J. Phys. Chem. B 1999, 103, 9743-9748
9743
Double-Layer Properties of an Ion-Penetrable Charged Membrane: Effect of Sizes of Charged Species Yung-Chih Kuo and Jyh-Ping Hsu* Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617 ROC ReceiVed: February 3, 1999
In this paper we have proposed a double-layer model, which takes the sizes of the charged species into account, for an ion-penetrable charged membrane immersed in an electrolyte solution. Analytical expressions for the electrical potential distribution, the potential drop across the membrane, and the net penetration charge into the membrane are derived for the case in which the linearized Gouy-Chapman model is employed. We will show that, if charged species have different sizes, the behavior of a positively charged membrane is unsymmetrical to that of a negatively charged membrane. Also, the larger the counterions, the greater the potential drop across a membrane and the smaller the net penetration charge into a membrane; the reverse is true for co-ions.
Introduction Over the past five decades, the interdisciplinary knowledge about colloid and surface science has become of great significance in both experimental study and theoretical development for macroscopic and microscopic systems. A particularly interesting problem in the interfacial phenomena is a hypothetical layer of liquid phase containing electrolyte ions near a charged surface, the so-called electrical double layer. The interest in the double-layer behavior is justified by the widespread applications of ionic systems in both chemical and biochemical technologies. Many of the ideas related to the double layer have been responsible for a considerable amount of theoretical activities, originating from the Gouy-Chapman theory (GCT),1,2 in which the equilibrium electrical potential and the spatial profile of ions are determined by solving the Poisson-Boltzmann equation (PBE) for a point-charge, continuumsolvent model of electrolyte solutions. This theory and many of the subsequent theoretical improvements are based on use of the PBE in a way analogous to that later invoked by Debye and Huckel3 in the theory of strong electrolytes. In liquid state physics, the PBE is a mean field approximation for the model used in the GCT in which all correlation among bath ions is neglected, and the only difference between cations and anions is in the charge they bear. Stern4 pointed out that a better fit between the Gouy-Chapman treatment and real electric double layers could be obtained by including a steric repulsion zone near the charged interface containing no mobile ions and having a lower dielectric constant than the bulk water. In this GouyChapman-Stern theory, the hard-core interactions among ions are neglected, but ionic sizes are included in the model in which ions are allowed to approach up to a specific distance from the charged surface. Grahame5 defined the position of closest approach between the mobile ions and the charged surface as the outer Helmholtz plane. The effective radii of the electrolyte ions were considered to have important influence on the double-layer behavior.6,7 The finite sizes for cations and anions are often characterized by the interaction of the ions with the solvent structure near the charged surface. In a study of the effect of ionic sizes on the * Corresponding author. Fax: 886-2-23623040. E-mail: t8504009@ ccms.ntu.edu.tw.
