Electrophoresis of a Colloidal Sphere in a Spherical Cavity with Arbitrary Zeta Potential Distributions

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Research Article

Electrophoresis of a Colloidal Sphere in a Spherical

Cavity with Arbitrary Zeta Potential Distributions

Huan J. Keh, and Tzu H. Hsieh

Langmuir, 2007, 23 (15), 7928-7935 • DOI: 10.1021/la7004002

Downloaded from http://pubs.acs.org on November 19, 2008

Articles

Electrophoresis of a Colloidal Sphere in a Spherical Cavity with

Arbitrary Zeta Potential Distributions

Huan J. Keh* and Tzu H. Hsieh

Department of Chemical Engineering, National Taiwan UniVersity, Taipei 10617, Taiwan, Republic of China

ReceiVed February 11, 2007. In Final Form: May 7, 2007

An analytical study is presented for the quasi-steady electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity when the surface potentials are arbitrarily nonuniform. The applied electric field is constant, and the electric double layers adjacent to the solid surfaces are assumed to be much thinner than the particle radius and the gap width between the surfaces. The presence of the cavity wall causes three basic effects on the particle velocity: (1) the local electric field on the particle surface is enhanced or reduced by the wall; (2) the wall increases the viscous retardation of the moving particle; and (3) a circulating electroosmotic flow of the suspending fluid exists because of the interaction between the electric field and the charged wall. The Laplace and Stokes equations are solved analytically for the electric potential and velocity fields, respectively, in the fluid phase, and explicit formulas for the electrophoretic and angular velocities of the particle are obtained. To apply these formulas, one has to calculate only the monopole, dipole, and quadrupole moments of theζ-potential distributions at the particle and cavity surfaces. It is found that the contribution from the electroosmotic flow developing from the interaction of the imposed electric field with the thin double layer adjacent to the cavity wall and the contribution from the wall-corrected electrophoretic driving force to the particle velocities can be superimposed as a result of the linearity of the problem.

1. Introduction

Electrophoresis refers to the motion of a charged particle in an electrolyte solution subject to an applied electric field. Most colloidal particles bear charges on their surfaces as a consequence of the dissociation of functional groups or crystal lattice defects when immersed in an ionic solution. The counterions in the solution are attracted by the surface charge of the particle so that their concentration becomes higher in the vicinity of the particle surface than the bulk value. However, the co-ions are repelled from the particle surface. Hence, a region of mobile ions that is not electrically neutral forms, surrounding the particle. The combination of this region and the fixed charge on the particle surface is well known as an electric double layer. When an external electric field is imposed, the interaction between the particle’s surface charge and this field drives the particle to migrate at an electrophoretic velocity in one direction, whereas the counterions in the double layer move in the opposite direction, inducing an ambient fluid flow field different from that caused by the sedimentation of the particle. Electrophoresis has long been used as an effective technique for the separation and identification of biologically active compounds in the biochemical and clinical fields.

A simple expression for the electrophoretic velocity of a dielectric particle of arbitrary shape is the Smoluchowski equation,1-3

whereζpis theζ potential on the particle surface, η is the fluid

viscosity, is the fluid permittivity, and E∞is the constant applied

electric field. This equation is valid on the basis of several assumptions: (i) the local radii of curvature of the particle are much larger than the thickness of the electric double layer; (ii) the fluid surrounding the particle is unbounded; and (iii) theζ

potential is uniform on the length scale of the particle. The first restriction also implies that the double layer remains ap-proximately in equilibrium despite the migrations of the particle and diffuse ions. Even though many colloidal particles undergoing electrophoresis fulfill this condition, electrophoresis of particles with thick or distorted double layers is encountered in certain cases so that relevant corrections to the Smoluchowski prediction in eq 1 are necessary and have been obtained.4-8

In many electrophoresis applications to particle analysis or separation, particles migrate in the vicinity of solid boundaries. For instance, electrophoresis in porous media is applied because the unwanted mix-up caused by natural convection due to Joule heating and nonuniform heat transfer can be avoided. Microporous gels or membranes could even be used to achieve high electric fields and permit separations based on both the size and the charge of the particles.9_{In capillary electrophoresis, gels in the}

capillary column can minimize particle diffusion, prevent particle adsorption to the capillary walls, and eliminate electroosmosis while serving as the anticonvective medium.10_Deep

electro-phoresis penetration and deposition of inert colloidal particles over the interstitial surfaces of porous composites has been

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

(1) Smoluchowski, M. V. Bull. Int. Acad. Sci. CracoVie 1903, 8, 182.

(2) Morrison, F. A. J. Colloid Interface Sci. 1970, 34, 210. (3) Anderson, J. L. Ann. ReV. Fluid Mech. 1989, 21, 61. (4) Henry, D. C. Proc. R. Soc. London, Ser. A 1931, 133, 106.

