gated theo-appli-rsion. Theess of the

of the dropr iswever, if

Journal of Colloid and Interface Science 282 (2005) 486–492www.elsevier.com/locate/jcis

Electrophoresis of a concentrated aqueous dispersionof non-Newtonian drops

Eric Leea, Chia-Jeng Changa, Jyh-Ping Hsub,∗

a Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Chinab Department of Chemical and Materials Engineering,National I-Lan University, I-Lan, Taiwan 26041, China

Received 5 April 2004; accepted 13 August 2004

Available online 11 November 2004

Abstract

The electrophoresis of a concentrated dispersion of non-Newtonian drops in an aqueous medium, which has not been investiretically in the literature, is analyzed under conditions of low zeta potential and weak applied electric field. The results obtained providetheoretical basis for the characterization of the nature of an emulsion and a microemulsion system. A Carreau fluid, which has wide acations in practice, is chosen for the non-Newtonian drops, and the unit cell model of Kuwabara is adopted to simulate a dispeeffects of the key parameters of a dispersion, including its concentration, the shear-thinning nature of the drop fluid, and the thickndouble layer, on the electrophoretic behavior of a drop are discussed. In general, the more significant the shear-thinning naturefluid is, the larger the mobility is, and this effect is pronounced as the thickness of the double layer decreases. However, if the double layesufficiently thick, this effect becomes negligible. In general, the higher the concentration of drops is, the smaller the mobility is; hothe double layer is either sufficiently thin or sufficiently thick, this effect becomes unimportant. 2004 Elsevier Inc. All rights reserved.

Keywords:Electrophoresis; Concentrated dispersion; Non-Newtonian drops; Carreau fluid; Cell model; Pseudo-spectral method

edrsedsiongedcol-yys asses

mi-asl to

r.

is isef-

u-a-of

onhe

ysese ofinly

pled,ingl-lly

-

1. Introduction

The stability of a colloidal dispersion is closely relatto the charged conditions on the surfaces of the dispeentities. Because the applications of a colloidal disperdepend largely on its stability, the estimation of the charnature of an entity becomes one of the key issues inloidal science. In practice,this is usually accomplished belectrophoretic measurements. Electrophoresis also plaimportant role as a separation method in industrial proceIn the United States, for example, about 4× 105 BTU ofenergy is consumed in separation or purification of checals [1], and a large fraction of this amount of energy wwasted due to low separation efficiency, which is harmfu

* On leave from National Taiwan University. Corresponding authoFax: +886-3-9353731.

E-mail address:jphsu@ntu.edu.tw(J.-P. Hsu).

0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2004.08.126

n.

both the economy and the environment. Electrophoresone of the potential techniques for improving separationficiency.

Under conditions of low surface potential and thin doble layer, Smoluchowski[2] was able to derive an anlytical expression relating the electrophoretic velocitya rigid, isolated particle in an infinite fluid as a functiof its surface potential and the applied electric field. Tsame problem was investigated by Hückel[3] for the casewhere the double layer is very thick. Subsequent analare ample in the literature; due to the complicated naturthe problem under consideration, however, they are malimited to drastically simplified cases[4,5]. The difficultyarises from the fact that the governing equations are couhighly nonlinear partial differential equations, and solvthese equations analyticallyunder general conditions is amost impossible. In fact, even solving them numericacan be challenging. O’Brien and White[6] proposed us

E. Lee et al. / Journal of Colloid and Interface Science 282 (2005) 486–492 487

f anarythe

ntse-

is

atis-bi-ar-

yeras-calent

cury

ee-

5 toioncallygingy oiln ofsiontion

ore-s ofuresiveicu-hastureericon-tem.ion,ted.ut-red.idern-urece-

que-eo-init-senctraper-

icerse,fre a

ri-ings

d

a’sy antric

ricaleium

sical

ing a shooting method to solve the electrophoresis oisolated sphere in an infinite fluid for the case of arbitrdouble-layer thickness and arbitrary surface potential;effect of double-layer polarization was taken into accoufor the case of a weak applied electric field. In a recentries of studies, Lee and co-workers[7–10] proposed usinga pseudo-spectral method[11] to solve the electrophoresunder the influence of a boundary[12]. The performanceof the numerical scheme adopted was found to be sfactory in solving complicated problems that involve artrary surface potential, arbitrary double-layer thickness,bitrary strength of applied electric field, and double-lapolarization. The electrophoresis of nonrigid entities woriginated by Craxford et al.[13], who investigated the behavior of a mercury drop. Due to their specific physinature, which is ideal for model construction, subsequanalyses on nonrigid entities were mainly based on merdrops[14–17].