electrical potential distribution, Valleau and Torrie8 showed that a diffuse double layer with zero charge on the surface and a concentration-dependent potential of zero charge could be caused by unequal ionic sizes. Bhuiyan et al.9 investigated the effects of asymmetric valences and unequal ionic sizes on the potential difference across a double layer. They found that the lower the charge density on an electrode, the more evident the effect of unequal ionic sizes. On the other hand, in a grand canonical ensemble study of unequally sized ions in spherical, charged cavities, Sorensen and Rivera10 found an increasingly stronger differentiation between unequally sized ions with increasing surface charge density. A Monte Carlo simulation was adopted by Torrie and Valleau11 to describe the effect of ionic sizes on the ionic concentration profile for a system of an infinite planar charged surface immersed in a strong electrolyte solution. It was concluded that the performance of the physical model of the double layer is quite satisfactory. Grand canonical ensemble Monte Carlo simulations of ions of finite size with different sizes and charges in spherical cavities with a fixed charge on the surface of the cavity have also been performed,12-14 and similar conclusions were reached. For biological cells and some artificial membranes, the outer surface layer usually carries fixed charges, which arise mainly from the dissociation of the functional groups in the layer itself or from the absorption of the ionic species in the nearby liquid solution.15 The fixed charges are spread over a surface layer rather than over a rigid surface, i.e., the fixed charge is distributed over a finite volume in a three-dimensional space, or the so-called soft polar interfaces. Levadny et al.16 considered the transport of ions across such interfaces. The different regions in phosholipid bilayers and their influence upon the permeability of species were discussed in detail by Disalvo.17 For biocolloids, since the charged surface layer usually has a capacitance of about 1 µF/cm2 and a specific resistance of 109-1011 Ω/ cm, the cell membrane can be regarded as a dielectric. Ohshima18 studied a system comprised of two dielectric membranes carrying no fixed charges in an electrolyte solution. It was concluded that the change in potential gradient within the cell membrane could be on the order of 103-105 V/cm when the distance between two cells is about 10-20 Å. Ohshima and Kondo19-21 analyzed a liquid system containing a symmetric
10.1021/jp9903965 CCC: $18.00 © 1999 American Chemical Society Published on Web 10/07/1999
9744 J. Phys. Chem. B, Vol. 103, No. 44, 1999
Kuo and Hsu uncharged membrane (Xc < X < Xca.), X being the scaled distance, (2) Xca.< X < Xan, which contains cations only, (3) Xan < X < Xi, which contains both cations and anions, (4) Xi < X < Xo, which contains all charged species, and (5) Xo< X, which includes the outer uncharged membrane (Xo < X < Xc + d) and the liquid phase (∞ > X > Xc + d). Here, Xi and Xo denote the locations of the inner and the outer planes of the fixed charges. The scaled symbols are defined by X ) κr, Xca ) Xc + κσca/2, Xan ) Xc + κσan/2, Xi ) Xc + κσf/2, and Xo ) Xc + d - κσf/2. κ and r are the reciprocal Debye length and the distance. The fixed charges are present in the region Xi e X e Xo. The spatial variation in the scaled electrical potential at equilibrium, φ, can be described by the Poisson-Boltzmann equation
Figure 1. Schematic representation of the system under consideration. Xc, Xca, Xan, Xi, Xo, and d are, respectively, the scaled location of the uncharged core-membrane interface, the effective most interior locations of cation and anion, the locations of inner and outer planes of fixed charge, and the scaled thickness of membrane. The scaled size of fixed group is Xi - Xc. Regions I, II, III, IV, and V denote, respectively, the charge-free region, the region in which only cations are present, the region in which both cations and anions are present, the region in which all charged species are present, and the region which comprises the outer uncharged membrane and the diffuse double layer.