(5) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607.

(6) O’Brien, R. W. J. Colloid Interface Sci. 1983, 92, 204. (7) Keh, H. J.; Chen, S. B. Langmuir 1993, 9, 1142.

(8) Keh, H. J.; Huang, T. Y. J. Colloid Interface Sci. 1993, 160, 354. (9) Jorgenson, J. W. Anal. Chem. 1986, 58, 743A.

(10) Ewing, A. G.; Wallingford, R. A.; Olefirowicz, T. M. Anal. Chem. 1989,

61, 292A.

U₀)ζp

η E∞ (1)

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suggested in the aerospace industry to protect the composites from burning or deterioration.11_{Another example is the}

elec-trophoresis of small particles through a Coulter counter designed not only to count and size the particles but also to determine their

ζ potentials.12_{Therefore, the boundary effects on electrophoresis}

are of great importance and have been studied extensively in the past for various cases of uniformly charged colloidal spheres and boundaries.13-24

However, many colloidal particles have heterogeneous surface structure or chemistry and are nonuniformly charged. For example, elementary clay particles are flat disks with edges having a different charge density or ζ potential from the faces.

Distributions of surface charge or potential for particles can also result from the aggregation of different species of colloids. Even if a particle is homogeneously charged on its surface, an applied electric field could cause the rearrangement of these charges if they are mobile.25_A_{ζ-potential distribution on particle surfaces}

has been found to lead to colloidal instability; even the average

ζ potential should be sufficiently high to keep the suspension

stable.26,27_{The electrophoretic motion of a dielectric sphere with}

a nonuniformζ potential and a thin electric double layer was

first analyzed thoroughly by Anderson,28_{although it had also}

been discussed to some extent earlier.29_{It was found that, in}

terms of the multipole moments of the ζ potential, the

electrophoretic mobility depends not only on the monopole moment (area-averagedζ potential) but also on the quadrupole

moment, and the dipole moment contributes to particle rotation, which tends to align the particle with the electric field. This analysis was later extended to cases of a nonuniformly charged spherical particle with a double layer of finite thickness30-33_and

a nonuniformly charged nonspherical particle.34-37_{Recently, that}

particles can have random charge nonuniformity has also been demonstrated experimentally.38,39

The electrophoretic motion of nonuniformly charged particles in the vicinity of confining walls could also be encountered in some real situations. In addition to the possible examples mentioned above, an electrophoretic positioning process has been employed in electronic applications for assembling very small individual devices, such as an InGaAs light-emitting diode or a nanowire, which must have all electric contacts available on one

surface, onto the contact electrodes of a silicon circuit by biasing the contacts to control the placement of these devices with the required precision.40,41 _{However, the boundary effects on the}

electrophoresis of a charged particle with a nonuniformζ potential

have not yet been investigated.

In this article, we examine the electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity with very thin electric double layers when the surface potentials are arbitrarily nonuniform. The motivation to study this problem arises from the technology of electric paper displays (known as Gyricon displays) where the translation and rotation of each member of an array of hemispherically bichromal, nonuniformly charged balls (about 100µm in diameter) in its own

elastomer-made and solvent-filled spherical cavity (which is only 10-40% larger than the ball) with either a monopole or a dipole on its wall between two thin, transparent plastic sheets are controlled by applying a voltage of either positive or negative polarity across the sheets.42,43_{Although the geometry of the concentric spherical}

cavity is an idealized abstraction of some other real systems, the result of boundary effects on the electrophoretic velocity of a uniformly charged sphere obtained in this geometry18_{has been}

shown to be in good agreement with that for a circular cylindrical pore.19_{The geometric symmetry in this model system allows an}

exact analytical solution to be obtained (as given by eqs 23-28 and illustrated in Figure 2a,b).

2. Analysis

We consider the quasi-steady electrophoretic motion of a nonconducting spherical particle of radius a andζ potential ζp

in a concentric spherical cavity (or pore) of radius b andζ potential ζwfilled with an electrolyte solution, as illustrated in Figure 1.

Bothζpandζwcan be nonuniform and are taken as arbitrary

functions of the position over the particle and cavity surfaces. The applied electric field (or the electric field in the absence of the particle) is constant and equals E∞ez, where ezis the unit

vector in the positive z (axial) direction. The rectangular co-ordinates (x, y, z) and spherical coco-ordinates (r,θ, φ) are established

with their origin at the particle and cavity center. The thickness of the electric double layers adjacent to the particle and cavity surfaces is assumed to be very small relative to the particle radius and the spacing between the solid surfaces. Gravitational effects are ignored. Our objective is to determine the electrophoretic velocity of the particle in the presence of the cavity.

Before determining the electrophoretic velocity of the confined particle with nonuniformζp, the electric potential and velocity

fields in the fluid phase must be solved.

