Dielectric Force and Relative Motion between Two Spherical Particles

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Langmuir 2006, 22, 1602-1608

Dielectric Force and Relative Motion between Two Spherical Particles in Electrophoresis Kwan Hyoung Kang† and Dongqing Li* Department of Mechanical and Industrial Engineering, UniVersity of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S 3G8 ReceiVed August 8, 2005. In Final Form: October 8, 2005 When two particles close to each other are in electrophoretic motion, each particle is under the influence of the nonuniform electric field generated by the other particle. Two particles may attract or repel each other due to the dielectric force, depending on their positions in the nonuniform electric field. In this work, the dielectric interaction and the subsequent relative motion of the two arbitrarily oriented spherical particles are analyzed. The dielectric force is obtained by integrating the Maxwell stress. The result is valid for arbitrary orientations of the particles under the thin electrical-double-layer assumption. The magnitude of the dielectric force is compared to the so-called inertiainduced force, which shows that the dielectric force is normally much greater than the inertia-induced force. The relative velocity of particles is determined by the force balance between the dielectric force and the Stokes drag. The regions of attraction and repulsion are defined. It is shown that a pair of particles eventually aligns parallel to the externally applied electric field, except in the case where the two particles are initially oriented perpendicular to the electric field. A closed-form analytical solution is obtained for the particle trajectory by using the approximate expression for the dielectric force valid for not-too-closely located particles.

Introduction Particles in electrophoretic motion often form aggregates, which may be caused by hydrodynamic or electrical interactions between particles. For example, in the capillary electrophoresis of microparticles, multiple narrow peaks (spikes) were often observed. Each spike was attributed to a large aggregate of particles passing by a detector position.1 Such phenomena occur for various particles such as sperm cells,2 chromosomes,3 rodshaped bacteria,4 and humic substances.5 It was observed that DNAs also form an aggregate under an AC electric field.6 For the particles to form aggregates in electrophoresis, first the particles should be brought very close to each other to allow short-ranged interactions to take effects. Many theoretical analyses on “hydrodynamic” interactions predicted that identical particles in electrophoretic motion may hardly show any interactions between them,7-11 even in the case of thick electrical double layers (EDLs).12 This implies that a particle would be unaware of the existence of other particles. This paradox is understandable because, within the framework of creeping flow, the pressure is * To whom all the correspondences should be addressed. Present address: Department of Mechanical Engineering, Vanderbilt University, Nashville, Tennessee 37235. E-mail: [email protected]. Telephone: (615) 322-8601. † Present address: Department of Mechanical Engineering, Pohang University of Science and Technology, San 31, Hyoja-dong, Pohang 790784, South Korea. E-mail: [email protected]. (1) Radko, S. P.; Chrambach, A. Electrophoresis 2002, 23, 1957-1972. (2) He, L.; Jepsen, R. J.; Evans, L. E.; Armstrong, D. W. Anal. Chem. 2003, 75, 825-834. (3) Roberts, M. A.; Locascio-Brown, L.; MacCrehan, W. A.; Durst, R. A. Anal. Chem. 1996, 68, 3434-3440. (4) Zheng, J.; Yeung, E. S. Anal. Chem. 2003, 75, 818-824. (5) U ¨ bner, M.; Lepane, V.; Lopp, M.; Kaljurand, M. J. Chromatogr., A 2004, 1045, 253-258. (6) Mitnik, L.; Heller, C.; Prost, J.; Viovy, J. L. Science 1995, 267, 219-222. (7) Reed, L. D.; Morrison, F. A., Jr. J. Colloid Interface Sci. 1976, 54, 117133. (8) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417-439. (9) Kang, S.-Y.; Sangani, A. S. J. Colloid Interface Sci. 1994, 165, 195-211. (10) Yariv, E.; Brenner, H. J. Fluid Mech. 2003, 484, 85-111. (11) Anderson, J. L. Annu. ReV. Fluid Mech. 1989, 21, 61-99. (12) Shugai, A. A.; Carnie, S. L.; Chan, D. Y. C.; Anderson, J. L. J. Colloid Interface Sci. 1997, 191, 357-371.