electrolyte and two ion-penetrable membranes with a uniform distribution of fixed charges. Hsu and Kuo22-24 extended this model to an arbitrary combination of electrolyte valences. Sorensen and Rivera10 conducted grand canonical ensemble Monte Carlo simulations of Swiss cheese ion exchange membranes in dilute solutions of primitive model 1:1 electrolytes with different sizes of ions or 2:1 electrolytes with different Bjerrum parameters. An ionic system comprising a lattice of fixed charges in polymer phase and a liquid solution of counterion was analyzed by Manning25-28 in a series of studies in which the charged species, including fixed charged groups and electrolyte ions, are regarded as point charges. Although relevant results for membrane systems are ample in the literature, the effect of finite ionic sizes in such systems on the behavior of a double-layer has not been discussed. Apparently, assuming point charges is an oversimplification, and a more realistic model should take the effect of the sizes of charged species into account. A further study on a physical model with various sizes of charged species seems to be of necessity. This is conducted in the present study. Here, we consider the case in which an ion-penetrable charged membrane is immersed in an electrolyte solution.12-14 In particular, we examined the effects of ionic sizes on the potential drop across a membrane and the net penetration charge into a membrane. Analysis A schematic representation of the system under consideration is shown in Figure 1. A planar particle which comprises a rigid, uncharged core and a membrane layer of scaled thickness d in an arbitrary electrolyte solution are considered. The membrane layer is ion-penetrable and contains uniformly distributed charges. Let σf, σan, and σca be the effective diameters of a fixed group, an anion, and a cation, respectively. Without loss of generality, we assume that σf > σan > σca. The fixed groups are arranged so that the margin of the leftmost one coincides with the core-membrane interface and that of the rightmost one coincides with the membrane-liquid interface. Referring to Figure 1, the system is divided into 5 regions: (1) X < Xca., which comprises the charge-free region (-∞ < X < Xc) and the inner
d2φ [-u exp(-aφ) + V exp(bφ) - wN] ) a+b dX2
(1)
where a and b are, respectively, the valences of the cation and anion, φ ) eφ/kBT, κ2 ) e2a(a + b)n0a /0rkBT, and N ) ZN0NA/an0a. In these expressions, φ is the electrical potential, N0 and Z are the density and valence of the fixed charges, respectively, NA and e represent the Avogadro number and the elementary charge, 0 and r are the permittivity of a vacuum and the relative permittivity, respectively, n0a is the number concentration of cations in the bulk liquid phase, kB and T denote, respectively, the Boltzmann constant and the absolute temperature. For convenience, a vector for the presence/absence of fixed charge, anion, and cation, (w,V,u), is defined, (w,V,u) ) (0,0,0), (0,0,1), (0,1,1), (1,1,1), or (0,1,1) represent, respectively, regions I, II, III, IV, or V. If the amount of fixed charge in a membrane is small, a considerable amount of counterions and coions may penetrate into the membrane. The effect of finite sizes of the charged species is, therefore, significant in the case of low membrane potential. In this case, eq 1 can be approximated by
2 d2φ ) [(ua + Vb)φ + V - u - wN] 2 ak dX
(2)
where k ) 2 + 2b/a. The boundary conditions associated with this equation are assumed as
dφ f 0 as X f -∞ dX dφ dφ ) r,m and φ(X+ r,m ca) ) dX X)Xca+ dX X)Xcar,c
( )
(2a)
( )
φ(Xca) at X ) Xca (2b) r,m
(dφ dX)
X)Xan+
) r,m
(dφ dX)
X)Xan-
and φ(X+ an) ) φ(Xan) at X ) Xan (2c)
r,m
r,m
(dφ dX)
(dφ dX)
X)Xi+
X)Xo+
) r,m ) r,m
(dφ dX)
(dφ dX)
r,dl
X)Xi-
X)Xo-
and φ(X+ i ) ) φ(Xi ) at X ) Xi
(2d) and φ(X+ o ) ) φ(Xo ) at X ) Xo