2.1. Electric Potential Distribution. The fluid outside the thin double layers is electrically neutral and of constant

(11) Haber, S.; Gal-Or, L. J. Electrochem. Soc. 1992, 139, 1071. (12) DeBlois, R. W.; Bean, C. P. ReV. Sci. Instrum. 1970, 41, 909. (13) Morrison, F. A.; Stukel, J. J. J. Colloid Interface Sci. 1970, 33, 88. (14) Keh, H. J.; Anderson, J. L. J. Fluid Mech. 1985, 153, 417. (15) Keh, H. J.; Chen, S. B. J. Fluid Mech. 1988, 194, 377. (16) Keh, H. J.; Lien, L. C. J. Fluid Mech. 1991, 224, 305. (17) Loewenberg, M.; Davis, R. H. J. Fluid Mech. 1995, 288, 103. (18) Zydney, A. L. J. Colloid Interface Sci. 1995, 169, 476. (19) Keh, H. J.; Chiou, J. Y. AIChE J. 1996, 42, 1397.

(20) Keh, H. J.; Jan, J. S. J. Colloid Interface Sci. 1996, 183, 458. (21) Ennis, J.; Anderson, J. L. J. Colloid Interface Sci. 1997, 185, 497. (22) Yariv, E.; Brenner, H. Phys. Fluids 2002, 14, 3354.

(23) Yariv, E.; Brenner, H. J. Fluid Mech. 2003, 484, 85.

(24) Chen, P. Y.; Keh, H. J. J. Colloid Interface Sci. 2005, 286, 774. (25) Bazant, M. Z.; Squires, T. M. Phys. ReV. Lett. 2004, 92, 066101. (26) Miklavic, S. J.; Chan, D. Y. C.; White, L. R.; Healy, T. W. J. Phys. Chem. 1994, 98, 9022.

(27) Grant, M. L.; Saville, D. A. J. Colloid Interface Sci. 1995, 171, 35. (28) Anderson, J. L. J. Colloid Interface Sci. 1985, 105, 45.

(29) Teubner, M. J. Phys. Chem. 1982, 76, 5564. (30) Yoon, B. J. J. Colloid Interface Sci. 1991, 142, 575.

(31) Solomentsev, Y. E.; Pawar, Y.; Anderson, J. L. J. Colloid Interface Sci. 1993, 158, 1.

(32) Velegol, D.; Feick, J. D.; Collins, L. R. J. Colloid Interface Sci. 2000,

230, 114.

(33) Kim, J. Y.; Yoon, B. J. J. Colloid Interface Sci. 2003, 262, 101. (34) Fair, M. C.; Anderson, J. L. J. Colloid Interface Sci. 1989, 127, 388. (35) Long, D.; Ajdari, A. Phys. ReV. Lett. 1998, 81, 1529.

(36) Feick, J. D.; Velegol, D. Langmuir 2000, 16, 10315.

(37) Kim, J. Y.; Yoon, B. J. J. Colloid Interface Sci. 2002, 251, 318. (38) Feick, J. D.; Velegol, D. Langmuir 2002, 18, 3454.

(39) Feick, J. D.; Chukwumah, N.; Noel, A. E.; Velegol, D. Langmuir 2004,

20, 3090.

(40) Edman, C. F.; Swint, R. B.; Gurtner, C.; Formosa, R. E.; Roh, S. D.; Lee, K. E.; Swanson, P. D.; Ackley, D. E.; Coleman, J. J.; Heller, M. J. IEEE Photonics

Technol. Lett. 2000, 12, 1198.

(41) Smith, P. A.; Nordquist, C. D.; Jackson, T. N.; Mayer, T. S.; Martin, B. R.; Mbindyo, J.; Mallouk, T. E. Appl. Phys. Lett. 2000, 77, 1399.

(42) Crawford, G. P. IEEE Spectrum 2000, 37, 40.

Figure 1. Geometric sketch of the electrophoresis of a colloidal

sphere in a concentric spherical cavity.

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conductivity, hence the electric potential distributionψ(r, θ) is

governed by the Laplace equation,

Because the particle is assumed to be perfectly insulating, the boundary condition forψ at the surface of the particle is

At the surface of the cavity, the electric potential distribution gives rise to the applied electric field when the particle does not exist. Thus, a reasonable choice of the boundary condition there is

Here, we have setψ ) 0 on the plane z ) 0 for convenience

without the loss of generality. The solution of eq 2 subject to these boundary conditions is

whereλ )a_/ b.

