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Electrophoresis of a Concentrated Dispersion of Spherical Particles in a Non-Newtonian Fluid Jyh-Ping Hsu,* Eric Lee, and Yu-Fen Huang Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617 Received August 14, 2003. In Final Form: December 30, 2003 Electrophoresis is one of the most widely used analytical tools for the quantification of the charged conditions on the surface of fine particles including biological entities. Although it has been studied extensively in the past, relevant results for the case when the dispersion medium is non-Newtonian are very limited. This may occur, for example, when the concentration of the dispersed phase is not low, which is not uncommon in practice. Here, the electrophoresis of a concentrated spherical dispersion in a Carreau fluid is analyzed theoretically under the conditions of low electric potential and weak external applied electrical field. A pseudospectral method coupled with a Newton-Raphson iteration procedure is used to solve the electrokinetic equations describing the phenomenon under consideration. We conclude that the more significant the shear thinning effect of the fluid, the larger the mobility, and this phenomenon is pronounced for the case when the double layer surrounding a particle is thin. We show that if the double layer is thin and the effect of shear thinning is significant, a second vortex can be observed in the neighborhood of a particle.
* To whom correspondence should be addressed. Fax: 886-223623040. E-mail:
[email protected].
that extending the conventional analyses based on a Newtonian fluid to a more general case of non-Newtonian fluid is desirable not only for theoretical interests but also for practical needs. Another problem, which deserves investigation from practical viewpoint, is the effect of the concentration of the dispersed phase. Most of the available results in the literature are based on an isolated entity in an infinite solution; that is, the interaction between adjacent entities is neglected. Also, many electrophoresis measuring instruments require predilution of a sample. It should be pointed out that this procedure might lead to a change in the surface properties of an entity, especially when it is of a charge-regulated nature. Kuwabara12 proposed a unit cell model to simulate the behavior of a concentrated dispersion of spheres. The unit cell comprises a representative sphere enveloped by a concentric spherical liquid shell. The size of the latter depends on the volume fraction of the dispersed phase. Kuwabara12 assumed that the vorticity of the flow field vanishes on a cell surface. Assuming the same boundary condition, Levine and Neale13 were able to derive the electrophoretic mobility for the case of low electrical potential and arbitrary double layer thickness. It was shown that in the limit of a thin double layer the electrophoretic mobility reduces to that predicted by Smoluchowski1 for an isolated particle. The analysis of Levine and Neale was extended by Kozak and Davis14 to the case of fibrous porous media. Kozak and Davis15,16 considered the electrophoresis of concentrated dispersions and highly charged, unconsolidated porous media for an arbitrary level of electrical potential for the case when the overlapping between adjacent double layers is negligible. Because the mobility relation derived by Levine and Neale13 involves a tedious numerical integration, it is not readily applicable. To circumvent this
(1) Smoluchowski, M. Z. Phys. Chem. 1918, 92, 129. (2) Dukhin, S. S.; Derjaguin, B. V. Surface and Colloid Science; Wiley: New York, 1974; Vol. 7. (3) O’Brien, R. W.; Hunter, R. J. Can. J. Chem. 1981, 59, 1878. (4) O’Brien, R. W. J. Colloid Interface Sci. 1983, 92, 204. (5) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 1978, 274, 1607. (6) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65. (7) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1999, 209, 240. (8) Lee, E.; Yen, F. Y.; Hsu, J. P. Electrophoresis 2000, 21, 475. (9) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Chem. Phys. 2000, 112, 6404.
(10) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, 1989; Vol. 1. (11) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, 1989; Vol. 2. (12) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (13) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (14) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1986, 112, 403. (15) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 127, 497. (16) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 129, 166.
