Concentrated Dispersions at Arbitrary Potentials - American Chemical

J. Phys. Chem. B 2002, 106, 4789-4798

4789

Dynamic Electrophoretic Mobility in Electroacoustic Phenomenon: Concentrated Dispersions at Arbitrary Potentials Jyh-Ping Hsu,* Eric Lee, and Fong-Yuh Yen Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617, R.O.C ReceiVed: September 7, 2001; In Final Form: February 13, 2002

Recent advancement in both fundamental theory and experimental technique makes electroacoustics devices powerful analytical tools in quantifying a dispersed system. Because properties of a concentrated sample can be measured directly without dilution, it is readily applicable to various applications of practical significance. The present work analyzes the dynamic electrophoresis of a concentrated dispersion for the case the applied electric field is weak. Available results in the literature are extended to a general condition of arbitrary electrical potential, double layer thickness, volume fraction of dispersed phase, and frequency of applied electric field, taking the effects of double layer polarization and interaction between neighboring double layers into account. The dynamic nonlinear problem leads to several unique features that are not observed in the corresponding linear (low surface potential) or static problems. For example, because of the effect of double layer polarization, the magnitude of the dynamic mobility may have a local maximum and the phase angle may have a negative (phase lead) local minimum as the frequency of the applied electric field varies.

1. Introduction Electroacoustics phenomena include two basic elements, namely, colloid vibration potential (CVP) and electrokinetic sonic amplitude (ESA). Recent developments in both fundamental theory and experimental techniques make electroacoustics device a powerful analytical tool in quantifying a dispersed system.1,2 In particular, because properties of a concentrated sample can be measured directly without dilution,3-7 a highly desirable nature in practice, it is readily applicable to various applications of practical significance. O’Brien8 showed that ESA (CVP) is proportional to the product of the volume fraction of the dispersed entity, the electrophoretic mobility, and the average electric field (supersonic field). Note that the average field is of dynamic nature, and both of its direction and strength are time-dependent. In this case, the inertia of the dispersed entity and ionic species lead to a nonsynchronized movement of a particle relative to the average field. The result of O’Brien8 is applicable to dilute spherical dispersion with a thin double layer. Results derived under more general conditions were reported by several workers.9-14 The results of Mangelsdorf and White,13 for example, take the effect of double layer distortion into account and are valid for levels of electric potential of particle and frequency of average electric field higher than that used by O’Brien. Because of the limitation of the numerical procedure adopted, however, their results diverge at high frequency. The approximate dynamic electrophoretic velocity derived by Ohshima15 is valid for arbitrary double layer thickness but is limited to low surface potential. These studies are limited to a dilute dispersion in which the volume fraction of the dispersed entity is below 5%. The problem of a concentrated dispersion at low electrical potential was discussed by Ohshima,17-19 where the cell model of Levine and Neale20 was adopted. This cell model, however, is found to be unsatisfactory in predicting experimental observations. Dukhin et al.21 suggested that this is not due to * To whom correspondence should be addressed. Fax: 886-2-23623040. E-mail: [email protected].

the structure of the cell model itself but to the condition specified at the outer boundary of a cell, where the micro electric field and the macro electric field are incorporated. They also showed that the boundary condition used by Shilov and Zharkikh22 is more appropriate than that used by Levine and Neale.20 This is also justified by Lee et al.23 in a study of the dynamic electrophoretic behavior of a concentrated dispersion at low surface potential. Here, the dynamic electrophoresis of a concentrated dispersion at an arbitrary electrical potential, double layer thickness, volume fraction of dispersed phase, and frequency of the average electric field is investigated for the case when the applied electric field is weak. Because the effects of double layer polarization and interaction between neighboring double layers are also taken into account, the present work is a generalization of all of the available results in the literature. 2. Theory Referring to Figure 1, we consider the electrophoretic behavior of a concentrated, positively charged, nonconductive, monodispersed spherical dispersion. An electric field is applied which yields an averaged electric field EZe-iωtez, where EZ is the magnitude of the averaged electric field, t is time, ez is the unit vector in the z direction, ω is the frequency of the averaged electric field, and i ) (-1)1/2. The system under consideration is simulated by the cell model of Kuwabara28 where a cell comprises a representative particle of radius a and a concentric liquid shell of radius b. The volume fraction of the particle is estimated by H ) (a/b)3. The spherical coordinates (r,θ,φ) are adopted with its origin located at the center of the representative particle. The liquid phase contains Z1:Z2 electrolyte, where Z1 and Z2 are respectively the valences of cations and anions with Z2 ) -RZ1. Let UEZe-iωtez be the velocity of particle, where U is the magnitude of the velocity per unit strength of the averaged electric field strength or the dynamic electrophoretic mobility. Because the velocity may not be synchronous with the averaged

10.1021/jp0134345 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/17/2002

4790 J. Phys. Chem. B, Vol. 106, No. 18, 2002

Hsu et al.

