Dielectrophoretic Force on a Sphere near a Planar Boundary

Langmuir 2005, 21, 12037-12046

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Dielectrophoretic Force on a Sphere near a Planar Boundary Edmond W. K. Young† and Dongqing Li*,†,‡ Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, ON M5S 3G8, Canada, and Department of Mechanical Engineering, Vanderbilt University, VU Station B 351592, 2301 Vanderbilt Place, Nashville, Tennessee 37235-1592 Received July 8, 2005. In Final Form: September 18, 2005 The small gap distance separating a spherical colloidal particle in electrophoretic motion from a planar nonconducting surface is a required parameter for calculating its electrophoretic mobility. In the presence of an externally applied electric field, this gap distance is determined by balancing the van der Waals, electrical double layer interaction, and gravitational forces with a dielectrophoretic (DEP) force. Here, the DEP force was determined analytically by integration of the Maxwell stress over the surface of the particle. The account of this force showed that its previous omission from the analysis always resulted in underpredicted gap distances. Furthermore, the DEP force dominated under conditions of low particle density or high electric field strength and led to much higher gap distances on the order of a few microns. In one particular case, a combination of low particle density and small particle size produced two possible equilibrium gap distances for the particle. However, the particle was unstable in the second equilibrium position when subjected to small perturbations. In general, larger particles had smaller gap sizes. The effects of four other parameters on gap distance were studied, and gap distances were found to increase with lower particle density, higher electric field strength, higher particle and wall zeta potentials, and lower Hamaker constants. Retardation effects on van der Waals attraction were considered.

Introduction Electrophoresis is defined as the motion in an unbounded quiescent fluid of a charged suspended particle under the influence of an externally applied electric field, while electroosmosis is the fluid flow past a stationary surface generated by a similar electric field.1 Both electrophoresis and electroosmosis occur simultaneously in typical electrokinetic flows through microchannels where charged particles are transported within an electrolyte solution. The impetus behind studying such electrokinetic phenomena is the ongoing desire to advance microfluidic devices for biomedical applications. The classical Smoluchowski formula for electrophoretic mobility of charged particles is simple and sufficient for describing the motion of particles in unbounded flow.1 However, it becomes invalid near boundaries where wall effects alter the electric and hydrodynamic influences on the particle. Boundary effects are important in a practical sense, because particles studied in microfluidic devices (polystyrene microspheres, latex beads, biological cells, or macromolecules) are typically denser (∼1020-1300 kg/ m3) than the surrounding fluid (1000 kg/m3 for water). As the particles are transported through the microchannels, gravity acts to pull them toward the bottom wall where they either attach or continue to move above the wall at an equilibrium height. Their eventual fate depends on competing attraction and repulsion forces that act on the particle in the presence of the boundary. Numerous theoretical studies have been conducted in relation to the boundary effects on electrophoretic motion of particles under various conditions. Keh and Anderson * To whom correspondence should be addressed. E-mail: [email protected]. † University of Toronto. ‡ Vanderbilt University. (1) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: New York, 1981.

used the thin electrical double layer (EDL) assumption and the method of reflections to obtain an expression for electrophoretic mobility of a colloidal sphere in the presence of several different boundary configurations.2 Ennis and Anderson used similar mathematical techniques to extend this work for arbitrary EDL thicknesses.3 Both these works assumed that gap sizes were small in relation to sphere radius yet sufficiently large with respect to the double layer thickness such that the method of reflections was applicable. Keh and Chen relaxed this assumption and solved the mobility problem for gap sizes as small as 0.5% of the sphere radius.4 Recently, Yariv and Brenner obtained a similar small-gap result using matched asymptotic expansions.5 The main lesson learned from these latter contributions is that, for very small gap sizes, electrophoretic mobility is actually enhanced with respect to the Smoluchowski equation because of a high local electric field generated within the narrow gap. Although these corrected electrophoretic mobility results are useful in describing particle motion near boundaries, it should be realized that their application hinges on the knowledge of the actual separation distance between the particle and the wall. This is obvious considering the mobility formulas presented in those works are all expressed in terms of a nondimensional gap size parameter equal to the ratio between the particle radius and the distance from wall to particle center. As mentioned previously, a particle reaches an equilibrium height above the wall under the influence of competing attraction and repulsion forces. Hence, given the nature and origin of these forces, a suitable equilibrium gap size can be predicted for any particle with specific surface and (2) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417-439. (3) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497-514. (4) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377-390. (5) Yariv, E.; Brenner, H. J. Fluid Mech. 2003, 484, 85-111.

10.1021/la0518546 CCC: $30.25 © 2005 American Chemical Society Published on Web 11/05/2005