almost uniform and the flow streamlines are like those of potential flows with no shear stress, resulting in no net “hydrodynamic force” pushing particles together. More detailed discussions on this issue can be found elsewhere in the literature.13-16 Several mechanisms have been suggested to account for some very long-range interactions that could bring particles very close to each other to induce an aggregation. (Here, we use the term “very long-range interaction” to distinguish it from the conventional long-range interactions such as the EDL interaction and the van der Waals interaction). These include the dipoledipole attraction, particle collision in heterogeneous systems, and the inertia-induced force. First, let us consider the dipoledipole attraction. When particles are in an electric field, each particle is under the influence of the nonuniform electric field generated by other particles. Then, a dielectric force may be exerted on each particle because the nonuniformity of the electric field may result in the asymmetric distribution of the electrical stress. This force truly has a long-range character, decaying proportionally to R-4 (R is a measure of the interparticle distance). As will be shown later, the dielectric force can have the same (or greater) order of magnitude compared to the electrophoretic force for typical conditions. Shilov and his colleague suggested that the dipole-dipole attraction between the particles could induce particle-chain formation and aggregation.17,18 They determined the dielectric force approximately by using the moment method19 for the parallel and vertical orientations with respect to the direction of the electric field17 as well as for arbitrary orientation.18 (13) Gamayunov, N. I.; Murtsovkin, V. A.; Duhkin, A. S. Kolloidn. Zh. 1984, 48, 233-239. (14) Yariv, E. Phys. Fluids 2004, 16, L24-L27. (15) Nichols, S. C.; Loewenberg, M.; Davis, R. H. J. Colloid Interface Sci. 1995, 176, 342-351. (16) Swaminathan, T. N.; Hu, H. H. J. Colloid Interface Sci. 2004, 273, 324330. (17) Simonova, T. S.; Shilov, V. N.; Shramko, O. A. Colloid J. 2001, 63, 108-115. (18) Shramko, O.; Shilov, V.; Simonova, T. Colloids Surf., A 1998, 140, 385393. (19) Jones, T. B. Electromechanics of Particles; Cambridge University Press: Cambridge, 1995; Chapters 2 and 3.

10.1021/la052162k CCC: $33.50 © 2006 American Chemical Society Published on Web 01/12/2006

Dielectric Force and RelatiVe Motion in Electrophoresis

Additionally, in a heterogeneous system, the ζ potentials of the particles are different, and therefore, the electrophoretic velocity will be different between different particles. The subsequent relative motion between the particles may provide a greater chance for particle contact. This mechanism was suggested by Zukoski and Saville20 and is still actively being investigated.4,15,21,22 Yariv14 and Swaminathan and Hu16 have recently shown analytically that another type of interaction force, the so-called “inertia-induced force”, can arise. In the analysis of Swaminathan and Hu,16 the particles are assumed to move without any relative motion between the particles to determine their inertia-induced force. They showed that the pressure field generated by the velocity gradient should generate a kind of “buoyancy” force on the particle. They derived a relationship by making use of the similarity between the hydrodynamic field and the electric field. Thereby, they were able to obtain the inertia-induced force without specifically analyzing the flow field. Yariv14 obtained an approximate expression for the inertia-induced force, relying on the successive-reflection method. He obtained an analytical expression for the trajectory of particles. This force decays in proportion to R-4, similar to the dielectric force. Some of the above-mentioned mechanisms may be dominant or coupled depending on the nature of media and particles and the stages of the interaction. To date, the mechanism of the whole process of particle interactions is still unclear; therefore, it is necessary to develop better understanding of very longrange particle interactions in the hope of ascertaining these interaction mechanisms. In the present investigation, we consider the dielectric interaction of the two identical spherical particles in electrophoretic motion in an unbounded domain. The present work is an extension of the dipole-dipole interaction mechanism of Shilov et al.17,18 for the case of the “not-too-far” separation. An exact result on the dielectric force is obtained, which is valid for arbitrary orientations of two particles. We also compare the relative importance of the dielectric interaction and the hydrodynamic interaction (i.e., the inertia-induced force) of Yariv14 and Swaminathan and Hu.16 The trajectory of the particles is obtained by balancing the dielectric force with the Stokes drag. In addition, an analytical expression for the particle trajectory is obtained by using an approximate expression of the dielectric force for the case of the “not-too-close” separation, and the validity of the result is discussed.

