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Potential distribution across a membrane with surface charge layers: Effects of nonuniform charge distribution
Potential Distribution across a Membrane with Surface Charge Layers: Effects of Nonuniform Charge Distribution HIROYUKI OHSHIMA, KIMIKO MAKINO, AND TAMOTSU KONDO 1 Faculty of Pharmaceutical Sciences and Institute of Colloid and Interface Science, Science University of Tokyo, Shinjuku-ku, Tokyo 162, Japan Received August 30, 1985; accepted December 31, 1985 A model for the potential distribution across a membrane with surface charge layers (H. Ohshima and S. Ohki, Biophys. J. 47, 673 (1985)) is extended to include the effects of nonuniform charge distribution in the surface layer. We treat the case in which the charge layer on the membrane surface consists of two (outer and inner) layers having different thicknesses and different densities of fixed charges, and assume that electrolyte ions can penetrate into these layers. Equations are derived which give the potential profile across the surface charge layers. The potential profile is found to depend strongly on the charge density and thickness of each layer as well as on the average charge density and total thickness of the surface charge layers. In particular, for the case where the fixed charges of the outer and inner surface layers are of opposite sign, the potential at the outer surface (i.e., the potential at the boundary between the outer surface layer and the surrounding solution) and the net charge of the whole surface layer may, in some cases, be of opposite sign. Also, in some cases, the potential at the outer surface, when plotted as a function of the concentration of electrolyte ions in the surrounding solution, is found to reverse its sign at large concentrations. © 1986AcademicPress, Inc. 1. INTRODUCTION
Biocolloids such as cells or their membranes are usually charged in an electrolyte solution. The membrane-fixed charges are often considered to be located only at the membrane surface (of zero thickness). Actually, however, the membrane-fixed charges may be distributed through some depth at the membrane surface. Recently, taking this fact into account, several studies have been made on the electrophoresis ofbiocolloids (1-5) and on the potential distribution across a charged membrane (6, 7). In the present paper, we extend the membrane model of Ohshima and Ohki (6) to include the effects of nonuniform charge distribution in the surface charge layer. We treat the case in which the charge layer on the
membrane surface consists of two (outer and inner) layers with different thicknesses and different densities of fixed charges. For simplicity, we confine ourselves to planar membranes immersed in a symmetrical electrolyte solution. 2. THEORY
Consider a planar charged membrane immersed in a solution containing a symmetrical electrolyte of valence v and bulk concentration n. Two (outer and inner) charge layers are formed on the membrane surface. In each of the two surface layers, charged groups are uniformly distributed. We denote by dl and d2 the thicknesses of the outer and inner layers, respectively. Let N~ and zl, respectively, be the density and the valence of fixed-charge groups contained in the outer surface layer, and let N2 and z2 be the corresponding quantities of the inner layer. We assume that electrolyte ions can penetrate into the surface
To whom correspondence should be addressed: Faculty of Pharrnaceutical Sciences, Science University of Tokyo, 12, Ichigaya Funagawara-machi, Shinjuku-ku, Tokyo 162, Japan. 369
JournalofColloidandInterfaceScience,Vol. 113,No. 2, October 1986
0021-9797/86 $3.00 Copyright© 1986by AcademicPress,Inc. All rightsof reproductionin any form reserved.
370
OHSHIMA, MAKINO, A N D K O N D O
~(x) -(dl.d2)
-d
f
/
~ou~
-
Membrane core
-
d2
dl
Inner layer
Outer layer
Solution
FIG. 1. Schematic representation of the potential distribution if(x) across a m e m b r a n e with two surface charge layers o f thicknesses dl and d2. to,t = if(0).
