Biophysical Chemistry of Biointerfaces (Ohshima/Biophysical Chemistry of Biointerfaces) || Potential Distribution Around a Charged Particle in a Salt-Free Medium

6 Potential DistributionAround a Charged Particle ina Salt-Free Medium

6.1 INTRODUCTION

The potential distribution around a charged colloidal particle in an electrolyte solu-

tion, which plays an essential role in various electric phenomena observed in colloi-

dal suspensions, can be described by the Poisson–Boltzmann equation. Consider a

particle carrying ionized groups on the particle surface in an electrolyte solution. In

the suspension, in addition to electrolyte ions, there are counterions produced by

dissociation of the particle surface groups. The following two assumptions are usu-

ally made: (i) the concentration of counterions from the surface groups can be ne-

glected as compared with the electrolyte concentration and (ii) the suspension is

assumed to be infinitely dilute so that any effects resulting from the finite particle

volume fraction can be neglected. The assumption (i) becomes invalid where the

electrolyte concentration is as low as or lower than that of counterions from

the particle. The assumption (ii) does not hold when the concentration of all ions

(electrolyte ions and counterions from the particle) is very low, since in this case

the potential around each particle becomes quite long range and the effects of the

finite particle volume fraction become appreciable.

The Poisson–Boltzmann equation for the potential distribution around a cylindri-

cal particle without recourse to the above two assumptions for the limiting case of

completely salt-free suspensions containing only particles and their counterions was

solved analytically by Fuoss et al. [1] and Afrey et al. [2]. As for a spherical parti-

cle, although the exact analytic solution was not derived, Imai and Oosawa [3,4]

studied the analytic properties of the Poisson–Boltzmann equation for dilute parti-

cle suspensions. The Poisson–Boltzmann equation for a salt-free suspension has re-

cently been numerically solved [5–8].

The above theories, however, is based on an idealized model assuming that

there are only counterions from the particle and no other ions. Actually, even

salt-free media may contain other ions rather than counterions from the particle

surface groups. Salt-free water at pH 7, for example, contains 10�7M H+ and OH�

ions. It is thus of interest to examine how the above behaviors of colloidal particles

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

132

in a salt-free medium are influenced if a small amount of salts is added to the me-

dium. In their second paper [4], Imai and Oosawa studied the analytic properties of

the Poisson–Boltzmann equation in the presence of added salts, which are found to

exhibit similar behaviors to those of completely salt-free suspensions.

In this chapter, we first discuss the case of completely salt-free suspensions of

spheres and cylinders. Then, we consider the Poisson–Boltzmann equation for the

potential distribution around a spherical colloidal particle in a medium containing

its counterions and a small amount of added salts [8]. We also deals with a soft

particle in a salt-free medium [9].

6.2 SPHERICAL PARTICLE

Consider a dilute suspension of spherical colloidal particles of radius a with a sur-

face charge density s or the total surface charge Q¼ 4pa2s in a salt-free medium

containing only counterions. We assume that each sphere is surrounded by a con-

centric spherical cell of radius R [5,7] (Fig. 6.1), within which counterions are dis-

tributed so that electrical neutrality as a whole is satisfied. The particle volume

fraction � is given by

� ¼ ða=RÞ3 ð6:1Þ

We treat the case of dilute suspensions, namely,

� � 1 or a=R � 1 ð6:2Þ

FIGURE 6.1 A spherical particle of radius a in a free volume of radius R. (a/R)3 equalsthe particle volume fraction �.

SPHERICAL PARTICLE 133

We denote the electric potential at a distance r from the center O of one particle by

c(r). Let the average number density and the valence of counterions be n and �z,respectively. Then we have

Q ¼ 4pa2s ¼ 4

3pðR3 � a3Þzen ð6:3Þ

We set the electric potential c(r) to zero at points where the volume charge density

r(r) resulting from counterions equals its average value (�zen). We assume that the

distribution of counterions obeys a Boltzmann distribution, namely,

rðrÞ ¼ �zen exp ��zecðrÞkT

� �¼ �zen exp

zecðrÞkT

� �ð6:4Þ

The Poisson equation for the electric potential c(r) around the particle is

given by

d2cðrÞdr2

þ 2

r

dcðrÞdr

¼ � rðrÞereo

ð6:5Þ

where er is the relative permittivity of the medium. By combining Eqs. (6.4) and

(6.5), we obtain the following spherical Poisson–Boltzmann equation for a salt-free

system:

d2y

dr2þ 2

r

dy

dr¼ k2ey ð6:6Þ

where

yðrÞ ¼ zecðrÞkT

ð6:7Þ

is the scaled potential and

k ¼ z2e2n

ereokT

� �1=2

¼ ze

ereokT� 3a2sðR3 � a3Þ

� �1=2¼ 3zeQ

4pereokTðR3 � a3Þ� �1=2

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3Q�aR3 � a3

r

ð6:8Þ

is the Debye–H€uckel parameter in the present system. Here

Q� ¼ ze

kT� Q

4pereoað6:9Þ

134 POTENTIAL DISTRIBUTION AROUND A CHARGED PARTICLE

is the scaled particle charge. For the dilute case (a/R� 1), Eq. (6.8) becomes

k ¼ 3zea2sereokTR3

� �1=2¼ 3zeQ

4pereokTR3

� �1=2¼

ffiffiffiffiffiffiffiffiffiffiffi3Q�aR3

rð6:10Þ

Note that the right-hand side of Eq. (6.6) contains only one term resulting from

counterions, unlike the usual Poisson–Boltzmann equation. Since s (or Q) and z areof the same sign, the product zs (or zQ) is always positive. Note that k depends on s(or Q), a, and R unlike the case of salt solutions, where k is essentially independent

of these parameters.

The boundary conditions are expressed as

dcdr

��r¼a

¼ � sereo

¼ � Q

4pereoa2¼ � kT

ze� Q

�

að6:11Þ

dcdr

��r¼R

¼ 0 ð6:12Þ

which can be rewritten in terms of y and k as

dy

dr

��r¼a

¼ �Q�

a¼ � k2ðR3 � a3Þ

3a2ð6:13Þ

dy

dr

��r¼R

¼ 0 ð6:14Þ

For the dilute case, Eq. (6.13) becomes

dy

dr

��r¼a

¼ � k2R3

3a2ð6:15Þ

Also from Eqs. (6.6) and (6.13) we have

Z R

a

eyr2 dr ¼ 1

3ðR3 � a3Þ ð6:16Þ

Equation (6.6) cannot be solved analytically but its approximate solution for the

case of dilute suspensions has been obtained by Imai and Oosawa [3,4]. They

showed that there are two distinct cases separated by a certain critical value of the

surface charge density s or the total surface charge Q, that is, case 1: low surface

charge density case and case 2: high surface charge density case, as schematically

shown in Fig. 6.2. For case 1, there are two regions I (R�� r�R) and II (a� r�R�),while for case II, there are three regions I (R�� r�R), II (a�� r�R�), and III


(a� r� a�). That is, for case 1, a� reduces to a so that region III disappears. Note

that counter-ion condensation occurs in the vicinity of the particle surface, that is,

in region III because of very strong electric field there. This is a characteristic of a

salt-free system.

