Electrophoresis of a Concentrated Dispersion of SphericalParticles in a Non-Newtonian Fluid

Jyh-Ping Hsu,* Eric Lee, and Yu-Fen HuangDepartment 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 chargedconditionsonthesurfaceof fineparticles includingbiological entities.Although ithasbeenstudiedextensivelyin 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 notuncommon in practice. Here, the electrophoresis of a concentrated spherical dispersion in a Carreau fluidis analyzed theoretically under the conditions of low electric potential and weak external applied electricalfield. A pseudospectral method coupled with a Newton-Raphson iteration procedure is used to solve theelectrokinetic equations describing the phenomenon under consideration. We conclude that the moresignificant the shear thinning effect of the fluid, the larger the mobility, and this phenomenon is pronouncedfor the case when the double layer surrounding a particle is thin. We show that if the double layer is thinand the effect of shear thinning is significant, a second vortex can be observed in the neighborhood of aparticle.

1. IntroductionThe surface properties of a charged entity of colloidal

size are often analyzed by electrophoresis measurements.Smoluchowski1 was able to show that the electrophoreticmobility, the electrophoretic velocity per unit applied elec-tric field, of a particle is proportional to its zeta potential.The derivation of this relation was based on an isolatedparticle in an infinite Newtonian fluid under the conditionsof constant low surface potential, weak applied electricfield, and insignificant local curvature. In calculation ofthe electrophoretic mobility of an entity, the electrokineticequations, i.e., equations governing the flow, the electric,and the concentration fields, need to be solved simulta-neously. In general, these equations are nonlinear, coup-led, partial differential equations, and solving them ana-lytically is extremely difficult, if not impossible. Often,this difficulty is circumvented by either considering asimplified problem1-4 or resorting to a numerical meth-od.5-9

Although relevant results for the electrophoretic be-havior of charged entities are ample in the literature, mostall of them are focused on the case when the liquid phaseis a Newtonian fluid. Apparently, solving the electrokineticequations for the case of a non-Newtonian fluid is evenmore difficult than that for the case of a Newtonian fluid.However, colloidal dispersions involving non-Newtonianfluid are not uncommon in practice. In preparation of acolloidal dispersion, for example, surfactant or polymer isoften introduced to improve its stability; this can yield ashear-thinning fluid.10 The dispersion medium may alsobecome non-Newtonian when the content of dispersedphase exceeds a certain level.11 These examples imply

that extending the conventional analyses based on aNewtonian fluid to a more general case of non-Newtonianfluid is desirable not only for theoretical interests butalso for practical needs.

Another problem, which deserves investigation frompractical viewpoint, is the effect of the concentration ofthe dispersed phase. Most of the available results in theliterature are based on an isolated entity in an infinitesolution; that is, the interaction between adjacent entitiesis neglected. Also, many electrophoresis measuring in-struments require predilution of a sample. It should bepointed out that this procedure might lead to a change inthe surface properties of an entity, especially when it isof a charge-regulated nature. Kuwabara12 proposed a unitcell model to simulate the behavior of a concentrateddispersion of spheres. The unit cell comprises a repre-sentative sphere enveloped by a concentric spherical liquidshell. The size of the latter depends on the volume fractionof the dispersed phase. Kuwabara12 assumed that thevorticity of the flow field vanishes on a cell surface.Assuming the same boundary condition, Levine andNeale13 were able to derive the electrophoretic mobilityfor the case of low electrical potential and arbitrary doublelayer thickness. It was shown that in the limit of a thindouble layer the electrophoretic mobility reduces to thatpredicted by Smoluchowski1 for an isolated particle. Theanalysis of Levine and Neale was extended by Kozak andDavis14 to the case of fibrous porous media. Kozak andDavis15,16 considered the electrophoresis of concentrateddispersions and highly charged, unconsolidated porousmedia for an arbitrary level of electrical potential for thecase when the overlapping between adjacent double layersis negligible. Because the mobility relation derived byLevine and Neale13 involves a tedious numerical integra-tion, it is not readily applicable. To circumvent this

* To whom correspondence should be addressed. Fax: 886-2-23623040. E-mail: jphsu@ntu.edu.tw.