Dielectric Force We consider the dielectric interaction and the resulting relative motion of two spherical particles submerged in an electrolyte medium in which a uniform electric field of E∞ is applied. The two particles have the same radius a and ζ potential ζp. The thickness of the electrical double layer is assumed to be very thin compared to particle radius. We exclude the DLVO-type interaction by assuming that particles are separated by a sufficiently large distance compared to the thickness of the EDL. Application of electric field can raise the temperature of the medium due to Joule heating, resulting in change of material properties. In this work, the material properties are assumed fixed, i.e., without being affected by Joule heating. This assumption is appropriate for the microfluidic chip devices. The microfluidic chip devices generally are less affected by the Joule (20) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1989, 132, 220229. (21) Wang, H.; Zeng, S.; Loewenberg, M.; Davis, R. H. J. Colloid Interface Sci. 1997, 187, 213-220. (22) Zeng, S.; Zinchenko, A. Z.; Davis, R. H. J. Colloid Interface Sci. 1999, 209, 282-301.

Langmuir, Vol. 22, No. 4, 2006 1603

Figure 1. A pair of identical spherical particles submerged in an electrolyte medium under an externally applied uniform electric field E∞.

heating because of effective heat dissipation due to large thermal mass and surface-to-volume ratio.23 We introduce two Cartesian coordinate systems, (X, Y) and (x, y), respectively, that are both moving with the particles (see Figure 1). The X- and Y-axes are parallel and normal, respectively, to the uniform electric field, while the x- and y-axes are parallel and normal, respectively, to the line connecting the center of the two particles. The origins of the two coordinate systems are set to the center point of the connecting line between the two particles. The angle θ and the distance R between the particles, shown in Figure 1, may vary with time due to the relative motion of the particles. The dielectric force is not affected by the motion. This is because the time required for the establishment of a steady electric field for a given orientation is very short compared to the time scale associated with the motion of particles. The analysis for the relative motion enforced by the dielectric force will be discussed next by introducing relevant approximations for the resisting hydrodynamic force. In the limit of thin-Debye-layer thickness, the electric field satisfies the following Laplace equation outside the EDL:

∇ 2φ ) 0

(1)

where φ is the electrical potential. We use the far-field condition of the inner domain (EDL) as the boundary condition for the outer (bulk) domain. Because of the thin EDL approximation, the normal component of the electric field at the particle surface vanishes, i.e., n‚∇φ ) 0, where n is the unit vector normal to the surface of the particle pointing into the fluid region. At sufficiently far distance from the particle, a uniform electric field E∞ is given as E ) E∞, where E ) -∇φ denotes the electric field. The total force F and torque Q acting on a particle are determined by integrating both the hydrodynamic and the Maxwell stress tensor over the particle surface S as follows:

F)

∫S Th‚n dS + ∫S Te‚n dS

(2)

Q)

∫S (Th‚n) × r′ dS + ∫S (Te‚n) × r′ dS

(3)

Here, Th and Te are the hydrodynamic and the Maxwell stress tensors, and r′ the vector connecting the center of a particle to a point on a particle surface. The stress tensors are defined as follows:

Th ) -pI + µ(∇u + ∇uT)

(4)

1 Te ) - E2I + EE 2

(5)

(23) Petersen, N. J.; Nikolajsen, R. P. H.; Morgensen, K. B.; Kutter, J. P. Electrophoresis 2004, 25, 253-269.

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Figure 2. Computed distribution of E2 around two spherical particles in a uniform electric field: (a) θ ) 0°, (b) θ ) 45°, (c) θ ) 90°. The electric field is directed from left to right for all the cases. The darkness level is proportional to the magnitude of E2 (i.e., the darker, the stronger), and the arrows schematically indicate the direction of the dielectric force.

where I is the second-order isotropic tensor, p the pressure, µ the fluid viscosity, u velocity vector, and  the electrical permittivity. In eqs 2 and 3, the first and the second terms on the right-hand side are called the hydrodynamic (Fh, Qh) and the dielectric (Fel,Qel) contributions, respectively. Because of the insulating condition on the surface, E‚n ) 0, the integration associated with the second term in eq 5 vanishes. Thus, the dielectric force is obtained by integrating only the first term in eq 5 such that

 Fel ) 2

∫S E2n dS

(6)