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material properties, as long as these properties determine the magnitude of the relevant forces explicitly. The pertinent forces involved in balancing a particle near a wall are as follows. A particle in electrophoresis within a microchannel is expected to migrate to the bottom wall and settle sufficiently close to it such that the van der Waals (vdW) and EDL interaction forces become important. These two forces constitute the well-known colloidal force in the classical DLVO (Derjaguin-LandauVerwey-Overbeek) theory,6 and they are the topics of many studies in colloid science such as colloidal stability and particle deposition.7,8 In addition to these two interaction forces, the gravitational force must also be considered in the presence of gravity, and this has also been done previously.9,10 However, in the presence of an applied electric field through a dielectric medium, an important consideration regarding the nonuniformity of the local electric field between the particle and the boundary has thus far been neglected.10 The local electric field in the gap between the particle and boundary becomes highly nonuniform as the particle nears contact with the wall. This nonuniformity is responsible for aligning electric dipoles in the fluid and generating the Maxwell stress on the particle surface. As a result, the particle experiences the so-called dielectrophoretic (DEP) force. Depending on the particle’s size and the strength of the applied electric field, the magnitude of this DEP force may be on the order of the other competing forces mentioned. Thus, it is important to account for its effect when determining, in this case, the equilibrium gap size. In this study, we include in the determination of separation distance the effect of the DEP force. We propose to calculate it by integrating the Maxwell stress over the particle surface. The simple first dipole moment approximation is inadequate for this calculation since the nonuniform electric field in the small gap is induced by the existence of the particle.11 The main objective is to use the new force balance to obtain equilibrium separation distances for a variety of different particle and wall conditions. Once the gap sizes are available, they can be used to predict the electrophoretic mobility of particles having specific surface and material properties. In the process, we reveal interesting effects previously hidden by neglecting this DEP effect. Although the current work focuses on theory, we note here that a well-documented experimental method exists for measuring the separation distance between particle and wall. Prieve and co-workers have developed a technique called total internal reflection microscopy (TIRM), which measures the light scattered from evanescent waves to determine gap sizes between spheres and planar boundaries in solution.12,13 The sphere is levitated above the surface by the same delicate balance of forces mentioned above, and through the use of this technique, forces on the order of 10-14 N have been detected and measured. However, aside from Brownian motion, TIRM requires the particle to be held in a fixed position during measurement. In contrast, the current problem contains (6) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: San Diego, CA, 1991. (7) Rajagopalan, R.; Kim, J. S. J. Colloid Interface Sci. 1981, 83, 428-448. (8) Chein, R.; Liao, W. J. Colloid Interface Sci. 2005, 288, 104-113. (9) Yang, C.; Dabros, T.; Li, D.; Czarnecki, J.; Masliyah, J. H. J. Colloid Interface Sci. 1998, 208, 226-240. (10) Ye, C.; Li, D. J. Colloid Interface Sci. 2002, 251, 331-338. (11) Liu, H.; Bau, H. H. Phys. Fluids 2004, 16, 1217-1228. (12) Walz, J. Y.; Prieve, D. C. Langmuir 1992, 8, 3073-3082. (13) Prieve, D. C. Adv. Colloid Interface Sci. 1999, 82, 93-125.

Young and Li

Figure 1. Illustration of a spherical particle a distance d away from a planar wall. Cartesian coordinates are fixed in the frame of reference of the particle, and it moves with particle velocity Vp.

an applied electric field that propels the particle at a constant velocity along the length of the boundary. Thus, to use TIRM for measuring gap distance in the present case, modifications to the current technique would be needed to track the nonstationary particle. The authors are unaware of any other techniques currently available for measuring gap distance between a boundary and a moving sphere. In the next section, the mathematical model for the force balance of a sphere above a planar wall is described in detail and the contributing forces are examined in turn. Results are then presented showing the effect of including the DEP force in the analysis of the particle-wall interaction. Plots of separation distance as a function of particle radius for a number of different conditions are also presented, and the interesting features are discussed. Theoretical Basis Consider a single spherical charged particle of radius a in an electrolyte solution moving above a planar wall, where both the particle and the wall are rigid and nonconducting (Figure 1). The particle moves under the influence of gravity and an external electric field E∞ applied parallel to the surface in the x-direction. Cartesian coordinates are fixed in the moving frame of reference of the particle, with the z-axis perpendicular to the planar wall and intersecting the particle center at height h above the wall. The electric field induces electrophoresis of the particle and electroosmosis of the surrounding fluid. The particle is assumed to be relatively far away from the surface to start, possibly several particle diameters in distance away from the wall. As time passes, the particle approaches the wall because of the gravitational force on the denser particle. It eventually reaches an equilibrium height d above the wall, where d ) h - a, as measured from the sphere surface. At this equilibrium separation distance, the buoyancy-corrected particle weight FBG is balanced by a collection of competing forces in the z-direction. In the subsequent analysis, the van der Waals force FvdW, the EDL interaction force FEDL, and the dielectrophoretic force FDEP are considered. As shown in Figure 2, FvdW is assumed to be always attractive (toward the wall) in this study. The force balance in the z-direction is, therefore, given by

FBG + FvdW ) FEDL + FDEP

(1)

4 FBG ) πa3g(Fp - Ff) 3

(2)

FBG is simply

where g ) 9.81 m/s2 is the gravitational acceleration, Fp

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Langmuir, Vol. 21, No. 25, 2005 12039

Figure 2. Force balance in z-direction on a spherical particle near a planar wall.

is the particle density, and Ff is the fluid density. Note that, when the equilibrium separation distance is reached, it is assumed the horizontal forces are likewise balanced and the particle has reached a constant velocity Vp equal to the vector sum of the bulk fluid electroosmotic velocity and the particle electrophoretic velocity. Thus, no force analysis in the x-direction is necessary to determine the final separation. First, the electric potential is derived for this geometry. Then, the relevant forces in the z-direction are discussed in sequence. Electric Potential. To obtain the DEP force, it is necessary first to solve the electric potential distribution φ and then to determine the local electric field E from E ) -∇φ. In the present study, we assume that the double layers of the particle and wall are orders of magnitude thinner than the particle radius. This permits the separation of the flow domain into customary inner and outer regions.2 We assume that this separation is valid even for very narrow gap distances between particle and wall, as long as EDL overlap is minimal. This assumption was used by Yariv and Brenner to derive the small-gap electrophoretic mobility of a sphere near a planar wall using matched asymptotic expansions.5 The thin EDL assumption is justified at the end of this paper. The inner region contains the EDLs of both the particle and the wall, and it is assumed that the applied electric field has no influence on the ion distribution within the inner region, i.e., there is no polarization of the EDLs. Furthermore, the outer region is electrically neutral, which means the electric potential distribution can be determined by solving the Laplace equation, 2

