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Three-Dimensional Interaction of a Large Number of Dense DEP Particles on a Plane Perpendicular to an AC Electrical Field^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Physical Description

#### 2.2. The Iterative Dipole Moment Method (IDM)

#### 2.3. The Modified Stokes Formula of a Large Number of Dense Particles

#### 2.4. The Governing Equation of the Particles and the Dimensionless Method

#### 2.5. The Validation of the Accuracy of the Modified Stokes Formula

## 3. Numerical Examples and Discussions

#### 3.1. The Interaction of Five Particles with Different Conductivities

#### 3.2. The Interaction of a Large Number of Particles with the Same Size

#### 3.3. The DEP Interaction of Large Numbers of Particles on a Bounded Circular Plate Chip

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**A number of particles randomly suspended on a flat chip filled with electrolyte and a uniform alternating current (AC) electric field ${\tilde{E}}_{0}$ is applied perpendicular to the chip plane.

**Figure 2.**(

**a**) The distribution of three particles; (

**b**–

**d**) the dimensionless hydrodynamic forces versus the ratio of the distance and the particle radius. ${F}_{h}^{\ast}$, ${F}_{s}^{\ast}$ and ${F}_{m}^{\ast}$ are the hydrodynamic forces, respectively calculated by integrating the hydrodynamic stress tensor, the traditional Stokes formula and the modified Stokes formula.

**Figure 3.**The real parts of the C–M factors of the two types of particles as a function of the frequency. In the low-frequency region, particle 1 is an nDEP particle, while particle 2 is a pDEP particle. In the transition region, particle 2 is transformed from a pDEP particle into an nDEP particle. In the high-frequency region, both particles are nDEP particles.

**Figure 4.**The DEP interaction of five particles initially positioned on a plane perpendicular to an AC electric field with ${\tilde{E}}_{0}=0.02$ and $f={10}^{3}\text{\hspace{0.17em}}\mathrm{Hz}$. (

**a**) The initial particle locations; (

**b**) the final cross-shaped particle chain. The blue and red colors denote pDEP and nDEP particles, respectively.

**Figure 5.**The particle trajectories and the final positions of five particles in a perpendicular AC field with ${\tilde{E}}_{0}=0.02$ and $f={10}^{3}\text{\hspace{0.17em}}\mathrm{Hz}$. The blue and red colors denote pDEP and nDEP particles, respectively.

**Figure 6.**The DEP interactive forces of particle 1 and particle 2 with time. The force of particle 1 is almost zero, the force of particle 2 increases rapidly as particle 2 get closer to particle 1, and finally reaches a constant value when the particle group is formed.

**Figure 7.**The DEP interaction of five particles with same properties and initial positions as Figure 4, except the conductivities of particle 2 and particle 4 (${\mathsf{\sigma}}_{p2}={\mathsf{\sigma}}_{p4}=6\times {10}^{-3}\text{\hspace{0.17em}}\mathrm{S}/\mathrm{m}$) are on a plane perpendicular to a uniform AC electric field with ${\tilde{E}}_{0}=0.02$ and $f={10}^{3}\text{\hspace{0.17em}}\mathrm{Hz}$. (

**a**) The initial positions of the particles; (

**b**) the final line-styled particle chain with alternately arranged pDEP and nDEP particles. The blue and red colors denote pDEP and nDEP particles, respectively.

**Figure 8.**The particle trajectories and the final positions of five particles under a perpendicular AC field with ${\tilde{E}}_{0}=0.02$ and $f={10}^{3}\text{\hspace{0.17em}}\mathrm{Hz}$. The blue and red colors denote pDEP and nDEP particles, respectively.

**Figure 9.**The DEP interaction of a large number of randomly distributed particles under a perpendicular field with ${\tilde{E}}_{0}\text{}=\text{}0.02$ and $f={10}^{3}\text{\hspace{0.17em}}\mathrm{Hz}$. (

**a**) The initial positions of 100 dissimilar particles; (

**b**) the initial positions of 200 dissimilar particles; (

**c**) particle chains of 100 dissimilar particles; (

**d**) particle chains of 200 dissimilar particles. The blue and red colors denote pDEP and nDEP particles, respectively.

**Figure 10.**The DEP interactions of randomly distributed particles on a bounded circular plane chip under a perpendicular AC field ${\tilde{E}}_{0}^{\ast}=0.02$ with different frequencies. (

**a**) The initial random distribution of 140 particles; (

**b**) the particle chains when $f={10}^{3}\text{\hspace{0.17em}}\mathrm{Hz}$; (

**c**) the particle chains when $f={10}^{6}\text{\hspace{0.17em}}\mathrm{Hz}$; (

**d**) the uniform distribution when $f={10}^{7}\text{\hspace{0.17em}}\mathrm{Hz}$.

**Figure 11.**The particle chains with different particle sizes and conductivities when $f={10}^{3}\text{\hspace{0.17em}}\mathrm{Hz}$. (

**a**) The initial particle positions of two different sizes, half of the particles with radius ${a}^{\ast}=1$ and the other half of the particles with radius ${a}^{\ast}=2$; (

**b**) the initial particle positions of random particle sizes from ${a}^{\ast}=1$ to ${a}^{\ast}=2$; (

**c**) the particle chains of two different sizes; (

**d**) the particle chains of random particle sizes.

Coordinate | Particle 1 | Particle 2 | Particle 3 | Particle 4 | Particle 5 |
---|---|---|---|---|---|

${x}^{\ast}$ | 0 | 2 | −2 | −2 | 2 |

${y}^{\ast}$ | 0 | 3 | 3 | −3 | −3 |

${z}^{\ast}$ | 0 | 0 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Xie, C.; Chen, B.; Wu, J.
Three-Dimensional Interaction of a Large Number of Dense DEP Particles on a Plane Perpendicular to an AC Electrical Field. *Micromachines* **2017**, *8*, 26.
https://doi.org/10.3390/mi8010026

**AMA Style**

Xie C, Chen B, Wu J.
Three-Dimensional Interaction of a Large Number of Dense DEP Particles on a Plane Perpendicular to an AC Electrical Field. *Micromachines*. 2017; 8(1):26.
https://doi.org/10.3390/mi8010026

**Chicago/Turabian Style**

Xie, Chuanchuan, Bo Chen, and Jiankang Wu.
2017. "Three-Dimensional Interaction of a Large Number of Dense DEP Particles on a Plane Perpendicular to an AC Electrical Field" *Micromachines* 8, no. 1: 26.
https://doi.org/10.3390/mi8010026