Effect of Ionic Sizes on Critical Coagulation Concentration: Particles

Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. ... Jyh-Ping Hsu , Cheng-Chi Chuang , Shiojenn Tseng...
0 downloads 0 Views 86KB Size
Download PDF
J. Phys. Chem. B 2002, 106, 4269-4275

4269

Effect of Ionic Sizes on Critical Coagulation Concentration: Particles Covered by a Charge-Regulated Membrane Jyh-Ping Hsu* and Shih-Wei Huang Department of Chemical Engineering, National Taiwan UniVersity, Taipei, Taiwan 10617, R.O.C.

Yung-Chih Kuo Department of Chemical Engineering, National Chung Cheng UniVersity, Chia-Yi, Taiwan 62102, R.O.C.

Shiojenn Tseng Department of Mathematics, Tamkang UniVersity, Tamsui, Taipei, Taiwan 25137, R.O.C. ReceiVed: June 18, 2001; In Final Form: February 14, 2002

The critical coagulation concentration (CCC) for the case planar particles covered by a charge-regulated membrane layer immersed in a mixed (a:b) + (c:b) electrolyte solution is evaluated theoretically. The present study extends previous analyses in that a general charged condition in the membrane layer, which mimics biological cells, is considered, and the sizes of the charged species are taken into account. We show that for particles carrying a net negative fixed charge the classic point charge model will overestimate both CCC and the electrical repulsive force between two particles. The electrical repulsive force and the total interaction energy between two particles are large if (1) the pH is high, (2) the membrane is thin, (3) the valence of counterions is low, (4) the fraction of multivalent counterions is low, and (5) the ionic strength is low. CCC is high if (1) the pH is high, (2) the valence of counterions is low, and (3) the size of counterions is large; these results are consistent with experimental observations in the literature.

1. Introduction Critical coagulation concentration (CCC) is one of the most important characteristics of a colloidal dispersion. Experimental observations reveal that the variation of CCC as a function of the valence of counterions follows roughly the inverse sixth power law, the so-called Schulze-Hardy rule. This rule was interpreted theoretically by the DLVO model,1 which considered the electrical repulsive force and the van der Waals attractive force between two particles. The original derivation of this model is based on rigid particles in a symmetric electrolyte solution. For nonrigid particles such as biocolloids and particles covered by an artificial membrane,2-7 the classic DLVO model needs to be modified accordingly. Terui et al.,8 for example, derived expressions for the electrical interaction potential and the interaction force between an ion-penetrable particle and a rigid particle for the case of low electrical potential, symmetric electrolyte, and uniformly distributed fixed charge in the former. The analysis was extended to various types of particles, and expressions for CCC were obtained.9 Hsu and Kuo10 derived an analytical expression for the CCC of counterions in an arbitrary a:b electrolyte. They considered the case where a particle comprises a rigid core and an ion-penetrable surface layer, which contains fixed charge because of either the dissociation of the functional groups or the adsorption of ions from the liquid phase. The electrical interaction between two dissimilar spherical particles covered by an ion-penetrable * To whom correspondence should be addressed. Fax: 886-2-23623040. E-mail: [email protected].

charged membrane in an a:b electrolyte solution was estimated by Hsu and Kuo.11 A perturbation method was applied to solve the governing nonlinear Poisson-Boltzmann equation, and approximate analytical expressions for potential distribution, stability ratio, and CCC of counterions were derived. The results discussed above are all based on the classic Gouy-Chapman model1 in which charged species are treated as point charges. Valleau and Torrie12 and Bhuiyan et al.13 adopted a modified Gouy-Chapman model to examine the property of electrical double layer, and they concluded that the effective radii of mobile ions plays a significant role. Both Monte Carlo simulation and statistical mechanics approaches were adopted by researchers14-18 to study the effect of ionic sizes on the behavior of electrical double layer, and they all concluded that it is significant. Hsu and Kuo19 and Kuo and Hsu20 investigated the effect of the sizes of charged species on the electrical properties of a particle covered by an ionpenetrable charged membrane. The electrical interaction between two particles, each is covered by an ion-penetrable charged membrane in an asymmetric electrolyte solution, was estimated by Kuo and Hsu21 by taking the effect of the sizes of charged species into account. Kuo et al.22 studied the electrical interaction force and the potential-energy barrier between a particle covered by an ion-penetrable charged membrane and a planar charged surface by taking the sizes of all of the charged species into account. In the present study, the classic DLVO model is extended to the case when particles are coated by a charge-regulated

