Biophysical Chemistry of Biointerfaces (Ohshima/Biophysical Chemistry of Biointerfaces) || Donnan Potential-Regulated Interaction Between Porous Particles

13 Donnan Potential-RegulatedInteraction BetweenPorous Particles

13.1 INTRODUCTION

The solution to the Poisson–Boltzmann equation for the system of two interacting

particles immersed in an electrolyte solution must satisfy the boundary conditions

on the surface of the respective particles. For this reason, the sum of the un-

perturbed potentials of the interacting particles in general cannot be the solution to

the corresponding Poisson–Boltzmann equation, even for the case where the linear-

ized Poisson–Boltzmann is employed. Only one exception is the linearized Pois-

son–Boltzmann equation for the interaction between porous particles (or, soft

particles without the particle core) (Fig. 13.1) provided that the relative permittivity

takes the same value in the solutions both outside and inside the particles. One can

thus calculate the interaction energy between two porous spheres or two parallel

cylinders without recourse to Derjaguin’s approximation. In such cases, the linear

superposition of unperturbed potentials of interacting particles always gives the

exact solution to the corresponding linearized Poisson–Boltzmann equation for all

particle separations. For interactions involving hard particles, the linear superposi-

tion holds approximately good only for large particle separations. In this sense,

porous particles can be regarded as a prototype of hard particles. It is also shown

that if the particle size is much larger than the Debye length 1/k, then the potential

in the region deep inside the particle remains unchanged at the Donnan potential in

the interior region of the interacting particles. We call this type of interaction the

Donnan potential-regulated interaction (the Donnan potential regulation model).

13.2 TWO PARALLEL SEMI-INFINITE ION-PENETRABLEMEMBRANES (POROUS PLATES)

Consider two parallel semi-infinite membranes 1 and 2 carrying constant densities

rfix1 and rfix2 of fixed charges separation h between their surfaces in an electrolyte

solution containing N ionic species with valence zi and bulk concentration (number

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

298

density) n1i (i¼ 1, 2, . . . , N) (in units of m�3) [1]. The fixed-charge density rfixi inmembrane i (i¼ 1, 2) is related to the density Ni of ionized groups of valence Zidistributed in membrane i by rfixi¼ ZieNi (i¼ 1, 2). Without loss of generality, we

may assume that membrane 1 is positively charged (Z1> 0) and membrane 2 may

be either positively or negatively charged and that

Z1N1 � Z2j jN2 > 0 ð13:1Þ

We also assume that the relative permittivity in membranes 1 and 2 tale the same

value er as that of the electrolyte solution. Suppose that membrane 1 is placed in

the region �1< x< 0 and membrane 2 is placed in the region x> h (Fig. 13.2).

The linearized Poisson–Boltzmann equations in the respective regions are

d2cdx2

¼ k2c� rfix1ereo

; �1 < x < 0 ð13:2Þ

d2cdx2

¼ k2c; 0 < x < h ð13:3Þ

d2cdx2

¼ k2c� rfix2ereo

; x > h ð13:4Þ

FIGURE 13.1 Various types of positively charged ion-penetrable porous particles with

fixed charges (large circles with plus sign). Electrolyte ions (small circles with plus or minus

signs) can penetrate the particle interior.

TWO PARALLEL SEMI-INFINITE ION-PENETRABLE MEMBRANES 299

with

k ¼ 1

ereokT

XNi¼1

z2i e2n1i

!1=2

ð13:5Þ

where k is the Debye–Huckel parameter. The boundary conditions are

c and dc=dx are continuous at x ¼ 0 and x ¼ h ð13:6Þ

dcdx

! 0 as x ! � 1 ð13:7Þ

It can be shown that the solution c(x) to Eqs. (13.2)–(13.4) subject to Eqs. (13.6)

and (13.7) is given by a linear superposition of the unperturbed potential cð0Þ1 ðxÞ

produced by membrane 1 in the absence of membrane 2 and the corresponding

unperturbed potential cð0Þ2 ðxÞ for membrane 2, which are obtained by solving the

linearized Poisson–Boltzmann equations for a single membrane, namely,

cðxÞ ¼ cð0Þ1 ðxÞ þ cð0Þ

2 ðxÞ ð13:8Þ

with

cð0Þ1 ðxÞ ¼ 2co1 1� 1

2ekx

� �; x < 0

co1 e�kx; x > 0

8<: ð13:9Þ

FIGURE 13.2 Schematic representation of the potential distribution c(x) across two par-

allel interacting ion-penetrable semi-infinite membranes (soft plates) 1 and 2 at separation h.The potentials in the region far inside the membrane interior is practically equal to the

Donnan potential cDON1 or cDON2.