charge layers and that there is no electric field within the membrane core. We take the x axis perpendicular to the membrane with its origin on the boundary between the electrolyte solution and the outer surface layer (Fig. 1). The P o i s s o n - B o l t z m a n n equations for the electric potential ~b(x) at a position x relative to the bulk solution phase x = + ~ are x(Z)
-2o
d2~ _ 2vne sinh vop dx 2 ~rl~O kT
zleN1 ErlEO
- d l < x < 0, d2~b _ 2 v n e dx 2
sinh
ve~
~r2~o
kT
z2eN2
~r2~O
- ( d l + d2) < x < - d l , 10
-lo
20
-~0
5 3 -20
-30 ~
Membrone
Sotution
FIG. 2. Potential distribution if(x) across a charged m e m b r a n e with d~ = d2 = 10 A calculated for several values of at and a2, both negative, with o.l + o.2 kept constant at - 4 . 8 × 10 -2 C / m 2. Curve 1: (o.~, a2) = ( - 4 . 0 X 10-2 C / m 2, - 0 . 8 X 10-2 C/m2). Curve 2: (al, a2) = ( - 3 . 2 X 10-2 C / m 2, - 1 . 6 X 10-2 C/m2). Curve 3: (o.1, a2) = ( - 2 . 4 X 10 -2 C / m 2, - 2 . 4 X 10 -2 C/m2). Curve 4: (o.t, o2) = ( - 1 . 6 X 10 -2 C / m E, - 3 . 2 X 10 -2 C/m2). Curve 5: (o'~, o.2) = ( - 0 . 8 X 10-2 C / m 2, - 4 . 0 X 10-2 C/m2). T = 298 K, Er~ = E r E = E r = 78.5, n = 0.1M. Journal of Colloid and InterfaceScience, Vol. 113,No. 2, October 1986
[1]
[2]
POTENTIAL
PROFILE A C R O S S M E M B R A N E
d2~
371
2vne sinh ve~
dx 2 -
~o
~--~,
x > 0,
[3]
where Erl and er2 are, respectively, the relative permittivities of the outer and inner surface charge layers, Eris the relative permittivity of the electrolyte solution, ~o is the permittivity of a vacuum, e is the elementary electric charge, k is the Boltzmann constant, and T is the absolute temperature. The second term on the right-hand side of each of Eqs. [1] and [2] arises from the membrane-fixed charges. The boundary conditions are
-2O 3
-40
-60
,g,(+O) = ,g,(-O), •(-dl -80
0.105
n(M)
0.1
+ 0) =
¢(-d,
0.115
~r
FIG. 3. The potential at the outer surface, flout = ~(0), as a function o f electrolyte concentration n. d~ = d2 = 10 A. Curve 1: (o-1, or2) = ( - 4 . 0 X 10 -2 C / m 2, - 0 . 8 X 10 -2 C/m2). Curve 2: (al, a2) = ( - 2 . 4 X 10 -2 C / m 2, - 2 . 4 X 10 -2 C/m2). Curve 3: (o'1, ~r2) = ( - 0 . 8 X 10 -2 C / m 2, - 4 . 0 x 10-2 C/m2). a~ + a2 is kept constant at - 4 . 8 X 10 -2 C / m 2. T = 298 K, E,l = ~2 = ~, = 78.5.
-20
~rl
~
- 0),
[5]
d¢l
-0
[6]
~d~
-a~-o ,
[7]
+0 = Erl ~
-a~+o =
~r2
[4]
-]-(a,+d2)+o = 0,
[8]
-I0
J
-5
:~ -I0
-IE
-2C
-2E
Y Membrane
Solution
FIG. 4. Potential distribution ~b(x) across a charged m e m b r a n e with di = 5 A and d2 = 15/~ calculated for several values o f a , ( > 0 ) and a2(<0) with a, + a2 kept constant at - 1 . 6 X 10 -2 C / m 2. Curve i: (al, ~r2) = ( + 3 . 2 X 10 -2 C / m 2, - 4 . 8 X 10 -2 C/m2). Curve 2: (trl, a2) = (+1.6 X 10 -2 C / m 2, - 3 . 2 X 10 -2 C/m2). Curve 3: (a~, a2) = (+0.8 x 10 -2 C / m 2, - 2 . 4 x 10 -2 C/m2). T = 298 K, ~r, = er2 = cr = 78.5, n = 0.1 M. Journal of Colloid and Interface Science, Vol. 113,No. 2, October 1986
372
OHSHIMA, MAKINO, AND KONDO
-2O -I-
0
10 ~
'
i
20
-I0
E
-9-20
3 Membrane
Solution
FIG. 5. Potential distribution ~b(x) across a charged m e m b r a n e with dl = 5 A and d2 = 15 A (Curve 1), 10/~ (Curve 2), and 5 A (Curve 3). trl = +1.6 × 10 -2 C / m 2, ~r2 = - 3 . 2 × 10 -2 C / m 2, T = 298 K, Er, = Er2 = ~r = 78.5, and n = 0.1 M.
