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195, 137–148 (1997)

CS975146

Dynamic Electrophoretic Mobility of Spherical Colloidal Particles in Concentrated Suspensions Hiroyuki Ohshima Faculty of Pharmaceutical Sciences and Institute of Colloid and Interface Science, Science University of Tokyo, Shinjuku-ku, Tokyo 162, Japan Received May 9, 1997; accepted August 19, 1997

A theory for the dynamic electrophoretic mobility m of spherical colloidal particles in concentrated suspensions in an oscillating electric field is proposed on the basis of Kuwabara’s cell model. The dynamic mobility depends on the frequency v of the applied electric field and the particle volume fraction f as well as on the reduced particle radius ka (where k is the Debye–Hu¨ckel parameter and a is the particle radius) and the zeta potential z . A mobility formula which involves numerical integration is obtained for particles with zero permittivity and low z . It is found that the mobility magnitude decreases with decreasing ka as in the static case ( v Å 0) and as in the single particle case ( f r 0) and that it decreases with increasing v as in the single particle case. However, the f dependence of the mobility magnitude is much more complicated. Namely, for small ka the mobility magnitude decreases with increasing f as in the static case. For large ka it increases with increasing f. For moderate ka and not very low v the mobility magnitude may exhibit a maximum. In all cases the v dependence of the mobility magnitude becomes less as f increases, that is, the dynamic mobility at any v approaches the static mobility as f increases. An accurate mobility formula without involving numerical integration applicable for all ka at zero particle permittivity and low z is also derived. This formula is applicable even for high z at ka r ` unless the dynamic relaxation effect becomes appreciable. q 1997 Academic Press Key Words: dynamic electrophoretic mobility; spherical particle; oscillating electric field; concentrated suspension.

1. INTRODUCTION

Theories of the dynamic electrophoretic mobility of spherical colloidal particles in an applied oscillating electric field have been developed by O’Brien (1), Babchin et al. (2, 3), Sawatzky and Babchin (4), and Fixman (5), who derived approximate mobility formulas. These approximations are compared by James et al. (6). Mangelsdorf and White (7, 8) obtained the full electrokinetic equations governing the dynamic electrophoresis of spherical colloidal particles as well as their numerical computer solutions and approximate analytic solutions applicable for low zeta potentials. Ohshima (9) derived a simpler approximate mobility formula

for the dynamic mobility of spherical particles which is applicable for all ka ( k is the Debye–Hu¨ckel parameter and a is the particle radius) at zero particle permittivity and low zeta potentials. The dynamic mobility has also been calculated for other types of particles than a spherical rigid particle. Lowenberg and O’Brien (10) derived a dynamic mobility formula of spheroidal particles with thin double layers. O’Brien (11) calculated the dynamic mobility of a porous particle. Ohshima (12) derived a general expression for the dynamic mobility of cylindrical colloidal particles as well as its approximate analytic mobility formula. The above theories, however, deal with single particles and thus they can be applied only to very dilute suspensions, i.e., suspensions having very small particle volume fractions. A few theoretical studies on concentrated suspensions have been made for the limiting case of the static electrophoresis problem (13–16). The theories of Levine and Neale (13) and Kozak and Davis (14–16) for electrokinetics of a swarm of identical particles take into account particle–particle interactions by means of Kuwabara’s cell model (17). This model assumes that each particle is surrounded by a virtual cell such that the particle/solution volume ratio in a unit cell is equal to the particle volume fraction throughout the entire system and the fluid vorticity is zero at the outer surface of the cell. Kozak and Davis (14) extended the theory of Levine and Neale (13) to the case of an array of circular cylinders. Kozak and Davis (15, 16) also developed a more general theory for electrokinetics of concentrated suspensions and porous media which is applicable to all zeta potential values but ignores double-layer overlap. Ohshima (18) derived a general mobility formula for a swarm of identical spherical particles in concentrated suspensions, which unites the mobility formulas of Levine and Neale (13) and of Kozak and Davis (15, 16), and obtained a simple analytical mobility formula. In this paper we combine the above two problems, i.e., dynamic electrophoresis of single particles and static electrophoresis in concentrated suspensions and try to derive a general theory of dynamic electrophoresis in concentrated suspensions on the basis of Kuwabara’s cell model (17). An approximate

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0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.

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HIROYUKI OHSHIMA

FIG. 1. Spherical particles of radius a in concentrated suspensions in the cell model (17). Each sphere is surrounded by a virtual shell of outer radius b. The particle volume fraction f is given by f Å (a/b) 3 . The nearest distance between the surfaces of two neighboring spheres is given by 2(b 0 a) Å 2a( f 01 / 3 0 1).

mobility formula is also derived on the basis of an approximation method devised in Refs. (9, 12, 18–20). 2. THEORY

Å 0) is set parallel to E. Let the electrolyte be composed of N ionic mobile species of valence zj and drag coefficient lj ( j Å 1, 2, . . . , N), and let n `j be the concentration (number density) of the jth ionic species in the electroneutral solution. The main assumptions in our analysis are as follows. (i) The applied field E is weak so that U is proportional to E and terms of higher order in E may be neglected. (ii) The slipping plane (at which the liquid velocity relative to the particle becomes zero) is located on the particle surface at r Å a (nonslip condition). Under this assumption, the zeta potential is equal to the surface potential of the particles and their surface charge is equal to the electrokinetic charge. An additional surface conductance due to the movement of ions behind the slipping plane is thus neglected in this paper. This effect is discussed in Refs. (21, 22) for the static field problem and in Refs. (23–25) for the dynamic field problem. (iii) No electrolyte ions can penetrate the particle surface. (iv) The relative permittivity ep of the particle is much smaller than that of the liquid er ( ep ! er ) so that ep is practically equal to zero. (v) In the absence of the applied electric field the particle surface is uniformly charged with a surface charge density s and has a corresponding zeta potential z . (vi) The fluid vorticity is zero at the outer surface of the unit cell (17). The flow velocity of the liquid u(r, t) at position r and time t and that of the jth mobile ionic species vj (r, t) may be written as