Liquid–liquid dispersion, or emulsion, is one of thimportant classes of colloidal dispersion in practice. Rcently, emulsions in which drop size ranged from about50 nm[18,19], or microemulsions, have drawn the attentof researchers in various areas. These thermodynamistable systems have many practical applications, ranfrom conventional ones such as detergents and tertiarrecovery to high-technology ones such as preparationanosized particles. The surface of a drop in an emulsystem is usually charged, which arises from the dissociaof the surfactant molecules. This implies that electrophsis can play a role in describing the physical propertiean emulsion system. A thorough review of the literatreveals that although experimental studies are extenrelevant theoretical analyses are very limited. In partlar, the case when a drop is of non-Newtonian naturenot been investigated. An emulsion system of this nais often observed in practice. The preparation of polymmicro- or nanosized particles through polymerization cducted in an emulsion, for example, involves such a sysCompared with the electrophoresis of a rigid dispersthe electrophoresis of an emulsion is more complicaThis is because in addition to solving the flow field oside an entity, that inside it also needs to be consideThe match of the flow field outside a drop and that insit can be nontrivial in the numerical treatment of the goveing equations of the flow field. The non-Newtonian natof a drop also adds some difficulty to the solution produre.

In this study, the electrophoresis of a concentrated aous dispersion of non-Newtonian drops is analyzed thretically. A Carreau fluid, which has wide applicationsmany areas[20,21], is chosen for the drop fluid, and the uncell model of Kuwabara[22] is adopted to simulate a dispersion. The electrokinetic equations describing the preboundary-valued problem are solved by a pseudo-spemethod, and the effects of the key parameters of a dission on its electrophoretic behavior are discussed.

,

tl

Fig. 1. Schematic representation of the problem considered. An electrfield E is applied to a concentrated spherical dispersion of monodispnon-Newtonian drops of radiusa, andU is the electrophoretic velocity odrops. The dispersion is simulated by Kuwabara’s unit cell model, whecell comprises a drop and a concentric spherical liquid shell of radiusb.

Fig. 2. Coordinates used in mathematical modeling, where (r, θ,φ) are thespherical coordinates with origin at the center of a drop.

2. Theory

Referring toFig. 1, we consider monodisperse, sphecal, non-Newtonian drops in an aqueous solution containz1:z2 electrolyte,z1 andz2 being respectively the valenceof cations and anions. Leta be the radius of a drop anz2 = −αz1. A uniform electric fieldE is applied, andU isthe electrophoretic velocity of drops. Adopting Kuwabarunit cell model[22], the dispersed system is simulated brepresentative cell, which contains a drop and a conceliquid shell of radiusb as shown inFig. 2. Let H = a/b,which measures the concentration of drops. The sphecoordinates(r, θ,ϕ) are chosen with its origin located at thcenter of the drop. We assume that the dispersion medis Newtonian and incompressible and has constant phy

488 E. Lee et al. / Journal of Colloid and Interface Science 282 (2005) 486–492

comhaso re

ian

-dsen

n bethe

on-

e

onicis

tem-

theby

-

-the-

nd

s

im-

-

ti-

nbe

properties, and the dispersed phase comprises an inpressible Carreau fluid, which is electrolyte-free andconstant physical properties. The system is assumed tmain in a quasi-steady state.

The constitutive equation of a generalized Newtonfluid can be expressed as

(1)τ = −η(γ̇ )γ̇ ,

(2)γ̇ = ∇v + (∇v)T.

