Dielectrophoresis (DEP) is a phenomenon where a force is exerted on a dielectric particle in a non-uniform electric field^{[1, 2, 3]}. The DEP phenomenon was first defined by Pohl^[4] based on the equivalent dipole method (EDM). Electrical polarization of finite-size particles can distort the local electric field, resulting in a further non-uniform field around particles. Consequently, an interactive DEP force between particles can be induced even in a uniform electric field. Traditionally, the EDM has been widely used for the DEP force calculation owing to its simplicity, but it is not accurate in some situations like a high number density or large size of particles^{[5, 6]}, where the gap between the particles is comparable to or even smaller than the particle size, and the interactions between the particles are non-negligible. The DEP force calculation based on the Maxwell stress tensor (MST)^[6] gives accurate results in theory. Kang and Li^[7] investigated relative motion of a pair of spherical particles in the DEP by applying a semi-analytical approximation on the solution of a dielectrophoretic force acting on particles and their trajectories. House et al. ^[8] studied the DEP particle-particle interactions for ellipsoidal particles using the boundary element method. Using an arbitrary Lagrangian-Eulerian (ALE) method, Ai et al. ^{[9, 10, 11]}, Ai and Qian^[12], and Xie et al. ^[13] performed direct numerical simulations of two-dimensional (2D) liquid-solid interaction of a pair of cylindrical particles in an externally applied uniform electric field. In their works, the flow, electric field, and particle motions were solved simultaneously. Later, Kang and Maniyeri^[14] performed direct numerical simulations to investigate the DEP interaction of three particles. They employed a finite difference method and a smoothed representation technique to solve for the electric field. Moreover, the particles were considered as a completely insulating medium, and therefore the field inside the particles could be ignored. Recently, Hossan et al. ^[15] investigated the DEP motions of multiple particles using a hybrid immersed interface-immersed boundary method with an aim to examine the DEP particle interactions including effects of permittivity difference between the particles and the electrolyte. Kang^[16] studied two-particle interactions and combined the dielectrophoretic and induced-charge electrophoretic motion of a pair of ideally polarizable (conducting) cylindrical particles with a finite-volume approach and an immersed-boundary method. Kurgan^{[17, 18]} made comparison of the dipole moment method and the MST method for calculating the dielectrophoretic forces in a direct current (DC) field. Washizus et al. ^{[19, 20]} presented an expansion method of Legendre series to calculate the electric field near a particle, including the applied field and the additional field resulting from the particle interactions. La Magna et al. ^[21] adopted the Monte Carlo-Poisson method to simulate particle interactions. Lee et al. ^[22] and Ai et al. ^[23] studied particle interactions in alternating current (AC) fields. Hector et al. ^[24] studied particle chains using dipole moment approximation. Local electrical fields around particle centers are corrected by additional electrical fields resulting from the dipole moments of the other particles. A correction of local fields is not sufficient for dense particle interactions. As a matter of fact, the corrected fields induce new dipole moments which interact each other again and then result in a second correction of local fields. The corrections of local fields can be repeatedly carried out in the same ways to obtain accurate local fields around all particles. The dielectrophoretic force calculation based on the MST method is theoretically accurate, but the numerical computation to solve the Laplace equation of the dense particles is cumbersome to implement. The numerical errors of the MST are inevitable and difficult to estimate in some cases. It is desirable to develop an effective and simple method for calculating the dielectrophoretic forces of dense particle interactions. Motivated by the above statements, we propose a new method to calculate dielectrophoretic forces, called the iterative dipole moment (IDM) method in this work. The IDM is able to calculate dielectrophoretic forces of particle interactions, and dispenses with numerical computations for electrical field when particles move. The IDM is an extended analytical algorithm of the equivalent dipole moment. Dipole-induced electrical fields of polarized particles are added to the applied field, leading to distortion of the local field around the particles and generating interaction forces between the dielectrophoretic particles. The local electric fields around the particles need to be iteratively corrected until a convergent electric field is obtained. A series of examples of two-particle interactions in a uniform field indicate that the IDM gives dielectrophoretic forces of particles in good agreement with the MST, and is easy to implement in applications.

