An Approximate Expression of Double Layer Interaction Between

Two Parallel Similar Plates with High Constant Surface

Potential in Asymmetric Electrolytes

Genxiang Luo,* Chunsheng Liu, Yun Ling, and Wang Hao Ping

Department of Chemistry, Liaoning University of Petroleum and Chemical Technology, Fushun, Liaoning, P.R. China

ABSTRACT

Using the extended Langmuir’s method, we derived the relation of surface potential,

potential midway, and the plate distance for asymmetric electrolytes in the case of

high constant surface potential. Thus, we obtained an approximate expression of the

interaction force and energy between two similar plates for asymmetric electrolytes

at constant surface potential. It is found that this approximation works quite well for

small plate separations for asymmetric electrolytes at high surface potentials.

Key Words: Poisson–Boltzmann equation; Double layer interaction; Asymmetrical

electrolyte; Langmuir’s method.

1. INTRODUCTION

The solution to the Poisson–Boltzmann equation for

the electric potential distribution is necessary to calculate

the electrical double layer interaction between colloidal

particles. For interactions involving symmetric electro-

lytes, the problem has been studied extensively. How-

ever, for mixed valence systems, only specific cases

have been analyzed. Gouy[1] gave explicit expressions

for the potential profiles in 1-2 and 2-1 electrolytes, a

problem revisited by Grahame.[2] deLevie[3] gave a rela-

tively simple approximation to the potential-distance

profile for l-l, 1-2, 2-1, and 2-2 electrolytes. Kuo and

Hsu[4] derived an analytical expressions for the electrical

potential, surface excess of co-ions, double layer free

energy, and entropy for a planar charged surface in

general electrolyte solution which was based on pertur-

bation method. It is therefore most desirable to have a

simple method of calculating electrical double layer

interactions for all electrolyte compositions. Recently,

Chen and Singh[5] proposed an exact solution for sym-

metrical and z2 2z/2z2 z asymmetrical electrolytes

and an approximate solution for other asymmetrical

electrolytes (z2 3z/3z2 z or 2z2 3z/3z2 2z) in

869

DOI: 10.1081/DIS-200035697 0193-2691 (Print); 1532-2351 (Online)

Copyright # 2004 by Marcel Dekker, Inc. www.dekker.com

*Correspondence: Genxiang Luo, Department of Chemistry, Liaoning University of Petroleum and Chemical Technology, Fushun,

Liaoning 113001, P.R. China; E-mail: genxiangluo@yahoo.com.cn.

JOURNAL OF DISPERSION SCIENCE AND TECHNOLOGY

Vol. 25, No. 6, pp. 869–874, 2004

semi-infinite planar symmetry. Chan[6] gave a simple

algorithm for calculating electrical double layer inter-

actions in asymmetric electrolytes. However, the solu-

tions are not simple to implement.

For high surface potentials, we have recently pro-

posed an extended Langmuir’s method, one integrating

the force between the two plates with respect to the dis-

tance, an approximate explicit expression for the inter-

action energy shall be derived. In previous papers,[7–9]

we showed that this method works quite well for the

case in which the high surface potential retains constant

during interaction in symmetric electrolytes. In this

paper, we applied this method to the cases of high

surface potential in asymmetric electrolytes.

2. THEORY

Considering two parallel plates are h apart in an

asymmetrical electrolyte of valence. An x-axis is perpen-

dicular to given plates, and one plate located its origin.

The distribution of scaled electrical potential y in the

electrical double layer near a planar charged surface in

an a : b electrolyte solution can be described by Ref.[10]

d2y

dj2¼ g

aþ bð1Þ

where

y ¼ ec

kT; j ¼ kx; g ¼ expðbyÞ � expð�ayÞ;

k2 ¼ e2abðaþ bÞNAn0

1r10kT

In these expressions, c is the potential, e is the

elementary electric charge, n0 is the bulk concentration,

10 is the permittivity of vacuum, 1r is the relative permit-

tivity of the solution, k is the Boltzmann constant, T is the

absolute temperature, NA is Avogadro’s number, and k is

the Debye–Huckel parameter.

For present purpose, y � 1 and obtained

g ¼ expðbyÞ � expð�ayÞ � expðbyÞ ð2Þ

Substituting Eq. (2) into Eq. (1) gives

d2y

dj2� expðbyÞ

aþ bð3Þ

Equation (3) is integrated:

dy

dj¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

bðaþ bÞ expðbyÞ þ C

s

ð4Þ

where C is integral constant.

