Research ArticleIterative Dipole Moment Method for the DielectrophoreticParticle-Particle Interaction in a DC Electric Field

Qing Zhang1 and Kai Zhang 2

1China Tobacco Standardization Research Center, Zhengzhou, China2Institute of Fluid Engineering of China Jiliang University, Hangzhou, China

Correspondence should be addressed to Kai Zhang; zkzb3026@yahoo.com

Received 1 February 2018; Accepted 12 March 2018; Published 9 May 2018

Academic Editor: Martin Seipenbusch

Copyright © 2018 Qing Zhang and Kai Zhang.+is is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Electric force is the most popular technique for bioparticle transportation and manipulation in microfluidic systems. In thispaper, the iterative dipole moment (IDM) method was used to calculate the dielectrophoretic (DEP) forces of particle-particleinteractions in a two-dimensional DC electric field, and the Lagrangian method was used to solve the transportation ofparticles. It was found that the DEP properties and whether the connection line between initial positions of particles per-pendicular or parallel to the electric field greatly affect the chain patterns. In addition, the dependence of the DEP particleinteraction upon the particle diameters, initial particle positions, and the DEP properties have been studied in detail. +econclusions are advantageous in elelctrokinetic microfluidic systems where it may be desirable to control, manipulate, andassemble bioparticles.

1. Introduction

Microfluidic systems are widely used for biochemical analysisover the past decade, and electric forces are often used as anefficient and effective transport mechanism, which does notinvolve the intervention of moving parts and offers goodmanipulation over sample handling. Deposition of bio-particles and colloids onto the surface [1] is very important formany biochemical processes, such as assembly of carbonnanotubes [2, 3] and trapping of nanoparticles and nano-particle synthesis [4–6]. In these processes, particles’ trans-lation and rotation should be controlled accurately forcounting and assembly of these particles [7, 8]. Inmicrofluidicsystems, electrical and hydrodynamic forces [9] dominateparticles’ transportation. In the bulk fluid, particle transport ismainly affected by the hydrodynamic interactions and itsneighboring particles.

As one of the important forces of particle manipulation,dielectrophoretic (DEP) force has been drawing much at-tention in recent years. Equivalent dipole moment (EDM) has

usually been used to calculate DEP force for single particle butcannot be used for the interaction between neighboringparticles in an electric field. Maxwell stress tensor (MST)method [10] is complicated for the calculation of a largenumber of particles’ interaction [11]. +e iterative dipolemoment (IDM) [12–14] method, which does not solvecomplicated differential equations of an electric field while theparticle is moving, was used in this work for calculating theDEP forces of particle interactions.

To control the assembling of bioparticles with highaccuracy, an in-depth study is necessary to the underlyingmechanisms of DEP particle-particle interaction in micro-fluidic systems.+e impetus for the present study came fromthe importance in understanding the mechanics of particleassembling, which is often used to produce specialized ma-terial in a lab on a chip. In the present study, the DEP in-teraction of particle-particle [15] in a DC electric field will bestudied, and a mathematical model based on the Lagrangianmethod incorporating DEP particle-particle interactions [16]will be presented to compute particle trajectories, and

HindawiJournal of NanotechnologyVolume 2018, Article ID 3539075, 7 pageshttps://doi.org/10.1155/2018/3539075

consequently, the assembling will be calculated. Finally, thedependence of particle assembling on the particle’s initial positionand radius will be further concluded and discussed in detail.

2. Computational Model

As shown in Figure 1, in a two-dimensional incompressiblestill media, DEP cylindrical particles are initially equispacedon a circle with radius R� 20 μm and an initial angleθ0 with respect to the applied electric �eld. Here, parti-cle diameter d� 5 μm, for negative DEP, the permittivityof the media and particles are εm � 6.9e− 10CV

−1·m−1and εp � 2.5∗ 8.8541878176e− 12CV

−1 ·m−1; for positive

dielectrophoresis, εm � 2.5 ∗ 8.8541878176e − 12 CV−1 ·m−1

and εp � 6.9e − 10 CV−1·m−1.

�e dielectrophoretic force, FDEP, acting on a spherical,homogeneous particle suspended in a local electric �eldgradient is given by the expression

FDEP � 2πr3pεmRe[K(w)]∇E2, (1)

where rp is the particle radius, εm is the permittivity of thesuspending medium, ∇ is the Del vector (gradient) operator,E is the electric �eld incorporating additional �elds due toparticle interactions, and Re[K(w)] is the real part of theClausius–Mossotti factor, which is given by

E

y

x

R

P1P2d2

Pi

d1

di

Oθi

θ0

θ1

Figure 1: Cylindrical particles are equispaced on a circle in a two-dimensional DC uniform electric �eld E� 1KV/m; here, red representspositive DEP and black represents negative DEP.