300 DONNAN POTENTIAL-REGULATED INTERACTION BETWEEN POROUS PARTICLES

cð0Þ1 ðxÞ ¼

co2 e�kðh�xÞ; x < h

2co2 1� 1

2ekðh�xÞ

� �; x > h

8<: ð13:10Þ

co1 ¼rfix1

2ereok2ð13:11Þ

co2 ¼rfix2

2ereok2ð13:12Þ

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

branes 1 and 2, which are half the Donnan potentials cDON1 and cDON2 for the low

potential case, namely,

cDON1 ¼ 2co1 ¼rfix1ereok2

ð13:13Þ

cDON2 ¼ 2co2 ¼rfix2ereok2

ð13:14Þ

The solution to Eqs. (13.2)–(13.4) can thus be written as

cðxÞ ¼ 2co1 1� 1

2ekx

� �þ co2 e

�kðh�xÞ; �1 < x < 0 ð13:15Þ

cðxÞ ¼ co1 e�kx þ co2 e

�kðh�xÞ; 0 < x < h ð13:16Þ

cðxÞ ¼ co1 e�kx þ 2co2 1� 1

2ekðh�xÞ

� �; h < x < �1 ð13:17Þ

Note that the boundary condition (13.6) states that membranes 1 and 2 are ‘‘electri-

cally transparent’’ to each other. As a result, the solution to Eqs. (13.2)–(13.4) is

obtained by a linear superposition approximation (LSA), which, for the case of rigid

membranes, holds only at large membrane separations.

In Fig. 13.3, we plot the potential distribution c(x) between two parallel similar

ion-penetrable membranes with cDON1¼cDON2¼c DON (or co1¼c o2¼co) for

kh¼ 0, 1, 2, and 1. In Fig. 13.3, we have introduced the following scaled potential

y, scaled unperturbed surface potential yo, and scaled Donnan potential yDON:

yðxÞ ¼ ecðxÞkT

ð13:18Þ

yo ¼eco

kTð13:19Þ


FIGURE 13.3 Scaled potential distribution y(x)¼ ec(x)=kT between two parallel similar

membranes 1 and 2 with yDON¼ ecDON=kT¼ 1 for kh¼ 0, 1, and 2. The dotted curves are

the scaled unperturbed potential distribution at kh¼1.


yDON ¼ ecDON

kTð13:20Þ

It follows from Fig. 13.3 and Eqs. (13.15) and (13.17) that the potentials in the

region deep inside the membranes are equal to the Donnan potential cDON in the

membranes, which is independent of the membrane separation h. That is, the poten-tial in the region deep inside the membrane remains unchanged at the Donnan po-

tential during interaction. Even for the interaction between membranes of finite

thickness, if the membrane thickness is much larger than the Debye length 1/k,then the potential in the region deep inside the membrane is always equal to the

Donnan potential. Note also that the values of the surface potentials c(0) and c(h)increase in magnitude from cDON/2 at infinite separation (h¼1) to cDON at h¼ 0

and that when the membranes are in contact with each other, the potential inside the

membranes is equal to the Donnan potential cDON everywhere in the membranes.

Figures 13.4 and 13.5 give the results for the interaction between two dissimilar

membranes, showing changes in the potential distribution y(x)¼ ec(x)/kT due to the

approach of two membranes for the cases when the two membranes are likely

charged, that is, Z1> 0 and Z2> 0 (Fig. 13.4) and when they are oppositely charged,

that is, Z1> 0 and Z2< 0 (Fig. 13.5). Here we have introduced the following scaled

unperturbed surface potentials yo, and scaled Donnan potentials yDON1 and yDON2:

yDON1 ¼ecDON1

kTð13:21Þ

yDON2 ¼ecDON2

kTð13:22Þ

Figure 13.4 shows that when Z1, Z1> 0, there is a potential minimum for large hand the surface potentials y(0) and y(h) increase with decreasing membrane

separation h. This potential minimum disappears at h¼ hc, where the surface poten-tial of the weakly charged membrane, y(h), coincides with its Donnan potential

yDON2. The value of hc can be obtained from Eq. (13.16), namely,

khc ¼ lnc01

c02

� �¼ ln

Z1N1

Z2N2

� �ð13:23Þ

At h< hc, y(h) exceeds yDON2. Note that both y(0) and y(h) are always less than

yDON1. Figure 13.5 shows that when Z1> 0 and Z2< 0, there is a zero point of the

potential y(x), 0< x< h, for large h. As h decreases, both y(0) and y(h) decrease inmagnitude. Then, at some point h¼ ho, the surface potential y(h) of the negativelycharged membrane becomes zero. The value of hc can be obtained from Eq. (13.16),

namely,

kh0 ¼ lnc01

c02j j� �

¼ lnZ1N1

Z2j jN2

� �ð13:24Þ


At h< ho, y(h) reverses its sign and both y(0) and y(h) become positive. Note that as

in the case where Z1> 0 and Z2> 0, both y(0) and y(h) never exceed yDON1.The interaction energy V(h) per unit area between two parallel membranes 1 and

2 is obtained from the free energy of the system, namely,

VðhÞ ¼ FðhÞ � Fð1Þ ð13:25Þ

FIGURE 13.4 Scaled potential distribution y(x)¼ ec(x)=kT between two parallel likely

charged dissimilar membranes 1 and 2 with yDON1¼ ecDON=kT¼ 2 and yDON2¼ ecDON2=kT¼ 0.8 (or yo1¼ eco1=kT¼ 1 and yo2¼ eco2=kT¼ 0.4) for kh¼ 0.5, 0.916 (¼ khc), and 2.