In these expressions,τ is the shear stress tensor,γ̇ is the rateof strain tensor,γ̇ is its magnitude,η is the apparent viscosity,v is the fluid velocity,∇ is the gradient operator, anthe superscript T denotes matrix transpose. For the preCarreau fluid[23,24]we assume that

(3)η(γ̇ ) = η0[1+ (λγ̇ )2](n−1)/2

.

In spherical coordinates,̇γ can be expressed as

(4)γ̇ =√

1

2

(γ̇ 2rr + γ̇ 2

θθ + γ̇ 2φφ

) + γ̇ 2rθ + γ̇ 2

rφ + γ̇ 2θφ,

whereη0 is the zero-shear-rate viscosity,λ is a relaxation-time constant, andn is a power-law index. Note that ifn = 1or λ = 0, Eq.(3) leads to a Newtonian fluid, and ifλγ̇ � 1,it yields a power-law fluid.

The electrophoresis problem under consideration cadescribed by the electrokinetic equations, which includeequations for the electric field, the flow field, and the ccentration field.

2.1. Electric field

According to Gauss’s law, the electrical potentialφ canbe described by the Poisson equation,

(5)∇2φ = −ρe

ε= −1

ε

2∑j=1

zj enj ,

where∇2 is the Laplace operator,ε is the permittivity of thedispersion medium,ρe = ∑2

j=1 zj enj is the space chargdensity,e is the elementary charge, andzj and nj are re-spectively the valence and the number concentration of ispeciesj . The spatial variation in the ionic concentrationdescribed by

(6)∇2nj + zj e

kT

(∇nj · ∇φ + nj∇2φ) − 1

Dj

v · ∇nj = 0,

whereDj is the diffusivity of ionic speciesj andk andT arerespectively the Boltzmann constant and the absoluteperature.

2.2. Flow field

Suppose that the flow field can be described byNavier–Stokes equation in the creeping flow regime and

-

-

t

the continuity equation. We have

(7)µ∇2v − ∇p − ρe∇φ = 0, a < r < b,

(8)−∇p − ∇ · τ = 0, r < a,

(9)∇ · v = 0,

wherep is the pressure, andµ is the viscosity of the dispersion medium.

For a simpler treatment,φ is partitioned into the electrical potential arising from the presence of a drop inabsence ofE, or the equilibrium potentialφ1, and the electrical potential outside the drop arising fromE, φ2, that is,φ = φ1 + φ2. For the case of low electrical potential aweakE, it can be shown that the governing equations forφ1andφ2 are

(10)∇2φ1 = −1

ε

∑j

zj enj0

(1− zj eφ1

kT

),

(11)∇2φ2 = 0,

wherenj0 is the bulk concentration of ionic speciesj . Theseequations can be rewritten in terms of scaled symbols a

(12)∇∗2φ∗1 = (κa)2φ∗

1,

(13)∇∗2φ∗2 = 0,

whereφ∗1 = φ1/ζa , φ∗

2 = φ2/ζa , ∇∗ = a∇. ζa is the surfacepotential of the drop, andκ = [∑j nj0(ezj )

2/εkT ]1/2 is thereciprocal Debye length.

The mathematical treatment of flow field can be splified by introducing the stream functionψ . In terms ofthis function, ther- and theθ -components of fluid velocity, vr and vθ , can be expressed respectively asvr =(−1/r2 sinθ)(∂ψ/∂θ) and vθ = (1/r sinθ)(∂ψ/∂r). Notethat the continuity equation of the flow field is automacally satisfied through the introduction ofψ . The pressureterms in Eqs.(7) and (8)can be eliminated by taking curl oboth sides of them. In terms of the stream function, it canshown that the flow field can be described by

