here

y,

and thethe doublemultipole

t

ldtiveent of a

Journal of Colloid and Interface Science 262 (2003) 101–106www.elsevier.com/locate/jcis

High-order field electrophoresis theory for a nonuniformly charged sp

Jae Young Kim and Byung Jun Yoon∗

Department of Chemical Engineering and Division of Mechanical and Industrial Engineering, Pohang University of Science and TechnologPohang 790-784, South Korea

Received 26 November 2002; accepted 20 February 2003

Abstract

An electrophoresis theory is developed for a rigid sphere in a general nonuniform electric field. The zeta potential distributiondouble-layer thickness are both arbitrary. The zeta potential of the sphere is assumed to be small so that the deformation oflayer can be neglected. Explicit expressions for the translational and rotational velocities of the sphere are derived in terms of themoments of the zeta potential distribution and the tensor coefficients of the applied electric field. The presence of thekth-order componenin the electrical potential field applied to the sphere results in a translation of the sphere only when the sphere possesses the(k − 1)th- or(k + 1)th-order multipole moments of the zeta potential distribution. In addition, thekth-order component in the electrical potential fiecauses a rotation of the sphere only when the sphere possesses thekth-order moment of the zeta potential distribution. As an illustraexample for the utility of our theory, we theoretically devise an electrophoresis analysis scheme for estimating the dipole momdipolar sphere by observing the electrophoretic translation of the sphere in a quadratic potential field. 2003 Elsevier Science (USA). All rights reserved.

Keywords:Dipolar particle; Electrophoresis; Nonuniform electric field; Nonuniform zeta potential

in-ar-tly,forof

ns-tion.ki-ted,extheec-riclec-i-

is ofc-een

letionpo-

erential.aticand

zetahe-zetarichere

the-ionis-areal-eld.d inhe

thets oflec-

1. Introduction

Conventional electrophoresis experiments for determing the zeta potential of colloidal particles are normally cried out under conditions of constant electric field. Recenwith the advent of microfabrication techniques availablemaking microdevices for various purposes, the utilitycomplex electric fields for manipulating the flow and traport of colloidal suspensions has received much attenAlthough the utility of such microdevices for the electronetic analysis of colloidal particles has yet to be investigathe analysis of motion of a colloidal particle in a complelectric field may reveal more detailed information onzeta potential distribution of the particle. Evidently, the eltrokinetic analysis of colloidal particles in a complex electfield requires electrophoresis theories that predict the etrokinetic motion of a colloidal particle in a highly nonunform electric field.

There are few theories available for the electrophorescolloidal particles in nonuniform electric fields. The eletrophoresis of a sphere in a linear electric field has b

* Corresponding author.E-mail address:bjyoon@postech.ac.kr (B.J. Yoon).

0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights rdoi:10.1016/S0021-9797(03)00219-4

analyzed by Anderson [1]. In the limit a of very thin doublayer Anderson showed that the electrophoretic translaof the sphere depends on the dipole moment of the zetatential and that the electrophoretic rotation of the sphdepends on the quadrupole moment of the zeta poteThe electrophoresis of spheroidal particles in a quadrelectric field has been also analyzed by SolomentsevAnderson [2], assuming that the spheroid has a uniformpotential distribution. Although various electrophoresis tories have been developed for particles with nonuniformpotential distributions under conditions of uniform electfield [3–7], extensions of such theories to the cases wthe electric fields are nonuniform are still lacking.

The aim of this paper is to develop an electrophoresisory for a sphere with a nonuniform zeta potential distributin a general nonuniform electric field. The zeta potential dtribution and the double-layer thickness of the sphereboth arbitrary. We first solve for the force and torque bances for the sphere in a general nonuniform electric fiThe zeta potential distribution of the sphere is expresseterms of the multipole moments of the zeta potential. Telectrophoretic translational and rotational velocities ofsphere are determined in terms of the multipole momenthe zeta potential and the coefficients of the external e

eserved.

