biophysical chemistry of biointerfaces (ohshima/biophysical chemistry of biointerfaces) ||...

15 Electrostatic InteractionBetween Soft Particles

15.1 INTRODUCTION

In this chapter, we give approximate analytic expressions for the force and potential

energy of the electrical double-layer interaction two soft particles. As shown in

Fig. 15.1, a spherical soft particle becomes a hard sphere without surface structures,

while a soft particle tends to a spherical polyelectrolyte when the particle core is

absent. Expressions for the interaction force and energy between two soft particles

thus cover various limiting cases that include hard particle/hard particle interaction,

soft particle/hard particle interaction, soft particle/porous particle interaction, and

porous particle/porous particle interaction.

15.2 INTERACTION BETWEEN TWO PARALLEL DISSIMILARSOFT PLATES

Consider two parallel dissimilar soft plates 1 and 2 at a separation h between their

surfaces immersed in an electrolyte solution containing N ionic species with va-

lence zi and bulk concentration (number density) n1i (i¼ 1, 2, . . . , N) [1]. We

assume that each soft plate consists of a core and an ion-penetrable surface charge

layer of polyelectrolytes covering the plate core and that there is no electric field

within the plate core. We denote by dl and d2 the thicknesses of the surface chargelayers of plates 1 and 2, respectively. The x-axis is taken to be perpendicular to the

plates with the origin at the boundary between the surface charge layer of plate 1

and the solution, as shown in Fig. 15.2. We assume that each surface layer is uni-

formly charged. Let Zl and Nl, respectively, be the valence and the density of fixed--

charge groups contained in the surface layer of plate 1, and let Z2 and N2 be the

corresponding quantities for plate 2. Thus, the charge densities rfix1 and rfix2 of thesurface charge layers of plates 1 and 2 are, respectively, given by

rfix1 ¼ Z1eN1 ð15:1Þrfix2 ¼ Z2eN2 ð15:2Þ

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

357

The Poisson–Boltzmann equations for the present system are then

d2cdx2

¼ � 1

ereo

XMi¼1

zien1i exp � ziec

kT

� �� rfix1

ereo; � d1 < x < 0 ð15:3Þ

FIGURE 15.2 Interaction between two parallel soft plates 1 and 2 at separation h and the

potential distribution c(x) across plates 1 and 2, which are covered with surface charge layersof thicknesses d1 and d2, respectively.

FIGURE 15.1 A soft sphere becomes a hard sphere in the absence of the surface layer of

polyelectrolyte while it tends to a spherical polyelectrolyte (i.e., porous sphere) when the

particle core is absent.

358 ELECTROSTATIC INTERACTION BETWEEN SOFT PARTICLES

d2cdx2

¼ � 1

ereo

XMi¼1

zien1i exp � ziec

kT

� �; 0 < x < h ð15:4Þ

d2cdx2

¼ � 1

ereo

XMi¼1

zien1i exp � ziec

kT

� �� rfix2

ereo; h < x < hþ d2 ð15:5Þ

where c(x) is the electric potential at position x relative to the one at a point in the

bulk solution far from the plates (the plates are actually surrounded by the electro-

lyte solution). We assume that the relative permittivity er in the surface layers takes

the same value as that in the electrolyte solution.

We consider the case where N1 and N2 are low. The Poisson–Boltzmann equa-

tions (15.3)–(15.5) can be linearized to give

d2cdx2

¼ k2c� rfix1ereo

; � d1 < x < 0 ð15:6Þ

d2cdx2

¼ k2c; 0 < x < h ð15:7Þ

d2cdx2

¼ k2c� rfix2ereo

; h < x < hþ d2 ð15:8Þ

where k is the Debye–Huckel parameter (Eq. (1.10)).

The boundary conditions are

cð0�Þ ¼ cð0þÞ ð15:9Þ

cðh�Þ ¼ cðhþÞ ð15:10Þ

dcdx

��x¼0�

¼ dcdx

��x¼0þ

ð15:11Þ

dcdx

��x¼h�

¼ dcdx

��x¼hþ

ð15:12Þ

dcdx

��x¼�dþ

1

¼ 0 ð15:13Þ

dcdx

��x¼hþd�2

¼ 0 ð15:14Þ

Integration of Eqs. (15.6)–(15.8) with the above boundary conditions gives

cðxÞ ¼ 1

ereok2rfix1 þ

f�rfix1 sinh½kðhþ d2Þ� þ rfix2 sinhðkd2Þgsinh½kðhþ d1 þ d2Þ� cosh kðxþ d1Þ½ �

� �;

