Biophysical Chemistry of Biointerfaces (Ohshima/Biophysical Chemistry of Biointerfaces) || Electrophoretic Mobility of Concentrated Soft Particles

22 Electrophoretic Mobilityof Concentrated Soft Particles

22.1 INTRODUCTION

In this chapter, we extend the electrokinetic theory of soft particles (Chapter 21),

which is applicable for dilute suspensions, to cover the case of concentrated suspen-

sions [1–3] on the basis of Kuwabara’s cell model [4], which has been applied to

theoretical studies of various electrokinetic phenomena in concentrated suspensions

of hard colloidal particles [5–23].

22.2 ELECTROPHORETIC MOBILITY OF CONCENTRATEDSOFT PARTICLES

Consider a concentrated suspension of charged spherical soft particles moving with

a velocity U in a liquid containing a general electrolyte in an applied electric field

E. We assume that the particle core of radius a is coated with an ion-penetrable

layer of polyelectrolytes with a thickness d. The polyelectrolyte-coated particle has

thus an inner radius a and an outer radius b� aþ d. We employ a cell model [4] in

which each particle is surrounded by a concentric spherical shell of an electrolyte

solution, having an outer radius c such that the particle/cell volume ratio in the unit

cell is equal to the particle volume fraction � throughout the entire dispersion

(Fig. 22.1), namely,

� ¼ ðb=cÞ3 ð22:1Þ

The origin of the spherical polar coordinate system (r, �, j) is held fixed at the

center of one particle and the polar axis (�¼ 0) is set parallel to E. Let the electro-lyte be composed of M ionic mobile species of valence zi and drag coefficient li(i¼ 1, 2, . . . ,M), and let n1i be the concentration (number density) of the ith ionicspecies in the electroneutral solution. We also assume that fixed charges are distrib-

uted with a density of rfix. We adopt the model of Debye–Bueche where the poly-

mer segments are regarded as resistance centers distributed in the polyelectrolyte

Biophysical Chemistry of Biointerfaces By Hiroyuki OhshimaCopyright# 2010 by John Wiley & Sons, Inc.

468

layer, exerting frictional forces on the liquid flowing in the polyelectrolyte layer

(Chapter 21).

The fundamental equations for the flow velocity of the liquid u(r) at position rand that of the ith ionic species vi(r) are the same as those for the dilute case (Chap-

ter 5) except that Eq. (5.10) applies to the region b< r< c (not to the region r> b).The boundary conditions for u(r) and vi(r) are also the same as those for the dilute

case, but we need additional boundary conditions to be satisfied at r¼ c. Accordingto Kuwabara’s cell model [4], we assume that at the outer surface of the unit cell

(r¼ c) the liquid velocity is parallel to the electrophoretic velocity U of the particle,

u � njr¼c� ¼ �U cos � ð22:2Þ

where n is the unit normal outward from the unit cell, U¼ jUj, and that the vorticityx is 0 at r¼ c:

x ¼ r� u ¼ 0 at r ¼ c ð22:3Þ

We also assume that at the outer surface of the unit cell (r¼ c), the gradient of theelectric potential c is parallel to the applied electric field [6], namely,

rc � njr¼c� ¼ �Ecos � at r ¼ c ð22:4Þ

FIGURE 22.1 Spherical particles in concentrated suspensions in a cell model. Each

sphere is surrounded by a virtual shell of outer radius c. The particle volume fraction � is

given by �¼ (b/c)3. The volume fraction of the particle core of radius a is given by �c¼ (a/c)3.

