Effective dipole moment method has been widely accepted as the de facto technique in predicting the dielectrophoretic force due to the non-uniform electric field. In this method, a finite-particle is modeled as an equivalent point-dipole that would induce a same electric field under the external electric field. This approach is only valid when the particle size is significantly smaller than the characteristic length of interest. This assumption is often violated in a microfluidic device, where the thickness or width of the microchannel can be as small as the particle. It is shown in this numerical study that when the dimensions of the particle were in the same order of magnitude as the characteristic length of the device, dielectrophoretic force can be induced even in a uniform electric field. This force arises due to the disturbance of the particle and the bounding wall.Singapore-MIT Alliance (SMA

Goh, S.

Isabel, R.

Kua, C.H.

Lam, Yee Cheong

Yang, C.

Youcef-Toumi, Kamal

DSpace@MIT

      Abstract — Effective dipole moment method has been widely accepted as the de facto technique in predicting the dielectrophoretic force due to the non-uniform electric field. In this method, a finite-particle is modeled as an equivalent point-dipole that would induce a same electric field under the external electric field. This approach is only valid when the particle size is significantly smaller than the characteristic length of interest. This assumption is often violated in a microfluidic device, where the thickness or width of the microchannel can be as small as the particle. It is shown in this numerical study that when the dimensions of the particle were in the same order of magnitude as the characteristic length of the device, dielectrophoretic force can be induced even in a uniform electric field. This force arose due to the disturbance of the particle and the bounding wall.  Index Terms — electrostatic force, dipole moment, dielectrophoresis  I. INTRODUCTION IELECTROPHORESIS has been demonstrated to be able to manipulate bio-particles having characteristic length in micrometers. For instance, the separation of viable and non-viable yeast cells [1], the separation of mixtures of Bacillus subtilis, Escherichia coli and Micrococcus luteus bacteria [2], the separation of human  Manuscript received November 29, 2005.  C. H. Kua is with the Singapore-MIT Alliance, Nanyang Technological University, Singapore 639798. (e-mail: r030001@ntu.edu.sg). C. Yang is with the School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798. (e-mail: mcyang@ntu.edu.sg). S. Goh is with the Institute of Material Research and Engineering, Singapore. (email: shireen-goh@imre.a-star.edu.sg) I. Rodriguez is with the Institute  of Material Research and Engineering, Singapore. (email: i-rodriguez@imre.a-star.edu.sg) K. Youcef-Toumi is with the Singapore-MIT Alliance. He is also with the Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. (email:  Youcef@mit.edu) Y. C. Lam is with the Singapore-MIT Alliance. He is also with the School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798. (e-mail: myclam@ntu.edu.sg). leukemia cells [3] and the separation of human breast cancer cells from blood [4].    Effective dipole moment method [5] has been commonly employed to predict the magnitude and direction of the dielectrophoretic forces in such devices. According to the effective dipole method, the dielectrophoretic force, F, on a particle can be approximated as the dot product of the equivalent dipole induced on the particle, p, with the gradient of the applied electric field, E, [5] F = (p⋅∇)E              (1) where ∇ is a del operator. For a more rigorous approach, multipolar moment method [6] can be used to find the dielectrophoretic force induced by the higher order electric field terms [7].  There are two major assumptions in using the effective moment method. Firstly, it is assumed that the characteristic length of interest is significantly larger than the particle size. Secondly, it is assumed that the equivalent dipole or multipoles induced on the particle under non-uniform electric field is the same as the dipole or multipoles induced under a uniform electric field. In microfluidic devices where the microchannel thickness or width can be as small as 100 µm, and the nominal cell diameter in the range of 1~20 µm, these assumptions are often violated. Some attempts [8] have been made in proposing Maxwell Stress Tensor analytical approach to replace the dipole moment method, which was later found to be the same exact solution as the multipolar moment method [9]. In addition, there were also numerical approaches using the Maxwell Stress Tensor [10,11]. This investigation extends the numerical approach to include the case of uniform electric field.  