~b(x)~0
as
x~+~.
[9]
constant at - 4 . 8 × 10-2 C / m 2, d~ = d2 = 10 A, and n = 0.1 M. It is seen that the potential Equations [4]-[7] state the continuity of ~b(x) profiles are quite different for different charge and of the electric displacement at x = 0 and distributions in the surface charge layer even x = - d l . Equation [8] corresponds to the asif the total charge is constant. When Jail sumption that there is no electric field within < [~2[, the potential ~k(x) changes monotonithe membrane core. cally with x; while, on the other hand, when The solution to the Poisson-Boltzmann 1~11 > [a21, the potential exhibits a maximum equations [1]-[3] subject to the boundary (in magnitude), because the potential should conditions [4]-[9] completely determines the be higher in the region where the fixed-charge potential profile across the membrane surface. density is larger. It is also seen from Fig. 2 that the potential 3. RESULTS AND DISCUSSION at the outer surface (i.e., at x = 0), flout = ~b(0), We introduce the total amount of mem- changes considerably (from - 10 to - 18 mV) brane-fixed charges contained in the outer depending on the charge distribution in the layer ( - d l < x < 0) per unit area, ~l, and that surface layer. We have calculated ~kout as a for the inner layer ( - ( d l + dE) < x < - d 0 , ~2, function of electrolyte concentration n for defined by several values of al and a2, both negative, ¢rl = Z l e N l d l , [10] where a~ + a2 is kept constant at - 4 . 8 × 10-2 C/m 2 and d~ = d2 = 10 A. The results are a2 = z2eN2d2. [ 11 ] given in Fig. 3. The difference in ~koutamong In numerical calculations of ~k(x) we put T curves 1-3 is seen to amount to 8 to 14 mV. = 298 K and v = 1 (1-1 electrolytes), and For the case where trl and #2 are of opposite assumed that the relative permittivities of the sign, the potential profile depends more reouter and inner surface charge layers have the markably on the charge distribution in the same value as that of the surrounding electro- surface layer. Figure 4 shows the potential dislyte solution, putting er~ = ~r2 = er = 78.5. tribution ~k(x) for several values of #1(>0) and Figure 2 shows the potential profile ~b(x) a2(<0), where #~ + a2 is kept constant at - 1 . 6 calculated for several values of ~1 and ~2, both × 10 - 2 C / m 2, d l = 5 , ~ , d E = 1 5 . ~ , a n d n negative, where the total charge ~ + a2 is kept = 0.1 M. Interestingly, the potential at the Journal of Colloid and Interface Science, Vol. 113, No. 2, October 1986
POTENTIAL PROFILE ACROSS MEMBRANE
also with increasing n. In Fig. 6 we plot ¢out as a function o f n for several values o f a l ( > 0 ) and a2(<0), where ai + a2 is kept constant at - 1 . 6 × 10 - 2 C / m 2 , d l = 5 A a n d d : = 15A, showing that ¢out changes its sign at high electrolyte concentrations in curves 1 a n d 2. Also, in the limit o f dl ---* 0% the influence o f the inner layer vanishes and ¢ ( - d l ) should tend to the D o n n a n potential (6), viz.,
I
3
E
373
-10
lim ¢ ( - d l ) dl~oo
k T lnFZlN1
=~e
-20
I(21NI] 2
L2-~-n + 0,2-~n!
