J. Phys. Chem. B 2001, 105, 7239-7245
7239
Dynamic Electrophoretic Mobility of Concentrated Spherical Dispersions Eric Lee, Fong-Yuh Yen, and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617, R.O.C. ReceiVed: January 25, 2001; In Final Form: April 30, 2001
The electrophoretic behavior of a concentrated spherical dispersion under an applied alternating electric field is investigated theoretically. We consider the case of low surface potential and the arbitrary thickness of the double layer, and the interaction between neighboring double layers may be significant. The boundary condition of the cell model of Shilov and Zharkikh (Colloid J. 1981, 43, 434) is adopted, which improves the performance of that used by Levine and Neale (J. Colloid Interface Sci. 1974, 47, 520). The difference between the magnitude of electrophoretic mobility based on these two cell models is found to increase with the volume fraction of particle; the phase lag, however, is the same regardless of which cell model is used. We show that the mobility decreases with an increase in the frequency of the applied electric field; the smaller the particle is the more sensitive the variation of the mobility to the frequency will be. The limiting mobility as double layer becomes infinitely thin depends on both the frequency of the applied electric field and the volume fraction of particles.
Introduction The application of electroacoustics on a colloidal dispersion involves the effect of ultrasonic vibration potential (UVP) and that of electrokinetic sonic amplitude (ESA).1 This technique can be used to measure both the surface potential of the dispersed entity and its size. Compared with the conventional techniques, it is more readily applicable for quantifying the physical properties of a colloidal system and is becoming more and more popular in recent years. In particular, it can be used directly to a concentrated dispersion. This is highly desirable since the dilution of a dispersion may lead to changes in the surface properties of the dispersed entities, and the data gathered become unreliable. O’Brien2 used the electroacoustics measurement to estimate the surface potential of a particle. On the basis of the analysis of the electrophoretic mobility, he was able to derive expressions for both UVP and ESA effects. The electrophoresis discussed here is different from that of the classic one since an alternating rather than a static electric field is applied. In this case, since both the strength and the phase of the applied electric field varies with time, the electrophoretic mobility of a particle varies accordingly. Furthermore, since the temporal variations in both flow field and concentration field are not synchronized with that of the applied electric field, a phase lag exists between the movement of a particle and the applied electric field, which also provides valuable information for the phenomenon under consideration. The description of electrophoretic phenomenon includes solving simultaneously the governing equations for electric field, flow field, and conservation of ions. These equations are highly nonlinear and coupled in general, and deriving an analytical solution is possible only under special conditions such as low surface potential and simple geometry. In practice, a numerical scheme is necessary. The problem becomes even more complicated when a dynamic case is considered in which the temporal variations of the dependent variable need to be determined. In a study of the behavior of a dilute spherical * Corresponding author. Fax: 886-2-23623040. E-mail: jphsu@ ccms.ntu.edu.tw.