dφ f 0 as X f -∞ dX
(2e) (2f)
where r,c, r,m, and r,dl denote, respectively, the relative
Effect of Sizes of Charged Species on Double Layer
J. Phys. Chem. B, Vol. 103, No. 44, 1999 9745
permittivities of hard core, membrane, and liquid solution. In general, r,c < r,m < r,dl. Equations 2a and 2f imply that the system under consideration is at electroneutrality. Region I is free of charges, and, therefore, eq 2 reduces to
Integrating this expression subject to eqs 2d and 8 we obtain
φ ) C5 sinh(X) + C6 cosh(X) +
(3) C5 ) C 3 +
2N sinh(Xi) ak
(10a)
C6 ) C4 -
2N cosh(Xi) ak
(10b)
Solving this equation subject to eq 2a yields
φ ) φc
(4)
where φc is the scaled electrical potential in the uncharged core of a particle or the scaled membrane potential. Since region II contains cations only, eq 2 becomes
Region V contains electrolyte ions, but is free of fixed charges. Therefore, eq 2 becomes
d 2φ )φ dX2
2
2 dφ 2 ) φak dX2 k
(5)
2 X + C2 cosh k
(x )
2 1 X + k a
(
) (x ) ) (x )
(
C2 ) φc -
1 cosh a
2 X k ca
2 X k ca
φ ) C7 exp(-X) + C8 exp(X)
(6) C7 )
(6b)
C8 ) 0
(7)
(8)
where
(x ) [x (x )] [x (x ) (x )] (x ) x [ (x )] [ (x ) x (x )]
2 2 cosh(Xan) cosh X - sinh(Xan) k k an 2 2 2 sinh X C + cosh(Xan) sinh X k an 1 k k an 2 1 sinh(Xan) cosh X C - sinh(Xan) (8a) k an 2 a
C3 )
2 sinh(Xan) k 2 2 2 X C + cosh(Xan) cosh X k an 1 k an k 2 1 sinh(Xan) sinh X C + cosh(Xan) (8b) k an 2 a
C4 ) cosh(Xan) sinh cosh
2 X k an
In region IV, since all charged species are present, eq 2 reduces to
2N d2φ )φak dX2
(9)
(12a) (12b)
It can be shown that N0 and φc have the following implicit relation (Appendix)
N)
C 3 + C4 ak 2 exp(-Xi) - exp(-X0)
(13)
The apparent scaled electric field due to the presence of a membrane, E, can be expressed by
Integrating this expression subject to eqs 2c and 6 yields
φ ) C3 sinh(X) + C4 cosh(X)
N 1 [exp(X0) - exp(Xi)] - (C3 - C4) ak 2
(6a)
Both cations and anions are present in region III, and eq 2 becomes
d2φ )φ dX2
(12)
where
where
1 C1 ) - φc - sinh a
(11)
Integrating this expression subject to eqs 2e and 2f yields (Appendix)
Integrating eq 5 subject to eqs 2b and 4 gives
(x )
(10)
where
d2φ )0 dX2
φ ) C1 sinh
2N ak
E)
φml - φc d
(14)
where φml ) C7 exp[-Xo + Xi - Xc)] and d ) (Xo + Xi 2Xc). It can be shown that the net penetration charge per unit area of a membrane-liquid interface, Qsl, is (Appendix)
Qsl ) -Qt + Ed + φc
(15)
where Qt is defined in eq A7. Discussion If the sizes of all the charged species are infinitely small, that is, Xca - Xc, Xan - Xc, and Xi - Xc all vanish, the present model reduces to the classic point-charge model (PCM) for an ion-penetrable charged membrane. In other words, the latter can be recovered as a special case of the former. Note that the region for fixed charges, (Xo - Xi < d), is narrower than that of the corresponding PCM, d. The properties of the system under consideration are examined through numerical simulation. Figure 2 shows the variation of the scaled fixed charge density as a function of the scaled electrical potential in the core region. As can be seen from this figure, φc increases with the increase in N. Also, for a fixed φc, |N| decreases with the increase in d, and for a fixed N, |φc| increases with d. This means that, for a thicker membrane, a smaller fixed charge density is sufficient to reach a specified
9746 J. Phys. Chem. B, Vol. 103, No. 44, 1999
Figure 2. Variation of the scaled fixed charged density as a function of the scaled electrical potential in the core region. Key: solid curves: present model; dashed curves: classic point-charge model, ionic strength ) 1 mol/m3, a ) 1, b ) 1, |Z| ) 1, Xca - Xc ) 0.025, Xan - Xc ) 0.05, Xi - Xc ) 0.1.
Figure 3. Variation of the apparent scaled electric field, E, as a function of the scaled electrical potential in the core region. Key: same as Figure 2.
