Basic Theory of Dielectrophoresis and Electrorotation

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Basic Theory of MICRO- & NANOELECTROKINETICS Dielectrophoresis and Electrorotation

Methods for Determining the Forces and Torques CIRCUIT © 1998 CORBIS CORP, DROPPER: © DIGITAL STOCK, 1997 Exerted by Nonuniform Electric Fields on Biological Particles Suspended in Aqueous Media THOMAS B. JONES

he forces exerted by nonuniform ac electric fields can be free, electrostatic field Eo(r) imposed by electrodes not shown harnessed to move and manipulate polarizable micropar- in the figure. To define the effective moment, it is convenient ticles—such as cells, marker particles, etc.—suspended to start with the electrostatic potential due to this electric Tin liquid media. Using rotating electric fields, controlled dipole [1]: rotation can be induced in these same particles. The ability to p(1) · r manipulate suspended particles remotely without direct con- (1) = (1) 3 tact has significant potential for applications in µTAS (micro 4 πε1r total-analysis systems) technology. The nonuniform fields for ¯ these particle manipulation and control operations are created where r is the radial vector distance measured from the center =| | by microelectrodes patterned on substrates using fabrication of the dipole and r r . Note the radial dependence of this (1) ∝ −2 techniques borrowed from MEMS (microelectromechanical potential, viz., r . If the dipole is small compared to systems) technology. A wide variety of structures, ranging the length scale of the nonuniformity of the imposed field Eo, from simple planar geometries to complex three-dimensional then the force and torque may be approximated as follows [2]: (3-D) designs, are now under investigation. The implications of these various schemes in certain fields of biotechnology are (1) ≈ (1) ·∇ ( ) far-reaching. For example, cells, cellular components, and F p Eo 2a (1) (1) synthetic marker particles treated with biochemical tags can be T ≈ p × Eo.(2b) collected, separated, concentrated, and transported using microelectrode structures having dimensions of the order of 1 to 100 µm (10−6 to 10−4 m). Furthermore, these forces can The dipole contribution to the total electric field cannot exert a manipulate DNA particles, which are several orders of magni- force on itself and therefore is not included in E o. tude smaller than cells. Now imagine replacing the dipole by a small dielectric This article presents a concise, unifying treatment of the sphere of radius R and permittivity ε2 at the same position in electromechanics of small particles under the influence of the structure, as shown in Figure 1(b). The particle has the electroquasistatic fields and offers a set of models useful in effect of perturbing the electric field. Expressed as an electro- calculating electrical forces and torques on biological particles static potential, this perturbation has the form: ∼ ∼ µ in the size range from 1 to 100 m. The theory is used to 3 ( ) (ε − ε ) R E · r consider DEP trapping, electrorotation, traveling-wave 1 ≈ 2 1 o (3) induced (ε + ε ) 3 induced motion, and orientational effects. The intent is to pro- 2 2 1 r vide a basic framework for understanding the forces and where it has been assumed that the particle radius is small torques exploited in the research represented by the other arti- compared to the length scale of the imposed field nonunifor- cles presented in this special issue of IEEE Engineering in mity. Equation (3) has the same form as (1) and the effective Medicine and Biology Magazine. moment is defined by comparing these two expressions. Effective Moment Method (1) ≡ πε (1) 3 ( ) The effective multipoles, including the dipole, the quadrupole, peff 4 1K R Eo 4 and other higher-order terms, facilitate a unified approach to (1) electric-field-mediated force and torque calculations on parti- where K ≡ (ε2 − ε1)/(ε 2 + 2ε1) is the Clausius- cles. We first introduce the effective dipole, leaving considera- Mossotti factor. Equation (4) defines the moment of the equiv- tion of the higher-order multipoles for later. Figure 1(a) alent, free-charge, electric dipole that would create a ( ) depicts a small electric dipole of vector moment p 1 = qd perturbation field identical to and indistinguishable from that located in a homogeneous, isotropic dielectric fluid of permit- of the dielectric sphere for all |r| > R. The only distinction tivity ε1. The dipole experiences a nonuniform, divergence- between this induced dipole and a general electric dipole is