(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 UniversityPress: Oxford, 1989; Vol. 1.

(11) Hunter, R. J. Foundations of Colloid Science; Oxford UniversityPress: 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.

2149Langmuir 2004, 20, 2149-2156

10.1021/la035490y CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 02/19/2004

difficulty, Ohshima17 derived an approximate simpleexpression for a concentrated spherical dispersion at lowelectric potential. In a discussion of the electrophoresis ofa concentrated spherical dispersion, Happel and Brenner18

assumed that both the shear stress and the radial velocityvanish on cell surface. Instead of assuming a Neumann-type boundary condition for the perturbation potential ona cell surface as is done in Kuwabara’s cell model, Shilovand Zharkikh19 used a Dirichilet-type boundary condition.In a recent study, Lee et al.20 analyzed the electrophoresisof a sphere in a spherical cavity filled with a non-Newtonian fluid.

In the present study the electrophoretic behavior of aconcentrated spherical dispersion in a Carreau fluid isanalyzed. This extends previous results for the electro-phoresis of a concentrated dispersion in a Newtonianfluid to a non-Newtonian case and that of an isolatedparticle in a non-Newtonian fluid to a concentratedcase. We consider the case when the electric potentialis low and the applied electric field is weak, and thegoverning equations are a set of coupled, linear electro-kinetic equations, which are solved by adopting a pseu-dospectral method coupled with a Newton-Raphsoniterative scheme.

2. TheoryReferring to Figure 1a, we consider a concentrated

dispersion of positively charged spherical particles ofradius a. An electric field E is applied, and the particlesmove with an electrophoretic velocity U. The dispersionis simulated by the unit cell model of Kuwabara12

illustrated in Figure 1b, where a cell comprises arepresentative particle and a concentric liquid shell ofradius b. Here, the ratio H ) (a/b) provides a measurefor theconcentrationofparticles.Thespherical coordinates(r,θ,æ) are chosen with its origin located at the center ofthe representative particle, and E is in the Z-direction.The liquid phase is a Carreau fluid containing z1:z2electrolyte, z1 and z2 being the valences of cations andanions, respectively. If we let z2/z1 ) -R, then electro-neutrality requires that n20 ) n10/R, n10 and n20 beingthe bulk concentrations of cations and anions, respective-ly.

At steady state, the equations governing the problemunder consideration, the so-called electrokinetic equations,comprise that for the electric field and that for the flowfield.

2.1. Electric Field. We assume that the applied electricfield is weak and the effect of double layer polarization isnegligible. It can be shown that, based on the Gauss law,the electric potential φ can be described by the Poissonequation

where ∇ is the gradient operator, ε is the permittivity ofthe liquid phase, Fe is the space charge density, e is theelementary charge, and n1 and n2 are the number con-centrations of cations and anions. We assume that thespatial variation of ionic concentration follows the Boltz-

mann distribution. Also, for a simpler mathematicaltreatment, φ is composed of two terms: the electricalpotential in the absence of the applied electric field or theequilibrium potential,φ1, and that outside a particle, whicharises from the applied electric field, φ2. That is

where kB and T are the Boltzmann constant and theabsolute temperature, respectively. On the basis of eqs 1,2, and 4, we obtain

(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.

Figure 1. (a) Schematic representation of the system underconsideration where an electric field E is applied to a concen-trated dispersion of spherical particles of radius a, and thedispersed particles move with an electrophoretic velocity U. (b)The system is simulated by a representative cell, whichcomprises a particle and a concentric liquid shell of radius b.The spherical coordinates (r,θ,æ) are chosen with its originlocated at the center of the representative particle.

