An Integral Expression for the Electrical Potential Distribution for Charged Surfaces in Electrolyte Solutions

ARTICLE NO.CS964743

An Integral Expression for the Electrical Potential Distribution

for Charged Surfaces in Electrolyte Solutions

JYH-PINGHSU1ANDMING-TSANTSENG

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China Received September 26, 1996; accepted December 13, 1996

associated boundary conditions, and the type of electrolyte.

An analytical procedure is suggested for the resolution of the _{In general, a nonlinear equation needs to be considered. If} linearized Poisson – Boltzmann equation governing the electrical

a surface is not highly charged, or if its surface potential is

potential distribution of a charged surface in an electrolyte solution

sufficiently low ( the Debye – Huckel condition ) , the

corre-under the Debye – Huckel condition. An Integral expression based

sponding PBE can be approximated by the Helmholtz

on the Green function of the linearized Poisson – Boltzmann

equa-tion, the solution of which may be expressed as a linear

tion is derived. The electrical potential distributions for the

follow-combination of harmonic functions ( 10 – 13 ) . The Debye –

ing cases are solved to illustrate the present method: an

arbitrary-Huckel condition is satisfactory for a surface potential on

shaped thin surface, a porous ( ion-penetrable ) sphere bearing fixed

the order of 25 mV. This may be violated for some of the

charges, a rigid, planar surface covered by a porous membrane

bearing fixed charges, and a rigid, nonuniformly charged planar charged surfaces in practice. Nevertheless, it is often adopted

surface. q 1997 Academic Press in the literature for a simpler treatment. Although reported

Key Words: Poisson – Boltzmann equation, linearized; Helmholtz _{results are ample in the literature, most of them are based}

equation; electrical potential distribution; Green function repre- _{on simple geometries such as planar ( 13 ) , cylindrical ( 9 ) ,} sentation; general expression; rigid charged surface; planar ;

non-spherical ( 10 – 11 ) , and spheroidal surfaces ( 14 – 15 ) , and

uniform surface condition; rigid charged surface; thin, arbitrary

uniform surface conditions, for example, uniform surface

shape; porous object, ion-penetrable, planar, or spherical.

potential or uniform surface charge density. Since a charged surface can take virtually any shape, and various surface conditions can be assumed in practice, a more general

treat-1. INTRODUCTION

ment is highly desirable.

In the present study the solution procedure for the electri-The electrical potential distribution for a charged surface

cal potential distribution around a charged surface in an in a vacuum is governed by the Poisson equation. It can

electrolyte solution under the Debye – Huckel condition is be solved by considering an equivalent integral equation

discussed. The types of charged surfaces examined include: containing the Green function of the Poisson equation and

an arbitrary-shaped thin rigid surface, a porous ( ion-penetra-the charge density of ion-penetra-the surface ( 1 ) . The integral equation

ble ) sphere bearing fixed charges, a planar surface covered is based on the principle that the electrical potential of the

by a porous membrane bearing fixed charges, and a rigid, system under consideration is the sum of the electrical

poten-nonuniformly charged planar surface. The mathematical tials contributed by each individual charges on the surface.

treatment is based on the Green function of the linearized For a charged surface immersed in an electrolyte solution,

PBE. A similar approach was adopted by Halle ( 16 ) for the the electrolyte provides an additional charge source. In this

case of two infinitely thin rodlike macroions. case, if the concentration of electrolyte follows the

Boltz-mann distribution, the electrical potential distribution of the

2. THEORY

system under consideration is governed by the Poisson – Boltzmann equation ( PBE ) ( 2 ) . For an infinite planar

sur-face in a symmetric electrolyte solution, the solution to the Consider first a unit positive fixed point charge located at PBE can be derived analytically ( 2 ) . Other than this case, _{the dimensionless position xV}Å[ x1, x2, x3] in an electrolyte the PBE needs to be solved either numerically ( 3 – 5 ) or solution. The dimensionless electrical potential distribution approximately ( 6 – 9 ) . The difficulty of solving the PBE induced by this charge, G , is described by the Helmholtz depends largely on the geometry of a charged surface, the equation