FIGURE 13.5 Scaled potential distribution y(x)¼ ec(x)=kT between two parallel oppo-

sitely charged dissimilar membranes 1 and 2 with yDON1¼ ecDON=kT¼ 2 and yDON2¼ecDON2=kT¼�0.8 (or yo1¼ eco1=kT¼ 1 and yo2¼ eco2=kT¼�0.4) for kh¼ 0.5, 0.916

(¼ kho), and 2.


with

FðhÞ ¼ 1

2

Z 0

�1rfix1c dxþ 1

2

Z 1

h

rfix2c dx ð13:26Þ

Substitution of Eqs. (13.15) and (13.17) into Eq. (13.26) gives the following expres-

sion for the interaction energy V(h) per unit area between two parallel dissimilar

membranes

VðhÞ ¼ 2ereokco1co2 e�kh ¼ rfix1rfix2

2ereok3e�kh ð13:27Þ

The interaction force P(h) per unit area between membranes 1 and 2 is thus

given by

PðhÞ ¼ 2ereok2co1co2 e�kh ¼ rfix1rfix2

2ereok2e�kh ð13:28Þ

We thus see that the interaction force P(h) is positive (repulsive) for the interactionbetween likely charged membranes and negative (attractive) for oppositely

charged membranes. It must be stressed that the sign of the interaction force P(h)remains unchanged even when the potential minimum disappears (for the case Z1,Z2> 0) or even when the surface potential of one of the membranes reverses its

sign (for the case Z1> 0 and Z2< 0).

13.3 TWO POROUS SPHERES

Consider two interacting ion-penetrable porous spheres 1 and 2 having radii a1 anda2, respectively, at separation R between their centers O1 and O2 (or, at separation

H¼R� a1� a2 between their closest distances (Fig. 13.6) [1–3].The linearized Poisson–Boltzmann equations in the respective regions are

Dc ¼ k2c; outside spheres 1 and 2 ð13:29Þ

Dc ¼ k2c� rfixiereo

; inside sphere i ði ¼ 1; 2Þ ð13:30Þ

The boundary conditions are

c ! 0 as r ! 1 ð13:31Þ

c and dc=dn are continuous at the surfaces of spheres 1 and 2 ð13:32Þ


The derivative of c being taken along the outward normal to the surface of each

sphere.

The solution to Eqs. (13.29) and (13.30) can be expressed as the sum

c ¼ cð0Þ1 þ cð0Þ

2 ð13:33Þ

where cð0Þ1 and cð0Þ

2 are, respectively, the unperturbed potentials for spheres 1 and 2.

This is because when the boundary conditions are given by Eq. (13.32), the un-

perturbed potential of one sphere automatically satisfies the boundary conditions at

the surface of the other sphere. The potential distribution for two interacting ion-

penetrable spheres are thus simply given by linear superposition of the unperturbed

potentials produced by the respective spheres, as in the case of ion-penetrable mem-

branes. Thus, one needs to solve only the potential distribution for a single isolated

sphere. Consider the unperturbed potential cð0Þ1 produced by sphere 1, for which

Eqs. (13.29) and (13.30) reduce to

d2cð0Þ1

dr21þ 2

r

dcð0Þ1

dr1¼ k2cð0Þ

1 ; r1 > a1 ð13:34Þ

d2cð0Þ1

dr21þ 2

r

dcð0Þ1

dr1¼ k2cð0Þ

1 � rfix1ereo

; 0 � r < a1 ð13:35Þ

and the boundary conditions to

c ! 0 as r1 ! 1 ð13:36Þ

cð0Þ1 ða�1 Þ ¼ cð0Þ

1 ðaþ1 Þ ð13:37Þ

dcð0Þ1

dr1

��r1¼a�

1

¼ dcð0Þ1

dr1

��r1¼aþ

1

ð13:38Þ

FIGURE 13.6 Interaction between two porous spheres 1 and 2 having radii a1 and a2,respectively, at separation R between their centers O1 and O2. H¼R� a1� a2. Q is a field

point.

TWO POROUS SPHERES 307

where r1 is the distance measured from the center O1 of sphere 1 (Fig. 13.6). The

solution to Eqs. (13.34) and (13.35) subject to Eqs. (13.36)–(13.38) are

cð0Þ1 ðr1Þ ¼

rfix1ereok2

coshðka1Þ � sinhðka1Þka1

� �a1

e�kr1

r1; r1 � a1

rfix1ereok2

1� 1þ 1

ka1

� �a1 e

�ka1 sinhðkr1Þr1

� �; 0 � r1 � a1

8>>><>>>:

ð13:39Þ

Similarly, we can derive the potential c2 produced by sphere 2 in the absence of

sphere 1, which c2 is obtained by replacing r1 with r2 and a1 with a2 in Eq. (13.39).