dispersion, O’Brien2 derived a relation between dynamic electrophoretic mobility and surface potential under the condition of thin double layer. The analysis was extended by several workers3-9 to more general conditions, such as a wider range for double-layer thickness, the frequency of applied electric field, and the level of surface potential. Mangelsdorf and White,7,8 for example, adopted the numerical procedure of O’Brien and White10 to estimate dynamic electrophoretic velocity. A wider level of surface potential and a higher frequency of the applied electric field than those assumed by O’Brien2 were considered, and the effect of the distortion of double layer was taken into account. Under the assumption of low surface potential, Ohshima9 was able to derive an approximation expression for dynamic electrophoretic mobility for an arbitrary double-layer thickness. These studies, however, are limited to a dilute dispersion, in which the volume fraction of particles is below 5%. The behavior of a concentrated dispersion was discussed by Ohshima11-13 through using the cell model adopted by Levine and Neale.14 The overlapping between adjacent double layers was considered, but the result obtained is limited to low surface potentials. Recent studies, however, show that the results based on this cell model deviate significantly from experimental observations. Dukhin et al.15 suggested that the idea of simulating a concentrated dispersion by a cell model is realistic provided that the boundary conditions are defined appropriately, and they proposed using the cell model of Shilov and Zharkikh.16 In the present study, the electrophoretic behavior of a concentrated spherical dispersion is investigated on the basis of the cell model of Shilov and Zharkikh.16 In particular, the effects of double-layer thickness, the frequency of applied electric field, and the volume fraction of particles on electrophoretic mobility are examined. The pseudospectral method based on Chebyshev polynomial used in our previous studies for the case of static electric field17,18 is adopted. Theory Let us consider a concentrated, monodispersed, positively charged spherical dispersion. The liquid phase contains a Z1:Z2 electrolyte solution with Z2 ) -RZ1. Referring to Figure 1, the
10.1021/jp010313w CCC: $20.00 © 2001 American Chemical Society Published on Web 07/10/2001
7240 J. Phys. Chem. B, Vol. 105, No. 30, 2001
Lee et al. the liquid phase can be treated as a Newtonian fluid with constant physical properties. Suppose that the flow field is in the creeping flow regime and can be described by
∇‚u ) 0
(4)
∂u ) -∇p + η∇2u - Fc∇φ ∂t
Ff
(5)
where Ff is the density of fluid, p is the pressure, and η is the viscosity of fluid. We assume that the applied electric field is weak so that each dependent variable can be expressed as the sum of its equilibrium value, that is, the value in the absence of the applied electric field, and a perturbed term. Also, since the electric field is weak, the magnitude of the perturbed term is proportional to its strength. We have
Figure 1. Schematic representation of the cell model used. A representative particle of radius a is surrounded by a concentric liquid shell of radius b. E and U are, respectively, the applied electric field and the velocity of fluid.
problem under consideration is simulated by a representative particle of radius a surrounded by a concentric liquid shell of radius b. The spherical coordinates (r,θ,φ) are adopted with its origin located at the center of the particle. An alternating electric field E of strength Ee-iωt (E ) Ee-iωtez) is applied, and the representative particle moves in the same direction as that of the applied electric field with velocity U (U ) UEe-iωtez), where i ) x-1, ez is a unit vector, and t and ω are, respectively, the time and the frequency of the applied electric field. Let U ) UR + iUI, where UR and UI are, respectively, the real and the imaginary parts of U. The equations governing the phenomenon under consideration include those for the electrical potential, the ionic concentration, and the flow field. Suppose that the variation of the electric field can be described by the Poisson equation