1

To whom correspondence should be addressed. Ç2_eVG ( e_{V, xV)}ÅG ( e_{V, xV)}0d ( e_{V 0 xV),} [1]

Consider next an infinitesimally thin charged surface im-whereÇ2

eV is the Laplace operator based on the coordinates

mersed in an electrolyte solution. If the surface is not highly

e_{V Å}[ e1, e2, e3] , and d the delta function. The dimensionless

charged, the potential distribution satisfies variables are defined as

Ç2

eVC( eV)ÅC( eV) , eV √ E , [ 7 ]

G ÅeF / kBT [1a ]

xi Åk xIi, i Å1, 2, 3 [1b ] _{where E is the domain excluding the charged surface. By}

referring to Fig. 1, the boundary condition associated with

k Å (

∑

e2 Z2 iCi ,0/ ere0kBT ) 1 / 2 , [1c ] Eq. [ 7 ] is where e is the elementary charge, F the electrical potential,

erand e0, respectively, the relative permittivity and the per- ÌC Ì_nV

Z

pV/

0ÌC Ì_nV

Z

pV0

Å_{r( pV), pV} √V, [ 8 ]

mittivity of a vacuum, Zi the valence of ion species i , kB the Boltzmann constant, T the absolute temperature, Ci ,0the

bulk concentration of ion species i , k01

the Debye length, _{where (ÌC /Ì}

nV)pV is the directional derivative of C at point

and x˜i, i Å1, 2, 3, the position variables. The solution to

pV, r(pV) the surface charge distribution at pV, V the domain

Eq. [1] is

of the charged surface, nV the unit outer normal, and pV/and

pV0the points immediately above and below pV, respectively. Define the function u ( e_V) as

G ( e_{V, xV)}Åexp (0r ( eV, xV))

4pr ( e_{V, xV)} , [ 2 ]

u ( e_V) Å

*

V

G ( e_{V, xV)r(xV)dV, xV}√V, [ 9 ] where r ( e_{V, xV) is the distance between an arbitrary point eV}

and xV. For a distribution of fixed charges r(eV) in the

electro-where the right-hand side of this expression denotes the lyte solution the electrical potential distribution, C, is

gov-surface integral over the charged gov-surface. It can be shown erned by

that u ( e_V) satisfies Eq. [ 7 ] , and Ç2 eVC( eV) ÅC( eV) 0r( eV) . [ 3 ] Ìu Ì_nV

Z

p_V/ Å

*

V ÌG ( e_{V, xV)} Ì_nV

Z

_eVÅpV r( xV)dV/ 1 2r( pV) [10 ] Define £_{( eV) Å}

*

V G ( e_{V, xV)r(xV)dxV,} [ 4 ] Ìu Ì_nV

Z

pV0 Å

*

V ÌG ( e_{V, xV)} Ì_nV

Z

_eVÅpV r( xV)dV0 1 2r( pV). [11] where the right-hand side of this expression denotes the These expressions imply that u satisfies Eq. [ 8 ] , and, there-volume integral over V . Employing the Laplace operator on fore, is the solution of Eq. [ 7 ] ; i.e.,

Eq. [ 4 ] yields Ç2 eV£( eV)Å

*

V Ç2 eVG ( eV, xV)r(xV)dxV Å

*

V [ G ( e_{V, xV)}0d ( e_{V, xV)]r(xV)dxV} Å

*

V G ( e_{V, xV)r(xV)dxV}0r( e_V) Å £( e_V) 0r( e_V) . [ 5 ] This implies that£_{satisfies Eq. [ 3 ] ; i.e.,}

C( e_V) Å

*

V

G ( e_{V, xV)r(xV)dxV.} [ 6 ]

FIG. 1. _{A schematic representation of an arbitrary thin surface. nV is the}

The expression was derived previously for the case the func- _{unit outer normal, pV the position of an arbitrary point on the surface, pV}/

and pV0

the points immediately above and below pV, respectively. tion G on the right-hand side of Eq. [1] vanishes ( 1 ) .

