# Inter-Particle Effects with a Large Population in Acoustofluidics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hybrid Algorithm for Massive Particle Interactions

#### 2.1. ARF Acting on Massive Cylindrically Symmetrical Targets

_{1}and V

_{1}represent the first-order pressure and velocity of the total sound field, respectively, $\overrightarrow{n}$ is the outward unit vector normal to surface Ω, and < > denotes the average operator over a time period. Equation (1) is derived from the time-average perturbation theory, which makes it possible to calculate the ARF accurately up to the second-order terms in the Mach number by using the velocity field potential satisfying the linear wave equation. With the first-order quantities derived from a finite element solution of the scattering problems, the time-averaged radiation stress tensor in Equation (1) is obtained. However, applying the surface integral to massive particles not only requires solving the multi-scattering problem efficiently, but also relies on the precise calculation across the fluid-particle interface. When using 3D FEM, the mesh elements on the particle surface should be fine enough to ensure the integral accuracy, which makes the computational cost unaffordable.

#### 2.2. Force in the y-Direction

#### 2.3. Force in the x-Direction

## 3. Method Validation

## 4. Acoustic Interactions Between Massive Particles

_{1}and l

_{2}(Figure 5a), Figure 5b,c shows the normalized y-axis secondary ARF acting on the upper and middle particle. The relative position of the lower particle does not significantly change the force on the upper one, but the middle particle is significantly affected. Both re-scattered waves from the upper and lower particles contribute to the secondary ARF of the middle one. It can be concluded that the effective secondary ARF acting on a target depends mainly on its direct neighbors, i.e., the acoustic interaction is non-transmissible and the indirect re-scattered waves from distant particles do not contribute much to the secondary ARF and can be neglected. When the particles are equidistant (l

_{1}= l

_{2}), the middle particle is balanced, and the critical distance for particle aggregation around the center one is about 0.66λ, and a slight change to this value is made by varying ka (Figure 5d). Since the relative position of the lower particle does not significantly change the force on the upper one (Figure 5b), the relative movement between the upper and middle particles resembles to the case of two particles where one is fixed. According to Silva and Bruce’s analytical model [49], the interaction force of Rayleigh scatters (ka << 1) only depends on distance between the source and the probe. Our numerical results show that the constraint of ka << 1 can be extended to ka < 2 for Rayleigh limit, which applies to the majority targets in acoustofuidics. Figure 5e shows the phase diagram of the secondary ARF acting on the upper sphere for various l

_{1}and l

_{2}. The critical distance for the upper sphere fluctuates slightly with l

_{2}. As for the interaction between two particles, ka has little effect on the critical distance for the interactions of three identical particles. Figure 5f shows the change of the normalized secondary ARF acting on the upper sphere when the size of lower particle varies. The particles are equidistant, and the dimensionless radii of the upper and middle targets are kept at ka = 0.17. Increasing ka of the lower sphere from 0.085 to 0.340, changes of the secondary ARF on the upper sphere increases from one to three times within a wavelength separation distance. Within the first attraction range, the secondary ARF acting on the upper sphere strengthens for the lower particle with larger ka, which is different from the effects of ka on particles with the same size. We may conclude that the absolute size of co-existing particles (same radius) does not affect the law of acoustic interactions, but the relative size of co-existing particles dominates the group behavior.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A large number of targets are distributed in the x-y plane, and incident waves enter from the x-. (

**b**) The distribution of sound pressure and velocity on the surface of any scatter in the sound field.

**Figure 2.**(

**a**) Calculation convergence for different values of ka. (

**b**) Calculation convergence for different incident angles.

**Figure 3.**(

**a**) Normalized force calculated by two methods. (

**b**) Difference between the results of the two methods

**.**

**Figure 4.**(

**a**) A pair of particles located in a standing wave field with a distance d to the pressure node and separated from each other by a distance l. (

**b**) Phase diagram of the secondary ARF acting on a pair of spheres in a one-dimensional standing wave. The dependence of the attraction-repulsion behavior on the position of the particle in the standing wave field in the two-particle interaction.

**Figure 5.**(

**a**) Schematic for the interaction of three identical particles located at the pressure node; (

**b**) The normalized secondary ARF on the upper sphere with ka = 0.170; (

**c**) The normalized secondary ARF acting on the middle sphere with ka = 0.170; (

**d**) Effects of ka on the secondary ARF of three equidistant spheres; (

**e**) Phase diagram of the secondary ARF acting on the upper sphere; (

**f**) Effect of the lower particle radius on the secondary ARF of the upper particle.

**Figure 6.**Interaction between four equidistant particles. (

**a**) Internal particles and (

**b**) external particles.

**Figure 7.**(

**a**) The schematic for acoustic interaction between massive equidistant particles. (

**b**) The phase diagram of the critical distance for the particle closest to the center.

**Figure 8.**(

**a**) The schematic for acoustic interaction between a cluster and a sphere. (

**b**) The effect of particle number and value of ka on critical distance.

Name | Value |
---|---|

fluid density | 998 kg/m^{3} |

fluid speed of sound | 1480 m/s |

particle density | 2000 kg/m^{3} |

particle speed of sound | 6559 m/s |

frequency | 1 MHz |

**Table 2.**Normalized force ${F}^{rad}/\left(k{a}^{3}{p}_{0}^{2}\right)$ on a sphere (unit 10

^{−10}*Pa

^{−1}).

ka | Analytical | 3D FEM | 2D FEM | Difference (3D/2D) | Difference (Analytical/2D) |
---|---|---|---|---|---|

0.0425 | 7.550 | 7.525 | 7.541 | 0.21% | 0.12% |

0.0849 | 7.550 | 7.503 | 7.515 | 0.13% | 0.47% |

0.170 | 7.550 | 7.408 | 7.412 | 0.054% | 1.9% |

0.340 | 7.550 | 7.013 | 7.016 | 0.043% | 7.6% |

Φ | 0 | π/9 | 2π/9 | π/3 | 4π/9 | |
---|---|---|---|---|---|---|

${F}_{}^{rad}/{p}_{0}^{2}$ | 3D | 5.480 | 5.503 | 5.558 | 5.618 | 5.660 |

$\left({10}^{-21}*N/P{a}^{2}\right)$ | 2D | 5.480 | 5.503 | 5.558 | 5.620 | 5.660 |

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**MDPI and ACS Style**

Jia, K.; Wang, Y.; Li, L.; Chen, J.; Yang, K. Inter-Particle Effects with a Large Population in Acoustofluidics. *Actuators* **2020**, *9*, 101.
https://doi.org/10.3390/act9040101

**AMA Style**

Jia K, Wang Y, Li L, Chen J, Yang K. Inter-Particle Effects with a Large Population in Acoustofluidics. *Actuators*. 2020; 9(4):101.
https://doi.org/10.3390/act9040101

**Chicago/Turabian Style**

Jia, Kun, Yulong Wang, Liqiang Li, Jian Chen, and Keji Yang. 2020. "Inter-Particle Effects with a Large Population in Acoustofluidics" *Actuators* 9, no. 4: 101.
https://doi.org/10.3390/act9040101