The boundary condition at the cavity wall may alternatively be taken that the local electric potential gradient is equal in magnitude to the prescribed electric field. In this case, the Dirichlet approach given by eq 4 becomes the following Neumann approach:18,44

Note that although the normal component of the electric potential gradient at the cavity wall given by this boundary condition is consistent with the applied electric field its tangential (angular) component is not specified. The solution of eq 2 subject to eq 3 and boundary condition 6 is given by

In fact, eq 5 predicts that the electric potential at the particle surface (r ) a) is decreased by the presence of the cavity by a factor of (1 +λ3_/2)-1_{, whereas eq 7 suggests that this potential}

is increased by a factor of(1 - λ3₎-1_{. In the limit}_{λ f 0, as}

expected, eqs 5 and 7 become identical and reduce to the potential distribution for a nonconducting sphere in an unbounded medium. 2.2. Fluid Velocity Distribution. Having obtained the solution for the electric potential distribution, we can now proceed to find the fluid velocity field. Because the Reynolds number is small, the fluid motion outside the thin electric double layers is governed by the Stokes equations,

The general solution of the above equations is given by45,46

where pn(r),Φn(r), andχn(r) are solid spherical harmonics of

order n and r is the position vector.

Because the electric field acting on the diffuse ions within the thin double layer at each solid surface produces a relative tangential fluid velocity at the outer edge of the double layer as given by the Helmholtz47_{expression for the electroosmotic flow,}

the boundary conditions for the fluid velocity require that

Here, U and Ω are the translational and angular velocities,

(43) Crowley, J. M.; Sheridon, N. K.; Romano, L. J. Electrost. 2002, 55, 247.

(44) Jackson, J. D. Classical Electrodynamics, 2nd ed.; John Wiley & Sons: New York, 1976; Chapter 1.

(45) Lamb, H. Hydrodynamics; Dover: New York, 1945.

(46) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Martinus Nijhoff: Dordrecht, The Netherlands, 1983.

(47) Helmholtz, H. Ann. 1879, 7, 337.

Figure 2. Plots of dimensionless electrophoretic mobility parameters R_p, Rw,γp, andγwas calculated from eqs 25-28 versus separation parameterλ. The solid curves represent the case using the Dirichlet boundary condition in eq 4, and the dashed curves denote the case using the Neumann boundary condition in eq 6.

∇2 ψ ) 0 (2) ∂ψ ∂r ) 0 at r ) a (3) ψ ) -E_∞r cosθ at r ) b (4) ψ ) - 2E∞ 2 +λ3

(

r + a3 2r2

)

cosθ (5) ∂ψ ∂r ) -E∞cosθ at r ) b (6) ψ ) - E∞ 1 -λ3

(

r + a3 2r2

)

cosθ (7) η∇2v -∇p ) 0 (8) ∇‚v ) 0 (9) v )

∑

n)-∞ ∞

[

∇ × (rχn) +∇Φn+ n + 3 2η(n + 1)(2n + 3) r2∇p_n -n η(n + 1)(2n + 3) rp_n

]

(10) p )

∑

n)-∞ ∞ p_n (11) v ) v_sp) U + aΩ × n +ζp η (I - nn)‚∇ψ at r ) a (12) v ) v_sw)ζw η (I - nn)‚∇ψ at r ) b (13)

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respectively, of the electrophoretic sphere to be determined, n is the unit normal vector on the particle surface pointing toward the fluid phase, I is the unit dyadic, and the expression forψ has

already been given by eq 5 or 7.

As it can be shown,46_{to get the solutions of eqs 8 and 9 in}

the form of eqs 10 and 11 with boundary conditions given by eqs 12 and 13 for a spherical particle in a concentric spherical cavity, the solid spherical harmonic functions can be calculated from the relationships

Here, X_np,w(θ, φ), Y_np,w(θ, φ), and Z_np,w(θ, φ) are the surface

spherical harmonics, the superscripts or subscripts p and w represent the surfaces of the particle and cavity, respectively, v_sp,w represents the fluid velocity distributions on the corre-sponding surfaces given by eqs 12 and 13, ap) a, and aw) b.

From the second part of eqs 14-16, the relations between the solid spherical harmonics (pn,Φn,χn) and the surface spherical

harmonics (X_np,w, Y_np,w, Z_np,w) can be obtained, and they are given by eqs A1-A9 in Appendix A.

Theζ potentials ζpandζware arbitrary functions of (θ, φ) and

can be expressed in terms of the multipole expansions,28,37

Here the monopole, dipole, and quadrupole moments Mp,w_{, D}p,w_,

and Qp,w_{, respectively, are defined by the following integrals}

over the particle and cavity surfaces Sp,w

and the higher-order moments (which make no contribution to the electrophoretic velocity of the particle, as will be discussed later) are neglected. Various nonuniformζ potentials ζp,wcan

result from eq 17 with appropriate choices of moments Mp,w_(i.e.,

area-averagedζ potentials), Dp,w_{, and Q}p,w_{(which are symmetric}

and traceless).