N particles are located at r_i (i = 1, 2, ···, N) in an applied electric field E₀(r), where r = (x, y, z) denotes the location vector. The particles are polarizable, and the polarized charges are approximated as a point-dipole moment at ri aligned in the direction of the local field E₀(r_i). The dipole moment P_i and the dipole-induced electric field E_d(r_i, r) can be expressed in analytical forms. The particle interaction changes the local electric field, resulting in a modified electric field around the particle location r_i, i. e. ,

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where E_i ⁽¹⁾ (r) denotes the first-time corrected electric field around the ith particle location r_i, E₀(r) is the applied field without particles, and E_d ⁽⁰⁾ (r_j , r) is the additional electric field resulting from the dipole moment of the jth particle at r_j , denoting interaction of the jth particle and the ith particle. The corrected local electric field E_i⁽¹⁾ (r) induces a new dipole moment at r_i and a new dipole-induced electric field at r_i as follows:

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where E_i⁽¹⁾ (r_i) is the local vector of the corrected electric field E_i⁽¹⁾ (r) at the location r_i, P_i⁽¹⁾ is the local dipole moment at the location r_i induced by the local electrical field E_i⁽¹⁾ (r_i), and E_d⁽¹⁾ (r_i, r) is the corrected dipole-induced electric field, originated at r_i. The local electrical field, dipole moment, and dipole-induced electric field can be corrected once again. In this way, iterative corrections of the electric field around all particle locations r_i are continuously carried out in the following way:

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where the superscript n = 0 indicates that the electric field is not corrected by the particle interactions, and n = 1, 2, · · · denotes the iterative number of the electric field correction. When a convergent electric field is obtained, the dielectrophoretic force of particles can be calculated as follows:

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It can be seen that the DEP forces of particle can be accurately calculated provided that the local fields around N particle locations are sufficiently accurate. The accurate local field is the sum of the applied field and the additional field of particle interactions. It is not necessary to accurately solve for the electrical field everywhere over the whole space. This is the core idea of the IDM which can save a lot of computational effort without loss of accuracy of DEP forces on particles in an electrical field.

N circular particles are located in a 2D electric field E₀(r) with the location vector r = (x, y). The polarized charges of the particles are approximated as a dipole moment at r_i in the 2D field, expressed as follows^[18]:

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where the Clausius-Mossotti factor i = 1, 2, ···, N, a_i is the ith particle radius, and ε_m and ε_i are the permittivities of the fluid and the ith particle. The particles are positive DEP particles when K_i > 0 and negative DEP particles when K_i < 0. P_i, E₀(r_i) are the dipole moment and the applied electric field at r_i, respectively. The 2D dipole-induced electric field can be written as

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where r′ is the local position vector originated at r_i = (x_i, y_i). The electric field is iteratively corrected in the following way:

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When the electric field is convergent, the dielectrophoretic force of the ith particle in a 2D DC electric field can be calculated as follows^{[17, 18]}:

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In an AC electrical field with the frequency ω, the time-dependent electric field can be expressed in a complex form, i. e. ,

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where is the complex amplitude of the electrical field, and The complex dipole moment can be written as follows:

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where E(r_i) is the amplitude of the local field, is the complex Clausius-Mossotti factor, and is the complex permittivity defined as where ε and σ are the permittivity and conductivity, respectively. The time-averaged DEP force in an AC field can be expressed as

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where E_i is the amplitude of the local AC field. The particles are positive DEP particles when and negative DEP particles when

The dielectrophoretic forces of the particle interactions in this work are calculated by both the IDM and the MST for comparison. The MST reads as follows:

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for the particle side of the surface, and

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for the fluid side of the surface, where I denotes the second-order unit tensor, and E_m and E_p are the electric fields on the interface of the fluid side and the particle side, respectively. The electrophoretic force of the particle is obtained by integrating the MST over the particle surface^{[17, 18]}:

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where n is the unit normal vector on the particle surface pointing to the fluid. The electric field is solved by COMSOL MULTIPHYSICS^{[11, 12]}.

In an AC electrical field with frequency ω, the time-dependent electric field can be expressed in a complex form as (9). The MST reads as follows^[25]:

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for the particle side of the surface, where is the complex amplitude of the electrical field, (*) indicates a complex conjugate, and

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for the fluid side of the surface. The time-averaged DEP force in an AC field can be expressed as

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When dielectric particles are suspended in a fluid, the electric potential φ is governed by the Laplace equation

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The local electric field E can be calculated as follows:

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The computational domain is a square (200 mm×200 mm), φ₀ = 20 V on the left boundary, φ = 0 on the right boundary, on top and bottom boundaries. The condition on the particle surface is where ε_p and ε_m are the permittivities of the particle and fluid, respectively. The grids in the vicinity of the particle surface have been sufficiently refined. The computational domain consists of 29 430 triangle meshes. A relative error criterion of 10⁻³ DEP forces of particle is used to obtain grid-independent numerical solutions in the present examples. COMSOL MULTI-PHYSICS is employed to compute the DEP forces of the particle. The numerical details can be found in a previously published paper^[13].

Two particle interactions in a 2D uniform electric field are carried out. It is well known that the DEP forces of the particles are zero in a uniform field on the basis of the classical DEP theory. This, in fact, is not true. The DEP forces of particles in a uniform field can be generated by the particle interaction when the particles are sufficiently close to each other. Two circular cylindrical particles are located in a 2D uniform electric field E₀, as shown in Fig. 2, where d is the distance between two particle centers, a is the particle radius, and θ is the inclined angle between the connecting lines of the particle centers and the applied electric field.