10.1021/jp012282a CCC: $22.00 © 2002 American Chemical Society Published on Web 03/29/2002

4270 J. Phys. Chem. B, Vol. 106, No. 16, 2002

Hsu et al. charges. Xi is the effective scaled radius of fixed charged group. The scaled symbols are defined by X ) κx, Xca ) κσca, Xan ) κσan, Xi ) κσf, and Xo ) d - κσf, with κ and x being respectively the reciprocal Debye length and the distance from the rigid core. 2.1. Dissociation of Functional Groups. We assume that the membrane layer contains both acidic and basic functional groups. The dissociation of these functional groups can be described by

AHZ(n-1)T AHZn-a-n + H+, Ka,n, n ) 1, 2, ..., Za (1) a-(n-1) -(n-1)]+ -n)+ BHZ[Zbb-(n-1) T BHZ(Zbb-n + H+, Kb,n, n ) 1, 2, ..., Zb (2)

Figure 1. Schematic representation of the system under consideration. Xca, Xan, Xi, Xo, XL, and d are respectively scaled locations of the effective most interior locations of positive and negative charges, the locations of inner and outer planes of fixed charge, the location of half separation distance between two particles, and the scaled thickness of the membrane. The scaled size of the fixed charged group is Xi. Regions I-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.

membrane layer, in a general electrolyte solution. In particular, the effect of the sizes of charged species on CCC is analyzed. The presence of multivalent counterions on the behavior of the system under consideration is also examined by allowing the liquid phase to contain two types of counterions.

In these expressions, Za is the number of dissociable protons of acidic group AHZa and Zb is the number of absorbable protons of basic group B. Ka,n, n ) 1, 2, ..., Za, and Kb,n, n ) 1, 2, ..., Zb, are equilibrium constants. It can be shown that the concentration of negative fixed charges, N-, and that of positive fixed charges, N+, are23

N- )

(

N + ) N b Zb -

(3)

)

Qb 1 + Pb

(4)

Here, Na and Nb are respectively the concentrations of acidic and basic functional groups in the membrane layer, and

[ ] ∑[ ∏ ] ZV

PV )

KV,j

m

∑∏ m)1 j)1 C

, V ) a,b

(5)

, V ) a,b

(6)

H+

ZV

QV )

2. Theory Referring to Figure 1, we consider two identical planar particles; each comprises a rigid, uncharged core, and an ionpenetrable membrane layer of scaled thickness d. These particles are immersed in a mixed (a:b) + (c:b) electrolyte solution, where a and c are the valences of cations and b is the valence of anions. The membrane layer contains uniformly distributed functional groups, the dissociation of which yields negative fixed charges. The symmetric nature of the present problem implies that only the interval (-∞, XL] needs to be considered, XL being the location of the middle plane between two particles. Let σca, σan, and σf be the effective radii of cations, anions, and fixed charged groups, respectively. Without loss of generality, we assume that σca < σan < σf. The fixed charged groups are arranged so that the margin of the leftmost one coincides with the core-membrane interface and the margin of the rightmost one coincides with the membrane-liquid interface. For convenience, the system is divided into five regions. Region I, X < Xca, which comprises the charge-free region (-∞ < X < 0) and the inner uncharged membrane (0 < X < Xca), X being the scaled distance. Region II, Xca < X < Xan, which contains cations only. Region III, Xan < X < Xi, which contains both cations and anions. Region IV, Xi < X < Xo, which contains all charged species. Region V, Xo < X, which includes the outer uncharged membrane (Xo < X < d) and the liquid phase (d < X < ∞). Here, Xi and Xo are respectively the locations of the inner and outer planes of fixed charge, and Xca and Xan represent respectively the most interior locations of positive and negative