Here the minus sign indicates that the dielectric force is always directed toward the region of lower electric-field strength. In the area of dielectrophoresis, this is called negative dielectrophoresis.19 Figure 2 illustrates the level of E2 computed numerically around two interacting particles in a uniform electric field that is directed from left to right. In Figure 2, the darkness level is proportional to the magnitude of E2. The arrow in the figure represents the direction of the dielectric force acting on each particle. When the particles are aligned parallel to the electric field, such as in Figure 2a, the lowest electric-field strength is in the gap between the two spheres. Thus, an attractive dielectric force will be exerted on the particles. In contrast, when the two particles are aligned orthogonally to the electric field, such as in Figure 2c, the highest electric-field strength is in the gap. In this case, the dielectric force is repulsive. In the case of θ ) 45° alignment shown in Figure 2b, the gap region has a higher electric-field strength in comparison to the electrical field strength around an isolated particle. The direction of the force for the case of Figure 2b is determined on the basis of the result of numerical simulation.

Comparison of Dielectric Force with Inertia-Induced Force In this section, we first determine the dielectric force for arbitrary orientations of the particles based on the superposition principle. We employ the known result of the electric-field distribution for the present configurations. We then describe the procedures for obtaining the inertia-induced force of Swaminathan and Hu16 and Yariv14 and compare its magnitude to that of the dielectric force. To evaluate the integration in eq 6, the electric field should be analyzed in advance. Because the electrical system (Laplace equation and the associated boundary conditions) is linear, the superposition principle can be used in the analysis. That is, the

electric field around a pair of particles can be analyzed by decomposing the uniform electric field (E∞) into two components, such as

E∞ ) (E∞)| + (E∞)⊥ ) E∞ cos θ xˆ - E∞ sin θ yˆ

(7)

Here, (E∞)| ) E∞ cos θxˆ and (E∞)⊥ ) -E∞ sin θyˆ represent the electric-field components parallel (x direction) and orthogonal (y direction) to the line connecting the two particles, and xˆ and yˆ represent the unit vector parallel to the x and y directions, respectively. Then, the resulting electric field satisfying the Laplace equation and the associated boundary conditions can be written as a sum of the two components E| and E⊥. These are the corresponding solutions for the parallel and the orthogonal components of E∞, i.e., (E∞)| and (E∞)⊥, respectively

E ) E| + E⊥ ) E∞ cos θ E ˜ | - E∞ sin θ E ˜⊥

(8)

where E ˜ | and E ˜ ⊥ are the nondimensionalized solutions for the parallel and the orthogonal components E∞. Hereinafter, the tilde represents the dimensionless variables. Because E2 ) E‚E ) E|‚E| + E⊥‚E⊥ + 2E|‚E⊥, the dielectric force in eq 6 for the particle in the first quadrant in Figure 1 can be written as

Fel ) -

Fa2E2∞ 2

∫S˜ {cos2 θE˜ |‚E˜ | + sin2 θE˜ ⊥‚E˜ ⊥ 2 sin 2θE ˜ |‚E ˜ ⊥}n dS˜ (9)

where S˜ represents the particle surface where the coordinate variables are normalized by particle radius. Note that E ˜ | and E ˜⊥ are symmetric with respect to the x axis, and accordingly, E ˜ |‚E ˜| and E ˜ ⊥‚E ˜ ⊥ are also symmetric with respect to the x axis. Therefore, the first two terms in braces of the above equation do not generate a force in the z direction, while they may generate nonzero electrical force in the x direction. On the other hand, E ˜ |‚E ˜ ⊥ is antisymmetric with respect to the x axis. Thus, its integration does not generate a force in the x direction but generates a nonvanishing force in the z direction. Then, the electrical force may have the following form in both x and y directions, respectively

Felx

FE2∞a2 )˜ ⊥ sin2 θ) (F ˜ | cos2 θ - F 2

Fely ) -

FE2∞a2 (F ˜ y sin 2θ) 2

(10a) (10b)

Dielectric Force and RelatiVe Motion in Electrophoresis

Langmuir, Vol. 22, No. 4, 2006 1605 Table 1. Sample Values of the Proportionality Constant E/(Gλ2) between the Dielectric Force and the Inertia-Induced Force Where G ) 103 Kg/m3, µ ) 10-3 kg/m/s, and E ) 6.9 × 10-10 C2/J/m ζp (mV)

-25

-50

-100

-150

/(Fλ2)

2.3 × 103

5.8 × 102

1.4 × 102

64

satisfying eqs 1, 12, and 13 and the relevant boundary conditions becomes7

u)

ζpE µ

(14)