∇ φ)2
Fc
N
)-
∑ j)1
zjeˆ nj
φ(r,θ,t) ) φe(r) + δφ(r,θ)Ee-iωt
(6)
nj(r,θ,t) ) nej (r,θ) + δnj(r,θ)Ee-iωt
(7)
u(r,θ,t) ) ue + δu(r,θ)Ee-iωt
(8)
p(r,θ,t) ) pe(r,θ) + δp(r,θ)Ee-iωt
(9)
In these expressions, subscript e denotes equilibrium property. Note that due the symmetric nature of the problem, the dependent variables in these expressions are functions of both time t and position variables (r,θ) but are φ-independent. Also, since a particle has uniform surface properties the equilibrium potential, φe is a function of r only. As illustrated in eqs 6-9, each perturbed term is expressed as the product of a positiondependent term and a time-dependent term. The latter has the form of wave function and can be treated independently of the former. This approach involves solving a differential equation with complex coefficients. Since the real and the imaginary parts of this equation require separate treatment, the number of equations is doubled. Nevertheless, it is more efficient than solving directly the original differential equation, which involves both time and space variables. Equilibrium System. In the absence of the applied electric field, particles remain fixed, and the spatial variation of electric potential (the equilibrium potential) can be described by
(1)
N
where ∇2 is the Laplace operator, φ is the electrical potential, Fc is the space charge density, is the permittivity of the liquid phase, eˆ is the elementary charge, and nj is the number concentration of ionic species j. Here, we assume that is independent of both position and time. A mass balance on the concentration of ionic species j yields
∂nj ) -∇‚fj ∂t
∇ φe ) 2
(2)
∑ j)1
zjeˆ nej (10)
Since fj ) 0 and ue ) 0, at equilibrium, eqs 2 and 3 yield
(
-Dj ∇nej +
)
nej eˆ zj ∇φ ) 0 kBT e
Integrating this expression yields
( ) zjeˆ φe kBT
nej ) nj0 exp -
with
(
fj ) -Dj ∇nj +
)
njeˆ zj ∇φ + nju kBT
(3)
where ∇ is the gradient operator, fj and Dj are, respectively, the flux and the diffusivity of ionic species j, kB is the Boltzmann constant, and u is the velocity of liquid phase. We assume that
(11)
(12)
where nj0 is the bulk concentration of ionic species j. Substituting this expression into eq 10, we obtain N
∇ φe ) 2
∑ j)1
e zeˆ nj0
( ) zjeˆ φe
exp -
kBT
(13)
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J. Phys. Chem. B, Vol. 105, No. 30, 2001 7241
If the surface potential is low, this equation can be approximated by the linear expression N
∇ φe ) 2
∑ j)1
(
e zeˆ nj0
1-
)
zjeˆ φe kBT
(14)
Equilibrium Boundary Conditions. Suppose that the surface potential is constant and a unit cell as a whole is electrically neutral. The latter implies that there is no net current between two adjacent cells. Therefore, we have
φe ) ζ,
r)a
(15)
dφe ) 0, dr
r)b
(16)
Perturbed System. Taking Laplacian on both sides of eq 6 yields
∇2(δφ(r,θ)Ee-iωt) ) ∇2φ(r,θ,t) - ∇2φe(r)
(17)
If the surface potential is low, the perturbed ion concentration due to the presence of the applied electric field can be neglected, that is, δnjEe-iωt ) 0, and we have
∇ φ(r,θ,t) ) ∇ φe(r) 2
2
In a study of the static mobility of a concentrated dispersion, Shilov and Zharkikh16 proposed using the boundary condition based on thermodynamic consideration that the perturbed electric potential is assigned the value of that induced by the applied electric field at the virtual surface of a cell (r ) b). For the present dynamic case this condition becomes