(14)D∗4ψ∗ = −(κa)2(

∂φ∗1

∂r∗∂φ∗

2

∂θ

)sinθ, 1 < r∗ < 1/H,

η∗D∗4ψ∗ + sinθ

[(∂η∗

∂r∗ γ̇ ∗rθ + r∗ ∂2η∗

∂r∗2γ̇ ∗rθ + r∗ ∂η∗

∂r∗∂γ̇ ∗

rθ

∂r∗

+ ∂2η∗

∂r∗∂θγ̇ ∗θθ + ∂η∗

∂θ

∂γ̇ ∗θθ

∂r∗

)−

(∂2η∗

∂r∗∂θγ̇ ∗rr + ∂η∗

∂r∗∂γ̇ ∗

rr

∂θ

+ 1

r∗∂2η∗

∂θ2 γ̇ ∗rθ + 1

r∗∂η∗

∂θ

∂γ̇ ∗rθ

∂θ

)+ ∂η∗

∂r∗

(1

sinθ

∂3ψ∗

∂r∗3

− cotθ

r∗2 sinθ

∂2ψ∗

∂r∗∂θ+ 1

r∗2 sinθ

∂3ψ∗

∂r∗∂θ2

− 2∗3

∂2ψ∗2

+ 2 cotθ∗3

∂ψ∗ )

r sinθ ∂θ r sinθ ∂θ

E. Lee et al. / Journal of Colloid and Interface Science 282 (2005) 486–492 489

the

s atthe

ativerms

er-

nonsecsentheted-

tressface

or.ese

ace:

:

neton-ell

− ∂η∗

∂θ

(− 1

r∗2 sinθ

∂3ψ∗

∂r∗2∂θ− 1

r∗4 sinθ

∂3ψ∗

∂θ3

− 1

r∗4 sin3 θ

∂ψ∗

∂θ+ cotθ

r∗4 sinθ

∂2ψ∗

∂θ2

)]

(15)= 0, 0 < r∗ < 1,

whereD∗4 = D∗2D∗2 with

(16)D∗2 = a2D2 = a2

(∂

∂r2 + sinθ

r2

∂

∂θ

(1

sinθ

∂

∂θ

)).

In these expressions,H = a/b, r∗ = r/a, ψ∗ = ψ/(UEa2),UE = εζ 2

a /µa, γ̇ ∗ = aγ̇ /UE , andη∗ = η/η0.

2.3. Boundary conditions

The following boundary conditions are assumed forequilibrium potentialφ1:

(17)φ1 = ζa, r = a,

(18)∂φ1

∂r= 0, r = b.

The first condition implies that the drop surface remaina constant potential. The second condition arises frombasic idea of the present cell model: a cell is a representof the whole dispersion; across the cell bounedary. In teof scaled symbols, these conditions become

(19)φ∗1 = 1, r∗ = 1,

(20)∂φ∗

1

∂r∗ = 0, r∗ = 1/H.

The following boundary conditions are assumed for the pturbed potentialφ2:

(21)∂φ2

∂r= 0, r = a,

(22)∂φ2

∂r= −Ez cosθ, r = b.

The first expression is based on the fact that drops areconductive and are impermeable to ionic species. Theond expression arises again from the nature of the precell model; that is, since the cell boundary corresponds tosystem boundary, the electric field there is that contribuby the applied electric fieldonly. In terms of scaled symbols, these conditions become

(23)∂φ∗

2

∂r∗ = 0, r∗ = 1,

(24)∂φ∗

2

∂r∗ = −E∗z cosθ, r∗ = 1/H,

whereE∗z = Ez/(ζa/a).

We assume that both the velocity and the shear sare continuous across the drop-dispersion medium interthat is,

--t

;

(25)vθ |r=a+ = vθ |r=a−,

(26)τrθ |r=a+ = τrθ |r=a−,

whereτrθ is the rθ -component of the shear stress tensIn terms of the stream function, it can be shown that thexpressions become

(27)∂ψ

∂r

∣∣∣∣r=a+

= ∂ψ

∂r

∣∣∣∣r=a−

,

η0[1+ (λγ̇ )2](n−1)/2

×[

1

r

∂2

∂r2 − 2

r2

∂

∂r− 1

r3

∂2

∂θ2 + cosθ

r3 sinθ

∂

∂θ

]ψ|r=a+

(28)

= µ

[1

r

∂2

∂r2− 2

r2

∂

∂r− 1

r3

∂2

∂θ2+ cosθ

r3 sinθ

∂

∂θ

]ψ|r=a− .