ELECTROPHORETIC MOBILITY OF CONCENTRATED SOFT PARTICLES 469

The boundary conditions for u(r) and vi(r) can be written in terms of h(r) and�i(r) (defined by Eqs. (21.18) and (21.19)) as

hðcÞ ¼ cU

2Eð22:5Þ

LhðrÞjr¼c ¼ 0 ð22:6Þ

Zd

drðrLhÞ

��r¼c

� rð0Þel ðcÞYðcÞ ¼ 0; ð22:7Þ

dY

dr

��r¼c

¼ 1 ð22:8Þ

where rel(c) is the equilibrium charge density at r¼ c.The electrophoretic mobility m of spherical soft particles in a concentrated sus-

pension is defined by m¼U/E. It must be mentioned here that the electrophoretic

mobility m in this chapter is defined with respect to the externally applied electric

field E so that the boundary condition (22.8) has been employed following Levine

and Neale [5]. There is another way of defining the electrophoretic mobility in the

concentrated case, where the mobility m� is defined as m�¼U/kEl, kEl being the

magnitude of the average electric field kEl within the suspension [8, 19–21]. It fol-

lows from the continuity condition of electric current that K�kEl ¼ K1E, where K�

and K1 are, respectively, the electric conductivity of the suspension and that of the

electrolyte solution in the absence of the particles. Thus, m and m� are related to eachother by m�=m ¼ K�=K1. For the dilute case, there is no difference between mand m�.

It follows from Eq. (22.5) that

m ¼ 2hðcÞc

ð22:9Þ

By evaluating h(r) at r¼ c, we obtain the following general expression for the elec-

trophoretic mobility m of soft particles in a concentrated suspension [1, 3]:

m ¼ b2

9

Z c

b

3 1� r2

b2

� �� 2L2

L11� r3

b3

� ��

�� 3� 2L2L1

� 9a

L1l2b3

� �1� r3

b3

� �� 3

51� r5

b5

� �� GðrÞdr

470 ELECTROPHORETIC MOBILITY OF CONCENTRATED SOFT PARTICLES

þ 2

3l2

Z c

a

ð1� �Þ L5L1

1þ r3

2b3

� �� 3�

� �GðrÞdr

� 2

3l2ð1� �Þ

Z b

a

1� 3a

2l2b3L1

(ðL5 þ L6 lrÞ cosh½lðr � aÞ�

"

�ðL6 þ L5lrÞsinh½lðr � aÞ��

GðrÞdr

� 2b2rð0Þel ðcÞYðcÞ9Zc

1þ 1

�� 9

5�2=3� �

5þ L7

�

� �ð22:10Þ

where L1–L4 are given by Eqs. (21.42)–(21.45) and L5–L7 are defined by

L5 ¼ L3 þ 3�

1� �cosh lðb� aÞ½ � � sinh½lðb� aÞ�

lb

� �ð22:11Þ

L6 ¼ L4 þ 3�

1� �sinh lðb� aÞ½ � � cosh½lðb� aÞ�

lb

� �ð22:12Þ

L7 ¼ ð1� �Þ2 L2L1

� 1

� �� 9�2

l2b2þ 27�2a

2l2b3L1þ 6ð�þ 1=2Þ2L3

l2b2L1ð22:13Þ

Consider several limiting cases.

1. In the limit �! 0 (i.e., c!1 so that rð0Þel ðcÞ becomes the bulk value and thus

rð0Þel ðcÞ ¼ 0), Eq. (6.11) tends to the mobility expression of a spherical soft

particle for the dilute case (Eq. (21.41)).

2. In the limit a! b, the polyelectrolyte layer vanishes and the soft particle be-

comes a rigid particle. Indeed, in this limit Eq. (22.10) tends to

m ¼ b2

9

Z c

b

1� 3r

b

� 2

þ 2r

b

� 3

� �2

5� r

b

� 3

þ 3

5

r

b

� 5� ��

GðrÞdr

� 2b2rð0Þel ðcÞYðcÞ9Zc

1þ 1

�� 9

5�2=3� �

5

� �ð22:14Þ

This agrees with the mobility expression for a rigid particle with a radius bin concentrated suspensions obtained in previous papers [5, 12].