∇‚u)0 Ff

(4)

∂u ) -∇p + η∇2u - Fc∇φ ∂t

(5)

where Ff and η are respectively the density and the viscosity of liquid and p is the pressure. Here, we assume that the liquid phase is incompressible and has constant physical properties. Suppose that the applied electric field is weak. In this case, each dependent variable can be approximated by the sum of its equilibrium value (i.e., the value in the absence of the applied electric field) and a perturbed term, which is linearly dependent on the averaged, applied electric field. We have

Figure 1. Schematic representation of the problem considered. Kuwabara’s cell model28 is used to simulate the present system where a cell comprises a representative particle of radius a and a concentric liquid shell of radius b. EZe-iωtez is the averaged dynamic electric field and UEZe-iωtez is the electrophoretic velocity. The spherical coordinates (r,θ,φ) are adopted with its origin located at the center of the representative particle.

electric field, we let U ) UR + iUI, where UR and UI are respectively the real and the imaginary parts of U. The behavior of the present problem is described by the electroacoustic equations, which include the governing equations for electric field, concentration field, and flow field. We assume that the electrical potential can be described by the Poisson equation

∇ 2φ ) -

Fc

2

)-

∑ j)1

zjeˆ nj (1)

∂nj ) - ∇‚fj ∂t

(

(6)

nj(r,θ,t) ) nje(r,θ) + δnj(r,θ)EZe-iωt

(7)

u(r,θ,t) ) 0 + δu(r,θ)EZe-iωt

(8)

p(r,θ,t) ) pe(r,θ) + δp(r,θ)EZe-iωt

(9)

At equilibrium, the liquid phase is static, and only electric and concentration field needed to be solved. Equations 1-3 yield

∑

)

φe ) ζ, r)a

(11)

dφe ) 0, r)b dr

(12)

Taking Laplacian on both sides of eq 6 gives

∇2(δφ(r,θ)EZe-iωt) ) ∇2φ(r,θ,t) - ∇2φe(r)

(13)

We assume that the governing equation for φ can be expressed as 2

∇ φ)2

∑ j)1

(

e zjeˆ nj0

exp -

(

(3)

)

zjeˆ (φ + gjEZe-iωt) kBT

where we apply the relation

In these expressions, ∇ is the gradient operator, fj and Dj are respectively the concentration flux and the diffusivity of ionic species j, u is the liquid velocity, T is absolute temperature, and kB is Boltzmann constant. Dj can be expressed as the ratio (kBT/λj), λj ) NAeˆ 2|Zj|/Λ∞j , where λj is the drag coefficient of ionic species j, NA is Avogadro number, and Λ∞j is the limiting conductance of ionic species j. We have Dj ) (Λ∞j kBT/NAeˆ 2|Zj|). Suppose that the flow field can be described by the NavierStokes equation in the creeping flow regime

(10)

e where nj0 is the bulk concentration of ionic species j. We assume that the surface potential of particle is constant and there is no net current passes through the surface of a cell. Therefore, the boundary conditions associated with eq 10 are

(2)

njeˆ zj ∇φ + nju fj ) -Dj ∇nj + kBT

( )

e zjeˆ nj0 zjeˆ φe ∇ φe(r) ) exp kBT j)1 2

2

where ∇2 is the Laplace operator, φ is the electrical potential, Fc is the space charge density, is the permittivity of liquid phase, eˆ is the elementary charge, and nj is the number density of ionic species j. The concentration field can be described by

φ(r,θ,t) ) φe(r) + δφ(r,θ)EZe-iωt

)


e exp nj ) nj0

(14)

(15)

Substituting eqs 10 and 14 into eq 13 and applying eq 6 gives -iωt

∇ (δφEZe 2

2

))-

∑ j)1

( (

e zjeˆ nj0

exp -

zjeˆ

(φe + δφEZe-iωt +

kBT

) ( ))

gjEZe-iωt) - exp Combining eqs 2 and 3 yields

zjeˆ

φe

kBT

(16)

Concentrated Dispersions at Arbitrary Potentials

(

J. Phys. Chem. B, Vol. 106, No. 18, 2002 4791

)

zjeˆ ∂nj ) - Dj ∇2nj + (∇nj‚∇φ + nj∇2φ) + u‚∇nj ∂t kBT

(17)

where eq 4 is applied. Substituting eqs 6-8 into this expression, we have, after dividing by [EZ]e-iωt