Also, since a drop moves with velocityU in thez-direction,the condition below needs to be satisfied on the drop surf

(29)ψ = −1

2Ur2 sinθ, r = a.

The following conditions are assumed on the cell surface

(30)vr = 0 and ∇ × v = 0, r = b.

The first expression arises from the fact that there is noflow of fluid across the cell boundary, and the second cdition is one of the key properties of Kuwabara’s unit cmodel. In terms of the stream function, we have

ψ =(

1

r sinθ

∂2

∂r2− cosθ

r3 sin2 θ

∂

∂θ+ 1

r3 sinθ

∂2

∂θ2

)ψ = 0,

(31)r = b.

Expressing Eqs.(27), (28), (29), and (31)in terms ofscaled symbols, we have

(32)∂ψ∗

∂r∗

∣∣∣∣r∗=1+

= ∂ψ∗

∂r∗

∣∣∣∣r∗=1−

,

η0

µ

[1+ (λ∗γ̇ ∗)2](n−1)/2

[1

r∗∂2

∂r∗2 − 2

r∗2

∂

∂r∗ − 1

r∗3

∂2

∂θ2

+ cosθ

r∗3 sinθ

∂

∂θ

]ψ∗|r∗=1+

=[

1

r∗∂2

∂r∗2 − 2

r∗2

∂

∂r∗ − 1

r∗3

∂2

∂θ2 + cosθ

r∗3 sinθ

∂

∂θ

](33)× ψ∗|r∗=1−,

(34)ψ∗ = −1

2U∗r∗2 sinθ, r∗ = 1,

ψ∗ =(

1

r∗ sinθ

∂2

∂r∗2 − cosθ

r∗3 sin2 θ

∂

∂θ+ 1

r∗3 sinθ

∂2

∂θ2

)ψ∗

(35)= 0, r∗ = 1/H,

490 E. Lee et al. / Journal of Colloid and Interface Science 282 (2005) 486–492

-ratevis-of

teds a

on,

g

l

a

for-l an

darytralas

inde-is ofeat-ularyed.

olve

un-am-ear-blenceo-

ab-ce isrop,

se,yersthe

, ase

ning-p is

o-d

whereλ∗ = UEλ/a andU∗ = U/UE . Here, the ratioη0/µ

is assigned the value of unity; that is, the zero-shearviscosity of the present Carreau fluid is the same as thecosity of the dispersion medium. The symmetric naturethe present problem also requires that

(36)∂φ1

∂θ= ∂φ2

∂θ= ψ = ∂ψ

∂θ= 0, θ = 0 andθ = π/2

or in terms of scaled symbols,

(37)∂φ∗

1

∂θ= ∂φ∗

2

∂θ= ψ∗ = ∂ψ∗

∂θ= 0, θ = 0 andθ = π/2.

2.4. Electrophoretic mobility

The electrophoretic mobility of a drop can be evaluabased on the fact that the net force acting on it vanishesteady state. For the present problem, only thez-componentof the net force, which includes the electrical contributiFEz, and the hydrodynamic contribution,FDz, needs to beconsidered. Thez-component of the electrical force actinon a drop can be evaluated by[25]

FEz =∫

σ(−∇φ)s · iz dA

(38)=∫

ε(∇φ · n)s(∇φ · iz)s dA,

whereσ is the surface charge density,n is the unit normavector on the drop surface, and the subscripts denotes thedrop surface. In spherical coordinates,dA = 2πr2 sinθ dθ ,and it can be shown that

FEz = 2πεζ 2a

π∫0

(∂φ∗

∂r∗

)r∗=1

(39)

×(

∂(φ∗)∂r∗ cosθ − 1

r∗∂(φ∗)∂θ

sinθ

)r∗=1

r∗2 sinθ dθ.

