,[5] and Bessel beam,[6,7] have been studied. Therefore, introducing an innovative type of beam for particle manipulation has to be taken into account. In optics, the Airy beam has drawn much attention of the researchers due to its unique properties,[8,9] which retains the shape of the intensity distribution during propagation (non-diffracting), and it can recover itself after encountering an obstacle (self-healing) and propagates along a parabolic trajectory (self-accelerating) similar to the ballistic trajectory under the effect of gravity, and there are some studies about the application of the Airy beam in the optical tweezers.[10–14] In addition, the Airy beam can also be applied to the study of the acoustical tweezers, which were investigated by Mitri in 2016.[15]
The Airy–Gaussian (AiG) beam, a generalized form of the Airy beam, has the similar propagation properties to the Airy beam: approximately non-diffracting, self-healing, and self-accelerating. The AiG beam was introduced by Bandres and Gutierrez-Vega,[16] who regarded the Airy beam[8] and the finite-energy Airy beam[17] as special cases of the AiG beam. In optics, the AiG beam has been analyzed extensively,[16,18] for example, the propagation properties of a single AiG beam in different media[19–21] and the interaction of two AiG beams in Kerr media[22] and nonlocal nonlinear media.[23] But in acoustics, there are few reports about the application of the AiG beam in particle manipulation, though Mitri has investigated the acoustical applications on a two-dimensional cylindrical particle in water[15] and an elastic medium[24] by an Airy beam. There is no investigation of the acoustic radiation force on a cylindrical particle induced by an AiG beam up to now.
The acoustic radiation forces induced by the traditional beams such as Bessel beam and Gaussian beam can accelerate the micro-particle along a straight line, while the acoustic radiation force induced by the Airy beam can pull, push, or accelerate the micro-particle along a parabolic trajectory,[15] in other words, the Airy beam not only exerts a force on the particle along its velocity direction but also exerts a force on the particle perpendicular to its velocity direction. The control of the particle under the effect of radiation force induced by acoustic beams is similar to the delivery of the drugs in the human’s blood vessels. Considering the complexity of the human’s blood vessels, the drugs need to be manipulated accurately in every direction. Therefore, the question is how to control the direction of an accelerated particle. In this paper, the acoustic radiation force induced by two AiG beams on a cylindrical particle is simulated and the effect of the radiation force on the direction of the accelerated particle is studied. We investigate the acoustic radiation force on a cylindrical particle immersed in water induced by two AiG beams using the angular spectrum decomposition and partial-wave series expansion methods. Most particle manipulations use surface acoustic wave devices which often deal with “acoustical sheets”, i.e., two-dimensional (2D) finite beams, therefore, here the two AiG beams and the cylindrical particle are both considered to be two-dimensional. The results displayed may contribute to the developments of cell screening, drug delivery, and the design of the acoustic tweezers.
Two incident Airy–Gaussian beams propagating along the x axis at normal incidence with respect to the z axis of a cylindrical particle are considered here, as shown in Fig.
The scattering velocity potential produced by the cylindrical particle can be expressed as
Based on the incident and scattering velocity potentials, the longitudinal (along the x axis) and transverse (along the y axis) radiation force functions can be obtained respectively as follows:[28]
This section presents several simulation results of the radiation force functions for two AiG beams. The material of the cylindrical particle used in the simulation is Lucite (with the density ρ1 = 1191 kg/m3, compressional velocity c1 = 2690 m/s, shear velocity c2 = 1340 m/s, normalized longitudinal absorption γ1 = 0.0035, and normalized shear absorption γ2 = 0.0053[29]) and the host fluid is water (ρ0 = 1000 kg/m3, c0 = 1500 m/s). In the following numerical simulation, we take ky0 = 4 and α = 0.01.
Figure
The computation of the motion trajectories of the cylindrical particle, only determined by the acoustic radiation force and the viscous drag force in the x–y plane here, is based on Newton’s second law of motion which is expressed as
The computation of the motion trajectories of the particle in MATLAB is accomplished by using the step-iterative Runge–Kutta method, the position and the velocity of the particle at the next time-step can be determined from the values at the previous time-step. In the simulations, the initial vector velocity and the initial time are all set to be zero, and the beam intensity is set to be 3.3 × 108 W/m2. There are four motion trajectories of the particle shown in Figs.
Figure
Next we investigate the influence of the beam width on the radiation force function. Here we take the phase Q = 0 with different beam widths as examples. The background of Fig.
Figure
In summary, we obtain the radiation force of two AiG beams on a cylindrical particle by using the angular spectrum decomposition and partial-wave series expansion methods and simulate the corresponding force functions and the motion trajectories of the particle with different parameters. We notice that the transverse direction of the accelerated particle can be controlled by turning the phase Q; the radiation forces become bigger and the distributions of the force become more complex when the beam width w0 increases; the control of the particle’s direction in the transverse direction can be achieved over a large frequency bandwidth range, which means we can find appropriate ka to manipulate the particle in experiments by adjusting the frequency. This work may provide a method to deliver particles to a target location by choosing suitable phases of two AiG beams and may contribute to the development for the design of acoustical tweezers.
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