2.1. Potential Distribution near a

Charged Planar Plate with

Asymmetric Electrolytes

For a single planar plate, the boundary conditions

for Eq. (4) are

x ¼ 0; y ¼ y0; x ! 1; y ¼ 0

The integral constant in Eq. (4) is

C ¼ � 2

bðaþ bÞ

Then, Eq. (4) gives

dy

dj¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

bðaþ bÞ expðbyÞ �2

bðaþ bÞ

s

ð5Þ

The general solution of Eq. (5) becomes

2

barctan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

expðbyÞ � 1p

� arctanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

expðby0Þ � 1p

h i

¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

bðaþ bÞ

s

� kj ð6Þ

Let us expand the arctangent in a power series:

arctanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

expðbyÞ � 1p

� p

2� 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

expðbyÞ � 1p ð7Þ

arctanffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

expðby0Þ � 1p

� p

2� 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

expðby0Þ � 1p ð8Þ

Substituting Eqs. (7) and (8) into Eq. (6) gives

y¼1

bln 1þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

expðby0Þ�1p þb

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

bðaþbÞ

s

�kj

!�22

4

3

5

ð9Þ

This is the general solution of the distribution of

scaled electrical potential y in the electrical double

layer near a planar charged surface in an a : b electrolyte

solution.

2.2. Strong Interaction Between

Two Similar Planar Plate with

Asymmetric Electrolytes

For two similar planar plate, the boundary con-

ditions for Eq. (4) are

x ¼ 0; y ¼ y0; x ¼ h; y ¼ y0

Luo et al.870

when

dy

dj¼ 0; C ¼ � 2

bðaþ bÞ expðbuÞ ð10Þ

where u is the dimensionless potential in the minimum

and u ¼ ecmin/(kT). For present purpose, y ¼ u at

x ¼ h/2. Substituting Eq. (10) into Eq. (4) gives

dy

dj¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

bðaþ bÞ expðbyÞ �2

bðaþ bÞ expðbuÞs

ð11Þ

The solution of Eq. (11) is

exp � bu

2

� �

� 4btan�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ebðy0�uÞ � 1p

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

bðaþ bÞ khs

ð12Þ

In Eq. (12),

ebðy0�uÞ � 1 � ebðy0�uÞ ð13Þ

and

tan�1 x � p

2� 1

xð14Þ

Equations (13) and (14) may then be substituted into Eq.

(12), we obtain

u ¼ 2

bln

2p

bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2=ðbðaþ bÞÞp

� khþ 4 expð�by0=2Þð15Þ

The repulsive force per unit area (or disjoining

pressure) between two similar plates for asymmetric

electrolyte, P, is given by Derjaguin and Levich[11] as

p ¼ kT �X

i

ni½expð�niuÞ � 1� ð16Þ

In Eq. (16), vi is the charge of the ion species i

including the sign, ni is its concentration at bulk. Substi-

tuting Eq. (15) into Eq. (16) gives

P ¼ n0kT bAkhþM

2p

� �2a=b

�1

" #(

þa2p

AkhþM

� �2

�1

" #)

ð17Þ

where A ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2=ðbðaþ bÞÞp

, M ¼ 4 exp(2by0/2), n0 is

the bulk concentration of a : b electrolyte at bulk.

A and M are constants for a given asymmetric

electrolyte at constant surface potential.

The free energy of interaction per unit area ought to be

VR ¼ð

1

h

p dh ð18Þ

Substituting Eq. (17) into Eq. (18) gets

VR ¼ð

1

h

n0kT bAkhþM

2p

� �2a=b

�1

" #(

þa2p

AkhþM

� �2

�1

" #)

dh ð19Þ

when h ! 1, u ! 0, then theP ! 0. Combined Eqs. (15)

and (17) under these condition, we have k h ¼ (2p2M)/A. This is upper limit, which is independent of the values of

y, of the integration of Eq. (19). Then, the integration of Eq.

(19) can converge.

Let G ¼ (2p2M)/A as the upper limit of inte-

gration and substituted it into Eq. (19), we have

VR ¼ n0kT

k

b2

Að4p2Þa=b�

ðA � GþMÞð2a=bÞþ1

2aþ b� ðA � khþMÞð2a=bÞþ1

2aþ b

" #

þ a � 4p2

A

1

A � khþM� 1

A � GþM

� �

�ðaþ bÞðG� khÞ�

ð20Þ

where A ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2=ðbðaþ bÞÞp

, M ¼ 4 exp (2 by0/2), andG ¼ (2p2M)/A.

3. RESULTS AND DISCUSSION

First, we check the accuracy of approximate solution

of potential distribution using the exact solution (z2 2z/2z2 z) and the approximate solutions (z2 3z/3z2 z

and 2z2 3z/3z2 2z) obtained by Chen,[5] respectively.

The error is defined as

Err ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P

ðy� y0Þ2P

y20

s

ð21Þ

where y0 is result obtained by Chen[5] and y is the present

result obtained from Eq. (9) at kh. The approximation

fd2 y/dj2g � exp (by)/(aþ b), which is applied to the

one-dimensional PB equation, is exact only in the

region near the plate surface where y � 1. However,

in the region far from the surface of the plate Eq. (3) is

not exact. Therefore, the errors were calculated using

equal length (0.000001) of kh in the range from 0 to 2.