�d1 � x � 0 ð15:15Þ

INTERACTION BETWEEN TWO PARALLEL DISSIMILAR SOFT PLATES 359

cðxÞ ¼ 1

ereok2rfix1 sinhðkd1Þcosh½kðhþ d2� xÞ�þrfix2 sinhðkd2Þcosh½kðxþ d1Þ�

sinh½kðhþ d1þ d2Þ��

;

0� x� h ð15:16Þ

cðxÞ¼ 1

ereok2rfix2þ

f�rfix2 sinh½kðhþd1Þ�þrfix1 sinhðkd1Þgsinh½kðhþd1þd2Þ� cosh kðhþd2�xÞ½ �

� �;

h� x� hþd2 ð15:17Þ

When the potential c(x) is low, the electrostatic force Ppl(h) between the

two parallel plates 1 and 2 at separation h per unit area can be calculated from

Eq. (10.18), namely,

PplðhÞ ¼ kTXNi¼1

n1i exp � ziecð0ÞkT

� �� 1

� �� 1

2ereo

dcdx

��x¼0

� �2

ð15:18Þ

which, for the low potential case, reduces to

PplðhÞ ¼ 1

2ereo k2c2ð0Þ � dc

dx

��x¼0

� �2" #

ð15:19Þ

The result is

PplðhÞ ¼ 1

8ereok2frfix1 sinhðkd1Þ þ rfix2 sinhðkd2Þg2

sinh2½kðhþ d1 þ d2Þ=2�

"

�frfix1 sinhðkd1Þ � rfix2 sinhðkd2Þg2cosh2½kðhþ d1 þ d2Þ=2�

#ð15:20Þ

Integrating Eq. (15.18) with respect to h gives the potential energy Vpl(h) of elec-trostatic interaction between the plates per unit area as a function of h:

VplðhÞ ¼ 1

4ereok3frfix1 sinhðkd1Þ þ rfix2 sinhðkd2Þg2 coth

kðhþ d1 þ d2Þ2

� �� 1

� ��

�(rfix1 sinhðkd1Þ � rfix2 sinhðkd2Þg2 1� tanh

kðhþ d1 þ d2Þ2

� ��

ð15:21Þ

For the special case of two similar soft plates carrying Z1¼ Z2¼ Z, N1¼N2¼N,and d1¼ d2¼ d so that rfix1¼ rfix2¼ rfix, Eqs. (15.20) and (15.21) reduce to

PplðhÞ ¼ r2fix2ereok2

sinh2ðkdÞsinh2½kðh=2þ dÞ� ð15:22Þ

VplðhÞ ¼ ðZeNÞ2!r2fixsinh2ðkdÞ

ereok3coth k

h

2þ d

� �� 1

� �ð15:23Þ


When the magnitude of c(x) is arbitrary, one must solve the original nonlinear

Poisson–Boltzmann equations (15.3)–(15.5). Consider the case of two parallel simi-

lar plates in a symmetrical electrolyte with valence z and bulk concentration n. Inthis case, we need consider only the region �d< x< h/2 so that Eqs. (15.3)–(15.5)

become

d2y

dx2¼ k2 sinh y; 0 < x < h=2 ð15:24Þ

d2y

dx2¼ k2ðsinh y� sinh yDONÞ; � d < x < 0 ð15:25Þ

with

y ¼ zeckT

ð15:26Þ

yDON ¼ zecDON

kTð15:27Þ

where y, cDON, and yDON are, respectively, the scaled potential, the Donnan poten-

tial given by Eq. (20.18), and the scaled Donnan potential. The boundary conditions

for y(x) similar to Eqs. (15.9)–(15.14) are

dy

dx

��x¼h=2

¼ 0 ð15:28Þ

yð0�Þ ¼ yð0þÞ ð15:29Þ

dy

dx

��x¼0�

¼ dy

dx

��x¼0þ

ð15:30Þ

dy

dx

��x¼�dþ

¼ 0 ð15:31Þ

Equation (15.31) follows from the symmetry of the system. The solution to

Eqs. (15.24) and (15.25) subject to the boundary conditions (15.28)–(15.29) takes a

complicated form, involving numerical integration. For the case where kd01, soft

plates can be approximated by porous membranes (see Chapter 13). In this case, the

value of y(h/2) can be calculated by solving the following coupled equations for twoparallel porous membranes (see Eqs. (13.94) and (13.95)):

2 cosh yDON þ ZNo

znyo � yDON½ � ¼ 2 cosh yðh=2Þ ð15:32Þ

coshyo2

� ¼ cosh

yðh=2Þ2

� �� dc kh

2� cosh yðh=2Þ

2

� �� 1=cosh yðh=2Þ

2

� �� ð15:33Þ

INTERACTION BETWEEN TWO PARALLEL DISSIMILAR SOFT PLATES 361

where dc is a Jacobian elliptic function with modulus 1/cosh(y(h/2)) and yo� y(0) isthe scaled unperturbed potential at the front edge x¼ 0 of the surface layer and is

given by Eq. (4.36).