]1/2-]
+1[
J,
[12]
and if~l = ~,, the limiting form Of¢o.t is found to be
-~°
0.;~ n(M)
0'.,
0.1~
FIG. 6. The potential at the outer surface, ~out = ~(0), as a function of electrolyte concentration n. d~ = 5 A, d2 = 15/;,. Curve 1: (al, 0-2) = (+3.2 × 10-2 C/m 2, -4.8 × 10-2 C/m2). Curve 2: (al, ~2) = (+1.6 × 10-2 C/m2, -3.2 × 10-2 C/m2). Curve 3: (al, a2) = (+0.8 × 10-2 C/ m2, -2.4 × 10-2 C/m2). al + a2 is kept constant at -1.6 × 10-2 C/m2. T = 298 K, ~rl = E,2 = ~ = 78.5.
lim ¢o.t =
dl--OO
kT(lnrz1N1
ve ~, L 2vn
2vn
+
i(Zlgl~2 }1/2] + 1 (k 2vn ]
(ZlN1]2~ 1/2
It can be shown that when ~dl ~> 1, ¢out can practically be given by Eq. [13], even if dl is o f the same order o f magnitude as d2. N o t e
0 outer surface, ~out, b e c o m e s positive in curves 1 and 2 despite the fact that the total fixed charges o f the whole surface layer is negative (al + a2 < 0). In other words, ~out m a y , in some cases, have the same sign as that o f the fixed charge in the outer surface layer even if the total fixed charges in the whole surface layer a n d the fixed charges in the outer layer only are o f opposite sign. Such behavior o f ¢out can be expected to occur when the influence o f the inner layer is sufficiently small. The contribution o f the inner layer b e c o m e s small as d2 increases, since the shielding effect o f electrolyte ions on ~r2 increases with increasing d2. The potential profiles for three different values o f d2 are c o m p a r e d in Fig. 5, where al = +1.6 × 10 -2 C / m 2, az = - 3 . 2 X 10 -2 C / m 2, dl - 5 A, and n = 0.1 M, showing that ¢out b e c o m e s positive in curve 1. The shielding effect increases directly with increasing electrolyte concentration n. T h u s the contribution from the inner layer decreases
-2(3 " ~
÷
\\
~ ~ - ( d
-80
/
-,oo[
I÷d2))
t\
-oi, Oil(C/m2) -o'.~
-o.'~
FIG. 7. tPo,t= if(0), ~(-d3, and ~b(-(d, + d2))as functions of al. o'2 = trl/2, dL = d2 = 10 A, T = 298 K, Erl = Er2 = er = 78.5, and n = 0.1 M. --, Exact results; ---, approximations (Eqs. [ 14]-[ 16]). Journal of Colloid and InterfaceScience,Vol.113,No.2, October1986
374
OHSHIMA, MAKINO, AND KONDO
that since Eq. [13] involves only the parameters for the outer layer, ~Poutgiven by Eq. [ 13] always has the same sign as 0"1,irrespective of the sign of the total charge 0"1 + 0"2. Finally we give an approximate expression for ¢(x) correct to the first order in N~ and N2 for the case where Erl = er2 = Er: kT ¢(X) = 4v2e-----~ [zlN1{2 - e Kx - (1 - e -2Kd2 + e-~(d'+2d2))e-~x+dl)}
where =
K
( 2 n v 2 e 2 ) 1/2 \ ~r~okT]
[171
is the Debye-Hiickel parameter. These approximate formulas are found to be in good agreement with the exact results when Ivop(x)/ kSq ~< 1; an example is shown in Fig. 7, which compares the approximate results for ¢out, ~b(-dl), and ~(-(d, + d2)) with the exact results when 0"2 = 0"1/2 and dl = dE = 10 A.