φc, and a higher potential in the core region is necessary to maintain a fixed charge density. Since for the same fixed charge density, the total amount of fixed charge contained in a thicker membrane is greater than that in a thinner membrane, the former yields a higher |φc|. Since the present model has a narrower fixed-charged region than the PCM, for fixed φc and d, |N(present model)| > |N(PCM)|, and for fixed N and d, |φc(present model)|> |φc(PCM)|; also, the smaller the d, the greater the difference between the two, as illustrated in Figure 2. The variation of the apparent scaled electric field due to the presence of a membrane as a function of the scaled electrical potential in the core region is illustrated in Figure 3. This figure reveals that the higher the |φc|, the greater the |E|. Figure 3 also suggests that for a fixed φc the greater the d, the smaller the |E|. Although for a fixed φc, the thicker the membrane, the smaller the |φml|, and the greater the |φc - φml|, the rate of increase in d is faster than that in |φc - φml|, and therefore |E| decreases, as can be seen from Figure 3 |E(present model)| > |E(PCM)| in general. Since for a fixed φc, the fixed charge density for the present model is greater than that for the corresponding PCM, the potential decay for the former is greater than that for the latter, and therefore |φml(present model)|
|E(PCM)|. If φc is negative and |φ| is small, then |E(present model)| < |E(PCM)|. Figure 4 presents the variation of the scaled net penetration charge per unit area of a membrane as a function of the scaled electrical potential in the core region. This figure shows that for a fixed d, the higher the |φc|, the greater the |Qsl|. Also, for a fixed φc, the greater the d, the greater the |Qsl|. The rationale behind this is elaborated as follows. First, as discussed in the case of Figure 3, for a fixed φc, the larger the d, the smaller the |E|, and the larger the |φc - φml|. Therefore |Ed| ) |φc - φml| increases with d. In other words, although the fact that |E| is small implies that the electrical attraction of a membrane for the counterion decreases, the permeable thickness for the mobile ion increases, however, and the net effect is that the potential drop across the membrane increases. Second, as discussed in the case of Figure 2, for a fixed φc, the greater the d the smaller the |N|. Since |Nd| increases with d, according to eq A7, the larger the |Xo - Xi| ) |d- (Xi - Xc)|, the larger the |Qt|. Therefore, the first term on the right-hand side of eq 15 also has a positive contribution to |Qsl|. Figure 4 also suggests that for a fixed φc, |Qsl(present model)| < |Qsl(PCM)| in general. Note that since the electrical potential of the system under consideration is low, the absorption of electrolyte ions in the membrane phase is proportional to |φ|. For a fixed φc, the decay in electrical potential for the present model is faster than that for the corresponding PCM. As explained in the discussion of Figure 3, the amount of ions penetrating into the membrane for the former is less than that for the latter. From the mathematical point of view, the first and second terms on the right-hand side of eq 15 are two competitive terms. For a fixed φc, |Ed(present model)| > |Ed(PCM)| as pointed out in the discussion of Figure 3. However, as stated in the discussion of Figure 2, for a fixed φc |Qt(present model)| < |Qt(PCM)|. The net effect is that Qt has a more significant effect on the value of Qsl than Ed. Figure 4 indicates that for a negative φc, |Qsl(present model)| > |Qsl(PCM)| if |φc| is low. Since |φ(present model)| > |φ(PCM)| for a negatively charged membrane and a small |φc|, as shown in the case of Figure 3, the |Qsl| of the present model is larger than that for the corresponding PCM. Figure 4 also suggests that the larger the d, the broader the range of φc in which |Qsl(present model)| > |Qsl(PCM)|. For instance, the range of φc is (-0.4,0) for d ) 0.5, and is (-1,0) for d ) 2. Figures 2 through 4 suggest that if N ) 0, φc, E, and Qsl all vanish for the case of PCM. For the present model, if N ) 0,
Effect of Sizes of Charged Species on Double Layer
Figure 5. Variation of the apparent scaled electric field, E, as a function of the relative magnitudes of cation and anion, (Xca - Xc)/(Xan - Xc). Key: Curve 1: Xan - Xc ) 0.05; 2: Xca - Xc ) 0.025, ionic strength ) 1 mol/m3, a ) 1, b ) 1, Z ) -1, d ) 1, φc ) - 0.5, Xi - Xc ) 0.1.