The interaction force between two parallel similar plates per unit area Ppl(h) isgiven by Eq. (13.96), namely,

PðhÞ ¼ 4nkT sinh2ym2

� ¼ 4nkT sinh2

yðh=2Þ2

� �ð15:34Þ

A simple approximate analytic expression for Ppl(h) can be obtained using

the linear superposition approximation (LSA) (Chapter 11). In this approximation,

y(h/2) in Eq. (15.34) is approximated by the sum of the asymptotic values of the two

scaled unperturbed potentials ys(x) that is produced by the respective plates in the

absence of interaction. For two similar plates,

yðh=2Þ � 2ysðh=2Þ ð15:35Þ

This approximation is correct in the limit of large kh. It follows from Eq. (1.37),

the value of the unperturbed potential of a single plate at x¼ h/2 is given by

ysðh=2Þ ¼ 4 arctanh tanhyo4

� e�kh=2

h ið15:36Þ

where the scaled unperturbed surface potential yo is given by the solution to

Eqs. (15.32) and (15.33). Equation (15.36) becomes, for large kh,

ysðh=2Þ � 4 tanhyo4

� e�kh=2 ð15:37Þ

Hence

yðh=2Þ ¼ 8 tanhyo4

� e�kh=2 ð15:38Þ

For large kh, Eq. (15.36) asymptotes

PplðhÞ � 4nkTyðh=2Þ

2

� �2ð15:39Þ

Substituting Eq. (15.38) into Eq. (15.39), we obtain

PelðhÞ ¼ 64 tanhyo4

� 2nkT expð�khÞ ð15:40Þ

The potential energy Vpl(h) can be obtained by integrating Eq. (15.40) with the

result

VelðhÞ ¼ 64

ktanh

yo4

� 2nkT expð�khÞ ð15:41Þ

which is of the same form as that for hard plates, although the expressions for yo aredifferent for hard and soft plates.


15.3 INTERACTION BETWEEN TWO DISSIMILAR SOFT SPHERES

Consider the electrostatic interaction between two dissimilar spherical soft spheres

1 and 2 (Fig. 15.3). We denote by dl and d2 the thicknesses of the surface charge

layers of spheres 1 and 2, respectively. Let the radius of the core of soft sphere 1 be

a1 and that for sphere 2 be a2. We imagine that each surface layer is uniformly

charged. Let Zl and Nl, respectively, be the valence and the density of fixed-charge

layer of sphere 1 and Z2 and N2 for sphere 2.

With the help of Derjaguin’s approximation [2] (Eq. (12.2)), namely,

VspðHÞ ¼ 2pa1a2a1 þ a2

Z 1

H

VplðhÞdh ð15:42Þ

which is a good approximation if

ka1 � 1; ka2 � 1; H � a1; and H � a2 ð15:43Þ

one can calculate the interaction energy Vsp(H) between two dissimilar soft spheres

1 and 2 separated by a distance H between there surfaces via the corresponding

interaction energy Vpl(h) between two parallel dissimilar plates. By substituting

Eq. (15.20) into Eq. (15.42) we obtain [3]

V spðHÞ ¼ 1

ereok4pa1a2a1 þ a2

� �frfix1 sinhðkd1Þ þ rfix2 sinhðkd2Þg2h

ln1

1� e�kðHþd1þd2Þ

� �

�frfix1 sinhðkd1Þ � rfix2 sinhðkd2Þg2lnð1þ e�kðHþd1þd2ÞÞi

ð15:44Þ

FIGURE 15.3 Interaction between two soft spheres 1 and 2 at separation H. Spheres 1 and2 are covered with surface charge layers of thicknesses d1 and d2, respectively. The core radiiof spheres 1 and 2 are a1 and a2, respectively.