NaQa 1 + Pa

n

n

n)1

KV,j

j)1 CH+

Assuming Boltzmann distribution, the concentration of H+, CH+, can be expressed by

CH+ ) CH0 + exp(-ψ)

(7)

where CH0 + is the bulk concentration of H+. 2.2. Interaction Energy. According to the DLVO theory, the total interaction energy between two particles, VT, is the sum of VR and VA, that is

V T ) VR + V A

(8)

where VR and VA are respectively the electrical repulsive energy and van der Waals attractive energy. VR can be calculated by

VR )

∫X∞ FR dXL

2 κ

(9)

L

where FR is the electrical interaction force between two particles. FR can be evaluated by (Appendix A)

FR bn0bkBT

1 1 - ξ -aψm ξ ) (ebψm - 1) + (e - 1) + (e-cψm - 1) b a c (10)

where ψm is the scaled electrical potential on the middle plane between two particles which can be calculated by the PoissonBoltzmann equation, which takes the sizes of all the charged

Effect of Ionic Sizes on CCC

J. Phys. Chem. B, Vol. 106, No. 16, 2002 4271

Figure 2. Variation of scaled potential ψ as a function of scaled distance XL. Solid curve, present model; dashed curve, PCM. Parameters used: T ) 298K, r ) 78.5, I ) 10-3 M, d ) 1.5, a ) 2, b ) 1, ξ ) 0, Za ) Zb ) 1, pH ) 7, pKa,1 ) 3, pKb,1 ) 5, Na ) Nb ) 8 × 10 - 3 M, Xca ) 0.05, Xan ) 0.1, Xi ) 0.2, and XL ) 2.0.

species into account and the associated boundary conditions (Appendix B). VA can be evaluated by10

VA ) -

A132κ2 48π(XL - d)

(11)

where A132 is the Hamaker constant. At CCC, both the total interaction energy and its derivative with respect to the distance between two particles vanish. We have

VT ) 0 and

dVT )0 dXL

(12)

These expressions can be used to determine CCC. 3. Results and Discussion The effect of ionic sizes on the electrical interaction force and the total interaction energy between two particles are examined through numerical simulation. For illustration, we assume that the net fixed charge in membrane layer is negative and Za ) Zb ) 1. The averaged thickness of the peripheral zone of human erythrocyte in a 1 mM saline solution is about 15 nm. Therefore, the scaled membrane thickness in the numerical simulations is assumed to be on the order of 1.5. Figure 2 shows the simulated variation in the scaled electrical potential ψ as a function of the scaled distance XL. For comparison, the corresponding result based on the classic point charge model (PCM) is also represented in this figure. We assume that the Hamaker constant, A132, is constant, and the van der Waals interaction energy is a function of the separation distance between two particles. Figure 2 reveals that |ψ| has a local maximum as XL varies, which does not present, however, in the corresponding PCM where |ψ| decreases monotonically with XL. In general, PCM will overestimate |ψ|. This is because fixed charge only exists in region IV in the present model, but it is present in the whole membrane layer in the corresponding PCM. Figure 3 illustrates the simulated variations in the scaled electrical repulsive force and the total interaction energy between two particles as a function of the scaled half separation distance between them (XL - d) at various pH. The result for the

Figure 3. Variation of scaled repulsive force (a) and total interaction energy (b) between two particles as a function of scaled half separation distance XL at various pH for the case of A132 ) 10-20 J. Solid curves, present model; dashed curves, PCM. Curve 1, pH ) 8; 2, pH ) 7; 3, pH ) 6; and 4, pH ) 5. Other parameters are the same as those of Figure 2.