1. Introduction The surface properties of a charged entity of colloidal size are often analyzed by electrophoresis measurements. Smoluchowski1 was able to show that the electrophoretic mobility, the electrophoretic velocity per unit applied electric field, of a particle is proportional to its zeta potential. The derivation of this relation was based on an isolated particle in an infinite Newtonian fluid under the conditions of constant low surface potential, weak applied electric field, and insignificant local curvature. In calculation of the electrophoretic mobility of an entity, the electrokinetic equations, i.e., equations governing the flow, the electric, and the concentration fields, need to be solved simultaneously. In general, these equations are nonlinear, coupled, partial differential equations, and solving them analytically is extremely difficult, if not impossible. Often, this difficulty is circumvented by either considering a simplified problem1-4 or resorting to a numerical method.5-9 Although relevant results for the electrophoretic behavior of charged entities are ample in the literature, most all of them are focused on the case when the liquid phase is a Newtonian fluid. Apparently, solving the electrokinetic equations for the case of a non-Newtonian fluid is even more difficult than that for the case of a Newtonian fluid. However, colloidal dispersions involving non-Newtonian fluid are not uncommon in practice. In preparation of a colloidal dispersion, for example, surfactant or polymer is often introduced to improve its stability; this can yield a shear-thinning fluid.10 The dispersion medium may also become non-Newtonian when the content of dispersed phase exceeds a certain level.11 These examples imply
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difficulty, Ohshima17 derived an approximate simple expression for a concentrated spherical dispersion at low electric potential. In a discussion of the electrophoresis of a concentrated spherical dispersion, Happel and Brenner18 assumed that both the shear stress and the radial velocity vanish on cell surface. Instead of assuming a Neumanntype boundary condition for the perturbation potential on a cell surface as is done in Kuwabara’s cell model, Shilov and Zharkikh19 used a Dirichilet-type boundary condition. In a recent study, Lee et al.20 analyzed the electrophoresis of a sphere in a spherical cavity filled with a nonNewtonian fluid. In the present study the electrophoretic behavior of a concentrated spherical dispersion in a Carreau fluid is analyzed. This extends previous results for the electrophoresis of a concentrated dispersion in a Newtonian fluid to a non-Newtonian case and that of an isolated particle in a non-Newtonian fluid to a concentrated case. We consider the case when the electric potential is low and the applied electric field is weak, and the governing equations are a set of coupled, linear electrokinetic equations, which are solved by adopting a pseudospectral method coupled with a Newton-Raphson iterative scheme. 2. Theory Referring to Figure 1a, we consider a concentrated dispersion of positively charged spherical particles of radius a. An electric field E is applied, and the particles move with an electrophoretic velocity U. The dispersion is simulated by the unit cell model of Kuwabara12 illustrated in Figure 1b, where a cell comprises a representative particle and a concentric liquid shell of radius b. Here, the ratio H ) (a/b) provides a measure for the concentration of particles. The spherical coordinates (r,θ,φ) are chosen with its origin located at the center of the representative particle, and E is in the Z-direction. The liquid phase is a Carreau fluid containing z1:z2 electrolyte, z1 and z2 being the valences of cations and anions, respectively. If we let z2/z1 ) -R, then electroneutrality requires that n20 ) n10/R, n10 and n20 being the bulk concentrations of cations and anions, respectively. At steady state, the equations governing the problem under consideration, the so-called electrokinetic equations, comprise that for the electric field and that for the flow field. 2.1. Electric Field. We assume that the applied electric field is weak and the effect of double layer polarization is negligible. It can be shown that, based on the Gauss law, the electric potential φ can be described by the Poisson equation
∇2φ ) -
Fe
)-
e
Figure 1. (a) Schematic representation of the system under consideration where an electric field E is applied to a concentrated dispersion of spherical particles of radius a, and the dispersed particles move with an electrophoretic velocity U. (b) The system is simulated by a representative cell, which comprises a particle and a concentric liquid shell of radius b. The spherical coordinates (r,θ,φ) are chosen with its origin located at the center of the representative particle.