102 J.Y. Kim, B.J. Yoon / Journal of Colloid and Interface Science 262 (2003) 101–106

lec-oryin a

yn be. Thetokeon-

t theque,8,9]

re.rquee othem

forcorcetion

-

ce o

ns-plicit

en

o-zeta

e ofor

by

trice ofriteas

-cro-

ingesis

n-

he-

tion

trical field. After discussing the general aspects of our etrophoresis theory we demonstrate the utility of the theby analyzing the electrophoresis of a dipolar spherequadratic potential field.

2. Theory

2.1. Balance equations

The translational velocityU and the rotational velocit of a particle that undergoes electrophoretic motion cadetermined from the force and torque balance equationsequations to be solved are the Poisson equation, the Sequation with an electrostatic body force term, and the cservation equations for the ionic species. Assuming thazeta potential of the particle is small, the force and torbalance equations for a spherical particle are given by [3

(1)6πµaU = −εκ2∫V

Ψ∇Φ dV + εκ2∫V

Ψ uF · ∇Φ dV,

(2)

8πµa3 = −εκ2∫V

Ψ x × ∇Φ dV + εκ2∫V

Ψ uT · ∇Φ dV.

Here,µ is the viscosity of the fluid,a the sphere radius,ε thepermittivity of the fluid,κ the inverse Debye length, andxthe position vector with its origin at the center of the spheThe first terms on the right hand sides of the force and tobalance equations are the electrostatic force and torquthe sphere. The divergence theorem is used to expressas volume integrals over the fluid volumeV outside thesphere. The second terms on the right hand sides of theand torque balance equations are the hydrodynamic fand torque on the sphere exerted by electroosmotic moof surrounding fluid. The potentialΨ denotes the equilibrium double-layer potential and the potentialΦ denotes thepotential field developed around the sphere in the presenthe external electric field. The second-order tensor fieldsuF

anduT represent the Stokes velocity fields around a tralating sphere and a rotating sphere, respectively. The exexpressions foruF anduT are given by

(3)uF = 1

4

[3

(a

r

)+(a

r

)3]δ + 3

4

[(a

r

)−(a

r

)3]

nn,

(4)uT = −a(a

r

)2

ε·n,

wherer = |x|, n = x/r, δ is the Kronecker delta, andε isthe permutation tensor.

2.2. Double-layer potential

The equilibrium double-layer potentialΨ around thesphere with a nonuniform zeta potential distribution is giv

s

n

e

f

by [3,10]

(5)Ψ =∞∑k=0

P(k)i1...ik

ni1 . . .nikRk(r).

Here we use the Einstein summation convention. Thekth-order tensorP(k) is the kth-order moment of the zeta ptential distribution over the sphere surface. When thepotential distribution is given byζ , its kth-order moment isdefined by

(6)P(k)i1...ik

= (−1)k2k+ 1

4πk!∫Ω

ζrk+1(∂

∂xi1. . .

∂

∂xik

)1

rdΩ.

HereΩ denotes the solid angle defined over the surfaca unit sphere. The tensorP(k) is traceless and symmetric feach pair of indices. The functionRk(r) is the radial solutionof the linearized Poisson–Boltzmann equation, as given

(7)Rk(r)=(a

r

)k+1Gk(κr)

Gk(κa)e−κ(r−a),

where

Gk(x)=k∑s=0

γ ks xs, γ ks = 2sk!(2k− s)!

s!(2k)!(k − s)! .

2.3. External potential field

As the sphere moves through the nonuniform elecfield developed inside an electrophoresis cell, the valuthe electric field acting on the sphere changes. We may wthe general form of the electric field acting on the sphere

(8)E(x)= A(1) + A(2) · x + A(3) : xx + · · · .Here,E = −∇Φ, and the value of eachA(k) does vary depending on the position of the sphere relative to the mielectrodes installed inside the electrophoresis cell. Assumthat the electric field developed inside the electrophorcell is given by

(9)E(X)= B(1) + B(2) · X + B(3) : XX + · · · ,the values ofA(k) can be determined by using the relatioshipX = Xc + x. As shown schematically in Fig. 1,X is theposition vector with its origin at some fixed point within telectrophoresis cell andXc is the position vector for the center of the sphere. The electrical potential fieldΦ is obtainedby solving the Laplace equation with the boundary condi

(10)

Φ|r→∞ = −A(1)i xi −1

2A(2)ij xixj − 1

3A(3)ijkxixj xk + · · ·

far away from the sphere and the boundary condition

(11)∂Φ

∂n

∣∣∣∣S

= 0

at the sphere surface. The solution forΦ is given by

(12)Φ = −∞∑k=1

[1

k+ 1

k + 1

(a

r

)2k+1]A(k)i1...ik

xi1 . . . xik .