II. BOUNDARY CONDITIONS The problem space consists of a dielectric rod with radius r, position at a distance a in a vacuum space, as shown in Figure 1. The length of the problem space is arbitrarily defined as twice the height. A potential with a normalized value 1 unit is applied to the left edge, and a zero potential is maintained at the right edge. The dielectric Generation of dielectrophoretic force under uniform electric field C.H. Kua1, C. Yang2, S. Goh4, R. Isabel4, K. Youcef-Toumi1,3, and Y.C. Lam1,2  1Singapore-MIT Alliance, N2-B2C-15, Nanyang Technological University 2School of Mechanical and Aerospace Engineering, Nanyang Technological University                3Department of Mechanical Engineering, Massachusetts Institute of Technology 4Institute of Materials Research and Engineering, Singapore  D   rod has a permittivity of ε2 = 7ε1, where ε1 is chosen as the permittivity of the free space, for simplicity. The normal components of the potential are zero at the top and bottom surfaces. The applied potential generated an electric field that induces a polarization charge on the dielectric rod surface. This surface charge interactes with the electric field to give rise to electrical forces acting on the dielectric rod. The problem is formulated and solved using a partial differential equation solver (FlexPDE).   Fig. 1. Two-dimensional model simulating the electric field and dielectrophoretic force. The potential is applied on the left and right edges to generate an uniform electric field.  III. ELECTROSTATIC  FORCE A. Electric field The calculation of the electric field often starts with the Gauss’ law, which relates the electric field intensity to its source. In the differential form, the Gauss’ law is [12]  ∇ ⋅ε0E = ρ                (2) where ε0 is the absolute permittivity of free space, and ρ is the total free charge in space. Equation (2) is readily valid for material having relative permittivity, κ, by taking the multiplication of κ on ε0. For convenient, the product is usually denoted as ε. In micrometer length scale, the total free charge can be assumed to be zero. In addition, the Faraday’s law relates the circulation of E to the time rate of change of the magnetic flux, H, which in differential form is [12] ∇ × E = ∂∂t µ0H              (3) where µ0 is the permeability of free space. In a microfluidic device, the characteristic length of the particle is much smaller than the electric field length, such that the right-hand side term is approximately zero. Thus, the vector electric field can be redefined as the gradient of a scalar Φ,  E = −∇Φ               (4) The governing equation is obtained by combining equation (4) into equation (2), taking into account the simplification for micrometer length scale,  ∇ ⋅ε(−∇Φ) = 0             (5) which is readily reduced to the Laplace equation for a homogenous material. In this study, the electric field on the problem space is obtained by solving the Laplace equation.  B. Dielectrophoretic force The electric field exerted a force on the surface charge. Employing the Maxwell Stress Tensor method, the total force acting on the rod surface, Fmst, is found by integrating the tangential and normal component of the electric field, Et2 and En2, respectively, over the dielectric rod surface, s, [10,13] Fmst = 12 (ε2 −ε1) (S∫ E t22 +ε2ε1 En 22 ) n ds     (6) where index 1 refers to the surrounding material, and index 2 refers to the object.  IV. NUMERICAL RESULT The electric field and electrostatic force acting on the dielectric rod are obtained by solving Equation (5) and (6) subjected to the boundary conditions in Figure 1. For the purpose of comparison, a simulation without the dielectric rod is also performed. The result in Figure 2 shows the electric field distribution when the dielectric rod is omitted. This condition corresponds to the dipole moment approach, where the dielectrophoretic force is obtained by taking the product of the equivalent dipole with the gradient of the electric field as depicted in equation (1). Using this approach, the dielectrophoretic force is predicted to be zero throughout the problem space, since the gradient of a uniform electric field is zero.  Fig. 2. Electric field distribution without considering the dielectric rod.  The electric field distribution when the dielectric rod is included in the calculation of the electric field is shown in Figure 3. Due to the existence of the dielectric rod, a non-uniform electric field is generated. The non-uniform electric field give rise to the dielectrophoretic force acting on the dielectric rod surface.  