∇ φ)0

Figure 3. Bispherical coordinates (ξ, η, φ). The particle surface is at η ) η0 and the planar wall is at η ) 0.

sphere to approach infinity. The analysis is facilitated by use of bispherical coordinates (ξ, η, φ), as shown in Figure 3, in conjunction with Cartesian coordinates. In addition, spherical coordinates (r, θ, φ), with origin at the sphere center, are needed for subsequent FDEP calculations. We summarize the transformations of these sets of coordinates as follows. For bispherical coordinates,

x)

c sin ξ cos φ , cosh η - cos ξ y)

c sin ξ sin φ , cosh η - cos ξ z)

c sinh η (6) cosh η - cos ξ

where c ) a sinh η0 and η0 ) cosh-1(h/a). For spherical coordinates,

r ) x(z - h)2 + x2 + y2, θ ) tan-1

xx2 + y2 z-h

,

(3)

φ ) tan-1

y (7) x

subject to the boundary conditions

n ˆ ‚∇φ ) 0

(4)

∇φ f -E∞

(5)

on the particle and wall surfaces, and in the far-field region away from the particle, respectively. Equation 4 is the insulating condition for non-conducting surfaces where n ˆ is the unit outward normal, and eq 5 is the far-field condition where the gradient of the electric potential distribution matches the applied electric field strength. Note that the particle and wall surfaces now refer to the interface between the inner and outer regions. To solve eqs 3-5, we draw upon the previous work by Keh and Chen, who determined the electric potential distribution analytically between two arbitrary spheres with the line of centers perpendicular to the applied electric field.14 The two-sphere solution can be applied to the current geometry by allowing the radius of the lower

The coordinate φ represents the angle of the meridian plane passing through the z-axis and is identical in both bispherical and spherical coordinates. Figure 3 shows the x-z plane, or equivalently, the φ ) 0 meridian plane. In its most general form, the solution to eqs 3-5 in the outer region is satisfied by the equation14

φ(ξ,η,φ) ) -cE∞ cos φ

[

sin ξ

cosh η - cos ξ ∞

-

( ( ) ( )) ]

∑ Rn sinh n)1

(cosh η - cos ξ)1/2 × sin ξ

Sn cosh n +

1

2

n+

1

2

η+

η P′n(cos ξ) (8)

(14) Keh, H. J.; Chen, S. B. J. Colloid Interface Sci. 1989, 130, 556567.

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where Pn is the nth order Legendre polynomial and the prime denotes differentiation with respect to cos ξ. The coefficients, Sn and Rn, are calculated from

3 3 η + Rn+1 cosh n + ηi 2 i 2 3 1 Sn n sinh n - ηi + (n + 1) sinh n + ηi 2 2 1 3 Rn n cosh n - ηi + (n + 1) cosh n + ηi + 2 2 1 1 (n - 1) Sn-1 sinh n - ηi + Rn-1 cosh n - ηi ) 2 2 1 25/2 exp - n + ηi sinh ηi (9) 2

[

(

(n + 2) Sn+1 sinh n +

[ [

( (

) ) (

[

)] )] )] ( )] )| |]

)

( ( (

)

[(

where ηi is either of the two spherical surfaces. For the current problem, the plane wall corresponds to η ) 0 while the sphere is at η ) η0. Substitution of η ) 0 into eq 9 shows the Rn coefficients all equate to zero. To solve the remaining Sn coefficients, Keh and Chen suggest choosing S0 ) 0 and rationalize that, for φ to remain finite, Sn must become negligible as n f ∞. Therefore, Sn can be solved as a set of (n + 2) linear equations. With the coefficients resolved, φ can then be used to determine FDEP. DEP Force. A DEP force on a particle arises in the presence of a nonuniform electric field, which exists in the present case (see Figure 1). The DEP force can be described from a microscopic point of view.15 Dipole moments (induced or permanent) in the dielectric medium tend to align themselves in the direction of the electric field. Electrical forces exerted on these dipoles generate a stress field similar to how hydrodynamic forces generate a stress field in the fluid. This electrical stress is known as the Maxwell stress, and is written in matrix form as15

1 Te ) - 0E2I + 0EE 2

(10)

where 0 ) 8.85 × 10-12 C/V/m is the permittivity of free space, is the relative permittivity of the fluid medium, and E is simply the magnitude of the local electric field E. Integration of the Maxwell stress over the surface of the particle ultimately yields the DEP force,

FDEP )

∫STe‚nˆ dS ) ∫S(- 210E2I + 0EE)‚nˆ dS

(11)

Because of the insulating condition given by eq 4, the second term of the Maxwell stress vanishes, and the above integration simplifies to

FDEP )

∫

∫

1 1 - 0E2n ˆ dS ) - 0 S 2 2

(∇φ)2n ˆ dS S

(12)

The integral can be evaluated in a fashion similar to that shown by Swaminathan and Hu, who performed the calculation in their study to determine an inertia-induced force.16 Using the spherical coordinates defined above, the integrand can be written as

(∇φ|r)a)2 )

∂φ (a1 ∂φ∂θ| ) + (a sin1 θ ∂φ | ) 2

r)a

2

r)a

(13)

From the chain rule, we have

∂φ ∂φ ∂ξ ∂φ ∂η ) + ∂θ ∂ξ ∂θ ∂η ∂θ

(14)

which simplifies to

∂φ ∂φ∂ξ ) ∂θ ∂ξ ∂θ

(15)

because ∂η/∂θ ) 0 on the particle surface. Since dS ) a2 sin θ dθ dφ, and since the component of FDEP in the z-direction can be obtained by noting that eˆ z‚n ˆ ) cos θ where eˆ z is the unit vector in the positive z-direction, we finally arrive at 2π π ∫φ)0 ∫θ)0 [(a1 ∂φ∂ξ ∂θ∂ξ)