(δφEe-iωt) ) Ee-iωtb cos θ,
r)b
(26)
The origin of the coordinates is located at the center of a representative particle, which moves with velocity (UR + iUI)Ee-iωt. Since the flow field is linear, we can assume that the particle is fixed and the liquid phase on the virtual surface is moving with velocity -(UR + iUI)Ee-iωt, for convenience. Suppose that the velocity vanishes on the surface of the representative particlesthat is, the surface is no-slip
δurEe-iωt ) 0 and δuθEe-iωt ) 0,
r)a
(27)
On the virtual surface, the vorticity vanishes, and the fluid velocity is -(UR + iUI)Ee-iωt, that is
∇ × δuEe-iωt ) 0 and δurEe-iωt ) -(UR + iUI)Ee-iωt cos θ,
r ) b (28)
(18)
Equations 17 and 18 suggest that
∇2(δφ(r,θ)Ee-iωt) ) 0
(19)
Since the flow field is induced by the applied electric field, we assume that the liquid-phase remains stationary at equilibrium, which implies that ∇pe ) 0 and ue ) 0. Therefore, only the perturbed variables needed to be determined. Equations 4, 5, 8, and 9 yield
∇‚(δu) ) 0
(20)
-iωFfδuEe-iωt ) -∇δpEe-iωt + η∇2δuEe-iωt - Fc∇φ
Solving eqs 19 and 24 simultaneously subject to the boundary conditions eqs 25-28 yields the solution for the perturbed system. For a simpler mathematical treatment, scaled quantities are used in the subsequent discussions. Here, the radius of a particle a, its surface potential ζ, and the bulk electrolyte concentration nj0 are chosen as the characteristic length, potential, and concentration, respectively. The scaled quantities are defined by r* ) r/a, φ/e ) φe/ζ, E* ) E/(ζ/a), δφ*E*e-iωt ) δφEe-iωt/ζ, U*E*e-iωt ) UEe-iωt/(ζ2/ηa), ψ*E*e-iωt ) ψEe-iωt/(ζ2a/η), and n/j ) nj/nj0. In terms of the scaled variables, eqs 14-16 become
∇*2φ/e ) (κa)2φ/e
(29)
(21) Instead of solving the flow field directly, the stream function representation is adopted in which eq 20 is satisfied automatically. Let ψ be the stream function and δur and δuθ be the rand the θ-components of the perturbed liquid velocity. Then
1 ∂ψ δur ) - 2 r sin θ ∂θ δuθ )
(22)
1 ∂ψ r sin θ ∂r
(23)
We have δuEe-iωt ) (δurrˆ + δuθθˆ )Ee-iωt. Taking curl on eq 21 to remove the pressure term and introducing the stream function, we obtain
(
)
1 ∂Fc ∂φ ∂Fc ∂φ sin θ E4ψ + iωFfE2ψ ) η ∂r ∂θ ∂θ ∂r
(24)
Perturbed Boundary Conditions. We assume that the surface conductivity of a particle is negligiblesthat is, the Dukhin number (Du ) kδ/Ka) is much smaller than unity. Therefore,
∇(δφEe-iωt)‚n ) 0,
r)a
(25)
φ/e ) 1, dφ/e ) 0, dr*
r* ) 1
(30)
r* ) b/a
(31)
Similarly, eqs 19, 25, and 26 become
∇*2(δφ*E*e-iωt) ) 0 ∂(δφ*E*e-iωt) ) 0, ∂r*
(32)
r* ) 1
δφ*E*e-iωt ) -E*e-iωt(b/a) cos θ,
(33) r* ) b/a (34)
If we let δφ* ) Φ cos θ, then eqs 32-34 reduce to
∂Φ 1 ∂ 2Φ r*2 - 2)0 2 ∂r* ∂r* r* r*
(
∂Φ ) 0, ∂r* Φ ) -b/a,
)
(35)
r* ) 1
(36)
r* ) b/a
(37)
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Similarly, eqs 24, 27, and 28 become
(
E*4 + i
)
ωFfa2 2 E* ψ*E*e-iωt ) η ∂φ/e ∂(δφ*E*e-iωt) sin θ (38) -(κa)2 ∂r* ∂θ
δu/r E*e-iωt ) 0 and δu/θE*e-iωt ) 0,
r* ) 1 (39)
δu/r E*e-iωt ) -(U/R + iU/I )E*e-iωt cos θ and ∇* × δu*E*e-iωt ) 0,
r* ) b/a (40)
If we let ψ* ) (ψR + iψI) sin2 θ, then since δφ* ) Φ cos θ, the real and the imaginary parts of eq 38 give, respectively
[
]
[
]
ωFfa2 d2 ∂φ/e 2 2 2 d2 2 ψ ) ψ + (κa) Φ (41) R η dr*2 r*2 I ∂r* dr*2 r*2
and
[
]
[
2
]
ωFfa d d 2 2 - 2 ψI ) - 2 ψR 2 2 η dr* dr* r* r* 2
2
2
(42)
Figure 2. Variation of the magnitude of scaled dynamic mobility as a function of κa for the case (Fp - Ff)/Ff ) 0.1 and (Ffωa2/η) ) 0 for various volume fractions of particles λ () a3/b3). Dashed lines, results based on the cell model of Levine and Neale;14 solid lines, results based on the cell model of Shilov and Zharkikh.16 Curve 1, λ ) 0.01; 2, λ ) 0.1; 3, λ ) 0.5.