As discussed by Yariv and Brenner10 and Swaminathan and Hu,16 the flow around a particle without a relative motion is a potential flow, and therefore, we can use the Bernoulli equation to obtain the pressure16 Figure 3. Dimensionless forces acting on interacting particles of Swaminathan and Hu.16 The dashed line is for F ˜ ⊥, the dash-dot line is for F ˜ |, and the dotted line is for F ˜ y. The two solid lines 1 and 2 are approximated ones that correspond to F ˜⊥ = F ˜ y = 6π/R ˜ 4 and F ˜| = 12π/R ˜ ,4 respectively. The variables F ˜ ⊥, F ˜ |, and F ˜ y, which are independent of θ, correspond respectively to F/⊥, F/| , and F/x in Figure 4 of Swaminathan and Hu.16

where

F ˜| )

∫S˜ E˜ |‚E˜ | nx dS˜

F ˜⊥ ) F ˜y )

∫S˜ E˜ ⊥‚E˜ ⊥ nx dS˜

∫S˜ E˜ |‚E˜ ⊥ ny dS˜

(11a) (11b) (11c)

Swaminathan and Hu16 analyzed the electric field around two interacting particles by introducing a bispherical coordinate system. They used the results to obtain an inertia-induced force (instead of the electrical force) by using the similarity between the electric field and the flow field. They defined F/⊥, F/| , and F/x (shown in Figure 4 of Swaminathan and Hu16) which correspond in a slightly different form to the forces of eq 11, i.e., F ˜ |, F ˜ ⊥, and F ˜ y, respectively. Figure 3 is a plot of the dimensionless forces F ˜ ⊥, F ˜ |, and F ˜ y, where the data shown had been extracted from Figure 4 of Swaminathan and Hu.16 Note that, in this figure, γ represents half of the gap distance, that is γ ) R/2 - a. Before we can compare magnitudes of the dielectric and inertiainduced forces, we first describe the procedure for determining the inertia-induced force. Swaminathan and Hu16 assumed in their derivation that the particles were uniformly charged, had the same ζ potential ζp, and flow was in a frozen state, i.e., without any relative motion. In the limit of creeping flow, the following continuity and incompressible momentum equations are satisfied:

∇‚u ) 0

(12)

µ∇ u - ∇p ) 0

(13)

2

The velocity at the particle surface is given by the HelmholtzSmoluchowski slip velocity of u ) ζpE/µ. It was considered that the mobility of the particles having the same ζ potential is not affected by the interaction with other particles.10 The particles are moving with the electrophoretic velocity of ζpE∞/µ. Therefore, in a moving frame of reference, the flow velocity is U∞ ) - ζpE∞/µ at a far distance from the particles. The solution

1 p ) p∞ - Fu2 2

(15)

where p∞ represents the hydrostatic pressure at a far distance from the particle, u ) |u|, and F the density of fluid. Because the flow is a potential flow, the viscous stress vanishes at the particle surface, and only the pressure force contributes to the hydrodynamic force. That is why the resulting hydrodynamic force is called the “inertia-induced force” by Swaminathan and Hu16 and Yariv.14 The velocity u can be represented by the local electric field by using eq 14. Then the hydrodynamic force becomes

Fh )

F 2

∫S u2n dS ) Fλ2 ∫S E2n dS 2

(16)

where λ )  ζp/µ represents the electrophoretic mobility of the particle. Because the right-hand side of eq 16 is similar to the dielectric force of eq 6, we can establish a direct relationship between the dielectric and inertia-induced (hydrodynamic) forces and compare their relative magnitudes. From eqs 6 and 16, we can derive the following relationship between the dielectric force and the inertiainduced force:

Fel ) -

 h µ2 h F )F 2 Fλ Fζ2p

(17)

It is obvious from the minus sign in eq 17 that the dielectric force is always opposite in direction to the inertia-induced force. Table 1 shows some typical values of /(Fλ2) for different ζ potentials in an aqueous solution. As shown, /(Fλ2) is always much greater than unity and typically in order O(102). This indicates that the dielectric force dominates the inertia-induced force by approximately 2 orders of magnitude. Note that the true inertiainduced force may be dependent on the relative motion of the particle.14 Although we considered the case of two spherical particles, the relationship given in eq 17 is valid, regardless of the number of particles and particle shape. Recently, the present authors measured the trajectory of particles around an electrically nonconducting rectangular hurdle in a rectangular microchannel under an applied DC field.24 The hurdle is placed at one side of the channel wall, and a narrow gap between the hurdle and the opposite channel wall is formed. In the gap region, the particle moved closer to the channel wall because of the contraction of streamlines in the gap region. It (24) Kang, K. H.; Xuan, X.; Kang, Y.; Li, D. J. Appl. Phys. Submitted.