With the substitution of the surface velocities given by eqs 12 and 13, in which the electric potential is given by eq 5 or 7 and

theζ potentials are expressed by eq 17, into the first part of eqs

14-16, the surface spherical harmonics X_np,w, Y_np,w, and Z_np,wcan be calculated in terms of the components of moments Mp,w_{, D}p,w_,

and Qp,w_{. It is found that X} n

p,w_{) 0 for all n, and the nonzero}

contributions of harmonic functions Y_np,wand Z_np,ware given by eqs A10-A14. For the electrophoresis of a sphere in a concentric spherical cavity with specifiedζ-potential distributions, the fluid

flow field can be obtained as an explicit function of the components of moments Mp,w_{, D}p,w_{, and Q}p,w_{using eqs 10, 11,}

and A1-A14.

The force F and torque T exerted by the fluid on the particle as a result of the electrokinetic motion can be determined from46

These equations indicate that only low-order solid harmonic functions p-2andχ-2contribute to the hydrodynamic force and torque on the particle. Note that p-2andχ-2are functions of monopole, dipole, and quadrupole moments Mp,w_{, D}p,w_{, and Q}p,w

of theζ-potential distributions but are independent of their

higher-order moments.

2.3. Derivation of the Particle Velocities. At the quasi-steady state, the net force and net torque acting on the electrophoretic particle must vanish. Applying these constraints to eqs 21 and 22 and using eqs A2 and A6 (taking n ) 1, X_np,w) 0, and Y_np,wand

Zn p,w

as given by eqs A10 and A13) for harmonic functions p-2

andχ-2, we obtain the translational and angular velocities of the particle in the cavity as

where

andν ) 2 + λ3_{if the Dirichlet boundary condition in eq 4 is}

employed, whereasν ) 2(1 - λ3_{) if the Neumann boundary}

condition in eq 6 is used. Note that all four parameters Rp, Rw, γp, andγwgiven by eqs 25-28 are functions of the ratioλ ) a_/

balone. Although the expressions for the particle velocities in

eqs 23 and 24 involve only the monopole, dipole, and quadrupole moments of the ζ potentials, they are exact for a

particle-in-cavity system with arbitraryζ potential distributions because the

M w I -1 2Q w

)

]

‚E_∞ (23) Ω ) 9 4ηa[γpD p_{- γ} wD w_]_{× E} ∞ (24) R p) 2 - 5λ3_{+ 3λ}5 ν(1 - λ5) (25) R w) 6 - 10λ2_{+ 3λ}3_{- λ}5_{+ 2λ}8 3ν(1 - λ5) (26) γp) 2_ν (27) γw) 2₃ νλ(2 + λ 3₎ ₍₂₈₎

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the particle velocities after the calculation of the multipole moments of theζ-potential distributions according to eqs

18-20. The translational velocity of the particle depends only on the monopole and quadrupole moments of theζ-potential distributions

at the particle and cavity surfaces (the existence of the quadrupole moments results in an anisotropic electrophoretic mobility of the particle), whereas the rotational velocity is affected only by the dipole moments. The effect of finite values of parameters Rwand γwis due to the electroosmotic flow that arises from the interaction

between the imposed electric field and the thin double layer adjacent to the cavity wall. Equations 23 and 24 indicate that the contributions from the wall-corrected electrophoretic driving force (involving parameters Rpandγp) and the electroosmotic flow to

the particle velocities can be superimposed, which is due to the linearity of the problem. Note that the dependence of Rpand Rw

onλ is different in order, and so is that for parameters γpand γw.

It can be found that all four parameters Rp, Rw,γp, andγw

given by eqs 25-28 are always positive as long as 0 <λ < 1.

In the limit ofλ f 0, eqs 25-28 reduce to Rp) Rw) γp) 1

andγw) 0, in which the electrophoretic and angular velocities

for an unconfined dielectric sphere with nonuniformζpobtained

by Anderson28

are reproduced (with the effect of the electroosmotic flow arising from the interaction of the prescribed electric field with the charged cavity at infinity given by Rw) 1 excluded). When ζp

is uniform over the particle surface, eqs 29 and 30 reduce to eq 1 andΩ0) 0. In the limit of λ f 1, eqs 25-28 become Rp) R_w_{) 0 and γ}_p_{) γ}_w)2_/

3if the boundary condition in eq 4 is

adopted for the electric potential at the cavity wall, and Rp) Rw )1_/

2andγp) γwf∞if the boundary condition in eq 6 is used.

Note that eqs 25 and 26 for Rpand Rwwithν ) 2(1 - λ3) are

identical to the corresponding equations derived by Zydney18_for

a uniformly charged spherical particle undergoing electrophoretic motion in a uniformly charged concentric spherical cavity using the boundary condition in eq 6. For a given finite value ofλ, the

values of parameters Rp, Rw,γp, andγwpredicted using eq 4 are

always smaller than their corresponding results obtained using eq 6.