Here r2 is the radial coordinate measured from the center O2 of sphere 2, which is

related to r1 via (Fig. 13.6)

r2 ¼ ðR2 þ r21 � 2Rr1 cos �Þ1=2 ð13:40Þ

The potential distribution for two interacting spheres 1 and 2 is given by the sum of

c1 and c2, namely,

cðr1; �Þ ¼ rfix1ereok2

1� 1þ 1

ka1

� �a1 e


� �

þ rfix2ereok2


� �a2

e�kr2

r2;

0 � r1 � a1ðinside sphere 1Þ

ð13:41Þ



� �a1

e�kr1

r1

þ rfix2ereok2


� �a2

e�kr2

r2;

r1 � a1; r2 � a2

ð13:42Þ


1� 1þ 1

ka2

� �a2 e


� �

þ rfix1ereok2


� �a1

e�kr1

r1;

0 � r1 � a1ðinside sphere 2Þ

ð13:43Þ

We see that when kai� 1, the potential in the region deep inside the spheres, that is,

around the sphere center, becomes the Donnan potential (Eqs. (13.13) and (13.14)).


The interaction energy V(R) is most easily obtained from the free energy of the

system, namely, V(R)¼F(R)�F(1), where

FðRÞ ¼ 1

2

ZV1

rfix1c dV1 þ 1

2

ZV2

rfix2c dV2 ð13:44Þ

Thus, we need only to obtain the unperturbed potential distribution produced within

sphere 1 by sphere 2. By using the following relation

ZV1

e�kr2

r2dV1 ¼ 2p

Z a1

0

Z p

0

exp�� k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ r21 � 2Rr1 cos �

q �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 þ r21 � 2Rr1 cos �

q r21 sin � d� dr1

¼ 4pk2


� �a1Re�kR ð13:45Þ

one can calculate the first term on the right-hand side of Eq. (13.44). Using this

result and adding the corresponding contribution from the integration over the vol-

ume V2 of sphere 2, we finally obtain

VðRÞ ¼ 4pereoa1a2co1co2

e�kðR�a1�a2Þ

Rð13:46Þ

where

coi ¼rfixi

2ereok21þ e�2kai � 1� e�2kai

kai

� �

¼ rfixiereok2

coshðkaiÞ � sinhðkaiÞkai

� � ð13:47Þ

is the unperturbed surface potential of sphere i (i¼ 1, 2). Equation (13.46) coincides

with the interaction energy obtained by the linear superposition approximation (see

Chapter .

We consider the limiting case of kai! 0 (i¼ 1, 2). In this case, Eq. (13.47)

becomes

coi ¼rfixia

2i

3ereok2¼ Qi

4pereoaið13:48Þ

where

Qi ¼4pa3

3rfixi ð13:49Þ

TWO POROUS SPHERES 309

is the total charge amount of sphere i. Thus, Eq. (13.46) tends to

VðRÞ ¼ Q1Q2

4pereoRð13:50Þ

which agrees with a Coulomb interaction potential between two point charges Q1

and Q2, as should be expected.

Consider the validity of Derjaguin’s approximation. In this approximation, the

interaction energy between two spheres of radii a1 and a2 at separation H between

their surfaces is obtained by integrating the corresponding interaction energy

between two parallel membranes at separation h via Eq. (13.28). We thus obtain

VðHÞ ¼ 2pereo2a1a2a1 þ a2

� �co1co2 e

�kH ð13:51Þ

This is the result obtained via Derjaguin’s approximation [4] (Chapter 12). On the

other hand, by expanding the exact expression (13.46), we obtain

VðHÞ ¼ 2pereo2a1a2a1 þ a2

� �co1co2 e

�kH 1� 1

ka1� 1

ka2� H

a1 þ a2þ � � �

� �ð13:52Þ

We see that the first term of the expansion of Eq. (13.52) indeed agrees with Derja-

guin’s approximation (Eq. (13.51)). That is, Derjaguin’s approximation yields the

correct leading order expression for the interaction energy and the next-order cor-

rection terms are of the order of 1/ka1, 1/ka2, and H/(a1 + a2).

13.4 TWO PARALLEL POROUS CYLINDERS

Consider the double-layer interaction between two parallel porous cylinders 1 and 2

of radii a1 and a2, respectively, separated by a distance R between their axes in an

electrolyte solution (or, at separation H¼R� a1� a2 between their closest dis-

tances) [5]. Let the fixed-charge densities of cylinders 1 and 2 be rfix1 and rfix2,respectively. As in the case of ion-penetrable membranes and porous spheres, the

potential distribution for the system of two interacting parallel porous cylinders is

given by the sum of the two unperturbed potentials

c ¼ cð0Þ1 þ cð0Þ

2 ð13:53Þwith

cð0Þi ðriÞ ¼

coi

K0ðkriÞK0ðkaiÞ ; ri � ai

rfixiereok2

� coi

K1ðkaiÞI0ðkriÞK0ðkaiÞI1ðkaiÞ ; 0 � ri � ai

8>>><>>>:

ð13:54Þ


where

coi ¼rfixiereok

aiK0ðkaiÞI1ðkaiÞ; ði ¼ 1; 2Þ ð13:55Þ

is the unperturbed surface potential of cylinder i and ri is the distance measured

from the axis of cylinder i. We calculate the interaction energy per unit length be-

tween soft cylinders 1 and 2 from V(R)¼F(R)�F(1), with the result that [5].