Fig. 1 Sketch of two particles in uniform electric field and numerical grids

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Fig. 2 Sketch of two particles in uniform electric field: (a) two identical negative DEP particles; (b) negative DEP particle and positive DEP particles; (c) two negative DEP particles with different sizes

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Example 1 Interaction of two identical negative DEP particles

a=5 mm, ε_p=2, ε_m=80, and E₀=10⁵ V/m, as shown in Fig. 2(a). The DEP forces of Particle 1 are shown in Fig. 3, and the DEP forces of Particle 2 are the same as those of Particle 1, but in the opposite directions. The numerical solutions of the MST are also presented in the same figures for comparison. It can be seen that the interaction forces of particles calculated by the IDM agree well with those by the MST method. The interaction forces of particles increase when particles are closer (d/a decreases). It is found that the two particles are purely attracted (F_y =0) when the inclined angle θ =0°, and purely repelled (F_x =0) when θ =90°. When the ratio d/a decreases to below 2. 5, the two particles almost contact with each other, and the difference of DEP forces between the IDM and the MST increases. The dipole moment approximation of the IDM and the numerical error of the MST may not be acceptable in this limiting case.

Fig. 3 (a) DEP force of Particle 1 in x-direction; (b) DEP force of Particle 1 in y-direction

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The particle interaction forces in the direction of the connecting lines of the particle centers are defined as the attracting/repelling forces in this work, denoted by F_n, positive for attracting forces, and negative for repelling forces. The interaction forces perpendicular to the connecting lines of particle centers are defined as revolving forces, denoted by F_τ , positive for clockwise, and negative for counterclockwise, as shown in Fig. 4.

Fig. 4 Attracting/repelling forces and revolving forces of two-particle interaction in uniform electrical field

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Attracting/repelling and revolving forces of particles are shown in Fig. 5 and Table1, where d/a = 4.

Fig. 5 Attracting/repelling and revolving forces of two identical negative DEP particles in uniform DC electric field, where d/a = 4

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Table 1 Attracting/repelling and revolving forces of particles, where ε_p1= ε_p2 = 2

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It can be seen that the interaction forces of particles calculated by the IDM agree well with the MST. Particles repel each other when θ > 41°, and attract each other when θ < 41°. The particles revolve clockwise in the region of 0° < θ < 90°. The maximum revolving force appears around θ = 45°. Convergence behaviors of the interactive forces of two particles calculated by the IDM are shown in Fig. 6.

Fig. 6 Convergence test of attracting force of particles Fx calculated by IDM, where θ = 0◦

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It can be seen that the convergence of the IDM is very good. In the case of n = 0, the particle interaction is not taken into account, and the interactive DEP forces are all zero. It is consistent with the traditional DEP theory. The dielectrophoretic forces calculated by the IDM are almost the same as those by the MST when the iterative number n ≥ 3 in the present example.

Example 2 Negative and positive dielectrophoretic particles

a₁=a₂=5 mm, ε_p1=2 (negative dielectrophoretic particle), ε_p2=160 (positive dielectrophoretic particle), ε_m=80, and E₀ = 10⁵ V/m, as shown in Fig. 2(b). The dielectrophoretic forces of the negative particle 1 are shown in Fig. 7, while the dielectrophoretic forces of Particle 2 are the same as those of Particle 1, but in the opposite direction.

Fig. 7 (a) Dielectrophoretic force of negative Particle 1 in x-direction; (b) Dielectrophoretic force of negative Particle 1 in y-direction

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It can be seen that the interaction forces of particles calculated by the IDM agree well with those by the MST in the case of d/a ≥ 3. The relative difference between the IDM and the MST is about 10% when d/a = 2. 5 in the present example. Two particles repel each other (F_y=0) when θ =0°, and attract each other (F_x= 0) when θ = 90°. Attracting/repelling and revolving forces of particles are shown in Fig. 8 and Table2, where d/a = 4. It can be seen that the interaction forces of particles calculated by the IDM agree well with those calculated by the MST. In contrast to Example 1, the particles attract each other when θ > 45°, and repel when θ < 45°. The particles always revolve counterclockwise when 0° < θ < 90°, while the repelling forces become attracting forces in the vicinity of θ = 45°.

Fig. 8 Attracting/repelling and revolving forces of negative and positive DEP particles in uniform DC electric field, where d/a = 4

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Table 2 Attracting/repelling and revolving forces, when ε_p1 = 2 and ε_p2 = 160

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Example 3 Two negative dielectrophoretic particles with different sizes

a₁=10 mm, a₂=5 mm, ε_p1=ε_p2=2, ε_m=80, and E₀=10⁵ V/m, as shown in Fig. 2(c). The dielectrophoretic forces of Particle 1 are shown in Fig. 9, while the dielectrophoretic forces of Particle 2 are the same as those of Particle 1, but in the opposite direction. It can also be seen that the particle interaction forces of the IDM are in good agreement with those of the MST.