1 FDEP ) - 0a2 2

2

+

∂φ (a sin1 θ ∂φ ) ]sin θ cos θ dθ dφ (16) 2

The scalar FDEP is understood in eq 16 and throughout this work as strictly the component in the z-direction. Equation 16 can be solved in conjunction with eqs 6-9 for a specific particle radius and separation distance using the Maple software package. The calculated value then contributes to the force balance of eq 1. Note that the integral is always negative, such that FDEP in the system studied here is always repulsive in nature. van der Waals Force. The vdW force is calculated using the Hamaker-De Boer approximation.6 For the limiting case of a sphere near a planar surface, the potential energy of interaction Uv is given by

Uv(d) ) -

a d Aa + + ln 6 d d + 2a d + 2a

[

]

(17)

where the subscript v denotes potential energy from van der Waals interaction. It has been noted that the retardation effect can become significant in some cases, particularly when the separation distance is large.6 Here, this retardation effect is included in the formulation through a correction factor suggested by Gu and Li,17

Uv(d) ) -

a d Aa λ + + ln 6 d d + 2a d + 2a λ + sd

[

](

)

(18)

where the term in parentheses is the correction factor, λ is termed the London characteristic wavelength, and s is a constant. Here, λ and s are given the commonly cited values 10-7 m and 11.116, respectively.8,17-19 To obtain the vdW force, eq 18 can be differentiated with respect to d to yield the following equation:

∂Uv ) ∂d A a λ a 2a + + 6 d2 (d + 2a)2 d(d + 2a) λ + sd a d A a λs + + ln (19) 6 d d + 2a d + 2a (λ + sd)2

FvdW )

(

(

)( ) )( )

Since FvdW is on the left-hand side of eq 1, a negative sign has been dropped in eq 19. This allows FvdW to represent an attractive force between the particle and surface for positive Hamaker constants. It can be shown easily that eq 19 simplifies to FvdW ) Aa/6d2 when d is (15) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, UK, 1989. (16) Swaminathan, T. N.; Hu, H. H. J. Colloid Interface Sci. 2004, 273, 324-330. (17) Gu, Y.; Li, D. J. Colloid Interface Sci. 1999, 217, 60-69. (18) Gregory, J. J. Colloid Interface Sci. 1981, 83, 138-145. (19) Adamczyk, Z.; van de Ven, T. G. M. J. Colloid Interface Sci. 1981, 84, 497-518.


Langmuir, Vol. 21, No. 25, 2005 12041

small and the retardation effect is suitably neglected. This result can be derived immediately upon differentiation of eq 17 when the last two terms on the right-hand side are omitted. It should be mentioned, however, that it is unreasonable to neglect these last two terms while at the same time including retardation. The account of retardation implies that d can be relatively large, which in turn means that these terms are as significant as the retardation correction factor. The inclusion of these effects complicates the formulation slightly compared to previous works,8,10 but the discussion in subsequent sections justifies this approach. EDL Interaction Force. The close proximity of the particle to the planar wall necessitates the consideration of the EDL interaction force. Hogg et al. derived an analytical expression for the potential energy of interaction between two spherical particles using Derjaguin’s method, and arrived at the equation20

Ue(d) )

[

(

)

π0a1a2(σ12 + σ22) 2σ1σ2 1 + exp(-κd) ln + 2 2 a1 + a 2 1 - exp(-κd) σ1 + σ 2

]

ln(1 - exp(-2κd)) (20) where a1 and a2 are the radii of the two spheres, ζ1 and ζ2 are the zeta potentials of the two spheres, and κ is the Debye-Hu¨ckel parameter. κ-1 represents the characteristic EDL thickness: small κd signifies EDL overlap while large κd means little or no overlap. For an electrolyte with symmetric ions, κ is given by1

(

)

2z2e2n∞ κ) 0kbT

1/2

(21)

where z is the valence of the ions, e ) 1.60 × 10-19 C is the elementary charge, n∞ is the bulk ionic concentration, kb ) 1.38 × 10-23 J/K is Boltzmann’s constant, and T is the absolute temperature, chosen to be T ) 293 K throughout this study. For the limiting case of a sphere near a planar wall, the EDL interaction potential can be obtained from eq 20 by setting a1 ) a, a2 ) ∞, ζ1 ) ζp, and ζ2 ) ζw:

/ Figure 4. Nondimensionalized DEP force FDEP versus nondimensionalized gap size δ*.

only valid for small zeta potentials. Hogg et al. showed that this approximation was valid for zeta potentials up to ∼75 mV.20 As mentioned previously, the vdW force and the EDL interaction force constitute the well-known colloidal force in the classical DLVO theory. An analysis of only this colloidal force is general to all particle-wall interactions, with or without an externally applied electric field. However, the applied field plays a crucial role in determining the DEP force. An analysis that includes the DEP force thus extends the model beyond the DLVO theory to include the effects of external influences. Note that, in this study, we are interested in particles ranging in size from a ) 0.5 µm to a ) 10 µm, which covers a large range of particle sizes that are typically studied in experiments relevant to electrokinetic flows and microfluidic applications. On the basis of this range of particle sizes, Brownian forces can be neglected.15,21 Results and Discussion

For small κd, this force is dominant, but it decays very quickly for large κd. Since eq 20 was originally derived using the Debye-Hu¨ckel linear approximation, eq 23 is