2

2

1

1

(a)

1

1

2

2

(b)

Figure 2: Two particles (red or black dashed line) are initially equispaced on a circle (blue dashed line) with θ0� π/2 in a uniformDC electric�eld with E� 1 kV/m along x direction, and then the �nal particle chains (solid particles) are, respectively, shown after DEP interactionunder di�erent statuses: (a) two negative (black) DEP particle, (b) one positive (red) and one negative DEP particle.

2 Journal of Nanotechnology

K(w) �ε∗p − ε∗m( )ε∗p + 2ε∗m( )

, (2)

where ε∗m and ε∗p are the complex permittivities of themedium and particle, respectively, and ε∗ � ε− (jσ/w) withσ is the conductivity, ε is the permittivity, and w is theangular frequency of the applied electric �eld. �e limitingdirect current (DC) case of the equation is

K(w � 0) �εp − εm( )εp + 2εm( )

. (3)

And the dielectrophoretic force can be modi�ed asfollows:

FDEP � 4πr3pεmεp − εm( )εp + 2εm( )

E · ∇E. (4)

Considering the in�uence of the dipole-induced �eld ofother particles surrounding particle i, the modi�ed electric�eld near the particle located at ri is shown below:

E(1)i (r) � E0(r) + ∑n

j�1,j≠ iE(0)d rj, r( ), i � 1, 2, 3, . . . , N,

(5)

2 2 1 1

(a)

2 2 11

(b)

22 11

(c)

Figure 3: �e initial position of two particles are initially equispaced on a circle with θ0� 0 in a uniform DC electric �eld with E� 1 kV/malong x direction, and the �nal particle chains after DEP interaction under di�erent status: (a) two negative DEP particle, (b) two positiveDEP particle, and (c) one positive and one negative DEP particle.

Journal of Nanotechnology 3

where E(1)i (r)denotes the modi�ed electric �eld andE(0)d (rj, r) is the in�uence of the dipole-induced �eld ofparticle j; here,

Ed( )x � −zφdzx

� −Pi∣∣∣∣∣∣∣∣

2πεm

cos αiΔx2i + Δy2i

−2Δxi Δxi cos αi + Δyi sin αi( )

Δx2i + Δy2i( )2 ,

Ed( )y � −zφdzy

� −Pi∣∣∣∣∣∣∣∣

2πε0

sin αiΔx2i + Δy2i

−2Δyi Δxi cos αi + Δyi sin αi( )

Δx2i + Δy2i( )2 .

(6)

�emodi�ed electric �eld induces a new dipole moment,which again induces a new electric �eld as follows:

E(1)i ri( )→P(1)i →E

(1)d ri, r( ), i � 1, 2, 3, . . . , N, (7)

where E(1)i (ri), P(1)i , and E(1)d (ri, r) are the modi�ed electric

�eld, the modi�ed dipole moment, and the modi�ed dipole-induced electric �eld, respectively. �ese parameters can beiteratively modi�ed until a converged value of the electric�eld E(n)i (ri) is obtained and the �nal dipole momentconsidering other particles’ in�uence can be achieved:

Pi �4πa3i εm εi − εm( )

εi + 2εmE(n)i ri( ). (8)

�ese particles are assumed to be far from boundaries.�e forces on particles are calculated by (1) while the dipolemoment have been modi�ed using IDM every time step, andparticle transportation can be easily solved for particletrajectories and �nal particle chains.

3. Results and Discussion

Imagine a particle suspended in a dielectric �uid and sub-jected to a uniform electric �eld, which will polarize thedielectric particle and induce a dipole moment in it, then thedipole moment will induce an electrostatic potential asshown below:

φd �Pi cos θ4πεmr2

. (9)

When particles are close to each other, one particle’sinduced electrostatic potential will distort its neighbor’selectric �eld and make it nonuniform, and then the DEPforce acting on its neighbor comes to a nonequilibrium state.�e nonzero DEP forces on particles in�uencing each othersuspended in a two-dimensional DC uniform electric �eldcan be calculated by the IDM method.

As shown in Figure 2, if two heterogeneous DEP par-ticles with same diameter are released at a small distancefrom each other and perpendicular to the electric �eld, theywill attract each other and are likely to cluster at the center ofline of their initial position. However, if all parametersexcept for the DEP property are the same, the results showthat two homogeneous DEP particles will repel each other tomove outwards symmetrically.

1

1

2

2

(a)

1

1

2

2

(b)

Figure 4: �e initial position of two particles is initially equispaced on a circle with θ0� π/2 in a uniform DC electric �eld with E� 1 kV/malong x direction, and the �nal particle chains after DEP interaction under di�erent status: (a) two heterogeneous particles with di�erentdiameters, (b) two negative DEP particles with di�erent diameters.

4 Journal of Nanotechnology

From Figure 3, it can be seen that if two homogeneousDEP particles with same diameter are released at a smalldistance from each other and parallel to the electric �eld,they will attract each other and cluster at the center of line oftheir initial position. However, if all parameters except forthe DEP property are the same, the results show that twoheterogeneous DEP particles will repel each other to moveoutwards symmetrically.