electrical repulsive force of the corresponding PCM is also presented in Figure 3a for comparison. Figure 3a indicates that for a fixed pH the scaled electrical repulsive force decreases with the increase in XL, as expected. As can be seen in Figure 3, both the scaled electrical repulsive force and the total interaction energy increase with the increase in pH for pH < 7 and becomes almost constant for pH > 7. This is because if pH is low the degree of dissociation of acidic functional groups is small, and at the same time, it is easy for H+ to bind to basic functional groups. These lead to a low concentration of negative fixed charge in the membrane layer, and, therefore, a small electrical repulsive force and small total interaction energy. On the other hand, if pH is sufficiently high, the dissociation of acidic functional groups is essentially complete, and the concentration of negative fixed charge remains constant and so are the electrical repulsive force and total interaction energy. Figure 3 also reveals that assuming PCM will overestimate the electrical repulsive force and the deviation increases with pH. The variations of the scaled electrical repulsive force and the total interaction energy between two particles as a function of the scaled half separation distance between them (XL - d) at various scaled membrane thickness d are illustrated in Figure 4. Here, the total number of functional groups in the membrane layer is held constant. Figure 4a suggests that for a fixed d, the

4272 J. Phys. Chem. B, Vol. 106, No. 16, 2002

Figure 4. Variation of scaled repulsive force (a) and total interaction energy (b) between two particles as a function of scaled half separation distance (XL - d) at various d for the case of A132 ) 10-20 J. Curve 1, d ) 1.2; 2, d ) 1.5; and 3, d ) 2.0. Other parameters are the same as those of Figure 2.

scaled repulsive force decrease with (XL - d) as expected. As can be seen from Figure 4, both the scaled repulsive force and the total interaction energy decrease with the increase in d. This is because, if the total number of functional groups is fixed, the thinner the membrane, the more concentrated the fixed charge and the greater the electrical interaction and the total interaction energy. Figure 5 shows the variations of the scaled electrical repulsive force and the total interaction energy between two particles as a function of the scaled half separation distance between them (XL - d) for various valances of cations (counterions). For simplicity, we assume that cations with various valences have the same size. Figure 5 reveals that the higher the valence of cations, the smaller the scaled repulsive force and the total interaction energy. This is because the membrane is negatively charged, and therefore, the higher the valence of cations, the greater its shielding effect, which leads to a lower absolute potential and smaller repulsive force and total interaction energy. The variations of the scaled electrical repulsive force and the total interaction energy between two particles as a function of the scaled half separation distance between them (XL - d) at various fraction of multivalent cations (counterions), ξ, are illustrated in Figure 6. This figure suggests that the larger the ξ, the lower the scaled repulsive force and the total interaction energy. Again, this is because the presence of multivalent cations in the liquid phase has the effect of raising the shielding effect

Hsu et al.

Figure 5. Variation of scaled repulsive force (a) and total interaction energy (b) between two particles as a function of scaled half separation distance XL at various valences of actions for the case of A132 ) 10-20 J. Curve 1, a ) 1 and b ) 1; 2, a ) 2 and b ) 1; and 3, a ) 3 and b ) 1. Other parameters are the same as those of Figure 2.

and reducing the scaled electrical repulsive force and the total interaction energy. Figure 7 shows the variations of the scaled electrical repulsive force and the total interaction energy between two particles as a function of the half separation distance between them (XL d) at various ionic strength I. Here, the total number of dissociable functional groups is fixed. As can be seen in Figure 7, for a fixed separation distance between two particles, the higher the ionic strength, the smaller the electrical repulsive force and the total interaction energy. This is because the increase in the concentration of electrolyte has the effects of increasing the degree of screening of the surface charge by counterions and decreasing the thickness of double layer. The variation of the scaled electrical repulsive force between two particles as a function of the scaled effective radius of cations (counterions) Xca for a fixed size of anions at various pH is illustrated in Figure 8. The leftmost point of each curve represents the result for the case when counterions are treated as point charges, and the rightmost point of each curve is that when both counterions and coions have the same size; that is, region II vanishes. Figure 8 suggests that the electrical repulsive force increases with Xca; that is, the larger the cations, the larger the electrical repulsive force. This is because the smaller the cations, the easier for them to bond to the negative fixed charge in membrane layer, which has the effect of lowering the absolute