Since in region I (far from the particle), the potential y(r) is almost equal to y(R),which is the potential at the outer boundary of the free volume of radius R, thePoisson–Boltzmann equation (6.6) can be approximated by

d2y

dr2þ 2

r

dy

dr¼ k2eyðRÞ ð6:17Þ

Rar

R*a*0

IIIIII

ψ (r)

Case 2

ψ (a)

ψ (R)

Counter-ion condensation

ψ (a)

Ra rψ (R)

R*0

III

ψ (r)

Case 1

ψ (r) ≅ constantψ (r)∝1/r

Surface

Surface

FIGURE 6.2 Three possible regions I, II, and III of r around a sphere of radius a. Forcase 1 (low surface charge densities), there are two regions I (R�� r�R) and II (a� r�R�),whereas for case 2 (high surface charge densities), there are three regions I (R�� r�R),II (a�� r�R�), and III (a� r� a�). Counterions are condensed in region III (counter-ion con-

densation). The particle surface potential co is defined by co�c(a)�c(R). From Ref. [5].


which, subject to Eq. (6.12), can be solved to give

yðrÞ ¼ Q� areyðRÞ 1þ r3

2R3� 3r

2R

� �þ yðRÞ; ðR� � r � RÞ ð6:18Þ

In the dilute case (a/R� 1), as r decreases to R�, approaching region II, the term

proportional to 1/r in Eq. (6.18) becomes dominant. Since y(r) must be continuous

at r¼R�, y(r) in region II (a� r�R� for case 1 and a�� r�R� for case 2) shouldbe of the form

yðrÞ ¼ Q� areyðRÞ þ yðRÞ ð6:19Þ

which satisfies

d2y

dr2þ 2

r

dy

dr¼ 0 ð6:20Þ

We now introduce

xðrÞ ¼ � 1

k2r� dydr

e�y ð6:21Þ

which is related to the ratio of the second term on the left-hand side to the right-

hand side of Eq. (6.6). Since Eq. (6.6) tends to Eq. (6.20) when x(r)� 1, we see

that the condition

xðrÞ � 1 ð6:22Þ

must hold in region II.

We evaluate x(r) at r¼ a� from Eqs. (6.19) and (6.21) and obtain

xða�Þ ¼ 1

3

R

a�

� �3

exp� yða�Þ � yðRÞf g½ � ð6:23Þ

which yields

yða�Þ � yðRÞ ¼ 3 lnR

a�

� �� lnð3xða�ÞÞ ð6:24Þ

Since x(a�)� 1 (Eq. (6.22)), we find that

yða�Þ � yðRÞ � 3 lnR

a�

� �ð6:25Þ


which implies that the value of y(a�)� y(R) cannot exceed 3 ln(R/a�) for the dilutecase. In other words, the maximum value of y(a�)� y(R) is 3 ln(R/a�). FromEq. (6.19), on the other hand, it follows that, for the dilute case,

yða�Þ � yðRÞ ¼ �rdy

dr

��r¼a�

ð6:26Þ

By combining Eqs. (6.25) and (6.26), we obtain

�dy

dr

��r¼a�

� 3

a�ln

R

a�

� �ð6:27Þ

We must thus consider two separate cases 1 and 2, depending on whether

�dy=drjr¼a is larger or smaller than (3/a)ln(R/a).

(a) Case 1 (low surface charge case). Consider first case 1, in which case the

following condition holds:

�dy

dr

��r¼a

� 3

aln

R

a

� �ð6:28Þ

By using Eq. (6.15) (for the dilute case), Eq. (6.28) can be rewritten as

ðkaÞ2 � 3� lnð1=�Þ ð6:29Þ

or by using Eq. (6.9)

Q� � lnð1=�Þ ð6:30Þ

In this case, region II can extend up to r¼ a, that is, a� can reduce to a. Thus,the entire region of r can be covered only with regions I and II and one does

not need region III. For the low surface charge case (small k), Eq. (6.6) canbe approximated by

d2y

dr2þ 2

r

dy

dr¼ k2 ¼ 3Q�

R3ð6:31Þ

with the solution

yðrÞ ¼ Q� ar

1þ r3

2R3� 3r

2R

� �þ yðRÞ; ða � r � RÞ ð6:32Þ

which is now applicable for the entire region of r. One can determine the

value of y(R) in Eq. (6.32) to satisfy Eq. (6.16) (in which ey may be approxi-

mated by 1 + y). The result for the dilute case is


yðRÞ ¼ � k2R2

10¼ �Q�a

10Rð6:33Þ

Thus, Eq. (6.32) becomes

yðrÞ ¼ Q� ar

1þ r3

2R3� 9r

5R

� �; ða � r � RÞ ð6:34Þ

and

yðRÞ 0 for � � 1 ð6:35Þ

Note that Eq. (6.34) satisfies Eq. (6.12). From Eq. (6.34) we obtain

yðaÞ ¼ Q� 1þ a3

2R3� 9a

5R

� � Q� ð6:36Þ

We define the particle surface potential co as the potential c(a) at r¼ aminus the potential c(R) at r¼R, that is, co¼c(a)�c(R). The scaled sur-

face potential of the particle yo� zeco/kT is thus given by yo¼ y(a)� y(R).From Eqs. (6.35) and (6.36), we obtain for the dilute case

yo ¼ Q� ð6:37Þ

that is,

co ¼Q

4pereoað6:38Þ

Note that Eq. (6.37) (or Eq. (6.38)) does not depend on the particle volume

fraction � and coincides with the surface potential of a sphere of radius acarrying a total charge Q¼ 4pa2s for the limiting case of k! 0, expressed

by an unscreened Coulomb potential. That is, the surface potential in this

case is the same as if the counterions were absent.