{

smaller than unity, where kδ and K∞ are respectively the surface conductivity and the macroscopic conductivity of ions. Also, because the dielectric constant of particle is usually much smaller than that of the liquid phase, we assume that particle surface is impenetrable to ions, that this

∇(δφEZe-iωt)‚r ) 0, r ) a

zjeˆ [∇nje‚∇(δnj) + - iωδnj ) - Dj ∇ (δnj) + kBT

(24)

2

∇(δnj)‚∇φe + ∇(δnj)‚∇(δφ EZe

-iωt

}

On the cell surface, the boundary condition of Shilov and Zharkikh22 is assumed, which correlates macroscopic electric field and microscopic electric field. We have

) + nj ∇ (δφ) + e

2

(δnj)∇2φe + (δnj)∇2(δφEZe-iωt)] + (δu)‚∇(nje + -iωt

δnjEZe

δφEZe-iωt ) EZe-iωtb cos θ, r)b

) (18)

The equilibrium potential can be determined by eq 10. The governing equation for the perturbed ionic concentration can be derived from eqs 7, 10, and 15 as

{ [

e exp δnjEZe-iωt ) nj0

]


(

exp -

)}

zjeˆ φe k BT

(20)

-iωFfδuEZe-iωt ) -∇δpEZe-iωt + η∇2δuEZe-iωt - Fc∇φ (21) The perturbed velocity can be expressed as δuEZe-iωt ) (δurr + δuθθ)EZe-iωt, where r and θ are respectively the unit vector in r and θ directions, δur and δuθ are the r and the θ components of the perturbed velocity. In terms of stream function, they can be expressed as δur ) - (1/r2 sin θ) (∂ψ/∂θ) and δuθ ) (1/r sin θ) (∂ψ/∂r). Note that eq 20 is satisfied automatically. If curl is taken on both sides of eq 21 and the stream function is introduced, the pressure term in eq 21 can be removed. We obtain

E4ψ + iωFfE2ψ )

(

)

(22)

)

(23)

1 ∂Fc ∂φ ∂Fc ∂φ sin θ η ∂r ∂θ ∂θ ∂r

where E4 ) E2E2 with

E2 )

( )

(

∂2 1 ∂ sin θ ∂ + 2 ∂r2 r ∂θ sin θ ∂θ

(fj - nju)‚rˆ ) 0, r ) a

Suppose that the conductivity of the particle surface is negligible; that is, the Dukhin number (Du ) kδ/K∞a) is much

(26)

Suppose that ion concentration reaches its equilibrium value at virtual surface. That is

(

For convenience, δnj will not be solved directly. Instead, gj is used to quantify the perturbed concentration. Substituting eq 19 into eq 18 yields the governing equation for gj. In the resolution of gj, the flow field needs to be determined simultaneously. Because the flow field is induced by the applied electric field, ∇pe ) 0 and ue) 0, with pe and ue being respectively the equilibrium pressure and flow velocity. Therefore, only the perturbed term of the flow field needs to be solved. On the basis of eqs 4, 5, 8, and 9, we have

∇‚(δuEZe-iωt) ) 0

The surface of particle is impenetrable to ions, therefore

nj ) nje exp (19)

(25)

)

zjeˆ φ , r)b kBT e

(27)

On the basis of eqs 3 and 15, eqs 26 and 27 can be rewritten as

∂gj E e-iωt ) 0, r ) a ∂r Z

(28)

gjEZe-iωt ) -δφEZe-iωt, r ) b

(29)

We assume that the velocity of the liquid at the particle surface is the same as that of the particle. This implies

δurEZe-iωt ) (UR + iUI)EZe-iωt cos θ, r ) a

(30)

δuθEZe-iωt ) -(UR + iUI)EZe-iωt sin θ, r ) a (31) We assume that the vorticity vanishes at the outer boundary of a cell (or virtual surface), and there is not net flow in the radial direction; that is

∇δuEZe-iωt ) 0, r ) b

(32)

δurEZe-iωt ) 0, r ) b

(33)

For a simpler mathematical treatment, the subsequent analyses are based on scaled variables. The following scaling variables are chosen for length, electrical potential, concentration, and velocity, respectively: radius a, surface (zeta) potential of particle ζ, bulk concentration of cations ne10, the velocity based on Smoluchoski’s result when a static electric field of strength (ζ/a) is applied UE ) ζ2/ηa. Symbols with an asterisk represent scaled variables, for example, r* ) r/a, φ/e ) φe/ζ, n/j ) nj/ne10, E /Z ) EZ/(ζ/a), U*E /Ze-iωt ) UEZe-iωt/UE, δφ*E /Ze-iωt ) δφEZe-iωt/ζ, g/j E /Ze-iωt ) gjEZe-iωt/ζ, and ψ*E /Ze-iωt ) ψEZe-iωt/UEa. Because the applied electric field is relatively weaker than that induced by a particle, |[EZ]e-iωt| , ζ/a, and, therefore, |δφEZe-iωt| , φe and |gjEZe-iωt| , φe. In this case, eq 15 leads to the following approximate expressions:

n/1 ) exp(- φrφ/e )[1 - φr(δφ* + g/1)E /Ze-iωt]

(34)

n/2 ) R exp(Rφrφ/e )[1 + Rφr(δφ* + g/2)E /Ze-iωt] (35)


Hsu et al.