Applying the relationsφ∗ = φ∗1 + φ∗

2, (∂φ∗1/∂θ)r∗=1 = 0,

and(∂φ∗2/∂r∗)r∗=1 = 0, this expression becomes

FEz = 2πεζ 2a

π∫0

(∂φ∗

1

∂r∗

)r∗=1

(40)×(

− 1

r∗∂(φ∗

2)

∂θsinθ

)r∗=1

r∗2 sinθ dθ.

The z-component of the hydrodynamic force acting ondrop includes the drag on a charge-free drop,Fz, and thatarising from the presence of charge on its surface. Themer can be evaluated based on the results of HappeBrenner[25]. It can be shown that

FDz = µπ

π∫0

(r4 sin3 θ

∂

∂r

E2ψ

r2 sin2 θ

)r=a

dθ

(41)− π

π∫ (r2 sin2 θ

∂φ

∂θρ

)r=a

dθ.

0

t

d

The first term on the right-hand side of this expression isFz.In terms of scaled symbols this expression becomes

FDz = πεζ 2a

π∫0

(r∗4 sin3 θ

∂

∂r∗E∗2ψ∗

r∗2 sin2 θ

)r∗=1

dθ

(42)+ πεζ 2a (κa)2

π∫0

(r∗2 sin2 θ

∂φ∗2

∂θφ∗

1

)r∗=1

dθ.

In steady state,FEz + FDz = 0.The governing equations and the associated boun

conditions are solved numerically by a pseudo-specmethod[11], which has many desirable properties sucha fast rate of convergence and convergence propertiespendent of boundary conditions. The present problema two-dimensional nature, and for a simpler numerical trment, the spherical domain is first mapped to a rectangdomain, and the pseudo-spectral method is then emploA Newton–Raphson iteration scheme is adopted to snonlinear algebraic equations.

3. Results and discussion

The influence of the key parameters of the systemder consideration on its electrophoretic behavior is exined through numerical simulation. These include the shthinning nature of the drop fluid, the thickness of the doulayer, and the concentration of a dispersion. The influeof the shear-thinning nature of the drop fluid on the mbility of a drop is illustrated inFig. 3, where the variationof the scaled mobilityµ∗

m as a function ofκa at variousnis presented for two levels ofλ∗. Note that ifn = 1.0, thedrop is Newtonian.Fig. 3 suggests thatµ∗

m increases withκa, in general. This is because a largeκa means that thedouble layer surrounding a drop is thin. In this case thesolute value of the potential gradient near the drop surfalarge, which yields a large electrical force acting on the dand the mobility becomes large accordingly. Ifκa is small,the double layer surrounding a drop is thick. In this cathe electrical interaction between neighboring double labecomes important, which has the effect of prohibitingmovement of a drop. It is expected thatµ∗

m → 0 asκa → 0,and the mobility is insensitive to the nature of the dropjustified inFig. 3. If κa is sufficiently large, the nature of thdrop becomes important; that is, the effect of shear thinis significant. In this case, the greater the effect of shear thinning is, the less the hydrodynamic drag acting on a droand, therefore, the larger its mobility is. This is whyFig. 3shows thatµ∗

m is large ifn is small and/orλ∗ is large.The influence of the concentration of drops on their m

bility is presented inFig. 4, where the variation of the scalemobility µ∗

m as a function ofκa at variousH is plotted. Thisfigure reveals that for both very thick (smallκa) and verythin double layers (largeκa), the mobility is insensitive to

E. Lee et al. / Journal of Colloid and Interface Science 282 (2005) 486–492 491

er,

l in-mald-ientto its

is isace

-tnd,

amin

therop.dis-n

ck-

thedn).,ed byn

r,q-

ed.ary

eeet ofsted

Fig. 3. Variation of scaled mobilityµ∗m as a function ofκa at variousn for

the caseH = 0.5, E∗z = 1.0, andα = 1: (a)λ∗ = 0.5, (b)λ∗ = 1.5.

the variation inH , and for a medium-thickness double laythe largerH is, the smaller the mobility is. As inFig. 3, ifthe double layer surrounding a drop is thick, the electricateraction between neighboring double layers leads to a smobility. On the other hand, if the double layer surrouning a drop is thin, the absolute value of the potential gradnear its surface becomes large, which is advantageousmovement. In this case, although a largeH will prohibit themovement of a drop hydrodynamically, its electrophoresmainly controlled by the potential gradient near its surfand therefore isH -insensitive. For the case of a mediumthickness double layer, the largerH is, the more significanthe hydrodynamic retardation of neighboring drops is atherefore, the smaller the mobility is.