+ z2N2(1 - e-Z~d2)e-K(x+dl)], -dl~
[14]
kT ¢(x) - 4vZe n [ZlNl(e ~a~ - 1) )< { e ~x + e-ZK(d'+d2)e -Kx} + zzN2{2 -- e K(x+dl) -
e-2~a2e-~(~+d~)}], -(dl+
d2) ~< x < - d l ,
[151 kT ¢(x) - 4v2e n [ZlNl(1 - e-Kd'){1 + e -K(a~+Ea2)} + z2N2e-~dL(1 -- e-2"a2)]e -~x,
x >~ O,
[16]
Journal of Colloid and InterfaceScience,VoL 113,No. 2, October1986
REFERENCES 1. Donath, E., and Pastushenko, V., Bioelectrochem. Bioenerg. 6, 543 (1979). 2. Wunderlich, R. W., J. Colloid Interface Sci. 88, 385 (1982). 3. Levine, S., Levine, M., Sharp, K. A., and Brooks, D. E., Biophys. J. 42, 127 (1983). 4. Sharp, K. A., and Brooks, D. E., Biophys. J. 47, 563
(1985). 5. Ohshima,H., and Kondo, T., J. Colloid Interface Sci., in press. 6. Ohshima,H., and Ohki, S., Biophys. J. ,17,673 (1985). 7. Ohshima,H., and Ohki, S., Bioelectrochem. Bioenerg., in press.
Biocolloids such as cells or their membranes are usually charged in an electrolyte solution. The membrane-fixed charges are often considered to be located only at the membrane surface (of zero thickness). Actually, however, the membrane-fixed charges may be distributed through some depth at the membrane surface. Recently, taking this fact into account, several studies have been made on the electrophoresis ofbiocolloids (1-5) and on the potential distribution across a charged membrane (6, 7). In the present paper, we extend the membrane model of Ohshima and Ohki (6) to include the effects of nonuniform charge distribution in the surface charge layer. We treat the case in which the charge layer on the
membrane surface consists of two (outer and inner) layers with different thicknesses and different densities of fixed charges. For simplicity, we confine ourselves to planar membranes immersed in a symmetrical electrolyte solution. 2. THEORY
Consider a planar charged membrane immersed in a solution containing a symmetrical electrolyte of valence v and bulk concentration n. Two (outer and inner) charge layers are formed on the membrane surface. In each of the two surface layers, charged groups are uniformly distributed. We denote by dl and d2 the thicknesses of the outer and inner layers, respectively. Let N~ and zl, respectively, be the density and the valence of fixed-charge groups contained in the outer surface layer, and let N2 and z2 be the corresponding quantities of the inner layer. We assume that electrolyte ions can penetrate into the surface
To whom correspondence should be addressed: Faculty of Pharrnaceutical Sciences, Science University of Tokyo, 12, Ichigaya Funagawara-machi, Shinjuku-ku, Tokyo 162, Japan. 369
JournalofColloidandInterfaceScience,Vol. 113,No. 2, October 1986
0021-9797/86 $3.00 Copyright© 1986by AcademicPress,Inc. All rightsof reproductionin any form reserved.
370
OHSHIMA, MAKINO, A N D K O N D O
~(x) -(dl.d2)
-d
f
/
~ou~
-
Membrane core
-
d2
dl
Inner layer
Outer layer
Solution
FIG. 1. Schematic representation of the potential distribution if(x) across a m e m b r a n e with two surface charge layers o f thicknesses dl and d2. to,t = if(0).
