Biophysical Chemistry of Biointerfaces (Ohshima/Biophysical Chemistry of Biointerfaces) || Series Expansion Representations for the Double-Layer Interaction Between Two Particles

14 Series ExpansionRepresentations for theDouble-Layer InteractionBetween Two Particles

14.1 INTRODUCTION

This chapter deals with a method for obtaining the exact solution to the linearized

Poisson–Boltzmann equation on the basis of Schwartz’s method [1] without

recourse to Derjaguin’s approximation [2]. Then we apply this method to derive

series expansion representations for the double-layer interaction between spheres

[3–13] and those between two parallel cylinders [14, 15].

14.2 SCHWARTZ’S METHOD

We start with the simplest problem of the plate–plate interaction. Consider two par-

allel plates 1 and 2 in an electrolyte solution, having constant surface potentials co1

and co2, separated at a distance H between their surfaces (Fig. 14.1). We take an

x-axis perpendicular to the plates with its origin 0 at the surface of one plate so that

the region 0< x< h corresponds to the solution phase. We derive the potential dis-

tribution for the region between the plates (0< x< h) on the basis of Schwartz’s

method [1]. The linearized Poisson–Boltzmann equation in the one-dimensional

case is

d2cdx2

¼ k2c; 0 < x < h ð14:1Þ

with the following boundary conditions at plate surfaces x¼ 0 and x¼ h:

cð0Þ ¼ co1 ð14:2ÞcðhÞ ¼ co2 ð14:3Þ

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

323

We write the solution to Eq. (14.1) as

cðxÞ ¼ c1ðxÞ þ c2ðxÞ; 0 < x < h ð14:4Þ

with

c1ðxÞ ¼ cð0Þ1 ðxÞ þ cð1Þ

1 ðxÞ þ cð2Þ1 ðxÞ þ � � � ð14:5Þ

c2ðxÞ ¼ cð0Þ2 ðxÞ þ cð1Þ

2 ðxÞ þ cð2Þ2 ðxÞ þ � � � ð14:6Þ

As the zeroth-order approximate solutions cð0Þ1 (x) and cð0Þ

2 (x), we choose the un-perturbed potentials produced by plates 1 and 2 in the absence of interaction (i.e.,

when they are isolated), which are (see Chapter 1)

cð0Þ1 ðxÞ ¼ co1 e

�kx ð14:7Þ

cð0Þ2 ðxÞ ¼ co2 e

�kðH�xÞ ð14:8Þ

Note that cð0Þ1 (x) and cð0Þ

2 (x), respectively, satisfy the boundary conditions (14.2)and (14.3). We construct the functions cðkÞ

1 (x) and cðkÞ2 (x) (k¼ 1, 2, . . . ) as fol-

lows. The unperturbed potential cð0Þ1 (x) satisfies the boundary condition (14.2) on

plate 1. The boundary condition (14.3) on plate 2, on the other hand, which has

been satisfied by cð0Þ2 (x), is now violated, since cð0Þ

1 (x) gives rise to the following

nonzero value on plate 2:

cð0Þ1 ðhÞ ¼ co1e

�kh ð14:9Þ

FIGURE 14.1 Interaction between two parallel dissimilar hard plates 1 and 2 at separa-

tion h.

324 SERIES EXPANSION FOR THE DOUBLE-LAYER INTERACTION

which is obtained from Eq. (14.7). We thus construct the first-order approximate

solution cð1Þ1 (x) so as to cancel cð0Þ

1 (h) on plate 2, namely,

cð0Þ1 ðhÞ þ cð1Þ

1 ðhÞ ¼ 0 ðon plate 2Þ ð14:10Þ

or

cð1Þ1 ðhÞ ¼ �cð0Þ

1 ðhÞ ¼ �co1e�kh ð14:11Þ

Therefore, the function cð1Þ1 (x) must take the form

cð1Þ1 ðxÞ ¼ ð�co1 e

�khÞe�kðh�xÞ ð14:12Þ

The function cð1Þ1 (x) can be interpreted as the ‘‘image’’ potential of cð0Þ

1 (x) withrespect to plate 2 by analogy with ‘‘the method of images’’ in electrostatics, as sche-

matically shown in Fig. 14.2.

The function cð1Þ1 (x) in turn gives rise to the following nonzero value on plate 1,

cð1Þ1 ð0Þ ¼ �co1e

�2kh ð14:13Þ

violating the boundary condition (Eq. (14.2)), which has already been satisfied

by cð0Þ1 (x). Therefore, the second-order approximate solution cð2Þ

1 (x) must be con-

structed so as to cancel cð1Þ1 (0) on plate 1, namely,

cð1Þ1 ð0Þ þ cð2Þ

1 ð0Þ ¼ 0 ðon plate 1Þ ð14:14Þ

or

cð2Þ1 ð0Þ ¼ �cð1Þ

1 ð0Þ ¼ þco1 e�2kh ð14:15Þ

Thus, cð2Þ1 (x), which can be interpreted as the ‘‘image’’ potential of cð1Þ

1 (x) withrespect to plate 1, is given by

cð1Þ1 ðxÞ ¼ ðþco1e

�2khÞe�kx ð14:16Þ

SCHWARTZ’S METHOD 325

The function cð3Þ1 (x) then must be constructed so as to satisfy

cð2Þ1 ðhÞ þ cð3Þ

1 ðhÞ ¼ 0 ðon plate 2Þ ð14:17Þ

In this way one can construct cðkÞ1 (x) (k¼ 1, 2, . . . ) that satisfy the boundary

condition (14.2) namely,

c1ð0Þ ¼ cð0Þ1 ðxÞ þ

X1n¼1

ncð2n�1Þ1 ð0Þ þ cð2nÞ

1 ð0Þo¼ co1 ðon plate 1Þ ð14:18Þ

c1ðhÞ ¼X1n¼0

ncð2nÞ1 ðhÞ þ cð2nþ1Þ

1 ðhÞo¼ 0 ðon plate 2Þ ð14:19Þ

FIGURE 14.2 The unperturbed potentials cð0Þ1 and cð0Þ

2 , and the correction terms cðkÞ1 and

cðkÞ2 (k¼ 1, 2, . . . ). Squares (dotted lines) mean that cð2nÞ

i is the image potential of cð2n�1Þi

with respect to plate i, while cð2n�1Þi is the image potential of cð2n�2Þ

i with respect to plate j

(n¼ 1, 2, . . . ; i, j¼ 1, 2; i 6¼ j). exp[�(kþ 1)kH) indicates that cðkÞ1 and cðkÞ

2 (k¼ 0, 1,

2, . . . ) are proportional to exp(�(kþ 1)kH).


Thus, we find that

c1ðxÞ ¼ co1 e�kx þ ð�co1 e

�khÞe�kðh�xÞ þ ðþco1 e�2khÞe�kx

þð�co1 e�3khÞe�kðh�xÞ þ � � �

¼ co1 e�kxð1þ e�2kh þ e�4kh þ � � �Þ � co1 e

�kðh�xÞðe�kh þ e�3kh þ � � �Þ

¼ co1 e�kx 1

1� e�2kh � e�kðh�xÞ e�kh

1� e�2kh

� �

¼ co1

sinhðkðh� xÞÞsinhðkhÞ ð14:20Þ

c2ðxÞ ¼ co2 e�kðh�xÞ þ ð�co2 e

�khÞe�kx

þðþco2 e�2khÞe�kðh�xÞð�co2 e

�3khÞe�kx þ � � �

¼ co2

sinhðkxÞsinhðkhÞ

ð14:21Þ

From Eqs. (14.20) and (14.21) we obtain

cðxÞ ¼ c1ðxÞ þ c2ðxÞ

¼ co1sinhðkðh� xÞÞ þ co1sinhðkxÞsinhðkhÞ ; 0 < x < h

ð14:22Þ

which agrees with an expression for the potential derived by directly solving

Eq. (14.1) to the boundary conditions (14.2) and (14.3).

14.3 TWO SPHERES

By applying the above method to two interacting spheres on the basis of the linear-

ized Poisson–Boltzmann equations, we can derive series expansion representations

for the double-layer interaction between two spheres 1 and 2 (Fig. 14.3).