3. In the limit l!1, the polyelectrolyte-coated particle behaves like a rigid

particle with a radius b and the slipping plane is shifted outward from r¼ a to

r¼ b. In this limit, Eq. (22.10) tends to Eq. (22.14).

4. In the limit l! 0, the polyelectrolyte layer vanishes and the particle becomes

a rigid particle with a radius a. In this limit, Eq. (22.10) tends to Eq. (22.14)

with b replaced by a, as expected.

We derive approximate mobility formulas for the simple but important case

where the double-layer potential remains spherically symmetrical in the presence

of the applied electric field (the relaxation effect is neglected). In this case, it can be

shown that Eq. (22.10) tends to, in the limit k!1,

m ! m1 ¼ rfixZl2

1þ a3=2b3

1� �c

� �ð1� �Þ L5

L1� 2�

� �; as k ! 1 ð22:15Þ

where

�c ¼ ða=cÞ3 ð22:16Þ

is the volume fraction of the particle core. The existence of the nonvanishing term

of the mobility at very high electrolyte concentrations is a characteristic of soft par-

ticles (Chapter 21).

We derive an approximate expression for the electrophoretic mobility of spheri-

cal polyelectrolytes for the case of low potentials. In this case, the equilibrium po-

tential c(0)(r) is given by

cð0ÞðrÞ ¼ rfixereok2

1� Oþ 1=kbcothðkbÞ þ O

� �b sinhðkrÞr sinhðkbÞ

� �; 0 � r � b ð22:17Þ


cothðkbÞ � 1=kbcothðkbÞ þ O

� �bfcoth½kðc� rÞ� � sinh½kðc� rÞ�=kcgrfcoth½kðc� bÞ� � sinh½kðc� bÞ�=kcg ;

b � r � c ð22:18Þ

where

O ¼ sinh½kðc� bÞ� � cosh½kðc� bÞ�=kccosh½kðc� bÞ� � sinh½kðc� bÞ�=kc ¼

1� kc � tanh½kðc� bÞ�tanh½kðc� bÞ� � kc

ð22:19Þ

With the help of Eqs. (22.17) and (22.18), we obtain


m ¼ rfixZl2

1þ 2

3

lk

� �2cothðkbÞ � 1=kbcothðkbÞ þ O

� �"1� ��

kbOþ 1

kb

� ��

þ 2

3ð1� �Þ l

k

� �21

ðl=kÞ2 � 1

Oþ 1=kbcothðkbÞ þ O

� �lk

� �cothðkbÞ � 1=kbcothðkbÞ � 1=lb

� 1

� �#

ð22:20Þ

This is the required expression for the mobility of spherical polyelectrolytes in con-

centrated suspension for low potentials.

When �! 0, Eq. (22.20) reduces to Eq. (21.75). When kb� 1 and lb� 1,

Eq. (22.20) becomes

m ¼ rfixZl2

1þ 2ð1� �Þ3

lk

� �21þ 2k=l1þ k=l

" #ð22:21Þ

which, for �! 0, reduces to Eq. (21.56).

For the case where

la � 1; ka � 1 ðand thus lb � 1; kb � 1Þ and

ld ¼ lðb� aÞ � 1; kd ¼ kðb� aÞ � 1 ð22:22Þ

we obtain from Eq. (22.10)

m ¼ ereoZ

co=km þ cDON=l1=km þ 1=l

fd

a; �

� �þ rfixZl2

; ð22:23Þ

where

fd

a; �

� �¼ 2

31þ 1

2

a

b

� 3� �

1� �

1� �c

� �¼ 2

31þ 1

2ð1þ d=aÞ3" #

ð1� �Þ1� �=ð1þ d=aÞ3

ð22:24Þ

Equation (22.23) is the required approximate expression for the electrophoretic mo-

bility of soft particles in concentrated suspensions when the condition [22.22]

(which holds for most cases) is satisfied. In the limit �! 0, Eq. (22.23) tends to Eq.