Fig. 5 shows the typical contours of the scaled strefunction ψ∗. Note that in the present problem, the dropa representative cell moves in thez-direction relative to thedispersion medium in the cell. For convenience, we letorigin of the coordinates be located at the center of the dThis is equivalent to saying that the drop is fixed and thepersion medium moves in the−z-direction. As can be seein Fig. 5, the movement of the drop in thez-direction yieldsa counterclockwise flow on its right-hand side and a clo

l

Fig. 4. Variation of scaled mobilityµ∗m as a function ofκa at variousH for

the case whenn = 0.9, λ∗ = 0.5, E∗z = 1.0, andα = 1.

Fig. 5. Contours of the scaled stream function for the case whenn = 0.9,λ∗ = 1.0, κa = 5.01187,H = 0.5, E∗

z = 1.0, andα = 1.

wise flow on its left-hand side (not shown). In contrast,flow of fluid inside the drop is clockwise on its right-hanside and counterclockwise on its left-hand side (not show

According to Saito[26], for the case of creeping flowthat is, as the Reynolds number,Re, approaches zero, thshape of a drop remains spherical. This is also confirmeTaylor and Acrivos[27], who showed that the deformatioof a drop is proportional toRe2, or to the Weber numbeWe= ρU2a/σ1, σ1 being the surface tension of the drop liuid. In our case, because a typical value ofReis on the orderof 10−10–10−8 and that ofWe is on the order of 10−17 (seeAppendix A), the deformation of a drop can be neglectIn a study of the behavior of a relatively large, stationdrop in an applied electric field, Eow et al.[28] observedthat breakup occurs to a drop of 1.2 mm diameter when thelectric Weber numberWee = 2εE2a/σ1 reaches 0.49. Thmechanism behind the breakup phenomenon is the onsinstability caused by the disruptive electric-induced stresover the stabilizing interfacial tension stress. As illustra

492 E. Lee et al. / Journal of Colloid and Interface Science 282 (2005) 486–492

fg-

rsiona-

eakofon-ica-l-theick-r to

thet ofo-

der, ii-the

ei-es

ncil

ly

, we

l

r

Wi-

78)

.0.

d

9.

1)

.

rk,

iq-

8.art-

79.

in Appendix A, typical Wee in our case is on the order o10−8–10−5, implying that the deformation of a drop is neligible.

4. Conclusion

The electrophoresis of a concentrated aqueous dispeof monodisperse drops containing a Carreau fluid is alyzed under conditions of low surface potential and welectric field. This is the first attempt at the descriptionthe electrophoretic behavior of a dispersion containing nNewtonian drops. A system of this nature has wide appltions in practice, especially for emulsions and microemusions. The results of numerical simulation reveal thatmobility of a drop increases with the decrease in the thness of the double layer surrounding it, which is similathe behavior of a rigid dispersion. In general, becauseshear-thinning nature of a Carreau fluid has the effecreducing the hydrodynamic drag acting on a drop, its mbility increases accordingly,and this effect is pronounceas the thickness of the double layer decreases. Howevthe double layer is sufficiently thick, this effect is negligble. In general, the higher the concentration of drops is,smaller the mobility is; however, if the double layer isther sufficiently thin or sufficiently thick, this effect becomunimportant.

Acknowledgment

This work is supported by the National Science Couof the Republic of China.