(b) Case 2 (high surface charge case). Consider next case 2, in which case the

condition

�dy

dr

��r¼a

>3

aln

R

a

� �ð6:39Þ

which can be rewritten as

ðkaÞ2 > 3� lnð1=�Þ ð6:40Þ


or

Q� > lnð1=�Þ ð6:41Þ

Thus, in this case, we find that

�dy

dr

��r¼a

> �dy

dr

��r¼a�

ð6:42Þ

That is, region II cannot extend up to r¼ a. Therefore, the entire region of rcannot be covered only with regions I and II and thus one needs region III

very near the particle surface between r¼ a and r¼ a�. In region III, the

term k2ey in Eq. (6.6) becomes very large so that x(r)� 1. The Poisson–

Boltzmann equation in region III thus becomes

d2y

dr2¼ k2ey ð6:43Þ

which is of the same form as the Poisson–Boltzmann equation for a planar

surface. Integration of Eq. (6.43) after multiplying dy/dr on both sides gives

1

2

dy

dr

� �2

¼ k2ey þ C ð6:44Þ

C being an integration constant. For high potentials, Cmay be ignored so that

Eq. (6.44) yields

dy

dr¼ �

ffiffiffi2

pkey=2 ð6:45Þ

By evaluating Eq. (6.45) at r¼ a and using Eq. (6.15), we obtain for the

dilute case

yðaÞ ¼ lnk2R6

18a4

� �¼ ln

1

6�

ze

kT

� � Q

4pereoa

� �� ¼ ln

Q�

6�

� �ð6:46Þ

The value of y(R) can be obtained as follows. In case 2, y(a�)� y(R) hasalready reached its maximum 3 ln(R/a�). Thus from Eq. (6.19) we have

k2R3

3a�exp yðRÞ½ � ¼ 3 ln

R

a�

� �ð6:47Þ


so that Eq. (6.19) becomes

yðrÞ ¼ 3 lnR

a�

� �a�

rþ yðRÞ; ða� � r � RÞ ð6:48Þ

The value of y(R) in case 2 can be obtained from Eq. (6.46). For the purpose

of determining y(R) from Eq. (6.47), one may replace a� by a in Eq. (6.47)

so that

yðRÞ ¼ �lnðkaÞ2

9 lnðR=aÞR

a

� �3" #

¼ �ln1

lnð1=�Þze

kT

� � Q

4pereoa

� ��

¼ �lnQ�

lnð1=�Þ� � ð6:49Þ

From Eqs. (6.46) and (6.49), we find that the scaled particle surface potential

ys¼ y(a)� y(R) in case 2 is given by

yo ¼ lnðkaÞ4

162 lnðR=aÞR

a

� �9" #

¼ ln1

6�lnð1=�Þze

kT

� �2 Q

4pereoa

� �2" #

¼ lnðQ�Þ2

6� lnð1=�Þ

" # ð6:50Þ

that is, co¼c(a)�c(R) in case 2 is given by

co ¼kT

zeln

1

6� lnð1=�Þze

kT

� �2 Q

4pereoa

� �2" #

¼ kT

zeln

ðQ�Þ26� lnð1=�Þ

" #ð6:51Þ

Note that Eq. (6.51) implies that c0 depends less on Q than Eq. (6.38) and

that co is a function of the particle volume fraction �.We introduce the effective surface potential ceff and the effective surface

charge Qeff defined by

ceff ¼kT

ze

� �lnð1=�Þ ð6:52Þ

Qeff ¼ 4pereoakT

ze

� �lnð1=�Þ ð6:53Þ

and the corresponding dimensionless quantities yeff and Q�eff

yeff ¼ lnð1=�Þ ð6:54Þ


Q�eff ¼

Qeff

4pereoaze

kT

� �¼ lnð1=�Þ ð6:55Þ

That is, yeff¼Q�eff . Then the potential distribution around the particle except

the region very near the particle surface is approximately given by

cðrÞ ¼ ceff

a

rþ yðRÞ ¼ Qeff

4pereorþ yðRÞ; ða� � r � RÞ ð6:56Þ

or

yðrÞ ¼ yeffa

rþ yðRÞ ¼ Q�

eff

a

rþ yðRÞ; ða� � r � RÞ ð6:57Þ

We thus see that for case 2 (high surface charge case), the particle behaves like a

sphere showing a Coulomb potential with the effective surface charge Qeff except

the region very near the particle surface (a�� r�R).We compare the exact numerical solution to the Poisson–Boltzmann equation

(6.6) and the approximate results, Eq. (6.37) for case 1 (low surface charge density

case) and Eq. (6.50) for case 2 (high surface charge density case) in Fig. 6.3, in

which the scaled surface potential yo� zeco/kT is plotted as a function of the scaled

FIGURE 6.3 Scaled surface potential yo¼ zeco/kT, defined by yo� y(a)� y(R) of a sphereof radius a, as a function of the scaled total surface charge Q� = (ze/kT)Q/4pereoa for various

values of the particle volume fraction �. Solid lines represent exact numerical results. The

dashed line, which passes through the origin (yo¼ 0 and Q�¼ 0), is approximate results calcu-

lated by an unscreened Coulomb potential, Eq. (6.37) (case 1: low surface charge densities)

and dotted lines by Eq. (6.50) (case 2: high surface charge densities). Closed circles corre-

spond to the approximate critical values of Q� separating cases 1 and 2, given by Eq. (6.58).


total surface charge Q� for various values of the particle volume fraction �. Notethat yo is always positive. The agreement between the exact results and the approxi-

mate results is excellent with negligible errors except in the transition region

between cases 1 and 2. It follows from the conditions for case 1 (Eq. (6.30)) and

case 2 (Eq. (6.41)) that an approximate expression for the critical value of Q sepa-

rating cases 1 and 2 for dilute suspensions is given by

Q� ¼ lnð1=�Þ ð6:58Þ

6.3 CYLINDRICAL PARTICLE

Consider a dilute suspension of parallel cylindrical particles of radius a with a sur-

face charge density s or total surface charge Q¼ 2pas per unit length in a salt-free

medium containing only counterions. We assume that each cylinder is surrounded

by a cylindrical free volume of radius R, within which counterions are distributed sothat electroneutrality as a whole is satisfied. We define the particle volume fraction

� as

� ¼ ða=RÞ2 ð6:59Þ

The cylindrical particles are assumed to be parallel and equidistant. We treat the

case of dilute suspensions, namely, �� 1 or a/R� 1. Let the average number den-

sity and the valence of counterions in the absence of the applied electric field be nand �z, respectively. Then we have