DEP Force. The main contribution from this work is the inclusion of the DEP force in the analysis of a spherical colloidal particle near a planar boundary. It is, therefore, of significant benefit to present the DEP force calculations in detail before examining the effects of different parameters on separation distance. The DEP force given by eq 16 can be represented in nondimensionalized form by dividing by the factor 0E∞2a2/2. Furthermore, the nondimensionalized DEP force, denoted F/DEP, can be shown to be a function of the nondimensional gap size, δ* ) d/a. Figure 4 shows a plot of F/DEP versus δ*, which reveals that F/DEP monotonically increases as δ* decreases. Since the DEP force is generated from a nonuniform electric field, the electric field is essentially more nonuniform when the gap is small. Nonuniformity in the electric field can be depicted as the departure from symmetry of the field lines around the sphere in the presence of a planar boundary; see Figure 1. As the gap is reduced, the electric field in the gap increases, leading to a higher DEP effect. We note here that the same nondimensionalized curve was obtained by Swaminathan and Hu, who investigated an inertia-induced force using a similar mathematical approach.16 Although the nature of the DEP and inertia-

(20) Hogg, R.; Healy, T.; Fuerstenau, D. W. Trans. Faraday Soc. 1966, 62, 1638-1651.

(21) Pohl, H. A. Dielectrophoresis; Cambridge University Press: Cambridge, UK, 1978.

Ue(d) ) π0a(σp2 + σw2)

[

2σpσw 2

2

σp + σ w

(

ln

)

1 + exp(-κd) + 1 - exp(-κd)

]

ln(1 - exp(-2κd)) (22) Accordingly, the EDL interaction force can be obtained by differentiating eq 22 with respect to d:

FEDL ) -

∂Ue ) ∂d

[

2π0aκ(σp2 + σw2)

2σpσw 2

( (

σp + σw

2

) )]

exp(-κd) 1 - exp(-2κd) exp(-2κd) 1 - exp(-2κd)

(23)

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Table 1. Nondimensionalized Values of FDEP for Different Gap Sizes δ* ) d/a

F/DEP

δ* ) d/a

F/DEP

δ* ) d/a

F/DEP

0.001 0.002 0.005 0.01 0.02 0.05 0.1

3.154662 3.002279 2.730313 2.459222 2.124311 1.587417 1.135621

0.2 0.3 0.4 0.5 0.6 0.8 1

0.695720 0.471388 0.336851 0.249348 0.189461 0.115966 0.075269

1.5 2 3 4 5 6 8

0.030462 0.014622 0.004612 0.001887 0.000910 0.000491 0.000180

induced forces are different, only the “dimensionalizing” factors differ in the derivation. Furthermore, by comparing these factors, it can be easily shown that the DEP force is several orders of magnitude larger than the inertiainduced force. This justifies the omission of the inertiainduced force in the present analysis. Table 1 supplements the plot of Figure 4 by listing the values of F/DEP for different δ*. Parameter Effects. The particle-wall separation distance d is dependent on the following parameters: the particle radius a, the density difference between the particle and the liquid medium ∆F ) Fp - Ff; the applied electric field strength E∞; the zeta potentials of the particle and wall, ζp and ζw, respectively; and the Hamaker constant A. In the following parameter study, we assumed the liquid medium was a symmetric aqueous electrolyte with Ff ) 1000 kg/m3, valence z ) 1, molar concentration c∞ ) 1 × 10-3 M (note that n∞ ) c∞ × 1000NA), and ) 80.1 at T ) 293 K. The particle radius, particle density, applied electric field strength, zeta potentials, and Hamaker constant were varied to determine their effects on d. The following values were chosen as reference values for all results presented: Fp ) 1050 kg/m3, E∞ ) 3.0 kV/m, ζw ) -25 mV, ζp ) -25 mV, and A ) 0.40 × 10-20 J. Unless otherwise specified, these values were held constant while one of the parameters was individually adjusted. Note that the Hamaker constant A ) 0.40 × 10-20 J was calculated using the classical Berthelot principle by considering the interaction between polystyrene and fused silica in water.6 Effect of DEP Force. Figure 5 shows the interaction forces acting on the spherical particle as a function of separation distance d for a particle with radius a ) 5.0 µm. Reference values were given to all other parameters. The thin solid line represents the sum of the EDL interaction, van der Waals, and gravitational forces. For convenience, we call the sum of these three forces the gravity-adjusted colloidal (GAC) force, i.e., FGAC ) FEDL FvdW - FBG. Note that the FGAC curve is shifted downward compared to the traditional colloidal force, which normally becomes asymptotic to the axis of the abscissa. The shift of the horizontal asymptote is clearly equal to FBG, which in this case is 2.57 × 10-13 N. The dashed line represents the DEP force only, while the thick solid line represents the total interaction force equal to the sum of the gravityadjusted colloidal and DEP forces. A positive force signifies repulsion between the particle and the wall, and a negative force signifies attraction. Note that the DEP force is always positive, or repulsive, in nature. The zeros of the force curve represent distances at which the forces are completely balanced, i.e., the equilibrium separation distance. The inclusion of the DEP force in the analysis moved the interaction force curve toward larger separation distances, as can be seen by comparing the zeros of the colloidal force and the total force. Previous omission of the DEP force therefore underestimated the equilibrium separation distance. In this case, the separation distance

Figure 5. Total interaction force versus separation distance: particle radius a ) 5.0 µm (all other parameters at reference values). The sum of the DEP force (dashed line) and the gravityadjusted colloidal force (thin solid line) equals the total interaction force (thick solid line).

