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18 Electrostatic InteractionBetween Ion-PenetrableMembranes in a Salt-FreeMedium

18.1 INTRODUCTION

Electric behaviors of colloidal particles in a salt-free medium containing counter-

ions only are quite different from those in electrolyte solutions, as shown in

Chapter 6. In this chapter, we consider the electrostatic interaction between two

ion-penetrable membranes (i.e., porous plates) in a salt-free medium [1].

18.2 TWO PARALLEL HARD PLATES

Before considering the interaction between two ion-penetrable membranes, we here

treat the interaction between two similar ion-impenetrable hard plates 1 and 2 carry-

ing surface charge density s at separation h in a salt-free medium containing coun-

terions only (Fig. 18.1) [2]. We take an x-axis perpendicular to the plates with its

origin on the surface of plate 1. As a result of the symmetry of the system, we need

consider only the region 0� x� h/2. Let the average number density and the va-

lence of counterions be no and �z, respectively. Then we have from electroneutral-

ity condition that

s ¼ zeno � h2

ð18:1Þ

or

no ¼ 2szeh

ð18:2Þ

Note that no, which is a function of h, is proportional to 1/h. We set the equilibrium

electric potential c(x) to zero at points where the volume charge density rel(x) re-sulting from counterions equals its average value (�zen).

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

388

The Poisson equation is thus given by

d2cdx2

¼ � relereo

; 0 < x < h=2 ð18:3Þ

Here we have assumed that the relative permittivity er is assumed to take the same

value inside and outside the membrane. We also assume that the distribution of

counterions n(x) obeys a Boltzmann distribution, namely,

nðxÞ ¼ no exp ��zecðxÞkT

� �¼ no exp

zecðxÞkT

� �ð18:4Þ

Thus, the charge density rel(x) is given by

relðxÞ ¼ zeno expzecðxÞkT

� �ð18:5Þ

Thus, we obtain the following Poisson–Boltzmann equation:

d2y

dx2¼ k2ey; 0 < x < h=2 ð18:6Þ

with

yðxÞ ¼ zecðxÞkT

ð18:7Þ

FIGURE 18.1 Schematic representation of the electrostatic interaction between two par-

allel identical hard plates separated by a distance h between their surfaces.

TWO PARALLEL HARD PLATES 389

and

k ¼ z2e2noereokT

� �1=2

¼ 2zesereokTh

� �1=2

ð18:8Þ

where y(r) is the scaled equilibrium potential, k is the Debye–Huckel parameter in

the present system. The boundary conditions for Eq. (18.6) are

dy

dx

��x¼0

¼ � ze

kT

� � sereo

¼ � k2h2

ð18:9Þ

dy

dx

��x¼h=2

¼ 0 ð18:10Þ

Equation (18.9), which implies that the influence of the electric field within the

plates can be neglected, is consistent with Eq. (18.1). Equation (18.10) comes from

the result of symmetry of the system.

Integration of Eq. (18.6) subject to Eq. (18.10) yields

dy

dx¼ �

ffiffiffi2

pk exp

ym2

� �tan exp

ym2

� � kffiffiffi2

p h

2� x

� �� ð18:11Þ

with

ym ¼ yðh=2Þ ð18:12Þ

where ym is the scaled potential at the midpoint between the plates. Equation (18.11)

is further integrated to give

yðxÞ ¼ �ln cos2 expym2

� � kffiffiffi2

p h

2� x

� � ��þ ym ð18:13Þ

By combining Eqs. (18.9) and (18.12), we obtain the following transcendental equa-

tion for ym:

tankh

2ffiffiffi2

p expym2

� �� ¼ kh

2ffiffiffi2

p exp � ym2

� �; ð18:14Þ

The electrostatic interaction force acting between plates 1 and 2 per unit area can

be calculated from an excess osmotic pressure at the midpoint between the plates,

namely,

PðhÞ ¼ nðh=2ÞkT ¼ nokT eym ¼ 2sh

kT

ze

� �eym ð18:15Þ

390 ELECTROSTATIC INTERACTION BETWEEN ION-PENETRABLE MEMBRANES

In the limit of small kh, it follows from Eq. (18.14) that

ym ! 0 as h ! 0 ð18:16Þ

so that Eq. (18.15) gives

PðhÞ ! 2sh

kT

ze

� �ð18:17Þ

In the opposite limit of large kh, it follows from Eq. (18.14) that

kh

2ffiffiffi2

p expym2

� �! p

2as h ! 1 ð18:18Þ

Thus by substituting Eq. (18.8) into Eq. (18.15), we have

PðhÞ ! 2p2ereoh2

kT

ze

� �2

as h ! 1 ð18:19Þ

Equations (18.17) and (18.19) show that P(h) is proportional to 1/h for small kh butto 1/h2 at large kh.