In terms of the electrophoretic and angular velocities for an isolated dielectric sphere given by eqs 29 and 30, it is convenient to express the bounded result of eqs 23 and 24 as the normalized velocity components in rectangular coordinates,

where

and subscript i represents the x, y, or z component for which the unbounded velocity Ui0orΩi0is nonzero. Parameters kiand li

correspond to the strength of the electroosmotic flow that develops from the interaction between the imposed electric field and the thin double layer adjacent to the cavity wall relative to the electrophoretic driving force when the cavity wall is at infinity. These two parameters can be either positive or negative (meaning that the contributions to the particle velocities from the cavity-induced electroosmotic flow can be in either the same or in the opposite direction to those from the electrophoretic driving force), depending on the combination of theζ-potential distributions at

the particle and cavity surfaces. Equations 31 and 32 show that the i component of the normalized electrophoretic velocity of the spherical particle depends only on parameters Rp, Rw, and kiand the i component of the normalized angular velocity depends

only onγp,γw, and li. Note that for a specified component i in

a bounded system it is possible that Ui(orΩi) is finite while U0i

(orΩ0i) vanishes. U₀)  η

(

M p I -1 2Q p

)

‚E_∞ (29) Ω0) 9₄_ηaD p_{× E} ∞ (30) U_i U_0i) Rp+ Rwki (31) Ωi Ω0i ) γ_p+ γ_wl_i (32) k_i)

[

(

MwI -1 2Q w

)

‚E_∞

]

i

[

(

MpI -1 2Q p

)

‚E_∞

]

i (33) l_i) -[D w_{× E} ∞]i [Dp× E_∞]_i (34)

Figure 3. Plots of normalized translational velocity Ui/U0iof the electrophoretic particle versus separation parameterλ: (a) the case using the Dirichlet boundary condition in eq 4 and (b) the case using the Neumann boundary condition in eq 6.

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3. Results and Discussion

The numerical values of dimensionless mobility parameters

R_p_{, R}_w_,_γ_p_{, and}_γ_w_{of a spherical particle undergoing}

electro-phoresis in a concentric spherical cavity, as calculated from eqs 25-28, are plotted versus separation parameterλ in Figure 2a,b.

It can be seen that both Rpand Rware monotonically decreasing

functions ofλ from unity at λ ) 0 to zero (if the Dirichlet boundary

condition in eq 4 is adopted for the electric potential at the cavity wall) or to1_/

2(if the Neumann boundary condition in eq 6 is

used) atλ ) 1. Thus, the net effect of the approach of the cavity

wall to the particle, dominated by the contribution from viscous retardation, is to reduce both the electrophoretic driving force and the cavity-induced electroosmotic sweeping force on the particle. However,γwis a monotonically increasing function of λ from zero at λ ) 0 to2_/

3(if eq 4 is adopted) or to infinity (if

eq 6 is used) at λ ) 1, whereas interestingly γp decreases

monotonically with an increase inλ from unity at λ ) 0 to2_/ 3

atλ ) 1 (if eq 4 is adopted) or increases monotonically with λ

from unity atλ ) 0 to infinity at λ ) 1 (if eq 6 is used). Note

that for any given value ofλ between 0 and 1, Rpis greater than R_w_and_γ_p_{is greater than}_γ_w_{, no matter whether eq 4 or 6 is used}

for the boundary condition of the electric potential at the cavity wall.

The normalized velocities Ui/U0iandΩi/Ω0icalculated from

eqs 31 and 32 as functions ofλ are depicted in Figures 3 and

4, respectively, for various values of kiand li. For a constant

value ofλ, as expected, the value of Ui/U0iincreases monotonically

with an increase in ki, and the value of Ωi/Ω0i increases

monotonically with an increase in li. As long as the values of ki

and li are greater than -1, the values of Ui/U0i andΩi/Ω0i,

respectively, are always positive. When the values of kiand li

are smaller than about -1, however, the values of Ui/U0iand Ωi/Ω0imay become negative, meaning that the translational and

rotational velocities of the particle may reverse their directions because of the relatively strong effect of the cavity-induced

Figure 4. Plots of normalized angular velocity Ωi/Ω0i of the electrophoretic particle versus separation parameterλ: (a) the case using the Dirichlet boundary condition in eq 4 and (b) the case using the Neumann boundary condition in eq 6.

sphere with an oddζ-potential distribution formed by one hemisphere of the constant valueζ₁pattached to a second hemisphere ofζ₂pin a concentric spherical cavity with an oddζ-potential distribution formed by one hemisphere ofζ1

w

attached to a second hemisphere of ζ₂w. The z-x plane is chosen to contain both unit vector ep defining the axis of rotational symmetry of the particle and applied electric field E∞.

sphere with an evenζ-potential distribution formed by two caps of the constant valueζ1

p

and a middle ofζ2 p

in a concentric spherical cavity with an evenζ-potential distribution formed by two caps of ζ1

w

and a middle ofζ2 w

. The z-x plane is chosen to contain both unit vector epdefining the axis of rotational symmetry of the particle and applied electric field E∞.

