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

^{1}

^{2}

^{*}

^{†}

## 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

- Jones, T.B. Electromechanics of Particles; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Morgan, H.; Green, N.G. AC Electrokinetics: Colloids and Nanoparticles; Research Studies Press: Philadelphia, PA, USA, 2002. [Google Scholar]
- Kang, Y.; Li, D. Electrokinetic motion of particles and cells in microchannels. Microfluid. Nanofluid.
**2009**, 6, 431–460. [Google Scholar] [CrossRef] - Piacentini, N.; Mernier, G.; Tornay, R.; Renaud, P. Separation of platelets from other blood cells in continuous-flow by dielectrophoresis field-flow-fractionation. Biomicrofluidics
**2011**, 5, 034122. [Google Scholar] [CrossRef] [PubMed] - Moon, H.S.; Kwon, K.; Kim, S.I.; Han, H.; Sohn, J.; Lee, S.; Jung, H.I. Continuous separation of breast cancer cells from blood samples using multi-orifice flow fractionation (MOFF) and dielectrophoresis (DEP). Lab Chip
**2011**, 11, 1118–1125. [Google Scholar] [CrossRef] [PubMed] - Alshareef, M.; Metrakos, N.; Perez, E.J.; Azer, F.; Yang, F.; Yang, X.; Wang, G. Separation of tumor cells with dielectrophoresis-based microfluidic chip. Biomicrofluidics
**2013**, 7, 011803. [Google Scholar] [CrossRef] [PubMed] - Salomon, S.; Leichlé, T.; Nicu, L. A dielectrophoretic continuous flow sorter using integrated microelectrodes coupled to a channel constriction. Electrophoresis
**2011**, 32, 1508–1514. [Google Scholar] [CrossRef] [PubMed] - Ben-Bassat, D.; Boymelgreen, A.; Yossifon, G. The influence of flow intensity and field frequency on continuous-flow dielectrophoretic trapping. J. Colloid Interface Sci.
**2015**, 442, 154–161. [Google Scholar] [CrossRef] [PubMed] - Liu, W.; Shao, J.; Jia, Y.; Tao, Y.; Ding, Y.; Jiang, H.; Ren, Y. Trapping and chaining self-assembly of colloidal polystyrene particles over a floating electrode by using combined induced-charge electroosmosis and attractive dipole–dipole interactions. Soft Matter
**2015**, 11, 8105–8112. [Google Scholar] [CrossRef] [PubMed] - Lumsdon, S.O.; Kaler, E.W.; Williams, J.P.; Velev, O.D. Dielectrophoretic assembly of oriented and switchable two-dimensional photonic crystals. Appl. Phys. Lett.
**2003**, 82, 949–951. [Google Scholar] [CrossRef] - Lumsdon, S.O.; Kaler, E.W.; Velev, O.D. Two-dimensional crystallization of microspheres by a coplanar AC electric field. Langmuir
**2004**, 20, 2108–2116. [Google Scholar] [CrossRef] [PubMed] - Gangwal, S.; Pawar, A.; Kretzschmar, I.; Velev, O.D. Programmed assembly of metallodielectric patchy particles in external AC electric fields. Soft Matter
**2010**, 6, 1413–1418. [Google Scholar] [CrossRef] - Giner, V.; Sancho, M.; Lee, R.S.; Martínez, G.; Pethig, R. Transverse dipolar chaining in binary suspensions induced by RF fields. J. Phys. D Appl. Phys.
**1999**, 32, 1182. [Google Scholar] [CrossRef] - Hermanson, K.D.; Lumsdon, S.O.; Williams, J.P.; Kaler, E.W.; Velev, O.D. Dielectrophoretic assembly of electrically functional microwires from nanoparticle suspensions. Science
**2001**, 294, 1082–1086. [Google Scholar] [CrossRef] [PubMed] - Gupta, S.; Alargova, R.G.; Kilpatrick, P.K.; Velev, O.D. On-chip electric field driven assembly of biocomposites from live cells and functionalized particles. Soft Matter
**2008**, 4, 726–730. [Google Scholar] [CrossRef] - Zhou, R.; Chang, H.C.; Protasenko, V.; Kuno, M.; Singh, A.K.; Jena, D.; Xing, H. CdSe nanowires with illumination-enhanced conductivity: Induced dipoles, dielectrophoretic assembly, and field-sensitive emission. J. Appl. Phys.
**2007**, 101, 073704. [Google Scholar] [CrossRef] - Schütte, J.; Hagmeyer, B.; Holzner, F.; Kubon, M.; Werner, S.; Freudigmann, C.; Benz, K.; Böttger, J.; Gebhardt, R.; Becker, H.; et al. “Artificial micro organs”—A microfluidic device for dielectrophoretic assembly of liver sinusoids. Biomed. Microdevices
**2011**, 13, 493–501. [Google Scholar] [CrossRef] [PubMed] - Aubry, N.; Singh, P. Control of electrostatic particle-particle interactions in dielectrophoresis. Europhys. Lett.