(43)
applied electric field, and in the second problem, the electric field is applied but the particle is remained fixed. We have
The associated boundary conditions are
ψR ) 0 and ψI ) 0,
r* ) 1
dψI dψR ) 0 and ) 0, dr* dr*
r* ) 1
1 1 ψR ) U/Rr*2 and ψI ) U/I r*2, 2 2
[
]
[
(44)
r* ) b/a
]
d2 d2 2 2 2 ψ ) 0 and ψI ) 0, R dr*2 r*2 dr*2 r*2
(45)
r* ) b/a (46)
( )
ωFfa2 Fp - Ff / (UR + iU/I )E*e-iωt (50) η Ff
F1 + F2 ) -i
where Fk, k ) 1 or 2, is the sum of the electric force and the drag force in problem k. On the basis of the linear nature of the problem under consideration, it can be inferred that
F1 ) R(U/R + iU/I )E*e-iωt
(51)
where R can be complex. Similarly,
Dynamic Electrophoretic Mobility. A force balance on the representative particle yields
F2 ) βE*e-iωt
4 du Fh + Fe ) πa3(Fp - Ff) 3 dt
where β can be complex. Substituting eqs 51 and 52 into eq 50 yields
(47)
where the drag force exerted on the particle by the flow filed, Fh, and the electric force exerted on the particle by the electric filed, Fe, can be expressed respectively by19,20
( ( (
)
))
∂ 1 d2 4 2 (ψ + iψI) × Fh ) πσ2 r*4 3 ∂r* r*2 dr*2 r*2 R r*)1 4 E*e-iωt - πσ2(κa)2(φ1Φ)r*)1E*e-ωt (48) 3 and
( )
∂φ1 8 Fe ) πσ2 3 ∂r
r*)1
(Φ)r*)1E*e-iωt
(49)
The electrophoretic mobility is defined by µ* ) |U*/E*|, where U ) (U/R + iU/I )e-iωtez and E ) E*e-iωtez. For convenience, the problem under consideration is decomposed into two problems. In the first problem a particle moves with scaled velocity U* ) -(U/R + iU/I )Ee-iωtez in the absence of the
|µ*| )
|
(52)
|
-β R + i(ωFfa /η)(Fp - Ff)/Ff 2
(53)
This is the magnitude of the dynamic mobility, and the corresponding phase angle is defined as tan-1 (µI/µR), µR and µI being, respectively, the real and the imaginary parts of the dynamic mobility. Results and Discussions The dynamic behavior, which is characterized by the magnitude and the phase lag of the electrophoretic mobility, of the phenomenon under consideration is examined through numerical simulation. In particular, the effects of the volume fraction of particles, the ionic strength, the thickness of double-layer, and the frequency of the applied electric field are investigated. Figure 2 shows the variation of the magnitude of the scaled electrophoretic mobility as a function of the scaled double-layer thickness κa for various volume fractions of particles λ () a3/ b3) for the case of a static applied electric field, that is (Ffωa2/
Concentrated Spherical Dispersions
J. Phys. Chem. B, Vol. 105, No. 30, 2001 7243
Figure 3. Variation of the magnitude of scaled dynamic mobility as a function of κa for the case of Figure 2, except that (Ffωa2/η) ) 1.