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electroosmotic flow in the opposite direction at the center of the cavity. For specified values of kiand li, the magnitude of Ui/U0i

in general decreases with an increase inλ (with exceptions when ki equals about -1), whereas the magnitude of Ωi/Ω0i - 1

increases with an increase inλ. For given finite values of kiand

λ, the magnitude of Ui/U0ipredicted from using the Neumann

boundary condition in eq 6 in general is greater than that predicted from using the Dirichlet boundary condition in eq 4.

As an example, we consider the electrophoresis of a colloidal sphere with an odd ζ-potential distribution formed by one

hemisphere of the constant value ζ₁p attached to a second hemisphere of ζ₂p in a concentric spherical cavity similarly composed of two hemispheres of different constantζ potentials ζ₁wandζ₂w, as shown in Figure 5. Unit vectors epand ewdefine

the axes of rotational symmetry of the particle and cavity, respectively, and both are allowed to orient arbitrarily with respect to the applied electric field E∞. The z-x plane is chosen to contain both epand E∞. After calculating the multipole moments according

to eqs 18-20 and substituting their result into eqs 23 and 24, we obtain

Because the quadrupole moments disappear and the higher-order moments do not contribute in this case, eq 35 shows that the electrophoretic velocity of the particle is always collinear with the applied electric field, irrespective of the orientations of ep

and ew. The cavity wall can induce an angular velocity of the

particle with both x and y components, although the particle rotates only about the y axis in the absence of the cavity.

For another example in contrast to the previous one, we consider the electrophoresis of a colloidal sphere with an evenζ-potential

distribution formed by two caps of the constant value ζ₁p

connected by a middle section ofζ₂pin a concentric spherical cavity with a similar evenζ-potential distribution formed by two

caps ofζ₁wand a middle ofζ₂w, as shown in Figure 6. Again, the orientations of the axes of rotational symmetry of the particle and cavity, given by unit vectors epand ew, respectively, can be

arbitrary relative to the imposed electric field E∞, and the z-x plane is chosen to contain both epand E∞. Using eqs 18-20 for

this case, we find that

and the dipole moments of theζ potentials at the particle and

cavity surfaces vanish. In this case, the electroosmotic flow that arises from the interaction between the applied electric field and the thin double layer adjacent to the cavity wall may contribute to the translational velocity of the particle in all three rectangular components, whereas the contribution from the electrophoretic driving force has only z and x components. The particle is irrotational without the contribution from the dipole moments. When the values of angles θp,w ) 0 or π/2, the ζ-potential

distributions are uniform, and the result of eqs 23 and 24 with eqs 37 and 38 reduces to that for a uniformly charged spherical

particle undergoing electrophoretic motion in a uniformly charged concentric spherical cavity.

4. Concluding Remarks

The quasi-steady electrophoretic motion of a dielectric sphere in a concentric spherical cavity with arbitrary ζ-potential

distributions at the particle and cavity surfaces has been theoretically investigated in this study. The applied electric field is constant, and the thickness of the electric double layers adjacent to the solid surfaces is assumed to be much smaller than the particle radius and the gap width between the surfaces. Both the Dirichlet-type and the Neumann-type boundary conditions are considered for the electric potential at the cavity wall. The Laplace and Stokes equations are solved analytically for the electric potential and velocity fields, respectively, in the fluid phase, and the translational and angular velocities of the electrophoretic particle are obtained in explicit expressions, eqs 23 and 24, with the relevant parameters (functions of the ratio of the particle-to-cavity radii only) given by eqs 25-28 and Figure 2a,b. Before using these equations, one has to evaluate only the monopole, dipole, and quadrupole moments of theζ-potential distributions

at the particle and cavity surfaces defined by eqs 18-20. The contributions from the electroosmotic flow arising from the interaction of the applied electric field with the thin double layer adjacent to the cavity wall and from the wall-corrected elec-trophoretic driving force to the particle velocities can be superimposed because of the linearity of the problem. Two examples of the particle-cavity system with odd and even

ζ-potential distributions, respectively, are given so that we may

discuss in detail the boundary effects of the cavity on the electrophoretic velocities of the particle.

We note that the two types of boundary conditions for the electric potential at the cavity wall lead to somewhat different results for the translational and rotational velocities of the electrophoretic particle. These two boundary conditions have also been used in the literature to study the electrophoresis of a suspension of colloidal spheres with thin electric double layers using the unit cell model.48-51_{The results of these studies indicate}

that the tendency of the dependence of the mean electrophoretic mobility on the volume fraction of the particles predicted by the Neumann type is not as correct as that predicted by the Dirichlet type in comparison with the ensemble-averaged results obtained by using the concept of statistical mechanics. Therefore, the boundary condition represented by eq 6 might not be as accurate as that represented by eq 4, probably because of the fact that the angular component of the electric potential gradient at the cavity wall is not specified in eq 6.