that, because the particle is a sphere and because it is lossless, best example of these is a coplanar, quadrupolar electrode the moment will always be parallel to Eo. We later take advan- structure. A particle with ε2 <ε1 can be levitated passively tage of the fact that no such restriction need be imposed for along the centerline, where the field intensity is zero. For a force or torque calculations, thus facilitating consideration of particle located on the centerline, the net dipole moment is particle inhomogeneity, anisotropy, and electrical loss. exactly zero irrespective of particle size. Thus, the particle is To evaluate the force on the dielectric particle, the effective actually levitated by the quadrupolar force. If the particle size moment of (4) is substituted directly into (2a). The validity of is increased, octupolar and other, higher-order moments can this procedure may be argued from the standpoint of energy. come into play [5]. Except for the planar quadrupolar elec- An even simpler approach is to note that, if the Maxwell stress trode geometry, such a situation is unusual. In far more cases tensor is used to calculate the force, then the cases of the phys- with micron-sized particles, quadrupolar corrections are negli- ical dipole and the dielectric sphere must yield the same result gible. Nevertheless, it is likely that, as µTAS structure sizes because, by definition, the fields are indistinguishable on any approach the 1-µm limit, higher-order multipolar contribu- surface enclosing the particle. Combining (2a) and (4) gives tions to the DEP force will become more influential for bio- the well-known expression for the DEP force on a dielectric logical particles. sphere in a dielectric medium [3], [4]: Appendix A summarizes a dyadic tensor formulation for the general multipoles. In just the same way that the effective ( ) dipole moment is identified by comparing the induced electro- 1 ≡ π 3ε (1)∇ 2.() F 2 R 1K E o 5 static potential due to a particle in an approximately uniform electric field to the electrostatic potential of a physical dipole, According to (5), a particle will be either attracted to or one may establish the induced multipoles [6]. For the dielec- repelled from a region of strong electric field intensity, tric sphere shown in Figure 1(b), the general, induced tensor (1) (1) depending on whether K > 0 (ε2 >ε1) or K < 0 moment of order n is (ε2 <ε1), respectively. Note that combining (2b) and (4) ˙ 2n+1 gives zero for the torque, because the dipole moment and elec- ˙( ) 4πε1R n ( ) − p n = K n (∇)n 1E .(6) tric field are always parallel. To escape this restriction, the (2n − 1)!! o particle must be electrically lossy, nonspherical, or possess a permanent dipole moment. The lossy and nonspherical cases In (6), (2n − 1)!! ≡ (2n − 1) · (2n − 3) · ...· 5 · 3 · 1, are considered below in the “Particle Models” and “Illustrative (∇)n−1E represents n − 1 del operations performed on the (n) Cases in Biological DEP” sections, respectively. vector field, and K ≡ (ε2 − ε1)/[nε2 + (n + 1)ε1] is the generalized Clausius–Mossotti factor. The quadrupole and all Multipolar Force Contributions higher-order moments (n ≥ 2) are tensors induced by spatial The accuracy of (5) for the DEP force depends on how parti- derivatives of Eo. Note that, in a uniform field, only the dipole cle size compares to the length scale of the nonuniformity of moment (n = 1) survives. Combining (6) with (A1) gives a the imposed electric field Eo. The dipole-only approximation vector expression for the nth multipolar contribution to the is quite robust, and in only a few electrode geometries are force on a dielectric sphere. multipolar correction terms needed for accurate modeling. The 2n+1 (n) 4πε R F = 1 K(n)(∇)n−1E [·]n (∇)n E.(7) (n − 1)!(2n − 1)!!

In (7), the dyadic operation [•]n means n dot multiplications [7]. An indicial notation provides a form often more useful for analysis. The first three terms of the xi-directed component of the total force vector on a dielectric sphere are written below [6]. ∂E K(2)R2 ∂E ∂2E _ ε (F ) = 4πε R3 K(1)E i + n i d 2 total i 1 m ∂x 3 ∂x ∂x ∂x m m n m (3) 4 ∂2 ∂3 + K R En Ei +··· . ε 1 30 ∂xl∂xm ∂xn∂xm∂xl (8)

Equation (8) uses the Einstein summation convention, (a) (b) according to which all repeated indices are summed. The first term in this expression is the same as (5), the dipolar Fig. 1. Definition of the effective dipole moment: (a) small DEP force. The second term, the quadrupole (n = 2), can be physical dipole in nonuniform electric field; (b) dielectric par- interpreted as the attraction of the particle to the region ticle in the same nonuniform electric field. Assuming that the where the gradient of the electric field is highest. To make physical scale of the nonuniformity of the imposed field is use of these expressions, it is best to consider specific classes much larger than the particle radius R, (4) may be used to of electrode geometries, which is a task taken up in the evaluate the effective moment for the dielectric sphere. “Illustrative Cases in Biological DEP” section.