VðRÞ ¼ 2pereoco1co2

K0ðkRÞK0ðka1ÞK0ðka2Þ ð13:56Þ

For large ka1 and ka2, Eq. (13.56) reduces to

VðRÞ ¼ 2ffiffiffiffiffiffi2p

pereoco1co2

ffiffiffiffiffiffiffiffiffiffiffiffika1a2

p e�kðR�a1�a2Þ

Rð13:57Þ

If, further, H� a1 and H� a1, then Eq. (13.61) becomes

VðRÞ ¼ 2ffiffiffiffiffiffi2p

pereoco1co2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffika1a2a1 þ a2

re�kðR�a1�a2Þ ð13:58Þ

which agrees with the result obtained via Derjaguin’s approximation [6]

(Chapter 12).

13.5 TWO PARALLEL MEMBRANESWITHARBITRARY POTENTIALS

13.5.1 Interaction Force and Isodynamic Curves

Consider two parallel planar dissimilar ion-penetrable membranes 1 and 2 at sep-

aration h immersed in a solution containing a symmetrical electrolyte of valence zand bulk concentration n. We take an x-axis as shown in Fig. 13.2 [7–9]. We

denote by N1 and Z1, respectively, the density and valence of charged groups in

membrane 1 and by N2 and Z2 the corresponding quantities of membrane 2. With-

out loss of generality we may assume that Z1> 0 and Z2 may be either positive or

negative and that Eq. (13.1) holds. The Poisson–Boltzmann equations (13.2)–

(13.4) for the potential distribution c(x) are rewritten in terms of the scaled

potential y¼ zec/kT as

d2y

dx2¼ k2 sinh y� Z1N1

2zn

� �; x < 0 ð13:59Þ

TWO PARALLEL MEMBRANES WITH ARBITRARY POTENTIALS 311

d2y

dx2¼ k2 sinh y; 0 < x < h ð13:60Þ

d2y

dx2¼ k2 sinh y� Z2N2

2zn

� �; x > h ð13:61Þ

where

k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2nz2e2

ereokT

sð13:62Þ

is the Debye–Huckel parameter for a symmetrical electrolyte solution. Equations

(13.59)–(13.61) are subject to the boundary conditions

yð0�Þ ¼ yð0þÞ ð13:63Þ

yðh�Þ ¼ yðhþÞ ð13:64Þ

dy

dx

��x¼0�

¼ dy

dx

��x¼0þ

ð13:65Þ

dy

dx

��x¼h�

¼ dy

dx

��x¼hþ

ð13:66Þ

y ! yDON1 as x ! �1 ð13:67Þ

y ! yDON2 as x ! þ1 ð13:68Þ

Equations (13.67) and (13.68) correspond to the assumption that the potential far

inside the membrane is always equal to the Donnan potential. Expressions for

yDON1 and yDON2 can be derived by setting the right-hand sides of Eqs. (13.59) and

(13.61) equal to zero, namely,

yDON1 ¼ arcsinhZ1N1

2zn

� �ð13:69Þ

yDON2 ¼ arcsinhZ2N2

2zn

� �ð13:70Þ

which, by Eq. (13.1), satisfy

yDON1 � yDON2j j > 0 ð13:71Þ


Equations (13.59)–(13.61) can be integrated once to give

dy

dx

� �2

¼ k2 2ðcosh y� cosh yDON1Þ �Z1N1

znðy� yDON1Þ

� �; x < 0 ð13:72Þ

dy

dx

� �2

¼ k2ð2 cosh yþ CÞ; 0 < x < h ð13:73Þ

dy

dx

� �2

¼ k2 2ðcosh y� cosh yDON2Þ �Z2N2

znðy� yDON2Þ

� �; x > h ð13:74Þ

where in deriving Eqs. (13.72) and (13.74), boundary conditions (13.67) and

(13.68) have been used and C is an integration constant. As will be shown later

(Eq. (13.88)), the interaction force is related to C.To obtain a relationship between C and h, one must further integrate Eq. (13.73).

If y(x) passes through a minimum ym at some point x¼ xm (0< xm< h), then furtherintegration of Eq. (13.73) for 0< x< xm yields

kx ¼Z yð0Þ

y

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 cosh yþ C

p ð13:75Þ

and for xm< x< h

kðx� xmÞ ¼Z y

ym


p ð13:76Þ

where xm must satisfy

kxm ¼Z yð0Þ

ym


p ð13:77Þ

It follows from Eq. (13.73) that

C ¼ �2 cosh ym ð13:78Þ

If there is no potential minimum, Eq. (13.75) is valid for any point in the region

0< x< h. In this case Eq. (13.78) does not hold.By evaluating integrals (13.75) and (13.76) at x¼ xm and x¼ h, we can obtain

relationships between C and h, as follows. If there exists a potential minimum, then

kh ¼Z yð0Þ

ym


p þZ yðhÞ

ym


p ð13:79Þ


If there is no potential minimum, then

kh ¼Z yð0Þ

yðhÞ


p ð13:80Þ

Note that y(0) and y(h), which are the membrane surface potentials, are not constant

but depend on C. Therefore one needs another relationship connecting y(0) and y(h)with C, which can be derived as follows. Substituting Eqs. (13.72)–(13.74) into