Appendix A

Relations among the Solid Spherical Harmonics, Surface Spherical Harmonics, and Multipole Moments of Zeta Potential Distributions Obtained from Equations 12-17. The relations between the nonzero solid spherical harmonics (pn,Φn, χn) and the surface spherical harmonics (X_np,w, Y_np,w, Z_np,w) obtained from eqs 14-16 are

(48) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (49) Zharkikh, N. I.; Shilov, V. N. Colloid J. USSR (English Translation) 1982, 43, 865.

(50) Kozak, M. W.; Davis, E. J. J. Colloid Interface Sci. 1989, 127, 497. (51) Wei, Y. K.; Keh, H. J. Langmuir 2001, 17, 1437.

U ) 2η[Rp(ζ1 p_{+ ζ} 2 P ) + R_w(ζ1 w_{+ ζ} 2 w )]E_∞ (35) Ω ) 9 16ηa[γp(ζ1 p_{- ζ} 2 p )e_p- γ_w(ζ1 w_{- ζ} 2 w )e_w]× E_∞ (36) Mp,w) ζ₂p,wcosθ_p,w+ ζ₁p,w(1 - cosθ_p,w) (37) Qp,w) 1 2(ζ1 p,w_{- ζ} 2 p,w )(cosθp,w- cos 3 θp,w)(3ep,wep,w- I) (38)

(9)

for n g 1, and

where

The expressions for the nonzero surface spherical harmonics

X_np,w, Y_np,w, and Z_np,w in terms of the components of multipole moments Mp,w_{, D}p,w_{, and Q}p,w_{of the}_{ζ-potential distributions at}

the particle and cavity surfaces in rectangular coordinates (x, y,

z) obtained from eqs 12-17 are

whereβp₎3_/

ν,βw) 1, and ν is defined right after eq 28.

Acknowledgment. This research was partially supported by the National Science Council of the Republic of China.

-n - 1 {λ[2n + (n - 1)(2n + 1)λ2n - 1- (n + 1)(2n - 1)λ2n+1)]X_np + λn_{[-(n + 1)(2n - 1) +} (n - 1)(2n + 1)λ2+ 2nλ2n+1]X_nw + λ[2 - (2n + 1)λ2n - 1_{+ (2n - 1)λ}2n+1 )]Y_np+ λn[2n - 1 - (2n + 1)λ2+ 2λ2n+1]Y_nw} (A4) χ_n)- λ 2n+1 Z_np+ λnZ_nw 1 -λ2n+1 (A5) χ-n - 1) Z_np- λnZ_nw 1 -λ2n+1 (A6) p_-1) - η r(2X0 w_{+ Y} 0 w ) (A7) Φ-1) a 2 2rλ2Y0 w (A8) Γ_n(λ) ) 4λ(1 + λ4n+2) -λ2n[(2n + 1)2(1 +λ4) -2(2n - 1)(2n + 3)λ2] (A9) Y₁p,w) βp,w[(-2Mp,w+ Q_zzp,w)cosθ +

(Q_yzp,wsin φ + Q_xzp,wcos φ)sinθ] (A10) Y₂p,w) -3βp,w[D_zp,w(3 cos2θ - 1) +

3(D_yp,wsin φ + D_xp,wcos φ)cosθ sin θ] (A11)

Y₃p,w) -βp,w[3Q_zzp,w(5 cos3θ - 3 cos θ) +

4(Q_yzp,wsin φ + Q_xzp,wcos φ)(5 cos2θ - 1)sin θ + 5(2Q_xyp,w

sin 2φ

-(Q_yyp,w- Q_xxp,w) cos 2φ)cosθ sin2θ] (A12)

Z₁p,w) 3βp,w(D_xp,wsin φ - D_yp,wcos φ)sinθ (A13)

Z₂p,w) 5βp,w{(Q_xzp,wsin φ - Q_yzp,wcos φ)cosθ sin θ -

[

1 2(Qyy p,w_{- Q} xx p,w )sin 2φ +

Q_xyp,wcos 2φ)

]

sin2θ} (A14)

Electrophoresis of a Colloidal Sphere in a Spherical Cavity with Arbitrary Zeta Potential Distributions

Research Article

Electrophoresis of a Colloidal Sphere in a Spherical

Cavity with Arbitrary Zeta Potential Distributions

Huan J. Keh, and Tzu H. Hsieh

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Electrophoresis of a Colloidal Sphere in a Spherical Cavity with

Arbitrary Zeta Potential Distributions

Huan J. Keh* and Tzu H. Hsieh

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