without DEP effect was d ) 87 nm, and that with DEP effect was d ) 96 nm. Note that the same gap sizes were obtained for the case where ζw ) -31 mV and ζp ) -20 mV. These differing values of zeta potential were chosen carefully in order to yield the same results; their relationship to the original values of ζw ) -25 mV and ζp ) -25 mV will be discussed in further detail below. We present a case with different wall and particle zeta potentials now in order to emphasize the implications of a change in δ* on particle electrophoretic mobility. It is well-known from the work of Keh and Chen4 that mobility depends on δ* only when the zeta potentials differ. The results of the current case, i.e., d ) 87 nm without DEP and d ) 96 nm with DEP, represent a difference of ∼10%. This difference is insignificant, however, since δ* has only marginally changed from δ* ) 0.017 to δ* ) 0.019 and ultimately has little effect on the electrophoretic mobility. At both these gap sizes, the normalized electrophoretic mobility µ j , as predicted by Keh and Chen4 and Yariv and Brenner,5 is roughly µ j ) 1.08, or 8% higher than in an unbounded flow. Therefore, under these conditions, the DEP force has little impact on predicted mobility. Interesting results were found for very small, lowdensity particles. Figure 6 shows interaction forces as a function of separation distance for Fp ) 1005 kg/m3 and a ) 0.5 µm. Under these conditions, the total interaction force was balanced at three different locations (as indicated by the vertical dotted lines), whereas only one forcebalanced position was found for Fp ) 1050 kg/m3 and a ) 5.0 µm (the case shown in Figure 5). To explain this finding, it is more convenient to examine the corresponding potential energy curves, shown in Figure 7. The smallest (first equilibrium position) and largest (second equilibrium position) zeros in Figure 6 correspond exactly to the first and second local energy minima, while the intermediate zero corresponds to an energy barrier between these two positions. Note that the gravity-adjusted colloidal force alone has only one local energy minimum. Once the DEP force was included in the analysis, however, the second minimum appeared. It is worth noting at this point that the first local energy minimum in fact corresponds to the secondary energy


Langmuir, Vol. 21, No. 25, 2005 12043

Table 2. Magnitudes of Forces and Total Force Balance for Different Separation Distancesa d

d*

FBG

FvdW

FEDL

FDEP

force balance

6.00e-008 8.00e-008 1.00e-007 1.20e-007 1.40e-007 1.60e-007 1.80e-007 2.00e-007 2.20e-007 2.40e-007 2.60e-007 2.80e-007 3.00e-007 5.00e-007 5.50e-007 6.00e-007 6.50e-007 7.00e-007 7.50e-007

0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 0.48 0.52 0.56 0.60 1.00 1.10 1.20 1.30 1.40 1.50

-2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017 -2.568e-017

-1.822e-014 -7.626e-015 -3.817e-015 -2.145e-015 -1.307e-015 -8.459e-016 -5.735e-016 -4.036e-016 -2.927e-016 -2.177e-016 -1.653e-016 -1.279e-016 -1.005e-016 -1.591e-017 -1.114e-017 -8.011e-018 -5.899e-018 -4.432e-018 -3.387e-018



5.514e-013 6.448e-014 5.677e-015 -5.744e-016 -7.871e-016 -5.037e-016 -2.914e-016 -1.603e-016 -8.096e-017 -3.263e-017 -3.164e-018 1.457e-017 2.487e-017 1.844e-017 1.226e-017 6.858e-018 2.332e-018 -1.470e-018 -4.777e-018

a

Particle density Fp ) 1005 kg/m3, particle radius a ) 0.5 µm.

Figure 6. Total interaction force versus separation distance: particle density Fp ) 1005 kg/m3, particle radius a ) 0.5 µm. Note the presence of a secondary equilibrium separation distance.

Figure 7. Total potential energy of interaction as a function of separation distance: particle density Fp ) 1005 kg/m3, particle radius a ) 0.5 µm. The colloidal force alone does not have a second local minimum, but the total interaction force that includes the DEP effect does have one.

minimum (SEM) in classical DLVO theory. The primary energy minimum is always defined as the global minimum that approaches negative infinity (not shown or discussed here). To avoid confusion, we continue to use the terminology of first and second local minima and refrain from using the primary and secondary designations of classical theory. To understand this result, it is pertinent to examine the magnitudes of the contributing forces to the total interaction force. Table 2 shows magnitudes of all four forces for different separation distances d for the present case of Fp ) 1005 kg/m3 and a ) 0.5 µm. The rows in boldface indicate the distances at which the interaction forces change sign, close to the actual equilibrium separation distance. At d ) 120 nm in Table 2 (the first minimum was actually at d ) 114 nm), the van der Waals, EDL interaction, and DEP forces have comparable magnitudes, with the DEP force being smaller by a factor of ∼3-4. Since these forces are similar in magnitude, the accurate account of these forces is clearly important at such small gap sizes. As d increases, the EDL interaction force is the first to drop off significantly, becoming negligible at ∼200

nm. Eventually, as the second minimum (d ) 680 nm) is approached, both the van der Waals and EDL interaction forces are negligible, such that the gravitational force is balanced essentially by DEP repulsion alone. It should be noted that the second local minimum is relatively shallow compared to the first local minimum (see Figure 7). This suggests that random perturbations, such as Brownian forces, can quite easily “knock” a particle from the second local minimum into the more stable first minimum. Therefore, it is conceivable that a very small particle with a density close to that of water only dwells in the second equilibrium position for a brief moment during its descent before eventually settling permanently at the first position. Nevertheless, the existence of the second position is a significant finding that was only possible due to the inclusion of the DEP force. As a final example, consider a slightly larger, lowdensity particle where Fp ) 1005 kg/m3 and a ) 2.0 µm. Figure 8 and Figure 9 show the interaction forces and potential energies, respectively. Under these conditions, only one energy minimum was found, and it existed at a much larger distance than previous minima, i.e., d ) 1.51

12044


Figure 8. Total interaction force as a function of separation distance: particle density Fp ) 1005 kg/m3, particle radius a ) 2.0 µm. Only one equilibrium position exists.