18.3 TWO PARALLEL ION-PENETRABLE MEMBRANES

Now consider two parallel identical ion-penetrable membranes 1 and 2 at separation

h immersed in a salt-free medium containing only counterions. Each membrane is

fixed on a planar uncharged substrates (Fig. 18.2). We obtain the electric potential

distribution c(x). We assume that fixed charges of valence Z are distributed in the

membrane of thickness d with a number density of N (m�3) so that the fixed-charge

density rfix within the membrane is given by

rfix ¼ ZeN ð18:20Þ

We take an x-axis perpendicular to the membranes with its origin on the surface of

membrane 1. Let the average number density and the valence of counterions be noand �v, respectively. Then we have from electroneutrality condition that

ZeNd ¼ zenoh

2þ d

� �ð18:21Þ

which is rewritten as

znoZN

¼ 2d

hþ 2dð18:22Þ

TWO PARALLEL ION-PENETRABLE MEMBRANES 391

Note that no becomes proportional to 1/h for h� d. We set the equilibrium electric

potential c(x) to zero at points where the volume charge density rel(x) resultingfrom counterions equals its average value (�zen).

The Poisson equations are thus given by

d2cdx2

¼ � rel þ rfixereo

; �d < x < 0 ð18:23Þ

d2cdx2

¼ � relereo

; 0 < x � h=2 ð18:24Þ

Here we have assumed that the relative permittivity er is assumed to take the same

value inside and outside the membrane. We also assume that the distribution of

counterions n(x) obeys Eq. (18.4) and thus the charge density rel(x) is given by

Eq. (18.5).Thus, we obtain the following Poisson–Boltzmann equations for the

scaled potential y(x)¼ zec(x)/kT:

d2y

dx2¼ k2 ey � ZN

zno

� �; �d < x < 0 ð18:25Þ

d2y

dx2¼ k2ey; 0 < x < h=2 ð18:26Þ

with

k ¼ z2e2noereokT

� �1=2¼ 2ze2ZN

ereokT� d

hþ 2d

� �1=2ð18:27Þ

FIGURE 18.2 Schematic representation of the electrostatic interaction between two par-

allel identical ion-penetrable membranes (porous plates) of thickness d separated by a dis-

tance h between their surfaces. Each membrane is fixed on an uncharged planar substrate.


where k is the Debye–Huckel parameter in the present system. With the help of

Eq. (18.21), Eq. (18.25) is rewritten as

d2y

dx2¼ k2 ey � hþ 2d

2d

� �; �d < x < 0 ð18:28Þ

Equation (18.27) shows that the Debye–Huckel parameter k depends on the fixed

charge density ZeN in the membrane and the membrane separation h (k is propor-

tional to 1/h1/2 for h� d and the h dependence of k becomes small for h� d), un-like the case of media containing electrolyte solution, in which case k is essentially

constant independent of ZeN and h. The boundary conditions for Poisson–Boltz-

mann equations (18.26) and (18.28) are

dy

dx

��x¼�d

¼ 0 ð18:29Þ

dy

dx

��x¼h=2

¼ 0 ð18:30Þ

yð0�Þ ¼ yð0þÞ ð18:31Þ

dy

dx

��x¼0�

¼ dy

dx

��x¼0þ

ð18:32Þ

As a result of symmetry of the system, we need only to consider the region �d�x� h/2. For the region inside the membrane �d� x� 0, Eq. (18.28) subject to

Eq. (18.29) is solved to give

dy

dx¼ �

ffiffiffi2

pk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexpðyÞ � expðyð�dÞÞ � hþ 2d

2d

� �y� yð�dÞf g

sð18:33Þ

Equation (18.33) can further be integrated to

�ffiffiffi2

pkðxþ dÞ ¼

Z yðxÞ

yð�dÞ

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexpðyÞ � expðyð�dÞÞ � ðhþ 2d=2dÞfy� yð�dÞgp

ð18:34Þ

which gives y(x) as an implicit function of x in the region �d� x� 0.

For the region 0� x� h/2, integration of Eq. (18.26) subject to Eq. (18.30) yields(see Eq. (18.11))

dy

dx¼ �

ffiffiffi2

pk exp

ym2

� �tan exp

ym2

� � kffiffiffi2

p h

2� x

� �� ð18:35Þ


where ym ¼ yðh=2Þis the potential at the midpoint between the membranes.