(21.51) for the dilute case. For low potentials, Eq. (22.23) reduces to

m ¼ rfixZl2

1þ lk

� �21þ 2k=l1þ k=l

fd

a; �

� �" #ð22:25Þ


In the limit of d� a, Eq. (22.25) reduces to Eq. (22.21) for spherical polyelectro-

lytes, as expected. In the limit of very high electrolyte concentrations (k!1), Eqs.

(22.20) and (22.25) give the following limiting value:

m ! m1 ¼ rfixZl2

; as k ! 1 ð22:26Þ

In Fig. 22.2, we plot the function f(d/a, �), given by Eq. (22.24), as a function of d/afor several values of �. The function f(d/a, �) tends to 1, as d/a decreases, whereas itbecomes 2(1��)/3 as d/a increases. We see that soft particles with d/a> 5 behave

like spherical polyelectrolytes (a¼ 0). The limiting forms of the mobility for the

two cases d/a 1 and d/a� 1 are given by

m ¼ ereoZ

co=km þ cDON=l1=km þ 1=l

þ rfixZl2

; d a ð22:27Þ

m ¼ 2ereoð1� �Þ3Z

co=km þ cDON=l1þ km þ 1=l

þ rfixZl2

; d � a ð22:28Þ

FIGURE 22.2 f(d/a, �), defined by Eq. (4.27), as a function of d/a. From Ref. [22].


22.3 ELECTROOSMOTIC VELOCITY IN AN ARRAY OF SOFTCYLINDERS

Consider the electroosmotic liquid velocity U in an array of parallel soft cylinders

in a liquid containing a general electrolyte in an applied electric field E[2](Fig. 22.3). The velocity of the liquid flow, U, is parallel to the applied field E. We

assume that the cylinder core of radius a is coated with an ion-penetrable layer of

polyelectrolytes with a thickness d. The polyelectrolyte-coated cylinder has thus an

inner radius a and an outer radius b� aþ d. We employ a cell model [4] in which

each cylinder is surrounded by a fluid envelope of an outer radius c such that the

cylinder/cell volume ratio in the unit cell is equal to the cylinder volume fraction �,namely,

� ¼ ðb=cÞ2 ð22:29Þ

The porosity e is thus given by

e ¼ 1� � ¼ 1� ðb=cÞ2 ð22:30Þ

The origin of the cylindrical coordinate system (r, �, z) is held fixed on the axis of

one cylinder. The polar axis (�¼ 0) is set parallel to the applied electric field E so

that E is perpendicular to the cylinder axis. Let the electrolyte be composed of Mionic mobile species of valence zi and drag coefficient li (i¼ 1, 2, . . . , M), and

n1i be the concentration (number density) of the ith ionic species in the

FIGURE 22.3 Electroosmosis in an array of parallel soft cylinders (polyelectrolyte-

coated cylinders), which consist of the core of radius a covered with a layer of polyelectro-

lytes of thickness d. Each cylinder is surrounded by a fluid envelope of outer radius c. Thevolume fraction of the cylinders is given by (b/c)2, where b¼ aþ d, and the porosity e isgiven by e¼ 1��¼ 1� (b/c)2. The volume fraction of the particle core of radius a is given

by �c¼ (a/c)3. The liquid flow U, which is parallel to the applied electric field E, is normal to

the axes of the cylinders.

ELECTROOSMOTIC VELOCITY IN AN ARRAY OF SOFT CYLINDERS 475

electroneutral solution. We assume that relative permittivity er takes the same value

both inside and outside the polyelectrolyte layer. We also assume that fixed charges

are distributed within the polyelectrolyte layer with a density of rfix(r). If dissoci-ated groups of valence Z are distributed with a constant density (number density) Nwithin the polyelectrolyte layer, then we have rfix ¼ ZeN, where e is the elemen-

tary electric charge.