2.1. Basic Equations for Liquid and Ionic Flows

u(r, t) Å u(r)exp( 0ivt),

Consider a swarm of identical spherical colloidal particles of radius a in a liquid containing a general electrolyte. All the particles move with the same velocity U exp( 0ivt) in an applied oscillating electric field E exp( 0ivt), where v is the frequency and t is time. The dynamic electrophoretic mobility m of the particle is defined by

vj (r, t) Å vj (r)exp( 0ivt), ( j Å 1, 2, . . . , N). [2.4]

U Å mE,

The fundamental electrokinetic equations are (7–9) r0

Ì [u(r, t) / U exp( 0ivt)] Ìt Å 0 hÇ 1 Ç 1 u(r, t) 0 Ç p(r, t)

[2.1]

0 rel (r, t)Çc(r, t),

where U Å ÉUÉ and E Å ÉEÉ. As shown in Fig. 1, we employ a cell model (17) in which each sphere is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius of b such that the particle/solution volume ratio in this unit cell is equal to the particle volume fraction f throughout the entire suspension, viz., f Å (a/b) 3 .

[2.3]

Çr u(r, t) Å 0,

[2.5] [2.6]

1 Çmj (r, t), lj

[2.7]

Ìn j (r, t) / Çr(n j (r, t)vj (r, t)) Å 0, Ìt

[2.8]

vj (r, t) Å u(r, t) 0

[2.2] with

The mobility m depends not only on the zeta potential z of the particle and ka ( k Å Debye–Hu¨ckel parameter) but also on the frequency v and the particle volume fraction f. The origin of the spherical polar coordinate system (r, u, w ) is held fixed at the center of the particle, and the polar axis ( u

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N

rel (r, t) Å

∑ zj enj (r, t),

[2.9]

j Å1

mj (r, t) Å m`j / zje c(r, t) / kT ln n j (r, t),

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[2.10]

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DYNAMIC MOBILITY IN CONCENTRATED SUSPENSIONS

Dc(r, t) Å 0

rel (r, t) , r ú a, ere0

[2.11]

with

S

N

r (el0 ) (r) Å

where r0 is the mass density of the liquid, h is the viscosity, p(r, t) is the pressure, rel (r, t) is the charge density resulting from the mobile charged ionic species, c(r, t) is the electric potential outside the sphere, mj (r, t) and n j (r, t) are, respectively, the electrochemical potential and the concentration (the number density) of the jth ionic species, m`j is a constant term in mj (r, t), e is the elementary electric charge, k is Boltzmann’s constant, T is the absolute temperature, and e0 is the permittivity of a vacuum. Equations [2.5] and [2.6] are the Navier–Stokes equations and the equation of continuity for an incompressible flow, where the term r0 (urÇ)u has been omitted (assumption i). The term involving the particle velocity U exp( 0ivt) in Eq. [2.5] arises from the fact that the particle has been chosen as the frame of reference for the coordinate system. Equation [2.7] expresses that the flow vj (r, t) of the jth ionic species is caused by the liquid flow u(r, t) and the gradient of the electrochemical potential mj (r, t), given by Eq. [2.10]. Equation [2.8] is the continuity equation for the jth ionic species. Equation [2.11] is the Poisson equation.

∑ zj en `j exp 0 j Å1

zj e c ( 0 ) kT

D

.

[2.18]

The boundary conditions for c ( 0 ) (r) at the particle surface r Å a and at the outer surface of the cell r Å b are as follows: c ( 0 ) (a) Å z , (0)

dc dr

dc ( 0 ) dr

Z Z

Å0 r Åa

/

[2.19] s , ere0

[2.20]

Å 0.

[2.21]

r Åb 0

Equations [2.20] and [2.21] state that the unit cell as a whole is electrically neutral. Equations [2.16] and [2.17] (with [2.18]) can be rewritten as n (j 0 ) (r) Å n `j exp( 0zj y),

1 d r 2 dr

S D r2

dy dr

Å0

[2.22]

k 2 ( NjÅ1 n `j zj exp( 0zj y) . ( NjÅ1 n `j z 2j

[2.23]

2.2. Linearized Equations Here For a weak field E (assumption i), the electrical double layer around the particle is only slightly distorted. Then we may write (7–9) n j (r, t) Å n (j 0 ) (r) / dnj (r)exp( 0ivt), c(r, t) Å c

(0)

(r) / dc(r)exp( 0ivt),

mj (r, t) Å m(j 0 ) / dmj (r)exp( 0ivt), rel (r, t) Å r

(0) el

(r) / drel (r)exp( 0ivt),

[2.12]

yÅ

kÅ

[2.14] [2.15]

S D

1 d r 2 dr

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S

r2

/

dc ( 0 ) dr

zj e c ( 0 ) kT

Å0

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D

,

r (el0 ) (r) , ere0

[2.16] [2.17]

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S

N

1 ∑ z 2j e 2n `j ere0kT j Å1

D

1/2

[2.25]

is the Debye–Hu¨ckel parameter. The small quantities are related to each other by dmj (r) Å zjedc(r) / kT

dnj (r) , n (j 0 ) (r)