J.Y. Kim, B.J. Yoon / Journal of Colloid and Interface Science 262 (2003) 101–106 103

The

Theed

es-the

on

thee

n.en itri-the

fy

the

thethe

rmssly

Fig. 1. Coordinate systems for a sphere in a nonuniform electric field.coordinates systemXYZ is fixed in space, and the coordinate systemxyzhas its origin at the center of the sphere. The spherical polar anglesθ andφare also defined.

3. Results and discussion

3.1. Electrophoretic velocity

Using the solutions forΨ , Φ, uF , and uT the volumeintegrals in Eqs. (1) and (2) can be evaluated readily.integration over the solid angle of the sphere is performusing the equalities listed in Appendix A. The final exprsions for the translational and rotational velocities ofsphere are given by

Uj = ε

µ

∞∑k=1

ak−1[H1(κa; k)P (k−1)i1...ik−1

A(k)ji1...ik−1

(13)+H2(κa; k)P (k+1)j i1...ik

A(k)i1...ik

],

(14)

Ωj = ε

µa

∞∑k=1

ak−1H3(κa; k)εjnmP (k)ni1i2...ik−1A(k)mi1...ik−1

.

The functionsH1,H2, andH3 are functions ofκa and aregiven by

H1(x; k)= − (k − 1)!x2

6(2k− 1)!!Gk−1(x)

×k−1∑s=0

γ k−1s xs

[3k

2k + 1E2k+3−s(x)

− 3k

2k+ 1E2k+1−s(x)− k − 1

2k+ 1E2−s(x)

(15a)+ 3(3k+ 1)

2k+ 1E−s (x)− 4E−1−s(x)

],

H2(x; k)= − (k + 1)!x2

6(2k+ 3)!!Gk+1(x)

×k+1∑s=0

γ k+1s xs

[k + 2

k + 1E2k+5−s(x)

− 3(3k+ 2)

k + 1E2k+3−s(x)

+ 4(2k+ 1)E2k+2−s(x)

k + 1

(15b)− 3E4−s(x)+ 3E2−s(x)],

H3(x; k)= − k!x2

2(2k+ 1)!!Gk(x)

×k∑s=0

γ ks xs

[k

k + 1E2k+3−s(x)

(15c)

− k

k + 1E2k−s(x)+E2−s(x)−E−1−s(x)

],

where

(16)En(x)=∞∫

1

t−nex(1−t ) dt.

Here the double factorialm!! denotesm(m− 2) . . . , wherethe last element in the product is either 2 or 1 dependingwhetherm is even or odd. The plots of threeHn functionsfor k = 1,2,3,4,5 are shown in Fig. 2.

As shown in Eq. (13), under an electric field wherekth-order potential componentA(k) is nonzero the sphertranslates only when it possesses either the(k − 1)th- or(k + 1)th-order moment of the zeta potential distributioUnder the same electric field the sphere rotates only whpossesses thekth-order moment of the zeta potential distbution. Equations (13) and (14) are the generalization ofresults previously obtained for the case ofk = 1 [3]. In thelimit of a very thin double layer Eqs. (13) and (14) simplito

Uj = ε

µ

[P (0)A

(1)j + a

3P(1)k A

(2)kj − 1

5P(2)jk A

(1)k

(17a)