Effect of Ionic Sizes on CCC

Figure 6. Variation of scaled repulsive force (a) and total interaction energy (b) between two particles as a function of scaled half separation distance XL at various ξ for the case of a ) 1, b ) 1, c ) 2, and A132 ) 10-20 J. Curve 1, ξ ) 0; 2, ξ ) 0.1; and 3, ξ ) 0.5. Other parameters are the same as those of Figure 2.

electrical potential. Colic et al.24 and Chapel25 discussed the effect of monovalent cations on the repulsive force between two silica surfaces. They found that for the common anion, Cl-, the strength of the repulsive force, from the weakest to the strongest, followed the order Li+ < Na+ < K+ < Cs+. They concluded that the repulsive force is related to the sizes of bare counterions. The same order was also observed for the case of alumina surfaces.26 These observations are consistent with the result illustrated in Figure 8. Figure 8 also shows that the electrical repulsive force increases with pH. Figure 9 presents the variation of the scaled electrical repulsive force between two particles as a function of the scaled effective radius of anions (coions) Xan for a fixed size of cations at various ξ. The leftmost point of each curve represents the result for the case when both counterions and coions have the same size; that is, region II vanishes. The rightmost point of each curve represents that when both coions and fixed charged groups have the same size; that is, region III vanishes. For a clearer illustration, curve 1 is magnified as curve 4. According to Figure 9, the electrical repulsive force decreases with the increase in Xan. The change in the repulsive force as Xan varies, however, is inappreciable. Figure 9 also suggests that the electrical repulsive force decreases with the increase in ξ. Table 1 shows the variation in the ratio [CCC/CCC(PCM)] as a function of the sizes of charged species, Xca, Xan, and Xi, where CCC(PCM) is the CCC predicted by the classic point

J. Phys. Chem. B, Vol. 106, No. 16, 2002 4273

Figure 7. Variation of scaled repulsive force (a) and total interaction energy (b) between two particles as a function of half separation distance x at various ionic strength I for the case of D ) 14.42 nm, σca ) 0.48 nm, σan ) 0.96 nm, σI ) 1.92 nm, and A132 ) 10-20 J. Curve 1, I ) 1.0 × 10-3 M; 2, I ) 1.5 × 10-3 M; and 3, I ) 2.0 × 10-3 M. Other parameters are the same as those of Figure 2.

Figure 8. Variation of scaled repulsive force between two particles as a function of scaled effective radius of cations Xca at various pH for the case Xan ) 0.1, Xi ) 0.2, XL ) 1.5, and d ) 1.0. Curve 1, pH ) 7; 2, pH ) 6; 3, pH ) 5. Other parameters are same as the same as those of Figure 2.

charge model. As can be seen from this table, the effect of the sizes of charged species on CCC is significant. For fixed sizes of anions (coions) and fixed charge, the larger the size of cations

4274 J. Phys. Chem. B, Vol. 106, No. 16, 2002

Hsu et al. 2 also reveals that the CCC ratio of cations of valences 3, 2, and 1 is roughly 1:2.5:10 at pH 7, which is different significantly from that based on the Schulze-Hardy rule. Acknowledgment. This work is supported by the National Science Council of the Republic of China. Appendix A For two identical, planar parallel particles separated by a scaled distance 2XL, a differential force balance yields

dFR ) -Fdφ

Figure 9. Variation of scaled repulsive force between two particles as a function of scaled effective radius of anions Xan at various ξ for the case Xca ) 0.05, Xi ) 0.2, XL ) 1.5, and d ) 1.0. Curves 1-3 use X-Y1 axes, and 4 uses X-Y2 axes. Curves 1 and 4, ξ ) 0; 2, ξ ) 0.1; and 3, ξ ) 0.2. Other parameters are the same as those of Figure 2.