Figure 6. Variation of the scaled net penetration charge per unit area of a membrane as a function of the relative magnitudes of cation and anion, (Xca - Xc)/(Xan - Xc). Key: same as Figure 5.
both φc and E are positive, and Qsl is negative. We conclude that an asymmetric double-layer behavior occurs for the present model. This is because the cation is the smallest one among all the charged species. In this case, the behavior of a negatively charged membrane is much more complicated than that of a positively charged membrane. It is worth noting that for the present model, φ decreases in region II and decreases first and then increases in region III as X increases. This implies that the penetrated ions have the tendency to raise the level of φ in region III. The variation of the apparent scaled electric field due to the presence of a membrane as a function of the relative magnitudes of cation and anion, measured by (Xca - Xc)/(Xan - Xc), is presented in Figure 5. Figure 6 illustrates the variation of the scaled net penetration charge per unit area of a membrane as a function of (Xca - Xc)/(Xan - Xc). Here, we consider a negatively charged membrane. The leftmost points for curves 1 and 2 represent, respectively, the result for the case in which counterions are treated as point charges and that for the case in which co-ions and fixed groups have the same size. The rightmost points of curves 1 and 2 are the result for the case in which the counterion and the co-ion have the same size with (Xca - Xc) ) (Xan - Xc) ) 0.05 and (Xca - Xc) ) (Xan - Xc) ) 0.025,
J. Phys. Chem. B, Vol. 103, No. 44, 1999 9747 respectively. As can be seen from Figures 5 and 6, curves 1 and 2 intersect at two points, one at (Xca - Xc)/(Xan - Xc) ) 0.5 and the other at (Xca - Xc)/(Xan - Xc) = 0.3. The former is due to the fact that the sizes of the cation and the anion in curve 1 are the same as those in curve 2. The latter implies that, for fixed relative magnitudes of counterion and anion, different ionic sizes may lead to the same double-layer properties. On the basis of Figures 5 and 6, we conclude the following (1) The larger the size of the counterion, the greater the |E| and the smaller the |Qsl|; the reverse is true for the co-ion. This is because the larger the size of the counterion, the narrower the range of region II, and both φ and φml are higher. Similar reasoning can be employed for the coion. (2) In both Figures 5 and 6 the absolute value of the slope at the leftmost point of curve 1 is smaller than that of curve 2. This is because the range of region II at the leftmost point for curve 1 is smaller than that for curve 2. The smaller the range of region II, the less significant the influence of the variation of range II on both |E|, and |Qsl|. (3) In both Figures 5 and 6 the absolute value of the slope at the rightmost point of curve 1 is greater than that of curve 2. This is because, in these figures, the range of region II at the rightmost point for curves 1 and 2 vanishes. The range of region III at the rightmost point for curve 1 is fixed, but that for curve 2 decreases with (Xca - Xc)/(Xan - Xc). In summary, the classic point-charge model for an ionpenetrable charged membrane in an electrolyte solution is modified to take the effect of finite sizes of the charged species on double-layer properties under the linearized Gouy-Chapman condition. The results of the numerical simulation reveal that the sizes of counterions and co-ions have a significant effect on both the potential drop across a membrane and the net penetration charge into a membrane. The larger the counterions, the higher the potential drop across a membrane and the smaller the amount of net penetration charges into a membrane; the reverse is true for co-ions. For unequal ionic sizes, the behavior of a positively charged membrane is unsymmetrical to that of a negatively charged membrane. It should be pointed out that since the linearized Poisson-Boltzmann equation is used with the electric potential as the potential of mean forces, the charged species are treated as point charges, except for the fact that there are different zones of closest approach between the ions and the membrane hard core. The size of the ions (and the excluded volume in ion-ion encounters) has not been taken into account between the mobile ions or between the mobile and the fixed ions. To obtain a consistent theory, more refined integral equation theories, such as mean spherical approximation or hypernetted chain or Monte Carlo simulations have to be applied. Appendix Substituting eqs 2e and 12b into eqs 10 and 12 yields
C7 exp(-Xo) ) C5 sinh(Xo) + C6 cosh(Xo) + 2N/ak
(A1)
-C7 exp(-Xo) ) C5 cosh(Xo) + C6 sinh(Xo)
(A2)
Combining these two expressions, we have
(C5 + C6) exp(Xo) + 2N/ak ) 0
(A3)
Equation A2 can be rewritten as
1 1 C7 ) - (C5 + C6) exp(2Xo) - (C5 - C6) 2 2
(A4)
9748 J. Phys. Chem. B, Vol. 103, No. 44, 1999
Kuo and Hsu
By substituting eqs 10a, 10b, and A3 into eq A4, eq 12a can be recovered. By substituting eqs 10a and 10b into eq A3, eq 13 can be recovered. The scaled amount of charges contained in the double-layer region per unit area of membrane-liquid interface, Qdl, is
Qdl )
dφ | dX Xo+Xi-Xc
(A5)
Substituting eq 12 into this expression yields
Qdl ) -C7 exp[-(Xo + Xi - Xc)]
(A6)
The scaled amount of fixed charges per unit area of a membrane-liquid interface, Qt, can be calculated by
Qt )
Ze2NAN0 0rkBTκ
2
∫XX i
o
dX )
ZNAN0(Xo - Xi) a(a + b)n0a
(A7)
The overall electroneutrality implies that Qsl ) -Qt - Qdl, where Qsl is the scaled amount of charges penetrated into a membrane per unit area of a membrane-liquid interface, i.e., the net penetration charge per unit area of a membrane-liquid interface. Equations 14, A6, A8, and the condition of overall electroneutrality lead to eq 15.