[2.26]

N

drel (r) Å

∑ zj ednj (r),

[2.27]

j Å1

Ddc(r) Å 0

n (j 0 ) Å n `j exp 0

[2.24]

is the scaled equilibrium potential outside the sphere, and

[2.13]

where the quantities with superscript (0) refer to those at equilibrium, i.e., in the absence of E, and m(j 0 ) is a constant independent of r. The distribution of electrolyte ions at equilibrium n (j 0 ) (r) obeys the Boltzmann equation and the equilibrium potential c ( 0 ) (r) outside the sphere satisfies the Poisson– Boltzmann equation, both being functions of r ( ÅÉrÉ) only, viz.,

e c (0) kT

drel (r) 1 Å0 ere0 ere0

N

∑ zj ednj (r). [2.28] j Å1

Substituting Eqs. [2.3], [2.4], and [2.12] – [2.15] into Eqs. [2.5], [2.7], and [2.8] and neglecting the products of small quantities u, dnj , dc, and dmj , we obtain after taking the curl to eliminate p(r, t)

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HIROYUKI OHSHIMA N

hÇ 1 Ç 1 Ç 1 u(r) 0 ivr0Ç 1 u(r)

G(r) Å 0

`

Å

∑ Çdmj (r) 1 Çn (j 0 ) (r), [2.29] j Å1

By multiplying zj en (j 0 ) (r) on both sides of Eq. [2.32], summing over j, and using Eqs. [2.28], [2.32], and [2.33], we obtain

and from Eqs. [2.7] and [2.8]

S

0ivdnj (r) / Çr n

(0) j

D

N

1 (r)u(r) 0 n (j 0 ) (r)Çdmj (r) Å 0. lj [2.30]

In addition, symmetry considerations permit us to write (7– 9) u(r) Å (ur , uu , uw)

S

e dy ∑ n `j z 2j exp( 0zj y) fj (r). [2.40] hr dr j Å1

1 d 2 hE cos u, (rh)E sin u, 0 r r dr

D

LY Å

1 ∑ z 2j e 2n (j 0 ) (r)(Y 0 fj ). ere0kT j Å1

The boundary conditions for h(r) are (i) u Å 0 at r Å a (assumption ii); (ii) ur at r Å b equals 0U cos u ( Å 0 mE cos u ); (iii) following Kuwabara (17), the vorticity v Å Ç 1 u Å (0, 0, LhE cos u ) is assumed to be zero at r Å b (assumption vi) (13, 15, 16, 18). These conditions are expressed in terms of h and fj as follows (15, 16, 18):

[2.31]

hÅ

dmj (r) Å 0zj efj (r)ErrP Å 0zj efj (r)E cos u,

[2.32]

h(b) Å

dc(r) Å 0Y (r)ErrP Å 0Y (r)E cos u,

[2.33]

Å

0

,

L(Lh / g 2h) Å G(r), Lfj 0 k 2gj ( fj 0 Y ) Å

dy dr

S

zj

dfj 2lj h 0 dr e r

D

h(r) Å 0

/

,

[2.35]

∑ zj ednj (r) Å 0 ere0Ddc(r)

1

j Å1

Å ere0 E cos u LY,

[2.36]

0

with

r gÅ

ivr0 Å (i / 1) h

gj Å 0

ivlj , k 2kT

Lå

r

1 vr0 Å (i / 1) , d 2h

[2.37] [2.38]

d 1 d 2 d2 2 d 2 Å / 0 2, r 2 2 dr r dr dr r dr r

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mb , 2

[2.43]

[2.39]

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r

G(r * )dr * /

a

C 1r C2 i / / g 2 g 2r 2 2g 3

FS S S

[2.44]

*

1 r2

* r * G(r * )dr * G r

3

a

r

a

D

r* 1 ir * i / 2 2/ 20 e ig ( r 0r = ) r g r gr gr

D

G S D

r* 1 ir * i / 2 20 2/ e 0ig ( r 0r = ) G(r * )dr r g r gr gr

/ C3

where d å (2h / vr0 ) 1 / 2 is the hydrodynamic penetration depth in the liquid, L is a differential operator, and G(r) is defined by

AID

[2.42]

Equation [2.34] can formally be integrated to give

[2.34]

N

drel (r) Å

dh Å 0 at r Å a, dr

Lh(r) Å 0 at r Å b.

with rˆ Å r/r. Here fj (r), Y (r), and h(r) are functions of r only and Eq. [2.31] automatically satisfies Eq. [2.6]. In terms of fj (r), Y (r), and h(r), Eqs. [2.29], [2.30], [2.27], and [2.28] can further be rewritten as

[2.41]

D

ig 1 ig 1 0 2 e ig ( r 0a ) / C4 / 2 e 0ig ( r 0a ) . r r r r [2.45]

In order to determine the integration constants C1 –C4 in Eq. [2.45] one needs one more boundary condition for h(r) in addition to Eqs. [2.42] – [2.44]. This condition can be derived from the equation of motion of the unit cell, as will be shown in the following section. 2.3. Equation of Motion of the Unit Cell Consider the forces acting on one unit cell. Since the net electric charge within the unit cell is zero, there is no net

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DYNAMIC MOBILITY IN CONCENTRATED SUSPENSIONS

electric force acting on the cell and one needs consider only the hydrodynamic force Fh . The equation of motion for the unit cell is thus given by

* * dtd [(u cos u 0 u sin u / U)e p

r0

where the relation Lh Å 0 at r Å b (Eq. [2.44]) has been employed. With the help of Eqs. [2.45] and [2.51] it follows from Eq. [2.46] that

b

0ivt

u

r

0

d [Ue 0ivt ] Å Fh , dt

1 2pr 2 sin u dr du / rp 43pa 3

* r G(r)dr / ivma (3rh 0 r ) 0 b r b

]

C2 Å 13

a

3

2

p

3

0

(0) el

a

(b)Y (b) . 3h

[2.46]

[2.52]

where the first term on the left is the contribution from the liquid contained in the unit cell and the second term is that from the particle. The hydrodynamic force Fh is parallel to E, and its magnitude Fh is given by

Equations [2.42] – [2.44] and [2.52] form a complete set of the boundary conditions for h(r).