∇2φ ) -

Fe

ε) -

e

ε∑j)1

2

zjnj (1)

φ ) φ1 + φ2 (2)

nj ) nj0 exp[-zje(φ1 + φ2)

kBT ] (3)

j ) 1, 2

∇2φ1 ) - ∑

j)1

2 zjenj0

εexp[-

zjeφ1

kBT ] (4)

2150 Langmuir, Vol. 20, No. 6, 2004 Hsu et al.

Suppose that the electrical potential is low so that thenonlinear terms in eqs 4 and 5 can be approximated bythe corresponding linear expressions. It can be shown that,in terms of scaled variables, these equations become

In these expressions, φ1* ) φ1/úa and φ2* ) φ2/úa, úa beingthe zeta potential of particle, κ-1 ) [εkBT/∑nj0(ezj)2]1/2, κ-1

being the Debye length, and ∇* ) a∇. It can be shown thatn10z1 ) [(κa)2εkBT/(1 + R)e2a2z1]. The condition of elec-troneutrality is applied in the derivation of eqs 6 and 7.

2.2. Flow Field. Compared with other properties of anon-Newtonian fluid, its shear-thinning nature is the mostimportant property to the present problem. This is becauseit is directly related to the drag force acting on a particle.On the basis of this consideration, choosing an appropriatemodel, which is capable of describing the shear-thinningnature of a non-Newtonian fluid, becomes crucial. In ouranalysis, the viscosity dependence on shear rate isdescribed by the Carreau model, which is the bestcorrelation model for various non-Newtonian systems.21

The constitutive equations for a Carreau fluid can bedescribed by21

where τ is the stress tensor, γ̆ is the rate of strain tensor,η is the apparent viscosity, v is the velocity, ∇ is thegradient operator, the superscript T denotes matrixtranspose, λ, n, and â are the relaxation time constant,the power-law exponent, and a dimensionless parameterthat describing the transition region between the zero-shear-rate region and the power-law region, respectively,and η0 and η∞ are the zero-shear-rate viscosity and infinite-shear-rate viscosity, respectively. In practice, â and η∞are roughly constant, and their effects are usuallyunimportant. Note that the special case of a Newtonianfluid can be recovered from eq 8 by letting either n f 1or λ f 0, and if λ is sufficiently large, eq 8 describes apower law fluid. We assume that the liquid is incom-pressible, and the steady-state flow field can be describedby the Cauchy momentum equation in the creeping flowregime

where p denotes the pressure. The last term on the left-hand side of eq 11 represents the body force term arisingfrom the effect of electric force. For illustration, we chooseâ ) 2 and η∞ ) 0 in eq 8. These values are typical underconditions of practical significance.

For a simpler mathematical treatment, the flow fieldis described by a stream function representation. If we letψ be the stream function of the flow field under consid-

eration, then the r- and the θ-components of the fluidvelocity, vr and vθ, can be expressed as vr ) -(1/r2 sinθ)(∂ψ/∂θ) and vθ ) (1/r sin θ)(∂ψ/∂r), respectively. Thegoverning equation for ψ can be obtained by taking curlon both sides of eq 11. Note that the pressure term doesnot exist in the resultant expression, which is nonlinear,however, because the apparent viscosity varies with theshear rate.

2.3. Boundary Conditions. We assume that particlesremained at constant surface potential, characterized bythe corresponding zeta potential úa, and there is no netflux for ionic species between adjacent cells. Thereforethe scaled boundary conditions associated with eq 6 are

where the scaled radial distance r* is defined by r* ) r/a,and φ1* ) φ1/úa. The particle is nonconductive, and theelectric field vanishes on its surface, which implies that

(21) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of PolymerLiquids; Wiley: New York, 1987; Vol. 1.