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Particle Models In biological DEP, electrical losses manifest themselves in Advantages of the effective moment method to determine terms of dramatic frequency dependence of force and torque; force and torque start to accrue when we seek realistic models thus, we assume sinusoidal time variation for the nonuniform for cells and other biological particles, such as concentric electric field imposed by the electrodes: shells and conductive or dielectric loss mechanisms. ( , ) = ( ) ( ω ) ( ) Eo r t Re Eo r exp j t 10 Spherical Shells Biological particles are complex, heterogeneous structures ( ) where Eo r √is a spatially dependent, rms electric field vector with multiple layers, each possessing distinct electrical prop- phasor, j = −1,ω is radian frequency, and t is time. erties. Reliable dielectric models for such particles are crucial Equation (10) accommodates any type of spatially varying, ac in biological dielectrophoresis. Consider the concentric, electric field, including linearly polarized, rotating (circularly dielectric shell subjected to an electric field Eo in Figure 2(a). or elliptically polarized), and traveling-wave fields. Assume As before, assume that the nonuniformity of this field is mod- that both the suspension medium and the particle are homoge- est on the scale of the particle’s dimensions. It may be shown neous dielectrics with ohmic electrical conductivities σ1 and that the induced electrostatic potential outside the particle, that σ , respectively. The method previously used to identify the | | > , 2 is, r R1 is indistinguishable from that of the equivalent, effective dipole can be employed again, this time using phasor ε homogeneous sphere of radius R1 with permittivity 2 shown quantities and the following substitutions: in Figure 2(b) [4], if ε1 → ε = ε1 + σ1/jω and 1 3 ε − ε ε → ε = ε + σ /jω. (11) ε = ε R1 + 3 2 2 2 2 2 2 2 2 R2 ε3 + 2ε2 The effective dipole moment, a vector phasor, becomes 3 R ε − ε ( ) ( ) 1 − 3 2 .() p 1 ≡ 4πε K 1 R3E .(12) 9 eff 1 o R2 ε3 + 2ε2 ε ε The multiplicative factor in (12) is 1, not 1. In the limit of a thin shell, i.e., (R1 − R2)/R1 1, (9) reduces The complex, frequency-dependent Clausius–Mossotti fac- to Maxwell’s mixture formula [8]. The identification proce- tor fits the Debye relaxation form: dure for the effective permittivity ε is the same one used to (1) (1) 2 (ε − ε ) K − K∞ identify the effective dipole moment, viz., an examination of K(1) ≡ 2 1 = K(1) + o (13) (ε + 2ε ) ∞ ωτ (1) + the external induced electrostatic potential function. The 2 1 j MW 1 effective, induced moment of the dielectric shell where ε → ε in Figure 2(a) is obtained by substituting 2 2 into the (1) Clausius–Mossotti factor, in other words, K → (ε − ε1)/ (1) 2 K∞ ≡ (ε2 − ε1)/(ε2 + 2ε1) and (ε + 2ε ). 2 1 (1) ≡ (σ − σ )/(σ + σ )() For a general multilayered shell, the procedure is to start Ko 2 1 2 2 1 14a τ (1) ≡ (ε + ε )/(σ + σ )() with the innermost sphere and the layer enclosing it, and to MW 2 2 1 2 2 1 14b define an effective permittivity using (9). The procedure is then repeated on the new, now homogeneous particle and the layer enclosing it. The same step is repeated until the outer- τ (1) most shell has been incorporated into the effective permittivi- MW is the dipolar Maxwell-Wagner relaxation time for ty. The particle is thus replaced by an equivalent, an ohmic, dielectric sphere suspended in a similar medium. (1) homogeneous sphere with a radius equal to that of the outer- Figures 3(a) and (b) contain typical plots of Re[K ] and most shell. Strictly speaking, it is not correct to use the above effective permittivity for the higher-order moments. In the case of an electric field with significant nonuniformity, (9) can not be substituted into K(2), K(3), etc., because the effective R permittivity differs for each multipolar moment. 2 R1 R1 Conductive Particles ε ε Accurate models for viable, biological cells suspended in 3 ´3 ε media must account for ionic charge conduction mechanisms 2 ε ε within and without the cell and, for frequencies above ∼1 1 1 MHz, dielectric losses. The simplest way to represent ionic charge transport is to employ an ohmic model. Largely (a) (b) because of their structure and because of the high internal, electrical conductivity, this approximation usually suffices Fig. 2. Multilayered shells: (a) spherical particle with one con- within the cell. On the other hand, an ohmic model is less suc- centric shell; (b) equivalent homogeneous particle. As long cessful in representing the aqueous, electrolytic media in as the physical scale of the nonuniformity of the imposed

which cells are commonly suspended. The problem is that field is much larger than the particle radius R1, the external double-layer phenomena introduce the complication of mobile field of the equivalent particle is indistinguishable from that of space charge outside but directly adjacent to the cell wall [9]. the heterogeneous particle.

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The effective multipoles, including the dipole, the quadrupole, and other higher-order terms, facilitate a unified approach to electric-field-mediated force and torque calculations on particles.

Im[K(1)] versus frequency. The rapid shift in the real part of Combining (12) with (2b), the time average torque is K(1) and the associated peak of the imaginary part at the relax- ( ) ation frequency, ω = 1/τ 1 , are of great importance in fre- (1) = (1) × ∗ c MW T Re p Eo quency-based dielectrometry of biological cells. eff = πε 3 (1) × ∗ .() The time-averaged, dipolar DEP force may now be written 4 1R Re K Eo Eo 18 using (12) in (2a): ∗ Equation (18) reveals that an electrical torque can be exerted F(1) = Re p(1) ·∇E .(15) eff o on a sphere only if i) loss is present (i.e., the Clausius- Mossotti factor is complex) and ii) the electric field has a Alternately, using indicial notation, the xi-directed vector com- spatially rotating component (i.e., the field is circularly or ponent of the time-average force is elliptically polarized). To incorporate dielectric loss—which is to be regarded as ∂ E∗ F = 4πε R3 Re K(1)E i .(16) distinct from ohmic loss—we replace the real permittivities total i 1 m ∂ xm ε = ε (ω) − ε (ω) + σ / ω with complex equivalents: 1 1 j 1 1 j and ε = ε (ω) − ε (ω) + σ / ω ε /ε 2 2 j 2 2 j . Note that tan( ) is the In (15) and (16), the effective dipole moment and all electric well-known dielectric loss tangent. When dielectric loss is field quantities take their rms magnitudes. incorporated in the force and torque expressions, ε replaces We can now reveal something of the nature of this time- 1 the multiplicative factor ε1 appearing in (16), (17), and (18) average, DEP force for a completely general alternating elec- [10], [11]. tric field by assuming E = ER + jEI.This form highlights the possibility that the electric field is circularly or elliptically More About Multilayered Particles polarized. Substituting into (15), one gets The introduction of loss mechanisms greatly enhances cell and bioparticle modeling capabilities. For example, (9) for the (1) = πε 3 (1) ∇ 2 effective permittivity of a layered particle can be modified to F 2 1R Re K Eo account for conductive loss using the substitution of (11). + (1) ∇× × ( ) 2Im K EI ER 17 3 R1 ε − ε ∗ ε = ε + 2 3 2 2 ≡ · 2 2 ε + ε where Eo Eo Eo . The first term in (17) looks just like (5) R2 3 2 2 and signifies that a particle will be attracted to or repelled from R 3 ε − ε regions of stronger electric field, depending on whether 1 − 3 2 .(19) (1) > (1) < ε + ε Re [K ] 0 or Re [ K ] 0, respectively. The second term R2 3 2 2 in (17) is a nonconservative force that exists only if the nonuniform field is rotating [6]. It is not useful to attempt to extract effective conductivity and permittivity expressions from (19). Rather, the best way to understand this expression 1.0 is to recognize that it adds a new Maxwell- DEP DEP Wagner interfacial charge relaxation mechanism ROT to K(ω) of the same form as (13) [12]. In general, 0.0 there will be one new relaxation frequency added ROT for each layer. Refer to Appendix E of Jones [4]. Particles with Thin Shells –1.0 0.01 0.11.0 10 100 0.01 0.11.0 10 100 Particles with thin, conductive, or insulative ω ω ω ω / MW / MW shells serve as useful models for some of the (a) (b) most important dielectric behavior of biological cells. Let R1 = R2 + δ, where δ R ≡ R1 . If Fig. 3. Examples of the DEP and ROT spectra, Re[K(ω)] and Im[K(ω)], respec- charge transport across the thin layer dominates, = ε /δ tively, for two limiting cases: (a) Ko = 1.0 and K∞ =−0.5. (b) Ko =−0.5 and one may define surface capacitance, cm 2 , K∞ = 1.0. A positive or negative peak of the ROT spectrum always accom- and transconductance, gm = σ2/δ. Taking proper panies a change in Re[K(ω)]. limits of (19) for δ/R 1, one obtains