Figure 9. Total potential energy of interaction as a function of separation distance: particle density Fp ) 1005 kg/m3, particle radius a ) 2.0 µm. The total interaction force has a minimum at a much larger separation distance than the minimum of just the colloidal force.

µm. In fact, this distance was more comparable to that of the second minimum found for a ) 0.5 µm. Close examination of the magnitudes of contributing forces revealed that the gravitational force on the particle was indeed almost entirely balanced by the DEP force alone. The effect of the DEP force in this case is clearly illustrated by the difference between the first local minima of the colloidal and total interaction potential energies. Without account of the DEP force, the equilibrium separation distance was d ) 108 nm, while inclusion of the DEP force led to d ) 1.51 µm. More importantly, recall that the same results for separate distance can be obtained for different wall and particle zeta potentials, namely, ζw ) -31 mV and ζp ) -20 mV, as presented previously. Then, according to the results from Keh and Chen,4 the normalized electrophoretic mobility would change from µ j ) 1.02 without DEP to µ j ) 0.99 with DEP. Therefore, neglect of the DEP force would lead to the erroneous conclusion that the mobility of this particle is enhanced

Young and Li

Figure 10. Effect of particle density on separation distance for varying particle radii, without DEP force.

Figure 11. Effect of particle density on separation distance for varying particle radii, with DEP force.

from that predicted for unbounded flows, i.e., µ j > 1. In fact, the mobility of this particle is reduced, i.e., µ j < 1. Note that, since this was the lone equilibrium position, it represented a stable configuration unaffected by minor perturbations. In the remainder of the discussion, the equilibrium separation distance is plotted as a function of particle radii ranging from a ) 0.5 µm to a ) 10 µm for a variety of parameter values. The effects of varying particle density, electric field strength, zeta potentials, and the Hamaker constant are examined, with and without the DEP force. Effect of Particle Density. Figures 10-12 show plots of separation distance versus particle radius for a set of different particle densities, varying from Fp ) 1005 to Fp ) 1300 kg/m3. The DEP force was neglected for the results of Figure 10 and included for those of Figures 11 and 12. The density curves were split into two figures in the latter case in order to display the salient features more clearly. When DEP was neglected, separation distance decreased with increasing particle size and with increasing particle density (Figure 10). This trend was also noticed when the DEP force was included, but only for higherdensity particles (Figure 11). A larger, denser particle is


Langmuir, Vol. 21, No. 25, 2005 12045

Figure 12. Effect of particle density on separation distance for varying particle radii, with DEP force; low particle densities.

Figure 13. Effect of applied electric field strength on separation distance for varying particle radii, with DEP force; high E∞.

simply heavier by eq 2, so larger repulsive forces are needed for all the forces to balance. The EDL interaction and DEP forces are both larger at smaller gap distances, thus leading to smaller separations. However, for lowdensity particles (Figure 12), separation distance decreased only with increasing particle density and not necessarily with increasing particle size. At low densities, the DEP force dominated the force balance. Since larger particles experience stronger DEP forces, they were repelled to a greater extent. Figure 12 shows the major increase in separation for low particle densities. For Fp ) 1005 and 1010 kg/m3, separation distance was on the order of 1-2 microns. Hence, neglect of the DEP force greatly underpredicts the separation distance for low-density particles. Note that, for Fp ) 1025 kg/m3, the curve appears to have two distinct regimes. For a < 6 µm, the curve resembles the low-density curves, whereas for a > 6 µm, the curve resembles the high-density curves of Figure 10. Effect of Electric Field Strength. Recall that E∞ only influences the DEP force and none of the other contributing forces. Therefore, changing E∞ only results in differences in separation distance when the DEP effect is included. At low E∞, separation distance decreased with increasing particle size and decreasing E∞. Curves at low E∞ from 1.0 kV/m to 3.0 kV/m were similar to the monotonically decreasing curves of Figure 10 and are, therefore, not shown. As mentioned previously, larger particles are simply heavier. Smaller applied electric field strengths lower the repulsive DEP force. To achieve equilibrium separation, a smaller gap distance must be reached to reacquire a large enough DEP force. Figure 13 shows plots of separation distance versus particle radius for a set of different E∞ varying from E∞ ) 4.0 kV/m to E∞ ) 7.5 kV/m. At such high E∞, the DEP force dominated, and major increases in separation distance were found. The trends at high E∞ resemble those at low densities (Figure 12). Clearly, when the DEP force dominates, separation distances on the order of microns are attainable. Note that for E∞ ) 4.0 kV/m, the curve is once again split into two regimes at a ≈ 6 µm. Effect of Zeta Potentials. As particle and wall zeta potentials increased, the separation distance increased, with or without account of the DEP force. Curves of separation distance versus particle radius for varying zeta