INTERACTION BETWEEN TWO DISSIMILAR SOFT SPHERES 363

For the special case of two similar soft spheres carrying Z1¼ Z2¼ Z, N1¼N2¼N, d1¼ d2¼ d so that rfix1¼ rfix2¼ rfix and a1¼ a2¼ a, Eq. (15.44) reduces to

VspðHÞ ¼ 2par2fix sinh2ðkdÞ

ereok4ln

1

1� e�kðHþ2dÞ

� �ð15:45Þ

We now introduce the quantities

s1 ¼ rfix1d1 ¼ Z1eN1d1; ð15:46Þ

s2 ¼ rfix2d2 ¼ Z2eN2d2 ð15:47Þ

which are, respectively, the amounts of fixed charges contained in the surface layers

per unit area on spheres 1 and 2. If we take the limit dl, d2! 0 and N1, N2!1,

keeping the products N1d1 and N2d2 constant, then s1 and s2 reduce to the surface

charge densities of two interacting hard plates without surface charge layers. In this

limit, Eqs. (15.20),(15.21) and (15.44) reduce to

PplðhÞ ¼ 1

8ereoðs1 þ s2Þ2cosech2 kh

2

� �� ðs1 � s2Þ2sech2 kh

2

� �� ð15:48Þ

VplðhÞ ¼ 1

4ereokðs1 þ s2Þ2 coth

kh2

� �� 1

� �� ðs1 � s2Þ2 1� tanh

kh2

� �� 1

� �� ð15:49Þ

V spðHÞ ¼ 1

ereok2pa1a2a1 þ a2

� �ðs1 þ s2Þ2ln 1

1� e�kH

� �� ðs1 � s2Þ2lnð1þ e�kHÞ

� �ð15:50Þ

Equations (15.49) and (15.50), respectively, agrees with the expression for the

electrostatic interaction energy between two parallel hard plates at constant surface

charge density and that for two hard spheres at constant surface charge density [4]

(Eqs. (10.54) and (10.55)).

In order to see the effects of the thickness of the surface charge layer, we calcu-

late the interaction energy Vsp(H) via Eq. (15.44) for the case of two dissimilar

soft spheres with fixed charges of like sign and that for spheres with fixed

charges of unlike sign and illustrate the results calculated for several values of kd1¼ kd1¼ kd with s1 and s2 kept constant at s2/s1¼ 0.5 (Fig. 15.4) and at s2/s1¼�0.5 (Fig. 15.5), showing a remarkable dependence of Vsp(H) upon kd. This isbecause electrolyte ions can penetrate the surface charge layer, exerting the shield-

ing effect on the fixed charges in the surface charge layers. Because of the ion pene-

tration, the increase in potential inside the interacting plates due to their approach is

much less than that for kd¼ 0. In particular, when kdl� 1 and kd2� 1, being ful-

filled for practical cases, the potential deep inside the plates remains constant, equal

to the Donnan potential for the respective surface charge layers, which are given by

Eq. (4.20), namely,


FIGURE 15.4 Scaled electrostatic interaction energy Vsp(H)¼ {ereok

4(a1þ a2)/pa1a2s21gV sp(H) between two dissimilar soft spheres with fixed charges of like sign as a func-

tion of scaled sphere separation kH calculated with Eq. (15.44) at various values of kdl¼ kd2¼ kd, where s1 and s2 (defined by Eqs. (15.46) and (15.47)) are kept constant at s2/sl¼ 0.5.

From Ref. [3].

FIGURE 15.5 Scaled electrostatic interaction energy Vsp(H)¼ {ereok

4(a1þ a2)/pa1a2s21gV sp(H) between two dissimilar soft spheres with fixed charges of unlike sign as a

function of scaled sphere separation kH calculated with Eq. (15.44) at various values of kdl¼kd2¼ kd, where s1 and s2 (defined by Eqs. (15.46) and (15.47)) are kept constant at s2/sl¼�0.5. From Ref. [3].


cDON1 ¼rfix1ereok2

ð15:51Þ

cDON2 ¼rfix2ereok2

ð15:52Þ

It is well known [4,5] that in the case of hard spheres (dl¼ d2¼ 0), the electro-

static force between two dissimilar spheres with charges of unlike sign is attractive

for large kH but becomes repulsive at small kH, that is, there is a minimum in the

interaction energy Vsp(H) except when s2/sl¼�1. The case of nonzero kdl and kd2,however, leads to quite different results. Figure 15.6 shows that except for very

small kdl and kd2, the interaction force is always attractive, that is, there is no mini-

mum in Vsp(H). On the other hand, it is repulsive for all kh when the fixed charges

of spheres 1 and 2 are like sign as is seen in Fig. 15.4. Figure 15.6 shows results for

Vsp(H) calculated with Eq. (15.44) at various values of kdl¼ kd2¼ kd when rfix1and rfix2 are kept constant at rfix1/rfix2¼ 0.5. It is seen that as kdl and kd2 increase,the dependence of Vsp(H) on kdl and kd2 becomes smaller. The limiting form of

Vsp(H) is given later by Eq. (15.55).Consider other limiting cases of Eqs. (15.20),(15.21) and (15.44).