charge layers and that there is no electric field within the membrane core. We take the x axis perpendicular to the membrane with its origin on the boundary between the electrolyte solution and the outer surface layer (Fig. 1). The P o i s s o n - B o l t z m a n n equations for the electric potential ~b(x) at a position x relative to the bulk solution phase x = + ~ are x(Z)
-2o
d2~ _ 2vne sinh vop dx 2 ~rl~O kT
zleN1 ErlEO
- d l < x < 0, d2~b _ 2 v n e dx 2
sinh
ve~
~r2~o
kT
z2eN2
~r2~O
- ( d l + d2) < x < - d l , 10
-lo
20
-~0
5 3 -20
-30 ~
Membrone
Sotution
FIG. 2. Potential distribution if(x) across a charged m e m b r a n e with d~ = d2 = 10 A calculated for several values of at and a2, both negative, with o.l + o.2 kept constant at - 4 . 8 × 10 -2 C / m 2. Curve 1: (o.~, a2) = ( - 4 . 0 X 10-2 C / m 2, - 0 . 8 X 10-2 C/m2). Curve 2: (al, a2) = ( - 3 . 2 X 10-2 C / m 2, - 1 . 6 X 10-2 C/m2). Curve 3: (o.1, a2) = ( - 2 . 4 X 10 -2 C / m 2, - 2 . 4 X 10 -2 C/m2). Curve 4: (o.t, o2) = ( - 1 . 6 X 10 -2 C / m E, - 3 . 2 X 10 -2 C/m2). Curve 5: (o'~, o.2) = ( - 0 . 8 X 10-2 C / m 2, - 4 . 0 X 10-2 C/m2). T = 298 K, Er~ = E r E = E r = 78.5, n = 0.1M. Journal of Colloid and InterfaceScience, Vol. 113,No. 2, October 1986
[1]
[2]
POTENTIAL
PROFILE A C R O S S M E M B R A N E
d2~
371
2vne sinh ve~
dx 2 -
~o
~--~,
x > 0,
[3]
where Erl and er2 are, respectively, the relative permittivities of the outer and inner surface charge layers, Eris the relative permittivity of the electrolyte solution, ~o is the permittivity of a vacuum, e is the elementary electric charge, k is the Boltzmann constant, and T is the absolute temperature. The second term on the right-hand side of each of Eqs. [1] and [2] arises from the membrane-fixed charges. The boundary conditions are
-2O 3
-40
-60
,g,(+O) = ,g,(-O), •(-dl -80
0.105
n(M)
0.1
+ 0) =
¢(-d,
0.115
~r
FIG. 3. The potential at the outer surface, flout = ~(0), as a function o f electrolyte concentration n. d~ = d2 = 10 A. Curve 1: (o-1, or2) = ( - 4 . 0 X 10 -2 C / m 2, - 0 . 8 X 10 -2 C/m2). Curve 2: (al, a2) = ( - 2 . 4 X 10 -2 C / m 2, - 2 . 4 X 10 -2 C/m2). Curve 3: (o'1, ~r2) = ( - 0 . 8 X 10 -2 C / m 2, - 4 . 0 x 10-2 C/m2). a~ + a2 is kept constant at - 4 . 8 X 10 -2 C / m 2. T = 298 K, E,l = ~2 = ~, = 78.5.
-20
~rl
~
- 0),
[5]
d¢l
-0
[6]
~d~
-a~-o ,
[7]
+0 = Erl ~
-a~+o =
~r2
[4]
-]-(a,+d2)+o = 0,
[8]
-I0
J
-5
:~ -I0
-IE
-2C
-2E
Y Membrane
Solution
FIG. 4. Potential distribution ~b(x) across a charged m e m b r a n e with di = 5 A and d2 = 15/~ calculated for several values o f a , ( > 0 ) and a2(<0) with a, + a2 kept constant at - 1 . 6 X 10 -2 C / m 2. Curve i: (al, ~r2) = ( + 3 . 2 X 10 -2 C / m 2, - 4 . 8 X 10 -2 C/m2). Curve 2: (trl, a2) = (+1.6 X 10 -2 C / m 2, - 3 . 2 X 10 -2 C/m2). Curve 3: (a~, a2) = (+0.8 x 10 -2 C / m 2, - 2 . 4 x 10 -2 C/m2). T = 298 K, ~r, = er2 = cr = 78.5, n = 0.1 M. Journal of Colloid and Interface Science, Vol. 113,No. 2, October 1986
372
OHSHIMA, MAKINO, AND KONDO
-2O -I-
0
10 ~
'
i
20
-I0
E
-9-20
3 Membrane
Solution
FIG. 5. Potential distribution ~b(x) across a charged m e m b r a n e with dl = 5 A and d2 = 15 A (Curve 1), 10/~ (Curve 2), and 5 A (Curve 3). trl = +1.6 × 10 -2 C / m 2, ~r2 = - 3 . 2 × 10 -2 C / m 2, T = 298 K, Er, = Er2 = ~r = 78.5, and n = 0.1 M.
~b(x)~0
as
x~+~.