References and Notes (1) Gouy, G. J. Phys. Radium 1910, 9, 457. (2) Chapman, D. L. Philos. Mag. 1913, 25, 475. (3) Debye, P.; Huckel, E. Z. Phys. 1923, 24, 133. (4) Stern, O. Z. Elektrochem. Angew. Phys. Chem. 1924, 30, 508. (5) Grahame, D. C. Chem. ReV. 1947, 41, 441. (6) Blum, L. J. Phys. Chem. 1977, 81, 136. (7) Henderson, D.; Blum, L. J. Chem. Phys. 1978, 69, 5441. (8) Valleau, J. P.; Torrie, G. M. J. Chem. Phys. 1982, 76, 4623. (9) Bhuiyan, L. B.; Blum, L.; Henderson, D. J. Chem. Phys. 1983, 78, 442. (10) Sorensen, T. S.; Rivera, S. R. Mol. Simul. 1995, 15, 79. (11) Torrie, G. M.; Valleau, J. P. Chem. Phys. Lett. 1979, 65, 343. (12) Sloth, P.; Sorensen, T. S. J. Chem. Phys. 1992, 96, 548. (13) Sorensen, T. S.; Sloth, P. J. Chem. Soc., Faraday Trans. 1992, 88, 571. (14) Rivera, S. R.; Sorensen, T. S. Mol. Simul. 1994, 13, 115. (15) Tatulian, S. A. In Surface Chemistry and Electrochemistry of Membranes; Sorensen, T. S., Ed.; Marcel Dekker: New York, 1999. (16) Levadny, V.; Aguilella, V.; Belaya, M. In Surface Chemistry and Electrochemistry of Membranes; Sorensen, T. S., Ed.; Marcel Dekker: New York, 1999. (17) Disalvo, E. A. In Surface Chemistry and Electrochemistry of Membranes; Sorensen, T. S., Ed.; Marcel Dekker: New York, 1999. (18) Ohshima, H. J. Theor. Biol. 1977, 65, 523. (19) Ohshima, H.; Kondo, T. J. Theor. Biol. 1987, 128, 187. (20) Ohshima, H.; Kondo, T. Biophys. Chem. 1988, 32, 161. (21) Ohshima, H.; Kondo, T. J. Colloid Interface Sci. 1989, 133, 521. (22) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1995, 171, 331. (23) Hsu, J. P.; Kuo, Y. C. J. Chem. Soc., Faraday Trans. 1995, 91, 1223. (24) Hsu, J. P.; Kuo, Y. C. J. Chem. Phys. 1995, 103, 465. (25) Manning, G. S. J. Chem. Phys. 1967, 46, 2324. (26) Manning, G. S. J. Chem. Phys. 1967, 46, 4976. (27) Manning, G. S. J. Chem. Phys. 1967, 47, 2010. (28) Manning, G. S. J. Chem. Phys. 1967, 47, 3377.