Case (a): spheres 1 and 2 both at constant surface potentialFor the interaction between two hard spheres 1 and 2 having radii a1 and a2,and constant surface potentials co1 and co2, respectively, at separation H be-

tween the two spheres, the interaction energy Vc(R) is given by [3, 6, 8, 10]

TWO SPHERES 327

VcðRÞ ¼ 4pereoco1co2a1a2e�kðR�a1�a2Þ

R

þ2pereoc2o1a

21

e2ka1

R

X1n¼0

ð2nþ 1ÞGnð2ÞK2nþ1=2ðkRÞ

þ2pereoc2o2a

22

e2ka2

R

X1n¼0

ð2nþ 1ÞGnð1ÞK2nþ1=2ðkRÞ

þ4peereoco1co2a1a2ekða1þa2Þ

R

�X1n¼0

X1m¼0

ð2nþ 1Þð2mþ 1ÞBnmGnð2ÞGmð1ÞKnþ1=2ðkRÞKmþ1=2ðkRÞþ � � �


R

�X1n1¼0

X1n2¼0

� � �X1n2n¼0

½L12ðn1;n2; . . . ;n2nÞþL21ðn1;n2; . . . ;n2nÞ�

�Kn1þ1=2ðkRÞKn2nþ1=2ðkRÞþ 2pereoX1n1¼0

X1n2¼0

� � �X1

n2n�1¼0

ð2n2n�1þ 1ÞBn2n�2n2n�1

��c2o1a

21

e2ka1

RL21ðn1;n2; . . . ;n2n�2ÞGn2n�1

ð2Þ

þc2o2a

22

e2ka2

RL12ðn1;n2; . . . ;n2n�2ÞGn2n�1

ð1Þ�

�Kn1þ1=2ðkRÞKn2n�1þ1=2ðkRÞþ � � �ð14:23Þ

FIGURE 14.3 Interaction between two charged hard spheres 1 and 2 of radii a1 and a2 ata separation R between their centers. H (¼R� a1� a2) is the closest distance between their

surfaces. From Ref. 6.


where

L21ðn1;n2; . . . ;n2n�2Þ ¼ ð2n1þ 1Þð2n2þ 1Þ � � � ð2n2nþ 1Þ�Bn1n2Bn2n3 � � �Bn2n�1n2nGn1ð2ÞGn2ð1Þ� � � ��Gn2n�1

ð2ÞGn2nð1Þ; ðv¼ 1;2; . . .Þð14:24Þ

L12ðn1;n2; . . . ;n2nÞ ¼ ð2n1þ 1Þð2n2þ 1Þ � � � ð2n2nþ 1Þ�Bn1n2Bn2n3 � � �Bn2n�1n2nGn1ð1ÞGn2ð2Þ� � � ��Gn2n�1

ð1ÞGn2nð2Þ; ðv¼ 1;2; . . .Þð14:25Þ

GnðiÞ ¼� Inþ1=2ðkaiÞKnþ1=2ðkaiÞ

; ði¼ 1;2Þ ð14:26Þ

Bnm ¼Xminfn;mg

r¼0

Anmrp

2kR

� �1=2Knþm�2rþ1=2ðkRÞ ð14:27Þ

Anmr ¼Gðn� rþ 1

2ÞGðm� rþ 1

2ÞGðrþ 1

2Þðnþm� rÞ!ðnþm� 2rþ 1

2Þ

pGðmþ n� rþ 32Þðn� rÞ!ðm� rÞ!r!

ð14:28ÞCase (b): spheres 1 and 2 both at constant surface charge density

For the interaction between two hard spheres 1 and 2 having radii a1 and a2,and constant surface charge densities s1 and s2, respectively, at separationH between the two spheres, the interaction energy Vs(R) is given by the

following equation [3, 8, 9, 10]:

VsðRÞ ¼ 4pereoco1co2a1a2e�kðR�a1�a2Þ

R

þ 2pereoc2o1a

21

e2ka1

R

X1n¼0

ð2nþ 1ÞHnð2ÞK2nþ1=2ðkRÞ

þ 2pereoc2o2a

22

e2ka2

R

X1n¼0

ð2nþ 1ÞHnð1ÞK2nþ1=2ðkRÞ

þ 4peereoco1co2a1a2ekða1þa2Þ

R

�X1n¼0

X1m¼0

ð2nþ 1Þð2mþ 1ÞBnmHnð2ÞHmð1ÞKnþ1=2ðkRÞKmþ1=2ðkRÞ þ � � �

TWO SPHERES 329


R

�X1n1¼0

X1n2¼0

� � �X1n2n¼0

½M12ðn1; n2; . . . ; n2nÞ þM21ðn1; n2; . . . ; n2nÞ�

�Kn1þ1=2ðkRÞKn2nþ1=2ðkRÞ þ 2pereoX1n1¼0

X1n2¼0

� � �X1

n2n�1¼0

ð2n2n�1 þ 1ÞBn2n�2n2n�1

��c2o1a

21

e2ka1

RM21ðn1; n2; . . . ; n2n�2ÞHn2n�1

ð2Þ

þc2o2a

22

e2ka2

RM12ðn1; n2; . . . ; n2n�2ÞHn2n�1

ð1Þ�

�Kn1þ1=2ðkRÞKn2n�1þ1=2ðkRÞ þ � � �ð14:29Þ

with

M21ðn1; n2; . . . ; n2n�2Þ ¼ ð2n1 þ 1Þð2n2 þ 1Þ � � � ð2n2n þ 1Þ� Bn1n2Bn2n3 � � �Bn2n�1n2nHn1ð2ÞHn2ð1Þ � � � ��Hn2n�1

ð2ÞHn2nð1Þ; ðv ¼ 1; 2; . . .Þð14:30Þ

M12ðn1; n2; . . . ; n2nÞ ¼ ð2n1 þ 1Þð2n2 þ 1Þ � � � ð2n2n þ 1Þ� Bn1n2Bn2n3 � � �Bn2n�1n2nHn1ð1ÞHn2ð2Þ � � � ��Hn2n�1

ð1ÞHn2nð2Þ; ðv ¼ 1; 2; . . .Þð14:31Þ

HnðiÞ ¼In�1=2ðkaiÞ � ðnþ 1þ nepi=erÞInþ1=2ðkaiÞ=kaiKn�1=2ðkaiÞ þ ðnþ 1þ nepi=erÞKnþ1=2ðkaiÞ=kai

; ði ¼ 1; 2Þ

ð14:32Þ

where epi is the relative permittivity of sphere i and coi is the unperturbed sur-

face potential of sphere i and is related to the surface charge density si ofsphere i as

coi ¼si

ereokð1þ kaiÞ ; ði ¼ 1; 2Þ ð14:33Þ

Case (c): sphere1 maintained at constant surface potential and sphere 2 atconstant surface charge density

For the interaction between two hard spheres 1 and 2 having radii a1 and a2for the case where sphere 1 is maintained at constant surface potential c1

and sphere 2 at constant surface charge density s2, at separation H between


the two spheres, the interaction energy Vc�sðRÞ is given by the following

equation [3, 10]:

Vc�sðRÞ ¼ 4pereoco1co2a1a2e�kðR�a1�a2Þ

R

þ2pereoc2o1a

21

e2ka1

R

X1n¼0

ð2nþ1ÞHnð2ÞK2nþ1=2ðkRÞ

þ2pereoc2o2a

22

e2ka2

R

X1n¼0

ð2nþ1ÞGnð1ÞK2nþ1=2ðkRÞ


R

�X1n¼0

X1m¼0

ð2nþ1Þð2mþ1ÞBnmHnð2ÞGmð1ÞKnþ1=2ðkRÞKmþ1=2ðkRÞþ ��


R

�X1n1¼0

X1n2¼0

� � �X1n2n¼0

½N12ðn1;n2; . . . ;n2nÞþN21ðn1;n2; . . . ;n2nÞ�

�Kn1þ1=2ðkRÞKn2nþ1=2ðkRÞþ2pereoX1n1¼0

X1n2¼0

� � �

�X1

n2n�1¼0

ð2n2n�1þ1ÞBn2n�2n2n�1

��c2o1a

21

e2ka1

RN21ðn1;n2; . . . ;n2n�2ÞHn2n�1

ð2Þ

þc2o2a

22

e2ka2

RN12ðn1;n2; . . . ;n2n�2ÞGn2n�1

ð1Þ�

�Kn1þ1=2ðkRÞKn2n�1þ1=2ðkRÞþ � � �ð14:34Þ

where

N21ðn1;n2; . . . ;n2n�2Þ ¼ ð2n1þ1Þð2n2þ1Þ � � �ð2n2nþ1Þ�Bn1n2Bn2n3 � � �Bn2n�1n2nHn1ð2ÞGn2ð1Þ� �� Hn2n�1