Eqs. (13.65) and (13.66), we obtain

yð0Þ ¼ yDON1 � tanhyDON12

� �� C þ 2

2 sinh yDON1ð13:81Þ

yðhÞ ¼ yDON2 � tanhyDON22

� �� C þ 2

2sinh yDON2ð13:82Þ

where Z1N1/zn and Z2N2/zn have been replaced by 2 sinh yDON1 and 2 sinh yDON2,respectively (via Eqs. (13.69) and (13.70). If we put C¼�2 (or P¼ 0) in

Eqs. (13.81) and (13.82), then these equations give the unperturbed surface poten-

tials co1 and co2 of membranes 1 and 2, respectively, in the absence of interaction

(or, the membranes are at infinite separation h¼1), namely,

yð0Þ ¼ yo1 ¼ yDON1 � tanhðyDON1=2Þ ð13:83Þ

yðhÞ ¼ yo2 ¼ yDON2 � tanhðyDON2=2Þ ð13:84Þ

with

yo1 ¼zeco1

kTð13:85Þ

yo2 ¼zeco2

kTð13:86Þ

where yo1 and yo2 are the scaled unperturbed surface potentials of membranes 1 and

2, respectively. Coupled equations (13.79) (or (13.80)), (13.81), and (13.82) can

provide y(0), y(h), and C as functions of h for given values of Z1N1/zn and Z2N2/zn(or yDON1 and yDON2).

The electrostatic force acting between the two membranes can be obtained by

integrating the Maxwell stress and the osmotic pressure over an arbitrary surface

enclosing any one of the membranes. We may choose two planes x¼�1 (in the

solution) and x¼ x at an arbitrary point in the region 0< x< h as a surface


enclosing membrane 1. On the plane x¼�1, Maxwell’s stress is zero. The force

driving the two membranes apart per unit area is thus given by (see Eq. (8.20))

PðhÞ ¼ 2nkT cosh yðxÞ � 1f g � 1

2ereo

dcdx

� �2

; 0 < x < h ð13:87Þ

which can be rewritten using Eq. (13.73) as

PðhÞ ¼ �nkTðC þ 2Þ ð13:88Þ

where P> 0 (C<�2) corresponds to repulsion and P< 0 (C>�2) to attraction.

As shown before, C(h) and thus P(h) are determined by yDON1 and yDON2.Consider the sign of P(h) when the values of yDON1 and yDON2 are given.

According to Eq. (13.73), we plot isodynamic curves (i.e., y as a function of x for

a given value of C) for the case C<�2 in Fig. 13.7 and for the case C>�2 in

Fig. 13.8. The idea of isodynamic curves has been introduced first by Derjaguin

[10]. The distance between two intersections of one of these isodynamic curves

with two straight lines y¼ y(0) (solid lines) and y¼ y(h) (dotted lines), both of

which must correspond to the same value of C, gives possible values for h. Threeisodynamic curves (C¼C1, C2, and C3) and the corresponding three values of h(h1, h2, and h3) are illustrated in Figs 13.7 and 13.8. It must be stressed that both y(0) and y(h) are not constant but change with C according to Eqs. (13.81) and

(13.82). (In the system treated by Derjaguin [10], y(0) and y(h) are constant.) Con-sequently, the pair of straight lines y¼ y(0) and y¼ y(h) moves with changing C. InFig. 13.7 (where C3<C2<C1<�2), both lines y¼ y(0) and y¼ y(h) shift upwardwith decreasing C, while in Fig. 13.8 (where C3>C2>C1>�2), y¼ y(0) moves

downward and y¼ y(h) moves upward with increasing C.

FIGURE 13.7 Schematic representation of a set of integral (isodynamic) curves for

C<�2 (see text).


Figure 13.7 shows that for the case C<�2 there exist intersections of the isody-

namic curve with y¼ y(0) and y¼ y(h) only when yDON1> 0 and yDON2> 0 (or

Z1> 0 and Z2> 0). In other words, if Z1> 0 and Z2> 0, then C<�2 (or P> 0).

Similarly, it can be proven by Fig. 13.8 that if Z1> 0 and Z2< 0, then C>�2 (or

P< 0). Figure 13.7 also shows that for the curve with C¼C2, y(h) coincides withyDON2. We denote the corresponding value of h (i.e., h2) by hc. In this situation,

from Eq. (13.82) we have C¼C2¼�2 cosh yDON2 so that Eq. (13.80) yields

khc ¼Z yð0Þ

yDON2

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðcosh y� cosh yDON2Þ

p ð13:89Þ

which, for low potentials, reduces to Eq. (13.23). Clearly, if h> hc, there exists

a potential minimum in the region 0< x< h, while if h� hc, there is no potential

minimum. Therefore, one must use Eq. (13.79) if Z1> 0, Z2> 0, and h> hc, andEq. (13.80) if Z1> 0, Z2> 0, and h� hc, or if Z1> 0 and Z2< 0. When Z1> 0 and

Z2< 0, at some point h¼ ho, the surface potential y(h) of the negatively charged

membrane becomes zero. It follows from Eqs. (13.80) and (13.82) that the value of

ho is given by

kho ¼Z yð0Þ

0


p ð13:90Þ

and

C ¼ �2ðcosh yDON2 � yDON2 sinh yDON2Þ ð13:91Þ

FIGURE 13.8 Schematic representation of a set of integral (isodynamic) curves for

C>�2.