Q ¼ 2pas ¼ pðR2 � a2Þzen ð6:60Þ

We set the equilibrium electric potential c(0)(r) (where r is the distance from the

axis of one cylinder) to zero at points where the volume charge density resulting

from counterions equals its average value (�zen). We assume that the distribution

of counterions obeys a Boltzmann distribution (Eq. (6.4)). The cylindrical Poisson–

Boltzmann equation thus becomes

d2y

dr2þ 1

r

dy

dr¼ k2ey ð6:61Þ

with

k ¼ z2e2n

ereokT

� �1=2

¼ ze

ereokT� 2asðR2 � a2Þ

� �1=2¼ zeQ

pereokTðR2 � a2Þ� �1=2

ð6:62Þ

where y(r)¼ zec(r)/kT is the scaled equilibrium potential and k is the Debye–

H€uckel parameter in the present system. The boundary conditions are

CYLINDRICAL PARTICLE 143

dcdr

��r¼a

¼ � sereo

¼ � Q

2pereoað6:63Þ

dcdr

��r¼R

¼ 0 ð6:64Þ

The Poisson–Boltzmann equation (6.61) subject to boundary conditions (6.63)

and (6.64) has been solved independently by Fuoss et al. [1] and Afrey et al. [2].

The results are given below.

(a) Case 1 (low surface charge case). If the condition

0 < Q� <lnða=RÞ

lnða=RÞ � 1ð6:65Þ

is satisfied, then we have

yðaÞ ¼ �lnk2a2

2� 1

ð1� Q�Þ2 � b2

" #ð6:66Þ

yðRÞ ¼ �lnk2R2

2� 1

1� b2

� �ð6:67Þ

where Q� is a scaled particle charge per unit length defined by

Q� ¼ Q

4pereo� zekT

ð6:68Þ

(Note that Q� is always positive, since Q and z are always of the same sign)

and b is given by the solution to the following transcendental equation:

Q� ¼ 1� b2

1� b cothfb lnða=RÞg ð6:69Þ

(b) Case 2 (high surface charge case). If the condition

Q� lnða=RÞlnða=RÞ � 1

ð6:70Þ


is satisfied, then we have

yðaÞ ¼ �lnk2a2

2� 1

ð1� Q�Þ2 þ b2

" #ð6:71Þ

yðRÞ ¼ �lnk2R2

2� 1

1þ b2

� �ð6:72Þ

where b is given by the solution to the following transcendental equation:

Q� ¼ 1þ b2

1� b cotfb lnða=RÞg ð6:73Þ

Now we confine ourselves to the dilute case a/R� 1, in which case the ln

(a/R)/(ln(a/R)� 1) on the right-hand side of Eqs. (6.65) and (6.70), which is

the critical value separating cases 1 (low charge case) and 2 (high charge

case), tends to unity. Also in this case we find that Eqs. (6.60) and (6.62) give

Q ¼ pR2zen ð6:74Þ

Q� ¼ k2R2

4ð6:75Þ

and that the above results for cases 1 and 2 can be rewritten as follows [2].


0 < Q� < 1 ð6:76Þ

is satisfied, then we have that

yðaÞ ¼ Q� lnð1=�Þ ð6:77Þ

yðRÞ ¼ ln 1� Q�

2

� �ð6:78Þ


Q� 1 ð6:79Þ

CYLINDRICAL PARTICLE 145

is satisfied, then we have that

yðaÞ ¼ lnQ�

2�

� �ð6:80Þ

yðRÞ ¼ �lnð2Q�Þ ð6:81Þ

Further, if we define the particle surface potential co as co¼c(a)�c(R)or the scaled surface potential yo� (ze/kT)co as yo¼ y(a)� y(R), then we

obtain for the dilute case

yo ¼ Q� lnð1=�Þ; for 0 < Q� < 1 ðcase 1: low surface charge caseÞð6:82Þ

yo ¼ lnQ�2

�

� �; forQ� 1 ðcase 2: high surface charge caseÞ ð6:83Þ

Equation (6.82) agrees with the unscreened Coulomb potential of a

charged cylinder. That is, the surface potential in this case is the same as

if the counterions were absent.

6.4 EFFECTS OF A SMALL AMOUNT OF ADDED SALTS

We consider a suspension of spherical colloidal particles of radius a with mono-

valent ionized groups on their surface in a medium which contains counterions pro-

duced by dissociation of the particle surface groups and a small amount of

monovalent symmetrical electrolytes [8]. We may assume that the particles are pos-

itively charged without loss of generality. We treat the case in which N ionized

groups of valence +1 are uniformly distributed on the surface of each particle and Ncounterions of valence �1 are produced by dissociation of the particle surface

groups. Each particle thus carries a charge Q¼ eN, where e is the elementary elec-

tric charge. We also assume that each particle is surrounded by a spherical free vol-

ume of radius R (Fig. 6.1), within which electroneutrality as a whole is satisfied.

The particle volume fraction � is given by �¼ (a/R)3. We treat the case of dilute

(but not infinitely dilute) suspensions, namely, �� 1 or a/R� 1. We assume that

the electrolyte is completely dissociated to give M cations of valence +1 and Manions of valence �1 in each free volume so that there are N+M counterions and

M coions in the free volume. Let the average concentration (number density) of

total counterions be n+m and that of anions be m, n and m being given by

n ¼ N

V; m ¼ M

Vð6:84Þ


where

V ¼ 4

3pðR3 � a3Þ ð6:85Þ

is the volume available for counterions and coions within each free volume. From

the condition of electroneutrality in each free volume, we have

Q ¼ eN ¼ Ven ð6:86Þ

We assume that the equilibrium distribution of ions obeys a Boltzmann distribu-

tion so that volume charge density r(r) at position r, r being the distance from the

particle center (r a), is given by

rðrÞ ¼ e �ðnþ mÞey V

4pR R

a eyr2 drþ mey

V

4pR R

a e�yr2 dr

" #ð6:87Þ

where y(r)¼ ec(r)/kT is the scaled potential. Note that Eq. (6.87) satisfies the elec-

troneutrality condition of a free volume, namely,

4pZ R

a

rðrÞr2dr ¼ ef�ðN þMÞ þMg ¼ �eN ¼ �Q ð6:88Þ

We set the electric potential c(r) to zero at points where the concentration of total

counterions equals its average value nþm so that

4pZ R

a

eyr2dr ¼ V ¼ 4p3ðR3 � a3Þ ð6:89Þ

and Eq. (6.87) becomes

rðrÞ ¼ en �ð1þ pÞey þ p

We�y

h ið6:90Þ

with

W ¼ 4pR Ra e�yr2 dr

V¼

R Ra e�yr2 dr

ðR3 � a3Þ=3 ð6:91Þ

and

p ¼ m

nð6:92Þ

EFFECTS OF A SMALL AMOUNT OF ADDED SALTS 147

where p is the ratio of the concentration of counterions (or coions) resulting from

the added electrolytes to that of counterions from the particle. The case of p¼ 0

corresponds to the completely salt-free case.