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c Rε Illustrative Cases in Biological DEP ε = m 3 , where c ≡ c + g /jω. (20) 2 + ε m m m cmR 3 DEP trapping and levitation, electrorotation, and traveling- wave particle transport exemplify important applications of On the other hand, if the principal mechanism of ion trans- dielectrophoresis in biotechnology. In this section, we apply port involves motion tangential to the particle surface, the theory of multipolar dielectrophoresis, introducing reason- then we have able models for the electric field to reduce (15) and (18), and their multipolar generalizations, (24a) and (24b), to useable ε = ε + ε , 2 3 2 R where forms in each of these cases. The approach facilitates compar- ison of the ordinarily dominant dipole terms to the higher- ε ≡ ε + σ /jω, ε = δε2, and order multipolar corrections. σ = δσ2.(21) DEP Trapping and Levitation Figure 4 provides an interpretation for (20) and (21) in terms The most prevalent applications envisioned for biological of a circuit model for a biological cell, such as a plant proto- dielectrophoresis involve selective trapping or levitation of plast. A plant protoplast consists of a cell membrane enclosing individual cells or particles. Related schemes take the form of cytoplasmic fluid. The cytoplasm contains intracellular parti- continuous flow or batch separation systems. Trapping and cles, which are ignored here, but can be incorporated readily separation often rely on the frequency-dependent, dielectric into a somewhat more complex model. responses of particles. Highly effective schemes may be real- Equation (20) has the form of series-connected conduc- ized if one subpopulation of cells to be separated expresses a tances, and in Figure 4, this complex conductance is depicted positive DEP effect, that is, Re[ K(1)] > 0, while the other as a combination of transmembrane polarization and conduc- exhibits a negative effect, i.e., Re[K(1)] < 0. In practice, the tion. cm is the familiar membrane capacitance, which ordinari- use of frequency as a control parameter offers an excellent ly dominates over the transconductance gm. Equation (21) has means to achieve selective, single-pass separations. the form of parallel conductance, consisting of surface (tan- It may be shown that the net DEP force on a particle can be ε gential) permittivity , which accounts for electric-field- expressed as the sum of the gradients of a set of electro- induced, out-of-phase motion of bound ions tangential to the mechanical potentials Un, where n = 1, 2, 3, . . . correspond to cell wall or membrane, and the more familiar surface conduc- the dipole, quadrupole, octupole, etc.; that is, tivity, σ , which accounts for in-phase, ionic motions. If surface polarization can be ignored, i.e., ε σ /ω, then F =−∇[U + U + U +···] (25) further simplification results. DEP 1 2 3

ε ≡ ε + σ / ω() and the potentials are related to the electrostatic potential as 2 3 eff j 22 follows [13]:

where σeff ≡ σ3 + 2σ /R is the effective ohmic conductivity of an equivalent, homogeneous particle.

The Multipolar Terms It is a straightforward matter to incorporate ohmic and dielec- g tric loss into the general, multipolar model using complex per- m c mittivities and phasors. The tensor expression for the nth m Electrolytic Medium effective moment is: δ

˙ 4πε R2n+1n p(n) = 1 K(n)(∇)n−1E (23) (2n − 1)!! o σΣ εΣ (n) = (ε − ε )/ ε + ( + )ε where K 2 1 [n 2 n 1 1]. Then, time-average expressions for force and torque are: Zint Cytoplasm   ˙ ∗  p(n) [·]n (∇)n E  (n)  o  F = Re  (24a) Cell n! Membrane

gm cm 1 ˙ ∗ T (n) = Re p(n)[·]n−1 (∇)n−1 × E . (n − 1)! o (24b) Fig. 4. Circuit model showing transconductance (gm, cm) and parallel (g , ε ) paths for current flow through and around a Refer to Appendix B, which provides expressions for the first cell. In general, both current paths exist, but usually one or few terms of (24a) and (24b) in less compact but more use- the other dominates in its influence on the effective dielectric able, indicial notation. response of a particle.