Fh Å

*

3. GENERAL MOBILITY EXPRESSION

p 2

( srr cos u 0 sr usin u )Ér Åb2pb sin u du,

[2.47]

0

where srr and srs are the normal and tangential components of the stress, respectively, given by Ìur 0ivt srr Å 0p / 2h e Ìr Å 0p / 4h

sr u Å h

F

F

The dynamic electrophoretic mobility m in concentrated suspensions can be obtained from Eq. [2.43] as mÅ

G G

h 1 dh 0 Ee 0ivt cos u, 2 r r dr

[2.48]

1 Ìur Ìuu uu 0ivt d 2h / 0 e Å h 2 Ee 0ivt sin u. r Ìu Ìr r dr [2.49]

F G F * G

0

r (el0 ) (b)Y (b) h

Å 0 hEe 0

0ivt

d r(L / g 2 )hÉr Åb 0 mg 2b dr

mÅ 1

2 3g [M(a) 0 G] 2

F*

b

a

b

a

M(r) Å [(1 / igb)e 0ig ( b0a ) H(r) 0 (1 0 igb)e /ig ( b0a ) H*(r)]/[{1 / [3f /( ga) 2 ]

/ u-independent terms.

1 H(a)}(1 / igb)e 0ig ( b0a ) 0 {1 / [3f /( ga) 2 ]

M(a) Å

p

p(b, u )cos u sin ud u RÅ

4p 3 b hEe 0ivt 3 1

F * 1 b3

b

r 3G(r)dr 0

a

3C2 r 0 b3

(0) el

(b)Y (b) bh

G

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1 H*(a)}(1 0 igb)e /ig ( b0a ) ],

[3.3]

H(a) / iga(1 0 R) , 1 0 f / [3f /( ga) 2 ](1 0 igaR)

[3.4]

(1 / igb)e 0ig ( b0a ) / (1 0 igb)e ig ( b0a ) , (1 / igb)e 0ig ( b0a ) 0 (1 0 igb)e ig ( b0a )

[3.5]

H(r) Å (1 0 igr)exp[ig(r 0 a)] 0 ,

[2.51]

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,

where 3C2 r G(r)dr 0 2 b

0

Å

G

3

Substituting Eqs. [2.48] – [2.50] into Eq. [2.47] yields

*

g 2b 2 ( 0 ) r el (b)Y (b) 3ha

[3.2]

[2.50]

Fh Å 02pb 2

[M(a) 0 M(r)]G(r)dr 0

/ u-independent terms

1 cos u 2 b

r (el0 ) (b)Y (b) h

[3.1]

After determining C1 –C4 in Eq. [2.45] subject to Eqs. [2.42] – [2.44] and [2.52] and using Eq. [3.1], we obtain the following general expression for the dynamic electrophoretic mobility m of spherical particles of radius a in concentrated suspensions:

The pressure p(r, u ) at r Å b is p(b, u ) Å 0 hEe 0ivt cos u

2h(b) . b

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g 2r 3 , 3a

H*(r) Å (1 / igr)exp[ 0ig(r 0 a)] 0

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g 2r 3 , 3a

[3.6] [3.7]

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HIROYUKI OHSHIMA

H(a) Å 1 0 iga 0

( ga) 2 , 3

[3.8]

H*(a) Å 1 / iga 0

( ga) 2 , 3

[3.9]

2( ga) 2 ( rp 0 r0 ) . 9r 0

[3.10]

GÅ

dmj (r) Å zjedc(r),

The mobility given by Eq. [3.2] is a complex quantity. The modulus (or the magnitude) of Eq. [3.2], ÉmÉ, gives the actual dynamic mobility. We consider two limiting cases. If b r ` , then f r 0, M(r) r H(r), R r 1, and r (el0 ) (b) r 0 so that Eq. [3.2] becomes the following general expression for the dynamic mobility of a single isolated sphere of radius a (9):

*

2 mÅ 2 3g [H(a) 0 G]

*F b

10

a

1 G(r)dr 0 1

S

1/

Y (r) Å fj (r).

[4.2]

Equation [2.35] becomes, if one neglects the second and higher order terms of z , Lfj Å 0 and LY Å 0.

[4.3]

[H(a) 0 H(r)]G(r)dr,

a

[3.11]

Consider next the case of v r 0 (or g r 0 and gj r 0). In this case the applied electric field becomes a static one so that the dynamic mobility m tends to the usual static mobility at v Å 0. Indeed, in this limit Eq. [3.2] tends to a2 9

[4.1]

and from Eqs. [2.32] and [2.33],

`

( f r 0).

mÅ

surface charge density on the particle surface is assumed constant and always equal to its equilibrium value s. In such a case the electrical double layer around the particle always remains spherically symmetrical (the relaxation effect becomes negligible) so that dnj (r) Å 0 and drel (r) Å 0. Thus we have from Eq. [2.26]

3r 2 2r 3 a 3 / 3 0 3 a2 a b

S

2 r3 3r 5 0 3/ 5 5 a 5a

DG

Y (b) 2a 2 ( 0 ) r el (b) 9h b

b 3 9b 2 a3 0 20 3 3 a 5a 5b

D

, ( v r 0).