By combining the Poisson equation (6.5) and Eq. (6.87), we obtain the following

Poisson–Boltzmann equation for the equilibrium electric potential c(r) at positionr, r being the distance from the particle center (r a):

d2y

dr2þ 2

r

dy

dr¼ k2

1þ 2pð1þ pÞey � p

We�y

h ið6:93Þ

where k is the Debye–H€uckel parameter defined by

k ¼ e2ðnþ 2mÞereokT

� �1=2¼ 3Q��ð1þ 2pÞ

a2ð1� �Þ� �1=2

ð6:94Þ

and

Q� ¼ Q

4pereoae

kT

� �¼ k2a2ð1� �Þ

3�ð1þ 2pÞ ð6:95Þ

is the scaled particle charge. Note that k generally depends on Q�, p, a, and � and

that when n�m, k becomes the usual Debye–H€uckel parameter of a 1–1 electro-

lyte of concentration m, given by

k ¼ 2e2m

ereokT

� �1=2

ð6:96Þ

The boundary conditions are the same as Eqs. (6.11) and (6.12).

We derive some approximate solutions to Eq. (6.93). It can be shown that as in

the case of completely salt-free case, there are three possible regions I, II, and III in

the potential distribution around the particle surface for the dilute case. As will be

seen later, region I (where y(r) y(R)) is found to be much wider than regions II

and III, and in region III the potential is very high so that there are essentially no

coions. We may thus approximate Eq. (6.91) as

W e�yðRÞ R R

a r2 dr

ðR3 � a3Þ=3 ¼ e�yðRÞ; for � � 1 ð6:97Þ

Thus Eq. (6.93) can be approximated by

d2y

dr2þ 2

r

dy

dr¼ k2

1þ 2p

ð1þ pÞeyðrÞ � p e�fyðrÞ�yðRÞg ð6:98Þ


or, equivalently

d2y

dr2þ 2

r

dy

dr¼ 3aQ�

R3ð1� �Þð1þ pÞeyðrÞ � p e�fyðrÞ�yðRÞg ð6:99Þ

For p¼ 0, Eq. (6.98) tends to the Poisson–Boltzmann equation for the completely

salt-free case (Eq. (6.61)). For large p� 1, y(R) approaches 0 (as shown later by

Eq. (6.123) and Eq. (6.99) tends to

d2y

dr2þ 2

r

dy

dr¼ k2 sinh y ð6:100Þ

with k given by Eq. (6.96). Equation (6.100), which ignores counterions from the

particle, is the familiar PB equation for media containing ample salts (Eq. (1.27)).

In region I, y(r) y(R) so that Eq. (6.99) can further be approximated by, for the

dilute case (�� 1),

d2y

dr2þ 2

r

dy

dr¼ 3aQ�

R3

ð1þ pÞeyðRÞ � p ð6:101Þ

which, subject to Eq. (6.12), can be solved to give

yðrÞ ¼ Q�ð1þ pÞeyðRÞ � p a

r

� �1þ r3

2R3� 3r

2R

� �þ yðRÞ; for R� � r � R

ð6:102Þ

In the dilute case (a/R� 1), as r decreases to R�, approaching region II, the term

proportional to 1/r in Eq. (6.102) becomes dominant, that is, y(r) becomes an un-

screened Coulomb potential. Since y(r) must be continuous at r¼R�, y(r) in region

II should be of the form

yðrÞ ¼ Q�ð1þ pÞeyðRÞ � p a

r

� �þ yðRÞ; for a� � r � R� ð6:103Þ

Note that Eq. (6.103) satisfies

d2y

dr2þ 2

r

dy

dr¼ 0 ð6:104Þ

and that Eq. (6.98) becomes Eq. (6.104), if the following two conditions are both

satisfied in addition to Eq. (6.2): (i) the terms on the right-hand side of Eq. (6.98)

are very small as compared with a2 and (ii) these terms are also much smaller than

the second term of the left-hand side of Eq. (6.98). The first condition may be

expressed as

ka � 1 ð6:105Þ


or equivalently,

ð1þ 2pÞQ�� 1 ð6:106Þ

and the second condition as

1

r

dy

dr

�� 3aQ�

R3

ð1þ pÞeyðrÞ � p e�fyðrÞ�yðRÞg ð6:107Þ

By evaluating Eq. (6.107) at r¼ a� using Eq. (6.103), we obtain

ð1þ pÞeyðRÞ � p � 3a�3

R3

ð1þ pÞeyða�Þ � p e�fyða�Þ�yðRÞg ð6:108Þ

which yields, by taking the logarithm of both sides,

yða�Þ � yðRÞ < ln 1� p

1þ pe�yðRÞ

� �þ ln

R3

a�3

� � lnð1=�Þ ð6:109Þ

Equation (6.109) implies that the value of y(a�)� y(R) cannot exceed ln(1/�), whichis thus the possible maximum value of y(a�)� y(R). From Eq. (6.103), on the other

hand, we obtain

yða�Þ � yðRÞ ¼ �a��dydr

��r¼a�

ð6:110Þ

By combining Eqs. (6.109) and (6.110), we obtain

�a�dy

dr

��r¼a�

< lnð1=�Þ; for � � 1 ð6:111Þ

We must thus consider two cases 1 and 2 separately, depending on whether is larger

or smaller than ln(1/�). This situation is the same as that for p¼ 0. The results are

summarized below.

(a) Case 1 (low surface charge case). Consider first case 1, in which the

following condition holds:

Q� � lnð1=�Þ ð6:112Þ

In this case, region II can extend up to r¼ a, that is, a� can reduce to a. Thus,the entire region of r can be covered only with regions I and II and one does

not need region III. For the low-charge case, Eq. (6.101), which is now


applicable essentially for the entire region of r, can further be simplified to

give

yðrÞ ¼ Q� ar

1þ r3

2R3� 9r

5R

� �; a � r � R ð6:113Þ

which is independent of p. For the dilute case (�� 1), Eq. (6.113) gives

yðRÞ 0 ð6:114Þ

and becomes essentially a Coulomb potential, namely,

yðrÞ ¼ Q� ar; a � r � R ð6:115Þ

The scaled particle surface potential yo¼ y(a)� y(R)¼ y(a) in case 1 is thus

given by

yo ¼ Q� ð6:116Þ

The above results do not depend on p and agree with those for the completely

salt- free case.