(i) Thick surface charge layersConsider the limiting case of kd1� 1 and kd2� 1. In this case, soft

plates and soft spheres become planar polyelectrolytes and spherical

FIGURE 15.6 Scaled electrostatic interaction energy Vsp (H)¼ {ereok

4(a1þ a2)/pa1a2r2fix1}Vsp(H) between two dissimilar soft spheres with fixed charges of like sign as a

function of scaled sphere separation kH calculated with Eq. (15.44) at various values of kdl¼kd2¼ kd, where rfix1 and rfix2 are kept constant at rfix2/rfix1¼ 0.5. From Ref. [3].


polyelectrolytes, respectively, and Eqs. (15.20),(15.21) and (15.44) reduce

to

PplðhÞ ¼ rfix1rfix22ereok2

e�kh ð15:53Þ

VplðhÞ ¼ rfix1rfix22ereok3

e�kh ð15:54Þ

V spðHÞ ¼ 1

ereok4pa1a2a1 þ a2

� �rfix1rfix2 e

�kH ð15:55Þ

It is seen that for thick surface charge layers, P(h), Vpl(h), and Vsp(H) arealways positive when Z1 and Z2 are of like sign while Vpl(h) and Vsp(H) arealways negative when Z1 and Z2 are of unlike sign

Note that the following exact expression for the electrostatic interaction

between two porous spheres (spherical polyelectrolytes) for the low charge

density case has been derived [5,6] (Eq. (13.46)):

V spðHÞ ¼ pa1a2rfix1rfix2ereok4

e�kH

H þ a1 þ a2

1þ e�2ka1 � 1� e�2ka1

ka1

� �1þ e�2ka2 � 1� e�2ka2

ka2

� �ð15:56Þ

or

VspðHÞ ¼ 4pereoa1a2co1co2

e�kH

H þ a1 þ a2ð15:57Þ

with

co1 ¼rfix1

2ereok21þ e�2ka1 � 1� e�2ka1

ka1

� �ð15:58Þ

co2 ¼rfix2

2ereok21þ e�2ka2 � 1� e�2ka2

ka2

� �ð15:59Þ

where co1 and co2 are, respectively, the unperturbed surface potentials of

soft spheres 1 and 2 at infinite separation. Equation (15.56), under the con-

dition given by Eq. (15.43), tends to Eq. (15.55).

As in the case of two interacting soft plates, when the thicknesses of the

surface charge layers on soft spheres 1 and 2 are very large compared with

the Debye length 1/k, the potential deep inside the surface charge layer is

practically equal to the Donnan potential (Eqs. (15.51) and (15.52)), inde-

pendent of the particle separation H. In contrast to the usual electrostatic

interaction models assuming constant surface potential or constant surface


charge density of interacting particles, the electrostatic interaction between

soft particles may be regarded as the Donnan potential-regulated interaction

(Chapter 13).

(ii) Interaction between soft sphere and porous sphere (sphericalpolyelectrolyte)

Consider the case where sphere 1 is a soft sphere and sphere 2 is a porous

sphere (spherical polyelectrolyte). By taking the limit kd2� 1, we obtain

from Eq. (15.44)

V spðHÞ¼ 2

ereok4pa1a2a1þa2

� �rfix1rfix2 sinhðkd1Þe�kðHþd1Þ þ1

4r2fix2 e

�2kðHþd1Þ� �

ð15:60Þ

In the limit of kd1� 1, Eq. (15.60) tends back to Eq. (15.55).

(iii) Interaction between a soft sphere and a hard sphereConsider the case where sphere 1 is a soft sphere and sphere 2 is a hard

sphere. By taking the limit d2! 0 and N2!1 with the product s2¼Z2eN2d2 kept constant, we obtain from Eq. (15.44)

VspðHÞ ¼ 1

ereok4pa1a2a1 þ a2

� � rfix1 sinhðkd1Þ þ s2kÞ

�2ln

1

1� e�kðHþd1Þ

� ��

�frfix1 sinhðkd1Þ � s2kÞg2lnð1þ e�kðHþd1ÞÞ#

ð15:61Þ

(iv) Interaction between a spherical polyelectrolyte and a hard sphereIf we further take the limit kd1� 1 in Eq. (15.61), then we obtain the

electrostatic interaction energy for the case where sphere 1 is a spherical

polyelectrolyte and sphere 2 is a hard sphere, namely,

V spðHÞ ¼ 2

ereok4pa1a2a1 þ a2

� �rfix1s2k e�kH þ 1

8r2fix1 e

�2kH� �

ð15:62Þ


between spherical polyelectrolyte 1 and hard sphere 2 has been derived

[5,7] (see Chapter 14):