ð2ÞGn2nð1Þ; ðv¼ 1;2; . . .Þð14:35Þ

TWO SPHERES 331

N12ðn1;n2; . . . ;n2nÞ ¼ ð2n1þ1Þð2n2þ1Þ � � � ð2n2nþ1Þ�Bn1n2Bn2n3 � � �Bn2n�1n2nGn1ð1ÞHn2ð2Þ� � � ��Gn2n�1

ð1ÞHn2nð2Þ; ðv¼ 1;2; . . .Þð14:36Þ

where co2 is the unperturbed surface potential of sphere 2 and is related to the

surface charge density s2 of sphere 2 as

co2 ¼s2

ereokð1þka2Þ ð14:37Þ

The interaction energy between spheres at constant surface potential involves

only the function Gn(i) (which depends only on the sphere radius ai), while theinteraction at constant surface charge density is characterized by the function

Hn(i) (which depends on both sphere radius ai and relative permittivity epi).The interaction energy in the mixed case involves both Gn(i) and Hn(i).

The first term (the leading term) of each of Eqs. (14.23), (14.29), and (14.34) is

independent of the type of the boundary conditions at the sphere surface and takes

the same form, namely,

VLSA ¼ Vð0ÞðRÞ ¼ 4pereoco1co2a1a2e�kðR�a1�a2Þ

Rð14:38Þ

This expression coincides with that obtained by the linear superposition of the

unperturbed potentials cð0Þ1 (r) and cð0Þ

2 (r) (Eq. (11.26)). Note that if spheres 1 and 2

were both ion-penetrable porous spheres (we here call this type of particles ‘‘soft’’

particles), the interaction energy would be given by only this term (see Chapter 13),

namely,

V soft=softðRÞ ¼ V ð0ÞðRÞ ð14:39Þ

The first-order correction to the linear superposition approximation VLSA is

given by the sum of the second and third terms on the right-hand side of

Eqs. (14.23), (14.29) and (14.34), each corresponding to the image interaction of

one sphere with respect to the other sphere. We denote this image interaction

by Vimage and expressed it as

VLSAðRÞ ¼ Vð1Þ1 ðRÞ þ V

ð1Þ2 ðRÞ ð14:40Þ

Here if sphere i is maintained at constant surface potential, then Vð0Þi (R) (i, j¼ 1,

2; i 6¼ j) is given by


Vcð1Þi ðRÞ ¼ 2pereoc

2oia

2i

e2kai

R

X1n¼0

ð2nþ 1ÞGnðiÞK2nþ1=2ðkRÞ ð14:41Þ

while if sphere J is maintained at constant surface charge density, then it is

given by

Vsð1Þi ðRÞ ¼ 2pereoc

2oia

2i

e2kai

R

X1n¼0

ð2nþ 1ÞHnðiÞK2nþ1=2ðkRÞ ð14:42Þ

If sphere i were not hard but ion-penetrable (a soft sphere), with sphere j (i, j¼ 1,

2; i 6¼ j) being a hard sphere with constant surface charge density, then the interac-

tion energy would be equal to the sum of only V(0)(R) and Vð1Þi (R), namely,

V soft=hardðRÞ ¼ Vð0ÞðRÞ þ Vð1Þi ðRÞ ði ¼ 1; 2Þ ð14:43Þ

The interaction Vð1Þi (R) depends only on the unperturbed surface potential coi of

sphere i (i¼ 1, 2) and can be interpreted as the interaction between sphere i and its

‘‘image’’ with respect to sphere j ( j¼ 1, 2; j 6¼ i). When both spheres are hard, the

interaction energy is given by

Vhardt=hardðRÞ ¼ V ð0ÞðRÞ þ Vð1Þ1 ðRÞ þ V

ð1Þ2 ðRÞ þ � � �

¼ VLSAðRÞ þ V imageðRÞ þ � � �ð14:44Þ

Figure 14.4 shows the linear superposition approximation (VLSA) and the image

interaction correction (Vimage) to VLSA as well as the full solutions, Eqs. (14.23) and

(14.29), (given as circles) for the interaction energy between two identical spheres

at ka1¼ ka2¼ 5 and co1¼co2 as functions of the scaled separation kH for two

cases where both spheres are maintained at constant surface potential or at constant

surface charge density. In the latter case the relative permittivity of spheres at con-

stant surface charge density is assumed to be zero (ep1¼ ep2¼ 0). We see that the

interaction energy is well approximated by the leading term plus the first-order cor-

rection especially for kH> 0.5. This is because the leading term and the first-order

correction are, respectively, of the order of e�kH and e�2kH and the higher correc-

tion terms are of O(e�3kH).

As is seen in Fig. 14.4, the first-order correction (the image interaction) is nega-

tive (attractive) for the constant surface potential case (a) and positive (repulsive)

for the constant surface charge density case (b) at ep1¼ ep2¼ 0. This explains the

following well-known behavior observed in the interaction between two unlike

spheres (see Fig. 14.5). Namely, in case (a) if the surface potentials of the two inter-

acting spheres are of the different magnitude and of the same sign (co1¼ 4co2 and

ka1¼ ka2¼ 5 in Fig. 14.5), then the interaction energy, when plotted as a function

TWO SPHERES 333

of kH, exhibits a maximum. In other words, the interaction force, which is repulsive

at large separations, may become attractive at small separations. In case (b), on the

other hand, if the surface potentials of the two interacting spheres are of the dif-

ferent magnitude and of the opposite sign (co1¼�4co2 and ka1¼ ka2¼ 5 in

FIGURE 14.4 Comparison of the linear superposition approximation VLSA¼V(0), the

image interaction correction Vimage¼V(1)þV(2), their sum VLSAþVimage, and the full

solutions, Eqs. (14.23) and (14.29), (represented by circles) as functions of the scaled separa-

tion kH at ka1¼ ka2¼ ka¼ 5 and c01¼c02¼c0 for the interaction of two similar spheres at

constant surface potential (a) and at constant surface charge density (b). The reduced quantit-

ies are given in the figure: V�LSA ¼VLSA/2pereoc0

2a and V�image ¼Vimage=2pereoc0

2a. FromRef. [10].


FIGURE 14.5 The linear superposition approximation VLSA, the image interaction correc-

tion Vimage, and their sum VLSAþVimage for the system of two dissimilar spheres as functions

of the scaled separation kH at ka1¼ ka2¼ ka¼ 5 for two cases. For the constant surface

potential case (a), the unperturbed surface potentials of the two spheres are of the different

magnitude and of the same sign (co1¼ 4co2¼co). For the constant surface charge density

case (b), the unperturbed surface potentials are of the different magnitude and of opposite

sign (co1¼�4co2¼�co). The spheres at constant surface charge density have ep¼ 0.

The reduced quantities are given in the figure: V�LSA ¼VLSA=2pereoc0

2a and V�image ¼

Vimage=2pereoc02a. From Ref. [10].

TWO SPHERES 335

Fig. 14.5), then the interaction energy shows a minimum. That is, the interaction

force, which is attractive at large separations, may become repulsive at small sepa-

rations. As Fig. 14.5 shows, the change in sign of the interaction force or the ap-

pearance of the extremum in the interaction energy occurs when the contribution

of the image interaction correction exceeds that of the leading term (or the linear

superposition approximation term).

In case (c), the image interaction energy carries both characters of cases (a) and

(b). When the unperturbed surface potentials and the radii of the two spheres be-

come similar, these two contributions from cases (a) and (b) tend to cancel each

other so that the total image interaction for case (c) becomes small, as shown

in Fig. 14.6, in which the interacting spheres are identical (ka1¼ ka2¼ 5 and

co1¼co2). In the opposite case where the difference in the two unperturbed poten-

tials is large, the image interaction for case (c) is determined almost only by the

larger unperturbed surface potential.