η) ) 0. For comparison, both the result based on the present model and that based on the cell model of Levine and Neale14 are illustrated. They assumed that the perturbed electric field can be assigned the value of that induced by the applied electric field on the virtual surface of a cell (r ) b). For the present dynamic case, we have
∂(δφEe-iωt) ) Ee-iωt cos θ, ∂r
r)b
(54)
Ohshima11 also assumed this boundary condition. A comparison between eqs 26 and 54 implies that the problem considered by Shilov and Zharkikh16 is of Dirichlet nature and that considered by Levine and Neale14 is of Neumann nature. As pointed out by Dukhin et al.,15 eq 54 does not lead to Smoluchowski’s result for mobility in the limiting case of dilute dispersion and appears to be unable to take the volume fraction of the dispersed phase into account. Figure 2 reveals that if a dispersion is dilute, the difference between the result based on eq 26 and that based on eq 54 is negligible. However, if the volume fraction of a
dispersed phase is on the order of 10%, the difference becomes significant, especially when the double layer is thin (κa large). If λ is large, both the electrical and the hydrodynamic interactions between neighboring particles are significant, which have a negative effect on particle movement, and therefore, the mobility is small. Since this effect presents even if the double layer surrounding a particle is thin, the Smoluchowski’s result for thin double layers (κa f ∞) should be adjusted by λ. The result of Levine and Neale14 is independent of λ, and as κa f ∞, it approaches ζ/η (scaled mobility approaches unity), the result of Smoluchowski for the case of dilute dispersions. Figure 2 suggests that the cell model of Levine and Neale will overestimate the mobility, in general. Figure 3 illustrates the variation of the magnitude of the scaled dynamic mobility as a function of the scaled double-layer thickness κa for various volume fractions of particles λ () a3/ b3) for the case when a dynamic electric field is applied. Similar conclusions as those obtained from Figure 2 can be drawn from Figure 3 for the magnitude of the scaled mobility. Although not shown, the phase lag predicted by the present model is the same as that predicted by the model of Levine and Neale14s that is, although the magnitudes of mobility predicted by these two models are different, they yield the same phase lag. Figures 4a, 5a, and 6a show the variations of the magnitude of the scaled dynamic electrophoretic mobility as a function of the scaled double-layer thickness κa for various volume fractions of particles λ ()a3/b3) and scaled frequency of applied electric field (Ffωa2/η). These figures suggest that for a fixed λ, the magnitude of the scaled dynamic mobility increases with an increase in κa, that is, the thinner the double layer, the greater the mobility. This is because if double layer is thick, the overlapping of adjacent double layers has the effect of slowing down the movement of particles. Figures 4a, 5a, and 6a reveal that as κa f ∞, the magnitude of the scaled mobility approaches a constant value, and the smaller the λ is, the larger this value will be. It can be shown that the result of Smoluchowski can be obtained by letting κa f ∞ and λ f 0. Figures 4a, 5a, and 6a also reveal that the smaller the λ is, the greater the influence of the frequency of the applied electric field will be. For fixed λ and κa, the higher the frequency is, the smaller the mobility will be. This is because as the frequency of the applied electric field increases, eq 21 is dominated by the inertial term, and others terms involving viscous force, pressure, and electric force
Figure 4. Variation of the magnitude of scaled dynamic mobility (a) and phase angle (b) as a function of κa for the case of Figure 4, except (Ffωa2/η) ) 1.
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Lee et al.
Figure 5. Variation of the magnitude of scaled dynamic mobility (a) and phase angle (b) as a function of κa for the case of Figure 4 except (Ffωa2/η) ) 5.
Figure 6. Variation of the magnitude of scaled dynamic mobility (a) and phase angle (b) as a function of κa for the case of Figure 4 except (Ffωa2/η) ) 10.
Figure 7. Variation of the magnitude of scaled dynamic mobility (a) and phase angle (b) as a function of scaled frequency (Ffωa2/η) for the case (Fp - Ff)/Ff ) 0.1 and κa ) 1 for various volume fraction of particles λ () a/b)3.