(b) Case 1 (high surface charge case). Consider next case 2, in which the condi-

tion

Q� > lnð1=�Þ ð6:117Þ

is satisfied so that

Q� > �a�dy

dr

��r¼a�

ð6:118Þ

That is, region II cannot extend up to r¼ a. Therefore, the entire region of rcannot be covered only with regions I and II and thus one needs region III

very near the particle surface between r¼ a and r¼ a�, where counterions

are condensed (counter-ion condensation). In region III, y(r) is very high so

that Eq. (6.99) becomes

d2y

dr2¼ 3�Q�

a2ð1þ pÞeyðrÞ ð6:119Þ

By integrating Eq. (6.119), we obtain for the dilute case

yðaÞ ¼ lnQ�

6�ð1þ pÞ� �

ð6:120Þ


The value of y(R) can be obtained as follows. In case 2, y(a�)� y(R) hasalready reached its maximum ln(1/�) so we find that Eq. (6.102) becomes

yðrÞ ¼ Q�eff

a

rþ yðRÞ; a� � r � R� ð6:121Þ

where

Q�eff ¼ yða�Þ � yðRÞ ¼ Q�ð1þ pÞeyðRÞ � p

¼ lnð1=�Þ ð6:122Þ

is the scaled effective particle charge and the value of y(R) is given by

yðRÞ ¼ lnlnð1=�Þ þ pQ�

Q�ð1þ �Þ� �

ð6:123Þ

which is obtained from the last equation of Eq. (6.122). We see from

Eq. (6.123) that y(R) tends to zero as p increases. Equation (6.121) implies

that in region II the potential distribution around the particle is a Coulomb

potential produced by the scaled particle charge Q�eff ¼ ln(1/�). In other

words, the particle behaves as if the particle charge were Q�eff instead of Q�

for regions I and II, in which regions Eq. (6.102) may be expressed as

yðrÞ ¼ Q�eff

a

r1þ r3

2R3� 3r

2R

� �þ yðRÞ; a� � r � R ð6:124Þ

The particle surface potential yo¼ y(a)� y(R) in case 2 is found to be, from

Eqs. (6.120) and (6.123),

yo ¼ lnðQ�Þ2

6�flnð1=�Þ þ pQ�g

" #ð6:125Þ

When p¼ 0, Eqs. (6.120), (6.123) and (6.125) all reduce back to results ob-

tained for the completely salt-free case. Also note that the critical value sepa-

rating cases 1 and 2 is ln(1/�), which is the same as that for the completely

salt-free case.

6.5 SPHERICAL SOFT PARTICLE

Consider a dilute suspension of polyelectrolyte-coated spherical colloidal particles

(soft particles) in a salt-free medium containing counterions only. We assume that

the particle core of radius a (which is uncharged) is coated with an ion-penetrable

layer of polyelectrolytes of thickness d. The polyelectrolyte-coated particle has thusan inner radius a and an outer radius b� a + d (Fig. 6.4). We also assume that ion-

ized groups of valence Z are distributed at a uniform density N in the polyelectrolyte


layer so that the charge density in the polyelectrolyte layer is given by ZeN, where eis the elementary electric charge. The total charge amount Q of the particle is thus

given by

Q ¼ 4

3pðb3 � a3ÞZeN ð6:126Þ

We assume that each sphere is surrounded by a spherical free volume of radius R(Fig. 6.4), within which counterions are distributed so that electroneutrality as a

whole is satisfied. The particle volume fraction � is given by

� ¼ ðb=RÞ3 ð6:127Þ

In the following we treat the case of dilute suspensions, namely,

� � 1 or b=R � 1 ð6:128Þ

We denote the electric potential at a distance r from the center O of one particle by

c(r). Let the average number density and the valence of counterions be n and �z,respectively. Then from the condition of electroneutrality in the free volume,

we have

Q ¼ 4

3pðb3 � a3ÞZeN ¼ 4

3pðR3 � a3Þzen ð6:129Þ

FIGURE 6.4 A polyelectrolyte-coated spherical particle (a spherical soft particle) in

a free volume of radius R containing counterions only. a is the radius of the particle core.

b¼ aþ d. d is the thickness of the polyelectrolyte layer covering the particle core. (b/R)3

equals the particle volume fraction �. From Ref. [9].

SPHERICAL SOFT PARTICLE 153

which gives

ZN

zn¼ R3 � a3

b3 � a3ð6:130Þ

We set c(r)¼ 0 at points where the counter-ion concentration equals its average

value n so that the volume charge density r(r) resulting from counterions equals

its average value (�zen). We assume that the distribution of counterions obeys a

Boltzmann distribution, namely,

rðrÞ ¼ �zen exp ��zecðrÞkT

� �¼ �zen exp

zecðrÞkT

� �ð6:131Þ

The Poisson equations for the electric potential c(r) in the region inside and outsidethe polyelectrolyte layer are then given by

d2cðrÞdr2

þ 2

r

dcðrÞdr

¼ � rðrÞereo

� ZeN

ereo; a < r < b ð6:132Þ

d2cðrÞdr2

þ 2

r

dcðrÞdr

¼ � rðrÞereo

; b < r < R ð6:133Þ

Here we have assumed that the relative permittivity er takes the same value in the

medium inside and outside the polyelectrolyte layer. By combining Eqs. (6.131)–

(6.133), we obtain the following Poisson–Boltzmann equations:

d2y

dr2þ 2

r

dy

dr¼ k2 ey � ZN

zn

� �; a < r < b ð6:134Þ

d2y

dr2þ 2

r

dy

dr¼ k2ey; b < r < R ð6:135Þ

where y(r)¼ zec(r)/kT is the scaled potential and

k¼ z2e2n

ereokT

� �1=2

¼ Zze2Nðb3� a3ÞereokTðR3� a3Þ

� �1=2¼ 3zeQ

4pereokTðR3� a3Þ� �1=2

¼ 3Q�bR3� a3

� �1=2

ð6:136Þis the Debye–H€uckel parameter in the present system. In Eq. (6.136), Q� is the