Figure 14.7 shows the dependence of the image interaction between two similar

spheres (ka1¼ ka2¼ ka and co1¼co2) on the value of ka at kH¼ 0.5 for three

cases: the constant surface potential case (a), the constant surface charge density

case (b), and the mixed case (c). In case (b), the relative permittivity values of

spheres at constant surface charge density are assumed to be zero (ep1¼ ep2¼ 0).

We see that in the limit of large ka, the image interaction energies for cases (a) and

(b) tend to have the same magnitude but the opposite sign so that the image interac-

tion becomes zero for case (c).

FIGURE 14.6 The reduced image interaction energy V�image ¼Vimage=2pereoc0

2a as a

function of the scaled separation kH at ka1¼ ka2¼ ka¼ 5 and co1¼co2¼co for the con-

stant surface potential case (a), the constant surface charge density (b), and the mixed cases

(c). The sphere at constant surface charge density has epi¼ 0. From Ref. [10].


In the limit of large ka1 and ka2, one can derive analytic approximate expres-

sions for the image interaction Vimage on the basis of the following approximations:

X1n¼0

ð2nþ 1ÞGnðiÞK2nþ1=2ðkRÞ � � ai e

�2kðHþajÞ

2ðH þ ajÞ ; ði ¼ 1; 2Þ ð14:45Þ

X1n¼0

ð2nþ 1ÞHnðiÞK2nþ1=2ðkRÞ �

ai e�2kðHþajÞ

2ðH þ ajÞ

� 1� epier

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðH þ a1 þ a2ÞkaiðH þ ajÞ

s( ); ði ¼ 1; 2Þ

ð14:46Þ

The results are given as follows.

Case (a): spheres 1 and 2 both at constant surface potential

Vcimage ¼ �pereoc

2o1a

21a2

e�2kH

Rða1 þ HÞ � pereoc2o2a1a

22

e�2kH

Rða2 þ HÞ ð14:47Þ

FIGURE 14.7 The reduced image interaction energy V�image ¼Vimage/2pereoc0

2a as a

function of the scaled sphere radius ka1¼ ka2¼ ka for co1¼co2¼co at kH¼ 0.5 for the

constant surface potential (a), the constant surface charge density (b), and the mixed cases

(c). The sphere at constant surface charge density has epi¼ 0. From Ref. [10].

TWO SPHERES 337

Case (b): spheres 1 and 2 both at constant surface charge density

Vsimage ¼ pereoc

2o1a

21a2

e�2kH

Rða1 þ HÞ 1� ep2er

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðH þ a1 þ a2Þka2ðH þ a1Þ

s( )

þpereoc2o2a

22a1

e�2kH



s( )

ð14:48Þ

Case (c): sphere 1 at constant surface potential and sphere 2 at constant sur-face charge density

Vc-simage ¼ pereoc

2o1a

21a2

e�2kH



s( )

�pereoc2o2a1a

22

e�2kH

Rða2 þ HÞ

ð14:49Þ

We thus again find that in the limit of large ka1 and ka2 (at finite ep1and ep2),the image interaction energies for cases (a) and (b) tend to have the same mag-

nitude but the opposite sign so that for case (c) Vc�simage becomes very small if

co1�co2.

In Figs 14.5–14.7, the image interaction for the constant surface charge density

case (b) is positive. Note, however, that this holds only when the relative permittiv-

ity is small. The image interaction in case (b) may become negative as the magni-

tude of the ratio of the relative permittivity of the sphere to that of the solution epi/erincreases. In the limit epi!1 (this case corresponds to metallic particles), in par-

ticular, the image interaction is always negative. Figure 14.8 shows this dependence

together with those in cases (a) and (c) for ep1¼ ep2, ka1¼ ka2¼ 5, and co1¼co2

at kH¼ 0.5. For two similar spheres with constant surface charge density (ka1¼ka2¼ ka, ep1¼ ep2¼ ep, and co1¼co2¼co) at ka� 1 and H� a, the image inter-

action energy (Eq. (14.48)) becomes

Vsimage ¼ pereoc

2oae

�2kH 1� ep2er

ffiffiffiffiffiffi2pka

r !ð14:50Þ

Thus, we see that for ka� 1 and H� a,

Vsimage 0 for

ep2er

ffiffiffiffiffiffika2p

rð14:51Þ


Vsimages < 0 for

ep2er

>

ffiffiffiffiffiffika2p

rð14:52Þ

For case (a), if, further,

H � a1 and H � a2 ð14:53Þ

that is, if ka1� 1 and ka2� 1, H� a1 and H� a2, then Eq. (14.47) tends to

Vcimage ¼ �pereo

a1a2a1 þ a2

ðc2o1 þ c2

o2Þe�2kH ð14:54Þ

so that Eq. (14.44) gives

VcðHÞ ¼ 4pereoa1a2

a1 þ a2co1co2 e

�kH � 1

4ðc2

o1 þ c2o2Þe�2kH

� �þ Oðe�3kHÞ

ð14:55Þ

FIGURE 14.8 The reduced image interaction energy V�image ¼Vimage=2pereoc0

2a as a

function of the ratio of the relative permittivity of the sphere to that of the solution ep=er forep1¼ ep2¼ ep, ka1¼ ka2¼ ka¼ 5, and co1¼co2¼co at kH¼ 0.5. The image interaction

energy at constant surface charge density changes its sign as the value of ep=er increases,while it does not depend on ep=er for the constant surface potential case. From Ref. [10].

TWO SPHERES 339

Equation (14.55) agrees with the HHF formula (Eq. (12.10)) [18], namely,

VcðHÞ ¼ pereoa1a2

a1 þ a2ðco1 þ co2Þ2 lnð1þ e�kHÞ þ ðco1 � co2Þ2 lnð1� e�kHÞn o

ð14:56Þ

which can be rewritten as


a1 þ a2co1co2

X1n¼0

e�ð2nþ1ÞkH

2nþ 1� 1

2ðc2

o1 þ c2o2ÞX1n¼1

e�2nkH

2n

( )

¼ 4pereoa1a2

a1 þ a2co1co2 e

�kH � 1

4ðc2

o1 þ c2o2Þe�2kH


ð14:57Þ

For case (b), if ka1� 1 and ka2� 1, H� a1 and H� a2, then Eq. (14.48)

becomes

Vsimage ¼ pereo

a1a2a1 þ a2

�c2o1 e

�2kH 1� ffiffiffip

p erp2er

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

ka1þ 1

ka2

r� �

þc2o2 e


p erp1er


ka1þ 1

ka2

r� �� ð14:58Þ


VsðHÞ ¼ 4pereoa1a2

a1 þ a2

�co1co2 e

�kH

þ 1

4c2o1 e


p ep2er


ka1þ 1

ka2

r� �

þ 1

4c2o2 e


p ep1er


ka1þ 1

ka2

r� ��þ Oðe�3kHÞ ð14:59Þ

If, further, the terms of the order of 1=ffiffiffiffiffiffiffikai

p(i¼ 1, 2) are neglected, Eq. (14.59)

tends to

Vc�simage ¼ pereo

a1a2a1 þ a2

ðc2o1 þ c2

o2Þe�2kH ð14:60Þ




a1 þ a2co1co2 e

�kH þ 1

4ðc2

o1 þ c2o2Þe�2kH


ð14:61Þ

which agrees with Wiese and Healy’s formula (Eq. (12.11)) [19], namely,

VsðHÞ ¼ pereoa1a2

a1 þ a2f�ðco1 þ co2Þ2 lnð1� e�kHÞ � ðco1 � co2Þ2 lnð1þ e�kHÞg

¼ 4pereoa1a2

a1 þ a2co1co2

X1n¼0

e�ð2nþ1ÞkH

2nþ 1þ 1

2ðc2

o1 þ c2o2ÞX1n¼1

e�2nkH

2n

( )

¼ 4pereoa1a2

a1 þ a2co1co2 e

�kH þ 1

4ðc2

o1 þ c2o2Þe�2kH


ð14:62Þ

Also we see from Eq. (14.59) that the next-order curvature correction to Derja-

guin’s approximation [2] is of the order of 1=ffiffiffiffiffiffiffikai

p(i¼ 1, 2). This has been sug-

gested first by Dukhin and Lyklema [20] and discussed by Kijlstra [21]. In the case

of the interaction between particles with constant surface potential, on the other

hand, the next-order curvature correction to Derjaguin’s approximation is of the

order of 1/kai (i¼ 1, 2), since in this case no electric fields are induced within the

interacting particles.