V spðHÞ ¼ 4pereoco1co2a1a2e�kH

H þ a1 þ a2

þ2pereoc2o1a

21 e

2ka1 1

H þ a1 þ a2

X1n¼0

ð2nþ 1Þ

In�1=2ðka2Þ � ðnþ 1ÞInþ1=2ðka2Þ=ka2Kn�1=2ðka2Þ þ ðnþ 1ÞKnþ1=2ðka2Þ=ka2

K2nþ1=2ðkðH þ a1 þ a2ÞÞ

ð15:63Þ


where the unperturbed surface potentials co1 and co2 are given by Eqs.

(15.58) and (1.76), respectively. Under the condition given by Eq. (15.43),

Eq. (15.63) becomes

V spðHÞ ¼ 4pereoa1a2

a1 þ a2co1co2 e

�kH þ 1

4c2o1 e

�2kH� �

ð15:64Þ

15.4 INTERACTION BETWEEN TWO DISSIMILAR SOFT CYLINDERS

Consider the electrostatic interaction between two parallel dissimilar cylindrical

soft particles 1 and 2. We denote by dl and d2 the thicknesses of the surface chargelayers of cylinders 1 and 2, respectively. Let the radius of the core of soft cylinder 1

be a1 and that for soft cylinder 2 be a2. We imagine that each surface layer is uni-

formly charged. Let Zl and Nl, respectively, be the valence and the density of fixed-

charge layer of cylinder 1, and Z2 and N2 for cylinder 2.

Consider first the case of two parallel soft cylinders (Fig. 15.7). With the help of

Derjaguin’s approximation for two parallel cylinders [8,9] (Eq. (12.38)), namely,

Vcy==ðHÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a1a2a1 þ a2

r Z 1

H

VplðhÞ dhffiffiffiffiffiffiffiffiffiffiffiffih� H

p ð15:65Þ

which is a good approximation for ka1� 1, ka2� 1, H� a1, and H� a2 (Eq.

(15.43)), one can calculate the interaction energy Vcy//(H) per unit length between

two dissimilar soft cylinders 1 and 2 separated by a distance H between there

FIGURE 15.7 Interaction between two parallel soft cylinders 1 and 2 at separation H.Cylinders 1 and 2 are covered with surface charge layers of thicknesses d1 and d2, respec-tively. The core radii of cylinders 1 and 2 are a1 and a2, respectively.

INTERACTION BETWEEN TWO DISSIMILAR SOFT CYLINDERS 369

surfaces via the corresponding interaction energy Vpl(h) per unit area between two

parallel dissimilar soft plates at separation h. By substituting Eq. (15.20) into Eq.

(15.65), we obtain [9]

Vcy==ðHÞ ¼ 1

2ereok7=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pa1a2a1 þ a2

rfrfix1 sinhðkd1Þ½

þrfix2 sinhðkd2Þg2Li1=2ðe�kðHþd1þd2ÞÞþfrfix1 sinhðkd1Þ � rfix2 sinhðkd2Þg2Li1=2ð�e�kðHþd1þd2ÞÞ�

ð15:66Þ

where Lis(z) is the polylogarithm function, defined by

LisðzÞ ¼X1k¼1

zk

ksð15:67Þ

For the special case of two similar soft cylinders carrying Z1¼ Z2¼ Z, N1¼N2¼N, a1¼ a2¼ a, d1¼ d2¼ d, and rfix1¼ rfix2¼ rfix, Eq. (15.66) reduces to

Vcy==ðHÞ ¼ 2ffiffiffiffiffiffipa

pereok7=2

r2fix sinh2ðkdÞLi1=2ðe�kðHþ2dÞÞ ð15:68Þ

The interaction force Pcy//(H) acting between two soft cylinders per unit length isgiven by Pcy//(H)¼�dVcy//(H)/dH, which gives [9]

Pcy==ðHÞ ¼ 1

2ereok5=2


rfrfix1 sinhðkd1Þ þ rfix2 sinhðkd2Þg2h

Li�1=2ðe�kðHþd1þd2ÞÞ

þ frfix1 sinhðkd1Þ � rfix2 sinhðkd2Þg2Li�1=2ð�e�kðHþd1þd2ÞÞi

ð15:69Þ

Consider next the case of two crossed soft cylinders (Fig. 15.8). Derjaguin’s ap-

proximation for two crossed cylinders under condition (15.43) is given by Eq.