The boundary conditions for dcj (r) (or fj (r)) and dm(r) (or Y (r)) are Ç dmj (r)r nP Ér Åa / Å 0,

[4.4]

Çdc(r)r nP Ér Åb 0 Å 0E,

[4.5]

where Eq. [4.4] follows from the assumption that the ionic species cannot penetrate the particle surface (vj (r)r nˆÉr Åa Å 0) (assumption iii); and Eq. [4.5] states that at the outer surface of the cell (r Å b) the local electric field is assumed to be parallel to E (13). It thus follows from Eqs. [4.4] and [4.5] that in the present approximation (Y Å fj ) Y and fj must satisfy the same boundary conditions, viz.,

[3.12]

dY dfj Å Å 0 at r Å a, dr dr

[4.6]

This is a static mobility expression for a swarm of spherical rigid particles with a radius a obtained in a previous paper (Eq. [24] of Ref. (18)). Equation [3.2] is a general expression for the dynamic electrophoretic mobility m of spherical colloidal particles in concentrated suspensions in an applied oscillating electric field. In order to calculate the mobility value via Eq. [3.2] one needs the function G(r), defined by Eq. [2.40], which in turn requires the function fj (r).

dY dfj Å Å 1 at r Å b. dr dr

[4.7]

The solution to Eq. [4.3] subject to Eqs. [4.6] and [4.7] is fi (r) Å Y (r) Å

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r/

1/

a3 2r 3

a3 2r 2

D

.

[4.8]

Thus Eq. [2.40] becomes

4. MOBILITY FORMULA FOR LOW z AND ZERO PARTICLE PERMITTIVITY

Consider the practically important case where the zeta potential z is low and the relative permittivity ep of the particle is very small compared to that of the liquid ( ep ! er ) so that ep is practically equal to zero and where the

S

1 10f

G(r) Å 0

ere0k 2 h(1 0 f )

S

D

dc ( 0 ) , dr

[4.9]

where the equilibrium potential distribution c ( 0 ) (r) can be obtained as follows. For low z , Eqs. [2.16] – [2.18] are linearized to give

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DYNAMIC MOBILITY IN CONCENTRATED SUSPENSIONS

r (el0 ) (b) Å 0 ere0k 2c ( 0 ) (b),

[4.10]

d 2c ( 0 ) 2 dc ( 0 ) / Å k 2c ( 0 ) . 2 dr r dr

[4.11]

z Å c ( 0 ) (a) Å

sa { kb cosh[ k(b 0 a)] 0 sinh[ k(b 0 a)]}/ ere0

{ k(b 0 a)cosh[ k(b 0 a)] By using Eqs. [4.8] – [4.10], Eq. [3.2] becomes

mÅ 1

ere0 h(1 0 f )[M(a) 0 G]

S

a3 1/ 3 2r

D

F

0

2k 2 3g 2

*

0 (1 0 k 2ab)sinh[ k(b 0 a)]}

b

Å

[M(a) 0 M(r)]

a

S

dc ( 0 ) 2( ka) 2 ( 0 ) f c (b) 1 / dr / dr 9f 2

DG

sa { kaf 01 / 3 cosh[ ka( f 01 / 3 0 1)] ere0 0 sinh[ ka( f 01 / 3 0 1)]}/

.

{ ka( f 01 / 3 0 1)cosh[ ka( f 01 / 3 0 1)] 0 {1 0 ( ka) 2f 01 / 3 }sinh[ ka( f 01 / 3 0 1)]}.

[4.12]

[4.16]

5. APPROXIMATE MOBILITY FORMULA

Further, substituting the solution to Eq. [4.11] subject to Eqs. [2.19] and [2.21], i.e., c ( 0 ) (r)

a kb cosh[ k(b 0 r)] 0 sinh[ k(b 0 r)] , r kb cosh[ k(b 0 a)] 0 sinh[ k(b 0 a)]

Åz

[4.13]

into Eq. [4.12] yields

mÅ

F * D HS D D J

2ere0z ( ka) 2 3h(1 0 f )P[M(a) 0 G] 1 /

S S

1/

a3 2r 3

kr 0

1 r2

10

1 g 2a

b

{M(a) 0 M(r)}

a

r cosh[ k(b 0 r)] b

1 1 / f /2 sinh[ k(b 0 r)] dr / kb 3f 2 / 3

G

,

[4.14] with

P Å cosh[ k(b 0 a)] 0

1 sinh[ k(b 0 a)] kb

Å cosh[ ka( f 01 / 3 0 1)] 0

Because Eq. [4.14] involves numerical integration, it is not always convenient for practical calculations. In this section, we derive a simple approximate mobility formula applicable for low z and zero particle permittivity from Eq. [4.12] using an approximation method proposed in our previous papers (9, 12, 18–20). This method was originally employed to derive an accurate approximate formula for Henry’s function (26) for the retardation effect in static electrophoresis (19) and later applied to other systems (9, 12, 18, 20). In the present problem this approximation is based on the observation that {M(a) 0 M(r)}dc ( 0 ) /dr in Eq. [4.12] has a sharp maximum around r Ç a / a / k, a being a number of the order of unity, and becomes zero at both limits of integration r Å a and r r ` and that the function (1 / a 3 /2r 3 ) in Eq. [4.12] varies slowly as compared with {M(a) 0 M(r)}dc ( 0 ) /dr. This is because the electrical double layer around the particle is confined in the narrow region between r Å a and r Ç a / 1/ k. Thus we may approximately replace r in the function (1 / a 3 /2r 3 ) (which is equal to Y (r)/r or fj (r)/r and expresses the distortion of the applied electric field by the presence of the particle) by a / a / k and take it out before the integral sign. We then obtain from Eq. [4.12] after some algebra mÅ

f 1/3 sinh[ ka( f 01 / 3 0 1)], ka

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J

G

2 1 1/ S1 / S2 , 3 2(1 / a / ka) 3

where S1 Å

AID

FH

[5.1]