[9]
constant at - 4 . 8 × 10-2 C / m 2, d~ = d2 = 10 A, and n = 0.1 M. It is seen that the potential Equations [4]-[7] state the continuity of ~b(x) profiles are quite different for different charge and of the electric displacement at x = 0 and distributions in the surface charge layer even x = - d l . Equation [8] corresponds to the asif the total charge is constant. When Jail sumption that there is no electric field within < [~2[, the potential ~k(x) changes monotonithe membrane core. cally with x; while, on the other hand, when The solution to the Poisson-Boltzmann 1~11 > [a21, the potential exhibits a maximum equations [1]-[3] subject to the boundary (in magnitude), because the potential should conditions [4]-[9] completely determines the be higher in the region where the fixed-charge potential profile across the membrane surface. density is larger. It is also seen from Fig. 2 that the potential 3. RESULTS AND DISCUSSION at the outer surface (i.e., at x = 0), flout = ~b(0), We introduce the total amount of mem- changes considerably (from - 10 to - 18 mV) brane-fixed charges contained in the outer depending on the charge distribution in the layer ( - d l < x < 0) per unit area, ~l, and that surface layer. We have calculated ~kout as a for the inner layer ( - ( d l + dE) < x < - d 0 , ~2, function of electrolyte concentration n for defined by several values of al and a2, both negative, ¢rl = Z l e N l d l , [10] where a~ + a2 is kept constant at - 4 . 8 × 10-2 C/m 2 and d~ = d2 = 10 A. The results are a2 = z2eN2d2. [ 11 ] given in Fig. 3. The difference in ~koutamong In numerical calculations of ~k(x) we put T curves 1-3 is seen to amount to 8 to 14 mV. = 298 K and v = 1 (1-1 electrolytes), and For the case where trl and #2 are of opposite assumed that the relative permittivities of the sign, the potential profile depends more reouter and inner surface charge layers have the markably on the charge distribution in the same value as that of the surrounding electro- surface layer. Figure 4 shows the potential dislyte solution, putting er~ = ~r2 = er = 78.5. tribution ~k(x) for several values of #1(>0) and Figure 2 shows the potential profile ~b(x) a2(<0), where #~ + a2 is kept constant at - 1 . 6 calculated for several values of ~1 and ~2, both × 10 - 2 C / m 2, d l = 5 , ~ , d E = 1 5 . ~ , a n d n negative, where the total charge ~ + a2 is kept = 0.1 M. Interestingly, the potential at the Journal of Colloid and Interface Science, Vol. 113, No. 2, October 1986
POTENTIAL PROFILE ACROSS MEMBRANE
also with increasing n. In Fig. 6 we plot ¢out as a function o f n for several values o f a l ( > 0 ) and a2(<0), where ai + a2 is kept constant at - 1 . 6 × 10 - 2 C / m 2 , d l = 5 A a n d d : = 15A, showing that ¢out changes its sign at high electrolyte concentrations in curves 1 a n d 2. Also, in the limit o f dl ---* 0% the influence o f the inner layer vanishes and ¢ ( - d l ) should tend to the D o n n a n potential (6), viz.,
I
3
E
373
-10
lim ¢ ( - d l ) dl~oo
k T lnFZlN1
=~e
-20
I(21NI] 2
L2-~-n + 0,2-~n!
]1/2-]
+1[
J,
[12]
and if~l = ~,, the limiting form Of¢o.t is found to be
-~°
0.;~ n(M)
0'.,
0.1~
FIG. 6. The potential at the outer surface, ~out = ~(0), as a function of electrolyte concentration n. d~ = 5 A, d2 = 15/;,. Curve 1: (al, 0-2) = (+3.2 × 10-2 C/m 2, -4.8 × 10-2 C/m2). Curve 2: (al, ~2) = (+1.6 × 10-2 C/m2, -3.2 × 10-2 C/m2). Curve 3: (al, a2) = (+0.8 × 10-2 C/ m2, -2.4 × 10-2 C/m2). al + a2 is kept constant at -1.6 × 10-2 C/m2. T = 298 K, ~rl = E,2 = ~ = 78.5.