[

/ Ffωa2 2 (κa)2 ∂g1 E*4ψ* + i E* ψ* ) exp(- φrφ/e ) + η 1 + R ∂θ

]

∂φ/e ∂g/2 / R exp(Rφrφe ) sin θ (39) ∂θ ∂r* 2 e where κ ) (kBT/∑j)1 nj0 (ezj)2)-1/2 is the reciprocal Debye length, φr ) z1eζ/kBT is the scaled surface potential of particle, Pej ) UEa2/Dj, j ) 1 and 2, is the electric Peclet number of ionic species j, and (Ffωa2/η) and ωa2/Dj, j)1 and 2, are scaled frequencies. The governing equations can be made one-dimensional by employing the method of separation of variables. The problem can further be simplified by partitioning a variable into real and imaginary parts. Separation of variables suggests that δφ* ) (ΦR + iΦI) cos θ, g/1 ) (G1R + iG1I) cos θ, g/2 ) (G2R + iG2I) cos θ, ψ* ) (ΨR + iΨI) sin2 θ, where a symbol with subscript R represents the real part, and that with subscript I, the imaginary part of a variable. On the basis of these expressions, it can be shown that

L12ΦR -

(κa)2 [exp(- φrφ/e ) + R exp(Rφrφ/e )]ΦR ) 1+R (κa)2 [exp(- φrφ/e )G1R + R exp(Rφrφ/e )G2R] (40) 1+R

L12ΦI -

(κa)2 [exp(-φrφ/e ) + R exp(Rφrφ/e )]ΦI ) 1+R (κa)2 [exp(-φrφ/e )G1I + R exp(Rφrφ/e )G2I] (41) 1+R

Figure 2. Variation of magnitude (a) and phase angle (b) of scaled dynamic mobility as a function of scaled frequency (Ffωa2/η) at various φr with κa ) 0.1 and H ) 0.1. b, result for φr f 0. Key: ) 6.9545 × 10-10 Fara/m, η ) 0.8904 cp, Ff ) 0.99704 g/cm3, Fp ) 1.05 g/cm3, ∞ 2 ΛK∞+ ) 73.5 cm2/Ω (g equiv), ΛCl - ) 76.3 cm /Ω (g equiv), Z1 ) 1, Z2 ) -1, T ) 298.15 K, (Fp - Ff)/Ff ) 0.053, Pe1 ) Pe2 ) 0.01, (ωa2/ D1) ) 454.88(Ffωa2/η), (ωa2/D2) ) 472.21(Ffωa2/η).

On the basis of these expressions, it can be shown that the governing equations for δφ*, g/1, g/2, and ψ* are respectively

(κa)2 ∇* δφ* [exp(-φrφ/e ) + R exp(Rφrφ/e )]δφ* ) 1+R (κa)2 [exp(- φrφ/e )g/1 + R exp(Rφrφ/e )g/2] (36) 1+R

L12G1R - φr

dφ/e dG1R dφ/e 1 + Pe1 2ΨR dr* dr* dr* r* ωa2 (Φ + G1I) ) 0 (42) D1 I

L12G1I - φr

dφ/e dG1I dφ/e 1 + Pe1 2ΨI + dr* dr* dr* r* ωa2 (Φ + G1R) ) 0 (43) D1 R

2

∇*2g/1 - φr∇*φ/e ‚∇*g/1 + Pe1δu*‚∇*φ/e +

L12G2R + Rφr

ωa2 (Φ + G2I) ) 0 (44) D2 I

ωa2 (δφ* + g/1) ) 0 (37) i D1 ∇*2g/2 + Rφr∇*φ/e ‚∇*g/2 + Pe2δu*‚∇*φ/e + ωa2 (δφ* + g/2) ) 0 (38) D2

i and

dφ/e dG2R dφ/e 1 + Pe2 2ΨR dr* dr* dr* r*

dφ/e dG2I dφ/e 1 L1 G2I + Rφr + Pe2 2ΨI + dr* dr* dr* r* 2

ωa2 (Φ + G2R) ) 0 (45) D2 R


L24ΨR -


Ffωa2 2 (κa)2 (G exp(-φrφ/e ) + L2 Ψ I ) η (1 + R) 1R RG2R exp(Rφrφ/e ))

dφ/e (46) dr*

Ffωa 2 (κa)2 (G exp(-φrφ/e ) + L 2 ΨI + L2 ΨR ) η (1 + R) 1I 2

4

RG2I exp(Rφrφ/e ))

dφ/e dr*

where L24 ) L22L22, and the linear operators L12 and L22 are defined by

(48)

d2 2 - 2 L2 ≡ 2 dr* r*

(49)