The same result can be obtained by solving the Poisson – Boltzmann equation for the present case

d2

C dp2 /

2 dC

pdp ÅC0r( p ) , p§0. [15 ]

This problem was discussed by Ohshima ( 12 ) , and he ar-rived at

FIG. 2. The coordinates used for a porous ( ion-penetrable ) sphere of radius a . The center of the sphere is located at the origin O . e_V denotes an arbitrary point, and L is the distance between e_V and O . _CÅ

C0a exp (0( L 0a ) ) / L , L§a

F

*

L 0 r*r0sinh ( r* 0L ) dr* / sinh ( L )

*

a 0 r*r0exp (0r*) dr*

GY

L , L£a , C( e_V) Å

*

S G ( e_{V, xV)r(xV)dS}x_V. [12 ] [16 ] Note that the function G defined above is the generalized

Green function of the Helmholtz equation, Eq. [1] . _{where r* is a dummy variable and}

3. EXAMPLES

C0Åexp (0a ) / a

*

a

0

r*r ( r*) sinh ( r*) dr*. Several examples are discussed to illustrate the

applicabil-ity of the present method.

A similar problem was examined by Hsu et al. ( 17 ) . Note that the present approach is applicable to an arbitrary-shaped

3.1. Porous ( Ion-Penetrable ) Sphere Bearing Fixed

porous object.

Charges

Let us consider a porous sphere bearing fixed charges 3.2. Planar Surface Covered by a Membrane Bearing

with radius a in an electrolyte solution. For illustration, we Fixed Charges

assume that the fixed charge distribution, r( e_V) , is

By referring to Fig. 3, we consider a planar rigid surface covered by an ion-penetrable membrane bearing fixed charges. Suppose that the dielectric constant of the rigid

r( e_V)Å

H

r0, q e2 1/e 2 2/e 2 3£ a 0, q e2 1/e 2 2/e 2 3úa . [13 ]

solid phase is small compared to that of the membrane phase. Let the charge density of the rigid surface and the fixed charge distribution in the membrane be rs( eV) and rV( eV) ,

For the coordinate system shown in Fig. 2, we have, on the basis of Eq. [ 6 ] , CÅ 1 4p

*

a 0a

*

qa2_0z2 0

*

2p 0 exp (0 q r2 /( z0L )2 ) q r2 /( z0L )2 r0rdudrdz Å0r0 2 L ( exp (0a ) ( a/1 ) )É aÅÉa0LÉ aÅÉa/LÉ /r0 2

*

a 0a exp (0Éz0LÉ_{) dz} Å r0exp ( a0L ) [ a01/exp (02a ) ( a/1 ) ] / 2 L , if Lúa r0( a/1 ) exp ( L0a ) [01/exp (02 L ) ] / 2 L/1,

if Lõa . FIG. 3. A schematic representation of a rigid surface covered by an ion-penetrable charged membrane. The rigid surface-membrane interface is located at e3Å0.

respectively. In this case, the electrical potential distribution the right-hand side of Eq. [ 25 ] vanishes. A similar problem was examined by Hsu et al. ( 19 ) .

is governed by

3.2.1. Nonuniformly charged planar surface. Let us

Ç2

eVC( eV) ÅC( eV) 0rV( eV) , e3ú0. [17 ] consider a planar surface with surface condition The associated boundary condition is ÌC

Ìe3

Z

e3Å0/

Å 0g ( e1, e2) . [ 26 ] ÌC

Ìe3

Å 0rs( eV) , e3Å0. [18 ]

Applying Eq. [ 25 ] yields The problem under consideration can be reformulated

C( e_V)Å

*

` 0`

*

` 0` G ( e_{V, xV)(2g(x}1, x2) ) dx1dx2. [ 27 ] through an image method ( 11 – 12 ) . Let us consider the

problem

As an example, let us consider the case Ç2

eVC( eV)ÅC( eV)0rV( eV) , e3ú0 [19 ]

g ( e1, e2)Åcos ( e1) cos ( e2)/1,0` õe1, e2õ`. [ 28 ] Ç2

eVC( eV)ÅC( eV)0r*V( eV) , e3õ0 [ 20 ]