The general expression for electroosmotic velocity is [2]

U

E¼ 1

2l2

Z c

a

L1ðlbÞL2ðlbÞ þ �

L1ðlbÞL2ðlbÞ þ

2

lbL2ðlbÞ� ��

ð1þ �Þ 1þ r2

b2

� �� 2�

� �GðrÞdr

þ 1

2l2

Z b

a

ð1þ �Þlr L1ðlrÞ � L1ðlbÞL2ðlbÞ L2ðlrÞ

� �þ 2� 1� rL1ðlrÞ

bL2ðlbÞ� ��

GðrÞdr

þ b2

8

Z c

b

1þ 2

lbL3

L2ðlbÞ � �L4

� �� 1� r2

b2

� �þ 2r2

b2ln

r

b

� �

��

21� r2

b2

� �2#GðrÞdr� b2rð0Þel ðcÞYðcÞ

4Zc1� 3

4��

4� ln �ð Þ

2�þ L5lb

� �ð22:31Þ

where L1–L3 are given by Eqs. (21.80)–(21.82) and L4 and L5 are defined as

L4 ¼ � L3L2ðlbÞ �

2L3lbL1ðlbÞL2ðlbÞ þ

2

L1ðlbÞ I0ðlaÞK0ðlbÞ � K0ðlaÞI0ðlbÞf g

ð22:32Þ

L5 ¼ �ð1� �ÞL3�L2ðlbÞ þ 2L1ðlbÞ

lbL2ðlbÞ1

�þ �þ 2

� �þ 4ð1þ �Þl2b2L2ðlbÞ

� 4�

lbþ ð1� �ÞL4

ð22:33Þ

where In(z) and Kn(z) are, respectively, the nth-order modified Bessel functions of

the first and second kinds.

Consider several limiting cases. In the limit �! 0, Eq. (22.31) becomes

U

E¼ b2

8

Z 1

b

1þ 2L3lbL2ðlbÞ

� �1� r2

b2

� �þ 2r2

b2ln

r

b

� � �GðrÞdr

þ L1ðlbÞ2l2L2ðlbÞ

Z 1

a

1þ r2

b2

� �GðrÞdr þ 1

2l2

Z b

a

lr L1ðlrÞ � L1ðlbÞL2ðlbÞ L2ðlrÞ

� �GðrÞdr

ð22:34Þ


which agrees with the general expression for the electrophoretic mobility m? of

a cylindrical soft cylinder in a transverse field (Eq. (21.79)).

In the limit a! b, the polyelectrolyte layer vanishes and the soft cylinder

becomes a rigid cylinder. Indeed, in this limit Eq. (22.31) tends to

U

E¼ b2

8

Z c

b

1� r2

b21� 2 ln

r

b

� n o� b2

2c21� r2

b2

� �2" #

GðrÞdr

� b2rð0Þel ðcÞYðcÞ4 Zc

1� 3c2

4b2� b2

4c2þ c2

b2ln

c

b

� � �:

ð22:35Þ

This agrees with the electrophoretic mobility expression m? for a rigid cylinder with

a radius b in concentrated suspensions in a transverse field obtained in a previous

paper [9, 17].

In the limit l!1, the polyelectrolyte-coated cylinder behaves like a rigid cylin-

der with a radius b and the slipping plane is shifted outward from r¼ a to r¼ b. Inthis limit, Eq. (22.31) tends also to Eq. (22.35). In the opposite limit of l! 0, the

polyelectrolyte layer vanishes and the cylinder becomes a rigid cylinder with a radius

a. In this limit, Eq. (22.31) tends to Eq. (22.35) with b replaced by a, as expected.Finally, in the limit a! 0, the cylinder core vanishes and the cylinder becomes a