+ 2a2

15P(2)kl A

(3)klj − 2a

21P(3)jklA

(2)kl + 2a3

35P(3)klmA

(4)klmj

],

Ωj = ε

µa

[3

4εjklP

(1)k A

(1)l + a

3εjklP

(2)kmA

(2)lm

(17b)+ 3a2

20εjklP

(3)kmnA

(3)lmn

],

assuming that the zeta potential of the sphere only hasmultipole moments up to the octupoleP (3). Here the firstterm on the right hand side of Eq. (17a) corresponds toSmoluchowski equation. The second and third terms onright hand side of Eq. (17a) and the first and second teon the right hand side of Eq. (17b) were obtained previouby Anderson [1]. Note that the definition ofP (k) is differentfrom that of Anderson by the factor(2k+ 1)/k!. In the limitof a very thick double layer Eqs. (13) and (14) simplify to

Uj = ε

µ

[2

3P (0)A

(1)j + 2a

3P(1)k A

(2)kj

(18a)+ 4a2P(2)kl A

(3)klj + 4a3

P(3)klmA

(4)klmj

],

9 15

104 J.Y. Kim, B.J. Yoon / Journal of Colloid and Interface Science 262 (2003) 101–106

ithtricmen

dingnedicaletae-

. Col-ontalat

t ini-Thes a

eldrig-rmtion

stri-

eticthat

tialerene.

icole,

sus-reticre insla-

n the

tioneri-t can

sing

such. 3.ofl

Fig. 2. Plots of (a)H1, (b)H2, and (c)H3 as functions ofκa.

Ωj = ε

µa

[1

2εjklP

(1)k A

(1)l + a

3εjklP

(2)kmA

(2)lm

(18b)+ a2

5εjklP

(3)kmnA

(3)lmn

].

First consider the electrophoretic motion of a sphere wa uniform zeta potential distribution in a nonuniform elecfield. Since the sphere only possesses the monopole moP (0), the translational velocity of the sphere is given by

(19)U = ε

µH1(κa;1)P (0)A(1),

and there is no electrophoretic rotation. The corresponresults in the limit of a thin double layer have been obtaiby Anderson (1). Equation (19) is essentially the classHenry equation derived for a sphere with a uniform zpotential in a uniform electric field [11]. The difference b

t

Fig. 3. The electrode system that generates a quadratic potential fieldloidal particles are suspended in the region between two parallel horizplates. The voltage difference of 2V is applied between the electrodes(X= ±L, Y = 0) and (X= 0, Y = ±L).

tween the original Henry equation and Eq. (19) is thaEq. (19) the electric fieldA(1) is the value of the nonunform electric field evaluated at the center of the sphere.electric field A(1) thus varies, as the sphere undergoetranslational motion through the nonuniform electric fideveloped inside the electrophoresis cell. Although the oinal Henry equation was derived under conditions of unifoelectric field, we can still use the same form of the equaeven for cases with nonuniform electric fields.

When the sphere has a nonuniform zeta potential dibution, the presence of thekth-order multipole moment inthe zeta potential distribution results in the electrophormotion of the sphere under the electric potential fieldcontains the(k − 1)th-, kth-, and(k + 1)th-order compo-nents. High-order multipole moments of the zeta potendistribution affect the electrophoretic motion of the sphonly when the external electric field is also a high-order oFigure 2 shows that the values ofHn functions are relativelysmall for large values ofk. Therefore, the electrophoretmotion of the sphere is governed mostly by the monopdipole, and quadrupole moments.

3.2. Electrophoresis of dipolar sphere

While the presence of the dipole moment for a spherepended in a uniform electric field causes an electrophorotation, the presence of the dipole moment for a sphea nonuniform electric field causes an electrophoretic trantion as well. As shown in Eq. (13), the dipole momentP(1)

causes an electrophoretic translation of the sphere wheexternal electric field contains,A(2) term. Since it can bemore convenient to measure the electrophoretic translathan the electrophoretic rotation in electrophoresis expments, we here devise an electrophoresis scheme thadetermine the dipole moment of a dipolar sphere by uelectrophoretic translation measurement.