lim ¢o.t =
dl--OO
kT(lnrz1N1
ve ~, L 2vn
2vn
+
i(Zlgl~2 }1/2] + 1 (k 2vn ]
(ZlN1]2~ 1/2
It can be shown that when ~dl ~> 1, ¢out can practically be given by Eq. [13], even if dl is o f the same order o f magnitude as d2. N o t e
0 outer surface, ~out, b e c o m e s positive in curves 1 and 2 despite the fact that the total fixed charges o f the whole surface layer is negative (al + a2 < 0). In other words, ~out m a y , in some cases, have the same sign as that o f the fixed charge in the outer surface layer even if the total fixed charges in the whole surface layer a n d the fixed charges in the outer layer only are o f opposite sign. Such behavior o f ¢out can be expected to occur when the influence o f the inner layer is sufficiently small. The contribution o f the inner layer b e c o m e s small as d2 increases, since the shielding effect o f electrolyte ions on ~r2 increases with increasing d2. The potential profiles for three different values o f d2 are c o m p a r e d in Fig. 5, where al = +1.6 × 10 -2 C / m 2, az = - 3 . 2 X 10 -2 C / m 2, dl - 5 A, and n = 0.1 M, showing that ¢out b e c o m e s positive in curve 1. The shielding effect increases directly with increasing electrolyte concentration n. T h u s the contribution from the inner layer decreases
-2(3 " ~
÷
\\
~ ~ - ( d
-80
/
-,oo[
I÷d2))
t\
-oi, Oil(C/m2) -o'.~
-o.'~
FIG. 7. tPo,t= if(0), ~(-d3, and ~b(-(d, + d2))as functions of al. o'2 = trl/2, dL = d2 = 10 A, T = 298 K, Erl = Er2 = er = 78.5, and n = 0.1 M. --, Exact results; ---, approximations (Eqs. [ 14]-[ 16]). Journal of Colloid and InterfaceScience,Vol.113,No.2, October1986
374
OHSHIMA, MAKINO, AND KONDO
that since Eq. [13] involves only the parameters for the outer layer, ~Poutgiven by Eq. [ 13] always has the same sign as 0"1,irrespective of the sign of the total charge 0"1 + 0"2. Finally we give an approximate expression for ¢(x) correct to the first order in N~ and N2 for the case where Erl = er2 = Er: kT ¢(X) = 4v2e-----~ [zlN1{2 - e Kx - (1 - e -2Kd2 + e-~(d'+2d2))e-~x+dl)}
where =
K
( 2 n v 2 e 2 ) 1/2 \ ~r~okT]
[171
is the Debye-Hiickel parameter. These approximate formulas are found to be in good agreement with the exact results when Ivop(x)/ kSq ~< 1; an example is shown in Fig. 7, which compares the approximate results for ¢out, ~b(-dl), and ~(-(d, + d2)) with the exact results when 0"2 = 0"1/2 and dl = dE = 10 A.
+ z2N2(1 - e-Z~d2)e-K(x+dl)], -dl~
[14]
kT ¢(x) - 4vZe n [ZlNl(e ~a~ - 1) )< { e ~x + e-ZK(d'+d2)e -Kx} + zzN2{2 -- e K(x+dl) -
e-2~a2e-~(~+d~)}], -(dl+
d2) ~< x < - d l ,
[151 kT ¢(x) - 4v2e n [ZlNl(1 - e-Kd'){1 + e -K(a~+Ea2)} + z2N2e-~dL(1 -- e-2"a2)]e -~x,
x >~ O,
[16]
Journal of Colloid and InterfaceScience,VoL 113,No. 2, October1986
REFERENCES 1. Donath, E., and Pastushenko, V., Bioelectrochem. Bioenerg. 6, 543 (1979). 2. Wunderlich, R. W., J. Colloid Interface Sci. 88, 385 (1982). 3. Levine, S., Levine, M., Sharp, K. A., and Brooks, D. E., Biophys. J. 42, 127 (1983). 4. Sharp, K. A., and Brooks, D. E., Biophys. J. 47, 563
(1985). 5. Ohshima,H., and Kondo, T., J. Colloid Interface Sci., in press. 6. Ohshima,H., and Ohki, S., Biophys. J. ,17,673 (1985). 7. Ohshima,H., and Ohki, S., Bioelectrochem. Bioenerg., in press.