Substituting Eq. [ 28 ] into Eq. [ 27 ] gives the electrical poten-ÌC

Ìe3

Z

e3Å0/ 0ÌC

Ìe3

Z

e3Å00

Å 02rs( eV) , e3Å0, [ 21]

tial distribution. In practice, the computational effort can be reduced significantly if the infinite domain in Eq. [ 27 ] is replaced by an adequately chosen finite domain S . Suppose where r*V( eV) is the image charge distribution satisfying

that g ( e1, e2) is bounded, i.e.,

r*V( e1, e2, e3) ÅrV( e1, e2,0e3) , e3õ0. [ 22 ]

Ég ( e₁, e₂)Éõ g_max, 0` õe<sub>1</sub>, e<sub>2</sub>õ` [ 29 ] Equations [19 ] – [ 22 ] suggest that the electrical potential

where gmax is a certain finite value. Then S can be chosen distribution is symmetric about e3Å0. Therefore,

as SÅ{ ( e1, e2)É q ( e10x1) 2 /( e20x2) 2 £Rmax, ÌC Ìe3

Z

e₃Å0/ Å 0ÌC Ìe3

Z

e₃Å00 [ 23 ] Rmaxú ln ( 4pgmax/ d ) } , [ 30 ]

and _{where d is a prespecified degree of accuracy.}

Figure 4 shows the spatial electrical potential distribution calculated by Eqs. [ 27 ] and [ 30 ] with dõ0.001, gmaxÅ2, ÌC

Ìe3

Z

e3Å0/

Å 0rs( eV) . [ 24 ]

and RmaxÅ10.13. In a discussion of the interaction between two nonuniformly charged surfaces, Kuin ( 13 ) assumed a periodic surface charge distribution. It is a special case of This implies that the solution to the reformulated problem, _{Eq. [ 27 ] .}

Eqs. [19 ] – [ 24 ] , in the region e3 ú 0, is the same as that _{3.2.2. Thin, planar-charged surface. Let us consider a} of Eqs. [17 ] and [18 ] . According to Eqs. [ 6 ] and [12 ] , the

thin, planar charged surface on the e10e2plane with surface former can be expressed as

condition C( e_V)Å

*

` 0`

*

` 0` G ( e_{V, xV)(2r}s( xV))dx1dx2 ÌC Ìe3

Z

e3Å0/ Å 0g ( e1, e2) . [ 31] /

*

` 0`

*

` 0`

*

00 0` G ( e_{V, xV)r}*V( xV)dx3dx1dx2

Here, we assume that the dielectric constant of the solid phase is small, and that the electrical potential distribution /

*

` 0`

*

` 0`

*

` 0/ G ( e_{V, xV)r}V( xV)dx3dx1dx2. [ 25 ]

inside is negligible. Employing Eq. [ 25 ] leads to The special case in which rV Å rV( e3) and rS Å 0 was

C( xV)Å

*

S

G ( e_{V, xV)(2g(e}1, e2) ) de1de2. [ 32 ] analyzed by Ohshima ( 18 ) . In this case, the first term on

FIG. 4. Simulated spatial electrical potential distribution for the case of a nonuniformly charged rigid planar surface. The rigid surface is represented by e3Å0, and its surface potential is g ( e1, e2)Åcos ( e1) cos ( e2)/1.

As an example, consider a thin disk of radius R with surface It can be shown that condition

*

` 0 l q 1/l2 J0( lRH )J0( lR ) exp (0 q 1/l2 x3) d l g ( e1, e2)Å

H

f ( R ) , R £Rm 0, R úRm , [ 33 ] Å

*

2p 0 exp (0 q ( R0 RH cos uH)2_/ ( RH sin uH )2_/ x2 3) 2p q ( R0 R_{H cos uH})2 /( R_{H sin uH} )2 / x2 3 duH, [ 36 ] where R Å q e2 1/e 2

2. The electrical potential distribution for e3ú0 is

and Eq. [ 34 ] is equivalent to Eq. [ 35 ] .

Figure 5 shows the electrical potential distribution for the case C( R , x3) Å 1 2p

*

R_m 0 R_{H f ( RH)}

*

2p 0 f ( R )Å 01, R £0.9 2, 0.9 õR £1 0, 1õR . [ 37 ] 1exp (0 q ( R0R_{H cos uH})2 /( R_{H sin uH})2 /x2 3) q ( R 0RH cos uH)2 /( RH sin uH)2 /x2 3 d R_{H du}H. [ 34 ]

This type of surface condition simulates that of a kaolin particle in a low pH aqueous solution ( 21 ) . In this case the In a study of the electrical interactions between two thin,

basal plane of a particle is negatively charged and its edge charged disks, Hsu and Tseng ( 20 ) derived the expression

positively charged ( 22 ) .