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∂ 2 cal means to solve for o. For purposes here, we restrict atten- 3 (1) o dipole: U1 =−2πR ε1K tion to azimuthally periodic structures. This class of geome- ∂x tries, amenable to analytical modeling, includes all ∂ 2 ∂ 2 cylindrically symmetric electrodes, plus many planar struc- + o + o (26a) ∂y ∂z tures of the type commonly used to trap and rotate particles. Refer to Figure 5(a), (b), (c). The net, electrostatic potential o of all such structures may be expressed in the following series form [13]: 4πR5ε 1 ∂2 2 quadrupole: U =− 1 K(2) o 2 3 2 ∂x2 (m) = + + ( + ) 2 − ρ2 V am bmz cm[ 2m 2 z ] 2 2 2 2 ∂ o ∂ o 3 2 m + + +dm[(2m + 2)z − 3zρ ] +··· ρ cos mϕ(27) ∂y2 ∂z2 2 2 2 2 ρ ϕ ∂ o ∂ o where V is the applied voltage and , , and z are cylindrical + + ... ∂y∂z ∂z∂x coordinates. The independent coefficients am , bm , , are determined by the electrode geometry. The index value m = 0 2 2 ∂ o > + (26b) covers the case of axisymmetric structures, while m 0 ∂x∂y accounts for axially periodic geometries, in which case m can be interpreted as the number of salient electrode pairs. If (27) is substituted into the electromechanical potential 2πR7ε 1 ∂3 2 expressions, the axial and radial force component can be octupole: U =− 1 K(3) o 3 ∂ 3 obtained. It is convenient to treat separately the axisymmetric 5 6 x = > (m 0) and azimuthally periodic (m 0) cases. ∂3 2 ∂3 2 + o + o Axisymmetric Electrodes: m = 0 ∂ 3 ∂ 3 y z Consider a particle in a cusped electric field as shown in 3 2 3 2 Figures 5(a) or 5(b). Assume that the particle, initially in equi- 1 ∂ o ∂ o + + librium at z = 0, ρ = 0, is displaced by some small amounts z ∂ 2∂ ∂ 2∂ 2 x y x z and ρ. Correct to quadrupolar terms, the axial and radial force ∂3 2 ∂3 2 components are: + o + o 2 2 3 (1) 2 2 ∂y ∂x ∂y ∂z Fz ≈ 16πε1R K boco + 4c + 3bo do z V o ∂3 2 ∂3 2 + 96πε R5K(2) c d + 3d2z V2 (28a) + o + o 1 o o o ∂z2∂x ∂z2∂y ≈ πε 3 (1) 2 − Fρ 8 1R K 2co 3bo do 2 3 ( ) ∂ o + πε 5 2 2 ρ 2 ( ) + (26c) 96 1R K do V 28b ∂x∂y∂z (28a) and (28b) may be used to determine the equilibrium condition, i.e., the voltage-dependent levitation position, and, just as important, the stability of the equilibrium with respect =−∇ where Eo o. to small displacements. It is only necessary to obtain the Most microstructures for trapping and separation of biologi- coefficients bo , co , do , etc., via appropriate analytical or cal particles are geometrically complicated, requiring numeri- numerical means. The relative magnitudes of the dipolar and quadrupolar terms may be compared using ( ) Upper F 96πR5K 2 c d 6d quad = 0 0 ≈ 0 R2.(29) Electrode 3 (1) Ring + Fdipole 16πR K b0c0 b0

VV For one axisymmetric DEP levitator where the –– particle diameter was less than 1/10 the electrode spacing, the quadrupolar correction was Lower + Lower Electrode estimated to be less than 1% [14]; however, the R2 dependence of (29) indicates that the correction will become significant as particle size is (a)(b) (c) increased [5].

Fig. 5. Some representative trapping and levitation electrode structures. (a) Azimuthally Periodic Electrodes: m > 0 Side view of cusped electric field for negative DEP levitation (m = 0); (b) For azimuthally periodic structures, it is conve- side view of electric field for positive DEP levitation (m = 0); (c) top view of nient to select a specific value for m. Consider azimuthally periodic electrodes for quadrupolar electric field (m = 2) with the 4-pole structure shown in Figure 5(c), for zero field magnitude along the central axis. which m = 2. The force terms are:

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Advantages of the effective moment method to determine force and torquestart to accrue when we seek realistic models for cells and other biological particles.

32 ( ) the opposite direction. Note that p always rotates with the F ≈ πε R3K 2 a b + b2 + 12a c z V2 (30a) eff z 3 1 2 2 2 2 2 electric field. Refer to Jones for a physical interpretation of 3 (1) 2 2 5 (2) 2 2 these distinct cases [4]. Fρ ≈ 16πε R K a + R K b − 6a c ρ V 1 2 3 2 2 2 The rotation of particles with multiple layers can be mod- (30b) eled by substituting the effective complex permittivity of (19) into (13). Particles with multiple layers usually display multi- Because the electric field is zero on the axis, there is no net ple peaks, each of which reveals useful information. induced dipole so the axial levitation force is provided by the quadrupolar term [5]. On the other hand, the radial restoring Traveling-Wave DEP force term (proportional to ρ ) contains both dipolar and Consider the planar, horizontal electrode array shown in quadrupolar contributions. Just as for the m = 0 case, one may Figure 7. If these electrodes are driven by polyphase ac, a investigate equilibria and their stability from (30a) and (30b), traveling wave of electrostatic potential is created that can if the coefficients a2, b2, and c2 are known. suspend a lossy dielectric sphere vertically while simultaneously propelling it along the array. To first order, the electric Electrorotation field may be represented by a simple harmonic wave traveling The electrode structure of Figure 6 can be excited with multi- from left to right. A convenient phasor form for the Laplacian phase ac voltage to create a rotating electric field. If the field electric field is rotates counter-clockwise, it will have the following vector