scaled particle charge defined by

Q� ¼ Q

4pereobze

kT

� �ð6:137Þ


By using Eq. (6.130), we can rewrite Eq. (6.134) as

d2y

dr2þ 2

r

dy

dr¼ k2 ey� R3� a3

b3� a3

� �� ¼ 3Q�b

R3ey� R3� a3

b3� a3

� �� ; a< r< b ð6:138Þ

which becomes for the dilute case

d2y

dr2þ 2

r

dy

dr¼ k2 ey� R3

b3� a3

� �¼ 3Q�b

R3ey� R3

b3� a3

� �; a< r< b ð6:139Þ

The boundary conditions for y are expressed as

dy

dr

��r¼a

¼ 0 ð6:140Þ

dy

dr

��r¼R

¼ 0 ð6:141Þ

yðb�Þ ¼ yðbþÞ ð6:142Þ

dy

dr

��r¼b�

¼ dy

dr

��r¼bþ

ð6:143Þ

We derive approximate solutions to Eqs. (6.135) and (6.139) for the dilute case by

using an approximation method described in Ref. [5], as follows.


Q� � lnð1=�Þ ð6:144Þ

is satisfied, Eq. (6.135) can be approximated by

d2y

dr2þ 2

r

dy

dr¼ k2; b < r < R ð6:145Þ

from which we have

yðrÞ ¼ Q� br

1þ r3

2R3� 9r

5R

� �; ðb � r � RÞ ð6:146Þ


or

cðrÞ ¼ Q

4pereor1þ r3

2R3� 9r

5R

� �; ðb � r � RÞ ð6:147Þ

For the dilute case (�� 1), Eq. (6.147) can be approximated well by an un-

screened Coulomb potential of a sphere carrying charge Q, namely,

cðrÞ ¼ Q

4pereor; ðb � r � RÞ ð6:148Þ

That is, in the low charge case, the potential is essentially the same as if con-

terions were absent. This suggests that the potential inside the polyelectrolyte

layer is also approximately the same as if conterions were absent. If counter-

ions are absent, then Eq. (6.139) becomes

d2y

dr2þ 2

r

dy

dr¼ � k2R3

b3 � a3¼ � 3bQ�

b3 � a3; ða < r < bÞ ð6:149Þ

with the solution

yðrÞ ¼ Q�b3

ðb3 � a3Þ3

2� r2

2b2� a3

b2r

� �; ða � r � bÞ ð6:150Þ

or

cðrÞ ¼ Qb2

4pereoðb3 � a3Þ3

2� r2

2b2� a3

b2r

� �; ða � r � bÞ ð6:151Þ

If the surface potential cs of the polyelectrolyte-coated particle is identified

as cs¼c(b)�c(R), then we find that for the dilute case

cðRÞ ¼ 0 ð6:152Þ

co ¼ cðbÞ ¼ Q

4pereobð6:153Þ

or

yðRÞ ¼ 0 ð6:154Þ

yo ¼ Q� ð6:155Þ


where

yo ¼zeco

kTð6:156Þ

is the scaled surface potential (yo¼ y(b)� y (R)).


Q� > lnð1=�Þ ð6:157Þ

is satisfied, then as in the case of hard particles, we find that for the outer

solution (b � r � R),

dy

dr¼ �

ffiffiffi2

pkey=2 ðr bÞ ð6:158Þ

yðrÞ ¼ ln1

�

� �b

r

� �1þ r3

2R3� 3r

2R

� �þ yðRÞ; ðb < r � RÞ ð6:159Þ

yðRÞ ¼ �lnQ�

lnð1=�Þ� �

ð6:160Þ

Note that Eqs. (6.158)–(6.160) are independent of a.For the inner region (a� r� b), by noting that the potential variation is very

small as compared with that in the outer region (b� r�R), we may replace

the spherical Poisson–Boltzmann equation [6.139] with the following planar

Poisson–Boltzmann equation:

d2y

dr2¼ k2 ey � R3

b3 � a3

� �� ; a < r < b ð6:161Þ

The solution to Eq. (6.161) is

dy

dr¼ �

ffiffiffi2

pk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexpðyÞ � expðyðaÞÞ � R3

b3 � a3

� �y� yðaÞf g

sð6:162Þ

By evaluating Eqs. (6.158) and (6.162) at r¼ b and using Eq. (6.143),

we have

yðbÞ ¼ yðaÞ � b3 � a3

R3

� �exp yðaÞ½ � ð6:163Þ


Equation (6.162) can further be integrated to yield

� ffiffiffi2

pkðr � aÞ ¼

Z y

yðaÞ

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexpðyÞ � expðyðaÞÞ � ðR3=ðb3 � a3ÞÞfy� yðaÞg

q

�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiyðaÞ � y

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR3=ðb3 � a3Þ � exp½yðaÞ�

qð6:164Þ

where we have expanded the integrand around y¼ y(a) and carried out the

integration, since the largest contribution to the integrand comes from the

region near y¼ y(a). Equation (6.164) yields

yðrÞ ¼ yðaÞ þ eyðaÞ � R3

ðb3 � a3Þ

� �k2ðr � aÞ2 ð6:165Þ

which may also be obtained by directly expanding y(r) around r¼ a on the

basis of Eq. (6.139). From Eqs. (6.163) and (6.165) (evaluated at r¼ b) weobtain

yðaÞ ¼ ln3Q�bR3

f3Q�bðb� aÞ þ 2ðb2 þ abþ a2Þgðb2 þ abþ a2Þ

� �ð6:166Þ

and

yðbÞ ¼ ln3Q�bR3

f3Q�bðb� aÞ þ 2ðb2 þ abþ a2Þgðb2 þ abþ a2Þ

� �

� 3Q�bðb� aÞf3Q�bðb� aÞ þ 2ðb2 þ abþ a2Þg

ð6:167Þ

Thus, we find that the scaled surface potential yo¼ y(b)� y(R) is given by

yo ¼ ln3Q�2bR3

lnð1=�Þf3Q�bðb� aÞ þ 2ðb2 þ abþ a2Þgðb2 þ abþ a2Þ

� �

� 3Q�bðb� aÞf3Q�bðb� aÞ þ 2ðb2 þ abþ a2Þg

ð6:168Þ

The Poisson–Boltzmann equations (6.135) and (6.139) subject to boundary condi-

tions (6.140)–(6.143) can be solved numerically with Mathematica to obtain

the potential distribution y(x). The results of some calculations for the potential


distribution are given in Figs 6.5 and 6.6, which demonstrate y(r) for Q�¼ 5 (low

charge case) and 50 (high charge case) at �¼ 10�6 and a/b¼ 0.9. Figure 6.5 shows

the exact numerical results for the inner solution (a � r � b) in comparison with

approximate results (Eq. (6.150) for Q�¼ 5 and Eq. (6.165) for Q�¼ 50), showing

FIGURE 6.6 Distribution of the scaled electric potential y(r) outside the polyelectrolyte

layer (b� r�R) at �¼ 10�6 and a/b¼ 0.9 for Q�¼ 5 (low charge case) and Q�¼ 50 (high

charge case). Approximate results (Eq. (6.146) for Q�¼ 5 and Eq. (6.159) for Q�¼ 50) agree

with exact results within the linewidth. From Ref. [9].