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Kang and Li

is observed that the distance between the particle and wall (opposite to the wall where the hurdle is attached) becomes larger after passing through the gap. The degree of shift from the wall is proportional to the electric-field strength. A dielectric force between the particle and the wall was obtained by integrating the Maxwell stress, such as shown in eq 6. A numerically computed trajectory considering the dielectric force showed a close match to the experimental results. If the inertia-induced force were truly important, the particle should move closer to the sidewall instead of moving far from the sidewall. Such a particle-wall interaction can be regarded, as far as the dielectric interaction is concerned, a special case of particle-particle interaction of the present investigation with θ ) π/2. Therefore, this result strongly supports that the dielectric force is significant, and certainly more important than other forces such as the inertiainduced force. The stress acting on a particle, either the hydrodynamic or the dielectric stress, Th‚n ) - (1/2)Fu2n and Te‚n ) - (1/2)E2n, is always normal to the surface. For a spherical particle, r′ ) an and r′ × n ) 0, and the torque therefore vanishes. It is uncertain if it is generally true for a particle of arbitrary geometry.

Trajectory of Interacting Particles In this section, we analyze the relative motion of a pair of particles caused by the dielectric force. Swaminathan and Hu16 and Yariv14 have investigated the relative motion driven by the inertia-induced force. The procedure to analyze the relative motion is similar to theirs, but the final results are strikingly different, which is mainly because the sign of the dielectric force is opposite to that of the inertia-induced force. The force components in the (x, y) coordinate system shown in eq 10 can be transformed to those in the nonrotating (X, Y) coordinate system as

[]

2 2

[

][

E∞a cos θ -sin θ F FelX ˜x el ) sin 2θF ˜y sin θ cos θ 2 FY

]

(18)

where F ˜x ) F ˜ | cos2 θ - F ˜ ⊥ sin2 θ. To determine the velocity of particles driven by the dielectric force, to the leading order, we assume that the Stokes force counteracts the dielectric force, as was done by Yariv.14 Then, the velocity of a particle is determined by up ) Fel/(6πµa), which becomes

up ) -

[

][

E2∞a cos θ -sin θ F ˜x sin 2θF ˜y 12πµ sin θ cos θ

]

E2∞a

µ Fel el , F ˜ , u ) ) c µ E2∞ E2∞a2

Then, the equation of particle motion can be written in dimensionless form as

[

][

dX ˜P ˜x 1 cos θ -sin θ F )sin 2θF ˜y dt˜ 12π sin θ cos θ

]

(19)

The instantaneous position of a particle can be numerically computed by integrating the above equation as follows:

X ˜P ) X ˜ P0 +

∫0˜t

( )

dX ˜P dt˜′ dt˜′

Figure 4 shows computed trajectories of two particles with the initial positions (X0/a, Y0/a) ) (0.1, 2) and (-0.1, -2), respectively. In the figure, the dotted circles show the initial position of the particle, while the solid circles correspond to the near-final positions. As can be inferred from Figure 2, a repulsion force initially acts on each particle (see Figure 2c), pushing the particles away from each other. However, because of the nonuniform electrical field shown in Figure 2b, there exists a force component that makes the particles rotate relative to each other. After the orientation angle reaches about θ ) 63.4°, the dielectric force becomes attractive and pulls the particles toward each other. When the interparticle distance becomes very small, say a few tens of nanometers, DLVO-type interactions may be involved, which are beyond the scope of the present investigation. Now we introduce an approximation for the dielectric force to derive an analytical solution for the trajectory. We follow the procedure of Yariv14 in which the trajectory of particles driven by the inertia-induced force was analyzed. Figure 3 shows that when γ is greater than unity, the force can be well predicted by

F ˜| =

12π 6π ,F ˜y ) F ˜⊥ = 4 4 R ˜ R ˜

(20)

By substituting these results to eq 10, the dielectric force for the particle in the first quadrant in Figure 1 can be approximately represented, in the (x, y) coordinate system, as

We choose the characteristic length scale l, time scale tc, and velocity scale uc as follows:

l ) a, tc )

Figure 4. Trajectory of a pair of particles in which (X0/a, Y0/a) ) (0.1, 2). The dotted and solid circles represent the initial and the final location of the particles. The electric field is from left to right.