Fig. 9 (a) DEP force of Particle 1 in x-direction; (b) DEP force of Particle 1 in y-direction

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Attracting/repelling and revolving forces of particles are shown in Fig. 10 and Table3, where d/a₁ = 3.

Fig. 10 Attracting/repelling and revolving forces of two negative dielectrophoretic particles with different sizes in uniform electric field, where d/a1 = 3

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Table 3 Attracting/repelling and revolving forces, where a₁=10 μm and a₂=5 μm

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Example 4 Interaction of two identical positive particles in a uniform AC field

a=5 mm, ε_p1=ε_p2=2, σ_p1=σ_p2=0. 01 S/m, ε_m=80, σ_m=0. 001 S/m, f=10³ Hz, d/a=4, and the applied field E0=10⁵ V/m. Time-averaged attracting/repelling forces and revolving forces of particles are shown in Fig. 11 and Table4.

Fig. 11 Attracting/repelling and revolving forces of two positive particles in AC field, where d/a = 4

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Table 4 Attracting/repelling and revolving forces, where ε_p1 = ε_p2 = 2, σ_p1 = σ_p2=0.01 S/m, and f=10³ Hz

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Similar to the cases of the DC field, it can be seen that the time-averaged DEP forces of particles calculated by the IDM are consistent with those calculated by the MST. Two particles repel each other when θ>48°, and attract each other when θ < 48°, while the particles experience clockwise revolving forces when 0° < θ < 90°. The maximum revolving force appears about θ = 48° in the present example. The two particles experience only attracting forces, no revolving forces when θ = 0° (parallel to the applied field), and only repelling forces when θ = 90° (perpendicular to the applied field).

Example 5 Interaction of a positive particle and a negative particle in a uniform AC field

The negative particle has a conductivity σ_p1=3×10⁻⁴ S/m, and the positive particle has a conductivity σ_p2=5×10⁻³ S/m. The rest of the parameters are the same as those of Example 4. Attracting/repelling and revolving forces of particles are shown in Fig. 12 and Table5, where d/a = 4.

Fig. 12 Attracting/repelling and revolving forces of particles with different conductivities in AC field, where d/a = 4

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Table 5 Attracting/repelling and revolving forces, where ε_p1 =ε_p2 = 2, σ_p1 =3×10⁻⁴ S/m, σ_p2 = 5 ×10⁻³ S/m, and f=10³ Hz

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It can be seen again that the DEP forces of particles by the IDM agree well with those calculated by the MST. Contrary to Example 4, the two particles attract each other when θ > 45°, and repel each other when θ < 45°, while the particles experience counter-clockwise revolving forces when 0° < θ < 90°. The maximum revolving force appears at θ = 45° in the present example. The two particles experience only repelling forces, no revolving forces when θ=0°, and only attracting forces when θ=90°.

Example 6 Interaction of two negative particles with different conductivities in an AC field

The parameters are the same as Example 5, but the frequency f=10⁷ Hz (high frequency). Two particles are negative DEP particles. Attracting/repelling and revolving forces of particles are shown in Fig. 13 and Table6 for d/a = 4. By comparing Example 6 (f =10⁷ Hz) with Example 5 (f =10³ Hz), it can be seen that the interaction forces of the DEP particles can easily be changed (both in magnitude and direction) via the frequency modulation of the AC field. It is of great significance for the cell manipulation in the fields of biology and medicine.

Fig. 13 Attracting/repelling and revolving forces of two particles with different conductivities in AC field, where d/a = 4

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Table 6 Attracting/repelling and revolving forces, where ε_p1 = ε_p2 =2, σ_p1 =3 × 10⁻⁴ S/m, σ_p2 = 5 × 10⁻³ S/m, and f=10⁷ Hz

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An IDM method for calculating the interaction forces of dielectrophoretic particles is pre- sented in this work. The accuracy, convergence, and simplicity of the IDM have been demon- strated by a series of examples of two-particle interaction in a 2D uniform DC/AC electric field. The IDM gives particle interaction forces which are in good agreement with those calculated by the MST method in the case of the ratio of the particle center distance to the radius is greater than 3. 0 (d/a ≥ 3). The IDM is automatically reduced to the traditional EDM model when the ratio d/a increases (sparse particles). The IDM can avoid complicated numerical computation for solving the differential equations of the electrical field during particles moving, and is much easier on implement than the MST. As the future work, it is straightforward to extend the IDM to the cases of dense particle interactions in the three-dimensional electric field.