Figure 4 shows the DEP interaction between two par-ticles with di�erent diameters. Figure 4(a) shows that whentwo heterogeneous DEP particles are released at a smalldistance from each other and perpendicular to the electric�eld, they will attract and cluster. However, as the di�erenceof two particles’ diameters becomes larger, the position of�nal particle chains deviates from the center of line of theirinitial position, and the chains will move towards the side of

112

2

3

3

(a)

1 1

2

2

3

3

(b)

Figure 6: �e initial position of three particles is initially equispaced on a circle with θ0� 0 in a uniform DC electric �eld with E� 1 kV/malong x direction, and the �nal particle chains after DEP interaction under di�erent status: (a) three negative DEP particle and (b) twonegative and one positive DEP particle.

22 1 1

(a)

2 21 1

(b)

Figure 5: �e initial position of two particles is initially equispaced on a circle with θ0� 0 in a uniform DC electric �eld with E� 1 kV/malong x direction, and the �nal particle chains after DEP interaction under di�erent status: (a) two negative DEP particle with di�erentdiameter and (b) two positive DEP particle with di�erent diameters.

Journal of Nanotechnology 5

the smaller particle. In addition, if all parameters except forthe DEP property are the same, Figure 4(b) shows that thetwo homogeneous DEP particles with di�erent diameterswill repel each other to move outwards asymmetrically, andthe smaller particle moves faster.

Figure 5 shows that when two homogeneous DEPparticles with di�erent diameters are released at a smalldistance from each other and parallel to the electric �eld,they will attract and cluster, and the chains will move to-wards the side of the smaller particle.

From Figure 6(a), it can be seen that if three homoge-neous dielectrophoretic particles with same diameters areinitially equispaced on a small circle, they will attract eachother and �nally cluster parallel to the electric �eld. How-ever, from Figure 6(b), it can found that when one of theparticles changed its DEP property, they still clustered butthe chains were perpendicular to the electric �eld.

From Figure 7(a), it can be seen that due to the sym-metric con�guration, homogeneous DEP particles 1 and 3are attracted to move inwards because their connection lineis parallel to the electric �eld; at the same time, positive DEPparticles 2 and 4 are attracted by the chains of negative DEPparticles 1 and 3 to move inward. While the connection lineof the initial position of homogeneous particles 2 and 4 isperpendicular to the electric �eld, they cannot contact thechains of negative DEP particles 1 and 3. However, when theconnection line between two homogeneous DEP particles 2and 4 has a little shift θ0� π/16 from the line perpendicularto the electric �eld, DEP interactions among particles causefour particles to cluster as shown in Figure 7(b).

As shown in Figure 8, three negative and two positiveDEP particles are initially equispaced on a circle with θ0� 0in a uniformDC electric �eld. From the above conclusions, itcan be found that the homogeneous adjacent particles whose

connection line is not perpendicular to the electric �eld willcluster parallel to the electric �eld, and homogeneous ad-jacent heterogeneous particles will cluster perpendicular tothe electric �eld, and eventually form the complex andasymmetric structure as shown in Figure 8.

According to these aforementioned conclusions, theregular aggregation patterns of large number of polarizableparticles can also be well understood and can be used tocontrol, manipulate, and assemble polarizable particles.

2

2

4

4

1 13 3

(a)

2

2

4

4

113

3

(b)

Figure 7:�e initial position of two negative and two positive DEP particles are initially equispaced on a circle in a uniformDC electric �eldwith E� 1 kV/m along x direction, and the �nal particle chains after DEP interaction under di�erent status: (a) θ0� 0, (b) θ0� π/16.

44

1

1

3

3

55

22

Figure 8:�e initial position of �ve heterogeneous DEP particles isinitially equispaced on a circle with θ0� 0 in a uniform DC electric�eld with E� 1 kV/m along x direction, and the �nal particle chainsafter DEP interaction.

6 Journal of Nanotechnology

4. Conclusions

+e IDM method was used to study multiple particle DEPinteractions in a uniform DC electric field, and the DEPinteractions among particles cause particles to cluster. It wasfound that the homogeneous adjacent particles whoseconnection line is not perpendicular to the electric field willcluster parallel to the electric field, and homogeneous ad-jacent heterogeneous particles will cluster perpendicular tothe electric field; however, as the difference of two particles’diameters becomes larger, the position of final particlechains deviates from the center of line of their initial po-sition, and the chains will move towards the side of thesmaller particle. In addition, the dependence of the DEPparticle interaction upon the particle diameters, initialparticle positions, and the DEP properties have been studiedin detail. +is can be advantageous in elelctrokineticmicrofluidic systems where it may be desirable to control,manipulate, and assemble cylindrical bioparticles.

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper.

Funding

+e authors received funding from the National NaturalScience Foundation of China (11472260).

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