2

1 ΨR ) U /Rr*2, r* ) 1 2

(62)

dΨR ) U /Rr*, r* ) 1 dr*

(63)

L22ΨR ) 0, r* ) b/a

(64)

r* ) b/a

(65)

1 ΨI ) U /I r*2, r* ) 1 2

(66)

dΨI ) U /I r*, r* ) 1 dr*

(67)

L22ΨI ) 0, r* ) b/a

(68)

ΨI ) 0, r* ) b/a

(69)

The scaled governing equation for the equilibrium electrical potential and the associated boundary conditions become

The corresponding boundary conditions are

dΦR ) 0, r* ) 1 dr*

(50)

b ΦR ) -[E /Z] , r* ) b/a a

(51)

dΦI ) 0, r* ) 1 dr*

(61)

ΨR ) 0,

(47)

d2 2 d 2 + 2 L12 ≡ - 2 2 dr* dr* r* r*

G2I ) -ΦI, r* ) b/a

(52)

b ΦI ) -[E /Z] , r* ) b/a a

(53)

dG1R ) 0, r* ) 1 dr*

(54)

(

)

dφ/e (κa)2 1 d 2 r* ) (exp(-φrφ/e ) dr* (1 + R)φr r*2 dr* exp(Rφrφ/e )) (70) φ/e ) 1, r* ) 1

(71)

dφ/e ) 0, r* ) b/a dr*

(72)

The dynamic electrophoretic mobility µ is defined by µ ) µR + iµI ) U/E, where E ) EZe-iωtez and U ) (UR + iUI)EZe-iωtez, with µR and µI being respectively the real and the imaginary parts of µ. The forces exerted on a particle include the hydrodynamic drag force contributed by the flow field Fh and the electric force contributed by the averaged electric field Fe. We have

4 dU Fh + Fe ) πa3(Fp - Ff) 3 dt

G1R ) -ΦR, r* ) b/a

(55)

dG1I ) 0, r* ) 1 dr*

(56)

G1I ) -ΦI, r* ) b/a

(57)

dG2R ) 0, r* ) 1 dr*

(58)

G2R ) -ΦR, r* ) b/a

(59)

dG2I ) 0, r* ) 1 dr*

(60)

(73)

Because U ) (UR + iUI)[EZ]e-iωtez, this expression can be rewritten in scaled form as

( )( )

Ffωa2 Fp - Ff 4 F /h + F /e ) - i πζ2 (U /R + iU /I )E /Ze-i$t 3 η Ff (74) where F /h and F /e are respectively the scaled magnitudes of Fh and Fe. The electrical force per unit surface experienced by a particle is σ(-∇φ)s, with σ being the surface charge density. The applied electric field is in the (Z direction and so is the electric force experienced by the particle, which can be evaluated by

Fe )

∫σ(-∇φ)s‚ez dS

(75)

where σ can be determined by Gauss’s law as

∫ ∫(E‚n) dS ) ∫ ∫σ dS

(76)


Hsu et al.

Figure 3. Variation of magnitude (a) and phase angle (b) of scaled dynamic mobility as a function of scaled frequency (Ffωa2/η) at various φr with κa ) 1.0 and H ) 0.1. b, result for φr f 0. Key: same as Figure 2.

Figure 4. Variation of magnitude (a) and phase angle (b) of scaled dynamic mobility as a function of scaled frequency (Ffωa2/η) at various φr with κa ) 5.0 and H ) 0.1. b, result for φr f 0. Key: same as Figure 2.

where n is the unit outer normal and E ) -∇φ. Substituting this expression and dS ) 2πr2 sinθ dθ into eq 75 and representing the resultant expression in scaled form, we obtain

It can be shown that this expression reduces to

F /e )

( ) (

/ π dφe

∫0

2πζ2

dr*

-

r*)1

)

1 ∂δφ* / -iωt E Ze sin θ r*2 r* ∂θ r*)1 sin θ dθ (77)