The magnitude of P(h) has its largest value at h¼ 0. The value of P(0) can be de-

rived by equating y(0) and y(h) in Eqs. (13.81) and (13.82), namely,

Pð0Þ ¼ � 2½yDON1 � yDON2 � ftanhðyDON1=2Þ � tanhðyDON2=2Þg�cosech yDON1 � cosech yDON2

nkT ð13:92Þ

For the special case of two similar membranes, that is, yDON1¼ yDON2¼ yDON, y(x)reaches a minimum ym¼ y(h/2) at the midpoint between the two membranes, that

is, x¼ xm¼ h/2 so that Eqs. (13.77) and (13.78) give

kh2

¼Z yo

ym

dyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðcosh y� cosh ymÞ

p ð13:93Þ

or equivalently

coshyo2

� �¼ cosh

ym2

� �dc

kh2cosh

ym2

� � 1

coshðym=2Þ� �

ð13:94Þ

where dc is a Jacobian elliptic function with modulus 1/cosh(ym/2). Also,

Eqs. (13.81) and (13.82) become

yo ¼ yDON � tanhðyDON=2Þ þ2 sinh2ðym=2Þsinh yDON

ð13:95Þ

where yo¼ y(0) is the scaled membrane surface potential. Equations (13.94) and

(13.95) form coupled equation that determine yo and ym for given values of kh and

yDON. These equations can be numerically solved. The interaction force P(h) perunit area between membranes 1 and 2 is calculated from Eqs. (13.88) and (13.88),

namely,

PðhÞ ¼ 4nkT sinh2ym2

� �¼ 4nkT sinh2

yðh=2Þ2

� �ð13:96Þ

which reaches its largest value P(0) at h¼ 0, that is,

Pð0Þ ¼ 4nkT sinh2yDON2

� �ð13:97Þ

13.5.2 Interaction Energy

We calculate the potential energy of the double-layer interaction between two par-

allel ion-penetrable dissimilar membranes. We imagine a charging process in which

all the membrane-fixed charges are increased at the same rate. Let n be a parameter

that expresses a stage of the charging process and varies from 0 to 1. Then, at stage


n, the fixed charges of membranes 1 and 2 are nZ1eN1 and nZ2eN2, respectively. We

denote by c(x, n) the potential at stage n. The double-layer free energy per unit area

of the present system is thus (see Eq. (5.90))

F ¼ Z1eN1

Z 1

0

dnZ 0

�1cðx; nÞdxþ Z2eN2

Z 1

0

dnZ 1

h

cðx; nÞdx ð13:98Þ

An alternative expression for the double-layer free energy can be derived by consid-

ering a discharging process in which all the charges in the system (both the mem-

brane-fixed charges and the charges of electrolyte ions) are decreased at the same

rate. Let l be a parameter that expresses a stage of the discharging process, varying

from 1 to 0, and c(x, l) be the potential at stage X. Then the free energy per unit

area is given by Eq. (5.86), namely,

F ¼Z 1

0

dl2

lEðlÞ ð13:99Þ

with

EðlÞ ¼ 1

2

Z 1

�1cðx; lÞrðx; lÞdX ð13:100Þ

where E(l) and r(x, l) are, respectively, the internal energy per unit area and the

volume charge density at stage l. The Poisson–Boltzmann equation at stage l is

@2cðx; lÞ@x2

¼ � rðx; lÞereo

ð13:101Þ

where

rðx; lÞ ¼ �2znle sinhzlecðx; lÞ

kT

� �� Z1N1

2zn

� �; x < 0 ð13:102Þ


kT

� �; 0 < x < h ð13:103Þ


kT

� �� Z2N2

2zn

� �; x > h ð13:104Þ

Substituting Eqs. (13.102)–(13.104) into (13.99), we obtain after lengthy algebra

F ¼ nkTðC þ 2Þh� ereo

Z 1

�1

dcdx

� �2

dxþ Fc ð13:105Þ


with

Fc ¼ 2nkT sinh yDON1 yDON1 � tanhyDON12

� �n oZ 0

�1dx

¼ þsinh yDON2 yDON2 � tanhyDON22

� �n oZ 1

h

dx

ð13:106Þ

Note that although Fc is infinitely large, it does not contribute to the potential

energy of interaction since it is independent of h. To calculate the integral in

Eq. (13.105), one must substitute Eqs. (13.72)–(13.74) separately in the three

regions x< 0, 0< x< h, and x> h. The potential energy V(h) of double-layer inter-action per unit area is given by


In Fig. 13.9, we give the reduced potential energy V¼ (k/64nkT)V of the double-

-layer interaction between two ion-penetrable semi-infinite similar membranes as a

FIGURE 13.9 The reduced potential energy V¼ (k/64nkT)V of the double-layer interac-

tion between two ion-penetrable semi-infinite similar membranes as a function of the scaled

membrane separation kh for yo1¼ yo2¼ yo¼ 1 and 2. Comparison is made with the results for

the two conventional models for hard plates given by Honig and Mul [11]. Curve 1, constant

surface charge density model; curve 2, Donnan potential regulation model; curve 3, constant

surface potential model. The dotted line shows the result for the linear superposition approxi-

mation. From Ref. [9].