[4.15] where b is replaced by af 01 / 3 in the second equation. Equation [4.14] is the required mobility expression for the case of low z and zero particle permittivity. Note that in the present approximation the zeta potential z is related to s, ka, and f as follows.

ere0z 1 h M(a) 0 G

F

1 ( ka) 2 0 10f 3f 2 / 3 P /

g 2 (1 / kaQ) / k 2 (1 0 igaR) ( g 2 / k 2 ){1 0 f / [3f /( ga) 2 ](1 0 igaR)}

G

, [5.2]

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HIROYUKI OHSHIMA

S2 Å

2(1 / f /2)( ka) 2 , 9f 2 / 3 (1 0 f )P

QÅ

1 0 kb tanh[ k(b 0 a)] tanh[ k(b 0 a)] 0 kb

Å

[5.3]

1 0 kaf 01 / 3 tanh[ ka( f 01 / 3 0 1)] , tanh[ ka( f 01 / 3 0 1)] 0 kaf 01 / 3

[5.4]

and P, M(r), M(a), R, and G are given by Eqs. [4.15], [3.3] – [3.5], and [3.10]. In the static and single particle case (19) we have shown that the relative error in Eq. [5.1] (with v Å 0 and f r 0) becomes less than 1%, if a is chosen to be aÅ

2.5 . 1 / 2 exp( 0 ka)

[5.5]

The same choice of a works quite well also for the dynamic mobility in concentrated suspensions, as will be seen later. For the limiting case of v Å 0, Eq. [5.1] reduces to mÅ

ere0z h

FH

J

G

2 1 1/ M1 / M2 , 3 2(1 / a / ka) 3

[5.6]

with M1 Å 1 0 1

S

Å10 1

M2 Å

S

3 f ( ka) 2 1 (1 / kaQ) / 2 ( ka) 1 0 f 5P f 2 / 3 1/

D

10f 3 3f 1 / 3 0 / 3 10f 10f

3 f ( ka) 2 (1 / k aQ) 0 ( ka) 2 1 0 f 3(1 0 f )P f 1/3 /

1 9 f 4/3 0 0 f 2 / 3 5f 1 / 3 5

2( ka) 2 1 / f /2 9P 10f

S

f 1/3 /

D

,

[5.7]

1 9 f 4/3 0 0 2/3 1/3 f 5f 5

D

,

[5.8] a result obtained in a previous paper (18).

FIG. 2. Magnitude of the reduced dynamic electrophoretic mobility ÉmÉ/ ms (where ms å ere0z / h is Smoluchowski’s mobility formula) as a function of ka for various values of the particle volume fraction f at ( rp 0 r0 )/ r0 Å 0.1 and the reduced frequency r0a 2v / h ( åÉg 2Éa 2 ) Å 1. The results calculated from Eq. [4.14] are represented by solid lines. Comparison is made with the approximate results calculated from Eq. [5.1], given as ( l ) f Å 0.5; ( j ) f Å 0.1; ( l ) f Å 0.01; ( n ) f Å 0.001; ( s ) f Å 0.0001; and ( h ) f r 0.

Figure 2 shows the dependence of the magnitude of the dynamic mobility ÉmÉ on the reduced particle radius ka for several values of the particle volume fraction f at r0a 2v / h Å Ég 2Éa 2 Å 1, where the limit f r 0 corresponds to the single particle case or very dilute suspensions. As in the static case ( v r 0) and in the single particle case ( f r 0), the magnitude of the dynamic mobility in concentrated suspensions decreases as decreasing ka (or increasing double-layer thickness 1/ k ) due to the double-layer overlap ‘‘(and the less distortion of the applied field)’’. In the limit of ka r 0, the double layers around the particles become infinitely thick so that the double-layer overlap inevitably occurs at any finite f and thus the mobility always decreases to zero. Only when f r 0 (or when the interparticle distance 2(b 0 a) becomes infinity), no double-layer overlap occurs even at ka r 0 and the mobility tends to a nonzero value given by mÅ

6. RESULTS AND DISCUSSION

The results of the calculation of the magnitude of the dynamic mobility via Eq. [4.14] are given in Figs. 2–10. All the calculations are performed at ( rp 0 r0 )/ r0 Å 0.1. Approximate results obtained via Eq. [5.1] are also shown for comparison in Fig. 2.