potentials were once again similar to the monotonically decreasing curves of Figure 10 and are, therefore, not shown. Higher zeta potentials result in higher EDL interaction forces. This, in turn, leads to larger repulsion between particle and wall such that larger separation is needed before forces can be balanced. Since the zeta potentials affect only the EDL interaction force and not the DEP force, major increases in separation distance did not occur when only the zeta potentials were varied. When we speak of increasing zeta potentials, we refer to an increase in the product of the wall and particle zeta potentials. For example, the product of [ζp, ζw] ) [-50 mV, -50 mV] is larger than the product of [ζp, ζw] ) [-75 mV, -25 mV], so larger separation is obtained in the former case. This is true only for thin EDLs, as can be seen by examining eq 23. When κd . 1, the second term in brackets becomes negligible, and the EDL interaction force becomes proportional to the product of the zeta potentials. We note that the cases of [ζp, ζw] ) [-100 mV, -25 mV] and [ζp, ζw] ) [-50 mV, -50 mV] produce essentially identical results for thin EDLs because their products are the same. This result was used above in discussing the implications on electrophoretic mobility due to the inclusion of the DEP effect, where the differing particle and wall zeta potentials, i.e., [ζp, ζw] ) [-20 mV, -31 mV] were carefully chosen to yield the same results as the case where [ζp, ζw] ) [-25 mV, -25 mV]. It is also interesting to note that, for zeta potentials of opposite signs, the EDL interaction force becomes attractive and no gap exists. That is, the combination of the van der Waals, EDL interaction, and gravitational forces toward the wall is too great for the DEP force to overcome on its own. The particle is, therefore, expected to attach to the wall (as long as van der Waals forces remain attractive as well). Effect of Hamaker Constant. Figure 14 shows plots of separation distance versus particle radius for a set of different Hamaker constants A, where the DEP force was included. Similar plots where the DEP force was omitted resulted, once again, in monotonically decreasing curves like those of Figure 10. Separation distances both with and without DEP decreased for increasing particle size and for increasing Hamaker constant for most cases. The explanation is straightforward: a larger Hamaker constant results in a larger attractive van der Waals force, so the particle gets pulled closer to the wall. However, as

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Figure 14. Effect of Hamaker constant on separation distance for varying particle radii, with DEP effect.

shown in Figure 14, certain conditions involving small Hamaker constants lead to much higher separation distances. In these instances, the van der Waals force is too small to overcome the combination of repulsive EDL interaction and DEP forces, and the particle is repelled from the wall. We verify at the end of this discussion the validity of the thin EDL assumption and the separation of the domain into inner and outer regions. According to the properties of the electrolyte listed above, and using eq 21, the EDL thickness throughout this study was κ-1 ) 9.6 nm. In all of the results here, the separation distances were always greater than d ) 60 nm. Furthermore, this extreme was only seen for the largest particles (a ) 10 µm) with the highest densities, the lowest applied electric fields, the lowest zeta potentials, and the largest Hamaker constants. Thus, the minimum κd for this study was κd ≈ 6, but it was found under extreme conditions. For most of the results, κd ≈ 10. Moreover, for the most interesting results where DEP forces dominated and separation distances reached 1-2 microns, κd > 100. Therefore, we conclude that the thin EDL assumption applied very well throughout the study, except under extreme conditions. As stated above, the EDL thickness used throughout this study was κ-1 ) 9.6 nm. In practice, results for any EDL thickness can be obtained similarly using the above approach and plotted in the same fashion. However, care must be taken to use results for particles large enough in size that the thin EDL assumption still applies. In the current results, the smallest particle studied was a ) 0.5 µm, which corresponds to κa ≈ 50. If a lower electrolyte concentration was used, e.g., c∞ ) 10-4 M where κ-1 ) 30 nm, no particles smaller than a ) 1.5 µm should be investigated with the above approach. Although we are not aware of an exact delineation for κa where the thin EDL assumption becomes inapplicable, we suggest these cautionary measures for the benefit of the reader so that they may take care in using their results.

Young and Li

As a further note, for the typical case of κd ≈ 10 and d ≈ 100 nm, a rough quantitative analysis shows that the strong electric field in the gap region does not have a significant impact on the EDL itself. At these gap sizes, the electric field strength through the gap roughly increases by a factor of 40 in comparison to the far-field strength, i.e., Egap/E∞ ≈ 40. Since 100-nm gap sizes occur typically under an applied field of E∞ ≈ 3 kV/m or lower, the expected local-field strength in the gap region is Egap ≈ 120 kV/m. For the common case of ζp ≈ -25 mV, the approximate electric field strength in the EDL region is estimated to be EEDL ≈ ζp/κ-1 ≈ 2500 kV/m. Therefore, Egap is roughly 20× smaller than EEDL. It is, thus, justifiable to assume the EDL is unaffected by Egap under the conditions presented in this work, even though the gap region has a significantly larger local electric field strength than the far field. Finally, we justify the consideration of retardation effects in the van der Waals force. Retardation effects only become important when the particle and wall are far apart. In our results, δ > 0.1 on many occasions, and retardation effects must be considered. Note that the account of retardation in all of the above results yielded slightly higher separation distances compared to those when retardation was neglected. The account of retardation lowers the van der Waals attraction force, so the repulsive forces push the particle farther away from the wall. Conclusions The repulsive dielectrophoretic force plays an important role in the force balance on a spherical particle in electrophoretic motion near a planar wall. Results showed that previous omission of the DEP force led to underestimated gap sizes. The first local minima in potential energy curves were shifted toward larger separation distances when DEP force was included. Under some conditions, these shifts were very significant, resulting in gap sizes on the order of a few microns. Furthermore, for small, low-density particles, and for high electric field strengths, second local minima were discovered, meaning two possible stable configurations were attainable for certain cases. It was noted that the second local minima were usually very shallow, such that minor perturbations would easily knock the particle into the more stable first equilibrium positions. In varying the different parameters that govern the equilibrium separation distance, it was found that larger, denser particles led to smaller gap sizes simply because they were heavier. Larger electric field strengths, higher zeta potentials, and lower Hamaker constants all resulted in larger gap sizes. Acknowledgment. The authors wish to acknowledge financial support from the Natural Sciences and Engineering Council of Canada (NSERC), in the form of a scholarship to E.W.K.Y., and a research grant from the Canadian Institute of Photonic Innovation (CIPI) to D.L. The authors also thank Dr. K. H. Kang (professor at Pohang University of Science and Technology in Korea) and Dr. X. C. Xuan for very insightful discussions and the reviewers for their helpful comments and suggestions. LA0518546

Dielectrophoretic Force on a Sphere near a Planar Boundary

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