According to eq 6, φe(r) is independent of θ, ∂φe/∂θ)0. Also, because φe is the equilibrium potential, (∂φe/∂r)r)a is constant. Equation 24 implies that (∂δφ/∂r)r)a)0. Applying these relations with ∫ πo (∂φe/∂r) cos θ dθ ) (∂φe/∂r)∫ πo cosθ dθ ) 0, eq 76 becomes

F /e )

∫0π

2πζ2

( ) ( dφ/e dr*

-

r*)1

)

1 ∂δφ* / -iωt sin θ r*2 E Ze r* ∂θ r*)1 sin θ dθ (78)

[

dφ/e 8 (Φ + iΦI)E /Ze-iωt F /e ) πζ2 r* 3 dr* R

]

r*)1

(79)

Following the derivation of Happel,29 the force experienced by the sphere in the Z direction is

Fh ) ηπ

∫0

π

[

(

)] [

2 -iωt ∂ E ψEZe r sin ∂r r2 sin2 θ 4

3

s

dθ - π

∫0π

r2 sin2 θ Fc

]

∂φ dθ (80) ∂θ s

The first and the second terms on the right-hand side of this expression represent respectively the drag force and the electric force exerted on a particle. Because ∂φ/e /∂θ ) 0, it can be


Figure 5. Variation of magnitude (a) and phase angle (b) of scaled dynamic mobility as a function of scaled frequency (Ffωa2/η) at various κa with φr ) 1.0 and H ) 0.1. Key: same as Figure 2.


Figure 6. Variation of magnitude (a) and phase angle (b) of scaled dynamic mobility as a function of scaled frequency (Ffωa2/η) at various κa with φr ) 3.0 and H ) 0.1. Key: same as Figure 2.

a particle in these two problems, then shown that eq 80 reduces to

[ (

) ]

∂ L2 (ΨR + iΨI) / -iωt 4 F /h ) πζ2 r*4 E Ze + 3 ∂r* r*2 r*)1 4 πζ2[r*2(exp(-φrφ/e ) 3 exp(Rφrφ/e ))(ΦR + iΦI)E /Ze-iωt]r*)1 (81) 2

The linear nature of the governing equations for the perturbed variables suggests that the problem under consideration can be decomposed into two virtual problems. This has the advantage that the tedious iterative procedure can be avoided in the evaluation of the electrophoretic velocity of a particle. In the first problem, a particle moves with speed (U /R + iU /I )E /Ze-iωt in the absence of the averaged applied electric field, and in the second problem, an averaged electric field of strength E /Ze-iωt is applied, but the particle is held fixed. If we let F /1 and F /2 be the scaled sum of electric force and drag force experienced by

F /1 ) χ(U /R + iU /I )E /Ze-iωt

(82)

F /2 ) βE /Ze-iωt

(83)

where χ and β are complex constant. Equation 74 can be rewritten as

F /h + F /e ) F /1 + F /2 )

( )( )

Ffωa2 Fp - Ff (U /R + iU /I )E /Ze-i$t (84) -i η Ff On the basis of eqs 83-85 and the definition of dynamic mobility, we have

µ* )

-β χ + i(Ffωa /η)[(Fp - Ff)/Ff] 2

(85)

where µ* ) µ/(ζ/η). µ* can also be expressed as µ* ) [µ/2 R + 1/2eiωγ, where |µ*| ) [µ/2 + µ/2]1/2 is its magnitude and ωγ ] µ/2 I R I


Hsu et al.

Figure 7. Variation of magnitude (a) and phase angles (b) of scaled dynamic mobility as a function of scaled frequency (Ffωa2/η) at various κa with φr ) 5.0 and H ) 0.1. Key: same as Figure 2.

Figure 8. Variation of magnitude (a) and phase angle (b) of scaled dynamic mobility as a function of scaled frequency (Ffωa2/η) at various H with φr ) 1.0 and κa ) 1.0. Key: same as Figure 2.

) tan-1(µ/I + µ/R) is its phase angle. Because E ) EZe-iωtez, the corresponding electrophoretic velocity of a particle is U ) [UR2 + UI2]1/2EZe-iωt+ωγez. Therefore, ωγ < 0 implies a phase lead (electrophoretic velocity leads the averaged electric field), and ωγ > 0 implies a phase lag.