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2ere0z 1 , ( ka r 0, f r 0). 3h H(a) 0 G

[6.1]

Further, in the static limit v Å 0, Eq. [6.1] becomes the following Hu¨ckel equation for the static mobility of a spherical particle with a very thick double layer as expected: mH Å

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2ere0z , ( ka r 0, f r 0, v Å 0). 3h

[6.2]

DYNAMIC MOBILITY IN CONCENTRATED SUSPENSIONS

145

Figure 2 shows that for small ka the mobility magnitude decreases as the volume fraction f increases (as in the case of the static mobility (13, 18)). For larger values of ka, however, the ka dependence of the mobility magnitude becomes different so that the mobility magnitude curves with different f intersect each other and at the limit of ka r ` (where the double layer becomes infinitesimally thin), the tendency inverses, i.e., the mobility magnitude increases with increasing f. This is in contrast to the static case ( v Å 0) where the mobility always decreases with increasing f for any ka. The large ka form of the dynamic mobility in concentrated suspensions is derived by taking the corresponding limit in Eq. [4.14]. That is, for infinitesimally thin double layers ( ka r ` , k(b 0 a) r ` , and k @ ÉgÉ), Eq. [5.1] becomes mÅ

ere0z [(1 0 igaR)/[{M(a) 0 G} h(1 0 f ) 1 [1 0 f / [3f /( ga) 2 ](1 0 igaR)]

Å

ere0z (1 0 igaR)/ {H(a) / iga(1 0 R) h(1 0 f ) 0 G[1 0 f / [3f /( ga) 2 ] 1 (1 0 igaR)]}, ( ka r ` ).

[6.3]

In the single particle case ( f r 0), Eq. [6.3] becomes the following O’Brien’s large ka formula for the dynamic mobility: mÅ

ere0z 1 0 iga , ( ka r ` , f r 0). h H(a) 0 G

[6.4]

Note that in the static problem ( v Å 0), the mobility becomes independent of f in the limit of ka r ` and is given by Smoluchowski’s formula (13): ms Å

ere0z . h

[6.5]

In the dynamic problem, however, as is shown in Eq. [6.3] the mobility still depends on f even in the limit of ka r ` . It should also be noted that Eq. [6.3] can be obtained directly from the general mobility expression [3.2] without recourse to low z approximation by using the fact that fj É 3a/2 at ka r ` (see Refs. (27) and (28) for the static field problem). Equation [6.3] is thus applicable even for high z unless the dynamic relaxation effect becomes appreciable. Figure 2 also shows a comparison of the results calculated with Eq. [4.14] (solid lines) with the approximate results calculated with Eq. [5.1] (represented by symbols). It is seen that this approximation is excellent with the maximum

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FIG. 3. Magnitude of the reduced dynamic electrophoretic mobility ÉmÉ/ ms (where ms å ere0z / h is Smoluchowski’s mobility formula) as a function of the reduced frequency r0a 2v / h ( åÉg 2Éa 2 ) for various values of the particle volume fraction f. Calculated from Eq. [4.14] at ( rp 0 r0 )/ r0 Å 0.1 and ka Å 1.

relative errors of about 3% for 0 £ f £ 0.74. Here the value 0.74 corresponds to the maximum attainable volume fraction for a swarm of identical spheres (13). Thus, Eq. [5.1], which does not involve numerical integration, serves as an accurate analytic approximation for the dynamic mobility in concentrated suspensions. Figures 3–5 exhibit the dependence of the magnitude of the dynamic mobility in concentrated suspensions on the frequency v of the applied external field for several values of f at ka Å 1, 10, and ` . In these figures the abscissa is given by the dimensionless frequency r0a 2v / h, which is equal to Ég 2Éa 2 Å 2a 2 / d 2 , where d å (2h / vr0 ) 1 / 2 is the penetration depth. All the figures show that the mobility magnitude decreases with increasing v as in the case of the single particle case ( f r 0). It can be seen from Fig. 3–5 that the v dependence is negligible for d ú 2(b 0 a) and becomes appreciable for d © 2(b 0 a). Namely, if the penetration depth is less than the interparticle distance, then the dynamic mobility depends significantly on v; in the opposite case the mobility depends little on v and is almost equal to the static value at v Å 0. This tendency is observed for all ka values. Note that in the limit of f r 0 (or, b r ` ), the condition d õ 2(b 0 a) always holds and the mobility depends significantly on v. From Figs. 3–5 we can again see that for small ka the mobility magnitude decreases as the volume fraction f increases, while for very large values of ka, it increases with increasing f. In order to see this more clearly, we plot in

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HIROYUKI OHSHIMA

FIG. 4. Same as Fig. 3 but for ka Å 10.

Figs. 6–10 the magnitude of the dynamic mobility as a function of the volume fraction f for several values of the dimensionless frequency r0a 2v / h and the dimensionless particle radius ka. We see that at small ka (Fig. 6), the mobility magnitude decreases with increasing f. For moderate values of ka (Figs. 7–9), there may be a maximum in the mobility magnitude curve except at very low v, where ÉmÉ decreases with increasing f. The position of the maximum is found to correspond to d É 2(b 0 a). At infinitely large ka (Fig. 10), the mobility magnitude increases with

FIG. 6. Magnitude of the reduced dynamic electrophoretic mobility ÉmÉ/ ms (where ms å ere0z / h is Smoluchowski’s mobility formula) as a function of the particle volume fraction f for various values of the reduced frequency r0a 2v / h ( åÉg 2Éa 2 ), calculated from Eq. [4.14] at ( rp 0 r0 )/ r0 Å 0.1 and ka Å 1. Note that the maximum attainable value of f is about 0.74 (13).

increasing f. Figures 6–10 show that for any values of ka the dependence of the mobility magnitude on v becomes less as f increases and thus the dynamic mobility at any v approaches the static mobility at v Å 0 as f increases. In the example as shown in Fig. 10 for ka r ` , the dynamic

FIG. 5. Same as Fig. 3 but for ka Å ` .

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FIG. 7. Same as Fig. 6 but for ka Å 5.

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DYNAMIC MOBILITY IN CONCENTRATED SUSPENSIONS

147

FIG. 8. Same as Fig. 6 but for ka Å 10.