phasor form on the axis: E(x, y) = E0(jxˆ +ˆy) exp(−jkx − ky)(35) ( , ) = (ˆ − ˆ)() E x y E0 x jy 31 where k = 2π/λ is the wavenumber and λ is the wavelength imposed by the center-to-center electrode spacing plus the where xˆ and yˆ are orthogonal unit vectors. If a spherical parti- electrical phasing. Note that, at any fixed point, the electric cle is introduced at the center, its induced dipole moment is field polarization is circular and counterclockwise. A lossy particle suspended above these electrodes will experience 3 (1) p = 4πε1R K E0(xˆ − jyˆ)(32) simultaneously x- and y-directed dielectrophoretic forces, as eff well as a torque. This dipole moment rotates in synchronism with the electric Using (35) in (B1) and (B2) for the x- and y-directed com- field but lags behind it by a phase factor associated with the ponents of the time-average DEP force, we obtain complex, frequency-dependent Clausius-Mossotti factor, K(1). It is this phase factor that makes electrorotation possible. From (18), the time-average electrorotational torque is:

( ) 1 =− πε 3 (1)(ω) 2.() ° T 4 1R Im K Eo 33 90

Note that the torque depends on the imaginary part of K(1), which is nonzero only if there is a loss mechanism. For a homogeneous sphere with complex permittivity ε = ε + σ /jω in a fluid with ε = ε + σ /jω, the time- 2 2 2 1 1 1 ° ° average torque is: 180 0

3 2 (1) 6πε R E (1 − τ /τ )ωτ T =− 1 o 1 2 MW (34) 2 (1 + 2ε1/ε2)(1 + σ2/2σ1)[1 + (ωτMW) ]

τ = ε /σ ,τ = ε /σ τ = τ 1 270° where 1 1 1 2 2 2 , and MW MW from (14b). As shown in Figures 3(a) and 3(b), the torque exhibits ω = τ −1 a peak at MW and the sign of this peak is positive or negative, depending on the relative magnitudes of τ1 and τ2. The possibility of positive or negative torque means that the Fig. 6. Four-pole electrode structure to create a rotating elec- particle can rotate, respectively, with the electric field or in tric field showing the phasing of the voltage excitation.

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=− πε 3 2 (− ) (36a) and (36b) becomes comparable to the dipolar term when Fx 4 1R kE0 exp 2ky R ≈ 0.19λ. 2k2R2 × Im K(1) + Im K(2) +··· (36a) Using (35) in (B4) and (B5) gives the time-average torque 3 for a particle in the traveling wave structure. =− πε 3 2 (− ) Fy 4 1R kE0 exp 2ky =− πε 3 2 (− ) Tz 4 1R E0 exp 2ky 2 2 (1) 2k R (2) 2 2 × Re K + Re K +··· (36b) ( ) 4k R ( ) 3 × Im K 1 + Im K 2 +··· .(37) 3

which is correct to the quadrupolar term. Note that the x- Comparing (36a) and (36b) to (37), we see that the quadru- directed, translational force depends on the imaginary parts of pole correction is more influential for the electrorotational the complex polarization coefficients. This force propels the torque than for the DEP force. It should be pointed out that particle in the +x or −x direction depending whether Im[K(1)] K(2) has almost the same Maxwell–Wagner relaxation fre- is negative or positive, respectively. The y-directed force will quency as K(1). As a result, the effect of the quadrupole either levitate the particle above the electrodes or hold it down terms in (36a), (36b), and (37) is to increase the magnitudes against them, depending on whether Re[K(1)] is negative or of the force and torque spectra but not significantly to alter positive, again respectively. The quadrupolar corrections in the frequency dependence. This analysis has been performed for a simple, harmonic, traveling electric field wave, with the electric field confined to two dimensions. Fy Substrate With Superposition may be employed to evaluate Embedded Electrodes forces and torques using (36a), (36b), and (37) for any electrode geometry, with the electric field y Fx obtained by analytical or numerical means. Numerical methods are in fact essential if one Fz wishes to account for the finite size of segmented x electrodes in real structures.

Alignment of Nonspherical Particles Nonspherical shapes are far more common forms of bioparticles than spheres. The most important cells in biomedical science are mammalian ery- Polyphase throcytes (red blood cells). Human erythrocytes Voltage Drive are essentially oblate spheroids with one side indented, while those of certain ruminants, e.g., the llama, happen to be fairly close to ideal Fig. 7. Traveling wave electrode structure. The traveling wave electric field spheroids with the three semi-major axes in the suspends the particle above the plane of the electrodes if Re[K(ω)] < 0. The approximate ratio of 4:2:1 [15]. Nonspherical induced motion is left to right or right to left depending on whether shape imparts geometric anisotropy to a particle, Im[K(ω)] < 0 or Im[K(ω)] > 0. In addition, the particle will rotate as it moves with the result that the induced dipole moment is along. parallel to the imposed electric field only if the particle is aligned with one of its principal axes parallel to the field. Refer to Figure 9. As a result, such particles experience an alignment torque in a uniform electric field. y The effective moment of a homogeneous spheroid having Eo semi-major axes a, b, and c may be written [1]:

4π abc − p = (ε − ε )E (38) eff 3 2 1 b − x where E is the uniform electric field internal to the particle, a which, in general, is not parallel to Eo. − c E = E , /[1 + (ε − ε )L /ε ], where x o x 2 1 x 1 abc ∞ ds z L ≡ (39) x 2 2 0 (s + a )Rs 2 2 2 Fig. 8. A spheroidal dielectric particle in a uniform electric where Rs ≡ (s + a )(s + b )(s + c ). Similar expressions field. The semimajor axes are a > b > c. The particle experi- for the y and z components of the internal field are readily ences an electrical torque that seeks to align the particle ascertained from (39). To determine the DEP force and torque with any of the axes, but only alignment along the longest on the ellipsoid, the effective moment from (38) is substituted axis (a) is stable. into (2a) and (2b), respectively.