immediately to the variation in the direction of the averaged electric field. As (Ffωa2/η) becomes high, the acceleration period of a particle is shortened, which yields a smaller mobility. Note that, because the phase angle in Figure 2 is positive, the latter implies that the phase lag of the electrophoretic velocity becomes more significant at a high (Ffωa2/η). Figure 2 also indicates that for a fixed (Ffωa2/η) the magnitude of mobility increase with φr and the phase angle decreases with φr. In Figure 3, the thickness of the double layer is comparable to the radius of a particle, and its role becomes significant. As can be seen from Figure 3, the magnitude of mobility has a local maximum and the phase angle has a negative local minimum if φr is sufficiently high; the latter implies that the mobility leads the averaged electric field. Also, except for very low (Ffωa2/η), the higher the φr the greater the magnitude of the mobility and the smaller the phase angle, in general. These behaviors can be explained as follows. For the case when a static electric field is applied, double layer polarization will induce an internal electric field, which is in the inverse direction as that of the averaged electric field, and, therefore, has the effect of retarding the movement of a particle. If a dynamic electric field is applied, its direction varies with time and the electric field induced by double layer polarization cannot respond immediately. In this case, double

3. Results and Discussion The behavior of the problem under consideration is investigated through numerical simulation. Figure 2 shows the variation of the magnitude and the phase angle of the scaled dynamic mobility as a function of the scaled frequency (Ffωa2/η) at various scaled surfaces φr for the case when the double layer is thick (κa ) 0.1). Those for thinner double layers are presented in Figures 3 and 4. For comparison, the corresponding results of Lee et al.,23 in which φr f 0 are also illustrated in these figures. Figure 2 reveals that, if the double layer is thick, the magnitude of the dynamic mobility decreases with the increase in (Ffωa2/η), and the phase angle of the dynamic mobility increases with the increase in (Ffωa2/η); these behaviors are the same as those for the case when φr f 0. The former can be explained by that the effect of the inertia of a particle is important as (Ffωa2/η) becomes high, and it is unable to respond





layer polarization has the effect of accelerating the movement of a particle. This is why the magnitude of mobility increases with the increase in (Ffωa2/η), and the phase angle is negative (phase lead) and decreases with the increases in (Ffωa2/η) in Figure 3 when (Ffωa2/η) has a low to medium value. If (Ffωa2/ η) is high, the frequency of the change in the direction of particle movement is high, too, and the effect of double layer polarization becomes inappreciable, the magnitude of mobility starts to decrease with the increase in (Ffωa2/η), and the phase angle becomes positive and increases with the increase in (Ffωa2/η), as in the case of Figure 2. The double layer in Figure 4 is thinner than that of Figure 3, and the general behavior of the mobility is similar to that of Figure 3, except that for a low to medium (Ffωa2/η) the higher the φr the smaller the magnitude of mobility, and the reverse is true if (Ffωa2/η) is high. Note that the magnitude of mobility depends on three basic factors: the frequency of the averaged electric field, the surface potential of the particle, and the effect of double layer polarization. In a study of the electrophoresis of a concentrated spherical dispersion under a static averaged electric field, Lee et al.25 concluded that if the double layer is thick (κa e 0.1) the mobility increases with φr, and assuming φr f 0 will underestimate the mobility. On the other hand, as the thickness of the double layer becomes

comparable to particle radius (κa = 1), because of the effect of double layer polarization, the mobility decreases with φr, and assuming φr f 0 will overestimate the mobility. These behaviors can also be seen in Figures 2-4. However, because a dynamic electric field is applied in the present study, its frequency also comes into play, and the situation becomes more complicated. The variation of the magnitude and the phase angle of the scaled dynamic mobility as a function of scaled frequency (Ffωa2/η) at various κa for the case of a lower surface potential of particle is illustrated in Figure 5. Those for higher surface potentials are presented in Figures 6 and 7. Figure 5 shows that for a fixed (Ffωa2/η) the magnitude of the dynamic mobility increases with the increase in κa and the phase angle decreases with the increase in κa. The former is because if κa is small (thick double layer) the overlapping of neighboring double layers has the effect of limiting the movement of particles. The latter is because the thinner the double layer the easier it is for it to follow the variation in the averaged electric field. Figure 5 also suggests that for a fixed κa the magnitude of mobility decreases with the increase in (Ffωa2/η), and the phase angle increases with (Ffωa2/η). However, as shown in Figure 6, if the surface potential of particle is higher, the magnitude of mobility has a local maximum and the phase angle has a negative (phase lead)