∇2φ2 ) -[∑j)1

2 zjenj0

εexp[-

zje(φ1 + φ2)

kBT ] -

∑j)1

2 zjenj0

εexp[-

zjeφ1

kBT ]] (5)

∇*2φ1* ) (κa)2

φ1* (6)

∇*2φ2* ) 0 (7)

τ ) -η(γ̆)γ̆ ) -[η∞ + (η0 - η∞)[1 + (λγ̆)â](n-1)/â]γ̆ (8)

γ̆ ) ∇v + (∇v)T (9)

∇‚v ) 0 (10)

∇‚τ + ∇p + Fe∇φ ) 0 (11)

Figure 2. Variation of scaled mobility µE* as a function of κafor 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.

φ1* ) 1, r* ) 1 (12)

∂φ1*∂r*

) 0, r* ) 1/H (13)

∂φ2*∂r*

) 0, r* ) 1 (14)

Electrophoresis of a Concentrated Dispersion Langmuir, Vol. 20, No. 6, 2004 2151

Also, we assume that the boundary condition for theelectrical potential arising from the applied electric fieldis of Neumann type, that is

where φ2* ) φ2/úa and Ez* ) Eza/úa, Ez is the Z-componentof the applied electric field and Ez is its magnitude. Itshould be emphasized that, according to O’Brien andWhite,5 if the electrokinetic equations are linearized andthe applied field is weak, the electrophoretic mobility isindependent of the electrostatic boundary conditionschosen on the particle surface. Since the geometry chosenis symmetric about the z-coordinate, we have

It can be shown that the governing equation for theflow field in terms of the scaled stream functionψ*, definedby ψ* ) ψ/UEa2, where UE ) εúa

2/ηa is the electrophoreticvelocity evaluated by Smoluchowski’s formula when anelectric field of strength úa/a is applied, for the case of lowelectrical potential is

where

with D ) D*/a, η ) η*η0, and γ̆ ) γ̆*UE/a.We assume that the vorticity vanishes on the virtual

surface of a cell, r ) b.12 Also, since the particle moveswith velocity U, the boundary conditions for the flow fieldare assumed as

Note that vr must vanish on the virtual surface, whichsimulates the boundary of the dispersion under consid-

eration. These conditions can be written as

where U* ) U/UE. Because the problem under consider-ation is symmetric about the z-axis, we have

The electric field and the flow field can be determined by

∂φ2*∂r*

) -Ez* cos θ, r* ) 1/H (15)

∂φ1*∂θ

)∂φ2*∂θ

) 0, θ ) 0 (16)

∂φ1*∂θ

)∂φ2*∂θ

) 0, θ ) π (17)

η*D*4ψ* + sin θ[(∂η*∂r*

γ̆*rθ + r* ∂2η*

∂r*2γ̆*rθ +

r* ∂η*∂r*

∂γ̆*rθ

∂r*+ ∂

2η*∂r*∂θ

γ̆*θθ + ∂η*∂θ

∂γ̆*θθ

∂r* ) - ( ∂2η*

∂r*∂θγ̆*rr +

∂η*∂r*

∂γ̆*rr

∂θ+ 1

r*∂

2η*∂θ2

γ̆*rθ + 1r*

∂η*∂θ

∂γ̆*rθ

∂θ ) +

∂η*∂r*( 1

sin θ∂

3ψ*∂r*3

- cot θr*2 sin θ

∂2ψ*

∂r*∂θ+ 1

r*2 sin θ∂

3ψ*∂r*∂θ2

-

2r*3 sin θ

∂2ψ*∂θ2

+ 2 cot θr*3 sin θ

∂ψ*∂θ ) -

∂η*∂θ (- 1

r*2 sin θ∂

3ψ*∂r*2

∂θ- 1

r*4 sin θ∂

3ψ*∂θ3

-

1r*4 sin3 θ

∂ψ*∂θ

+ cot θr*4 sin θ

∂2ψ*∂θ2 )] )

-(κa)2 ∂φ1*∂r*

∂φ2*∂θ

sin θ (18)

D*2 ) ∂2

∂r*2+ sin θ

r*2∂

∂θ( 1sin θ

∂

∂θ) (18a)

νr ) U cos θ, r ) a (19)

νθ ) -U sin θ, r ) a (20)

∇ × v ) 0 and νr ) 0, r ) b (21)

Figure 3. Variation of scaled mobility µE* as a function of κafor 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.