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Of interest here is the torque, the x component of which is Acknowledgments The author has benefited from research collaborations extend- 2 4π abc(ε − ε ) (L − L )E , E , ing over many years with K.V.I.S. Kaler at the University of e = 2 1 z y o y o z . Tx Calgary and M. Washizu at Tokyo University. He is pleased ε2 − ε1 ε2 − ε1 3ε1 1 + Ly 1 + Lz to acknowledge past financial support from the Eastman ε1 ε1 ( ) Kodak Company, the National Science Foundation, and the 40 Japan Society for the Promotion of Science. With no loss of generality, we may assume that the electric field components are all positive and that a > b > c, in which Thomas B. Jones received the Ph.D. < < < < e < , e > case 0 Lx Ly Lz 1. Thus, Tx 0 Ty 0, and degree in electrical engineering from the e > Tz 0, which means that the electrical torque always tends to Massachusetts Institute of Technology, align the particle with one of its axes parallel to the electric Cambridge, in 1970. He began his profes- field. Only one of these alignments, viz., the one where the sional career in 1970 at Colorado State longest axis is parallel to the field, is stable. Prolate and oblate University in Ft. Collins, where he taught spheroids are readily treated as special cases [4]. and did research for 11 years. In 1981, he For a lossy ellipsoid, alignment and stability assume a fre- joined the technical staff at Xerox quency-dependent aspect. It is no longer true that only the Corporation in Webster, New York, to conduct research on longest axis is stable. In fact, depending on relative conductivity xerographic physics. In 1984, he left Xerox to become profes- and permittivity values, all three orientations become possible, sor of electrical engineering at the University of Rochester. He each in a different frequency range. The stable orientation in the focused his research for many years on the behavior of small low- and high-frequency limits is always with the long axis par- particles in electric and magnetic fields, and in 1995, he allel to the field, but for intermediate values, a particle will flip authored the monograph Electromechanics of Particles. His spontaneously to new orientation as each of several critical fre- present interests are in the exploitation of electrical forces to quencies is reached. The set of turnover frequencies is called microfluidics systems for application in the laboratory on a the orientational spectrum [15], and the cause of this behavior is chip. He was a member of the Electrostatic Processes the distinct Maxwell-Wagner charge relaxation time constants Committee of the IAS for many years and served as chair from for each of the three orientations. Ellipsoids with multiple layers 1985 to 1987. From 1996 to 1999, he was a member of the can be treated in a way analogous to the method explicated in IAS/IEEE Fellow Nominating Committee and served as chair above; however, there arises the difficulty that the effective during the last year. Prof. Jones was editor-in-chief of the complex permittivity is different for each axis. Journal of Electrostatics from 1992 to 2003. He has consulted extensively in industrial electrostatic hazards and nuisances Conclusion and co authored a book on that subject titled Powder Handling In this article, the effective dipole method, and its generaliza- and Electrostatics in 1991. tion to effective multipoles, has been exploited to evaluate forces and torques exerted on small particles by electric fields. Address for Correspondence: T.B. Jones, Department of The method makes it possible to treat multilayered concentric Electrical and Computer Engineering, University of shells and particles exhibiting ohmic and dielectric loss. This Rochester, Rochester, NY 14627 USA. E-mail: method may be extended further to the case of nonspherical [email protected]. particles, where alignment torques can be considered. These capabilities are well suited to modeling DEP behavior of bio- References logical particles including cells. [1] J.A. Stratton, Electromagnetic Theory. New York: McGraw-Hill, 1941, section 3.9. [2] P. Lorrain and D.R. Corson, Electromagnetic Fields and Waves, 2nd ed.. San The models and methods presented in this review are suffi- Francisco, CA: W.H. Freeman, 1970, section 3.12. ciently general to be of use in a broad range of applications for [3] H.A. Pohl, Dielectrophoresis. Cambridge, UK: Cambridge University Press, 1978. biological dielectrophoresis and particle electrokinetics. The [4] T.B. Jones, Electromechanics of Particles. New York: Cambridge University Press, 1995. range of validity can be stated confidently to cover particles hav- [5] T. Schnelle, T. Müller, and G. Fuhr, “Manipulation of particles, cells, and liquid ing diameters approximately 1 µm and larger. However, droplets by high frequency electric fields,” Biomethods, vol. 10, pp. 417–452, 1999. advances in fabrication techniques for nanostructures coupled [6] T.B. Jones and M. Washizu, “Multipolar dielectrophoretic and electrorotational theory,” J. Electrostatics, vol. 37, pp. 121–134, 1996. with ever-increasing demands for new capabilities for manipula- [7] R.B. Bird, R.C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids: tion and detection of biomolecules are inexorably pushing parti- Fluid Mechanics (vol. 1). New York: Wiley, 1977. [8] J.C. Maxwell, A Treatise on Electricity and Magnetism, New York: Dover cle size limits downward into the nanometer range. While the Press, 1954, art. 314. scaling laws of DEP and electrorotation would suggest that elec- [9] G. Schwarz, “A theory of the low-frequency dielectric dispersion of colloidal trostatic forces should become more and more dominant as size particles in electrolyte solution,” J. Chem. Phys., vol. 66, pp. 2636–2642, 1962. µ [10] F.A. Sauer, “Interactions forces between microscopic particles in an external is reduced below 1 m [16], it is well to bear in mind that other electromagnetic field,” in Interactions Between Electromagnetic Fields and Cells, forces, usually ignored for a 10-µm particle , become influential A. Chiabrera, C. Nicolini, and H.P. Schwan, Eds. New York: Plenum, 1985, pp. for a 100-nm particle. Furthermore, the discernible size scale of 181–202. 2 [11] F.A. Sauer and R.W. Schlögel, “Torques exerted on cylinders and spheres by distributed electrical charges in proteins starts at ∼10 nanome- external electromagnetic fields,” in Interactions Between Electromagnetic Fields ter. The implication of such charge distributions for the electrical and Cells, A. Chiabrera, C. Nicolini, and H.P. Schwan, Eds. New York: Plenum, 1985, pp. 203–251. forces and torques is that the higher-order multipoles, usually [12] X-B. Wang, R. Pethig, and T.B. Jones, “Relationship of dielectrophoresis and small corrections for a 10-µm particle, may become dominant in electrorotational behavior exhibited by polarized particles,” J. Phys. D: Appl. biomolecules. The lesson of these observations is that the simple Phys., vol. 25, pp. 905–912, 1992. [13] T.B. Jones and M. Washizu, “Equilibria and dynamics of DEP-levitated parti- models reviewed in this article will require modification and cor- cles: multipolar theory,” J. Electrostatics, vol. 33, pp. 199–212, 1994. rection as biotechnology moves into the nanoscale. [14] T.B. Jones, “Multipole corrections to dielectrophoretic force,” IEEE Trans.