FIG. 10. Same as Fig. 6 but for ka Å ` .

mobility at very large f ( É fmax Å 0.74) becomes close to Smoluchowski’s static mobility ms (Eq. [6.5]) even at high frequencies ( r0a 2v / h Å 10 2 ). It is of interest to compare with O’Brien’s work (11) which shows that a porous sphere moves with the static mobility at high frequencies. The above complicated behaviors are a consequence of the fact that the dynamic electrophoretic mobility in concentrated suspensions is characterized by four quantities having the dimension of length, that is, the particle radius a, the

double layer thickness 1/ k, the penetration depth d (which stands for the distance over which the amplitude of the fluid velocity relative to the particle falls off by a factor of e) and the distance between the surfaces of two neighboring spheres 2(b 0 a) Å 2a( f 01 / 3 0 1). Note that in the static problem ( v Å 0) the penetration length d becomes infinity and in the single particle problem ( f r 0) the interparticle distance 2(b 0 a) is infinity. As 1/ k increases the mobility magnitude decreases. As d decreases, the mobility magnitude also decreases. The increase in f (or the decrease in the interparticle distance) has two effects: the first effect corresponds relatively to an effective increase in 1/ k, leading to a decrease in ÉmÉ; the second corresponds to an effective increase in the penetration depth d, leading to an increase in ÉmÉ. For small ka, the former effect is greater, and for large ka, the latter is dominant. Thus, for intermediate ka, the latter effect is greater at small f and the former is greater at large f so that there may occur a maximum in the ÉmÉ versus f curve. 7. CONCLUSIONS

FIG. 9. Same as Fig. 6 but for ka Å 50.

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We have derived an expression for the dynamic mobility m for a swarm of spherical identical particles in concentrated suspensions (Eq. [4.14]) on the basis of Kuwabara’s cell model (17) (Fig. 1). This expression is applied to the case where the particle zeta potential z is low and the particle permittivity is practically zero. We have discussed the dependence of the mobility magnitude ÉmÉ on the reduced particle radius ka, the frequency v of the external applied field, and the particle volume fraction f (Figs. 2–10). An accurate approximate mobility formula (Eq. [5.1]) has been

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HIROYUKI OHSHIMA

derived that does not involve tedious numerical integration and is suitable for practical calculations of the dynamic mobility in concentrated suspensions.

11. 12. 13. 14.

REFERENCES

15.

1. O’Brien, R. W., J. Fluid Mech. 190, 71 (1988). 2. Babchin, A. J., Chow, R. S., and Sawatzky, R. P., Adv. Colloid Interface Sci. 30, 111 (1989). 3. Babchin, A. J., Sawatzky, R. P., Chow, R. S., Isaacs, E. E., and Huang, H., ‘‘International Symposium on Surface Charge Characterization: 21st Annual Meeting of the Fine Particle Society, August, 1990, San Diego, CA,’’ (Oka, K., Ed.), p. 49. Fine Particle Society, Tulsa, OK, 1990. 4. Sawatzky, R. P., and Babchin, A. J., J. Fluid Mech. 246, 321 (1993). 5. Fixman, M., J. Chem. Phys. 78, 1483 (1983). 6. James, R. O., Texter, J., and Scales, P. J., Langmuir 7, 1993 (1991). 7. Mangelsdorf, C. S., and White, L. R., J. Chem. Soc., Faraday Trans. 88, 3567 (1992). 8. Mangelsdorf, C. S., and White, L. R., J. Colloid Interface Sci. 160, 275 (1993). 9. Ohshima, H., J. Colloid Interface Sci. 179, 431 (1996). 10. Lowenberg, M., and O’Brien, R. W., J. Colloid Interface Sci. 150, 158 (1992).

16.

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17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

O’Brien, R. W., J. Colloid Interface Sci. 171, 495 (1995). Ohshima, H., J. Colloid Interface Sci. 185, 131 (1997). Levine, S., and Neale, G. H., J. Colloid Interface Sci. 47, 520 (1974). Kozak, M. W., and Davis, E. J., J. Colloid Interface Sci. 112, 403 (1986). Kozak, M. W., and Davis, E. J., J. Colloid Interface Sci. 127, 497 (1989). Kozak, M. W., and Davis, E. J., J. Colloid Interface Sci. 129, 166 (1989). Kuwabara, S., J. Phys. Soc. Jpn. 14, 527 (1959). Ohshima, H., J. Colloid Interface Sci. 188, 481 (1997). Ohshima, H., J. Colloid Interface Sci. 168, 269 (1994). Ohshima, H., J. Colloid Interface Sci. 180, 299 (1996). Dukhin, S. S., and Semenikhin, N. M., Kolloid Zh. 32, 360 (1970). Dukhin, S. S., and Derjaguin, B. V., in ‘‘Surface and Colloid Science’’ (Matijevic, E., Ed.), Vol. 7, Chap. 3. Wiley, New York, 1974. Zukoski, C. F., IV., and Saville, D. A., J. Colloid Interface Sci. 114, 32 (1986). Mangelsdorf, C. S., and White, L. R., J. Chem. Soc., Faraday Trans. 86, 2859 (1990). Ennis, E., and White, L. R., J. Colloid Interface Sci. 178, 446 (1989). Henry, D. C., Proc. R. Soc. London, Ser. A 133, 106 (1931). O’Brien, R. W., and Hunter, R. J., Can. J. Chem. 59, 1878 (1981). Ohshima, H., Healy, T. W., and White, L. R., J. Chem. Soc., Faraday Trans. 2 79, 1613 (1983).

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