Appendix A

The Reynolds numberRe, the Weber numberWe, andthe electrical Weber numberWee are defined respectiveby Re= ρUa/µ, We= ρU2a/σ1, andWee = 2εE2a/σ1, σ1being the surface tension of the drop liquid. In our casehaveµ∗

m = U∗/E∗, E∗ = E/(ζa/a), and E = E∗(ζa/a).Also, U∗ = U/UE = U/(εζ 2

a /µa), U = U∗(εζ 2a /µa),

U∗ = µ∗mE∗ = U/(εζ 2

a /µa), andU = µ∗mE∗(εζ 2

a /µa). Fora sunflower oil drop at 1 atm and 20◦C [28], ε = 4.34×

n

f

10−11 F/m, µ = 4.72× 10−2 Pa s,ρ = 992 kg/m3, andσ1(oil–water)= 1.6 × 10−2 N/m. If we choose the typicavaluesµ∗

m = 1, E∗ = 1, a = 10−7 m, ζa = 25.6 mV, E =E∗ζa/a = 256 kV/m, andU = 1.0 × 10−7 m/s, thenRe=2.10× 10−10, We= 6.20× 10−17, andWee1= 3.55× 10−5.Similarly, for ann-heptane drop at 1 atm and 20◦C [28], ε =1.77× 10−11 F/m, µ = 4.21× 10−4 Pa s,ρ = 682 kg/m3,σ1 and (HPLC-water)= 2.6 × 10−2 N/m. Therefore, usingthe same set ofµ∗

m, E∗, a, ζa , E, andU as for a sunfloweoil drop, we obtainRe= 1.62× 10−8, We= 2.62× 10−17,andWee = 8.92× 10−8.

References

[1] C.H. Byers, A. Amarnath, Chem. Eng. Prog. 91 (1995) 63.[2] M. Smoluchowski, Z. Phys. Chem. 92 (1918) 129.[3] E. Hückel, Phys. Z. 25 (1924) 204.[4] S.S. Dukhin, B.V. Derjaguin, Surface and Colloid Science, vol. 7,

ley, New York, 1974.[5] R.W. O’Brien, R.J. Hunter, Can. J. Chem. 59 (1981) 1878.[6] R.W. O’Brien, L.R. White, J. Chem. Soc. Faraday Trans. 274 (19

1607.[7] E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 205 (1998) 65[8] E. Lee, J.W. Chu, J.P. Hsu, J. Colloid Interface Sci. 209 (1999) 24[9] E. Lee, F.Y. Yen, J.P. Hsu, Electrophoresis 21 (2000) 475.

[10] J.P. Hsu, E. Lee, F.Y. Yen, J. Chem. Phys. 112 (2000) 6404.[11] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Metho

in Fluid Dynamics, Springer-Verlag, Berlin, 1987.[12] A.L. Zydney, J. Colloid Interface Sci. 169 (1995) 476.[13] S.R. Craxford, O. Gatty, H.A.C. McCay, Phil. Mag. 23 (1937) 107[14] S. Levine, R.W. O’Brien, J. Colloid Interface Sci. 43 (1973) 616.[15] H. Ohshima, J. Colloid Interface Sci. 218 (1999) 535.[16] J.C. Baygents, D.A. Saville, J. Chem. Soc. Faraday Trans. 87 (199

1883.[17] E. Lee, J.K. Hu, J.P. Hsu, J. Colloid Interface Sci. 257 (2003) 250[18] T.P. Hoar, J.H. Schulman, Nature 152 (1943) 102.[19] J.Th.G. Overbeek, Faraday Discuss. 65 (1978) 7.[20] L.M. Prince, Emulsions and Emulsion Technology, Dekker, New Yo

1974.[21] J. Danielsson, B. Lindman, Colloids Surf. 3 (1981) 391.[22] S. Kuwabara, J. Phys. Soc. Jpn. 14 (1959) 527.[23] R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymer L

uids, vol. I, Wiley, New York, 1987.[24] P.J. Carreau, Ph.D. thesis, University of Wisconsin, Madison, 196[25] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, M

inus Nijhoff, 1983.[26] S. Saito, Science Rep. Tohoku Imp. Univ. Sendai Jpn. 2 (1913) 1[27] T.D. Taylor, A. Acrivos, J. Fluid Mech. 18 (1964) 466.[28] J.S. Eow, M. Ghadiri, A. Sharif, Colloids Surf. A 225 (2003) 193.