function of the scaled membrane separation kh for yo1¼ yo2¼ yo¼ 1 and 2 calcu-

lated via Eqs. (13.105) and (13.106), which become

F ¼ �4nkT sinh2yðh=2Þ

2

� �� h� ereo

Z h=2

�1

dcdx

� �2

dxþ Fc ð13:108Þ

with

Fc ¼ 4nkT sinh yDON yDON � tanhyDON2

� �n oZ 0

�1dx

ð13:109Þ

Comparison is made with the results for the two conventional models for hard plates

given by Honig and Mul [11]. We see that the values of the interaction energy

calculated on the basis of the Donnan potential regulation model lie between those

calculated from the conventional interaction models (i.e., the constant surface

potential model and the constant surface charge density model) and are close to the

results obtained the linear superposition approximation.

In the linear superposition approximation (see Eqs. (11.11) and (11.14)), the in-

teraction force P(h) and potential energy V(h) per unit area between membranes 1

and 2 are, respectively, given by

PðhÞ ¼ 64 tanhyo4

� �2nkT expð�khÞ ð13:110Þ

VðhÞ ¼ 64

ktanh

yo4

� �2nkT expð�khÞ ð13:111Þ

Note that for the low potential case, the Donnan potential regulation model agrees

exactly with the LSA results.

13.6 pH DEPENDENCE OF ELECTROSTATIC INTERACTIONBETWEEN ION-PENETRABLE MEMBRANES

Consider two parallel planar similar ion-penetrable membranes 1 and 2 at sepa-

ration h immersed in a solution containing a symmetrical electrolyte of valence

z and bulk concentration n [12]. We take an x-axis as shown in Fig. 13.2. The

surface is in equilibrium with a monovalent electrolyte solution of bulk concen-

tration n. Note here that n represents the total concentration of monovalent cat-

ions including Hþ ions and that of monovalent anions including OH� ions. Let

nH be the Hþ concentration in the bulk solution phase. In the membrane phase,

monovalent acidic groups of dissociation constant Ka are distributed at a density

Nmax. The mass action law for the dissociation of acidic groups AH (AH ()A�þHþ) gives the number density N(x) of dissociated groups at position x in


membrane 1, namely,

NðxÞ ¼ Nmax

1þ nH=Ka expð�ecðxÞ=kTÞ ð13:112Þ

Here nH exp(�ec(x)/kT) is the H+ concentration at position x. The charge densityrfix(x) resulting from dissociated acidic groups at position x in the membrane is

thus given by

rfixðxÞ ¼ �eNðxÞ ¼ � eNmax

1þ nH=Ka expð�ecðxÞ=kTÞ ð13:113Þ

The Poisson–Boltzmann equations for c(x) to be solved are then given by

d2cdx2

¼ 2en

ereosinh

eckT

� �� rfixðxÞ

ereoð13:114Þ

which is

d2cdx2

¼ 2en

ereosinh

eckT

� �þ e

ereo

Nmax

1þ nH=Ka expð�ecðxÞ=kTÞ ; x < 0 ð13:115Þ

For the solution phase (x> 0) we have

d2cdx2

¼ 2en

ereosinh

eckT

� �; 0 < x � h=2 ð13:116Þ

The Donnan potential in the membrane 2, which we denote by cDON is obtained

by setting the right-hand side of the Poisson–Boltzmann equation (13.115) equal

to zero. That is, cDON is the solution to the following transcendental equation:

sinhecDON

kT

� �þ Nmax

2n

1

1þ nH=Ka expð�ecDON=kTÞ¼ 0 ð13:117Þ

The double-layer free energy per unit area of the present system is given by

FðhÞ ¼ 2

Z 0

�1f ðxÞdx ð13:118Þ

where the factor 2 corresponds to two membranes 1 and 2, and f(x) is the free

energy density at position x in membrane 1, given by

f ðxÞ ¼Z rfixðxÞ

0

cðxÞdxe� rfixðxÞcðxÞ

�NkT ln 1þ nH e�yo

Kd

exp � ecðxÞkT

� �� ð13:119Þ

ph dependence of electrostatic interaction between ion-penetrable membranes 321

The potential energy V(h) of the double-layer interaction per unit area between

two membranes 1 and 2 is given by


REFERENCES

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

Press, 2006, Chapter 16.

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

3. H. Ohshima, Adv. Colloid Interface Sci. 53 (1994) 77.

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

5. H. Ohshima, Colloid Polym. Sci. 274 (1998) 1176.

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

7. H. Ohshima and T. Kondo, J. Theor. Biol. 128 (1987) 187.



10. B. V. Derjaguin, Discuss. Faraday Soc. 18 (1954) 85.

11. E. P. Honig and P. M. Mul, J. Colloid Interface Sci. 36 (1973) 258.

12. H. Ohshima and T. Kondo, Biophys. Chem. 32 (1988) 131.


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Biophysical Chemistry of Biointerfaces (Ohshima/Biophysical Chemistry of Biointerfaces) || Donnan Potential-Regulated Interaction Between Porous Particles