**2006**, 74, 623. [Google Scholar] [CrossRef] - Aubry, N.; Singh, P. Influence of particle-particle interactions and particles rotational motion in traveling wave dielectrophoresis. Electrophoresis
**2006**, 27, 703–715. [Google Scholar] [CrossRef] [PubMed] - Ai, Y.; Qian, S. DC dielectrophoretic particle–particle interactions and their relative motions. J. Colloid Interface Sci.
**2010**, 346, 448–454. [Google Scholar] [CrossRef] [PubMed] - Ai, Y.; Zeng, Z.; Qian, S. Direct numerical simulation of AC dielectrophoretic particle–particle interactive motions. J. Colloid Interface Sci.
**2014**, 417, 72–79. [Google Scholar] [CrossRef] [PubMed] - Xie, C.; Chen, B.; Ng, C.O.; Zhou, X.; Wu, J. Numerical study of interactive motion of dielectrophoretic particles. Eur. J. Mech. B Fluids
**2015**, 49, 208–216. [Google Scholar] [CrossRef][Green Version] - Xie, C.; Liu, L.; Chen, B.; Wu, J.; Chen, H.; Zhou, X. Frequency effects on interactive motion of dielectrophoretic particles in an AC electrical field. Eur. J. Mech. B Fluids
**2015**, 53, 171–179. [Google Scholar] [CrossRef] - Kang, S.; Maniyeri, R. Dielectrophoretic motions of multiple particles and their analogy with the magnetophoretic counterparts. J. Mech. Sci. Technol.
**2012**, 26, 3503–3513. [Google Scholar] [CrossRef] - Kang, S. Two-dimensional dipolophoretic motion of a pair of ideally polarizable particles under a uniform electric field. Eur. J. Mech. B Fluids
**2013**, 41, 66–80. [Google Scholar] [CrossRef] - Kang, S. Dielectrophoretic motion of two particles with diverse sets of the electric conductivity under a uniform electric field. Comput. Fluids
**2014**, 105, 231–243. [Google Scholar] [CrossRef] - Kang, S. Dielectrophoretic motions of multiple particles under an alternating-current electric field. Eur. J. Mech. B Fluids
**2015**, 54, 53–68. [Google Scholar] [CrossRef] - Hossan, M.R.; Dillon, R.; Roy, A.K.; Dutta, P. Modeling and simulation of dielectrophoretic particle–particle interactions and assembly. J. Colloid Interface Sci.
**2013**, 394, 619–629. [Google Scholar] [CrossRef] [PubMed] - Kretschmer, R.; Fritzsche, W. Pearl chain formation of nanoparticles in microelectrode gaps by dielectrophoresis. Langmuir
**2004**, 20, 11797–11801. [Google Scholar] [CrossRef] [PubMed] - Sancho, M.; Martínez, G.; Muñoz, S.; Sebastián, J.L.; Pethig, R. Interaction between cells in dielectrophoresis and electrorotation experiments. Biomicrofluidics
**2010**, 4, 022802. [Google Scholar] [CrossRef] [PubMed] - Lee, D.H.; Yu, C.; Papazoglou, E.; Farouk, B.; Noh, H.M. Dielectrophoretic particle–particle interaction under AC electrohydrodynamic flow conditions. Electrophoresis
**2011**, 32, 2298–2306. [Google Scholar] [CrossRef] [PubMed] - Zhao, Y.; Hodge, J.; Brcka, J.; Faguet, J.; Lee, E.; Zhang, G. Effect of electric field distortion on particle-particle interaction under DEP. In Proceedings of the COMSOL Conference 2013, Boston, MA, USA, 9–11 October 2013.
- Liu, L.; Xie, C.; Chen, B.; Wu, J. Iterative dipole moment method for calculating dielectrophoretic forces of particle-particle electric field interactions. Appl. Math. Mech.
**2015**, 36, 1499–1512. [Google Scholar] [CrossRef] - Liu, L.; Xie, C.; Chen, B.; Chiu-On, N.; Wu, J. A new method for the interaction between multiple DEP particles: Iterative dipole moment method. Microsyst. Technol.
**2015**, 22, 2223–2232. [Google Scholar] [CrossRef] - Liu, L.; Xie, C.; Chen, B.; Wu, J. Numerical study of particle chains of a large number of randomly distributed DEP particles using iterative dipole moment method. J. Chem. Technol. Biotechnol.
**2015**, 91, 1149–1156. [Google Scholar] [CrossRef] - Xie, C.; Chen, B.; Liu, L.; Chen, H.; Wu, J. Iterative dipole moment method for the interaction of multiple dielectrophoretic particles in an AC electrical field. Eur. J. Mech. B Fluids
**2016**, 58, 50–58. [Google Scholar] [CrossRef] - White, F.M.; Corfield, I. Viscous Fluid Flow; McGraw-Hill: New York, NY, USA, 2006. [Google Scholar]

**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 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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