TABLE 1: Variation of CCC/CCC(PCM) as a Function of the Sizes of Charged Species, XCa, Xan, and Xia σca. (nm)

σan (nm)

σi (nm)

CCC/CCC(PCM)

0.09616 0.96156 1.92312 2.88467 3.84623 0.00962

3.84623 3.84623 3.84623 3.84623 3.84623 0.01923

4.80779 4.80779 4.80779 4.80779 4.80779 0.03846

0.3621 0.3695 0.3830 0.3916 0.4052 0.9877

Parameters used: T ) 298 K, r ) 78.5, a ) 2, b ) 1, ξ ) 0, Za ) Zb ) 1, pH ) 7, pKa,1 ) 3, pKb,1 ) 5, Na ) Nb ) 8 × 10-3 M, D ) 14.42 nm, and A132 ) 10-20 J.

(A1)

where dFR is the differential electrical repulsive force per unit area between two particles and F is the space charge density. Integrating equation (A1) from ∞ to XL, gives the electrical repulsive force FR. We have

FR ) -

∫0φ

L

(aena - benb + cenc) dφ

(A2)

where φL is the potential on the middle plane between two particles, and

na ) n0a exp(-aψ)

(A3)

nb ) n0b exp(bψ)

(A4)

nc ) n0c exp(-cψ)

(A5)

a

TABLE 2: Variation of CCC (mM) as a Function of pH and the Valence of Cationsa

In these expressions, na, nc, and nb represent respectively the number concentrations of cations of valence a and c and anions of valence b. When eqs A3-A5 are substituted into eq A2 and the resultant expression is integrated, eq 10 can be obtained.

pH a:b

5

6

7

1:1 2:1 3:1

1.77 0.485 0.2

2.9 0.833 0.325

3.6 0.91 0.353

a Parameters used: T ) 298 K, r ) 78.5, ionic strength ) 10-3 M, d ) 1.5, ξ ) 0, Za ) Zb ) 1, pKa,1 ) 3, pKb,1 ) 5, Na ) Nb ) 8 × 10-3 M, Xca ) 0.05, Xan ) 0.1, Xi ) 0.2, and A132 ) 10-20 J.

(counterions), the higher the CCC. This is because the electrical repulsive force increases with Xca, and the potential barrier of the total interaction energy increases with Xca also, and this leads to a higher CCC. In a study of the stability of R-haematite hydrosols in the presence of various monovalent cations, Amhamdi et al.27 concluded that the CCC for cations followed the order Li+ < Na+ < K+ at a pH higher than PZC. The same trend was also observed by Ghenne et al.28 for the case of TiO2 hydrosols. Ohki and Ohshima29 examined the interaction between two phosphatidylserine vesicles. They showed that the effectiveness of monovalent cations to induce the aggregation of phosphatidylserine vesicles followed the order Na+ > K+ > Cs+. This is consistent with the result presented in Table 1. If all of the sizes of charged species are small, the result predicted by the present model approaches that predicted by PCM. Table 2 shows the variation of CCC as a function of pH and valence of ions. This table suggests that CCC increases with pH. This is consistent with experimental observations.27,28 Table

Appendix B We assume that the scaled electrical potential ψ can be described by the scaled Poisson-Boltzmann equation

d 2ψ ) dX2 β exp(bψ) - R[(1 - ξ) exp(-aψ) + ξ exp(-cψ)] + γN (a + b) + (c - a)ξ (B1) where ψ ) eφ/kBT, ξ ) cn0c /bn0b, κ2 ) e2[a(a + b)n0a + c(b + c)n0c ]/0rkBT, bn0b ) an0a + cn0c , and N ) NA(N- - N+ )/bn0b. Here, φ is the electrical potential and ξ is the fraction of cations of valence c in the bulk liquid phase. n0a, n0b, and n0c are respectively the number concentrations of ionic species of valences a, -b, and c in the bulk liquid phase. e and NA are respectively the elementary charge and the Avogadro number, and r and 0 are the relative permittivity of the liquid phase and the permittivity of a vacuum, respectively. kB and T denote respectively the Boltzmann constant and the absolute temperature. A region index vector (R,β,γ) is defined, with (R,β,γ) ) (0,0,0), (1,0,0), (1,1,0), (1,1,1), and (1,1,0) represent respectively regions I through V. We assume the following: (i) the electric field is absent in the rigid core of the particle and (ii) both the electrical potential and its gradient are continuous at the