For the mixed case where sphere 1 has a constant surface potential and sphere 2

has a constant surface charge density (case (c)), if ka1� 1 and ka2� 1, H� a1 andH� a2, then Eq. (14.48) becomes

Vc�simage ¼ pereo

a1a2a1 þ a2

c2o1 e


p erp2er


ka1þ 1

ka2

r� �� c2

o2 e�2kH

� �ð14:63Þ


Vc�sðHÞ ¼ 4pereoa1a2

a1 þ a2

�co1co2 e

�kH

þ 1

4c2o1 e


p ep2er


ka1þ 1

ka2

r� �� 1

4c2o2 e

�2kH�þ Oðe�3kHÞ ð14:64Þ

If, further, the terms of the order of 1=ffiffiffiffiffiffiffikai

p(i¼ 1, 2) are neglected, Eq. (14.64)

tends to

TWO SPHERES 341


a1 þ a2co1co2 e

�kH þ 1

4ðc2

o1 � c2o2Þ e�2kH


ð14:65Þ

Equation (14.65) agrees with Kar et al’s equation (Eq. (12.12)) [22]


a1 þ a2co1co2 arctanðe�kHÞ þ 1

4ðc2

o1 � c2o2Þ lnð1þ e�2kHÞ

� �ð14:66Þ

which is rewritten as


a1 þ a2

(co1co2

X1n¼0

ð�1Þn e�ð2nþ1ÞkH

2nþ 1

� 1

2ðc2

o1 � c2o2ÞX1n¼1

ð�1Þn e�2nkH

2n

)

¼ 4pereoa1a2

a1 þ a2co1co2 e

�kH þ 1

4ðc2

o1 � c2o2Þe�2kH


ð14:67Þ

14.4 PLATE AND SPHERE

Although when the radius of either of the two interacting spheres is very large, the

interaction energy between two spheres gives the interaction energy between a

sphere and a plate, the following alternative expression is more convenient for prac-

tical calculations of the interaction energy.

Case (a): plate 1 and sphere 2 both at constant surface potentialThe result for the interaction energy Vc between a plate 1 carrying a con-

stant surface potential co1 and a sphere 2 of radius a2 carrying a constant

surface potential co2, separated by a distance H between their surfaces, im-

mersed in an electrolyte solution is (Fig. 14.9)

VcðHÞ¼4pereoco1co2a2 e�kH þp2ereoc

2o1

1

ke�2kðHþa2Þ

X1n¼0

ð2nþ1ÞGnð2Þ

�4ereoc2o2ka

22 e

2ka2b00�2pereoco1co2a2 e�kHX1n¼0

ð2nþ1Þðb0nþbn0ÞGnð2Þ

�p2ereoc2o1

1

ke�2kðHþa2Þ

X1n¼0

X1m¼0

ð2nþ1Þð2mþ1ÞbnmGnð2ÞGmð2Þ

þ4ereoc2o2ka

22 e

2ka2X1n¼0

ð2nþ1Þb0nbn0Gnð2Þ


þ2pereoco1co2a2 e�kHX1n¼0

X1m¼0

ð2nþ1Þð2mþ1Þbnmðb0nþbm0ÞGnð2ÞGmð2Þþ � � �

�2pereoco1co2a2 e�kH

X1n1¼0

X1n2¼0

� � �X1nn¼0

ð�1Þn�1ð2n1þ1Þð2n2þ1Þ � � �ð2nnþ1Þ

bn1n2bn2n3 � � �bnn�1nnðb0n1 þbnn0ÞGn1ð2ÞGn2ð2Þ � � �Gnnð2Þ

þp2ereoc2o1

1

ke�2kðHþa2Þ

X1n1¼0

X1n2¼0

� � �X1nn¼0


�bn1n2bn2n3 � � �bnn�1nnGn1ð2ÞGn2ð2Þ � � �Gnnð2Þ

þ4ereoc2o2ka

22 e

2ka2X1n1¼0

X1n2¼0

� � �X1nn¼0


�bn1n2bn2n3 � � �bnn�1nnb0n1bnn0Gn1ð2ÞGn2ð2Þ � � �Gnnð2Þþ �� ð14:68Þ

where

bnm ¼ p2k

Z 1

0

expð�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2þk2

pðHþa2ÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2þk2p Pn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2þk2

pk

!Pm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2þk2

pk

!kdk

¼ p2

Z 1

1

e�2ktðHþa2ÞPnðtÞPmðtÞdtð14:69Þ

FIGURE 14.9 Interaction between a plate 1 and a sphere 2 of radius a2 at a separation R.H (=R� a2) is the closest distance between their surfaces.

PLATE AND SPHERE 343

where Pn(t) are Legendre polynomials. Or, alternatively

bnm ¼Xminfm;ng

r¼0

Anmr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip

4kðH þ a2Þr

Knþm�2rþ1=2ð2kðH þ a2ÞÞ ð14:70Þ

Case (b): plate 1 and sphere 2 both at constant surface potentialFor the case where the surface charge densities (instead of the surface

potentials) of plate 1 (of relative permittivity ep1) and sphere 2 (of relative

permittivity ep2) remain constant, the corresponding expression for the in-

teraction energy Vs is given by

VsðHÞ ¼ 4pereoco1co2a2 e�kH þ p2ereoc

2o1

1

ke�2kðHþa2Þ

X1n¼0

ð2nþ 1ÞHnð2Þ

þ 4ereoc2o2ka

22 e

2ka2g00 þ 2pereoco1co2a2 e�kH

�X1n¼0

ð2nþ 1Þðg0n þ gn0ÞHnð2Þ

þ p2ereoc2o1

1

ke�2kðHþa2Þ

X1n¼0

X1m¼0

ð2nþ 1Þð2mþ 1ÞgnmHnð2ÞHmð2Þ

þ 4ereoc2o2ka

22 e

2ka2X1n¼0

ð2nþ 1Þg0ngn0Hnð2Þ

þ 2pereoco1co2a2 e�kHX1n¼0

X1m¼0

ð2nþ 1Þð2mþ 1Þgnm

�ðg0n þ gm0ÞHnð2ÞHmð2Þ þ � � �

þ 2pereoco1co2a2 e�kH

X1n1¼0

X1n2¼0

� � �X1nn¼0

ð2n1 þ 1Þð2n2 þ 1Þ � � � ð2nn þ 1Þ

� gn1n2gn2n3 � � � gnn�1nnðg0n1 þ gnn0ÞHn1ð2ÞHn2ð2Þ � � �Hnnð2Þ

þ p2ereoc2o1

1

ke�2kðHþa2Þ

X1n1¼0

X1n2¼0

� � �X1nn¼0

ð2n1 þ 1Þð2n2 þ 1Þ � � � ð2nn þ 1Þ

� gn1n2gn2n3 � � � gnn�1nnHn1ð2ÞHn2ð2Þ � � �Hnnð2Þ

þ 4ereoc2o2ka

22 e

2ka2X1n1¼0

X1n2¼0

� � �X1nn¼0

ð2n1 þ 1Þð2n2 þ 1Þ � � � ð2nn þ 1Þ

� gn1n2gn2n3 � � � gnn�1nn g0n1gnn0Hn1ð2ÞHn2ð2Þ � � �Hnnð2Þ þ � � �ð14:71Þ


where gnm is defined by

gnm ¼ � p2k

ep1k� erffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ k2

pep1kþ er

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ k2

p !