3.3. Thin, Arbitrary Surface C( R , x3) Å

*

R_m 0 R_{H f ( RH)}

*

` 0 l q

1/l2 J0( lRH )J0( lR ) _{For illustration, let us consider the axis-symmetric, thin} charged surface formed by rotating the curve defined below about the z axis:

1exp (0q1/l2

FIG. 5. Simulated electrical potential distribution over an infinitesimally thin, charged disk for the case the surface condition is defined in Eq. [ 37 ] . potential distribution on the r0z plane for the case rsÅ1

H

rÅf ( t )

zÅg ( t ) , 0£t £tmax. [ 38 ] are shown in Fig. 7. Since the electrical potential distribution

is axis-symmetric, only those on the half r 0 z plane are

presented. Suppose that f ( t ) § 0, f ( 0 ) Å f ( tmax) Å 0, and d g ( t ) / dt

§0, 0£t£tmax. Under these conditions the surface defined

4. CONCLUSION

is a simple closed surface with surface charge density

rs( eV) Å rs( t ) . The potential distribution can be calculated _{We show that the exact solution to the linearized Poisson –}

by Eq. [12 ] . We have

Boltzmann equation governing the electrical potential distri-bution of a charged surface in an electrolyte solution under the Debye – Huckel condition can be expressed as

C( r , z )Å

*

t_max 0

*

2p 0 exp (0R ) rs( t ) 1f ( t ) q ( f*( t ) )2 _/ ( g*( t ) )2 / ( 4pR ) dudt , [ 39 ] CÅ

*

V G ( e_{V, xV)r(xV)deV}, [ 41] where R Å (( r 0 f ( t ) cos ( u ) )2 _/ ( f ( t ) sin ( u ) )2 _/ ( z 0 g ( t ) )2 )1/2 . _{where G ( e}

V, xV) is the generalized Green function of the linear-Figure 6 shows the surface defined by rotating the curve _{ized Poisson – Boltzmann equation. Equation [ 41] is valid} for the case the fixed charges are distributed in the liquid phase; e.g., they are carried by an ion-penetrable membrane.

H

r Åt ( 50t ) ( sin2

( t )/0.3 ) / ( t /3 )

zÅ t ( t/1 ) / 15 , 0£t£ 5 If the fixed charges are distributed over a rigid surface, an

image method can be applied which transforms the original problem to a form that Eq. [ 41] is applicable. The distribu-[ 40 ]

tion r( xV) in both cases can be arbitrary. Most of the reported results for the linearized Poisson – Boltzmann equation in the about the z axis ( or e3axis ) . The contours for the electrical

FIG. 6. An arbitrary thin surface obtained by rotating the curve defined by Eq. [ 40 ] about the e3axis.

FIG. 7. The contours of the electrical potential distribution on the r0z plane for the case of Fig. 6. The surface charge density is rsÅ1. The dashed line denotes the curve described by Eq. [ 40 ] .

7. Stokes, A. N., J. Chem. Phys. 65, 261 ( 1976 ) . literature can be recovered as special cases of the present

8. Hsu, J. P., and Kuo, Y. C., J. Chem. Soc. Faraday Trans. 89, 1229 analysis.

( 1993 ) .

9. van Aken, G. A., Lekkerkerker, H. N. W., Overbeek, J. Th. G., and de

ACKNOWLEDGMENT Bruyn, P. L., J. Phys. Chem. 94, 8468 ( 1990 ) .

10. Glendinning, A. B., and Russel, W. B., J. Colloid Interface Sci. 93, 95 ( 1983 ) .

This work is supported by the National Science Council of the Republic

11. Carnie, S. L., Chan, D. Y. C., and Gunning, J. S., Langmuir 10, 2993 of China under project NSC85-2214-E002-023.

( 1994 ) .

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

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