ψ* ) - 12

U*r*2 sin2 θ, r* ) 1 (22)

∂ψ*∂r*

) -U*r* sin2 θ, r* ) 1 (23)

( 1r* sin θ

∂2

∂r*2- cos θ

r*3 sin2 θ∂

∂θ+ 1

r*3 sin θ∂

2

∂θ2)ψ* ) 0

(24)

r* ) 1H

ψ* ) ∂ψ*∂θ

) 0, θ ) 0 (25)

ψ* ) ∂ψ*∂θ

) 0, θ ) π (26)

2152 Langmuir, Vol. 20, No. 6, 2004 Hsu et al.

solving eqs 6, 7, and 18 simultaneously subject to eqs 12-17 and 22-26. To this end, the pseudospectral methoddescribed by Canuto et al.22 based on Chebyshev poly-nomials, coupled with a Newton-Raphson’s iterativeprocedure, is chosen. This numerical scheme is found tobe efficient for the present type of electrokinetic phen-omenon.6 Once the electric field and the flow field areknown, the electrophoretic mobility of particles can beevaluated by employing the condition that the net forceacting on a particle vanishes at the steady state.

3. Results and DiscussionsThe behaviors of the system under consideration are

examined through numerical simulation. Figure 2 showsthe variation of the scaled mobility µE* (µE* ) U*/Ez*) asa function of the thickness of double layer surrounding aparticle, κa, at various n at two levels of λU/a, the scaledrelaxation time constant. For comparison, the corre-sponding 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, thesmaller its mobility. This is because if the double layer isthick, the viscous retardation is serious, and the interac-

(22) Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A. SpectralMethods in Fluid Dynamics; Springer-Verlag: New York, 1986.

Figure 4. Contours of stream function for various n for thecase when κa ) 12.2284, Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, anda/b ) 0.5.

Figure 5. Contours of stream function for various n for thecase when κa ) 18.2858, Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, anda/b ) 0.5.

Electrophoresis of a Concentrated Dispersion Langmuir, Vol. 20, No. 6, 2004 2153

tions between neighboring particles becomes significant,both of these factors are disadvantageous to the movementof particle. Figure 2 indicates that if κa is small (thickdouble layer), the effect of the non-Newtonian nature ofthe fluid on the mobility of a particle is negligible.Appreciable increase in µE* is observed for both Newtonianand non-Newtonian fluids when κa increases to aboutunity. For a further increase in κa, µE* for the case of aNewtonian fluid approaches a constant value, but thatfor the case of a non-Newtonian fluid still increases, andthe smaller the n or the larger the λU/a, the larger theµE*. This is because according to eq 8, the smaller the nor 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 thescaled mobility µE* as a function of κa at various λU/a fortwo 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 shownin Figure 2.

Figures 4 and 5 show the contours for the streamfunction of the electrophoretic phenomenon for various nat two levels of κa. As a response to the applied electricfield, the representative particle in a cell moves upward,and the adjacent fluid flows downward. This yields aclockwise vortex on one right-hand side of the particle

Figure 6. Contours of shear rate for various n for the casewhen 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 casewhen κa ) 18.2858, Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, and a/b )0.5.