Equation (18.35) is further integrated to give

yðxÞ ¼ �ln cos2 expym2

� � kffiffiffi2

p h

2� x

� � ��þ ym ð18:36Þ

Equations (18.34) and (18.36) contain unknown potential values ym¼ y(h/2), y(�d),and y(0) that can be determined by continuity conditions (18.31) and (18.32) at

x¼ 0, namely,

yð0Þ ¼ �ln cos2 expym2

� � kh

2ffiffiffi2

p ��

þ ym ð18:37Þ

�ffiffiffi2

pkd ¼

Z yð0Þ

yð�dÞ

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexpðyÞ � expðyð�dÞÞ � ðhþ 2d=2dÞfy� yð�dÞgp ð18:38Þ

eym � eyð�dÞ � hþ 2d

2d

� �yð0Þ � yð�dÞf g ¼ 0 ð18:39Þ

where the following condition must be satisfied:

0 � kh

2ffiffiffi2

p expym2

� �<

p2

ð18:40Þ

The solution to coupled Eqs. (18.37)–(18.39) determines the values of ym¼ y(h/2),y(�d), and y(0) as functions of the membrane separation h.

The potential distribution can be calculated with Eqs. (18.34) and (18.36) with

the help of the values of y(�d) and ym. The electrostatic interaction force acting

between membranes 1 and 2 per unit area can be calculated from

PðhÞ ¼ nðh=2ÞkT ¼ nokT eym

¼ ZN

z

2d

hþ 2d

� �kT eym

ð18:41Þ

where the value of ym can be obtained by solving Eqs. (18.37)–(18.39). Note that

both no and ym depend on h.In Figs 18.3–18.5, the results of some calculations are given for the counter-ion

concentration n(x) (Fig. 18.3), potential distribution c(x) (Fig. 18.4), and the inter-

action force per unit area P(h) (Fig. 18.5). Calculations are made for water at 25�C(er¼ 78.55), Z¼ z¼ 1 and the membrane thicknesses d¼ 0, 1, and 5 nm with the

amount of fixed charges s in the membrane per unit area kept constant at 0.2 C/m2,

with s defined by

s ¼ ZeNd ð18:42Þ


FIGURE 18.3 Distributions of counter-ion concentration n(x) across two parallel identi-

cal ion-penetrable membranes of thickness d separated by a distance h between their surfacesin a salt-free medium. Distributions were calculated at h¼ 0, 2, 6, and 10 nm for Z¼ z¼ 1,

d¼ 5 nm, and the charge amount per unit area s¼ ZeNd¼ 0.2 C/m2 in water at 25�C(er¼ 78.55). From Ref. [1].

FIGURE 18.4 Distribution of electric potential c(x) across two parallel identical ion-

penetrable membranes of thickness d separated by a distance h between their surfaces in a

salt-free medium. Calculated at h¼ 0, 2, 6, and 10 nm for Z¼ z¼ 1, d¼ 0, 1, and 5 nm and

the charge amount per unit area s¼ ZeNd kept constant at s¼ 0.2 C/m2 in water at 25�C(er¼ 78.55). From Ref. [1].


Figures 18.3–18.5 demonstrate that n(x), c(x), and P(h) depend significantly on hand/or d. In the limiting case in which d! 0 and N!1 with s¼ ZeNd kept con-

stant so that the membrane becomes a planar surface carrying a charge density s. Inthis limit, Eq. (18.27) for the Debye–Huckel parameter tends to Eq. (18.8) and the

electroneutrality condition (Eq. (18.21)) tends to Eq. (18.1). Thus, Eqs. (18.37)–

(18.39) reduce to the single transcendental Eq. (18.14) for two parallel planar sur-

faces. Figure 18.4 demonstrate how the potential distribution c(x) changes with the

membrane separation h. Figure 18.4 shows that the potential in the region

deep inside the membrane is almost equal to the Donnan potential cDON,

which is given by setting the right-hand side of Eq. (18.25) or Eq. (18.28) equal

zero:

cDON ¼ kT

zeln

ZN

zno

� �¼ kT

zeln

hþ 2d

2d

� �ð18:43Þ

Equation (18.43) shows that cDON depends on h and decreases in magnitude with

decreasing h, tending to zero in the limit h! 0. Note that in the limiting case of

h! 0, the potential c(x) becomes zero everywhere outside and inside the mem-

branes, as is seen in Fig. 18.4. In the region where the potential is equal to the