�Z 1

0

expð�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ k2

pðH þ a2ÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 þ k2p Pn


pk

!Pm


pk

!k dk

¼ p2

Z 1

1

ert� ep1ffiffiffiffiffiffiffiffiffiffiffiffit2 � 1

p

ertþ ep1ffiffiffiffiffiffiffiffiffiffiffiffit2 � 1

p !

e�2ktðHþa2ÞPnðtÞPmðtÞdt

ð14:72Þ

Case (c): plate 1 at constant surface potential and sphere 2 at constant surfacecharge density

When plate 1 has a constant surface potential and sphere 2 has a constant

surface charge density, the interaction energy Vc�s is given by

Vc�sðHÞ ¼ 4pereoco1co2a2 e�kH þ p2ereoc

2o1

1

ke�2kðHþa2Þ

X1n¼0

ð2nþ 1ÞHnð2Þ

� 4ereoc2o2ka

22 e

2ka2b00 � 2pereoco1co2a2 e�kH

�X1n¼0

ð2nþ 1Þðb0n þ bn0ÞHnð2Þ

� p2ereoc2o1

1

ke�2kðHþa2Þ

X1n¼0

X1m¼0

ð2nþ 1Þð2mþ 1ÞbnmHnð2ÞHmð2Þ

þ 4ereoc2o2ka

22 e

2ka2X1n¼0

ð2nþ 1Þb0nbn0Hnð2Þ


X1m¼0

ð2nþ 1Þð2mþ 1Þbnm

�ðb0n þ bm0ÞHnð2ÞHmð2Þ þ � � �

� 2pereoco1co2a2 e�kH

X1n1¼0

X1n2¼0

� � �X1nn¼0

ð�1Þn�1ð2n1 þ 1Þ

�ð2n2 þ 1Þ � � � ð2nn þ 1Þbn1n2bn2n3 � � � bnn�1nnðb0n1 þ bnn0ÞHn1ð2ÞHn2ð2Þ � � �Hnnð2Þ

þ p2ereoc2o1

1

ke�2kðHþa2Þ

X1n1¼0

X1n2¼0

� � �X1nn¼0


�ð2n2 þ 1Þ � � � ð2nn þ 1Þ


�bn1n2bn2n3 � � � bnn�1nnHn1ð2ÞHn2ð2Þ � � �Hnnð2Þ

þ 4ereoc2o2ka

22 e

2ka2X1n1¼0

X1n2¼0

� � �X1nn¼0


�ð2n2 þ 1Þ � � � ð2nn þ 1Þ� bn1n2bn2n3 � � � bnn�1nnb0n1bnn0Hn1ð2ÞHn2ð2Þ � � �Hnnð2Þ þ � � �

ð14:73Þ

Case (d): plate 1 at constant surface charge density and sphere 2 at constantsurface potential

When plate 1 has a constant surface charge density and sphere 2 has a con-

stant surface potential, the interaction energy Vs�c is given by

Vs��ðHÞ ¼ 4pereoco1co2a2 e�kH þ p2ereoc

2o1

1

ke�2kðHþa2Þ

X1n¼0

ð2nþ 1ÞGnð2Þ

þ 4ereoc2o2ka

22 e

2ka2g00 þ 2pereoco1co2a2 e�kH

�X1n¼0

ð2nþ 1Þðg0n þ gn0ÞGnð2Þ

þ p2ereoc2o1

1

ke�2kðHþa2Þ

X1n¼0

X1m¼0

ð2nþ 1Þð2mþ 1ÞgnmGnð2ÞGmð2Þ

þ 4ereoc2o2ka

22 e

2ka2X1n¼0

ð2nþ 1Þg0ngn0Gnð2Þ


X1m¼0

ð2nþ 1Þð2mþ 1Þgnm�ðg0n þ gm0ÞGnð2ÞGmð2Þ þ � � �

þ 2pereoco1co2a2 e�kH

X1n1¼0

X1n2¼0

� � �X1nn¼0

ð2n1 þ 1Þ

�ð2n2 þ 1Þ � � � ð2nn þ 1Þ� gn1n2gn2n3 � � � gnn�1nnðg0n1 þ gnn0ÞGn1ð2ÞGn2ð2Þ � � �Gnnð2Þ

þ p2ereoc2o1

1

ke�2kðHþa2Þ

X1n1¼0

X1n2¼0

� � �X1nn¼0

ð2n1 þ 1Þ

�ð2n2 þ 1Þ � � � ð2nn þ 1Þ� gn1n2gn2n3 � � � gnn�1nnGn1ð2ÞGn2ð2Þ � � �Gnnð2Þ

þ 4ereoc2o2ka

22 e

2ka2X1n1¼0

X1n2¼0

� � �X1nn¼0

ð2n1 þ 1Þð2n2 þ 1Þ � � � ð2nn þ 1Þ

� gn1n2gn2n3 � � � gnn�1nn g0n1gnn0Gn1ð2ÞGn2ð2Þ � � �Gnnð2Þ þ � � �ð14:74Þ


We compare the image interactions appearing in the present theory with the

usual image interaction. For this purpose we consider the case where the surface

charge density of plate 1 is always zero cð0Þo1 ¼ 0 and ka! 0, since the usual image

interaction refers to a point charge interacting with a uncharged plate. In the case of

cð0Þo1 ¼ 0, the interaction energy (Eq. (14.71) becomes

VsðHÞ ¼ 4ereoc2o2ka

22 e

2ka2g00

þ4ereoc2o2ka

22 e

2ka2X1n¼0

ð2nþ 1Þg0ngn0Hnð2Þ þ � � �

þ4ereoc2o2ka

22 e

2ka2X1n1¼0

X1n2¼0

� � �X1nn¼0

ð2n1 þ 1Þð2n2 þ 1Þ � � � ð2nn þ 1Þ

�gn1n2gn2n3 � � � gnn�1nn g0n1gnn0Hn1ð2ÞHn2ð2Þ � � �Hnnð2Þ þ � � �ð14:75Þ

In the limit ka! 0, all the terms on the right-hand side except the first term van-

ish and g00 tends to

g00 !p

4kHer � ep1er þ ep1

ð14:76Þ

We introduce the total charge Q� 4pa2s on sphere 2, which is related to the

unperturbed surface potential co2 of sphere 2 by

co2 ¼Q

4pereoa2ð1þ ka2Þ ð14:77Þ

Then Eq. (14.75) tends to

VsðHÞ ¼ Q2e�kH

16pereoHer � ep1er þ ep1

ð14:78Þ

This is the screened image interaction between a point charge and an uncharged

plate, both immersed in an electrolyte solution of Debye–Huckel parameter k. Fur-ther, in the absence of electrolytes (k! 0), Eq. (14.78) becomes

VsðHÞ ¼ Q2

16pereoHer � ep1er þ ep1

ð14:79Þ

which is the usual image interaction energy [23].

We can thus conclude that Eq. (14.75) is a generalization of the usual image

interaction of point charge to a colloidal particle of finite size (Fig. 14.9).


14.5 TWO PARALLEL CYLINDERS

Similarly, series expansion representations for the double-layer interaction between

two parallel cylinders can be obtained [13, 14]. The results are given below.

Case (a): cylinders 1 and 2 both at constant surface potentialConsider a cylinder of radius a1 carrying a constant surface potential co1

(cylinder 1) and a cylinder of radius a2 carrying a constant surface potential

co2 (cylinder 2), separated by a distance R between their axes, immersed in

an electrolyte solution (Fig. 14.11). The interaction energy Vc(R) betweencylinders 1 and 2 per unit length is given by [13]

VcðRÞ¼2pereoco1co2

K0ðkRÞK0ðka1ÞK0ðka2Þ

þpereoc2o1

1

K20ðka1Þ

X1n¼�1

Gnð2ÞK2nðkRÞþpereoc

2o2

1

K20ðka2Þ

X1n¼�1

Gnð1ÞK2nðkRÞ

þ2pereoco1co2

1

K0ðka1ÞK0ðka2Þ

FIGURE 14.10 Reduced potential energy V�¼ 16pereoV=kQ2 of the image interaction be-

tween a hard plate (plate 1) and a hard sphere (sphere 2) of radius a2 with e2¼ 0 as a function

of kH for several values of the reduced radius ka2 of sphere 2. Solid lines: e1¼ 0; dashed

lines: e1¼1 (plate 1 is a metal). From Ref. [14].