The normalized dielectrophoretic force for different value of dielectric rod radius and position is shown in Φ=1 Φ=0∂Φ/∂y =0xyar+2hhε1∂Φ/∂y =0ε2-0.5-0.3-0.10.10.30.5-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Dimensionless width, 2hNormalized electric field (E)  Figure 4. The result predicts that the force on the dielectric rod increases significantly when the rod is near to the wall. The force is small when the particle size is an order of magnitude smaller than the channel thickness, and zero when the dielectric rod is at the center of the channel.  Fig. 3. Electric field distribution taking into account of the dielectric rod. All variables have been normalized.  Fig. 4. The dependence of electrostatic force on the particle diameter to microchannel height, a/h, and the particle radius, r. All variables have been normalized.  V. DISCUSSIONS AND CONCLUSION This simulation shows that the particle has a significant effect on the electric field distribution when the particle diameter is in the same order of magnitude as the microchannel thickness. More specifically, the result highlighted the shortfall of the current effective dipole method in determining the dielectrophoretic force when the size of the particle cannot be ignored. The dielectrophoretic force can be generated even under an initially uniform electric field. Hitherto, the biggest hindrance in using the numerical approach is that it is not able to account for the case of a lossy material. The simulation is only readily available for a perfect conductor or perfect insulator. Without this capability, the dependence of dielectrophoretic force on the applied voltage frequency cannot be included in the simulation. REFERENCES [1] X.B. Wang, Y. Huang, J.P.H. Burt, G.H. Markx and R. Pethig, “Selective dielectrophoretic confinement of bioparticles in potential-energy wells,” Journal of Physics D: Applied Physics 26 (1993) 1278-1285 [2] G.H. Markx, P.A. Dyda and R. Pethig, “Dielectrophoretic separation of bacteria using a conductivity gradient,” Journal of Biotechnology 51 (1996) 175-180 [3] F.F. Becker, X-B. Wang, Y. Huang, R. Pethig and J. Vykoukal, “The removal of human leukaemia cells from blood using interdigitated microelectrodes,” Journal of  Physics D: Applied Physics 27 (1994) 2659-2662 [4] F.F. Becker, X-B. Wang, Y. Huang, R. Pethig, J. Vykoukal and P.R.C. Gascoyne, “Separation of human breast cancer cells from blood by differential dielectric affinity,” Proceeding of the National Academy of Sciences, USA. 92 (1995) 860-864 [5] T.B. Jones, Electromechanics of particles, Cambridge University [6] T.B. Jones, and M. Washizu, “Multipolar dielectrophoretic and electrorotation theory,” Journal of Electrostatics 37 (1996) 121-134 [7] T. Schnelle, T. Müller, S. Fiedler and G. Fuhr, “The influence of higher moments on particle behaviour in dielectrophoretic field cages,” Journal of Electrostatics 46 (1999) 13-28 [8] R. Paul and K.V.I.S. Kaler, “ Comparison of the effective dipole method for the computation of the torque with the Maxwell’s stress tensor method in non-lossy systems.” Conference Record of the 1990 IEEE Industry Applications Society Annual Meeting, 1990.  [9] X. Wang, X.-B. Wang, and P.R.C. Gascoyne, “General expressions for dielectrophoretic force and electrorotational torque derived using the Maxwell stress tensor method,” Journal of Electrostatics 39 (1997) 277-295 [10] K. Goossens, and L. Van Biesen, “A computer-aided analysis of dielectrophoretic force calculations,” Fifth International Conference on Dielectric Materials, Measurements and Applications (1988) 131-134. [11] M. Akyuz, M. Berg, A. Larsson and R. Thottappillil, “Calculation of the electrostatic force acting on axisymmetric objects in non-uniform electric fields.” High Voltage Engineering Symposium, 1999 [12] H.A. Haus and J.R. Melcher, Electromagnetic Fields and Energy, Prentice Hall, 1989.  [13] G. Backstrom, Fields of Physics by Finite Element Analysis: Electricity, Magnetism, and Heat in 2D and 3D, GB Publishing, Sweden.  0.00.20.40.60.81.00.0 0.1 0.2 0.3 0.4 0.5Dimensionless Position, a/h0.050.10.2r =0.30.01-0.5-0.3-0.10.10.30.5-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Dimensionless width, 2hNormalized electric field (E)

Generation of Dielectrophoretic Force under Uniform Electric Field

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Generation of Dielectrophoretic Force under Uniform Electric Field

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