mann distribution. Also, for a simpler mathematical treatment, φ is composed of two terms: the electrical potential in the absence of the applied electric field or the equilibrium potential, φ1, and that outside a particle, which arises from the applied electric field, φ2. That is
φ ) φ1 + φ2
(2)
2
zjnj ∑ j)1
[
(1)
where ∇ is the gradient operator, is the permittivity of the liquid phase, Fe is the space charge density, e is the elementary charge, and n1 and n2 are the number concentrations of cations and anions. We assume that the spatial variation of ionic concentration follows the Boltz(17) Ohshima, H. J. Colloid Interface Sci. 1997, 188, 481. (18) Happel, J.; Brenner, H. Low-Reynolds Number Hydrodynamics; Martinus: Nijhoof, 1983. (19) Shilov, V. N.; Zharkikh, N. I.; Borkovskaya, Yu. B. Colloid J. 1981, 43, 434. (20) Lee, E.; Huang, Y. F.; Hsu, J. P. J. Colloid Interface Sci. 2003, 258, 283.
nj ) nj0 exp -
]
zje(φ1 + φ2) kBT
(3)
j ) 1, 2 2
∇2φ1 ) -
∑ j)1
zjenj0
[ ]
exp -
zjeφ1 kBT
(4)
where kB and T are the Boltzmann constant and the absolute temperature, respectively. On the basis of eqs 1, 2, and 4, we obtain
Electrophoresis of a Concentrated Dispersion
[∑ 2
∇2φ2 ) -
j)1
zjenj0
[
exp -
]
zje(φ1 + φ2) kBT 2 z en j j0
∑ j)1
-
[ ]]
exp -
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zjeφ1 kBT
(5)
Suppose that the electrical potential is low so that the nonlinear terms in eqs 4 and 5 can be approximated by the corresponding linear expressions. It can be shown that, in terms of scaled variables, these equations become
∇*2φ1* ) (κa)2φ1* 2
∇* φ2* ) 0
(6) (7)
In these expressions, φ1* ) φ1/ζa and φ2* ) φ2/ζa, ζa being the zeta potential of particle, κ-1 ) [kBT/∑nj0(ezj)2]1/2, κ-1 being the Debye length, and ∇* ) a∇. It can be shown that n10z1 ) [(κa)2kBT/(1 + R)e2a2z1]. The condition of electroneutrality is applied in the derivation of eqs 6 and 7. 2.2. Flow Field. Compared with other properties of a non-Newtonian fluid, its shear-thinning nature is the most important property to the present problem. This is because it is directly related to the drag force acting on a particle. On the basis of this consideration, choosing an appropriate model, which is capable of describing the shear-thinning nature of a non-Newtonian fluid, becomes crucial. In our analysis, the viscosity dependence on shear rate is described by the Carreau model, which is the best correlation model for various non-Newtonian systems.21 The constitutive equations for a Carreau fluid can be described by21
τ ) -η(γ˘ )γ˘ ) -[η∞ + (η0 - η∞)[1 + (λγ˘ )β](n-1)/β]γ˘ (8) γ˘ ) ∇v + (∇v)T
(9)
where τ is the stress tensor, γ˘ is the rate of strain tensor, η is the apparent viscosity, v is the velocity, ∇ is the gradient operator, the superscript T denotes matrix transpose, λ, n, and β are the relaxation time constant, the power-law exponent, and a dimensionless parameter that describing the transition region between the zeroshear-rate region and the power-law region, respectively, and η0 and η∞ are the zero-shear-rate viscosity and infiniteshear-rate viscosity, respectively. In practice, β and η∞ are roughly constant, and their effects are usually unimportant. Note that the special case of a Newtonian fluid can be recovered from eq 8 by letting either n f 1 or λ f 0, and if λ is sufficiently large, eq 8 describes a power law fluid. We assume that the liquid is incompressible, and the steady-state flow field can be described by the Cauchy momentum equation in the creeping flow regime
∇‚v ) 0
(10)
∇‚τ + ∇p + Fe∇φ ) 0
(11)
where p denotes the pressure. The last term on the lefthand side of eq 11 represents the body force term arising from the effect of electric force. For illustration, we choose β ) 2 and η∞ ) 0 in eq 8. These values are typical under conditions of practical significance. For a simpler mathematical treatment, the flow field is described by a stream function representation. If we let ψ be the stream function of the flow field under consid(21) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Polymer Liquids; Wiley: New York, 1987; Vol. 1.