F =el

6πE2∞a2 (R/a)

4

[xˆ (cos θ - 21 sin θ) + yˆ sin θ cos θ] 2

2

(21)

The same approximate relation can be derived by using the method of reflection, as was done by Yariv.14 It should be noted that a similar relation has been derived by Parthasarathy and Klingenberg25 to investigate the particle interaction in electrorheological (ER) fluids. Let us compare the order of magnitude of the above dielectric force and the electrophoretic force. From eq 21, it is evident that the dielectric force acting on a particle is on the order of Fel ∼ 6πE2∞a2(R/a)-4, while the electrophoretic force on a spherical particle is represented by FEP ) 6πζpaE∞.24,26 It follows then (25) Parthasarathy, M.; Klingenberg, D. J. Mater. Sci. Eng. 1996, R17, 57103.

Dielectric Force and RelatiVe Motion in Electrophoresis

Langmuir, Vol. 22, No. 4, 2006 1607

Figure 5. Schematic diagram for region of attractive and repulsive force.

Fel/FEP ∼ (E∞a/ζp) × (R/a)-4. For example, when E∞ ) 10 kV/ m, a ) 5 µm, ζp ) -50 mV, and R ) 5 µm, the two forces may have similar magnitudes of about 3.2 × 10-11 N. By using the force balance between the dielectric force and the Stokes force, the velocity is obtained in dimensionless form as

u˜ p = -

1 1 xˆ cos2 θ - sin2 θ + yˆ sin θ cos θ 2 R ˜4

[(

)

]

Accordingly, we can obtain the following equations in a (R,θ) polar coordinate system:

dR ˜ 2 cos2 θ - sin2 θ )dt˜ R ˜4

(22)

dθ 2 sin θ cos θ )dt˜ R ˜5

(23)

From eq 22, it is obvious that the sign (direction) of the force will be changed at θ ) 63.4° at which the right-hand side becomes zero. Figure 5 shows schematically the regions of attractive and repulsive forces. In the figure, one particle is at the center of the coordinate axes. When the other particle is in the shaded region, the force is attractive. Conversely, when the particle is in the nonshaded region, the force is repulsive. According to eq 23, the dielectric force always tends to align the particles in parallel direction to the electric field. For example, if we consider the range 0 eθ e π/2, the right-hand side of eq 23 is always negative. That is, the angle θ is always decreasing to zero. It means that the stable configuration is formed only when the particles are in the parallel orientation. This is certainly in contrast to the result (the inertia-induced force) of Yariv,14 because the sign of the dielectric force and the inertia-induced force is opposite. That is, the force is repulsive when the particles are aligned in normal to the electric field, while the force becomes attractive when the particles are in parallel to the electric field. It is reversed for the case of the inertia-induced force. By combining eq 22 with eq 23, we can derive the following particle-trajectory equation:

˜ 2 cos2 θ - sin2 θ 1 dR ) R ˜ dθ 2 sin θ cos θ

(24)

Yariv14 has obtained the closed form solution of the preceding equation in the following form:

sin θ R ) R0 sin θ0

x

cos θ cos θ0

(25)

where R0 and θ0 represent the initial value of R and θ, respectively. The similarity in the trajectory between the inertia-induced case

Figure 6. Trajectory of a pair of particles. The dotted line is for the analytical result of eq 23, the dashed line is for (X0/a, Y0/a) ) (0.1, 1), and the solid line is for (X0/a, Y0/a) ) (0.1, 2). The electric field is from left to right and the particle coordinates are scaled by R′ ) R0/(sin θ0 xcos θ0).