2154 Langmuir, Vol. 20, No. 6, 2004 Hsu et al.

and a counterclockwise vortex on the other side, asillustrated in Figure 4 (only the clockwise vortex is shown).These vortexes are near the outer boundary of a cell. Notethat the corresponding streamlines are not compressedappreciably because the outer boundary of a cell plays therole of a free surface. The vortex is closer to the particlesurface if n is smaller (fluid is more shear-thinning) andκa is larger (double layer is thinner). Note that if n issufficiently small and κa is sufficiently large, in additionto a clockwise vortex, a counterclockwise vortex may alsooccur simultaneously on the right-hand side of a particle,as can be seen in Figure 5. The contours for the shear ratedistribution for the cases of Figures 4 and 5 are presentedin Figures 6 and 7, respectively, and the correspondingvariations in the shear rate on the equator (θ ) π/2) of therepresentative particle are illustrated in Figure 8. Theviscosity contours for the cases of Figures 4 and 5 areillustrated in Figures 9 and 10, respectively. Figure 8reveals that a sharp reduction in the shear rate occursnear the particle surface, and a local minimum appearsat the center of the clockwise vortex. The location at whichthe local minimum of shear rate occurs depends on both

the magnitude of κa and that of n. After the local minimumis passed, the shear rate increases with the increase inthe distance away from particle surface and then exhibitsa local maximum. Figure 8 reveals that a small n and alarge κa yield a large local maximum. Also, if n issufficiently small and κa is sufficiently large, a secondlocal minimum appears, which occurs at the center of thecounterclockwise vortex. This phenomenon arises mainlyfrom the shear-thinning nature of the fluid. The increasein the shear rate after the first local minimum leads toa decrease in the viscosity and an increase in the velocityat the boundary of the clockwise vortex, which, in turn,induces a counterclockwise vortex. Figures 9 and 10 showthat the minimum viscosity occurs on the particle surface,

Figure 8. Variation of shear rate on the equator (θ ) π/2) ofa 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.

Figure 9. Contours of viscosity for various n for the case whenκa ) 12.2284, Ez* ) 1.0, λU/a ) 0.1, and a/b ) 0.5.

Electrophoresis of a Concentrated Dispersion Langmuir, Vol. 20, No. 6, 2004 2155

and this minimum decreases with the increase in κa andthe decrease in n. The effect of shear thinning can beappreciable if n is sufficiently small and κa sufficiently

large. For instance, if n ) 0.3 and κa ) 18.2858, theviscosity is 1 cP on cell surface and 0.1526 cP on particlesurface. A comparison between Figures 9 and 10 withFigures 6 and 7 shows that the shapes of the contours ofshear rate are the same as those of viscosity, except thatthe minimum of the former occurs on the cell surface butthat of the latter occurs on the particle surface.

The influence of the concentration of particles, measuredby the ratio a/b, on their electrophoretic mobility ispresented in Figure 11. This figure indicates that thescaled electrophoretic mobility µE* decreases with theincrease in a/b. This is expected because the higher theconcentration of particles, the more significant the in-teraction between nearby particles, and the more impor-tant the steric hindrance. Note that µE* is roughly linearlycorrelated with a/b, with some positive deviation fromthis relation when a/b is large.

In summary, the electrophoretic behavior of a concen-trated spherical dispersion is investigated for the case oflow electrical potential and weak electric field. In par-ticular, we consider the case where the liquid phase is ofshear-thinning nature, which is not uncommon in practice.The electrophoretic behavior of a particle is found tocorrelate with the key factors of the system underconsideration, including the thickness of double layer, theconcentration of particles, and the degree of deviation ofthe fluid from a Newtonian fluid. The electrophoreticmobility of a particle is different both quantitatively andqualitatively with that of the corresponding Newtoniancase.

Acknowledgment. This work is supported by theNational Science Council of the Republic of China.

LA035490Y

Figure 10. Contours of viscosity for various n for the casewhen κa ) 18.2858. Ez* ) 1.0, R ) 1.0, λU/a ) 0.1, and a/b )0.5.

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.

2156 Langmuir, Vol. 20, No. 6, 2004 Hsu et al.