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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
360 ELECTROSTATIC INTERACTION BETWEEN SOFT PARTICLES
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.
362 ELECTROSTATIC INTERACTION BETWEEN SOFT PARTICLES
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,
364 ELECTROSTATIC INTERACTION BETWEEN SOFT PARTICLES
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].
INTERACTION BETWEEN TWO DISSIMILAR SOFT SPHERES 365
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].
366 ELECTROSTATIC INTERACTION BETWEEN SOFT PARTICLES
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
INTERACTION BETWEEN TWO DISSIMILAR SOFT SPHERES 367
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
Note that the following exact expression for the electrostatic interaction
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
368 ELECTROSTATIC INTERACTION BETWEEN SOFT PARTICLES
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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pa1a2a1 þ a2
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
370 ELECTROSTATIC INTERACTION BETWEEN SOFT PARTICLES
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
s1 ¼ rfix1d1 ¼ Z1eN1d1 ð15:73Þ
s2 ¼ rfix2d2 ¼ Z2eN2d2 ð15:74Þ
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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pa1a2a1 þ a2
rs1 þ s2
2
� 2
Li1=2ðe�kHÞ � s1 � s22
� 2
Li1=2ð�e�kHÞ� �
ð15:75Þ
Vcy?ðHÞ ¼ 4pffiffiffiffiffiffiffiffiffia1a2
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.
INTERACTION BETWEEN TWO DISSIMILAR SOFT CYLINDERS 371
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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pa1a2a1 þ a2
rrfix1rfix2 e
�kH ð15:77Þ
Vcy?ðHÞ ¼ pffiffiffiffiffiffiffiffiffia1a2
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.
372 ELECTROSTATIC INTERACTION BETWEEN SOFT PARTICLES
(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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pa1a2a1 þ a2
rrfix1rfix2 sinhðkd1Þe�kðHþd1Þh
þ 1
4ffiffiffi2
p r2fix2 e�2kðHþd1Þ
ið15:81Þ
Vcy?ðHÞ ¼ 2pffiffiffiffiffiffiffiffiffia1a2
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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pa1a2a1 þ a2
r½frfix1 sinhðkd1Þ þ s2kÞg2Li1=2ð�e�kðHþd1ÞÞ
þfrfix1 sinhðkd1Þ � s2kÞg2Li1=2ð�e�kðHþd1ÞÞ�ð15:83Þ
Vcy?ðHÞ ¼ pffiffiffiffiffiffiffiffiffia1a2
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
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pa1a2a1 þ a2
rrfix1s2k e�kH þ 1
4ffiffiffi2
p r2fix1 e�2kH
� �ð15:85Þ
INTERACTION BETWEEN TWO DISSIMILAR SOFT CYLINDERS 373
Vcy?ðHÞ ¼ 2pffiffiffiffiffiffiffiffiffia1a2
pereok4
rfix1s2k e�kH þ 1
8r2fix1 e
�2kH� �
ð15:86Þ
Note that the following exact expression for the electrostatic interaction
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.
374 ELECTROSTATIC INTERACTION BETWEEN SOFT PARTICLES
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