of Yariv14 and the present investigation is due to the similarity of the dielectric force and the inertia-induced force. What determines the trajectory is the direction of the force rather than the magnitude of the force. The only difference between these two cases, in nondimensionalized form, is that the direction of particle movement is reversed because of the difference in the direction of the forces. Figure 6 plots the analytical solution of eq 25 (dotted line), together with the numerical solution for (X0/a, Y0/a) ) (0.1, 1) (dashed line) and (X0/a, Y0/a) ) (0.1, 2) (solid line). Note that the dotted line is overlapped with the solid line. In the former case of (X0/a, Y0/a) ) (0.1, 1), the relative distance between the particles is closer, and the analytical result shows some deviation from the numerical prediction. It is because the force obtained by the method of reflection may have some error when the particles are very close. The latter case of (X0/a, Y0/a) ) (0.1, 2) corresponds to the case already shown in Figure 4. Note that X and Y are normalized by R′ ) R0/(sin θ0 xcosθ0) in Figure 6. The dotted line is essentially overlapped with the solid line. Thus, the analytical result shows an excellent agreement with the numerical result for (X0/a, Y0/a) ) (0.1, 2) when R ˜ is greater than unity. When the particles are aligned normal to the electric field (θ ) π/2) so that they are on the Y-axis, the particles move away from each other along the Y-axis as R ˜5 ) R ˜ 50 + 5t˜. When the particles are in the X-axis (θ ) 0), they approach each other as R ˜5 ) R ˜ 50 - 10t˜. Note that when the particles are very close, i.e., R ˜ ∼ 2, the assumption introduced to derive the particle velocity may not be valid because the hydrodynamic interaction may become important. The time required for the two particles to become nearly in contact can be predicted approximately as ˜tc ) (R50 - 2)/10. This time can be written in dimensional form as follows:

tc )

( )[( ) ]

1 µ 10 E2 ∞

R0 5 -2 a

(26)

Table 2 shows sample values of tc for different initial relative distance and the electric-field strength in an aqueous electrolyte (26) Probstein, R. F. Physicochemical Hydrodynamics, 2nd ed.; John Wiley & Sons: New York, 1994; Chapter 7.

1608 Langmuir, Vol. 22, No. 4, 2006

Kang and Li

Table 2. Sample Value of tc in an Aqueous Electrolyte Solution E∞

R

gap

tc (sec)

5 kV/m

4a 6a 8a

d 2d 3d

5.9 × 100 4.5 × 101 1.9 × 102

10 kV/m

4a 6a 8a

d 2d 3d

1.5 × 100 1.1 × 101 4.7 × 101

15 kV/m

4a 6a 8a

d 2d 3d

6.6 × 10-1 5.0 × 100 2.1 × 101

20 kV/m

4a 6a 8a

d 2d 3d

3.7 × 10-1 2.8 × 100 1.2 × 101

solution. The range of the electric-field strength is chosen considering the typical electric-field strengths in electrokinetic experiments in microchannels. In this work, we considered only the case in which the size of the two interacting particles is the same. Under the point dipole approximation, however, the dielectric force on a particle is generated because of the perturbed electric field by the neighboring particle. That is, direction of the dielectric force will be determined only by orientation of the particles, but the magnitude of the force will be affected by particle sizes. Consequently, the particles having different sizes may show a similar character in stability to the case considered here, although the approaching or repelling speed may be affected by the size of the particles.

Conclusion The dielectric force, under the thin EDL assumption, acting on two identical spherical particles in electrophoresis in a uniform

electric field, is analyzed. The result is accurate for any orientation of particles. The dielectric force is derived by making use of the similarity between the dielectric force and the inertia-induced force.14,16 As the interparticle distance becomes larger, the dielectric force shows R-4 decay, in excellent agreement with the dielectric force obtained by the dipole-approximation method. It is shown for the particles having an identical ζ potential that the dielectric force is always much greater than the inertia-induced force. We analyzed the relative motion of two particles caused by the dielectric force, both numerically and analytically. These results show that the two particles will align parallel to the external electric field regardless of the initial orientations of the particles. One exceptional configuration is the case in which the two particles are initially oriented perpendicularly to the electric field (θ ) 90°). However, such an alignment may not be stable, as thermal fluctuation may change the perpendicular alignment. Consequently, the particles will attract each other and orient with the applied field. The accuracy of the trajectory obtained on the basis of the dipole approximation may somehow degrade as the interparticle distance becomes smaller. Nevertheless, the trajectory obtained by the dipole approximation gives a good result.

Acknowledgment. We are thankful for the financial support of the Canadian Institute of Photonic Innovation through a research grant to D. Li. We appreciate inspiring discussions with Dr. Xiangchun Xuan and the careful review of the manuscript by Mr. Edmond Young in the University of Toronto. LA052162K