�X1n¼�1

X1m¼�1

Gnð2ÞGmð1ÞKnðkRÞKnþmðkRÞKmðkRÞ

þ�� þpereoco1co2

1

K0ðka1ÞK0ðka2Þ

�X1

n1¼�1

X1n2¼�1

�� X1

n2n¼�1fL21ðn1;n2; . . . ;n2nÞþL12ðn1;n2; . . . ;n2nÞg

�Kn1ðkRÞKn2nðkRÞ

þpereoX1

n1¼�1

X1n2¼�1

�� X1

n2n�1¼�1

�c2o1

K20ðka1Þ

L21ðn1;n2; . . . ;n2n�2ÞG2n�1ð2Þ

þ c2o2

K20ðka2Þ

L12ðn1;n2; . . . ;n2n�2ÞG2n�1ð1Þ�

�Kn1ðkRÞKn2n�2þn2n�1ðkRÞKn2n�1

ðkRÞþ ��

ð14:80Þ

with

L21ðn1;n2; . . . ;n2n�2Þ ¼ Kn1þn2ðkRÞKn2þn3ðkRÞ� �� Kn2n�2þn2n�1

ðkRÞKn2n�1þn2nðkRÞ�Gn1ð2ÞGn2ð1Þ� �� Gnn�1ð2ÞGn2nð1Þ

ð14:81Þ

L12ðn1;n2; . . . ;n2n�2Þ ¼ Kn1þn2ðkRÞKn2þn3ðkRÞ� � � ��Kn2n�2þn2n�1

ðkRÞKn2n�1þn2nðkRÞ�Gn1ð1ÞGn2ð2Þ� � � ��Gnn�1ð1ÞGn2nð2Þ

ð14:82Þ

FIGURE 14.11 Interaction between two infinitely long charged hard cylinders 1 and 2 of

radii a1 and a2 at a separation R between their axes. H (¼R� a1� a2) is the closest distancebetween their surfaces.

TWO PARALLEL CYLINDERS 349

GnðiÞ¼� InðkaiÞKnðkaiÞ ; ði¼ 1;2Þ ð14:83Þ

The leading term of Vc(R) and Vs(R) agrees with Eq. (11.82) obtained by

the linear superposition approximation.

Case (b): cylinders 1 and 2 both at constant surface charge densityIt can be shown that the interaction energy Vs(R) per unit length between

cylinder 1 (of relative permittivity ep1) and cylinder 2 (of relative permittiv-

ity ep2) at constant surface charge density is obtained by the interchange

Gn(i)$Hn(i) with the result that

VsðRÞ ¼ 2pereoco1co2


þ pereoc2o1

1

K20ðka1Þ

X1n¼�1

Hnð2ÞK2nðkRÞ þ pereoc

2o2

1

K20ðka2Þ

�X1n¼�1

Hnð1ÞK2nðkRÞ

þ 2pereoco1co2

1

K0ðka1ÞK0ðka2Þ

�X1n¼�1

X1m¼�1

Hnð2ÞHmð1ÞKnðkRÞKnþmðkRÞKmðkRÞ

þ � � � þ pereoco1co2

1

K0ðka1ÞK0ðka2Þ

�X1

n1¼�1

X1n2¼�1

� � �X1

n2n¼�1fM21ðn1;n2; . . . ;n2nÞ þM12ðn1;n2; . . . ;n2nÞg


þ pereoX1

n1¼�1

X1n2¼�1

� � �X1

n2n�1¼�1

�c2o1

K20ðka1Þ

M21ðn1;n2; . . . ;n2n�2ÞH2n�1ð2Þ

þ c2o2

K20ðka2Þ

M12ðn1;n2; . . . ;n2n�2ÞH2n�1ð1Þ�


ðkRÞ þ � � �ð14:84Þ

with

M21ðn1;n2; . . . ;n2n�2Þ ¼ Kn1þn2ðkRÞKn2þn3ðkRÞ � � � ��Kn2n�2þn2n�1

ðkRÞKn2n�1þn2nðkRÞ�Hn1ð2ÞHn2ð1Þ � � � � �Hnn�1ð2ÞHn2nð1Þ

ð14:85Þ


M12ðn1;n2; . . . ;n2n�2Þ ¼ Kn1þn2ðkRÞKn2þn3ðkRÞ � � � ��Kn2n�2þn2n�1

ðkRÞKn2n�1þn2nðkRÞ�Hn1ð1ÞHn2ð2Þ � � � � �Hnn�1ð1ÞHn2nð2Þ

ð14:86Þ

where

HnðiÞ ¼ � I0nðkaiÞ � ðepi nj j=erkaiÞInðkaiÞK 0

nðkaiÞ � ðepi nj j=erkaiÞKnðkaiÞ ; ði ¼ 1;2Þ ð14:87Þ

Case (c): cylinder 1 maintained at constant surface potential and cylinder 2 atconstant surface charge density

It can also be shown that when cylinder 1 has a constant surface potential

and cylinder 2 has a constant surface charge density, the interaction energy

Vc�s between plates 1 and 2 per unit length s given by [13]

Vc�sðRÞ ¼ 2pereoco1co2


þpereoc2o1

1

K20ðka1Þ

X1n¼�1

Hnð2ÞK2nðkRÞþpereoc

2o2

1

K20ðka2Þ

�X1n¼�1

Gnð1ÞK2nðkRÞ

þ 2pereoco1co2

1

K0ðka1ÞK0ðka2ÞX1n¼�1

X1m¼�1

Hnð2ÞGmð1ÞKnðkRÞ

�KnþmðkRÞKmðkRÞ

þ � � �þpereoco1co2

1

K0ðka1ÞK0ðka2Þ

�X1

n1¼�1

X1n2¼�1

� � �X1

n2n¼�1fN21ðn1;n2; . . . ;n2nÞþN12ðn1;n2; . . . ;n2nÞg


þpereoX1

n1¼�1

X1n2¼�1

� � �X1

n2n�1¼�1

�c2o1

K20ðka1Þ

N21ðn1;n2; . . . ;n2n�2ÞH2n�1ð2Þ

þ c2o2

K20ðka2Þ

N12ðn1;n2; . . . ;n2n�2ÞG2n�1ð1Þ�


ðkRÞþ � � �ð14:88Þ

TWO PARALLEL CYLINDERS 351

with

N21ðn1;n2; . . . ;n2n�2Þ ¼Kn1þn2ðkRÞKn2þn3ðkRÞ� � � ��Kn2n�2þn2n�1

ðkRÞKn2n�1þn2nðkRÞ�Hn1ð2ÞGn2ð1Þ� � � ��Hnn�1ð2ÞGn2nð1Þ

ð14:89Þ

N12ðn1;n2; . . . ;n2n�2Þ ¼Kn1þn2ðkRÞKn2þn3ðkRÞ� � � ��Kn2n�2þn2n�1

ðkRÞKn2n�1þn2nðkRÞ�Gn1ð1ÞHn2ð2Þ� � � ��Gnn�1ð1ÞHn2nð2Þ

ð14:90Þ

Consider the case where ka1� 1 and ka2� 1. For the constant surface potential

case, Vc becomes

VcðRÞ ¼ 2ffiffiffiffiffiffi2p

pereoco1co2

ffiffiffiffiffiffiffiffiffiffiffiffika1a2

p e�kHffiffiffiR

p

� ffiffiffip

pereo

e�2kH

Rc2o1a1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffika2RR� a2

rþ c2

o2a2


r� �þ Oðe�3kHÞ

ð14:91Þ

For the constant surface charge density case, if ep1 and ep2 are finite, we obtain

VsðRÞ ¼ 2ffiffiffiffiffiffi2p

pereoco1co2


p e�kHffiffiffiR

p

þ ffiffiffip

pereo

e�2kH

R

�c2o1a1


r1� 2ep2

er

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR

pka2ðR� a2Þ

s( )

þc2o2a2


r1� 2ep1

er


pka1ðR� a1Þ

s( )�þ Oðe�3kHÞ

ð14:92Þ

For the mixed case,

Vc�sðRÞ ¼ 2ffiffiffiffiffiffi2p

pereoco1co2


p e�kHffiffiffiR

p

þ ffiffiffip

pereo

e�2kH

R

�c2o1a1


r1� 2ep2

er


pka2ðR� a2Þ

s( )