Effect of Ionic Sizes on CCC

J. Phys. Chem. B, Vol. 106, No. 16, 2002 4275

intersection of two adjacent regions. These assumptions yield the following boundary conditions:

dψ ) 0, X ) 0 dX

(dψ dX )

)

(dψ dX )

)

X)X+ ca

(dψ dX )

X)X+ i

(dψ dX )

X)X+ o

(dψ dX )

and ψ(X+ ca) ) ψ(Xca)

(B3)

(dψ dX )

and ψ(X+ an) ) ψ(Xan)

(B4)

and ψ(X+ i ) ) ψ(Xi )

(B5)

and ψ(X+ o ) ) ψ(Xo )

(B6)

X)Xca

X)X+ an

)

)

(B2)

X)Xan

(dψ dX )

X)Xi

(dψ dX )

X)Xo

dψ ) 0, X ) XL dX

(B7)

References and Notes (1) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: Oxford, U.K., 1989; Vol. 1. (2) Donath, E.; Pastushenko, V. Bioelectrochem. Bioenerg. 1979, 6, 543. (3) Wunderlich, R. W. J. Colloid Interface Sci. 1982, 88, 385.

(4) Levine, S.; Levine, M.; Sharp, K. A.; Brooks, D. E. Biophys. J. 1983, 42, 127. (5) Sharp, K. A.; Brooks, D. E. Biophys. J. 1985, 47, 563. (6) Reiss, H.; Bassidnana, I. C. J. Membr. Sci. 1982, 11, 219. (7) Selvey, C.; Reiss, H. J. Membr. Sci. 1985, 23, 11. (8) Terui, H.; Taguchi, T.; Ohshima, H.; Kondo, T. Colloid Polym. Sci. 1990, 268, 76. (9) Taguchi, T.; Terui, H.; Ohshima, H.; Kondo, T. Colloid Polym. Sci. 1990, 268, 83. (10) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1995, 174, 250. (11) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1996, 183, 184. (12) Valleau, J. P.; Torrie, G. M. J. Chem. Phys. 1982, 76, 4623. (13) Bhuiyan, L. B.; Blum, L.; Henderson, D. J. Chem. Phys. 1983, 78, 442. (14) Torrie, G. M.; Valleau, J. P. Chem. Phys. Lett. 1979, 65, 343. (15) Blum, L. J. Phys. Chem. 1977, 81, 136. (16) Sloth, P.; Sorensen, T. S. J. Chem. Phys. 1992, 96, 548. (17) Sorensen, T. S.; Sloth, P. J. Chem. Soc., Faraday Trans. 1992, 88, 571. (18) Rivera, S. R.; Sorensen, T. S. Mol. Simul. 1994, 13, 115. (19) Hsu, J. P.; Kuo, Y. C. J. Chem. Phys. 1999, 111, 4807. (20) Kuo, Y. C.; Hsu, J. P. J. Phys. Chem. B 1999, 103, 9743. (21) Kuo, Y. C.; Hsu, J. P. Langmuir 2000, 16, 6233. (22) Kuo, Y. C.; Hsieh, M. Y.; Hsu, J. P. Langmuir accepted for publication. (23) Hsu, J. P.; Kuo, Y. C. J. Colloid Interface Sci. 1996, 183, 194. (24) Colic, M.; Fisher, M. L.; Franks, G. V. Langmuir 1998, 14, 6107. (25) Chapel, J. P. Langmuir 1994, 10, 4237. (26) Colic, M.; Franks, G. V.; Fisher, M. L.; Lange F. F. Langmuir 1997, 13, 3129. (27) Amhamdi, H.; Dumont, F.; Buess-Herman, C. Colloids Surf. A 1997, 125, 1. (28) Ghenne, E.; Dumont, F.; Buess-Herman, C. Colloids Surf. A 1998, 131, 63. (29) Ohki, S.; Ohshima, H. Colloids Surf. B 1999, 14, 27.