Figures 1–3 are the comparisons of the exact solutions

Double Layer Interaction Between Two Parallel Similar Plates 871

obtained by Chen[5] and our approximate solutions when

the surface potential is 10 and the electrolyte is 2 : 1/1 : 2,1-3/3-1, and 2-3/3-2 cases, respectively. Exact solutionsand approximate solutions are indistinguishable. From

Figs. 1–3, we find that the larger the b/a, the smaller

the errors become for all asymmetric electrolytes. The

errors are 3.3%, 2.11%, 2.67%, 3.38%, 2.80%, and

3.00% for 2 : 1, 1 : 2, 1 : 3, 3 : 1, 2 : 3, and 3 : 2 cases,

respectively. It shows that our general solution gives

much better results. Its error is ,3.4% in all the cases

we considered here. Besides, the general solution does

not involve complicated formulation.

Recently, Chan[6] gave a simple algorithm for calcu-

lating electrical double layer interactions in asymmetric

electrolytes. In order to evaluate the accuracy of our

approximate solution, we obtained the dimensionless

force and energy from Eqs. (17) and (20) which corre-

spond to the forms of the dimensionless force and

energy in Ref.[6], respectively.

P1 ¼ 1

ðaþ bÞ1

a

AkhþM

2p

� �2a=b

�1

" #(

þ 1

b

2p

AkhþM

� �2

�1

" #)

; ð22Þ

V1 ¼ 1

aþ b

b

A � a1

4p2

� �a=b ðA � GþMÞð2a=bÞþ1

2aþ b

"(

� ðA � khþMÞð2a=bÞþ1

2aþ b

#

þ 1

A � b 4p2

� 1

A � khþM� 1

A � GþM

� �

� 1

aþ 1

b

� �

ðG� khÞ�

ð23Þ

where A ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2=ðbðaþ bÞÞp

, M ¼ 4 exp(2by0/2), and

G ¼ (2p2M)/A.Figures 4 and 5 make the comparisons of Eqs. (22)

and (23) with the values of the interaction of similar

plates given by Chan[6] for asymmetric electrolytes.

The percentage relative error has been defined by

Err ¼ zi � z

z� 100

Figure 1. The comparison of potential distributions of present

approximate solution (------) with the exact solution (z2 2z/2z2 z) (——) by Chen.[5] The scaled potential is fixed at 10.

Figure 2. The comparison of potential distributions of present

approximate solution (------) with the approximate solutions

(z2 3z/3z2 z) (——) by Chen.[5] The scaled potential is

fixed at 10.

Figure 3. The comparison of potential distributions of present

approximate solution (------) with the approximate solutions

(2z2 3z/3z2 2z) (——) by Chen.[5] The scaled potential is

fixed at 10.

Luo et al.872

whereas zi is the value calculated with Eq. (22) or (23), z

is the exact numerical value for force or energy from

Ref.[6].

The agreement of the approximation, Eqs. (22) and

(23), with the exact numerical results is good.

For y1 ¼ y2 ¼ 10, the relative errors of the dimen-

sionless force are less than 20.69% for the electrolyte

of 1 : 2 at kh , 1.21769, 2 : 1 electrolyte 20.72% at

kh, 3.61088, 1 : 3 electrolyte 20.19% at kh, 0.5411,

3 : 1 electrolyte 20.14% at kh, 4.16128, 2 : 3 electrolyte

20.4% at kh, 1.28211, and 3 : 2 electrolyte 0.03%

at kh , 1.56745, respectively. From the above results,

we find that the applicable range of Eq. (21) is larger

for a : b electrolyte if b/a becomes smaller.

For the interaction energy, when y1 ¼ y2 ¼ 10, the

relative errors of the dimensionless energy are less than

22.1% for the electrolyte of 1 : 2 at kh , 0.44657, 2 : 1

electrolyte 23.84% at kh , 2.17199, 1 : 3 electrolyte

23.61% at kh , 0.5411, 3 : 1 electrolyte 24.52% at

kh , 3.39871, 2 : 3 electrolyte 23.17% at kh ,

0.75732, and 3 : 2 electrolyte 23.36% at kh , 1.56745,

respectively. Beyond these ranges, the errors increases

rapidly. The applicable range of Eq. (23) has the same

behavior with Eq. (22) for a : b electrolyte.

4. CONCLUSIONS

We have presented a simple method of calculating

the interaction force and energy per unit area between

two similar plates with high potentials at a constant

surface potential in asymmetric electrolytes. It is seen

that the present approximation works quite well for

small separations for high surface potential but

becomes less accurate at larger separations.

ACKNOWLEDGMENT

We are grateful to the National Science Foundation

of China (NSFC) for financial support no. 20273028.

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Figure 4. The relative error of the interaction force at

y1 ¼ y2 ¼ 10 for different asymmetric electrolytes.

Figure 5. The relative error of the interaction energy at

y1 ¼ y2 ¼ 10 for different asymmetric electrolytes.

Double Layer Interaction Between Two Parallel Similar Plates 873

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Received November 3, 2003

Accepted July 27, 2004

Luo et al.874