Figure 2. Variation of scaled mobility µE* as a function of κa for various n at two different λU/a for the case when a/b ) 0.5, Ez* ) 1.0, and R ) 1.0: (a) λU/a ) 0.1; (b) λU/a ) 0.5.
eration, then the r- and the θ-components of the fluid velocity, vr and vθ, can be expressed as vr ) -(1/r2 sin θ)(∂ψ/∂θ) and vθ ) (1/r sin θ)(∂ψ/∂r), respectively. The governing equation for ψ can be obtained by taking curl on both sides of eq 11. Note that the pressure term does not exist in the resultant expression, which is nonlinear, however, because the apparent viscosity varies with the shear rate. 2.3. Boundary Conditions. We assume that particles remained at constant surface potential, characterized by the corresponding zeta potential ζa, and there is no net flux for ionic species between adjacent cells. Therefore the scaled boundary conditions associated with eq 6 are
φ1* ) 1, ∂φ1* ) 0, ∂r*
r* ) 1
(12)
r* ) 1/H
(13)
where the scaled radial distance r* is defined by r* ) r/a, and φ1* ) φ1/ζa. The particle is nonconductive, and the electric field vanishes on its surface, which implies that
∂φ2* ) 0, ∂r*
r* ) 1
(14)
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Also, we assume that the boundary condition for the electrical potential arising from the applied electric field is of Neumann type, that is
∂φ2* ) -Ez* cos θ, ∂r*
r* ) 1/H
(15)
where φ2* ) φ2/ζa and Ez* ) Eza/ζa, Ez is the Z-component of the applied electric field and Ez is its magnitude. It should be emphasized that, according to O’Brien and White,5 if the electrokinetic equations are linearized and the applied field is weak, the electrophoretic mobility is independent of the electrostatic boundary conditions chosen on the particle surface. Since the geometry chosen is symmetric about the z-coordinate, we have
∂φ1* ∂φ2* ) ) 0, ∂θ ∂θ ∂φ1* ∂φ2* ) ) 0, ∂θ ∂θ
θ)0
(16)
θ)π
(17)
It can be shown that the governing equation for the flow field in terms of the scaled stream function ψ*, defined by ψ* ) ψ/UEa2, where UE ) ζa2/ηa is the electrophoretic velocity evaluated by Smoluchowski’s formula when an electric field of strength ζa/a is applied, for the case of low electrical potential is
[(
η*D*4ψ* + sin θ r*
∂2η* ∂η* γ˘ *rθ + r* γ˘ *rθ + ∂r* ∂r*2
) (
∂2η* ∂η* ∂γ˘ *θθ ∂2η* ∂η* ∂γ˘ *rθ + γ˘ *θθ + γ˘ * + ∂r* ∂r* ∂r*∂θ ∂θ ∂r* ∂r*∂θ rr *
(
)
1 ∂2η* ∂η* ∂γ˘ rr 1 ∂η* ∂γ˘ *rθ + γ˘ *rθ + + ∂r* ∂θ r* ∂θ2 r* ∂θ ∂θ 3
2
1 ∂3ψ* ∂η* 1 ∂ ψ* cot θ ∂ ψ* + ∂r* sin θ ∂r*3 r*2 sin θ ∂r*∂θ r*2 sin θ ∂r*∂θ2
)
∂2ψ* 2 cot θ ∂ψ* 2 + 3 3 2 r* sin θ ∂θ r* sin θ ∂θ ∂3ψ* 1 ∂3ψ* ∂η* 1 - 4 - 2 2 ∂θ r* sin θ ∂r* ∂θ r* sin θ ∂θ3
(
Figure 3. Variation of scaled mobility µE* as a function of κa for various λU/a at two different n for the case when a/b ) 0.5, Ez* ) 1.0, and R ) 1.0: (a) n ) 0.5; (b) n ) 0.3.