�c2o2a2


r �þ Oðe�3kHÞ

ð14:93Þ


Further, if H� a1 and H� a2, Eqs. (14.91)–(14.93) reduce to

VcðRÞ ¼ 2ffiffiffiffiffiffi2p

pereoco1co2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffika1a2a1 þ a2

re�kH � ffiffiffi

pp

ereo


re�2kHðc2

o1 þ c2o2Þ

ð14:94Þ

VsðRÞ ¼ 2ffiffiffiffiffiffi2p

pereoco1co2


re�kH

þ ffiffiffip

pereo


re�2kH c2

o1 1� 2ffiffiffip

p ep2er


ka1þ 1

ka2

r� �

þc2o2 1� 2ffiffiffi

pp ep1

er


ka1þ 1

ka2

r� �ð14:95Þ

V��sðRÞ ¼ 2ffiffiffiffiffiffi2p

pereoco1co2


re�kH

þ ffiffiffip

pereo


re�2kH c2

o1 1� 2ffiffiffip

p ep2er


ka1þ 1

ka2

r� � c2

o2

� �ð14:96Þ

Equations (14.94)–(14.96) are consistent with the results obtained via Derja-

guin’s approximation. Equation (14.94) shows that the next-order curvature correc-

tion to Derjaguin’s approximation is of the order of 1/ffiffiffiffiffiffiffikai

p(i¼ 1, 2), as in the case

of sphere–sphere interaction.

14.6 PLATE AND CYLINDER

One can also obtain the interaction energy between a cylinder and a hard plate, both

having constant surface potential for the case where the cylinder axis is perpendicu-

lar to the plate surface (Fig. 14.12). The result is [14]

VcðHÞ ¼ 2pereoco1co2

e�kðHþa2Þ

K0ðka2Þ þ pereoc2o1 e

�2kðHþa2ÞX1n¼�1

Gnð2Þ

�pereoc2o2

b00fK0ðka2Þg2

� pereoco1co2

e�kðHþa2Þ

K0ðka2ÞX1n¼�1

ðb0n þ bn0ÞGnð2Þ

�pereoc2o1 e

�2kðHþa2ÞX1n¼�1

X1n¼�1

bnmGnð2ÞGmð2Þ

PLATE AND CYLINDER 353

þpereoc2o2

1

fK0ðka2Þg2X1n¼�1

b0nbn0Gnð2Þ

þpereoco1co2

e�kðHþa2Þ

K0ðka2ÞX1n¼�1

X1n¼�1

bnmðb0n þ bn0ÞGnð2ÞGmð2Þ þ � � �

�pereoco1co2

e�kðHþa2Þ

K0ðka2ÞX1

n1¼�1

X1n2¼�1

� � �X1

nn¼�1ð�1Þn�1

�bn1n2bn2n3 � � � bnn�1nnðb0n1 þ bnn0ÞGn1ð2ÞGn2ð2Þ � � �Gnnð2Þ

þpereoc2o1 e

�2kðHþa2ÞX1

n1¼�1

X1n2¼�1

� � �X1


�bn1n2bn2n3 � � � � � bnn�1nnGn1ð2ÞGn2ð2Þ � � � Gnnð2Þ

þpereoc2o2

1

fK0ðka2Þg2X1

n1¼�1

X1n2¼�1

� � �X1


�bn1n2bn2n3 � � � bnn�1nnb0n1bnn0Gn1ð2ÞGn2ð2Þ � � � Gnnð2Þ þ � � �

ð14:97Þ

with

bnm ¼Z 1

0

exp½�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ k2

pðH þ a2Þ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 þ k2p Tn


pk

!Tm


pk

!dk

¼Z 1

1

exp½�2ktðH þ a2Þ�ffiffiffiffiffiffiffiffiffiffiffiffit2 � 1

p TnðtÞTmðtÞdt

ð14:98Þwhere Tn(x) is the nth-order Tchebycheff’s polynomial.

FIGURE 14.12 Interaction between a hard plate 1 and an infinitely long charged hard

cylinder 2 of radius a2 at a separation R. H (¼R� a2) is the closest distance between their

surfaces.


When plate 1 and cylinder 2 have constant surface charge densities, the interac-

tion energy Vs is obtained by Eq. (18.14) with Gn(2) and bnm replaced by Hn(2) and

�gnm, respectively, where gnm is defined by

gnm ¼ �Z 1

0

e1k � effiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ k2

pe1k þ e


p exp½�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ k2

pðH þ a2Þ�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 þ k2p

�Tn


pk

!Tm


pk

!dk

¼Z 1

1

et � e1ffiffiffiffiffiffiffiffiffiffiffiffit2 � 1

p

et þ e1ffiffiffiffiffiffiffiffiffiffiffiffit2 � 1

p exp½�2ktðH þ a2Þ�ffiffiffiffiffiffiffiffiffiffiffiffit2 � 1

p TnðtÞTmðtÞdt

ð14:99Þ

When plate 1 has a constant surface potential and cylinder 2 has a constant sur-

face charge density, the interaction energy Vc�s is given by Eq. (14.97) with Gn(2)

replaced by Hn(2). When plate 1 has a constant surface charge density and cylinder

2 has constant surface potential, the interaction energy is given by Eq. (14.97) with

bnm replaced by �gnm.Finally, we compare the image interactions between a hard cylinder and a hard

plate with the usual image interaction between a line charge and a plate by taking

the limit of ka2! 0 for the case where the surface charge density of plate 1 is al-

ways zero (co1¼ 0). In this limit, we have

VsðHÞ ¼Q2

4pereoK0ð2kHÞ; ðep1 ¼ 0Þ

� Q2

4pereoK0ð2kHÞ; ðep1 ¼ 0Þ

8>>><>>>:

ð14:100Þ

where we have introduced the total charge Q� 2pas on cylinder 2 per unit length

(i.e., the line charge density of cylinder 2), which is related to the unperturbed sur-

face potential co2 of cylinder 2 by

co2 ¼Q

2pereoka2

K0ðka2ÞK1ðka2Þ ð14:101Þ

Equation (14.101) is the screened image interaction between a line charge and an

uncharged plate, both immersed in an electrolyte solution of Debye–Huckel param-

eter k. We see that in the former case (ep1¼ 0), the interaction force is repulsion and

the latter case (ep1¼1) attraction. Further, in the absence of electrolytes (k! 0),

we can show from Eq. (14.101) that the interaction force �@Vs/@H per unit length

between plate 1 and cylinder 2 with a2! 0 is given by

� @Vs

@H¼ Q2

4peeo

er � ep1er þ ep1

� 1

Hð14:102Þ

PLATE AND CYLINDER 355

which exactly agrees with usual image force per unit length between a line charge

and an uncharged plate [23].

REFERENCES

1. V. I. Smirnov, Course of Higher Mathematics, Vol. 4, Pergamon, Oxford, 1958,

Chapter 4.

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

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

Press, Amsterdam, 2006.

4. H. Ohshima and T. Kondo, J. Colloid Interface Sci. 157 (1993) 504.

5. H. Ohshima and T. Kondo, Colloid Polym. Sci. 271 (1993) 1191.

6. H. Ohshima, J. Colloid Interface Sci. 162 (1994) 487.


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



11. H. Ohshima, E. Mishonova, and E. Alexov, Biophys. Chem. 57 (1996) 189.


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

14. H. Ohshima, Colloid Polym. Sci 277 (1999) 563.

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

16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover,

New York, 1965, p. 445.

17. D. Langbein, Theory of van der Waals Attraction, Springer, Berlin, 1974, p. 82.

18. R. Hogg, T. W. Healy, and D. W. Fuerstenau, Trans. Faraday Soc. 62 (1966) 1638.

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

20. S. S. Dukhin and J. Lyklema, Colloid J. USSR Eng. Trans. 51 (1989) 244.

21. J. Kijlstra, J. Colloid Interface Sci. 153 (1992) 30.

22. G. Kar, S. Chander, and T. S. Mika, J. Colloid Interface Sci. 44 (1973) 347.

23. L. D. Landau and E. M. Lifshitz, Electrodynamics in Continuous Media, Pergamon,

New York, 1963.


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