Donnan potential, the electroneutrality condition holds so that the membrane-

fixed charges (ZeN) are completely neutralized by the charges of counterions pen-

etrating the membrane interior (�zen(x)). Figure 18.3 indeed shows that in the

region deep inside the membrane the concentration n(x) of counterions is

FIGURE 18.5 Repulsive force P(h) per unit area between two parallel identical ion-

penetrable membranes of thickness d separated by a distance h between their surfaces in a

salt-free medium. Calculated for Z¼ z¼ 1, d¼ 0, 1, and 5 nm with the charge amount per

unit area s¼ ZeNd kept constant at s¼ 0.2 C/m2 in water at 25�C (er¼ 78.55). From Ref. [1].


practically equal to the concentration N of membrane-fixed charges (which is in-

dependent of h). It is seen from Fig. 18.4, on the other hand, that the potential

deep inside the membrane (which is almost equal to the Donnan potential)

changes significantly with h. This is because the Donnan potential cDON is a func-

tion of h (Eq. (18.43)).

Figure 18.5 shows the dependence of the interaction force P(h) on the membrane

separation h. The interaction force is shown to be always repulsive and decreases in

magnitude with increasing h, tending to zero as h!1. It can be shown that in the

case where d! 0 and N!1 with s¼ ZeNd kept constant, P(h) becomes propor-

tional to 1/h as h! 0, namely,

PðhÞ ! 2sh

kT

ze

� �as h ! 0 ð18:44Þ

which agrees with the case of two charged planar surfaces in a salt-free medium

(Eq. (16.17)). It is of interest to note that Eq. (18.44) agrees also with the limiting

force expression for the case of electrolyte solutions (Eq. (9.203)). This is because

for very small h, the interaction force is determined only by counterions irrespective

of whether coions are present (salt-containing media) or absent (salt-fee media). For

the case of finite d, the interaction force P(h) remains finite in the limit of h! 0.

The value of P(0) can easily be obtained from Eq. (18.41) by noting that in this limit

zno¼ ZN and ym¼ 0, namely,

PðhÞ ! nokT ! ZN

zkT as h ! 0 ð18:45Þ

For large values of h, on the other hand, it follows from Eq. (18.40) that (kh/2ffiffiffi2

p)

exp[ym/2] must tend to p/2. Thus in this limit we have from Eq. (18.41) that

PðhÞ ! 2p2ereoh2

kT

ze

� �2

as h ! 1 ð18:46Þ

It is of interest to note that this limiting expression for P(h) is independent of themembrane-fixed charges ZeN and the membrane thickness d. This limiting form

thus agrees with that for the interaction between two charged planar surfaces in a

salt-free medium (Eq. 18.19).

Equation (18.38) involves numerical integration. Here we give an approximate

expression without numerical integration for Eq. (18.38). Since the largest contribu-

tion to the integrand of Eq. (18.38) comes from the region near y¼ y(�d), weexpand the integrand around y¼ y(�d) so that the right-hand side of Eq. (18.38)

can be approximated by

Z yð0Þ

yð�dÞ

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiey � eyð�dÞ � ðhþ 2d=2dÞfy� yð�dÞg

p � �2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiyð�dÞ � yð0Þp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðhþ 2d=2dÞ � eyð�dÞp

ð18:47Þ


By using this result, Eqs. (18.37)–(18.41) reduce to the single transcendental equa-

tion for ym,

Q

1þQ

� �1� 2d

hþ 2d

� �eym

¼ ln1þ h=2dþ 1ð ÞQe�ym

1þQ

cos2

2Qd

hþ 2d

� �1=2h

2d

� �exp

ym2

� �)(" #ð18:48Þ

with

0� 2Qd

hþ 2d

� �1=2h

2d

� �exp

ym2

� �<p2

ð18:49Þ

where Q is defined by

Q¼ zZe2Nd2

2ereokTð18:50Þ

and is related to k by

2Qd

hþ 2d

� �1=2h

2d

� �¼ kh

2ffiffiffi2

p

Note that in the limit of d! 0 (and N!1) with s kept constant, Eq. (18.48) re-

duces correctly to Eq. (18.14).

REFERENCES

1. H. Ohshima, J. Colloid Interface Sci. 260 (2003) 339.

2. J.N. Israelachivil, Intermolecular Forces, Academic Press, New York, 1991, Sec. 12.7.


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