)]
cot θ ∂2ψ* ∂ψ* 1 + ) r*4 sin3 θ ∂θ r*4 sin θ ∂θ2 ∂φ1* ∂φ2* sin θ (18) -(κa)2 ∂r* ∂θ
eration. These conditions can be written as
ψ* ) -
∂ψ* ) -U*r* sin2 θ, ∂r*
where
D*2 )
∂2 sin θ ∂ 1 ∂ + 2 ∂r* r*2 ∂θ sin θ ∂θ
(
)
(18a)
with D ) D*/a, η ) η*η0, and γ˘ ) γ˘ *UE/a. We assume that the vorticity vanishes on the virtual surface of a cell, r ) b.12 Also, since the particle moves with velocity U, the boundary conditions for the flow field are assumed as
νr ) U cos θ, νθ ) -U sin θ,
r)a r)a
∇ × v ) 0 and νr ) 0, r ) b
1 U*r*2 sin2 θ, 2
(21)
Note that vr must vanish on the virtual surface, which simulates the boundary of the dispersion under consid-
(22)
r* ) 1
(
(23)
)
∂2 1 ∂ ∂2 1 cos θ + ψ* ) 0 r* sin θ ∂r*2 r*3 sin2 θ ∂θ r*3 sin θ ∂θ2 (24) r* )
1 H
where U* ) U/UE. Because the problem under consideration is symmetric about the z-axis, we have
(19) (20)
r* ) 1
ψ* )
∂ψ* ) 0, ∂θ
θ)0
(25)
ψ* )
∂ψ* ) 0, ∂θ
θ)π
(26)
The electric field and the flow field can be determined by
Electrophoresis of a Concentrated Dispersion
Figure 4. Contours of stream function for various n for the case when κa ) 12.2284, Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, and a/b ) 0.5.
solving eqs 6, 7, and 18 simultaneously subject to eqs 1217 and 22-26. To this end, the pseudospectral method described by Canuto et al.22 based on Chebyshev polynomials, coupled with a Newton-Raphson’s iterative procedure, is chosen. This numerical scheme is found to be efficient for the present type of electrokinetic phenomenon.6 Once the electric field and the flow field are known, the electrophoretic mobility of particles can be evaluated by employing the condition that the net force acting on a particle vanishes at the steady state. (22) Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A. Spectral Methods in Fluid Dynamics; Springer-Verlag: New York, 1986.
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Figure 5. Contours of stream function for various n for the case when κa ) 18.2858, Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, and a/b ) 0.5.
3. Results and Discussions The behaviors of the system under consideration are examined through numerical simulation. Figure 2 shows the variation of the scaled mobility µE* (µE* ) U*/Ez*) as a function of the thickness of double layer surrounding a particle, κa, at various n at two levels of λU/a, the scaled relaxation time constant. For comparison, the corresponding result for the case of a Newtonian fluid (n ) 1.0) is also presented. As can be seen from Figure 2, µE* increases monotonically with the increase in κa, that is, the thicker the double layer surrounding a particle, the smaller its mobility. This is because if the double layer is thick, the viscous retardation is serious, and the interac-
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Figure 6. Contours of shear rate for various n for the case when κa ) 12.2284, Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, and a/b ) 0.5.
Figure 7. Contours of shear rate for various n for the case when κa ) 18.2858, Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, and a/b ) 0.5.
tions between neighboring particles becomes significant, both of these factors are disadvantageous to the movement of particle. Figure 2 indicates that if κa is small (thick double layer), the effect of the non-Newtonian nature of the fluid on the mobility of a particle is negligible. Appreciable increase in µE* is observed for both Newtonian and non-Newtonian fluids when κa increases to about unity. For a further increase in κa, µE* for the case of a Newtonian fluid approaches a constant value, but that for the case of a non-Newtonian fluid still increases, and the smaller the n or the larger the λU/a, the larger the µE*. This is because according to eq 8, the smaller the n or the larger the λ, the greater the effect of shear thinning,
the smaller the viscous force acting on a particle, and, therefore, the larger the mobility. The variations of the scaled mobility µE* as a function of κa at various λU/a for two levels of n are illustrated in Figure 3. In general, µE* increases if κa increases, λU/a increases, or n decreases. These observations are consistent with the results shown in Figure 2. Figures 4 and 5 show the contours for the stream function of the electrophoretic phenomenon for various n at two levels of κa. As a response to the applied electric field, the representative particle in a cell moves upward, and the adjacent fluid flows downward. This yields a clockwise vortex on one right-hand side of the particle
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Figure 8. Variation of shear rate on the equator (θ ) π/2) of a particle for various n at two levels of κa for the case when Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, and a/b ) 0.5: (a) κa ) 12.2284; (b) κa ) 18.2858.