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Industry Applicat., vol. IA-21, pp. 930–934, 1985. The summation in (A3) guarantees that all tensor moments are [15] R.D. Miller and T.B. Jones, “Electro-orientation of ellipsoidal erythrocytes: Theory and experiment,” Biophys. J., vol. 64, pp. 1588–1595, 1993. symmetric [17]. This is a requirement because force and [16] T.B. Jones, “Electrostatics on the Microscale,” presented at Electrostatics torque cannot depend upon the order in which the displace- 2003, Institute of Physics (UK), Edinburgh, April, 2003. ments creating the multipole are taken. The general expression [17] C-T. Tai, Generalized Vector and Dyadic Analysis, New York: IEEE Press, 1992, ch. 7. for the electrostatic potential due to the multipole of order n is ˙ 1 Appendix A p(n)[·]n(∇)n Figure A1 illustrates a straightforward way to generate the ( ) r n = (−1)n .(A4) general multipoles, starting with a point charge qo, and then 4πε1n! progressively moving to the dipole, quadrupole, etc. According to this construction, the dipole is formed from two Appendix B opposite sign charges displaced by d ; the quadrupole is then 1 Using (23), the individual force components of (24a) can be formed from two opposite sign dipoles displaced by d . For 2 expressed in a convenient, indicial form. The i-directed com- generality, each vector displacement must be independent. ponents of the n =1, 2, and 3 terms of the force are: The force and torque on a multipole of order n are [6]: ∂ ( = ) (1) = πε 3 (1) ∗ ( ) ˙ dipole n 1 : Fi 4 1R Re K Em Ei B1 ˙(n) n n ∂ (n) p [·] (∇) E xm F = o (A1) n! ( ) 4 ( = ) 2 = πε 5 ˙ quadrupole n 2 : Fi 1R (n) 1 ˙( ) − − 3 T = p n [·]n 1(∇)n 1 × E (A2) (n − 1)! o ∂E ∂2E∗ × Re K(2) n i (B2) ∂xm ∂xn∂xm and the multipoles are described in dyadic tensor form [6]

( ) 2 ( = ) 3 = πε 7 ˙ octupole n 3 : Fi 1R ˙( ) 15 p n = q d d ...d .() n i j k A3 ∂2 ∂3 ∗ (3) En Ei all permutations of × Re K (B3) ∂x ∂x ∂x ∂x ∂x i= j...=k with 1≤i≤n, l m n m l 1≤ j≤n,... ,1≤k≤n In the above, we employ the Einstein summary convention (ESC), according to which one sums over all repeated indices. Equation (24b) for the torque also can be reduced to more convenient and easily interpreted equations in indicial form. q 1 All electric field variables take their rms magnitudes.

d q0 1 ( = ) (1) =− πε 3 ∗ (1) ( ) dipole n 1 : Ti 4 1R Im Ej Ek Im K B4 n = 0 –q1

n = 1 − πε 5 ( = ) (2) = 8 1R q quadrupole n 2 : Ti 3 3 ∂E ∂E∗ × j k (2) ( ) –q3 Im Im K B5 ∂xm ∂xm q2 –q3 7 (3) 2πε1R –q d –q3 ( = ) =− 2 q 3 octupole n 3 : Ti 3 5 2 ∗ d –q q ∂ E ∂2E 2 2 3 × Im j k Im K(3) (B6) ∂xm∂xn ∂xm∂xn q2 q3 n = 2 In (B4), (B5), and (B6), indices i, j, and k must be in right- –q3 hand sequence: xyz, yzx, or zxy. These equations share two n = 3 common requirements for nonzero electrical torque: i) Im[K(n)] = 0, i.e., the particle (or the suspension medium) Fig. A1. Systematic generation of multipoles starting from a must be lossy, and ii) the orthogonal electric field components point charge. In the case of the general multipole, each dis- Ej and Ek must be out of phase, i.e., the field vector must placement must be independent. rotate in time.

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