FIGURE 6.5 Distribution of the scaled electric potential y(r) within the polyelectrolyte

layer (a� r� b) at �¼ 10�6 and a/b¼ 0.9 for Q�¼ 5 (low charge case) and Q�¼ 50 (high

charge case). Solid curves, exact results; dotted curves, approximate results (Eq. (6.150) for

Q�¼ 5 and Eq. (6.165) 6. for Q�¼ 50). From Ref. [9].


good agreement between numerical and approximate results. Figure 6.6 shows the

exact numerical results for the outer region b� r�R, which agree with approxi-

mate results (Eq. (6.146) for Q�¼ 5 and Eq. (6.159) for Q�¼ 50) within the line-

width. It is seen that for the high charge case (Q�¼ 50) the potential outside

the polyelectrolyte layer decreases very sharply near the surface of the poly-

electrolyte layer (r¼ b) and essentially constant (y(r) y(R)) except in the region

very near the polyelectrolyte layer.

In Fig. 6.7, the scaled surface potential ys¼ y(b)� y(R) is plotted as a func-

tion of Q� at a/b¼ 0.5 for �¼ 10�3, 10�6, and 10�9. We see that as in the case

of a rigid particle, there is a certain critical value separating the low charge

case and the high charge case. Approximate results (Eq. (6.155) for the low

charge case and Eq. (6.168) for the high charge case) are also shown in Fig.

6.7. An approximate expression for the critical value Q�cr is given by (see Eqs.

(6.144) and (6.157))

Q�cr ¼ lnð1=�Þ ð6:169Þ

When Q��Q�cr (low charge case), the surface potential may be approximated by

Eq. (6.155), which is the unscreened Coulomb potential for the case where coun-

terions are absent. The surface potential ys is proportional to Q� in this case. When

Q�>Q�cr (high charge case), the surface potential can be approximated by

FIGURE 6.7 Scaled surface potential yo¼ zeco/kT, defined by yo� y(b)� y(R), as a func-tion of the scaled charge Q�= (ze/kT)Q/4pereob at a/b¼ 0.5 for various values of the particle

volume fraction �. Solid lines represent exact numerical results. The dashed line, which

passes through the origin (yo¼ 0 and Q�¼ 0), is approximate results calculated by an un-

screened Coulomb potential, Eq. (6.155) (low charge case) and dotted lines by Eq. (6.168)

(high charge case). Closed circles correspond to the approximate critical values Qcr� of

the scaled particle charge Q� separating cases, the low and high charge cases, given by

Qcr�¼ ln(1/�). From Ref. [9].


Eq. (6.168). The dependence of the surface potential ys upon the particle charge

Q� is considerably suppressed since the counterions are accumulated within and

near the polyelectrolyte layer (counter-ion condensation).

For the limiting case of a¼ b (in which case the polyelectrolyte-coated particle

becomes a rigid particle with no polyelectrolyte layer), Eq. (6.168) tends to

yo ¼ lnQ�2

6� lnð1=�Þ� �

ð6:170Þ

which agrees with Eq. (6.50) for the surface potential of a rigid particle in a salt-free

medium. For the case where a � b and Q is high, Eq. (6.168) tends to

yo ¼ lnQ�

expð1Þ� lnð1=�Þð1� a3=b3Þ

� �ð6:171Þ

When a¼ 0, in particular, the polyelectrolyte-coated particle becomes a spherical

polyelectrolyte with no particle core. In this case, Eq. (6.171) tends to

ys ¼ lnQ�

expð1Þ� lnð1=�Þ� �

ð6:172Þ

FIGURE 6.8 Scaled surface potential yo¼ zeco/kT, defined by yo� y(b)� y(R), as a func-tion of the ratio d/a of the thickness d¼ b� a of the polyelectrolyte layer to the radius a of

the particle core at �¼ 10�6 for Q�¼ 5 (low charge case) and Q�¼ 50 (high charge case).

Solid lines represent exact numerical results. The dotted lines are approximate results calcu-

lated by an unscreened Coulomb potential (Eq. (6.155)) for the low charge case and by Eq.

(6.168) for the high charge case. From Ref. [9]


Figure 6.8 shows yo as a function of the ratio d/a of the polyelectrolyte layer thick-

ness d to the core radius a for two values of Q� (5 and 50) at �¼ 10�6. Note that as

d/a tends to zero, the polyelectrolyte-coated particle becomes a hard sphere with

no polyelectrolyte layer, while as d/a tends to infinity, the particle becomes a

spherical polyelectrolyte with no particle core. Approximate results calculated with

Eq. (6.155) for Q�¼ 5 (low charge case) and Eq. (6.168) for Q�¼ 50 (high charge

case) are also shown in Fig. 6.8. Agreement between exact and approximate results

is good. For the low charge case, the surface potential is essentially independent

of d and is determined only by the charge amount Q�. In the example given in

Fig. 6.8, for the high charge case, the particle behaves like a hard particle with no

polyelectrolyte layer for d/a � 10�3 and the particle behaves like a spherical poly-

electrolyte for d/a � 1.

REFERENCES

1. R. M. Fuoss, A. Katchalsky, and S. Lifson, Proc. Natl. Acad. Sci. USA 37 (1951) 579.

2. T. Afrey, P. W. Berg, and H. Morawetz, J. Polym. Sci. 7 (1951) 543.

3. N. Imai and F. Oosawa, Busseiron Kenkyu 52 (1952) 42; 59 (1953) 99 (in Japanese).

4. F. Oosawa, Polyelectrolytes, Dekker, New York, 1971.

5. H. Ohshima, J. Colloid Interface Sci. 247 (2002) 18.


7. H. Ohshima, Theory of Colloid and Interfacial Electric Phenomena, Elsevier/Academic

Press, Amsterdam, 2006.

8. H. Ohshima, Colloid Polym. Sci. 282 (2004) 1185.



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