and a counterclockwise vortex on the other side, as illustrated in Figure 4 (only the clockwise vortex is shown). These vortexes are near the outer boundary of a cell. Note that the corresponding streamlines are not compressed appreciably because the outer boundary of a cell plays the role of a free surface. The vortex is closer to the particle surface if n is smaller (fluid is more shear-thinning) and κa is larger (double layer is thinner). Note that if n is sufficiently small and κa is sufficiently large, in addition to a clockwise vortex, a counterclockwise vortex may also occur simultaneously on the right-hand side of a particle, as can be seen in Figure 5. The contours for the shear rate distribution for the cases of Figures 4 and 5 are presented in Figures 6 and 7, respectively, and the corresponding variations in the shear rate on the equator (θ ) π/2) of the representative particle are illustrated in Figure 8. The viscosity contours for the cases of Figures 4 and 5 are illustrated in Figures 9 and 10, respectively. Figure 8 reveals that a sharp reduction in the shear rate occurs near the particle surface, and a local minimum appears at the center of the clockwise vortex. The location at which the local minimum of shear rate occurs depends on both
Figure 9. Contours of viscosity for various n for the case when κa ) 12.2284, Ez* ) 1.0, λU/a ) 0.1, and a/b ) 0.5.
the magnitude of κa and that of n. After the local minimum is passed, the shear rate increases with the increase in the distance away from particle surface and then exhibits a local maximum. Figure 8 reveals that a small n and a large κa yield a large local maximum. Also, if n is sufficiently small and κa is sufficiently large, a second local minimum appears, which occurs at the center of the counterclockwise vortex. This phenomenon arises mainly from the shear-thinning nature of the fluid. The increase in the shear rate after the first local minimum leads to a decrease in the viscosity and an increase in the velocity at the boundary of the clockwise vortex, which, in turn, induces a counterclockwise vortex. Figures 9 and 10 show that the minimum viscosity occurs on the particle surface,
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Figure 11. Variation of scaled mobility µE* as a function of (a/b) for the case when Ez* ) 1.0, R ) 1.0, n ) 0.5, λU/a ) 0.5, and κa ) 1.5848.
Figure 10. Contours of viscosity for various n for the case when κa ) 18.2858. Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, and a/b ) 0.5.
and this minimum decreases with the increase in κa and the decrease in n. The effect of shear thinning can be appreciable if n is sufficiently small and κa sufficiently
large. For instance, if n ) 0.3 and κa ) 18.2858, the viscosity is 1 cP on cell surface and 0.1526 cP on particle surface. A comparison between Figures 9 and 10 with Figures 6 and 7 shows that the shapes of the contours of shear rate are the same as those of viscosity, except that the minimum of the former occurs on the cell surface but that of the latter occurs on the particle surface. The influence of the concentration of particles, measured by the ratio a/b, on their electrophoretic mobility is presented in Figure 11. This figure indicates that the scaled electrophoretic mobility µE* decreases with the increase in a/b. This is expected because the higher the concentration of particles, the more significant the interaction between nearby particles, and the more important the steric hindrance. Note that µE* is roughly linearly correlated with a/b, with some positive deviation from this relation when a/b is large. In summary, the electrophoretic behavior of a concentrated spherical dispersion is investigated for the case of low electrical potential and weak electric field. In particular, we consider the case where the liquid phase is of shear-thinning nature, which is not uncommon in practice. The electrophoretic behavior of a particle is found to correlate with the key factors of the system under consideration, including the thickness of double layer, the concentration of particles, and the degree of deviation of the fluid from a Newtonian fluid. The electrophoretic mobility of a particle is different both quantitatively and qualitatively with that of the corresponding Newtonian case. Acknowledgment. This work is supported by the National Science Council of the Republic of China. LA035490Y