4798 J. Phys. Chem. B, Vol. 106, No. 18, 2002 local minimum as (Ffωa2/η) varies. As can be seen in Figure 7, the variations of the magnitude and the phase angle of mobility are more complicated as the surface potential of particle becomes even higher. Again, these behaviors arise from the effect of double layer polarization. As illustrated in Figures 6 and 7, both the frequency at which the local maximum of the magnitude of mobility and the frequency at which the local minimum of phase angle occur increase with κa. This is because the thinner the double layer (larger κa) the smaller the range of its polarization, and it is easier for ions in the double layer to respond to the variation in the averaged electrical field. In this case, the effect of the frequency is appreciable only if it is sufficiently high. Figure 8 shows the variation of the magnitude and the phase angle of the scaled dynamic mobility as a function of scaled frequency (Ffωa2/η) at various volume fraction of particle H for the case of a lower surface potential of particle. Those for higher surface potentials are illustrated in Figures9 and 10. As can be seen from Figure 8, if φr is low, both the magnitude and the phase angle of the mobility decrease with the increase in H. This is due to the fact that the interaction between neighboring particles has the effect of retarding their movement. The phase angle of the mobility is positive (phase lag) in Figure 8. Figures 9 and 10 suggest that, if φr is high, the magnitude of mobility has a local maximum and the phase angle of mobility has a negative (phase lead) local minimum. As in the previous cases, these are due to the effect of double layer polarization. Note that if H is too large the phase lead of the electrophoretic velocity no longer exists. This is because the effect of double layer polarization is significant only if H is small, because as the volume fraction of the particle becomes large the extent of double layer polarization is limited and its effect is confined. According to Figures 9 and 10, the degree of phase lag at high frequency decreases with the increase in H. Again, this is due to the fact that the double layer polarization is confined by neighboring particles, and it is easier for the ionic cloud to respond to the variation in the average electric field. Figures 8-10 also show that both of the frequencies at which the magnitude of mobility starts to decrease and that at which the phase angle starts to increase increase with H. This is mainly due to the effect of the inertia of a particle, which tends to reduce

Hsu et al. the magnitude of mobility and increase the phase lag. As H increases, the inertia of a particle is confined. Acknowledgment. This work is supported by the National Science Council of the Republic of China. References and Notes (1) Hunter, R. J. Colloid Surf., A 1998, 141, 37. (2) O’Brien, R. W.; Cannon, D. W.; Rowlands, W. N. J. Colloid Interface Sci. 1995, 173, 406. (3) Gibb, S. E.; Hunter, R. J. J. Colloid Interface Sci. 2000, 224, 99. (4) Dukhin, A. S.; Goetz, P. J.; Wines, T. H.; Somasundaran, P. Colloids Surf. A 2000, 173, 127. (5) Hunter, R. J.; O’Brien, R. W. Colloids Surf. A 1997, 126, 123. (6) Guffond, M. C.; Hunter, R. J.; O’Brien, R. W.; Beattie, J. K. J. Colloid Interface Sci. 2001, 235, 371. (7) Mizuno, D.; Kimura, Y.; Hayakawa, R. Langmuir 2000, 16, 9547. (8) O’Brien, R. W. J. Fluid Mech. 1988, 190, 71. (9) Babchin, A. J.; Chow, R. S.; Sawatzky, R. P. AdV. Colloid Interface Sci. 1989, 30, 111. (10) Sawatzky, R. P.; Babchin, A. J. J. Fluid Mech. 1993, 246, 321. (11) Fixman, M. J. Chem. Phys. 1983, 78, 1483. (12) James, R. O.; Texter, J.; Scales, P. J. Langmuir 1991, 7, 1993. (13) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 1992, 88, 3567. (14) Mangelsdorf, C. S.; White, L. R. J. Colloid Interface Sci. 1993, 160, 275. (15) Ohshima, H. J. Colloid Interface Sci. 1996, 179, 431. (16) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1607. (17) Ohshima, H. J. Colloid Interface Sci. 1997, 195, 137. (18) Ohshima, H. Colloids Surf. A 1999, 149, 5. (19) Ohshima, H. Colloids Surf. A 1999, 159, 293. (20) Levine, S.; Neale, G. H. J. Colloid Interface Sci. 1974, 47, 520. (21) Dukhin, A. S.; Shilov, V.; Borkovskaya, Y. Langmuir 1999, 15, 3452. (22) Shilov, V. N.; Zharkikh, N. I.; Borkovskaya, Y. B. Colloid J. 1981, 43, 434. (23) Lee, E.; Yen, F. Y.; Hsu, J. P. J. Phys. Chem. B 2001, 105, 7239. (24) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 205, 65. (25) Lee, E.; Chu, J. W.; Hsu, J. P. J. Colloid Interface Sci. 1998, 209, 240. (26) Lee, E.; Yen, F. Y.; Hsu, J. P. Electrophoresis 2000, 21, 475. (27) Hsu, J. P.; Lee, E.; Yen, F. Y. J. Chem. Phys. 2000, 112, 6404. (28) Kuwabara, S. J. Phys. Soc. Jpn. 1959, 14, 527. (29) Happel, J. Low Reynolds Number Hydrodynamics; Noordhoff: Leyden, The Netherlands, 1973.

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