(12.48), namely,

Vcy?ðHÞ ¼ 2pffiffiffiffiffiffiffiffiffia1a2

p Z 1

H

VplðhÞdh ð15:70Þ

By substituting Eq. (15.21) into Eq. (15.70), we obtain [9]

Vcy?ðHÞ ¼ pffiffiffiffiffiffiffiffiffia1a2

pe1e0k4

(rfix1 sinhðkd1Þ þ rfix2 sinhðkd2Þgln

1

1� e�kðHþd1þd2Þ

�"

�frfix1 sinhðkd1Þ � rfix2 sinhðkd2Þg2lnð1þ e�kðHþd1þd2ÞÞ#

ð15:71Þ


For the special case of two similar soft cylinders carrying Z1¼ Z2¼ Z, N1¼N2¼N, a1¼ a2¼ a, d1¼ d2¼ d, and rfix1¼ rfix2¼ rfix, Eq. (15.71) reduces to

Vcy?ðHÞ ¼ 4paereok4

r2fix sinh2ðkdÞln 1

1� e�kðHþ2dÞ

� �ð15:72Þ

We introduce the quantities



which are, respectively, the amounts of fixed charges contained in the surface layers

per unit area on cylinders 1 and 2. If we take the limit dl, d2! 0 and N1, N2!1,

keeping the products N1d1 and N2d2 constant, then s1 and s2 reduce to the surface

charge densities of two interacting hard cylinders without surface charge layers. In

this limit, Eqs. (15.66) and (15.71) reduce to

Vcy==ðHÞ ¼ 2

ereok3=2


rs1 þ s2

2

� 2

Li1=2ðe�kHÞ � s1 � s22

� 2

Li1=2ð�e�kHÞ� �

ð15:75Þ


pereok2

s1 þ s22

� 2

ln1

1� e�kH

� �� s1 � s2

2

� 2

lnð1þ e�kHÞ� �

ð15:76Þ

FIGURE 15.8 Interaction between two crossed soft cylinders 1 and 2 at separation H.Cylinders 1 and 2 are covered with surface charge layers of thicknesses d1 and d2, respec-tively. The core radii of cylinders 1 and 2 are a1 and a2, respectively.


Equations (15.75) and (15.76), respectively, agree with the expression for the

electrostatic interaction energy between two parallel hard cylinders at constant sur-

face charge density and that for two crossed hard cylinders at constant surface

charge density (Chapter 12).

When kdl� 1 and kd2� 1, being fulfilled for practical cases, the potential deep

inside the plates remains constant, equal to the Donnan potential for the respective

surface charge layers, which are given by Eqs. (15.51) and (15.52).

Where co1 and co2 are, respectively, the unperturbed surface potentials of hard

cylinders 1 and 2 at infinite separation and In(z) and Kn(z) are, respectively, modi-

fied Bessel functions of the first and second kinds, epi is the relative permittivity of

cylinder i (i¼ 1 and 2).

Consider other limiting cases of Eqs. (2.101) and (2.105).

(i) Thick surface charge layersIn the limiting case of kd1� 1 and kd2� 1, soft cylinders become cylin-

drical polyelectrolytes and Eqs. (2.101) and (2.105) reduce to

Vcy==ðRÞ ¼ 1

2ereok7=2


rrfix1rfix2 e

�kH ð15:77Þ


pereok4

rfix1rfix2 e�kH ð15:78Þ

It is seen that for thick surface charge layers, Vcy//(H) and Vcy?(H) arealways positive when Z1 and Z2 are of like sign while they are always nega-

tive when Z1 and Z2 are of unlike sign.Note that the following exact expression for the electrostatic interaction

energy per unit area Vcy//(H) between two porous cylinders for the low

charge density case has been derived [5,10] (Chapter 13):

Vcy==ðHÞ ¼ 2pereoco1co2

K0ðkðH þ a1 þ a2ÞÞK0ðka1ÞK0ðka2Þ ð15:79Þ

with

coi ¼rfixiereok

aiK0ðkaiÞI1ðkaiÞ; ði ¼ 1; 2Þ ð15:80Þ

where co1 and co2 are, respectively, the unperturbed surface potentials of

cylindrical polyelectrolytes 1 and 2 at infinite separation. Equation (15.79),

under the condition given by Eq. (15.43), tends to Eq. (15.77).