Concentrated Spherical Dispersions
J. Phys. Chem. B, Vol. 105, No. 30, 2001 7245
Figure 8. Variation of the magnitude of scaled dynamic mobility (a) and phase angle (b) as a function of scaled frequency (Ffωa2/η) for the case (Fp - Ff)/Ff ) 0.1 and κa ) 10 for various volume fraction of particles λ () a3/b3).
become negligible. Note that the direction of the movement of particles varies with that of the applied electric field. Therefore, the higher the frequency of the applied electric field is, the shorter the time interval will be for particles to move in a certain direction, that is, the shorter the time available for particles to be accelerated in that direction which leads to a smaller mobility. Apparently, if (Ffωa2/η) f ∞, particles remain essentially still. Figures 4b, 5b, and 6b show the variation of phase lag as a function of the scaled double-layer thickness κa for various volume fractions of particles λ () a3/b3) and scaled frequency of applied electric field (Ffωa2/η) for the cases of Figures 4a, 5a, and 6a, respectively. The phase lag arises from the fact that when a particle moves as a response to the applied electric field, a certain period of time is needed for the adjustment of the ionic cloud surrounding it. Figures 4b, 5b, and 6b reveal that for fixed volume fraction and frequency, the phase lag decreases with the increase in κa (i.e., decreases in double-layer thickness). This is because the thinner the double layer the shorter the time required for its relaxation. Figures 4b, 5b, and 6b also reveal that the phase lag decreases with the increase in the volume fraction of particles. This is because the larger the volume fraction is, the more compact the spatial arrangement of particles will be. This implies that the double layer surrounding a particle is confined in a relative small space, and therefore, the the time required for its relaxation is shorter. Figures 4b, 5b, and 6b suggest that for fixed volume fraction of particles and κa, the phase lag increases with the frequency of the applied electric field. This is because the higher the frequency is, the shorter the time interval for particles to reverse their direction of movement will be. Figure 7a shows the variation of the magnitude of the scaled dynamic mobility as a function of the scaled frequency (Ffωa2/ η) for various volume fractions of particles λ; that for a larger κa is presented in Figure 8a. These figures reveal that the larger the volume fraction of particles is, the less significant the effect of the frequency on the behavior of a dispersion will be. This is because the larger the volume fraction, the less the free space between neighboring particles. Therefore, the effect of the
frequency of the applied electric field will become significant only if it is sufficiently high so that the displacement of a particle is shorter than the mean distance between particles. Otherwise, this effect will be masked by the presence of the neighboring particles. Figure 7b illustrates the variation of the phase lag as a function of the scaled frequency for the case of Figure 7a; that for the case of Figure 8a is presented in Figure 8b. As in the case of Figure 4b, 5b, and 6b for fixed λ and κa, the phase lag increases with the frequency of the applied electric field. Acknowledgment. This work is supported by the National Science Council of the Republic of China. References and Notes (1) Hunter, R. J. Colloids Surf., A 1998, 141, 37. (2) O’Brien, R. W. J. Fluid Mech. 1988, 190, 71. (3) Babchin, A. J.; Chow, R. S.; Sawatzky, R. P. AdV. Colloid Interface Sci. 1989, 30, 111. (4) Sawatzky, R. P.; Babchin, A. J. J. Fluid Mech. 1993, 246, 321. (5) Fixman, M. J. Chem. Phys. 1983, 78, 1483. (6) James, R. O.; Texter, J.; Scales, P. J. Langmuir 1991, 7, 1993. (7) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1992, 88, 3567. (8) Mangelsdorf, C. S.; White, L. R. J. Colloid Interface Sci. 1993, 160, 275. (9) Ohshima, H. J. Colloid Interface Sci. 1996, 179, 431. (10) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (11) Ohshima, H. J. Colloid Interface Sci. 1997, 195, 137. (12) Ohshima, H. Colloids Surf., A 1999, 149, 5. (13) Ohshima, H. Colloids Surf., A 1999, 159, 293. (14) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (15) Dukhin, A. S.; Shilov, V.; Borkovskaya, Y. Langmuir 1999, 15, 3452. (16) Shilov, V. N.; Zharkikh, N. I.; Borkovskaya, Yu. B. Colloid J. 1981, 43, 434. (17) Lee, E.; Chu,. J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65. (18) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 209, 240. (19) Lee, E.; Yen, F. Y.; Hsu, J. P. Electrophoresis 2000, 21, 475. (20) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Chem. Phys. 2000, 112, 6404.