The electric field that contains anA(2) term is a quadraticpotential field. A microelectrodes system that generatesa quadratic potential field is schematically shown in FigThe voltage difference 2V is applied across each setelectrodes that are very small relative toL. The potentiafield developed inside the square region,−L < X < L by

J.Y. Kim, B.J. Yoon / Journal of Colloid and Interface Science 262 (2003) 101–106 105

l

tem

c

eodeposgovweenttion

nott re-

aec-the

n of

f-ithm-

how

esic

ipo-n of

Af-s too-ion

-

thepeeded oft offe ists.

mlyeld.rmsionld.

tionle toby

s ofntsbeetaandesusslarans-

−L< Y <L, is given by

(20)Φ = V

L2 (Y2 −X2).

Here, the potential field is independent ofZ, so the potentiafield is two-dimensional. We set the origin of theXYZ co-ordinate system at the center of the microelectrodes sysIn this setup the only nonzeroB(n) in Eq. (9) isB(2), whichis given by

(21)B(2)ij = 2V

L2

(1 0 00 −1 00 0 0

).

Nonzero coefficientsA(n) in Eq. (8) for the ambient electrifield for the sphere are thenA(1) = B(2) ·Xc andA(2) = B(2).In addition to the quadratic termA(2), the linear termA(1)

also exists. Since the magnitude ofA(1) increases as thsphere moves away from the origin of the microelectrsystem, the electrophoretic translation of a sphere thatsesses both monopole and dipole moments would beerned mostly by the monopole moment. In this workassume that the dipolar sphere has only a dipole momBased on Eqs. (13) and (14), the electrophoretic translaand rotation of the dipolar sphere are then given by

(22)U = aε

µH1(κa;2)P(1) · A(2),

(23) = ε

µH3(κa;1)P(1)× A(1).

The translational velocity of the dipolar sphere doeschange as long as the orientation of the dipole momenmains constant.

The electrophoretic motion of the dipolar sphere inquadratic potential field is analyzed by solving the trajtory equations. The orientation of the dipole moment ofsphere relative to the coordinates systemxyz is expressed interms of the spherical polar anglesθ andφ. The trajectoryequations for the center of the sphere and the orientatiothe dipole moment are given by

(24a)Xc = αH1 sinθ cosφ,

(24b)Y c = −αH1 sinθ sinφ,

(24c)θ = αH3

a(Xc cosφ − Y c sinφ)cosθ,

(24d)φ = − αH3

a sinθ(Xc sinφ + Y c cosφ).

Hereα is 2V aεP (1)/µL2, whereP (1) is the magnitude othe dipole momentP(1). By integrating the trajectory equations the path of the dipolar sphere is calculated along wthe orientation of the dipole moment. Figure 4 shows coputational examples. In these examples we seta = 5 µm,κa = 100,L = 50 µm,V = 1 V, andP (1) = 25 mV. Eachsolid line represents a path of the sphere, and arrows sthe orientation of the dipole momentP(1). In all cases theinitial orientation of the dipole moment is set by the valuθ = π/2 andφ = −π/4. Equipotential lines of the quadrat

.

--

.

Fig. 4. Trajectories of a dipolar sphere in a quadratic potential field. Equtential lines are shown by dotted lines. In all cases the initial orientatiothe dipole moment is given byθ = π/2 andφ = −π/4. The arrow repre-sents the orientation of the dipole moment.

potential field, Eq. (20), are also shown by dotted lines.ter a short transitional motion, the dipolar sphere tendtranslate following a straight line without any further rtation. According to Eqs. (24c) and (24d), the orientatof the dipolar sphere remains unchanged whenθ = π/2andφ = − tan−1(Y c/Xc). Note, that this orientation is exactly the same orientation of the electric field vector atXc.After the dipolar sphere attains the steady orientationsphere translates along a straight line at a constant sU = αH1(κa;2). Sinceα is proportional to the magnitudof the dipole moment, by measuring the constant speethe dipolar sphere we can determine the dipole momenthe dipolar sphere. In this particular example the value oUis about 20 µm/s. It should be stressed that this schemrestricted to spherical particles near their isoelectric poin

4. Summary

We develop an electrophoresis theory for a nonuniforcharged sphere suspended in a nonuniform electric fiElectrophoretic velocity of the sphere is expressed in teof the multipole moments of the zeta potential distributand the high-order coefficients of the applied electric fieAlthough we assume an arbitrary zeta potential distribuover the sphere surface, the results are also applicaba sphere with an arbitrary particle charge distributionusing the relationships between the multipole momentthe zeta potential distribution and the multipole momeof the particle charge distribution [3]. Our theory canuseful for determining the multipole moments of the zpotential distribution of a spherical particle by employingelectrophoresis cell equipped with a set of microelectrothat generates a particular type of electric field. We discone such example in which the dipole moment of a diposphere is determined by measuring the electrophoretic trlation of the sphere in a quadratic potential field.