As in the case of soft spheres, when the thicknesses of the surface charge

layers on soft cylinders 1 and 2 are very large compared with the Debye

length 1/k, the potential deep inside the surface charge layer is practically

equal to the Donnan potential (Eqs. (15.51) and (15.52)), independent of the

particle separation H.


(ii) Interaction between soft cylinder and cylindrical polyelectrolyteConsider the case where cylinder 1 is a soft cylinder and cylinder 2 is a

cylindrical polyelectrolyte. By taking the limit kd2� 1, we obtain from

Eqs. (15.66) and (15.71)

Vcy==ðHÞ ¼ 1

ereok7=2


rrfix1rfix2 sinhðkd1Þe�kðHþd1Þh

þ 1

4ffiffiffi2

p r2fix2 e�2kðHþd1Þ

ið15:81Þ


pereok4

rfix1rfix2 sinhðkd1Þe�kðHþd1Þ þ 1

8r2fix2 e

�2kðHþd1Þ� �

ð15:82ÞIn the limit of kd1� 1, Eqs. (15.81) and (15.82) tend back to Eqs.

(15.77) and (15.78), respectively.

(iii) Interaction between a soft cylinder and a hard cylinderConsider the case where cylinder 1 is a soft cylinder and cylinder 2 is a

hard cylinder. By taking the limit d2! 0 and N2!1 with the product s2¼Z2eN2d2 kept constant, we obtain from Eqs. (15.66) and (15.71)

Vcy==ðHÞ ¼ 1

2ereok7=2


r½frfix1 sinhðkd1Þ þ s2kÞg2Li1=2ð�e�kðHþd1ÞÞ

þfrfix1 sinhðkd1Þ � s2kÞg2Li1=2ð�e�kðHþd1ÞÞ�ð15:83Þ


pereok4

nrfix1 sinhðkd1Þ þ s2kÞ

o2

ln1

1� e�kðHþd1Þ

� ��

�frfix1 sinhðkd1Þ � s2kÞg2lnð1þ e�kðHþd1ÞÞ�

ð15:84Þ

(iv) Interaction between a porous cylinder and a hard cylinderIf we further take the limit kd1� 1 in Eqs. (15.82) and (15.84), then we

obtain the electrostatic interaction energies for the case where cylinder 1 is a

cylindrical polyelectrolyte and cylinder 2 is a hard cylinder, namely,

Vcy==ðHÞ ¼ 1

ereok7=2


rrfix1s2k e�kH þ 1

4ffiffiffi2

p r2fix1 e�2kH

� �ð15:85Þ



pereok4

rfix1s2k e�kH þ 1

8r2fix1 e

�2kH� �

ð15:86Þ


between porous cylinder (cylindrical polyelectrolyte) 1 and hard cylinder 2

has been derived [5,10]:

Vcy==ðHÞ ¼ 2pereoco1co2

K0ðkðH þ a1 þ a2ÞÞK0ðka1ÞK0ðka2Þ

�pereoc2o1a1 e

2ka1

X1n¼�1

I0nðka2Þ � ðep2jnj=eka2ÞInðka2ÞK 0

nðka2Þ � ðep2jnj=eka2ÞKnðka2ÞK2nðkðH þ a1 þ a2ÞÞ ð15:87Þ

with

co1 ¼rfix1ereok

aiK0ðka1ÞI1ðka1Þ ð15:88Þ

co2 ¼s2

ereokK0ðka2ÞK1ðka2Þ ; ði ¼ 1; 2Þ ð15:89Þ

where ep2 is relative permittivity of hard cylinder 2. Under the condition

given by Eq. (15.43), Eq. (15.87) becomes Eq. (15.85).

REFERENCES

1. H. Ohshima, K. Makino, and T. Kondo, J. Colloid Interface Sci. 116 (1987) 196.

2. B. V. Derjaguin, Kolloid Z. 69 (1934) 155.

3. H. Ohshima, J. Colloid Interface Sci. 328 (2008) 3.

4. G. R. Wiese and T. W. Healy, Trans. Faraday Soc. 66 (1970) 490.

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

Press, 1968.

6. H. Ohshima and T. Kondo, J. Colloid Interface Sci. 155 (1993) 499.

7. H. Ohshima, J. Colloid Interface Sci. 168 (1994) 255.

8. M. J. Sparnaay, Recueil 78 (1959) 680.

9. H. Ohshima and A. Hyono, J. Colloid Interface Sci. 332 (2009) 251.

10. H. Ohshima, Colloid Polym. Sci. 274 (1996) 1176.


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