cylindrical polyelectrolyte. In this limit, Eq. (22.35) becomes

U

E¼ 1

2l2

Z c

0

ð1þ�Þ 1þ r2

b2

� �� 2�

� �GðrÞdrþ ð1� �Þ

2l2

Z b

0

rI1ðlrÞbI1ðlbÞ � 1

� �GðrÞdr

þb2

8

Z c

b

1� 2ð1� �ÞI0ðlbÞlbI1ðlbÞ

� �1� r2

b2

� �þ 2r2

b2ln

r

b

� � �

21� r2

b2

� �2" #

GðrÞdr

�b2rð0Þel ðcÞYðcÞ4Zc

1� 3

4��

4� ln �ð Þ

2�þ 2

l2b21

�� þ 2

� �þ I0ðlbÞlbI1ðlbÞ

1

�þ �� 2

� �� ð22:36Þ

In this case, the equilibrium potential c(0)(r) can be obtained from the linearized

Poisson–Boltzmann equation Dc(0)¼ k2c(0), namely,


1� kb K1ðkbÞ �K1ðkcÞI1ðkcÞ I1ðkbÞ

� �I0ðkrÞ

� �; 0� r � b ð22:37Þ

cð0ÞðrÞ ¼ kbrfix

ereok2I1ðkbÞ K0ðkrÞ þK1ðkcÞ

I1ðkcÞ I0ðkrÞ� �

; b� r � c ð22:38Þ

ELECTROOSMOTIC VELOCITY IN AN ARRAY OF SOFT CYLINDERS 477

Substituting the above equations into the general formula (22.36) yields

U

E¼ rfix

Zl2þ rfixð1� �Þb

2ZkI1ðkbÞ K0ðkbÞ þK1ðkcÞ

I1ðkcÞ I0ðkbÞ� �

þrfixð1� �Þkb2Zðk2 � l2Þ K1ðkbÞ �K1ðkcÞ


I0ðkbÞ � lI0ðlbÞkI1ðlbÞ I1ðkbÞ

� �

�rfixZ

�l2

k2I1ðkbÞ K1ðkbÞ �K1ðkcÞ


:

ð22:39Þ

When �! 0, Eq. (22.39) reduces to

U

E¼ rfixZl2

þ rfixb2Zk

I1ðkbÞK0ðkbÞ þ k2

k2 � l2I0 ðkbÞ � lI0ðlbÞ

kI1ðlbÞ I1ðkbÞ� �

K1ðkbÞ� �

ð22:40Þ

which agrees with the electrophoretic mobility expression m? for a cylindrical poly-

electrolyte in a transverse field for the dilute case (Eq. (21.83)). If, further, kb� 1

and lb� 1, then Eq. (22.40) becomes

U

E¼ rfixZl2

1þ ð1� �Þ2

lk

� �21þ l=2k1þ l=k

" #ð22:41Þ

Finally, we consider the important case where the double-layer potential remains

cylindrically symmetrical in the presence of the applied electric field (the relaxation

effect is neglected) and where

la� 1; ka� 1 ðand thus lb� 1; kb� 1Þ and

ld ¼ lðb� aÞ � 1; kd ¼ kðb� aÞ � 1 ð22:42Þ

In this case, we obtain [23]

U

E¼ ereo

Zco=km þcDON=l

1=km þ 1=lf

d

a; �

� �þ rfixZl2

ð22:43Þ

where

fd

a; �

� �¼ 1

21þ a

b

� 2� �

1� �

1� �c

� �¼ 1

21þ 1

ð1þ d=aÞ2" #

ð1� �Þ1� �=ð1þ d=aÞ2 :

ð22:44Þ


In the limit �! 0, Eq. (22.43) tends to Eq. (21.84) for m? of soft cylinders for the

dilute case. For low potentials, Eq. (22.43) further reduces to

U

E¼ rfixZl2

1þ lk

� �21þ 2k=l1þ k=l

fd

a; �

� �" #ð22:45Þ

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Biophysical Chemistry of Biointerfaces (Ohshima/Biophysical Chemistry of Biointerfaces) || Electrophoretic Mobility of Concentrated Soft Particles