106 J.Y. Kim, B.J. Yoon / Journal of Colloid and Interface Science 262 (2003) 101–106

omo by

nces:

rsr of

231.

0)

.

cs

Acknowledgments

This work was supported by Grant R01-2001-00410 frthe Korea Science and Engineering Foundation and alsthe Ministry of Education under the BK21 program.

Appendix A

The volume integrals in the force and torque balaequations are evaluated by using the following equation

(A.1)∫Ω

ni1 . . .ni2k dΩ = 4π

(2k+ 1)!!I(2k)i1...i2k

;

∫Ω

[ ∞∑m=0

P(m)i1...im

ni1 . . .nimA(k)j1...jk

nj1 . . .nj(k−1)

]dΩ

=∫Ω

[P(k−1)i1...i(k−1)

ni1 . . .ni(k−1)A(k)j1...jk

nj1 . . .nj(k−1)

]dΩ

(A.2)= 4π(k− 1)!(2k − 1)!! P

(k−1)j1...j(k−1)

A(k)j1...jk

;∫Ω

[ ∞∑m=0

P(m)i1...im

ni1 . . .nimA(k)j1...jk

nj1 . . .nj(k+1)

]dΩ

=∫Ω

[P(k−1)i1...i(k−1)

ni1 . . .ni(k−1) + P (k+1)i1...i(k+1)

ni1 . . .ni(k+1)

]

×A(k)j1...jk nj1 . . .nj(k+1) dΩ

= 4πk!(2k+ 1)!!P

(k−1)j1...j(k−1)

A(k)j1...j(k−1)j(k+1)

(A.3)+ 4π(k+ 1)!(2k + 3)!! P

(k+1)j1...j(k+1)

A(k)j1...jk

;∫Ω

[ ∞∑m=0

P(m)i1...im

ni1 . . .nimεlmnnm

×A(k)nj1...j(k−1)nj1 . . .nj(k−1)

]dΩ

=∫Ω

[P(k)i1...ik

ni1 . . . nik εlmnnm

×A(k)nj1...j(k−1)nj1 . . .nj(k−1)

]dΩ

(A.4)= 4πk!(2k+ 1)!!εlmnP

(k)mj1...j(k−1)

A(k)nj1...j(k−1)

.

HereI(2k) is the 2kth-order unit isotropic tensor. The tensoP(k) andA(k) are symmetric and traceless for each paiindices.

References

[1] J.L. Anderson, J. Colloid Interface Sci. 105 (1985) 45.[2] Y. Solomentsev, J.L. Anderson, Ind. Eng. Chem. Res. 34 (1995) 3[3] B.J. Yoon, J. Colloid Interface Sci. 142 (1991) 575.[4] D. Velegol, J.D. Feick, L. Collins, J. Colloid Interface Sci. 230 (200

114.[5] M.C. Fair, J.L. Anderson, J. Colloid Interface Sci. 127 (1989) 388[6] J.D. Feick, D. Velegol, Langmuir 16 (2000) 10315.[7] D. Long, A. Ajdari, Phys. Rev. Lett. 81 (1998) 1529.[8] M. Teubner, J. Chem. Phys. 76 (1982) 5564.[9] J.Y. Kim, B.J. Yoon, in: A.V. Delgado (Ed.), Interfacial Electrokineti

and Electrophoresis, Dekker, New York, 2002, p. 173.[10] J.G. Kirkwood, J. Chem. Phys. 2 (1934) 8.[11] D.C. Henry, Proc. R. Soc. London Ser. A 133 (1931) 106.